Mathematics Batch 05 - Applied Mathematics - Programming Framework Analysis
This document presents applied mathematics processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.
1. Numerical Methods Process
graph TD
A1[Numerical Problem Input] --> B1[Method Selection]
C1[Error Analysis] --> D1[Convergence Study]
E1[Algorithm Design] --> F1[Implementation Strategy]
B1 --> G1[Root Finding Methods]
D1 --> H1[Convergence Criteria]
F1 --> I1[Numerical Algorithm]
G1 --> J1[Bisection Method]
H1 --> K1[Tolerance Setting]
I1 --> L1[Iterative Process]
J1 --> M1[Newton Raphson Method]
K1 --> N1[Error Estimation]
L1 --> O1[Convergence Check]
M1 --> P1[Secant Method]
N1 --> Q1[Stability Analysis]
O1 --> R1[Solution Refinement]
P1 --> S1[Fixed Point Iteration]
Q1 --> T1[Error Bounds]
R1 --> U1[Numerical Solution]
S1 --> V1[Integration Methods]
T1 --> W1[Accuracy Assessment]
U1 --> X1[Solution Validation]
V1 --> Y1[Trapezoidal Rule]
W1 --> Z1[Numerical Result]
X1 --> AA1[Numerical Output]
Y1 --> BB1[Simpson Rule]
Z1 --> CC1[Numerical Analysis]
AA1 --> DD1[Numerical Methods Output]
style A1 fill:#ff6b6b,color:#fff
style C1 fill:#ff6b6b,color:#fff
style E1 fill:#ff6b6b,color:#fff
style B1 fill:#ffd43b,color:#000
style D1 fill:#ffd43b,color:#000
style F1 fill:#ffd43b,color:#000
style G1 fill:#ffd43b,color:#000
style H1 fill:#ffd43b,color:#000
style I1 fill:#ffd43b,color:#000
style J1 fill:#ffd43b,color:#000
style K1 fill:#ffd43b,color:#000
style L1 fill:#ffd43b,color:#000
style M1 fill:#ffd43b,color:#000
style N1 fill:#ffd43b,color:#000
style O1 fill:#ffd43b,color:#000
style P1 fill:#ffd43b,color:#000
style Q1 fill:#ffd43b,color:#000
style R1 fill:#ffd43b,color:#000
style S1 fill:#ffd43b,color:#000
style T1 fill:#ffd43b,color:#000
style U1 fill:#ffd43b,color:#000
style V1 fill:#ffd43b,color:#000
style W1 fill:#ffd43b,color:#000
style X1 fill:#ffd43b,color:#000
style Y1 fill:#ffd43b,color:#000
style Z1 fill:#ffd43b,color:#000
style AA1 fill:#ffd43b,color:#000
style BB1 fill:#ffd43b,color:#000
style CC1 fill:#ffd43b,color:#000
style DD1 fill:#ffd43b,color:#000
style M1 fill:#51cf66,color:#fff
style N1 fill:#51cf66,color:#fff
style O1 fill:#51cf66,color:#fff
style P1 fill:#51cf66,color:#fff
style Q1 fill:#51cf66,color:#fff
style R1 fill:#51cf66,color:#fff
style S1 fill:#51cf66,color:#fff
style T1 fill:#51cf66,color:#fff
style U1 fill:#51cf66,color:#fff
style V1 fill:#51cf66,color:#fff
style W1 fill:#51cf66,color:#fff
style X1 fill:#51cf66,color:#fff
style Y1 fill:#51cf66,color:#fff
style Z1 fill:#51cf66,color:#fff
style AA1 fill:#51cf66,color:#fff
style BB1 fill:#51cf66,color:#fff
style CC1 fill:#51cf66,color:#fff
style DD1 fill:#51cf66,color:#fff
style DD1 fill:#b197fc,color:#fff
Triggers & Inputs
Numerical Methods
Numerical Operations
Intermediates
Products
Figure 1. Numerical Methods Process. This applied mathematics process visualization demonstrates root finding and integration methods. The flowchart shows numerical problem inputs and error analysis, numerical methods and algorithms, numerical operations and calculations, intermediate results, and final numerical solutions.
2. Statistical Analysis Process
graph TD
A2[Data Set Input] --> B2[Statistical Method Selection]
C2[Hypothesis Testing] --> D2[Test Design]
E2[Regression Analysis] --> F2[Model Selection]
B2 --> G2[Descriptive Statistics]
D2 --> H2[Null Hypothesis H0]
F2 --> I2[Linear Regression]
G2 --> J2[Mean Median Mode]
H2 --> K2[Alternative Hypothesis H1]
I2 --> L2[Multiple Regression]
J2 --> M2[Variance Standard Deviation]
K2 --> N2[Significance Level Alpha]
L2 --> O2[Model Fitting]
M2 --> P2[Correlation Analysis]
N2 --> O2
O2 --> Q2[Parameter Estimation]
P2 --> R2[Inferential Statistics]
Q2 --> S2[Residual Analysis]
R2 --> T2[Confidence Intervals]
S2 --> U2[Model Validation]
T2 --> V2[Statistical Tests]
U2 --> W2[Statistical Result]
V2 --> X2[Statistical Analysis]
W2 --> Y2[Statistical Output]
X2 --> Z2[Statistical Analysis Output]
Y2 --> AA2[Statistical Analysis Final Result]
Z2 --> BB2[Statistical Analysis Complete]
AA2 --> CC2[Statistical Analysis Output]
style A2 fill:#ff6b6b,color:#fff
style C2 fill:#ff6b6b,color:#fff
style E2 fill:#ff6b6b,color:#fff
style B2 fill:#ffd43b,color:#000
style D2 fill:#ffd43b,color:#000
style F2 fill:#ffd43b,color:#000
style G2 fill:#ffd43b,color:#000
style H2 fill:#ffd43b,color:#000
style I2 fill:#ffd43b,color:#000
style J2 fill:#ffd43b,color:#000
style K2 fill:#ffd43b,color:#000
style L2 fill:#ffd43b,color:#000
style M2 fill:#ffd43b,color:#000
style N2 fill:#ffd43b,color:#000
style O2 fill:#ffd43b,color:#000
style P2 fill:#ffd43b,color:#000
style Q2 fill:#ffd43b,color:#000
style R2 fill:#ffd43b,color:#000
style S2 fill:#ffd43b,color:#000
style T2 fill:#ffd43b,color:#000
style U2 fill:#ffd43b,color:#000
style V2 fill:#ffd43b,color:#000
style W2 fill:#ffd43b,color:#000
style X2 fill:#ffd43b,color:#000
style Y2 fill:#ffd43b,color:#000
style Z2 fill:#ffd43b,color:#000
style AA2 fill:#ffd43b,color:#000
style BB2 fill:#ffd43b,color:#000
style CC2 fill:#ffd43b,color:#000
style M2 fill:#51cf66,color:#fff
style N2 fill:#51cf66,color:#fff
style O2 fill:#51cf66,color:#fff
style P2 fill:#51cf66,color:#fff
style Q2 fill:#51cf66,color:#fff
style R2 fill:#51cf66,color:#fff
style S2 fill:#51cf66,color:#fff
style T2 fill:#51cf66,color:#fff
style U2 fill:#51cf66,color:#fff
style V2 fill:#51cf66,color:#fff
style W2 fill:#51cf66,color:#fff
style X2 fill:#51cf66,color:#fff
style Y2 fill:#51cf66,color:#fff
style Z2 fill:#51cf66,color:#fff
style AA2 fill:#51cf66,color:#fff
style BB2 fill:#51cf66,color:#fff
style CC2 fill:#51cf66,color:#fff
style CC2 fill:#b197fc,color:#fff
Triggers & Inputs
Statistical Methods
Statistical Operations
Intermediates
Products
Figure 2. Statistical Analysis Process. This applied mathematics process visualization demonstrates hypothesis testing and regression analysis. The flowchart shows data set inputs and hypothesis testing, statistical methods and algorithms, statistical operations and calculations, intermediate results, and final statistical analysis outputs.
3. Cryptographic Algorithms Process
graph TD
A3[Plaintext Message] --> B3[Cryptographic Method Selection]
C3[Key Generation] --> D3[Key Management]
E3[Encryption Algorithm] --> F3[Algorithm Implementation]
B3 --> G3[RSA Algorithm]
D3 --> H3[Public Key Generation]
F3 --> I3[Encryption Process]
G3 --> J3[Prime Number Generation]
H3 --> K3[Private Key Generation]
I3 --> L3[Message Encoding]
J3 --> M3[Modular Exponentiation]
K3 --> L3
L3 --> N3[Ciphertext Generation]
M3 --> O3[Elliptic Curve Cryptography]
N3 --> P3[Decryption Process]
O3 --> Q3[Point Multiplication]
P3 --> R3[Hash Function Application]
Q3 --> S3[Digital Signature]
R3 --> T3[Message Authentication]
S3 --> U3[Cryptographic Hash]
T3 --> V3[Key Exchange Protocol]
U3 --> W3[Cryptographic Result]
V3 --> X3[Cryptographic Analysis]
W3 --> Y3[Cryptographic Output]
X3 --> Z3[Cryptographic Analysis Output]
Y3 --> AA3[Cryptographic Analysis Final Result]
Z3 --> BB3[Cryptographic Analysis Complete]
AA3 --> CC3[Cryptographic Analysis Output]
style A3 fill:#ff6b6b,color:#fff
style C3 fill:#ff6b6b,color:#fff
style E3 fill:#ff6b6b,color:#fff
style B3 fill:#ffd43b,color:#000
style D3 fill:#ffd43b,color:#000
style F3 fill:#ffd43b,color:#000
style G3 fill:#ffd43b,color:#000
style H3 fill:#ffd43b,color:#000
style I3 fill:#ffd43b,color:#000
style J3 fill:#ffd43b,color:#000
style K3 fill:#ffd43b,color:#000
style L3 fill:#ffd43b,color:#000
style M3 fill:#ffd43b,color:#000
style N3 fill:#ffd43b,color:#000
style O3 fill:#ffd43b,color:#000
style P3 fill:#ffd43b,color:#000
style Q3 fill:#ffd43b,color:#000
style R3 fill:#ffd43b,color:#000
style S3 fill:#ffd43b,color:#000
style T3 fill:#ffd43b,color:#000
style U3 fill:#ffd43b,color:#000
style V3 fill:#ffd43b,color:#000
style W3 fill:#ffd43b,color:#000
style X3 fill:#ffd43b,color:#000
style Y3 fill:#ffd43b,color:#000
style Z3 fill:#ffd43b,color:#000
style AA3 fill:#ffd43b,color:#000
style BB3 fill:#ffd43b,color:#000
style CC3 fill:#ffd43b,color:#000
style M3 fill:#51cf66,color:#fff
style N3 fill:#51cf66,color:#fff
style O3 fill:#51cf66,color:#fff
style P3 fill:#51cf66,color:#fff
style Q3 fill:#51cf66,color:#fff
style R3 fill:#51cf66,color:#fff
style S3 fill:#51cf66,color:#fff
style T3 fill:#51cf66,color:#fff
style U3 fill:#51cf66,color:#fff
style V3 fill:#51cf66,color:#fff
style W3 fill:#51cf66,color:#fff
style X3 fill:#51cf66,color:#fff
style Y3 fill:#51cf66,color:#fff
style Z3 fill:#51cf66,color:#fff
style AA3 fill:#51cf66,color:#fff
style BB3 fill:#51cf66,color:#fff
style CC3 fill:#51cf66,color:#fff
style CC3 fill:#b197fc,color:#fff
Triggers & Inputs
Cryptographic Methods
Cryptographic Operations
Intermediates
Products
Figure 3. Cryptographic Algorithms Process. This applied mathematics process visualization demonstrates RSA, elliptic curve cryptography, and hash functions. The flowchart shows plaintext inputs and key generation, cryptographic methods and algorithms, cryptographic operations and calculations, intermediate results, and final cryptographic analysis outputs.