Add files using upload-large-folder tool

-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e90ea8f3dda621f75520b27bc1d69b08510e6796592ce1192593b7c24dfdb07b
+size 3134596

-tAzT4oBgHgl3EQfvf0n/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d23f77aac179c197e996bc906fc32a98347361d23626ff09a9700c6de2e7a8f8
+size 194483

.gitattributes

@@ -4127,3 +4127,57 @@ wdFRT4oBgHgl3EQfgjd8/content/2301.13580v1.pdf filter=lfs diff=lfs merge=lfs -tex
 jtFST4oBgHgl3EQfHTjm/content/2301.13725v1.pdf filter=lfs diff=lfs merge=lfs -text
 UNAzT4oBgHgl3EQfJvtM/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf filter=lfs diff=lfs merge=lfs -text
+CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf filter=lfs diff=lfs merge=lfs -text
+A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf filter=lfs diff=lfs merge=lfs -text
+F9E1T4oBgHgl3EQfqwX2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+INA0T4oBgHgl3EQfB__n/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf filter=lfs diff=lfs merge=lfs -text
+utFAT4oBgHgl3EQfhx3p/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf filter=lfs diff=lfs merge=lfs -text
+9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf filter=lfs diff=lfs merge=lfs -text
+bdAzT4oBgHgl3EQfLfsG/content/2301.01113v1.pdf filter=lfs diff=lfs merge=lfs -text
+mNE4T4oBgHgl3EQftw2O/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wdFRT4oBgHgl3EQfgjd8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf filter=lfs diff=lfs merge=lfs -text
+ytFRT4oBgHgl3EQfjDex/content/2301.13589v1.pdf filter=lfs diff=lfs merge=lfs -text
+JtE4T4oBgHgl3EQf7Q7_/content/2301.05339v1.pdf filter=lfs diff=lfs merge=lfs -text
+ddE1T4oBgHgl3EQfeARB/content/2301.03201v1.pdf filter=lfs diff=lfs merge=lfs -text
+EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf filter=lfs diff=lfs merge=lfs -text
+FNFRT4oBgHgl3EQfyzjn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+U9AzT4oBgHgl3EQfJvv5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+9dAzT4oBgHgl3EQfFPqV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+mtFAT4oBgHgl3EQfch1G/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+jtFST4oBgHgl3EQfHTjm/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+U9E1T4oBgHgl3EQfbAQs/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ddE1T4oBgHgl3EQfeARB/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+z9E0T4oBgHgl3EQf_QJq/content/2301.02824v1.pdf filter=lfs diff=lfs merge=lfs -text
+hdE1T4oBgHgl3EQfzQWw/content/2301.03443v1.pdf filter=lfs diff=lfs merge=lfs -text
+ttE_T4oBgHgl3EQf9hxa/content/2301.08381v1.pdf filter=lfs diff=lfs merge=lfs -text
+ytFAT4oBgHgl3EQfBBwz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+MdE2T4oBgHgl3EQfBQZc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+mtFAT4oBgHgl3EQfch1G/content/2301.08564v1.pdf filter=lfs diff=lfs merge=lfs -text
+iNE2T4oBgHgl3EQfHwaf/content/2301.03672v1.pdf filter=lfs diff=lfs merge=lfs -text
+4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf filter=lfs diff=lfs merge=lfs -text
+U9E1T4oBgHgl3EQfbAQs/content/2301.03167v1.pdf filter=lfs diff=lfs merge=lfs -text
+59FKT4oBgHgl3EQfTC3B/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf filter=lfs diff=lfs merge=lfs -text
+xtFAT4oBgHgl3EQfARxu/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ytFRT4oBgHgl3EQfjDex/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ltFJT4oBgHgl3EQfZSy-/content/2301.11531v1.pdf filter=lfs diff=lfs merge=lfs -text
+0tE4T4oBgHgl3EQfZwwm/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+oNE1T4oBgHgl3EQfOwM_/content/2301.03018v1.pdf filter=lfs diff=lfs merge=lfs -text
+xtFAT4oBgHgl3EQfARxu/content/2301.08397v1.pdf filter=lfs diff=lfs merge=lfs -text
+F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf filter=lfs diff=lfs merge=lfs -text
+4dE2T4oBgHgl3EQfjwe5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ltFLT4oBgHgl3EQfeC95/content/2301.12089v1.pdf filter=lfs diff=lfs merge=lfs -text
+XtFRT4oBgHgl3EQf-zgt/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+odE1T4oBgHgl3EQf1wVk/content/2301.03471v1.pdf filter=lfs diff=lfs merge=lfs -text
+-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf filter=lfs diff=lfs merge=lfs -text
+vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf filter=lfs diff=lfs merge=lfs -text
+z9E0T4oBgHgl3EQf_QJq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+lNE1T4oBgHgl3EQfNwNu/content/2301.03006v1.pdf filter=lfs diff=lfs merge=lfs -text
+hdE1T4oBgHgl3EQfzQWw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf filter=lfs diff=lfs merge=lfs -text
+btAzT4oBgHgl3EQfLfs5/content/2301.01114v1.pdf filter=lfs diff=lfs merge=lfs -text
+gNE0T4oBgHgl3EQfpAFc/content/2301.02533v1.pdf filter=lfs diff=lfs merge=lfs -text
+odE1T4oBgHgl3EQf1wVk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text

0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b86d9c7bf4f776ef130d4cf643591f03f5696b9b6a9974ff26136aa0ee6f977f
+size 1080329

0tE4T4oBgHgl3EQfZwwm/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7464fc9fcef18bd8dda16c6b6052bf9b93a32fe91ca63034c96bebea9cdba920
+size 5177389

0tE4T4oBgHgl3EQfZwwm/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bf2d47be2b22c04eb85f90b4db970cdbc15723f80326c804c91c5dfe654182a2
+size 172830

29E4T4oBgHgl3EQf0A2b/content/tmp_files/2301.05279v1.pdf.txt

@@ -0,0 +1,659 @@
+Characterization of Fast Ion Transport via Position-Dependent Optical Deshelving
+Craig R. Clark,∗ Creston D. Herold, James T. Merrill, Holly N. Tinkey, Wade
+Rellergert, Robert Clark, Roger Brown, Wesley D. Robertson, Curtis Volin, Kara
+Maller, Chris Shappert, Brian J. McMahon, Brian C. Sawyer,† and Kenton R. Brown‡
+Georgia Tech Research Institute, Atlanta, Georgia 30332, USA
+Ion transport is an essential operation in some models of quantum information processing, where
+fast ion shuttling with minimal motional excitation is necessary for efficient, high-fidelity quan-
+tum logic. While fast and cold ion shuttling has been demonstrated, the dynamics and specific
+trajectory of an ion during diabatic transport have not been studied in detail. Here we describe
+a position-dependent optical deshelving technique useful for sampling an ion’s position through-
+out its trajectory, and we demonstrate the technique on fast linear transport of a 40Ca+ ion in a
+surface-electrode ion trap. At high speed, the trap’s electrode filters strongly distort the transport
+potential waveform. With this technique, we observe deviations from the intended constant-velocity
+(100 m/s) transport: we measure an average speed of 83(2) m/s and a peak speed of 251(6) m/s
+over a distance of 120 µm.
+Systems of trapped atomic ions represent some of the
+most promising platforms for quantum information pro-
+cessing, benefiting from long qubit coherence times and
+from the highest operational fidelities demonstrated to
+date [1–3]. In the QCCD ion-trap architecture [4], ions
+are transported between various regions within the pro-
+cessor to reconfigure which ions can interact at any given
+step of an algorithm.
+Such an architecture demands
+finely tuned control of ion shuttling in order to imple-
+ment reconfigurations as quickly as possible without also
+degrading the fidelities of subsequent logic gates [1, 5, 6].
+Performing logic operations on the ions at the same time
+as they are transported within the trap also requires well
+characterized and reproducible trajectories for success
+[7, 8].
+Recent work has shown that ion transport contributes
+substantially to the latency of current QCCD systems [9].
+Rapid linear ion transport has been demonstrated previ-
+ously [10–12] but has not been widely adopted likely due
+to the demands it places on waveform control hardware
+and on motional characterization and calibration. Trans-
+port at slower speeds has also been achieved through
+more complicated structures such as junctions of linear
+sections [13–17]. While prior results have demonstrated
+transport between two locations with sub-quanta mo-
+tional excitation, the ion trajectory during transport has
+not been measured. In fact, all prior reported transport
+speeds assumed an average speed calculated from the dis-
+tance between static well positions and the designed play-
+back speed of the trapping voltage waveforms. Some ex-
+periments have proved that an ion was transported to an
+intended location, e.g. by applying a focused laser pulse
+at that location [10, 12]; however, those experiments did
+not verify the exact time of arrival.
+In this work, we present a method to measure the loca-
+tion of an ion throughout the entire arc of its transport,
+∗ craig.clark@gtri.gatech.edu
+† brian.sawyer@gtri.gatech.edu
+‡ kenton.brown@gtri.gatech.edu
+which enables us to reliably extract both instantaneous
+and average velocity during transport. This is achieved
+by measuring the probability for the ion to undergo a
+spontaneous irreversible transition from a metastable ex-
+cited state to a ground state when illuminated by a laser
+beam having a spatial intensity gradient. Similar ideas
+have been explored in the context of neutral atoms in
+optical cavities [18, 19], as well as in the field of ultrafast
+physics [20]. The method presented here contrasts with
+previous schemes employing Fourier-limited coherent op-
+tical interactions to extract in-flight Doppler shifts [7, 8].
+Here, we leverage the technique to show that an ion’s
+trajectory deviates significantly from a simple constant-
+velocity path as the trap’s electrode filters distort the ap-
+plied potentials. Although the intended trajectory has a
+constant speed of 100 m/s, corresponding to a displace-
+ment of 120 µm (two electrodes) in 1.2 µs, the ion ac-
+tually achieves a maximum speed of 251(6) m/s in the
+middle of its path due to this distortion. These trajec-
+tory deviations lead also to a large coherent displacement
+of the ion’s motional state.
+We superimpose an addi-
+tional compensating sinusoidal potential onto the wave-
+form to remove this displacement [12, 21] and achieve a
+final transport-induced excitation of 0.7(2) quanta.
+Our experimental system employs a GTRI/Honeywell
+Ball Grid Array (BGA) trap [22], shown schematically in
+Fig. 1a, mounted in a room-temperature ultrahigh vac-
+uum chamber and confining a single 40Ca+ ion.
+The
+trap radiofrequency (rf) electrode is driven at 55 MHz
+and realizes radial frequencies of 3.7 and 4.2 MHz. We
+determine the control electrode voltages necessary to
+produce axial confinement at different positions along
+the trap symmetry axis using an in-house boundary ele-
+ment method electrostatic solver [23]. We employ NIST-
+designed PDQ digital-to-analog converters (DACs) [21]
+to vary the potentials on the axial control electrodes.
+Each control electrode’s potential is filtered with a two-
+pole low-pass filter having a bandwidth of 608 kHz [24].
+The axial trap frequency (parallel to the axis of symme-
+try) is approximately 1.7 MHz. The axial heating rate
+arXiv:2301.05279v1  [quant-ph]  12 Jan 2023
+
+2
+854 nm 
+Intensity
+a)
+𝑧
+𝑧�
+time
+Cool & 
+Prep
+Detect
+Adiabatic Transport
+b)
+𝒛𝒊  → 𝒛
+π
+𝒛 → 𝒛𝒊
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+50
+100
+150
+200
+Deshelve Probability
+Position (µm)
+c)
+𝒛𝒊
+𝒛𝒇
+𝑧�
+Optical operations
+𝑺𝟏/𝟐
+𝑫𝟓/𝟐
+𝑷𝟑/𝟐
+729 nm
+854 nm
+393 nm
+FIG. 1.
+(a) (Left) Schematic of the BGA trap with a graph of
+the nominal 854 nm repumper intensity profile superimposed.
+The initial and final positions of the ion for the characterized
+transport are zi and zf.
+(Right) Energy level diagram of
+40Ca+ ion showing the shelving transition driven by 729 nm
+light in red, the deshelving transition driven by 854 nm light
+in orange, and spontaneous decay through emission of 393 nm
+light in purple. (b) Ion trajectory calibration sequence. An
+ion in a stationary potential at zi is cooled and prepared in
+the S1/2 level and then shelved into the D5/2 level with a
+resonant 729 nm π-pulse (shown in red in the timeline). The
+ion is adiabatically transported to another position along the
+trap axis, and the 854 nm beam is pulsed for 200 ns, thereby
+deshelving a portion of the D5/2 population. This deshelving
+fraction is in one-to-one correspondence with the local beam
+intensity. The ion is shuttled adiabatically back to its initial
+location zi. (c) Deshelving probability as a function of ion
+position (black points) with error bars representing the 68%
+confidence interval assuming binomial statistics. The red line
+represents a polynomial fit to the data.
+for a stationary potential is 210 quanta/s.
+For Doppler cooling we use a 397 nm laser beam, nearly
+resonant with the S1/2 − P1/2 transition, in combina-
+tion with another beam at 866 nm which is used to re-
+turn population from the D3/2 level into the cooling cy-
+cle. Pulses from a beam at 729 nm, resonant with the
+S1/2−D5/2 transition, coherently populate the D5/2 level
+and are employed for sideband cooling and motional-
+state characterization. Another beam at 854 nm is re-
+sponsible for deshelving population from the D5/2 to the
+S1/2 level via the intermediate P3/2 level (see Fig.1a) [25].
+We distinguish between an ion in the S1/2 level (bright)
+and one in the D5/2 level (dark) via observation of state-
+dependent 397 nm fluorescence as recorded on a photo-
+multiplier tube.
+After many experimental repetitions,
+the bright and dark state populations are estimated via
+maximum likelihood (see Supplemental Material of Ref.
+[1]).
+To characterize the ion’s position throughout the full
+arc of its roughly linear trajectory, we implement a
+position-sensitive optical deshelving technique [26]. For
+this, we prepare the ion in the D5/2 state before transport
+and then apply a pulse (200 ns, shorter than the trans-
+port duration) of 854 nm light at a later time, thereby
+deshelving the D5/2 level with a probability dependent on
+the laser beam’s local intensity. By choosing the Gaus-
+sian waist w0 of the 854 nm beam such that its intensity
+varies significantly and monotonically along the ion’s tra-
+jectory (w0 ∼ 100 µm), we achieve a position-dependent
+deshelving probability (Pd). We can therefore invert spa-
+tially monotonic measurements of this probability, ac-
+quired at various times after the start of the transport,
+to determine the ion’s position at these instants. The
+beam is directed perpendicular to the linear transport
+axis to remove velocity dependence (first-order Doppler
+shifts, e.g.) from the deshelving probability.
+The
+above
+technique
+requires
+calibration
+of
+the
+deshelving probability for a given ion location using a
+procedure diagrammed in Fig. 1b. For this, we Doppler
+and sideband cool the ion nearly to its axial motional
+ground state (¯n < 1) and prepare it in S1/2. We then
+shelve it into the D5/2 level with a resonant 729 nm π-
+pulse and subsequently move it adiabatically to a given
+position along the trap axis (as determined from an elec-
+trostatic model of the trapping potential). We illuminate
+the ion with a 200 ns, 854 nm deshelving pulse. After
+adiabatically returning the ion to its initial location, we
+determine its state by collecting fluorescence. Repeating
+this experiment 400 times each at 2 µm intervals yields
+a measurement of deshelving probability Pd(z) for each
+location as shown in Fig. 1c. We fit Pd(z) with a polyno-
+mial (red curve of Fig. 1c) rather than with a Gaussian
+to allow for possible distortion of the 854 nm laser beam
+mode shape, and we ensure that Pd(z) is single-valued by
+having previously displaced the beam such that its inten-
+sity varies monotonically through the calibration region.
+It is important that any movement of ions from one
+place to another in the trap be accomplished with only a
+small degree of motional excitation. For the experiments
+outlined below, we design the waveform as a simple lin-
+ear time interpolation of the potential minimum through
+a 120 µm displacement. We choose a transport duration
+(1.2 µs) that is an integer multiple of the ion’s harmonic
+motional period (0.6 µs, nominally held constant during
+the transport). In the absence of filter distortion, such a
+waveform is expected to coherently excite ion motion at
+the beginning of transport but subsequently to suppress
+this excitation at the end [10, 27]. Our filter bandwidths
+lie below the ion motional frequencies by design. At slow
+enough speeds, the action of the filters is to smooth out
+
+3
+the beginning and ending accelerations, so that the ion’s
+motion is not excited as much during transport as would
+otherwise be the case. At faster speeds such as are used
+here, filter distortion is great enough that simple scal-
+ing of the waveform confinement strength can no longer
+fully suppress the final excitation [27]. However, provided
+that the waveform is performed identically for each ex-
+perimental repetition and that the confinement remains
+harmonic, the final excitation corresponds to a coherent
+displacement.
+We remove it by superimposing on the
+transport waveform a sinusoidal rf pulse applied to four
+of the trap electrodes, and we optimize the phase, fre-
+quency, and amplitude of this pulse to achieve minimum
+mode occupation.
+With the Pd(z) calibration complete, we then trans-
+port the ion through the full trajectory of 120 µm. Fig. 2a
+gives a diagram of the sequence. We pulse the 854 nm
+deshelving laser for 200 ns while the ion is in motion,
+with a configurable delay between the start of the trans-
+port waveform and the start of the pulse. The ion is then
+returned adiabatically to its initial location for state de-
+tection [28]. With 100 repetitions of this experiment at
+each delay, we obtain a map Pd(t) of deshelving probabil-
+ity in time following the start of the waveform (Fig. 2b).
+To obtain the corresponding position map z(t), we in-
+vert the polynomial fit of Fig. 1c to obtain z(Pd) and
+then compute z(Pd(t)), producing the points in Fig. 2c.
+Finally, we fit the z(t) data of Fig. 2c with the follow-
+ing phenomenological expression to extract the mean and
+maximum linear velocities:
+z(t) = zi +
+√π
+2 vmaxtσ
+�
+erf
+�t − tc
+tσ
+�
+− erf
+�t0 − tc
+tσ
+��
+.
+(1)
+Equation 1 is derived assuming that the speed follows
+a Gaussian profile in time, an empirical assumption justi-
+fied by its agreement with the z(t) data in Fig. 2c. Here,
+zi represents the initial ion position, t0 is the initial time
+(time when z(t) = zi), tσ is the 1/e temporal half-width
+of the ion speed, and tc is the time of maximum speed.
+We use the following standard definition for the error
+function:
+erf(x) =
+2
+√π
+� x
+0
+e−y2dy.
+(2)
+We note that a naive estimate of the mean speed, ob-
+tained from the 120 µm and 1.2 µs waveform displace-
+ment and duration, would be 100 m/s. In contrast, the
+fit of Fig. 2c (red curve) yields a much higher maximum
+slope of 251(6) m/s.
+To determine an effective mean
+speed, we estimate the beginning and ending times of
+the transport by determining when the ion is within a
+given distance of its asymptotic positions. Such a choice
+must always be made if we are to take into account the
+influence of the electrode filters on the results, just as
+similar cutoffs must be chosen when studying the time
+response of such analog filters more generally. In partic-
+ular, here we choose a distance from the fitted asymptotes
+that equals the ground-state extent of the 1.7 MHz trap
+potential, approximately 8 nm. This rather arbitrary de-
+cision, as well as our selection of Eq. 1 as a model for ion
+position, both play an outsized role in our determination
+of the mean ion speed and highlight the need to define
+these terms with sufficient detail in studies of ion trans-
+port. With these choices we determine a mean speed of
+83(2) m/s, slightly lower than the naive estimate.
+To avoid additional gate errors within a quantum al-
+gorithm, fast transport must not excite excessive ion mo-
+tion. For trapped ions in thermal states of motion probed
+in the Lamb-Dicke regime, one can measure the ratio of
+the first red and first blue sideband excitations to de-
+termine the mean thermal mode occupation (⟨nth⟩) [29].
+However, characterization of non-thermal (e.g. coherent)
+distributions is more complicated since the ratio of first
+sidebands can vary with the probe duration. Given that
+fast transport can leave the ion with a large coherent exci-
+tation (⟨ncoh⟩) [10, 11], we directly fit the time-dependent
+excitation of the first blue sideband assuming a convolu-
+tion of coherent and thermal distributions [11].
+To measure the ion’s axial motional excitation after the
+full transport (diagram in Fig. 1b), we apply the same
+waveform as before but do not shelve the ion with an ini-
+tial π-pulse. Instead, following the ion’s adiabatic return
+to its initial position, we drive the blue axial motional
+sideband of the S1/2−D5/2 transition and we analyze the
+dependence of S1/2 state populations PS on pulse dura-
+tion (sideband flopping curves). This probability is sen-
+sitive to the motional state and yields information about
+both the average mode occupation and its statistical dis-
+tribution [30]. Neglecting the radial modes, it is given
+by
+PS(t) = 1
+2
+�
+1 + e−γt
+∞
+�
+n=0
+pn cos(2Ωn,n+1t)
+�
+(3)
+where pn is the mode population fraction in the Fock
+state |n⟩, γ is a phenomenological decoherence rate, and
+Ωn,n+1 is the Rabi frequency for the first blue sideband
+transition for an ion starting in |n⟩.
+We use the full
+expression for the first blue sideband Rabi frequency [31],
+Ωn,n+1 = ηΩ0e−η2/2
+�
+1
+n + 1L1
+n(η2),
+(4)
+where Ω0 is the optical carrier Rabi frequency, η is the
+Lamb-Dicke parameter, and L1
+n is the nth associated La-
+guerre polynomial of order 1.
+As a simple model, we
+assume that any excitation can be represented as a con-
+volution of thermal and coherent contributions [11, 30],
+and we fit the measured probabilities to Eq. 3 under this
+assumption.
+Figure 3a shows a fit to the time-dependent blue side-
+band excitation, revealing a purely thermal (⟨ncoh⟩ = 0)
+
+4
+a)
+b)
+c)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Deshelve Probability
+Time (µs)
+Time (µs)
+Position (µm)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+0
+50
+100
+150
+200
+time
+Cool & 
+Prep
+Detect
+Transport
+Optical operations
+  → 
+π
+τ
+  → 
+FIG. 2.
+(a) Ion trajectory measurement sequence. An ion
+in a stationary potential at zi is cooled and prepared in the
+S1/2 level and then shelved into the D5/2 level with a reso-
+nant 729 nm π-pulse. The transport waveform to zf begins
+to play, and following a delay τ the 854 nm beam is pulsed for
+200 ns, thereby deshelving a portion of the D5/2 population.
+The ion is shuttled adiabatically back to its initial location
+zi for final detection. (b) Experimental time-dependence of
+deshelving probability sampled at 10 ns intervals after the
+start of the fast transport waveform; error bars represent the
+68% confidence interval in state populations assuming bino-
+mial statistics.
+(c) Time-dependence of ion position, with
+experimental data (black points) and empirical fit (red line,
+see main text). Here, the experimental points are obtained by
+inverting the polynomial curve in Fig. 1c with the measured
+points in (b). The fit yields a maximum speed of 251(6) m/s,
+while the waveform was designed with 30 samples at 40 ns
+intervals to shuttle the ion across a 120 µm displacement in
+1.2 µs (100 m/s).
+−20
+0
+20
+Frequency (kHz)
+0.4
+0.6
+0.8
+1.0
+Probability
+¯nth = 1.0 ± 0.2
+0
+100
+200
+300
+400
+Time (µs)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Probability
+¯nth = 1.0 ± 0.1
+¯nth = 1.0 ± 0.1
+a)
+b)
+FIG. 3.
+Characterization of ion temperature after fast trans-
+port. (a) We perform a pulse on the axial blue motional side-
+band for a variable duration after the fast transport operation
+described in Fig. 2. The red trace represents a fit of the data
+(black points) to Eq. 3, which yields a purely thermal axial
+mode excitation of 1.0(1) quanta. Here we have added to the
+transport waveform a sinusoidal oscillation near the ion ax-
+ial frequency with appropriate amplitude and phase to remove
+transport-induced coherent excitation. (b) Red and blue side-
+band lineshapes, also measured after optimized fast transport,
+confirm that the ion is nearly in the ground state. Fits (solid
+curves) to the data (individual points) confirm the low ion
+temperature: the ratio of sideband amplitudes corresponds
+to 1.0(2) quanta.
+Error bars represent the 68% confidence
+interval in state populations assuming binomial statistics
+excitation of ⟨nth⟩ = 1.0(1).
+Having determined that
+the excitation is thermal, we verify the mode temper-
+ature through a comparison of red- and blue-sideband
+transition amplitudes [29] (Fig. 3b). This yields a post-
+transport temperature of ⟨nth⟩ = 1.0(2) compared to
+⟨nth⟩ = 0.3(1) measured before transport, and we con-
+clude that the optimized transport induces an additional
+0.7(2) quanta of motional excitation.
+We note that,
+without the resonant de-excitation of motion during the
+transport operation, we measure an additional coherent
+excitation of ⟨ncoh⟩ = 61.7(6).
+In conclusion, we have developed a general method
+for experimentally characterizing ion transport trajecto-
+
+5
+ries using position-dependent optical deshelving, and we
+verified the technique in a surface-electrode ion trap by
+shuttling an ion along a linear trajectory of 120 µm (two
+electrode widths) with a 1.2 µs waveform. Owing to the
+impact of filter distortion on the transport potentials,
+the ion reaches instantaneous speeds significantly higher
+than might be naively assumed from the waveform de-
+sign. We characterized the final motional state using two
+complementary methods to fit blue sideband flop curves
+as well as red and blue sideband lineshapes.
+Even at
+this high speed the transport incurs only 0.7(2) quanta
+of axial excitation, small enough to have minimal impact
+within a quantum algorithm.
+Beyond single-ion transport through linear sections,
+this technique could also be applied to optimize fast
+merging and separation of ions into chains [10, 32]. With
+the incorporation of multiple deshelving wavelengths, the
+positions of disparate ion species could be tracked simul-
+taneously [9]. The method might prove particularly use-
+ful when optimizing the paths of ions through junctions
+of linear sections, where trajectories deviate significantly
+from straight lines both horizontally and vertically. With
+multiple deshelving beams at complementary angles one
+could isolate an ion’s position in all three dimensions.
+This work was done in collaboration with Los Alamos
+National Laboratory.
+[1] C. R. Clark, H. N. Tinkey, B. C. Sawyer, A. M. Meier,
+K. A. Burkhardt, C. M. Seck, C. M. Shappert, N. D.
+Guise, C. E. Volin, S. D. Fallek, H. T. Hayden, W. G. Rel-
+lergert, and K. R. Brown, Phys. Rev. Lett. 127, 130505
+(2021).
+[2] R. Srinivas, S. C. Burd, H. M. Knaack, R. T. Sutherland,
+A. Kwiatkowski, S. Glancy, E. Knill, D. J. Wineland,
+D. Leibfried, A. C. Wilson, D. T. C. Allcock, and D. H.
+Slichter, Nature 597, 209 (2021).
+[3] C. Ryan-Anderson, N. C. Brown, M. S. Allman, B. Arkin,
+G. Asa-Attuah, C. Baldwin, J. Berg, J. G. Bohnet,
+S. Braxton, N. Burdick, J. P. Campora, A. Chernoguzov,
+J. Esposito, B. Evans, D. Francois, J. P. Gaebler, T. M.
+Gatterman, J. Gerber, K. Gilmore, D. Gresh, A. Hall,
+A. Hankin, J. Hostetter, D. Lucchetti, K. Mayer, J. My-
+ers, B. Neyenhuis, J. Santiago, J. Sedlacek, T. Skripka,
+A. Slattery, R. P. Stutz, J. Tait, R. Tobey, G. Vittorini,
+J. Walker, and D. Hayes, Implementing Fault-tolerant
+Entangling Gates on the Five-qubit Code and the Color
+Code (2022), arXiv:2208.01863 [quant-ph].
+[4] D. Kielpinski, C. Monroe, and D. J. Wineland, Nature
+417, 709 (2002).
+[5] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol,
+and D. M. Lucas, Phys. Rev. Lett. 117, 060504 (2016).
+[6] A. Sørensen and K. Mølmer, Phys. Rev. A 62, 022311
+(2000).
+[7] L. E. de Clercq, H.-Y. Lo, M. Marinelli, D. Nadlinger,
+R. Oswald, V. Negnevitsky, D. Kienzler, B. Keitch, and
+J. P. Home, Phys. Rev. Lett. 116, 080502 (2016).
+[8] H. N. Tinkey, C. R. Clark, B. C. Sawyer, and K. R.
+Brown, Phys. Rev. Lett. 128, 050502 (2022).
+[9] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler,
+S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig,
+D. Hayes, K. Mayer, C. Ryan-Anderson, and B. Neyen-
+huis, Nature 592, 209 (2021).
+[10] R. Bowler, J. Gaebler, Y. Lin, T. R. Tan, D. Hanneke,
+J. D. Jost, J. P. Home, D. Leibfried, and D. J. Wineland,
+Phys. Rev. Lett. 109, 080502 (2012).
+[11] A. Walther,
+F. Ziesel,
+T. Ruster,
+S. T. Dawkins,
+K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and
+U. Poschinger, Phys. Rev. Lett. 109, 080501 (2012).
+[12] J. D. Sterk, H. Coakley, J. Goldberg, V. Hietala, J. Lecht-
+enberg, H. McGuinness, D. McMurtrey, L. P. Parazzoli,
+J. Van Der Wall, and D. Stick, npj Quantum Inf. 8, 68
+(2022).
+[13] W. K. Hensinger, S. Olmschenk, D. Stick, D. Hucul,
+M. Yeo, M. Acton, L. Deslauriers, C. Monroe, and
+J. Rabchuk, Appl. Phys. Lett. 88, 034101 (2006).
+[14] R. B. Blakestad, C. Ospelkaus, A. P. VanDevender,
+J. H. Wesenberg, M. J. Biercuk, D. Leibfried, and D. J.
+Wineland, Phys. Rev. A 84, 032314 (2011).
+[15] K. Wright, J. M. Amini, D. L. Faircloth, C. Volin, S. C.
+Doret, H. Hayden, C.-S. Pai, D. W. Landgren, D. Deni-
+son, T. Killian, R. E. Slusher, and A. W. Harter, New J.
+Phys. 15, 033004 (2013).
+[16] C. Decaroli, R. Matt, R. Oswald, C. Axline, M. Ernzer,
+J. Flannery, S. Ragg, and J. P. Home, Quantum Sci.
+Technol. 6, 044001 (2021).
+[17] W. C. Burton, B. Estey, I. M. Hoffman, A. R. Perry,
+C. Volin, and G. Price, Transport of multispecies ion
+crystals through a junction in an RF Paul trap (2022),
+arXiv:2206.11888 [physics.atom-ph].
+[18] T. Puppe, P. Maunz, T. Fischer, P. W. H. Pinkse, and
+G. Rempe, Phys. Scr. 2004, 7 (2004).
+[19] J. Du, W. Li, R. Wen, G. Li, P. Zhang, and T. Zhang,
+Appl. Phys. Lett. 103, 083117 (2013).
+[20] M. Maiuri, M. Garavelli, and G. Cerullo, J. Am. Chem.
+Soc. 142, 3 (2020).
+[21] R. Bowler, U. Warring, J. W. Britton, B. C. Sawyer, and
+J. Amini, Rev. Sci. Instrum. 84, 033108 (2013).
+[22] N. D. Guise, S. D. Fallek, K. E. Stevens, K. R. Brown,
+C. Volin, A. W. Harter, J. M. Amini, R. E. Higashi, S. T.
+Lu, H. M. Chanhvongsak, T. A. Nguyen, M. S. Marcus,
+T. R. Ohnstein, and D. W. Youngner, J. Appl. Phys. 117,
+174901 (2015).
+[23] S. Charles Doret, J. M. Amini, K. Wright, C. Volin,
+T. Killian, A. Ozakin, D. Denison, H. Hayden, C.-S. Pai,
+R. E. Slusher, and A. W. Harter, New J. Phys. 14, 073012
+(2012).
+[24] The trap filters consist of two low-pass stages: the first is
+a series inductor with shunt capacitor to ground, and the
+second is a series resistor with a shunt capacitor tuned
+to damp voltage ‘ringing’ from the first stage.
+[25] C. Roos, T. Zeiger, H. Rohde, H. C. N¨agerl, J. Eschner,
+D. Leibfried, F. Schmidt-Kaler, and R. Blatt, Phys. Rev.
+Lett. 83, 4713 (1999).
+[26] The use of a single optical deshelving laser beam does
+not allow us to directly distinguish between different tra-
+jectories perpendicular to the laser beam propagation di-
+rection. Additional orthogonal beam orientations would
+
+6
+permit a full characterization of the three-dimensional
+ion trajectory.
+[27] S. L. Todaro, Scalable State Detection and Fast Transport
+of Trapped-Ion Qubits for Quantum Computing, Ph.D.,
+University of Colorado at Boulder, United States – Col-
+orado (2020).
+[28] Final state detection is independent of the return speed
+after deshelving provided that the ion is not sufficiently
+excited during the return to modify the its fluorescence.
+[29] Q. A. Turchette, Kielpinski, B. E. King, D. Leibfried,
+D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sack-
+ett, C. S. Wood, W. M. Itano, C. Monroe, and D. J.
+Wineland, Phys. Rev. A 61, 063418 (2000).
+[30] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano,
+and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).
+[31] D. Wineland,
+C. Monroe,
+W. Itano,
+D. Leibfried,
+B. King, and D. Meekhof, J. Res. Natl. Inst. Stand. Tech-
+nol. 103, 259 (1998).
+[32] T. Ruster,
+C. Warschburger,
+H. Kaufmann,
+C. T.
+Schmiegelow,
+A. Walther,
+M. Hettrich,
+A. Pfister,
+V. Kaushal, F. Schmidt-Kaler, and U. G. Poschinger,
+Phys. Rev. A 90, 033410 (2014).
+

3NAzT4oBgHgl3EQf9P6r/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:02b10055ca8e0c10b5510da5b3c1c64fdd235766cf4f4715612beb1e29a0206a
+size 145904

3dE1T4oBgHgl3EQf5wXA/content/tmp_files/2301.03516v1.pdf.txt

@@ -0,0 +1,671 @@
+Coherent Stokes Raman scattering microscopy
+(CSRS)
+SANDRO HEUKE1,* AND HERVÉ RIGNEAULT1,*
+1Aix Marseille Univ, CNRS, Centrale Marseille, Turing Center for Living Systems, Institut Fresnel,
+Marseille, France.
+*Corresponding authors: Sandro.Heuke@fresnel.fr & herve.rigneault@fresnel.fr
+Abstract:
+We report the first implementation of laser scanning Coherent Stokes Raman
+scattering (CSRS - pronounced "sCiSsoRS") microscopy. To overcome the major challenge in
+CSRS imaging, we show how to suppress the fluorescence background by narrow bandpass
+filter and a lock-in based demodulation. Near background free CSRS imaging of polymer beads,
+human skin, onion cells, avocado flesh and the wing disc of a drosphila larva are presented.
+Finally, we explain and demonstrate numerically that CSRS solves a major obstacle of other
+coherent Raman techniques by sending a significant part (up to 100%) of the CSRS photons into
+the backward direction under tight focusing conditions. We believe that this discovery will pave
+the way for numerous technological advances, e.g. in epi-detected coherent Raman multi-focus
+imaging, real-time laser scanning based spectroscopy or efficient endoscopy.
+© 2023 Optica Publishing Group under the terms of the Optica Publishing Group Publishing Agreement
+1.
+Introduction
+Conventional bright-field microscopy provides information about the refractive index and
+absorption properties, but cannot elucidate the sample’s chemical composition.
+Infra-red
+absorption and linear Raman scattering retrieve the chemical fingerprint [1,2], but are incompatible
+with high spatial resolution or real-time imaging. Coherent Raman imaging (CRI) fills this
+technological gab joining a chemical bond specific contrast with signal levels that permit
+video-rate image acquisition. Well established CRI microscopy techniques are the coherent
+anti-Stokes Raman scattering (CARS) [3, 4] and stimulated Raman scattering (SRS) [5–7].
+CARS owes its wide-range application to the blue-shifted anti-Stokes radiation which greatly
+facilitates its separation from linear fluorescence. When working with near infra-red excitation
+wavelength, the blue-shifted CARS radiation is readily detected using photo-election multiplier
+tubes (PMT) of standard laser scanning microscopes. SRS’s popularity arises from the homodyne
+signal amplification that frees SRS images from an omnipresent non-resonant four-wave-mixing
+background and allows for measurements under daylight conditions.
+Overshadowed by CARS and SRS until now, there exists a 3rd four-wave-mixing process termed
+coherent Stokes Raman scattering (CSRS, "Scissors") [8–10] which is always appearing within
+any CARS or SRS experiment and provides near identical mapping of molecular oscillators [11] -
+see Fig.1. In analogy to the Stokes emission in linear Raman microscopy, the CSRS radiation
+(2𝜔𝑆 − 𝜔𝑝) is red-shifted with respect to the excitation frequencies of the pump (𝜔𝑝) and
+Stokes beams (𝜔𝑆). Surprisingly, CSRS was not yet implemented for laser scanning microscopy.
+Presumably, this neglect must be attributed to the high degree of resemblance of CARS and
+CSRS spectra [11] rendering CSRS - prima facie - to be either CARS with an added fluorescence
+background when working with visible light sources or, using near infra-red (NIR) excitation,
+CARS with a radiation wavelength offside high quantum yields of common detectors. CSRS
+provides, however, some unique properties that are of high interest for imaging. (1) The CSRS
+spectrum differs from CARS in the presence of accessible electronic resonances. For example,
+pre-resonant CSRS will offer complementary information in application to alkyne-labeled
+dyes [12] and standard dyes used in microbiology [13]. (2) The red-shifted radiation of CSRS
+arXiv:2301.03516v1  [physics.optics]  9 Jan 2023
+
+Fig. 1. Coherent Raman imaging techniques in energy diagrams, relative radiation
+wavelength and energy conservation under plane-wave illumination.
+imaging becomes an advantage for UV or near-UV excitation where CARS photons [14] would
+be too far blue-shifted to be detected efficiently while any SRS image [15] is likely to be
+compromised by various artifacts such as multi-photon absorption [16,17]. Thus, UV excited
+CSRS holds the potential to achieve the highest possible spatial resolution (𝜆Stokes/[
+√
+8𝑁 𝐴])
+in coherent Raman imaging. (3) NIR-excitation wavelength combined with CSRS may allow
+for deeper tissue imaging due to the reduced scattering and absorption of its radiation [18]. (4)
+Last but most important: Due to a modified phase-matching geometry, CSRS microscopy can
+be configured to radiate more light into the backward direction which will add game-changing
+benefits for the investigation of thick samples, real-time spectroscopy, multi-focus imaging and
+endoscopy [19]. Within this contribution, we want to open up the field of laser scanning CSRS
+imaging by demonstrating CSRS microscopy within the visible excitation spectrum. To remove
+the major obstacle, we will show how linear fluorescence can be suppressed by a set of bandpass
+filter and nearly nullified in combination with a lock-in based detection scheme as a premise
+for near-UV excited CSRS imaging with a lateral resolution < 100 nm. Furthermore, we shall
+investigate numerically CSRS’ spatial radiation behavior under NIR excitation paving the way
+towards CSRS experiments with an efficient epi-detection.
+2.
+Experimental result and discussion
+The CSRS signal of biomedical samples is readily overwhelmed by linear fluorescence. Time-
+gating [20], a time-resolved detection using streak cameras [21] or polarization filtration can be
+used to reduce or suppress any fluorescence signal. These methods require, however, either a
+substantial alteration of standard coherent Raman microscopes or do not work in the presence of
+large quantities of fluorescence light. Here, we exploit the fact that the CSRS is spectrally narrow
+under ps-excitation. Thus, the majority of fluorescence is readily suppressed by the choice of a
+the narrow-band filter. Filters with a spectral width below < 1nm are commercially available but
+the selection of a specific center wavelength requires expensive costume solutions. This is the
+reason why we use a combination of two inexpensive bandpass filter with a width of about 15 nm,
+but different center wavelength. In a addition, we fine-tune the filter transmission by a tilt (<20◦)
+with respect to the incident beam. Thus, two tilt-adjusted bandfilter create a sharp transmission
+line (FWHM<3nm) for the CSRS signal while rejecting significant parts of the autofluorescence.
+
+CARS
+SRS
+CSRS
+kas+ks=kp1+kp2
+111
+CARS
+ks1+kp1=ks2+kp2
+SRS
+dwnd
+Stokes
+SRS
+kcs+k,=ks1+ks2
+Intensity
+CARS
+CSRS
+CSRS
+2Fig. 2. CSRS experimental implementation and characterization. Bottom left: The
+CSRS signal is separated from fluorescence by means of 2 angle-tuned narrow bandpass
+filter. Bottom right: Additional suppression of fluorescence is achieved by intensity
+modulating the Stokes and pump beam at the radio frequencies f1 and f2, respectively.
+Fluorescence free CSRS signal is obtained at f1-f2. Right center: Time separation of
+the pump and Stokes pulses as well as blocking the excitation highlights the superior
+suppression of fluoresence background at the demodulation frequency f1-f2 compared
+to CSRS signal obtained at f1, f2 or the DC frequency. Top right: The intensity
+profile at the interface of a PMMA bead and olive oil indicates a lateral resolution
+of <400nm. Top left: scheme of the CSRS experiment. 1 Yb-fiber laser, 2 optical
+parametric oscillator (OPO), 3 Second harmonic generation (SHG), 4 acousto-optic
+modulator (AOM), Laser scanning microscope (LSM), 6 photo-electron multiplier
+(PMT), 7 Lock-in amplifier.
+
+IcSRS
+PSF.
+CSRS demodulated at
+20μm
+DC, f1, f2, f1-f2
+PMMA bead
+<400nm
+G
+5 μm
+88
+0
+2
+(
+1
+3
+x / μm
+④ f2
+Lock-in
+DC
+f1
+3
+f2
+f1-f2
+lp=0
+④f1
+Is=0
+△t>>3ps
+T
+CSRS
+Intensity
+ump2
+f1-f2
+Fluorescence
+SRS
+concefiFig. 3. LSM-CSRS at 2850 cm−1. The left and right column show the CSRS image
+demodulated at the frequencies f1-f2 = 1.47 MHz and 0 Hz (DC). To estimate the
+remaining fluorescence level, images without temporal overlap of the pump and Stokes
+pulses are displayed to the right. a) Mixture of polystyrene (PS, 30µm) and Poly-methyl-
+methacrylate (PMMA, 20µm) beads in olive oil. b) and c) Epithelium and dermis of a
+20µm thick human skin section d) Cells of an onion. e) Lipid droplets within the flesh
+of an avocado. d) Wing disc of a Drosophila larva. The white and blue scale bar equals
+20µm and 5µm, respectively.
+
+CSRS at f1-f2
+△t >> 3ps
+CSRS at DC
+△t >> 3ps
+PS
+PMMA
+olive.oil
+Human skin
+Epithelium
+Dermis
+Onion
+avocado
+Drosophila larva
+wing disc
+notum
+hinge
+pouchAs a second method for fluorescence discrimination, we take advantage of CSRS intensity
+dependence on both excitation colors while linear fluorescence follows either the intensity of
+the pump or the Stokes laser. Consequently, modulating the pump and Stokes beams at f1 and
+f2 while demodulation the signal at f1-f2 (or f1+f2) yields exclusively nonlinear signals that
+depend on both excitation colors. The f1-f2 demodulation, therefore, also discriminates the CSRS
+signal against 2-photon excited fluorescence (2PEF) under single-color excitation. It shall be
+noted that the double modulation is also sensitive to two-color 2-photon fluorescence (2C-2PEF).
+Nevertheless, we will find experimentally, that the emission strength of native 2C-2PEF is
+negligible within our CSRS approach.
+For the experimental implementation of CSRS into laser scanning microscopy, we chose visible
+excitation wavelengths at 445nm (pump) and 515nm (Stokes) for the following reasons: (1)
+CSRS under near UV excitation is a potentially important application area since the CARS signal
+falls into the UV range while SRS artifacts are increased due the high concentration of matching
+chromophores. (2) The red-shifted CSRS radiation is readily detected by ordinary PMTs. (3)
+Fluorescence artifacts are enhance compared to a near infra-red (NIR) excitation. Thus, our
+approach will be viable as well for CSRS under NIR excitation, if pure CSRS signals can be
+obtained under VIS excitation. The experimental implementation, the spectral filtration and
+the double modulation are schematically shown in Fig.2a. Our implementation resembles a
+standard SRS setup with the difference that we use visible excitation wavelengths, we modulate
+not one but both beams and the photo-diode is replaced by a PMT which is connected to a
+lock-in amplifier. More information about the setup can be found within the part Methods:
+Experimental setup. To quantify the level of fluorescence rejection, we investigated the signal
+of native olive oil at 2850 cm−1 when blocking the Stokes or pump beams and when the
+temporal pulse overlap is removed. The output signal of the lock-in is plotted as functions
+of the demodulation frequencies at 0 Hz (DC), f1, f2 and f1-f2 in Fig.2. It can be observed
+that the DC channel contains significant amounts of fluorescence while this artifact is already
+reduced within the channels f1 and f2. Nevertheless, only the difference frequency channel at
+f1-f2 becomes dark, when the excitation pulses do not overlap in time. In a second experiment,
+we imaged the interface of olive oil and a 20µm sized Plexiglas (PMMA) bead to obtain an
+estimation of the lateral resolution for an excitation objective featuring an NA of 1.45 - see
+Fig.2. From this "knife-edge" CSRS intensity profile, we can infer a lateral resolution below
+400nm. The difference to the expected 𝜆𝑆𝑡𝑜𝑘𝑒𝑠/[
+√
+8𝑁 𝐴]= 515nm/[
+√
+81.49]=120nm can be
+attributed to underfilling of the excitation objective lens and the bent oil/bead interface. Having
+confirmed a high-resolved, fluorescence-free CSRS image contrast, we investigated the suitability
+of LSM-CSRS for vibrational imaging of various objects featuring non-negligible fluorescence
+levels. Within Fig. 3, we show the CSRS images of test and biomedical samples demodulated at
+the DC and f1-f2 frequencies for (non-)overlapping pump and Stokes pulses. The images were
+organized along the ratio of the CSRS to fluorescence signal starting from the highest at the
+top. Comparing the DC and f1-f2 images in Fig. 3a, it obvious that a narrow spectral filtering is
+already sufficient for CSRS imaging of polymer beads in oil. The first artifacts become visible for
+the DC CSRS images of the epithelium and dermis of a 20µm thick section of human skin - see
+Figs. 3b and c. For the epithelium, a pronounced fluorescence artifact arises from melanin within
+the Stratum basale. Artifacts within the Dermis can be attributed to the auto-fluorescence of
+collagen and elastin [22]. The quantity of fluorescence observed within the DC channel increases
+stepwise further for CSRS imaging of onion cells, lipid droplets within the flesh of an avocado
+and the wing disc of a Drosophilia larva. From the second row of Fig. 3, it is reconfirmed that
+almost no fluorescence is leaking into the f1-f2 CSRS channel as an important condition for the
+estimation of the true concentration of the targeted molecular group. The origin of fluorescence
+for these 3 samples, however, cannot be attributed with certainty, but might arise from NADH,
+flavins and chlorophyll.
+
+In a broader context, we would like to point out that other nonlinear microscopy techniques would
+also greatly benefit from the narrow-band filter plus demodulation combination for rejection of
+spurious background signals. For example, the 2PEF signal of chlorophyll in plant leaves readily
+overwhelms any CARS or second harmonic generation (SHG) image contrast even under NIR
+excitation. A double modulation of the excitation combined with a lock-in based demodulation
+will purify the signal, reduce the sensitivity against other light sources such as room light
+and reestablish the reliability of the following image analysis. Having removed the why-not
+argument for the CSRS image contrast, we shall introduce in the next section a non-intuitive
+but game-changing argument for CSRS microscopy : the increased backwards radiation as the
+prerequisite of an effective epi-CSRS detection.
+3.
+Numerical results
+In this section, we shall show and explain CSRS’ superior backward radiation properties. Before
+entering into the calculations, we want to consider CSRS from a heuristic viewpoint investigating
+the momentum conservation laws for CSRS and compare it to CARS. Under plane illumination,
+the momentum conservation laws can be written as K = k𝑝 − k𝑆 + k𝑝 − k𝑎𝑆 for CARS [23] and
+K = k𝑆 − k𝑝 + k𝑆 − k𝑐𝑆 for CSRS with K, k𝑝, k𝑆, k𝑎𝑆 and k𝑐𝑆 representing the wavevectors
+of the object, the pump(probe) and Stokes beam as well as the anti-Stokes and coherent Stokes
+radiation, respectively. Note that for homogeneous samples these laws are also referred to as
+phase-matching condition and simplify to k𝑝 + k𝑝 = k𝑆 + k𝑎𝑆 (CARS) and k𝑆 + k𝑆 = k𝑝 + k𝑐𝑆
+(CSRS). Under focusing conditions, the single wavevectors are replaced by the distribution of
+incident wavevectors which are distributed over a cap of a sphere. To identify those object
+frequencies (K) that are effectively probed, every combination of excitation and emission
+wavevector must be identified. This operation is equivalent to the convolution of the caps of
+the illumination and detection Ewald spheres. Neglecting polarization effects, the result of this
+convolution (simplified to 3 points per arc) is shown in 2D within Fig. 4a.
+Evidently, there exist no vector combination for epi-scattered CARS photons which would
+cover the origin K(0,0,0) of the object space. Thus, a homogeneous sample, such as olive oil,
+does not provide any backward radiation. On the contrary, structures that feature high object
+frequencies, such as small polymer beads or layered materials, generate Epi-CARS radiation.
+In the past, Epi-CARS was occasionally considered to be a size selective contrast that would
+highlight exclusively small objects [24]. While this statement holds for the majority of biomedical
+samples, there do exist large structures, e.g. multi-layered lipids in vesicles that also emit a strong
+CARS radiation into the backward direction. Hence, it is more appropriate to refer to Epi-CARS
+as a technique that probes high object frequencies instead of been considered as size selective.
+Switching the detection wavelength to the red-shifted coherent Stokes radiation changes the
+covered object support significantly and includes now the origin at K(0,0,0). Due to the reduced
+size of the detection wavevector (|k𝑐𝑆| ≪ |k𝑎𝑆|) and the pump vector entering as complex
+conjugated, see Eq. 3, it is now possible to find vector combinations that cover the origin
+at K(0,0,0). Consequently, even a homogeneous object will radiate considerable amounts of
+Epi-CSRS. Nevertheless, since the the centroid of the Epi-CSRS object support, i.e. the gray
+cloud within Fig. 4a, does not coincidence with the K-space origin, Epi-CSRS images will also
+highlight objects containing higher frequencies.
+To address the question of how to increase the ratio of Epi versus forward Epi-CSRS, and which
+object frequencies are most efficiently probed using Epi-CSRS, we performed finite element
+simulations whose results are summarized in Fig. 4b-e. The equations implemented numerically
+as well as important parameters are found in the annex - numerical calculation. From the
+momentum conservation law and the vector diagrams in Fig. 4a, it is readily comprehensible
+that a larger wavelength difference in between the pump and coherent Stokes wavelength relaxes
+greatly the necessity for extreme incident illumination angles of the Stokes beam. Furthermore,
+
+Fig. 4. Object frequency support and radiation behavior of CSRS versus CARS. a)
+The object K-support for Epi-CSRS(CARS) is found by convolving the illumination
+Ewald spheres of the Stokes (pump), pump (Stokes), and Stokes (probe) with the cap
+of detection Ewald sphere at (anti-)Stokes frequency. Note that vector combinations
+covering the frequency of a homogeneous sample K(0,0,0) are only found for CSRS
+but not for CARS. A single wavevector combination that phase-matches K(0,0,0) is
+highlighted to the left while a similar approach for CARS leads to a large phase-mismatch
+(ΔK). b) CSRS and CARS radiation behavior of a homogeneous sample under standard
+illumination condition, i.e. the pump and Stokes beam fill the objective aperture
+homogeneously (𝜃𝑚𝑎𝑥=80◦). c) same as in b) but with an annular pupil filter applied
+to the Stokes beam for CSRS covering 50% of area of the objective back-aperture. For
+an equitable comparison with CARS, the same pupil filter was applied to the pump
+beam. d) same as for b) but the homogeneous sample was replaced by a frequency
+object whose scatter density is described as 1 + cos(2𝜋𝑧/𝜆𝑜) and 𝜆𝑜=1µm. e) Plot of
+the ratio of backward/forward radiation (Rb/f) as a function of the object frequency 𝜆𝑜.
+
+Ks1
+Ks2
+kcs
+Kp
+(a)
+个 Kz
+个Kz
+个Kz
+Epi
+Ks1
+Epi-CSRS
+Ks2
+Kcs
+0
+0
+0
+0
+0
+Kx
+0
+Kx
+Kp1
+ks
+Kp2
+kas
+Epi
+个Kz
+个 Kz
+个 Kz
+Epi-CARS
+0
+Kp2
+AK
+kas
+0
+0
+o
+Kx
+0
+Kx
+0
+(b)
+(c)
+(d)
+(e)
+Rb/f = 0.79*10-2
+Rb/f = 1.54
+Rp/f = 0.25
+%
+30
+ratio forward/backward in
+5N
+IN
+z
+1.5
+20
+CSRS
+0.5 
+10
+5
+5
+0.5
+20
+0.66
+0.5
+x
+y
+y
+y
+Rb/f = 0.18*10-3
+Rb/f = 0.33*10-2
+z
+z
+z
+1.5
+1.5
+4
+CARS
+0.5 -
+2
+0.5
+1
+20
+0.66
+0.5
+y
+y
+2。 / μm
+Stokes
+pump
+Stokes
+pump
+z
+z
+50%
+"standard CRS"
+pupil filtering
+frequency objectsince most of the coherent Raman experiments apply NIR instead of VIS excitation wavelength,
+we used for within our simulations the wavelength 𝜆𝑝 = 797𝑛𝑚 and 𝜆𝑆 = 1030𝑛𝑚 which matches
+the most commonly targeted Raman shift in CRI imaging at 2850cm−1. For these conditions, the
+coherent Stokes radiation will be observed at 𝜆𝑐𝑆 = 1450𝑛𝑚. It shall be noted that our results
+equally apply for the visible excitation wavelength with gently higher excitation angle or thinner
+annular masks.
+To start with, we computed the radiation pattern of CSRS and CARS of a homogeneous object
+using an NA of 1.49 (oil immersion) corresponding to a maximum illumination angle of 80◦.
+From Fig. 4b, it is evident that both CARS and CSRS are predominately forward directed
+though the CSRS’ radiation distribution features a larger radiation cone. Considering the ratio of
+backward versus forward directed photons Rb/ 𝑓 , we find numerically that less than 1 photon in
+105 is backward directed for CARS. Note that the momentum conservation actually law predicts
+Rb/ 𝑓 =0 for CARS. Thus, the resulting deviation must be attributed to the finite number of voxels
+of the numerical model. For CSRS, Rb/ 𝑓 increase dramatically to about 1 in 100 photons.
+Since common surfaces within biomedical samples scatter more than 1%, we have to assume,
+however, that also epi detected CSRS will be just forward generated CSRS that was redirected
+by linear scattering at an interface. Still, using a confocal detection, i.e. a pinhole in front of
+the detector placed at the conjugated plane of the excitation focus, might already yield true
+Epi-CSRS images of homogeneous samples where Epi-CARS images would remain dark. To
+find an approach that increases the proportion of CSRS’ epi radiation, we shall consider the
+CSRS vector diagram matching K(0,0,0) on the left of Fig. 4a. The ratio of backward versus
+forward radiation is readily increased by reducing the impact of vectors combinations probing
+higher frequencies and favoring those covering the origin. This boost of epi-CSRS radiation
+can be achieved using an annular illumination of the Stokes beam. Experimentally, such an
+annular illumination is generated, without power-loss, using 2 axicons within the Stokes beam
+path [25,26]. Numerically, we restricted the incident angles for the Stokes between 𝜃𝑚𝑖𝑛=56.5◦
+and 𝜃𝑚𝑎𝑥=80◦, which corresponds to covering 50% of the area of the objective lens’ back-focal
+plane. With this pupil filtering, the radio of backward to forward radiation increased for CARS to
+2 in 104 photons while the majority of all CSRS radiation is backward directed (Rb/ 𝑓 =1.5) when
+focusing the pump and Stokes beam into a homogeneous object - see Fig. 4c.
+As a second important result from the heuristic derivation of CSRS’ object support, we found that
+the presence of high object frequencies increases the amount of backward radiation. To confirm
+this prediction, we investigated in Fig. 4d and e an object whose nonlinear scatterer density,
+i.e. concentration of molecular groups, is modulated along the optical axis as 1+cos(𝐾𝑧𝑧) with
+𝐾𝑧 = 2𝜋/𝜆𝑜 being the object frequency. As an example, Fig. 4d outlines the radiation behavior
+of a wave-like structured object with K𝑧=2𝜋/1µm. It is found that R𝑏/ 𝑓 increases to one forth
+for Epi-CSRS while Epi-CARS remains negligible weak. To identify those object frequencies
+which are most efficiently probed by Epi-CSRS, we computed R𝑏/ 𝑓 as a function of K𝑧. From
+Fig. 4e, we find that Epi-CSRS peaks at K𝑧=2𝜋/1µm whereas Epi-CARS R𝑏/ 𝑓 still increases at
+𝐾𝑧 = 2𝜋/0.25µm. It shall be note that the Rb/ 𝑓 never reaches 1 which arise from the 1+ within
+the definition of the frequency object (1+cos(𝐾𝑧𝑧)). The 1+ implies that the wave-object always
+features twice the amplitude at K(0,0,0), which corresponds to a homogeneous predominantly
+forward scattering object, compared to the scatterer density modulation K(0,0,±K𝑧). Our
+simulation results in a nutshell: we have found that CSRS features a non-negligible backward
+radiation from a homogenous sample under tight-focusing conditions while this is not the case for
+CARS. The amount of backward radiated CSRS can be enhanced by increasing the illumination
+power of the Stokes beam with high incident angles. Furthermore, the natural structure of
+biomedical samples, which are usually not homogeneous, will also elevate the CSRS backward
+radiation.
+
+Conclusion
+We have demonstrated the first laser scanning microscopy CSRS experiment. As the major
+challenge, we were able to reduce the fluorescence background significantly using a pair of tilted
+bandpass filter. The remaining fluorescence contribution is removed by intensity modulating the
+Stokes and pump beams at the radio frequencies f1 and f2 and a lock-in based demodulation of
+the CSRS signal. Taking advantage of CSRS’ characteristic dependence on both excitation colors,
+the best fluoresence background suppression is obtained when demodulating the CSRS signal at
+f1-f2. Background-free LSM-CSRS imaging was demonstrated for samples of decreasing ratio
+of CSRS to fluoresence signal, namely: polymer beads, the epithelium and dermis of human
+skin, onion cells, avocado flesh and the wing disc of a Drosophila larva. Having removed the
+major obstacle for CSRS imaging, we introduced and quantified numerically the major interest of
+CSRS which is its unique backward radiation property in combination with high NA objective
+lenses. CSRS’ backward radiation and its distinction from CARS is readily understood from the
+momentum conservation laws when considering all incident k-vectors forming the excitation focal
+spots. Using dynadic Green functions, we show numerically that the CSRS is predominantly
+forward directed for a homogeneous object, but the backward CSRS contribution rises to 1/4 for
+objects that are structured axially. Moreover, backward CSRS signal can even dominate forward
+CSRS (up to 100%) if an annular Stokes illumination is applied. With an efficient Epi-CARS
+radiation at hand, various coherent Raman experiments become feasible which were impossible
+before. Just to name a few: Epi-detected confocal multi-focus CSRS; Epi-detected LSM-CSRS
+with a spectrometer at the descanned position; Epi-detected CSRS image scanning microscopy.
+Thus, we believe that this contribution is just the first milestone in CSRS microscopy with many
+others to follow.
+Methods: Experimental setup
+A Yb-based fiber laser (APE Emerald engine, 80 MHz, 2–3 ps) is frequency doubled yielding 7 W
+of 515 nm output power. Parts of the emissions is used directly as Stokes beam to drive the CSRS
+process. The major part (4 W) of the 515 nm is employed to pump an optical parametric oscillator
+(OPO, APE Emerald). The OPO’s signal beam is tunable to 660-950 nm and coupled into an
+external SHG unit. The latter generates up to 50 mW within the spectral range of 330-475 nm
+serving as pump beam for the CSRS four wave mixing. Thus, the 330-475 nm pump combined
+with the 515 nm Stokes beam allows addressing a Raman shift range from 1630-11000cm−1. The
+pump and Stokes beams are superimposed in space and time via a dichroic beam splitter (Semrock,
+FF470-Di01-25x36) and a delay stage. Both beams are coupled into a home-built laser scanning
+microscope and focused by a 40x water objective lens (Nikon, Plan, NA = 1.15, immersion:
+water) into the sample. The excitation objective lens was replaced for a 60x objective (Nikon, Plan
+Apo TIRF, NA 1.45, immersion:oil) to generate the bead-oil interface image within Fig. 2. The
+CSRS radiation is collected by a condenser lens (Nikon, Achr-Apl, NA 1.4) in forward direction,
+spectrally separated from the broadband fluorescence background by means of 2 tilted bandpass
+filter (Semrock FF01-620/14-25 + FF01-605/15-25) and detected by a photo-electron multiplier
+(PMT, Thorlabs, PMT1001). For an enhanced suppression of the linear fluorescence background,
+2 acousto-optic modulators (AOM, AA, MT200-A0.5-VIS) were applied to modulate the intensity
+of the Stokes and pump beams and at the frequencies f1 = 2.28 MHz and f2 = 3.75 MHz,
+respectively. The PMT output was demodulated simultaneously at the DC frequency, f1, f2 and
+at f1-f2 = 1.47 MHz using a lock-in amplifier (Zürich instruments, HF2LI). The lock-in time
+constant was set to 30 µs. All CSRS-images shown were recorded with a pixel dwell time of
+40 µs.
+
+Annex - numerical calculation
+In the following, we shall summarize the equations used to generate Fig. 4b-e. The meaning of
+the variables is summarized in Fig. 5.
+The focused field at the sample is given by the angular spectrum representation [27]:
+���������
+𝐸𝑥(𝜌, 𝜙, 𝑧)
+𝐸𝑦(𝜌, 𝜙, 𝑧)
+𝐸𝑧(𝜌, 𝜙, 𝑧)
+���������
+= 𝑖𝑘 𝑓
+2 exp(−𝑖𝑘 𝑓 )
+���������
+𝐼00 + 𝐼02 cos(2𝜙)
+𝐼02 sin(2𝜙)
+−𝑖2𝐼01 cos(𝜙)
+���������
+(1)
+Here 𝑓 denotes the focal length of the objective lens and the integrals 𝐼0𝑚 are provided by
+𝐼0𝑚 =
+∫
+𝜃𝑚𝑎𝑥
+𝜃𝑚𝑖𝑛
+𝐸𝑖𝑛𝑐(𝜃) sin(𝜃)[cos(𝜃)]1/2𝑔𝑚(𝜃)Jm[𝑘𝜌 sin(𝜃)]d𝜃
+(2)
+where 𝑔𝑚 equals 1 + cos(𝜃), sin(𝜃) and 1 − cos(𝜃) for 𝑚 = 0, 1, 2, respectively. 𝐽𝑚 is the
+𝑚𝑡ℎ order Bessel function while 𝐸𝑖𝑛𝑐 is the incoming electric field which we assumed to be
+x-polarized and constant within the (annular) aperture angles 𝜃𝑚𝑖𝑛 ≤ 𝜃 ≤ 𝜃𝑚𝑎𝑥. The nonlinear
+polarization at anti-Stokes and coherent Stokes wavelength is given by:
+𝑃(3)
+𝑎𝑆,𝑎(𝑟) = 3𝜒(3)
+𝑎𝑏𝑐𝑑(𝑟)𝐸 𝑝,𝑏𝐸∗
+𝑆,𝑐𝐸 𝑝,𝑑
+𝑃(3)
+𝑐𝑆,𝑎(𝑟) = 3𝜒(3)
+𝑎𝑏𝑐𝑑(𝑟)𝐸𝑆,𝑏𝐸∗
+𝑝,𝑐𝐸𝑆,𝑑
+(3)
+Fig. 5. Declaration of variables
+Where a,b,c,d represent the polarization
+coordinates x, y or z. Using an x-polarized ex-
+citation, it was noticed that 𝜒(3)
+𝑥𝑥𝑥𝑥 dominates
+all other tensor components even under tight
+focusing conditions while filling the objective
+lens homogeneously [27]. Nevertheless, for
+the generation of Fig. 4c an annular mask with
+𝜃𝑚𝑖𝑛 = 56.5◦ and 𝜃𝑚𝑎𝑥 = 80◦ was applied
+which does necessitate the inclusion of other
+tensor elements. For simplicity, we consider
+here only isotropic samples reducing the 81
+susceptibility tensor elements to 21 which
+are nonzero [28].
+Within isotropic media,
+these nonzero elements follow certain sym-
+metry rules which are, 𝜒1111 = 𝜒2222 = 𝜒3333,
+𝜒1122 = 𝜒1133 = 𝜒2211 = 𝜒2233 = 𝜒3311 =
+𝜒3322, 𝜒1212 = 𝜒1313 = 𝜒2323 = 𝜒2121 = 𝜒3131 = 𝜒3232, 𝜒1221 = 𝜒1331 = 𝜒2112 = 𝜒2332 = 𝜒3113 =
+𝜒3223. Further, it applies 𝜒1111 = 𝜒1122 + 𝜒1212 + 𝜒1221 [28]. Within our simulations we were
+setting 𝜒1122 = 𝜒1212 = 𝜒1221 = 1 and, hence, 𝜒1111 = 3. The nonlinear far-field radiation
+distributions is obtained using a dyadic Green function approach:
+
+���������
+𝐸𝑞,𝑅(𝑅, Θ, Φ)
+𝐸𝑞,Θ(𝑅, Θ, Φ)
+𝐸𝑞,Φ(𝑅, Θ, Φ)
+���������
+= −
+𝜔2
+𝑞
+𝑐2
+exp(𝑖𝑘𝑞|𝑅|)
+|𝑅|
+∭ ∞
+−∞
+𝜌d𝜌d𝜙d𝑧 exp(𝑖𝑘𝑞rR)
+|𝑅|
+×
+���������
+0
+0
+0
+cos(Θ) cos(Φ)
+cos(Θ) sin(Φ)
+− sin(Θ)
+− sin(Φ)
+cos(Φ)
+0
+���������
+���������
+𝑃(3)
+𝑞,𝑥(r)
+𝑃(3)
+𝑞,𝑦(r)
+𝑃(3)
+𝑞,𝑧(r)
+���������
+(4)
+where q is replaced by aS or cS to calculate either the anti-Stokes or coherent Stokes radiation.
+Within the simulations, we segmented the focal area into (121×121×121≈) 1.77 mio elements
+of a width of 50 nm equally spaced into the x, y and z direction. The far-field radiation sphere
+was discretized into (ΔΘ=1◦, ΔΦ=2◦) 32400 elements. The coherent (anti-)Stokes radiation
+was qualified as either forward or backward directed if falling into the range Θ.. 0-80◦ or Θ..
+100-180◦, respectively.
+Funding Information
+We acknowledge financial support from the Centre National de la Recherche Scientifique (CNRS),
+Aix-Marseille University (A-M-AAP-ID-17-13-170228-15.22-RIGNEAULT), A*Midex (ANR-
+11-IDEX-0001-02), Cancéropôle Provence-Alpes Côte d’Azur, French National Cancer institute
+(INCa), Région Sud, ANR grants (ANR-10-INSB-04-01, ANR-11-INSB-0006, ANR-16-CONV-
+0001), INSERM PC201508 and 18CP128-00.
+Data availability
+The data that support the findings of this study are available from the corresponding author upon
+reasonable request.
+Disclosure
+The authors declare no conflict of interest.
+References
+1.
+A. Bunaciu, S. Fleschin, and H. Aboul-Enein, “Infrared microspectroscopy applications - review,” Curr. Anal. Chem.
+10, 132–139 (2013).
+2.
+K. A. Antonio and Z. D. Schultz, “Advances in biomedical raman microscopy,” Anal. Chem. 86, 30–46 (2013).
+3.
+M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-stokes raman microscope,” Opt. Lett. 7, 350
+(1982).
+4.
+A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman
+scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
+5.
+P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated raman scattering microscopy,”
+New J. Phys. 11, 033026 (2009).
+6.
+C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free
+biomedical imaging with high sensitivity by stimulated raman scattering microscopy,” Science 322, 1857–1861
+(2008).
+7.
+Y. Ozeki, F. Dake, S. Kajiyama, K. Fukui, and K. Itoh, “Analysis and experimental assessment of the sensitivity of
+stimulated raman scattering microscopy,” Opt. Express 17, 3651 (2009).
+8.
+P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric
+field strength,” Phys. Rev. 137, A801–A818 (1965).
+9.
+J. B. Zheng, A. Leipertz, J. B. Snow, and R. K. Chang, “Simultaneous observation of rotational coherent stokes
+raman scattering and coherent anti-stokes raman scattering in air and nitrogen,” Opt. Lett. 8, 350 (1983).
+10. M. Cui, B. R. Bachler, and J. P. Ogilvie, “Comparing coherent and spontaneous raman scattering under biological
+imaging conditions,” Opt. Lett. 34, 773 (2009).
+
+11. S. A. Druet and J.-P. E. Taran, “Cars spectroscopy,” Prog. Quantum Electron. 7, 1–72 (1981).
+12. L. Wei, Z. Chen, L. Shi, R. Long, A. V. Anzalone, L. Zhang, F. Hu, R. Yuste, V. W. Cornish, and W. Min,
+“Super-multiplex vibrational imaging,” Nature 544, 465–470 (2017).
+13. L. Wei and W. Min, “Electronic preresonance stimulated raman scattering microscopy,” The J. Phys. Chem. Lett. 9,
+4294–4301 (2018).
+14. R. C. Prince and E. O. Potma, “Going visible: high-resolution coherent raman imaging of cells and tissues,” Light.
+Sci. & Appl. 8 (2019).
+15. Y. Bi, C. Yang, Y. Chen, S. Yan, G. Yang, Y. Wu, G. Zhang, and P. Wang, “Near-resonance enhanced label-free
+stimulated raman scattering microscopy with spatial resolution near 130 nm,” Light. Sci. & Appl. 7 (2018).
+16. P. Berto, E. R. Andresen, and H. Rigneault, “Background-free stimulated raman spectroscopy and microscopy,” Phys.
+Rev. Lett. 112 (2014).
+17. S. Heuke, A. Lombardini, E. Büttner, and H. Rigneault, “Simultaneous stimulated raman gain and loss detection
+(srgal),” Opt. Express 28, 29619–29630 (2020).
+18. D. Kobat, M. E. Durst, N. Nishimura, A. W. Wong, C. B. Schaffer, and C. Xu, “Deep tissue multiphoton microscopy
+using longer wavelength excitation,” Opt. Express 17, 13354 (2009).
+19. A. Lombardini, V. Mytskaniuk, S. Sivankutty, E. R. Andresen, X. Chen, J. Wenger, M. Fabert, N. Joly, F. Louradour,
+A. Kudlinski, and H. Rigneault, “High-resolution multimodal flexible coherent raman endoscope,” Light. Sci. &
+Appl. 7 (2018).
+20. M. Kögler and B. Heilala, “Time-gated raman spectroscopy – a review,” Meas. Sci. Technol. 32, 012002 (2020).
+21. T. Tahara and H.-O. Hamaguchi, “Picosecond raman spectroscopy using a streak camera,” Appl. Spectrosc. 47,
+391–398 (1993).
+22. S. Heuke, N. Vogler, T. Meyer, D. Akimov, F. Kluschke, H.-J. Röwert-Huber, J. Lademann, B. Dietzek, and J. Popp,
+“Multimodal mapping of human skin,” Br. J. Dermatol. 169, 794–803 (2013).
+23. S. Heuke, K. Unger, S. Khadir, K. Belkebir, P. C. Chaumet, H. Rigneault, and A. Sentenac, “Coherent anti-stokes
+raman fourier ptychography,” Opt. Express (2019).
+24. A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent
+anti-stokes raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
+25. S. Heuke, J. Zheng, D. Akimov, R. Heintzmann, M. Schmitt, and J. Popp, “Bessel beam CARS of axially structured
+samples,” Sci. Reports 5 (2015).
+26. S. Heuke, F. B. Legesse, D. Akimov, U. Hübner, J. Dellith, M. Schmitt, and J. Popp, “Bessel beam coherent anti-stokes
+raman scattering microscopy,” J. Opt. Soc. Am. B 32, 1773 (2015).
+27. J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-stokes
+raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363 (2002).
+28. J.-X. Cheng and X. S. Xie, eds., Coherent Raman Scattering Microscopy (Series in Cellular and Clinical Imaging)
+(CRC Press, 2016).
+

4NE2T4oBgHgl3EQfOQYe/content/tmp_files/2301.03745v1.pdf.txt

@@ -0,0 +1,1344 @@
+arXiv:2301.03745v1  [math.AG]  10 Jan 2023
+FOURIER–MUKAI TRANSFORMS FOR NON-COMMUTATIVE COMPLEX
+TORI
+NOBUKI OKUDA
+Abstract. Let X be a complex torus of dimension g and ˆX be the dual torus. For any g(g − 1)/2-
+tuple λ of complex numbers of absolute value 1, we define a non-commutative complex torus Xλ as
+a sheaf of algebras on a real torus of dimension g. We prove that if all components of λ are roots of
+unity, then the category of coherent sheaves on Xλ is abelian and derived-equivalent to the category
+of coherent sheaves on ˆX twisted by an element of the Brauer group of ˆX determined by λ.
+1. Introduction
+Let X and ˆX be smooth projective varieties. An integral functor
+Φ: Db(X) → Db( ˆX)
+(1.1)
+between derived categories of coherent sheaves is said to be a Fourier–Mukai functor if it is an
+equivalence. A Fourier–Mukai functor Φ induces an isomorphism
+HH(Φ): HH•(X) → HH•( ˆX)
+(1.2)
+of Hochschild cohomologies. For any u ∈ HH2(X), Toda [Tod09] gave a C[ε]/(ε2)-linear category
+Db(X, u) of first order deformations of X along u and an equivalence
+Φ† : Db(X, u) → Db( ˆX, HH(Φ)(u))
+(1.3)
+extending Φ.
+The integral functor FM with the Poincar´e line bundle as the integral kernel is the first example
+of a Fourier–Mukai functor given by Mukai [Muk81] for an abelian variety X and the dual abelian
+variety ˆX. Under the Hochschild–Kostant–Rosenberg isomorphism
+HH2(X) ∼= H0(∧2TX) ⊕ H1(TX) ⊕ H2(OX),
+(1.4)
+the induced map HH(FM) sends H0(∧2TX) to H2(O ˆ
+X). This suggests that non-commutative de-
+formations of abelian varieties are Fourier–Mukai partners of gerby deformations of dual abelian
+varieties.
+While the notion of gerby deformations is well-established in terms of twisted sheaves, non-
+commutative deformations are much harder to define in general. At the formal level, the Moyal
+deformation quantizations give formal non-commutative deformations of complex tori, and derived
+equivalences to formal gerby deformations are proved in [BBBP07]. The next problem is to construct
+non-commutative tori and derived equivalences for non-formal parameters (or even as a family over
+a complex manifold).
+The first attempt to define non-commutative complex tori is given by Schwarz [Sch01], who intro-
+duced the notion of complex structures on non-commutative tori, which are irrational rotation alge-
+bras (see [Rie81], for example) regarded as non-commutative spaces in the sense of Connes[Con94].
+Categories of holomorphic vector bundles on them are studied in [PS03].
+In the paper [Blo10] and the preprint [Blo], Block discussed Fourier–Mukai functors between non-
+commutative complex tori and gerby deformations of dual complex tori, using DG categories which
+include objects corresponding to quasi-coherent sheaves.
+In [Blo], he also announced to discuss
+coherent sheaves in a paper in preparation.
+We now explain the results of this paper. Let X = T/Γ be a complex torus, where T := (C∗)g
+and Γ is a discrete subgroup of T isomorphic to Zg. The dual complex torus ˆX := Pic0 X can
+naturally be identified with ˆΓ/ ˆT where ˆΓ := Hom(Γ, C∗) ∼= (C∗)g and ˆT := Hom(T, C∗) ∼= Zg. Let
+λ ∈ H2( ˆT, U(1)) ∼= U(1)g(g−1)/2 be an element of the second group cohomology of ˆT with values in
+1
+
+U(1). We construct a non-commutative deformation of X with parameter λ. When λ takes values
+in roots of unity, we give an equivalence of the derived category of coherent sheaves on the non-
+commutative deformation of X with parameter λ and the derived category of coherent sheaves on
+the gerby deformation of ˆX with the same parameter λ.
+Our construction of non-commutative complex tori is different from those in [Sch01], [PS03] and
+[Blo].
+Ours can be regarded as a patching of Archimedean analog of quantum analytic tori in
+[Soi09]. To prove the equivalence of derived categories, we use the idea of equivariant Fourier–Mukai
+transforms developed in [Sos12].
+This paper is organized as follows: In Section 2, we discuss q-Weyl algebras as toy models to
+motivate constructions in later sections. In Section 3, we recall Fourier–Mukai transforms for complex
+tori.
+In Section 4, we define non-commutative complex tori by deforming sheaves of convergent
+Laurent series rings on real tori. It can be regarded as an Archimedean analog of the construction
+of quantum analytic tori in [Soi09]. In Section 5, we discuss a dual pair X → Y and ˆY → ˆX of
+finite coverings of tori associated with a deformation parameter λ with values in roots of unity. In
+Section 6, we describe non-commutative complex tori whose deformation parameters take values in
+roots of unity in terms of a finite sheaf of algebras. In Section 7, we collect basic definitions on
+twisted sheaves on complex manifolds. In Section 8, we introduce Fourier–Mukai transforms from
+gerby complex tori to non-commutative complex tori at roots of unity and state Theorem 8.1, which
+is the main result in this paper. In Section 9, we recall basic definitions and results on finite group
+actions on abelian and derived categories, and discuss twistings by group cocycles. In Section 10, we
+discuss group actions on categories appearing in our construction. In Section 11, we prove Theorem
+8.1.
+Notations and conventions. We fix the complex number field C as the ground field. All modules
+(resp. actions) are right modules (resp. actions) unless otherwise specified. In contrast, all group ac-
+tions on categories are left actions. The word ‘non-commutative’ is synonymous with ‘not necessarily
+commutative’.
+For a ringed space X = (Z, OX), we write the ringed space (Z, OX op := (OX)op) as X op. The
+category of OX-modules will be denoted by Mod X . For an element a of a complex abelian Lie group
+A, we write the right translation map A → A, x �→ xa as Ra. For two or three complex manifolds
+Z1, Z2 or Z1, Z2, Z3, the projection to the first (resp. second) component Z1 (resp. Z2) will be denoted
+by pZ1,Z2 or pZ1,Z2,Z3 (resp. qZ1,Z2 or qZ1,Z2,Z3).
+Acknowledgment. The author is deeply grateful to his advisor, Kazushi Ueda, for a lot of guidance,
+useful comments, and encouragement.
+The author also thanks Masahiro Futaki for many useful
+questions, one of which lead to the formula (6.9). Finally, the author has deep gratitude to his
+parents for their various supports throughout his life.
+2. q-Weyl algebras
+The category of OX-modules is equivalent to the category of Γ-equivariant OT -modules, and the
+category of Γ-equivariant C[T]-modules can be regarded as a toy model for it. The latter can be
+identified with the category of modules over the crossed product algebra C[T] ⋊ Γ.
+For a parameter λ = (λi,j)1≤i<j≤g ∈ (C∗)g(g−1)/2, the q-Weyl algebra is the non-commutative
+deformation of C[T] defined by
+Wλ := C⟨t±
+1 , t±
+2 , . . . , t±
+g ⟩/ (titj − λi,jtjti)1≤i<j≤g ,
+(2.1)
+which gives C[T] if λ = 1 := (1, . . . , 1).
+Let Q = (qi,j)g
+i,j=1 be a multiplicative period matrix, i.e. a matrix so that {γj := (qi,j)g
+i=1}g
+j=1 is a
+free generator of Γ. The group Γ acts on the q-Weyl algebra by
+ti · γj := q−1
+i,j ti,
+(2.2)
+2
+
+which reduces to the natural action on C[T] when λ = 1. The crossed product algebra Wλ ⋊ Γ is
+isomorphic to another q-Weyl algebra
+(2.3)
+Wλ,Q,1 := C⟨t±
+1 , t±
+2 , . . . , t±
+g , γ±
+1 , γ±
+2 , . . . , γ±
+g ⟩
+/({titj − λi,jtjti}1≤i<j≤g, {tiγj − q−1
+i,j γjti}g
+i,j=1, {[γi, γj]}g
+i,j=1)
+generated by 2g elements.
+The parameter λ describing a non-commutative deformation of X can be used to describe a gerby
+deformation of the dual torus ˆX:
+Definition 2.1. A λ-twisted O ˆ
+X-module is a pair (M, (ρˆγ)ˆγ∈ ˆT) consisting of an OˆΓ-module M and a
+family (ρˆγ)ˆγ∈ ˆT of morphisms ρˆγ : M → R∗
+ˆγM satisfying ρˆγj ◦ ρˆγi = λi,jρˆγi ◦ ρˆγj for all i, j ∈ {1, . . . , g}
+such that i < j.
+As a toy model of λ-twisted O ˆ
+X-modules, we consider modules over the q-Weyl algebra
+(2.4)
+W1,Q,λ := C⟨ˆt±
+1 , ˆt±
+2 , . . . , ˆt±
+g , ˆγ±
+1 , ˆγ±
+2 , . . . , ˆγ±
+g ⟩
+/({[ˆti, ˆtj]}g
+i,j=1, {ˆtiˆγj − q−1
+j,i ˆγjˆti}i,j, {ˆγiˆγj − λi,jˆγjˆγi}1≤i<j≤g),
+which contains the ring of regular functions C[ˆΓ] := C[ˆt±
+1 , ˆt±
+2 , . . . , ˆt±
+g ]. Note that the roles of ti and
+γi are interchanged between (2.3) and (2.4).
+A toy model for the deformed Poincar´e line bundle, which should give the integral kernel of the
+deformed Fourier–Mukai transform, is the (W1,Q,λ)op ⊗ Wλ,Q,1-module Pλ such that
+(1) Pλ = C[ˆΓ] ⊗C Wλ as a
+�
+C[ˆΓ]
+�op
+⊗ Wλ-module,
+(2) actions of γi are given by
+�
+ψ(ˆt1, ˆt2, . . . , ˆtg) ⊗ φ(t1, t2, . . . , tg)
+�
+· γi = ψ(ˆt1, ˆt2, . . . , ˆtg)ˆt−1
+i
+⊗ φ(t1q−1
+1,i , t2q−1
+2,i , . . . , tgq−1
+g,i ),
+(3) actions of ˆγi are given by
+ˆγi ·
+�
+ψ(ˆt1, ˆt2, . . . , ˆtg) ⊗ φ(t1, t2, . . . , tg)
+�
+= ψ(ˆt1qi,1, ˆt2qi,2, . . . , ˆtgqi,g) ⊗ tiφ(t1, t2, . . . , tg).
+Note that the action of ˆγi satisfies ˆγiˆγj = λi,jˆγjˆγi since titj = λi,jtjti in Wλ.
+While q-Weyl algebras are only toy models and one has to work with analytic functions (such as
+theta functions) rather than regular functions, they provide intuition behind constructions in later
+sections.
+The duality between non-commutative deformations and gerby deformations is clearly
+visible in this toy model. Note also that if all components of λ are roots of unity, then Wλ is finite
+over its center, which is the ring of functions on a finite quotient of T.
+3. Fourier–Mukai transforms
+We identify OX-modules with Γ-equivariant OT-modules.
+An element ˆx ∈ ˆΓ determines a Γ-
+equivariant OT-module Lˆx as the trivial OT-module equipped with the Γ-action
+φ(x) · γ = φ(xγ−1)ˆx(γ)−1
+(3.1)
+for γ ∈ Γ. Since any t ∈ ˆT gives an isomorphism
+Lˆx
+∼−→ Lˆxt−1,
+φ(x) �→ t(x)φ(x)
+(3.2)
+of Γ-equivariant OT-modules, the map ˆx �→ L��x descends to a map ˆΓ/ ˆT
+∼−→ ˆX := Pic0 X, which is an
+isomorphism of groups because of the isomorphisms
+Lˆx ⊗ Lˆx′
+∼−→ Lˆxˆx′,
+φ(x) ⊗ ψ(x) �→ φ(x)ψ(x).
+(3.3)
+The Poincar´e line bundle P is the Γ × ˆT op-equivariant OT׈Γ-module, defined as the trivial OT׈Γ-
+module equipped with the left ˆT-action
+t · φ(x, ˆx) = t(x)φ(x, ˆxt)
+(3.4)
+3
+
+and the right Γ-action
+φ(x, ˆx) · γ = φ(xγ−1, ˆx)ˆx(γ)−1.
+(3.5)
+Let s be the map given by
+s: ˆΓ × ˆΓ × T → ˆΓ × T,
+(ˆx1, ˆx2, x) �→ (ˆx1ˆx2, x).
+(3.6)
+The map s: ˆX × ˆX × X → ˆX × X is defined similarly. Since the isomorphism
+m: (p
+ˆΓ,ˆΓ,T)∗P ⊗ (q
+ˆΓ,ˆΓ,T)∗P → s∗P,
+φ ⊗ φ′ �→ φφ′
+(3.7)
+of OˆΓ׈Γ×T-modules is ˆT op × ˆT op × Γ-equivariant, it defines an isomorphism
+m: (p
+ˆ
+X, ˆ
+X,X)∗P ⊗ (q
+ˆ
+X, ˆ
+X,X)∗P
+∼−→ s∗P
+(3.8)
+os O ˆ
+X× ˆ
+X×X-modules, which restricts to an isomorphism
+mˆx1,ˆx2 : Lˆx1 ⊗ Lˆx2
+∼−→ Lˆx1ˆx2
+(3.9)
+of OX-modules on {ˆx1} × {ˆx2} × X where Lˆx ≃ P|X×{ˆx} by definition.
+The Fourier–Mukai transform is the integral functor with the Poincar´e line bundle as the integral
+kernel;
+FMP : Db( ˆX) → Db(X),
+M �→ R(pX, ˆ
+X)∗((qX, ˆ
+X)∗M ⊗OX× ˆ
+X P).
+(3.10)
+It sends the skyscraper sheaf Oˆx of a point ˆx ∈ ˆX to the line bundle Lˆx.
+Theorem 3.1 ([Muk81]). The Fourier–Mukai transform FMP is an equivalence, whose quasi-inverse
+is given by the integral functor FMP−1[g] with P−1[g] as the integral kernel.
+To be more precise, Mukai proved this theorem for abelian varieties. A proof for complex tori can
+be found in [BBBP07].
+4. Non-commutative deformations of complex tori
+Set
+̟: T → |T| := (R>0)g,
+(x1, x2, . . . , xg) �→ (|x1|, |x2|, . . . , |xg|).
+(4.1)
+For a product D = �g
+i=1(ri, Ri) of open intervals, the ring ̟∗OT (D) consists of Laurent series in g
+variables with radii of convergence (ri, Ri) for 1 ≤ i ≤ g.
+Definition 4.1. A unitary deformation parameter is an element of Z2( ˆT, U(1)), i.e., a map λ: ˆT ×
+ˆT → U(1) satisfying
+λ(t2, t3)λ(t1t2, t3)−1λ(t1, t2t3)λ(t1, t2)−1 = 1
+(4.2)
+for all t1, t2, t3 ∈ ˆT.
+Given a unitary deformation parameter λ, the star product ∗λ on ̟∗OT (D) is defined by
+
+�
+t∈ ˆT
+att
+
+ ∗λ
+
+�
+t∈ ˆT
+btt
+
+ =
+�
+t∈ ˆT
+
+
+�
+t1,t2∈ ˆT, t1t2=t
+λ(t1, t2)at1bt2
+
+ t,
+(4.3)
+which is easily seen to be associative by using (4.2). The convergence of the right hand side follows
+from
+������
+�
+t1,t2∈ ˆT, t1t2=t
+λ(t1, t2)at1bt2
+������
+≤
+�
+t1,t2∈ ˆT, t1t2=t
+|at1||bt2|
+(4.4)
+and
+�
+t∈ ˆT
+
+
+�
+t1,t2∈ ˆT,t1t2=t
+|at1||bt2|
+
+ t =
+
+�
+t∈ ˆT
+|at|t
+
+
+
+�
+t∈ ˆT
+|bt|t
+
+ ,
+(4.5)
+4
+
+which depends on the unitarity of λ. The resulting sheaf of associative algebras on |T| will be denoted
+by OTλ which turns |T| into a non-commutative ringed space Tλ := (|T|, OTλ).
+A cochain α ∈ Z1( ˆT, U(1)) bounding λ, λ′ ∈ Z2( ˆT, U(1)) is a map α: ˆT → U(1) satisfying
+λ′(t1, t2) = λ(t1, t2)α(t1)α(t2)α(t1t2)−1.
+(4.6)
+It gives an isomorphism
+�
+t∈ ˆT
+att �→
+�
+t∈ ˆT
+α(t)att
+(4.7)
+of the ring of sections, which ensures that the isomorphism class of the sheaf OTλ of associative
+algebras depends only on the cohomology class [λ] ∈ H2( ˆT, U(1)).
+The natural T-action on T induces a T-action on |T|, which lifts to a T-action on the ringed space
+Tλ in such a way that the morphism ρa : OTλ → (R̟(a)−1)∗OTλ of sheaves of associative algebras for
+a ∈ T is given by
+�
+t∈ ˆT
+att �→
+�
+t∈ ˆT
+att(a)−1t.
+(4.8)
+The action of Γ on |T| is free since ̟(γ) = 1 for 1 ̸= γ ∈ Γ contradicts the freeness or the properness
+of Γ-action on T.
+Definition 4.2. The non-commutative complex torus associated with a complex torus X = T/Γ and
+a unitary deformation parameter λ ∈ Z2( ˆT, U(1)) is the non-commutative ringed space Xλ := Tλ/Γ.
+Recall that a sheaf M of OX-modules on a ringed space X = (X, OX) is said to be coherent if
+(1) M is finitely generated, i.e., for any point x ∈ X, there exists an open neighborhood U of x
+and an epimorphism O⊕m
+X |U → M|U → 0 for some m ∈ N, and
+(2) for any open set U and any m ∈ N, the kernel of any morphism O⊕m
+X |U → M|U is finitely
+generated.
+The full subcategory of Mod X consisting of coherent modules will be denoted by coh X .
+Lemma 4.3. The sheaf OT1 is coherent.
+Proof. It is clear that OT1 is finitely generated. For any open subset U of |T|, a morphism α: O⊕m
+T1 |U →
+OT1|U is the same as a morphism ˜α: O⊕m
+T
+|̟−1(U) → OT|̟−1(U). For any x ∈ U, let V ⊂ U be an
+open neighborhood of x obtained as the product of intervals. Then ̟−1(V ) is Stein and hence there
+is an epimorphism ˜β from O⊕m′
+̟−1(V ) to the kernel of ˜α|̟−1(V ) for some integer m′. We ensure that ̟∗
+is exact (and hence the morphism β := ̟∗ ˜β is also an epimorphism) by applying [Tay02, Corollary
+11.5.4] for every fiber of ̟. This shows that ker α is finitely generated, and Lemma 4.3 is proved.
+□
+Non-commutative complex tori at λ = 1 are usual complex tori:
+Propositon 4.4. The adjunction ̟∗ ⊣ ̟∗ induces an equivalence coh T ≃ coh T1.
+Proof. Oka coherence theorem implies that an object of Mod T is coherent if and only if it is finitely
+presented. Similarly, Lemma 4.3 implies that an object of Mod T1 is coherent if and only if it is
+finitely presented.
+The functors ̟∗ and ̟∗ induce mutually inverse functors on categories of finitely presented mod-
+ules since
+(1) the functor ̟∗ is exact (and hence preserves cokernels in particular),
+(2) the functor ̟∗ preserves cokernels since it is a left adjoint, and
+(3) ̟∗ and ̟∗ interchanges OT and OT1.
+□
+Corollary 4.5. The adjunction (̟X)∗ ⊣ (̟X)∗ induces an equivalence coh X ≃ coh X1, where
+̟X : X → |X| is the map induced by ̟.
+The category coh OXλ is abelian if Problem 4.6 below has an affirmative answer:
+5
+
+Problem 4.6 (Oka coherence for non-commutative tori). Is OTλ coherent?
+Yet another problem is a generalization to non-unitary deformation parameters, which would be
+needed for the duality with general gerby deformations.
+5. Deformation parameters at roots of unity
+As one can see (e.g. by noting that ˆT-modules are equivalent to Z[T] ∼= Z[t±1
+1 , t±
+2 , . . . , t±
+g ]-modules,
+and the trivial ˆT-modules Z has the Koszul resolution associated to t1 −1, t2 −1, . . . , tg −1) that the
+cohomology Hi( ˆT, A) with coefficients in an abelian group A with the trivial ˆT-action is isomorphic to
+Hom(∧i ˆT/(∧i ˆT)tors, A) ∼= A(g
+i) (note that (∧i ˆT)tors is generated by torsion elements a∧a∧t1∧· · ·∧ti−2
+(a, t1, . . . , ti−2 ∈ ˆT) of order 2). Let Λ ∈ Hom(∧2 ˆT, U(1)) be the element corresponding to the class
+[λ] ∈ H2( ˆT, U(1)) of a unitary deformation parameter λ ∈ Z2( ˆT, U(1)), and set
+ˆH :=
+�
+t ∈ ˆT
+��� Λ(t ∧ t′) = 1 for all t′ ∈ ˆT
+�
+,
+(5.1)
+so that one has an exact sequence
+1 → ˆH → ˆT → ˆK → 1,
+(5.2)
+and [λ] ∈ H2( ˆT, U(1)) descends to an element of H2( ˆK, U(1)), which can be represented by a bilinear
+cochain in Z2( ˆK, U(1)).
+For the rest of this paper and unless otherwise specified, we will assume that a unitary deformation
+parameter λ ∈ Z2( ˆT, U(1)) is contained in Z2( ˆT, µN) for some positive integer N where µN :=
+�
+ζ ∈ U(1)
+�� ζN = 1
+�
+. This implies that ˆK is a finite abelian group, and we will also fix a bilinear
+map
+λ: ˆK ⊗Z ˆK → µN
+(5.3)
+representing [λ].
+The dual group K := Hom( ˆK, C∗) can be identified with the kernel of the map T → H dual to
+the inclusion ˆH → ˆT, so that one has an exact sequence
+1 → K → T
+πT
+−→ H → 1.
+(5.4)
+Since Γ∩K is the trivial group, the free action of K on T descends to a free action of K on X := T/Γ,
+so that one has an exact sequence
+1 → K → X
+π−→ Y → 1
+(5.5)
+where Y is a complex torus. We also have an exact sequence
+1 → ˆK → ˆY
+ˆπ−→ ˆX → 1
+(5.6)
+Since K goes to the identity under the homomorphism ̟: T → |T|, there exists ̟Y : Y → |X|
+making the diagram
+X
+Y
+|X|
+�
+π
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+̟X
+�✤✤✤✤✤✤✤
+̟Y
+(5.7)
+commute.
+6
+
+6. Non-commutative deformations at roots of unity
+We have an isomorphism
+π∗OX ∼=
+�
+ˆk∈ ˆ
+K
+Lˆk
+(6.1)
+of sheaves of OY -algebras on ˆY := Pic0 Y . The summand Lˆk is locally generated by a monomial
+function t ∈ ˆT representing ˆk. We define a star product ⋆λ on π∗OX by
+⋆λ:
+�
+ˆk∈ ˆ
+K
+Lˆk ×
+�
+ˆk∈ ˆ
+K
+Lˆk →
+�
+ˆk∈ ˆ
+K
+Lˆk
+(6.2)
+((φˆk)ˆk, (ψˆk)ˆk) �→
+
+ �
+ˆk1ˆk2=ˆk
+λ(ˆk1, ˆk2)mˆk1,ˆk2(φˆk1 ⊗ ψˆk2)
+
+
+ˆk
+.
+(6.3)
+We write the resulting sheaf (π∗OX, ⋆λ) of OY -algebras as OXλ, and the ringed space (Y, OXλ) as
+Xλ.
+Propositon 6.1. One has an isomorphism (̟Y )∗OXλ ∼= OXλ of sheaves of algebras.
+Proof. A direct calculation shows that
+
+�
+t1∈ ˆT
+at1t1
+
+ ∗λ
+
+�
+t2∈ ˆT
+bt2t2
+
+
+(6.4)
+=
+
+ �
+t∈ ˆT/ ˆ
+H
+
+
+�
+t1 mod ˆ
+H=t
+at1t1
+
+
+
+ ∗λ
+
+ �
+t′∈ ˆT/ ˆ
+H
+
+
+�
+t2 mod ˆ
+H=t′
+bt2t2
+
+
+
+
+(6.5)
+=
+�
+t∈ ˆT/ ˆ
+H
+
+ �
+t′∈ ˆT/ ˆ
+H
+
+
+�
+t1 mod ˆH=t
+
+
+�
+t2 mod ˆ
+H=t′
+λ(t1, t2)at1bt2t1t2
+
+
+
+
+
+
+(6.6)
+=
+�
+t∈ ˆT/ ˆ
+H
+
+ �
+t′∈ ˆT/ ˆ
+H
+
+
+�
+t1 mod ˆH=t
+
+
+�
+t2 mod ˆ
+H=t′
+λ(t, t′)at1bt2t1t2
+
+
+
+
+
+
+(6.7)
+=
+�
+t∈ ˆT/ ˆ
+H
+
+ �
+t′∈ ˆT/ ˆ
+H
+λ(t, t′)mt,t′
+
+
+�
+t1 mod ˆ
+H=t
+at1t1,
+�
+t2 mod ˆ
+H=t′
+bt2t2
+
+
+
+
+(6.8)
+coincide with ⋆λ.
+□
+Propositon 6.2. The adjunction (̟Y )∗ ⊣ (̟Y )∗ induces an equivalence coh Xλ ≃ coh Xλ.
+Proof. We write the full subcategory of Mod Xλ (resp. Mod Xλ) consisting of coherent OY -modules
+(resp. OY1-modules) as (coh Y ) ˆ
+K,λ (resp. (coh Y1) ˆ
+K,λ). Since OXλ is a finite OY -algebra by defi-
+nition and OXλ is a finite OY1-algebra by Proposition 6.1, the category coh Xλ (resp. coh Xλ) is
+equivalent to (coh Y ) ˆ
+K,λ (resp. (coh Y1) ˆ
+K,λ). It follows from Corollary 4.5 and Proposition 6.1 that
+OXλ ∈ ob(coh Y1) and the adjunction (̟Y )∗ ⊣ (̟Y )∗ induces an equivalence (coh Y ) ˆ
+K,λ ≃ (coh Y1) ˆ
+K,λ.
+□
+In particular, OXλ is a coherent sheaf on Xλ and hence coh OXλ is an abelian category.
+Remark 6.3. It follows from the definition of ˆK that there exists an isomorphism λ♯ : ˆK → K such
+that λ(ˆk1, ˆk2) = ˆk2(λ♯(ˆk1)). If we write the inverse of λ♯ as λ♭ and define ω : K ⊗Z K → U(1) by
+ω(k1, k2) = λ(λ♭(k1), λ♭(k2)), then the star product on OXλ can alternatively be described as
+φ(x) ⋆λ ψ(x) = 1
+♯K
+�
+k1,k2∈K
+ω(k1, k2)φ(xk1)ψ(xk2).
+(6.9)
+7
+
+A coherent sheaf on Xλ consists of an OY -module M and a collection of morphisms {mˆk : M⊗Lˆk →
+M}ˆk∈ ˆ
+K such that the diagram
+M ⊗ Lˆk1 ⊗ Lˆk2
+M ⊗ Lˆk2
+M ⊗ Lˆk1ˆk2
+M
+�
+mˆk1⊗id
+�
+λ(ˆk1,ˆk2) id ⊗mˆk1ˆk2
+�
+mˆk2
+�
+mˆk1ˆk2
+(6.10)
+commutes. The dual OY -module M−1 := HomOY (M, OY ) equipped with the transposes of mt gives
+a coherent sheaf on Xop
+λ .
+7. Twisted sheaves
+Let M be a complex manifold and α ∈ H2(M, O∗
+M) be a second ´etale cohomology class of O∗
+M.
+Take an ´etale covering U = (Ui)i and a representative (αi,j,k) of α on U. We write the projections
+from Ui ×M Uj to the first (resp. second) component as Ii,j (resp. Ji,j).
+Definition 7.1. An α-twisted sheaf on M is a collection ((Fi)i, (ρi,j)i,j) of OUi-modules Fi and
+isomorphisms ρi,j : I∗
+i,jFi → J∗
+i,jFj satisfying ρ−1
+i,k ◦ ρj,k ◦ ρi,j = αi,j,k id . An α-twisted sheaf is coherent
+if all Fi are coherent.
+The category of α-twisted sheaves on M and the full subcategory consisting of α-twisted coherent
+sheaves will be denoted by Mod Mα and coh Mα respectively. Note that for an OM-algebra A and the
+resulting ringed space X := (M, A), we similarly can define the notion of α-twisted sheaves on X and
+categories Mod X α, coh X α. They do not depend on the choice of U and (αi,j,k) up to equivalence.
+One has the tensor product functor
+⊗: Mod Mα × Mod Mα′ → Mod Mαα′,
+(7.1)
+�
+((Fi)i, (ρi,j)i,j), ((F ′
+i)i, (ρ′
+i,j)i,j)
+�
+�→ ((Fi ⊗ F ′
+i)i, (ρi,j ⊗ ρ′
+i,j)i,j)
+(7.2)
+and the duality functor
+(−)−1 : Mod Mα → Mod Mα−1,
+(7.3)
+((Fi)i, (ρi,j)i,j) �→ ((HomOUi(Fi, OUi))i, ((ρ−1
+i,j )∗)i,j).
+(7.4)
+If U consists of a principal G-bundle P on M for some discrete group G, then the isomorphism
+P × Gp → P ×M P ×M · · · ×M P,
+(y, g1, . . . , gp) �→ (y, yg1, yg1g2, . . . , yg1 · · · gp)
+(7.5)
+induces an isomorphism from the ˇCech complex C•(U, O∗
+M) to the standard complex C•(G, O∗
+P(P))
+for group cohomology.
+The composite of the resulting map H2(G, O∗
+P(P)) → H2(M, O∗
+M) with
+the map H2(G, C∗) → H2(G, O∗
+P(P)) will be denoted by ιP : H2(G, C∗) → H2(M, O∗
+M). For any
+λ ∈ H2(G, C∗), an ιP(λ)-twisted sheaf will simply be called a λ-twisted sheaf. If G is a finite abelian
+group, then ιP(λ) is a torsion element since any group cohomology of finite abelian group is torsion.
+In other words, ιP(λ) is an element of the cohomological Brauer group Br(M) := H2(M, O∗
+M)tors. A
+λ-twisted sheaf consists of an OP-module F and a λ-twisted G-linearization of F, i.e., a collection
+(ρg)g∈G of morphisms ρg : F → R∗
+gF satisfying R∗
+g1ρg2 ◦ ρg1 = λ(g1, g2)ρg1g2.
+A λ-twisted sheaf (F, (ρg)g∈G) will also be called a λ-twisted G-equivariant OP-module; it reduces
+to a G-equivariant OP-module if λ = 1 (which in turn is equivalent to an OM-module).
+8. Deformed Fourier–Mukai transforms
+Let Q be the Poincar´e line bundle on Y × ˆY . For a complex manifold Z, a sheaf of associative
+algebras (pY,Z)∗OXλ (resp. (pY,Z)∗Oop
+Xλ) will denoted by OXλ×Z (resp. OXop
+λ ×Z), and the resulting
+ringed space will be denoted by Xλ ×Z (resp. Xop
+λ ×Z). Symbols Y × ˆXλ (resp. Xλ × ˆXλ, Xop
+λ × ˆXλ)
+denote (Y × ˆX)1×λ (resp. (Xλ × ˆX)1×λ, (Xop
+λ × ˆX)1×λ).
+8
+
+The deformed Poincar´e line bundle Pλ is an object of coh Xλ × ˆXλ−1 defined as the OY × ˆY -module
+�
+ˆk∈ ˆ
+K
+Q ⊗ (pY, ˆY )∗Lˆk ∼=
+�
+ˆk∈ ˆ
+K
+R∗
+ˆkQ
+(8.1)
+equipped with the OXλ× ˆY ∼= �
+ˆk∈ ˆ
+K(pY, ˆY )∗Lˆk-action
+�
+ˆk∈ ˆ
+K
+Q ⊗ (pY, ˆY )∗Lˆk ×
+�
+ˆk∈ ˆ
+K
+(pY, ˆY )∗Lˆk →
+�
+ˆk∈ ˆ
+K
+Q ⊗ (pY, ˆY )∗Lˆk
+(8.2)
+((ψˆk ⊗ φˆk)ˆk, (φ′
+ˆk)ˆk) �→
+
+
+�
+ˆk1,ˆk2∈ ˆ
+K, ˆk1ˆk2=ˆk
+ψˆk1 ⊗ (φˆk1 ⋆λ φ′
+ˆk2)
+
+
+ˆk
+,
+(8.3)
+and the λ−1-twisted ˆK-action (i.e. the λ-twisted left ˆK-action)
+ρˆk :
+�
+ˆk′∈ ˆ
+K
+R∗
+ˆk′Q →R∗
+ˆk−1
+�
+ˆk′∈ ˆ
+K
+R∗
+ˆk′Q
+(8.4)
+(φˆk′)ˆk′ �→(λ(ˆk, ˆk′ˆk−1)φˆk′ˆk−1)ˆk′.
+(8.5)
+The deformed Fourier–Mukai transform
+FMλ: Db( ˆXλ) → Db(Xλ)
+(8.6)
+is the integral functor with the deformed Poincar´e line bundle as the integral kernel, i.e., the composite
+of the pull-back
+(qY, ˆY )∗: Db( ˆXλ) → Db(Y × ˆXλ),
+(8.7)
+the tensor product
+(−) ⊗ Pλ : Db(Y × ˆXλ) → Db(Xλ × ˆX),
+(8.8)
+and the push-forward
+R(pY, ˆ
+X)∗: Db(Xλ × ˆX) → Db(Xλ).
+(8.9)
+Its right adjoint is the integral functor FM−1
+λ
+with the g-shift of
+P−1
+λ
+:= HomOY × ˆ
+Y (Pλ, OY × ˆY ) ∈ ob(coh Xop
+λ × ˆXλ)
+(8.10)
+as the kernel, since
+• the push-forward R(qY, ˆY )∗ is right adjoint to the pull-back (qY, ˆY )∗,
+• the tensor product P−1
+λ
+⊗ (−) is right adjoint to the tensor product (−) ⊗ Pλ, and
+• the pull-back (pY, ˆ
+X)∗[g] shifted by g is right adjoint to the push-forward R(pY, ˆ
+X)∗ because
+Db(Xλ) is Calabi–Yau of dimension g and Db(Xλ × ˆX) is Calabi–Yau of dimension 2g (it will
+be proved in Section 10).
+Theorem 8.1 below is the main result in this paper:
+Theorem 8.1. The deformed Fourier–Mukai transform FMλ is an equivalence of derived categories.
+Remark 8.2. Let OTλ×ˆΓ be the sheaf associative algebras (̟ × id)∗OT׈Γ, with a non-commutative
+associative product defined by the formula similar to (4.3), but at and bt are functions on ˆΓ. For
+any λ ∈ H2( ˆT, U(1)) not necessarily at roots of unity, it is natural to expect that coh ˆXλ is derived-
+equivalent to coh Xλ. Although the latter is not known to be abelian, we can define the deformed
+Poincar´e line bundle Pλ as an object of Mod
+�
+Xλ × ˆXλ−1�
+, i.e. a OTλ׈Γ-module equipped with a
+λ-twisted left ˆT-action and a right Γ-action.
+Pλ is defined by a free OTλ׈Γ-module of rank 1 equipped with a λ-twisted left ˆT-action
+ˆγ · φ(x, ˆx) = ˆγ(x) ∗λ φ(x, ˆxˆγ)
+(8.11)
+9
+
+and the right Γ-action
+φ(x, ˆx) · γ = φ(xγ−1, ˆx) ∗λ ˆx(γ)−1.
+(8.12)
+9. Finite group actions on abelian and derived categories
+It is natural to examine group actions on DG-categories in relation to group actions on derived
+categories and equivariant Fourier-Mukai transforms. However, coherent data for group actions on
+DG-categories are more intricate than those for group actions on abelian categories. As such, we will
+concentrate on group actions on abelian categories and the actions they induce on derived categories.
+A weak action of a finite group G on a category C is a family (g∗)g∈G of autoequivalences g∗: C → C
+such that the functor (g1)∗◦(g2)∗ is isomorphic to (g1g2)∗ for any g1, g2 ∈ G. An action is a weak action
+equipped with a coherence data, i.e., a family (cg1,g2)g1,g2∈G of isomorphisms cg1,g2 : (g1)∗ ◦ (g2)∗
+∼−→
+(g1g2)∗ of functors such that the diagram
+(g1)∗ ◦ (g2)∗ ◦ (g3)∗
+cg1,g2
+−−−→ (g1g2)∗ ◦ (g3)∗
+cg2,g3
+�
+cg1g2,g3
+�
+(g1)∗ ◦ (g2g3)∗
+cg1,g2g3
+−−−−→
+(g1g2g3)∗
+(9.1)
+commutes for any g1, g2, g3 ∈ G (cf. e.g. [Del97]). An action is strict if the coherence data consists
+of identities.
+Let C be a category equipped with an action of a finite group G. The following definition is taken
+from [Sos12]:
+Definition 9.1 ([Sos12, Definition 3.1]). A linearization of A ∈ ob(C) is a family (ρg)g∈G of isomor-
+phisms ρg : A
+∼−→ g∗A such that the diagram
+A
+(g1)∗A
+(g1g2)∗A
+�
+ρg1
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+ρg1g2
+�✤✤✤✤✤✤✤✤
+cg1,g2◦(g1)∗(ρg2)
+(9.2)
+commutes for any g1, g2 ∈ G. An equivariant object is an object equipped with a linearization. A
+morphism of equivariant objects from (A, (ρg)g∈G) to (A′, (ρ′
+g)g∈G) is a morphism φ: A → A′ such
+that the diagram commute
+A
+φ
+−−−→
+A′
+ρg
+�
+ρ′
+g
+�
+g∗A
+g∗φ
+−−−→ g∗A′
+(9.3)
+commutes.
+The category of G-equivariant objects in C will be denoted by CG. For the rest of this paper and
+unless otherwise specified, we will assume that C is a C-linear category and weak actions consists of
+C-linear functors.
+Propositon 9.2 ([Sos12, Proposition 3.2]). If C is abelian, then so is CG.
+We extend the above constructions to twisted group actions. Let φ be a second cocycle of G with
+values in C∗.
+10
+
+Definition 9.3. A φ-twisted linearization of A ∈ ob(C) is a family (ρg)g∈G of isomorphisms ρg : A
+∼−→
+g∗A such that the diagram
+A
+(g1)∗A
+(g1g2)∗A
+�
+ρg1
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+���
+❄
+φ(g1,g2)ρg1g2
+�✤✤✤✤✤✤✤✤
+cg1,g2◦(g1)∗(ρg2)
+(9.4)
+commutes for any g1, g2 ∈ G. Morphisms of φ-twisted equivariant objects are defined in the same
+way as in CG.
+A φ-twisted linearization of A ∈ ob(C) is equivalent to a linearization of A for G-action {ρg}g∈G
+equipped with a coherence data (φ(g1, g2)−1cg1,g2)g1,g2∈G. The category of φ-twisted equivariant ob-
+jects will be denoted by CG,φ. The cocycle condition on φ ensures the equality of
+ρg3 ◦ ρg2 ◦ ρg1 = ρg3 ◦ (φ(g1, g2)ρg1g2) = φ(g1, g2)φ(g1g2, g3)ρg1g2g3
+(9.5)
+and
+ρg3 ◦ ρg2 ◦ ρg1 = (φ(g2, g3)ρg2g3) ◦ ρg1 = φ(g2, g3)φ(g1, g2g3)ρg1g2g3,
+(9.6)
+where we have omitted (g1)∗ and so on. If a pair of cocycles φ and φ′ differ by the coboundary of
+α ∈ C1(G, C∗), then there exists an equivalence CG,φ → CG,φ′ sending (A, (ρg)g∈G) to (A, (α(g)ρg)g∈G).
+Explanations of relations between group cohomology of G in low degrees and (weak) G-actions are
+found in [BO20].
+Corollary 9.4 below is obtained by applying Proposition 9.2 to the G-action equipped with a
+coherence data (φ(g1, g2)−1cg1,g2)g1,g2∈G:
+Corollary 9.4. If C is abelian, then so is CG,φ.
+By applying the free-forgetful adjunction
+Free ⊣ Forget
+(9.7)
+between
+Free: C → CG,φ,
+A �→
+
+�
+g′∈G
+(g′)∗A,
+�
+ρg :=
+�
+g′∈G
+φ(g, g′) (id: (gg′)∗A → g∗(g′)∗A)
+�
+g∈G
+
+
+(9.8)
+and
+Forget: CG,φ → C,
+(A, (ρg)g∈G) �→ A
+(9.9)
+to the opposite categories and using the equivalence (CG,φ)op ≃ (Cop)G,φ−1, one obtains an adjunction
+Forget ⊣ Free .
+(9.10)
+For any (A, (ρg)g∈G), (A′, (ρ′
+g)g∈G) ∈ CG,φ, the space HomC(A, A′) comes with a natural linear action
+of G in such a way that the diagram
+A
+χ
+−−−→
+A′
+ρg
+�
+ρ′
+g
+�
+g∗A
+g∗(χ·g)
+−−−−→ g∗A′
+(9.11)
+11
+
+commutes since
+χ · g1 · g2 = ((g1∗)−1(ρ′
+g1 ◦ χ ◦ ρ−1
+g1 )) · g2
+(9.12)
+= (g2∗)−1(ρ′
+g2 ◦ ((g1∗)−1(ρ′
+g1 ◦ χ ◦ ρ−1
+g1 )) ◦ ρ−1
+g2 ))
+(9.13)
+= (g2∗)−1(g1∗)−1(g1∗ρ′
+g2 ◦ (ρ′
+g1 ◦ χ ◦ ρ−1
+g1 ) ◦ g1∗ρ−1
+g2 )
+(9.14)
+= ((g1g2)∗)−1((φ(g1, g2)ρ′
+g1g2) ◦ χ ◦ (φ(g1, g2)ρg1g2)−1)
+(9.15)
+= χ · g1g2.
+(9.16)
+It follows from the definition that
+HomCG((A, (ρg)g∈G), (A′, (ρ′
+g)g∈G)) = HomC(A, A′)G.
+(9.17)
+A functor Φ: C → C′ between categories with G-actions is said to be G-equivariant if it is equipped
+with a family (ag)g∈G of natural isomorphisms ag : Φ ◦ g∗
+∼−→ g∗ ◦ Φ of functors such that the diagram
+Φ ◦ (g1)∗ ◦ (g2)∗
+(g1)∗ ◦ Φ ◦ (g2)∗
+(g1)∗ ◦ (g2)∗ ◦ Φ
+Φ ◦ (g1g2)∗
+(g1g2)∗ ◦ Φ
+�
+ag1
+�✤✤✤✤✤✤✤✤
+cg1,g2
+�
+ag2
+�♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦
+cg1,g2
+�
+ag1g2
+(9.18)
+commutes. A G-equivariant functor Φ: C → C′ induces a functor ΦG,φ : CG,Φ → C′G,φ sending an
+object (A, (ρg)g∈G) to (Φ(A), (ag ◦ Φ(ρg))g∈G) and a morphism f : (A, (ρg)g∈G) → (A′, (ρ′
+g)g∈G) to
+Φ(f): Φ(A) → Φ(A′). It is straightforward to show that ΦG,φ send G-equivariant objects to G-
+equivariant objects.
+Propositon 9.5. If Φ is right (resp. left) exact, then so is ΦG,φ.
+Proof. Since Free and Forget are mutually both left and right adjoint to each other, they are exact,
+so that a sequence A → B → C in CG,φ is exact if and only if Forget(A) → Forget(B) → Forget(C)
+is exact in C.
+□
+Now we discuss the derived category of CG,φ.
+Propositon 9.6. An object (I, (ρg)g∈G) ∈ ob(CG,φ) is injective if and only if so is I ∈ ob(C). The
+category CG,φ has enough injectives if and only if so is C.
+Proof. If I is injective, then the functor
+A �→ HomCG,φ(A, (I, (ρg))) = HomC(Forget(A), I)G
+(9.19)
+is exact, since Forget is exact, I is injective, and taking the G-invariant part is exact.
+Conversely, if (I, (ρg)) is injective, then the functor
+A �→ HomC(A, I) ∼= HomCG,φ(Free(A), (I, (ρg)))
+(9.20)
+is exact.
+Let (A, (ρg)g) be an object in CG,φ. If C has enough injectives, then a monomorphism A → I into
+an injective object I ∈ ob(C) gives a monomorphism A → Free I into Free I, which is injective in
+CG,φ.
+Conversely, if CG,φ has enough injectives, then for any A ∈ ob(C), a monomorphism Free A → I
+into an injective I ∈ ob(CG,φ) gives a monomorphism A → Forget I into an injective Forget I.
+□
+Assume that C has enough injectives. For any pair A, B ∈ ob(D+(CG,φ)) of objects, the space
+Hom(Forget(A), Forget(B)) has a natural G-action in such a way that
+Hom(A, B) ∼= Hom(Forget(A), Forget(B))G.
+(9.21)
+Propositon 9.7. If Db(C) is Calabi–Yau of dimension n, then so is Db(CG,φ).
+12
+
+Proof. There exists a natural isomorphism
+HomDb(C)(A, B) ∼= HomDb(C)(B, A[n])∗
+(9.22)
+for A, B ∈ ob(Db(CG,φ)), since Db(C) is Calabi–Yau of dimension n.
+It is G-equivariant by the
+naturality, so it induces an isomorphism on G-invariant part.
+This means that Db(CG,φ) is also
+Calabi–Yau of dimension n by (9.21).
+□
+A G-equivariant left exact functor Φ: C → C′ induces functors RΦ: D+(C) → D+(C′) and
+RΦG,φ : D+(CG,φ) → D+(C′G,φ). If RΦ is fully faithful, then so is RΦG,φ by (9.21).
+10. Group actions on coherent sheaves
+An action of a finite group G on a complex manifold Z induces a strict G-action (R∗
+g)g∈G on coh Z.
+In the case of the ˆK-action on ˆY , one obtains (coh ˆY ) ˆ
+K,λ ≃ coh ˆXλ.
+Another example of a finite group action on the category of coherent sheaves comes from a coherent
+injection from a finite abelian group G to the Picard group Pic0 Z of a complex manifold Z, i.e., a
+family {Lg}g∈G of line bundles and a family (mg1,g2)g1,g2∈G of isomorphisms mg1,g2 : Lg1 ⊗Lg2
+∼−→ Lg1g2
+such that the diagrams
+Lg1 ⊗ Lg2 ⊗ Lg3
+mg1,g2⊗id
+−−−−−−→ Lg1g2 ⊗ Lg3
+id ⊗mg2,g3
+�
+mg1g2,g3
+�
+Lg1 ⊗ Lg2g3
+mg1,g2g3
+−−−−−→
+Lg1g2g3
+(10.1)
+Lg1 ⊗ Lg2
+Lg1g2
+Lg2 ⊗ Lg1
+�
+mg1,g2
+�
+�t
+t
+t
+t
+t
+t
+t
+t
+t
+t
+t
+mg2,g1
+(10.2)
+commutes, inducing a G-action
+�
+(−) ⊗ L−1
+g )
+�
+g∈G on coh Z. Here, the vertical arrow in (10.2) comes
+from the canonical symmetric monoidal structure in coh Z. If Z is compact and connected, then one
+has Aut L = C∗ for any line bundle L, and an example of a coherence data (mg1,g2)g1,g2∈G comes from
+a choice of a collection (ϕg)g∈G of linear isomorphisms ϕ: (Lg)z
+∼−→ C from the fibers (Lg)z of Lg at
+an arbitrarily chosen and fixed base point z ∈ Z to the complex line.
+A φ-twisted G-linearization (ρg)g∈G of M is equivalent to a family (mg)g∈G of morphisms
+mg = (g∗)−1ρg : M ⊗ Lg → M
+(10.3)
+such that diagram
+M ⊗ Lg1 ⊗ Lg2
+M ⊗ Lg2
+M
+�
+mg1
+�❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+φ(g1,g2)mg1g2
+�✤✤✤✤✤✤✤
+mg2
+(10.4)
+commutes. The category (coh Z)G,φ is equivalent to coh Aφ, where Aφ is the sheaf of OZ-algebras ob-
+tained as the OZ-module �
+g∈G Lg equipped with the multiplication given by �
+g1,g2∈G φ(g1, g2)mg1,g2.
+If φ = 1, Aφ is commutative by the commutativity of (10.2). In particular, coh Xλ is equivalent to
+(coh Y ) ˆ
+K,λ. By Proposition 9.7, Db(Xλ) is Calabi–Yau of dimension g and Db(Xλ× ˆX) is Calabi–Yau
+of dimension 2g, which were needed to prove that FM−1
+λ
+is right adjoint to FMλ.
+13
+
+11. Proof of Theorem 8.1
+Functors FMλ and FMQ are the right derived functors of functors
+FMab
+λ : coh ˆXλ → coh Xλ
+(11.1)
+M �→ (pY, ˆ
+X)∗((qY, ˆY )∗M ⊗OY × ˆY Pλ)
+(11.2)
+and
+FMab
+Q : coh ˆY → coh Y
+(11.3)
+M �→ (pY, ˆY )∗((qY, ˆY )∗M ⊗OY × ˆ
+Y Q)
+(11.4)
+of abelian categories.
+Since
+FMab
+Q (R∗
+ˆyM) = (pY, ˆY )∗((qY, ˆY )∗R∗
+ˆyM ⊗ Q)
+(11.5)
+∼= (pY, ˆY )∗R∗
+(1,ˆy−1)((qY, ˆY )∗R∗
+ˆyM ⊗ Q)
+(11.6)
+∼= (pY, ˆY )∗((qY, ˆY )∗M ⊗ R∗
+(1,ˆy−1)Q)
+(11.7)
+∼= (pY, ˆY )∗((qY, ˆY )∗M ⊗ Q ⊗ (pY, ˆY )∗L−1
+ˆy )
+(11.8)
+∼= (pY, ˆY )∗((qY, ˆY )∗M ⊗ Q) ⊗ L−1
+ˆy
+(11.9)
+= FMab
+Q (M) ⊗ L−1
+ˆy ,
+(11.10)
+for any ˆy ∈ ˆK and M ∈ ob(coh ˆY ), FMab
+Q commutes with the weak ˆK-action. The commutativity
+of the diagram (9.18) in this case is a straightforward diagram chasing. This turns FMab
+Q into a
+ˆK-equivariant functor, inducing a functor
+(FMab
+Q )
+ˆ
+K,λ: coh ˆXλ → coh Xλ.
+(11.11)
+Lemma 11.1. The functor (FMab
+Q ) ˆ
+K,λ is isomorphic to FMab
+λ .
+Proof. The squares on the left and the right of the diagram
+(coh ˆY ) ˆ
+K,λ
+((qY, ˆ
+Y )∗) ˆ
+K,λ
+�
+�
+(coh(Y × ˆY )) ˆ
+K,λ (π∗(−⊗Q)) ˆ
+K,λ
+�
+�
+(coh(Y × ˆX)) ˆ
+K,λ ((pY, ˆ
+X)∗) ˆ
+K,λ
+�
+�
+(coh Y ) ˆ
+K,λ
+�
+coh ˆXλ
+(qY, ˆ
+Y )∗
+� coh(Y × ˆXλ)
+−⊗Pλ
+� coh(Xλ × ˆX)
+(pY, ˆ
+X)∗
+� coh ˆXλ
+commute by definition, and the natural isomorphism
+⊕ˆy′∈ ˆ
+Kρ−1
+ˆy′ ⊗ id:
+�
+ˆy′∈ ˆ
+K
+R∗
+(1,ˆy′)(M ⊗OY × ˆY Q) →
+�
+ˆy′∈ ˆ
+K
+M ⊗OY × ˆY R∗
+(1,ˆy′)Q,
+(11.12)
+whose ˆK-equivariance can be checked by a straightforward computation, gives the commutativity of
+the square in the middle.
+□
+Therefore R(FMab
+Q ) ˆ
+K,λ and FMλ are isomorphic. Hence FMλ is fully faithful as explained at the
+end of Section 9. Similarly, the right adjoint FM−1
+λ
+of FMλ, whose kernel is given by the g-shift of
+(8.10), is also fully faithful, and Theorem 8.1 is proved.
+14
+
+References
+[BBBP07] O. Ben-Bassat, J. Block, and T. Pantev, Non-commutative tori and Fourier-Mukai duality, Compos. Math.
+143 (2007), no. 2, 423–475. MR 2309993 1, 4
+[Blo]
+Jonathan Block, Duality and equivalence of module categories in noncommutative geometry II: Mukai du-
+ality for holomorphic noncommutative tori, arXiv:math/0604296. 1, 2
+[Blo10]
+, Duality and equivalence of module categories in noncommutative geometry, A celebration of the
+mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI,
+2010, pp. 311–339. MR 2648899 1
+[BO20]
+Thorsten Beckmann and Georg Oberdieck, On equivariant derived categories, arxiv:2006.13626v2, 2020. 11
+[Con94]
+Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779 1
+[Del97]
+P. Deligne, Action du groupe des tresses sur une cat´egorie, Invent. Math. 128 (1997), no. 1, 159–175.
+MR 1437497 10
+[Muk81]
+Shigeru Mukai, Duality between D(X) and D( ˆX) with its application to Picard sheaves, Nagoya Math. J.
+81 (1981), 153–175. MR 607081 1, 4
+[PS03]
+A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on noncommutative two-tori,
+Comm. Math. Phys. 236 (2003), no. 1, 135–159. MR 1977884 1, 2
+[Rie81]
+Marc A. Rieffel, C∗-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429.
+MR 623572 1
+[Sch01]
+Albert Schwarz, Theta functions on noncommutative tori, Lett. Math. Phys. 58 (2001), no. 1, 81–90.
+MR 1865115 1, 2
+[Soi09]
+Y. Soibelman, On non-commutative analytic spaces over non-Archimedean fields, Homological mirror sym-
+metry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 221–247. MR 2596639 2
+[Sos12]
+Pawel Sosna, Linearisations of triangulated categories with respect to finite group actions, Math. Res. Lett.
+19 (2012), no. 5, 1007–1020. MR 3039826 2, 10
+[Tay02]
+Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, Graduate
+Studies in Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2002. MR 1900941 5
+[Tod09]
+Yukinobu Toda, Deformations and Fourier-Mukai transforms, J. Differential Geom. 81 (2009), no. 1, 197–
+224. MR 2477894 1
+Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
+Tokyo, 153-8914, Japan.
+Email address: hiokuc8h18@gmail.com
+15
+

4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:782760ab5b7a7b4e244539c5814686366eb5e6bd4befe4128f96e8b334b68290
+size 941775

4dE2T4oBgHgl3EQfjwe5/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:05932f185a9994aeb5a733eef2887eb3adb178f7941be27e118a54ad81e795f7
+size 4390957

4dE2T4oBgHgl3EQfjwe5/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f4a0e58e406b25482248f30621d20c479a4e75ede141ffdd4729a1bca3e18af7
+size 178035

59AzT4oBgHgl3EQff_xu/content/tmp_files/2301.01461v1.pdf.txt

@@ -0,0 +1,2241 @@
+A Novel Koopman-Inspired Method for the Secondary
+Control of Microgrids with Grid-Forming and
+Grid-Following Sources
+Xun Gong, Xiaozhe Wang∗
+Department of Electrical and Computer Engineering, McGill University, 3480 Rue
+University, Montreal,H3A 0E9, Quebec, Canada
+Abstract
+This paper proposes an online data-driven Koopman-inspired identification and
+control method for microgrid secondary voltage and frequency control. Unlike
+typical data-driven methods, the proposed method requires no warm-up train-
+ing yet with guaranteed bounded-input-bounded-output (BIBO) stability and
+even asymptotic stability under some mild conditions. The proposed method
+estimates the Koopman state space model adaptively so as to perform effective
+secondary voltage and frequency control that can handle microgrid nonlinearity
+and uncertainty. Case studies in the 4-bus and 13-bus microgrid test systems
+(with grid-forming and grid-following sources) demonstrate the effectiveness and
+robustness of the proposed identification and control method subject to the
+change of operating conditions and large disturbances (e.g., microgrid mode
+transitions, generation/load variations) even with measurement noises and time
+delays.
+Keywords:
+data-driven control, adaptive Koopman-inspired identification,
+microgrid secondary control, grid-forming, grid-following, Koopman operator
+control, observer Kalman filter identification
+1. Introduction
+The microgrids (MGs) are small local grids that can disconnect from the
+bulk grid to operate independently. The MGs facilitate the integration of sus-
+tainable distributed energy resources (DERs) like wind, solar as well as energy
+storage. Nonetheless, the DERs are interfaced with microgrids by power con-
+verters, making MGs low-inertia or even inertia-less [1, 2]. In addition, MGs
+are characterized by frequency-voltage dependence due to low X/R ratios as
+∗Corresponding author
+1This work was supported by the Fonds de Recherche du Quebec-Nature et technologies
+under Grant FRQ-NT PR-298827 and NSERC ALLRP 571554 - 21.
+Preprint submitted to Applied Energy
+January 5, 2023
+arXiv:2301.01461v1  [cs.SY]  4 Jan 2023
+
+opposed to conventional power systems [2, 3]. Therefore, the frequency and
+voltage of MGs tend to experience coupled large deviations subject to volatile
+operation conditions of generation and load, transitions between islanded and
+grid-connected modes, etc.
+The hierarchical control is commonly adopted to maintain the MG’s voltage
+and frequency stability. The hierarchical control includes primary control at
+the individual DER level, and secondary/tertiary control at the systemwide
+level. Even though the droop-based primary control at individual DERs can
+coordinate the power of DERs in a decentralized manner and improve the local
+stability, the frequency and voltage deviation at the system level may not be
+eliminated by merely using primary control [4]. The system stability may even
+be compromised when the droop gains are improperly designed to high values
+[5].
+Hence, the secondary control is essential to achieve stable voltage and
+frequency restoration.
+The scope of the paper lies in the secondary control, aiming to restore fre-
+quency and voltage for islanded MGs under larger disturbances or MGs in mode
+transitions(e.g., from grid-connected to islanded).
+The secondary control of
+MGs can be classified as model-based and model-free. There have been many
+research papers on model-based control. For example, the small-signal models
+have been used in [3, 6] to regulate droop gains and improve the systemwide
+small-signal stability. To handle large disturbance, multi-agent distributed co-
+operative control with feedback linearization was proposed in [7, 8] to deal with
+the nonlinearity.
+Despite advancement, all the aforementioned methods rely
+heavily on accurate physical models that may not always be available to MG
+operators due to time-varying topologies and operating conditions, as well as
+high uncertainty introduced by volatile renewables.
+To relax the pre-knowledge of accurate models, researchers have designed
+various model-free control methods. A common model-free method is Propor-
+tional and Integral (PI) control [4, 9, 10], which nevertheless may lack online
+adaptiveness to compensate for uncertainty. Besides, the MG may suffer from
+high starting overshoot, high sensitivity to controller gains, and sluggish re-
+sponse to disturbances if the PI control is not properly tuned [9].
+Another
+category of MG secondary control method is the averaging/consensus-based
+secondary droop control [11–13] that targets on accurate power sharing in quasi
+steady-state rather than voltage and frequency stability under large distur-
+bances.
+To improve systemwide voltage and frequency stability under both
+small and big disturbances, machine learning based methods were proposed
+[9, 14–18] for secondary voltage and frequency control. However, the universal
+learning machines such as artificial neural network (ANN) and reinforcement
+learning (RL) may lack physical interpretability and thus reliability of repre-
+senting the system’s dynamics in diverse topologies and operating conditions.
+Obtaining adequate offline training data that can sufficiently represent the sys-
+tem dynamics is challenging too.
+Moreover, individual DERs can be either controllable (e.g., energy storage
+systems (ESSs), renewable energy with ESSs), or non-controllable (e.g., renew-
+able generation operating under maximum power point tracking (MPPT)) at the
+2
+
+secondary level. They possess diverse modes of primary control (e.g., conven-
+tional isochronous grid-forming, power-based grid-forming and grid-following).
+The resulting model complexity may affect the performance of the secondary
+control. Yet all the aforementioned works (both model-based and model-free)
+on secondary voltage and frequency control assume that all DERs in islanded
+MGs work under the grid-forming or voltage control mode [5, 7–10, 15–17, 19].
+However, in existing MGs, the mix of grid-forming and grid-following control
+with diverse control structures and parameters introduces uncertainty that chal-
+lenges MG secondary control. Particularly, when large disturbances occur, the
+interaction among diverse grid-forming and grid-following converters and the
+dynamics of the affiliated phasor-locked loops (PLLs) may deteriorate the sys-
+tem stability and control performance.
+In this paper, we propose a new data-driven secondary voltage and frequency
+control method for MGs with both grid-forming and grid-following DERs. The
+method is able to handle MG nonlinearity and uncertainty (e.g., MG mode
+transitions from grid-connected to islanded, generation and load variations)
+in an adaptive data-driven fashion. The proposed method requires no offline
+training and uses only a small window of phasor angle and voltage data from
+synchrophasors (e.g., micoPMUs) at the DER output ends. In the proposed
+method, Koopman operator theory [20] is leveraged to convert the nonlinear
+dynamical system into a linear one under Koopman embedding mapping. As
+such, the system can be identified and controlled with mature and powerful
+linear system techniques. Particularly, we tailor the OKID (Observer Kalman-
+filter IDentification)-based algorithm so that the Koopman-based linear dynam-
+ical system can be identified optimally. Then, the discrete-time linear quadratic
+regulator (LQR) is applied to the identified Koopman-based linear dynamical
+system with well-characterized stability properties. It is noteworthy that M.
+Korda et al [21] utilized Koopman operator control for power system transient
+stability and control, while the method required offline training data. Besides,
+the identification based on the brute-force least-squares estimation, could lead
+to unsatisfactory identification results. Gong et al [22] presented a combined ap-
+plication of the Koopman operator and identification method for MG secondary
+control. However, the method assumes that the droop parameters of DERs are
+known by the secondary controller, whereby the control matrix in the Koopman
+state space can be directly obtained. In this paper, we lift the assumption that
+the local control mechanism and parameters are fully unknown. In short, the
+advantages of the proposed method are summarized as below:
+(i) The proposed Koopman-inspired enhanced OKID method can help identify
+the system dynamics accurately and adaptively due to the capacity of dealing
+with nonlinearity and uncertainty under large disturbances.
+(ii) The proposed Koopman-inspired identification and control method is purely
+data-driven using only a small window of synchrophasor data. It requires no
+knowledge of network information and primary controllers, and no offline train-
+ing.
+(iii) The MG system with the proposed Koopman-inspired identification and
+control is guaranteed to be bounded-input-bounded-output (BIBO) stable. On
+3
+
+Figure 1: Microgrid control architecture
+top of the BIBO stability, the sufficient condition under which the MG system
+is asymptotically stable is also developed.
+(iv) The proposed control method is robust to measurement noises and time
+delays as tested in numerical studies.
+The remainder of the paper is organized as follows: Section II describes
+the MG hierarchical control and the interfaces between secondary and primary
+control. Section III details the proposed Koopman-inspired identification and
+control method.
+Section IV presents case studies for validation.
+Section V
+concludes the paper.
+2. Microgrid System Description
+A MG can be controlled hierarchically with the secondary and primary con-
+trol as shown in Fig. 1. The primary controllers enable fast response of the
+individual DERs to guarantee local stability, while the secondary controller
+globally dictates the primary controllers of controllable DERs according to the
+data collected from microPMUs, whereby the systemwide interaction dynamics
+of MGs can be handled and the voltage and frequency can be restored.
+The local primary control modes can be different, such as grid-forming or
+grid-following [23]. In an islanded MG or future power system without syn-
+chronous generators, at least one DER is required to work under the grid-forming
+mode to actively form the grid voltage and frequency; then the rest of DERs can
+remain operating under the grid-following modes [23]. As discussed in [24, 25],
+grid-forming DERs define the voltage magnitude and frequency. In contrast,
+grid-following DERs follow the measured frequency and voltage magnitude in
+the grid via PLL, which represents the prevalent type of control strategy for
+grid-connected PV and wind converters in existing power grids. As droop is
+commonly used in MGs, we consider the droop-based grid-forming [26, 27] and
+4
+
+Control Command
+Microgrid Node/Bus
+Measurement Data Sensing
+Microgrid Distribution Line
+Microgrid Network
+Secondary Control
+Primary
+Control
+Control Center
+DER 1
+Adaptive Online
+MicroPMUData
+Identification
+Real-Time
+MicroPMU
+Control
+Cvber
+Data
+Algorithms
+Network
+DER 2
+Primary
+Control
+Primary
+Control
+MicroPMUDatainverse-droop-based grid-following control [28]. Specifically, consider a DER at
+the bus i. As shown in Fig. 2(a), the droop for grid-forming converter control
+is defined as:
+Droop :
+� ωi − ω∗
+i
+Vi − V ∗
+i
+�
+=
+� −σω(Pi − P ∗
+i )
+−σV (Qi − Q∗
+i )
+�
+(1)
+where ωi and Vi denote the frequency and voltage magnitude for the grid-
+forming control. ω∗
+i and V ∗
+i are the rated frequency and voltage. The parameters
+σω and σV are frequency and voltage droop gains, respectively. P ∗
+i and Q∗
+i are
+the reference power in the droop, which can be the steady-state power without
+the secondary control or an augmented reference power after the secondary
+control. Pi and Qi are the active and reactive power, which are measured with
+a low-pass filter embedded in the power measurement block in Fig. 2. The filter
+is in the form of [29, 30]:
+Pi =
+1
+Tfs + 1P (IN)
+i
+,
+Qi =
+1
+Tfs + 1Q(IN)
+i
+(2)
+where Tf is the time constant of the first-order low-pass filter; P (IN)
+i
+and Q(IN)
+i
+represent the active and reactive power before filtering. Generally, the filter is
+required to attenuate high-frequency dynamics (e.g., harmonics) and preserve
+low-frequency dynamics (e.g., sub-synchronous components which can be fur-
+ther managed by secondary control). With the secondary control, the reference
+power P ∗
+i and Q∗
+i of the droop in Fig. 2 was updated in discrete time as:
+P ∗(+)
+i
+= P ∗
+i + ∆P ∗
+i ,
+Q∗(+)
+i
+= Q∗
+i + ∆Q∗
+i
+(3)
+To distinguish P ∗
+i and Q∗
+i before and after secondary control, the superscription
+(+) is added in Eq.(3) to denote the values of P ∗
+i and Q∗
+i after considering the
+secondary control.
+Similarly, if the DER at the bus i is grid-following as shown in Fig. 2(b),
+the inverse droop control is defined as:
+Inverse
+Droop :
+� ¯Pi
+¯Qi
+�
+=
+� − 1
+σω (ωi − ω∗
+i )
+− 1
+σV (Vi − V ∗
+i )
+�
+(4)
+where ¯Pi and ¯Qi are the power generated by the inverse droop. The eventual real
+power Pi and reactive power Qi sent to the grid-following control as references
+are:
+Pi = ¯Pi + P ∗
+i ,
+Qi = ¯Qi + Q∗
+i
+(5)
+where P ∗
+i and Q∗
+i are reference power guided by the secondary control. In a
+similar form to Eq. (3), P ∗
+i and Q∗
+i was updated in discrete time as: P ∗(+)
+i
+=
+P ∗
+i + ∆P ∗
+i and Q∗(+)
+i
+= Q∗
+i + ∆Q∗
+i .
+Consequently, the droop for grid-forming control and the inverse droop for
+the grid-following control can be represented as:
+Droop:
+�ωi − ω∗
+i
+Vi − V ∗
+i
+�
+=
+�
+−σω(Pi − P ∗(+)
+i
+)
+−σV (Qi − Q∗(+)
+i
+)
+�
+=
+�−σω(Pi − P ∗
+i )
+−σV (Qi − Q∗
+i )
+�
++
+�σω
+σV
+�
+ui, with ui =
+�∆P ∗
+i
+∆Q∗
+i
+�
+(6a)
+5
+
+(a)
+(b)
+Figure 2: Different primary control modes: (a) droop-based grid-forming control; (b) inverse-
+droop-based grid-following control.
+Inverse Droop:
+�
+Pi − P ∗(+)
+i
+Qi − Q∗(+)
+i
+�
+=
+�− 1
+σω (ωi − ω∗
+i )
+− 1
+σV (Vi − V ∗
+i )
+�
+⇒
+�ωi − ω∗
+i
+Vi − V ∗
+i
+�
+=
+�−σω(Pi − P ∗
+i )
+−σV (Qi − Q∗
+i )
+�
++
+�σω
+σV
+�
+ui
+(6b)
+According to (6a)-(6b), both the droop and the inverse droop take the same
+form:
+� ˙θi
+˙Vi
+�
+=
+� −σω(Pi − P ∗
+i )
+− σV
+τV (Qi − Q∗
+i )
+�
++
+�σω
+σV
+τV
+�
+ui
+(7)
+where
+˙θi
+=
+ωi − ω∗
+i
+(8)
+Pi
+=
+n
+�
+j=1
+ViVj(Gij cos(θi − θj) + Bij sin(θi − θj))
+(9)
+Qi
+=
+n
+�
+j=1
+ViVj(Gij cos(θi − θj) − Bij sin(θi − θj))
+(10)
+τV is the equivalent time constant of voltage magnitude dynamics due to the
+grid-forming or the grid-following control loops, which can be treated as a first-
+order inertia system when properly tuned; u denotes the external control inputs
+due to secondary control; θ is the voltage phasor angle; j denotes the bus
+number; Gij and Bij represent the equivalent conductance and susceptance
+between bus i and j.
+As shown in Fig. 2, the droop generates the voltage reference for the grid-
+forming control system, and the inverse droop generates the power references
+for the grid-following control system. Note that the grid-forming control based
+on a PLL is adopted in this paper to mitigate negative impacts on systemwide
+stability [26]. The dynamics of the grid-forming and grid-following loops are not
+presented in detail but will be considered in all simulations presented in Section
+4. Interested readers are referred to [24, 27] for the detailed modeling.
+Uncertainty from Grid-Forming Control: As the local converter control is
+much faster than the secondary control, the voltage reference fed to grid-forming
+control in Fig. 2(a) is approximately equal to ω and V assuming that the grid-
+forming control loop is well tuned. Nonetheless, when large disturbances (e.g.,
+6
+
+Primary
+Power
+io
+Control I
+Measurement
+Droop
+Local
+3
+Vo
+Grid-Forming
+Bus
+w = *-O(P- P*)
+DER
+V
+V = V* - v(Q - Q*)
+Control
+p*Q* I Updated by Secondary ControlPrimary Control
+Voltage Magnitude
+Inverse droop
+w*
+Measurement
+30
+w (estimated by PLL)
+V*
+0
+*A-
+7
+40
+io
+Local
+P
+Grid-Following
+Bus
+P=P+P*
+Control
+DER
+Q
+Q=Q +Q*
+(with PLL)
+p* 0*
+ Updated by Secondary ControlMG transitions from the grid-connected mode to the islanded mode, volatile
+generation and load) occur that cause large power perturbations, the nonlin-
+earity driven by the system power flows (8)-(10) and by the control interaction
+between the droop module and the grid-forming control loops can emerge, thus
+leading to modeling uncertainty in Eq. (7).
+Uncertainty from Grid-Following Control: Likewise, the control interaction
+between the inverse-droop module and the grid-following control loops may
+emerge when there are system disturbances causing big perturbations to the
+angle and voltage. In addition, when there are large disturbances or measure-
+ment noises that make the grid voltage measurement distorted, the uncertainty
+due to the PLL can also directly introduce the modeling error to Eq. (7) [31].
+To describe the uncertainty from either the grid-forming or the grid-following
+control, Eq. (7) can be modified as
+� ˙θi
+˙Vi
+�
+=
+� −σω(Pi − P ∗
+i )
+− σV
+τV (Qi − Q∗)
+�
++
+�σω
+σV
+τV
+�
+ui +
+�fω(P, Q, θ, V )
+fV (P, Q, θ, V )
+�
+(11)
+where fω(.) and fV (.) are unknown nonlinear functions to describe the residual
+dynamics for the voltage phasor angle and magnitude.
+The aforementioned nonlinearity and uncertainty pose challenges to the con-
+ventional secondary control of MGs (e.g., model-based ones and PI) especially
+under large disturbances. To address these challenges, we propose a Koopman-
+inspired method that can help identify the system accurately and adaptively
+using data despite nonlinearity and uncertainty such that effective control can
+be designed.
+3. Koopman-Inspired Identification and Control
+3.1. Koopman Operator Theory
+Koopman operator theory [20] shows that a nonlinear dynamical system can
+be transformed into an infinite-dimensional linear system under a Koopman em-
+bedding mapping. The Koopman-enabled linear model is valid for global non-
+linearity with the infinite-dimensional representation as opposed to traditional
+locally linearized small-signal models. However, in practice, one can consider
+finite-dimensional Koopman invariant subspaces where dominant dynamics can
+be described. Particularly, given a nonlinear dynamical system with external
+control xk+1 = F(xk, uk), where x ∈ M and u ∈ U with M and U being the
+manifolds of state and control input, we consider the Koopman embedding map-
+ping Φ from the two manifolds to a new Hilbert space Φ : M×U → H, which lies
+within the span of the eigenfunctions ϕj. That is, Φ(x, u) = �Nϕ
+j=1 ϕj(x, u)vj,
+where Φ(x, u) = [Φ1(x, u), Φ2(x, u), . . . , Φi(x, u), . . . , Φp(x, u)]T is a set of
+Koopman observables, vj are the vector-valued coefficients called Koopman
+modes.
+The Koopman operator K, acting on the span of ϕj, advances the
+embeddings Φ(x, u) linearly in the Hilbert space H as [20]:
+Φ(xk+1, uk+1) = KΦ(xk, uk)
+= K
+Nϕ
+�
+j=1
+ϕj(x, u)vj =
+Nϕ
+�
+j=1
+(ρjϕj(xk, uk)vj)
+(12)
+7
+
+where ρj are the eigenvalues satisfying Kϕj(x, u) = ρjϕj(x, u). To be con-
+sistent with the linear form of control inputs in Eq.
+(11), we assume that
+Φi(x, u) = gi(x)+li(u) where gi(x) is a nonlinear observable function and li(u)
+is linear with li(0) = 0 [32]. In addition, we assume Φi(xk+1, 0) = KΦi(xk, uk)
+for all k. Then, gi(xk+1) + li(0) = Kgi(xk) + Kli(uk) ⇒ gi(xk+1) = Kgi(xk) +
+Kli(uk). This assumption means that the Koopman operator is only attempt-
+ing to propagate the observable functions at the current state xk and inputs
+uk to the future observable functions on the state xk+1 but not on future in-
+puts uk+1 (i.e., [∆P ∗, ∆Q∗]T are not state-dependent) [20].
+Let us define
+z := g(x) = [g1(x), g2(x), . . . , gi(x), . . . gp(x)]T . Then we have an approxima-
+tion of Eq. (11) in a form of extended dynamic mode decomposition with control
+(EDMDc) [32] as below
+(Process
+model)
+zk+1 = Azk + Buk + δk
+(13a)
+(Observation
+model)
+yk = Czk + ek
+(13b)
+where yk are the outputs of the Koopman state space model.
+We define
+yk = [dθk, dVk]T = [θk − θ∗
+L, Vk − V ∗
+L]T in Eq. (13b) as the PMU-measured
+phasor angle and voltage magnitude deviations from the local operation points
+[θ∗
+L, V ∗
+L]T that are the first data sample from a window of collected PMU data.
+A and B are the state transition matrix and control matrix, satisfying that
+Azk = Kg(xk) and Buk = Kl(uk). δk is the Koopman modeling error associ-
+ated with the EDMDc approximation. ek is the observation model error. Given
+proper Koopman observables z, the Koopman state space model (13a)-(13b)
+can describe large signal-driven nonlinear dynamics. That being said, under
+the Koopman embedding z = g(x), the nonlinear dynamical system (11) can
+be represented by the linear dynamical system (13a)-(13b) that is valid under
+both small and large perturbations. There are three consecutive tasks to use
+this model for control: determination of Koopman observables, online identifi-
+cation of the Koopman state space model, and implementation of linear control
+(illustrated in Sections 3.2, 3.3 and 3.4, respectively).
+3.2. Koopman Observables for MG Secondary Control
+The selection of Koopman observables is important for realizing accurate
+modeling.
+The observables can be selected either empirically [21, 22, 33] or
+with the help of machine learning techniques [34–36], while it remains an open
+question to obtain the best possible observables.
+In this paper, we selected
+the Koopman observables based on our experience and domain knowledge of
+power systems and microgrids. According to Eq. (7)-Eq. (10), sinusoidal-driven
+interaction dynamics may emerge when subject to large perturbations and low
+inertia (i.e., the general solution for the droop-control differential equations
+contains trigonometric patterns). Inspired by this, we include the functions sin θ
+and cos θ into the Koopman embedding to describe such underlying dynamics,
+which were shown effective to describe interaction transients of power grids [21].
+Thus, let us define the MG original states xk = [θk, Vk]T and the Koopman real-
+valued observables zk = g(xk) as:
+zk = g(xk) = [∆Vk, sin θk − sin(θ∗
+L,k), cos θk − cos(θ∗
+L,k), ∆ωk]T
+(14)
+8
+
+where ∆V and ∆ω are voltage and angular frequency deviations from the
+nominal values. θ∗
+L,k represents the approximate underlying operation point of
+voltage phasor angle at time step k. The Koopman observables z constitute the
+Koopman state space in the form of Eq. (13a)-Eq. (13b), where the parameter
+matrices A, B and C are to be determined by an advanced system identification
+method online as described in the next section.
+3.3. Online Identification: A Koopman-Inspired Enhanced OKID Algorithm
+Considering the Koopman-based linear dynamical system model (13a)-(13b),
+we propose an observer Kalman filter identification (OKID)–based optimization
+algorithm to optimally identify the MG Koopman state space model (i.e., the
+matrix parameters A,B and C).
+The OKID Algorithm. Belonging to the category of closed-loop subspace
+methods, the conventional OKID algorithm is commonly used to identify linear
+systems [37]. It is free of the bias problem that most typical closed-loop subspace
+methods have [37], and has been applied in many areas such as aircraft control
+and autonomous underwater vehicles [38]. In OKID, the impulse response of the
+system is estimated in a least-squares fashion with data. Then, a state space
+model of the system is obtained with the eigensystem realization algorithm
+(ERA). Specifically, let Y and U represent the matrix stacking the time series
+data of the outputs y and the control inputs u in a matrix form. Let Yi and
+Ui represent the observation outputs and the control inputs at the ith time step
+in the data matrix, and consider the length of the sliding window is N. By
+observing Eq. (13a)-Eq. (13b) and assuming zero initial conditions, yk can
+be expressed with iterations in a form of yk = Czk = C(Azk−1 + Buk−1) =
+C(A(Azk−2+Buk−2)+Buk−1) = CAk−1Bu0+CAk−2Bu1+...CBuk−1 =
+�k−1
+i=0 CAk−i−1Bui, whereby we obtain
+Y =
+�
+CB
+· · ·
+CAN−1B
+�
+�
+����
+U0
+U1
+...
+UN−1
+0
+U0
+...
+UN−2
+...
+...
+...
+...
+. . .
+. . .
+. . .
+U0
+�
+����
+(15)
+Let h denote the impulse response of the Koopman state space model (13a)-
+(13b) in the sliding window of size N (from k = 1 to k = N) with zero initial
+conditions (x0 = 0) and impulse inputs (u0 = 1 and uk = 0 when k > 0), we
+have
+h =
+�
+h1
+h2
+...
+hN
+�
+=
+�
+CB
+CAB
+...
+CAN−1B
+�
+(16)
+Then according to Eq. (15)- (16) and with the knowledge of the observation
+matrix Y and the control input matrix U, one can estimate the impulse response
+in a least-squares fashion
+�
+h1
+h2
+...
+hN
+�
+= Y
+�
+����
+U0
+U1
+...
+UN−1
+0
+U0
+...
+UN−2
+...
+...
+...
+...
+0
+0
+. . .
+U0
+�
+����
+†
+(17)
+9
+
+where the operator † represents the Moore-Penrose pseudo-inverse. Note that
+the noise is not optimally filtered by the least-squares inverse as presented in Eq.
+(17). To address the issue, the conventional OKID can be designed based on an
+optimal observer system whereby optimal system parameters can be identified.
+For simplicity, we refer readers to [39] (Pages 340-343) for detailed explanation
+and implementation.
+Next, with the obtained impulse response, the Hankel matrix H and the
+next-step Hankel matrix H
+′ can be written as follows:
+H =
+�
+����
+h1
+h2
+...
+hN
+0
+h1
+...
+hN−1
+...
+...
+...
+...
+0
+0
+. . .
+h1
+�
+���� , H
+′ =
+�
+����
+h2
+h3
+...
+hN+1
+0
+h2
+...
+hN
+...
+...
+...
+...
+0
+0
+. . .
+h2
+�
+����
+(18)
+The Hankel matrix H could be truncated with Singular Value Decomposition
+(SVD):
+H = UΣVT = [�U, U tr]
+��Σ
+0
+0
+Σtr
+� �
+�V
+T
+VT
+tr
+�
+≈ �
+U �Σ �V
+T
+(19)
+Let
+O = [C, CA, CA2, ..., CAN−1]T
+(20)
+be the observability matrix, and
+C = [B, AB, A2B, ..., AN−1B]
+(21)
+be the controllability matrix. Then, by observing Eq. (16) and Eq. (18), we
+have
+H = OC,
+H
+′ = OAC
+(22)
+Furthermore, considering Eq. (19), we can assume that O = �U �Σγ and C =
+�Σ1−γ �V
+T , where γ is an arbitrary real value.
+Conventional OKID algorithm. For the conventional OKID algorithm, ERA
+is thereafter used to identify the matrix A and B, with γ set to a constant
+1
+2 for a special balanced realization. That is, one can assume O = �U �Σ
+1
+2 and
+C = �Σ
+1
+2 �V
+T , whereby a state space model with balanced Grammians is realized
+(i.e., the same degree of controllability and observability) that agrees with the
+control input and the observation data. As such, with γ = 1
+2 and by Eq. (20) -
+Eq. (22), the matrices A and B can be identified by the conventional OKID
+as follows [39]:
+�
+A = �Σ− 1
+2 �U
+T H
+′ �V �Σ− 1
+2
+(23a)
+�
+B = CNS×NU =
+�
+�Σ
+1
+2 �V
+T �
+NS×NU
+(23b)
+where the operator
+�.�
+NS×NU represents the first NS rows and the first NU
+columns of the matrix in the bracket; NS is the dimension of Koopman embed-
+ding space and NU is the dimension of control inputs.
+10
+
+In this paper, to better identify the Koopman-based process dynamics, we
+propose a Koopman-inspired algorithm to find an optimal γ rather than assum-
+ing γ = 1
+2 as in the conventional OKID. Consider a general form with γ unfixed
+�U �Σγ = O = [C, CA, CA2, ..., CAN−1]T
+= IN×N ⊗ C · [I, A, A2, ..., AN−1]T
+(24a)
+�Σ1−γ �V
+T = C = [B, AB, A2B, ..., AN−1B]
+= [I, A, A2, ..., AN−1](IN×N ⊗ B)
+(24b)
+where IN×N is the identity matrix with the dimension N × N, and ⊗ denotes
+the Kronecker product which is
+IN×N ⊗ C =
+�
+��
+C
+...
+C
+�
+�� ,
+IN×N ⊗ B =
+�
+��
+B
+...
+B
+�
+��
+(25)
+Then
+(IN×N ⊗ C)† �U �Σγ = [I, A, A2, ..., AN−1]T
+(26a)
+�Σ1−γ �V
+T (IN×N ⊗ B)† = [I, A, A2, ..., AN−1]
+(26b)
+By observing Eq. (26a) and Eq. (26b), we have
+�Σγ �U
+T ((IN×N ⊗ C)†)T = �Σ1−γ �V
+T (IN×N ⊗ B)†
+⇒ �Σ2γ−1 �U
+T �
+(IN×N ⊗ C)†�T = �V
+T (IN×N ⊗ B)†
+(27)
+Treating Eq. (27) as a soft constraint for the parameter γ, one can formulate a
+quadratic optimization problem to solve the optimal parameter γopt:
+γopt = arg min
+γ
+∥�Σ2γ−1 �U
+T �
+(IN×N ⊗ C)†�T − �V
+T (IN×N ⊗ B)∥F
+subject to:
+0 ≤ γ ≤ 1
+(28)
+where ∥.∥F represents the Frobenius norm of a matrix. The inequality 0 ≤ γ ≤ 1
+is added to constrain problem complexity. The novel OKID-based algorithm for
+parameter estimation is summarized below. The flowchart of the algorithm is
+also presented in Fig. 3.
+The Proposed Online Koompan-Inspired Enhanced OKID Algorithm
+Algorithm Initialization. Initialize γopt = γopt,0, the smoothing factor η,
+and the time step TOP T between two updates of γopt. The selection of these
+parameters will be discussed in Remarks after the presentation of the algorithm.
+At each time step of identification and the secondary control, i.e., for k =
+1, 2, ..., conduct Step 1 -Step 5.
+Step 1: Data preparation. Collect the last N data samples from microPMUs
+to obtain the data matrices of phasor angle Θ, voltage deviation ∆V and
+angular frequency deviation ∆Ω from the nominal values. Collect control input
+11
+
+data U from the secondary controller.
+For example, the phasor angle Θ is
+stacked in a form of
+Θ =
+�
+�
+|
+|
+|
+Θ1
+...
+ΘN
+|
+|
+|
+�
+�
+(29)
+∆V , ∆Ω and U are formed in the same way. The approximated operation
+points of voltage phasor angles and magnitudes Θ∗
+L and V ∗
+L are defined as the
+first data sample from a window of collected PMU data, prepared in a matrix
+form as follows:
+Θ∗
+L =
+�
+�
+|
+|
+|
+Θ1
+...
+Θ1
+|
+|
+|
+�
+� ,
+V ∗
+L =
+�
+�
+|
+|
+|
+V1
+...
+V1
+|
+|
+|
+�
+�
+(30)
+Prepare the data matrices for y and z as follows: Y = [Θ − Θ∗
+L, V − V ∗
+L ]T ,
+and Z = [∆V, sin (Θ) − sin (Θ∗
+L), cos (Θ) − cos (Θ∗
+L), ∆Ω]T .
+Step 2: Hankel matrix preparation and SVD. Estimate the impulse re-
+sponse and prepare the Hankel matrices according to Eq. (17)-Eq. (18). Con-
+duct the SVD on the obtained Hankel matrix H ≈ �U �Σ �V
+T .
+Step 3: Estimation of C. Ignoring the error term in Eq. (13b), we have
+Y = CZ. Thus, one can estimate the observation matrix C at each time step
+k in a least-squares fashion by multiplying the pseudo-inverse on both sides of
+the equation, which is
+�
+Ck = Y Z†
+(31)
+Step 4: Optimization for γopt. Check if the run time of optimization between
+the last update of γopt is larger than TOP T . If no, γopt,k = γopt,k−1, go to Step
+5; otherwise, solve the optimization problem in Eq. (28) for γ−
+opt. To do so, by
+Eq. (23b), replace B with
+�
+�Σ1−γ �V
+T �
+NS×NU
+and replace C with �
+Ck from Step
+3 in Eq. (28). Then, adaptively update γopt by
+γopt,k =ηγ−
+opt + (1 − η)γopt,k−1,
+for
+k = TOP T , 2TOP T , 3TOP T , ...
+(32)
+where γ−
+opt is the optimal value of the realization parameter γ according to Eq.
+(28). That is, once γ−
+opt is updated, we update γopt with the weighted sum of the
+old γopt at last time step and the updated value γ−
+opt. η is the weight to smooth
+online learning. The role of η is to smooth the online learning of γ. As the small
+piece of online data used for identification is characterized by stochasticity,
+the smoothing factor η can mitigate aggressive change to make the learning
+process more reliable. This is so because the estimation is equivalent to the
+Robbins–Monro form [40], which is γopt,k = ηγ−
+opt + (1 − η)γopt,k−1 = γopt,k−1 +
+η(γ−
+opt−γopt,k−1). The larger the value of γ is, the smoother the learning process
+tends to be, whereas the adaptiveness of learning is compromised.
+Step 5: Estimation of A and B. By Eq. (23a)-(23b)
+�
+Ak =
+�
+η �Σ−γopt,k �U
+T H
+′ �V �Σγopt,k−1 + (1 − η) �
+Ak−1
+if k ≥ 1
+�Σ−γopt,k �U
+T H
+′ �V �Σγopt,k−1
+if k = 0
+(33)
+12
+
+�
+Bk =
+�
+�
+�
+�
+�
+η
+�
+�Σ1−γopt,k �V
+T �
+NS×NU
++ (1 − η) �
+Bk−1
+if k ≥ 1
+�
+�Σ1−γopt,k �V
+T �
+NS×NU
+if k = 0
+(34)
+After implementing the identification algorithm, the identified Koopman state
+space model at the time step k is obtained as:
+zk+1 = �
+Akzk + �
+Bkuk
+(35a)
+yk = �
+Ckzk
+(35b)
+Compared to the traditional EDMDc used in power systems [21], the pro-
+posed Koopman-inspired OKID can use the observation data y as in Eq. (13b)
+to help learn the Koopman state space model in Eq. (13a), while the traditional
+EDMDc only estimates the Koopman state space in Eq. (13a) in a least-squares
+fashion without the incorporation of observation data. The fusion of the infor-
+mation from the observation data provides extra opportunities to enhance the
+modeling efficacy.
+Remarks
+• γopt: in this paper, γopt,0 = 1
+2. Thus the enhanced OKID is initially equiv-
+alent to the conventional one while it gradually learns the optimized value
+for γopt with the online OKID and the periodically enabled optimization
+in Eq. (28).
+• The smoothing factor η: it is used to weigh the past estimations and
+the latest one, and set to
+1
+N in this paper with the assumption that all
+estimations have the same weight independent on the time of occurrence.
+A larger η means the estimation put more weight on the newest data, and
+vice versa.
+• The time step TOP T for updating γopt: it is set to 0.6s, which is longer
+than the run time of the proposed Koopman-inspired enhanced OKID
+and the time step of secondary control (30ms) as detailed in Section 4.
+A small TOP T is favorable as a fast update of γ to compensate for the
+uncertainty of the Koopman process model (13a), while it should be longer
+than the run time of the optimization (28) to ensure the feasibility of online
+implementation.
+3.4. The Linear Control Based on the Koopman-Inspired Enhanced OKID
+After obtaining the identified model (35a) - (35b), a discrete-time linear
+quadratic regulator (LQR) is applied at each time step of secondary control,
+aiming to reduce the voltage and frequency deviations by minimizing the cost
+J(u) =
+∞
+�
+k=0
+zT
+k Qzk + uT
+k Ruk,
+subject to
+zk+1 = �
+Azk + �
+Buk
+(36)
+13
+
+Figure 3: Algorithm flowchart of the proposed Koopman-inspired enhanced OKID
+where Q and R are cost matrices defined as:
+Q =
+�
+���
+qV
+qsin θ
+qcos θ
+qω
+�
+��� ,
+R =
+�rP
+rQ
+�
+(37)
+where qV , qsin θ, qcos θ and qω are cost submatrices for the Koopman observ-
+ables presented in (14). rP and rQ are cost submatrices for the control signals
+∆P ∗ and ∆Q���. They are basically selected empirically in this paper based on
+which factor is treated to be more important. The optimal control input can be
+obtained by:
+uk =
+�
+�
+�
+ULB
+uk < ULB
+−Kzk
+ULB ⩽ uk ⩽ UUB
+UUB
+uk > UUB
+with
+K = ( �
+BT S �
+B + R)−1 �
+BT S �
+A
+(38)
+where K is the control gain matrix. S is the solution of Riccati equation [41].
+UUB and ULB are the upper and lower saturation limits that can bound the
+uncertainty introduced by control inputs. The bounds are user-defined values,
+which are determined empirically in the paper. Usually, large bounds can lead
+to faster response whereas the uncertainty introduced through control input
+channels could be increased to an unmanageable level that degrades the dynamic
+control performance or even stability. On the other hand, the bounds cannot be
+set to too small values, otherwise, the response could be slow and the capability
+of the controller cannot be fully taken use of.
+The Koopman-inspired enhanced OKID illustrated in Section 3.3 and the
+LQR illustrated in Section 3.4 can be respectively applied to the identification
+block and the control algorithm block of secondary control in Fig. 1. Specifi-
+cally, Fig. 4 presents the proposed online identification and control structure.
+The stability of such Koopman-inspired identification and control is guaranteed,
+which is proved in what follows.
+3.5.
+Stability Analysis
+MG dynamics can be expressed in a Koopman-based structure and can be
+approximated with the online Koopman-inspired identification in Section 3.3.
+The approximation error is bounded but often not quantifiable as it depends on
+14
+
+At each time
+Steps 1-2
+Step 3
+Step 4
+Step 5
+step k:
+No
+Keep Old opt
+Data Matrix
+Initialization
+Estimate Ck
+If k = ToPT, 2ToPT, :
+Preparation & SVD
+OKID (update Ak and Bk)
+Yes.
+Update Yopt
+EndFigure 4: Online structure of the proposed Koopman-inspired enhanced OKID and control
+the appropriateness of Koopman observables and the online parameter identi-
+fication algorithm. In what follows, we aim to prove stability properties in a
+general sense.
+3.5.1. Proof of BIBO Stability
+We prove that the proposed Koopman-inspired OKID-based control is BIBO
+(bounded-input-bounded-output) stable. Denoted by ˆxk+1 the one-step-ahead
+prediction of the state vector x at the time step k with the OKID-based es-
+timation.
+Denoted by ˆKk the estimated Koopman operator at time step k.
+According to Eq. (12), we have
+g(ˆxk+1) = Φ(ˆxk+1, 0) = ˆKkΦ(xk, uk) = ˆKk
+Nϕ
+�
+j=1
+ϕj(xk, uk)vj =
+Nϕ
+�
+j=1
+(ρj,kϕj(xk, uk)vj)
+(39)
+where ρj,k is the eigenvalue corresponding to the jth eigenfunction ϕj for the
+estimated Koopman operator ˆKk. Recall that Φ(x, u) = g(x) + l(u) discussed
+in Section III.A, where l(u) = [l1(u), l2(u), . . . lp(u)]T and l(0) = 0. Then
+g(xk+1) = g(ˆxk+1) + δk = ˆKkΦ(xk, uk) + δk
+= ˆKk(g(xk) + l(uk)) + δk
+= ˆKk( ˆKk−1Φ(xk−1, uk−1) + δk−1 + l(uk)) + δk
+= ˆKk( ˆKk−1(g(xk−1) + l(uk−1)) + δk−1 + l(uk)) + δk
+= ˆKk( ˆKk−1( ˆKk−2Φ(xk−2, uk−2) + δk−2 + l(uk−2)) + l(uk−1)) + δk−1 + l(uk)) + δk
+= · · · =
+k
+�
+h=0
+ˆKk−hΦ(x0, u0) +
+k
+�
+h=1
+k
+�
+i=h
+ˆKk−i+h(δh−1 + l(uh)) + δk
+=
+Nϕ
+�
+j=1
+(
+k
+�
+h=0
+ρj,h)ϕj(x0, u0)vj +
+k
+�
+h=1
+k
+�
+i=h
+ˆKk−i+h(δh−1 + l(uh)) + δk
+15
+
+Microgrid Network
+Control Command
+Microgrid Node/Bus
+(e.g., four-bus, thirteen-bus microgrids)
+Measurement Data Sensing
+Microgrid Distribution Line
+Action:(control inputs)u=[△P",Q*jr
+Primary
+Control
+Secondary Controller:
+DER 1
+Proposed Koopman-Inspired Enhanced OKID and LQR Control
+Control
+MicroPMU Data
+Action Generator (LQR)
+Online
+Minimize the Cost Function
+Identification
+J(u) =z Qz +utRui
+MicroPMUl
+Control
+Measurement:
+ak = jox, VaJT
+Cyber
+Koopman Embedding Mapping g :
+Data
+[Section 3.4]
+Inputs
+Network
+DER 2
+(Action)
+Zk =g(ak[Section 3.2]
+Data
+Primary
+Koopman-inspired Enhanced OKID
+Control
+[Section 3.3]
+3
+Primary
+States
+DER
+Control
+MicroPMU Data
+zk
+Estimated System Model:
+Estimated Parameters
+Zk+1 = Akzk + Buk
+Ak
+Bk
+Yk = Ckzk
+4where δk is the Koopman modeling error which has been defined in Eq. (13a),
+and vj is the jth Koopman mode associated with the Koopman eigenfunction
+ϕj. Apparently,
+0 ⩽ ∥
+Nϕ
+�
+j=0
+(
+k
+�
+j=0
+ρj,h)ϕj(x0, u0)vj∥2 ⩽ lim
+k→∞(maxj,h|ρj,h|)k+1
+Nϕ
+�
+j=1
+∥ϕj(x0, u0)vj∥2
+(40)
+With LQR in the Koopman invariant subspace, assume the MG secondary
+controller can optimally make the magnitudes of all system eigenvalues smaller
+than 1 (if the system is stabilizable). That is ˆKkϕj = ρj,kϕj with |ρj,k| < 1.
+Due to the online rolling-based estimation in the proposed method, we can
+assume the global error ∥ �k
+h=1 Πk
+i=h ˆKk(δh−1 + l(uh))∥2 is bounded by ζg, and
+the modeling error is ∥δk∥2 bounded by ϵm. According to (40), we have
+lim
+k→∞ ∥g(xk+1)∥2 ⩽ lim
+k→∞(maxj,h|ρj,h|)k+1
+Nϕ
+�
+j=1
+∥ϕj(x0, u0)vj∥2 + ζg + lim
+k→∞ ∥δk∥2 ⩽ ζg + ϵm
+(41)
+Based on (41), g(x) converges till reaching the area Ξ =
+�
+g(x)|∥g(x)∥2 ⩽
+ζg + ϵm
+�
+. Thus, the system is BIBO stable. Besides, the Koopman-based LQR
+can guarantee asymptotic stability subject to the disturbance in control input
+channels under mild conditions. See Section 3.5.2.
+3.5.2. Stability Margins of Koopman-Enabled LQR
+The discrete-time LQR used in this paper has analytical disc stability mar-
+gins [42], within which asymptotic stability subject to the disturbance in control
+input channels is guaranteed. Specifically, consider the identified Koopman state
+space model described as below:
+g(xk+1) = Ag(xk) + Buk + BMuk = Ag(xk) + BKg(xk) + BMKg(xk)
+= Ag(xk) + B(I + M)Kg(xk)
+(42)
+where M = diag([m1, m2, . . . m2NDER]) is an introduced diagonal matrix to
+represent model uncertainty in control input channels.
+In other words, the
+introduced matrix parameter M can be used to quantify the uncertainty from
+control input channels, whereby one can provide the stability analysis based
+on the disc margin for each channel (which will be provided below). K is the
+control gain matrix such that uk = Kg(xk) in line with the LQR.
+Consider M and g(xk) to be complex-valued to reflect both gain and phase
+disturbances. Define a Lyapunov function V (x) = g(x)∗Sg(x) (where S is the
+solution of Riccati equation). Based on the Lyapunov function and following
+the steps in [42], we provide the disk stability margin for the ith control input
+channel in (43) without further explanation (also see Fig. 5). Interested readers
+can refer to [42] for the derivation of the disc margin.
+1 + mi =
+�
+αi + jβi :
+�
+αi − (1 + ri
+µ )
+�2
++ β2
+i
+< (1 + ri
+µ )2 + ρ − ri
+µ
+− 1
+�
+,
+where
+i = 1, 2, ..., 2NDER
+(43)
+where ρ = σmin[Q]/(σmax[K])2 and µ = σmax[BT SB]. σmax[.] and σmin[.]
+represent the matrix operation to obtain the maximum and minimum singular
+16
+
+Figure 5: Disk stability margin for the discrete-time LQR
+values, respectively. ri is the ith diagonal element of the cost matrix R. Fig.
+5 shows the disc margin, within which the system is asymptotically stable.
+Specifically, according to Eq.
+(43) and Fig.
+5 , the sufficient conditions of
+asymptotic convergence against the model uncertainty is: 1 + GL,i < αi <
+1 + GU,i for the gain margin and PML,i < arctan βi
+αi < PMU,i for the phase
+margin, with
+GL,i = ri
+µ −
+�
+(1 + ri
+µ )2 + ρ − ri
+µ
+− 1,
+GU,i = ri
+µ +
+�
+(1 + ri
+µ )2 + ρ − ri
+µ
+− 1 (44)
+and
+PML,i = − arccos(1/(1 + ri
+µ )) = − arccos
+µ
+µ + ri
+PMU,i = arccos(1/(1 + ri
+µ )) = arccos
+µ
+µ + ri ,
+for
+i = 1, 2, ..., 2NDER.
+(45)
+4. Case Studies
+This section presents case studies based on two MG test systems, namely
+a four-bus MG as shown in Fig. 6 and a thirteen-bus MG as shown in Fig.
+7, to verify the effectiveness of the proposed Koopman-inspired identification
+and control.
+The two test systems were established in MATLAB Simulink
+2021b. The DERs in the test systems are primary-controlled in different control
+modes (grid-forming converters, grid-following converters, and an isochronous-
+controlled diesel generator as given in Fig. 6 and Fig. 7) with the inner control
+loops modeled in detail. Therefore, the interaction of primary and secondary
+control is preserved in simulation to test the effectiveness of secondary control
+in realistic setups. The implementation of the converter voltage and current
+control inner-loops can be found [9, 24].
+Besides, randomized measurement noises, control time delays, and ambient
+perturbations were incorporated into the test systems to mimic practical oper-
+ation. The simulation parameters of the two test systems are summarized in
+17
+
+βi
+Disc Stability Margin (1+m;=α;+jβ):
+When the model uncertainty is within the
+disc (grey area), the system is Lyapunov
+asymptotically stable.
+Gain Margin
+1
+αi
+11+
+Phase
+1
+Disc Radius :
+Margin
+μTable 1 and Table 2, respectively. The readers can find more information about
+the test systems at https://github.com/nash13123/MG-Test-System.git.
+4.1. Identification and Control in the 4-Bus MG Test System
+The small 4-bus MG test system was used to test the proposed Koopman-
+inspired enhanced OKID with control under load variations and the MG transi-
+tion from the grid-connected mode to islanded mode. The DERs at Bus 1 and 3
+are droop-based grid-forming, and the DERs at Bus 2 and 4 are inverse-droop-
+based grid-following. At 0.7s, the MG was disconnected from the main grid by
+turning off the switch SW, which causes sudden voltage drops and consequent
+dynamics. After detecting the sudden change, the secondary control was en-
+abled and kept online from 0.8s, i.e., approximately 0.1s lag to mimic a time
+delay of islanding event detection in practical applications.
+Modeling accuracy of the Koopman-inspired OKID. First, we eval-
+uate the modeling accuracy with the one-step-ahead prediction error of the
+Figure 6: The MG 4-bus test system
+Figure 7: The MG 13-bus test system
+18
+
+@Utility Grid
+R14
+Local
+L14
+Local
+Busl
+DER4
+DER1
+Bus1
+Bus4
+sw
+Battery
+Rf1
+Vo1
+Bus4
+Re1
+Vo4
+DC/
+R
+4
+AC/
+Rfel
+n
+AC
+DC
+dc
+R
+Droop Control
+Inverse Droop-Based
+(Grid Forming)
+Cf1
+ Grid-Following Control
+P13
+DER2
+Local
+Local
+DER3
+Battery
+Bus2
+Bus3
+Bus2
+Bus3
+Vo2
+R
+V。
+Rf3
+R
+c3
+AC/
+DC/
+AC
+DC
+dc
+R
+Droop Control
+Inverse Droop-Based
+(Grid Forming)
+Grid-Following ControlSecondary Uncontrollable Distributed Resources
+PCC: Point of Common Coupling
+Utility Grid
+MPPT: Maximum Power Point Tracking
+Secondary Controllable Distributed Resources
+PV: Photovoltaic
+Transformer
+BESS: Battery Energy Storage System
+Load 1
+T1
+DER: Distributed Energy Resource
+3
+Grid-Feeding PV
+Switch SW1
+Diesel
+Farm 1 (MPPT)
+PCC
+2
+23
+9
+Grid-Following BESS 1
+Load 6
+Grid-Forming DER 2
+SW2
+13
+4
+Load 2
+Load 4
+5
+10
+2
+Grid-Forming DER 1
+Grid-Feeding PV
+119
+Grid-Following BESS 2
+6
+Farm 2 (MPPT)
+120
+Load 5
+Load 3
+7Table 1: Parameters of the 4-Bus MG Test System
+Parameters
+Value
+Power base Sbase
+30kVA
+Voltage Base Vbase
+480V
+Primary control time step Tsp
+0.1ms
+Secondary control time step Ts
+30ms
+Sliding window length for estimation N
+9 (270ms)
+Local Voltage proportional gain KP
+0.5
+Local Voltage integral gain KS
+523
+Local current proportional gain KP
+0.3
+Local current integral gain KS
+635
+Frequency droop parameters for DERs 1,2: σω
+2.14 × 10−3rad/(W · s)
+Voltage droop parameters for DERs 1: σV
+1.0 × 10−3V/V ar
+Voltage droop parameters for DERs 2: σV
+6.3 × 10−3V/V ar
+Frequency droop parameters for DERs 3,4: σω
+2.83 × 10−3rad/(W · s)
+Voltage droop parameters for DERs 3: σV
+1.5 × 10−3V/V ar
+Voltage droop parameters for DERs 4: σV
+9.4 × 10−3V/V ar
+PMU measurement noise
+N(0, 0.00562)
+Control Time delay
+N(0.05, 0.0022)s
+Ambient perturbation level added to the reference
+of DER output voltage and angle:
+N(0, 0.012)
+Filter resistance Rf1,2,3,4(Ω)
+0.1
+Filter inductance Lf1,2,3,4, Lc1,2(mH)
+1.35
+Filter capacitance Cf1,2,3,4(µF)
+50
+Filter capacitor resistance Rfc1,2,3,4(Ω)
+1
+Line resistance Rc1,2(Ω)
+0.08
+Line resistance Rc3,4(Ω)
+0.09
+Line inductance Lc1,2(mH)
+0.35
+Line inductance Lc3,4(mH)
+0.45
+Line Resistance Rl1,2,3,4(Ω)
+0.15, 0.35, 0.23, 0.17
+Line inductance Ll1,2,3,4(mH)
+0.42, 0.33, 0.55, 2.40
+Load PL1,2,3 (active power in kW)
+20, 16, 12
+Load QL1,2,3 (reactive powe in kVar)
+9, 9, 6
+LQR control parameter qV
+1 × 103I
+LQR control parameter qsin , qcos
+0
+LQR control parameter qω
+1 × 10−6I
+LQR control parameter rP , rQ
+1 × 10−6I
+Control input lower bounds ULB
+-1.0 kVA
+Control input upper bounds UUB
+1.0 kVA
+Time period for the optimization (28)
+TOP T
+0.6 s
+Time constant of the power low-pass filter
+0.02857s
+* N(a, b) is the normal distribution with mean of a and variance of b. Control
+parameters are designed based on Per Unit.
+19
+
+Table 2: Parameters of the 13-Bus MG Test System
+Parameters
+Value
+Power base Sbase
+150kVA
+Voltage Base Vbase
+4.16kV
+Sliding window length for estimation N
+14(420ms)
+Droop parameters for all DERs: σω
+3.14 × 10−4rad/(W · s)
+Droop parameters for all DERs: σV
+1.5 × 10−3V/V ar
+Ambient perturbation level
+N(0, 0.022)
+LQR control parameter qV
+1 × 103I
+LQR control parameter qsin , qcos
+0
+LQR control parameter qω
+0.01I
+LQR control parameter rP , rQ
+1 × 10−6I
+* Other control parameters are the same to the values in Table 1.
+Figure 8: Comparison of the prediction error
+voltage magnitude, which is defined as
+e(pred)
+k+1
+=
+1
+dim(∆V )∥∆Vk+1 − ∆ ˆVk+1∥
+(46)
+where dim[.] represents the dimension of the vector in the bracket, and ∆ ˆVk+1
+represents the predicted voltage magnitude at time step k + 1 by the identified
+model of interest. In Fig. 8, we compared the prediction error of two different
+ways of modeling: (i) the proposed Koopman-inspired enhanced OKID with
+the basis z = [∆V , sin θ − sin(θ∗
+L), cos θ − cos(θ∗
+L), ∆ω]T ; (ii) the conventional
+OKID (i.e., linearize the system model in Eq. (11) and apply OKID with γopt
+fixed at 1
+2). It was found that the proposed Koopman-inspired enhanced OKID
+leads to smaller prediction error than the conventional OKID. These results
+show that the salient features of the proposed Koopman-inspired OKID, i.e., the
+Koopman nonlinear basis and the adaptive γopt, can ensure a good modeling
+accuracy regardless of nonlinearity and uncertainty during large disturbances.
+Control results comparison. Fig. 9 compares the voltage and frequency
+trajectories with different secondary control methods. As Fig. 9(a)-(c) show,
+the bus voltage suddenly drops with incurred transients after the disturbance at
+0.7s, which triggers the secondary control to restore the voltage and frequency
+to their nominal values (1p.u and 60Hz). Fig. 9(a) shows the control results
+20
+
+0.5
+(n'
+Koopman-InspiredEnhancedOKID(proposed)
+0.4
+Conventional OKID
+Error
+?
+0.3
+UO
+0.2
+0.
+0
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Time (seconds)(a) The proposed Koopman-inspired enhanced OKID with LQR control
+(b) The secondary PI control
+(c) The conventional OKID with LQR control (γ = 0.5)
+(d) The classical EDMDc (least-squares-based Koopman operator control with
+the Koopman observables z)
+Figure 9: Voltage and frequency trajectories of the 4-bus MG test system with different
+secondary control methods
+21
+
+ (p.u.)
+1.05
+0.95
+DER 1
+DER 2
+Voltage
+0.9
+DER 3
+DER 4
+0.85
+1
+2
+3
+4
+Time (seconds)61
+59
+DER 1
+DER 2
+58
+DER 3
+DER 4
+57
+1
+2
+3
+4
+Time(seconds)(p.u.)
+1.05
+0.95
+DER 1
+DER 2
+Voltage
+0.9
+DER 3
+DER 4
+0.85
+1
+3
+4
+Time (seconds)61
+DER 1
+DER 2
+60
+DER 3
+DER 4
+59
+58
+57
+1
+2
+3
+4
+Time(seconds) (p.u.)
+1.05
+0.95
+DER 1
+DER 2
+Voltage
+0.9
+DER 3
+DER 4
+0.85
+1
+2
+3
+4
+Time (seconds)61
+59
+DER 1
+DER 2
+58
+DER 3
+DER 4
+57
+1
+2
+3
+4
+Time(seconds)Voltage Magnitude (p.u.)
+1.05
+0.95
+DER
+DER 2
+0.9
+DER 3
+DER 4
+0
+1
+2
+3
+4
+5
+Time (seconds)61
+60
+Frequency
+59
+DER
+DER
+2
+58
+DER
+3
+DER
+4
+57
+2
+3
+4
+Time (seconds)with the proposed Koopman-inspired enhanced OKID with LQR control; both
+voltage and frequency are corrected approximately to the nominal values. For
+comparison, Fig. 9(b) presents the voltage and frequency trajectories using sec-
+ondary PI control that is tuned with the best effort. The PI control with respect
+to the voltage magnitude and the frequency shows a slower response for voltage
+restoration compared to the proposed control, and has non-zero steady-state
+errors.
+It also suffers from a larger frequency deviation as it cannot handle
+the voltage-frequency dependence properly. Fig. 9(c) shows the results of con-
+ventional OKID with LQR control. In contrast to the proposed method, the
+conventional OKID with LQR cannot realize the same fast voltage restoration.
+Fig. 9(d) shows the voltage and frequency trajectories of the classical ED-
+MDc (i.e., LQR with pseudo-inverse least-squares identification based on the
+Koopman observables z). By comparing Fig. 9(d) with Fig. 9(a), we found
+that the LQR with the least-squares-based identification cannot perform as well
+as the LQR with the proposed Koopman-inspired Koopman-inspired identifica-
+tion. These results indicate the effectiveness of the proposed Koopman-inspired
+enhanced OKID that possibly results from the two ingredients: (i) the nonlin-
+ear basis functions of the Koopman observables proposed in Eq. (14); (ii) the
+OKID with adaptive γopt. Both ingredients help better describe the MG sys-
+temwide dynamics under big disturbances, realizing more effective control for
+both voltage and frequency.
+Fig. 10 presents the optimized parameters γopt during control, and Fig. 11
+shows the run time of the proposed Koopman-inspired enhanced OKID at each
+Figure 10: Estimated γopt of the proposed Koopman-inspired enhanced OKID
+Figure 11: Run time of the proposed Koopman-inspired enhanced OKID
+22
+
+0.56
+0.54
+opt
+0.52
+0.5
+0.48
+0.46
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+Time (seconds)600
+(su)
+400
+Time
+Run
+200
+20ms
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+Time (seconds)time step of secondary control. The run time of the proposed method is about
+20ms in case that γopt is not updated, less than the time step of secondary
+control (30ms). The run time of the proposed method is around 250-500ms in
+case that γopt is updated, which is still less than the time period TOP T = 0.6s
+between two updates of γopt. These indicate the feasibility to implement the
+proposed Koopman-inspired identification and control online.
+4.2. Identification and Control in the 13-Bus MG Test System
+To show the performance of the Koopman-inspired enhanced OKID with
+LQR control in larger systems for generality, we consider the 13-bus MG test
+system presented in Fig. 7, which is adapted from the IEEE 13-node test feeder
+[43].
+The DERs at Bus 6 and 9 are droop-based grid-forming.
+The BESSs
+at Bus 1 and 11 are inverse-droop-based grid-following.
+The solar farms at
+Bus 3 and 5 are grid-following under MPPT, which are not controllable for
+secondary control. The MG system is under transition from the grid-connected
+to the islanded modes, and under generation/load variations. At 0.4s, the MG
+is disconnected from the main grid by turning off the switch SW1, causing the
+sudden drop of voltage with incurred transient. After detecting the islanding
+transient, the secondary control is triggered and kept online from 0.5s, i.e., 0.1s
+lag to mimic a time delay of islanding detection in practical application. Next,
+an active power perturbation of the two solar farms (around 80kW for each)
+occurs at 1.0s due to a drop of the solar irradiation from 1000 to 200 W/m2.
+Then, a load perturbation happens at the Bus 4 at 1.05s: the consumed active
+power increases by 150kW and the consumed reactive power increased by 50kVar
+by turning on the switch SW2.
+Fig. 12 compares the voltage and frequency trajectories with different sec-
+ondary control methods. Fig. 12(a) shows that the proposed Koopman-inspired
+OKID with LQR control can correct the voltage and frequency approximately to
+the nominal values. For comparison, the voltage and frequency trajectories with
+the secondary PI control are shown in Fig. 12(b), which illustrates that the PI
+control with the best effort of tuning still fails to realize the stable and accurate
+voltage and frequency restoration. Fig. 12(c) shows the results of the conven-
+tional OKID with LQR control, which suffers from larger voltage and frequency
+oscillations after 1.0s. Fig. 12(d) shows the voltage and frequency trajectories
+of classical EDMDc (i.e., LQR with pseudo-inverse least-squares identification
+based on the Koopman observables z).
+By comparing Fig.
+12(d) with Fig.
+12(a), we found that the classical EDMDc cannot perform as well as the LQR
+with the proposed Koopman-inspired Koopman-inspired identification. These
+results further demonstrate the advantages of the proposed Koopman-inspired
+OKID with LQR control. Because of the nonlinear Koopman embeddings and
+the adaptive γopt, the proposed method can effectively restore the voltage and
+frequency to their nominal values despite nonlinearity and uncertainty due to
+large disturbances.
+23
+
+(a) The proposed Koopman-inspired enhanced OKID with LQR control
+(b) The secondary PI control
+(c) The conventional OKID with LQR control (γ = 0.5)
+(d) The classical EDMDc (least-squares-based Koopman operator control with
+the Koopman observables z)
+Figure 12: Voltage and frequency trajectories of the 13-bus MG test system with different
+secondary control methods
+24
+
+(p.u.)
+1.1
+Magnitude
+0.9
+DER 1
+Voltagel
+0.8
+DER 2
+0.7
+DER 3
+DER 4
+0.6
+0.5
+1
+1.5
+Time (seconds)60.5
+(zH)
+60
+DER 1
+59.5
+DER 2
+DER 3
+DER 4
+59
+0.5
+1.5
+Time (seconds)(p.u.)
+1.2
+Magnitude
+0.8
+0.6
+DER 1
+DER 2
+DER 3
+0.2
+DER 4
+0.5
+1.5
+Time (seconds)60.5
+(zH)
+60E
+59.5
+59
+DER 1
+DER 2
+58.5
+DER 3
+DER 4
+58
+0.5
+1.5
+Time (seconds)(p.u.)
+1.1
+Magnitude
+0.9
+DER 1
+Voltagel
+0.8
+DER 2
+0.7
+DER 3
+DER 4
+0.6
+0.5
+1.5
+Time (seconds)60.5
+(ZH)
+59.5
+DER 1
+DER 2
+DER 3
+DER 4
+59
+0.5
+1.5
+Time(seconds)MA
+0.9
+0.8
+DER
+DER 2
+0.7
+DER 3
+DER 4
+0.6
+0
+0.5
+1
+1.5
+Time (seconds)60.5
+(ZH)
+60
+Frequency (
+59.5
+DER
+DER
+2
+DER 3
+DER
+4
+59
+0
+0.5
+1
+1.5
+Time (seconds)5. Conclusions
+This paper proposed a data-driven Koopman-inspired identification and con-
+trol method for MG secondary voltage and frequency control. The proposed
+method requires no knowledge of network information and primary controllers.
+It requires no warm-up training yet with guaranteed BIBO stability and even
+asymptotic stability under some mild conditions. In this method, a Koopman
+operator-inspired enhanced OKID (observer Kalman filter identification) algo-
+rithm is proposed, whereby the Koopman state space model is estimated online
+and used for control to handle microgrid nonlinearity and uncertainty adap-
+tively. Case studies in the 4-bus and 13-bus MG test systems (with different
+converter control modes) demonstrate the effectiveness and robustness of the
+proposed Koopman-inspired identification and control method subject to mode
+transitions, varying operating conditions, measurement noises and time delays.
+References
+[1] K. P. Schneider, N. Radhakrishnan, Y. Tang, F. K. Tuffner, C.-C. Liu,
+J. Xie, D. Ton, Improving primary frequency response to support networked
+microgrid operations, IEEE Transactions on Power Systems 34 (1) (2019)
+659–667. doi:10.1109/TPWRS.2018.2859742.
+[2] M. Farrokhabadi, C. A. Ca˜nizares, J. W. Simpson-Porco, E. Nasr, L. Fan,
+P. A. Mendoza-Araya, R. Tonkoski, U. Tamrakar, N. Hatziargyriou, D. La-
+gos, R. W. Wies, M. Paolone, M. Liserre, L. Meegahapola, M. Ka-
+balan, A. H. Hajimiragha, D. Peralta, M. A. Elizondo, K. P. Schnei-
+der, F. K. Tuffner, J. Reilly, Microgrid stability definitions, analysis, and
+examples, IEEE Transactions on Power Systems 35 (1) (2020) 13–29.
+doi:10.1109/TPWRS.2019.2925703.
+[3] Y. Zhang, L. Xie, Q. Ding, Interactive control of coupled microgrids for
+guaranteed system-wide small signal stability, IEEE Transactions on Smart
+Grid 7 (2) (2016) 1088–1096. doi:10.1109/TSG.2015.2495233.
+[4] J. W. Simpson-Porco, Q. Shafiee, F. D¨orfler, J. C. Vasquez, J. M. Guerrero,
+F. Bullo, Secondary frequency and voltage control of islanded microgrids via
+distributed averaging, IEEE Transactions on Industrial Electronics 62 (11)
+(2015) 7025–7038. doi:10.1109/TIE.2015.2436879.
+[5] R. Majumder, B. Chaudhuri, A. Ghosh, R. Majumder, G. Ledwich, F. Zare,
+Improvement of stability and load sharing in an autonomous microgrid us-
+ing supplementary droop control loop, IEEE Transactions on Power Sys-
+tems 25 (2) (2010) 796–808. doi:10.1109/TPWRS.2009.2032049.
+[6] M. Honarvar Nazari, M. Ilic, J. Pe¸cas Lopes, Small-signal stability and de-
+centralized control design for electric energy systems with a large penetra-
+tion of distributed generators, Control Engineering Practice 20 (9) (2012)
+823–831. doi:https://doi.org/10.1016/j.conengprac.2012.04.006.
+25
+
+[7] A. Bidram, F. L. Lewis, A. Davoudi, Distributed control systems for small-
+scale power networks: Using multiagent cooperative control theory, IEEE
+Control Systems Magazine 34 (6) (2014) 56–77. doi:10.1109/MCS.2014.
+2350571.
+[8] A. Bidram, A. Davoudi, F. L. Lewis, J. M. Guerrero, Distributed co-
+operative secondary control of microgrids using feedback linearization,
+IEEE Transactions on Power Systems 28 (3) (2013) 3462–3470.
+doi:
+10.1109/TPWRS.2013.2247071.
+[9] Z. Ma, Z. Wang, Y. Guo, Y. Yuan, H. Chen, Nonlinear multiple models
+adaptive secondary voltage control of microgrids, IEEE Transactions on
+Smart Grid 12 (1) (2021) 227–238. doi:10.1109/TSG.2020.3023307.
+[10] C. Ahumada, R. C´ardenas, D. S´aez, J. M. Guerrero, Secondary control
+strategies for frequency restoration in islanded microgrids with consider-
+ation of communication delays, IEEE Transactions on Smart Grid 7 (3)
+(2016) 1430–1441. doi:10.1109/TSG.2015.2461190.
+[11] I. U. Nutkani, W. Peng, L. P. Chiang, F. Blaabjerg, Secondary droop
+for frequency and voltage restoration in microgrids, in: 2015 17th Euro-
+pean Conference on Power Electronics and Applications (EPE’15 ECCE-
+Europe), 2015, pp. 1–7. doi:10.1109/EPE.2015.7309457.
+[12] L.-Y. Lu, C.-C. Chu, Consensus-based secondary frequency and voltage
+droop control of virtual synchronous generators for isolated ac micro-grids,
+IEEE Journal on Emerging and Selected Topics in Circuits and Systems
+5 (3) (2015) 443–455. doi:10.1109/JETCAS.2015.2462093.
+[13] L.-Y. Lu, C.-C. Chu, Consensus-based droop control of isolated micro-grids
+by admm implementations, IEEE Transactions on Smart Grid 9 (5) (2018)
+5101–5112. doi:10.1109/TSG.2017.2680445.
+[14] M. Jafari, V. Sarfi, A. Ghasemkhani, H. Livani, L. Yang, H. Xu, R. Koosha,
+Adaptive neural network based intelligent secondary control for microgrids,
+in: 2018 IEEE Texas Power and Energy Conference (TPEC), 2018, pp. 1–6.
+doi:10.1109/TPEC.2018.8312064.
+[15] X. Shen, H. Wang, J. Li, Q. Su, L. Gao, Distributed secondary voltage con-
+trol of islanded microgrids based on rbf-neural-network sliding-mode tech-
+nique, IEEE Access 7 (2019) 65616–65623. doi:10.1109/ACCESS.2019.
+2915509.
+[16] D. O. Amoateng, M. Al Hosani, M. S. Elmoursi, K. Turitsyn, J. L. Kirtley,
+Adaptive voltage and frequency control of islanded multi-microgrids, IEEE
+Transactions on Power Systems 33 (4) (2018) 4454–4465. doi:10.1109/
+TPWRS.2017.2780986.
+26
+
+[17] F.-J. Lin, C.-I. Chen, G.-D. Xiao, P.-R. Chen, Voltage stabilization control
+for microgrid with asymmetric membership function-based wavelet petri
+fuzzy neural network, IEEE Transactions on Smart Grid 12 (5) (2021)
+3731–3741. doi:10.1109/TSG.2021.3071357.
+[18] A. Younesi, H. Shayeghi, Q-learning based supervisory pid controller for
+damping frequency oscillations in a hybrid mini/micro-grid, Iranian Journal
+of Electrical and Electronic Engineering 15 (1) (2019) 126–141.
+[19] R. Zhang, B. Hredzak, Distributed control system with aperiodic sampled
+time-delayed data for batteries and renewable energy sources in microgrid,
+IEEE Transactions on Sustainable Energy 11 (2) (2020) 1013–1022. doi:
+10.1109/TSTE.2019.2916206.
+[20] J. L. Proctor, S. L. Brunton, J. N. Kutz, Generalizing koopman theory to
+allow for inputs and control, SIAM Journal on Applied Dynamical Systems
+17 (1) (2018) 909–930. doi:10.1137/16M1062296.
+[21] M. Korda, Y. Susuki, I. Mezi´c, Power grid transient stabilization using
+koopman model predictive control, IFAC-PapersOnLine 51 (28) (2018) 297–
+302. doi:https://doi.org/10.1016/j.ifacol.2018.11.718.
+[22] X. Gong, X. Wang, G. Joos, An online data-driven method for microgrid
+secondary voltage and frequency control with ensemble koopman model-
+ing, IEEE Transactions on Smart Grid (2022) 1–1doi:10.1109/TSG.2022.
+3190237.
+[23] Y. e. a. Lin, Research Roadmap on Grid-Forming Inverters, Tech. rep.,
+National Renewable Energy Lab, Golden, CO, United States, report #
+NREL/TP-5D00-73476 (2020).
+[24] J. Rocabert, A. Luna, F. Blaabjerg, P. Rodr´ıguez, Control of power con-
+verters in ac microgrids, IEEE Transactions on Power Electronics 27 (11)
+(2012) 4734–4749. doi:10.1109/TPEL.2012.2199334.
+[25] B. Kroposki, B. Johnson, Y. Zhang, V. Gevorgian, P. Denholm, B.-M.
+Hodge, B. Hannegan, Achieving a 100% renewable grid: Operating electric
+power systems with extremely high levels of variable renewable energy,
+IEEE Power and Energy Magazine 15 (2) (2017) 61–73. doi:10.1109/
+MPE.2016.2637122.
+[26] E. Rokrok, T. Qoria, A. Bruyere, B. Francois, X. Guillaud, Effect of using
+pll-based grid-forming control on active power dynamics under various scr,
+in: IECON 2019 - 45th Annual Conference of the IEEE Industrial Elec-
+tronics Society, Vol. 1, 2019, pp. 4799–4804. doi:10.1109/IECON.2019.
+8927648.
+[27] W. Du, Z. Chen, K. P. Schneider, R. H. Lasseter, S. Pushpak Nandanoori,
+F. K. Tuffner, S. Kundu, A comparative study of two widely used grid-
+forming droop controls on microgrid small-signal stability, IEEE Journal of
+27
+
+Emerging and Selected Topics in Power Electronics 8 (2) (2020) 963–975.
+doi:10.1109/JESTPE.2019.2942491.
+[28] D. J. Ryan, R. Razzaghi, H. D. Torresan, A. Karimi, B. Bahrani, Grid-
+supporting battery energy storage systems in islanded microgrids: A data-
+driven control approach, IEEE Transactions on Sustainable Energy 12 (2)
+(2021) 834–846. doi:10.1109/TSTE.2020.3022362.
+[29] N. Pogaku, M. Prodanovic, T. C. Green, Modeling, analysis and testing of
+autonomous operation of an inverter-based microgrid, IEEE Transactions
+on Power Electronics 22 (2) (2007) 613–625.
+doi:10.1109/TPEL.2006.
+890003.
+[30] H. Bevrani, S. Shokoohi, An intelligent droop control for simultaneous volt-
+age and frequency regulation in islanded microgrids, IEEE Transactions on
+Smart Grid 4 (3) (2013) 1505–1513. doi:10.1109/TSG.2013.2258947.
+[31] S. Golestan, M. Monfared, F. D. Freijedo, J. M. Guerrero, Performance
+improvement of a prefiltered synchronous-reference-frame pll by using a
+pid-type loop filter, IEEE Transactions on Industrial Electronics 61 (7)
+(2014) 3469–3479. doi:10.1109/TIE.2013.2282607.
+[32] M. Korda, I. Mezi´c, Linear predictors for nonlinear dynamical systems:
+Koopman operator meets model predictive control, Automatica 93 (2018)
+149–160. doi:https://doi.org/10.1016/j.automatica.2018.03.046.
+[33] M. Netto, Y. Susuki, V. Krishnan, Y. Zhang, On analytical construction of
+observable functions in extended dynamic mode decomposition for nonlin-
+ear estimation and prediction, IEEE Control Systems Letters 5 (6) (2021)
+1868–1873. doi:10.1109/LCSYS.2020.3047586.
+[34] E. Yeung, S. Kundu, N. Hodas, Learning deep neural network repre-
+sentations for koopman operators of nonlinear dynamical systems, in:
+2019 American Control Conference (ACC), 2019, pp. 4832–4839.
+doi:
+10.23919/ACC.2019.8815339.
+[35] H. Shi, M. Q.-H. Meng, Deep koopman operator with control for nonlinear
+systems, IEEE Robotics and Automation Letters 7 (3) (2022) 7700–7707.
+doi:10.1109/LRA.2022.3184036.
+[36] Y. Han, W. Hao, U. Vaidya, Deep learning of koopman representation for
+control, in: 2020 59th IEEE Conference on Decision and Control (CDC),
+2020, pp. 1890–1895. doi:10.1109/CDC42340.2020.9304238.
+[37] S. J. Qin, An overview of subspace identification, Computers & chemical
+engineering 30 (10-12) (2006) 1502–1513.
+[38] A. Alenany, D. Westwick, H. Shang, A modified observer/kalman filter
+identification (okid) algorithm employing output residuals, International
+Journal of Dynamics and Control 7 (2) (2019) 636–643.
+28
+
+[39] S. L. Brunton, J. N. Kutz, Data-driven science and engineering: Machine
+learning, dynamical systems, and control, Cambridge University Press,
+2022.
+[40] S. Haykin, Neural networks and learning machines, 3/E, Pearson Education
+India, 2009.
+[41] A. Shaiju, I. R. Petersen, Formulas for discrete time lqr, lqg, leqg and
+minimax lqg optimal control problems, IFAC Proceedings Volumes 41 (2)
+(2008) 8773–8778.
+[42] J. Lee, J.-S. Kim, H. Shim, Disc margins of the discrete-time lqr and its
+application to consensus problem, International Journal of Systems Science
+43 (10) (2012) 1891–1900. doi:10.1080/00207721.2011.555012.
+[43] K. P. Schneider, B. A. Mather, B. C. Pal, C.-W. Ten, G. J. Shirek,
+H. Zhu, J. C. Fuller, J. L. R. Pereira, L. F. Ochoa, L. R. de Araujo,
+R. C. Dugan, S. Matthias, S. Paudyal, T. E. McDermott, W. Kerst-
+ing, Analytic considerations and design basis for the ieee distribution test
+feeders, IEEE Transactions on Power Systems 33 (3) (2018) 3181–3188.
+doi:10.1109/TPWRS.2017.2760011.
+29
+

59E2T4oBgHgl3EQfOwac/content/tmp_files/2301.03752v1.pdf.txt

@@ -0,0 +1,2396 @@
+Draft version January 11, 2023
+Typeset using LATEX default style in AASTeX631
+Magnetic Fields, Star Formation Rates and Gas Densities at Sub-kpc Scales in a Pilot Sample of
+Nearby Galaxies
+Souvik Manna1 and Subhashis Roy1
+1National Center for Radio Astrophysics, TIFR,
+Pune University Campus, Ganeshkhind, Pune 411007, India
+ABSTRACT
+We have estimated the magnetic field strengths of a sample of seven galaxies using their non-thermal
+synchrotron radio emission at metre wavelengths, and assuming energy equipartition between magnetic
+fields and cosmic ray particles. We tested for deviation of magnetic fields from energy equipartition with
+cosmic ray particles, and found that deviations of ∼25% are typical for the sample galaxies. Spatially
+resolved star formation rates (SFR) were estimated for the seven galaxies along with five galaxies
+studied previously. For the combined sample of twelve galaxies, the equipartition magnetic fields (Beq)
+are correlated with the SFR surface densities (ΣSFR) at sub-kpc scales with Beq ∝ Σ0.31±0.06
+SFR
+, consistent
+with model predictions. We estimated gas densities (ρgas) for a sub-sample of seven galaxies using
+archival observations of the carbon monoxide (CO) rotational transitions and the atomic hydrogen
+(Hi) 21 cm line and studied the spatially-resolved correlation between the magnetic fields and ρgas.
+Magnetic fields and gas densities are found to be correlated at sub-kpc scale as Beq ∝ ρ0.40±0.09
+gas
+. This
+is broadly consistent with models, which typically predict B ∝ ρ0.5
+gas.
+Keywords: Radio continuum emission — Interstellar medium — Star formation — Magnetic fields
+1. INTRODUCTION
+Magnetic fields are believed to influence several physical processes in a galaxy at almost every scale (e.g. Elmegreen
+1981; Niklas & Beck 1997; Groves et al. 2003; Price & Bate 2008; Adebahr et al. 2013). Magnetic fields have been
+found to consist of two main components: a small-scale turbulent magnetic field up to a few hundred parsecs (e.g.
+Batchelor 1950; Groves et al. 2003) and a large-scale “ordered” or “regular” magnetic field component at scales of a few
+kpcs (e.g. Moss & Shukurov 1996; Shukurov et al. 2006; Kulsrud & Zweibel 2008). Magnetic fields in galaxies can be
+measured using their effects on different radiation processes like Zeeman splitting of emission lines, polarized emission
+from dust, the polarization of starlight, Faraday rotation of polarized radio emission, and intensity of synchrotron
+emission which we use in this work. Measurement of the line-of-sight component of the magnetic field via the Zeeman
+effect in galaxies other than the Milky Way has been possible for only a few systems (Kazes et al. 1991; Sarma et al.
+2005; Robishaw et al. 2008); a significant expansion of such studies is very difficult with current-generation telescopes.
+Magnetic fields in galaxies can be measured and studied using synchrotron emission at radio frequencies, at scales
+larger than the resolution of the radio observation. For example, a Very Large Array (VLA) polarization study of
+NGC 4736 at 8.46 and 4.86 GHz found that the magnetic field in the galaxy was ordered in a spiral shape (Chy˙zy
+& Buta 2008). An X-shaped structure of the magnetic field in the galactic halo region was observed by stacking the
+Karl G. Jansky VLA polarized emission maps of 16 nearly edge-on spiral galaxies, obtained as part of the CHANG-ES
+survey (Krause et al. 2020); such structures had also been observed in individual spiral galaxies (e.g. Krause et al.
+2006; Krause 2009; Heesen et al. 2009). However, polarized radio emission from external individual galaxies is difficult
+to study at low radio frequencies due to Faraday depolarization (e.g. Sokoloff et al. 1998).
+Corresponding author: Souvik Manna
+souvik@ncra.tifr.res.in
+arXiv:2301.03752v1  [astro-ph.GA]  10 Jan 2023
+
+2
+Manna and Roy
+The average magnetic field strength can also be estimated from the total intensity of synchrotron radio emission,
+assuming energy equipartition between magnetic fields and cosmic ray particles (e.g. Miley 1980; Beck & Krause 2005).
+Equipartition magnetic fields have been studied in several nearby galaxies, but primarily at frequencies >1 GHz (e.g.
+Chy˙zy et al. 2000; Soida et al. 2001; Heesen et al. 2009; Fletcher et al. 2011; Adebahr et al. 2013). Vargas et al. (2018)
+studied a sample of three nearly edge-on galaxies from the CHANG-ES survey to separate the thermal Bremsstrahlung
+from the non-thermal synchrotron emission at 1.5 and 6 GHz. At these frequencies, the thermal component is large and
+hence the correction for the thermal emission can be as large as ∼ 20%, making the derived magnetic field strengths
+prone to errors. Conversely, the steep spectral index of synchrotron emission implies that it will dominate the total
+emission at frequencies < 1 GHz, with ∼ 95% contribution (Basu et al. 2012b; Roy & Manna 2021). Thus, magnetic
+field strengths derived using observations at <1 GHz are very robust to any correction for thermal emission.
+Magnetic fields are believed to play an important role at various stages of the star-formation process - from the
+fragmentation of clouds at the few kpc scales to the final collapse of gas into stars (e.g. Elmegreen 1981; Crutcher
+1999; Price & Bate 2008; Van Loo et al. 2015). To understand the influence of magnetic fields and star-formation
+activities on different physical processes in the ISM at different physical scales, several studies on radio-infrared
+correlations have been carried out in the past (e.g. Murphy et al. 2006a,b, 2008; Tabatabaei et al. 2013). Magnetic
+fields (B) and star formation rate surface densities (SFRSD) are expected to be correlated (Niklas & Beck 1997). Semi-
+analytical model also predicts a strong correlation between B and SFRSDs (ΣSFR) as B ∝ Σ1/3
+SFR at sub-kpc scales
+to explain the local radio-FIR correlation (Schleicher & Beck 2013, 2016). Observational studies of the correlation
+between B and star formation rates (SFR) have been done primarily in samples of nearby dwarf galaxies. For example,
+Chy˙zy et al. (2011) studied 12 local group dwarf galaxies to find that the galaxy-averaged magnetic field and the SFR
+follow B ∼ SFR0.30±0.04, consistent with the prediction of B ∝ Σ1/3
+SFR. However, Jurusik et al. (2014) found the same
+power-law index in a sample of Magellanic type dwarf galaxies to be 0.25±0.02, somewhat lower than the expectation.
+Recently, a study of the dwarf galaxy IC 10 by Basu et al. (2017) provides the only study of the correlation between
+spatially-resolved magnetic fields and SFRSDs; these authors found that the SFRSD is related to the magnetic field
+as B ∝ Σ0.35±0.03
+SFR
+. Therefore, it is important to test such predictions by carrying out systematic spatially-resolved
+studies of magnetic fields in galaxies and their connection to the star formation rate in nearby large galaxies.
+The energy density of magnetic fields and gas in galaxies are expected to be in equipartition, which implies B
+∝ √ρgas (e.g. Chandrasekhar & Fermi 1953; Groves et al. 2003). The observed Radio-FIR correlation can be explained
+based on such equipartition between the energy density of magnetic fields and gas (Niklas & Beck 1997). Several other
+numerical magnetohydrodynamic (MHD) simulations of the ISM have predicted the coupling constant (k) between
+magnetic fields and gas (B ∝ ρk
+gas) to be in the range of ≈0.4−0.6 (Fiedler & Mouschovias 1993; Kim et al. 2001;
+Thompson et al. 2006). Niklas & Beck (1997) studied the correlation between galaxy-integrated equipartition magnetic
+fields and gas densities for a sample of 43 galaxies to find a power-law index of 0.48 ± 0.05; the observed correlation is
+consistent with B ∝ √ρgas. Although the correlation between gas surface densities and SRFSDs has been extensively
+studied in the nearby Universe (Kennicutt-Schmidt law; e.g. Kennicutt 1998a; Onodera et al. 2010; Roychowdhury
+et al. 2015), systematic studies of spatially-resolved correlations between magnetic fields, SFRs and gas densities in
+nearby galaxies are yet to be carried out. It is thus important to carry out a systematic investigation of both the
+B-ρ and the B-SFR correlations, at high-spatial resolutions (≈ sub-kpc scales), using direct estimates of the magnetic
+fields, gas densities, and star-formation rates, in a sample of nearby galaxies. In this paper, we present a pilot study
+of the connection between spatially resolved magnetic fields, SFRSDs and gas densities in a sample of nearby galaxies.
+We have selected a sample of 46 galaxies (Sample 0; Table 2) from the Spitzer Local Volume Legacy (LVL) sample
+of 258 galaxies within 11 Mpc (Dale et al. 2009). As a pilot project, seven (Sample 1; Table 2) of these 46 galaxies
+have been observed with the Giant Metrewave Radio Telescope (GMRT) at 0.33 GHz (Roy & Manna 2021). Six of
+our seven sample galaxies are spirals and the other one is a dwarf irregular Magellanic-type galaxy.
+In this paper, we present spatially resolved equipartition magnetic field (Beq) maps of the seven galaxies in Sample
+1 (Table 2). We also incorporate the magnetic field maps of five galaxies studied by Basu et al. (2012a) from previous
+GMRT observations in our study. We derived SFRSD maps of all 12 galaxies (Sample 2; Table 2) using extinction-
+free diagnostics and used these maps to study the relation between SFRSDs and Beq at sub-kpc scales in our pilot
+study. We used available archival CO and Hi 21 cm data to measure the gas densities (ρgas) of seven (Sample 3;
+Table 2) of the combined sample of 12 galaxies and studied the correlation between ρgas and Beq in these galaxies.
+We also studied the magnetic field-gas connection through an indirect measurement of their coupling coefficient using
+radio−FIR correlations of the galaxies in Sample 1.
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+3
+Table 1. Details of the seven sample galaxies. Note that the images at 0.33 GHz were obtained from observations with the
+GMRT reported in Roy & Manna (2021) while those at 1.4 GHz were obtained from archival VLA data. The distances to
+the galaxies were taken from Dale et al. (2009). Galaxies with an asterisk are those for which spatially-resolved CO data are
+available.
+Name
+Class
+Distance
+Inclination
+Position
+uv
+Angular
+Spatial
+RMS
+RMS
+VLA
+(Mpc)
+angle
+angle
+range
+resolution
+resolution
+(0.33 GHz)
+(1.4 GHz)
+Project ID
+(deg)
+(deg)
+(kλ)
+(arcsec2)
+(pc)
+(µJy/beam)
+(µJy/beam)
+(1.4 GHz)
+NGC 2683
+Sb
+7.7
+83
+43
+0.19 - 15
+19 × 13
+670
+200
+40
+AI23
+NGC 3627∗
+SAB
+10.
+65
+170
+0.26 - 25
+16 × 11
+760
+800
+370
+AS541, AP462
+NGC 4096
+SABc
+8.3
+76
+20
+0.14 - 17
+14 × 12
+730
+100
+25
+16A-013
+NGC 4449
+Irregular
+4.2
+0
+0
+0.15 - 15
+26 × 15
+360
+300
+180
+AB167
+NGC 4490
+SBm
+8.0
+60
+126
+0.13 - 14
+19 × 18
+560
+230
+100
+AA181
+NGC 4826∗
+SAab
+7.5
+60
+120
+0.22 - 20
+15 × 14
+650
+280
+70
+AS541
+NGC 5194∗
+Sbc
+8.0
+20
+10
+0.15 - 10
+23 × 18
+740
+310
+30
+AB505, AN57
+Table 2.
+List of different samples studied in this paper.
+Sample Name
+Galaxies
+Sample 0
+Full sample containing 46 galaxies from Spitzer LVL survey
+Sample 1
+Pilot sample containing 7 galaxies from Sample 0; galaxies listed in Table 1
+Sample 2
+Sample 1 + 5 galaxies (NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946)
+from Basu et al. (2012b) = 12 galaxies; used to probe the Beq-SFRSD correlations
+Sample 3
+A subset of 7 galaxies (NGC 3627, NGC 4826, NGC 5194, NGC 4736, NGC 5055, NGC 5236 and
+NGC 6946) from Sample 2 which have archival CO data; used to study the Beq-gas density correlations
+The paper is organized as follows. The analysis of the data is discussed in Section 2. In Section 3, we present
+the results of our analysis, including the correlation between magnetic fields, SFRSDs and gas densities of the seven
+galaxies in Sample 1. In Section 4, we have extended our study to include a sample of five galaxies of Basu et al.
+(2012a) . We discuss the results in Section 5. A summary of this paper is presented in Section 6.
+2. DATA ANALYSIS
+As can be seen in Table 1, six of the seven galaxies in Sample 1 are spirals of varying inclination angles. The seventh
+galaxy NGC 4449 is a dwarf irregular galaxy. Basic information about the seven sample galaxies, including their
+types, distances, inclination angles, position angles, angular resolutions, spatial resolutions, and RMS noise obtained
+on the GMRT and VLA images are also listed in Table 1. The distances, inclination angles, and position angles of
+the galaxies were taken from Dale et al. (2009). Radio observations and the data reduction procedures are discussed
+in detail in Roy & Manna (2021). Briefly, we used GMRT 0.33 GHz observations (covering 0.309−0.342 GHz) and
+archival VLA observations at 1.4 and ∼6 GHz to derive non-thermal emission maps for each galaxy. We used Hα and
+24µm observations of the seven galaxies to model free-free emission from them and subsequently, we subtracted the
+modelled free-free emission from the observed radio emission to get the non-thermal radio maps at 0.33, 1.4 and ∼6
+GHz (Roy & Manna 2021). To generate the non-thermal spectral index maps, we used the non-thermal radio maps
+at 0.33 and ∼6 GHz for NGC 2683, NGC 3627, NGC 4096, and NGC 4449. For the rest of the galaxies (NGC 4490,
+NGC 4826, and NGC 5194), we used non-thermal images at 0.33 and 1.4 GHz to generate the non-thermal spectral
+index maps (Roy & Manna 2021). In the following subsections, we present the analysis of other ancillary data and
+relevant measurements.
+2.1. Magnetic Field Strengths
+The average magnetic field strengths can be estimated from the observed synchrotron flux densities, assuming energy
+equipartition between cosmic ray particles and magnetic fields (“Classical Equipartition Formula”, e.g. Pacholczyk
+1970; Miley 1980; Longair 2011). The equipartition condition is achieved when the total energy in magnetic fields and
+cosmic ray particles is minimum.
+
+4
+Manna and Roy
+The classical equipartition formalism has shortcomings that lead to an overestimation of the magnetic field strength
+(B) at regions of steep spectral indices and underestimation of B at flat spectral index regions. To overcome these
+shortcomings of the classical equipartition formula, Beck & Krause (2005) proposed a revised formula to estimate the
+average magnetic field strength. The formula is expressed as
+Beq = [4π(K0 + 1)E1−2αnt
+p
+f(αnt)
+C4(i)
+Iνναnt
+l
+]
+1
+αnt+3
+(1)
+K0, Ep, Iν, and αnt are the number density ratio of cosmic ray protons to electrons, the proton rest mass energy,
+the intensity of the synchrotron emission at frequency ν, and the spectral index of synchrotron emission, respectively.
+f(αnt) is a function of αnt given as f(αnt) = (2αnt + 1)[2(αnt − 1)c2(αnt)cαnt
+1
+] (Beck & Krause 2005). C4(i) is a
+constant that depends on the inclination angle (i) of the galaxy and is expressed as C4(i) = [cos(i)](γ+1)/2, where
+γ = (2αnt +1). l is the path length of the synchrotron emission. The path length was assumed to be 1 kpc for a galaxy
+with an inclination angle of 0 degree (face-on). For galaxies with low- and moderate- inclination angles (< 75◦),
+the assumed path length was corrected for the inclinations of the galaxies as l/cos(i). For the two nearly edge-on
+galaxies in Sample 1, NGC 2683 and NGC 4096, we have assumed an oblate spheroidal shape of the synchrotron
+emission, such that the diameter on the plane of the galaxy is equal to its major axis. The path lengths (l) were then
+appropriately calculated, with the path length being maximum (equal to the galaxy’s major axis) at the optical centre
+of the galaxy and gradually declining to the edge of the galaxy. We note that Beq has only a weak dependence on l as
+Beq(r) = l(r)
+−1
+αnt+3 and hence is less sensitive to the exact choice of l. Values of K0 and Ep were assumed to be 100
+and 938.28 MeV, respectively, the same as used by Beck & Krause (2005). Finally, we used non-thermal radio maps at
+0.33 GHz (Iν) and spectral index maps (αnt) made using 0.33 and 1.4 or ∼6 GHz radio observations (Roy & Manna
+2021) to produce magnetic field maps of the sample galaxies using Equation 1.
+The revised equipartition formula diverges for spectral index values ≤ 0.5 because such flat spectra indicate energy
+loss of electrons through ionizations or Coulomb interactions (Sarazin 1999). The central bulge and arm regions have a
+mostly flatter spectrum due to the association of star-forming regions and the estimates of equipartition magnetic fields
+in such regions might be affected by systematic uncertainties. This issue affects the derived magnetic field strengths
+for 8%, 12%, 3%, 70%, 17%, 7%, and 6% of the projected total surface area of NGC 2683, NGC 3627, NGC 4096,
+NGC 4449, NGC 4490, NGC 4826, and NGC 5194, respectively. We note that a large fraction of the derived magnetic
+field values are affected for NGC 4449 due to its non-thermal spectral indices being predominantly flat. This could
+bias the Beq for NGC 4449.
+2.1.1. Uncertainties on Magnetic Field Maps
+The procedure we used to estimate the uncertainties on our magnetic field maps is similar to that of Basu & Roy
+(2013). We used a Monte Carlo method that generated 104 random flux density values for each pixel in a galaxy map
+at 0.33 GHz and either 1.4 GHz or 6 GHz. These flux density values have Gaussian probability distributions with rms
+values equal to the measured rms of each of the 0.33 and 1.4/6 GHz maps. For each of the 104 intensity maps, we
+computed a magnetic field map using the procedure described in the beginning of Section 2.1. The rms of these 104
+magnetic field maps provided us with the magnetic field uncertainty maps for each of the seven galaxies in sample 1.
+2.2. Star Formation Rates
+Rest frame Hα and ultraviolet (UV) observations are the best tracers of recent SFRs as the radiation from these
+predominantly originate in newly formed massive stars. However, the observations are affected by extinction caused by
+interstellar dust in both the host galaxy as well as the Milky Way. SFRs estimated from Hα and UV observations are
+therefore corrected for the extinction. Dust-corrected SFRs can be estimated by combining far-ultraviolet (FUV) and
+Hα data with infrared (IR) data to exploit the complementary strengths at different wavelengths (e.g. Kennicutt &
+Evans 2012a; Buat 1992; Meurer et al. 1995, 1999; Cortese et al. 2008; Leroy et al. 2012). In addition to the FUV+IR
+and Hα+IR tracers, the low-frequency radio emission from galaxies, which is predominantly optically thin synchrotron
+emission, can be used to estimate their dust-unobscured SFRs via the radio-FIR correlation (e.g. Yun et al. 2001).
+We estimated the spatially-resolved star formation rates of our Sample 1 galaxies using FUV+24µm, Hα+24µm,
+and 1.4 GHz data, which are discussed, respectively, in the following Sections, 2.2.1, 2.2.2, and 2.2.3. We used data
+of these different frequencies as tracers in order to (1) get a fair comparison between different SFR diagnostics and
+(2) for studying star-formation history at different timescales. All SFRs in this paper assume a Kroupa IMF (Kroupa
+2001).
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+5
+2.2.1. SFRs using FUV and 24µm Observations
+To estimate SFRSD maps of the seven galaxies in Sample 1 (Table 2) using FUV+24µm emission, we used SPITZER
+24 µm IR data (Dale et al. 2009) and GALEX FUV data (11HUGS survey; Kennicutt et al. 2008). We first convolved
+both the 24 µm and the FUV maps of all galaxies to the same resolutions as our magnetic field maps. The FUV data
+were corrected for extinction due to dust in the Milky Way (see Section 2.2.4). The FUV images were in units of
+counts/sec/pixel and were converted to flux-density units of MJy Sr−1. We also converted the 24µm images to units of
+MJy Sr−1 and used the following calibration from Leroy et al. (2012) to derive SFRSD maps for the sample galaxies:
+ΣSFR[M⊙yr−1kpc−2] = 0.081 IFUV[MJy sr−1] + 0.032 I24µm[MJy sr��1]
+(2)
+The uncertainties of the coefficients are ∼10-30%. Note that the uncertainty in SFR estimates arises from issues
+such as the error in sampling the stellar IMF of different star-forming regions, determining the contribution of different
+emission which are not associated with recent star formation, etc. (e.g Kennicutt & Evans 2012b; Leroy et al. 2012).
+2.2.2. SFRs using Hα and 24µm Observations
+To estimate SFRSD maps using Hα+24µm as a tracer, we used 24 µm emission along with Hα emission from
+11HUGS (Kennicutt et al. 2008), for all but NGC 5194, for which we used data from the SINGS survey (Kennicutt
+et al. 2003). All the maps were convolved and regridded to the resolution and pixel size of the magnetic field maps. For
+the Hα maps from 11HUGS and SINGS, the flux density units were converted to erg/s/cm−2. We used the following
+calibration from Leroy et al. (2012) to estimate SFRSDs of the galaxies in Sample 1.
+ΣSFR[M⊙yr−1kpc−2] = 634.0 IHα[erg s−1 sr−1] + 0.0025 I24µm[MJy sr−1]
+(3)
+2.2.3. SFRs using 1.4 GHz Observations
+Our 1.4 GHz non-thermal maps of the galaxies (Sample 1) (Roy & Manna 2021) and an SFR calibration from
+Murphy et al. (2011) were used to derive SFRSD maps (Equation 4). The calibration is based on the observed radio-
+FIR correlation in a sample of nearby star-forming galaxies (Bell 2003) and has a scatter of 0.26 dex. We used this
+galaxy-integrated calibration (Equation 4) to derive the formula for spatially-resolved radio-ΣSFR calibration.
+SFR1.4GHz
+M⊙yr−1
+= 6.35 × 10−29
+L1.4GHz
+erg Hz−1s−1
+(4)
+The spatially-resolved calibration is consistent with the calibration of Heesen et al. (2014). We used the above
+relation to estimate the SFRSD maps of the sample galaxies from the measured 1.4 GHz surface brightness.
+2.2.4. Galactic Extinction Correction for FUV Emission
+We corrected for the extinction of FUV emission due to dust in the Milky Way using the E(B-V) values along the
+line of sight to the sample galaxies from Bianchi et al. (2017). The extinction coefficients (AFUV) of the GALEX FUV
+bands were measured using Table 1 from Bianchi et al. (2017) and intrinsic fluxes (Fintrinsic) were estimated from the
+following formula:
+AFUV = −2.5 × log[Fobserved
+Fintrinsic
+]
+(5)
+The extinction percentage of the FUV emission is listed in Table 3.
+2.3. Gas Densities
+Atomic hydrogen (Hi) and molecular hydrogen (H2) predominantly contribute to the total gas mass of galaxies.
+H2 is best traced using rotational transitions in CO (e.g. Bolatto et al. 2013). Spatially-resolved observations of CO
+transitions exist for only three of our seven galaxies in Sample 1 (Table 2). We have used CO J=2-1 line data of NGC
+3627 and NGC 5194 from the HERA CO-Line Extragalactic Survey (HERACLES; Leroy et al. 2009) and CO J=1-0
+data of NGC 4826 from the BIMA Survey of Nearby Galaxies (BIMA SONG; Regan et al. 2001). The HERACLES
+and BIMA survey have a spatial resolution of 13′′ and 6′′, respectively. The velocity resolution of the HERACLES and
+BIMA spectral cubes are ∼5 and 6 km/s, respectively. We restrict our study of the connection between gas densities
+and magnetic fields to only these three galaxies for which spatially-resolved CO data are available.
+
+6
+Manna and Roy
+Table 3. FUV extinction values of the Sample 1 galaxies due to the Milky Way foreground dust. The extinctions were computed
+using E(B-V) values along the line of sight to the sample galaxies from Bianchi et al. (2017).
+Name
+Percentage extinction
+NGC 2683
+22
+NGC 3627
+23
+NGC 4096
+21
+NGC 4449
+15
+NGC 4490
+15
+NGC 4826
+13
+NGC 5194
+27
+The HI Nearby Galaxy Survey (THINGS; Walter et al. 2008) used VLA observations to obtain very high spectral
+(≤ 5.2 km/s) and spatial (∼ 6
+′′) resolution maps of nearby galaxies at 21cm. We used the publicly available 21cm
+moment maps from this THINGS survey to estimate the distribution of Hi in the three galaxies for which CO data
+are available. All CO and Hi 21 cm maps were convolved and regridded to a common resolution and pixel size of the
+non-thermal radio maps. Gas densities were estimated (for NGC 3627, NGC 4826 and NGC 5194) following Basu &
+Roy (2013) assuming CO to H2 conversion factor of 2 ×1020 (K km s−1)−1 (e.g. Bolatto et al. 2013). A line ratio of 0.8
+was assumed to convert COJ=2-1 to COJ=1-0 (e.g. Leroy et al. 2009). We accounted for the contribution of helium to
+the gas density using ρgas=1.36 × (ρHi + ρH2). Line of sight depths were assumed to be 300 and 400 pc for molecular
+and atomic gas, respectively (Basu & Roy 2013).
+3. RESULTS
+3.1. Magnetic Fields in the Galaxies
+We have estimated spatially resolved revised equipartition magnetic field maps for seven galaxies in Sample 1, using
+the procedures of Section 2.1; these maps are shown in Figures 1 & 2. Flux density contours of 1.4 GHz observations
+are overlaid on magnetic field maps. The resolution of these maps corresponds to spatial scales of ∼ 0.4−0.8 kpc
+(see Table 1). The bottom right panel of Figure 2 shows the radial variation of the magnetic field with galactocentric
+radius of all the seven galaxies where both the axes are normalized by their maximum values. Here, we have averaged
+the magnetic field strengths over an annular elliptical region of width equal to the beam size of the corresponding
+map. Position and inclination angle (Table 1) of each galaxy were used while selecting the elliptical regions. We find
+magnetic fields to be stronger at the central region and at the star formation sites (arm regions) with field strengths
+up to 50 µG. Field strengths fall by ∼50% at the edges of the magnetic field maps. The Milky Way also shows such a
+trend in the variation of magnetic field strengths (Beck et al. 1996). We note that our analysis was limited to distances
+where the signal-to-noise ratio in spectral index maps is > 5; the magnetic field strengths at these distances are thus
+likely to be reliable.
+We note that, compared to the magnetic field strengths obtained using the classical equipartition expression, these
+values are higher by ∼ 1.3−1.5 for a non-thermal spectral index of -0.6, and they match for a spectral index of -0.75
+(Beck & Krause 2005).
+Figure 7 shows the uncertainties in the magnetic field values for Sample 1 derived using the Monte Carlo method
+described in Section 2.1.1. Statistical uncertainties on mean magnetic fields for these seven galaxies are provided in
+Table 4.
+3.2. Star Formation Rates in the Galaxies
+We have estimated the global, galaxy-averaged SFRs of Sample 1 galaxies using 1.4 GHz, FUV+24µm, and Hα+24µm
+emission using calibrations discussed in Section 2.2. Globally integrated star formation rates of the sample galaxies
+are given in Table 5. No systematic offset was found in the SFR values estimated using these tracers. The differences
+in the SFR values for our galaxies are much less than the calibration uncertainty except for NGC 4490. For NGC
+4490, SFR calculated from 1.4 GHz emission is higher than the same from FUV+24µm emission by a factor of 2.2.
+As discussed in Section 2.2, we have estimated SFRSD maps of the seven galaxies (Sample 1) using FUV+24µm,
+Hα+24µm and 1.4GHz emission. We show SFRSD maps of the seven galaxies in the Appendix (Figures 8-9), where
+SFRSDs estimated using 1.4 GHz and FUV+24µm emission are shown in contours and colors, respectively. In the
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+7
+Figure 1. The equipartition magnetic field maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096 (clockwise from top
+left) (Sample 1). Non-thermal radio contours at 1.4 GHz are overlaid on magnetic field maps. The magnetic field strengths are
+shown in color with non-thermal emission at 1.4 GHz shown as overlaid contours. Contour levels are presented below each panel
+in the figure. The circle in the bottom-left corner of the panels indicates the angular resolution of the maps. The uncertainties
+on mean magnetic fields are 0.06µG, 0.17µG, 0.04µG and 0.18µG for the above galaxies, respectively.
+
+COLOR:NGC26832683.B.final.TH.SUB.1
+CONT:NGC2683IPOL1490.572MHz2683.L.Ths.TH.SUB.1
+10
+20
+30
+40
+33 29
+28
+27
+Declination (J2000)
+26
+25
+24
+23
+22
+21
+085305
+00
+52 55
+50
+45
+40
+35
+30
+25
+20
+RightAscension(J2000)
+Colorscalerange=5.0040.00uG
+Contpeakflux=4.8560E-03JY/BEAM
+Levs = 1.600E-04 * (-2, -1, 1, 2, 4, 8, 16, 32,
+64, 128, 256, 512)COLOR:NGC36273627.B.final.TH.SUB.1
+CONT:N3627LIPOL1430.389MHz3627.L.Ths.TH.SUB.1
+10
+20
+30
+40
+1302
+01
+Declination (J2000)
+00
+1259
+58
+57
+1120 25
+20
+15
+10
+05
+RightAscension(J2000)
+Colorscale ranqe=5.0040.00uG
+Contpeakflux=1.4880E-02JY/BEAM
+Levs = 1.480E-03 * (-2, -1, 1, 2, 4, 8, 16, 32
+64,128,256,512COLOR:NGC40964096.B.final.TH.SUB.1
+CONT:NGC4096IPOL1432.873MHz4096.L.Ths.TH.SUB.1
+5
+10
+15
+20
+25
+4734
+32
+Declination (J2000)
+30
+28
+26
+24
+可
+120630
+15
+00
+05 45
+30
+RightAscension(J2000)
+Colorscalerange=5.0025.00uG
+Contpeakflux=1.9020E-03JY/BEAM
+Levs = 1.000E-04 * (-2, -1, 1, 2, 4, 8, 16, 32
+64,128,256,512)COLOR:NGC44494449.B.final.TH.SUB.1
+CONT:N49IPOL1489.900MHz4449.L.Ths.TH.SUB.1
+10
+20
+30
+4412
+10
+08
+Declination (J2000)
+06
+04
+02
+00
+4358
+12 28 45
+30
+15
+00
+27 45
+30
+RightAscension(J2000)
+Colorscalerange=5.0035.00uG
+Contpeakflux=2.6366E-01JY/BEAM
+Levs = 7.200E-04 *(-2, -1, 1,2, 4, 8, 16, 32.
+64, 128, 256,512)8
+Manna and Roy
+0.2
+0.4
+0.6
+0.8
+1.0
+Radial distance (Normalised)
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Magnetic field (Normalised)
+NGC 2683
+NGC 3627
+NGC 4096
+NGC 4449
+NGC 4490
+NGC 4826
+NGC 5194
+Figure 2. The equipartition magnetic field maps of NGC 4490 (top left), NGC 4826 (top right) and NGC 5194 (bottom left).
+The magnetic field strengths are shown in color with non-thermal emission at 1.4 GHz shown as overlaid contours. Contour
+levels are presented below each panel in the figure. The circle in the bottom-left corner of the panels indicates the angular
+resolution of the maps. The uncertainties on mean magnetic fields are 0.06µG, 0.11µG and 0.02µG, respectively. The bottom
+right panel presents the radial variation of magnetic field strengths with galactocentric distance for all seven galaxies in Sample
+1.
+
+COLOR:NGC44904490.B.final.TH.SUB.1
+CONT:N4490IPOL1435.114MHz4490.L.ThS.TH.SUB.1
+10
+20
+30
+40
+4144
+42
+1
+Declination (J2000)
+40
+38
+36
+34
+123100
+30 45
+30
+15
+RightAscension(J2000)
+Colorscalerange=5.0045.00uG
+Contpeakflux=3.3720E-02JY/BEAM
+Levs = 4.000E-04 * (-2, -1, 1, 2, 4, 8, 16, 32,
+64,128, 256, 512)COLOR:NGC48264826.B.final.TH.SUB.1
+CONT:N4826LIPOL1425.677MHz4826.L.Ths.TH.SUB.1
+10
+20
+30
+21 45
+44
+43
+Declination (J2000)
+42
+41
+40
+39
+38
+37
+Q
+12 57 00
+56 55
+50
+45
+40
+35
+30
+25
+RightAscension(J2000)
+Colorscalerange=5.0035.00uG
+Contpeakflux=2.5190E-02JY/BEAM
+Levs = 2.800E-04 * (-2, -1, 1, 2, 4, 8, 16, 32,
+64, 128,256,512)COLOR:NGC51945194.B.final.OHGSPX.1
+CONT:M51IPOL1664.900MHz5194.L.Ths.TH.SUB.1
+10
+20
+30
+40
+4718
+16
+14
+Declination (J2000)
+12
+10
+08
+06
+13 30 30
+15
+00
+29 45
+30
+15
+RightAscension(J2000)
+Colorscale range=5.0040.00 uG
+Contpeakflux=4.6656E-02JY/BEAM
+Levs = 1.200E-04 *(-2, -1, 1,2, 4, 8, 16, 32
+64,128,256,512)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+9
+Table 4. Statistical uncertainties on mean magnetic fields for galaxies in Sample 1.
+Name
+Statistical uncertainty
+on mean magnetic fields
+(µG)
+NGC 2683
+0.06
+NGC 3627
+0.17
+NGC 4096
+0.04
+NGC 4449
+0.18
+NGC 4490
+0.06
+NGC 4826
+0.11
+NGC 5194
+0.02
+Table 5. Galaxy-averaged star formation rates of the galaxies in Sample 1, using 1.4 GHz, FUV+24µm, and Hα+24µm data.
+The uncertainties on the SFR values are ≈ 30%.
+Name
+SFR from 1.4 GHz (M⊙yr−1)
+SFR from FUV+24µm (M⊙yr−1)
+SFR from Hα+24µm (M⊙yr−1)
+NGC 2683
+0.28
+0.25
+0.33
+NGC 3627
+1.56
+2.00
+1.84
+NGC 4096
+0.42
+0.35
+0.38
+NGC 4449
+0.37
+0.38
+0.32
+NGC 4490
+4.63
+2.13
+2.30
+NGC 4826
+0.63
+0.73
+0.78
+NGC 5194
+4.16
+3.88
+3.65
+Appendix (Figures 10-11), we also present the SFRSD maps estimated using Hα+24µm and 1.4 GHz emission in colors
+and contours, respectively. The SFRSD maps of each galaxy in Figures 8−11 are shown in the same color scale and
+contours. To determine the radial variation of SFRSDs, we have averaged the SFRSD maps of our sample galaxies
+over tilted rings centred on the optical centre of each galaxy using their inclinations and position angles. The width of
+the tilted rings was taken to be equal to the beam size of the corresponding image. Figure 3 shows the radial variation
+of the average SFRSD, derived using FUV+24µm and Hα+24µm emission, with galactocentric distance where both
+the axes are normalized to their maximum values. We also derived the radial variation of SFRSDs for the galaxies
+using 1.4 GHz emission and it is consistent within 1σ statistical uncertainties, with those derived using FUV+24µm
+and Hα+24µm data. Azimuthally averaged SFRSDs of all the seven galaxies decrease gradually towards the outer
+region and drop by a factor of 6 to 8 at the edge.
+3.3. Details of the Individual Galaxies of Sample 1
+(i) NGC 2683: In this galaxy, Krause et al. (2020) found very weak linear polarisation using C-band and L-band
+VLA observations. Based on the optical image, we could separate the central region from the disk. The average
+magnetic field in the central region is found to be ≈31 µG and the outer region of the disk has an average value of
+≈19 µG (see Figure 1 and Table 6).
+Wiegert et al. (2015) used WISE 22 µm data to estimate a galaxy-averaged SFR of ≈0.09 M⊙yr−1 for NGC 2683.
+From our analysis, integrated SFR was measured to be ∼0.24 M⊙yr−1 and ∼0.28 M⊙yr−1 using FUV+24µm and 1.4
+GHz radio emission, respectively. However, we note that Wiegert et al. (2015) used a distance of 6.27 Mpc for this
+galaxy, but we have used a distance of 7.7 Mpc. The SFR is estimated to be 0.16 M⊙yr−1 using FUV+24µm emission,
+assuming the same distance as used by Wiegert et al. (2015). Taking the calibration uncertainties and the assumed
+distance into account, our estimated SFR is hence consistent with that of Wiegert et al. (2015). We note that the
+contours on the background sources (Figure 8) are not real SFRSDs, as these are likely to be background AGNs.
+(ii) NGC 3627: NGC 3627 was observed at 8.46 GHz and 4.85 GHz using the VLA in its D-configuration (Soida
+et al. 2001). These authors estimated the magnetic field strengths using the classical equipartition formula (Longair
+2011) and found an average equipartition magnetic field strength of 11±2 µG, assuming a constant non-thermal spectral
+
+10
+Manna and Roy
+0.2
+0.4
+0.6
+0.8
+1.0
+Radial distance (Normalised)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+SFRSD from FUV+24μm (Normalised)
+NGC 2683
+NGC 3627
+NGC 4096
+NGC 4449
+NGC 4490
+NGC 4826
+NGC 5194
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Radial distance (Normalised)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+SFRSD using Hα+24μm (Normalised)
+NGC 2683
+NGC 3627
+NGC 4096
+NGC 4449
+NGC 4490
+NGC 4826
+NGC 5194
+Figure 3.
+The variation of SFRSDs (normalized), estimated using FUV+24µm (left panel) and Hα+24µm (right panel)
+emission as a function of galactocentric distance (normalized) for all seven galaxies in Sample 1.
+index of 0.9 and a disk thickness of 2 kpc. Soida et al. (2001) also studied the polarized emission at these frequencies
+to find a regular magnetic field of 4±1 µG. They suggested two distinct magnetic field components of NGC 3627: one
+for the spiral arms and another for the inter-arm regions. We have separately studied equipartition magnetic fields
+in the arm and interarm regions of the galaxy. We find that the central region and the edges of the extended bar
+have magnetic field strengths of ≈ 34 µG (see Figure 1). The arm region has a field strength of ≈28 µG (see Table
+6). However, the magnetic field strength in the interarm regions has values ≈21 µG. We note that our estimates
+of the equipartition magnetic field strengths in the galaxy are higher than those found by Soida et al. (2001); this
+difference likely arises from the fact that Soida et al. (2001) estimated the magnetic field strengths using the classical
+equipartition formula, which is known to significantly underestimate the magnetic field in the star-forming regions.
+We measured a galaxy-averaged SFR of ≈2.0 M⊙yr−1 and ≈1.56 M⊙yr−1 from FUV+24µm and 1.4 GHz emission,
+respectively.
+Our measurements of spatially resolved SFRs in different regions are consistent, within calibration
+uncertainties, with the SFR estimates of Watanabe et al. (2011).
+(iii) NGC 4096: Our estimate of the equipartition magnetic field in NGC 4096 varies from ≈21 µG at the centre
+to ≈12 µG at the edge (table 6). The magnetic field strength in both the central region and northern periphery is
+quite similar, with typical field strengths of ≈ 20 µG; this is presumably due to its high inclination. The outer part of
+the galaxy has an average field strength of ≈14 µG. NGC 4096 was observed (Irwin et al. 2012; Wiegert et al. 2015)
+with its B-field and further studied by Krause et al. (2020) who found very little polarized emission from the galaxy.
+Wiegert et al. (2015) used the 22 µm−SFR calibration to measure a galaxy-averaged SFR of 0.27±0.02 M⊙yr−1.
+Our measurement of the galaxy-averaged SFR is ≈0.35 M⊙yr−1 and ≈0.43 M⊙yr−1 using FUV+24µm and 1.4 GHz
+emission, respectively. Considering the calibration uncertainties, our estimates are consistent with that of Wiegert
+et al. (2015).
+(iv) NGC 4449: This is an optically bright irregular starburst galaxy. Chy˙zy et al. (2000) used VLA 4.86 and
+8.46 GHz observations to find a galaxy-averaged equipartition magnetic field of ≈14 µG. These authors also used
+polarization emission to estimate a regular field of ≈8 µG. The equipartition magnetic field map of NGC 4449 from
+our study is shown in Figure 1. As noted in Section 2.1, about 70 % of the total projected area of this galaxy has
+spectral index values of less than 0.55. We have replaced the pixel values with αnt < 0.55 with αnt = 0.55 while
+computing the magnetic field for NGC 4449 (see Section 2.1). The average magnetic field strength is ≈17 µG in this
+galaxy, which is comparable to the findings of Chy˙zy et al. (2000).
+Our measurements of the galaxy-averaged SFR are ≈0.38 M⊙yr−1 and ≈0.37 M⊙yr−1 using FUV+24µm and 1.4
+GHz emission, respectively, which are consistent with the SFR of 0.47 M⊙yr−1 estimated by Chy˙zy et al. (2011).
+(v) NGC 4490: Nikiel-Wroczy´nski et al. (2016) observed NGC 4490 at 0.61 GHz using the GMRT, and at 4.86 &
+8.44 GHz using VLA + Effelsberg. The authors used these observations to find a mean equipartition magnetic field
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+11
+Table 6. Magnetic field strengths in different regions of the galaxies in Sample 1. For the irregular galaxy NGC 4449, we could
+only measure the galaxy-integrated magnetic field. We have separated the two nearly face-on galaxies (NGC 3627 and NGC
+5194) into arm and inter-arm regions. For the rest of the galaxies, we could not separate the arm and inter-arm region due to
+their higher inclinations.
+Galaxy
+Galaxy-average
+Beq in
+Beq in
+Beq in
+Beq in
+name
+Beq
+central region
+disk region
+arm region
+inter-arm region
+(µG)
+(µG)
+(µG)
+(µG)
+(µG)
+NGC 2683
+24±6
+31±3
+19±5
+–
+–
+NGC 3627
+25±4
+34±8
+–
+28±5
+21±4
+NGC 4096
+16±4
+21±5
+14±3
+–
+–
+NGC 4449
+17±6
+–
+–
+–
+–
+NGC 4490
+23±10
+40±6
+17±7
+–
+–
+NGC 4826
+23±9
+38±8
+20±5
+–
+–
+NGC 5194
+16±6
+34±6
+–
+25±5
+18±4
+of 21.9±2.9 µG, with typical field strengths in the range of 18 µG to 40 µG. We have found a typical equipartition
+magnetic field strength of ≈40 µG in the central region, which decreases to ≈17 µG in the outer region (see Figure 2);
+these values are consistent with the estimates of Nikiel-Wroczy´nski et al. (2016). We find a relatively lower magnetic
+field strength of ≈15 µG in both the interacting region and the companion galaxy NGC 4485. Therefore, a gradual
+decrease in the average magnetic field strength occurs from the center to the outer region.
+Clemens et al. (1999) used radio observations to find a galaxy-averaged SFR of 4.7 M⊙yr−1. We found a similar SFR
+(≈4.63 M⊙yr−1) using 1.4 GHz radio emission but a factor of ∼2 lower SFR (2.13 M⊙yr−1) using the FUV+24µm
+emission (Table 5). Extinction corrections for NGC 4490 are believed to be higher than those typically assumed and
+this may lead to an underestimation of the SFR while using the FUV+24µm diagnostics (Clemens et al. 1999).
+(vi) NGC 4826: No spatially resolved maps of magnetic fields and SFRSDs are available in the literature. We
+measure the central and outer regions of the galaxy to have an average equipartition magnetic field strength of ≈38
+µG and ≈20 µG, respectively (see Figure 2 and Table 6). We find galaxy-averaged SFR of ≈0.73 M⊙yr−1 and ≈0.63
+M⊙yr−1 using FUV+24µm and 1.4 GHz data, respectively.
+(vii) NGC 5194: Fletcher et al. (2011) used VLA C-band observations of the galaxy and assumed a constant
+thermal and non-thermal spectral index of 0.1 and 1.1 to find an average equipartition magnetic field strength of 20
+µG using the revised formula by Beck & Krause (2005). They found a magnetic field of 20−25 µG in the spiral arms,
+higher than the 15−20 µG typical in the interarm regions. Using VLA observations at S-band (2−4 GHz) frequencies,
+Kierdorf et al. (2020) found the field strength of turbulent and regular components of the magnetic field in the arm
+regions of 18−24 µG and 8−16 µG, respectively. We find an equipartition magnetic field strength of ≈25 µG in the
+arm region and ≈18 µG in the interarm region (see Table 6). The peripheral region has a magnetic field of ≈12 µG,
+while the overlapping region between NGC 5194 and NGC 5195 has an average Beq of ≈16 µG. Considering our use
+of Equation 1 (Beck & Krause 2005), measurements are roughly consistent with the earlier study of Fletcher et al.
+(2011) and Kierdorf et al. (2020).
+Spatially resolved SFRs were measured in several star-forming regions of NGC 5194 using Hα+24µm and Hα+Paα
+emission (Kennicutt et al. 2007).
+SFRSDs in different regions were found to be in the range of 0.10 to 0.46
+M⊙yr−1kpc−2. Our estimates using the two tracers are consistent with the estimates of Kennicutt et al. (2007) (See
+Figures 9 & 11). Furthermore, we find that the galaxy-integrated SFR derived using FUV+24µm (≈3.88 M⊙yr−1)
+and 1.4 GHz data (≈4.16 M⊙yr−1) are consistent with each other, within 1-sigma statistical uncertainty.
+3.4. Is the Minimum Energy Condition Valid for the Sample Galaxies?
+We have estimated magnetic fields for the galaxies in Sample 1 assuming the “minimum energy condition” or
+“equipartition condition”, i.e. by assuming that the energy density in the magnetic field is approximately equal to
+the energy density in cosmic ray particles. Therefore, it is important to verify the validity of this assumption in our
+sample galaxies. The tightness of the spatially-resolved radio−FIR correlation can be used to estimate the deviation
+of the energy densities from the minimum energy condition (Hummel 1986; Basu & Roy 2013). According to the
+simplified model of Hummel (1986), when the minimum energy condition is satisfied, the distribution of Int/IFIR will
+be similar to the distribution of B1+αnt. The model assumes the following to be constant across galaxies: (a) the ratio
+of the number densities of relativistic electrons and dust-heating stars, (b) the volume ratio of radio and FIR emitting
+
+12
+Manna and Roy
+regions, and (c) the ratio of efficiency factors for both the radio and FIR emission. In this model, the cumulative
+distribution function (CDF) of the quantity Int/IFIR and B1+αnt
+eq
+is expected to follow each other if Beq is close to B.
+To verify the validity of the minimum energy condition in our sample galaxies, we have followed the procedure as
+in Hummel (1986) and Basu & Roy (2013). The CDF of Int/IFIR and B1+αnt
+eq
+were estimated using our radio maps
+of the sample galaxies at both 0.33 and 1.4 GHz. We used an ensemble of spatially-resolved values of αnt, Int (both
+at 0.33 and 1.4 GHz), IFIR (70 µm) and magnetic fields (Beq), which are averaged over the beam size from all the
+galaxies in Sample 1 (Table 2) to generate these distributions. The CDFs of all quantities were normalized by their
+median values. The top panels in Figure 4 show the median-normalized CDFs of Int/IFIR and B1+αnt
+eq
+at both 0.33
+and 1.4 GHz.
+We find that the CDFs of Int/IFIR and B1+αnt
+eq
+at both 0.33 and 1.4 GHz broadly follow each other but with slight
+deviations at high and low ends (see top panels in Figure 4). This implies that the minimum energy condition is
+broadly valid and is consistent with earlier findings. For example, Hummel (1986) found the distribution of the two
+quantities is similar in a sample of Sbc galaxies while Basu & Roy (2013) reached similar conclusions in a study of 5
+nearby large spiral galaxies, but with slight deviations observed in the CDFs of Int/IFIR and B1+αnt
+eq
+in the interarm
+regions of the galaxies.
+The observed deviation in the CDFs of Int/IFIR and B1+αnt
+eq
+for our sample galaxies imply a corresponding deviation
+from the minimum-energy condition. In order to quantify this deviation, we performed a Monte Carlo simulation orig-
+inally proposed by Hummel (1986). In this simulation, random numbers (X) were drawn from a Gaussian distribution
+with standard deviation σ. Thereafter, we multiplied 10X with the observed equipartition magnetic fields to introduce
+deviations from the minimum-energy condition. We thus constructed the CDF of B1+αnt
+eq
+using the deviated magnetic
+field values. The CDF of B1+αnt
+eq
+were then compared to the observed CDF of Int/IFIR via a Kolmogorov-Smirnov
+(KS) test. This procedure was repeated for a range of σ from 0 to 0.2. We find that the p-values for the KS test
+comparing the distributions are maximized when σ = 0.1. Indeed, B1+αnt
+eq
+derived after deviating the magnetic field
+using σ = 0.1 and Int/IFIR are consistent with being derived from the same distribution, with a KS test p-value of
+0.41 and 0.55, when using Int at 0.33 and 1.4 GHz, respectively. The bottom panels in Figure 4 show the CDFs of the
+two quantities for σ = 0.1 at 0.33 and 1.4 GHz; it is clear that the CDFs follow each other. This implies the actual
+magnetic field values may deviate from the equipartition values by ∼ 25% in our galaxies in Sample 1. We note that
+any violation of the assumptions made by Hummel (1986) may also lead to the observed deviation in the CDFs.
+3.5. Correlation Between Magnetic Fields and SFRSDs
+We have studied the correlation between the spatially-resolved equipartition magnetic field and SFRSDs for the
+galaxies in Sample 1 (Table 2) at scales of ≈360−760 pc (Table 1). For the seven sample galaxies, we used the SFRSD
+maps estimated using the FUV+24µm emission. The correlations between magnetic fields and SFRSDs for the seven
+galaxies are shown in Figure 5. Each point represents the logarithms of equipartition magnetic fields and SFRSD values
+that are averaged over the beam size of the corresponding maps. da Silva et al. (2014) found that SFR calibrations
+could be biased and strongly affected by stochasticity at small spatial scales where the star formation rate is low (≤
+10−2.5 M⊙yr−1); we have therefore excluded regions of low star formation rates from the correlation study.
+We find that the equipartition magnetic field and the SFSRD are correlated in all seven sample galaxies. We use
+orthogonal distance regression in Scipy (Virtanen et al. 2020) to fit a power law of the form B = B0 (ΣSFR)η to the
+magnetic field − SFRSD data points; the spatially-resolved uncertainty maps of equipartition magnetic fields and rms
+noise on the SFRSD maps were used to estimate the uncertainties on each data point during the fitting procedure. The
+best-fit parameters of the power-law are given in Table 7. We have also estimated the scatter (rms of the data points
+along the y-axis) of the correlations which are presented in Table 7 and are shown in dashed lines in the corresponding
+plots (Figure 5). We find that six of the seven galaxies have slopes (η) in the range of ≈ 0.27 − 0.40 but that the slope
+is relatively lower for NGC 4449 with η ≈ 0.18. Averaging over the slope of all galaxies in Sample 1, we find a mean
+slope of 0.32 ± 0.06.
+3.6. Correlation Between Magnetic Fields and Gas Densities
+We have studied the correlation between spatially-resolved equipartition magnetic fields and gas densities for three
+of the galaxies in Sample 1, NGC 3627, NGC 4826, and NGC 5194, for which spatially resolved CO observations
+were available (see Section 2.3). Similar to the study of correlations between Beq and SFRSDs, we have studied the
+correlations between Beq and gas density values, both averaged over the beam size of the corresponding maps. The
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+13
+Figure 4. The top panels show the cumulative distribution function (CDF) of Int,radio/I70µm (in red) and B1+αnt
+eq
+(in blue),
+where Int is the non-thermal emission at 0.33 GHz (top left) and 1.4 GHz (top right) (Sample 1). The variables are normalized
+by their median values. The bottom panels show the same but now with the magnetic field perturbed from its measured value
+using σ=0.1 (see Section 3.4); the CDFs of the Int,radio/I70µm and B1+αnt
+eq
+are now consistent with being derived from the same
+distribution.
+Table 7. Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the seven galaxies in
+Sample 1. The data were fitted with a power law of the form B=B0(ΣSFR)η.
+Name
+Slope (η)
+Intercept (B0) (log(µG))
+Intercept (B0) (µG)
+Scatter
+NGC 2683
+0.34 ± 0.04
+2.10 ± 0.07
+125 ± 1.2
+0.05
+NGC 3627
+0.31 ± 0.03
+1.71 ± 0.03
+51 ± 1.1
+0.04
+NGC 4096
+0.33 ± 0.04
+1.80 ± 0.08
+63 ± 1.2
+0.05
+NGC 4449
+0.18 ± 0.03
+1.64 ± 0.04
+43 ± 1.1
+0.03
+NGC 4490
+0.27 ± 0.02
+1.90 ± 0.03
+79 ± 1.1
+0.06
+NGC 4826
+0.38 ± 0.02
+1.80 ± 0.02
+63 ± 1.0
+0.05
+NGC 5194
+0.40 ± 0.01
+2.00 ± 0.02
+100 ± 1.0
+0.07
+
+1.0
+0.8
+X
+0.6
+=）
+P
+0.4
+0.2
+lo.33GHz/170μm
+0.0
+0
+2
+4
+6
+X1.0
+0.8
+X
+0.6
+=）
+v
+P
+0.4
+0.2
+eq
+l1.4GHz/l70μm
+0.0
+0
+1
+2
+3
+4
+5
+X1.0
+0.8
+X
+0.6
+=）
+V
+P
+0.4
+0.2
+lo.33GHz/170μm
+0.0
+0
+2
+4
+6
+X1.0
+0.8
+X
+0.6
+=）
+v
+P
+0.4
+0.2
+eq
+l1.4GHz/l70μm
+0.0
+0
+1
+2
+3
+4
+5
+X14
+Manna and Roy
+2.25
+2.00
+1.75
+1.50
+1.2
+1.3
+1.4
+1.5
+1.6
+log(Magnetic Field in G)
+NGC 2683
+1.0
+0.8
+0.6
+0.4
+1.3
+1.4
+1.5
+1.6
+1.7
+NGC 3627
+2.0
+1.8
+1.6
+1.1
+1.2
+1.3
+1.4
+NGC 4096
+1.6
+1.4
+1.2
+1.0
+1.25
+1.30
+1.35
+1.40
+1.45
+1.50
+log(Magnetic Field in G)
+NGC 4449
+2.0
+1.5
+1.2
+1.4
+1.6
+NGC 4490
+2.0
+1.5
+1.0
+1.2
+1.4
+1.6
+NGC 4826
+2.0
+1.5
+1.0
+1.0
+1.2
+1.4
+1.6
+log(Magnetic Field in G)
+NGC 5194
+2.0
+1.5
+1.0
+1.0
+1.1
+1.2
+1.3
+NGC 1097
+2.5
+2.0
+1.5
+1.0
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+NGC 4736
+1.00
+0.75
+0.50
+0.25
+log(SFRSD in M
+ yr
+1 kpc
+2)
+1.05
+1.10
+1.15
+1.20
+log(Magnetic Field in G)
+NGC 5055
+2.0
+1.5
+1.0
+log(SFRSD in M
+ yr
+1 kpc
+2)
+1.0
+1.2
+1.4
+1.6
+NGC 5236
+2.0
+1.5
+1.0
+log(SFRSD in M
+ yr
+1 kpc
+2)
+1.1
+1.2
+1.3
+1.4
+NGC 6946
+Figure 5. The correlation between magnetic fields and SFRSD for the combined sample of 12 galaxies (Sample 2, Table 2).
+For the seven galaxies in Sanple 1, the SFRSD estimates shown in the plots were derived using FUV + 24µm (Section 3.5). The
+SFRSD estimates for the five galaxies from Basu et al. (2012a) (Sample 2) were derived using Hα + 24µm (Section 4) The red
+line shows a linear fit to the data points. The black dashed lines show the ±1σ vertical scatter.
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+15
+23.2
+23.0
+1.3
+1.4
+1.5
+1.6
+log(Magnetic Field in G)
+NGC 3627
+23.0
+22.5
+1.2
+1.4
+1.6
+NGC 4826
+23.0
+22.5
+1.0
+1.2
+1.4
+1.6
+NGC 5194
+23.5
+23.0
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+log(Magnetic Field in G)
+NGC 4736
+23.6
+23.4
+23.2
+log(Gas density in gm/cm3)
+1.05
+1.10
+1.15
+1.20
+NGC 5055
+23.5
+23.0
+22.5
+log(Gas density in gm/cm3)
+1.1
+1.2
+1.3
+1.4
+1.5
+NGC 5236
+23.0
+22.5
+log(Gas density in gm/cm3)
+1.1
+1.2
+1.3
+log(Magnetic Field in G)
+NGC 6946
+Figure 6.
+The correlations between magnetic fields (µG) and gas densities (gm/cm−3) for seven galaxies of Sample 3 (Table 2).
+The red line shows a linear fit to the data points. The black dashed lines show the ±1σ vertical scatter.
+correlations between magnetic fields and gas densities of NGC 3627, NGC 4826, and NGC 5194 are shown in Figure
+6. We have again used orthogonal distance regression in Scipy (Virtanen et al. 2020) to fit a power-law to the Beq and
+gas density data points. The scatters of the three correlations are shown in dashed lines in all the figures.
+The measured best-fit power-law indices are 0.40±0.02, 0.49±0.03 and 0.53±0.02 (Table 9) for NGC 3627, NGC
+4826 and NGC 5194, respectively. The mean of the power-law indices is 0.47±0.05.
+4. EXTENDING THE SAMPLE WITH 5 GALAXIES FROM EXISTING GMRT OBSERVATIONS
+As mentioned earlier, a study of Beq and radio-FIR correlations for a sample of five large nearly face-on galaxies was
+carried out by Basu et al. (2012a,b); Basu & Roy (2013), using low-radio frequency observations at 0.33 and 1.4 GHz
+
+16
+Manna and Roy
+Table 8. Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the five galaxies in
+Basu et al. (2012a) (Sample 2). The data were fitted with a power law of the form B=B0(ΣSFR)η.
+Name
+Slope (η)
+Intercept (B0) (log(µG))
+Intercept (B0) (µG)
+Scatter
+NGC 1097
+0.27 ± 0.01
+1.61 ± 0.01
+41 ± 1.0
+0.02
+NGC 4736
+0.32 ± 0.02
+1.78 ± 0.05
+60 ± 1.1
+0.04
+NGC 5055
+0.27 ± 0.04
+1.26 ± 0.02
+18 ± 1.0
+0.02
+NGC 5236
+0.38 ± 0.07
+1.91 ± 0.02
+81 ± 1.0
+0.08
+NGC 6946
+0.25 ± 0.03
+1.62 ± 0.05
+42 ± 1.1
+0.05
+Table 9. Best-fit parameters and the scatter of the correlation between spatially-resolved magnetic fields and gas densities for
+the seven galaxies in Sample 3. Galaxies with an asterisk are from the sample of Basu et al. (2012a).
+Name
+Exponent
+Scatter
+NGC 3627
+0.40 ± 0.02
+0.03
+NGC 4826
+0.49 ± 0.03
+0.05
+NGC 5194
+0.53 ± 0.02
+0.06
+NGC 4736∗
+0.44 ± 0.03
+0.03
+NGC 5055∗
+0.25 ± 0.02
+0.02
+NGC 5236∗
+0.40 ± 0.03
+0.04
+NGC 6946∗
+0.31 ± 0.03
+0.04
+at sub-kpc linear resolutions. In this paper, we expand our study of spatially-resolved correlations between magnetic
+fields, gas densities, and SFRSDs by including these five galaxies.
+We refer readers to Basu et al. (2012a) for a detailed discussion of their sample, GMRT observations, data reduction
+procedures, and estimation of non-thermal spectral indices. It is to be noted that the modelling of the thermal free-free
+emission from these galaxies is performed in the same way as was done for our seven galaxies in Sample 1.
+We have estimated the SFRSD maps of these five galaxies using Hα data along with 24 µm IR data. We obtained
+Hα maps of four of the galaxies, NGC 1097, NGC 4736, NGC 5055, and NGC 6946 from the ancillary data at the
+SINGS website1 and obtained the Hα map of NGC 5236 from 11HUGS (Kennicutt et al. 2008). We used Hα and MIPS
+24 µm data in combination to derive the SFRSD maps of these galaxies using the calibration from Leroy et al. (2012)
+(Equation 3, Section 2.2). To estimate the equipartition magnetic field strengths of these five galaxies, we have used
+the non-thermal radio maps at 0.33 and 1.4 GHz from Basu et al. (2012a). The correlations between equipartition
+magnetic fields and SFRSDs are shown in Figure 5 where, similar to the previous correlation studies, each point
+represents the logarithms of magnetic fields and SFRSD values that are averaged over the beam size. Similar to the
+previous correlations (Section 3.5), we used orthogonal distance regression in Scipy to fit a power law to the data. We
+have provided the best-fit parameters of the power-law fit in Table 8. The scatters of all five correlations (presented
+in Table 8) are shown in dashed lines in all the figures. We find a mean exponent of 0.30±0.05 for the five galaxies
+where the exponent of individual galaxies varies from ≈0.25 to ≈0.38.
+We have computed maps of cold gas densities of four out of the five galaxies; NGC 4736, NGC 5055, NGC 5236
+and NGC 6946, using the atomic and molecular gas surface density maps from Basu & Roy (2013). The assumed
+parameters are taken to be the same as described in Section 2.3. For the remaining galaxy, NGC 1097, we could not
+measure gas densities as there are no archival CO data available for the galaxy. Following the procedures of Section 3.5,
+we have also studied the spatially-resolved correlation between equipartition magnetic fields and gas densities for the
+four sample galaxies, which are shown in Figure 6. The best-fit parameters are presented in Table 9. The exponents
+of the individual galaxies vary between ≈0.25 to ≈0.44 where the mean exponent is found to be 0.35±0.07.
+5. DISCUSSION
+Understanding the relationship between the physical condition of the interstellar medium (ISM) and the star
+formation process is crucial to understand galaxy evolution.
+Gas and magnetic fields are key constituents of the
+1 https://irsa.ipac.caltech.edu/data/SPITZER/SINGS/
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+17
+ISM and therefore it is important to study the interrelations between gas, magnetic fields, and SFRs. Though the
+Kennicutt−Schmidt relation, i.e. the relation between gas densities and SFRs, has been extensively studied at high
+spatial resolutions in various types of nearby galaxies (e.g. Onodera et al. 2010; Roychowdhury et al. 2015; Filho et al.
+2016; Miettinen et al. 2017), similar high-resolution observations of how the magnetic fields are related to SFRs and
+gas densities are yet to be systematically investigated. Such observations are critical to understand the validity of
+several models that predict strong correlations between the magnetic fields and gas densities (e.g. Chandrasekhar &
+Fermi 1953; Fiedler & Mouschovias 1993; Cho & Vishniac 2000; Groves et al. 2003) as well as magnetic fields and
+SFRSDs (e.g. Niklas & Beck 1997; Schleicher & Beck 2013, 2016). Here, we have studied these correlations in a sample
+of twelve galaxies (Sample 3) at sub-kpc scales (see Sections 3.5, 3.6 & 4). To our knowledge, this is the first spatially
+resolved study of the above correlations in nearby large galaxies. In this section, we place these findings in the light
+of predictions made by various models and in the process attempt to provide physical insights into the interrelation
+between magnetic fields, gas densities, and star formation rates at sub-kpc scales.
+5.1. Magnetic Fields and SFRSDs
+Several Magneto-Hydrodynamical simulations find that galactic magnetic fields are amplified by gas turbulence in
+very short timescales (i.e. ∼100 Myr) (e.g. Brandenburg & Subramanian 2005; Beresnyak 2012; Schober et al. 2012;
+Schleicher & Beck 2013; Bovino et al. 2013). The primary driver of gas-turbulence in the ISM of galaxies is supernova
+explosion (Bacchini et al. 2020), the rate of which is in turn directly coupled to the SFR in the galaxy. Therefore,
+it is expected that the star formation rates and the magnetic fields in a galaxy will be correlated. Indeed, using
+semi-analytical models, Schleicher & Beck (2013, 2016) found that in order to explain the radio-FIR correlation at
+sub-kpc scales, magnetic fields and SFRSDs, again at sub-kpc scales, must be related as B ∝ Σ1/3
+SFR.
+Studies in the literature on the correlation between magnetic fields and SFRSDs have focused on dwarf galaxies and
+those studies were carried out using galaxy-integrated magnetic fields and SFRSDs. As mentioned in Section 1, to our
+knowledge, there is only one published work of the spatially-resolved study of the correlation between magnetic fields
+and SFRSDs (Basu et al. 2017).
+For the 12 galaxies in Sample 2 (Table 2), we find that the mean value of the power-law index of the correlation
+between Beq and SFRSDs is 0.31±0.06, i.e Beq ∝ Σ0.31±0.06
+SFR
+(2), consistent (at < 1σ error) with the model of Schleicher
+& Beck (2013, 2016). Thus, it appears that the semi-analytical models that are based on the amplification of magnetic
+fields due to supernova-driven gas turbulence work remarkably well for the pilot sample, in predicting the correlation
+between magnetic fields and SFRSDs down to sub-kpc scales.
+We note that the power-law index for the correlation between Beq and SFRSDs for NGC 4449 was found to be
+0.18 ± 0.03, significantly lower than for the remaining galaxies (Table 7) as well as lower than the model prediction
+of B ∝ Σ1/3
+SFR Schleicher & Beck (2013) (at > 5σ significance). For the case of NGC 4449, the relatively flat spectral
+index values (αnt ≤ 0.55) in ≈ 70% of the galaxy meant that the magnetic field values could not be estimated reliably
+for a large part of the galaxy (see Section 2.1 and 3.5). This could lead to biases in the correlation and therefore, the
+low value of the power-law index for NGC 4449 should be taken with caution.
+5.1.1. Intercept of the Correlation
+According to the model proposed by Schleicher & Beck (2013), the intercept of the B-ΣSFR correlation depends on
+several ISM parameters such as gas density (ρ0), the fraction of turbulent kinetic energy converted into magnetic energy
+(fsat), the injection rate of turbulent supernova energy (C) and the intercept of Kennicutt-Schmidt (KS) relation (C1)
+(Equation 5).
+B ∼
+�
+fsat8π ρ1/6
+0
+( C
+C1
+)1/3 Σ1/3
+SFR.
+(6)
+Schleicher & Beck (2013) predicted the intercept of the B-ΣSFR correlation to be ∼ 26 µG assuming ρ0 = 10−24
+g cm−3 and fsat ∼ 5 percent.
+We have found an average intercept at 65±25 µG of the Beq-ΣSFR correlation of
+the 12 galaxies in sample 2 (see Table 7 & 8). Although the mean value is a factor of ≈2.5 higher than the value
+predicted by Schleicher & Beck (2013), this value is consistent with the predicted value, within the scatter (at ≈1.6σ).
+Future follow-up studies, such as using our full survey (Sample 0 which consists of 46 galaxies), are required to draw
+statistically robust conclusions about the value of the intercept.
+2 The uncertainty quoted is the scatter of the measured value of η across the galaxies in Sample 2.
+
+18
+Manna and Roy
+If the value of fsat is indeed higher, this would imply a higher than assumed value of one or more of ρ0, C, and fsat.
+The intercept is broadly insensitive to the assumed value of ρ0 (Equation 6) and therefore, in order to explain a factor
+of ≈ 2.5 higher value of the intercept, the actual value ρ0 has to be higher than the assumed value of 10−24 g cm−3
+by a factor of ≈ 240; such high gas densities are unphysical and are not observed in typical regions of a galaxy. The
+other possibility that the assumed value of the injection rate of turbulent supernova energy (C) is higher by a factor
+of ≈ 16 is also contrary to expectation; Basu et al. (2017) found that under reasonable conditions the value of C can
+be higher by at most a factor of 1.4. Therefore, fsat must be higher than 0.05 to explain a significantly higher value
+of the intercept. An understanding of how galaxies can achieve such efficient amplification of magnetic fields with fsat
+much greater than 5% requires detailed MHD simulations. We note that Basu et al. (2017) found that the value of
+the intercept for B-ΣSFR for the dwarf galaxy IC 10 is 51 µG, similar to our findings of a higher than predicted value
+of the intercept.
+5.2. Magnetic Fields and Gas
+Magnetic fields and gas are expected to be correlated as B ∝ √ρgas (e.g. Chandrasekhar & Fermi 1953; Groves et al.
+2003). We find that equipartition magnetic fields are correlated with gas densities for the seven galaxies (Sample 3)
+with an average power-law index, k=0.40±0.09 (see Section 3.6 & 4)3. This value of k is consistent with the numerical
+simulations that predict k ≈ 0.4−0.6 and also consistent with the theories that predict B ∝ ρ0.5
+gas. The power-law index
+of the correlation between Beq and gas densities is found to be 0.25±0.02 and 0.31±0.03 for NGC 5055 and NGC
+6946 respectively, significantly lower than the model predictions and as compared to the other galaxies in Sample 3.
+A lower value of k could mean that either the efficiency of the amplification of the magnetic field is less or that the
+magnetic field strengths derived assuming the “minimum energy condition” are underestimated (Dumas et al. 2011).
+Strong synchrotron or inverse Compton losses of cosmic-ray electrons could suppress the radio synchrotron emission
+which would then cause the equipartition magnetic fields to be underestimated.
+5.2.1. Magnetic Fields, Gas Densities and the Radio-FIR Correlations
+Energy equipartition between the magnetic field (B) and the gas density (ρgas), and between magnetic fields and
+cosmic ray particles implies that the non-thermal emission is related to the gas density as Int ∝ ρk(3+αnt)
+gas
+where k
+is the power-law index relating magnetic fields and gas densities (Beq ∝ ρk
+gas) (Niklas & Beck 1997). Further, the
+Kennicutt-Schmidt law and the radio-FIR correlation imply that Int is related to gas densities as (1) Int ∝ ρm(n+1)
+gas
+for
+optically thin dust to UV photons and (2) Int ∝ ρmn
+gas for optically thick dust to UV photons, where m is the power-law
+index of the radio-FIR correlation and n is the power-law index of the Kennicutt-Schmidt law. Therefore, we can
+obtain the following relation between the power-law index of all four correlations (Dumas et al. 2011):
+k = (n + 1)m
+3 + αnt
+; Optically thin dust
+(7)
+k =
+nm
+3 + αnt
+; Optically thick dust
+(8)
+We can use the above equations to indirectly estimate the power-law index, k, of the correlation between magnetic
+fields and gas densities. For the three galaxies, NGC 3627, NGC 4826, and NGC 5194 (Roy & Manna 2021), we have
+estimated gas densities using CO and Hi observations. Now we can compare the direct measurement of k with an
+indirect estimate of k using Equations 7 and 8; this will provide additional information on the validity of both the
+minimum energy conditions that were assumed between magnetic fields and the gas densities as well as the magnetic
+fields and cosmic ray particles.
+For the galaxies from Basu et al. (2012a), this study was already presented and
+discussed in Basu et al. (2012b).
+We have estimated k for all the seven sample galaxies from Roy & Manna (2021) (Sample 1), using the assumption of
+optically thin dust to UV photons, using (i) the slope of radio-FIR correlation (m) as derived in Roy & Manna (2021),
+(ii) the measured galaxy-averaged spectral index (αnt) from Roy & Manna (2021), and (iii) a Kennicutt-Schmidt
+power-law index of 1.4±0.15 (Kennicutt 1998b).
+Table 10 provides the relevant values as well as estimated values of k derived using the measured value of m using
+radio emission at both 0.33 and 1.4 GHz. For two of the galaxies, NGC 3627 & NGC 5194, the value of k estimated
+3 The uncertainty quoted is the scatter of the measured value of k across the galaxies in Sample 3.
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+19
+Table 10. Power law index (k) of the relation between magnetic fields and gas densities (B ∝ ρk) of galaxies in Sample 1,
+indirectly estimated using the slope of radio-FIR correlation (m) and the slope of the Kennicutt−Schmidt law. See Section 5.2.1
+for a discussion on these.
+Name
+m
+m
+αnt
+k (Optically thin)
+k (Optically thin)
+0.33 GHz
+1.4 GHz
+0.33 GHz
+1.4 GHz
+NGC 2683
+0.54±0.06
+0.91±0.07
+-0.84±0.08
+0.33 ± 0.04
+0.57 ± 0.06
+NGC 3627
+0.55±0.03
+0.85±0.13
+-1.10±0.07
+0.32 ± 0.03
+0.50 ± 0.08
+NGC 4096
+0.74±0.05
+0.90±0.04
+-0.78±0.06
+0.47 ± 0.04
+0.57 ± 0.05
+NGC 4449
+0.77±0.05
+0.65±0.04
+-0.48±0.06
+0.53 ± 0.05
+0.45 ± 0.04
+NGC 4490
+0.68±0.02
+0.75±0.02
+-0.59±0.07
+0.45 ± 0.03
+0.50 ± 0.04
+NGC 4826
+1.39±0.1
+1.47±0.08
+-0.49±0.06
+0.95 ± 0.09
+1.00 ± 0.09
+NGC 5194(arm)
+0.50±0.05
+0.65±0.04
+-0.63±0.05
+0.33± 0.04
+0.43 ± 0.04
+NGC 5194(interarm)
+0.73±0.11
+1.03±0.05
+-0.85±0.10
+0.46± 0.08
+0.64 ± 0.05
+using Equation 7 is comparable to the direct measurement of k. This broadly validates the assumption of energy
+equipartition between magnetic fields and cosmic ray particles in these two galaxies.
+For the optically thin case, the mean of indirectly-estimated k values of the sample of seven galaxies are 0.59 ± 0.16
+and 0.53 ± 0.19 at 1.4 and 0.33 GHz, respectively. However, this includes the galaxy NGC 4826, which shows an
+anomalously high value of k=1.0 and 0.95 derived at 1.4 and 0.33 GHz, respectively. Excluding this galaxy from the
+mean calculation, we find that k=0.52±0.04 and 0.47±0.09 at 1.4 and 0.33 GHz, respectively. Remarkably, for all the
+galaxies except NGC 4826, the k value at 1.4 GHz, for the optically thin case, is consistent with 0.5 within error bars.
+Thus, the indirectly estimated values of k are consistent with equipartition between magnetic fields and gas energy
+densities (Chandrasekhar & Fermi 1953; Fiedler & Mouschovias 1993; Cho & Vishniac 2000; Groves et al. 2003). This
+is similar to the findings of Niklas & Beck (1997) for their sample of 43 galaxies and Basu et al. (2012b) for their
+sample of four galaxies.
+The value of k derived for NGC 4826, for the optically thin case, is a consequence of the anomalously high value
+of the power-law index of the radio-FIR correlation (≈1.39 and ≈1.47 for 0.33 and 1.4 GHz respectively, Table 10)
+which is different from the other six galaxies in the sample. NGC 4826 has been classified as a Seyfert 2 galaxy in
+the past (Malkan et al. 2017) and therefore the emission from the core contributes to the observed power-law index of
+the radio-FIR correlation (Roy & Manna 2021). It is likely that the significant contribution of the AGN to the radio
+emission makes the estimate of k for NGC 4826 unreliable.
+6. SUMMARY
+1. We made spatially resolved maps of equipartition magnetic fields in seven galaxies (Sample 1): NGC 2683,
+NGC 3627, NGC 4096, NGC 4449, NGC 4490, NGC 4826, and NGC 5194 and find that the magnetic fields are
+strongest near the central region and go down by a factor of ∼2 at the edge of the magnetic field maps.
+2. We have used the tightness of the spatially-resolved radio-FIR correlations to verify the validity of the equipar-
+tition condition between magnetic fields and cosmic ray particles for the sample galaxies. We find that the
+magnetic field values may deviate from the equipartition values by ∼25%.
+3. We have estimated spatially resolved maps of SFRSDs of the galaxies in Sample 1 using FUV+24µm, Hα+24µm,
+and 1.4 GHz data. Azimuthally averaged SFRSDs drop by a factor of 6 to 8 at the edge of the galaxies, where
+SFRSD values are 5 times the rms of the maps.
+4. We also included five additional galaxies: NGC 1097, NGC 4736, NGC 5055, NGC 5236, and NGC 6946 from
+previous GMRT observations of Basu et al. (2012a) and estimated their equipartition magnetic field, SFRSD
+and gas density maps.
+5. We studied the spatial correlation between magnetic fields and star formation rates at < 1 kpc resolution for
+the 12 galaxies (Sample 2) and find that magnetic field strengths and SFRSDs are correlated with an average
+power-law index of 0.31±0.06. This result is in remarkable agreement (at < 1σ error) with semi-analytical model
+predictions of B ∝ Σ1/3
+SFR (Schleicher & Beck 2013, 2016).
+
+20
+Manna and Roy
+6. We measure an average intercept of ≈ 65 µG from the B-ΣSFR correlations of our galaxies in Sample 2. This is
+higher than the predictions of Schleicher & Beck (2013) by a factor of ≈ 2.5, and, if confirmed with a larger sample,
+would imply a significantly higher efficiency of magnetic field amplification than what is typically assumed.
+7. We used spatially resolved gas density maps for seven (Sample 3) of the 12 galaxies, for which archival CO
+data was available, to find that magnetic fields are correlated with gas densities as B ∝ ρ0.40±0.09
+gas
+. This result
+is consistent with numerical simulations that predict k ≈ 0.4−0.6 and broadly consistent (within ≈1 sigma
+uncertainty) with theories that predict B ∝ ρ0.5
+gas.
+8. We have indirectly estimated the power-law index (k) of the correlation between the magnetic fields and the
+gas densities using the slope of the radio-FIR correlation, the slope of the Kennicutt-Schmidt law, and the non-
+thermal spectral index. The mean value of k, for optically thin dust, was found to be 0.52±0.04 and 0.47±0.09
+at 1.4 and 0.33 GHz respectively for the six galaxies in Sample 1, with NGC 4826 excluded due to its high value
+of k. This is consistent with the equipartition between magnetic fields and gas. The anomalously high values
+of k (1.0 and 0.95 at 1.4 and 0.33 GHz respectively) for NGC 4826 are possibly due to the contribution of the
+central AGN to the radio emission.
+We have started to follow up these pilot study results with a survey of a much larger sample of galaxies (Sample
+0, Table 2). For this, we have already observed another 24 galaxies using the upgraded GMRT (uGMRT), a Square
+Kilometer Array (SKA) pathfinder facility. Sensitivities of the images from these uGMRT observations are significantly
+better (≈ 3 times) than those of the observations presented here and the result will be part of a future publication.
+In addition, SKA precursors such as the MeerKAT will also provide very deep images of the diffuse radio-continuum
+emission around nearby galaxies. Eventually, the dramatic increase in sensitivity and ∼arc-sec resolution of the SKA
+has the potential to significantly advance our understanding of magnetic fields in nearby galaxies. For example, the
+SKA is expected to provide sensitive images of polarised synchrotron emission from nearby galaxies at a few GHz
+frequencies which would provide information on the large-scale ordered fields on the plane of the sky (e.g. Johnston-
+Hollitt et al. 2015). Further, polarised emission from nearby galaxies at <∼1 GHz, where significant depolarisations
+take place, could be modelled through Faraday tomography (e.g. Heald et al. 2015).
+A combination of the two
+approaches could eventually allow us to infer the three-dimensional structure of the magnetic fields in nearby galaxies.
+SKA observations will also provide detailed images of star formation with resolutions of tens of parsecs. These will
+help to identify any dependence of SFR and IMF on galaxy type, evolution and environment within the local volume
+(Beswick et al. 2015).
+We would like to thank Aditya Chowdhury for his help at various stages of this research. We thank Yogesh Wadadekar,
+Preeti Kharb, and Dipanjan Mitra for reading the manuscript and providing useful comments. Aritra Basu provided
+their earlier published images and also suggested checking the B vs SFRSD relation for our sample galaxies. We
+thank him for the above. We also thank the anonymous referee whose comments helped significantly improve the
+presentation of the paper. We thank the staff of GMRT that allowed these observations to be made. GMRT is run by
+National Centre for Radio Astrophysics of the Tata Institute of fundamental research. We acknowledge the support
+of the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.02-0700.
+1
+2
+3
+4
+5
+6
+7
+REFERENCES
+Adebahr, B., Krause, M., Klein, U., et al. 2013, A&A, 555,
+A23, doi: 10.1051/0004-6361/201220226
+Bacchini, C., Fraternali, F., Iorio, G., et al. 2020, A&A,
+641, A70, doi: 10.1051/0004-6361/202038223
+Basu, A., Mitra, D., Wadadekar, Y., & Ishwara-Chandra,
+C. H. 2012a, MNRAS, 419, 1136,
+doi: 10.1111/j.1365-2966.2011.19764.x
+Basu, A., & Roy, S. 2013, MNRAS, 433, 1675,
+doi: 10.1093/mnras/stt845
+Basu, A., Roy, S., & Mitra, D. 2012b, ApJ, 756, 141,
+doi: 10.1088/0004-637X/756/2/141
+Basu, A., Roychowdhury, S., Heesen, V., et al. 2017,
+MNRAS, 471, 337, doi: 10.1093/mnras/stx1567
+Batchelor, G. K. 1950, Proceedings of the Royal Society of
+London Series A, 201, 405, doi: 10.1098/rspa.1950.0069
+Beck, R., Brandenburg, A., Moss, D., Shukurov, A., &
+Sokoloff, D. 1996, ARA&A, 34, 155,
+doi: 10.1146/annurev.astro.34.1.155
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+21
+Beck, R., & Krause, M. 2005, Astronomische Nachrichten,
+326, 414, doi: 10.1002/asna.200510366
+Bell, E. F. 2003, ApJ, 586, 794, doi: 10.1086/367829
+Beresnyak, A. 2012, PhRvL, 108, 035002,
+doi: 10.1103/PhysRevLett.108.035002
+Beswick, R., Brinks, E., Perez-Torres, M., et al. 2015, in
+Advancing Astrophysics with the Square Kilometre
+Array (AASKA14), 70. https://arxiv.org/abs/1412.5810
+Bianchi, L., Shiao, B., & Thilker, D. 2017, ApJS, 230, 24,
+doi: 10.3847/1538-4365/aa7053
+Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA&A,
+51, 207, doi: 10.1146/annurev-astro-082812-140944
+Bovino, S., Schleicher, D. R. G., & Schober, J. 2013, New
+Journal of Physics, 15, 013055,
+doi: 10.1088/1367-2630/15/1/013055
+Brandenburg, A., & Subramanian, K. 2005, PhR, 417, 1,
+doi: 10.1016/j.physrep.2005.06.005
+Buat, V. 1992, A&A, 264, 444
+Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113,
+doi: 10.1086/145731
+Cho, J., & Vishniac, E. T. 2000, ApJ, 539, 273,
+doi: 10.1086/309213
+Chy˙zy, K. T., Beck, R., Kohle, S., Klein, U., & Urbanik, M.
+2000, A&A, 355, 128.
+https://arxiv.org/abs/astro-ph/0001205
+Chy˙zy, K. T., & Buta, R. J. 2008, ApJL, 677, L17,
+doi: 10.1086/587958
+Chy˙zy, K. T., We˙zgowiec, M., Beck, R., & Bomans, D. J.
+2011, A&A, 529, A94, doi: 10.1051/0004-6361/201015393
+Clemens, M. S., Alexander, P., & Green, D. A. 1999,
+MNRAS, 307, 481, doi: 10.1046/j.1365-8711.1999.02594.x
+Cortese, L., Boselli, A., Franzetti, P., et al. 2008, MNRAS,
+386, 1157, doi: 10.1111/j.1365-2966.2008.13118.x
+Crutcher, R. M. 1999, in Science with the Atacama Large
+Millimeter Array (ALMA), 12
+da Silva, R. L., Fumagalli, M., & Krumholz, M. R. 2014,
+MNRAS, 444, 3275, doi: 10.1093/mnras/stu1688
+Dale, D. A., Cohen, S. A., Johnson, L. C., et al. 2009, ApJ,
+703, 517, doi: 10.1088/0004-637X/703/1/517
+Dumas, G., Schinnerer, E., Tabatabaei, F. S., et al. 2011,
+AJ, 141, 41, doi: 10.1088/0004-6256/141/2/41
+Elmegreen, B. G. 1981, ApJ, 243, 512, doi: 10.1086/158616
+Fiedler, R. A., & Mouschovias, T. C. 1993, ApJ, 415, 680,
+doi: 10.1086/173193
+Filho, M. E., S´anchez Almeida, J., Amor´ın, R., et al. 2016,
+ApJ, 820, 109, doi: 10.3847/0004-637X/820/2/109
+Fletcher, A., Beck, R., Shukurov, A., Berkhuijsen, E. M., &
+Horellou, C. 2011, MNRAS, 412, 2396,
+doi: 10.1111/j.1365-2966.2010.18065.x
+Groves, B. A., Cho, J., Dopita, M., & Lazarian, A. 2003,
+PASA, 20, 252, doi: 10.1071/AS03016
+Heald, G., Beck, R., de Blok, W. J. G., et al. 2015, in
+Advancing Astrophysics with the Square Kilometre Array
+(AASKA14), 106. https://arxiv.org/abs/1501.00408
+Heesen, V., Beck, R., Krause, M., & Dettmar, R. J. 2009,
+A&A, 494, 563, doi: 10.1051/0004-6361:200810543
+Heesen, V., Brinks, E., Leroy, A. K., et al. 2014, AJ, 147,
+103, doi: 10.1088/0004-6256/147/5/103
+Hummel, E. 1986, A&A, 160, L4
+Irwin, J., Beck, R., Benjamin, R. A., et al. 2012, AJ, 144,
+44, doi: 10.1088/0004-6256/144/2/44
+Johnston-Hollitt, M., Govoni, F., Beck, R., et al. 2015, in
+Advancing Astrophysics with the Square Kilometre Array
+(AASKA14), 92. https://arxiv.org/abs/1506.00808
+Jurusik, W., Drzazga, R. T., Jableka, M., et al. 2014, A&A,
+567, A134, doi: 10.1051/0004-6361/201323060
+Kazes, I., Troland, T. H., & Crutcher, R. M. 1991, A&A,
+245, L17
+Kennicutt, Robert C., J. 1998a, ApJ, 498, 541,
+doi: 10.1086/305588
+—. 1998b, ARA&A, 36, 189,
+doi: 10.1146/annurev.astro.36.1.189
+Kennicutt, Robert C., J., Lee, J. C., Funes, J. G., et al.
+2008, ApJS, 178, 247, doi: 10.1086/590058
+Kennicutt, Robert C., J., Armus, L., Bendo, G., et al. 2003,
+PASP, 115, 928, doi: 10.1086/376941
+Kennicutt, Robert C., J., Calzetti, D., Walter, F., et al.
+2007, ApJ, 671, 333, doi: 10.1086/522300
+Kennicutt, R. C., & Evans, N. J. 2012a, ARA&A, 50, 531,
+doi: 10.1146/annurev-astro-081811-125610
+—. 2012b, ARA&A, 50, 531,
+doi: 10.1146/annurev-astro-081811-125610
+Kierdorf, M., Mao, S. A., Beck, R., et al. 2020, A&A, 642,
+A118, doi: 10.1051/0004-6361/202037847
+Kim, J., Balsara, D., & Mac Low, M.-M. 2001, Journal of
+Korean Astronomical Society, 34, 333,
+doi: 10.5303/JKAS.2001.34.4.333
+Krause, M. 2009, in Revista Mexicana de Astronomia y
+Astrofisica Conference Series, Vol. 36, Revista Mexicana
+de Astronomia y Astrofisica Conference Series, 25–29.
+https://arxiv.org/abs/0806.2060
+Krause, M., Wielebinski, R., & Dumke, M. 2006, A&A,
+448, 133, doi: 10.1051/0004-6361:20053789
+Krause, M., Irwin, J., Schmidt, P., et al. 2020, A&A, 639,
+A112, doi: 10.1051/0004-6361/202037780
+Kroupa, P. 2001, MNRAS, 322, 231,
+doi: 10.1046/j.1365-8711.2001.04022.x
+
+22
+Manna and Roy
+Kulsrud, R. M., & Zweibel, E. G. 2008, Reports on
+Progress in Physics, 71, 046901,
+doi: 10.1088/0034-4885/71/4/046901
+Leroy, A. K., Walter, F., Bigiel, F., et al. 2009, AJ, 137,
+4670, doi: 10.1088/0004-6256/137/6/4670
+Leroy, A. K., Bigiel, F., de Blok, W. J. G., et al. 2012, AJ,
+144, 3, doi: 10.1088/0004-6256/144/1/3
+Longair, M. S. 2011, High Energy Astrophysics
+Malkan, M. A., Jensen, L. D., Rodriguez, D. R., Spinoglio,
+L., & Rush, B. 2017, ApJ, 846, 102,
+doi: 10.3847/1538-4357/aa8302
+Meurer, G. R., Heckman, T. M., & Calzetti, D. 1999, ApJ,
+521, 64, doi: 10.1086/307523
+Meurer, G. R., Heckman, T. M., Leitherer, C., et al. 1995,
+AJ, 110, 2665, doi: 10.1086/117721
+Miettinen, O., Delvecchio, I., Smolˇci´c, V., et al. 2017, A&A,
+602, L9, doi: 10.1051/0004-6361/201731157
+Miley, G. 1980, ARA&A, 18, 165,
+doi: 10.1146/annurev.aa.18.090180.001121
+Moss, D., & Shukurov, A. 1996, MNRAS, 279, 229,
+doi: 10.1093/mnras/279.1.229
+Murphy, E. J., Helou, G., Kenney, J. D. P., Armus, L., &
+Braun, R. 2008, ApJ, 678, 828, doi: 10.1086/587123
+Murphy, E. J., Braun, R., Helou, G., et al. 2006a, ApJ,
+638, 157, doi: 10.1086/498636
+Murphy, E. J., Helou, G., Braun, R., et al. 2006b, ApJL,
+651, L111, doi: 10.1086/509722
+Murphy, E. J., Condon, J. J., Schinnerer, E., et al. 2011,
+ApJ, 737, 67, doi: 10.1088/0004-637X/737/2/67
+Nikiel-Wroczy´nski, B., Jamrozy, M., Soida, M., Urbanik,
+M., & Knapik, J. 2016, MNRAS, 459, 683,
+doi: 10.1093/mnras/stw414
+Niklas, S., & Beck, R. 1997, A&A, 320, 54
+Onodera, S., Kuno, N., Tosaki, T., et al. 2010, ApJL, 722,
+L127, doi: 10.1088/2041-8205/722/2/L127
+Pacholczyk, A. G. 1970, Radio astrophysics. Nonthermal
+processes in galactic and extragalactic sources
+Price, D. J., & Bate, M. R. 2008, MNRAS, 385, 1820,
+doi: 10.1111/j.1365-2966.2008.12976.x
+Regan, M. W., Thornley, M. D., Helfer, T. T., et al. 2001,
+ApJ, 561, 218, doi: 10.1086/323221
+Robishaw, T., Quataert, E., & Heiles, C. 2008, ApJ, 680,
+981, doi: 10.1086/588031
+Roy, S., & Manna, S. 2021, MNRAS, 507, 4734,
+doi: 10.1093/mnras/stab2441
+Roychowdhury, S., Huang, M.-L., Kauffmann, G., Wang,
+J., & Chengalur, J. N. 2015, MNRAS, 449, 3700,
+doi: 10.1093/mnras/stv515
+Sarazin, C. L. 1999, ApJ, 520, 529, doi: 10.1086/307501
+Sarma, A. P., Momjian, E., Troland, T. H., & Crutcher,
+R. M. 2005, AJ, 130, 2566, doi: 10.1086/497637
+Schleicher, D. R. G., & Beck, R. 2013, A&A, 556, A142,
+doi: 10.1051/0004-6361/201321707
+—. 2016, A&A, 593, A77,
+doi: 10.1051/0004-6361/201628843
+Schober, J., Schleicher, D., Federrath, C., et al. 2012, ApJ,
+754, 99, doi: 10.1088/0004-637X/754/2/99
+Shukurov, A., Sokoloff, D., Subramanian, K., & Brand
+enburg, A. 2006, A&A, 448, L33,
+doi: 10.1051/0004-6361:200600011
+Soida, M., Urbanik, M., Beck, R., Wielebinski, R., &
+Balkowski, C. 2001, A&A, 378, 40,
+doi: 10.1051/0004-6361:20011185
+Sokoloff, D. D., Bykov, A. A., Shukurov, A., et al. 1998,
+MNRAS, 299, 189, doi: 10.1046/j.1365-8711.1998.01782.x
+Tabatabaei, F. S., Berkhuijsen, E. M., Frick, P., Beck, R.,
+& Schinnerer, E. 2013, A&A, 557, A129,
+doi: 10.1051/0004-6361/201218909
+Thompson, T. A., Quataert, E., Waxman, E., Murray, N.,
+& Martin, C. L. 2006, ApJ, 645, 186, doi: 10.1086/504035
+Van Loo, S., Tan, J. C., & Falle, S. A. E. G. 2015, ApJL,
+800, L11, doi: 10.1088/2041-8205/800/1/L11
+Vargas, C. J., Mora-Partiarroyo, S. C., Schmidt, P., et al.
+2018, ApJ, 853, 128, doi: 10.3847/1538-4357/aaa47f
+Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020,
+Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
+Walter, F., Brinks, E., de Blok, W. J. G., et al. 2008, AJ,
+136, 2563, doi: 10.1088/0004-6256/136/6/2563
+Watanabe, Y., Sorai, K., Kuno, N., & Habe, A. 2011,
+MNRAS, 411, 1409,
+doi: 10.1111/j.1365-2966.2010.17746.x
+Wiegert, T., Irwin, J., Miskolczi, A., et al. 2015, AJ, 150,
+81, doi: 10.1088/0004-6256/150/3/81
+Yun, M. S., Reddy, N. A., & Condon, J. J. 2001, ApJ, 554,
+803, doi: 10.1086/323145
+
+A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+23
+200
+0
+200
+Relative RA (arcseconds)
+300
+200
+100
+0
+100
+200
+300
+Relative DEC (arcseconds)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Error on B ( G)
+200
+100
+0
+100
+200
+Relative RA (arcseconds)
+200
+100
+0
+100
+200
+Relative DEC (arcseconds)
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Error on B ( G)
+400
+200
+0
+200
+400
+Relative RA (arcseconds)
+400
+200
+0
+200
+400
+Relative DEC (arcseconds)
+0.2
+0.4
+0.6
+0.8
+Error on B ( G)
+400
+200
+0
+200
+400
+Relative RA (arcseconds)
+400
+200
+0
+200
+400
+Relative DEC (arcseconds)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Error on B ( G)
+200
+0
+200
+Relative RA (arcseconds)
+200
+0
+200
+Relative DEC (arcseconds)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Error on B ( G)
+200
+0
+200
+Relative RA (arcseconds)
+300
+200
+100
+0
+100
+200
+300
+Relative DEC (arcseconds)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Error on B ( G)
+250
+0
+250
+Relative RA (arcseconds)
+400
+200
+0
+200
+400
+Relative DEC (arcseconds)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Error on B ( G)
+Figure 7. The magnetic field uncertainty maps (in µG) of NGC 2683 (top left), NGC 3627 (top centre), NGC 4096 (top right),
+NGC 4449 (middle left), NGC 4490 (middle centre), NGC 4826 (middle right) and NGC 5194 (bottom) (Sample 1), shown in
+colour scale. Blanked regions (in white colour) in the centre of each galaxy correspond to regions with spectral index values ≤
+0.55.
+APPENDIX
+A. MAGNETIC FIELD UNCERTAINTY MAPS
+We present here (Figure 7) magnetic field uncertainty maps of the galaxies in Sample 1, generated using the procedure
+described in Section 2.1.1.
+B. STAR FORMATION RATE SURFACE DENSITY MAPS
+We show SFRSD maps of the seven galaxies (Sample 1) in Figures 8 and 9, where SFRSDs estimated using 1.4 GHz
+and FUV+24µm emission are shown in contours and colors, respectively. In Figures 10 and 11, we have also shown
+the SFRSD maps estimated using Hα+24µm and 1.4GHz data in colors and contours, respectively. The SFRSD maps
+of each galaxy in Figures 8, 9, 10, and 11 have been shown in the same color scale and contours.
+
+24
+Manna and Roy
+Figure 8. SFRSD (M⊙yr−1kpc−2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096(clockwise from top left) (Sample
+1). SFRSDs estimated using 1.4 GHz radio and FUV+24µm emission are shown in contours and colors, respectively. Contour
+levels are listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular resolution
+of the maps.
+
+COLOR:NGC26832683sfrhyb.OHGSPX.1
+CONT:NGC2683IPOL1490.572MHz2683.sfr.L.TH.SUB.1
+0
+10
+20
+30
+33 29
+28
+27
+Declination (J2000)
+26
+25
+24
+23
+22
+21
+085300
+5255
+50
+45
+40
+35
+30
+25
+20
+Right Ascension (J2000)
+Colorscale ranqe=-0.1039.85MilliSolarmass/yr/kpc^2
+Contpeakflux=5.4958E-02Solarmass/yr/kpc^2
+Levs = 2.327E-03 * (-1,1,2, 4,8, 16,25, 50,
+80,100,130,160)COLOR:NGC36273627sfrhyb.OHGSPX.1
+CONT:N3627LIPOL1430.389MHz3627.sfr.L.TH.SUB.1
+200
+400
+600
+13 02
+01
+00
+Declination (J2000)
+1259
+58
+57
+56
+11 20 25
+20
+15
+10
+05
+RightAscension(J2000)
+Colorscalerange=-0.0604.9MilliSolarmass/yr/kpc^2
+Contpeakflux=2.4733E-01Solarmass/yr/kpc^2
+Levs = 2.000E-02 * (-1, 1,2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)COLOR:NGC40964096sfrhyb.OHGSPX.1
+CONT:NGC4096IPOL1432.873MHz4096.sfr.L.TH.SUB.1
+0
+10
+20
+30
+40
+4731
+30
+Declination (J2000)
+29
+28
+0
+27
+26
+25
+120620
+15
+10
+05
+00
+0555
+50
+45
+40
+Right Ascension (J2000)
+Colorscale range=-0.1546.02MilliSolarmass/yr/kpc^2
+Contpeakflux=3.1588E-02Solarmass/yr/kpc^2
+Levs = 1.511E-03 * (-1, 1, 2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)COLOR:NGC44494449sfrFUV.OHGSPX.1
+CONT:N49IPOL1489.984MHz4449sfrL.TH.SUB.2
+100
+200
+4412
+10
+08
+Declination (J2000)
+06
+04
+02
+00
+12 28 45
+30
+15
+00
+27 45
+RightAscension(J2000)
+Colorscalerange=-0.1251.0MilliSolarmass/yr/kpc^2
+Contpeakflux=7.4433E+00Solarmass/yr/kpc^2
+Levs = 5.421E-03 * (-1, 1,2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+25
+Figure 9. SFRSD (M⊙yr−1kpc−2) maps of NGC 4490, NGC 4826 and NGC 5194 (clockwise from top left) (Sample 1). SFRSDs
+estimated using 1.4 GHz radio and FUV+24µm emission are shown in contours and colors, respectively. Contour levels are
+listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular resolution of the
+maps.
+
+COLOR:NGC44904490sfrFUV.OHGSPX.1
+CONT:N4490IPOL1435.114MHz4490sfrL.TH.SUB.2
+0
+50
+100
+150
+200
+41 44
+42
+Declination (J2000)
+40
+38
+36
+34
+0
+12 31 00
+30 45
+30
+15
+RightAscension(J2000)
+Colorscale range=-0.1204.4MilliSolarmass/yr/kpc^2
+Contpeakflux=2.9951E-01Solarmass/yr/kpc^2
+Levs = 7.000E-03 * (-1, 1, 2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)COLOR:NGC48264826sfrhyb.OHGSPX.1
+CONT:N4826LIPOL1425.677MHz4826.sfr.L.TH.SUB.1
+0
+100
+200
+300
+214300
+42 30
+00
+Declination (J2000)
+41 30
+00
+40 30
+00
+39 30
+00
+12 56 50
+45
+40
+35
+RightAscension (J2000)
+Color scalerange=-0.1354.2Milli Solarmass/yr/kpc^2
+Contpeakflux=3.4554E-01Solarmass/yr/kpc^2
+Levs=3.758E-03 *(-1, 1,2, 4,8, 16,25, 50,
+80,100,150,200,400,800)COLOR:NGC51945194sfrhyb.OHGSPX.1
+CONT: M51 iPOL 1664.900 MHz 5194.sfr.L.TH.SUB.1
+100
+200
+4718
+16
+14
+Declination (J2000)
+12
+10
+08
+06
+13 30 15
+00
+29 45
+30
+RightAscension (J2000)
+Color scale range= -0.0293.2'Milli Solar mass/yr/kpc2
+Contpeakflux=3.1707E-01Solarmass/yr/kpc^2
+Levs = 1.215E-03 * (-1, 1, 2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)26
+Manna and Roy
+Figure 10.
+SFRSD (M⊙yr−1kpc−2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096 (clockwise from top left)
+(Sample 1). SFRSDs estimated using 1.4 GHz radio and Hα+24µm emission are shown in contours and colors, respectively.
+Contour levels are listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular
+resolution of the maps.
+
+COLOR:N2683652683sfrha.OHGEO.1
+CONT:NGC2683 IPOL 1490.572MHz 2683 sfr L.TH.SUB.1
+0
+10
+20
+30
+3329
+28
+27
+Declination (J2000)
+26
+25
+24
+23
+22
+085300
+52 55
+50
+45
+40
+35
+30
+25
+RightAscension(J2000)
+Colorscalerange=-0.1039.85MilliSolarmass/yr/kpc^2
+Contpeakflux=5.4958E-02Solarmass/yr/kpc^2
+Levs=2.327E-03 *(-1,1,2, 4,8,16,25,50,
+80,100,130,160)COLOR:nqc36273627sfrha.OHGEO.1
+CONT:N3627LIPOL1430.389MHz3627sfrL.TH.SUB.1
+200
+400
+600
+13 02
+01
+Declination (J2000)
+00
+1259
+58
+57
+56
+11 20 25
+20
+15
+10
+05
+RightAscension(J2000)
+Colorscalerange=0.0604.9MilliSolarmass/yr/kpc^2
+Contpeak flux= 2.4733E-01Solarmass/yr/kpc^2
+Levs = 2.000E-02 *(-1, 1,2,4, 16,25, 50, 80,
+100,150,200,400,800)COLOR:N4096654096sfrha.OHGEO.1
+CONT:NGC4096IPOL1432.873MHz4096sfrL.TH.SUB.1
+0
+10
+20
+30
+40
+47 32
+31
+30
+Declination (J2000)
+29
+28
+27
+26
+25
+12.0620
+15
+10
+05
+0555
+50
+45
+40
+RightAscension(J2000)
+Colorscalerange=-0.1546.02MilliSolarmass/yr/kpc^2
+Contpeak flux=3.1588E-02Solarmass/yr/kpc^2
+Levs = 1.511E-03 *(-1, 1,2, 4,8, 16, 25, 50,
+80,100,150,200,400,800)COLOR:N4449654449sfrha.OHGEO.1
+CONT:N49IPOL1489.984MHz4449sfrL.TH.SUB.2
+100
+200
+4412
+10
+08
+?
+Declination (J2000)
+06
+04
+02
+00
+12 28 30
+15
+00
+27 45
+RightAscension(J2000)
+Colorscalerange=-0.1251.0MilliSolarmass/yr/kpc^2
+Contpeakflux=7.4433E+00Solarmass/yr/kpc^2
+Levs = 5.421E-03 * (-1, 1, 2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities
+27
+Figure 11. SFRSD (M⊙yr−1kpc−2) maps of NGC 4490, NGC 4826 and NGC 5194 (clockwise from top left) (Sample 1).
+SFRSDs estimated using 1.4 GHz radio and Hα+24µm emission are shown in contours and colors, respectively. Contour levels
+are listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular resolution of
+the maps.
+
+COLOR:N4485/904490sfrha.OHGEO.1
+CONT:N4490IPOL1435.114MHz4490sfrL.TH.SUB.2
+50
+100
+150
+200
+41 44
+42
+Declination (J2000)
+40
+38
+36
+34
+0
+12 31 00
+30 45
+30
+15
+RightAscension(J2000)
+Colorscale range=-0.1204.4MilliSolarmass/yr/kpc^2
+Contpeakflux=2.9951E-01Solarmass/yr/kpc^2
+Levs=7.000E-03 *(-1,1,2,4,8,16,25,50,
+80,100,150,200,400,800)COLOR:N4826654826sfrha.OHGEO.1
+CONT:N4826LIPOL1425.677MHz4826sfrL.TH.SUB.1
+100
+200
+300
+21 43 30
+00
+42 30
+Declination (J2000)
+00
+41 30
+00
+40 30
+00
+39 30
+00
+38 30
+125655
+50
+45
+40
+35
+30
+RightAscension(J2000)
+Colorscalerange=-0.1354.2MilliSolarmass/yr/kpc^2
+Contpeakflux=3.4554E-01Solarmass/yr/kpc^2
+Levs = 3.758E-03 * (-1, 1, 2, 4, 8, 16, 25, 50,
+80,100,150,200,400,800)COLOR:NGC5194+5194sfrha.OHGEO.1
+CONT:M51IPOL1664.900MHz5194sfrL.TH.SUB.1
+100
+200
+4718
+16
+D
+14
+Declination (J2000)
+12
+10
+08
+06
+Q
+13 30 30
+15
+00
+29 45
+30
+15
+Right Ascension (J2000)
+Color scale range=0.0293.2Mili Solar mass/yr/kpc^2
+Contpeakflux=3.1707E-01Solarmass/yr/kpc^2
+Levs=1.215E-03 *(-1,1,2, 4,8, 16,25,50,
+80,100,150,200,400,800)

59FKT4oBgHgl3EQfTC3B/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f106d8145848555566972b2dc548a78eb30d6c9a90996e1a95cc410f0a9eb4b2
+size 3866669

5NFKT4oBgHgl3EQfSS3g/content/tmp_files/2301.11775v1.pdf.txt

@@ -0,0 +1,2988 @@
+arXiv:2301.11775v1  [math.AP]  11 Jan 2023
+REGULARITY IN THE TWO-PHASE BERNOULLI PROBLEM FOR THE
+p-LAPLACE OPERATOR
+MASOUD BAYRAMI AND MORTEZA FOTOUHI
+Abstract. We show that any minimizer of the well-known ACF functional (for the
+p-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness
+decay at the two-phase free boundary points to improve the flatness, that boils
+down to C1,η regularity of the flat part of the free boundary. This result, in turn, is
+used to prove the Lipschitz regularity of minimizers by a dichotomy argument.
+1. Introduction and main result
+We study the problem of minimizing the following two-phase functional
+JTP(v, D) :=
+�
+D
+|∇v|p + (p − 1)λp
++χ{v>0} + (p − 1)λp
+−χ{v<0} dx,
+v ∈ K,
+where D is a bounded and smooth domain in Rn, χA is the characteristic function
+of the set A, 1 < p < ∞, and λ± > 0 are given constants. The class of admissible
+functions K, consists of all functions v ∈ g + W1,p
+0 (D), where g ∈ W1,p(D).
+Any minimizer u satisfies, in a certain weak sense, the following system of
+equations
+(1)
+
+∆pu := div(|∇u|p−2∇u) = 0,
+in
+Ω+
+u ∪ Ω−
+u,
+|∇u+|p − |∇u−|p = λp
++ − λp
+−,
+on
+�∂Ω+
+u ∩ ∂Ω−
+u
+� ∩ D,
+|∇u+| ≥ λ+, |∇u−| ≥ λ−,
+on
+�∂Ω+
+u ∩ ∂Ω−
+u
+� ∩ D,
+|∇u+| = λ+,
+on
+�∂Ω+
+u \ ∂Ω−
+u
+� ∩ D,
+|∇u−| = λ−,
+on
+�∂Ω−
+u \ ∂Ω+
+u
+� ∩ D,
+where Ω±
+u = {x ∈ D : ±u(x) > 0}, u± := max{±u, 0}, and ∆pu = div(|∇u|p−2∇u) is the
+p-Laplace operator; see Lemma 3.1.
+These types of problems are known as Bernoulli-type free boundary problems
+which appear in various models of fluid mechanics or heat conduction (see e.g.
+[2, 4, 5, 7, 21, 18]). For the admissible functions in K+ := {v ∈ K : v ≥ 0}, the
+analogous one-phase functional and the corresponding overdetermined problem
+called the one-phase Bernoulli problem, was first studied in [1] for the case p = 2,
+and then in [6] for the two-phase problem. Also, the case of uniformly elliptic
+quasilinear equations in the one-phase case has been treated in [3]. The difficulty
+of the problem (1) is that the governing operator, ∆pu = div(|∇u|p−2∇u), is not
+Date: January 30, 2023.
+1991 Mathematics Subject Classification. 35R35, 35J92.
+Key words and phrases. Free boundary regularity, Two-phase Bernoulli problem, p-Laplacian.
+M. Bayrami and M. Fotouhi was supported by Iran National Science Foundation (INSF) under
+project No. 4001885.
+1
+
+2
+M. BAYRAMI AND M. FOTOUHI
+uniformly elliptic. Obviously, close to regular free boundary points one expects
+that |∇u| > 0 implying uniform ellipticity of the p-Laplacian. However, without
+such a regularity assumption, it is difficult to prove non-degeneracy up to the free
+boundary. In [10], the authors circumvent this issue by simultaneously showing
+the non-degeneracy of the gradient and the regularity of the free boundary.
+Here below we list terminologies and definitions that are frequently used in this
+paper:
+• A function u : D → R is said to be a minimizer of JTP in D if and only if
+JTP(u, D) ≤ JTP(v, D),
+for all v ∈ K.
+• F(u) := �∂Ω+
+u ∪ ∂Ω−
+u
+� ∩ D, denotes the free boundary of the minimizer u.
+• The set ΓTP := ∂Ω+
+u ∩ ∂Ω−
+u ∩ D is the two-phase points of the free boundary
+F(u).
+• The boundary of positive and negative phases, i.e. ∂Ω±
+u ∩ D can be decom-
+posed as
+∂Ω±
+u ∩ D = Γ±
+OP ∪ ΓTP,
+where Γ+
+OP := �∂Ω+
+u \ ∂Ω−
+u
+�∩D and Γ−
+OP := �∂Ω−
+u \ ∂Ω+
+u
+�∩D are the one-phase
+parts of F(u).
+• We will say that x0 ∈ ΓTP is an interior two-phase point and will denote it
+by x0 ∈ Γint
+TP, if
+|Br(x0) ∩ {u = 0}| = 0,
+for some
+r > 0.
+• We will say that x0 ∈ ΓTP is a branching point and will denote it by x0 ∈ Γbr
+TP,
+if
+|Br(x0) ∩ {u = 0}| > 0,
+for every
+r > 0.
+• We denote by Hα,e the following one-dimensional function
+Hα,e(x) = α (x · e)+ − β (x · e)− ,
+with a unit vector e ∈ Sn−1 and the constants α and β satisfying the condi-
+tions
+(2)
+α ≥ λ+,
+β ≥ λ−,
+αp − βp = λp
++ − λp
+−.
+Hα,e is the so-called two-plane solution to (1).
+Our goal is to study the regularity of the free boundary F(u) = �∂Ω+
+u ∪ ∂Ω−
+u
+�∩D,
+for minimizers of JTP in D, around the two-phase points. More precisely, we prove
+that in a suitable neighborhood of the two-phase points, the sets Ω+
+u and Ω−
+u are
+two C1,η-regular domains touching along the closed set of two-phase points ΓTP.
+For the special case p = 2, this result has been recently obtained in [12], by invoking
+the linearization technique and we will closely follow this technique in order to
+generalize this result to any 1 < p < ∞.
+As is usual for problems of this type, prior to applying any method to determine
+the regularity of the free boundary, the Lipschitz continuity of the minimizers
+across the free boundary is required. Our partial result for the regularity of the
+free boundary, however, gives us the Lipschitz regularity of the solution as well.
+We first show C1,η-regularity of the free boundary with a flatness assumption in
+the following theorem.
+
+3
+Theorem 1.1 (Flatness implies C1,η). Let u : D → R be a minimizer of JTP in D. For
+any positive constants Λ0 and Λ1, there exists a constant ¯ǫ = ¯ǫ(n, p, Λ0, Λ1) such that if
+(3)
+∥u − Hα,e∥L∞(B1) ≤ ¯ǫ,
+for some e ∈ Sn−1 and max(Λ0, λ+) ≤ α ≤ Λ1, then ∂Ω±
+u ∩ Br0 are C1,η graphs for some
+r0 > 0 and for any η ∈ (0, 1
+3).
+We need to remark that the critical flatness to obtain the regularity does not
+depend on λ±. Indeed, as long as we are close enough to a two-plane solution with
+coefficient α ∈ [Λ0, Λ1], we obtain the regularity of the free boundary. This result
+in turn is crucial to obtain the Lipschitz regularity of minimizers in the following
+theorem.
+Theorem 1.2 (Lipschitz regularity). Let u : D → R be a minimizer of JTP in D. Then
+u is locally Lipschitz continuous; u ∈ C0,1
+loc(D).
+2. Basic properties of minimizers
+In this section, we gather some basic properties of minimizers of JTP.
+Theorem 2.1 (Existence). If the admissible set K is nonempty, then there exists a mini-
+mizer u of JTP over K. Moreover, every minimizer satisfies
+
+∆pu = 0,
+in
+Ω+
+u ∪ Ω−
+u,
+∆pu± ≥ 0,
+in
+D,
+∥u∥L∞(D) ≤ ∥g∥L∞(D).
+Proof. The existence of a bounded minimizer u of the functional JTP can be easily
+established using the semi-continuity of the p-Dirichlet energy and the weak con-
+vergence in W1,p, and can be obtained in the standard way. See e.g. [6, 23] for
+the details. Also, notice that by comparison of u and u + tϕ, where ϕ is a suitable
+smooth that supp ϕ ⊂ Ω+
+u ∪ Ω−
+u, it is easy to find that ∆pu = 0 in Ω+
+u ∪ Ω−
+u in the
+sense of distributions.
+To prove that u+ are p-subharmonic, we first note that since ∆pu = 0 in Ω+
+u, we
+may choose ǫk → 0 such that {u = ǫk} to be a C1 manifold by the Sard’s Theorem,
+resulting in −∇u
+|∇u| to be the outer normal vector on ∂{u > ǫk}. Now take 0 ≤ ϕ ∈ C∞
+c (D),
+the integration by parts implies that
+�
+D
+|∇u+|p−2∇u+ · ∇ϕ dx =
+�
+{u>0}
+|∇u+|p−2∇u+ · ∇ϕ dx
+= lim
+ǫk→0
+�
+{u>ǫk}
+|∇u+|p−2∇u+ · ∇ϕ dx
+= lim
+ǫk→0
+�
+{u=ǫk}
+|∇u+|p−2
+�
+∇u+ · −∇u
+|∇u|
+�
+ϕ dx
+−
+�
+{u>ǫk}
+∆pu ϕ dx
+= − lim
+ǫk→0
+�
+{u=ǫk}
+|∇u+|p−1ϕ dx ≤ 0.
+The proof of ∆pu− ≥ 0 is the same. Finally, the last estimate
+∥u∥L∞(D) ≤ ∥g∥L∞(D)
+
+4
+M. BAYRAMI AND M. FOTOUHI
+is the consequence of the p-subharmonicity of u± in D.
+□
+In the following proposition we show the non-degeneracy property for the
+minimizers. It reveals the fact that each of the two phases Ω+
+u and Ω−
+u are optimal
+with respect to one-sided inwards perturbations. The proof is the same as the proof
+of non-degeneracy for one-phase problems; see [10, Lemma 4.2]. We postpone the
+proof to Appendix C.
+Proposition 2.2 (Non-degeneracy). Let D ⊂ Rn be an open set, and u be a minimizer
+of JTP. Then, u is non-degenerate; i.e. there is a constant C = C(n, λ±, p) > 0 such that
+⧸
+�
+Br(x0)
+�u±�p dx ≥ Crp,
+for every x0 ∈ Ω±
+u ∩ D and every 0 < r < dist (x0, ∂D).
+The next proposition concerns the Lipschitz regularity of the minimizers around
+the one-phase points.
+Proposition 2.3 (Lipschitz regularity at one-phase points). Let u : D → R be a
+minimizer of JTP in D. There there is constant C = C(n, p, ±λ) such that if x0 ∈ Γ+
+OP (or
+x0 ∈ Γ−
+OP) is one-phase point and Br(x0) ∩ Ω−
+u = ∅ (Br(x0) ∩ Ω+
+u = ∅), then
+∥∇u∥L∞(B r
+2 (x0)) ≤ C.
+We remark that the condition Br(x0) ∩ Ω−
+u = ∅ always holds for some r > 0 by
+the definition of one-phase points.
+Proof. We know that ux0,r(x) = u(x0+rx)
+r
+is a minimizer of the following one phase
+functional in B1, i.e. minimizer of
+JOP(v, B1) :=
+�
+B1
+|∇v|p dx + (p − 1)λp
++|{v > 0} ∩ B1|,
+over the class of nonnegative functions. Then the boundedness of the gradient
+∥∇u∥L∞(B r
+2 (x0)) = ∥∇ux0,r∥L∞(B 1
+2 ) ≤ C(n, p, λ+),
+follows from [10, Theorem 3.3]. We shall remark that the Lipschitz constant for
+one-phase problems does not depend on the boundary values of the minimizer as
+long as we stay uniformly far from the boundary.
+□
+Next, we mention the following continuity result for minimizers.
+Proposition 2.4 (BMO estimates for the gradient). Let u be a minimizer of JTP and
+D′ ⋐ D. Then,
+(i) for 1 < p < 2, we have that |∇u|
+p−2
+2 ∇u ∈ BMO(D′), and consequently u ∈ Cσ(D′)
+for any σ ∈ (0, 1);
+(ii) for 2 < p < ∞, we have that ∇u ∈ BMO(D′) and thus u is locally log-Lipschitz
+continuous.
+In particular, ∇u ∈ Lq(D′) for any 1 < q < ∞ and for any 1 < p < ∞.
+Proof. The proof is the same as the proof of Lemma 3.1 in [16].
+□
+
+5
+The BMO estimate for the gradient of minimizers is sufficient to obtain the fol-
+lowing compactness result. Since we have not yet proved the Lipschitz continuity,
+this result will be extremely valuable for our argument in the next section. We
+postpone the proof to Appendix A.
+Proposition 2.5. Let uj be a bounded minimizer of JTP in B2 with the points xj ∈ B1 such
+that uj(xj) = 0. Also, set vj(x) =
+uj(xj+rjx)
+Sj
+, for any x ∈ BR, with 0 < R < 1
+rj , where rj → 0,
+as j → +∞ and Sj > 0. Then, vj is the minimizer (according to its own boundary values)
+of the following scaled functional
+(4)
+ˆJTP(v) :=
+�
+BR
+|∇v|p + (p − 1)σp
+jλp
++χ{v>0} + (p − 1)σp
+jλp
+−χ{v<0} dx,
+where σj :=
+rj
+Sj . Moreover, if |vj| ≤ M in BR, for any fixed 0 < R <
+1
+rj and for some
+M = M(R) > 0, then up to a subsequence, the followings hold:
+(i) For any q > 1, and some α ∈ (0, 1) (if q > n, one can take α = 1 − n
+q), vj converges
+to some function v0 as j → +∞ in Cα(BR) and weakly in W1,q(BR);
+(ii) vj → v0 strongly in W1,p(BR);
+(iii) If moreover, σj :=
+rj
+Sj → σ, as j → +∞, then v0 is a minimizer of
+ˆJTP(v) :=
+�
+BR
+|∇v|p + (p − 1)σpλp
++χ{v>0} + (p − 1)σpλp
+−χ{v<0} dx.
+In particular, if σ = 0, then v0 is p-harmonic in BR.
+The following lemma states that u+ and u− have coherent growth at two-phase
+points. This is essential to show that the minimizers are the viscosity solution of
+(1).
+Lemma 2.6. Let u be a bounded minimizer of JTP. Let x0 ∈ ΓTP and r0 > 0 be small
+such that Br0(x0) ⊂ D. Assume that supBr(x0) u− ≤ C0r (resp. supBr(x0) u+ ≤ C0r) for
+all r ∈ (0, r0]. Then there exist constant C1 > 0 such that supBr(x0) u+ ≤ C1r (resp.
+supBr(x0)u− ≤ C1r) for all r ∈ (0, r0].
+Proof. We will just demonstrate one of the claims; the other can be demonstrated
+similarly. By the assumption of the lemma
+(5)
+sup
+Br(x0)
+u− ≤ C0r,
+∀r ∈ (0, r0].
+We claim that there is ˜C1 > 0 such that
+(6)
+S(k + 1) ≤ max
+� ˜C1
+2k+1 , 1
+2S(k)
+�
+,
+where S(k) := ∥u∥L∞(B2−k(x0)), for any k ∈ N that 2−k ≤ r0. To prove this, we argue
+by contradiction and suppose that (6) fails. Then there is a sequence of integers kj,
+with j = 1, 2, · · · such that
+(7)
+S(kj + 1) > max
+�
+j
+2kj+1 , 1
+2S(kj)
+�
+.
+
+6
+M. BAYRAMI AND M. FOTOUHI
+Observe that since u is a bounded minimizer, then (7) implies that kj → +∞ as
+j → +∞. Also, notice that (7) implies that
+(8)
+σj :=
+2−kj
+S(kj + 1) ≤ 2
+j → 0
+as
+j → +∞.
+Now, we introduce the scaled functions vj(x) := u(x0+2−kjx)
+S(kj+1) , for any x ∈ B1. Then,
+from (5) and (8), it follows that vj(0) = 0 and
+(9)
+v−
+j (x) = u−(x0 + 2−kjx)
+S(kj + 1)
+≤
+2−kjC0
+S(kj + 1) ≤ 2C0
+j
+→ 0,
+as
+j → +∞.
+Furthermore, it is simple to show that (7) implies that
+(10)
+sup
+B1
+|vj| ≤ 2,
+and
+sup
+B 1
+2
+|vj| = 1.
+Also, Proposition 2.5 entails that vj is a minimizer of the scaled functional (4) for
+R = 3
+4 and we can extract a converging subsequence such that vj → v0 uniformly
+in B 3
+4 that v0 is p-harmonic. The uniform convergence of vj to v0 along with (9),
+(10), give that
+∆pv0(x) = 0,
+v0(x) ≥ 0 if x ∈ B 3
+4 ,
+v(0) = 0,
+sup
+B 1
+2
+v0 = 1,
+which is in contradiction with the strong minimum principle. Thus (6) obtains.
+Now we show how (6) implies the lemma. Assume that k0 is the smallest integer
+k that 2−k ≤ r0. Let ¯C1 = max( ˜C1, 2k0S(k0)). It is not difficult to observe from (6) that
+S(k) ≤ ¯C12−k. For an arbitrary r ∈ (0, r0] choose k ≥ k0 such that 2−(k+1) < r ≤ 2−k,
+then
+∥u∥L∞(Br(x0)) ≤ ∥u∥L∞(B2−k(x0)) = S(k) ≤ ¯C12−k ≤ 2 ¯C1r.
+Thus the statement in the lemma holds for C1 = 2 ¯C1.
+□
+The following theorem roughly says that, in a very weak sense,the free boundary
+conditions (1) hold.
+Proposition 2.7. Suppose that u is a minimizer of JTP in D and D′ ⊂ D be such that
+|D′ ∩ {u = 0}| = 0. Then, we have the following free boundary condition in the very weak
+sense
+lim
+ǫ→0+
+�
+∂{u>ǫ}∩D′
+�
+|∇u+|p − λp
++
+�
+η · ν + lim
+δ→0+
+�
+∂{u<−δ}∩D′
+�
+|∇u−|p − λp
+−
+�
+η · ν = 0,
+for any η ∈ W1,p
+0 (D′, Rn), and where ν is the outward normal.
+Proof. The proof can be established precisely as in [6, Theorem 2.4].
+□
+Corollary 2.8. Suppose u(x) = α (x · e)+ − β (x · e)− is a global minimizer of JTP for some
+unite vector e ∈ Sn−1 and the positive constants α and β. Then α and β satisfy conditions
+(2).
+Proof. The equality αp − βp = λp
++ − λp
+− is obvious by invoking Proposition 2.7.
+Besides, conditions
+α ≥ λ+,
+and
+β ≥ λ−,
+
+7
+can be obtained by a smooth variation of the free boundary {u = 0} = {x · e = 0}.
+Indeed, by considering competitors of the form ut(x) = u+(x) − u−(x + tξ(x)) for
+vector fields ξ ∈ C∞
+c (Rn; Rn) with ξ · e ≤ 0 so that it moves negative phase only
+inwards, that is, {ut < 0} ⊂ {u < 0}, and taking the derivative of JTP(ut, BR) at t > 0
+and letting t → 0 (where R is sufficiently large such that supp ξ ⊂ BR), we get
+�
+{u=0}∩BR
+(ξ · e)
+�
+|∇u−|p − λp
+−
+�
+≤ 0,
+which gives β ≥ λ−. The estimate on α is analogous.
+□
+3. Free boundary conditions in the viscosity sense
+Let u : D → R be a local minimizer of JTP. In this section, we will show that u
+satisfies the free boundary conditions (1) in a weak (viscosity) sense.
+Definition 3.1. Let D be an open set. We say that a function Q : D → R touches a
+function w : D → R from below (resp. from above) at a point x0 ∈ D if Q(x0) = w(x0) and
+Q(x) − w(x) ≤ 0
+(resp. Q(x) − w(x) ≥ 0),
+for every x in a neighborhood of x0. We will say that Q touches w strictly from below (resp.
+above) if the above inequalities are strict for x � x0.
+A function Q is an (admissible) comparison function in D if
+(a) Q ∈ C1({Q > 0} ∩ D) ∩ C1({Q < 0} ∩ D);
+(b) Q ∈ C2({Q > 0} ∩ D) ∩ C2({Q < 0} ∩ D);
+(c) ∂{Q > 0} and ∂{Q < 0} are smooth manifolds in D.
+We should remark that if ∇Q � 0 on ∂{Q > 0} ∪ ∂{Q < 0}, the condition (c) above holds.
+Lemma 3.1. Let u be a local minimizer of JTP in the open set D ⊂ Rn. Then the following
+optimality conditions on the free boundary F(u) hold.
+(A) Suppose that Q is a comparison function that touches u from below at x0.
+(A.1) If x0 ∈ Γ+
+OP, then |∇Q+(x0)| ≤ λ+;
+(A.2) if x0 ∈ Γ−
+OP, then Q+ ≡ 0 in a neighborhood of x0 and |∇Q−(x0)| ≥ λ−;
+(A.3) if x0 ∈ ΓTP, then |∇Q−(x0)| ≥ λ− and
+|∇Q+(x0)|p − |∇Q−(x0)|p ≤ λp
++ − λp
+−.
+(B) Suppose that Q is a comparison function that touches u from above at x0.
+(B.1) If x0 ∈ Γ+
+OP, then Q− ≡ 0 in a neighborhood of x0 and |∇Q+(x0)| ≥ λ+;
+(B.2) if x0 ∈ Γ−
+OP, then |∇Q−(x0)| ≤ λ−;
+(B.3) if x0 ∈ ΓTP, then |∇Q+(x0)| ≥ λ+ and
+|∇Q+(x0)|p − |∇Q−(x0)|p ≥ λp
++ − λp
+−.
+Proof. First, we will prove the gradient bounds in (A.1) and (B.1). The case x0 ∈ Γ−
+OP,
+and the proofs of (A.2) and (B.2) can be obtained similarly.
+Let x0 ∈ Γ+
+OP be a one-phase point and let Q touches u from below at x0. Then, Q+
+touches u from below at x0, too. Consider ux0,rk(x) = u(x0+rkx)
+rk
+and Q+
+x0,rk(x) = Q+(x0+rkx)
+rk
+as the blow-up sequences of u and Q+ at x0. By virtue of Proposition 2.3, the
+functions ux0,rk are uniformly Lipschitz for sufficiently rk small and up to extracting
+a subsequence, we can assume that ux0,rk converges uniformly to a blow-up limit
+v. The limit v is a minimizer of one-phase functional JOP and so ∆pv = 0 in {v > 0}.
+
+8
+M. BAYRAMI AND M. FOTOUHI
+On the other hand, since Q+ is differentiable at x0 in Ω+
+Q, we get that Q+
+x0,rk
+converges to the function
+(11)
+HQ+(x) = (x · e′)+
+with
+e′ = ∇Q+(x0).
+If ∇Q+(x0) = 0, (A.1) is trivially valid. We assume that e′ � 0, and since HQ+ touches
+v from below at x = 0, we get
+v(x) = α(x · e′)+ + o(|x|),
+α ≥ 1,
+for some α; see [19, Lemma B.1]. We get that any blow-ups of v will be v0(x) =
+α(x ·e′)+ which is also a minimizer of JOP. Thus α|e′| = λ+ due to the free boundary
+condition for one-phase minimizers (Proposition 2.7 or see [10, Theorem 2.1]) and
+so
+|∇Q+(x0)| ≤ λ+.
+Similarly, when Q touches u from above at x0, then also Q+ touches u from above
+at x0, and the claim Q− ≡ 0 in (B.1) is trivially true. Again, consider ux0,rk, which up
+to extracting a subsequence, converges uniformly to a blow-up limit v and Q+
+x0,rk,
+as the blow-up sequences of Q+ at x0, which converges to the function (11). Now,
+we argue similar to the proof of (A.1) to get
+|∇Q+(x0)| ≥ λ+.
+Now, we prove (A.3). Suppose x0 ∈ ΓTP and assume that Q touches u from
+below at x0. Then u− ≤ Q− and u−(x) ≤ C0|x − x0| for C0 = 2|∇Q−(x0)| if |x − x0| is
+sufficiently small. Now we employ Lemma 2.6 to deduce that |u(x)| ≤ C1|x − x0| in
+a neighborhood of x0.
+Let ux0,rk and Qx0,rk be the blow-up sequences of u and Q at x0. Then, by using
+Proposition 2.5, up to extracting a subsequence, we can assume that ux0,rk converges
+uniformly to some function v which is also a minimizer of JTP. Moreover, it satisfies
+(12)
+|v(x)| ≤ C1|x|.
+On the other hand, since Q+ and Q− are differentiable at x0 (respectively in Ω+
+Q
+and Ω−
+Q), we get that Qx0,rk converges to the function
+HQ(x) = (x · ˜e+)+ − �x · ˜e−�− ,
+where ˜e± = ∇Q±(x0). Since HQ touches v from below at x = 0, we have ([19, Lemma
+B.1])
+v+(x) = α(x · ˜e−)+ + o(|x|),
+|˜e+| ≤ α|˜e−|,
+v−(x) = β(x · ˜e−)− + o(|x|),
+β ≤ 1,
+for some α, β ≥ 0. Note that by virtue of the non-degeneracy, Proposition 2.2,
+v− � 0 and so ˜e− � 0. If v0 is a blowup of v (recall (12) and Proposition 2.5), it
+will be v0(x) = α(x · ˜e−)+ − β(x · ˜e−)− which is also a minimizer of JTP. Now apply
+Corollary 2.8, we get
+(αp − βp)|˜e−|p = λp
++ − λp
+−,
+β|˜e−| ≥ λ−.
+Hence,
+|∇Q+(x0)|p − |∇Q−(x0)|p ≤ αp − βp = λp
++ − λp
+−,
+as well as |∇Q−(x0)| ≥ λ−. The proof of (B.3) is analogous.
+□
+
+9
+If u : D → R is a continuous function such that the claims (A) and (B) hold for
+every comparison function Q, then we say that u satisfies the boundary condition
+(1) on the free boundary in viscosity sense.
+We need the following straightforward consequence of the definition of viscosity
+solution. It emphasizes what happens when a function is touching only one of the
+two phases.
+Lemma 3.2. Let u : D → R be a continuous function that satisfies (1).
+(i) Assume that Q is a comparison function touching u+ from above at x0 ∈ ∂Ω+
+u
+(resp. −u− from below at x0 ∈ ∂Ω−
+u), then
+|∇Q+(x0)| ≥ λ+
+�resp. |∇Q−(x0)| ≥ λ−
+� .
+(ii) Assume that Q is a comparison function touching u+ from below at x0 ∈ Γ+
+OP
+(resp. −u− from above at x0 ∈ Γ−
+OP), then
+|∇Q+(x0)| ≤ λ+
+�resp. |∇Q−(x0)| ≤ λ−
+� .
+Proof. Statement (i) will be obtained directly from (B). The proof of (ii) follows the
+same lines of arguments as the proof of Lemma 3.1.
+□
+4. Flatness decay at two-phase points
+In this section, we will follow the method of improvement of flatness. In fact,
+we will prove that at two-phase points x0 ∈ ΓTP, there is a constant ǫ0 > 0 such that
+if u is ǫ0-flat in Br(x0) with respect to H = Hα,e, then it has excess flatness in smaller
+scales with respect to another ˜H = H ˜α,˜e.
+Theorem 4.1. For every 1 < p < ∞, 0 < L0, L1 and γ ∈ (0, 1
+2), there exist ǫ0 > 0, C > 0
+and ρ > 0 such that if the function u : B1 → R satisfies:
+(a) the origin is on the two-phase free boundary, 0 ∈ ΓTP;
+(b) u is p-harmonic in Ω+
+u ∪ Ω−
+u;
+(c) u satisfies the free boundary condition (1) in viscosity sense;
+(d) u is ǫ0-flat in B1, that is,
+(13)
+∥u − Hα,en∥L∞(B1) ≤ ǫ0,
+for some
+max(λ+, L0) ≤ α ≤ L1,
+then, there are e ∈ Sn−1 and ˜α ≥ max(λ+, L0), such that
+(14)
+|e − en| + | ˜α − α| ≤ C∥u − Hα,en∥L∞(B1),
+and
+(15)
+∥uρ − H ˜α,e∥L∞(B1) ≤ ργ∥u − Hα,en∥L∞(B1),
+where uρ(x) denotes u0,ρ(x) = u(ρx)
+ρ .
+Theorem 4.1 is an easy consequence of the two upcoming lemmas. In the first
+one, we deal with the situation where the two-plane is, roughly, Hλ+,e for some
+e ∈ Sn−1. Note that this is the case where one might expect the presence of branching
+points and it is indeed in this setting that we will obtain the two membrane
+problems as ”linearization” (see e.g. [12, Subsection 1.3] for a presentation of the
+linearization method in studying the regularity of free boundaries). In the second
+lemma, we deal with the case when the closest half-plane solution has a gradient
+much larger than λ+. In this case, the origin will be an interior two-phase point.
+In fact, in one-phase problems, it is possible to obtain universal interior bounds, in
+
+10
+M. BAYRAMI AND M. FOTOUHI
+the sense that, if u is a solution in a ball B1 and 0 ∈ F(u), then |∇u| is bounded in B 1
+2
+by a universal constant, no matter what the boundary data are. However, in two-
+phase problems, this is generally not possible. For instance, in the one-dimensional
+minimization scenario, increasing the boundary data leads to the appearance of a
+solution with a large gradient near the origin, see [9, Section 1.1].
+Lemma 4.2 (Improvement of flatness: branching points). For every 1 < p < ∞,
+0 < L0, L1, γ ∈ (0, 1
+2), and M > 0, there exist ǫ1 = ǫ1(p, γ, n, L0, L1, M), C1 =
+C1(p, γ, n, L0, L1, M) and ρ = ρ(p, γ, n, L0, L1, M) such that if function u : B1 → R
+satisfies (a) − (b) − (c) of Theorem 4.1 and furthermore
+∥u − Hα,en∥L∞(B1) ≤ ǫ1,
+with
+L0 ≤ λ+ ≤ α ≤ λ+ + M∥u − Hα,en∥L∞(B1),
+then there exist e ∈ Sn−1 and ˜α ≥ λ+, for which (14) and (15) hold.
+Lemma 4.3 (Improvement of flatness: non-branching points). For every 1 < p < ∞,
+0 < L0, L1 and γ ∈ (0, 1), there exist ǫ2 = ǫ2(p, γ, n, L0, L1), M = M(p, γ, n, L0, L1),
+ρ = ρ(p, γ, n, L0, L1) and C2 = C2(p, γ, n, L0, L1) such that if function u : B1 → R satisfies
+(a) − (b) − (c) of Theorem 4.1 and furthermore
+∥u − Hα,en∥L∞(B1) ≤ ǫ2,
+with
+α ≥ max(λ+, L0) + M∥u − Hα,en∥L∞(B1),
+then there exist e ∈ Sn−1 and ˜α ≥ max(λ+, L0), for which (14) and (15) hold.
+Proof of Theorem 4.1. The proof follows easily by combining the Lemmas 4.2 and
+4.3.
+□
+In order to prove Lemma 4.2 and Lemma 4.3, we will argue by contradiction.
+Hence in the following, we consider a sequence uk of minimizers such that
+(16)
+ǫk := ∥uk − Hαk,en∥L∞(B1) → 0
+and
+λ+ ≤ αk ≤ L.
+We also set
+(17)
+ℓ := λp
++ lim
+k→∞
+αp
+k − λp
++
+pαp
+kǫk
+= λp
+− lim
+k→∞
+βp
+k − λp
+−
+pβp
+kǫk
+,
+which we can assume to exist up to a subsequence. It might be useful to keep in
+mind that ��� = ∞ will correspond to Lemma 4.3 while 0 ≤ ℓ < ∞ (so αk → λ+ and
+λ+ ≥ L0) to Lemma 4.2.
+We first show that the sequence
+(18)
+vk(x) =
+
+v+,k(x) := uk(x) − αkx+
+n
+ǫkαk
+x ∈ Ω+
+uk ∩ B1
+v−,k(x) := uk(x) + βkx−
+n
+ǫkβk
+x ∈ Ω−
+uk ∩ B1
+is compact in some suitable sense. This will be mentioned in Lemma 4.4 below and
+the proof will come in Subsection 4.1. Then, in Lemma 4.5, we obtain the limiting
+problem which is solved by v, the limit of vk. Finally, in Subsection 4.3 we show
+how to deduce Lemma 4.3 and Lemma 4.2 from Lemma 4.4 and Lemma 4.5.
+In the following, we will denote with
+B±
+r := Br ∩ {x±
+n > 0},
+for every r > 0.
+
+11
+Lemma 4.4 (Compactness of the linearizing sequence vk). Let uk be a sequence of
+functions satisfying (a) − (b) − (c) of Theorem 4.1 uniformly in k and let ǫk and αk be as in
+(16) and let vk be defined by (18). Then there are H¨older continuous functions
+v+ : B+
+1
+2 → R
+and
+v− : B−
+1
+2 → R,
+with
+v+ ≤ v−
+on
+B 1
+2 ∩ {xn = 0},
+v+(0) = v−(0) = 0,
+and such that the sequence of closed graphs
+Γ±
+k :=
+�
+(x, v±,k(x)) : x ∈ Ω±
+uk ∩ B 1
+2
+�
+,
+converge, up to a subsequence, in the Hausdorff distance to the closed graphs
+Γ± =
+�
+(x, v±(x)) : x ∈ B±
+1
+2
+�
+.
+In particular, the following claims hold.
+(i) For every δ > 0, v±,k converges uniformly to v± on B 1
+2 ∩ {±xn > δ}.
+(ii) For every sequence xk ∈ Ω±
+uk ∩ B1 converging to x ∈ B±
+1
+2 , we have
+v±(x) = lim
+k→∞ v±,k(xk).
+(iii) For every x ∈ {xn = 0} ∩ B 1
+2 , we have
+v±(x) = − lim
+k→∞
+xk · en
+ǫk
+for any sequence
+∂Ω±
+uk ∋ xk → x.
+In particular, {xn = 0} ∩ B 1
+2 decomposes into an open jump set
+J = {v+ < v−} ∩ {xn = 0} ∩ B 1
+2 ,
+and its complementary contact set
+C = {v+ = v−} ∩ {xn = 0} ∩ B 1
+2 .
+Furthermore, if x ∈ J, then
+(19)
+lim inf
+k→∞ dist
+�
+x, ∂Ω+
+uk ∩ ∂Ω−
+uk
+�
+> 0.
+In particular for all x ∈ J, there exists two sequences x±
+k ∈ Γ±
+k,OP such that x±
+k → x.
+Now, in the next lemma, we determine the limiting problem for the function v
+which is defined as
+(20)
+v(x) =
+
+v+(x)
+for x ∈ B+
+1
+2 ,
+v−(x)
+for x ∈ B−
+1
+2 ,
+where v+ and v− are the functions defined in Lemma 4.4.
+In what follows, we will denote with
+Lp(u) := ∆u + (p − 2)∂nnu,
+the frequently used operator which appears in the linearized problem.
+
+12
+M. BAYRAMI AND M. FOTOUHI
+Lemma 4.5 (The ”linearized” problem). Let uk, ǫk and αk be as in (16), vk be defined
+by (18) and ℓ as in (17). Let also v± be as in Lemma 4.4:
+If ℓ = ∞, then J = ∅ and v± are viscosity solutions of the following transmission
+problem:
+(21)
+
+Lp(v±) = ∆v± + (p − 2)∂nnv± = 0,
+in
+B±
+1
+2 ,
+αp
+∞∂nv+ = βp
+∞∂nv−,
+on
+B±
+1
+2 ∩ {xn = 0},
+where α∞ = limk→∞ αk and β∞ = limk→∞ βk, which we can assume to exist up to extracting
+a further subsequence.
+If 0 ≤ ℓ < ∞, then v± are viscosity solutions of the following two membranes problem:
+(22)
+
+Lp(v±) = ∆v± + (p − 2)∂nnv± = 0,
+in
+B±
+1
+2 ,
+λp
+±∂nv± + ℓ ≥ 0,
+in
+B 1
+2 ∩ {xn = 0},
+λp
+±∂nv± + ℓ = 0,
+in
+J,
+λp
++∂nv+ = λp
+−∂nv−,
+in
+C,
+v+ ≤ v−,
+in
+B 1
+2 ∩ {xn = 0}.
+Remark 4.6. Here by viscosity solution of (21) and (22), we mean a function v as in
+(20) such that v± are continuous in B±
+1
+2 , Lp(v±) = 0 in B±
+1
+2 (in viscosity or equivalently the
+classical sense) and such that the following holds.
+• If we are in case (21), let s, t ∈ R and let ˜P be a quadratic polynomial such that
+∂n ˜P = 0. Suppose that Lp( ˜P) ≥ 0 (Lp( ˜P) ≤ 0) and that the function
+P := sx+
+n − tx−
+n + ˜P,
+touches v strictly from below (above) at a point x0 ∈ B 1
+2 ∩ {xn = 0}, then
+αp
+∞s ≤ βp
+∞t,
+�
+αp
+∞s ≥ βp
+∞t
+�
+.
+• If we are in case (22) then
+(1) if P± is a quadratic polynomial with Lp(P±) ≤ 0 in B±
+1
+2 touching v± strictly
+from above at x0 ∈ B 1
+2 ∩ {xn = 0}, then λp
+±∂nP± ≥ 0;
+(2) if P± is a quadratic polynomial with Lp(P±) ≥ 0 in B±
+1
+2 touching v± strictly
+from below at x0 ∈ J, then λp
+±∂nP± ≤ 0;
+(3) if s, t ∈ R and ˜P is a quadratic polynomial with Lp(P±) ≥ 0 (Lp(P±) ≤ 0)
+such that ∂n ˜P = 0 and the function
+P := sx+
+n − tx−
+n + ˜P,
+touches v strictly from below (above) at a point x0 ∈ B 1
+2 ∩ {xn = 0}, then
+λp
++s ≤ λp
+−t,
+�
+λp
++s ≥ λp
+−t
+�
+.
+4.1. Compactness of the linearizing sequence. As explained in [12, Subsection
+3.1] for the case of classical two-phase Bernoulli problem, the authors declare that
+the key point in establishing suitable compactness for vk is a ”partial Harnack”
+inequality. We will follow the same approach and start with the following useful
+lemma.
+
+13
+Lemma 4.7. There is a constant τ = τ(n, p) > 0 such that the following holds. Assume
+that v : B1 → R is a continuous function with ∆pv = 0 in {v > 0} and
+λ (xn + b)+ ≤ v(x) ≤ λ (xn + a)+ ,
+x ∈ B1,
+for some λ > 0 and a, b ∈ (− 1
+100,
+1
+100). Let P = (0, · · · , 0, 1
+2), then for all ǫ ∈ (0, 1
+2)
+v(P) ≤ λ(1 − ǫ)
+�1
+2 + a
+�+
+=⇒
+v(x) ≤ λ(1 − τǫ) (xn + a)+
+in
+B 1
+4 (0),
+and
+v(P) ≥ λ(1 + ǫ)
+�1
+2 + b
+�+
+=⇒
+v(x) ≥ λ(1 + τǫ) (xn + b)+
+in
+B 1
+4 (0).
+Proof. We prove only the first implication since the second statement can be ob-
+tained by the same arguments. First, we notice that, since |b| <
+1
+100, both v and
+λ(xn + a)+ are positive and p-harmonic in B 1
+4 (P). Thus,
+λ(xn + a)+ − v(x) ≥ 0,
+x ∈ B 1
+4 (P),
+and
+λ
+�1
+2 + a
+�+
+− v(P) ≥ λǫ
+�1
+2 + a
+�+
+≥ 49
+100λǫ.
+Now, we distinguish two cases:
+Case (i). Suppose |∇v(P)| < λ
+4 . Therefore, there exists r1 = r1(n, p) > 0 such that
+|∇v(x)| ≤ λ
+2 in B4r1(P) (note that v
+λ is universally bounded and p-harmonic in B 1
+4 (P)).
+It is easy to find that for ˜v := (xn + a)+ − 1
+λv, we have
+div
+�
+|∇˜v − en|p−2(∇˜v − en)
+�
+= 0,
+in
+B 1
+20 (P).
+We now apply Harnack’s inequality for the above operator (see e.g. [17, Lemma
+4.1]) in B4r1(P), to deduce that
+(xn + a)+ − 1
+λv(x) ≥ C−1
+��1
+2 + a
+�+
+− 1
+λv(P)
+�
+− r1,
+in
+Br1(P),
+for an appropriate universal constant C = C(n, p) > 0. On the other hand, for all
+x ∈ Br1(P), we obtain
+C−1 49
+100ǫ − r1 ≤ (xn + a)+ − 1
+λv(x)
+≤ (xn + 2r1 + a)+ − 2r1 − 1
+λv(x + 2r1en) + 2r1
+λ ∥∇v∥L∞(B4r1(P))
+≤ (xn + 2r1 + a)+ − 2r1 − 1
+λv(x + 2r1en) + r1
+≤ (xn + 2r1 + a)+ − 1
+λv(x + 2r1en) − r1.
+Thus, with ˜P = P + 2r1en, we get
+(23)
+C−1 49
+100ǫ ≤ (xn + a)+ − 1
+λv (x) ,
+for all
+x ∈ Br1( ˜P).
+Hence, by considering the inequality (23) and also using the bound |a| ≤
+1
+100, there
+is a constant c = c(n, p) such that
+v(x) ≤ λ(1 − cǫ)(xn + a)+,
+for all
+x ∈ Br1( ˜P).
+
+14
+M. BAYRAMI AND M. FOTOUHI
+We now let w be the solution to the following problem
+
+∆pw = 0
+in
+�
+B1(0) \ Br1( ˜P)
+�
+∩ {xn > −a}
+w = 0
+on
+B1 ∩ {xn = −a}
+w = (xn + a)+
+on
+∂B1(0) ∩ {xn > −a}
+w = (1 − cǫ)(xn + a)+
+on
+∂Br1( ˜P) ∩ {xn > −a}.
+By the Hopf boundary lemma ([24, Proposition 3.2.1]),
+w(x) ≤ (1 − τǫ)(xn + a)+,
+for every x ∈ B 1
+4 ∩ {xn > −a},
+for a suitable constant τ = τ(n, p). On the other hand, by the comparison principle,
+we have v ≤ λw in {v > 0} ∩ B1 \ Br1( ˜P), which concludes the proof in Case (i).
+Case (ii). Suppose |∇v(P)| ≥ λ
+4 . By the interior gradient estimate, we know that
+∇v is bounded in B 1
+40 (P), and there exist a constant 0 < r0 = r0(n, p), with 8r0 ≤
+1
+40
+such that
+λ
+8 ≤ |∇v(x)| ≤ Cλ,
+for all
+x ∈ B8r0(P),
+for an appropriate universal constant C = C(n, p) > 0. Now, v will be the weak
+solution to the following uniformly elliptic equation
+n
+�
+i,j=1
+θij∂xixjv = 0
+in
+B4r0(P),
+with θij = δij + (p − 2)|∇v|−2∂xiv∂xjv. Then, applying Harnack’s inequality (see e.g.
+[20, Chapter 9]), we get
+(24)
+C−1 49
+100ǫ ≤ (xn + a)+ − 1
+λv (x) ,
+for all
+x ∈ Br0(P).
+Now, we can repeat the same argument of Case (i), by considering the inequality
+(24) in the ball Br0(P) instead of inequality (23). This completes the proof of the
+lemma.
+□
+We next prove the two partial Harnack inequalities. The proof of these inequal-
+ities is based on a comparison with suitable test functions. In order to build these
+”barriers”, we will often use the following function ϕ. Let Q = (0, · · · , 0, 1
+5) and
+define ϕ : B1 → R by
+(25)
+ϕ(x) =
+
+1,
+if x ∈ B 1
+100 (Q),
+κn
+�
+|x − Q|−n − ( 3
+4)−n�
+,
+if x ∈ B 3
+4 (Q) \ B 1
+100 (Q),
+0,
+otherwise,
+where the dimensional constant κn is chosen in such a way that ϕ is continuous.
+One can check that ϕ has the following properties:
+(ϕ.1) 0 ≤ ϕ ≤ 1 in Rn, and ϕ = 0 on ∂B1;
+(ϕ.2) For s > 0 small,
+−div
+����en − s∇ϕ
+���
+p−2 �en − s∇ϕ��
+≥ c(n, p, s) > 0,
+in
+{ϕ > 0} \ B 1
+100 (Q),
+(with fairly simple computations same as the ones which have been done
+in [17, Lemma 4.2]);
+(ϕ.3) ∂nϕ > 0 in {ϕ > 0} ∩ {|xn| ≤
+1
+100};
+
+15
+(ϕ.4) ϕ ≥ cn > 0 in B 1
+6 ;
+where c(n, p) and cn are constants.
+Lemma 4.8 (Partial Boundary Harnack I). Given 1 < p < ∞ and λ+ ≥ λ− > 0, there
+exist constants ǫ = ǫ(n, λ±, p) > 0 and c = c(n, λ±, p) ∈ (0, 1) such that, for every function
+u : B4 → R satisfying (b) − (c) in Theorem 4.1, the following properties hold true.
+Let a±, b± ∈ (− 1
+100,
+1
+100) be such that
+b+ ≤ b− ≤ a− ≤ a+,
+and
+(a− − b−) + (a+ − b+) ≤ ǫ.
+Assume that for x ∈ B4
+λ+(xn + b+)+ �� u+(x) ≤ λ+(xn + a+)+,
+and
+−λ−(xn + b−)− ≤ −u−(x) ≤ −λ−(xn + a−)−.
+Then, one can find new constants a±, b± ∈ (− 1
+100,
+1
+100), with
+b+ ≤ b− ≤ a− ≤ a+,
+and
+a− − b− ≤ c(a− − b−),
+a+ − b+ ≤ c(a+ − b+)
+such that for x ∈ B 1
+6
+λ+(xn + b+)+ ≤ u+(x) ≤ λ+(xn + a+)+,
+and
+−λ−(xn + b−)− ≤ −u−(x) ≤ −λ−(xn + a−)−.
+Remark 4.9. We need to remark that the assumption λ+ ≥ λ− is not restrictive as one can
+always replace u by −u in JTP. Also, when λ+ ≤ λ− the similar result holds if we replace
+the order of a±, b± with a+ ≤ a− ≤ b− ≤ b+.
+Proof of Lemma 4.8. Let us show how to improve the positive part. More precisely,
+given a+, a−, b+, b− we will show how we can find a+ and b+. The proof for b− and
+a− follows in the same way. We let
+P = (0, · · · , 0, 2),
+and distinguish two cases:
+Case 1. Improvement from above. Assume that, at the point P, u+ is closer to
+λ+(2 + b+)+ than to the upper barrier λ+(2 + a+)+. Precisely that
+u+(P) ≤ λ+(2 + a+)+ − λ+(a+ − b+)
+2
+.
+In this case, we will show that u(x) is less than λ+(xn+a+)+ in a smaller ball centered
+at the origin for a+ strictly smaller than a+.
+We start by setting
+ǫ := a+ − b+ ≤ ǫ.
+Then
+u+(P) ≤ λ+(2 + a+)+ − λ+ǫ
+2
+≤ λ+(1 − cǫ)(2 + a+)+,
+
+16
+M. BAYRAMI AND M. FOTOUHI
+for a suitable (universal) constant c. We can thus apply (the scaled version of)
+Lemma 4.7 to u+, to infer the existence of a constant τ = τ(n, p) such that
+(26)
+u+(x) ≤ λ+(1 − τǫ)(xn + a+)+,
+in
+B1.
+For ϕ as in (25) and t ∈ [0, 1], we set
+ft = λ+
+�
+1 − τǫ
+2
+�
+(xn + a+ − tcǫϕ)+,
+where c = c(n, p) is a small constant chosen such that for all x ∈ B 1
+100 (Q) and t ∈ [0, 1),
+(27)
+u(x) ≤ λ+(1 − τǫ)(xn + a+)+
+≤ λ+
+�
+1 − τǫ
+2
+�
+(xn + a+ − cǫ)+ < ft(x),
+where we have used that (xn + a+) is within two universal constant for x ∈ B 1
+100 (Q).
+We now let t ∈ (0, 1] the largest t such that ft ≥ u in B1 and we claim that t = 1.
+Indeed assume that t < 1, then there exists x ∈ B1 such that
+(28)
+u(x) − ft(x) ≤ u(x) − ft(x) = 0,
+for all
+x ∈ B1.
+Note that by (27), x � B 1
+100 (Q), while, by (26), x ∈ {ϕ > 0}. Moreover, if u(x) = ft(x) >
+0, by (ϕ.2) we will have
+∆p ft(x) =
+�
+λ+
+�
+1 − τǫ
+2
+��p−1
+div
+����en − tcǫ∇ϕ(x)
+���
+p−2 �
+en − tcǫ∇ϕ(x)
+��
+< 0,
+but, since ∆pu(x) = 0, we reach a contradiction with (28) and the definition of
+viscosity solution for the p-harmonic function u. Hence, u(x) = ft(x) = 0. Now
+recall the free boundary condition (1) and apply (ϕ.3) to get
+λp
++ ≤ |∇ft(x)|p = λp
++
+�
+1 − τǫ
+2
+�p �
+1 − pctǫ∂nϕ(x) + O(ǫ2)
+�
+< λp
++,
+provided ǫ ≤ ǫ(n, λ+, p) ≪ 1 (note that necessarily u(x) = 0 and thus x ∈ {|xn| ≤
+1
+100}).
+This contradiction implies that t = 1. Hence, by (ϕ.4), we get for all x ∈ B 1
+6
+u(x) ≤ λ+
+�
+1 − τǫ
+2
+�
+(xn + a+ − cǫϕ)+ ≤ λ+(xn + a+ − cǫ)+,
+for a suitable constant c = c(n, p). Setting
+a+ = a+ − cǫ,
+b+ = b+,
+and recalling that ǫ = a+ − b+ we finish the proof in this case.
+Case 2. Improvement from below. We now assume that, at point P, u+ is closer to
+λ+(2 + a+)+ than to λ+(2 + b+)+. Hence, we have
+u+(P) ≥ λ+(2 + b+)+ + λ+(a+ − b+)
+2
+,
+and we set again
+ǫ := a+ − b+ ≤ ǫ.
+Arguing as in Case 1, by Lemma 4.7, there exists a constant τ = τ(n, p) such that
+(29)
+u+(x) ≥ λ+(1 + τǫ)(xn + b+)+,
+in
+B1.
+
+17
+We need now to distinguish two further sub-cases:
+Case 2.1: Suppose that
+ηǫ ≤ b− − b+,
+where η ≪ τ is a small universal constant which we will choose at the end of the
+proof. In this case, for x ∈ B1,
+(30)
+u(x) ≥ λ+(1 + τǫ)(xn + b+)+ − λ−(xn + b−)−
+≥ λ+(1 + τǫ)(xn + b+)+ − λ−(1 − c1ηǫ)(xn + b+)−,
+for a suitable universal constant c1. We now take ϕ as in (25) and set, for t ∈ [0, 1],
+ft(x) = λ+
+�
+1 + τǫ
+2
+�
+(xn + b+ + c2tǫϕ)+ − λ−(1 − c1ηǫ)(xn + b+ + c2tǫϕ)−,
+for a suitably small universal constant 0 < c2 ≪ τ, chosen so that for all x ∈ B 1
+100 (Q)
+(1 + τǫ)(xn + b+)+ ≥
+�
+1 + τǫ
+2
+�
+(xn + b+ + c2ǫ)+.
+This together with (29) implies that
+(31)
+u(x) ≥ λ+(1 + τǫ)(xn + b+)+ ≥ λ+
+�
+1 + τǫ
+2
+�
+(xn + b+ + c2ǫ)+
+≥ f1(x) ≥ ft(x),
+for all
+x ∈ B 1
+100 (Q), t ∈ [0, 1].
+Furthermore u ≥ f0 in B1 thanks to (30). Similar to Case 1, let t be the biggest t such
+that ft ≤ u in B1 and x be the first contact point, so that
+u(x) − ft(x) ≥ u(x) − ft(x) = 0,
+for all x ∈ B1.
+Since, by using (ϕ.2), it can be checked that
+∆p ft > 0,
+on
+{ft � 0} \ B 1
+100 (Q),
+therefore, as in Case 1, x is a free boundary point. Moreover, since ft changes sign
+in a neighborhood of x:
+either
+x ∈ Γ+
+OP = ∂Ω+
+u \ ∂Ω−
+u,
+or
+x ∈ ΓTP = ∂Ω+
+u ∩ ∂Ω−
+u.
+In the first case, by definition of viscosity solution and (ϕ.3),
+λp
++ ≥ |∇f +
+t (x)|p = λp
++
+�
+1 + τǫ
+2
+�p �
+1 + pc2tǫ∂nϕ(x) + O(ǫ2)
+�
+> λp
++,
+a contradiction for ǫ ≪ 1. In the second case, we have a contradiction as well, since
+(recall also the assumption λ+ − λ− ≥ 0)
+λp
++ − λp
+− ≥ |∇f +
+t |p − |∇f −
+t |p
+=
+�
+λp
++
+�
+1 + τǫ
+2
+�p
+− λp
+−(1 − c1ηǫ)p
+� �
+1 + pc2tǫ∂nϕ(x) + O(ǫ2)
+�
+> λp
++ − λp
+−,
+provided ǫ ≪ 1 (only depending on n, λ+ and p). Hence, t = 1, u ≥ f1 (so u+ ≥ f +
+1 )
+which implies the desired conclusion by setting
+a+ = a+,
+b+ = b+ + c2ǫ,
+for a suitable constant c2 = c2(n, p) and by recalling that ǫ = a+ − b+.
+
+18
+M. BAYRAMI AND M. FOTOUHI
+Case 2.2: Assume instead that:
+0 ≤ b− − b+ ≤ ηǫ,
+where η = η(n, p) will be determined later. In this case we consider the family of
+functions
+ft(x) = λ+
+�
+1 + τǫ
+2
+�
+(xn + b+ + ηtǫϕ)+ − λ−(xn + b−)−.
+Since ϕ ≤ 1, this function is well defined due to b− ≤ b+ + ηǫ. Moreover, u ≥ f0 and,
+thanks to (29) and by assuming η is sufficiently small (this can also be determined
+universally depending only on the dimension and p) we will have,
+u(x) ≥ f1(x) ≥ ft(x),
+for all
+x ∈ B 1
+100 (Q), t ∈ [0, 1].
+We consider again the first touching time t and the first touching point x.
+By
+arguing as in the previous cases, we get x ∈ {u = 0} ∩ {|xn| ≤
+1
+100}.
+Also, the
+definition of ft yields that x ∈ ∂{ft > 0}. This infer that x ∈ ∂Ω+
+u \ ∂Ω−
+u (note that
+ϕ(x) < 1). However, again by arguing as in Case 2.1, this is in contradiction with u
+being a viscosity solution. We now conclude as in the previous cases.
+□
+The following lemma addresses the situation in which the origin is not a branch-
+ing point.
+Lemma 4.10 (Partial Boundary Harnack II). Given 1 < p < ∞ and 0 < L0, L1
+and assume that 0 < λ− ≤ λ+ ≤ L1, then there exist constants ǫ = ǫ(n, L0, L1, p) > 0,
+M = M(n, L0, L1, p) and c = c(n, L0, L1, p) ∈ (0, 1) such that for every function u : B4 → R
+satisfying (b) − (c) in Theorem 4.1 the following property holds true. If there are constants
+a, b ∈ (− 1
+100,
+1
+100) with
+0 ≤ a − b ≤ ǫ,
+such that for x ∈ B4
+Hα,en(x + ben) ≤ u(x) ≤ Hα,en(x + aen),
+and
+max(λ+, L0) + Mǫ ≤ α ≤ L1,
+then there are constants a, b ∈ (− 1
+100,
+1
+100) with
+0 ≤ a − b ≤ c(a − b),
+such that for x ∈ B 1
+6
+Hα,en(x + ben) ≤ u(x) ≤ Hα,en(x + aen).
+Proof. We consider the point P = (0, · · · , 0, 2) and distinguish two cases (note that
+one of these inequalities is always satisfied):
+either
+Hα,en (P + ben) + α(a − b)
+2
+≤ u(P),
+or
+Hα,en (P + aen) − α(a − b)
+2
+≥ u(P).
+Since the argument in both cases is completely symmetric we only consider the
+second case. If we set
+ǫ = a − b,
+
+19
+by Lemma 4.7 and by arguing as in Lemma 4.8 we deduce the existence of a
+constant τ = τ(n, p) such that
+u(x) ≤ α(1 − τǫ)(xn + a)+ − β(xn + a)−,
+in B1. We let ϕ as in (25) and set
+ft(x) = α
+�
+1 − τǫ
+2
+�
+(xn + a − ctǫϕ)+ − β(xn + a − ctǫϕ)−,
+where c = c(n, p) is a constant chosen such that
+u(x) ≤ f1(x) ≤ ft(x),
+for all
+x ∈ B 1
+100 (Q), t ∈ [0, 1],
+where, Q = (0, · · · , 0, 1
+5). As in Lemma 4.8, we let t and x be the first contact time
+and the first contact point and we aim to show that t = 1. For this purpose, we note
+that, by the same arguments as in Lemma 4.8, necessarily x ∈ {u = 0}. We claim
+that
+x ∈ ΓTP = ∂Ω+
+u ∩ ∂Ω−
+u.
+Indeed, otherwise x ∈ ∂Ω−
+u \ ∂Ω+
+u (the case x ∈ ∂Ω+
+u \ ∂Ω−
+u will be impossible since
+ft is negative in a neighborhood of x). And by definition of viscosity solution, this
+along with (2) would imply
+λp
+− ≥ |∇f −
+t (x)|p = βp(1 − pctǫ∂nϕ(x) + O(ǫ2))
+≥ (λp
+− − λp
++ + αp)(1 − pctǫ∂nϕ(x) + O(ǫ2))
+≥ (λp
+− − λp
++ + (max(λ+, L0) + Mǫ)p)(1 − pctǫ∂nϕ(x) + O(ǫ2))
+= λp
+− + p(Lp−1
+0
+M − ct∂nϕ(x))ǫ + O(ǫ2),
+where the implicit constants in O(ǫ2) can control by L1, p and n. This inequality is
+impossible if M is chosen sufficiently large.
+Hence x ∈ ∂Ω+
+u ∩ ∂Ω−
+u. This however implies:
+λp
++ − λp
+− ≤ |∇f +
+t (x)|p − |∇f −
+t (x)|p
+=
+�
+αp
+�
+1 − τǫ
+2
+�p
+− βp
+� �
+1 − pctǫ∂nϕ(x) + O(ǫ2)
+�
+< αp − βp = λp
++ − λp
+−,
+provided ǫ and as a consequence of ǫ = a − b ≤ ǫ, ǫ is chosen small enough, where
+we have used (ϕ.3) and the equality
+0 ≤ λp
++ − λp
+− = αp − βp.
+This contradiction shows that t = 1 and as in Lemma 4.8, this completes the
+proof.
+□
+With Lemmas 4.7 and 4.8 at hand the proof of Lemma 4.4 is as follows.
+Proof of Lemma 4.4. We distinguish two cases:
+Case 0 ≤ ℓ < +∞: By triangular inequality we have
+∥uk − Hλ+,en∥L∞(B1) ≤ ǫk
+�
+1 + 2ℓ max(λ1−p
++ , λ1−p
+− )
+�
+,
+
+20
+M. BAYRAMI AND M. FOTOUHI
+for k sufficiently large. Define the bounded sequence wk by
+wk(x) =
+
+w+,k(x) := uk(x) − λ+x+
+n
+αkǫk
+x ∈ Ω+
+uk ∩ B1,
+w−,k(x) := uk(x) + λ−x−
+n
+βkǫk
+x ∈ Ω−
+uk ∩ B1.
+Now we can repeatedly apply Lemma 4.8 to deduce that wk satisfies
+(32)
+|wk(x) − wk(y)| ≤ C|x − y|γ,
+when x, y ∈ B 1
+2 , and |x − y| ≥ ǫk
+ǫ ,
+for some universal exponent 0 < γ < 1 and constant C; see [13, Corollary 4.2]. This
+gives that the graphs of
+˜Γ±
+k := {(x, w±,k(x)) : x ∈ Ω±
+uk ∩ B 1
+2 },
+converge, up to a subsequence, in the Hausdorff distance to the closed graphs
+˜Γ± := {(x, w±(x)) : x ∈ B±
+1
+2 },
+where w ∈ C0,α for some α > 0. Since
+hk(x) := Hαk,en − Hλ+,en
+ǫk
+→
+
+λ1−p
++ ℓxn
+xn > 0,
+λ1−p
+− ℓxn
+xn < 0,
+the original sequence vk satisfies that their graphs, converges to the graph of a
+limiting function v as we wanted, this in particular proves (i), (ii), and (iii).
+Since 0 ∈ ∂Ω+
+uk ∩ ∂Ω−
+uk then 0 is in the domain of v±,k and
+v±,k(0) = 0,
+which implies that v±(0) = 0. To show that v+(x) ≤ v−(x) for x = (x′, 0) ∈ B 1
+2 , we
+simply exploit (iii) at the points x±
+k = (x′, t±
+k ) where
+t+
+k = sup{t : (x′, t) ∈ ∂Ω+
+uk}
+and
+t−
+k = inf{t : (x′, t) ∈ ∂Ω−
+uk},
+and by noticing that t−
+k ≤ t+
+k .
+Finally, to see the last claim, (19), it is enough to note that if xk ∈ ∂Ω+
+uk ∩ ∂Ω−
+uk is
+converging to x then v+,k(xk) = v−,k(xk) and thus v+(x) = v−(x), yielding x ∈ C.
+Case ℓ = ∞: In this case, the conclusion follows exactly with a similar argument
+by using repeatedly Lemma 4.10 for function vk to obtain a relation similar to (32)
+for functions vk.
+□
+4.2. The linearized problem: proof of Lemma 4.5. Lemma 4.5 proves through the
+following technical lemma, whose proof is easily obtained by adapting the one in
+[12, Lemma 3.10] exactly. Then we present the statement without proof.
+Lemma 4.11. Let uk, ǫk and αk be as in the statement of Lemma 4.4, vk be defined by (18)
+and v± be as in Lemma 4.4. Then:
+(1) Let P+ be a quadratic polynomial with Lp(P+) > 0 (or Lp(P+) < 0) on B+
+1
+2 touching
+v+ strictly from below (above) at a point x0 ∈ {xn = 0} ∩ B 1
+2 . Then, there exists
+
+21
+a sequence of points ∂Ω+
+uk ∋ xk → x0 and a sequence of comparison functions Qk
+such that Qk touches from below (above) u+
+k at xk, and such that
+(33)
+∇Q+
+k (xk) = αken + ǫkαk∇P+(x0) + o(ǫk).
+(2) Let P− be a quadratic polynomial with Lp(P−) > 0 (Lp(P−) < 0) on B−
+1
+2 touching
+v− strictly from below (above) at a point x0 ∈ {xn = 0} ∩ B 1
+2 . Then, there exists
+a sequence of points ∂Ω−
+uk ∋ xk → x0 and a sequence of comparison functions Qk
+such that Qk touches from below (above) −u−
+k at xk, and such that
+(34)
+∇Q−
+k (xk) = −βken + ǫkβk∇P−(x0) + o(ǫk).
+(3) Let s, t ∈ R and ˜P be a quadratic polynomial on B 1
+2 such that ∂n ˜P = 0. Suppose
+that Lp( ˜P) ≥ 0 (Lp( ˜P) ≤ 0) and that the function
+P := sx+
+n − tx−
+n + ˜P,
+touches v strictly from below (above) at a point x0 ∈ C. Then, there exists a
+sequence of points xk → x0 and a sequence of comparison functions Qk such that
+Qk touches from below (above) the function uk at xk ∈ ∂Ωuk, and such that
+(35)
+∇Q+
+k (xk) = αk(1 + ǫks)en + o(ǫk),
+∇Q−
+k (xk) = −βk(1 + ǫkt)en + o(ǫk).
+In particular, if s > 0 and Qk touches uk from below then xk � ∂Ω−
+uk \ ∂Ω+
+uk, while
+if t < 0 and Qk touches uk from above then xk � ∂Ω+
+uk \ ∂Ω−
+uk.
+Proof of Lemma 4.5. Step 1: In this step, we prove Lp(v±) = 0 in B±
+1
+2 .
+Let P(x) be a quadratic polynomial touching v = v+ at x ∈ B+
+1
+2 strictly from below.
+We need to show that at this point
+Lp(P) = ∆P + (p − 2)∂nnP ≤ 0.
+Since v+,k → v+, there exist points xk ∈ Ω+
+uk ∩ B 1
+2 , xk → x and constants ck → 0 such
+that
+(36)
+v+,k(xk) = P(xk) + ck,
+and
+(37)
+v+,k ≥ P + ck,
+in a neighborhood of xk.
+From the definition of v+,k, (36) and (37) read
+uk(xk) = Qk(xk),
+and
+uk(x) ≥ Qk(x),
+in a neighborhood of xk,
+where
+Qk(x) = ǫkαk(P(x) + ck) + αkx+
+n.
+Note that
+(38)
+∇Qk = ǫkαk∇P + αken,
+thus,
+(39)
+∇Qk(xk) � 0,
+for k large.
+
+22
+M. BAYRAMI AND M. FOTOUHI
+Since uk is p-harmonic and Qk touches uk from below at xk, and ∇Qk(xk) � 0, by
+the equivalence of weak and viscosity solutions of p-harmonic functions, we get
+0 ≥ ∆pQk(xk)
+= div
+�
+|∇Qk(xk)|p−2∇Qk(xk)
+�
+= |∇Qk(xk)|p−2 ∆Qk(xk) + (p − 2) |∇Qk(xk)|p−4
+n
+�
+i,j=1
+Qkxi(xk)Qkxj(xk)Qkxixj(xk)
+= ǫk |∇Qk(xk)|p−2 ∆P(xk) + ǫk(p − 2) |∇Qk(xk)|p−4
+n
+�
+i,j=1
+Qkxi(xk)Qkxj(xk)Pxixj(xk).
+Now, dividing both sides by ǫk, and passing to the limit k → ∞, and recalling that
+∇Qk(xk) → αken,
+we conclude that
+∆P(x) + (p − 2)∂nnP(x) ≤ 0.
+Touching from above and reaching the opposite inequality is similar. Also, the
+reasoning of the case v = v− in the negative half ball B−
+1
+2 can be done similarly.
+Step 2: In this step, we show that J = ∅, when ℓ = ∞.
+Assume the contrary, since the set {v− > v+} is open in {xn = 0}, it contains a
+(n − 1)-dimensional ball
+B′
+ǫ(y′) := Bǫ((y′, 0)) ∩ {xn = 0} ⊂ J.
+Next, let P be the polynomial
+P(x) = A
+�
+n − 1
+2
+�
+x2
+n − |x′ − y′|2 − Bxn,
+where
+x = (x′, xn),
+for some constants A, B. We first choose suitable A = A(p) so that Lp(P) > 0. Notice
+that
+P < v+
+on
+{|x′ − y′| = ǫ} ∩ {xn = 0}.
+Moreover, we choose B ≫ A so that
+P < v+
+on
+Bǫ((y′, 0)).
+Now we can translate P first down and then up to find that there exists C such
+that P + C is touching v+ from below at a point x0 ∈ Bǫ((y′, 0)) ∩ {xn ≥ 0}. Since
+Lp(P) > 0, the touching point can not be in the interior of the (half) ball, and thus
+x0 ∈ B′
+ǫ(y′) ⊂ J.
+By using Lemma 4.11, there exists a sequence of points ∂Ω+
+uk ∋ xk → x0 and of
+functions Qk touching u+
+k from below at xk such that
+∇Q+
+k (xk) = αken + ǫkαk∇P(x0) + o(ǫk).
+Since x0 ∈ J, by (19) in Lemma 4.4, xk ∈ ∂Ω+
+uk \ ∂Ω−
+uk. Hence, by (ii) in Lemma 3.2
+λp
++ ≥ |∇Q+
+k (xk)|p ≥ αp
+k + pαp
+kǫk∂nP(x0) + o(ǫk).
+Now recalling (17), the definition of ℓ,
+−B = ∂nP(x0) ≤
+λp
++ − αp
+k
+pαp
+kǫk
++ o(1) → −∞.
+This contradiction proves that J = ∅.
+
+23
+Step 3: In this step, we check the transmission condition in (21) when ℓ = ∞.
+Let us show that
+αp
+∞∂nv+ − βp
+∞∂nv− ≤ 0,
+the opposite inequality can then be proved in a similar way. Suppose that there
+exist s and t with αp
+∞s > βp
+∞t and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such
+that
+P = sx+
+n − tx−
+n + ˜P,
+touches v strictly from below at a point x0 ∈ {xn = 0}∩B 1
+2 (note that {xn = 0}∩B 1
+2 = C
+due to the previous step and Lemma 4.4). By Lemma 4.11 there exists a sequence of
+points ∂Ω+
+uk ∪ ∂Ω−
+uk ∋ xk → x0 and a sequence of comparison functions Qk touching
+uk from below at xk and satisfying (35). In particular, xk � ∂Ω−
+uk \ ∂Ω+
+uk. We claim
+that xk ∈ ∂Ω+
+uk ∩ ∂Ω−
+uk. Indeed, otherwise by (A.1) in Lemma 3.1,
+λp
++ ≥ |∇Q+
+k (xk)|p,
+and, by arguing as Step 2, this contradicts ℓ = +∞. Hence, by (A.3) in Lemma 3.1
+λp
++ − λp
+− ≥ |∇Q+
+k (xk)|p − |∇Q−
+k (xk)|p
+= αp
+k − βp
+k + pǫk(αp
+ks − βp
+kt) + o(ǫk)
+= λp
++ − λp
+− + pǫk(αp
+ks − βp
+kt) + o(ǫk).
+Dividing by ǫk and letting k → ∞, we obtain the desired contradiction.
+Step 4: Here, we show that λp
+±∂nv± ≥ −ℓ on B 1
+2 ∩ {xn = 0}, when 0 ≤ ℓ < ∞.
+We focus on v− since the argument is symmetric. Let us assume that there exists
+t ∈ R with λp
+−t < −ℓ and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such that
+function
+P = txn + ˜P = tx+
+n − tx−
+n + ˜P,
+touches v− strictly from below at a point x0 ∈ {xn = 0} ∩ B 1
+2 . Let now xk and Qk be
+as in Lemma 4.11-(2). By optimality conditions
+λp
+− ≤ |∇Q−
+k (xk)|p = βp
+k + pǫkβp
+kt + o(ǫk).
+Since ℓ < ∞, we have βk = λ− + O(ǫk) and so the above inequality leads to
+− ℓ
+λp
+−
+= lim
+k→∞
+λp
+− − βp
+k
+pǫkβp
+k
+≤ t < − ℓ
+λp
+−
+,
+which is a contradiction.
+Step 5: We now show that λp
+±∂nv± = −ℓ on J, when 0 ≤ ℓ < ∞.
+By the previous step, it is enough to show that if there exists a polynomial ˜P with
+Lp( ˜P) < 0 and ∂n ˜P = 0 such that
+P = txn + ˜P = tx+
+n − tx−
+n + ˜P,
+touches v− strictly from above at a point x0 ∈ J, then λp
+−t ≤ −ℓ. Again, by Lemma
+4.11, we find points xk → x0 and functions Qk satisfying (34) and touching −u−
+k
+from below at xk. Since x0 ∈ J, by (19) in Lemma 4.4, xk ∈ ∂Ω−
+uk \ ∂Ω+
+uk. Hence, by
+Lemma 3.1,
+λp
+− ≥ |∇Q−
+k (xk)|p = βp
+k + pβp
+kǫkt + o(ǫk),
+which by arguing as above implies that λp
+−t ≤ −ℓ.
+
+24
+M. BAYRAMI AND M. FOTOUHI
+Step 6: In the last step, we show the transmission condition in (22) at points in
+C.
+Again by the symmetry of the arguments, we will only show that
+λp
++∂nv+ − λp
+−∂nv− ≤ 0
+on
+C.
+Let us hence assume that there exist s and t with λp
++s > λp
+−t and a polynomial ˜P
+with Lp( ˜P) > 0 and ∂n ˜P = 0 such that
+P = sx+
+n − tx−
+n + ˜P,
+touches v+ and v− strictly from below at x0 ∈ C. By Lemma 4.11, we find points
+xk → x0 and functions Qk satisfying (35). In particular xk � ∂Ω−
+uk \ ∂Ω+
+uk. By the
+previous step we know that λp
+−t ≥ −ℓ and thus λp
++s > −ℓ, since we are assuming
+λp
++s > λp
+−t ≥ 0. We now distinguish two cases:
+1) xk is one-phase point, namely xk ∈ ∂Ω+
+uk \ ∂Ω−
+uk. In this case
+λp
++ ≥ |∇Q+
+k (xk)|p = αp
+k + pαp
+kǫks + o(ǫk),
+which implies that
+λp
++s + ℓ = λp
++ lim
+k→��
+
+s +
+αp
+k − λp
++
+pαp
+kǫk
+
+ ≤ 0,
+in contradiction with λp
++s > −ℓ.
+2) xk is two-phase point, namely xk ∈ ∂Ω+
+uk ∩ ∂Ω−
+uk. Arguing as in Case 1), we
+have that, by Lemma 3.1,
+λp
++ − λp
+− ≥ |∇Q+
+k (xk)|p − |∇Q−
+k (xk)|p
+= αp
+k − βp
+k + pǫk(αp
+ks − βp
+kt) + o(ǫk)
+= λp
++ − λp
+− + pǫk(λp
++s − λp
+−t) + o(ǫk),
+which gives a contradiction with λp
++s > λp
+−t, as ǫk → 0.
+□
+4.3. Proof of Lemmas 4.2 and 4.3. We recall the following regularity results for
+the limiting problems.
+Lemma 4.12 (Regularity for the transmission problem). There exists a universal
+constant C = C(α∞, β∞, n, p) > 0 such that if v ∈ C0(B 1
+2 ) is a viscosity solution of (21)
+with ∥v∥L∞(B 1
+2 ) ≤ 1 then there exist v ∈ Rn−1, s, t ∈ R with αp
+∞s = βp
+∞t such that
+sup
+x∈Br
+���v(x) − v(0) − (v · x′ + sx+
+n − tx−
+n)
+��� ≤ Cr2,
+for every r ≤ 1
+4.
+Proof. For the proof when p = 2, we refer to [13, Theorem 3.2]. This result can be
+extended easily to the general case (for any p) by changing the coordinate such that
+the operator Lp = ∆ + (p − 2)∂nn transfer to the Laplacian.
+□
+Lemma 4.13 (Regularity for the two-membrane problem). There exists a universal
+constant C = C(λ±, n, p) > 0 such that if v is a viscosity solution of (22) with ∥v∥L∞(B 1
+2 ) ≤ 1
+then there exist v ∈ Rn−1, s, t ∈ R with λp
++s = λp
+−t ≥ −ℓ such that
+sup
+x∈B±
+r
+���v(x) − v(0) − (v · x′ + sx+
+n − tx−
+n)
+��� ≤ C(1 + ℓ)r
+3
+2 ,
+for every r ≤ rp,
+
+25
+where rp = 1
+4 for 1 < p ≤ 2 and rp =
+1
+4√
+p−1 for 2 < p.
+The proof of this lemma can be found in [12, Lemma 3.12] with a minor changes.
+To keep the paper self-contained we will provide a complete proof for our case in
+Appendix A.
+Now, the proof of Lemmas 4.2 and 4.3 by the regularity theory for the limiting
+problems and a classical compactness argument is available:
+Proof of Lemma 4.2. Toward a contradiction assume that for fixed γ ∈ (0, 1
+2) and M,
+we have a sequences of functions uk and numbers αk such that
+ǫk = ∥uk − Hαk,en∥L∞(B1) → 0,
+and
+λ+ ≤ αk ≤ λ+ + Mǫk,
+and fail (14) and (15) for some ρ and C which will be determined later. Note that
+by the second assumption above
+ℓ < Mλp−1
++
+< ∞.
+We let (vk)k be the sequence of functions defined in (18) and assume that they
+converge to a function v as in Lemma 4.4, note that ∥v∥L∞(B 1
+2 ) ≤ max( 1
+λ+ , 1
+λ− ). By
+Lemma 4.5, v solves (22) and thus by Lemma 4.13 there exist v ∈ Rn−1, s, t ∈ R
+satisfying λp
++s = λp
+−t ≥ −ℓ such that for all r ∈ (0, rp)
+sup
+x∈Br
+���v(x) − (v · x′ + sx+
+n − tx−
+n)
+��� ≤ C(1 + M)r
+3
+2 .
+Hence, we can fix ρ = ρ(λ±, γ, L, M, p, n) < rp such that C(1 + M)ρ
+1
+2 −γ ≤ 1
+2, so
+(40)
+sup
+x∈Bρ
+���v(x) − (v · x′ + sx+
+n − tx−
+n)
+��� ≤ ρ1+γ
+2L .
+We now set
+˜αk := αk(1 + ǫks) + δkǫk
+and
+ek :=
+en + ǫkv
+�
+1 + ǫ2
+k|v|2
+,
+where δk → 0 is chosen so that ˜αk ≥ λ+; note that the existence of such sequence is
+due to the condition λp
++s ≥ −ℓ since
+αk(1 + ǫks) =
+
+λ+ +
+ℓ
+λp−1
++
+ǫk + o(ǫk)
+
+ (1 + ǫks) ≥ λ+ + o(ǫk).
+We let Hk := H ˜αk,ek and note that
+| ˜αk − αk| + |ek − en| ≤ Cǫk,
+for a universal constant C > 0; we also have used (40) to find out that s is universally
+bounded. By the contradiction assumption we have
+ρ1+γ < 1
+ǫk
+sup
+Bρ
+|uk(x) − Hk(x)|
+≤ max
+�
+αk∥v+
+k − Hk − Hαk,en
+ǫkαk
+∥L∞(Ω+uk ∩Bρ), βk∥v−
+k − Hk − Hαk,en
+ǫkβk
+∥L∞(Ω−
+uk ∩Bρ)
+�
+.
+
+26
+M. BAYRAMI AND M. FOTOUHI
+To close the argument, we need to recall (40), the convergence of vk to v in the sense
+of Lemma 4.4 and the convergence of (again in the sense of Lemma 4.4)
+
+Hk(x) − Hαk,en(x)
+αkǫk
+xn > 0,
+Hk(x) − Hαk,en(x)
+βkǫk
+xn < 0,
+to the function
+(v · x′) + sx+
+n − tx−
+n.
+□
+Proof of Lemma 4.3. Arguing by contradiction one assume for fixed γ ∈ (0, 1) the
+existence of a sequence of functions uk and numbers αk, Mk → ∞ such that
+ǫk = ∥uk − Hαk,en∥L∞(B1) → 0,
+and
+αk − λ+
+ǫk
+≥ Mk → ∞,
+and fail (14) and (15) for some ρ and C which will be determined later. This implies
+that ℓ = ∞ and that the limiting function v obtained in Lemma 4.4 is a solution
+of (21). One then concludes the proof similar to the proof of Lemma 4.2 by using
+Lemma 4.12.
+□
+5. Regularity of the free boundary
+The last step in achieving the desired regularity result is to demonstrate that
+|∇u±| are C0,η for a suitable η > 0 up to the boundary, in the viscosity sense.
+Indeed, this shows that u± are solutions to the classical one-phase free boundary
+problem in its viscosity formulation and that the regularity will follow form [17].
+The arguments are similar to the ones in [25, Section 8] (see also [12, Section 4]).
+Therefore we only sketch the main steps and refer the reader to that paper for more
+details.
+Before stating the main results, we introduce some notation. For every x0 ∈ F(u)
+and every 0 < r < dist(x0, ∂D), we consider the function
+ux0,r(x) := u(x0 + rx)
+r
+,
+which is well-defined for |x| <
+1
+r dist(x0, ∂D) and vanishes at the origin. When
+x0 = 0, we denote u0,r by ur. Given a sequence rk > 0 such that rk → 0, we say that
+the sequence of functions ux0,rk is a blow-up sequence of u at x0. If a subsequence of
+ux0,rk convergs to v on every ball BR ⊂ Rn, we say that v is a blow-up limit of u at x0.
+Lemma 5.1. There exists ¯ǫ > 0 such that if the minimizer u satisfies (3), then at every
+point x0 ∈ ΓTP ∩ Br0 for a universal radius r0 > 0, there is a unique blow-up. Moreover, u
+is Lipschitz in Br0/2 and there exists η > 0 and a constant C0(n, p, Λ0, Λ1) > 0 such that
+for every x0, y0 ∈ ΓTP ∩ Br0/2, we have
+(41)
+|α(x0) − α(y0)| ≤ C0|x0 − y0|η
+and
+|e(x0) − e(y0)| ≤ C0|x0 − y0|η,
+for any η ∈ (0, 1
+3), where Hα(x0),e(x0) and Hα(y0),e(y0) are the blow-ups at x0 and y0, respec-
+tively.
+
+27
+Proof. Let L0 = Λ0 and L1 = 2Λ1 in Theorem 4.1 and find the universal constants
+ǫ0 > 0, ρ0 > 0 and C > 0. Choose ¯ǫ < min{(1 − ργ) Λ1
+2C, ǫ
+2} and r0 <
+¯ǫ
+Λ1 , then if the
+minimizer u satisfies (3) for some e ∈ Sn−1 and λ+ ≤ α ≤ Λ1, then
+∥ux0, 1
+2 − Hα,e∥L∞(B1) ≤ ¯ǫ + |Hα,e(x0)| ≤ 2¯ǫ,
+for any x0 ∈ ΓTP ∩ Br0. Now we can thus repeatedly apply Theorem 4.1 to obtain
+the sequences ux0, ρk
+2 (x) = 2
+ρk u(x0 + ρk
+2 x), max(L0, λ+) ≤ αk ≤ L1 and ek ∈ Sn−1 that
+∥ux0, ρk
+2 − Hαk,ek∥L∞(B1) ≤ 2¯ǫρkγ,
+|ek+1 − ek| + |αk+1 − αk| ≤ 2C¯ǫρkγ.
+This implies that αk and ek converge to some α = α(x0) and e = e(x0), respectively.
+Now let r ≤ 1
+2 be arbitrary and choose k ∈ N such that ρk+1 ≤ 2r ≤ ρk, then
+∥ux0,r − Hα(x0),e(x0)∥L∞(B1) ≤ 1
+ρ∥ux0, ρk
+2 − Hα(x0),e(x0)∥L∞(B1)
+≤ 1
+ρ
+�
+∥ux0, ρk
+2 − Hαk,ek∥L∞(B1) + ∥Hαk,ek − Hα(x0),e(x0)∥L∞(B1)
+�
+≤ C¯ǫρkγ.
+Therefore, there is ˜C = ˜C(n, p, Λ0, Λ1) such that for every r ≤ 1
+2 and x0 ∈ Br0,
+(42)
+∥ux0,r − Hα(x0),e(x0)∥L∞(B1) ≤ ˜Crγ,
+where γ ∈ (0, 1
+2).
+According to (42),
+∥u∥L∞(Br(x0)) ≤ (L1 + ˜C)r,
+r ≤ 1
+2,
+for every x0 ∈ ΓTP ∩ Br0. From this and the Lipschitz regularity around the one-
+phase points, Proposition 2.3, we conclude that u is Lipschitz in B r0
+2 ; see [3, Theorem
+2.3] or [11, Theorem 2.1].
+Next, for x0, y0 ∈ ΓTP ∩ B r0
+2 set r := |x0 − y0|1−η and η :=
+γ
+1+γ, and recall that u is
+Lipschitz (with a constant ˜L) to get
+∥Hα(x0),e(x0) − Hα(y0),e(y0)∥L∞(B1)
+≤ ∥ux0,r − Hα(x0),e(x0)∥L∞(B1) + ∥ux0,r − uy0,r∥L∞(B1) + ∥uy0,r − Hα(y0),e(y0)∥L∞(B1)
+≤
+�
+C0rγ +
+˜L
+r |x0 − y0| + C0rγ
+�
+= (˜L + 2C0)|x0 − y0|η.
+The conclusion now follows easily from this inequality; see e.g. [25, Lemma 8.8]
+for the details.
+□
+Lemma 5.2. Under the same assumptions of Lemma 5.1, there are C0,η continuous func-
+tions α : ∂Ω+
+u → R and β : ∂Ω−
+u → R such that α ≥ λ+, β ≥ λ− and u± are viscosity
+solutions of the one-phase problems
+∆pu+ = 0
+in
+Ω+
+u,
+|∇u+| = α
+on
+∂Ω+
+u,
+and
+∆pu− = 0
+in
+Ω−
+u,
+|∇u−| = β
+on
+∂Ω−
+u.
+
+28
+M. BAYRAMI AND M. FOTOUHI
+Proof. We will sketch the argument for u+. The proof of the case u− is similar.
+Clearly ∆pu+ = 0 in Ω+
+u. By (42) we have that, if x0 ∈ ΓTP ∩ D′, then
+(43)
+���u+(x) − α(x0) ((x − x0) · e(x0))+��� ≤ C0|x − x0|1+γ,
+for every x ∈ Br0(x0) ∩ Ω+
+u where r0 and C0 depends only on D′. In particular, u+ is
+differentiable on Ω+
+u up to x0 (in the classical sense) and |∇u+(x0)| = α(x0). On the
+other hand if x0 ∈ Γ+
+OP, then |∇u+(x0)| = λ+ is constant, in the viscosity sense.
+To close the argument, we only need to prove that α ∈ C0,η(∂Ω+
+u). Since α is
+η-H¨older continuous on ΓTP by Lemma 5.1, and constant on Γ+
+OP (in the viscosity
+sense), we just need to show that if x0 ∈ ΓTP is such that there is a sequence xk ∈ Γ+
+OP
+converging to x0, then α(x0) = λ+. To this end, let yk ∈ ΓTP be such that
+dist(xk, ΓTP) = |xk − yk|,
+and denote
+rk = |xk − yk|
+and
+uk(x) = 1
+rk
+u+(xk + rkx),
+and note that uk is a viscosity solution of the free boundary problem
+∆puk = 0
+in
+Ω+
+uk ∩ B1,
+|∇uk| = λ+
+on
+∂{uk > 0} ∩ B1.
+Since uk are uniformly Lipschitz in B 1
+2 (Proposition 2.3) they converge to a function
+u∞ which is also a viscosity solution of the same problem (see e.g. [17]). On the
+other hand, by (43) for two-phase point yk ∈ ΓTP and letting zk := xk−yk
+rk , we have
+that
+���uk(x) − α(yk) �(x − zk) · e(yk)�+��� ≤ C0rγ
+k|x − zk|1+γ.
+Suppose zk → z0 and passing to the limit
+u∞(x) = α(x0) ((x − z0) · e(x0))+ ,
+in B 1
+2 ,
+which gives that α(x0) = |∇u∞(0)| = λ+.
+□
+Proof of Theorem 1.1. Let x0 ∈ ΓTP = ∂Ω+
+u ∩ ∂Ω−
+u and let ǫ be the constant satisfies
+in [17, Theorem 1.1] and Lemma 5.1. By virtue of Lemma 5.2, we can apply [17,
+Theorem 1.1] to conclude that locally at x0 ∈ ΓTP the free boundaries ∂Ω±
+u are C1,η
+graphs. Since x0 is arbitrary, we conclude the proof.
+□
+6. Lipschitz regularity of solutions
+In this section, we are going to prove Theorem 1.2. We will follow the idea in
+[14].
+Proposition 6.1. Let u : D → R be a minimizer of JTP that 0 ∈ F(u) ∩ B1 ⊂ D. Then
+there exists constants L and δ such that one of the following alternative holds:
+(1) u is Lipschitz in Bδ and
+|∇u| ≤ C max(∥u∥L∞(B1), L),
+in Bδ,
+for some universal constant C.
+(2)
+1
+δ∥u∥L∞(Bδ) ≤ 1
+2 max(∥u∥L∞(B1), L).
+
+29
+Proof. Let δ be fixed, to be specified later. Assume by contradiction that there exist
+a sequence of Lj → ∞ and a sequence of solutions uj such that does not satisfy
+either (1) nor (2). Let Cj := max(∥uj∥L∞(B1), Lj) and define
+˜uj :=
+uj
+Cj
+,
+which satisfy
+∥ ˜uj∥L∞(B1) ≤ 1,
+and
+∥ ˜uj∥L∞(Bδ) ≥ δ
+2.
+By using Proposition 2.5, we get that ˜uj is a minimizer of the scaled functional (4)
+for σj =
+1
+Cj → 0. Thus up to a subsequence, ˜uj converges uniformly to a p-harmonic
+function u0. Hence by C1,α regularity for p-harmonic functions we get that
+(44)
+sup
+Br
+|u0(x) − ∇u0(0) · x| ≤ ˜Cr1+α,
+for all r ≤ 1,
+where the constant C is universal and also |∇u0(0)| ≤ ˜C. Now we distinguish two
+cases:
+Case I: |∇u0(0)| ≤ 1
+4.
+In this case, from (44) we deduce that
+1
+δ∥u0∥L∞(Bδ) ≤ 1
+4 + ˜Cδα ≤ 1
+3,
+if we choose δ small enough. Thus all uj for sufficiently large j will satisfy (2),
+which is a contradiction.
+Case II: |∇u0(0)| ≥ 1
+4.
+In this case we will use our flatness result in Theorem 1.1. Put ˜r = 2δ
+r0 in (44) where
+r0 is the radius obtained in Theorem 1.1 (we have also assumed 2δ ≤ r0)
+sup
+B1
+��� ˜uj,˜r(x) − ∇u0(0) · x
+��� ≤
+˜C
+rα
+0
+δα.
+Now let e = ∇u0(0)
+|∇u0(0)|, α = |∇u(0)| and βj =
+1
+Cj (λp
+− − λp
++ + αpCp
+j)
+1
+p , then
+∥uj,˜r − Hα,e∥L∞(B1) ≤ ∥uj,˜r − ∇u0(0) · x∥L∞(B1) + |α − βj| ≤ 2 ˜C
+rα
+0
+δα,
+for sufficiently large j.
+Applying Theorem 1.1 for some Λ0 ≤
+1
+4 and Λ1 ≥ ˜C
+and notice that uj,˜r is a minimizer of JTP for coefficients 1
+Cj λ±. Note that the critical
+flatness in Theorem 1.1 or Lemma 5.1 depends on Λ0 and Λ1 rather than coefficients
+λ±. Then we can find δ universally small such that uj,˜r satisfy in Lemma 5.1. In
+particular, uj,˜r is Lipschitz in B r0
+2 with a universal constant. It proves that uj is
+Lipschitz in Bδ.
+□
+Proof of Theorem 1.2. Let δ, C and L be the universal constants in Proposition 6.1.
+Assume 0 ∈ F(u) and let ˜L := max(∥u∥L∞(B1), L). We first show
+(45)
+∥u∥L∞(Bδk) ≤ C˜Lδk,
+∀k ≥ 0.
+By Proposition 6.1 either (1) or (2) holds. In the first case, u is Lipschitz in Bδ and
+|∇u| ≤ C˜L
+in Bδ.
+Thus (45) holds for all k ≥ 1.
+
+30
+M. BAYRAMI AND M. FOTOUHI
+If (2) holds, then
+∥u∥L∞(Bδ) ≤
+˜L
+2 δ.
+We now rescale and iterate. Define
+uk(x) := u(δkx)
+δk
+,
+which is also a minimizer of JTP and we can apply Proposition 6.1. If k0 is the
+smallest k for which uk satisfies (1), then for 0 ≤ k < k0 the item (2) holds and so
+∥u∥L∞(Bδk) ≤ ˜Lδk,
+for 0 ≤ k ≤ k0.
+Moreover, uk0 is Lipschitz in Bδ, with
+|∇uk0| ≤ C max(∥uk0∥L∞(B1), L) ≤ C max(˜L, L) = C˜L,
+in Bδ.
+Hence, (45) holds for all k ≥ k0. If uk satisfy the alternative (2) for all k, the estimate
+(45) will be obtained easily.
+Now for an arbitrary r choose k such that δk+1 ≤ r ≤ δk, then by (45) we get
+∥u∥L∞(Br) ≤ ∥u∥L∞(Bδk) ≤ C˜Lδk ≤ C˜L
+δ r.
+This is enough to obtain the Lipschitz continuity locally in D.
+□
+Appendix A. Proof of Proposition 2.5
+Proof of Proposition 2.5. By definition of vj and an easy computation, we get
+∇vj(x) =
+rj
+Sj
+∇uj(xj + rjx) = σj∇uj(xj + rjx).
+In order to show that vj is a minimizer of ˆJTP in BR, consider w that
+ˆJTP(w; BR) < ˆJTP(vj; BR),
+and
+w = v
+on ∂BR.
+Then ˆw(x) = w(
+x−xj
+rj ) will satisfy ˆw = uj on ∂Brj(xj) and by a simple calculation, we
+get that
+JTP( ˆw; Brj) =
+rn
+j
+σp
+j
+ˆJTP(w; BR) <
+rn
+j
+σp
+j
+ˆJTP(vj; BR) = ˆJTP(uj; Brj).
+This is a contradiction with the minimality of uj.
+Moreover, using |vj| ≤ M in B 4R
+3 and Caccioppoli’s inequality, we conclude that
+�
+BR
+|∇v±
+j |p dx ≤ 4pC(n)
+�
+B 4R
+3
+(v±
+j )p dx ≤ (4M)pC(n),
+for some C(n) > 0, indicating that ∥vj∥W1,p(BR) are uniformly bounded. Hence, from
+Proposition 2.4, i.e. the BMO estimate for the gradient, we obtain that for any q > 1
+and 0 < R < 1
+rj there exists a constant C = C(R, q) > 0 independent of j such that
+max
+�
+∥vj∥Cα(BR), ∥∇vj∥Lq(BR)
+�
+≤ C,
+for some α ∈ (0, 1) (if q > n, one can take α = 1 − n
+q by the Morrey’s inequality).
+Therefore, by a standard compactness argument, we have that, up to a subse-
+quence, vj converges to some function v0 as j → +∞ in Cα(BR) and weakly in
+W1,q(BR) for any q > 1, and for any fixed R. This completes the proof of (i).
+
+31
+For obtaining (ii), firstly, we prove that ∆pv0 = 0 in the positivity set of v0. Let
+E ⋐ {v0 > 0}. Then, there exists c > 0 such that v0 ≥ 2c in E. By the uniform
+convergence of vj to v0, we will have vj > c in E for large j. This implies that v0 is
+p-harmonic in E. Since E was arbitrary, we are done. Now, take 0 ≤ ϕ ∈ C1
+c(Rn)
+and s > 0. By using (v0 −s)+ϕ as a test function in the weak formulation of ∆pv0 = 0
+in the set {v0 > 0}, we have
+�
+{v0>s}
+|∇v0|pϕ dx = −
+�
+{v0>s}
+|∇v0|p−2 �∇v0 · ∇ϕ� v0 dx + s
+�
+{v0>s}
+|∇v0|p−2∇v0 · ∇ϕ dx.
+Letting s → 0 gives that
+�
+{v0>0}
+|∇v0|pϕ dx = −
+�
+{v0>0}
+|∇v0|p−2 �∇v0 · ∇ϕ� v0 dx.
+Similar argument holds for using test function (v0 + s)−ϕ and finally we get
+(46)
+�
+Rn |∇v0|pϕ dx = −
+�
+Rn |∇v0|p−2 �∇v0 · ∇ϕ� v0 dx.
+On the other hand since vjϕ∆pvj ≥ 0 (Theorem 2.1), we have
+(47)
+�
+Rn |∇vj|pϕ dx ≤ −
+�
+Rn |∇vj|p−2 �
+∇vj · ∇ϕ
+�
+vj dx.
+Usingtheuniformconvergenceofvj tov0 andtheweakconvergenceof|∇vj|p−2∇vj ⇀
+|∇v0|p−2∇v0 in L
+p
+p−1
+loc (Rn) (see [15]), we infer from (46) and (47) that
+(48)
+lim sup
+j→+∞
+�
+Rn |∇vj|pϕ dx ≤
+�
+Rn |∇v0|pϕ dx.
+Since also ∇vj ⇀ ∇v0 weakly in Lp
+loc(Rn), we have
+(49)
+�
+Rn |∇v0|pϕ dx ≤ lim inf
+j→+∞
+�
+Rn |∇vj|pϕ dx.
+It follows from (48), (49), and a simple compactness argument that
+(50)
+|∇vj|p → |∇v0|p,
+strongly in
+L1
+loc(Rn),
+and we get (ii).
+Finally, we prove the claim (iii). For this, notice that for any ψ ∈ C∞
+c (BR)
+(51)
+�
+BR
+|∇vj|p + σp
+j(p − 1)λp
++χ{vj>0} + σp
+j(p − 1)λp
+−χ{vj<0} dx
+≤
+�
+BR
+|∇(vj + ψ)|p + σp
+j(p − 1)λp
++χ{vj+ψ>0} + σp
+j(p − 1)λp
+−χ{vj+ψ<0} dx,
+because vj is a minimizer for ˆJTP defined in (4). Recall the strong convergence (50)
+along with the following standard inequality
+|∇(vj + ψ)|p ≤ 2p−1 �
+|∇vj|p + |∇ψ|p�
+,
+we get
+�
+BR
+|∇(vj + ψ)|p dx →
+�
+BR
+|∇(v0 + ψ)|p dx,
+
+32
+M. BAYRAMI AND M. FOTOUHI
+as j → +∞. Thus passing (51) to limit, we have
+�
+BR
+|∇v0|p + σp(p − 1)λp
++χ{v0>0} + σp(p − 1)λp
+−χ{v0<0} dx
+≤
+�
+BR
+|∇(v0 + ψ)|p + σp(p − 1)λp
++χ{v0+ψ>0} + σp(p − 1)λp
+−χ{v0+ψ<0} dx,
+for any ψ ∈ C∞
+c (BR). This implies (iii) and the proof of proposition finishes.
+□
+Appendix B. Proof of the regularity for two-membrane problem
+Proof of Lemma 4.13. For the given solution v, define w
+w±(x) = v±(x′, (p − 1)
+1
+2 xn) + ℓ(p − 1)
+1
+2
+λp
+±
+xn,
+x ∈ B±
+2rp.
+It is straightforward to check that w± is a viscosity solution of
+
+Lp(w±) = 0,
+in
+B±
+2rp,
+∂nw± ≥ 0,
+in
+B2rp ∩ {xn = 0},
+∂nw± = 0,
+in
+J = {w+ < w−} ∩ {xn = 0},
+λp
++∂nw+ = λp
+−∂nw−,
+in
+C = {w+ = w−} ∩ {xn = 0},
+w+ ≤ w−,
+in
+B2rp ∩ {xn = 0}.
+Furthermore one can easily check that
+(52)
+w±(x′, xn) = ˜w(x′, ∓xn) ∓ 1
+λp
+±
+wS(x′, ∓xn),
+where ˜w solves the following Neumann problem
+
+∆ ˜w = 0,
+on
+B−
+2rp,
+∂n ˜w = 0,
+on
+B−
+2rp ∩ {xn = 0},
+and wS is a solution to the thin obstacle (the Signorini) problem
+
+∆wS = 0,
+on
+B−
+2rp,
+wS ≥ 0,
+on
+B−
+2rp ∩ {xn = 0},
+∂nwS ≥ 0,
+on
+B−
+2rp ∩ {xn = 0},
+wS∂nwS = 0,
+on
+B−
+2rp ∩ {xn = 0}.
+The boundary data of ˜w and wS on ∂B2rp ∩ {xn < 0} will be obtained uniquely from
+(52). Clearly ˜w ∈ C∞(B−
+rp) with
+∥ ˜w∥Ck(B−
+rp) ≤ Ck∥ ˜w∥L∞(B−
+2rp).
+On the other hand, by [8], wS ∈ C1, 1
+2 (B−
+rp) with
+∥wS∥C1, 1
+2 (B−
+rp) ≤ C∥wS∥L∞(B−
+2rp).
+
+33
+From the last two estimates and the definition of w, it is easy to deduce the con-
+clusion of the lemma for (note that the positivity of wS along with its regularity
+necessitates that ∇′wS(0) = 0)
+v := ∇′ ˜w(0)
+and
+s± := (p − 1)− 1
+2
+λp
+±
+∂nwS(0) − ℓ
+λp
+±
+.
+□
+Appendix C. Proof of non-degeneracy
+Proof of Proposition 2.2. We will prove that for any k ∈ (0, 1), there exists a constant
+ck > 0 such that for any local minimizer of JTP and for any small ball Br(x0) ⊂ D
+if
+1
+r
+�
+⧸
+�
+Br(x0)
+(u±)p dx
+� 1
+p
+< ck
+then u± ≡ 0 in Bkr(x0).
+By symmetry of the problem, we prove only the case u+.
+Also, by the scale
+invariance, we can take r = 1 and x0 = 0 for simplicity. Now, let define
+ε :=
+1√
+k
+sup
+B √
+k
+u+.
+Since u+ is p-subharmonic, then by [22, Theorem 3.9]
+ε ≤
+1√
+k
+C(n, p)
+(1 −
+√
+k)
+n
+p
+�
+⧸
+�
+B1
+(u+)p dx
+� 1
+p
+.
+Also, let
+v(x) :=
+
+C1ε
+�
+e−µ|x|2 − e−µk2�
+in
+B √
+k \ Bk,
+0
+in
+Bk,
+where µ > 0 and C1 are such that
+(53)
+v���
+∂B √
+k
+:=
+√
+kε = sup
+B √
+k
+u+ ≥ u���
+∂B √
+k
+.
+By direct computation, it is straightforward to check that
+∇v(x) = −2C1εµxe−µ|x|2
+in
+B √
+k \ Bk,
+and
+∆pv(x) = C1ε(p − 1)(2µ)2|∇v|p−2e−µ|x|2
+�
+|x|2 − n + p − 2
+2µ(p − 1)
+�
+.
+Thus v is nonnegative p-superharmonic in B √
+k \ Bk, if µ is sufficiently small, say,
+(54)
+µ < n + p − 2
+2k(p − 1).
+On the other hand, since
+w := min(u, v) = u on ∂B √
+k,
+thanks to (53), by invoking the minimality of u we get
+(55)
+JTP(u, B √
+k) ≤ JTP(w, B √
+k).
+
+34
+M. BAYRAMI AND M. FOTOUHI
+Now, since we have
+JTP(w, B √
+k) =
+�
+Bk
+|∇w|p + (p − 1)λp
++χ{w>0} + (p − 1)λp
+−χ{w<0} dx
++
+�
+B √
+k\Bk
+|∇w|p + (p − 1)λp
++χ{w>0} + (p − 1)λp
+−χ{w<0} dx
+=
+�
+Bk∩{u≤0}
+|∇u|p + (p − 1)λp
++χ{u>0} + (p − 1)λp
+−χ{u<0} dx
++
+�
+B √
+k\Bk
+|∇w|p + (p − 1)λp
++χ{w>0} + (p − 1)λp
+−χ{w<0} dx.
+Therefore, from (55), {w > 0} ⊆ {u > 0} and {w < 0} = {u < 0}, we have that
+�
+Bk∩{u>0}
+|∇u|p + (p − 1)λp
++χ{u>0} + (p − 1)λp
+−χ{u<0} dx
+≤
+�
+B √
+k\Bk
+|∇w|p + (p − 1)λp
++χ{w>0} + (p − 1)λp
+−χ{w<0} dx
+−
+�
+B √
+k\Bk
+|∇u|p + (p − 1)λp
++χ{u>0} + (p − 1)λp
+−χ{u<0} dx
+≤
+�
+B √
+k\Bk
+|∇w|p − |∇u|p dx =
+�
+(B √
+k\Bk)∩{u>v}
+|∇v|p − |∇u|p dx
+≤ − p
+�
+B √
+k\Bk
+|∇v|p−2∇v · ∇ max(u − v, 0) dx
+= − p
+�
+B √
+k\Bk
+−∆pv max(u − v, 0) + div
+�
+|∇v|p−2∇v max(u − v, 0)
+�
+dx
+≤ − p
+�
+B √
+k\Bk
+div
+�
+|∇v|p−2∇v max(u − v, 0)
+�
+dx
+=p
+�
+∂Bk
+|∇v|p−2(∇v · ν)u+,
+where to get the last inequality we have used the fact that v is a p-superharmonic
+in B √
+k \ Bk. Moreover, by (54), we have that |∇v| = 2C1εµke−µk2 ≤ Cε on ∂Bk, for
+some C > 0. Thus
+(56)
+�
+Bk∩{u>0}
+|∇u|p + (p − 1)λp
++χ{u>0} dx ≤ p(Cε)p−1
+�
+∂Bk
+u+.
+
+35
+On the other hand, from trace estimate, Young’s inequality, we get
+(57)
+�
+∂Bk
+u+ ≤ C(n, k)
+��
+Bk
+u+ dx +
+�
+Bk
+|∇u+| dx
+�
+≤ C(n, k)
+
+sup
+Bk
+u+
+�
+Bk
+χ{u>0} dx +
+�
+Bk
+1
+p|∇u+|p + 1
+p′ χ{u>0} dx
+
+
+≤ C(n, k)
+�
+(ε
+√
+k + 1
+p′ )
+�
+Bk
+χ{u>0} dx + 1
+p
+�
+Bk
+|∇u+|p dx
+�
+≤ C0
+�
+Bk∩{u>0}
+|∇u|p + (p − 1)λp
++χ{u>0} dx,
+where p′ is the conjugate of p and
+C0 := C(n, k)
+�
+ε
+√
+k + 1
+�
+.
+Finally, putting together (56) and (57), we reach to
+�
+Bk(x0)∩{u>0}
+|∇u|p + (p − 1)λp
++χ{u>0} dx
+≤ p(Cε)p−1C0
+�
+Bk(x0)∩{u>0}
+|∇u|p + (p − 1)λp
++χ{u>0} dx,
+which implies that u ≡ 0 in Bk(x0) if ε is small enough. This completes the proof of
+the non-degeneracy property.
+□
+Declarations
+Data availability statement: All data needed are contained in the manuscript.
+Funding and/or Conflicts of interests/Competing interests: The authors declare
+that there are no financial, competing or conflict of interests.
+References
+[1] Alt, H. W., and Caffarelli, L. A. Existence and regularity for a minimum problem with free
+boundary. J. Reine Angew. Math. 325 (1981), 105–144.
+[2] Alt, H. W., Caffarelli, L. A., and Friedman, A. Axially symmetric jet flows. Arch. Rational Mech.
+Anal. 81, 2 (1983), 97–149.
+[3] Alt, H. W., Caffarelli, L. A., and Friedman, A. A free boundary problem for quasilinear elliptic
+equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11, 1 (1984), 1–44.
+[4] Alt, H. W., Caffarelli, L. A., and Friedman, A. Jets with two fluids. I. One free boundary. Indiana
+Univ. Math. J. 33, 2 (1984), 213–247.
+[5] Alt, H. W., Caffarelli, L. A., and Friedman, A. Jets with two fluids. II. Two free boundaries.
+Indiana Univ. Math. J. 33, 3 (1984), 367–391.
+[6] Alt, H. W., Caffarelli, L. A., and Friedman, A. Variational problems with two phases and their
+free boundaries. Trans. Amer. Math. Soc. 282, 2 (1984), 431–461.
+[7] Alt, H. W., Caffarelli, L. A., and Friedman, A. Compressible flows of jets and cavities. J.
+Differential Equations 56, 1 (1985), 82–141.
+[8] Athanasopoulos, I., and Caffarelli, L. A. Optimal regularity of lower dimensional obstacle
+problems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310, Kraev. Zadachi
+Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34] (2004), 49–66, 226.
+[9] Caffarelli, L., and Salsa, S. A geometric approach tofree boundary problems,vol. 68 of Graduate
+Studies in Mathematics. American Mathematical Society, Providence, RI, 2005.
+[10] Danielli, D., and Petrosyan, A. A minimum problem with free boundary for a degenerate
+quasilinear operator. Calc. Var. Partial Differential Equations 23, 1 (2005), 97–124.
+
+36
+M. BAYRAMI AND M. FOTOUHI
+[11] Danielli, D., Petrosyan, A., and Shahgholian, H. A singular perturbation problem for the p-
+Laplace operator. Indiana Univ. Math. J. 52, 2 (2003), 457–476.
+[12] De Philippis, G., Spolaor, L., and Velichkov, B. Regularity of the free boundary for the two-phase
+Bernoulli problem. Invent. Math. 225, 2 (2021), 347–394.
+[13] De Silva, D., Ferrari, F., and Salsa, S. Two-phase problems with distributed sources: regularity
+of the free boundary. Anal. PDE 7, 2 (2014), 267–310.
+[14] De Silva, D., and Savin, O. Lipschitz regularity of solutions to two-phase free boundary problems.
+Int. Math. Res. Not. IMRN, 7 (2019), 2204–2222.
+[15] DiBenedetto, E., and Manfredi, J. On the higher integrability of the gradient of weak solutions
+of certain degenerate elliptic systems. Amer. J. Math. 115, 5 (1993), 1107–1134.
+[16] Dipierro, S., and Karakhanyan, A. L. Stratification of free boundary points for a two-phase
+variational problem. Adv. Math. 328 (2018), 40–81.
+[17] Ferrari, F., and Lederman, C. Regularity of flat free boundaries for a p(x)-Laplacian problem with
+right hand side. Nonlinear Anal. 212 (2021), Paper No. 112444, 25.
+[18] Ferrari, F., and Lederman, C. Regularity of lipschitz free boundaries for a p(x)-laplacian problem
+with right hand side. Journal de Math´ematiques Pures et Appliqu´ees (2022).
+[19] Fotouhi, M., and Shahgholian, H. A minimization problem with free boundary for p-laplacian
+weakly coupled system. (2023) arxiv:2301.02236 (preprint).
+[20] Gilbarg, D., and Trudinger, N. S. Elliptic partial differential equations of second order. Classics
+in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
+[21] Karakhanyan, A. Regularity for the two-phase singular perturbation problems. Proceedings of
+the London Mathematical Society 123, 5 (2021), 433–459.
+[22] Mal´y, J., and Ziemer, W. P. Fine regularity of solutions of elliptic partial differential equations,
+vol. 51 of Mathematical Surveys and Monographs. American Mathematical Society, Providence,
+RI, 1997.
+[23] Petrosyan, A., and Valdinoci, E. Geometric properties of Bernoulli-type minimizers. Interfaces
+Free Bound. 7, 1 (2005), 55–77.
+[24] Tolksdorf, P. On the Dirichlet problem for quasilinear equations in domains with conical boundary
+points. Comm. Partial Differential Equations 8, 7 (1983), 773–817.
+[25] Velichkov, B. Regularity of the one-phase free boundaries. Lecture notes available at http://cvgmt.
+sns. it/paper/4367 (2019).
+Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
+Email address: masoud.bayrami1990@sharif.edu
+Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
+Email address: fotouhi@sharif.edu
+

69FKT4oBgHgl3EQf_y40/content/tmp_files/load_file.txt

@@ -0,0 +1,508 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf,len=507
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='11963v1  [hep-th]  27 Jan 2023  On 10 dimensional Exceptional Drinfel’d Algebras  Sameer Kumar1, Edvard T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Musaev2  Moscow Institute of Physics and Technology,  Institutskii pereulok 9, Dolgoprudny, 141700, Russia  Abstract  Based on the Mubarakzyanov’s classification of four-dimensional real Lie Algebras,  we classify ten-dimensional Exceptional Drinfel’d Algebras (EDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The classifica-  tion is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot  be represented as a product of a 3D Lie algebra and a 1D abelian factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' We show  that all obtained EDAs are inequivalent and conclude that there are no Nambu-Lie  U-dualities between 11D supergravity backgrounds within 10D EDAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  1kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='samip@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='edu  2musaev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='et@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='edu  1  Introduction  String theory is a background-dependent theory meaning that dynamics of the string is  defined on a fixed background of space-time fields including the metric, the dilaton, Kalb-  Ramond 2-form field, and Ramond-Ramond p-form fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The moduli space of these vacua  appears to be highly degenerate due to duality symmetries of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Some of them, such  as (abelian) T-dualities are exact perturbative symmetries of the superstring partition function  at all orders in α′ and gs [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' This implies that physics of the string does does not change if the  underlying space-time background is transformed by T-duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Given a non-abelian algebra of  isometries of a string background, abelian T-duality transformation rules can be generalized to  what is called non-abelian T-duality (NATD) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In contrast to the abelian case NATD is not  an exact quantum symmetry of the conformal theory due to problems with definition of winding  modes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' However, the NATD transformation map can be corrected to be a valid symmetry  at the leading order in α′ [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Using the notion of non-commutative currents, the non-abelian  T-duality transformations can be extended to Poisson-Lie T-dualities that are symmetries of  string theory in the same sense [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' While abelian T-duality starts from a background with  certain abelian isometries and preserves them, non-abelian T-duality breaks the non-abelian  algebra of initial isometries naively preventing from performing the inverse transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  The algebraic structure behind non-abelian T-duality symmetries, that is classical Drinfeld  algebras, reveals that the initial isometry becomes hidden inside the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' More specifically  classical Drinfeld algebra D is defined in terms of Manin triple (D, g, ˜g), where D is a Lie algebra  with non-degenerate quadratic form η, and g and ˜g are subalgebras maximally isotropic with  respect to the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The algebra g is commonly referred to as the geometric subalgebra, and  is responsible for the background space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  a group manifold or a coset space, while ˜g is  commonly referred to as the dual algebra and it is responsible for conservation laws of the  sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' To illustrate that, denote fabc and ˜fabc as structure constants of the algebras - g  and ˜g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Then the following holds,  [va, vb] = fab  cvc,  dJa = ˜fa  bcJb ∧ Jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1)  Here, vectors va define action of G = exp g on itself or on a coset space as δxi = vaiǫa, where  xi denote coordinates on the group (coset) manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Noether currents Ja = Ja idxi satisfy  the non-commutative conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  When ˜fabc = 0, the currents are conserved in the  usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Non-abelian T-duality simply maps g ↔ ˜g, hence vanishing ˜fabc get replaced by  2  non-vanishing fabc and the conservation law becomes non-commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The initial isometry  becomes hidden in g′ = ˜g and is no longer manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In this language the condition for classical  equations of motion for the string to satisfy is simply the Leibniz identity  [X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]],  X, Y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='2)  Here, the brackets are given by the following relations in terms of the generators (Ta, ˜T a) =  bas D:  [Ta, Tb] = fab  cTc,  [ ˜T a, ˜T a] = fc  ab ˜T c,  [ ˜T a, Tb] = ˜fc  ab ˜T c + fab  cTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='3)  In terms of structure constants, Leibniz identity is equivalent to Jacobi identities for fabc and  ˜fabc along with the following mixed identity  ˜fl  jkfmi  l + ˜fm  klfli  j + ˜fi  jlflm  k + ˜fm  jlfil  k + ˜fi  klflm  j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='4)  For a review of the algebraic construction behind Poisson-Lie T-dualities see [10], for a review of  applications of NATD see [11,12], for formulation of Poisson-Lie T-dualities in the supergravity  language see [13,14], for geometric aspects see [15,16]  In the most general case when both sets of structure constants are non-zero, one is able  to define the so-called Poisson-Lie duality transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' When dim g = d, these are such  O(d, d) maps CAB that preserve the structure of classical Drinfel’d double:  TA → CA  BTB,  TA = (Ta, ˜T a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='5)  There is a distinguished set of such transformations called Poisson-Lie (PL) T-dualities (plu-  ralities) when the map CAB relates different realization of the same Drinfeld algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The  simplest example is the swapping g ↔ ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For lower dimensional Lie algebras full classification  of all possible Poisson-Lie T-dualities or likewise of all equivalent Manin triples is available [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  This is based on classification of all possible dual algebras ˜g for each g belonging to the Bianchi  classification of three-dimensional real Lie algebras (for more on classification of Lie Algebras,  see for example [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' More generally, one may have maps CAB that relate different Drinfeld  algebras, for example, Yang-Baxter deformations that draw the interest since they preserve  integrability of the underlying sigma-model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  When extending abelian T-duality symmetries by S-dualities that are non-perturbative  3  transformations exchanging gs with g−1  s , one arrives at U-duality transformations that are  symmetries of M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Speaking more concretely, U-duality is a symmetry of classical field  equations of 11D supergravity compactified on a d-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' These are known as Cremmer-Julia  symmetries and are given by the exceptional groups Ed(d) [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In M-theory, whose low-energy  approximation is given by 11D supergravity, U-duality can be thought of as symmetries of BPS  states [22] or in terms of a Buscher-like procedure for M2-brane wrapping a 4-torus [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The  algebraic structure behind Poisson-Lie T-dualities can be extended to the so-called Exceptional  Drinfeld Algebras (EDA), that include the usual abelian U-dualities (Cremmer-Julia symme-  tries) [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Keeping the more detailed description of EDAs to the next section, we mention  that these are Leibniz algebras with generators TA on which exceptional group Ed(d) acts in the  same sense as the orthogonal group O(d, d) acts on generators of the classical Drinfeld double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Nambu-Lie U-dualities are then transformations that preserve the structure of the EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' What  differs these from the PL T-duality case is that there is no naturally defined analogue of the  swapping g ↔ ˜g, simply due to the following two facts: i) dimension of the geometric subalgebra  g of an EDA is never half of dimension of the EDA itself, ii) orthogonal completion of g inside  the EDA is not an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For this reason, searching for pairs of 11D geometries related by a  Nambu-Lie U-duality is an extremely complicated task for a general EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' At the moment few  examples of such dualities between 11D backgrounds and solutions to Type IIB supergravity  equations are known [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In [30] a general procedure has been suggested similar to the  natural swapping g ↔ ˜g based on external automorphisms of Ed(d) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Further it has been  used to generate few examples of mutually dual backgrounds in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  In this work, we elaborate further on the results of [30,31] that in particular state that there  are no non-abelian U-dualities in the defined sense between 11D background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The narrative  we follow is along the same lines as [32] where a full classification of 6D Exceptional Drinfeld  Doubles based on 3D geometric algebras has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Starting from the classification  of four-dimensional real Lie algebras [33], we construct all possible EDAs for a representative  of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For each pair of such obtained EDAs we search for an SL(5) transformation  relating them, that would mean existence of a Nambu-Lie U-duality between backgrounds that  geometrically realize the corresponding geometric algebras g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Restricting ourselves to only such  4D real Lie algebras that do not contain a 1d (abelian) factor we find no such transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  The restriction is motivated by the interest only in dualities between 11D background as maps  from 11D→IIB are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In the beginning of Section 2 we briefly review the con-  struction of Exceptional Drinfel’s Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1 we discuss the geometric realization  of EDAs and Nambu-Lie U-dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='2, we present classification of 10D EDAs,  4  given the conditions stated in the preceding section and state the main results of the paper  2  Exceptional Drinfel’d Algebras  Before proceeding with the classification of 10d EDAs, let us briefly review the algebraic  construction following [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' We will be focusing on the 10d case where generators of the  exceptional Drinfeld algebra ED4 are collected into the 10-dimensional representation of the  SL(5) group basED4 = {TAB}, where A, B = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Multiplication table is then given by  TAB ◦ TCD = i  2FAB,CD  GHTGH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1)  The structures constants FAB,CDGH are defined by the following relations  FAB,CD  GH = 4FAB,[C  [GδH]  D]  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='2)  FAB,C  D = 1  2ǫABCGHZGHD + 1  2δD  [ASB]C + 1  3δD  [AτB]C + 1  6δD  C τAB,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='3)  where τ is antisymmetric and S is a symmetric tensor, while Z[ABC] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For the algebra to be an  EDA, components of the constants ZABC, SAB and τAB under decomposition SL(5) ←֓ GL(4)  must be defined as  Zabc = 1  6ǫabcdfde  e + 1  4ǫabeffef  c,  S5a = fab  b − 3Za,  τ5a = 9  2Za − 1  2fab  b  Z5[a,b] = 1  6  ˜fc  abc,  Sab = 1  3  ˜f(a  cdeǫb)cde,  τab = −1  6  ˜f[a  cdeǫb]cde  Zab,5 = −Z5a,b + Z5b,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='4)  The constants FAB,CD have the same structure as the embedding tensor of [34], and in this  language the above construction implies that only the geometric flux (anholonomy coefficients)  and Q-flux are turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The former is given by the structure constants fabc of the geometric  subalgebra g and the latter is given by ˜fabcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The algebra is Leibniz with the fundamental  identity given by the quadratic relations analogous to those of 7d maximal gauged SUGRA [34]:  2F G  AB[CF I  GD],H − F I  ABGF G  CDH + F G  ABHF I  CDG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='5)  5  In terms of structure constants fabc and dual constants fabcd, the conditions become  6ff[a  [c ˜fb]  de]f + fab  f ˜ff  cde − 1  3  ˜f[a  cdefb]f  f = 0  ˜fc  abcfbd  d = 0,  fde  a ˜f bde  c  − 1  3  ˜fc  abdfde  e = 0  ˜fc  abg ˜fg  def − 3 ˜fc  g[de ˜fg  f]ab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='6)  The last of the above equations is also referred to as the dual Jacobi condition, just as the  dual conditions in the Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' It describes the internal (isolated) relations between the  structure constants of the dual algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  As in the case of Classical Drinfeld Algebra, in general, there might exist multiple equivalent  choices of the geometric subalgebra g inside an EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Proper generalization of the isometry  condition to the case of exceptional structures has been given in [25,26] and can be written as  follows  ǫABCDETAB ⊗ TCD  ����  g⊗g  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='7)  In other words, for a given EDA, its geometric subalgebra g is spanned by such a subset of the  whole set of generators {TAB} that satisfy the above condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For Classical Drinfel’d Double  the condition is ηABTA ⊗ TB = 0, implying that one may, for example, take bas g = {Ta},  or bas g = { ˜T a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For EDAs, one choice is self-evident - bas g = {T5a}, while presenting an  alternative choice is usually a hard task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' This implies that there is no natural generalization  of the Non-Abelian T-duality transformation swapping g ↔ ˜g in the case of EDAs, although  certain progress in defining an analogue of these swappings has been done in [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1  Geometric realization and dualities  The algebraic structure of EDAs stands behind Nambu-Lie U-dualities of supergravity so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' These can map solutions to 11D supergravity equations into each other or into Type  IIB supergravity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Such duality transformations map the group manifolds correspond-  ing to different choices of the geometric subalgebra g into each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For more detailed and  concrete algorithm of constructing mutually dual backgrounds see [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Below, we will briefly  recall the overall construction and highlight relations to Exceptional Field Theory (ExFT) that  provide convenient variables for writing such duality maps [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' These are Ed(d)-covariant  field theories defined in 11-dimensional space-time with an explicit split - 11 = D + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The  D-dimensional space-time is usually referred to as the external, the d-dimensional space is  6  usually referred to as internal, although no compactification is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In the d = 4 case  relevant to the present discussion, field content of the theory includes the external metric gµν,  ten vector fields AµMN, five 2-form fields BµνM, and 14 scalar fields parametrized by a coset  element MMN ∈ SL(5)/SO(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The indices µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' , 6 parameterize directions of the external  space-time whereas the indices M, N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' , 5 belong to the 5 of SL(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For more details of  the construction see [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Here we are interested in the special case where all fields transform-  ing in irreps of SL(5) can be decomposed in terms of matrices EABMN (generalized vielbeins)  geometrically realizing an EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In compact notation one writes  [EAB, ECD] = FAB,CD  EFEEF,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='8)  where the constants FAB,CDEF are precisely the structure constants of the EDA and the brackets  denote the so-called generalized Lie derivative of ExFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Generalized vielbeins are parametrized by fields of 11D supergravity in the 11 = 7 + 4 split  transforming as scalars under 7-dimensional diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Introducing a unity matrix MAB  compose  MMN,KL = 2EMN  ABEKL  CDMACMBD = MMKMNL − MMLMNK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='9)  The symmetric matrix  mMN = e− φ  2  �  |g|− 1  2gij  Vi  Vj  |g|  1  2(1 + V 2)  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='10)  is then defined in terms of the 4d metric gmn on the group manifold, the vector V m = 1  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='ǫmnklCnkl  and a scalar field eφ = |g7|1/7 which is the determinant |g7| of external 7 dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  The metric gmn on the group manifold is defined as usual in terms of Maurer-Cartan forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Let g ∈ G = exp g be an element of the group G whose Lie algebra is g, then 1-forms on the  group manifold g−1dg ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In components we have  g−1dg = rm  aTadxm,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='11)  where xm are some coordinates on the group manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Given an EDA and a choice of the isotropic subalgebra g one can explicitly construct  the corresponding generalized vielbein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' A step-by-step algorithm of this procedure based on  constructing adjoint action of eh ∈ G for some h ∈ g on an element of EDA can be found  in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' An alternative choice of the isotropic subalgebra, if exists, is related to the given one  by an SL(5) transformation  T ′  AB = CA  CCB  DTCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='12)  7  If this transformation respects the structure of EDA, then the alternative isotropic subalgebra  is spanned by T ′  5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Structure constants of the EDA then transform as  F ′  A′B′,C′D′ = CA′ACB′BCC′CCD  D′FAB,C  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='13)  Note that not any such matrix corresponds to a Nambu-Lie U-duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Indeed,  one can always perform a GL(4) transformation on generators of a given algebra g thus changing  explicit realization of the corresponding EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Two EDA’s related by such transformation then  correspond to 11D backgrounds related by a coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Another trivial choice  is  CA  B =  �  14×4  λm  0  1  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='14)  that corresponds to simply a gauge transformation of the 3-form Cmnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' To avoid counting  of EDA’s related by a rotation of the basis of their isotropic subalgebras we first classify  Exceptional Drinfeld Algebras using classification of 4D real Lie Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='2  Classification of 10 dimensional EDA’s  The main goal of this work is to investigate relations between 10d EDAs that correspond  to Nambu-Lie U-duality transformations of 11-dimensional supergravity backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For this  purpose, we start with a classification of 10D EDAs of certain class based on the classification of  4-dimensional real Lie Algebras by Mubarakzyanov [33] (for a review in English see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Since  explicit examples of Nambu-Lie U-dualities between 11D and Type IIB backgrounds are known  in the literature, we are interested here only in EDAs constructed on 4d real Lie Algebras g4  that cannot be decomposed into a sum g4 = g4 ⊕ g1, where g3 is a 3d Lie algebra and g1 is  1-dimensional Abelan factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' We list all relevant 4d real Lie Algebras in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T1  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='5  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = AT1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = BT2  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = CT3  ABC̸= 0  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='9  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T3] = T1  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = 2AT1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = AT2 − T3  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2 + AT3  A > 0  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='2  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = βT1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2 + T3  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='6  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = AT1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = BT2 − T3  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2 + BT3  A > 0  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='10  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T3] = T1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T3] = T2  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = −T2  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T1  8  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='3  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T1  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='7  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T3] = T1  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = 2T1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2 + T3  2g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T2] = T1  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T3  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='4  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T1 + T2  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2 + T3  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='8  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T3] = T1  [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = (1 + β)T1  [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = T2  [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T4] = βT3  β ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 1]  Table 1: Classification of 4-dimensional indecomposable  real Lie algebras g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='n with n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The algebra  2g2,1 is decomposable, however does not have a u(1) fac-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  To arrive at the corresponding classification of 10d EDAs, we solve quadratic constraints  for each class in the table above to find all possible sets of the dual structure coefficients ˜fdabc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  To solve the equations we use mathematical software Mathematica , that gives us all the 4  dimensional EDAs in the chosen class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  The result is listed in Table 2, where only unique  combinations of indices are explicitly given in the coefficients of the underlying algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The  rest of the indices are obtained by the antisymmetric property of the structure coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  EDA  Structure Constants ˜f abcd  g4,1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1232 = ˜f 1344, ˜f 1242 = ˜f 1343  ˜f 1234 =  ˜f 1233 ˜f 1344 − ˜f 1244 ˜f 1344  2 ˜f 1343  ˜f 1243 = ( ˜f 1244 − ˜f 1233) ˜f 1343  2 ˜f 1344  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1232 = − ˜f 1344, ˜f 1244 = ˜f 1233, ˜f 2344 = ˜f 1231  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1242 = ˜f 1343 ˜f 1244 = ˜f 1233  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1244 = ˜f 1233, ˜f 2344 = ˜f 1231  g4,2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1244 = −1  3 (1 + 2β) ˜f 1233  6  ˜f 2344 =  1  3β(β − 4) ˜f 1231  g4,3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 2344 = 1  3 ˜f 1231  g4,4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1244 = − ˜f 123  3  g4,5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1244 = −2A−2B+C  3C  ˜f 1233  9  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1344 =  1  3B(2A − B + 2C) ˜f 1232  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 2344 =  1  3A(A − 2B − 2C) ˜f 1231  g4,6  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 2344 =  1  3A(A − 2B − 2C) ˜f 1231  g4,7  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1244 = −5  3 ˜f 1233  g4,8  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1244 = − 1  3β(4 − β) ˜f 1233  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1344 = 1  3(1 + 4B) ˜f 1232  g4,9  ˜f abcd = 0 or imaginary  g4,10  ˜f abcd = 0  2g2,1  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  ˜f 1234 = ˜f 1232, ˜f 1342 = ˜f 1344  Table 2: All possible structure constant of 10d EDAs for  each g4,n with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' , 10 and 2g2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The constants  A, B, C, β are the same as in the previous table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Hence, given we are interested only in real non-trivial EDAs, we end up with 16 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  A natural question would be: whether there exists a pair of EDAs in this set that are equivalent  up to an SL(5) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' This would mean that the same EDA can be generated by two  4d Lie Algebras that belong to different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In the supergravity language this would mean  existence of a Nambu-Lie U-duality between 11D backgrounds geometrically realizing this pair  of 4d Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Result of our calculations is that there are no such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' To arrive at this  statement we used Mathematica software and explicitly solve equations on components of the  matrix CAB for each pair of 16 algebras with no further restrictions on the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' This  means, that although in Table 2 we list algebras as though all explicitly written dual structure  constants are non-vanishing, our code does not assume that [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  3  Discussion  In this work we obtain a classification of 10-dimensional EDA based on the classification of  4-dimensional real Lie Algebras by Mubarakzyanov [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' We intentionally restrict only to such  4d algebras that cannot be decomposed into a 3d algebra and a 1d abelian factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='e, we are  interested in Nambu-Lie U-dualities between 11d backgrounds, rather than dualities between  11D and Type IIB solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' More specifically, we look only at EDAs whose isotropic (geomet-  ric) subalgebra is given by g4,n with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=', 10 and 2g2,1 in terms of the Mubarakzyanov’s  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Given these restrictions the classification of EDAs is summarized in Table 2,  10  where 16 non-trivial EDAs are listed in terms of dual structure constants ˜fabcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  The important question we were interested in is whether there exists a Nambu-Lie U-duality  between 11D solutions and supergravity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Equivalently, in the algebraic language:  whether any of the sixteen exceptional Drinfeld algebras are equivalent up to an SL(5) trans-  formation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' For that we computed the explicit form of all possible transformations between all  possible pairs of EDA listed in Table 2 of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In our findings, we discovered that  none of the EDA pairs except the (ED2, ED4) possess transformation matrices, taking the basis  of one EDA to another, with a non-zero determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Moreover, the transformation relating the  aforesaid algebras - ED2 and ED4 is simply a GL(4) transformation rotating the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hence,  these two solutions are equivalent and their geometric realizations can be mapped into each  other by a 4D coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hence, there are no Nambu-Lie U-dualities inside  SL(5) exceptional Drinfeld algebras relating 11D backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Note however, this does not  rule out transformations between 11D and Type IIB backgrounds, explicit examples of which  are known [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Previously in [30] the same has been shown for transformations involving  external automorphisms of the algebra sl(5), suggested as the natural analogue of Non-Abelian  T-duality transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Here we complete the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  There are further directions to extend this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' The most obvious task is to complete  the classification including all 4D real Lie algebras and list sets of EDA’s mutually Nambu-  Lie U-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Less straightforward is to increase the dimension of the geometric subalgebra g  by one and consider 16D Exceptional Drinfeld Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Unfortunately, there is no ready to  use classification of 5D real Lie algebras, but certain restricted classifications are present in  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Some useful examples can be found in [40–43], for a review see [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Another  interesting direction of further research is to list those EDAs from our classification that can be  obtained as generalized Yang-Baxter deformations of the trivial EDA when all dual structure  constants are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In other words, to answer the question: for which algebras in Table 2 dual  structre constants can be represented in the form  ˜fa  bcd = re[bcfae  d],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1)  where rabc is completely antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' In the case of classical Drinfeld algebras such trans-  formations are known to preserve integrability of the 2d sigma-model on the corresponding  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  There is no analogous statement for 3d sigma-models describing membranes  propagating on 11d supergravity backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' However, such defined generalized Yang-Baxter  deformations are of certain interest (see [45] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  11  Acknowledgments  This work has been supported by the Foundation for the Advancement of Theoretical  Physics and Mathematics “BASIS”, grant No 21-1-2-3-1 and by Russian Ministry of Education  and Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Buscher, “A Symmetry of the String Background Field Equations,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  B194 (1987) 59–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Buscher, “Path Integral Derivation of Quantum Duality in Nonlinear Sigma  Models,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B201 (1988) 466–472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Giveon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Porrati, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Rabinovici, “Target space duality in string theory,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 244 (1994) 77–202, arXiv:hep-th/9401139 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [4] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' de la Ossa and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Quevedo, “Duality symmetries from nonAbelian isometries in  string theory,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B 403 (1993) 377–394, arXiv:hep-th/9210021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Giveon and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Rocek, “On Nonabelian duality,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B421 (1994) 173–190,  arXiv:hep-th/9308154 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hassler and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Rochais, “α′-Corrected Poisson-Lie T-Duality,” Fortsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 68  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 9, (2020) 2000063, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='07897 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Borsato and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Wulff, “Quantum Correction to Generalized T Dualities,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 125 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 20, (2020) 201603, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='07902 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Klimcik and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Severa, “Dual nonAbelian duality and the Drinfeld double,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B 351 (1995) 455–462, arXiv:hep-th/9502122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Klimˇc´ık and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Severa, “Poisson-Lie T duality and loop groups of Drinfeld doubles,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B372 (1996) 65–71, arXiv:hep-th/9512040 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Klimcik, “Poisson-Lie T duality,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 46 (1996) 116–121,  arXiv:hep-th/9509095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  12  [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Sfetsos, “Recent developments in non-Abelian T-duality in string theory,” Fortsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 59 (2011) 1149–1153, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='0537 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Thompson, “An Introduction to Generalised Dualities and their Applications to  Holography and Integrability,” PoS CORFU2018 (2019) 099, arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='11561  [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hassler, “Poisson-Lie T-Duality in Double Field Theory,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B 807 (2020)  135455, arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='08624 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Demulder, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hassler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Thompson, “An invitation to Poisson-Lie T-duality  in Double Field Theory and its applications,” PoS CORFU2018 (2019) 113,  arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='09992 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Bugden, “Non-abelian T-folds,” JHEP 03 (2019) 189, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='03782 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hlavat´y and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Petr, “T-folds as Poisson-Lie plurals,” arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='08387 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hlavaty and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Snobl, “Classification of 6-dimensional manin triples,”  arXiv:math/0202209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Popovych, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Boyko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Nesterenko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lutfullin, “Realizations of  real low-dimensional Lie algebras,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' A 36 (2003) 7337–7360,  arXiv:math-ph/0301029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Klimˇc´ık, “On integrability of the Yang-Baxter sigma-model,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 50  (2009) 043508, arXiv:0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='3518 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Cremmer and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Julia, “The SO(8) supergravity,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B159 (1979) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Cremmer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Julia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Pope, “Dualization of dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=',” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  B523 (1998) 73–144, arXiv:hep-th/9710119 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [22] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Obers and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Pioline, “U duality and M theory,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 318 (1999) 113–225,  arXiv:hep-th/9809039 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Duff, “Duality rotations in string theory,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B335 (1990) 610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Duff and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Lu, “Duality rotations in membrane theory,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B347 (1990)  394–419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  13  [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Sakatani, “U-duality extension of Drinfel’d double,” PTEP 2020 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 2, (2020)  023B08, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='06320 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Malek and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Thompson, “Poisson-Lie U-duality in Exceptional Field Theory,”  JHEP 04 (2020) 058, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='07833 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Malek, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Sakatani, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Thompson, “E6(6) exceptional Drinfel’d algebras,”  JHEP 01 (2021) 020, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='08510 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Blair, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Thompson, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Zhidkova, “Exploring Exceptional Drinfeld  Geometries,” JHEP 09 (2020) 151, arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='12452 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Blair and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Zhidkova, “Generalised U-dual solutions in supergravity,” JHEP  05 (2022) 081, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='01838 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [30] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Musaev, “On non-abelian U-duality of 11D backgrounds,” arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='01213  [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [31] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Musaev and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Sakatani, “Non-Abelian U duality at work,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' D 104  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 4, (2021) 046015, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='13263 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [32] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hlavaty, “Classification of 6D Leibniz algebras,” PTEP 2020 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 7, (2020) 071B01,  arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='06164 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Mubarakzyanov, “On solvable Lie algebras,” Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Vys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Ucheb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Zaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Matematika 1  (32) (1963) 114–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Samtleben and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Weidner, “The Maximal D=7 supergravities,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B725  (2005) 383–419, arXiv:hep-th/0506237 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [35] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hohm and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Samtleben, “U-duality covariant gravity,” JHEP 1309 (2013) 080,  arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='0509 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [36] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Hohm and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Samtleben, “Exceptional Form of D=11 Supergravity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  111 (2013) 231601, arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1673 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [37] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Musaev, “Exceptional field theory: SL(5),” JHEP 02 (2016) 012,  arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='02163 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [38] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Sakatani, “Type II DFT solutions from Poisson-Lie T-duality/plurality,”  arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='12175 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' [PTEP,073B04(2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  14  [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Kumar and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Musaev, “eda10,” 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='com/emusaev/eda10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [40] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Darijani and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Usefi, “Classification of 5-dimensional restricted Lie algebras over  perfect fields, I,” Journal of Algebra 464 (2016) 97–140, arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='8377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Andrada and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Fino, “A class of Sasakian 5-manifolds,” Transformation Groups 14  (2009) 493–512, arXiv:0807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1800 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='dg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [42] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Bieszk, “Classification of Five-Dimensional Lie Algebras of Class T2,” Demonstratio  Mathematica 30 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' 2, (1997) 403–424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='1515/dema-1997-0219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Calvaruso and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Perrone, “Five-dimensional paracontact Lie algebras,” Differential  Geometry and its Applications 45 (2016) 115–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='com/science/article/pii/S0926224516000024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [44] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Prieto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Martel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Vald´es, and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Tenorio, “A historical review of  the classifications of Lie algebras,” Revista De La Union Matematica Argentina 54  (2013) 75–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  [45] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Gubarev and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content=' Musaev, “Integrability structures in string theory,”  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='06486 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}
+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'}

6tAyT4oBgHgl3EQfQfaP/content/tmp_files/2301.00047v1.pdf.txt

@@ -0,0 +1,375 @@
+The Simplest Proof of Parikh’s Theorem via
+Derivation Trees
+Alexander Rubtsov � �
+National Research University Higher School of Economics
+Moscow Institute of Physics and Technology
+Abstract
+Parikh’s theorem is a fundamental result of the formal language’s theory. There had been published
+many proofs and many papers claimed to provide a simplified proof, but most of them are long and
+still complicated. We provide the proof that is really short, simple and discloses the nature of this
+fundamental result. We follow the technique closed to the original Parikh’s paper and our proof is
+similar to the proof by Ryoma Sin’ya 2019, but we provide more detailed exposition and pretend to
+more simplicity as well. We achieve the simplicity via nonconstructivenes that allows us avoiding
+many difficulties met by other proofs.
+2012 ACM Subject Classification Theory of computation → Grammars and context-free languages
+Keywords and phrases Formal Languages, Context-Free Languages, Parikh’s Theorem
+Funding The paper was supported by RFBR grant 20-01-00645
+Acknowledgements I want to thank Dmitry Chistikov for the helpfull discussion of the paper
+1
+Introduction
+Parikh’s theorem [6] is a fundamental theorem of the formal language’s theory. There had
+been published many proofs (see [1], [4] for the survey of different proofs and detailed
+exposition on the topic). Many papers claimed to provide a simplified proof ([2] looks to
+be the most known), but most of them are long and still complicated. Despite really short
+and simple proofs are already known (e.g., [7]), papers with another proofs continues to be
+published [5]. We provide the proof that is really short, simple and discloses the nature of
+this fundamental result. Our proof is based on derivations trees so as the original Parikh’s
+proof [6]. As Parikh, we decompose derivation trees into small ones; we consider similar
+kinds of trees: ordinary derivation trees and auxiliary ones (with only nonterminal in the
+crown that is the same as the root). We get rid of duplicates in auxiliary trees and pump
+them. While our technique is similar to the original proof in general, it is simpler since we
+do not have restrictions on grammar and other technical issues (like considering derivation
+trees that contains nonterminals from a fixed subset). The simplicity of our construction is
+based on nonconstructivenes that allows us avoiding many technical issues. Also trees (in
+the decomposition) in our construction have linear height (in the number of nonterminals)
+while in the Parikh’s construction trees have quadratic height. In our proof we generalize the
+idea of derivation from words to trees and translate the idea of the pumping lemma to the
+trees as well. It makes our construction clear enough to explain students during a lecture in
+a basic course of formal languages and automata theory, and so we hope that the detailed
+version will help to spread it in the community.
+Our technique is similar to [8]. Unlike concise exposition in [8], we provide more detailed
+exposition and explain the intuition of the proof. And we do not use auxiliary results for our
+proof (thanks to nonconstructivenes).
+arXiv:2301.00047v1  [cs.FL]  30 Dec 2022
+
+2
+The Simplest Proof of Parikh’s Theorem via Derivation Trees
+2
+Definitions
+We denote by N non-negative integers. Let Σk = {a1, . . . , ak} be an alphabet, k ⩾ 1. We
+denote by Ψ : Σ∗
+k → Nk the Parikh mapping that maps a word w to its Parikh’s image the
+vector (|w|a1, |w|a2, . . . , |w|ak), where |w|aj is the number of letters aj in w. We denote the
+Parikh’s image of a language L ⊆ Σ∗
+k in the natural way:
+Ψ(L) = {Ψ(w) | w ∈ L}.
+A set S ⊆ Nk is linear if there exist vectors v0, v1, . . . , vm ∈ Nk such that
+S = {v0 + v1t1 + v2t2 + . . . vmtm | t1, . . . tm ∈ N}.
+A set is semi-linear if it is a union of finitely many linear sets.
+▶ Theorem 1 (Parikh). For each context-free language L the set Ψ(L) is semi-linear.
+We use classical notation for context-free grammars [3]. We denote by N the set of
+nonterminals, and denote nonterminals by capital letters. Small letters from the end of
+the alphabet denote words and Greek letters denote words over the alphabet Σ ∪ N, called
+sentential forms.
+Our proof uses derivation trees and exploits the idea similar to the classical proof of the
+pumping lemma. We use derivation trees not only for words, but also for sentential forms of
+a special kind. We call a tree that corresponds to a derivation of the form A
+∗=⇒ uAv a block
+tree. We call derivation trees for words ground trees. We say that a tree T (a ground or a
+block one) is minimal if it does not have a block tree as a subtree, i.e. T does not have the
+form (S
+∗=⇒ xAz
+∗=⇒ xuAvz
+∗=⇒ xuβvz). Hereinafter, we describe a tree by any derivation
+corresponding to the tree.
+S
+A
+A
+x
+u
+y
+v
+z
+S
+A
+y
+x
+z
+A
+A
+u
+v
+Figure 1 Decomposition of tree T into the pair of trees (T1, T2)
+A non-minimal ground tree has the form S
+∗=⇒ xAz
+∗=⇒ xuAvz
+∗=⇒ xuyvz. It can be
+decomposed into a pair of the ground tree S
+∗=⇒ xAz
+∗=⇒ xyz and the block tree A
+∗=⇒ uAv,
+see Fig. 1. Formally, we say that a tree T is decomposed to the pair of trees (T1, T2) if
+for some B ∈ N : T = A
+∗=⇒ xBz
+∗=⇒ xuBvz
+∗=⇒ xuβvz, T1 = A
+∗=⇒ xBz
+∗=⇒ xβz and
+T2 = B
+∗=⇒ uBv; here β is either A if T is a block tree or β ∈ Σ∗ if T is a ground tree (and
+in this case A is the axiom). We say that T can be composed from T1 and T2. Note that T1
+and T2 can be composed in several ways if T1 has several B nodes. We denote
+T1 ◦ T2 = {T | T can be decomposed into (T1, T2)}
+
+A. Rubtsov
+3
+In addition to derivation of words we consider derivation of trees as follows.
+▶ Definition 2. A derivation of a tree starts with a minimal ground tree and each derivation
+step leads to a ground tree as well. At a derivation step we choose a node A and a minimal
+block tree A
+∗=⇒ xAy, replace the chosen node by the block tree; the subtree of the chosen
+node is glued into A from the crown of the block tree.
+So in the trees derivation a minimal ground tree has the role of the axiom (there can
+be several ones), and a replacement A → Ti where Ti = (A
+∗=⇒ xAz) has the role of the
+production rule.
+▶ Definition 3. Let S be a multiset of trees. We say that S′ is derived from S if S′ is
+obtained from S by the replacement of two trees T1 and T2 by a tree T ∈ T1 ◦ T2. We denote
+it as S ⊢ S′. We say that a multiset of trees S is well-formed such that S ⊢
+∗ {T} for some
+ground tree T.
+It is easy to see, that if S contains only minimal trees and S ⊢
+∗ {T}, then there exists a
+derivation of T such that S = {T0, T1, . . . , Tn} where T0 is the ground tree and Ti, i ⩾ 1 is
+the block tree that was chosen at the i-th derivation step.
+We define the Parikh image for the trees in a natural way. Ψ(T) = Ψ(w) if T = (S
+∗=⇒ w)
+and Ψ(T) = Ψ(xz) if T = (A
+∗=⇒ xAz). The following lemma directly follows from the
+definitions.
+▶ Lemma 4.
+If T, T ′ ∈ T1 ◦ T2, then Ψ(T) = Ψ(T ′) = Ψ(T1) + Ψ(T2).
+If S ⊢
+∗ {T} and S ⊢
+∗ {T ′} then Ψ(T) = Ψ(T ′).
+Lemma 4 implies that we can extend the definition of the Parikh map to well-formed
+multisets: Ψ(S) = �
+T ∈S Ψ(T).
+By the pigeonhole principle each minimal tree has depth at most |N| − 1 (otherwise on
+the longest path from the root to a leaf there would be a repetition of nonterminals). Since
+(for a fixed grammar) each node of a tree has a bounded degree, there is only finitely many
+minimal trees. Let us enumerate them and denote the number of minimal trees by m. So,
+each multiset of minimal trees S has a corresponding vector ⃗v ∈ Nm where ⃗vi is the number
+of occurrences of Ti in S.
+3
+Proof
+We begin with the proof idea. Any word w derived from a grammar G has some derivation
+tree Tw and Ψ(Tw) = Ψ(w). We decompose Tw into a multiset Sw of minimal trees (Sw ⊢
+∗ Tw).
+Denote by Tg the ground tree from Sw. Obtain the set S′
+w from Sw by removing repetitions.
+Note that S′
+w ⊆ MG where MG is the finite set of minimal trees of grammar G. So we have
+defined a mapping w �→ S′
+w with a finite codomain 2MG. For any multiset S obtained from
+S′
+w by repetition of non-ground trees, there exists a derivation tree T (of G) such that S ⊢
+∗ T
+(S is well-formed). Therefore Ψ(T) ∈ Ψ(L(G)). In other words,
+�
+�Ψ(S′
+w) +
+�
+T ′∈S′w\{Tg}
+tT ′ · Ψ(T ′)
+�
+� ∈ Ψ(L(G))
+
+4
+The Simplest Proof of Parikh’s Theorem via Derivation Trees
+no matter how we choose tT ′ ∈ N for each T ′. Let S be a well-formed multiset with a ground
+tree Tg ∈ S. Denote by
+Lin(S) =
+�
+�
+�Ψ(S) +
+�
+T ′
+i ∈S\{Tg}
+ti · Ψ(T ′
+i)
+�����
+ti ∈ N
+�
+�
+� .
+(1)
+The set Lin(S) is linear by the definition. Since there are only finitely many different sets S′
+w,
+the set Ψ(L(G)) is a finite union of linear sets Lin(S′
+w), so it is semilinear by the definition.
+The scheme of the proof is depicted in Fig. 2
+L(G)
+Ψ(L(G))
+{Tw : w ∈ L(G)}
+{Sw : w ∈ L(G)}
+{S′
+w : w ∈ L(G)}
+{Lin(S′
+w) : w ∈ L(G)}
+Ψ
+∪
+Figure 2 Scheme of the proof
+3.1
+Auxiliary Lemmas
+▶ Lemma 5. Let S be a well-formed multiset of minimal trees and S′′ be a multiset. Denote
+by ⃗v and ⃗v′′ the corresponding vectors of S and S′′, respectively. Suppose ⃗vi > 0 iff ⃗v′′
+i > 0;
+then S′′ is a well-formed multiset as well.
+Proof. The statement is equivalent to the conjunction of two claims. The first one: if ⃗vi > 0
+and ⃗v′′
+i = ⃗vi + 1 while ⃗v′′
+j = ⃗vj for j ̸= i, then S′′ is well-formed. The second one: if ⃗vi > 1
+and ⃗v′′
+i = ⃗vi − 1 while ⃗v′′
+j = ⃗vj for j ̸= i, then S′′ is well-formed. Starting with the vector ⃗v
+and subsequently increasing or decreasing its components by 1 we can obtain any vector ⃗v′′
+satisfying the conditions of the lemma. Since at each step the condition of one of the claims
+hold, we obtain that the condition of the lemma holds in the result.
+Recall Definitions 2 and 3 of trees and multiset derivations. Denote by Ti = (A
+∗=⇒ uAv)
+the i-th block-tree. Fix a derivation tree T such that S ⊢
+∗ {T} with a derivation of the tree
+T as well.
+Proof of the first claim. Due to the form of Ti ∈ S, the tree T contains the non-terminal A.
+So we can glue Ti into the place of some occurrence of the non-terminal A and obtain the
+tree T ′ as the result. S′′ is well-formed, since T ′ is a derivation tree of G by the construction.
+Proof of the second claim. If Ti is a subtree of T, than it can be removed from T (as
+in Fig. 1) and the resulting tree T ′ would also be a derivation tree of G. Otherwise, let us
+consider the steps of the fixed derivation of T. We change this derivation as follows. Recalling
+that ⃗vi > 1, fix two copies T 1
+i and T 2
+i of the tree Ti in the multiset S such that T 1
+i was used
+
+A. Rubtsov
+5
+earlier than T 2
+i in the derivation of T. When T 2
+i must be composed with the ground tree, we
+skip this step. If at some step a tree Tj = (B
+∗=⇒ u′Bv′) is glued into a node B of T 2
+i , then
+we glue it into the corresponding node B of T 1
+i (that is already in the ground tree). So, this
+modification yields a derivation S ⊢
+∗ {T ′, T 2
+i }, where T ′ is some ground tree. But then we
+could do the same starting from S′′ and obtain S′′ ⊢
+∗ {T ′}. Therefore S′′ is well-formed.
+◀
+▶ Corollary 6. For any w ∈ L(G) : Lin(S′
+w) ⊆ Ψ(L(G)).
+▶ Lemma 7. For any w ∈ L(G):
+Ψ(w) ∈ Lin(S′
+w).
+Proof. From the definitions it follows that Ψ(w) = Ψ(Sw) ∈ Lin(Sw).
+We prove that
+Lin(Sw) ⊆ Lin(S′
+w).
+Note that Ψ(Sw) = Ψ(S′
+w) + �m
+i=1(⃗vi − 1)Ψ(Ti), where ⃗v is the
+corresponding vector of Sw and Ti is the i-th minimal tree (in the enumeration above). By
+the definition a vector ⃗u ∈ Lin(Sw) has the form
+⃗u = Ψ(Sw) +
+�
+T ′
+j∈Sw\{Tg}
+t′
+j · Ψ(T ′
+j) = Ψ(Sw) +
+n
+�
+i=1
+ti · Ψ(Ti),
+where in the second equality we put together all T ′
+j’s that are copies of the same tree Ti.
+Note that each T ′
+j is a minimal block tree, so T ′
+j = Ti for some i. Thus,
+⃗u =
+�
+�Ψ(S′
+w) +
+�
+i:⃗vi>0
+(ti + ⃗vi − 1)Ψ(Ti)
+�
+� ∈ Lin(S′
+w)
+◀
+3.2
+Proof of Parikh’s theorem
+Let G be a context-free grammar generating L. For a word w ∈ L denote by T(w) the set of
+all derivation trees Tw. Note that T(w) ̸= ∅ and it can be even infinite if there are ε-rules
+in G. Denote by S(w) = {Sw | ∃Tw ∈ T(w) : Sw ⊢
+∗ {Tw}} where Sw is a multiset consisting
+only of minimal trees (as before). Finally S′(w) = {S′
+w | Sw ∈ S(w)}; recall that S′
+w is
+the set obtained from Sw by deleting the duplicates. Now we show that S′(w) is finite for
+each w and moreover the union ∪w∈LS′(w) = S′(L) is finite as well. Recall that there are
+finitely many minimal trees, and we denote them by T1, . . . , Tm. Therefore, each Sw has a
+corresponding m-dimensional vector ⃗v such that
+Ψ(w) = Ψ(Sw) =
+m
+�
+i=1
+⃗vi · Ψ(Ti).
+The corresponding vector ⃗v′ of S′
+w is a 0-1 vector such that ⃗v′
+i = 1 iff ⃗vi > 0. So since there
+are only finitely many 0-1 vectors of length m and each S′
+w ∈ S′(L) has a corresponding 0-1
+vector ⃗v′, then the set S′(L) is finite as well.
+Putting everything together, by Lemma 7
+Ψ(L) ⊆
+�
+S′∈S′(L)
+Lin(S′)
+and by Corollary 6
+�
+S′∈S′(L)
+Lin(S′) ⊆ Ψ(L).
+
+6
+The Simplest Proof of Parikh’s Theorem via Derivation Trees
+Since the set S′(L) is finite and each set Lin(S′) is linear by Eq. (1), the equality
+Ψ(L) =
+�
+S′∈S′(L)
+Lin(S′)
+proves Parikh’s theorem.
+◀
+References
+1
+Javier Esparza, Pierre Ganty, Stefan Kiefer, and Michael Luttenberger. Parikh’s theorem: A
+simple and direct automaton construction. Information Processing Letters, 111(12):614–619,
+2011.
+URL: https://www.sciencedirect.com/science/article/pii/S0020019011000822,
+doi:https://doi.org/10.1016/j.ipl.2011.03.019.
+2
+Jonathan Goldstine. A simplified proof of Parikh’s theorem. Discrete Mathematics, 19(3):235–
+239, 1977.
+3
+John E Hopcroft, Rajeev Motwani, and Jeffrey D Ullman. Introduction to automata theory,
+languages, and computation. Acm Sigact News, 32(1):60–65, 2001.
+4
+Caleb Koch. A friendly tour of Parikh’s theorem. 2018. URL: http://cakoch10.github.io/
+papers/Kleene_Algebra.pdf.
+5
+Toshihiro Koga. A proof of Parikh’s theorem via Dickson’s lemma. International Journal of
+Foundations of Computer Science, 32(02):163–173, 2021. doi:10.1142/S012905412150009X.
+6
+Rohit J. Parikh.
+On Context-Free Languages.
+J. ACM, 13(4):570–581, oct 1966.
+doi:
+10.1145/321356.321364.
+7
+Jeffrey O. Shallit. A Second Course in Formal Languages and Automata Theory. Cambridge
+University Press, 2008.
+8
+Ryoma Sin’ya. Simple proof of Parikh’s theorem a la Takahashi. CoRR, abs/1909.09393, 2019.
+

6tAyT4oBgHgl3EQfQfaP/content/tmp_files/load_file.txt

@@ -0,0 +1,215 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf,len=214
+page_content='The Simplest Proof of Parikh’s Theorem via  Derivation Trees  Alexander Rubtsov � �  National Research University Higher School of Economics  Moscow Institute of Physics and Technology  Abstract  Parikh’s theorem is a fundamental result of the formal language’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' There had been published  many proofs and many papers claimed to provide a simplified proof, but most of them are long and  still complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We provide the proof that is really short, simple and discloses the nature of this  fundamental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We follow the technique closed to the original Parikh’s paper and our proof is  similar to the proof by Ryoma Sin’ya 2019, but we provide more detailed exposition and pretend to  more simplicity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We achieve the simplicity via nonconstructivenes that allows us avoiding  many difficulties met by other proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  2012 ACM Subject Classification Theory of computation → Grammars and context-free languages  Keywords and phrases Formal Languages, Context-Free Languages, Parikh’s Theorem  Funding The paper was supported by RFBR grant 20-01-00645  Acknowledgements I want to thank Dmitry Chistikov for the helpfull discussion of the paper  1  Introduction  Parikh’s theorem [6] is a fundamental theorem of the formal language’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' There had  been published many proofs (see [1], [4] for the survey of different proofs and detailed  exposition on the topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Many papers claimed to provide a simplified proof ([2] looks to  be the most known), but most of them are long and still complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Despite really short  and simple proofs are already known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=', [7]), papers with another proofs continues to be  published [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We provide the proof that is really short, simple and discloses the nature of  this fundamental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Our proof is based on derivations trees so as the original Parikh’s  proof [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' As Parikh, we decompose derivation trees into small ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' we consider similar  kinds of trees: ordinary derivation trees and auxiliary ones (with only nonterminal in the  crown that is the same as the root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We get rid of duplicates in auxiliary trees and pump  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' While our technique is similar to the original proof in general, it is simpler since we  do not have restrictions on grammar and other technical issues (like considering derivation  trees that contains nonterminals from a fixed subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' The simplicity of our construction is  based on nonconstructivenes that allows us avoiding many technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Also trees (in  the decomposition) in our construction have linear height (in the number of nonterminals)  while in the Parikh’s construction trees have quadratic height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' In our proof we generalize the  idea of derivation from words to trees and translate the idea of the pumping lemma to the  trees as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' It makes our construction clear enough to explain students during a lecture in  a basic course of formal languages and automata theory, and so we hope that the detailed  version will help to spread it in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Our technique is similar to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Unlike concise exposition in [8], we provide more detailed  exposition and explain the intuition of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' And we do not use auxiliary results for our  proof (thanks to nonconstructivenes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='00047v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='FL]  30 Dec 2022  2  The Simplest Proof of Parikh’s Theorem via Derivation Trees  2  Definitions  We denote by N non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Let Σk = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' , ak} be an alphabet, k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We  denote by Ψ : Σ∗  k → Nk the Parikh mapping that maps a word w to its Parikh’s image the  vector (|w|a1, |w|a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' , |w|ak), where |w|aj is the number of letters aj in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We denote the  Parikh’s image of a language L ⊆ Σ∗  k in the natural way:  Ψ(L) = {Ψ(w) | w ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  A set S ⊆ Nk is linear if there exist vectors v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' , vm ∈ Nk such that  S = {v0 + v1t1 + v2t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' vmtm | t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' tm ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  A set is semi-linear if it is a union of finitely many linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ▶ Theorem 1 (Parikh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' For each context-free language L the set Ψ(L) is semi-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  We use classical notation for context-free grammars [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We denote by N the set of  nonterminals, and denote nonterminals by capital letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Small letters from the end of  the alphabet denote words and Greek letters denote words over the alphabet Σ ∪ N, called  sentential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Our proof uses derivation trees and exploits the idea similar to the classical proof of the  pumping lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We use derivation trees not only for words, but also for sentential forms of  a special kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We call a tree that corresponds to a derivation of the form A  ∗=⇒ uAv a block  tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We call derivation trees for words ground trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We say that a tree T (a ground or a  block one) is minimal if it does not have a block tree as a subtree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' T does not have the  form (S  ∗=⇒ xAz  ∗=⇒ xuAvz  ∗=⇒ xuβvz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Hereinafter, we describe a tree by any derivation  corresponding to the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  S  A  A  x  u  y  v  z  S  A  y  x  z  A  A  u  v  Figure 1 Decomposition of tree T into the pair of trees (T1, T2)  A non-minimal ground tree has the form S  ∗=⇒ xAz  ∗=⇒ xuAvz  ∗=⇒ xuyvz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' It can be  decomposed into a pair of the ground tree S  ∗=⇒ xAz  ∗=⇒ xyz and the block tree A  ∗=⇒ uAv,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Formally, we say that a tree T is decomposed to the pair of trees (T1, T2) if  for some B ∈ N : T = A  ∗=⇒ xBz  ∗=⇒ xuBvz  ∗=⇒ xuβvz, T1 = A  ∗=⇒ xBz  ∗=⇒ xβz and  T2 = B  ∗=⇒ uBv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' here β is either A if T is a block tree or β ∈ Σ∗ if T is a ground tree (and  in this case A is the axiom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We say that T can be composed from T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Note that T1  and T2 can be composed in several ways if T1 has several B nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We denote  T1 ◦ T2 = {T | T can be decomposed into (T1, T2)}  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Rubtsov  3  In addition to derivation of words we consider derivation of trees as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ▶ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' A derivation of a tree starts with a minimal ground tree and each derivation  step leads to a ground tree as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' At a derivation step we choose a node A and a minimal  block tree A  ∗=⇒ xAy, replace the chosen node by the block tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' the subtree of the chosen  node is glued into A from the crown of the block tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  So in the trees derivation a minimal ground tree has the role of the axiom (there can  be several ones), and a replacement A → Ti where Ti = (A  ∗=⇒ xAz) has the role of the  production rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ▶ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Let S be a multiset of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We say that S′ is derived from S if S′ is  obtained from S by the replacement of two trees T1 and T2 by a tree T ∈ T1 ◦ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We denote  it as S ⊢ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We say that a multiset of trees S is well-formed such that S ⊢  ∗ {T} for some  ground tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  It is easy to see, that if S contains only minimal trees and S ⊢  ∗ {T}, then there exists a  derivation of T such that S = {T0, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' , Tn} where T0 is the ground tree and Ti, i ⩾ 1 is  the block tree that was chosen at the i-th derivation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  We define the Parikh image for the trees in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Ψ(T) = Ψ(w) if T = (S  ∗=⇒ w)  and Ψ(T) = Ψ(xz) if T = (A  ∗=⇒ xAz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' The following lemma directly follows from the  definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  If T, T ′ ∈ T1 ◦ T2, then Ψ(T) = Ψ(T ′) = Ψ(T1) + Ψ(T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  If S ⊢  ∗ {T} and S ⊢  ∗ {T ′} then Ψ(T) = Ψ(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Lemma 4 implies that we can extend the definition of the Parikh map to well-formed  multisets: Ψ(S) = �  T ∈S Ψ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  By the pigeonhole principle each minimal tree has depth at most |N| − 1 (otherwise on  the longest path from the root to a leaf there would be a repetition of nonterminals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Since  (for a fixed grammar) each node of a tree has a bounded degree, there is only finitely many  minimal trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Let us enumerate them and denote the number of minimal trees by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' So,  each multiset of minimal trees S has a corresponding vector ⃗v ∈ Nm where ⃗vi is the number  of occurrences of Ti in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  3  Proof  We begin with the proof idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Any word w derived from a grammar G has some derivation  tree Tw and Ψ(Tw) = Ψ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We decompose Tw into a multiset Sw of minimal trees (Sw ⊢  ∗ Tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Denote by Tg the ground tree from Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Obtain the set S′  w from Sw by removing repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Note that S′  w ⊆ MG where MG is the finite set of minimal trees of grammar G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' So we have  defined a mapping w �→ S′  w with a finite codomain 2MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' For any multiset S obtained from  S′  w by repetition of non-ground trees, there exists a derivation tree T (of G) such that S ⊢  ∗ T  (S is well-formed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Therefore Ψ(T) ∈ Ψ(L(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' In other words,  �  �Ψ(S′  w) +  �  T ′∈S′w\\{Tg}  tT ′ · Ψ(T ′)  �  � ∈ Ψ(L(G))  4  The Simplest Proof of Parikh’s Theorem via Derivation Trees  no matter how we choose tT ′ ∈ N for each T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Let S be a well-formed multiset with a ground  tree Tg ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Denote by  Lin(S) =  �  �  �Ψ(S) +  �  T ′  i ∈S\\{Tg}  ti · Ψ(T ′  i)  �����  ti ∈ N  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  (1)  The set Lin(S) is linear by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Since there are only finitely many different sets S′  w,  the set Ψ(L(G)) is a finite union of linear sets Lin(S′  w), so it is semilinear by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  The scheme of the proof is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' 2  L(G)  Ψ(L(G))  {Tw : w ∈ L(G)}  {Sw : w ∈ L(G)}  {S′  w : w ∈ L(G)}  {Lin(S′  w) : w ∈ L(G)}  Ψ  ∪  Figure 2 Scheme of the proof  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='1  Auxiliary Lemmas  ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Let S be a well-formed multiset of minimal trees and S′′ be a multiset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Denote  by ⃗v and ⃗v′′ the corresponding vectors of S and S′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Suppose ⃗vi > 0 iff ⃗v′′  i > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  then S′′ is a well-formed multiset as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' The statement is equivalent to the conjunction of two claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' The first one: if ⃗vi > 0  and ⃗v′′  i = ⃗vi + 1 while ⃗v′′  j = ⃗vj for j ̸= i, then S′′ is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' The second one: if ⃗vi > 1  and ⃗v′′  i = ⃗vi − 1 while ⃗v′′  j = ⃗vj for j ̸= i, then S′′ is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Starting with the vector ⃗v  and subsequently increasing or decreasing its components by 1 we can obtain any vector ⃗v′′  satisfying the conditions of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Since at each step the condition of one of the claims  hold, we obtain that the condition of the lemma holds in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Recall Definitions 2 and 3 of trees and multiset derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Denote by Ti = (A  ∗=⇒ uAv)  the i-th block-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Fix a derivation tree T such that S ⊢  ∗ {T} with a derivation of the tree  T as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Proof of the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Due to the form of Ti ∈ S, the tree T contains the non-terminal A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  So we can glue Ti into the place of some occurrence of the non-terminal A and obtain the  tree T ′ as the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' S′′ is well-formed, since T ′ is a derivation tree of G by the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Proof of the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' If Ti is a subtree of T, than it can be removed from T (as  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' 1) and the resulting tree T ′ would also be a derivation tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Otherwise, let us  consider the steps of the fixed derivation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' We change this derivation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Recalling  that ⃗vi > 1, fix two copies T 1  i and T 2  i of the tree Ti in the multiset S such that T 1  i was used  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Rubtsov  5  earlier than T 2  i in the derivation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' When T 2  i must be composed with the ground tree, we  skip this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' If at some step a tree Tj = (B  ∗=⇒ u′Bv′) is glued into a node B of T 2  i , then  we glue it into the corresponding node B of T 1  i (that is already in the ground tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' So, this  modification yields a derivation S ⊢  ∗ {T ′, T 2  i }, where T ′ is some ground tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' But then we  could do the same starting from S′′ and obtain S′′ ⊢  ∗ {T ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Therefore S′′ is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ◀  ▶ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' For any w ∈ L(G) : Lin(S′  w) ⊆ Ψ(L(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ▶ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' For any w ∈ L(G):  Ψ(w) ∈ Lin(S′  w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' From the definitions it follows that Ψ(w) = Ψ(Sw) ∈ Lin(Sw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  We prove that  Lin(Sw) ⊆ Lin(S′  w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Note that Ψ(Sw) = Ψ(S′  w) + �m  i=1(⃗vi − 1)Ψ(Ti), where ⃗v is the  corresponding vector of Sw and Ti is the i-th minimal tree (in the enumeration above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' By  the definition a vector ⃗u ∈ Lin(Sw) has the form  ⃗u = Ψ(Sw) +  �  T ′  j∈Sw\\{Tg}  t′  j · Ψ(T ′  j) = Ψ(Sw) +  n  �  i=1  ti · Ψ(Ti),  where in the second equality we put together all T ′  j’s that are copies of the same tree Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Note that each T ′  j is a minimal block tree, so T ′  j = Ti for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Thus,  ⃗u =  �  �Ψ(S′  w) +  �  i:⃗vi>0  (ti + ⃗vi − 1)Ψ(Ti)  �  � ∈ Lin(S′  w)  ◀  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='2  Proof of Parikh’s theorem  Let G be a context-free grammar generating L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' For a word w ∈ L denote by T(w) the set of  all derivation trees Tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Note that T(w) ̸= ∅ and it can be even infinite if there are ε-rules  in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Denote by S(w) = {Sw | ∃Tw ∈ T(w) : Sw ⊢  ∗ {Tw}} where Sw is a multiset consisting  only of minimal trees (as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Finally S′(w) = {S′  w | Sw ∈ S(w)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' recall that S′  w is  the set obtained from Sw by deleting the duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Now we show that S′(w) is finite for  each w and moreover the union ∪w∈LS′(w) = S′(L) is finite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Recall that there are  finitely many minimal trees, and we denote them by T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' , Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Therefore, each Sw has a  corresponding m-dimensional vector ⃗v such that  Ψ(w) = Ψ(Sw) =  m  �  i=1  ⃗vi · Ψ(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  The corresponding vector ⃗v′ of S′  w is a 0-1 vector such that ⃗v′  i = 1 iff ⃗vi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' So since there  are only finitely many 0-1 vectors of length m and each S′  w ∈ S′(L) has a corresponding 0-1  vector ⃗v′, then the set S′(L) is finite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  Putting everything together, by Lemma 7  Ψ(L) ⊆  �  S′∈S′(L)  Lin(S′)  and by Corollary 6  �  S′∈S′(L)  Lin(S′) ⊆ Ψ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  6  The Simplest Proof of Parikh’s Theorem via Derivation Trees  Since the set S′(L) is finite and each set Lin(S′) is linear by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' (1), the equality  Ψ(L) =  �  S′∈S′(L)  Lin(S′)  proves Parikh’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  ◀  References  1  Javier Esparza, Pierre Ganty, Stefan Kiefer, and Michael Luttenberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Parikh’s theorem: A  simple and direct automaton construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Information Processing Letters, 111(12):614–619,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='com/science/article/pii/S0020019011000822,  doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='ipl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  2  Jonathan Goldstine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' A simplified proof of Parikh’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Discrete Mathematics, 19(3):235–  239, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  3  John E Hopcroft, Rajeev Motwani, and Jeffrey D Ullman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Introduction to automata theory,  languages, and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Acm Sigact News, 32(1):60–65, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  4  Caleb Koch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' A friendly tour of Parikh’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' URL: http://cakoch10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='io/  papers/Kleene_Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  5  Toshihiro Koga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' A proof of Parikh’s theorem via Dickson’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' International Journal of  Foundations of Computer Science, 32(02):163–173, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='1142/S012905412150009X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  6  Rohit J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Parikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  On Context-Free Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' ACM, 13(4):570–581, oct 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='1145/321356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='321364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  7  Jeffrey O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Shallit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' A Second Course in Formal Languages and Automata Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Cambridge  University Press, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='  8  Ryoma Sin’ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' Simple proof of Parikh’s theorem a la Takahashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content=' CoRR, abs/1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}
+page_content='09393, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'}

7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b908d8cf809055aa9e72e21e2b848022e555aaf4670378c05810e2a56fb951a7
+size 1028429

7NE4T4oBgHgl3EQfcgzB/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1966df9c186d3f2f0793a0df30f9a680df1fb69151e3627b6f16a83496a194d4
+size 394391

9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2abea2ad74c8d8d04b25c9245121b3225e204d72e75fde4d405f9c99fbbb60c3
+size 342927

9dAzT4oBgHgl3EQfFPqV/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:21b38421cee1558ad96b1f826a8ff9b0a6b2cb459baf29ab2427d3a20d1c699a
+size 3211309

9dAzT4oBgHgl3EQfFPqV/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:103465a3ca236fa1d9883c112a5347d21b046693790f51542729e81313c07a64
+size 114529

A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a0707afbf0b7ad350c0b5a32e072ccc8da28c0dffc1628711e45322cb777cd9a
+size 2030513

AdE4T4oBgHgl3EQf4w7g/content/tmp_files/2301.05317v1.pdf.txt

@@ -0,0 +1,1506 @@
+arXiv:2301.05317v1  [gr-qc]  12 Jan 2023
+Next-to-leading-order Solution to Kerr-Newman Black Hole Superradiance
+Shou-Shan Bao,1, ∗ Qi-Xuan Xu,2, † and Hong Zhang1, ‡
+1Institute of Frontier and Interdisciplinary Science,
+Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong University, QingDao 266237, China
+2Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2AZ, United Kingdom
+(Dated: January 16, 2023)
+The superradiant instabilities of Kerr-Newman black holes with charged or uncharged massive
+spin-0 fields are calculated analytically to the next-to-leading order in the limit of α ∼ rgµ ≪ 1.
+A missing factor of 1/2 in the previous leading-order result is identified. The next-to-leading order
+result has a compact form and is in good agreement with existing numerical calculations.
+The
+percentage error increases with α, from a few percent for α ∼ 0.1 to about 50% for α ∼ 0.4.
+Massive neutral scalars too heavy to be produced with Kerr black hole superradiance may exist in
+the superradiant region of Kerr-Newman black holes.
+I.
+INTRODUCTION
+Ultralight boson condensate could form around a rotat-
+ing black hole (BH) if the boson’s Compton wavelength is
+comparable to the size of the BH horizon. With proper
+choice of parameters, such scalar condensate can con-
+tinuously extract energy and angular momentum from
+the BH until the BH spin is below some critical value
+and/or nonlinear effects become important[1–3].
+This
+phenomenon is known as BH superradiance [4, 5]. There
+exist numerous works on various bosons, including spin-
+0 [1, 6–26], spin-1 [25–38] and spin-2 [39, 40] fields. In
+this work, we focus on the ultralight scalars. The super-
+radiance of other types of bosons could be found in the
+comprehensive review [41].
+The scalar superradiance, especially with a Kerr BH, is
+important in phenomenology. Such BH-condensate sys-
+tems have been widely studied for constraining the scalar
+properties and for the possible observation of the GW
+emission. It has been shown that the BH evolves along
+the Regge trajectories on the mass-spin plot if the su-
+perradiant effect is strong [1, 17]. Consequently, there
+are “holes” on the Regge plot in which BHs cannot re-
+side. Combing with the observed BH spin distribution,
+favored and unfavored scalar mass ranges can be iden-
+tified [42–44]. On the other hand, with the continuous
+GW generated by the BH-condensate, works has been
+done to study the possibility of resolving these systems
+from the backgrounds [1, 13–15, 18, 19, 45]. The pos-
+itive frequency drift [13, 27] and the beat-like pattern
+[46] have been proposed to distinguish them from other
+monochromatic GW sources, such as neutron stars. The
+unresolved BH-condensate systems have also been care-
+fully studied as stochastic backgrounds for GW detectors
+[18, 19].
+The phenomenological study of BH superradiance de-
+pends on the accurate determination of the bound state’s
+∗ ssbao@sdu.edu.cn
+† qixuan.xu22@imperial.ac.uk
+‡ hong.zhang@sdu.edu.cn
+eigenfrequency. For Kerr BHs, the numerical continued
+fraction method was first proposed by Leaver for mass-
+less scalars [47]. It is later developed for massive scalars
+in Ref. [8] and further refined in Ref. [10]. In the small
+α ∼ rgµ limit, an analytic approximation was obtained
+by Detweiler [6].
+Nonetheless, these two solutions are
+not consistent with each other. The problem is recently
+resolved in our previous work by including the next-to-
+leading-order (NLO) contribution to the analytic approx-
+imation [21]. A power-counting strategy is also proposed
+which facilitates the NLO calculation.
+In Ref. [48], Damour et.al. have shown that the su-
+perradiance can also be realized with a charged massive
+scalar field in Kerr-Newman spacetime.
+Comparably,
+it does not attract as much attention as that for Kerr
+BHs. It may be because the Kerr-Newman BH (KNBH)
+is unlikely to play important roles in astrophysics [49–
+51]. Nonetheless, as pointed out in Ref. [52], the KNBH
+provides an ideal testing ground for studying the inter-
+play between gravity and electrodynamics. In the pre-
+vious studies of scalar superradiance with KNBHs, De-
+tweiler’s method has been applied to obtain the leading-
+order (LO) analytic approximation at the α ≪ 1 limit
+[53–55]. The numerical solution has also been achieved
+using the 3-term continued fraction method [56]. The pa-
+rameter space of the KNBH superradiance is also probed
+by analyzing the existence of the potential well [57–59].
+In this work, we refine the power-counting strategy in
+our previous work and apply it to calculate the NLO con-
+tribution of the KNBH superradiance. A compact NLO
+expression for α ≪ 1 is obtained which could be straight-
+forwardly applied to phenomenological study. The scalar
+field can be either neutral or charged.
+By comparing
+to the existing numerical results, the percentage error of
+the analytic approximation increases with α, from a few
+percent for α ∼ 0.1 to about 50% for α ∼ 0.4.
+This paper is organized as follows. In Sec. II, we briefly
+review the Klein-Gordon equation to be solved and ob-
+tain the superradiance condition from its solution at the
+outer horizon. Detweiler’s method is applied to derive
+the LO and NLO analytic expressions in Sec. III.
+In
+Sec. IV, the obtained analytic expressions are compared
+
+2
+to the existing numerical calculation. Some effects rel-
+evant to phenomenology are also discussed. Finally, we
+summarize our results in Sec. V.
+II.
+SCALARS IN KERR-NEWMAN SPACETIME
+The spacetime around a KNBH with mass M, angular
+momentum J and charge Q can be expressed in Boyer-
+Lindquist coordinates [60],
+ds2 = −
+�
+1 − 2rgr − Q2
+Σ2
+�
+dt2 + Σ2
+∆ dr2 + Σ2dθ2
++
+�
+(r2 + a2) + (2rgr − Q2)a2 sin2 θ
+Σ2
+�
+sin2 θdϕ2
+− 2(2rgr − Q2)a sin2 θ
+Σ2
+dtdϕ,
+(1)
+with
+a = J/M,
+(2a)
+rg = GM,
+(2b)
+Σ2 = r2 + a2 cos2 θ,
+(2c)
+∆ = r2 − 2rgr + a2 + Q2.
+(2d)
+The equation ∆ = 0 gives two event horizons at r± =
+rg ± b with b =
+�
+r2g − a2 − Q2. In this work, we only
+consider the KNBHs with r2
+g − a2 − Q2 ≥ 0.
+To study the superradiance of a scalar field close to a
+BH, one needs to solve the combined Einstein and Klein-
+Gordon field equations, which is a very difficult task. Es-
+pecially, the existence of the scalar perturbs the space-
+time around the BH. Nonetheless, it has been shown that
+this perturbation could be safely ignored due to the tiny
+energy-stress tensor of the scalar cloud for Kerr BH [17].
+We assume the same situation happens for the KNBHs.
+We further assume the self-interaction of the scalar field
+can also be ignored. Then the problem reduces to solv-
+ing the Klein-Gordon equation on the stationary Kerr-
+Newman background,
+(∇α − iqAα)(∇α − iqAα)φ − µ2φ = 0,
+(3)
+where µ and q are the mass and electric charge of the
+scalar field, respectively.
+The vector Aα is the back-
+ground electromagnetic potential,
+Aα = Qr
+Σ2
+�
+−1, 0, 0, a sin2 θ
+�
+.
+(4)
+For complex scalars, φ can be written with the separation
+of variables,
+φ(t, r, θ, ϕ) =
+�
+l,m
+�
+dωRlm(r)Slm(θ)eimϕe−iωt.
+(5)
+Inserting it into Eq. (3), one obtains the angular equa-
+tion,
+1
+sin θ
+d
+dθ
+�
+sin θdSlm
+dθ
+�
++
+�
+−a2(µ2 − ω2) cos2 θ −
+m2
+sin2 θ + Λlm
+�
+Slm = 0,
+(6)
+where Λlm is the eigenvalue. Its solution Slm(θ) is called
+the spheroidal harmonic function, whose properties can
+be found in Ref. [61]. The corresponding radial equation
+is [62],
+∆ d
+dr
+�
+∆dRlm
+dr
+�
++ U(r)Rlm = 0,
+(7)
+with
+U(r) =[ω(a2 + r2) − am − qQr]2
++ ∆[2amω − µ2r2 − a2ω2 − Λlm].
+(8)
+These are the equations for the complex scalar field. For
+real scalars, one should set q = 0 in Eq. (3) and choose
+only the real part on the right side of Eq. (5).
+The
+obtained angular and radial equations are the same as
+Eqs. (6) and (7) with q = 0. In the rest of this paper,
+we focus on the equations for the complex scalars. The
+case for the real scalars can then be simply obtained by
+choosing q = 0.
+To obtain a constraint of parameters that allow super-
+radiance, we change to the tortoise coordinates,
+dr∗ = r2 + a2
+∆
+dr,
+(9)
+with which the interesting region r ∈ (r+, +∞) corre-
+sponds to r∗ ∈ (−∞, +∞). We also define,
+R∗(r∗) =
+�
+r2 + a2R(r).
+(10)
+Then Eq. (7) can be rewritten into a Schr¨odinger-like
+equation,
+d2R∗(r∗)
+dr2∗
+− V (r)R∗(r∗) = 0,
+(11)
+where the effective potential is,
+V (r) = −
+�
+ω − am + qQr
+a2 + r2
+�2
++
+∆µ2
+a2 + r2
+−
+∆
+(a2 + r2)2
+�
+2amω − Λlm + a2(µ2 − ω2)
+�
++ ∆[∆ + 2r(r − rg)]
+(a2 + r2)3
+−
+3∆2r2
+(a2 + r2)4 .
+(12)
+In the region close to the outer horizon r+, the potential
+has the asymptotic form,
+lim
+r→r+ V (r) = −(ω − ωc)2 + O(r − r+),
+(13)
+
+3
+where the critical frequency is defined as,
+ωc = ma + qQr+
+r2
++ + a2
+= ma + qQr+
+2rgr+ − Q2 .
+(14)
+Inserting
+this
+asymptotic
+expression
+of
+V (r)
+into
+Eq. (11), one gets the solution at the outer horizon,
+lim
+r∗→−∞R∗(r∗) = d1e−i(ω−ωc)r∗ + d2ei(ω−ωc)r∗,
+(15)
+where the first term is the wave falling into the outer hori-
+zon, and the second term is the wave escaping from the
+outer horizon, d1 and d2 are their amplitudes. Physically,
+nothing can escape from the horizon, indicating d2 = 0.
+The superradiance requires the phase velocity and the
+group velocity to be in opposite directions, which leads
+to the superradiance condition for a KNBH,
+Re(ω) < ωc.
+(16)
+With Q fixed, it is more relaxed (strict) compared to the
+superradiance condition of a Kerr BH if the charges of the
+scalar and the BH have the same sign (different signs).
+III.
+ANALYTIC SOLUTION AT α ≪ 1
+In the small α limit, the asymptotic matching method
+first proposed in Ref. [6] gives a reasonable approxima-
+tion of the complex eigenfrequency ω. In a previous work,
+we have further calculated the NLO contribution for Kerr
+BH superradiance [21]. The NLO result has a much bet-
+ter agreement with the numerical solutions compared to
+the LO approximation. In the current work, we apply
+the method to KNBHs. In this section, we first repeat
+the LO approximation in Ref. [56]. A missing factor of
+1/2 is identified. We then continue to calculate the NLO
+contribution. The calculation is valid for both real and
+complex scalar fields. For a real scalar field, one simply
+sets q = 0 throughout.
+A.
+Leading-order Approximation
+We first formally introduce the power-counting param-
+eter α ∼ rgµ for the expansion. The scaling of other pa-
+rameters are Re ω ∼ µ ∼ q and a ∼ Q ∼ r+ ∼ r− ∼ rg.
+Unlike some previous calculations in which α is defined
+to be rgµ, here we leave α as a power-counting parame-
+ter, which could be rgµ or any other quantity with the
+same scaling. In the limit r → +∞, the derivative term
+in Eq. (7) divided by ∆2 can be written into a familiar
+form,
+1
+∆
+d
+dr
+�
+∆dR
+dr
+�
+≈ d2R
+dr2 + 2
+r
+dR
+dr = 1
+r
+d2
+dr2 (rR).
+(17)
+The second term on the left side of Eq. (7) divided by
+∆2 can be expanded in powers of rg/r. Keeping terms
+up to r2
+g/r2, the radial function at large r limit (r ≫ rg)
+can be simplified as,
+d2
+dr2 (rR) +
+�
+(ω2 − µ2) + 2(2rgω2 − rgµ2 − qQω)
+r
+− l′(l′ + 1)
+r2
++ O(r−3)
+�
+rR = 0,
+(18)
+where
+l′(l′ + 1) =Λlm + 4r2
+g(µ2 − 3ω2) + a2(ω2 − µ2)
++ Q2(2ω2 − q2 − µ2) + 8rgqQω.
+(19)
+The l′ is related to the orbital angular number by,
+l′ = l + ǫ.
+(20)
+Here ǫ ∼ O(α2) plays the role of a regulator and cannot
+be simply dropped.
+For a confined profile, the real part of ω is less than
+the boson mass µ. The physical solution is the one that
+decays exponentially at large r. It is more convenient to
+define,
+κ =
+�
+µ2 − ω2,
+(21)
+λ = 2rgω2 − rgµ2 − qQω
+κ
+,
+(22)
+y = κr,
+(23)
+u(y) = yR
+�y
+κ
+�
+.
+(24)
+Then Eq. (18) can be rewritten as,
+d2u(y)
+dy2
++
+�
+−1 + 2λ
+y − l′(l′ + 1)
+y2
+�
+u(y) = 0.
+(25)
+The two solutions are Whittaker functions, and only one
+of them has the correct behavior at r → +∞ required by
+the bound states. The solution with the correct behavior
+can be further written in terms of confluent hypergeo-
+metric functions. Finally, the radial function at large r
+
+4
+is,
+R(r) = e−κr(2κr)l′U(l′ + 1 − λ, 2l′ + 2; 2κr),
+(26)
+up to an arbitrary normalization.
+The bound states only exist if λ > 0.
+The super-
+radiance conditon in Eq. (16) gives another constraint
+2rgω < (ma + qQr+)/r+. Combining these two inequal-
+ities, one can obtain,
+0 < 2rgω2 − rgµ2 − qQω
+<
+�ma
+r+
++ qQ
+�
+ω − rgµ2 − qQω
+= ma
+r+
+ω − rgµ2.
+(27)
+So there is no superradiant bound state if m ≤ 0.
+It also shows that Reissner-Nordstr¨om BHs could not
+hold bounded scalar clouds [55]. The minimum KNBH
+spin a allowing superradiant instability is approximately
+rgr+µ/m.
+Next, we look at Eq. (7) in the small r limit. For BH
+superradiance, the inner boundary is the outer horizon
+r = r+. It is more convenient to write the radial function
+in terms of z = (r − r+)/2b,
+z(z + 1) d
+dz
+�
+z(z + 1)dR
+dz
+�
++ U(z)R = 0,
+(28)
+where U(z) can be written as an expansion of z,
+U(z) = p2 + z
+�4rgr+ω
+b
+�
+r+ω − am
+2r+
+− Q2ω
+2rg
+�
+− (Λlm + r2
++µ2 + a2ω2) + qQ
+b (am + r+qQ − a2ω − 3r2
++ω)
+�
++ z2(a2ω2 − Λlm + 2µ2a2 − 3µ2r2
++ + 6r2
++ω2 + 2Q2µ2 + q2Q2 − 6r+qQω)
++ 4z3b [rgµ2 + 2r+(ω2 − µ2) − qQω] + 4z4b2(ω2 − µ2),
+(29)
+in which,
+p = (r2
++ + a2)
+2b
+(ω − ωc).
+(30)
+Note that both p and rgωc scale as O(α0).
+In the limit of small α, the Λlm has the expanded form
+Λlm = l(l+1)+O(α4). At the LO of α, we get the radial
+equation in limit (r − r+) ≪ max(1/ω, 1/µ),
+z(z + 1) d
+dz
+�
+z(z + 1)dR
+dz
+�
++
+�
+p2 − l′(l′ + 1)z(1 + z)
+�
+R = 0.
+(31)
+In principle, the l′ should be replaced by l in this order.
+Nonetheless, the ǫ in l′ plays the role of a regulator in
+the intermediate steps. It will be set to zero at the end.
+The general solution of Eq. (31) is a linear combination
+of two associated Legendre functions, and the physical
+solution is the one with the ingoing wave at r → r+.
+After changing the variable back to r, the solution of the
+radial function is,
+R(r) =
+�r − r+
+r − r−
+�−ip
+2F1
+�
+−l′, l′ + 1; 1 − 2ip; −r − r+
+2b
+�
+,
+(32)
+up to an arbitrary normalization.
+Next, we apply the matching method first proposed
+in [6] and further developed recently in Ref. [21]. The
+solution of Eq. (26) is only valid in r ≫ rg limit, while
+the solution in Eq. (32) requires r ≪ rgα−2 from the ig-
+norance of terms proportional to z3 and z4. They have
+an overlapped region in the limit α ≪ 1. In this region,
+the two solutions are expected to have the same behav-
+ior. The behavior of Eq. (26) in the overlapped region is
+obtained by looking at its small r limit, which is,
+(2κ)l′Γ(−2l′ − 1)
+Γ(−l′ − λ)
+rl′ + (2κ)−l′−1Γ(2l′ + 1)
+Γ(l′ + 1 − λ)
+r−l′−1. (33)
+On the other hand, the behavior of Eq. (32) in the over-
+lapped region is obtained by looking at its large r limit,
+which is,
+(2b)−l′Γ(2l′ + 1)
+Γ(l′ + 1)Γ(l′ + 1 − 2ip)rl′ + (2b)l′+1Γ(−2l′ − 1)
+Γ(−l′ − 2ip)Γ(−l′) r−l′−1.
+(34)
+The ratio of the coefficients of the rl′ and r−l′−1 should be
+the same for the two solutions in the overlap region. The
+obtained equation is the eigenequation of ω. It can be
+
+5
+solved perturbatively by the observation that the second
+term in the expression (33) must be suppressed at small
+r, indicating l′+1−λ is very close to zero or some negative
+integer,
+l′ + 1 − λ = −n − δλ,
+(35)
+where |δλ| ≪ 1 and n is zero or a positive integer.
+Following the convention in literature, we also define
+¯n = n + l + 1. Then the above relation is re-expressed as
+λ = ¯n + ǫ + δλ. At LO of α, it reduces to λ = ¯n + δλ.
+Combining with the definition of λ in Eq. (22), the rgκ
+scales as α2, which is important in power-counting. Since
+|δλ| ≪ 1, one could solve for δλ perturbatively with ex-
+pressions (33) and (34).
+The LO calculation of δλ for Kerr BHs was completed
+in Ref. [6], with the regulator ǫ set to zero from the be-
+ginning. Recently, we have confirmed a missing factor
+of 1/2 in that result [21], which was first identified in
+Ref. [34]. The missing factor is conjectured to be from
+mistreatments of Γ functions with negative integer argu-
+ments. The correct formula is provided in the appendix of
+Ref. [21]. This subtle calculation turns out to be straight-
+forward with the regulator ǫ kept in the intermediate
+steps.
+More details could be found in Ref. [21].
+For
+KNBHs, the first LO calculation of δλ was completed
+in Ref. [56]. It followed the same steps in Ref. [6] and
+missed the factor 1/2 as well. After the correction, the
+LO result of δλ is,
+δλ(0) = − ip (4κb)2l+1 (n + 2l + 1)!(l!)2
+n! [(2l)!(2l + 1)!]2
+l�
+j=1
+(j2 + 4p2),
+(36)
+where the superscript (0) indicates that it is the LO re-
+sult. It scales as O(α4l+2).
+The eigen-frequency ω can be expressed in terms of δλ
+with Eqs. (22) and (35). Defining ω = ω0 + ω1δλ(0) in
+Eq. (22) and expanding it to the linear term of δλ(0), one
+arrives at,
+λ = rg(2ω2
+0 − µ2) − qQω0
+�
+µ2 − ω2
+0
++ rgω0ω1(3µ2 − 2ω2
+0) − qQµ2ω1
+(µ2 − ω2
+0)3/2
+δλ(0) + O
+�
+(δλ(0))2�
+.
+(37)
+On the other hand, we have λ = ¯n + δλ(0) from Eq. (35).
+Then it is straightforward to get,
+rg(2ω2
+0 − µ2) − qQω0
+�
+µ2 − ω2
+0
+= ¯n,
+(38a)
+rgω0ω1(3µ2 − 2ω2
+0) − qQµ2ω1
+(µ2 − ω2
+0)3/2
+= 1.
+(38b)
+Note that in getting Eq. (38a), we have ignored the ǫ
+which could be traced back to the l′ in Eq. (35). This
+omission leads to an error in rgω0 at the order of O(α5).
+Solving ω0 perturbatively from Eq. (38a), one arrives at,
+ω(0)
+0
+µ
+= 1 − 1
+2
+�rgµ − qQ
+¯n
+�2
++ O(α4).
+(39)
+Then the ω1 could be expressed in terms of ω0 from
+Eq. (38b) and expanded in powers of α,
+ω(0)
+1
+µ
+= (rgµ − qQ)2
+¯n3
++ O(α4).
+(40)
+Since both ω0 and ω1 are real, ω0 and ω1δλ(0) are the
+leading terms of the real and imaginary parts of ω, re-
+spectively.
+Especially, the imaginary part of ω scales
+as O(α4l+5).
+B.
+Next-to-leading-order Approximation
+In a previous work, we have carefully studied the su-
+perradiance of a real scalar field around a Kerr BH [21].
+The LO eigenfrequency ω obtained in Ref. [6] has an er-
+ror as large as 160% compared to the numerical result.
+After correcting the missing factor 1/2, the convergence
+is improved, with the error ≲ 80%. Except for the large
+discrepancy, the LO result also has some strange behav-
+iors. Since the LO result is the leading term in the Taylor
+series of the exact ω at α = 0, it is expected to converge
+to the exact ω with α approaching zero. Nonetheless, the
+relative error seems to be a nonzero constant for small α,
+reaching as large as 30% at α = 0.07 for a = 0.99. This
+discrepancy at small α puts the question on the power-
+counting strategy. Moreover, the discrepancy at small α
+increases quickly with the BH spin parameter a.
+These problems are solved by adding the NLO correc-
+tion of ω [21]. Below we follow the same steps for the
+KNBHs. The key observation is that the first term in
+the square bracket in Eq. (29), which scales as α2, is
+enhanced by a factor of 1/b. For BHs with large spin
+a and/or charge Q, this term can be as important as
+the LO contribution. Other NLO contributions are also
+added for consistency.
+The first NLO correction appears as ǫ in the asymp-
+totic radial wave function at large r, which is given in
+Eq. (26). It can be calculated from the definition of l′ in
+Eq. (19),
+ǫ = −8r2
+gµ2 + Q2µ2 + 8rgqQµ − q2Q2
+2l + 1
++ O(α4).
+(41)
+The second NLO contribution is from the asymptotic ra-
+dial wave function at small r.
+The potential U(z) in
+Eq. (29) can be approximated by p2−l′(l′+1)z(1+z)+zd,
+where d is defined as
+d =(4rgµ − 2qQ)p − 2(4rg − r+)rgµ2
++ 2µqQ(4rg − r+) − q2Q2 + O(α3).
+
+6
+Up to an arbitrary normalization, the corresponding ra-
+dial function at the NLO is,
+R(r) =(r − r−)
+√
+d−p2
+(r − r+)ip
+2F1
+�
+− l′ − ip +
+�
+d − p2,
+l′ + 1 − ip +
+�
+d − p2; 1 − 2ip; −r − r+
+2b
+�
+.
+(42)
+In the r → +∞ limit, the asymptotic behavior of this
+function is,
+(2b)−l′−ip+√
+d−p2Γ(2l′ + 1)Γ(1 − 2ip)
+Γ(l′ + 1 − ip −
+�
+d − p2)Γ(l′ + 1 − ip +
+�
+d − p2)
+rl′
++
+(2b)l′+1−ip+√
+d−p2Γ(−2l′ − 1)Γ(1 − 2ip)
+Γ(−l′ − ip −
+�
+d − p2)Γ(−l′ − ip +
+�
+d − p2)
+r−l′−1.
+(43)
+Following similar matching steps above, the NLO contri-
+bution of δλ could be obtained after some algebra,
+δλ(1) =
+� d
+2ǫ − ǫ
+2 − ip
+� (4κb)2l′+1 Γ(n + 2l′ + 2)Γpd
+n! [Γ(2l′ + 1)Γ(2l′ + 2)]2
+,
+(44)
+where the superscript (1) indicates it is the NLO result,
+and the Γpd is defined as,
+Γpd =
+���Γ(l′ + 1 + ip +
+�
+d − p2)Γ(l′ + 1 + ip −
+�
+d − p2)
+���
+2
+Γ(1 + 2ǫ)Γ(1 − 2ǫ)
+Γ(1 − ip −
+�
+d − p2 − ǫ)Γ(1 + ip +
+�
+d − p2 + ǫ)Γ(1 − ip +
+�
+d − p2 − ǫ)Γ(1 + ip −
+�
+d − p2 + ǫ)
+.
+(45)
+The last NLO contribution is from ω0 and ω1. Defining
+ω = ω(1)
+0
++ ω(1)
+1 δλ(1), the expansion of λ in Eq. (37) is
+still valid, only with δλ(0) replaced by δλ(1). Combining
+with λ = ¯n + ǫ + δλ(1), one could follow the same steps
+as in the LO calculation and obtain,
+ω(1)
+0
+µ
+=1 − 1
+2
+�rgµ − qQ
+¯n
+�2
++ (rgµ − qQ)2
+8¯n4
+[3(rgµ − qQ)(5rgµ − qQ) + 8¯nǫ]
++ O(α6),
+(46a)
+ω(1)
+1
+µ
+=(rgµ − qQ)2
+¯n3
+− 3(rgµ − qQ)2
+2¯n5
+[(rgµ − qQ)(5rgµ − qQ) + 2¯nǫ]
++ O(α6).
+(46b)
+Finally, we discuss a subtle problem related to the ω
+dependence in the definition of p. In the calculation of the
+δλ(1), the ω in p should be replaced by ω(0)
+0 , rather than
+ω(1)
+0 . Here we explain the reason. In deriving the small-r
+asymptotic form of the radial function, we approximate
+U(z) in Eq. (29) by p2 − l′(l′ + 1)z(z + z) + zd. The
+coefficient of z and z2 are accurate at O(α2) and O(α0),
+respectively. At z ∼ O(α), this two terms are at the same
+order of O(α4). Consequently, we only need to keep the
+terms in p2 up to O(α4), which then leads to ω = ω(0)
+0
+in p. In comparison to the numerical calculation, this
+choice of ω gives a satisfactory NLO result. Using ω(1)
+0
+in p is not as satisfactory, due to partially including the
+high-order contributions.
+IV.
+RESULTS
+The eigenfrequency of the Kerr BH superradiance has
+been studied in Refs. [8, 10, 21]. In comparison, the case
+for Kerr-Newman BH has two more parameters, the BH
+charge Q and the scalar charge q. In this section, we first
+study the superradiance of a neutral scalar field, focusing
+on the effect of Q. Then we consider the superradiance of
+a charged scalar field. Comparisons with the numerical
+calculations in the literature are also provided.
+A.
+Neutral Scalar Fields
+In the following study of neutral scalar superradiance,
+we adopt the NLO δλ(1) in Eq. (44), where the scalar
+charge q is set to zero. The ω(1)
+0
+and ω(1)
+1
+in Eqs. (46)
+are used. Then the NLO eigen-frequency is ω = ω(1)
+0
++
+ω(1)
+1 δλ(1).
+The BH charge Q cannot be chosen arbitrarily. In our
+derivation, we have implicitly assumed the KNBH has
+
+7
+horizons, which requires |Q| ≤
+�
+r2g − a2. In addition,
+neutral scalars could not distinguish the sign of the BH
+charge. Mathematically, it means the BH charge Q can
+only appear in the formulas as Q2. So it is sufficient to
+only consider positive Q.
+The superradiance condition in Eq. (16) with q = 0
+has the same form as the Kerr BH. The effect of the BH
+charge Q is hidden in r+ = rg +
+�
+r2g − a2 − Q2. Keeping
+the BH mass M and spin a fixed, larger charge Q results
+in a larger upper limit of Re(ω). Thus massive scalars
+too heavy to be produced with Kerr BH superradiance
+may exist in the superradiant region of KNBHs.
+Q=0
+Q=0.1
+Q=0.2
+Q=0.3
+Q=0.4
+0.0
+0.5
+1.0
+1.5
+10-13
+10-11
+10-9
+10-7
+rg
+�
+Im (
+�
+) /
+�
+l=m=1
+l=m=2
+l=m=3
+a=0.9
+Q=0
+Q=0.2
+Q=0.4
+Q=0.6
+Q=0.7
+0.0
+0.5
+1.0
+1.5
+10-13
+10-11
+10-9
+10-7
+rg
+�
+Im (
+�
+) /
+�
+l=m=1
+l=m=2
+l=m=3
+a=0.7
+FIG. 1. The imaginary part of NLO eigenfrequency with q =
+0 as a function of rgµ. Only the curves with n = 0 are shown.
+In the top (bottom) panel, the BH spin a is 0.9 (0.7).
+In
+both panels, from left to right, the three bunches correspond
+to l = m = 1, 2, 3, respectively. In each bunch, the curves
+with different colors correspond to different values of the BH
+charge Q.
+Fig. 1 shows the imaginary part of �� as a function
+of rgµ.
+For comparison, the curves for Kerr BHs are
+also shown, labeled with Q = 0.
+All curves have the
+same qualitative behavior. With an increasing value of
+rgµ, they first increase, then drop rapidly to below zero
+after reaching the maxima.
+There are three effects of
+the BH charge Q. Firstly, the superradiant region of rgµ
+is enlarged with larger Q. Correspondingly, the peak of
+Q=0.1
+Q=0.2
+Q=0.3
+Q=0.4
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+rg μ
+s (Q)
+a=0.9
+Q=0.2
+Q=0.4
+Q=0.6
+Q=0.7
+0.00
+0.05
+0.10
+0.15
+0.20
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+rg μ
+s (Q)
+a=0.7
+FIG. 2. Factor s(Q) with q = 0 as a function of rgµ for BH
+spin a = 0.9 (upper panel) and a = 0.7 (lower panel). The
+vertical dashed line in each panel labels the value of rgµ where
+Im ω(Q = 0) reaches its maximum value for the corresponding
+spin parameter a.
+the curve moves to the right with increasing Q.
+The
+maximum rgµ with positive Im(ω) is quite accurately
+determined by µ = ωc. Secondly, the maximum Im(ω)
+increases with larger Q. Fixing the BH spin to be a =
+0.9, the maximum values of rgIm(ω) with Q = 0 are
+2.088 × 10−8, 2.427 × 10−9 and 1.029 × 10−10 for l =
+m = 1, 2, 3, respectively. The numbers for Q = 0.43 are
+1.476 × 10−7, 2.006 × 10−8 and 8.760 × 10−10, which are
+larger than the Q = 0 cases by factors of 7.07, 8.26 and
+8.51. For BHs with spin a = 0.7, the maximum Q is 0.71.
+The enhancement factors are 90.29, 269.91, and 707.16,
+for l = m = 1, 2, 3, respectively. Finally, in the ranges of
+small rgµ before reaching the round peaks of the Q = 0
+curves, the charge Q turns out to impede the growth of
+the scalar clouds. We define a factor s(Q) as
+s(Q) =
+Im ω(Q)
+Im ω(Q = 0).
+(47)
+In Fig. 2, we show s(Q) as a function of rgµ, for two
+different BH spins and several values of Q. Interestingly,
+the suppression factor varies slowly with rgµ. It decreases
+with increasing Q, reaching the minimum value ∼ 0.8 for
+
+8
+a = 0.9 and ∼ 0.5 for a = 0.7.
+In Ref. [56], the authors claim that when a ≳ 0.997rg,
+the maximum value of Im ω decreases as Q grows. We do
+not observe the same behavior. For any spin parameter
+a, the peak value of Im ω from the NLO approximation
+increases monotonically with Q.
+B.
+Charged Scalar Fields
+In this part, we study the superradiance of KNBHs
+under charged scalar perturbation. The NLO eigenfre-
+qency is given by ω = ω(1)
+0
++ ω(1)
+1 δλ(1), with the NLO
+δλ(1) in Eq. (44), and the ω(1)
+0
+and ω(1)
+1
+in Eqs. (46).
+Note that the ω in p should take the form of ω(0)
+0
+in
+Eq. (39), as explained at the end of Sec. III B. We also
+compare the NLO results to the LO ones. The latter is
+given by ω = ω(0)
+0
++ ω(0)
+1 δλ(0), with the expressions de-
+fined in Eqs. (36), (39) and (40). The ω in p is replaced
+by µ for consistency.
+-60
+-40
+-20
+0
+20
+10-9
+10-8
+10-7
+q
+Im (ω)
+μ=0.1
+μ=0.2
+μ=0.3
+μ=0.41
+FIG. 3. Comparison of the numerical result and the analytic
+approximations for n = 0, l = m = 1, a = 0.98, and Q = 0.01,
+with rg chosen to be 1 for compacity. The imaginary part of
+ω is plotted as a function of the scalar field charge q. The
+dashed (solid) curves are the LO (NLO) approximations and
+the scattered dots are numerical results taken from Fig. 6
+in Ref. [53]. The curves with different colors correspond to
+different values of µ, labeled above the corresponding curves
+with the same color.
+Fig. 3 shows the comparison of the LO and NLO ap-
+proximations to the numerical results taken from Fig. 6
+in Ref. [53]. The NLO approximation agrees much better
+with the numerical results. In particular, the average per-
+centage errors of the NLO results for the points in Fig. 3
+are 6.7%, 9.9%, 20.7% and 48.3% for rgµ = 0.1, 0.2, 0.3
+and 0.41, respectively. These numbers can be used as
+estimates of the NLO results for different values of α.
+Moreover, the convergence of NLO results is better for a
+smaller value of rgµ, qualifying the power-counting strat-
+egy. To the contrary, the LO results do not seem to con-
+verge to the numerical result at small rgµ, which is also
+observed for Kerr BHs [21]. The reason for the bad con-
+vergence of the LO result is explained at the beginning of
+Sec. III B. A caveat is that the curves for the LO approx-
+imations in Fig. 3 are not the same as those in Ref. [53].
+The latter misses a factor of 1/2.
+Table. I shows the comparison of the NLO results and
+the numerical solutions for five more parameter sets in
+the literature. They are the most unstable modes with
+different parameters. The percentage uncertainty of the
+NLO approximation varies from 14% to 29% compared
+to the numerical results.
+TABLE I. Comparison of the NLO approximations of Im(ω)
+with the numerical results from Ref. [56] (cases A to D) and
+from Ref. [53] (case E). All cases are with n = 0 and l =
+m = 1. The numbers below assume rg = 1 for compacity.
+The percentage error is calculated by taking the difference
+between the approximation and the numerical result, then
+dividing it by the numerical result.
+Case A: a = 0.9, Q = 0.2 , q = −0.264, µ = 0.282;
+Case B: a = 0.99, Q = 0.1105 , q = −0.6335, µ = 0.397;
+Case C: a = 0.997, Q = 0.004 , q = −18.91, µ = 0.39822;
+Case D: a = 0.997, Q = 0.0001, q = −756.68, µ = 0.39816;
+Case E: a = 0.98, Q = 0.01, q = −8, µ = 0.35.
+Case
+Type
+Im(ω)
+% error
+LO
+5.623×10−9
+74.9%
+A
+NLO
+2.882×10−8
+28.5%
+Numerical
+2.243×10−8
+-
+LO
+1.224×10−8
+92.9%
+B
+NLO
+1.981×10−7
+14.1%
+Numerical
+1.736×10−7
+-
+LO
+1.264×10−8
+92.9%
+C
+NLO
+2.041×10−7
+14.1%
+Numerical
+1.788×10−7
+-
+LO
+1.263×10−8
+92.9%
+D
+NLO
+2.041×10−7
+14.1%
+Numerical
+1.788×10−7
+-
+LO
+1.27×10−8
+88.8%
+E
+NLO
+1.39×10−7
+22.7%
+Numerical
+1.13×10−7
+-
+Next, we analyze the effect of q. In the formulas, the
+q and Q appears as qQ and Q2.
+So it is sufficient to
+consider the case with Q > 0, and with q being either
+positive or negative. There are two constraints for the
+existence of the superradiant bound states. The superra-
+diance requires ω < ωc in Eq. (16). And the existence of
+the bound states gives the second constraint λ > 0 from
+Eq. (22), which is approximately rgµ − qQ > 0.
+If the scalar and the KNBH at the center have opposite
+charges, i.e.
+qQ < 0, the scalar cloud is more tightly
+bounded.
+In this case, the second constraint above is
+automatically satisfied. Fig. 4 shows the imaginary part
+of ω as a function of rgµ in the n = 0, l = m = 1
+bound state, with BH spin a = 0.9 and charge Q =
+0.01.
+The scalar charge q varies from −45 to 0.
+The
+region of superradiance shrinks when q is more negative,
+which is a consequence that ωc decreases with q for fixed
+
+9
+q=0
+q=-5
+q=-10
+q=-15
+q=-30
+q=-45
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+10-20
+10-17
+10-14
+10-11
+10-8
+rg μ
+Im (
+�
+) / μ
+FIG. 4. The imaginary part of NLO eigenfrequency as a func-
+tion of rgµ with different negative values of q. Other param-
+eters are n = 0, l = m = 1, a = 0.9 and Q = 0.01.
+TABLE II. The maximum value of Im(ω) obtained by varying
+q, with a and Q fixed.
+(a,Q)
+q
+Im(ω)
+-2.5
+2.10313×10−8
+(0.9, 0.01)
+-2.25
+2.10329×10−8
+-2.2
+2.10329×10−8
+-2
+2.10268×10−8
+-1.25
+2.10814×10−8
+(0.9, 0.02)
+-1.1
+2.10831×10−8
+-1
+2.10815×10−8
+-0.75
+2.10682×10−8
+-3
+4.14247×10−10
+(0.7, 0.01)
+-2.8
+4.14270×10−10
+-2.75
+4.14260×10−10
+-2.5
+4.14104×10−10
+-1.5
+4.14863×10−10
+(0.7, 0.02)
+-1.4
+4.14888×10−10
+-1.25
+4.14726×10−10
+-1
+4.13927×10−10
+Q. The peak value of Im(ω) seems to be smaller with
+decreasing q. Nonetheless, a more careful study shows
+that the maximum Im(ω) happens at some small but
+nonzero |q| (see Table. II).
+If the charges of the scalar and the KNBH have the
+same sign, i.e. qQ > 0, the scalar cloud is less bounded.
+The second constraint above gives rgµ > qQ for the ex-
+istence of bound states. Fig. 5 shows the imaginary part
+of ω as a function of rgµ in the n = 0, l = m = 1
+bound state, with BH spin a = 0.9 and charge Q = 0.01.
+With larger value of positive q, the superradiance region
+shrinks and the peak is lower as well.
+q=0
+q=5
+q=10
+q=15
+q=30
+0.0
+0.1
+0.2
+0.3
+0� �
+0.5
+10-20
+10-17
+10-14
+10-11
+10-8
+rg μ
+Im (
+�
+) / μ
+FIG. 5. The imaginary part of NLO eigenfrequency as a func-
+tion of rgµ with different positive values of q. Other parame-
+ters are n = 0, l = m = 1, a = 0.9 and Q = 0.01.
+V.
+CONCLUSION
+In this work, we have studied the scalar superradi-
+ant instability of the KNBH and obtained the LO and
+NLO expressions of the superradiant rate in the regime
+of α ≪ 1.
+The calculation is based on the matching
+method which is proposed by Detweiler for Kerr BHs in
+Ref. [6] and developed in our previous work [21]. In this
+manuscript, we further refine the power-counting strat-
+egy and apply it to the KNBH.
+The LO scalar superradiant rate for KNBH has been
+calculated previously in Ref. [53].
+With our refined
+power-counting strategy, a similar result is obtained but
+with an extra overall factor of 1/2. We conjecture the
+factor is from the mistreatment of the Γ functions with
+negative integer arguments, similar to the case of Kerr
+BHs. More analysis could be found in our previous work
+[21].
+We compare the LO and NLO results with the existing
+numerical calculations in the literature. The LO results
+are smaller than the numerical solutions by an order of
+magnitude. To the contrary, the percentage error of the
+NLO result ranges from a few percent to about 50%, de-
+pending on the value of α (see Fig. 3 and Table I). In
+particular, the error of the NLO result decreases for a
+smaller value of α, qualifying our power-counting strat-
+egy.
+The obtained NLO expression has a compact form
+and can be straightforwardly applied to phenomenolog-
+ical studies of the KNBH superradiance as well as the
+ultralight scalars, either neutral or charged. Besides the
+superradiance condition Re(ω) < mΩH as the Kerr BHs,
+there is another condition rgµ > qQ for the existence
+of bound states. For neutral scalars, larger BH charge
+Q leads to a larger superradiant range of rgµ as well as
+the maximum superradiant rate (see Fig. 1). Thus mas-
+sive neutral scalars too heavy to be produced with Kerr
+
+10
+BH superradiance may exist in the superradiant region
+of KNBHs. The situation is different for charged scalars.
+For fixed BH spin a and charge Q, increasing the scalar
+charge q always leads to narrower superradiant range of
+rgµ (see Figs. 4 and 5). Interestingly, the maximum su-
+perradiant rate happens at a small negative scalar charge
+q (see Table II). We have no explanation for this obser-
+vation.
+ACKNOWLEDGMENTS
+This work is supported in part by the National Nat-
+ural Science Foundation of China (NSFC) under Grant
+No.
+12075136 and the Natural Science Foundation of
+Shandong Province under Grant No. ZR2020MA094.
+[1] A. Arvanitaki and S. Dubovsky, Phys. Rev. D 83, 044026
+(2011) [arXiv:1004.3558 [hep-th]].
+[2] H. Yoshino and H. Kodama, Prog. Theor. Phys. 128,
+153-190 (2012) [arXiv:1203.5070 [gr-qc]].
+[3] M. Baryakhtar, M. Galanis, R. Lasenby and O. Simon,
+Phys. Rev. D 103, no.9, 095019 (2021) [arXiv:2011.11646
+[hep-ph]].
+[4] R. Penrose, Riv. Nuovo Cim. 1, 252-276 (1969)
+[5] C. W. Misner, Phys. Rev. Lett. 28, 994-997 (1972)
+[6] S. L. Detweiler, Phys. Rev. D 22, 2323-2326 (1980)
+[7] T. J. M. Zouros and D. M. Eardley, Annals Phys. 118,
+139-155 (1979)
+[8] V. Cardoso and S. Yoshida, JHEP 07, 009 (2005)
+[arXiv:hep-th/0502206 [hep-th]].
+[9] R. A. Konoplya and A. Zhidenko, Phys. Rev. D 73,
+124040 (2006) [arXiv:gr-qc/0605013 [gr-qc]].
+[10] S.
+R.
+Dolan,
+Phys.
+Rev.
+D
+76,
+084001
+(2007)
+[arXiv:0705.2880 [gr-qc]].
+[11] R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83,
+793-836 (2011) [arXiv:1102.4014 [gr-qc]].
+[12] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper
+and J. March-Russell, Phys. Rev. D 81, 123530 (2010)
+[arXiv:0905.4720 [hep-th]].
+[13] A. Arvanitaki, M. Baryakhtar and X. Huang, Phys. Rev.
+D 91, no.8, 084011 (2015) [arXiv:1411.2263 [hep-ph]].
+[14] A.
+Arvanitaki,
+M.
+Baryakhtar,
+S.
+Dimopoulos,
+S.
+Dubovsky and
+R. Lasenby,
+Phys. Rev.
+D
+95,
+no.4, 043001 (2017) [arXiv:1604.03958 [hep-ph]].
+[15] H. Yoshino and H. Kodama, PTEP 2014, 043E02 (2014)
+[arXiv:1312.2326 [gr-qc]].
+[16] H. Yoshino and H. Kodama, Class. Quant. Grav. 32,
+no.21, 214001 (2015) [arXiv:1505.00714 [gr-qc]].
+[17] R. Brito, V. Cardoso and P. Pani, Class. Quant. Grav.
+32, no.13, 134001 (2015) [arXiv:1411.0686 [gr-qc]].
+[18] R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso,
+I. Dvorkin, A. Klein and P. Pani, Phys. Rev. D 96, no.6,
+064050 (2017) [arXiv:1706.06311 [gr-qc]].
+[19] R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso,
+I. Dvorkin, A. Klein and P. Pani, Phys. Rev. Lett. 119,
+no.13, 131101 (2017) [arXiv:1706.05097 [gr-qc]].
+[20] G. Ficarra, P. Pani and H. Witek, Phys. Rev. D 99,
+no.10, 104019 (2019) [arXiv:1812.02758 [gr-qc]].
+[21] S. Bao, Q. Xu and H. Zhang, Phys. Rev. D 106, no.6,
+064016 (2022) [arXiv:2201.10941 [gr-qc]].
+[22] R. Roy, S. Vagnozzi and L. Visinelli, Phys. Rev. D 105,
+no.8, 083002 (2022) [arXiv:2112.06932 [astro-ph.HE]].
+[23] Y. Chen, R. Roy, S. Vagnozzi and L. Visinelli, Phys. Rev.
+D 106, no.4, 043021 (2022) [arXiv:2205.06238 [astro-
+ph.HE]].
+[24] L. Hui, Y. T. A. Law, L. Santoni, G. Sun, G. M. Tomaselli
+and E. Trincherini, [arXiv:2208.06408 [gr-qc]].
+[25] H. Witek, V. Cardoso, A. Ishibashi and U. Sperhake,
+Phys. Rev. D 87, no.4, 043513 (2013) [arXiv:1212.0551
+[gr-qc]].
+[26] S.
+Endlich
+and
+R.
+Penco,
+JHEP
+05,
+052
+(2017)
+[arXiv:1609.06723 [hep-th]].
+[27] M. Baryakhtar, R. Lasenby and M. Teo, Phys. Rev. D
+96, no.3, 035019 (2017) [arXiv:1704.05081 [hep-ph]].
+[28] S. R. Dolan, Phys. Rev. D 98, no.10, 104006 (2018)
+[arXiv:1806.01604 [gr-qc]].
+[29] W. E. East, Phys. Rev. D 96, no.2, 024004 (2017)
+[arXiv:1705.01544 [gr-qc]].
+[30] W. E. East and F. Pretorius, Phys. Rev. Lett. 119, no.4,
+041101 (2017) [arXiv:1704.04791 [gr-qc]].
+[31] W. E. East, Phys. Rev. Lett. 121, no.13, 131104 (2018)
+[arXiv:1807.00043 [gr-qc]].
+[32] V. P. Frolov, P. Krtouˇs, D. Kubizˇn´ak and J. E. Santos,
+Phys. Rev. Lett. 120, 231103 (2018) [arXiv:1804.00030
+[hep-th]].
+[33] P.
+Pani,
+V.
+Cardoso,
+L.
+Gualtieri,
+E.
+Berti
+and
+A. Ishibashi,
+Phys. Rev. Lett. 109,
+131102 (2012)
+[arXiv:1209.0465 [gr-qc]].
+[34] P.
+Pani,
+V.
+Cardoso,
+L.
+Gualtieri,
+E.
+Berti
+and
+A.
+Ishibashi,
+Phys.
+Rev.
+D
+86,
+104017
+(2012)
+[arXiv:1209.0773 [gr-qc]].
+[35] N. Siemonsen and W. E. East, Phys. Rev. D 101, no.2,
+024019 (2020) [arXiv:1910.09476 [gr-qc]].
+[36] J. Percival and S. R. Dolan, Phys. Rev. D 102, no.10,
+104055 (2020) [arXiv:2008.10621 [gr-qc]].
+[37] A. Caputo, S. J. Witte, D. Blas and P. Pani, Phys. Rev.
+D 104, no.4, 043006 (2021) [arXiv:2102.11280 [hep-ph]].
+[38] W. E. East, Phys. Rev. Lett. 129, no.14, 141103 (2022)
+[arXiv:2205.03417 [hep-ph]].
+[39] R. Brito, V. Cardoso and P. Pani, Phys. Rev. D 88, no.2,
+023514 (2013) [arXiv:1304.6725 [gr-qc]].
+[40] R. Brito, S. Grillo and P. Pani, Phys. Rev. Lett. 124,
+no.21, 211101 (2020) [arXiv:2002.04055 [gr-qc]].
+[41] R. Brito, V. Cardoso and P. Pani, Physics,” Lect.
+Notes Phys. 906, pp.1-237 (2015) 2020, ISBN 978-3-
+319-18999-4, 978-3-319-19000-6, 978-3-030-46621-3, 978-
+3-030-46622-0 [arXiv:1501.06570 [gr-qc]].
+[42] K. K. Y. Ng, O. A. Hannuksela, S. Vitale and T. G. F. Li,
+Phys. Rev. D 103, no.6, 063010 (2021) [arXiv:1908.02312
+[gr-qc]].
+[43] K.
+K.
+Y.
+Ng,
+S.
+Vitale,
+O.
+A.
+Hannuksela
+and
+T. G. F. Li, Phys. Rev. Lett. 126, no.15, 151102 (2021)
+[arXiv:2011.06010 [gr-qc]].
+[44] L. d. Cheng, H. Zhang and S. s. Bao, [arXiv:2201.11338
+[gr-qc]].
+[45] H. Yoshino and H. Kodama, PTEP 2015, no.6, 061E01
+
+11
+(2015) [arXiv:1407.2030 [gr-qc]].
+[46] Y. d. Guo, S. s. Bao and H. Zhang, [arXiv:2212.07186
+[gr-qc]].
+[47] E. W. Leaver, Proc. Roy. Soc. Lond. A 402, 285-298
+(1985)
+[48] T. Damour, N. Deruelle and R. Ruffini, Lett. Nuovo Cim.
+15, 257-262 (1976)
+[49] G. W. Gibbons, Commun. Math. Phys. 44, 245-264
+(1975)
+[50] R. D. Blandford and R. L. Znajek, Mon. Not. Roy. As-
+tron. Soc. 179, 433-456 (1977)
+[51] E. Barausse, V. Cardoso and P. Pani, Phys. Rev. D 89,
+no.10, 104059 (2014) [arXiv:1404.7149 [gr-qc]].
+[52] M. Zilh˜ao, V. Cardoso, C. Herdeiro, L. Lehner and
+U. Sperhake, Phys. Rev. D 90, no.12, 124088 (2014)
+[arXiv:1410.0694 [gr-qc]].
+[53] H. Furuhashi and Y. Nambu, Prog. Theor. Phys. 112,
+983-995 (2004) [arXiv:gr-qc/0402037 [gr-qc]].
+[54] S.
+Hod,
+Phys.
+Rev.
+D
+90,
+no.2,
+024051
+(2014)
+[arXiv:1406.1179 [gr-qc]].
+[55] S.
+Hod,
+Phys.
+Rev.
+D
+94,
+no.4,
+044036
+(2016)
+[arXiv:1609.07146 [gr-qc]].
+[56] Y. Huang, D. J. Liu, X. h. Zhai and X. z. Li, Phys. Rev.
+D 98, no.2, 025021 (2018) [arXiv:1807.06263 [gr-qc]].
+[57] Y. S. Myung, Eur. Phys. J. C 82, no.6, 518 (2022)
+[arXiv:2201.06706 [gr-qc]].
+[58] Y. S. Myung, Int. J. Mod. Phys. D 31, no.12, 2250087
+(2022) [arXiv:2204.04789 [gr-qc]].
+[59] Y. S. Myung, Phys. Rev. D 105, no.12, 124015 (2022)
+[arXiv:2204.06750 [gr-qc]].
+[60] R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265
+(1967)
+[61] E. Berti, V. Cardoso and M. Casals, Phys. Rev. D 73,
+024013 (2006) [erratum: Phys. Rev. D 73, 109902 (2006)]
+[arXiv:gr-qc/0511111 [gr-qc]].
+[62] W. H. Press and S. A. Teukolsky, Astrophys. J. 185,
+649-674 (1973)
+

CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:862674715e49f648eedff8260f650d36c2fb85f9c69905e8b953860772c1d1ee
+size 1572542

DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:013c3d29422c41d8fb4f1d6f9138d64b1399ac35cc5599cdd51d495728bfe0c7
+size 689865

DNE2T4oBgHgl3EQfoQhP/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9b86249700479b49b8b7b1d36d49a46e0c24515a2ac74c9f268a99fdf3e9c02c
+size 436988

EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b1f6f52dba627d68736d6ed9c5b66bf484b65084a151b382a0d2f569e778a640
+size 1458908

EtE1T4oBgHgl3EQfEgOL/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:35bf7837152b4f6266694160e268b8b4c064167af659ff1aafd5f6c898358118
+size 121859

F9E0T4oBgHgl3EQfhQEq/content/tmp_files/2301.02428v1.pdf.txt

@@ -0,0 +1,965 @@
+Sensitivity analysis using Physics-informed neural
+networks
+John M. Hannaa,b, Jos´e V. Aguadoa, Sebastien Comas-Cardonaa, Ramzi
+Askrib, Domenico Borzacchielloa
+aNantes Universit´e, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, 1 Rue de la No¨e,
+44300 Nantes, France
+bNantes Universit´e, IRT Jules Verne, 44340 Bouguenais, France
+Abstract
+The paper’s goal is to provide a simple unified approach to perform sensitiv-
+ity analysis using Physics-informed neural networks (PINN). The main idea lies
+in adding a new term in the loss function that regularizes the solution in a small
+neighborhood near the nominal value of the parameter of interest. The added
+term represents the derivative of the residual with respect to the parameter of
+interest. The result of this modification is a solution to the problem along with
+the derivative of the solution with respect to the parameter of interest (the sen-
+sitivity). We call the new technique to perform sensitivity analysis within this
+context SA-PINN. We show the effectiveness of the technique using 3 examples:
+the first one is a simple 1D advection-diffusion problem to show the methodol-
+ogy, the second is a 2D Poisson’s problem with 9 parameters of interest and the
+last one is a transient two-phase flow in porous media problem.
+Keywords:
+Physics-informed neural networks, sensitivity analysis, two-phase
+flow in porous media
+1. Introduction
+Sensitivity analysis is a technique to measure the effect of uncertainties in
+one or more input parameters on output quantities. Sensitivity can be regarded,
+quantitatively, as the derivative of output quantities with respect to input pa-
+rameters that might have some uncertainties. It has great importance in several
+Preprint submitted to Computer Methods in Applied Mechanics and Engineering
+arXiv:2301.02428v1  [math.NA]  6 Jan 2023
+
+engineering applications. Examples of these applications include aerodynamic
+optimization [1], shape optimization in solid mechanics [2], injection molding [3]
+and biomedical applications [4, 5]. This is just to name few applications.
+Several methods exist to calculate the sensitivities; the simplest one is the
+finite difference approach. It includes solving the system repeatedly, using any
+numerical technique, while varying the parameter of interest. Afterwards, the
+derivatives can be approximated by calculating the finite differences.
+This
+method becomes impractical when the number of the parameters of interest
+increases because the number of times the full system needs to be solved will
+grow exponentially, thus getting the sensitivities will become computationally
+intractable [6].
+The adjoint method is becoming the state of the art for performing sensi-
+tivity analysis specially in CFD applications [7]. The method attempts to solve
+an adjoint system of equations that is usually derived from the primal system.
+By solving the adjoint system, one obtains the gradient values with respect to
+parameters of interest. Despite the success and increasing widespread of the
+method, some issues still exist. One of these issues is related to the differentia-
+bility of the solution with discontinuities appearing in shock wave problems or
+two-phase flow problems that leads to instabilities [8].
+Machine learning based techniques have been growing rapidly to solve prob-
+lems governed by partial differential equations (PDEs). Physics-informed neu-
+ral networks (PINN) is one of the fast growing fields that attempts to solve
+forward and inverse problems governed by PDEs [9]. PINN gained wide in-
+terest due to the lack of need to a big data set since the physics governed by
+PDEs are used to regularize the model. PINN is powerful due to the use of feed
+forward neural networks, that are universal approximators [10], as the approx-
+imation space and the recent advances in automatic differentiation capabilities
+[11] that facilitated the derivative calculations. The method has been applied
+to several fields including solid mechanics [12], fluid mechanics [13, 14], addi-
+tive manufacturing [15], two-phase flow in porous media [16] and many others
+[17, 18, 19, 20, 21, 22, 23, 24].
+2
+
+In this article, PINN is used as the base framework to develop the new sensi-
+tivity analysis technique. The main objectives of the article can be summarized
+as follows:
+• Introducing a new technique to perform sensitivity analysis based on the
+framework of PINN and we call it SA-PINN.
+• Showing the ease and effectiveness of getting the sensitivity with respect
+to multiple parameters of interest at once.
+• Displaying the ability of SA-PINN to get the sensitivities for problems
+with discontinuities such as moving boundaries or sharp gradients.
+The paper is organized in the following manner. In chapter 2, a quick intro-
+duction to PINN is given followed by explanation to SA-PINN. Chapter 3 intro-
+duces three problems presented in the paper: 1D advection-diffusion problem,
+2D Poisson’s problem with multiple parameters of interest and a 1D unsteady
+two-phase flow in porous media problem. Chapter 4 gives the results to the
+three problems which includes the calculation of some sensitivities of interest.
+Chapter 5 offers a summary to the method and a conclusion.
+2. Sensitivity Analysis-PINN (SA-PINN)
+We consider a general partial differential equation of the form
+ut + µL(u) = 0,
+x ∈ Ω, t ∈ [0, T]
+(1)
+where ut is the time derivative, L a general differential operator and µ a ma-
+terial parameter. Initial and boundary conditions for the problems are defined
+as
+u(0, x) = u0
+(2)
+u(t, xD) = uD
+(3)
+B(u(t, xN)) = f(xN)
+(4)
+3
+
+where B is a differential operator, xD the boundary where Dirichlet bound-
+ary condition is enforced and xN the boundary where Neumann boundary con-
+dition is applied.
+2.1. PINN
+The first step to solve this problem with PINN is to choose the approxima-
+tion space. The choice is a feed forward neural network. Through automatic
+differentiation, a combined loss is formed of the residual of the PDE defined
+at spatio-temporal points called collocation points and the error in the ini-
+tial/boundary conditions’ enforcement.
+A solution can be obtained through
+updating the weights and biases of the neural network by minimizing the loss
+function using algorithms such as: gradient-descent, Adam, BFGS, etc. The
+loss function can be written as:
+Loss = λ0 loss0 + λD lossD + λN lossN + λ1 lossr
+(5)
+where λi are the weights for each loss term which play an important role in
+the optimization process and:
+loss0 = 1
+N0
+N0
+�
+i=1
+r2
+0(ti
+0, xi
+0) = 1
+N0
+N0
+�
+i=1
+||u(ti
+0, xi
+0) − ui
+0||2
+(6)
+lossD =
+1
+ND
+ND
+�
+i=1
+r2
+D(ti
+D, xi
+D) =
+1
+ND
+ND
+�
+i=1
+||u(ti
+D, xi
+D) − ui
+D||2
+(7)
+lossN =
+1
+NN
+NN
+�
+i=1
+r2
+N(ti
+N, xi
+N) =
+1
+NN
+NN
+�
+i=1
+||B(u(ti
+N, xi
+N)) − f i
+N||2
+(8)
+lossr = 1
+Nr
+Nr
+�
+i=1
+||r(ti
+r, xi
+r)||2 = 1
+Nr
+Nr
+�
+i=1
+||ut + µL(u)||2
+(ti
+r,xi
+r)
+(9)
+(10)
+lossi respectively are the losses representing the initial condition, Dirichlet,
+Neumann boundary conditions and the PDE residual.
+4
+
+2.2. SA-PINN
+The main objective of PINN is to find a solution that minimizes the residual
+of the PDE within the spatiotemporal domain that is represented by the collo-
+cation points while respecting the initial and boundary conditions. The result
+is a solution ˆu(t, x; ˆµ) to the PDE at a specific value ˆµ. To perform sensitivity
+analysis with µ as the input parameter of interest, we would like to find not
+only the solution ˆu but also the derivative of the solution with respect to µ (the
+sensitivity) at a given nominal value ˆµ.
+One way to obtain the sensitivity in PINN is to build a parametric model.
+This is done by changing the structure of the neural network to accommodate for
+another input which is the parameter of interest µ. Then, one adds collocation
+points in the spatiotemporal-parametric space. Then, the residual is minimized
+in the whole parametric domain while respecting the initial and boundary con-
+ditions.
+Afterwards, the derivative of the solution with respect to µ can be
+easily obtained through automatic differentiation. The main issue of building
+such parametric models is that the number of collocation points grows exponen-
+tially with the number of parameters of interest. The problem can, then, easily
+become computationally intractable if we have several parameters of interest
+which is common in most engineering applications.
+To overcome this issue, we thought that instead of only minimizing the
+residual of the PDE, we can also minimize the derivative of the residual with
+respect to the parameter of interest. First, the structure of the neural network
+should accommodate for µ as an input. Then, the loss function is formed as the
+sum of the residual and the derivative of the residual with respect to µ along
+with the terms to respect the initial and boundary conditions. This way we
+make sure that the solution is accurate within a small neighborhood of ˆµ, thus
+sensitivity can be calculated. We call this technique SA-PINN. The technique
+can be summarized in points as follows:
+• Choose the neural network to have inputs related to space, time and pa-
+rameter of interest.
+5
+
+• Sample the collocation points only in space and time, however the points
+will be living in a higher dimension but without adding more points.
+• Create the loss function having terms related to PDE residual, the residual
+derivative with respect to the parameter of interest and the terms related
+to the initial and boundary conditions.
+The modified loss function will then be
+Lossm = Loss + λ0µ loss0µ + λDµ lossDµ + λNµ lossNµ + λ1µ lossrµ
+(11)
+where
+loss0µ = 1
+N0
+N0
+�
+i=1
+����
+∂r0(ti
+0, xi
+0, µi
+0)
+∂µ
+����
+(12)
+lossDµ =
+1
+ND
+ND
+�
+i=1
+����
+∂rD(ti
+D, xi
+D, µi
+D)
+∂µ
+����
+(13)
+lossNµ =
+1
+NN
+NN
+�
+i=1
+����
+∂rN(ti
+N, xi
+N, µi
+N)
+∂µ
+����
+(14)
+lossrµ = 1
+Nr
+Nr
+�
+i=1
+����
+∂r(ti
+r, xi
+r, µi
+r)
+∂µ
+����
+(15)
+(16)
+Figure 1 shows a diagram that summarizes the methodology of SA-PINN.
+The parts in orange are the added parts from classical PINN. The u − ˆu term
+represents the mismatch of the solution from the initial and boundary condi-
+tions. It must be noted that we sample the collocation points only in space and
+time, but the points have another coordinate µ and all have a nominal value ˆµ.
+6
+
+Figure 1: Diagram explaining the methodology of SA-PINN.
+3. Model problems
+In this section, we introduce the models of the three examples that are used
+to show the effectiveness of the technique.
+3.1. 1D diffusion-advection equation
+The first example is a steady one-dimensional diffusion-advection equation
+where we would like to study the effect of perturbations in the diffusion term ϵ
+on the solution. The strong form of the problem can be written as follows:
+ϵ yxx − yx + 1 = 0,
+x ∈ [0, 1],
+y(0) = 1,
+y(1) = 3
+(17)
+The chosen nominal value for ϵ is 0.1. yxx and yx are respectively the second
+and first order derivatives of the solution y. The weights for the different terms
+in the loss function are set to 1 for the original PINN terms and 0.1 for the
+added sensitivity terms.
+7
+
+X
+dr
+ne
+t
+NN
+LosS
+u
+u-a
+a(u-a)
+ne3.2. 2D Poisson’s problem
+The next example is a 2-dimensional Poisson’s problem where we have mul-
+tiple parameters to study their effect on the solution. The domain is shown in
+figure 2 where there exists 9 subdomains each having different diffusivity value.
+k1
+k3
+k2
+k4
+k9
+k6
+k5
+k8
+k7
+Figure 2: 2D Poisson’s problem domain
+The strong form of the problem can be written as:
+k ∆u = −1,
+in Ω,
+u = 0,
+on ∂Ω
+(18)
+where Ω is a square with unit sides and k is the diffusivity. The 9 subdomains
+have equal areas. The nominal values for the diffusivity is 1; k1 = k2 = ... =
+k9 = 1.
+The main PINN terms weights are set to 1 and 0.1 for the added
+sensitivity terms.
+3.3. 1D two-phase flow in porous media
+In this section, we introduce a 1D two-phase flow in porous media problem.
+The problem is faced in Liquid Transfer Molding composite manufacturing pro-
+cesses, where resin is injected in a mold that has prepositioned fibrous matrix.
+The problem is shown in figure 3. At t = 0, the domain is initially saturated
+with one fluid (fluid 1). Another fluid (fluid 2) is being injected from the left
+end at constant pressure pin, while the pressure at the other end is fixed to pout.
+8
+
+pin
+pout
+Flow front
+fluid 2
+fluid 1
+Figure 3: One-dimensional domain (filling problem)
+The momentum equation can be approximated with Darcy’s law that can
+be written in 1D as follow:
+v = − k
+φµpx
+(19)
+where v is the volume average Darcy’s velocity, µ the viscosity, and px the
+pressure gradient, and φ the porosity. Both fluids are assumed to be incom-
+pressible, therefore, the mass conservation equation reduces to
+vx = 0
+(20)
+Pressure boundary conditions can prescribed on the inlet and oulet:
+p(xinlet, t) = pin,
+p(xoutlet, t) = pout
+(21)
+To track the interface between the two fluids, the Volume Of Fluids (VOF)
+technique is used; a fraction function c is introduced which takes a value 1 for
+the resin and 0 for the air. The viscosity µ is redefined as
+µ = cµ2 + (1 − c)µ1
+(22)
+where µ2 and µ1 are the two fluids’ viscosities. c evolves with time according
+to the following advection equation
+ct + vcx = 0
+(23)
+9
+
+where ct and cx are the time and spatial derivative of the fraction function
+c, respectively.
+Initial and boundary conditions are defined to solve the advection of c.
+c(x, t = 0) = c0(x),
+c(xinlet, t) = 1
+(24)
+To sum up, the strong form of the problem can be written as:
+ct + v cx = 0,
+x ∈ [0, l],
+t ∈ [0, T],
+v = − k
+φµpx,
+x ∈ [0, l],
+t ∈ [0, T],
+vx = 0,
+x ∈ [0, l],
+t ∈ [0, T],
+µ = cµ2 + (1 − c)µ1
+p(0, t) = pin,
+p(l, t) = pout
+c(0, t) = 1,
+c(x, 0) = 0
+(25)
+The parameters of the problem are shown in table 1.
+Table 1: Parameters used for the two-phase flow problem.
+Parameter
+Value
+k
+1
+µ1
+10−5
+µ2
+1
+pin
+1
+pout
+0
+l
+1
+φ
+1
+The main PINN terms weights are set to 1 and 0.01 for the added sensitivity
+terms. The adaptivity algorithm presented in [16] is used to get a better sharper
+solution.
+10
+
+4. Results
+4.1. 1D diffusion-advection equation
+The solution u using PINN and SA-PINN is shown in figure 4 along with
+the analytical solution for ϵ = 0.1.
+Figure 4: Solution u at ϵ = 0.1 using PINN and SA-PINN along with the analytical solution
+of the 1D advection-diffusion problem.
+From figure 4, we can see that PINN and SA-PINN acturetly captures the
+analytical solution to the problem. The derivative of the solution with respect
+to ϵ at to ϵ = 0.1 is shown in figure 5.
+11
+
+3.00
+2.75-
+2.50 
+2.25 -
+u
+2.00
+1.75 -
+1.50
+Analytical
+PINN
+1.25-
+SA-PINN
+1.00
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+XFigure 5:
+∂u
+∂ϵ at ϵ = 0.1 using PINN and SA-PINN along with the finite difference solution of
+the 1D advection-diffusion problem.
+The reference finite difference solution in figure 5 is obtained by obtaining
+different PINN solutions near ϵ = 0.1 and then calculating the derivative. We
+can see that classical PINN fails to predict the derivative, while, SA-PINN
+accurately predicts the derivative due to the added regularization term in the
+loss function. The loss function for different values of ϵ is plotted in figure 6 for
+PINN and SA-PINN.
+12
+
+4
+-2 
+3e/ne
+-4 
+-6 
+Finite difference
+-8 -
+PINN
+SA-PINN
+-10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+XFigure 6: Loss function for different ϵ values using PINN and SA-PINN of the 1D advection-
+diffusion problem.
+As seen in figure 6, SA-PINN has the effect of greatly flattening the loss
+curve in a neighborhood near the nominal value of ϵ = 0.1. This leads to better
+solutions that PINN in the neighborhood and accurate derivative calculation at
+ϵ = 0.1.
+4.2. 2D Poisson’s problem
+The PINN solution of the boundary value problem is shown in figure 7. The
+solution appears to be accurate and agrees with the analytical solution of the
+problem.
+13
+
+3.0
+PINN
+SA-PINN
+2.5
+2.0
+Loss funcion
+1.5
+1.0
+0.5
+0.06
+0.08
+0.10
+0.12
+0.14Figure 7: PINN solution of the 2D Poisson’s boundary value problem.
+The sensitivity terms
+∂u
+∂ki can then be plotted to see the effect of the diffu-
+sivity on the solution.
+14
+
+1.0
+0.08
+0.07
+0.8
+0.06
+0.05
+0.6
+0.04
+0.4 -
+0.03
+0.02
+0.2 -
+0.01
++0'0
+0.00
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+XFigure 8: Different derivatives of the solution with respect to ki of the 2D Poisson’s problem.
+The computational time is plotted versus the number of parameters with
+respect to which sensitivity terms are added in figure 9.
+15
+
+au/aki
+au/ak2
+au/ak3
+1.0 -
+ 0.002
+1.0 
+0.0025
+1.0 
+0.002
+ 0.000
+0.0000
+0.000
+0.8 
+0.8 
+0.0025
+0.8 
+0.002
+0.002
+0.0050
+0.6 
+0.004
+0.6 -
+0.6 
+0.004
+0.0075
+y
+0.006
+y
+y
+0.006
+0.0100
+0.4 
+0.008
+0.4 
+0.4 
+0.0125
+0.008
+0.010
+0.2 
+0.2 
+0.0150
+0.010
+0.2 
+0.012
+0.0175
+0.012
+0.0 
+0.014
+0.0 
+0.0200
+0'0
+0.2
+0.4
+0.6
+8'0
+1.0
+0'0
+0.2 
+0.4
+0.6
+8'0
+1.0
+0.0 +
+0.2
+0.4 
+0.8
+0.014
+0'0
+0.6
+1.0
+x
+x
+x
+au/ak4
+au/aks
+au/ak6
+1.0 
+0.002
+1.0 
+0.000
+1.0 
+0.0000
++0.000
+0.003
+0.0025
+80
+0.002
+0.8
+0.8 
+0.006
+0.0050
+0.004
+0.6
+0.009
+0.6 
+0.6
+0.0075
+0.006
+y
+y
+0.012
+y
+0.0100
+0.008
+0.4 
+0.4 
+0.015
+0.4
+0.0125
+0.010
+0.018
+0.2 
+0.0150
+0.2 -
+0.012
+0.2 
+0.014
+0.021
+0.0175
+0.0 +
+0.016
+0.0
+0.2
+0.4
+0.6
+8'0
+1.0
+0.024
+0.0+
+0.4
+0.6
+80
+0.0200
+0.0
+0.2 
+0.4 
+9'0
+8:0
+1.0
+0.0
+0.0
+0.2
+1.0
+x
+x
+x
+au/ak7
+au/akg
+au/akg
+1.0 
+T 0.000
+1.0 
+0.0000
+1.0 -
+0.002
+0.0025
+0.002
+0.000
+0.8
+0.8
+80
+0.0050
+0.004
+0.002
+0.6 
+0.6 
+0.0075
+0.6
+0.004
+0.006
+y
+y
+0.0100
+y
+0.006
+0.4 
+0.008
+0.4
+0.0125
+0.4 
+0.008
+0.010
+0.0150
+0.010
+0.2 
+0.2 
+0.2 -
+0.012
+0.0175
+0.012
+0.0 +
+0.014
+0.0 +
+0.0200
+0.0+
+0.014
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+8'0
+1.0
+x
+x
+xFigure 9: Computational time vs. no of sensitivity parameters.
+It can be seen from the figure that the computational time grows linearly
+when increasing the number of parameters the sensitivity is calculated with
+respect to. This happens because the number of collocation points is the same
+when adding a new term to the loss function; the added cost is the same when
+adding new sensitivity terms.
+4.3. 1D transient two-phase flow in porous media
+First, we plot the front location for three different values of k by taking the
+0.5 level set of the fraction function c in figure 10. We compare SA-PINN with
+classical PINN along with the analytical solution.
+16
+
+1000
+computation time (s)
+800
+600 
+400 
+200 -
+0
+2
+4
+6
+8
+no of parametersFigure 10: Flow front location vs. time for three different values of k (k = 1, 0.5 and 2) of
+the transient two-phase flow in porous media problem.
+We can notice that SA-PINN provides good results for values of k away from
+the nominal value k = 1. Classical PINN accurately predicts the solution only
+at the nominal values, however, away from that values, random solutions were
+obtained which is clear from the two red lines.
+In figure 11, we plot the time the flow front reaches x = 0.5 vs. k. We
+compare the solution from SA-PINN with the analytical solution.
+17
+
+1.0 
+1.0
+1.0 
+0.8
+0.8 -
+0.8 
+0.6
+0.6 -
+0.6 .
+0.4 
+0.4 
+0.4 -
+0.2 .
+0.2 1/
+0.2 
+Analytical
+PINN
+SA-PINN
+0.0 +
+0.0
+0.1
+0.2
+0
+0.4
+0.5
+0.1
+0.2 
+0.3
+0.4
+0.5
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.0
+0.0
+TimeFigure 11: Time at which the flow front reaches x = 0.5 vs. k for the transient two-phase flow
+in porous media problem.
+We can see a good estimation of the filling time at different values of k using
+SA-PINN. This result can be useful in applications of injection processes to
+estimate the filling time as a function of a parameter of interest.
+5. Conclusion
+In the article, we presented a new method to perform sensitivity analysis
+based on the paradigm of PINN. The method is easy to implement using any
+of the machine learning libraries as TensorFlow or PyTorch. We show, through
+the examples, that the technique is easy to use when sensitivity with respect to
+multiple parameters of interest are studied at the same time. The computation
+time grows linearly as the parameters increase which is an advantage to the
+method. We also show through the last example that the method is working for
+a problem where a discontinuity exists (flow front) and VOF method is used.
+18
+
+0.5
+Analytical
+SA-PINN
+0.4 -
+0.3 
+time
+0.2
+0.1
+0.0
+0.6
+0.8
+1.0
+1.2
+1.4
+kAcknowledgements
+This study was funded under the PERFORM Thesis program of IRT Jules
+Verne.
+References
+[1] J. E. Peter, R. P. Dwight, Numerical sensitivity analysis for aerodynamic
+optimization: A survey of approaches, Computers & Fluids 39 (3) (2010)
+373–391.
+[2] L.-J. Leu, S. Mukherjee, Sensitivity analysis and shape optimization in
+nonlinear solid mechanics, Engineering analysis with boundary elements
+12 (4) (1993) 251–260.
+[3] D. E. Smith, D. A. Tortorelli, C. L. Tucker III, Analysis and sensitivity
+analysis for polymer injection and compression molding, Computer Meth-
+ods in Applied Mechanics and Engineering 167 (3-4) (1998) 325–344.
+[4] G. L. Pishko, G. W. Astary, T. H. Mareci, M. Sarntinoranont, Sensitivity
+analysis of an image-based solid tumor computational model with hetero-
+geneous vasculature and porosity, Annals of biomedical engineering 39 (9)
+(2011) 2360–2373.
+[5] L. Santos, M. Martinho, R. Salvador, C. Wenger, S. R. Fernandes,
+O. Ripolles, G. Ruffini, P. C. Miranda, Evaluation of the electric field in the
+brain during transcranial direct current stimulation: a sensitivity analysis,
+in: 2016 38th Annual International Conference of the IEEE Engineering in
+Medicine and Biology Society (EMBC), IEEE, 2016, pp. 1778–1781.
+[6] M. Koda, A. H. Dogru, J. H. Seinfeld, Sensitivity analysis of partial differen-
+tial equations with application to reaction and diffusion processes, Journal
+of Computational Physics 30 (2) (1979) 259–282.
+19
+
+[7] A. Lincke, T. Rung, Adjoint-based sensitivity analysis for buoyancy-driven
+incompressible navier-stokes equations with heat transfer, in: Proceedings
+of the Eighth Internat. Conf. on Engineering Computational Technology,
+Dubrovnik, Croatia, 2012.
+[8] N. K¨uhl, J. Kr¨oger, M. Siebenborn, M. Hinze, T. Rung, Adjoint comple-
+ment to the volume-of-fluid method for immiscible flows, Journal of Com-
+putational Physics 440 (2021) 110411.
+[9] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural net-
+works: A deep learning framework for solving forward and inverse problems
+involving nonlinear partial differential equations, Journal of Computational
+Physics 378 (2019) 686–707.
+[10] K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks
+are universal approximators, Neural networks 2 (5) (1989) 359–366.
+[11] A. G. Baydin, B. A. Pearlmutter, A. A. Radul, J. M. Siskind, Automatic
+differentiation in machine learning: a survey, Journal of machine learning
+research 18 (2018).
+[12] E. Haghighat, M. Raissi, A. Moure, H. Gomez, R. Juanes, A deep learning
+framework for solution and discovery in solid mechanics, arXiv preprint
+arXiv:2003.02751 (2020).
+[13] S. Cai, Z. Mao, Z. Wang, M. Yin, G. E. Karniadakis, Physics-informed
+neural networks (pinns) for fluid mechanics: A review, Acta Mechanica
+Sinica (2022) 1–12.
+[14] M. Raissi, A. Yazdani, G. E. Karniadakis, Hidden fluid mechanics: Learn-
+ing velocity and pressure fields from flow visualizations, Science 367 (6481)
+(2020) 1026–1030.
+[15] Q. Zhu, Z. Liu, J. Yan, Machine learning for metal additive manufactur-
+ing: predicting temperature and melt pool fluid dynamics using physics-
+informed neural networks, Computational Mechanics 67 (2) (2021) 619–635.
+20
+
+[16] J. M. Hanna, J. V. Aguado, S. Comas-Cardona, R. Askri, D. Borzacchiello,
+Residual-based adaptivity for two-phase flow simulation in porous media
+using physics-informed neural networks, Computer Methods in Applied
+Mechanics and Engineering 396 (2022) 115100.
+[17] A. D. Jagtap, G. E. Karniadakis, Extended physics-informed neural net-
+works (xpinns):
+A generalized space-time domain decomposition based
+deep learning framework for nonlinear partial differential equations, Com-
+munications in Computational Physics 28 (5) (2020) 2002–2041.
+[18] A. D. Jagtap, E. Kharazmi, G. E. Karniadakis, Conservative physics-
+informed neural networks on discrete domains for conservation laws: Ap-
+plications to forward and inverse problems, Computer Methods in Applied
+Mechanics and Engineering 365 (2020) 113028.
+[19] G. Pang, L. Lu, G. E. Karniadakis, fpinns: Fractional physics-informed
+neural networks, SIAM Journal on Scientific Computing 41 (4) (2019)
+A2603–A2626.
+[20] K. Shukla, P. C. Di Leoni, J. Blackshire, D. Sparkman, G. E. Karniadakis,
+Physics-informed neural network for ultrasound nondestructive quantifica-
+tion of surface breaking cracks, Journal of Nondestructive Evaluation 39 (3)
+(2020) 1–20.
+[21] S. A. Niaki, E. Haghighat, T. Campbell, A. Poursartip, R. Vaziri, Physics-
+informed neural network for modelling the thermochemical curing process
+of composite-tool systems during manufacture, Computer Methods in Ap-
+plied Mechanics and Engineering 384 (2021) 113959.
+[22] L. Sun, H. Gao, S. Pan, J.-X. Wang, Surrogate modeling for fluid flows
+based on physics-constrained deep learning without simulation data, Com-
+puter Methods in Applied Mechanics and Engineering 361 (2020) 112732.
+[23] A. M. Tartakovsky, C. O. Marrero, P. Perdikaris, G. D. Tartakovsky,
+D. Barajas-Solano, Physics-informed deep neural networks for learning pa-
+21
+
+rameters and constitutive relationships in subsurface flow problems, Water
+Resources Research 56 (5) (2020) e2019WR026731.
+[24] S. Goswami, C. Anitescu, S. Chakraborty, T. Rabczuk, Transfer learning
+enhanced physics informed neural network for phase-field modeling of frac-
+ture, Theoretical and Applied Fracture Mechanics 106 (2020) 102447.
+22
+

F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6b008cbdfc9dbd622e2912a82a9b5f261e8f1f41cb8a53b45c96ddb697bbfc14
+size 7590582

F9E1T4oBgHgl3EQfqwX2/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1ddaed406db83455db408d8c6fe71250220ca8722b150595bc184161635765a1
+size 5111853

FNE1T4oBgHgl3EQfEwMv/content/tmp_files/2301.02893v1.pdf.txt

@@ -0,0 +1,1278 @@
+J. Bio. & Env. Sci. 2022 
+ 
+109 | Javier et al. 
+ 
+    
+RE
+RE
+RE
+RESEARCH
+SEARCH
+SEARCH
+SEARCH    PAPER
+PAPER
+PAPER
+PAPER                                                     
+                                                     
+                                                     
+                                                                                                              
+                 
+                 
+                                      
+         
+         
+         OPEN ACCESS
+OPEN ACCESS
+OPEN ACCESS
+OPEN ACCESS 
+ 
+MangngalApp- An integrated package of technology for 
+COVID- 19 response and rural development: Acceptability and 
+usability using TAM 
+ 
+Billy S. Javier*, Leo P. Paliuanan, James Karl A. Agpalza, Jesty S. Agoto 
+ 
+College of Information and Computing Sciences, Cagayan State University, Aparri, Philippines 
+ 
+Article published on  October 20, 2022 
+Key words: Acceptability, COVID-19, Fishers, POTs, Technology acceptance model, ISO 25010 
+Abstract 
+The COVID19 pandemic has challenged universities and organizations to devise mechanisms to uplift the well-
+being and welfare of people and communities. In response, the design and development of an integrated package 
+of technologies, MangngalApp- A web-based portal and mobile responsive application for rural development 
+served as an opportunity. It showcases different packets of technologies that were outputs of R&D in the field of 
+fisheries and aqua-culture, innovations that were IP-protected, and technologies that harness locally available 
+resources for post-harvest development and aiding in sustaining growth and development in the communities. 
+This paper focused on the usability and acceptability of the MangngalApp implementing a descriptive research 
+design using the Technology Acceptance Model or TAM and ISO 25010 software quality standards. Constrained 
+by government health restrictions due to COVID- 19, a Google form-based questionnaire was forwarded to 
+consented participants via an email with the attached consent and evaluation form. Results revealed that the 
+MangngalApp was found to be very acceptable and usable, and compliant to ISO 25010 software quality 
+characteristics to the higher extent. From the results, it is concluded that the developed MangngalApp will be a 
+usable and responsive technology that aids to rural development especially among target users- fishers, 
+gatherers, processors, traders, and farmers. Considering compatibility and usefulness, the MangngalApp is 
+expected to provide greater social development in the community. 
+*Corresponding Author: Billy S. Javier  billyjavier@csu.edu.ph 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Journal of Biodiversity and Environmental Sciences (JBES) 
+ISSN: 2220-6663 (Print) 2222-3045 (Online) 
+Vol. 21, No. 4, p. 109-117, 2022 
+http://www.innspub.net 
+ 
+
+J. Bio. & Env. Sci. 2022 
+ 
+110 | Javier et al. 
+Introduction 
+The COVID 19 pandemic has disrupted many 
+organizations, 
+government 
+and 
+non-government 
+institutions, 
+schools, 
+companies, 
+and 
+various 
+communities. As a result, more displaced workers and 
+job losses increased, more families sent home, locked 
+down due to COVID19 restrictions and uncertain of 
+how and where to obtain immediate income for the 
+family. The government may have provided financial 
+assistance to erring families and fed empty stomachs. 
+However, resources deplete as no concrete measure to 
+total stop the threat of the on-going pandemic the 
+Filipino people is enjoying. The Cagayan State 
+University is mandated to transforming the lives of 
+people and communities through high quality 
+instruction, innovative research, development, and 
+production. Through the years, CSU has been 
+working hard on innovating technologies that could 
+help alleviate poverty, increase productivity and 
+improve socioeconomic status of the communities, 
+and 
+help 
+in 
+sustaining 
+and 
+protecting 
+the 
+environment. However, no matter how promising 
+these 
+technologies 
+are 
+if 
+these 
+packages 
+of 
+technologies are not widely accessible to target 
+communities, to its intended stakeholders: fishers, 
+farmers, gatherers, and processors. In fact, Sharma A, 
+and Kiranmayi, D (2019) was unable to find in many 
+literature and studies pertaining to a package of 
+technologies as an IEC initiative to adopting and 
+utilizing research-based fisheries technologies, post-
+harvest technologies, and aquaculture techniques. 
+Most of the 124 applications reported focused on 
+mobile apps for angling, aquaculture management, 
+aquarium management, marine fisheries, and fisheries 
+governance, marketing and biology.  
+  
+Research project generating innovative technologies 
+and products has been funded and curated by experts 
+in 
+the 
+various 
+fields 
+leading 
+to 
+technology 
+commercialization. These then has to be extended to 
+communities via available and relevant technologies 
+so that as an academic institution, it really radiates its 
+mantra of improving the lives of people and 
+communities. The MangngalApp research program 
+was generally geared at providing a solution for a 
+well-informed 
+utilization 
+of 
+the 
+packets 
+of 
+technologies (POTs) developed as results of scientific 
+inquiries and experiments of the University and 
+collaborating agencies. It has been said that 
+technologies should be utilized by the communities, 
+adopted via technology-transfer, generating income 
+from them. However, access to POTs may have not 
+deliberately 
+reaching 
+the 
+realms 
+of 
+coastal 
+communities. Lack of or limited access to POTs 
+among fishers, farmers, gatherers, and processors 
+may cause inefficiency, increased cost for production, 
+and lower productivity among fishers, fish processors 
+and gatherers, as well as farmers in the coastal 
+communities in northern Philippines.  
+  
+With aqua-marine as banner program in the Aparri 
+Campus, a multi-disciplinary research program was 
+proposed with the hope of generating a package of 
+technology showcasing the science-based packages of 
+technologies of university along fishing activities, 
+seaweed farming, post-harvest, product development 
+and more. The research is expected to benefit the 
+coastal communities through provision of mobile-
+ready and friendly application accessible to users 
+aiding to improve productivity, increased awareness 
+and protection for the environment, and providing 
+livelihood for women and differently-able persons. 
+Packages of technologies developed will be best 
+adopted or utilized in the community once an 
+integrated package or technology is made available. 
+Hence, the potential benefits expand from the fishers 
+in the conduct of and management of their fisheries 
+activities to any other intended users. Coastal farmers 
+will be able to uncover scientific ways to conservation 
+and management of marine species or seaweeds. Fish 
+processors will have the potential to improve 
+productivity, creation of jobs, and increased revenues.  
+  
+Adapting the vision of the Food and Agriculture 
+Organization of the United Nations (FAO) on 
+enhancing the role of small-scale fisheries in 
+contributing to poverty alleviation and food security, 
+the project also focused on understanding the 
+technology awareness, technology adoption practices, 
+the information needs and seeking behaviors, media 
+
+J. Bio. & Env. Sci. 2022 
+ 
+111 | Javier et al. 
+literacy and media adoption of various stakeholders 
+in the fishing communities of Northeastern Cagayan 
+Philippines. In the academe, students and teachers 
+may benefit from the having obtained the scientific 
+packages of technologies for instruction purposes, and 
+an opportunity for more relevant research formulation. 
+The results of the study hope to provide and cultivate 
+new knowledge for students, researchers, and teachers. 
+In so doing, students and teachers may devise projects, 
+programs, and studies that could add up to the 
+packages of technologies. Institutions or organization 
+may have devise appropriate strategies, programs and 
+plans from data mining and knowledge data discovery 
+thru the program.  
+ 
+The emergence of an information, communication, 
+and education platform through varied technologies 
+is a must especially in the dissemination of scientific 
+results and innovations from rigid experiments and 
+research. Digital visibility is considered an efficient 
+and reasonable way to publicize the outputs of 
+innovative 
+developments 
+and 
+research 
+results 
+(Magdalinou, 2019). The Technology Acceptance 
+Model (TAM) is a theory in information systems that 
+explain how consumers come to embrace the use of a 
+technology. When consumers are introduced with 
+new technology, the model argues that a variety of 
+factors influence their decision on how and when to 
+use it. TAM has been critiqued for a variety of 
+reasons, but it is a useful overall framework that is 
+compatible with several studies examining the 
+elements that influence older individuals' willingness 
+to utilize new technology (Braun, 2013).  
+ 
+This paper generally aims to describe the usability 
+and acceptability of the developed mobile responsive 
+web project known as MangngalApp - an integrated 
+package of technology using open-source web 
+development platform.  
+ 
+The assessment of the usability and acceptability of 
+the MangngalApp using the Technology Acceptance 
+Model (TAM) focused on (a) Perceived Ease of Use, 
+(b) Perceived usefulness, (c)Attitudes towards usage, 
+(d) Behavioral intention to use, and Relevance to the 
+present job. In addition, the assessment of the 
+developed MangngalApp based on ISO 25010 
+software quality characteristics has been reported. 
+ 
+Materials and methods 
+The descriptive research design was implemented in 
+this part of the project. The assessment of the 
+usability and acceptability of the developed Mangngal 
+App using the Technology Acceptance Model or TAM 
+was participated by 200 non-technical respondents. 
+These included fishers, farmers, fish processors, 
+gatherers, and households involved in post-harvest. A 
+listing of which was taken from municipal agriculture 
+office 
+thru 
+communications. 
+Meanwhile, 
+the 
+assessment of the 20 technical respondents applying 
+ISO 25010 software quality standards, provided proof 
+of the compliance in terms of compatibility, 
+reliability, user-friendliness, security, portability, and 
+functional suitability. The profile of the technical 
+respondents is presented herein in table 1. The 
+survey-questionnaire included some profile data of 
+respondents, their assessment of the MangngalApp, 
+and an optional remark or comment part. A consent 
+form was part of the questionnaire, while prior 
+presentation or orientation on its use was provided 
+via Google Meet.  
+ 
+The researchers took the assistance of partner-
+students and community leaders handled by the team 
+in the locality to share the MangngalApp project and 
+guide intended users including those involved in 
+actual fishing, post-harvest development, processing, 
+gathering, as well as those who are trading. This is a 
+COVID-19 initiative of the project team in order to 
+gather sentiments and assessment of those greater 
+users. On the other hand, the technical respondents 
+were 
+communicated 
+formally 
+requesting 
+their 
+expertise, and provided the team consent to 
+participate in the assessment.  
+ 
+The respondents in the evaluation of the technical 
+compliance, usability and acceptability standards 
+using TAM included 10 industry practitioners, 10 ICT 
+teachers 
+with 
+experiences 
+in 
+databases, 
+web 
+development and design, and programming.  
+
+J. Bio. & Env. Sci. 2022 
+ 
+112 | Javier et al. 
+Table 1. Profile of the Technical Respondents. 
+Participants 
+Male 
+Female 
+Total 
+% 
+Classification  
+ 
+ 
+ 
+ 
+Industry 
+Practitioners  
+6 
+4 
+10 
+50.0 
+ICT Teachers  
+5 
+5 
+10 
+50.0 
+Area of Interest  
+ 
+ 
+ 
+ 
+Web Design  
+2 
+3 
+5 
+25.0 
+Web Programming  
+3 
+2 
+5 
+25.0 
+Databases  
+2 
+2 
+4 
+20.0 
+Programming  
+2 
+2 
+4 
+20.0 
+Networks  
+2 
+ 
+2 
+10.0 
+Years of Relevant ICT Experience 
+ 
+ 
+1 to 3  
+4 
+5 
+9 
+45.0 
+4 to 6  
+6 
+3 
+9 
+45.0 
+More than 6  
+1 
+1 
+2 
+10.0 
+  
+  
+  
+The participants were notified via email on their 
+participation in the assessment. A brief orientation 
+via Google Meet was conducted to provide them 
+overview of the project. The project team provided the 
+link of web based MangngalApp project. They were 
+given at least 2 to 5 weeks to access the web project 
+and were requested to fill out the evaluation forms via 
+Google Forms. Treating the assessment of the 
+Usability and Acceptability of the MangngalApp using 
+the Technology Acceptance Model, the 4-point Likert 
+scale was used: 1 being not acceptable and usable to 4 
+being very acceptable and usable. The MangngalApp 
+web portal was developed applying the Design 
+Science Research (DSR) for Information Systems. The 
+Design Science Research creates and evaluates IT 
+artifacts intended to solve identified organizational 
+problems, (Peffers, 2007). Accessible thru http://cics-
+csuaparri.org.ph/mangngalapp, 
+the 
+XAMPP 
+development framework was mainly used. XAMPP is 
+a cross-platform development tool involving the use 
+of the PHP scripting language, My SQL database 
+engine, and Apache web service. Other tools used 
+included CSS3, HTML5 and JavaScript. 
+ 
+ 
+Fig. 1. The MangngalApp Ecosystem. 
+The MangngalApp Web Project is an ecosystem that 
+involves people in the research and development, 
+technologies for development and dissemination of 
+research outputs, people and communities that are 
+the main reason for this project towards rural 
+development. The research outputs of the researchers 
+and scientific organization that were IP-registered are 
+highlighted for dissemination towards adoption 
+strategy. Bridging the gap is maximizing the use of 
+web tools and technologies that are accessible to the 
+communities. The package of technology available in 
+the 
+current 
+version 
+contains 
+14 
+IP-registered 
+technologies showcasing most of the CSU Aparri-
+based research and innovations.  
+ 
+Permissions were sought through the Knowledge and 
+Technology Management Office and the Office of the 
+Research and Development, and Extension. End-
+users of the project may click on the view process to 
+see the detailed descriptions, as well as the steps 
+involved in making, producing, or utilizing the 
+technology. The project is scalable, it will still house 
+other registered post-harvest technologies, fisheries-
+based products, and technologies supporting the 
+different 
+arrays 
+of 
+fisheries 
+and 
+aquaculture 
+development for rural use.  
+ 
+ 
+Fig. 2. Mobile View of the MangngalApp. 
+ 
+Results and discussions  
+Assessment of the Usability and Acceptability of the 
+MangngalApp using the Technology Acceptance 
+Model (TAM)  
+Table 3 presents the results of the assessment made by 
+the technical respondents along the aspects of perceived 
+ease of use, perceived usefulness, attitudes towards 
+usage, behavioral intention to use, and job relevance.  
+
+Fishersfarmers,gatherersandprocessors
+Inaddition,currenttechnologyadoption
+practices,accesstorelevantdata,
+preferencesonfishingandfarming
+NOODLESEINRICHEDWITHARAMANG
+DeVEIOPCDRLENMFAPMOLNA
+technologiesthemobileinternetand
+VwProcOSS
+medialiteracyandtheneedtosupport
+activitiesofthefishers,farmers,and
+processorswillbeobtained.TheprojectMangngalApp
+MangngalApp
+Home
+PackagesofTechnology
+Packs ofTechnology
+AboutMangngalApp
+NewsandUpdates
+describestheinformation seeking
+ARAMANG-ENRICHEDPOLVORON
+DeVelODer:DR.LENMFARANOLINA
+practices,technologyawareness
+WewProc心5出
+RESEARCH
+TECHNOLOGY
+MangngalAppll
+APP
+Aeelicotlon
+DATABASE
+mamgalapn.cics顺
+csuanarrlorg.hJ. Bio. & Env. Sci. 2022 
+ 
+113 | Javier et al. 
+Table 1. Assessment of the Usability and Acceptability using TAM. 
+                                                                                                                        VAU      AU       SAU     NAU    Weighted   DV 
+Aspects of the Technology Acceptance  
+f 
+f 
+f 
+f 
+Mean 
+ 
+Perceived Ease of Use  
+ 
+ 
+ 
+ 
+3.14 
+AU 
+I feel that using MangngalApp would be easy for me  
+I feel that my interaction with MangngalApp would be clear  
+8 
+12 
+0 
+0 
+3.40 
+ 
+VAU 
+and understandable  
+I feel that it would be easy to become skillful at using  
+6 
+12 
+1 
+0 
+3.25 
+VAU 
+MangngalApp  
+3 
+15 
+1 
+0 
+3.10 
+AU 
+I would find MangngalApp to be flexible to interact with  
+6 
+12 
+1 
+0 
+3.25 
+VAU 
+Learning to operate MangngalApp would be easy for me  
+It would be easy for me to get MangngalApp to do what I  
+6 
+12 
+2 
+0 
+3.20 
+AU 
+want to do  
+I feel that my ability to determine MangngalApp ease of use  
+2 
+13 
+4 
+0 
+2.90 
+AU 
+is limited by my lack of experience  
+4 
+11 
+3 
+2 
+2.85 
+AU 
+Perceived Usefulness  
+ 
+ 
+ 
+ 
+3.32 
+VAU 
+Using MangngalApp in disseminating technologies to intended 
+users would enable me or users to accomplish  
+ 
+ 
+ 
+ 
+ 
+ 
+tasks more quickly  
+Using MangngalApp would improve my skills and is useful  
+11 
+8 
+1 
+0 
+3.50 
+ 
+VAU 
+in the fishers and user's needs.  
+7 
+12 
+1 
+0 
+3.30 
+VAU 
+Using MangngalApp would increase my productivity  
+Using MangngalApp would enhance other users' capabilities  
+7 
+12 
+1 
+0 
+3.30 
+VAU 
+adopting the technology shared.  
+Using MangngalApp would make it easier to know new 
+technological updates in fishing, postharvest and related  
+7 
+12 
+1 
+0 
+3.30 
+VAU 
+activities.  
+I would find MangngalApp useful in helping the fishers and  
+6 
+12 
+2 
+0 
+3.20 
+AU 
+related sectors towards rural development.  
+7 
+12 
+1 
+0 
+3.30 
+VAU 
+Attitudes towards Usage  
+ 
+ 
+ 
+ 
+3.43 
+VAU 
+I believe it is a good idea to use the MangngalApp web  
+ 
+ 
+ 
+ 
+ 
+ 
+project  
+8 
+12 
+0 
+0 
+3.40 
+VAU 
+I like the idea of using the MangngalApp web project  
+8 
+12 
+0 
+0 
+3.40 
+VAU 
+Using the MangngalApp is a positive idea  
+10 
+10 
+0 
+0 
+3.50 
+VAU 
+Behavioural Intention to Use  
+ 
+ 
+ 
+ 
+3.22 
+AU 
+I tend to use the MangngalApp web project for seeking new 
+innovations in fisheries post-harvest and technologies. 
+6 
+13 
+1 
+0 
+ 
+3.25 
+VAU 
+I tend to use MangngalApp to enhance my interest in related 
+fishing, aqua-culture, and post-harvest activities 
+I tend to use the MangngalApp to provide multi-approaches on 
+sharing and obtaining technological and innovations in 
+6 
+12 
+2 
+0 
+3.20 
+AU 
+fisheries, aqua-marine and post-harvest activities. 
+6 
+12 
+2 
+0 
+3.20 
+AU 
+Relevance of the MangngalApp to Current Job 
+ 
+ 
+3.35 VAU 
+In disseminating new packets of technologies along fisheries and 
+aqua-marine, the usage of MangngalApp is important  
+8 
+11 
+1 
+0 
+3.35 
+VAU 
+In disseminating new packets of technologies along fisheries and 
+aqua-marine, the usage of MangngalApp is timely relevant 
+8 
+11 
+1 
+0 
+3.35 
+VAU 
+Overall Weighted Mean  
+ 
+3.29  
+VAU 
+3.25 – 4.00 >> Very acceptable and usable (VAU)  
+1.75 – 2.49 >> 
+Somewhat acceptable and 
+usable (SAU) 
+2.50 – 3.24 >> Acceptable and usable (AU)  
+1.00 – 1.74 >> Not acceptable and usable (NAU) 
+ 
+With an overall mean of 3.29, the assessment of the 
+MangngalApp along the usability and acceptability 
+aspects were found to be “very acceptable and usable” 
+(table 3). Specifically, the assessment of perceived 
+usefulness (3.32), their attitude towards usage (3.43), 
+and relevance (3.45) were rated very acceptable and 
+usable. The perceived usefulness could be associated 
+to their perceived attitude towards its usage as well as 
+how relevant the MangngalApp web project specially 
+to intended users. For the purpose of clarity and 
+understanding, the project team intended to have the 
+MangngalApp project be assessed by the fishers, 
+processors, farmers, traders, and gathers. However, 
+the team was constrained to do the actual 
+demonstration due to restrictions of the COVID-19 
+virus and high-risk alert levels of cases in the locality. 
+The team also tried to meet the all intended 
+participants via virtual setup in a video conferencing 
+
+J. Bio. & Env. Sci. 2022 
+ 
+114 | Javier et al. 
+tool 
+as 
+well 
+as 
+used 
+other 
+strategies 
+like 
+communicating with students and leaders in the area. 
+Feed backs from the students who were parents of the 
+fishers and farmers as well as processors; said most of 
+their parents prefer to have the project demonstrated 
+in face-to-face setup so they could easily grasp the 
+technology. The team decided to conduct the actual 
+dissemination and training in the actual users in the 
+ground upon notice of approval from relevant office 
+still confirming to minimum health protocols. It is 
+one of the key future directions the team is looking 
+forward.  
+ 
+As presented, the group of non-technical respondents 
+generally assessed the usability and acceptability of 
+the MangngalApp as “very acceptable and usable” 
+with a mean of 3.39. This rating is associated to the 
+very acceptable and usable descriptive values for 
+perceived usefulness, attitude towards usage, and job 
+relevance. Interestingly, more male respondents 
+perceived higher valuation of the Mangngal App 
+compared to their female counterparts. Meanwhile, 
+the technical respondents rated the aspects of TAM as 
+“acceptable and usable” with a mean of 3.21. Higher 
+assessment has been made by female industry 
+practitioners with a mean of 3.38, especially along 
+usefulness, attitudes towards usage, behavioral 
+intention to use and job relevance. There were 40 
+percent 
+of 
+the 
+respondents 
+who 
+rated 
+the 
+MangngalApp as overall very acceptable and usable.  
+ 
+Table 2. Detailed presentation of the assessment of the usability and acceptability 
+Aspects of TAM 
+Technical Respondents 
+Non-Technical Respondents 
+Male 
+Female 
+Weighted 
+Mean 
+Descriptive 
+Value 
+Male 
+Female 
+Weighted 
+Mean 
+Descriptive 
+Value 
+1. Perceived ease of use 
+3.07 
+3.14 
+3.10 
+AU 
+3.23 
+3.11 
+3.17 
+AU 
+2. Perceived usefulness 
+3.00 
+3.50 
+3.27 
+VAU 
+3.47 
+3.27 
+3.37 
+VAU 
+3. Attitude towards usage 
+3.33 
+3.50 
+3.40 
+VAU 
+3.47 
+3.47 
+3.47 
+VAU 
+4. Behavioral intention to use 
+3.00 
+3.50 
+3.20 
+AU 
+3.40 
+3.07 
+3.23 
+AU 
+5. Job Relevance 
+3.17 
+3.50 
+3.30 
+VAU 
+3.40 
+3.40 
+3.40 
+VAU 
+Overall 
+3.12 
+3.38 
+3.32 
+AU 
+3.37 
+3.23 
+3.30 
+VAU 
+ 
+ 
+3.21 
+ 
+AU 
+ 
+3.39 
+VAU 
+ 
+Percentage of those who rated the 
+MangngalApp as overall “very acceptable 
+and usable” 
+ 
+ 
+40% 
+ 
+ 
+40% 
+ 
+Compliance to ISO 25010 software quality characteristics of the developed MangngalApp 
+  
+Table 3. Summary table of the assessment of the developed MangngalApp based on ISO 25010 software quality 
+characteristics. 
+Indicator 
+Technical Evaluators 
+Non-Technical (Fisher) 
+Overall 
+WM 
+DV 
+WM 
+DV 
+WM 
+DV 
+Accuracy 
+3.47 
+VHE 
+3.87 
+VHE 
+3.67 
+VHE 
+Reliability 
+3.53 
+VHE 
+3.90 
+VHE 
+3.72 
+VHE 
+Security  
+3.50 
+VHE 
+4.00 
+VHE 
+3.75 
+VHE 
+Functional Suitability 
+3.60 
+VHE 
+3.93 
+VHE 
+3.77 
+VHE 
+Portability  
+3.67 
+VHE 
+3.87 
+VHE 
+3.77 
+VHE 
+Usability  
+3.60 
+VHE 
+3.9 
+VHE 
+3.75 
+VHE 
+Maintainability  
+3.57 
+VHE 
+3.87 
+VHE 
+3.72 
+VHE 
+Efficiency  
+3.60 
+VHE 
+3.90 
+VHE 
+3.75 
+VHE 
+Overall Weighted Mean 
+3.57 
+VHE 
+3.91 
+VHE 
+3.74 
+VHE 
+Legend: 
+WM– Weighted Mean; DV– Descriptive Value 
+3.25-4.00 >> Very High Extent (VHE, 1.75-2.49 >> Fair Extent (FE) 
+2.50-3.24 >> High Extent (HE), 1.00-1.74>> Poor Extent (PE) 
+ 
+Presented in table the summary table of the 
+assessment 
+of 
+the 
+MangngalApp 
+web 
+project 
+following 
+the 
+ISO 
+25010 
+software 
+quality 
+characteristics. The assessment of the technical and 
+non-technical respondents revealed an overall remark 
+of excellent with an overall mean of 3.74. Notably, 
+both groups made a high remark or excellent 
+highlighting functionality and portability aspects.  
+
+J. Bio. & Env. Sci. 2022 
+ 
+115 | Javier et al. 
+The functionality can be associated to the fact that the 
+MangngalApp follows a WYSWYG approach making 
+ease of access and functional. Meanwhile, the 
+portability aspect could be associated to the project 
+being compatible to varied devices making it 
+convenient to users. 
+ 
+The participants were asked about their problems and 
+challenges associated to the use of the MangngalApp. 
+Although the participants are technical evaluators, it 
+is believed that common issues will be experienced by 
+the intended users. This includes but not limited to:  
+a. Internet connectivity issues  
+b. Not very good using via tablets PC  
+c. Limited contents only focused to fisheries and 
+aquaculture  
+d. Cannot visualize from just an image  
+  
+There were comments and suggestions highlighted by 
+the respondents. This includes but not limited to:  
+a. Strengthen internet connection in the area  
+b. Share more techno guides that are easily 
+understood by intended users  
+c. Produce video of the steps which are visibly 
+understood by intended users  
+d. Add more contents not only along post-harvest 
+and processing.  
+e. Translation of contents to Filipino or vernaculars 
+if possible  
+  
+  
+Moreover, the overall impressions made by the 
+participants include:  
+a. MangngalApp as a good project for rural 
+development  
+b. The project is impressive  
+c. Great project especially if with more contents for 
+the intended users  
+d. Very good one-stop IEC mechanism  
+  
+Considering the above-mentioned, the project team is 
+looking way forward to scale up the project, fast-track 
+the translation to Filipino, as well as integrating other 
+technologies that would benefit the communities for 
+rural development. The translation is in coordination 
+with owners of the technology. 
+Conclusions  
+The MangngalApp project was found to be very 
+acceptable and usable based on the assessment of the 
+technical respondents. There were uncontrolled 
+issues or problems in the use of the MangngalApp, 
+the constructive comments and suggestions, as well as 
+the overall impressions over the project. Based on the 
+ISO 25010 software quality characteristics, the 
+respondents generally remark it as “excellent” with an 
+overall mean of 3.74.  
+ 
+From the results, it is concluded that the developed 
+MangngalApp will be a usable and responsive 
+technology that aids to rural development especially 
+among target users- fishers, gatherers, processors, 
+traders, and farmers. Considering compatibility and 
+usefulness, the MangngalApp is expected to provide 
+greater social development in the community.  
+ 
+Social Implications  
+The use of the MangngalApp would offer greater 
+opportunity for local users to livelihood development 
+adopting the technologies being shared from the 
+output of scientific undertakings at the University and 
+with collaborators. Meanwhile, the adoption of the 
+technologies 
+may 
+be 
+undertaken 
+providing 
+opportunities for small to medium organizations 
+towards livelihood development – forging partnership 
+with the University and other stakeholders and 
+private institutions.  
+ 
+Project Limitations 
+The researchers acknowledge the technical challenge 
+that may have encountered by the participants as 
+there were very limited face-to-face presentations 
+made with intended users, thus may affect the results 
+in the study. There is a need to perform actual 
+demonstration 
+with 
+them 
+upon 
+approval 
+of 
+authorities and observing minimum health protocols.  
+  
+Recommendations  
+From the results of the study, it is recommended to 
+integrate the fully translated content and additional 
+technologies geared towards full utilization of the 
+MangngalApp especially creating opportunities for 
+
+J. Bio. & Env. Sci. 2022 
+ 
+116 | Javier et al. 
+livelihood development. Further, the conduct of 
+extension activities to adopt and utilize the project 
+accessible in the web is highly encouraged thru 
+demonstration activities forging collaboration with 
+fishers and women organizations. In addition, there is 
+a need to constantly update and make the project 
+scalable providing other opportunities for rural 
+development 
+in 
+general 
+especially 
+when 
+new 
+innovations are IP-registered from the research 
+innovations in fisheries and aqua-marine. The 
+development of a video production is suggested for 
+actual demonstration of the processes involved 
+especially in post-harvest or product development.  
+ 
+Acknowledgement  
+The research project would not be a success without 
+the support of the administration of the Cagayan 
+State University headed by Dr. Urdujah G. Alvarado, 
+the kind assistance and support of the RDE for the 
+funding thru VP for RDE Dr. Junel Guzman, as well 
+as the commitment and leadership of the Campus 
+Executive Officer Dr. Simeon R. Rabanal, Jr. The 
+project team is ever grateful for the usual and 
+unparalleled support and drive of the Coordinator for 
+Research and Development Dr. Lenimfa Molina for 
+sharing the technologies and helping us in the project 
+contents. Special mention to Ms. Eunice Daluddung 
+for her patience and assistance to the project team. 
+Kind appreciation is extended to Dr. Corazon T. 
+Talamayan for supporting us in the project. Morever, 
+the assessment of the project as well as how could we 
+better improve the MangngalApp is greatly attributed 
+to the self-less sharing of time, effort and expertise of 
+the industry practitioners and ICT teachers despite 
+being very busy also. To all the fishers, farmers, 
+processors, gatherers, and small-scale merchants – we 
+owe this project to you, as our inspiration of doing the 
+project towards rural development. Special mention 
+goes to the member of the review committee in the 2 
+in-house reviews conducted – Engr. Gil Mark Hizon of 
+DOST RO2 and Dr. Emma Ballad of BFAR RO2 for 
+their constructive comments, guidance and inspiration: 
+GAD-Focal Person Prof Kristine Lara, Extension 
+coordinator Josie Bas-ong and KTM Coordinator Dr. 
+Gilbert Magulod Jr for the inputs and support. 
+References 
+Abdelaziz T, Elammari M, Bani W. 2015. 
+Applying the ISO Standard in Assessing the Quality of 
+Software Systems. American Journal of Computer 
+Science and Information Engineering 2(3), 28-32. 
+Retrieved from www.aascit.org/journal/ajcsie  
+ 
+Davis FD. 1989. Perceived usefulness, perceived 
+ease of use, and user acceptance of information 
+technology. 
+MIS 
+Quarterly 
+13(3), 
+319-340. 
+https:/doi.org/10.2307/249008  
+ 
+Department 
+of 
+Agriculture 
+- 
+Bureau 
+of 
+Agricultural 
+Research. 
+2016. 
+Research 
+and 
+Development, and Extension Agenda and Programs 
+2016-2022. Retrieved April 2018, from DA-BAR 
+Website: http://www.bar.gov.ph/downloadables.  
+ 
+Dhaka BL. 2016. Farmers’ experience with ICTs on 
+transfer 
+of 
+technology 
+in 
+changing 
+agri-rural 
+environment. Indian Research Journal of Extension 
+Education 10(3), 114-118.  
+ 
+FAO. 2018. Fishery and Aquaculture Country 
+Profiles. 
+Retrieved 
+March 
+2018, 
+from 
+FAO: 
+http://www.fao.org/fishery/facp/PHL/en  
+ 
+Gyaneshwar 
+Singh 
+Kushwaha 
+DB. 
+2010. 
+Development of a theoretical framework of supply 
+chain quality management. Serbian Journal of 
+Management 
+5(1), 
+127-142. 
+Retrieved 
+from 
+http://www.sjm06.com/SJM%20ISSN1452-
+4864/5_1_2010_May_1188/5_1_127-142.pdf  
+ 
+Hossain 
+MI. 
+2021. 
+COVID-19 
+Impacts 
+on 
+Employment and Livelihood of Marginal People in 
+Bangladesh: Lessons Learned and Way Forward. 
+SAGE Journals South Asian Survey 28(1), 57-71. 
+DOI: https://doi.org/10.1177/0971523121995072  
+ 
+Magdalinou AM. 2019. Disseminating Research 
+Outputs: The Crowd Health Project. Acta informatica 
+medica: AIM: journal of the Society for Medical 
+Informatics of Bosnia & Herzegovina :Casopis Drustva 
+za medicinsku informatiku BiH 27(5), 348-354. 
+DOI: https://doi.org/10.5455/aim.2019.27.348-354  
+
+J. Bio. & Env. Sci. 2022 
+ 
+117 | Javier et al. 
+Patel N. 2020. Future of On-Demand Economy | Rise of 
+On-DemandApps. Retrieved from globalvincitore.com: 
+https://www.globalvincitore.com/rise-of-on-demand  
+ 
+Patel 
+R. 
+2019. 
+On-demand 
+App 
+Benefits, 
+Applications 
+and 
+Future. 
+Retrieved 
+from 
+yourstory.com: 
+https://yourstory.com/mystory/on-
+demand-app/amp  
+ 
+Robbert-Jan 
+van 
+der 
+Burg 
+KA. 
+2019. 
+Investigating the on-demand service characteristics: 
+an empirical study. Journal of Service Management. 
+DOI: 10.1108/JOSM-01-2019-0025  
+The Strait Times Asia. 2020. Philippines Suffers 
+worst job losses in 15 years due to Covid-19 and 
+lockdown. Retrieved June 2021, from The Strait 
+Times Asia:  
+ 
+Truong T, Rothschild BJ, Azadivar F. 2005. 
+Decision Support System for Fisheries Management. 
+DOI: 10.1145/1162708.1163075 
+ 
+ 
+ 
+ 
+ 
+ 
+

FNE1T4oBgHgl3EQfEwMv/content/tmp_files/load_file.txt

@@ -0,0 +1,435 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf,len=434
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  109 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  RE RE RE RESEARCH SEARCH SEARCH SEARCH    PAPER PAPER PAPER PAPER  OPEN ACCESS  OPEN ACCESS  OPEN ACCESS  OPEN ACCESS  MangngalApp- An integrated package of technology for COVID- 19 response and rural development: Acceptability and usability using TAM  Billy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Javier*, Leo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Paliuanan, James Karl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Agpalza, Jesty S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Agoto  College of Information and Computing Sciences, Cagayan State University, Aparri, Philippines  Article published on  October 20, 2022 Key words: Acceptability, COVID-19, Fishers, POTs, Technology acceptance model, ISO 25010 Abstract The COVID19 pandemic has challenged universities and organizations to devise mechanisms to uplift the well- being and welfare of people and communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' In response, the design and development of an integrated package of technologies, MangngalApp- A web-based portal and mobile responsive application for rural development served as an opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' It showcases different packets of technologies that were outputs of R&D in the field of fisheries and aqua-culture, innovations that were IP-protected, and technologies that harness locally available resources for post-harvest development and aiding in sustaining growth and development in the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' This paper focused on the usability and acceptability of the MangngalApp implementing a descriptive research design using the Technology Acceptance Model or TAM and ISO 25010 software quality standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Constrained by government health restrictions due to COVID- 19, a Google form-based questionnaire was forwarded to consented participants via an email with the attached consent and evaluation form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Results revealed that the MangngalApp was found to be very acceptable and usable, and compliant to ISO 25010 software quality characteristics to the higher extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  From the results, it is concluded that the developed MangngalApp will be a usable and responsive technology that aids to rural development especially among target users- fishers, gatherers, processors, traders, and farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Considering compatibility and usefulness, the MangngalApp is expected to provide greater social development in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' *Corresponding Author: Billy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Javier \uf02a billyjavier@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='ph  Journal of Biodiversity and Environmental Sciences (JBES) ISSN: 2220-6663 (Print) 2222-3045 (Online) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 21, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 109-117, 2022 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='innspub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='net  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  110 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Introduction The COVID 19 pandemic has disrupted many organizations, government and non-government institutions, schools, companies, and various communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' As a result, more displaced workers and job losses increased, more families sent home, locked down due to COVID19 restrictions and uncertain of how and where to obtain immediate income for the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The government may have provided financial assistance to erring families and fed empty stomachs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' However, resources deplete as no concrete measure to total stop the threat of the on-going pandemic the Filipino people is enjoying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The Cagayan State University is mandated to transforming the lives of people and communities through high quality instruction, innovative research, development, and production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Through the years, CSU has been working hard on innovating technologies that could help alleviate poverty, increase productivity and improve socioeconomic status of the communities, and help in sustaining and protecting the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' However, no matter how promising these technologies are if these packages of technologies are not widely accessible to target communities, to its intended stakeholders: fishers, farmers, gatherers, and processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  In fact, Sharma A, and Kiranmayi, D (2019) was unable to find in many literature and studies pertaining to a package of technologies as an IEC initiative to adopting and utilizing research-based fisheries technologies, post- harvest technologies, and aquaculture techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Most of the 124 applications reported focused on mobile apps for angling, aquaculture management, aquarium management, marine fisheries, and fisheries governance, marketing and biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Research project generating innovative technologies and products has been funded and curated by experts in the various fields leading to technology commercialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' These then has to be extended to communities via available and relevant technologies so that as an academic institution, it really radiates its mantra of improving the lives of people and communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The MangngalApp research program was generally geared at providing a solution for a well-informed utilization of the packets of technologies (POTs) developed as results of scientific inquiries and experiments of the University and collaborating agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' It has been said that technologies should be utilized by the communities, adopted via technology-transfer, generating income from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' However, access to POTs may have not deliberately reaching the realms of coastal communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Lack of or limited access to POTs among fishers, farmers, gatherers, and processors may cause inefficiency, increased cost for production, and lower productivity among fishers, fish processors and gatherers, as well as farmers in the coastal communities in northern Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  With aqua-marine as banner program in the Aparri Campus, a multi-disciplinary research program was proposed with the hope of generating a package of technology showcasing the science-based packages of technologies of university along fishing activities, seaweed farming, post-harvest, product development and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The research is expected to benefit the coastal communities through provision of mobile- ready and friendly application accessible to users aiding to improve productivity, increased awareness and protection for the environment, and providing livelihood for women and differently-able persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Packages of technologies developed will be best adopted or utilized in the community once an integrated package or technology is made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Hence, the potential benefits expand from the fishers in the conduct of and management of their fisheries activities to any other intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Coastal farmers will be able to uncover scientific ways to conservation and management of marine species or seaweeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Fish processors will have the potential to improve productivity, creation of jobs, and increased revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Adapting the vision of the Food and Agriculture Organization of the United Nations (FAO) on enhancing the role of small-scale fisheries in contributing to poverty alleviation and food security, the project also focused on understanding the technology awareness, technology adoption practices, the information needs and seeking behaviors, media  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  111 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' literacy and media adoption of various stakeholders in the fishing communities of Northeastern Cagayan Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' In the academe, students and teachers may benefit from the having obtained the scientific packages of technologies for instruction purposes, and an opportunity for more relevant research formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The results of the study hope to provide and cultivate new knowledge for students, researchers, and teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' In so doing, students and teachers may devise projects, programs, and studies that could add up to the packages of technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Institutions or organization may have devise appropriate strategies, programs and plans from data mining and knowledge data discovery thru the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  The emergence of an information, communication, and education platform through varied technologies is a must especially in the dissemination of scientific results and innovations from rigid experiments and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Digital visibility is considered an efficient and reasonable way to publicize the outputs of innovative developments and research results (Magdalinou, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The Technology Acceptance Model (TAM) is a theory in information systems that explain how consumers come to embrace the use of a technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' When consumers are introduced with new technology, the model argues that a variety of factors influence their decision on how and when to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=" TAM has been critiqued for a variety of reasons, but it is a useful overall framework that is compatible with several studies examining the elements that influence older individuals' willingness to utilize new technology (Braun, 2013)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  This paper generally aims to describe the usability and acceptability of the developed mobile responsive web project known as MangngalApp - an integrated package of technology using open-source web development platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  The assessment of the usability and acceptability of the MangngalApp using the Technology Acceptance Model (TAM) focused on (a) Perceived Ease of Use, (b) Perceived usefulness, (c)Attitudes towards usage, (d) Behavioral intention to use, and Relevance to the present job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' In addition, the assessment of the developed MangngalApp based on ISO 25010 software quality characteristics has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Materials and methods The descriptive research design was implemented in this part of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The assessment of the usability and acceptability of the developed Mangngal App using the Technology Acceptance Model or TAM was participated by 200 non-technical respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' These included fishers, farmers, fish processors, gatherers, and households involved in post-harvest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' A listing of which was taken from municipal agriculture office thru communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Meanwhile, the assessment of the 20 technical respondents applying ISO 25010 software quality standards, provided proof of the compliance in terms of compatibility, reliability, user-friendliness, security, portability, and functional suitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The profile of the technical respondents is presented herein in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The survey-questionnaire included some profile data of respondents, their assessment of the MangngalApp, and an optional remark or comment part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' A consent form was part of the questionnaire, while prior presentation or orientation on its use was provided via Google Meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  The researchers took the assistance of partner- students and community leaders handled by the team in the locality to share the MangngalApp project and guide intended users including those involved in actual fishing, post-harvest development, processing, gathering, as well as those who are trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' This is a COVID-19 initiative of the project team in order to gather sentiments and assessment of those greater users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' On the other hand, the technical respondents were communicated formally requesting their expertise, and provided the team consent to participate in the assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  The respondents in the evaluation of the technical compliance, usability and acceptability standards using TAM included 10 industry practitioners, 10 ICT teachers with experiences in databases, web development and design, and programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  112 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Profile of the Technical Respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Participants Male Female Total % Classification  Industry  Practitioners  6  4  10  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  ICT Teachers  5  5  10  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  Area of Interest  Web Design  2  3  5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  Web Programming  3  2  5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  Databases  2  2  4  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  Programming  2  2  4  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  Networks  2  2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0 Years of Relevant ICT Experience  1 to 3  4  5  9  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  4 to 6  6  3  9  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  More than 6  1  1  2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='0  The participants were notified via email on their participation in the assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' A brief orientation via Google Meet was conducted to provide them overview of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The project team provided the link of web based MangngalApp project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' They were given at least 2 to 5 weeks to access the web project and were requested to fill out the evaluation forms via Google Forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Treating the assessment of the Usability and Acceptability of the MangngalApp using the Technology Acceptance Model, the 4-point Likert scale was used: 1 being not acceptable and usable to 4 being very acceptable and usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The MangngalApp web portal was developed applying the Design Science Research (DSR) for Information Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The Design Science Research creates and evaluates IT artifacts intended to solve identified organizational problems, (Peffers, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Accessible thru http://cics- csuaparri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='ph/mangngalapp, the XAMPP development framework was mainly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' XAMPP is a cross-platform development tool involving the use of the PHP scripting language, My SQL database engine, and Apache web service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Other tools used included CSS3, HTML5 and JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The MangngalApp Ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The MangngalApp Web Project is an ecosystem that involves people in the research and development, technologies for development and dissemination of research outputs, people and communities that are the main reason for this project towards rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The research outputs of the researchers and scientific organization that were IP-registered are highlighted for dissemination towards adoption strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bridging the gap is maximizing the use of web tools and technologies that are accessible to the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The package of technology available in the current version contains 14 IP-registered technologies showcasing most of the CSU Aparri- based research and innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Permissions were sought through the Knowledge and Technology Management Office and the Office of the Research and Development, and Extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' End- users of the project may click on the view process to see the detailed descriptions, as well as the steps involved in making, producing, or utilizing the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The project is scalable, it will still house other registered post-harvest technologies, fisheries- based products, and technologies supporting the different arrays of fisheries and aquaculture development for rural use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Mobile View of the MangngalApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Results and discussions Assessment of the Usability and Acceptability of the MangngalApp using the Technology Acceptance Model (TAM) Table 3 presents the results of the assessment made by the technical respondents along the aspects of perceived ease of use, perceived usefulness, attitudes towards usage, behavioral intention to use, and job relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Fishersfarmers,gatherersandprocessors Inaddition,currenttechnologyadoption practices,accesstorelevantdata, preferencesonfishingandfarming NOODLESEINRICHEDWITHARAMANG DeVEIOPCDRLENMFAPMOLNA technologiesthemobileinternetand VwProcOSS medialiteracyandtheneedtosupport activitiesofthefishers,farmers,and processorswillbeobtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='TheprojectMangngalApp MangngalApp Home PackagesofTechnology Packs ofTechnology AboutMangngalApp NewsandUpdates describestheinformation seeking ARAMANG-ENRICHEDPOLVORON DeVelODer:DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='LENMFARANOLINA practices,technologyawareness WewProc心5出 RESEARCH TECHNOLOGY MangngalAppll APP Aeelicotlon DATABASE mamgalapn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='cics顺 csuanarrlorg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='hJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  113 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Assessment of the Usability and Acceptability using TAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' VAU      AU       SAU     NAU    Weighted   DV Aspects of the Technology Acceptance f f f f Mean  Perceived Ease of Use  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='14 AU I feel that using MangngalApp would be easy for me I feel that my interaction with MangngalApp would be clear 8 12 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40  VAU and understandable I feel that it would be easy to become skillful at using 6 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='25 VAU MangngalApp 3 15 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='10 AU I would find MangngalApp to be flexible to interact with 6 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='25 VAU Learning to operate MangngalApp would be easy for me It would be easy for me to get MangngalApp to do what I 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='20 AU want to do I feel that my ability to determine MangngalApp ease of use 2 13 4 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='90 AU is limited by my lack of experience 4 11 3 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='85 AU Perceived Usefulness  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='32 VAU Using MangngalApp in disseminating technologies to intended users would enable me or users to accomplish  tasks more quickly Using MangngalApp would improve my skills and is useful 11 8 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content="50  VAU in the fishers and user's needs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 7 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content="30 VAU Using MangngalApp would increase my productivity Using MangngalApp would enhance other users' capabilities 7 12 1 0 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='30 VAU adopting the technology shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Using MangngalApp would make it easier to know new technological updates in fishing, postharvest and related 7 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='30 VAU activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' I would find MangngalApp useful in helping the fishers and 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='20 AU related sectors towards rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 7 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='30 VAU Attitudes towards Usage  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='43 VAU I believe it is a good idea to use the MangngalApp web  project 8 12 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 VAU I like the idea of using the MangngalApp web project 8 12 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 VAU Using the MangngalApp is a positive idea 10 10 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 VAU Behavioural Intention to Use  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='22 AU I tend to use the MangngalApp web project for seeking new innovations in fisheries post-harvest and technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 6 13 1 0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='25 VAU I tend to use MangngalApp to enhance my interest in related fishing, aqua-culture, and post-harvest activities I tend to use the MangngalApp to provide multi-approaches on sharing and obtaining technological and innovations in 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='20 AU fisheries, aqua-marine and post-harvest activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='20 AU Relevance of the MangngalApp to Current Job  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='35 VAU In disseminating new packets of technologies along fisheries and aqua-marine, the usage of MangngalApp is important 8 11 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='35 VAU In disseminating new packets of technologies along fisheries and aqua-marine, the usage of MangngalApp is timely relevant 8 11 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='35 VAU Overall Weighted Mean  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='29 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='25 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00 >> Very acceptable and usable (VAU) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='75 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='49 >> Somewhat acceptable and usable (SAU) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='24 >> Acceptable and usable (AU) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='74 >> Not acceptable and usable (NAU)  With an overall mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='29, the assessment of the MangngalApp along the usability and acceptability aspects were found to be “very acceptable and usable” (table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Specifically, the assessment of perceived usefulness (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='32), their attitude towards usage (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='43), and relevance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='45) were rated very acceptable and usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The perceived usefulness could be associated to their perceived attitude towards its usage as well as how relevant the MangngalApp web project specially to intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' For the purpose of clarity and understanding, the project team intended to have the MangngalApp project be assessed by the fishers, processors, farmers, traders, and gathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' However, the team was constrained to do the actual demonstration due to restrictions of the COVID-19 virus and high-risk alert levels of cases in the locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The team also tried to meet the all intended participants via virtual setup in a video conferencing  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  114 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' tool as well as used other strategies like communicating with students and leaders in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Feed backs from the students who were parents of the fishers and farmers as well as processors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' said most of their parents prefer to have the project demonstrated in face-to-face setup so they could easily grasp the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The team decided to conduct the actual dissemination and training in the actual users in the ground upon notice of approval from relevant office still confirming to minimum health protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' It is one of the key future directions the team is looking forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  As presented, the group of non-technical respondents generally assessed the usability and acceptability of the MangngalApp as “very acceptable and usable” with a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' This rating is associated to the very acceptable and usable descriptive values for perceived usefulness, attitude towards usage, and job relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Interestingly, more male respondents perceived higher valuation of the Mangngal App compared to their female counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Meanwhile, the technical respondents rated the aspects of TAM as “acceptable and usable” with a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Higher assessment has been made by female industry practitioners with a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='38, especially along usefulness, attitudes towards usage, behavioral intention to use and job relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' There were 40 percent of the respondents who rated the MangngalApp as overall very acceptable and usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Detailed presentation of the assessment of the usability and acceptability Aspects of TAM Technical Respondents Non-Technical Respondents Male Female Weighted Mean Descriptive Value Male Female Weighted Mean Descriptive Value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Perceived ease of use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='10 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='17 AU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Perceived usefulness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='27 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='37 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Attitude towards usage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='47 VAU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Behavioral intention to use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='20 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='23 AU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Job Relevance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='30 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='40 VAU Overall 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='32 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='30 VAU  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='21  AU  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='39  VAU  Percentage of those who rated the MangngalApp as overall “very acceptable and usable”  40%  40%  Compliance to ISO 25010 software quality characteristics of the developed MangngalApp  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Summary table of the assessment of the developed MangngalApp based on ISO 25010 software quality characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Indicator Technical Evaluators Non-Technical (Fisher) Overall WM DV WM DV WM DV Accuracy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='47 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='87 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='67 VHE Reliability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='53 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='90 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='72 VHE Security 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50 VHE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='75 VHE Functional Suitability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='60 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='93 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='77 VHE Portability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='67 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='87 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='77 VHE Usability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='60 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='9 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='75 VHE Maintainability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='57 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='87 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='72 VHE Efficiency 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='60 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='90 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='75 VHE Overall Weighted Mean 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='57 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='91 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='74 VHE Legend: WM– Weighted Mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' DV– Descriptive Value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='25-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00 >> Very High Extent (VHE, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='75-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='49 >> Fair Extent (FE) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='50-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='24 >> High Extent (HE), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='00-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='74>> Poor Extent (PE)  Presented in table the summary table of the assessment of the MangngalApp web project following the ISO 25010 software quality characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The assessment of the technical and non-technical respondents revealed an overall remark of excellent with an overall mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Notably, both groups made a high remark or excellent highlighting functionality and portability aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  115 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The functionality can be associated to the fact that the MangngalApp follows a WYSWYG approach making ease of access and functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Meanwhile, the portability aspect could be associated to the project being compatible to varied devices making it convenient to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  The participants were asked about their problems and challenges associated to the use of the MangngalApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Although the participants are technical evaluators, it is believed that common issues will be experienced by the intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' This includes but not limited to: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Internet connectivity issues b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Not very good using via tablets PC c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Limited contents only focused to fisheries and aquaculture d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Cannot visualize from just an image  There were comments and suggestions highlighted by the respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' This includes but not limited to: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Strengthen internet connection in the area b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Share more techno guides that are easily understood by intended users c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Produce video of the steps which are visibly understood by intended users d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Add more contents not only along post-harvest and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Translation of contents to Filipino or vernaculars if possible  Moreover, the overall impressions made by the participants include: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' MangngalApp as a good project for rural development b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The project is impressive c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Great project especially if with more contents for the intended users d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Very good one-stop IEC mechanism  Considering the above-mentioned, the project team is looking way forward to scale up the project, fast-track the translation to Filipino, as well as integrating other technologies that would benefit the communities for rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The translation is in coordination with owners of the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Conclusions The MangngalApp project was found to be very acceptable and usable based on the assessment of the technical respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' There were uncontrolled issues or problems in the use of the MangngalApp, the constructive comments and suggestions, as well as the overall impressions over the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Based on the ISO 25010 software quality characteristics, the respondents generally remark it as “excellent” with an overall mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  From the results, it is concluded that the developed MangngalApp will be a usable and responsive technology that aids to rural development especially among target users- fishers, gatherers, processors, traders, and farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Considering compatibility and usefulness, the MangngalApp is expected to provide greater social development in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Social Implications The use of the MangngalApp would offer greater opportunity for local users to livelihood development adopting the technologies being shared from the output of scientific undertakings at the University and with collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Meanwhile, the adoption of the technologies may be undertaken providing opportunities for small to medium organizations towards livelihood development – forging partnership with the University and other stakeholders and private institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Project Limitations The researchers acknowledge the technical challenge that may have encountered by the participants as there were very limited face-to-face presentations made with intended users, thus may affect the results in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' There is a need to perform actual demonstration with them upon approval of authorities and observing minimum health protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Recommendations From the results of the study, it is recommended to integrate the fully translated content and additional technologies geared towards full utilization of the MangngalApp especially creating opportunities for  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  116 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' livelihood development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Further, the conduct of extension activities to adopt and utilize the project accessible in the web is highly encouraged thru demonstration activities forging collaboration with fishers and women organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' In addition, there is a need to constantly update and make the project scalable providing other opportunities for rural development in general especially when new innovations are IP-registered from the research innovations in fisheries and aqua-marine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The development of a video production is suggested for actual demonstration of the processes involved especially in post-harvest or product development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Acknowledgement The research project would not be a success without the support of the administration of the Cagayan State University headed by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Urdujah G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Alvarado, the kind assistance and support of the RDE for the funding thru VP for RDE Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Junel Guzman, as well as the commitment and leadership of the Campus Executive Officer Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Simeon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Rabanal, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' The project team is ever grateful for the usual and unparalleled support and drive of the Coordinator for Research and Development Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Lenimfa Molina for sharing the technologies and helping us in the project contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Special mention to Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Eunice Daluddung for her patience and assistance to the project team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Kind appreciation is extended to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Corazon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Talamayan for supporting us in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Morever, the assessment of the project as well as how could we better improve the MangngalApp is greatly attributed to the self-less sharing of time, effort and expertise of the industry practitioners and ICT teachers despite being very busy also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' To all the fishers, farmers, processors, gatherers, and small-scale merchants – we owe this project to you, as our inspiration of doing the project towards rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Special mention goes to the member of the review committee in the 2 in-house reviews conducted – Engr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Gil Mark Hizon of DOST RO2 and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Emma Ballad of BFAR RO2 for their constructive comments, guidance and inspiration: GAD-Focal Person Prof Kristine Lara, Extension coordinator Josie Bas-ong and KTM Coordinator Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Gilbert Magulod Jr for the inputs and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' References Abdelaziz T, Elammari M, Bani W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Applying the ISO Standard in Assessing the Quality of Software Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' American Journal of Computer Science and Information Engineering 2(3), 28-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Retrieved from www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='aascit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='org/journal/ajcsie  Davis FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Perceived usefulness, perceived ease of use, and user acceptance of information technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' MIS Quarterly 13(3), 319-340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' https:/doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='2307/249008  Department of Agriculture - Bureau of Agricultural Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Research and Development, and Extension Agenda and Programs 2016-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Retrieved April 2018, from DA-BAR Website: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='ph/downloadables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Dhaka BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Farmers’ experience with ICTs on transfer of technology in changing agri-rural environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Indian Research Journal of Extension Education 10(3), 114-118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  FAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Fishery and Aquaculture Country Profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Retrieved March 2018, from FAO: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='fao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='org/fishery/facp/PHL/en  Gyaneshwar Singh Kushwaha DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Development of a theoretical framework of supply chain quality management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Serbian Journal of Management 5(1), 127-142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Retrieved from http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='sjm06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='com/SJM%20ISSN1452- 4864/5_1_2010_May_1188/5_1_127-142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='pdf  Hossain MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' COVID-19 Impacts on Employment and Livelihood of Marginal People in Bangladesh: Lessons Learned and Way Forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' SAGE Journals South Asian Survey 28(1), 57-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='1177/0971523121995072  Magdalinou AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Disseminating Research Outputs: The Crowd Health Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Acta informatica medica: AIM: journal of the Society for Medical Informatics of Bosnia & Herzegovina :Casopis Drustva za medicinsku informatiku BiH 27(5), 348-354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='5455/aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='348-354  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2022  117 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Patel N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Future of On-Demand Economy | Rise of On-DemandApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Retrieved from globalvincitore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='com: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='globalvincitore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='com/rise-of-on-demand  Patel  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  On demand  App  Benefits,  Applications  and  Future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='  Retrieved  from  yourstory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='com:  https://yourstory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='com/mystory/on demand app/amp  Robbert-Jan van der Burg KA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Investigating the on-demand service characteristics: an empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Journal of Service Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='1108/JOSM-01-2019-0025 The Strait Times Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Philippines Suffers worst job losses in 15 years due to Covid-19 and lockdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Retrieved June 2021, from The Strait Times Asia:  Truong T, Rothschild BJ, Azadivar F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' Decision Support System for Fisheries Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='1145/1162708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}
+page_content='1163075' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'}

FNFRT4oBgHgl3EQfyzjn/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:29b45c06bb95179ea87a4b292a3ec93c71ebbedfb4219b40e74f1c7f812d8469
+size 1703981