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| <title>Mathematics Batch 05 - Applied Mathematics - Programming Framework Analysis</title> | |
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| <h1>Mathematics Batch 05 - Applied Mathematics - Programming Framework Analysis</h1> | |
| <p>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.</p> | |
| <h2>1. Numerical Methods Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| 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 | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Numerical Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Numerical Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 1.</strong> 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. | |
| </div> | |
| </div> | |
| </div> | |
| <h2>2. Statistical Analysis Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| 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 | |
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| 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 | |
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| 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 | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Statistical Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Statistical Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 2.</strong> 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. | |
| </div> | |
| </div> | |
| </div> | |
| <h2>3. Cryptographic Algorithms Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| 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 | |
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| style V3 fill:#ffd43b,color:#000 | |
| style W3 fill:#ffd43b,color:#000 | |
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| 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 | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Cryptographic Methods | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Cryptographic Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 3.</strong> 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. | |
| </div> | |
| </div> | |
| </div> | |
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| <h3>Navigation</h3> | |
| <div class="nav-links"> | |
| <a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a> | |
| <a href="mathematics_batch_04.html" class="nav-link">← Previous: Geometry & Topology</a> | |
| <a href="mathematics_batch_06.html" class="nav-link">Next: Discrete Mathematics →</a> | |
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| </div> | |
| </div> | |
| <div class="footer"> | |
| <p><strong>Generated using the Programming Framework methodology</strong></p> | |
| <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p> | |
| <div class="contact-info"> | |
| <p><strong>Gary Welz</strong></p> | |
| <p>Retired Faculty Member</p> | |
| <p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p> | |
| <p>Borough of Manhattan Community College, CUNY</p> | |
| <p>CUNY Graduate Center (New Media Lab)</p> | |
| <p>Email: gwelz@jjay.cuny.edu</p> | |
| </div> | |
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