Mathematics Examples

Natural language prompts for mathematical computation and analysis.

Calculus & Analysis

Compute the limit of (1 + 1/n)^n as n approaches infinity

Find the Taylor series of sin(x) around x=0 to 10th order

Evaluate the integral of x^2 * e^(-x) from 0 to infinity

Solve the differential equation y'' + y = cos(x) with y(0)=1, y'(0)=0

Find the critical points and classify them for f(x,y) = x^3 - 3xy + y^3

Compute the gradient, divergence, and curl of F = (xy, yz, zx)

Evaluate the line integral of F around a circle

Find the Fourier series of a square wave

Compute the Laplace transform of t^2 * e^(-t)

Solve the wave equation with given initial and boundary conditions

Linear Algebra

Find the eigenvalues and eigenvectors of a 4x4 rotation matrix

Compute the SVD of a matrix and reconstruct with rank-k approximation

Solve the least squares problem Ax = b for an overdetermined system

Find the null space and column space of a matrix

Compute the matrix exponential e^A

Determine if a matrix is positive definite

Find the QR decomposition and use it to solve a linear system

Compute the condition number and analyze numerical stability

Find the pseudoinverse of a non-square matrix

Diagonalize a symmetric matrix

Differential Equations

Solve the Lorenz system and visualize the strange attractor

Find the general solution to y''' - 6y'' + 11y' - 6y = 0

Solve the heat equation on a finite rod with insulated ends

Model predator-prey dynamics with Lotka-Volterra equations

Solve the Schrodinger equation for a harmonic oscillator

Find equilibrium points and their stability for a nonlinear system

Solve the Van der Pol oscillator and show limit cycles

Model the SIR epidemic spreading dynamics

Solve the Black-Scholes PDE for option pricing

Analyze the stability of a system using Lyapunov functions

Number Theory

Find all prime factors of a large number using Pollard's rho

Compute the greatest common divisor using the extended Euclidean algorithm

Check if a number is prime using Miller-Rabin

Find modular inverses and solve linear congruences

Compute Euler's totient function for various n

Implement RSA encryption with small primes

Find primitive roots modulo p

Compute the Legendre symbol and quadratic residues

Solve the Chinese Remainder Theorem system

Find Pythagorean triples with a given constraint

Combinatorics & Discrete Math

Count the number of ways to partition n into distinct parts

Generate all permutations with specific constraints

Compute binomial coefficients using Pascal's triangle

Find the chromatic polynomial of a graph

Count the number of spanning trees in a graph

Solve a recurrence relation using generating functions

Find the shortest path in a weighted graph

Compute the permanent of a matrix

Count lattice paths with obstacles

Solve a combinatorial optimization problem using dynamic programming

Probability & Statistics

Compute the probability of getting exactly 3 heads in 10 coin flips

Find the expected value and variance of a mixed distribution

Perform Bayesian inference with a conjugate prior

Compute the correlation matrix for multivariate data

Test for normality using Shapiro-Wilk and Q-Q plots

Perform ANOVA to compare multiple groups

Compute confidence intervals using bootstrapping

Fit a mixture of Gaussians using EM algorithm

Perform hypothesis testing with corrections for multiple comparisons

Compute mutual information between random variables

Numerical Methods

Find roots of a nonlinear equation using Newton-Raphson

Integrate a function numerically using adaptive quadrature

Solve a stiff ODE using implicit methods

Interpolate data using cubic splines

Compute the FFT of a signal and analyze frequency content

Solve a PDE using finite difference methods

Find eigenvalues of a large sparse matrix using Arnoldi iteration

Perform numerical optimization using BFGS

Solve a boundary value problem using shooting method

Compute special functions (Bessel, Legendre, etc.) numerically

Abstract Algebra

Find the subgroups of the symmetric group S4

Determine if two groups are isomorphic

Compute the character table of a finite group

Find the Galois group of a polynomial

Check if an ideal is prime or maximal

Compute the quotient ring and its properties

Find the center and commutator subgroup

Determine the structure of a finitely generated abelian group

Check if a ring is a unique factorization domain

Compute the automorphism group of a graph

Topology & Geometry

Compute the fundamental group of a torus

Find the Euler characteristic of a surface

Compute geodesics on a sphere

Determine if two knots are equivalent

Calculate the curvature and torsion of a space curve

Find the homology groups of a simplicial complex

Compute the de Rham cohomology of a manifold

Visualize a Mobius strip and Klein bottle

Calculate the volume of an n-dimensional sphere

Find the shortest path on a curved surface

Complex Analysis

Find all zeros and poles of a complex function

Compute contour integrals using the residue theorem

Map a region using a conformal transformation

Find the Laurent series around a singularity

Compute the analytic continuation of a function

Evaluate real integrals using complex methods

Find the Riemann surface of a multi-valued function

Compute the argument principle and count zeros

Solve a complex ODE in the complex plane

Visualize phase portraits of complex functions

Optimization

Minimize a function using gradient descent with momentum

Solve a linear programming problem using simplex

Find the global minimum using simulated annealing

Solve a quadratic programming problem

Implement constraint optimization using Lagrange multipliers

Find the Pareto frontier for multi-objective optimization

Solve a mixed-integer programming problem

Implement genetic algorithms for combinatorial optimization

Find saddle points using min-max optimization

Solve an optimal control problem using Pontryagin's principle

Dynamical Systems

Find the fixed points and classify their stability

Compute Lyapunov exponents for a chaotic system

Generate a bifurcation diagram varying a parameter

Find periodic orbits and their stability

Compute the basin of attraction for multiple attractors

Analyze the Henon map and its strange attractor

Find invariant manifolds of a saddle point

Compute the fractal dimension of an attractor

Analyze synchronization in coupled oscillators

Study the route to chaos through period doubling