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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