garywelz/programming_framework
0
1<!DOCTYPE html>2<html lang="en">3<head>4 <meta charset="UTF-8" />5 <meta name="viewport" content="width=device-width, initial-scale=1" />6 <title>Mathematics Batch 05 - Applied Mathematics - Programming Framework Analysis</title>7 <style>8 body { 9 font-family: 'Times New Roman', Times, serif, 'Arial Unicode MS'; 10 margin: 0; 11 background: #ffffff; 12 color: #000000; 13 line-height: 1.6; 14 font-size: 12pt; 15 }16 .container { 17 max-width: 1000px; 18 margin: 0 auto; 19 padding: 1.5rem; 20 }21 h1, h2, h3 { 22 color: #000000; 23 margin-top: 1.5rem; 24 margin-bottom: 0.75rem; 25 }26 h1 { 27 font-size: 18pt; 28 text-align: center; 29 }30 h2 { 31 font-size: 16pt; 32 border-bottom: 2px solid #000; 33 padding-bottom: 0.5rem; 34 }35 h3 { 36 font-size: 14pt; 37 }38 p { 39 margin-bottom: 1rem; 40 text-align: justify; 41 }42 .figure { 43 margin: 2rem 0; 44 text-align: center; 45 border: 1px solid #ccc; 46 padding: 1rem; 47 background: #f9f9f9; 48 }49 .figure-caption { 50 margin-top: 1rem; 51 font-style: italic; 52 text-align: left; 53 }54 .mermaid { 55 background: white; 56 padding: 1rem; 57 border-radius: 4px; 58 }59 .navigation {60 margin: 3rem 0;61 padding: 1rem;62 background: #f8f9fa;63 border-radius: 8px;64 }65 .nav-links {66 display: flex;67 flex-wrap: wrap;68 gap: 1rem;69 justify-content: center;70 }71 .nav-link {72 color: #007bff;73 text-decoration: none;74 padding: 0.5rem 1rem;75 border: 1px solid #007bff;76 border-radius: 4px;77 transition: all 0.3s ease;78 }79 .nav-link:hover {80 background: #007bff;81 color: white;82 }83 .footer {84 margin-top: 3rem;85 padding: 1rem;86 background: #f8f9fa;87 border-radius: 8px;88 text-align: center;89 }90 .contact-info {91 margin-top: 1rem;92 }93 .contact-info p {94 margin: 0.25rem 0;95 text-align: center;96 }97 </style>98 <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>99 <script>100 mermaid.initialize({ 101 startOnLoad: true, 102 theme: 'default', 103 flowchart: { 104 useMaxWidth: false, 105 htmlLabels: true,106 curve: 'linear',107 nodeSpacing: 30,108 rankSpacing: 30,109 padding: 10110 },111 themeVariables: {112 fontFamily: 'Arial Unicode MS, Arial, sans-serif'113 }114 });115 </script>116</head>117<body>118 <div class="container">119 <h1>Mathematics Batch 05 - Applied Mathematics - Programming Framework Analysis</h1>120 121 <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>122 123 <h2>1. Numerical Methods Process</h2>124 <div class="figure">125 <div class="mermaid">126graph TD127 A1[Numerical Problem Input] --> B1[Method Selection]128 C1[Error Analysis] --> D1[Convergence Study]129 E1[Algorithm Design] --> F1[Implementation Strategy]130 131 B1 --> G1[Root Finding Methods]132 D1 --> H1[Convergence Criteria]133 F1 --> I1[Numerical Algorithm]134 135 G1 --> J1[Bisection Method]136 H1 --> K1[Tolerance Setting]137 I1 --> L1[Iterative Process]138 139 J1 --> M1[Newton Raphson Method]140 K1 --> N1[Error Estimation]141 L1 --> O1[Convergence Check]142 143 M1 --> P1[Secant Method]144 N1 --> Q1[Stability Analysis]145 O1 --> R1[Solution Refinement]146 147 P1 --> S1[Fixed Point Iteration]148 Q1 --> T1[Error Bounds]149 R1 --> U1[Numerical Solution]150 151 S1 --> V1[Integration Methods]152 T1 --> W1[Accuracy Assessment]153 U1 --> X1[Solution Validation]154 155 V1 --> Y1[Trapezoidal Rule]156 W1 --> Z1[Numerical Result]157 X1 --> AA1[Numerical Output]158 159 Y1 --> BB1[Simpson Rule]160 Z1 --> CC1[Numerical Analysis]161 AA1 --> DD1[Numerical Methods Output]162 163 style A1 fill:#ff6b6b,color:#fff164 style C1 fill:#ff6b6b,color:#fff165 style E1 fill:#ff6b6b,color:#fff166 167 style B1 fill:#ffd43b,color:#000168 style D1 fill:#ffd43b,color:#000169 style F1 fill:#ffd43b,color:#000170 style G1 fill:#ffd43b,color:#000171 style H1 fill:#ffd43b,color:#000172 style I1 fill:#ffd43b,color:#000173 style J1 fill:#ffd43b,color:#000174 style K1 fill:#ffd43b,color:#000175 style L1 fill:#ffd43b,color:#000176 style M1 fill:#ffd43b,color:#000177 style N1 fill:#ffd43b,color:#000178 style O1 fill:#ffd43b,color:#000179 style P1 fill:#ffd43b,color:#000180 style Q1 fill:#ffd43b,color:#000181 style R1 fill:#ffd43b,color:#000182 style S1 fill:#ffd43b,color:#000183 style T1 fill:#ffd43b,color:#000184 style U1 fill:#ffd43b,color:#000185 style V1 fill:#ffd43b,color:#000186 style W1 fill:#ffd43b,color:#000187 style X1 fill:#ffd43b,color:#000188 style Y1 fill:#ffd43b,color:#000189 style Z1 fill:#ffd43b,color:#000190 style AA1 fill:#ffd43b,color:#000191 style BB1 fill:#ffd43b,color:#000192 style CC1 fill:#ffd43b,color:#000193 style DD1 fill:#ffd43b,color:#000194 195 style M1 fill:#51cf66,color:#fff196 style N1 fill:#51cf66,color:#fff197 style O1 fill:#51cf66,color:#fff198 style P1 fill:#51cf66,color:#fff199 style Q1 fill:#51cf66,color:#fff200 style R1 fill:#51cf66,color:#fff201 style S1 fill:#51cf66,color:#fff202 style T1 fill:#51cf66,color:#fff203 style U1 fill:#51cf66,color:#fff204 style V1 fill:#51cf66,color:#fff205 style W1 fill:#51cf66,color:#fff206 style X1 fill:#51cf66,color:#fff207 style Y1 fill:#51cf66,color:#fff208 style Z1 fill:#51cf66,color:#fff209 style AA1 fill:#51cf66,color:#fff210 style BB1 fill:#51cf66,color:#fff211 style CC1 fill:#51cf66,color:#fff212 style DD1 fill:#51cf66,color:#fff213 214 style DD1 fill:#b197fc,color:#fff215 </div>216 217 <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">218 <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;">219 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs220 </div>221 <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;">222 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Numerical Methods223 </div>224 <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;">225 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Numerical Operations226 </div>227 <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;">228 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates229 </div>230 <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;">231 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products232 </div>233 </div>234 235 <div class="figure-caption">236 <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.237 </div>238 </div>239 </div>240 241 <h2>2. Statistical Analysis Process</h2>242 <div class="figure">243 <div class="mermaid">244graph TD245 A2[Data Set Input] --> B2[Statistical Method Selection]246 C2[Hypothesis Testing] --> D2[Test Design]247 E2[Regression Analysis] --> F2[Model Selection]248 249 B2 --> G2[Descriptive Statistics]250 D2 --> H2[Null Hypothesis H0]251 F2 --> I2[Linear Regression]252 253 G2 --> J2[Mean Median Mode]254 H2 --> K2[Alternative Hypothesis H1]255 I2 --> L2[Multiple Regression]256 257 J2 --> M2[Variance Standard Deviation]258 K2 --> N2[Significance Level Alpha]259 L2 --> O2[Model Fitting]260 261 M2 --> P2[Correlation Analysis]262 N2 --> O2263 O2 --> Q2[Parameter Estimation]264 265 P2 --> R2[Inferential Statistics]266 Q2 --> S2[Residual Analysis]267 R2 --> T2[Confidence Intervals]268 269 S2 --> U2[Model Validation]270 T2 --> V2[Statistical Tests]271 U2 --> W2[Statistical Result]272 273 V2 --> X2[Statistical Analysis]274 W2 --> Y2[Statistical Output]275 X2 --> Z2[Statistical Analysis Output]276 277 Y2 --> AA2[Statistical Analysis Final Result]278 Z2 --> BB2[Statistical Analysis Complete]279 AA2 --> CC2[Statistical Analysis Output]280 281 style A2 fill:#ff6b6b,color:#fff282 style C2 fill:#ff6b6b,color:#fff283 style E2 fill:#ff6b6b,color:#fff284 285 style B2 fill:#ffd43b,color:#000286 style D2 fill:#ffd43b,color:#000287 style F2 fill:#ffd43b,color:#000288 style G2 fill:#ffd43b,color:#000289 style H2 fill:#ffd43b,color:#000290 style I2 fill:#ffd43b,color:#000291 style J2 fill:#ffd43b,color:#000292 style K2 fill:#ffd43b,color:#000293 style L2 fill:#ffd43b,color:#000294 style M2 fill:#ffd43b,color:#000295 style N2 fill:#ffd43b,color:#000296 style O2 fill:#ffd43b,color:#000297 style P2 fill:#ffd43b,color:#000298 style Q2 fill:#ffd43b,color:#000299 style R2 fill:#ffd43b,color:#000300 style S2 fill:#ffd43b,color:#000301 style T2 fill:#ffd43b,color:#000302 style U2 fill:#ffd43b,color:#000303 style V2 fill:#ffd43b,color:#000304 style W2 fill:#ffd43b,color:#000305 style X2 fill:#ffd43b,color:#000306 style Y2 fill:#ffd43b,color:#000307 style Z2 fill:#ffd43b,color:#000308 style AA2 fill:#ffd43b,color:#000309 style BB2 fill:#ffd43b,color:#000310 style CC2 fill:#ffd43b,color:#000311 312 style M2 fill:#51cf66,color:#fff313 style N2 fill:#51cf66,color:#fff314 style O2 fill:#51cf66,color:#fff315 style P2 fill:#51cf66,color:#fff316 style Q2 fill:#51cf66,color:#fff317 style R2 fill:#51cf66,color:#fff318 style S2 fill:#51cf66,color:#fff319 style T2 fill:#51cf66,color:#fff320 style U2 fill:#51cf66,color:#fff321 style V2 fill:#51cf66,color:#fff322 style W2 fill:#51cf66,color:#fff323 style X2 fill:#51cf66,color:#fff324 style Y2 fill:#51cf66,color:#fff325 style Z2 fill:#51cf66,color:#fff326 style AA2 fill:#51cf66,color:#fff327 style BB2 fill:#51cf66,color:#fff328 style CC2 fill:#51cf66,color:#fff329 330 style CC2 fill:#b197fc,color:#fff331 </div>332 333 <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">334 <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;">335 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs336 </div>337 <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;">338 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Statistical Methods339 </div>340 <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;">341 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Statistical Operations342 </div>343 <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;">344 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates345 </div>346 <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;">347 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products348 </div>349 </div>350 351 <div class="figure-caption">352 <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.353 </div>354 </div>355 </div>356 357 <h2>3. Cryptographic Algorithms Process</h2>358 <div class="figure">359 <div class="mermaid">360graph TD361 A3[Plaintext Message] --> B3[Cryptographic Method Selection]362 C3[Key Generation] --> D3[Key Management]363 E3[Encryption Algorithm] --> F3[Algorithm Implementation]364 365 B3 --> G3[RSA Algorithm]366 D3 --> H3[Public Key Generation]367 F3 --> I3[Encryption Process]368 369 G3 --> J3[Prime Number Generation]370 H3 --> K3[Private Key Generation]371 I3 --> L3[Message Encoding]372 373 J3 --> M3[Modular Exponentiation]374 K3 --> L3375 L3 --> N3[Ciphertext Generation]376 377 M3 --> O3[Elliptic Curve Cryptography]378 N3 --> P3[Decryption Process]379 O3 --> Q3[Point Multiplication]380 381 P3 --> R3[Hash Function Application]382 Q3 --> S3[Digital Signature]383 R3 --> T3[Message Authentication]384 385 S3 --> U3[Cryptographic Hash]386 T3 --> V3[Key Exchange Protocol]387 U3 --> W3[Cryptographic Result]388 389 V3 --> X3[Cryptographic Analysis]390 W3 --> Y3[Cryptographic Output]391 X3 --> Z3[Cryptographic Analysis Output]392 393 Y3 --> AA3[Cryptographic Analysis Final Result]394 Z3 --> BB3[Cryptographic Analysis Complete]395 AA3 --> CC3[Cryptographic Analysis Output]396 397 style A3 fill:#ff6b6b,color:#fff398 style C3 fill:#ff6b6b,color:#fff399 style E3 fill:#ff6b6b,color:#fff400 401 style B3 fill:#ffd43b,color:#000402 style D3 fill:#ffd43b,color:#000403 style F3 fill:#ffd43b,color:#000404 style G3 fill:#ffd43b,color:#000405 style H3 fill:#ffd43b,color:#000406 style I3 fill:#ffd43b,color:#000407 style J3 fill:#ffd43b,color:#000408 style K3 fill:#ffd43b,color:#000409 style L3 fill:#ffd43b,color:#000410 style M3 fill:#ffd43b,color:#000411 style N3 fill:#ffd43b,color:#000412 style O3 fill:#ffd43b,color:#000413 style P3 fill:#ffd43b,color:#000414 style Q3 fill:#ffd43b,color:#000415 style R3 fill:#ffd43b,color:#000416 style S3 fill:#ffd43b,color:#000417 style T3 fill:#ffd43b,color:#000418 style U3 fill:#ffd43b,color:#000419 style V3 fill:#ffd43b,color:#000420 style W3 fill:#ffd43b,color:#000421 style X3 fill:#ffd43b,color:#000422 style Y3 fill:#ffd43b,color:#000423 style Z3 fill:#ffd43b,color:#000424 style AA3 fill:#ffd43b,color:#000425 style BB3 fill:#ffd43b,color:#000426 style CC3 fill:#ffd43b,color:#000427 428 style M3 fill:#51cf66,color:#fff429 style N3 fill:#51cf66,color:#fff430 style O3 fill:#51cf66,color:#fff431 style P3 fill:#51cf66,color:#fff432 style Q3 fill:#51cf66,color:#fff433 style R3 fill:#51cf66,color:#fff434 style S3 fill:#51cf66,color:#fff435 style T3 fill:#51cf66,color:#fff436 style U3 fill:#51cf66,color:#fff437 style V3 fill:#51cf66,color:#fff438 style W3 fill:#51cf66,color:#fff439 style X3 fill:#51cf66,color:#fff440 style Y3 fill:#51cf66,color:#fff441 style Z3 fill:#51cf66,color:#fff442 style AA3 fill:#51cf66,color:#fff443 style BB3 fill:#51cf66,color:#fff444 style CC3 fill:#51cf66,color:#fff445 446 style CC3 fill:#b197fc,color:#fff447 </div>448 449 <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">450 <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;">451 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs452 </div>453 <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;">454 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Cryptographic Methods455 </div>456 <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;">457 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Cryptographic Operations458 </div>459 <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;">460 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates461 </div>462 <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;">463 <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products464 </div>465 </div>466 467 <div class="figure-caption">468 <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.469 </div>470 </div>471 </div>472 473 <div class="navigation">474 <h3>Navigation</h3>475 <div class="nav-links">476 <a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a>477 <a href="mathematics_batch_04.html" class="nav-link">← Previous: Geometry & Topology</a>478 <a href="mathematics_batch_06.html" class="nav-link">Next: Discrete Mathematics →</a>479 <a href="index.html" class="nav-link">Programming Framework Home</a>480 </div>481 </div>482 483 <div class="footer">484 <p><strong>Generated using the Programming Framework methodology</strong></p>485 <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p>486 <div class="contact-info">487 <p><strong>Gary Welz</strong></p>488 <p>Retired Faculty Member</p>489 <p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p>490 <p>Borough of Manhattan Community College, CUNY</p>491 <p>CUNY Graduate Center (New Media Lab)</p>492 <p>Email: gwelz@jjay.cuny.edu</p>493 </div>494 </div>495 </div>496</body>497</html>498 