Spaces:
Running
Running
File size: 30,948 Bytes
ab71acf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 | <!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Mathematics Processes - Programming Framework Analysis</title>
<style>
body {
font-family: 'Times New Roman', Times, serif, 'Arial Unicode MS';
margin: 0;
background: #ffffff;
color: #000000;
line-height: 1.6;
font-size: 12pt;
}
.container {
max-width: 1000px;
margin: 0 auto;
padding: 1.5rem;
}
h1, h2, h3 {
color: #000000;
margin-top: 1.5rem;
margin-bottom: 0.75rem;
}
h1 {
font-size: 18pt;
text-align: center;
}
h2 {
font-size: 16pt;
border-bottom: 2px solid #000;
padding-bottom: 0.5rem;
}
h3 {
font-size: 14pt;
}
p {
margin-bottom: 1rem;
text-align: justify;
}
.figure {
margin: 1rem 0;
text-align: center;
border: 1px solid #ccc;
padding: 1rem;
background: #f9f9f9;
}
.figure-caption {
margin-top: 1rem;
font-style: italic;
text-align: left;
}
.mermaid {
background: white;
padding: 1rem;
border-radius: 4px;
}
</style>
<script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>
<script>
mermaid.initialize({
startOnLoad: true,
theme: 'default',
flowchart: {
useMaxWidth: false,
htmlLabels: true,
curve: 'linear',
nodeSpacing: 50,
rankSpacing: 50,
padding: 20
},
themeVariables: {
fontFamily: 'Arial Unicode MS, Arial, sans-serif'
}
});
</script>
</head>
<body>
<div class="container">
<h1>Mathematics Processes - Programming Framework Analysis</h1>
<p>This document presents 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. Mathematical Induction Proof Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
%% Initial Setup
%% Axioms and Given Conditions
A1[Peano Axioms] --> B1[Axiom Processing]
C1[Given n in Natural Numbers] --> D1[Input Validation]
E1[Goal: Prove P(n)] --> F1[Target Identification]
%% Logical Framework Setup
B1 --> G1[Mathematical Universe Setup]
D1 --> H1[Variable Declaration]
F1 --> I1[Proof Strategy Selection]
%% Proof Structure
G1 --> J1[Induction Hypothesis P(k)]
H1 --> K1[Base Case Analysis]
I1 --> L1[Inductive Step Planning]
%% Base Case Processing
K1 --> M1[P(0) Verification]
M1 --> N1[Base Case Success]
N1 --> O1[Induction Foundation]
%% Inductive Step Processing
L1 --> P1[Assume P(k) for k in Natural Numbers]
P1 --> Q1[Show P(k+1) follows]
Q1 --> R1[Inductive Step Execution]
%% Mathematical Operations
R1 --> S1[Algebraic Manipulation]
S1 --> T1[Logical Deduction]
T1 --> U1[Theorem Application]
%% Intermediate Calculations
U1 --> V1[Sub-proof Construction]
V1 --> W1[Lemma Application]
W1 --> X1[Contradiction Analysis]
%% Proof Validation
X1 --> Y1[Logical Consistency Check]
Y1 --> Z1[Mathematical Rigor Verification]
Z1 --> AA1[Proof Completeness Assessment]
%% Decision Points
AA1 --> BB1{Proof Complete?}
BB1 -->|No| CC1[Identify Gap]
BB1 -->|Yes| DD1[Proof Validated]
%% Gap Resolution
CC1 --> EE1[Additional Lemma Needed]
EE1 --> FF1[Sub-proof Construction]
FF1 --> GG1[Gap Resolution]
GG1 --> Y1
%% Final Output
DD1 --> HH1[Theorem P(n) Proven]
HH1 --> II1[Mathematical Truth Established]
II1 --> JJ1[Proof Tree Complete]
%% Styling - Mathematical Color Scheme
%% Styling - Biological Color Scheme
style A1 fill:#ff6b6b,color:#fff
style C1 fill:#ff6b6b,color:#fff
style E1 fill:#ff6b6b,color:#fff
style J1 fill:#ffd43b,color:#000
style P1 fill:#ffd43b,color:#000
style Q1 fill:#ffd43b,color:#000
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 B1 fill:#74c0fc,color:#fff
style D1 fill:#74c0fc,color:#fff
style F1 fill:#74c0fc,color:#fff
style G1 fill:#74c0fc,color:#fff
style H1 fill:#74c0fc,color:#fff
style I1 fill:#74c0fc,color:#fff
style K1 fill:#74c0fc,color:#fff
style L1 fill:#74c0fc,color:#fff
style M1 fill:#74c0fc,color:#fff
style N1 fill:#74c0fc,color:#fff
style O1 fill:#74c0fc,color:#fff
style R1 fill:#74c0fc,color:#fff
style Y1 fill:#74c0fc,color:#fff
style Z1 fill:#74c0fc,color:#fff
style AA1 fill:#74c0fc,color:#fff
style BB1 fill:#74c0fc,color:#fff
style CC1 fill:#74c0fc,color:#fff
style DD1 fill:#74c0fc,color:#fff
style EE1 fill:#74c0fc,color:#fff
style FF1 fill:#74c0fc,color:#fff
style GG1 fill:#74c0fc,color:#fff
style HH1 fill:#b197fc,color:#fff
style II1 fill:#b197fc,color:#fff
style JJ1 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>Logical Structures & Hypotheses
</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>Deductions & Theorem Applications
</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> Mathematical Induction Proof Process. This mathematics process visualization demonstrates formal mathematical reasoning. The flowchart shows axioms and given conditions, logical structures and hypotheses, deduction steps and theorem applications, intermediate calculations and sub-proofs, and final proven theorems.
</div>
</div>
<h2>2. Euclidean Algorithm Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
%% Initial Setup
%% Input Conditions
A2[Integer a] --> B2[Input Validation]
C2[Integer b] --> D2[Input Validation]
E2[Goal: Find GCD(a,b)] --> F2[Problem Statement]
%% Algorithm Initialization
B2 --> G2[Set r₀ = a]
D2 --> H2[Set r₁ = b]
F2 --> I2[Algorithm Selection]
%% Division Process
G2 --> J2[Division Algorithm]
H2 --> K2[Division Algorithm]
I2 --> L2[Iterative Process]
%% First Division
J2 --> M2[r₀ = q₁r₁ + r₂]
K2 --> N2[Calculate q₁ and r₂]
L2 --> O2[Initialize iteration counter]
%% Iterative Loop
M2 --> P2{Is r₂ = 0?}
N2 --> Q2[Store r₂]
O2 --> R2[Increment counter]
%% Continue or Terminate
P2 -->|No| S2[Set r₀ = r₁, r₁ = r₂]
P2 -->|Yes| T2[GCD Found: r₁]
Q2 --> U2[Update remainders]
R2 --> V2[Track iterations]
%% Next Iteration
S2 --> W2[Next Division Step]
U2 --> X2[Prepare for next iteration]
V2 --> Y2[Check termination condition]
%% Final Result
T2 --> Z2[GCD(a,b) = r₁]
W2 --> AA2[Repeat division process]
X2 --> BB2[Update variables]
Y2 --> CC2{Continue?}
%% Algorithm Completion
Z2 --> DD2[Result Validation]
AA2 --> P2
BB2 --> P2
CC2 -->|Yes| AA2
CC2 -->|No| T2
%% Output
DD2 --> EE2[GCD Calculation Complete]
EE2 --> FF2[Mathematical Proof of Correctness]
FF2 --> GG2[Algorithm Efficiency Analysis]
%% Styling - Mathematical Color Scheme
%% Styling - Biological Color Scheme
style A2 fill:#ff6b6b,color:#fff
style C2 fill:#ff6b6b,color:#fff
style E2 fill:#ff6b6b,color:#fff
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:#51cf66,color:#fff
style M2 fill:#51cf66,color:#fff
style N2 fill:#51cf66,color:#fff
style O2 fill:#51cf66,color:#fff
style S2 fill:#51cf66,color:#fff
style W2 fill:#51cf66,color:#fff
style AA2 fill:#51cf66,color:#fff
style B2 fill:#74c0fc,color:#fff
style D2 fill:#74c0fc,color:#fff
style F2 fill:#74c0fc,color:#fff
style P2 fill:#74c0fc,color:#fff
style Q2 fill:#74c0fc,color:#fff
style R2 fill:#74c0fc,color:#fff
style T2 fill:#74c0fc,color:#fff
style U2 fill:#74c0fc,color:#fff
style V2 fill:#74c0fc,color:#fff
style X2 fill:#74c0fc,color:#fff
style Y2 fill:#74c0fc,color:#fff
style CC2 fill:#74c0fc,color:#fff
style DD2 fill:#74c0fc,color:#fff
style Z2 fill:#b197fc,color:#fff
style EE2 fill:#b197fc,color:#fff
style FF2 fill:#b197fc,color:#fff
style GG2 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>Mathematical Methods & Algorithms
</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>Computational 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> Euclidean Algorithm Process. This mathematics process visualization demonstrates algorithmic computation. The flowchart shows integer inputs, mathematical methods and algorithms, computational operations, intermediate calculations, and final GCD results.
</div>
</div>
<h2>3. Calculus Integration Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
%% Initial Setup
%% Input Conditions
A3[Function f(x)] --> B3[Function Analysis]
C3[Integration Limits [a,b]] --> D3[Boundary Definition]
E3[Goal: ∫f(x)dx] --> F3[Problem Statement]
%% Function Classification
B3 --> G3[Function Type Classification]
D3 --> H3[Domain Analysis]
F3 --> I3[Integration Strategy Selection]
%% Integration Methods
G3 --> J3{Function Type?}
H3 --> K3[Continuity Check]
I3 --> L3[Method Selection]
%% Direct Integration
J3 -->|Polynomial| M3[Power Rule Application]
J3 -->|Trigonometric| N3[Trig Identity Application]
J3 -->|Exponential| O3[Exponential Rule]
J3 -->|Rational| P3[Partial Fractions]
%% Substitution Method
K3 --> Q3[Substitution Detection]
L3 --> R3[Substitution Method]
M3 --> S3[Direct Integration]
N3 --> T3[Trig Integration]
%% Integration by Parts
O3 --> U3[Integration by Parts]
P3 --> V3[Partial Fraction Decomposition]
Q3 --> W3[Substitution Application]
R3 --> X3[Variable Change]
%% Definite Integration
S3 --> Y3[Antiderivative F(x)]
T3 --> Z3[Trig Antiderivative]
U3 --> AA3[Parts Integration]
V3 --> BB3[Fraction Integration]
%% Evaluation
W3 --> CC3[Substituted Integral]
X3 --> DD3[New Variable Integration]
Y3 --> EE3[F(b) - F(a)]
Z3 --> FF3[Definite Trig Result]
%% Final Results
AA3 --> GG3[Parts Result]
BB3 --> HH3[Fraction Result]
CC3 --> II3[Substitution Result]
DD3 --> JJ3[Variable Back-Substitution]
%% Verification
EE3 --> KK3[Result Verification]
FF3 --> LL3[Trigonometric Verification]
GG3 --> MM3[Parts Verification]
HH3 --> NN3[Fraction Verification]
%% Output
KK3 --> OO3[Definite Integral Value]
LL3 --> PP3[Trigonometric Integral Value]
MM3 --> QQ3[Parts Integral Value]
NN3 --> RR3[Fractional Integral Value]
%% Styling - Mathematical Color Scheme
%% Styling - Biological Color Scheme
style A3 fill:#ff6b6b,color:#fff
style C3 fill:#ff6b6b,color:#fff
style E3 fill:#ff6b6b,color:#fff
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:#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 AA3 fill:#51cf66,color:#fff
style BB3 fill:#51cf66,color:#fff
style B3 fill:#74c0fc,color:#fff
style D3 fill:#74c0fc,color:#fff
style F3 fill:#74c0fc,color:#fff
style Y3 fill:#74c0fc,color:#fff
style Z3 fill:#74c0fc,color:#fff
style AA3 fill:#74c0fc,color:#fff
style BB3 fill:#74c0fc,color:#fff
style CC3 fill:#74c0fc,color:#fff
style DD3 fill:#74c0fc,color:#fff
style EE3 fill:#74c0fc,color:#fff
style FF3 fill:#74c0fc,color:#fff
style GG3 fill:#74c0fc,color:#fff
style HH3 fill:#74c0fc,color:#fff
style II3 fill:#74c0fc,color:#fff
style JJ3 fill:#74c0fc,color:#fff
style EE4 fill:#b197fc,color:#fff
style FF4 fill:#b197fc,color:#fff
style GG4 fill:#b197fc,color:#fff
style HH4 fill:#b197fc,color:#fff
style KK4 fill:#b197fc,color:#fff
style LL4 fill:#b197fc,color:#fff
style MM4 fill:#b197fc,color:#fff
style NN4 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>Mathematical Methods & Theorems
</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>Integration 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> Calculus Integration Process. This mathematics process visualization demonstrates integral calculus computation. The flowchart shows function inputs, mathematical methods and theorems, integration operations, intermediate calculations, and final integral values.
</div>
</div>
<h2>4. Linear Algebra Matrix Operations</h2>
<div class="figure">
<div class="mermaid">
graph TD
%% Initial Setup
%% Input Conditions
A4[Matrix A] --> B4[Matrix Analysis]
C4[Matrix B] --> D4[Matrix Analysis]
E4[Operation Type] --> F4[Operation Selection]
%% Matrix Classification
B4 --> G4[Matrix Dimensions Check]
D4 --> H4[Matrix Properties Analysis]
F4 --> I4[Operation Feasibility]
%% Operation Types
G4 --> J4{Operation Type?}
H4 --> K4[Matrix Properties]
I4 --> L4[Compatibility Check]
%% Matrix Addition
J4 -->|Addition| M4[Dimension Matching]
J4 -->|Multiplication| N4[Inner Product Dimensions]
J4 -->|Inverse| O4[Square Matrix Check]
J4 -->|Determinant| P4[Square Matrix Check]
%% Addition Process
K4 --> Q4[Element-wise Addition]
L4 --> R4[Compatibility Verification]
M4 --> S4[Add Corresponding Elements]
N4 --> T4[Matrix Multiplication Algorithm]
%% Multiplication Process
O4 --> U4[Inverse Calculation]
P4 --> V4[Determinant Calculation]
Q4 --> W4[Result Matrix C]
R4 --> X4[Operation Validation]
%% Inverse Calculation
S4 --> Y4[Addition Result]
T4 --> Z4[Multiplication Result]
U4 --> AA4[Gauss-Jordan Elimination]
V4 --> BB4[Determinant Expansion]
%% Determinant Calculation
W4 --> CC4[Result Verification]
X4 --> DD4[Properties Check]
Y4 --> EE4[Matrix C = A + B]
Z4 --> FF4[Matrix C = A × B]
%% Final Results
AA4 --> GG4[Inverse Matrix A⁻¹]
BB4 --> HH4[Determinant |A|]
CC4 --> II4[Result Validation]
DD4 --> JJ4[Properties Verification]
%% Output
EE4 --> KK4[Addition Complete]
FF4 --> LL4[Multiplication Complete]
GG4 --> MM4[Inverse Found]
HH4 --> NN4[Determinant Calculated]
%% Styling - Mathematical Color Scheme
%% Styling - Biological Color Scheme
style A4 fill:#ff6b6b,color:#fff
style C4 fill:#ff6b6b,color:#fff
style E4 fill:#ff6b6b,color:#fff
style G4 fill:#ffd43b,color:#000
style H4 fill:#ffd43b,color:#000
style I4 fill:#ffd43b,color:#000
style J4 fill:#ffd43b,color:#000
style K4 fill:#ffd43b,color:#000
style L4 fill:#ffd43b,color:#000
style M4 fill:#51cf66,color:#fff
style N4 fill:#51cf66,color:#fff
style O4 fill:#51cf66,color:#fff
style P4 fill:#51cf66,color:#fff
style Q4 fill:#51cf66,color:#fff
style R4 fill:#51cf66,color:#fff
style S4 fill:#51cf66,color:#fff
style T4 fill:#51cf66,color:#fff
style U4 fill:#51cf66,color:#fff
style V4 fill:#51cf66,color:#fff
style AA4 fill:#51cf66,color:#fff
style BB4 fill:#51cf66,color:#fff
style B4 fill:#74c0fc,color:#fff
style D4 fill:#74c0fc,color:#fff
style F4 fill:#74c0fc,color:#fff
style W4 fill:#74c0fc,color:#fff
style X4 fill:#74c0fc,color:#fff
style Y4 fill:#74c0fc,color:#fff
style Z4 fill:#74c0fc,color:#fff
style CC4 fill:#74c0fc,color:#fff
style DD4 fill:#74c0fc,color:#fff
style II4 fill:#74c0fc,color:#fff
style JJ4 fill:#74c0fc,color:#fff
style EE4 fill:#b197fc,color:#fff
style FF4 fill:#b197fc,color:#fff
style GG4 fill:#b197fc,color:#fff
style HH4 fill:#b197fc,color:#fff
style KK4 fill:#b197fc,color:#fff
style LL4 fill:#b197fc,color:#fff
style MM4 fill:#b197fc,color:#fff
style NN4 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>Mathematical Structures & 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>Matrix 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 4.</strong> Linear Algebra Matrix Operations. This mathematics process visualization demonstrates matrix algebra computation. The flowchart shows matrix inputs, mathematical structures and methods, matrix operations, intermediate calculations, and final matrix results.
</div>
</div>
<h2>5. Probability Theory Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
%% Initial Setup
%% Input Conditions
A5[Sample Space S] --> B5[Space Analysis]
C5[Event E] --> D5[Event Analysis]
E5[Probability Measure P] --> F5[Measure Definition]
%% Probability Framework
B5 --> G5[Sample Space Properties]
D5 --> H5[Event Properties]
F5 --> I5[Probability Axioms]
%% Axiomatic Foundation
G5 --> J5[Kolmogorov Axioms]
H5 --> K5[Event Algebra]
I5 --> L5[Measure Theory]
%% Probability Calculation
J5 --> M5[P(S) = 1]
K5 --> N5[Event Operations]
L5 --> O5[Probability Functions]
%% Event Operations
M5 --> P5[Complement Rule]
N5 --> Q5[Union Rule]
O5 --> R5[Conditional Probability]
%% Conditional Probability
P5 --> S5[P(E') = 1 - P(E)]
Q5 --> T5[P(A∪B) = P(A) + P(B) - P(A∩B)]
R5 --> U5[P(A|B) = P(A∩B)/P(B)]
%% Bayes' Theorem
S5 --> V5[Probability Calculation]
T5 --> W5[Set Operations]
U5 --> X5[Bayes' Theorem]
%% Independence
V5 --> Y5[Result Verification]
W5 --> Z5[Venn Diagram Analysis]
X5 --> AA5[P(A|B) = P(B|A)P(A)/P(B)]
%% Final Results
Y5 --> BB5[Probability Value]
Z5 --> CC5[Set Relationships]
AA5 --> DD5[Posterior Probability]
%% Output
BB5 --> EE5[Probability Calculated]
CC5 --> FF5[Event Relationships]
DD5 --> GG5[Bayesian Update]
%% Styling - Mathematical Color Scheme
%% Styling - Biological Color Scheme
style A5 fill:#ff6b6b,color:#fff
style C5 fill:#ff6b6b,color:#fff
style E5 fill:#ff6b6b,color:#fff
style G5 fill:#ffd43b,color:#000
style H5 fill:#ffd43b,color:#000
style I5 fill:#ffd43b,color:#000
style J5 fill:#ffd43b,color:#000
style K5 fill:#ffd43b,color:#000
style L5 fill:#ffd43b,color:#000
style M5 fill:#51cf66,color:#fff
style N5 fill:#51cf66,color:#fff
style O5 fill:#51cf66,color:#fff
style P5 fill:#51cf66,color:#fff
style Q5 fill:#51cf66,color:#fff
style R5 fill:#51cf66,color:#fff
style S5 fill:#51cf66,color:#fff
style T5 fill:#51cf66,color:#fff
style U5 fill:#51cf66,color:#fff
style X5 fill:#51cf66,color:#fff
style AA5 fill:#51cf66,color:#fff
style B5 fill:#74c0fc,color:#fff
style D5 fill:#74c0fc,color:#fff
style F5 fill:#74c0fc,color:#fff
style V5 fill:#74c0fc,color:#fff
style W5 fill:#74c0fc,color:#fff
style Y5 fill:#74c0fc,color:#fff
style Z5 fill:#74c0fc,color:#fff
style BB5 fill:#74c0fc,color:#fff
style CC5 fill:#74c0fc,color:#fff
style DD5 fill:#74c0fc,color:#fff
style EE5 fill:#b197fc,color:#fff
style FF5 fill:#b197fc,color:#fff
style GG5 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>Probability Axioms & Theorems
</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>Probability Calculations
</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 5.</strong> Probability Theory Process. This mathematics process visualization demonstrates probabilistic reasoning. The flowchart shows sample space inputs, probability axioms and theorems, probability calculations, intermediate computations, and final probability values.
</div>
</div>
<p><strong>Generated using the Programming Framework methodology</strong></p>
<p>This collection demonstrates the computational nature of mathematical processes and systems</p>
<p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p>
</div>
</body>
</html> |