mathcode

A Python library containing reusable mathematical models and methods from coursework, including modules on stochastic processes, optimization, graph theory, abstract algebra, and more.

Features

Abstract Algebra (mathcode.abstract_algebra)

• Groups: Cyclic groups, symmetric groups, permutation groups
• Rings: Integer rings, polynomial rings, quotient rings
• Fields: Finite fields (Galois fields), field extensions

Graph Theory (mathcode.graph_theory)

• Data Structures: Graph, DirectedGraph, WeightedGraph, WeightedDirectedGraph
• Traversal: BFS, DFS (iterative and recursive), level-order traversal
• Properties: Connectivity, cycle detection, bipartiteness, bridges, articulation points, strongly connected components
• Shortest Paths: Dijkstra's algorithm, Bellman-Ford, A* search
• Spanning Trees: Kruskal's algorithm, Prim's algorithm, Union-Find
• Network Flow: Ford-Fulkerson, Edmonds-Karp, min-cost max-flow
• Centrality: Degree, closeness, betweenness (Brandes' algorithm), PageRank
• Advanced: Hamiltonian paths/cycles, graph coloring, planarity testing, graph isomorphism

Optimization (mathcode.optimization)

• Gradient Descent: Standard, momentum, Nesterov, AdaGrad, RMSProp, Adam
• Linear Programming: Simplex method, dual conversion, Dantzig-Wolfe decomposition, Benders decomposition

Reinforcement Learning (mathcode.reinforcement_learning)

• Environments: GridWorld, abstract Environment base class
• Dynamic Programming: Value iteration, policy iteration, policy evaluation, Q-value iteration
• Monte Carlo Methods: First-visit/every-visit prediction, exploring starts, epsilon-greedy control, off-policy with importance sampling
• Policy/Value Functions: Deterministic and stochastic policies, value functions V(s), action-value functions Q(s,a)

Stochastic Processes (mathcode.stochastics)

• Brownian Motion: Standard, geometric, drifted, Ornstein-Uhlenbeck
• Markov Chains: Discrete-time Markov chains, transition matrices, stationary distributions
• Poisson Processes: Event simulation, counting processes
• Stochastic Differential Equations: Euler-Maruyama, Milstein methods
• Stochastic Partial Differential Equations: Heat equation, wave equation, Allen-Cahn, Burgers

Installation

pip install mathcode

Requirements

• Python >= 3.8
• numpy
• matplotlib
• networkx
• scipy

Usage Examples

Abstract Algebra

from mathcode.abstract_algebra import CyclicGroup, FiniteField

# Create cyclic group Z_12
g = CyclicGroup(12)
print(g.order())  # 12

# Create finite field GF(8)
f = FiniteField(8)
# ... field operations

Graph Theory

from mathcode.graph_theory import Graph, dijkstra, bfs, pagerank

# Create a graph
g = Graph()
g.add_edge(0, 1, weight=4.0)
g.add_edge(0, 2, weight=1.0)
g.add_edge(1, 3, weight=1.0)

# Find shortest paths
distances, predecessors = dijkstra(g, start=0)

# Traverse the graph
visited = bfs(g, start=0)

# Compute PageRank
scores = pagerank(g)

Optimization

from mathcode.optimization import Adam, SimplexLP

# Optimize with Adam
optimizer = Adam(learning_rate=0.001)
# ... training loop with optimizer.update(params, gradients)

# Solve linear program
lp = SimplexLP(c=[1, 2], A=[[1, 1], [2, 1]], b=[4, 5])
solution, value = lp.solve()

Reinforcement Learning

from mathcode.reinforcement_learning import GridWorld, value_iter, mc_es

# Create a grid world environment
env = GridWorld(height=4, width=4, start=(0, 0), goal=(3, 3))

# Solve with Value Iteration (Dynamic Programming)
V, policy = value_iter(env, gamma=0.9)
print(f"Optimal action at start: {policy.get_action((0, 0))}")

# Solve with Monte Carlo Exploring Starts
Q, policy = mc_es(env, num_episodes=5000, gamma=0.9)

Stochastic Processes

from mathcode.stochastics import GBM, MarkovChain
import numpy as np

# Simulate Geometric Brownian Motion
gbm = GBM(T=1.0, dt=0.01, mu=0.05, sigma=0.2, X0=100)
paths = gbm.sample(n_paths=1000)

# Create Markov chain
P = np.array([[0.7, 0.3], [0.4, 0.6]])
mc = MarkovChain(P)
stationary = mc.stationary_distribution()

Development

Running Tests

pytest tests/ -v

Project Structure

mathcode/
├── abstract_algebra/        # Group theory, ring theory, field theory
├── graph_theory/            # Graph algorithms and data structures
├── optimization/            # Optimization algorithms
├── reinforcement_learning/  # RL algorithms (DP, Monte Carlo)
└── stochastics/             # Stochastic processes and SDEs
tests/                       # Unit tests for all modules

Author

Ishaan Goel

Links

• Homepage: https://github.com/igoeldc/mathcode
• Issues: https://github.com/igoeldc/mathcode/issues