Herta.js

An advanced mathematics framework for Node.js providing powerful tools for mathematical computation, symbolic mathematics, and scientific computing. Designed specifically for scientists, researchers, and advanced mathematical applications.

Framework Architecture

Herta.js is organized in a modular, intuitive folder structure for better organization and easier navigation:

/src
├── core/        # Core mathematical operations
├── algebra/     # Algebraic operations
├── calculus/    # Calculus operations
├── discrete/    # Discrete mathematics (including graph theory)
├── statistics/  # Statistical operations
├── geometry/    # Geometric operations
├── optimization/# Optimization algorithms 
├── physics/     # Physics models and simulations
├── crypto/      # Cryptography algorithms
├── utils/       # Utility functions
├── applied/     # Applied mathematics
└── advanced/    # Advanced specialized modules

Feature Highlights

Units Conversion System

Complete system for converting between various units of measurement:

// Convert between different length units
const meters = herta.utils.units.length(5, 'feet', 'meter');  // 1.524 meters
const miles = herta.utils.units.length(10, 'kilometer', 'mile');  // 6.21371 miles

// Temperature conversion
const fahrenheit = herta.utils.units.temperature(100, 'celsius', 'fahrenheit');  // 212°F

// Other unit types
const kilograms = herta.utils.units.mass(16, 'ounce', 'kilogram');  // 0.45359 kg
const seconds = herta.utils.units.time(2, 'hour', 'second');  // 7200 seconds
const sqMeters = herta.utils.units.area(1, 'acre', 'squareMeter');  // 4046.86 m²
const joules = herta.utils.units.energy(100, 'calorie', 'joule');  // 418.4 joules

Advanced Fraction Operations

Full-featured fraction class with arithmetic and comparison methods:

const { Fraction } = herta.core.fraction;

// Create fractions
const frac1 = new Fraction(3, 4);  // 3/4
const frac2 = new Fraction(2, 5);  // 2/5

// Arithmetic operations
const sum = frac1.add(frac2);              // 23/20
const difference = frac1.subtract(frac2);  // 7/20
const product = frac1.multiply(frac2);     // 6/20 (simplified to 3/10)
const quotient = frac1.divide(frac2);      // 15/8

// Comparison methods
frac1.equals(new Fraction(6, 8));         // true (both simplify to 3/4)
frac1.greaterThan(frac2);                 // true
frac2.lessThan(frac1);                    // true

// Conversions
frac1.toDecimal();                        // 0.75
frac1.toString();                         // "3/4"

// Create fractions from decimals
const frac3 = Fraction.fromDecimal(0.333333);  // Approximates to 1/3

Random Number Generation

Enhanced random number generation with multiple probability distributions:

// Basic random generation
const randomInteger = herta.utils.random.randomInt(1, 100);       // Random integer between 1-100
const randomDecimal = herta.utils.random.randomFloat(0, 1);       // Random float between 0-1
const randomTrueOrFalse = herta.utils.random.randomBoolean(0.7);  // 70% chance of true

// Random selections
const array = [10, 20, 30, 40, 50];
const randomElement = herta.utils.random.randomItem(array);       // Random item from array
const shuffledArray = herta.utils.random.shuffle(array);          // Randomly shuffled array

// Advanced distributions
const normalRandom = herta.utils.random.randomNormal(50, 10);      // Normal distribution (μ=50, σ=10)
const exponentialRandom = herta.utils.random.randomExponential(2); // Exponential distribution (λ=2)
const poissonRandom = herta.utils.random.randomPoisson(5);         // Poisson distribution (λ=5)

// Other utilities
const uuid = herta.utils.random.uuid();                           // Generate UUID v4
const randomHexColor = herta.utils.random.randomColor();           // Random hex color like #FF5733

Mathematical Sequence Generators

Functions for generating mathematical sequences:

// Common number sequences
const fibSeq = herta.utils.generators.fibonacci(10);          // [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
const primeSeq = herta.utils.generators.primes(20);           // [2, 3, 5, 7, 11, 13, 17, 19]
const triangular = herta.utils.generators.triangularNumbers(7); // [1, 3, 6, 10, 15, 21, 28]
const squares = herta.utils.generators.squareNumbers(5);      // [1, 4, 9, 16, 25]

// Pattern-based sequences
const arithmetic = herta.utils.generators.arithmeticSequence(2, 3, 5);  // [2, 5, 8, 11, 14]
const geometric = herta.utils.generators.geometricSequence(1, 2, 6);    // [1, 2, 4, 8, 16, 32]

// Advanced mathematical sequences
const pascalTri = herta.utils.generators.pascalsTriangle(4);          // Triangle with 4 rows
const catalan = herta.utils.generators.catalanNumbers(6);             // [1, 1, 2, 5, 14, 42]
const collatz = herta.utils.generators.collatzSequence(12);           // [12, 6, 3, 10, 5, 16, 8, 4, 2, 1]

Graph Theory Module

The graph module has been completely rewritten with enhanced functionality and performance:

// Create a new graph (directed or undirected)
const graph = new herta.discrete.graph.Graph(true);  // directed graph

// Add vertices and weighted edges
graph.addVertex('A', { label: 'Start' });
graph.addVertex('B', { label: 'Checkpoint' });
graph.addVertex('C', { label: 'Junction' });
graph.addVertex('D', { label: 'Station' });
graph.addVertex('E', { label: 'End' });

graph.addEdge('A', 'B', { weight: 2 });
graph.addEdge('B', 'C', { weight: 1 });
graph.addEdge('C', 'D', { weight: 3 });
graph.addEdge('D', 'E', { weight: 1 });
graph.addEdge('A', 'E', { weight: 5 });

// Find shortest path using Dijkstra's algorithm
const shortestPath = graph.findShortestPath('A', 'E');
// Returns { path: ['A', 'B', 'C', 'D', 'E'], distance: 7 }

// Generate minimum spanning tree using Kruskal's algorithm
const mst = graph.minimumSpanningTreeKruskal();
// Returns a new Graph representing the MST

// Alternative: use Prim's algorithm
const mstPrim = graph.minimumSpanningTreePrim();

// Find all-pairs shortest paths using Floyd-Warshall algorithm
const distances = graph.floydWarshall();
// Returns distance matrix between all pairs of vertices

// Community detection using Louvain method
const communities = graph.detectCommunities();
// Returns array of communities (groups of vertices)

// Network analysis
const degreeCentrality = graph.degreeCentrality('C');
const betweennessCentrality = graph.betweennessCentrality();
const closenessCentrality = graph.closenessCentrality('A');

// Detect critical points in the network
const articulationPoints = graph.findArticulationPoints();
const bridges = graph.findBridges();

// For directed acyclic graphs (DAGs)
const sorted = graph.topologicalSort();

The graph module integrates seamlessly with other Herta.js features:

// Analyze network data using the graph module and statistics
const networkDiameter = graph.diameter();
const avgPathLength = graph.averagePathLength();
const densityValue = graph.density();

// Generate graph visualizations (returns data for plotting)
const layout = graph.forceDirectedLayout();

Features

Core Mathematical Foundations

• Pure Mathematics: Arithmetic, algebra, calculus, and complex analysis
• Symbolic Mathematics: Integration, differentiation, and tensor calculus
• Number Theory: Prime factorization, modular arithmetic, Diophantine equations, and quadratic residues
• Abstract Algebra: Group theory, ring theory, field theory, and Galois theory

Advanced Mathematics

• Differential Geometry: Riemannian metrics, curvature, geodesics, Lie derivatives, and parallel transport
• Category Theory: Categories, functors, natural transformations, and universal constructions
• Algebraic Geometry: Elliptic curves, toric varieties, blowups, and sheaf cohomology
• Topology: Persistent homology, manifold operations, Betti numbers, and simplicial complexes

Scientific Computing

• Numerical Methods: ODEs, PDEs, spectral methods, and stochastic differential equations
• Linear Algebra: Large-scale matrix operations and eigenvalue computations
• Optimization: Gradient descent, genetic algorithms, simulated annealing, and constrained optimization
• Signal Processing: FFT, filter design, convolution, and wavelet transforms

Physics and Dynamics

• Quantum Mechanics: State vectors, density matrices, quantum gates, and measurements
• Chaos Theory: Lyapunov exponents, fractals, bifurcation diagrams, and strange attractors
• Fluid Dynamics: Reynolds numbers, Navier-Stokes solvers, and flow analysis
• Relativistic Physics: Black hole physics, gravitational waves, and spacetime mathematics

Computer Science

• Graph Theory: Network analysis, community detection, flow algorithms, and graph coloring
• String Algorithms: Pattern matching (KMP, Boyer-Moore), suffix arrays, and sequence alignment
• Computer Vision: Image processing, edge detection, feature matching, and segmentation
• Machine Learning: Neural networks, reinforcement learning, and deep learning primitives

Applied Mathematics

• Probability & Statistics: Distributions, hypothesis testing, and Monte Carlo methods
• Financial Mathematics: Risk metrics, portfolio optimization, and trading strategies
• Cryptography: Encryption, zero-knowledge proofs, and cryptoeconomic models
• Information Theory: Entropy, coding theory, and data compression

Getting Started

Installation

Create a new Herta.js project (recommended):

# Using npm
npm install -g herta
herta erudition make project MyMathApp
cd my-math-app

# Or directly with npx
npx herta erudition make project MyMathApp
cd my-math-app

Add to an existing project:

# Using npm
npm install herta

# Using yarn
yarn add herta

Project Structure

A typical Herta.js project has the following structure:

my-math-app/
├── node_modules/
├── src/
│   ├── models/           # Data models
│   ├── services/         # Business logic
│   ├── controllers/      # Route handlers
│   └── utils/            # Helper functions
├── test/                 # Test files
├── config.js             # Application configuration
├── herta.config.js       # Herta.js framework configuration
├── package.json
└── README.md

Quick Start

// app.js
const herta = require('herta');

// Initialize the framework
const app = herta.createApplication({
  // Configuration options
  debug: process.env.NODE_ENV !== 'production',
  modules: ['algebra', 'calculus', 'statistics']
});

// Use framework components
const matrix = app.core.matrix.create([[1, 2], [3, 4]]);
const determinant = matrix.determinant();

console.log(`Matrix determinant: ${determinant}`);

// Start the application
app.start();

Herta Erudition CLI

Herta.js comes with a powerful command-line interface called "Erudition" that helps you scaffold, analyze, test, document, and understand the framework.

CLI Usage

Once Herta.js is installed globally (see Installation section above), the CLI is automatically available. Or you can use it directly with npx from within your project:

npx herta erudition <command>

Available Commands

make - Generate Code Components

Create boilerplate code for various components:

herta erudition make module MyModule           # Create a new module
herta erudition make controller DataController  # Create a new controller
herta erudition make service AnalysisService    # Create a new service
herta erudition make test QuantumModule         # Create a new test file
herta erudition make api UserManagement         # Create a complete REST API
herta erudition make rest-controller Products   # Create a REST controller
herta erudition make graphql BookSchema         # Create GraphQL schema and resolver

The make command generates proper file structure and boilerplate code with appropriate naming conventions (PascalCase, camelCase, snake_case) depending on the component type.

API Generator Options

The API generator creates fully-functional RESTful endpoints:

# Create a complete User Management API with all required files
herta erudition make api UserManagement --with-auth        # Include authentication
herta erudition make api DataAnalytics --with-validation   # Include validation
herta erudition make api SensorData --with-docs            # Include Swagger docs

REST Controller Options

# Generate specialized REST controllers
herta erudition make rest-controller Products --crud         # Basic CRUD operations
herta erudition make rest-controller Orders --with-relations # Include related resources
herta erudition make rest-controller Analytics --read-only   # Read-only endpoints

GraphQL Generator Options

# Generate GraphQL components
herta erudition make graphql BookSchema --with-resolvers    # Include resolvers
herta erudition make graphql UserSchema --with-mutations    # Include mutations
herta erudition make graphql OrderSchema --with-directives  # Include custom directives

analyze - Analyze Code Quality and Patterns

Perform static analysis on your code:

herta erudition analyze src/advanced/             # Default analysis (stats)
herta erudition analyze --complexity src/core/     # Analyze code complexity
herta erudition analyze --dependencies src/utils/  # Analyze module dependencies
herta erudition analyze --stats src/algorithms/    # Analyze code statistics

The analyze command examines your code and provides insights on complexity metrics, dependency graphs, and code statistics to help improve code quality.

doc - Generate Documentation

Automatically generate documentation from JSDoc comments:

herta erudition doc matrix                  # Generate docs for a specific component
herta erudition doc --all                    # Generate docs for all components
herta erudition doc --format html            # Generate in HTML format (default is markdown)
herta erudition doc --output custom-docs/    # Specify output directory

The doc command extracts JSDoc comments from your code and generates comprehensive documentation in markdown or HTML format.

test - Run Tests with Detailed Reporting

Execute tests with comprehensive reports:

herta erudition test core/matrix           # Run tests matching a specific pattern
herta erudition test --unit                 # Run only unit tests
herta erudition test --integration          # Run only integration tests
herta erudition test --coverage             # Generate test coverage report
herta erudition test --verbose              # Show detailed test output

The test command runs tests and provides detailed summaries of test results, including pass/fail counts, execution time, and coverage metrics.

explain - Get Plain English Explanations

Receive explanations of framework concepts in plain language:

herta erudition explain config                # Explain configuration options
herta erudition explain algorithms            # Explain general algorithms concepts
herta erudition explain eigenvalues           # Explain mathematical concepts
herta erudition explain --detailed quantum    # Get more detailed explanation

The explain command provides clear, plain-English explanations of Herta.js concepts, configuration options, and advanced mathematical and scientific topics. It's especially useful for understanding complex modules like Quantum Mechanics, Chaos Theory, and Number Theory.

api - Create and Manage APIs

Build and manage APIs powered by Herta.js:

herta erudition api create --name math-api --type rest     # Create a new REST API
herta erudition api add-endpoint calculus --module calculus  # Add module endpoint
herta erudition api generate-docs --format openapi          # Generate API docs
herta erudition api test --endpoint graph-analysis          # Test an endpoint
herta erudition api deploy --provider aws                   # Deploy the API

The api command helps you build APIs that leverage Herta.js's mathematical capabilities, enabling powerful computation services.

web - Web Development Tools

Create web applications and components with Herta.js:

herta erudition web create-app --name matrix-explorer    # Create a web app
herta erudition web add-component matrix-visualizer       # Add a component
herta erudition web build --optimize                      # Build for production
herta erudition web serve --port 3000                     # Start dev server
herta erudition web generate-ssr --framework next         # Add SSR support

The web command facilitates creating rich web applications that utilize Herta.js's mathematical modules for visualization, computation, and analysis.

Additional CLI Functions

Global Options

All Erudition commands support these common options:

--help, -h        # Display command-specific help
--version, -v     # Display Erudition version information
--quiet, -q       # Reduce output verbosity
--json            # Output results in JSON format
--config <file>   # Use custom configuration file

Sub-command Details

Make Command Templates

The make command supports various templates for different component types:

herta erudition make module --functional MyModule    # Create a functional module
herta erudition make module --class MyModule         # Create a class-based module
herta erudition make test --unit MyTest              # Create a unit test
herta erudition make test --integration MyTest       # Create an integration test
herta erudition make test --performance MyTest       # Create a performance test

Analyze Command Options

The analyze command provides specialized analysis modes:

herta erudition analyze --complexity-threshold 10   # Flag methods over threshold
herta erudition analyze --visualize                  # Generate visual reports
herta erudition analyze --trends                     # Compare to historical data
herta erudition analyze --unused                     # Find unused code

Documentation Format Options

The doc command supports different output formats and templates:

herta erudition doc --format html --theme dark      # Dark themed HTML docs
herta erudition doc --format markdown --toc          # Markdown with table of contents
herta erudition doc --template custom-template.ejs   # Use custom template
herta erudition doc --diagrams                       # Include UML diagrams

Test Reporting Options

The test command has additional reporting capabilities:

herta erudition test --reporters mocha,junit        # Multiple report formats
herta erudition test --watch                         # Watch mode for auto-reruns
herta erudition test --parallel 4                    # Run tests in parallel
herta erudition test --snapshot                      # Run snapshot tests
herta erudition test --perf                          # Include performance benchmarks

Explain Command Topics

The explain command covers numerous specialized topics:

herta erudition explain architecture                # System architecture overview
herta erudition explain modules                      # Module system details
herta erudition explain patterns                     # Design patterns used
herta erudition explain best-practices               # Coding best practices
herta erudition explain api-stability                # API stability guarantees

Module-Specific Commands

Erudition includes specialized commands for working with Herta.js's advanced mathematical modules:

Quantum Mechanics Helper

# Visualize quantum states and gates
herta erudition quantum visualize --state "[0.7071, 0.7071]" --output quantum-state.png
herta erudition quantum circuit --gates "H,X,CNOT" --qubits 2 --output circuit.svg

# Generate common quantum circuits
herta erudition quantum generate --type bell-state --qubits 2
herta erudition quantum generate --type qft --qubits 4

# Analyze quantum properties
herta erudition quantum analyze-entanglement --state "[0.7071, 0, 0, 0.7071]"

Graph Theory Utilities

# Generate and analyze graphs
herta erudition graph generate --type erdos-renyi --nodes 20 --probability 0.3
herta erudition graph generate --type scale-free --nodes 50 --output graph.json

# Analyze graph properties
herta erudition graph analyze --file graph.json --metrics "centrality,connectivity,communities"
herta erudition graph visualize --file graph.json --layout force-directed --output graph.svg

Optimization Playground

# Test optimization algorithms on standard problems
herta erudition optimize --algorithm gradient-descent --problem rosenbrock
herta erudition optimize --algorithm genetic --problem traveling-salesman --cities 15

# Benchmark optimization performance
herta erudition optimize benchmark --algorithms "gradient-descent,genetic,simulated-annealing" --problem sphere

Data Science Toolkit

# Statistical analysis and probability tools
herta erudition stats analyze --data data.csv --tests "normality,correlation,t-test"
herta erudition stats generate --distribution normal --params "0,1" --samples 1000 --output samples.csv

# Monte Carlo simulations
herta erudition stats monte-carlo --simulation coin-toss --trials 10000 --probability 0.5

Number Theory Explorer

# Prime number utilities
herta erudition number-theory primes --range "1,1000" --output primes.json
herta erudition number-theory factorize --number 12345678 --algorithm "pollard-rho"

# Explore number properties
herta erudition number-theory properties --number 42 --checks "prime,perfect,abundant,deficient"

# Modular arithmetic operations
herta erudition number-theory solve-congruence --equation "3x ≡ 5 (mod 7)"
herta erudition number-theory chinese-remainder --remainders "2,3,2" --moduli "3,5,7"

Fluid Dynamics Simulator

# Calculate fluid properties
herta erudition fluid reynolds --velocity 2 --diameter 0.05 --fluid water
herta erudition fluid pressure-drop --length 10 --diameter 0.05 --velocity 2 --roughness 0.0001

# Run fluid simulations
herta erudition fluid simulate --model "1d-advection" --domain-length 10 --nodes 100 --time-steps 1000
herta erudition fluid simulate --model "2d-incompressible" --resolution "100x100" --time 10 --output flow.mp4

# Visualize fluid results
herta erudition fluid visualize --data simulation.json --type "velocity-field" --output velocity.png
herta erudition fluid visualize --data simulation.json --type "pressure-contour" --output pressure.png

Computer Vision Toolkit

# Image processing operations
herta erudition vision process --input image.jpg --operations "grayscale,gaussian-blur:5,canny:30:100" --output processed.jpg
herta erudition vision histogram --input image.jpg --channel rgb --equalize --output histogram.png

# Feature detection and analysis
herta erudition vision detect-features --input image.jpg --algorithm "fast" --threshold 20 --output features.json
herta erudition vision match-features --image1 scene.jpg --image2 object.jpg --output matches.jpg

# Advanced computer vision operations
herta erudition vision segment --input image.jpg --algorithm "kmeans" --clusters 5 --output segmented.jpg
herta erudition vision detect-objects --input image.jpg --model "herta-detector" --confidence 0.7 --output detected.jpg

String Algorithms Workshop

# Pattern matching
herta erudition strings search --text file.txt --pattern "important text" --algorithm "kmp" --output matches.json
herta erudition strings multi-pattern --text file.txt --patterns patterns.txt --algorithm "aho-corasick" --output results.json

# String similarity and distance
herta erudition strings compare --string1 "kitten" --string2 "sitting" --metrics "levenshtein,lcs,similarity"
herta erudition strings cluster --input strings.txt --method "edit-distance" --threshold 3 --output clusters.json

# Suffix structures and compression
herta erudition strings build-suffix-array --text "banana" --output suffix-array.json
herta erudition strings compress --input file.txt --method "run-length" --output compressed.txt

Chaos Theory Explorer

# Generate fractal visualizations
herta erudition chaos fractal --type "mandelbrot" --bounds "-2:1:-1:1" --resolution "1000x1000" --output mandelbrot.png
herta erudition chaos julia --c "-0.7:0.27" --bounds "-2:2:-2:2" --resolution "1000x1000" --output julia.png

# Simulate chaotic systems
herta erudition chaos simulate --system "lorenz" --params "10,28,8/3" --duration 100 --output lorenz.json
herta erudition chaos simulate --system "logistic-map" --param 3.9 --iterations 1000 --output bifurcation.json

# Analyze chaotic properties
herta erudition chaos lyapunov --data timeseries.json --dimension 3 --delay 2 --output lyapunov.json
herta erudition chaos recurrence-plot --data timeseries.json --threshold 0.1 --output recurrence.png

Advanced Mathematics Explorer

# Differential Geometry operations
herta erudition differential-geometry manifold --dimension 2 --curvature constant --output sphere.json
herta erudition differential-geometry geodesic --manifold sphere.json --start "0,0" --end "pi/2,pi/2" --output geodesic.json

# Category Theory visualization
herta erudition category diagram --objects "A,B,C,D" --morphisms "f:A->B,g:B->C,h:A->D,i:D->C" --output diagram.svg
herta erudition category functor --source diagram1.json --target diagram2.json --output functor.json

# Complex Analysis tools
herta erudition complex-analysis contour-plot --function "z^2" --region "-2:2:-2:2" --resolution 500 --output contour.png
herta erudition complex-analysis laurent-series --function "1/(z^2-1)" --point 0 --terms 10

Cross-Domain Integration Tools

# Machine Learning with Herta modules
herta erudition ml train --algorithm "neural-network" --data dataset.csv --features "x,y,z" --target "output" --model-save model.json
herta erudition ml optimize-hyperparams --algorithm "svm" --data dataset.csv --param-grid params.json --output best-params.json

# Scientific computing pipelines
herta erudition pipeline create --name "data-analysis" --steps "load:data.csv,clean,normalize,analyze:correlation,visualize:heatmap" --output pipeline.json
herta erudition pipeline run --file pipeline.json --params params.json --output results/

# Interactive explorations
herta erudition explore --module quantum --interactive  # Opens an interactive jupyter-like notebook
herta erudition explore --module graph --dataset social-network.json --interactive

Developer Utilities

# Code quality and maintenance
herta erudition dev audit-dependencies --security --outdated
herta erudition dev lint --fix --style standard

# Performance profiling
herta erudition dev profile --function "matrix.multiply" --size "1000x1000" --iterations 10 --output profile.json
herta erudition dev benchmark --suite "matrix-operations" --compare-with v1.0.0 --output benchmark.html

# Versioning and publishing
herta erudition dev prepare-release --bump minor --changelog --tag
herta erudition dev publish --dry-run  # Verify everything before actual publish

API & Web Development

# API integration with mathematical modules
herta erudition api create --type rest --name mathematical-api
herta erudition api add-endpoint optimization --methods "POST,GET" --module optimization
herta erudition api add-endpoint quantum --methods "POST" --module quantum
herta erudition api add-endpoint graph-analysis --methods "POST,GET" --module graph

# Generate client libraries for mathematical APIs
herta erudition api client-gen --language javascript --output ./clients/js
herta erudition api client-gen --language python --output ./clients/python
herta erudition api client-gen --language typescript --with-types --output ./clients/ts

# Web component generation for visualization
herta erudition web component graph-visualizer --module graph --output ./components
herta erudition web component matrix-editor --module matrix --output ./components
herta erudition web component fractal-explorer --module chaos --output ./components

# Create interactive web dashboards for mathematical modules
herta erudition web dashboard optimization --algorithms "gradient-descent,genetic" --output ./dashboards
herta erudition web dashboard quantum --features "state-visualization,gate-simulation" --output ./dashboards
herta erudition web dashboard statistics --features "distribution-explorer,regression" --output ./dashboards

Extending Erudition

You can create custom Erudition plugins by adding files to the commands/erudition/plugins directory:

// commands/erudition/plugins/myPlugin.js
module.exports = function(args) {
  // Plugin implementation
  console.log('My custom plugin is running!');
};

Then use your plugin with:

herta erudition myPlugin [args]

Basic Usage

const herta = require('herta');

// Basic arithmetic and constants
console.log(herta.round(herta.e, 3));                // 2.718
console.log(herta.atan2(3, -3) / herta.pi);          // 0.75
console.log(herta.log(10000, 10));                   // 4
console.log(herta.sqrt(-4).toString());              // 2i

// Matrix operations
const matrix = [[-1, 2], [3, 1]];
console.log(herta.pow(matrix, 2));                   // [[7, 0], [0, 7]]
console.log(herta.det(matrix));                      // -7

// Expression evaluation
console.log(herta.evaluate('1.2 * (2 + 4.5)'));     // 7.8
console.log(herta.evaluate('12.7 cm to inch'));     // 5 inch
console.log(herta.evaluate('sin(45 deg) ^ 2'));     // 0.5
console.log(herta.evaluate('9 / 3 + 2i'));          // 3 + 2i

// Chaining operations
const result = herta.chain(3)
    .add(4)
    .multiply(2)
    .done();                                         // 14
console.log(result);

Advanced Features

Symbolic Integration

// Advanced symbolic integration
const integral = herta.integrate('sin(x^2)', 'x');
console.log(integral);  // Complex symbolic result

// Definite integration
const definiteIntegral = herta.integrate('sin(x^2)', 'x', 0, 1);
console.log(definiteIntegral);  // Numerical approximation

Differential Equations

// Solving differential equations
const solution = herta.solveDifferentialEquation('dy/dx = y^2 + x', 'y', 'x');
console.log(solution);  // Symbolic solution

// Numerical solution with initial conditions
const numericalSolution = herta.solveODE('y\'=y^2+x', 'y', 'x', 0, 1, [0, 5]);
console.log(numericalSolution);  // Array of solution points

Large-Scale Linear Algebra

// Create a large sparse matrix
const sparseMatrix = herta.createSparseMatrix(1000, 1000);

// Efficient eigenvalue computation
const eigenvalues = herta.eigenvalues(sparseMatrix);
console.log(eigenvalues.length);  // 1000

Tensor Calculus

// Create tensors for General Relativity calculations
const metric = herta.createMetricTensor(4, 4);  // 4D spacetime metric

// Compute Christoffel symbols
const christoffel = herta.christoffelSymbols(metric);
console.log(christoffel);  // 3D tensor

// Compute Riemann curvature tensor
const riemann = herta.riemannTensor(metric);
console.log(riemann);  // 4D tensor

Automatic Differentiation

// Define a function for automatic differentiation
const f = herta.autodiff.createFunction('x^2 + sin(y) + z*x', ['x', 'y', 'z']);

// Compute gradient at a point
const gradient = f.gradient([2, Math.PI, 3]);
console.log(gradient);  // [7, 0, 2]

// Compute Hessian matrix
const hessian = f.hessian([2, Math.PI, 3]);
console.log(hessian);  // [[2, 0, 1], [0, -1, 0], [1, 0, 0]]

Module Examples

Advanced Algebra

// Working with group theory
const elements = [0, 1, 2, 3, 4];  // Z5 elements
const operation = (a, b) => (a + b) % 5;  // Addition modulo 5

// Create a cyclic group
const Z5 = herta.advancedAlgebra.createCyclicGroup(5);

// Group operations
console.log(Z5.apply(2, 3));  // 0 (2+3 mod 5)
console.log(Z5.inverse(3));   // 2 (since 3+2=0 mod 5)
console.log(Z5.orderOf(2));   // 5

// Create a modular ring
const modRing = herta.advancedAlgebra.createModularRing(7);
console.log(modRing.power(3, -1));  // 5 (modular inverse of 3 mod 7)

// Polynomial operations
const F = herta.advancedAlgebra.createModularRing(5); // Field Z/5Z
const poly1 = herta.advancedAlgebra.createPolynomial([1, 0, 1], F); // x^2 + 1
const poly2 = herta.advancedAlgebra.createPolynomial([2, 1], F);    // x + 2
const product = poly1.multiply(poly2); // (x^2 + 1)(x + 2) = x^3 + 2x^2 + x + 2

Reinforcement Learning

// Create a GridWorld environment
const gridWorld = herta.reinforcementLearning.GridWorld(5, 5, {
  start: { x: 0, y: 0 },
  terminals: [
    { x: 4, y: 4, reward: 1 },  // Goal state with positive reward
    { x: 4, y: 0, reward: -1 }, // Negative reward state
  ],
  obstacles: [
    { x: 2, y: 1 }, { x: 2, y: 2 }, { x: 2, y: 3 }, // Wall
  ],
  defaultReward: -0.04, // Small penalty for each step
});

// Create a Q-learning agent
const agent = herta.reinforcementLearning.QLearning(gridWorld, {
  learningRate: 0.1,
  discountFactor: 0.9,
  epsilon: 0.2,
  epsilonDecay: 0.995,
});

// Train the agent
const trainingResults = agent.train(1000); // Train for 1000 episodes

// Get the learned policy
const policy = agent.getPolicy();
console.log(policy); // Maps states to optimal actions

// Multi-armed bandit example
const banditRewards = (armIndex) => {
  const means = [0.3, 0.5, 0.7, 0.9]; // Mean rewards for 4 arms
  return means[armIndex] + 0.1 * (Math.random() - 0.5); // Add noise
};

// Create a UCB bandit solver
const bandit = herta.reinforcementLearning.ucbBandit(banditRewards, 4);
const results = bandit.run(1000); // Run for 1000 steps

Text Analysis

// Basic text tokenization and processing
const text = "Natural language processing is fascinating. It has many applications in AI."
const tokens = herta.textAnalysis.tokenize(text);
console.log(tokens); // ['Natural', 'language', 'processing', ...]

// Sentence tokenization
const sentences = herta.textAnalysis.tokenize(text, 'sentence');
console.log(sentences); // ['Natural language processing is fascinating.', ...]

// TF-IDF calculation
const documents = [
  "This is a document about dogs.",
  "This document discusses cats and their behavior.",
  "Dogs and cats are popular pets."
];

const idf = herta.textAnalysis.inverseDocumentFrequency(documents);
const tfidf = herta.textAnalysis.tfidf(documents[0], idf);
console.log(tfidf); // { 'document': 0.1053..., 'dogs': 0.4054... }

// Sentiment analysis
const sentiment = herta.textAnalysis.analyzeSentiment("I love this product, it's amazing!");
console.log(sentiment); // { positive: 0.2, negative: 0, score: 0.2, magnitude: 0.2 }

// Create a text classifier
const classifier = herta.textAnalysis.createNaiveBayesClassifier();

// Train the classifier
classifier.train("This product is excellent", "positive");
classifier.train("I'm very happy with my purchase", "positive");
classifier.train("This doesn't work at all", "negative");
classifier.train("Poor quality and expensive", "negative");

// Classify new text
const result = classifier.classify("I'm happy with this product");
console.log(result.category); // 'positive'

Cryptoeconomics

// Create a tokenomics model for a cryptocurrency
const tokenModel = herta.cryptoeconomics.createTokenomicsModel({
  initialSupply: 100000000, // 100 million tokens
  inflationRate: 0.02, // 2% annual inflation
  stakingRewardRate: 0.05, // 5% annual staking rewards
  burnRate: 0.001, // 0.1% tokens burned per year
  distributionRatios: { 
    team: 0.15, 
    investors: 0.15, 
    community: 0.6, 
    reserves: 0.1 
  }
});

// Project supply after 10 years
const supplyProjection = tokenModel.projectSupply(10);
console.log(supplyProjection.finalSupply); // Future token supply

// Calculate staking rewards
const stakingRewards = tokenModel.calculateStakingRewards(1000000, 5);
console.log(stakingRewards); // Returns staking rewards over 5 years

// Create a bonding curve pricing model
const curve = herta.cryptoeconomics.createBondingCurve({
  curveType: 'linear',
  slope: 0.1,
  initialPrice: 1
});

// Calculate token price impact
const impact = curve.simulatePriceImpact(1000, true);
console.log(impact); // Shows price before and after 1000-token purchase

Zero Knowledge Proofs

// Create a range proof system
const rangeProver = herta.zeroKnowledgeProofs.rangeProof();

// Generate a proof that a value is within range [0, 1000]
const proof = rangeProver.prove(750, 1000);

// Verify the proof
const isValid = rangeProver.verify(proof, 1000);
console.log(isValid); // true - confirms value is in range without revealing it

// Set membership proof
const allowlist = ["address1", "address2", "address3"];
const membershipProof = herta.zeroKnowledgeProofs.setMembership(allowlist);

// Prove membership without revealing which element
const addressProof = membershipProof.proveInSet("address2");

// Verify membership
const isMember = membershipProof.verifyInSet(addressProof);
console.log(isMember); // true - confirms address is in the allowlist

Language Model Math

// Calculate self-attention scores in a transformer model
const queries = [[1, 0, 1], [0, 1, 0]]; // Query vectors
const keys = [[1, 1, 0], [0, 1, 1]];    // Key vectors

// Calculate attention weights
const attentionWeights = herta.languageModelMath.selfAttention(queries, keys);
console.log(attentionWeights); // Attention distribution

// Apply attention to values
const values = [[1, 2], [3, 4]]; // Value vectors
const attentionOutput = herta.languageModelMath.applyAttention(attentionWeights, values);

// Apply position encoding to embeddings
const embeddings = [[1, 2, 3, 4], [5, 6, 7, 8]];
const positionEncoded = herta.languageModelMath.positionEncoding(embeddings);

// Sample from a distribution using nucleus sampling (top-p)
const logits = [2.5, 1.2, 0.8, 0.3, -0.5]; // Raw model outputs
const sampledToken = herta.languageModelMath.nucleusSampling(logits, 0.9);
console.log(sampledToken); // Token ID selected by sampling

Neural Networks

// Create a feedforward neural network
const model = herta.neuralNetworks.feedForward(
  [784, 128, 64, 10], // Layer sizes (MNIST classification example)
  { activation: 'relu', outputActivation: 'softmax' }
);

// Forward pass
const input = [/* flattened image data */];
const prediction = model.forward([input]);

// Create a convolutional layer
const convLayer = herta.neuralNetworks.convLayer2d({
  inputChannels: 1,
  outputChannels: 16,
  kernelSize: 3,
  stride: 1,
  padding: 1,
  activation: 'relu'
});

// Create an LSTM layer
const lstm = herta.neuralNetworks.lstmLayer(100, 64, { returnSequences: true });

// Process a sequence through the LSTM
const sequence = [/* sequence of embedding vectors */];
const { output, hidden, cell } = lstm.forward([sequence]);

Relativistic Astrophysics

// Calculate the Schwarzschild radius of a black hole
const solarMass = 1.989e30; // kg
const blackHoleMass = 10 * solarMass; // 10 solar masses
const radius = herta.relativisticAstrophysics.schwarzschildRadius(blackHoleMass);
console.log(radius); // ~29.5 km

// Calculate gravitational time dilation
const distance = radius * 3; // 3 times the Schwarzschild radius
const timeDilation = herta.relativisticAstrophysics.gravitationalTimeDilation(
  blackHoleMass, 
  distance
);
console.log(timeDilation); // Time runs slower by this factor

// Calculate properties of a binary black hole merger
const mass1 = 30 * solarMass;
const mass2 = 25 * solarMass;
const separation = 1000000; // 1000 km

// Gravitational wave frequency
const freq = herta.relativisticAstrophysics.gravitationalWaveFrequency(
  mass1, 
  mass2, 
  separation
);

// Time until merger
const timeToMerge = herta.relativisticAstrophysics.timeToMerger(
  mass1,
  mass2,
  separation
);
console.log(timeToMerge); // Time in seconds until the black holes merge

Technical Analysis

// Calculate Simple Moving Average
const prices = [10.5, 11.2, 10.8, 11.5, 11.7, 12.1, 12.5, 12.0, 11.8, 12.2];
const sma = herta.technicalAnalysis.sma(prices, 5);
console.log(sma); // [11.14, 11.46, 11.72, 11.96, 12.12, 12.1]

// Calculate Relative Strength Index
const priceHistory = [45.34, 45.67, 46.12, 46.82, 46.45, 46.89, 47.23, 47.56, 47.32, 47.65, 48.12, 48.45, 48.23, 47.98];
const rsi = herta.technicalAnalysis.rsi(priceHistory, 14);
console.log(rsi); // Returns RSI values

// Calculate Bollinger Bands
const priceSeries = [26.10, 25.78, 26.01, 26.35, 26.57, 26.42, 26.35, 26.70, 26.89, 26.95, 27.12, 27.42, 27.67, 27.85];
const { upperBand, middleBand, lowerBand } = herta.technicalAnalysis.bollingerBands(priceSeries, 10, 2);
console.log('Upper Band:', upperBand);
console.log('Middle Band:', middleBand);
console.log('Lower Band:', lowerBand);

// Detect support and resistance with Pivot Points
const pivots = herta.technicalAnalysis.pivotPoints(27.85, 26.35, 27.45);
console.log('Pivot:', pivots.pivot);
console.log('Resistance 1:', pivots.resistance.r1);
console.log('Support 1:', pivots.support.s1);

// Calculate MACD
const { macdLine, signalLine, histogram } = herta.technicalAnalysis.macd(priceSeries);
console.log('MACD line:', macdLine);
console.log('Signal line:', signalLine);
console.log('Histogram:', histogram);

Trading Strategies

// Calculate optimal position size using Kelly Criterion
const winRate = 0.55; // 55% of trades are profitable
const winLossRatio = 1.5; // Average win is 1.5x the average loss
const capital = 10000; // $10,000 trading capital
const positionSize = herta.tradingStrategies.kellyPositionSize(winRate, winLossRatio, capital);
console.log('Optimal position size:', positionSize); // $1,666.67

// Calculate stop loss and take profit based on volatility
const priceData = [
  { high: 150.5, low: 149.2, close: 150.1 },
  { high: 151.2, low: 149.8, close: 151.0 },
  // ... more price history
];

const entryPrice = 151.5;
const levels = herta.tradingStrategies.volatilityBasedLevels(entryPrice, priceData, 2, 3);
console.log('Entry price:', levels.entryPrice);
console.log('Stop loss:', levels.stopLoss);
console.log('Take profit:', levels.takeProfit);

// Implement a trend following strategy
const btcPrices = [
  // Daily price data for Bitcoin
];

const signals = herta.tradingStrategies.trendFollowing(btcPrices, {
  shortPeriod: 10,
  longPeriod: 30,
  stopLossPercent: 0.05
});

console.log('Strategy signals:', signals);

// Evaluate trading performance
const trades = [
  { entryPrice: 10000, exitPrice: 10500, type: 'long', size: 0.1 },
  { entryPrice: 10500, exitPrice: 10300, type: 'short', size: 0.1 },
  { entryPrice: 10300, exitPrice: 11000, type: 'long', size: 0.15 }
];

const performance = herta.tradingStrategies.evaluatePerformance(trades, 5000);
console.log('Total return:', performance.totalReturns);
console.log('Sharpe ratio:', performance.sharpeRatio);
console.log('Win rate:', performance.winRate);

Risk Management

// Calculate Value at Risk (VaR) using historical simulation
const portfolioValue = 1000000; // $1 million portfolio
const returns = [-0.02, -0.015, -0.01, -0.005, 0, 0.005, 0.01, 0.015, 0.02]; // Historical returns
const var95 = herta.riskManagement.historicalVaR(returns, 0.95, portfolioValue);
console.log('95% VaR:', var95); // Amount you could lose with 95% confidence

// Calculate parametric Value at Risk
const mean = 0.001; // Mean daily return
const stdDev = 0.015; // Standard deviation of returns
const parametricVaR = herta.riskManagement.parametricVaR(mean, stdDev, 0.99, portfolioValue);
console.log('99% Parametric VaR:', parametricVaR);

// Calculate portfolio volatility
const weights = [0.4, 0.3, 0.2, 0.1]; // Asset allocation weights
const covarianceMatrix = [
  [0.04, 0.02, 0.01, 0.01],
  [0.02, 0.09, 0.01, 0.02],
  [0.01, 0.01, 0.16, 0.01],
  [0.01, 0.02, 0.01, 0.04]
];

const volatility = herta.riskManagement.portfolioVolatility(weights, covarianceMatrix);
console.log('Portfolio Volatility:', volatility);

// Calculate Sharpe Ratio
const expectedReturn = 0.12; // 12% annual expected return
const annualVolatility = 0.18; // 18% annual volatility
const riskFreeRate = 0.03; // 3% risk-free rate
const sharpeRatio = herta.riskManagement.sharpeRatio(expectedReturn, annualVolatility, riskFreeRate);
console.log('Sharpe Ratio:', sharpeRatio);

// Perform a stress test
const assetValues = [400000, 300000, 200000, 100000]; // Values of assets in portfolio
const stressShocks = [0.25, 0.15, 0.35, 0.10]; // Stress scenario percentage losses
const stressTest = herta.riskManagement.stressTest(assetValues, stressShocks);
console.log('Stressed Portfolio Value:', stressTest.stressedValue);
console.log('Total Impact:', stressTest.impactAmount);
console.log('Impact Percentage:', stressTest.impactPercent);

Tabular Analysis

// Analyze a dataset of sales transactions
const salesData = [
  { date: '2023-01-01', product: 'A', region: 'North', sales: 1200, units: 5, returns: 0 },
  { date: '2023-01-01', product: 'B', region: 'North', sales: 900, units: 3, returns: 1 },
  { date: '2023-01-02', product: 'A', region: 'South', sales: 1500, units: 6, returns: 0 },
  { date: '2023-01-02', product: 'B', region: 'East', sales: 1100, units: 4, returns: 0 },
  { date: '2023-01-03', product: 'C', region: 'West', sales: 1800, units: 7, returns: 2 },
  { date: '2023-01-03', product: 'A', region: 'East', sales: 2000, units: 8, returns: 1 },
  { date: '2023-01-04', product: 'B', region: 'West', sales: 1300, units: 5, returns: 0 },
  { date: '2023-01-04', product: 'C', region: 'North', sales: 950, units: 3, returns: 0 },
];

// Generate summary statistics for numeric columns
const summary = herta.tabularAnalysis.summarize(salesData);
console.log('Sales Summary:', summary.sales);
console.log('Units Summary:', summary.units);

// Calculate correlation matrix
const { correlationMatrix } = herta.tabularAnalysis.correlationMatrix(salesData);
console.log('Correlation between sales and units:', correlationMatrix.sales.units);

// Detect outliers
const outliers = herta.tabularAnalysis.detectOutliers(salesData, ['sales', 'units']);
console.log('Sales outliers:', outliers.sales);

// Group data by product and calculate aggregates
const salesByProduct = herta.tabularAnalysis.groupBy(salesData, 'product', {
  sales: 'sum',
  units: 'sum',
  avgSale: (values) => values.reduce((sum, val) => sum + val, 0) / values.length
});
console.log('Sales by Product:', salesByProduct);

// Normalize the sales data
const { data: normalizedData } = herta.tabularAnalysis.normalize(salesData, ['sales', 'units']);
console.log('Normalized Data:', normalizedData);

// One-hot encode categorical variables
const encodedData = herta.tabularAnalysis.oneHotEncode(salesData, ['region', 'product']);
console.log('Encoded Data (first row):', encodedData[0]);

Herta Erudition CLI

Herta.js comes with a powerful CLI tool called "Erudition" that helps you scaffold, analyze, test, document, and understand the framework.

Installation

The CLI is automatically available when you install herta:

npm install herta

After installation, you can access the CLI globally:

herta erudition <command> [options]

Available Commands

Scaffolding with make

# Create a new module
herta erudition make module QuantumPhysics

# Create a new controller
herta erudition make controller UserController

# Create a new service
herta erudition make service DataService

# Create a new test
herta erudition make test Vector

Analyzing Code with analyze

# General analysis
herta erudition analyze

# Specific analysis types
herta erudition analyze --complexity
herta erudition analyze --dependencies

Generating Documentation with doc

# Document a specific module
herta erudition doc matrix

# Generate all documentation
herta erudition doc --all

Running Tests with test

# Run all tests
herta erudition test

# Run with coverage
herta erudition test --coverage

# Run specific tests
herta erudition test matrix.test.js

Understanding the Framework with explain

# Explain configuration options
herta erudition explain config

# Explain algorithms used
herta erudition explain algorithms

# Explain specific modules
herta erudition explain module:neuralNetworks

Project Structure

herta/
├── bin/              # CLI tools
├── commands/         # CLI command implementations
│   └── erudition/    # Erudition CLI commands
├── src/
│   ├── core/         # Core mathematical functionality
│   ├── advanced/     # Advanced mathematical capabilities
│   ├── utils/        # Utility functions
│   └── index.js      # Main entry point
├── test/             # Test suite
└── examples/         # Usage examples

Advanced Mathematical Capabilities

Herta.js offers a comprehensive suite of mathematical capabilities that set it apart from other frameworks:

1. Advanced Symbolic Integration: Solve complex integrals with sophisticated techniques
2. Differential Equation Solving: Analytical and numerical solutions for ODEs and PDEs
3. Large-Scale Linear Algebra: Optimized for high-performance numerical computing
4. Symbolic Tensor Calculus: Support for tensor operations used in physics
5. Automatic Differentiation: Compute gradients for machine learning applications
6. Numerical Methods: Advanced root-finding, optimization, and interpolation algorithms
7. Statistical Analysis: Comprehensive statistical functions for data analysis
8. Quantum Computing: Full quantum circuit simulation and quantum algorithm implementation
9. Symbolic Computation: Powerful symbolic mathematics for algebraic manipulation
10. Scientific Mode: Specialized functions for research with higher precision
11. Optimization: Gradient descent, Newton's method, linear programming, and particle swarm optimization
12. Geometry: Computational geometry, vector operations, coordinate transformations, and convex hull algorithms
13. Signal Processing: FFT implementations, window functions, filter design, convolution, and wavelet transforms
14. Machine Learning: Activation functions, loss functions, dimensionality reduction, and clustering algorithms
15. Topology: Topological space operations, homology groups, and simplicial complex generation
16. Financial Mathematics: Option pricing models, portfolio metrics, bond calculations, and risk analysis
17. Discrete Mathematics: Combinatorial functions, permutations, combinations, and recurrence relation solvers
18. Dynamical Systems: Iteration, bifurcation diagrams, fixed points, and fractal dimensions
19. Group Theory: Group operations, symmetry groups, representation theory, and algebraic structures
20. Information Theory: Shannon entropy, mutual information, channel capacity, and coding algorithms
21. Game Theory: Nash equilibria, Shapley values, evolutionary dynamics, and cooperative game solutions
22. Algebraic Geometry: Polynomial systems, varieties, Gröbner bases, and resultant calculations
23. Differential Geometry: Riemannian metrics, curvature tensors, Christoffel symbols, geodesic equations, Lie derivatives, holonomy calculations, and frame fields
24. Category Theory: Categories, functors, natural transformations, adjunctions, universal properties, limits, and colimits
25. Complex Analysis: Analytic functions, contour integration, residue theory, conformal mappings, and Laurent series expansion
26. Group Theory: Group operations, representations, automorphisms, crystallographic wallpaper groups, and quaternion groups
27. Algebraic Geometry: Varieties, Gröbner bases, elliptic curves with group law, toric varieties, blowups, and sheaf cohomology
28. Numerical Methods: Adaptive integration, Gaussian quadrature, Monte Carlo methods, finite difference PDE solvers, spectral methods, and stochastic differential equations
29. Number Theory: Advanced prime factorization with Pollard's rho, Diophantine equation solvers, Chinese remainder theorem, continued fractions, quadratic residues, Euler's totient function, and Möbius function
30. Graph Theory: Directed and undirected graph operations, minimum spanning trees, shortest paths, centrality measures, community detection, maximum flow, graph coloring, cycle detection, and topological sorting
31. Topology: Topological spaces, equivalence relations, homology groups, persistent homology, Betti numbers, manifold operations, and triangulations
32. Group Theory: Abstract groups, group homomorphisms, Cayley tables, matrix representations, crystallographic wallpaper groups, automorphisms, and quaternion groups
33. Optimization: Gradient descent, newton methods, genetic algorithms, simulated annealing, differential evolution, particle swarm optimization, and constrained optimization with COBYLA
34. Chaos Theory: Lyapunov exponents, fractal dimensions, bifurcation diagrams, Mandelbrot and Julia sets, Lorenz attractor, and recurrence plots
35. Quantum Mechanics: Quantum state vectors, density matrices, von Neumann entropy, common quantum gates, and tensor products
36. Fluid Dynamics: Reynolds number calculations, Navier-Stokes solvers, pressure drop, head loss, advection-diffusion equations, and Mach number
37. String Algorithms: Knuth-Morris-Pratt, Boyer-Moore, and Rabin-Karp pattern matching, suffix arrays, Levenshtein distance, and LCS
38. Probability Theory: Binomial, Poisson, normal, and exponential distributions, statistical tests, Monte Carlo integration, and regression analysis
39. Computer Vision: Image processing filters, edge detection, feature extraction, keypoint matching, Hough transforms, and image segmentation
40. Cryptography: Secure encryption, hashing, and cryptographic primitives
41. Category Theory: Category operations and functorial mappings
42. Combinatorics: Permutation, combination, and advanced counting techniques
43. Time Series Analysis: Tools for analyzing temporal data patterns
44. Chaos Theory: Fractal generation and chaotic system analysis

Scientific Computing Features

Herta.js provides specialized features for scientific computing:

Scientific Mode

// Create a scientific instance with higher precision
const scientific = herta.scientific();

// Use Fourier transform capabilities
const signal = [1, 2, 3, 4, 5, 6, 7, 8];
const spectrum = scientific.fourier.dft(signal);
const reconstructed = scientific.fourier.idft(spectrum);

Numerical and Advanced Methods

// Root finding with Newton's method
const root = herta.numerical.roots.newton('x^2 - 4', 1);

// Solve differential equations
const solution = herta.numerical.ode.rk4('y - x', 0, 1, 10, 100);

// Optimization with gradient descent
const costFunction = (x) => x[0]**2 + x[1]**2;
const gradientFunction = (x) => [2*x[0], 2*x[1]];
const minimum = herta.optimization.gradientDescent(costFunction, gradientFunction, [1, 1]);

// Fast Fourier Transform for signal processing
const signal = [1, 2, 3, 4, 5, 6, 7, 8];
const spectrum = herta.signalProcessing.fft(signal);
const reconstructed = herta.signalProcessing.ifft(spectrum);

// Computational geometry
const points = [[0, 0], [1, 0], [0, 1], [1, 1], [0.5, 0.5]];
const convexHull = herta.geometry.convexHull(points);

// Machine learning with k-means clustering
const data = [[1, 2], [1, 4], [1, 0], [4, 2], [4, 4], [4, 0]];
const clusters = herta.machineLearning.kmeans(data, 2);

// Dynamical systems analysis
const logisticMap = x => 3.9 * x * (1 - x);
const orbit = herta.dynamicalSystems.iterate(logisticMap, 0.5, 100);
const lyapunov = herta.dynamicalSystems.lyapunovExponent(logisticMap, 0.5, 1000);

// Group theory operations
const cyclicGroup = herta.groupTheory.cyclicGroup(8);
const cayleyTable = herta.groupTheory.cayleyTable(cyclicGroup.elements, cyclicGroup.operation);
const quaternions = herta.groupTheory.quaternionGroup();
const representation = herta.groupTheory.createRepresentation(cyclicGroup.elements, cyclicGroup.operation, 
  element => [[Math.cos(2*Math.PI*element/8), -Math.sin(2*Math.PI*element/8)], 
              [Math.sin(2*Math.PI*element/8), Math.cos(2*Math.PI*element/8)]]);
const wallpaper = herta.groupTheory.wallpaperGroup('p4m');

// Information theory calculations
const probabilities = [0.5, 0.25, 0.125, 0.125];
const entropy = herta.informationTheory.entropy(probabilities);
const huffmanCode = herta.informationTheory.huffmanCoding(['A', 'B', 'C', 'D'], probabilities);

// Game theory analysis
const prisonersDilemma = [
  [[3, 3], [0, 5]],
  [[5, 0], [1, 1]]
];
const equilibria = herta.gameTheory.findPureNashEquilibria(prisonersDilemma[0], prisonersDilemma[1]);

// Algebraic geometry computations
const circle = herta.algebraicGeometry.polynomial({'x^2': 1, 'y^2': 1, '': -1}, ['x', 'y']);
const line = herta.algebraicGeometry.polynomial({'x': 1, 'y': 1, '': -1}, ['x', 'y']);
const intersections = herta.algebraicGeometry.curvesIntersection(circle, line);
const ellipticCurve = herta.algebraicGeometry.ellipticCurve(-3, 2); // y^2 = x^3 - 3x + 2
const P = [1, 0];
const Q = [2, 3];
const R = ellipticCurve.addPoints(P, Q); // Point addition on elliptic curve
const toricRays = [[1,0], [0,1], [-1,0], [0,-1]];
const toricCones = [[0,1], [1,2], [2,3], [3,0]];
const toricVariety = herta.algebraicGeometry.toricVariety(toricRays, toricCones);

// Advanced numerical methods
const integral = herta.numerical.integrate.adaptiveSimpson(x => Math.sin(x) * Math.exp(-x), 0, 10);
const mcIntegral = herta.numerical.integrate.monteCarlo(function(x, y) { 
  return Math.sin(x) * Math.cos(y); 
}, [0, 0], [Math.PI, Math.PI], 10000);

// PDE solution using finite differences
const heat2d = herta.numerical.pde.heat2d(
  (x, y) => Math.exp(-(x*x + y*y)), // Initial temperature
  (x, y, t) => 0, // Boundary condition
  -2, 2, -2, 2, 0.5, 0.1, 0.1, 0.001, 0.5
);

// Solve stochastic differential equation (geometric Brownian motion)
const gbm = herta.numerical.sde.eulerMaruyama(
  (x, t) => 0.1 * x, // Drift term
  (x, t) => 0.2 * x, // Diffusion term
  1.0, // Initial value
  0, 1, 0.01, 5 // From t=0 to t=1 with 5 sample paths
);

// Advanced number theory operations
const primes = herta.numberTheory.generatePrimes(1000);
const largeNumberFactors = herta.numberTheory.fastFactorize(987654321);
const phi = herta.numberTheory.eulerTotient(120); // Euler's totient function

// Solving a Diophantine equation: 5x + 7y = 31
const dioSolution = herta.numberTheory.solveDiophantine(5, 7, 31);

// Chinese remainder theorem solving system of congruences
const crSolution = herta.numberTheory.chineseRemainderTheorem([2, 3, 2], [3, 5, 7]);

// Converting a fraction to continued fraction
const contFrac = herta.numberTheory.toContinuedFraction(415, 93); // 4.462...
const convergents = herta.numberTheory.continuedFractionConvergents(contFrac);

// Generating primitive Pythagorean triples
const pythTriples = herta.numberTheory.primitivePythagoreanTriples(100);

// Differential geometry calculations
const sphereMetric = [[1, 0], [0, Math.sin(u)*Math.sin(u)]];
const christoffel = herta.differentialGeometry.christoffelSymbols(sphereMetric, ['u', 'v']);
const gaussianCurv = herta.differentialGeometry.ricciScalar(sphereMetric, ['u', 'v']);
const geodesicPath = herta.differentialGeometry.geodesic(sphereMetric, [0, 0], [Math.PI/2, Math.PI/2]);
const frame = herta.differentialGeometry.createFrameField(sphereMetric, ['u', 'v']);
const transportedVector = herta.differentialGeometry.parallelTransport(sphereMetric, [1, 0], geodesicPath);

// Category theory structures
const setCategory = herta.categoryTheory.createCategory(['A', 'B'], {
  'A->B': { domain: 'A', codomain: 'B', compose: f => f }
});
const functorF = herta.categoryTheory.createFunctor(setCategory, setCategory, 
  obj => obj, morphism => morphism);

// Complex analysis
const z = herta.complexAnalysis.complex(3, 4);
const w = z.exp();
const mobius = herta.complexAnalysis.conformalMappings.mobius(
  herta.complexAnalysis.complex(1, 0),
  herta.complexAnalysis.complex(0, 1),
  herta.complexAnalysis.complex(0, 0),
  herta.complexAnalysis.complex(1, 0)
);

// Topological data analysis
const simplicialComplex = herta.topology.simplicialComplex([[0, 1], [1, 2], [2, 0]]);

// Financial mathematics
const optionPrice = herta.financialMath.blackScholes('call', 100, 100, 0.05, 0.2, 1);

// Discrete mathematics
const combinations = herta.discreteMath.combinations([1, 2, 3, 4, 5], 3);

Statistical Analysis

// Descriptive statistics
const data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];
const mean = herta.statistics.descriptive.mean(data);
const stdDev = herta.statistics.descriptive.standardDeviation(data);

// Hypothesis testing
const tTest = herta.statistics.hypothesis.tTest(data, 5);

// Time series analysis
const movingAvg = herta.statistics.timeSeries.movingAverage(data, 3);

// Bayesian inference
const posterior = herta.statistics.bayesian.posteriorDistribution(prior, likelihood, data);

Quantum Computing

// Create a quantum register with 3 qubits
const qreg = herta.quantum.register.create(3);

// Apply quantum gates to create a GHZ state
const ghz = herta.quantum.circuit.createGHZ(qreg);

// Apply quantum gates
const hadamardState = herta.quantum.gates.hadamard(qreg, 0);
const cnotState = herta.quantum.gates.cnot(hadamardState, 0, 1);
const toffoliState = herta.quantum.gates.toffoli(cnotState, 0, 1, 2);

// Run Shor's algorithm for integer factorization
const factors = herta.quantum.algorithms.shor(15);

// Run Grover's search algorithm
const database = [0, 1, 2, 3, 4, 5, 6, 7];
const searchResult = herta.quantum.algorithms.grover(database, x => x === 3);

// Quantum Fourier Transform
const qft = herta.quantum.transforms.qft(qreg);

// Measure the quantum state
const measurement = herta.quantum.measurement.measure(qreg);

Symbolic Computation

// Symbolic differentiation
const derivative = herta.symbolic.calculus.diff('x^2 + 2*x + 1', 'x');

// Symbolic integration
const integral = herta.symbolic.calculus.integrate('x^2', 'x');

// Solve equations symbolically
const solution = herta.symbolic.solve.equations('x^2 - 4 = 0', 'x');

// Number theory operations
const primes = herta.numberTheory.generatePrimes(1000);
const factors = herta.numberTheory.factorize(120);
const fastFactors = herta.numberTheory.fastFactorize(987654321); // Using Pollard's rho algorithm
const gcd = herta.numberTheory.gcd(48, 18);
const [d, x, y] = herta.numberTheory.extendedGcd(17, 23); // 17x + 23y = 1
const phi = herta.numberTheory.eulerTotient(36); // φ(36) = 12

// Solving a Diophantine equation: 5x + 7y = 31
const solution = herta.numberTheory.solveDiophantine(5, 7, 31);
console.log(solution.particular); // {x: -2, y: 6}

// Chinese remainder theorem
const crt = herta.numberTheory.chineseRemainderTheorem([2, 3, 2], [3, 5, 7]); // x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7)

// Continued fractions
const contFrac = herta.numberTheory.toContinuedFraction(415, 93); // [4, 2, 6, 7]
const convergents = herta.numberTheory.continuedFractionConvergents(contFrac); // Approximations converging to 415/93

// Finding quadratic residues modulo a prime
const qResidues = herta.numberTheory.quadraticResidues(11); // [0, 1, 3, 4, 5, 9]

// Graph theory operations
// Create a directed graph
const directedGraph = herta.graph.create({ directed: true, weighted: true });
directedGraph.addEdge('A', 'B', 5);
directedGraph.addEdge('B', 'C', 3);
directedGraph.addEdge('A', 'C', 10);

// Find shortest paths from A to all other vertices
const paths = herta.graph.algorithms.dijkstra(directedGraph, 'A');
console.log(paths.distances); // {A: 0, B: 5, C: 8}

// Calculate centrality measures
const centrality = herta.graph.algorithms.centrality(directedGraph);
console.log(centrality.betweenness); // Betweenness centrality for each vertex

// Create an undirected graph for a social network
const socialGraph = herta.graph.create({ directed: false });
socialG