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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
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style Q1 fill:#ffd43b,color:#000
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style Y1 fill:#ffd43b,color:#000
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style CC1 fill:#ffd43b,color:#000
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style O1 fill:#51cf66,color:#fff
style P1 fill:#51cf66,color:#fff
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</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]
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style E2 fill:#ff6b6b,color:#fff
style B2 fill:#ffd43b,color:#000
style D2 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
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style W2 fill:#ffd43b,color:#000
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style Y2 fill:#ffd43b,color:#000
style Z2 fill:#ffd43b,color:#000
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style BB2 fill:#ffd43b,color:#000
style CC2 fill:#ffd43b,color:#000
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style X2 fill:#51cf66,color:#fff
style Y2 fill:#51cf66,color:#fff
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style CC2 fill:#51cf66,color:#fff
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</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
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style N3 fill:#ffd43b,color:#000
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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
</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>
<div class="navigation">
<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>
<a href="index.html" class="nav-link">Programming Framework Home</a>
</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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