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Mathematics concept index

Explore 540 published mathematical concepts by subject. Start with a definition, then follow reviewed questions, related ideas and available video explanations.

Mathematical modeling

  • Agent-based models

    Individual rules and interactions generate collective behavior in simulation.

  • Analytic hierarchy process

    Build pairwise comparison matrices over criteria, then read weights from the dominant eigenvector and check consistency.

  • ARIMA and differencing

    Difference until stationary, fit autoregressive and moving average terms, and forecast with error growing in the horizon.

  • Assumptions and simplification

    State which effects are ignored and why, so the model stays solvable while remaining defensible against the data.

  • Branch and bound

    Solve relaxed linear problems, branch on fractional integers and prune nodes whose bound cannot beat the incumbent.

  • Cluster analysis

    Group objects by a distance metric; the metric and the linkage choice decide the partition as much as the algorithm does.

  • Compartmental differential equation models

    Split the system into compartments with flows between them; the SIR model is the standard epidemic example.

  • Copying templates without assumptions

    Reusing a famous model without stating its preconditions hides why the answer stops being valid.

  • Cross-validation

    Repeated training and held-out evaluation estimate generalization while respecting dependence and leakage risks.

  • Data provenance

    Data records should preserve origin, collection conditions and transformations.

  • Decision and state variables

    Name every symbol with its unit and range before writing equations, because undefined symbols make results unverifiable.

  • Difference equation models

    Describe discrete-step dynamics such as generations or quarters, and study stability through the multiplier magnitude.

  • Dimensional analysis

    Check that both sides of every equation share the same units, and derive scaling groups such as the Reynolds number.

  • Discrete-event simulation

    A system's state changes at scheduled or randomly occurring events.

  • Discretization

    Replace continua by grids, time steps or samples so a numerical solver can work, and study the resulting truncation error.

  • Discriminant analysis

    Find linear combinations that separate labelled groups and classify new records by the closest group centre.

  • Duality and shadow price

    Every linear program has a dual; the dual value of a binding constraint is the marginal worth of relaxing it.

  • Dynamic programming recursions

    Split a sequential decision into stages and store subproblem values so each is computed once.

  • Entropy weighting

    Let the data set criterion weights from dispersion: a more uniform column carries less information and gets a smaller weight.

  • Feasible region and graphical method

    Plot the constraints to see the feasible polygon; the optimum sits at a vertex unless the objective is flat on an edge.

  • Fuzzy comprehensive evaluation

    Map graded judgements through membership functions and a fuzzy relation to a single comparable score.

  • Global sensitivity analysis

    Global sensitivity explores how input variation across a range influences model outputs.

  • Integer programming

    Force some variables to be whole numbers or binary, which models choosing, counting and sequencing decisions.

  • Interpolation versus curve fitting

    Interpolation passes through every sample, while fitting smooths noise by trading bias against variance.

  • Knapsack problem

    Choose a subset under a capacity limit; dynamic programming solves it pseudo-polynomially.

  • Least squares and residual diagnostics

    Minimise squared residuals, then read residual plots for curvature, heteroscedasticity and outliers before trusting R-squared.

  • Linear programming model

    Maximise or minimise a linear objective subject to linear inequalities, the workhorse of allocation problems.

  • Markov chain prediction

    Advance a state distribution by a transition matrix; its stationary vector is the long-run share of states.

  • Mathematical modeling process

    Turn a real question into variables, assumptions, a solvable model, a solution and a check against data.

  • Minimum spanning tree

    Connect all nodes at least total cost with Kruskal or Prim; the base of routing and pipeline design.

  • Model validation and error measures

    Hold out data, compare units and magnitudes, and report errors that a reader can act on.

  • Modeling report structure

    Answer with assumptions, model, solution, validation and limitations, so a reviewer can retrace each choice.

  • Monte Carlo simulation

    Sample the uncertain inputs many times to obtain an output distribution and confidence intervals for the answer.

  • Multi-objective optimization and Pareto front

    Trade off competing objectives; the Pareto front collects solutions that no longer can be improved on one goal without hurting another.

  • Network flow models

    Maximise flow through capacities; the min-cut theorem turns a bottleneck argument into a computable bound.

  • Nonlinear programming and KKT conditions

    For smooth nonlinear models, optimality requires stationarity, primal and dual feasibility and complementary slackness.

  • Outlier handling

    Distinguish data errors from genuine extremes; robust fits and winsorising change conclusions in different ways.

  • Overfitting and leakage

    A model tuned on the test set or fed future information looks excellent and fails on new data.

  • Population and growth models

    Write per-capita birth and death rates into a differential equation; the logistic term caps growth by carrying capacity.

  • Principal component analysis

    Rotate correlated variables onto orthogonal directions of maximal variance and keep the leading components.

  • Problem formulation

    A modeling question is translated into objectives, variables, assumptions and constraints.

  • Queueing and inventory models

    Balance arrival and service rates for waiting times, and trade ordering cost against holding cost for order size.

  • Regression modeling

    Fit a conditional mean function to data, then interpret coefficients only within the range the design covers.

  • Reproducible modeling

    Data, assumptions, code and evaluation details permit checking a modeling result.

  • Robust optimization

    Decisions are optimized against an explicitly specified uncertainty set.

  • Sensitivity and robustness analysis

    Re-run the model with parameters perturbed to see which assumptions actually drive the conclusion.

  • Shortest path models

    Dijkstra solves non-negative networks; Bellman-Ford tolerates negative edges and underpins route pricing.

  • Simplex method

    Move vertex to vertex along the feasible polytope, each step improving the objective until no improving edge remains.

  • Standardization and scaling

    Rescale columns before distance-based methods, otherwise the largest unit dominates the result.

  • Structural identifiability

    Ideal observations must uniquely determine parameters before noisy-data estimation can identify them.

  • Time series and exponential smoothing

    Model level, trend and seasonality over time; smoothing weights recent observations more heavily than old ones.

  • TOPSIS ranking

    Score each alternative by its relative closeness to an ideal point and distance from an anti-ideal point.

  • Transportation and assignment problems

    Ship supplies to demands or match workers to jobs at least cost; special structures beat generic solvers.

  • Traveling salesman routing

    Visit each site once at least tour length; heuristics plus bounds are the practical modelling answer.

  • Types of uncertainty

    Random variability and limited knowledge require different uncertainty representations.

  • Uncertainty propagation

    Input uncertainty induces output uncertainty through a model.

  • Unit and scale inconsistency

    Mixing kilometres with hours or percent with proportion is the most common silent modelling error.

  • Verification and validation

    Verification checks implementation against specifications; validation checks fitness for an intended use.

Category Theory

  • Abelian categories

    An abelian category supports additive structure, kernels, cokernels and exactness with compatible axioms.

  • Adjunctions

    An adjunction relates morphism sets naturally between two functors.

  • Categorical isomorphisms

    A morphism is an isomorphism when it has a two-sided compositional inverse.

  • Categories

    A category has objects, morphisms, associative composition and identity morphisms.

  • Equivalence of categories

    Functors inverse up to natural isomorphism express equivalence without literal equality of categories.

  • Functors

    A functor maps objects and morphisms while preserving identities and composition.

  • Limits and colimits

    Limits and colimits generalize universal constructions over diagrams of objects and arrows.

  • Monoidal categories

    A tensor-like product and unit object obey coherent associativity and unit constraints.

  • Natural transformations

    Compatible morphisms between functor values commute with every source-category morphism.

  • Products and coproducts

    Products receive universal compatible maps; coproducts reverse the direction.

  • Universal properties

    A universal property characterizes an object by unique compatible maps.

  • Yoneda lemma

    Maps from a representable functor into a functor correspond naturally to an element at its representing object.

Elementary Algebra and Functions

  • Absolute value

    Absolute value measures distance from zero on the real line.

  • Algebraic expressions

    Expressions combine constants, variables and operations without necessarily asserting an equation.

  • Arithmetic operations

    Addition, subtraction, multiplication and division follow precedence rules; division by zero is undefined.

  • Exponent laws

    Power identities simplify products and quotients where bases and exponents satisfy the required domain conditions.

  • Exponential functions

    An exponential with positive base other than one models multiplicative change.

  • Factorization

    Factorization rewrites an expression as a product, helping locate zeros and simplify equations.

  • Fractions and ratios

    Fractions express quotients with nonzero denominators; ratios compare quantities with compatible units.

  • Function composition

    Composition applies one function after another where the output fits the next domain.

  • Function transformations

    Translations, scalings and reflections change a graph through controlled changes of its formula.

  • Inequalities

    Inequalities compare ordered values; multiplying by a negative number reverses the inequality.

  • Inverse functions

    An inverse reverses a bijection; domain restrictions can make a function invertible.

  • Linear equations

    A one-variable linear equation has the form ax+b=0b=0 with a nonzero for a unique solution.

  • Logarithms

    A logarithm is the inverse of an exponential, defined for positive arguments over the reals.

  • Number systems

    Natural, integer, rational, real and complex numbers support progressively broader operations.

  • Percentages

    A percentage expresses a ratio per hundred; percentage-point changes differ from relative percentage changes.

  • Polynomials

    A polynomial is a finite sum of nonnegative integer powers with coefficients.

  • Quadratic equations

    The discriminant determines the real-root structure of a quadratic with nonzero leading coefficient.

Algebraic Geometry

  • Affine varieties

    Common polynomial zero sets define affine algebraic sets over a chosen field.

  • Algebraic dimension

    Chains of prime ideals define Krull dimension and connect algebraic structure to geometric dimension.

  • Algebraic singularities

    Singular points fail suitable smoothness conditions measured by local geometry or tangent behavior.

  • Coordinate rings

    Polynomial functions on an affine algebraic set form a quotient by its vanishing ideal.

  • Divisors

    Divisors encode codimension-one geometric data with multiplicities under specified definitions.

  • Localization

    Localization formally inverts a chosen multiplicatively closed set of ring elements.

  • Morphisms of varieties

    Regular maps preserve algebraic structure through locally appropriate polynomial or rational expressions.

  • Prime ideals

    For a prime ideal in a commutative ring, a product inside the ideal forces a factor inside.

  • Projective space

    Nonzero vectors modulo nonzero scalar multiplication represent projective points.

  • Schemes

    Schemes glue locally ringed spaces modeled on spectra of commutative rings.

  • Sheaves

    A sheaf organizes local data that restrict consistently and glue uniquely when compatible.

  • Spectrum of a ring

    Prime ideals form a topological space whose basic structure encodes a commutative ring.

Number Theory

  • Algebraic numbers

    An algebraic number is a root of a nonzero polynomial with rational coefficients.

  • Euler's totient

    Euler's totient counts positive residues coprime to a positive integer modulus.

  • Finite fields

    A finite field has prime-power size; fields of each such size exist and are unique up to isomorphism.

  • Prime number theorem

    The number of primes up to x is asymptotic to x divided by log x.

  • Quadratic residues

    A residue is quadratic when it is congruent to a square modulo the chosen integer.

  • Riemann zeta function

    A Dirichlet series over positive integers defines zeta in a convergent half-plane and admits analytic continuation.

Discrete Mathematics: Counting and Number Theory

  • Algorithmic complexity

    Time and space requirements are expressed as functions of input size under a computational model.

  • Diophantine equations

    Diophantine equations seek integer or rational solutions rather than unrestricted real solutions.

  • Euler and Hamilton paths

    Euler paths traverse each edge once; Hamilton paths visit each vertex once.

  • Fermat and Euler theorems

    Coprime residues satisfy power congruences controlled by prime moduli or Euler's totient.

  • Generating functions

    A sequence encoded as series coefficients supports algebraic counting and recurrence calculations.

  • Graphs

    Vertices and edges model connections, with directed and undirected variants.

  • Inclusion–exclusion

    Alternating intersection counts correct double-counting in unions of finite sets.

  • Pigeonhole principle

    More objects than boxes force at least one box to contain multiple objects.

  • Prime factorization

    Every integer greater than one factors uniquely into primes up to their order.

  • Recurrence relations

    A recurrence specifies sequence terms using earlier terms and initial data.

  • Trees

    A finite undirected tree is connected and has no cycles.

Measure and Integration

  • Almost everywhere

    A property holds almost everywhere if its exceptional set has measure zero.

  • Dominated convergence theorem

    An integrable dominating function permits exchanging almost-everywhere limits and integrals.

  • Fubini and Tonelli theorems

    Product integrals can be iterated under integrability or nonnegativity and measure-space hypotheses.

  • Lebesgue integral

    Integration via simple-function approximation handles measurable functions beyond Riemann integration.

  • Lebesgue measure

    Lebesgue measure extends interval length to a complete measure on a class of real sets.

  • Lp spaces

    Integrable powers define function spaces modulo equality almost everywhere.

  • Measurable functions

    Inverse images of appropriate Borel sets must be measurable.

  • Measures

    A measure assigns nonnegative size and is countably additive on disjoint measurable sets.

  • Monotone convergence theorem

    Increasing nonnegative measurable functions allow exchanging limit and integral.

  • Outer measures

    An outer measure is monotone and countably subadditive on all subsets.

  • Radon–Nikodym theorem

    Under suitable sigma-finiteness, absolute continuity of measures yields an integral density.

  • Sigma-algebras

    A sigma-algebra is closed under complements and countable unions.

Probability and statistics

  • Analysis of variance

    ANOVA partitions variation to test mean differences under model assumptions.

  • Autocorrelation

    Autocorrelation measures dependence between a process and its lagged values.

  • Bayes theorem

    Bayes theorem relates conditional probabilities in opposite directions using prior probabilities and a nonzero evidence probability.

  • Bayesian inference

    Prior and likelihood combine into a posterior distribution through Bayes' rule.

  • Bernoulli distribution

    A Bernoulli variable records one binary outcome with success probability p.

  • Binomial distribution

    A binomial variable counts successes in a fixed number of independent Bernoulli trials with a common success probability.

  • Bootstrap

    Resampling observed data estimates sampling variability under appropriate conditions.

  • Central limit theorem

    For independent identically distributed variables with finite positive variance, standardized sums approach a normal distribution.

  • Chi-squared distribution

    A sum of squared independent standard normal variables has a chi-squared distribution.

  • Conditional probability

    When P(B)>0P(B)>0, conditional probability restricts attention to B and renormalizes the probabilities.

  • Confidence intervals

    A confidence procedure has a repeated-sampling coverage rate; the realized interval is not a posterior probability statement about a fixed parameter.

  • Credible intervals

    A credible interval contains a specified posterior probability under the Bayesian model.

  • Distributions

    A distribution records how probability is assigned to the values of a random variable, using masses, densities or a cumulative distribution function.

  • Expectation

    Expectation is a probability-weighted average when defined; not every random variable has a finite expectation.

  • Experimental design

    Randomization, controls and replication structure studies to distinguish effects from confounding.

  • Exponential distribution

    An exponential waiting time has a constant hazard rate and the memoryless property.

  • F distribution

    The ratio of independent chi-squared variables divided by their degrees of freedom has an F distribution.

  • Gamma and beta distributions

    Gamma distributions model positive quantities; beta distributions model values on the unit interval.

  • Hypothesis testing

    A hypothesis test compares data to a null model while controlling a specified error rate; a p-value is not the probability that the null is true.

  • Independence

    Two events are independent when their joint probability equals the product of their probabilities; this differs from being disjoint.

  • Law of large numbers

    Under suitable assumptions, sample averages approach the population expectation in a specified mode of convergence.

  • Likelihood

    A likelihood treats observed data as fixed and parameters as varying.

  • Linear regression

    A response's conditional mean is modeled as linear in coefficients with assumptions about residuals.

  • Logistic regression

    A logistic link models binary-event probability through a linear predictor.

  • Maximum likelihood

    Parameters maximize the likelihood of observed data within a specified model.

  • Multiple testing

    Testing many hypotheses requires controlling an appropriate combined error measure.

  • Normal distribution

    A normal distribution is specified by its mean and positive variance, with a symmetric bell-shaped density.

  • Parameter estimation

    An estimator uses sample data to estimate a model parameter; bias, variance and consistency describe different properties.

  • Poisson distribution

    A Poisson variable counts nonnegative integer events with a specified mean parameter.

  • Probability

    A probability measure assigns nonnegative weights to events, with total weight one and countable additivity on disjoint events.

  • Random variables

    A random variable is a measurable function on a probability space, allowing outcomes to be described numerically.

  • Sampling

    Sampling selects observations from a population; the design determines what inferences the observed sample can support.

  • Stationary time series

    Stationarity imposes time-shift invariance on distributions or specified moments.

  • Student's t distribution

    A standard normal divided by an independent scaled chi-squared root gives a t distribution.

  • Uniform distribution

    A uniform distribution assigns equal probabilities to finite outcomes or constant density on a finite interval.

  • Variance

    Variance measures expected squared deviation from the mean when the relevant moments exist.

Complex Analysis

  • Analytic continuation

    Compatible holomorphic functions extend an analytic function to larger connected regions.

  • Cauchy integral formula

    Contour values determine interior values and derivatives of a holomorphic function under the contour hypotheses.

  • Cauchy–Riemann equations

    Complex differentiability imposes relations between real and imaginary partial derivatives.

  • Complex contour integrals

    Complex integration accumulates function values along an oriented parameterized curve.

  • Complex numbers

    A complex number has real and imaginary parts and is represented on the complex plane.

  • Conformal maps

    A holomorphic map with nonzero derivative locally preserves oriented angles.

  • Holomorphic functions

    A function is holomorphic when complex differentiable throughout an open neighborhood.

  • Isolated singularities

    A punctured neighborhood can contain a removable singularity, a pole or an essential singularity.

  • Laurent series

    A Laurent series uses positive and negative powers on an annulus.

  • Residue theorem

    A contour integral equals two pi i times enclosed residues under the theorem's hypotheses.

  • Residues

    The coefficient of power minus one in a Laurent expansion controls a local contour integral.

Trigonometry

  • Angle addition formulas

    Addition formulas express trigonometric functions of summed angles using individual angles.

  • Complex exponentials

    Euler's formula links complex exponentials to cosine and sine for real angles.

  • Inverse trigonometric functions

    Restricted domains make trigonometric functions invertible with specified principal-value ranges.

  • Law of cosines

    The cosine law relates one side to the other two sides and their included angle.

  • Law of sines

    In a triangle, each side divided by its opposite-angle sine has the same value.

  • Radians

    Radian measure is arc length divided by radius.

  • Sine and cosine

    Sine and cosine are the vertical and horizontal coordinates on the unit circle.

  • Sinusoidal models

    Amplitude, frequency and phase describe a sinusoidal oscillation.

  • Tangent function

    Tangent is sine divided by cosine where cosine is nonzero.

  • Trigonometric equations

    Solutions must respect periodicity, domains and any restricted angle interval.

  • Trigonometric identities

    Trigonometric identities hold throughout their common domains, unlike equations requiring selected solutions.

  • Trigonometric periodicity

    Sine and cosine repeat after a full turn; tangent repeats after a half turn on its domain.

  • Unit circle

    The unit circle relates an angle to cosine and sine coordinates.

Geometry

  • Angles

    An angle measures rotation between rays sharing an endpoint.

  • Area and volume

    Area measures planar regions and volume measures spatial regions in corresponding squared or cubed units.

  • Circle geometry

    Chord, tangent and inscribed-angle relations connect angles and distances in a circle.

  • Conic sections

    Ellipses, parabolas and hyperbolas arise from planar sections of a cone.

  • Coordinate geometry

    Coordinates translate geometric conditions into equations and algebraic calculations.

  • Distance formula

    Euclidean distance is the square root of the sum of squared coordinate differences.

  • Geometric loci

    A locus consists of all points satisfying specified geometric conditions.

  • Geometric projections

    A projection maps spatial objects to a lower-dimensional view according to a chosen rule.

  • Lines and planes

    Linear equations and vector representations describe straight lines and planes.

  • Non-Euclidean geometry

    Changing the parallel-postulate setting yields geometries with different angle and distance behavior.

  • Polyhedra

    A polyhedron has polygonal faces meeting along edges and vertices.

  • Pythagorean theorem

    In a right triangle, squared leg lengths sum to the squared hypotenuse length.

  • Rigid motions

    Translations, rotations and reflections preserve Euclidean distances.

  • Solid geometry

    Spatial geometry studies incidence, distance and angles among three-dimensional objects.

  • Triangle congruence

    Congruent triangles have matching sides and angles under a rigid motion.

  • Triangle similarity

    Similar triangles have equal corresponding angles and proportional corresponding sides.

Calculus

  • Antiderivatives

    An antiderivative has the given function as its derivative; on an interval, any two antiderivatives differ by a constant.

  • Chain rule

    The derivative of a differentiable composition combines the derivative of the outer function with that of the inner function.

  • Change of variables

    A suitable coordinate change transforms volume elements through the absolute Jacobian determinant.

  • Continuity

    Continuity at a point means that the function value equals its limit there; continuity on an interval requires this at each relevant point.

  • Definite integrals

    A Riemann integral describes signed accumulation as a limit of sums, for functions satisfying the integrability conditions.

  • Derivatives

    A derivative is the limit of a difference quotient, measuring local rate of change when that limit exists.

  • Directional derivatives

    A directional derivative measures the local rate of change along a specified direction.

  • Divergence and curl

    Divergence measures local flux expansion; curl measures local circulation in three dimensions.

  • Double and triple integrals

    Integration over planar or spatial regions accumulates density into total quantities.

  • Functions

    A function assigns exactly one output to each input in its domain; its domain matters when comparing expressions.

  • Fundamental theorem of calculus

    For a continuous integrand, differentiation of its accumulated integral recovers the integrand and antiderivatives evaluate definite integrals.

  • Gradient

    In Euclidean coordinates, partial derivatives form a vector describing directional change.

  • Green, Stokes and divergence theorems

    Under smoothness and orientation hypotheses, boundary integrals equal corresponding interior derivative integrals.

  • Implicit differentiation

    Differentiate an equation relating variables and use the chain rule, while checking when the resulting relation determines a derivative.

  • Improper integrals

    Integrals over unbounded intervals or near singularities are defined through limits; convergence must be established.

  • Integration by parts

    The product rule yields an integration identity that trades one integral for another, with suitable differentiability and integrability.

  • Jacobian matrix

    The derivative matrix of a multivariable map captures its local linear action.

  • Limits

    A limit describes how function values approach a value as the input approaches a point, without requiring the function to be defined at that point.

  • Line integrals

    A line integral accumulates a scalar density or vector-field work along a curve.

  • Mean value theorem

    For a function continuous on a closed interval and differentiable inside it, some interior derivative equals the average rate of change.

  • Multivariable functions

    A multivariable function assigns an output to a tuple of inputs in its domain.

  • Optimization

    Derivatives help identify candidate extrema; endpoints, domain restrictions and the distinction between local and global extrema must also be checked.

  • Partial derivatives

    A partial derivative varies one coordinate while holding the others fixed.

  • Series

    An infinite series converges when its sequence of partial sums has a finite limit; terms tending to zero alone do not suffice.

  • Surface integrals

    Surface integrals accumulate scalar densities or oriented flux over a surface.

  • Taylor series

    Taylor coefficients come from derivatives at a point; equality of a function to its series requires additional convergence and remainder conditions.

  • Total differential

    Differentiability gives a linear first-order approximation in all input directions.

Combinatorial Optimization

  • Approximation algorithms

    An approximation algorithm provides a provable ratio to the optimum under stated assumptions.

  • Assignment problem

    A costed one-to-one pairing can be optimized using matching or linear programming structure.

  • Bipartite matching

    Matching in a two-part graph pairs vertices across parts without shared endpoints.

  • Graph matchings

    A matching is a set of edges with no shared endpoints.

  • Integer relaxations

    Removing integrality yields easier problems whose optima can bound the discrete optimum.

  • Matroids

    A matroid abstracts independence through hereditary and exchange axioms.

  • Max-flow min-cut theorem

    For a finite capacitated network, the maximum flow value equals minimum separating cut capacity.

  • NP-completeness

    An NP-complete decision problem lies in NP and admits polynomial reductions from every NP problem.

Differential Geometry

  • Arc length

    Integrating speed measures a sufficiently regular curve's length.

  • Covariant derivatives

    A connection differentiates vector fields while accounting for changing tangent spaces.

  • Curvature and torsion

    Curvature measures tangent turning; torsion describes departure from a plane in space.

  • Differential forms

    Alternating multilinear objects encode integration over oriented geometric pieces.

  • Gauss–Bonnet theorem

    For a closed oriented surface, total Gaussian curvature relates to its Euler characteristic.

  • Gaussian curvature

    The product of principal curvatures measures intrinsic surface curvature.

  • Geodesics

    Geodesics have zero covariant acceleration and locally extremize length under suitable conditions.

  • Regular curves

    A regular curve has a nonzero velocity, allowing local tangent geometry.

  • Regular surfaces

    Regular surface charts have linearly independent coordinate tangent vectors.

  • Riemannian metrics

    A smoothly varying positive-definite inner product on tangent spaces measures lengths and angles.

  • Smooth manifolds

    Compatible smooth coordinate charts support differentiation on a manifold.

  • Tangent spaces

    Tangent vectors at a point form a linear space describing infinitesimal directions.

Dynamical Systems and Chaos

  • Attractors

    An attractor is an invariant set approached by a specified collection of trajectories.

  • Bifurcations

    Changing a parameter can qualitatively change equilibria or periodic behavior.

  • Chaotic dynamics

    Chaos combines forms of sensitive dependence and complex recurrence under specified definitions.

  • Continuous flows

    An autonomous differential equation generates time-indexed state transformations where solutions exist uniquely.

  • Discrete dynamical systems

    Iteration of a map evolves a state through discrete time steps.

  • Ergodicity

    In measure-preserving systems, ergodicity means invariant measurable sets have trivial measure.

  • Fixed and periodic points

    A fixed point returns after one iterate; a periodic point returns after finitely many.

  • Logistic map

    The quadratic iteration rx(1-x) exhibits fixed points, periodic cycles and chaotic regimes as r varies.

  • Lyapunov exponents

    Asymptotic growth rates quantify separation of nearby trajectories.

  • Lyapunov functions

    A suitably decreasing scalar function can establish stability without solving trajectories explicitly.

  • Orbits

    An orbit is the collection of states visited from a chosen initial state.

Mathematical Foundations and Proofs

  • Axiomatic systems

    An axiomatic theory derives theorems from stated axioms and inference rules.

  • Bijections

    A bijection is both injective and surjective and has a two-sided inverse.

  • Cartesian products

    A Cartesian product consists of ordered pairs with one component from each set.

  • Countability

    A set is countable if it is finite or can be enumerated by natural numbers.

  • Counterexamples

    One valid counterexample disproves a universal statement but does not establish its converse.

  • Direct proof

    A direct proof derives the conclusion from assumptions using justified logical steps.

  • Equivalence relations

    A reflexive, symmetric and transitive relation partitions a set into equivalence classes.

  • Injective and surjective functions

    Injectivity distinguishes inputs by outputs; surjectivity reaches every element of the codomain.

  • Logical implication

    An implication is false only when its premise is true and conclusion false.

  • Mathematical induction

    A base case and an inductive step establish a statement for all subsequent integers.

  • Necessary and sufficient conditions

    A sufficient condition implies a conclusion; a necessary condition is implied by that conclusion.

  • Negation and De Morgan laws

    Negation reverses truth values and exchanges universal with existential quantification.

  • Order relations

    A partial order is reflexive, antisymmetric and transitive; not every pair must be comparable.

  • Proof by contradiction

    Assume a claim fails and derive a contradiction with established assumptions.

  • Proof by contraposition

    Proving not Q implies not P establishes the implication P implies Q.

  • Propositions

    A proposition has a truth value; logical connectives build compound statements.

  • Quantifiers

    Universal and existential quantifiers specify how a statement ranges over a domain.

  • Relations

    A binary relation is a subset of a Cartesian product; its properties constrain related pairs.

  • Set operations

    Union, intersection, difference and complement combine sets relative to a specified universe.

  • Sets

    A set is a collection of distinct objects; membership is different from inclusion.

Game and Decision Theory

  • Backward induction

    Reasoning backward from terminal choices solves finite perfect-information games under appropriate preferences.

  • Bayesian games

    Players' private types and beliefs extend strategic analysis to incomplete information.

  • Best responses

    A best response maximizes a player's payoff given opponents' strategies.

  • Dominant strategies

    A dominant strategy is at least as good against every opponent choice, with strength depending on the definition.

  • Expected utility

    Expected utility evaluates uncertain outcomes using probabilities and a utility function.

  • Extensive-form games

    A game tree represents the order of decisions, information and terminal payoffs.

  • Mechanism design

    Rules are chosen to induce desired outcomes from strategic participants.

  • Mixed strategies

    A mixed strategy is a probability distribution over pure strategies.

  • Nash equilibrium

    At equilibrium no player gains by unilateral deviation while others keep their strategies.

  • Normal-form games

    Players, strategies and payoff functions define a normal-form game.

  • Social choice

    Voting rules aggregate individual preferences and face fairness tradeoffs.

  • Zero-sum games and minimax

    For finite two-player zero-sum games, optimal mixed strategies equalize maximin and minimax values.

Functional Analysis

  • Banach spaces

    A Banach space is a normed vector space complete in its induced metric.

  • Bounded linear operators

    A linear operator bounded by a constant times input norm is continuous.

  • Compact operators

    A compact operator maps bounded sets into relatively compact sets.

  • Dual spaces

    Continuous linear functionals on a normed space form its continuous dual.

  • Hahn–Banach theorem

    Appropriately bounded linear functionals extend from subspaces while preserving their bounds.

  • Hilbert spaces

    A Hilbert space is complete under a norm induced by an inner product.

  • Normed spaces

    A norm measures vector size with homogeneity, separation and the triangle inequality.

  • Operator spectrum

    The spectrum consists of scalars where subtracting the scalar identity fails to have a bounded inverse.

  • Riesz representation in Hilbert spaces

    Each continuous linear functional on a Hilbert space is represented by an inner product with a unique vector.

  • Self-adjoint operators

    For bounded Hilbert-space operators, self-adjointness means equality with the adjoint.

  • Spectral theorem

    Suitable self-adjoint or normal operators admit spectral decompositions; the precise form depends on the setting.

  • Weak convergence

    A sequence converges weakly when every continuous linear functional converges on it.

Linear algebra

  • Basis and dimension

    A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.

  • Condition numbers

    A matrix condition number describes sensitivity of linear-system solutions to perturbations.

  • Determinants

    The determinant of a square matrix tracks signed volume scaling and is nonzero exactly when the matrix is invertible.

  • Diagonalization

    A matrix is diagonalizable when it has a basis of eigenvectors; having repeated eigenvalues does not by itself decide diagonalizability.

  • Eigenvalues and eigenvectors

    A nonzero eigenvector is scaled by a linear operator; the scale factor is its eigenvalue.

  • Gaussian elimination

    Elementary row operations simplify a system while preserving its solution set, revealing pivots and free variables.

  • Gram–Schmidt process

    Independent vectors are orthogonalized by subtracting projections onto earlier directions.

  • Inner products

    An inner product defines lengths and orthogonality through a positive-definite bilinear or sesquilinear structure.

  • Least squares

    Least squares minimizes the squared residual norm; a minimizer corresponds to projecting the target onto the column space.

  • Linear independence

    A collection is linearly independent when the zero vector has only the trivial linear combination using that collection.

  • Linear systems

    A system Ax=b asks whether b is a linear combination of the columns of A and, if so, with what coefficients.

  • Linear transformations

    A linear transformation preserves vector addition and scalar multiplication; its matrix depends on the chosen bases.

  • Matrices

    A matrix is a rectangular array representing data or a linear map in chosen bases; matrix multiplication represents composition when dimensions agree.

  • Matrix norms

    Matrix norms quantify matrix size; induced norms measure maximum vector amplification.

  • Orthogonal complements

    Vectors orthogonal to every vector of a subspace form its orthogonal complement.

  • Orthogonal projections

    Orthogonal projection splits a vector into a component in a subspace and a component perpendicular to that subspace.

  • Orthonormal bases

    An orthonormal basis consists of mutually orthogonal unit vectors spanning the space.

  • Positive-definite matrices

    A symmetric or Hermitian matrix is positive definite when its quadratic form is positive on nonzero vectors.

  • Pseudoinverse

    The Moore–Penrose inverse generalizes matrix inversion to least-squares and rank-deficient problems.

  • QR factorization

    A matrix factors into an orthonormal-column factor and an upper-triangular factor under suitable dimensions.

  • Quadratic forms

    A quadratic form assigns a homogeneous degree-two scalar expression to a vector.

  • Rank and nullity

    The rank is the dimension of the image; rank plus nullity equals the dimension of the domain for a finite-dimensional linear map.

  • Singular value decomposition

    Unitary or orthogonal factors and a rectangular diagonal factor describe any finite matrix.

  • Singular values

    Singular values are nonnegative square roots of eigenvalues of the conjugate-transpose product.

  • Span

    The span of a set consists of all finite linear combinations of its vectors.

  • Tensor products

    Tensor products linearize bilinear maps and combine vector spaces.

  • Vector spaces

    A vector space specifies addition and scalar multiplication satisfying the vector-space axioms over a chosen field.

  • Vectors

    Vectors can be added and scaled; coordinate lists represent vectors relative to a choice of basis.

Ordinary Differential Equations

  • Bernoulli equations

    A power substitution converts a suitable nonlinear first-order Bernoulli equation into a linear one.

  • Characteristic equations

    An exponential trial converts a constant-coefficient homogeneous linear ODE into a polynomial equation.

  • Equilibria and stability

    Stability describes whether nearby trajectories remain close to or approach an equilibrium.

  • Exact differential equations

    A differential expression is exact when it is the differential of a potential function on the domain.

  • Existence and uniqueness for ODEs

    Continuity and suitable local Lipschitz conditions ensure a locally unique initial-value solution.

  • Initial value problems

    Initial data select solutions of a differential equation when existence and uniqueness conditions hold.

  • Integrating factors

    An integrating factor turns a linear first-order ODE into a product derivative.

  • Laplace transform

    An integral transform converts time-domain derivatives into algebraic relations under convergence conditions.

  • Linearization

    The Jacobian approximates a smooth system near an equilibrium; zero-real-part eigenvalues need further analysis.

  • Ordinary differential equations

    An ODE relates an unknown function of one variable to its derivatives.

  • Phase portraits

    A phase portrait displays trajectories in state space rather than against time.

  • Resonance

    Forcing near a natural frequency can produce large oscillatory responses depending on damping.

  • Second-order linear ODEs

    A second-order linear ODE combines a function and its first two derivatives linearly.

  • Separable equations

    Separation places dependent and independent variables on opposite sides before integration.

  • Systems of ODEs

    Coupled differential equations describe interacting state variables.

  • Undetermined coefficients

    A forcing-compatible trial form finds particular solutions of suitable constant-coefficient linear equations.

  • Variation of parameters

    Variable coefficients multiplying homogeneous solutions construct a particular solution of a forced linear ODE.

  • Wronskian

    A Wronskian tests independence of solutions under differential-equation hypotheses; arbitrary functions require care.

Abstract Algebra

  • Binary operations

    A binary operation combines two elements into one in a specified set.

  • Cosets and Lagrange's theorem

    Cosets partition a finite group, so a subgroup's order divides the group's order.

  • Cyclic groups

    A cyclic group is generated by repeated powers of a single element.

  • Field extensions

    A field extension contains a smaller field and can be viewed as its vector space.

  • Fields

    A field is a commutative ring where every nonzero element has a multiplicative inverse.

  • Galois theory

    Galois theory relates suitable field extensions to groups of automorphisms fixing the base field.

  • Group actions

    A group action assigns compatible transformations of a set to group elements.

  • Group homomorphisms

    A homomorphism preserves group multiplication; its kernel measures failure of injectivity.

  • Groups

    A group has an associative operation, an identity and an inverse for every element.

  • Ideals

    An ideal is an additive subgroup closed under multiplication by ring elements on the required sides.

  • Integral domains

    An integral domain is a nonzero commutative ring with identity and no nonzero zero divisors.

  • Modules

    Modules generalize vector spaces by allowing scalars from a ring rather than a field.

  • Normal subgroups

    A subgroup invariant under conjugation allows consistent multiplication of cosets.

  • Polynomial rings

    Polynomials over a ring form a ring with coefficientwise addition and polynomial multiplication.

  • Principal ideal domains

    Every ideal in a principal ideal domain is generated by one element.

  • Quotient groups

    Cosets of a normal subgroup inherit a well-defined group operation.

  • Quotient rings

    Reducing ring elements modulo an ideal gives a well-defined quotient ring.

  • Rings

    A ring supports addition and multiplication linked by distributive laws.

  • Subgroups

    A subgroup is a subset that forms a group under the inherited operation.

  • Sylow theorems

    For finite groups, Sylow theorems constrain existence, conjugacy and number of maximal prime-power subgroups.

  • Unique factorization domains

    Nonzero nonunits factor into irreducibles uniquely up to order and units.

Financial Mathematics

  • Binomial pricing models

    Up-and-down price moves support backward valuation under no-arbitrage assumptions.

  • Black–Scholes model

    Option pricing follows a diffusion market with restrictive volatility, trading and friction assumptions.

  • Compound interest

    Interest accrues on both principal and previously accumulated interest.

  • Geometric Brownian motion

    A multiplicative stochastic differential model keeps positive initial asset prices positive.

  • Mean–variance optimization

    Expected return and covariance describe a constrained portfolio optimization model.

  • Model risk

    Incorrect assumptions, calibration or use can invalidate model-based conclusions.

  • No-arbitrage

    A market model excludes zero-cost trades with nonnegative and sometimes positive payoff.

  • Portfolio risk

    Covariances link individual asset variability to the variability of weighted portfolios.

  • Replicating portfolios

    A trading strategy replicates a claim when its terminal payoff matches the claim under the model.

  • Risk-neutral measures

    Discounted traded-price processes are martingales under a suitable pricing measure.

  • Time value of money

    Discounting compares amounts at different dates under a specified rate model.

  • Value at Risk

    VaR is a loss quantile at a stated horizon and confidence level, not a maximum possible loss.

Combinatorics

  • Binomial theorem

    Binomial coefficients give the expansion of an integer nonnegative power of a sum.

  • Burnside's lemma

    For a finite group action, orbit count equals the average number of fixed points.

  • Catalan numbers

    Catalan numbers count several recursively structured objects, including balanced parenthesis strings.

  • Integer partitions

    A partition writes a positive integer as an unordered sum of positive integers.

  • Ramsey theory

    Sufficiently large structures force specified regular substructures under suitable coloring conditions.

Biomathematics

  • Biological parameter inference

    Data constrain model parameters, with identifiability and uncertainty affecting interpretation.

  • Evolutionary dynamics

    Selection, mutation and interaction change trait or strategy frequencies.

  • Exponential growth

    A constant per-capita growth rate yields exponential population change.

  • Logistic growth

    Density-dependent growth slows toward a carrying capacity in the basic logistic model.

  • Lotka–Volterra models

    Coupled equations represent predator–prey or competition interactions under simplified assumptions.

  • Michaelis–Menten kinetics

    A saturating enzyme-rate law arises under appropriate kinetic and approximation assumptions.

  • Population matrix models

    Age or stage structure produces matrix updates of population counts.

  • Reaction kinetics

    Reaction rates determine concentration changes through stoichiometry and kinetic assumptions.

  • Reaction–diffusion systems

    Local reactions and spatial diffusion jointly evolve concentrations or populations.

  • Reproduction number

    A reproduction number summarizes secondary infections in a specified epidemic setting.

  • SIR models

    Susceptible, infectious and recovered compartments evolve through transmission and recovery rules.

  • Turing patterns

    Diffusion can destabilize a reaction equilibrium and create spatial patterns under suitable parameter conditions.

Numerical Analysis and Scientific Computing

  • Bisection method

    Continuity and opposite endpoint signs allow repeated halving to bracket a root.

  • Conditioning

    Conditioning measures sensitivity of a problem's exact solution to perturbations in its data.

  • Euler method

    Euler's method advances an ODE solution using its current derivative and a step size.

  • Finite differences

    Differences of nearby samples approximate derivatives, balancing truncation and roundoff errors.

  • Floating-point arithmetic

    Finite-precision arithmetic introduces rounding and has bounded representable ranges.

  • Iterative linear solvers

    Repeated approximations solve large linear systems when the iteration satisfies convergence conditions.

  • LU factorization

    With appropriate pivoting, triangular factors reduce linear-system solution to substitution.

  • Newton's method

    A tangent-based iteration approximates a root but convergence depends on the initial point and local hypotheses.

  • Numerical quadrature

    Weighted function samples approximate definite integrals with error depending on smoothness and sampling.

  • Numerical stability

    A stable algorithm controls amplification of rounding and data errors relative to the problem.

  • Polynomial interpolation

    Distinct nodes determine a unique polynomial of bounded degree passing through their data.

  • Runge–Kutta methods

    Multiple derivative evaluations per step improve ODE approximation accuracy.

  • Splines

    Piecewise polynomials joined with specified smoothness approximate data and curves.

  • Stiff equations

    Widely separated decay scales can force explicit methods to take very small stable steps.

Mathematical and Real Analysis

  • Bolzano–Weierstrass theorem

    Every bounded sequence in finite-dimensional Euclidean space has a convergent subsequence.

  • Cauchy sequences

    Terms of a Cauchy sequence eventually become arbitrarily close to each other.

  • Compactness

    A space is compact when every open cover admits a finite subcover.

  • Completeness of the real numbers

    Every nonempty real set bounded above has a least upper bound.

  • Epsilon–delta limits

    Every prescribed output tolerance must follow from a sufficiently small nonzero input displacement.

  • Extreme value theorem

    A continuous real function on a nonempty compact set attains its maximum and minimum.

  • Interchanging limits

    Exchanging limits, derivatives or integrals requires hypotheses such as suitable uniform convergence.

  • Intermediate value theorem

    A continuous real function on an interval takes every value between two of its values.

  • Limit superior and inferior

    Tail suprema and infima describe the extreme limiting behavior of a sequence.

  • Metric spaces

    A metric assigns nonnegative distances satisfying separation, symmetry and the triangle inequality.

  • Pointwise convergence

    A sequence of functions converges separately at each point, with tolerances allowed to depend on the point.

  • Power series

    A power series converges inside its convergence radius; boundary points require separate tests.

  • Riemann integrability

    Riemann integrability requires upper and lower sums to approach the same value for bounded functions.

  • Sequence convergence

    A sequence converges when its terms eventually lie within every prescribed distance of a limit.

  • Uniform continuity

    A single input tolerance controls output error uniformly across the domain.

  • Uniform convergence

    One index bounds the approximation error at every point of the domain.

Partial Differential Equations

  • Boundary value problems

    Boundary data restrict a differential equation's solution on the edge of its domain.

  • Energy estimates

    Bounds on integrated solution quantities help establish stability and uniqueness.

  • Finite element method

    Piecewise basis functions approximate weak formulations over a subdivided domain.

  • Heat equation

    The heat equation models diffusion through time and spatial second derivatives.

  • Laplace and Poisson equations

    The Laplacian equals zero for harmonic fields or a source term in Poisson's equation.

  • Method of characteristics

    Characteristic curves reduce suitable first-order PDEs to equations along paths.

  • Partial differential equations

    A PDE relates a multivariable unknown function to its partial derivatives.

  • PDE classification

    Second-order principal terms distinguish elliptic, parabolic and hyperbolic behavior under appropriate assumptions.

  • Separation of variables

    Product-form trial solutions reduce suitable PDEs to ordinary differential equations.

  • Wave equation

    The wave equation links temporal and spatial second derivatives to describe propagation.

  • Weak solutions

    Integral identities define solutions whose classical derivatives may not exist.

  • Well-posedness

    A well-posed problem has a solution, uniqueness and continuous dependence on data.

Stochastic Processes

  • Brownian motion

    Brownian motion has continuous paths and independent Gaussian increments with variance equal to elapsed time.

  • Filtrations

    An increasing family of sigma-algebras represents information available over time.

  • Itô calculus

    Stochastic integration against Brownian motion includes a quadratic-variation correction in change-of-variable formulas.

  • Markov chains

    A Markov chain's next-state distribution depends on the present state rather than the full history.

  • Martingales

    An integrable adapted process has conditional expected future value equal to its present value.

  • Poisson processes

    A homogeneous Poisson process counts events with stationary independent increments at a constant rate.

  • Queueing processes

    Arrival and service mechanisms determine random queue lengths and waiting times.

  • Renewal processes

    Independent identically distributed waiting times generate repeated renewal events.

  • Stationary distributions

    A stationary distribution remains unchanged after applying the chain's transition rule.

  • Stochastic processes

    A stochastic process is a time-indexed collection of random variables.

  • Stopping times

    Whether a stopping time has occurred by a time is determined by information available then.

  • Transition matrices

    A finite Markov chain uses a nonnegative matrix with row sums one to describe transitions.

Mathematical Physics

  • Calculus of variations

    Variational calculus studies extrema of functionals depending on functions.

  • Dimensional similarity

    Dimensionless groups compare physical systems while respecting the dimensions of quantities.

  • Distributions as generalized functions

    A distribution acts linearly and continuously on test functions, allowing generalized derivatives.

  • Euler–Lagrange equations

    Stationarity of a smooth variational problem gives differential conditions with appropriate boundary constraints.

  • Fourier series

    Periodic functions can be expanded in harmonic components with convergence interpreted under suitable hypotheses.

  • Fourier transform

    A Fourier transform represents a suitable function through continuous frequency components.

  • Green's functions

    A Green's function is a kernel representing a linear operator's response to a point source with specified boundary conditions.

  • Hamiltonian systems

    A Hamiltonian function generates evolution of paired position and momentum variables.

  • Noether's theorem

    Continuous variational symmetries lead to conservation laws under the theorem's hypotheses.

  • Special functions

    Functions such as Bessel and Legendre functions solve recurring differential equations in geometry and physics.

  • Sturm–Liouville problems

    Self-adjoint second-order boundary problems yield eigenvalues and orthogonal eigenfunctions under suitable conditions.

  • Tensors

    Tensors encode multilinear quantities with consistent transformation laws under coordinate changes.

Algebraic Topology

  • Chain complexes

    Successive boundary maps compose to zero in a graded sequence of groups or modules.

  • Cohomology

    Dual cochain constructions yield invariants and additional operations such as cup products.

  • Covering spaces

    A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.

  • Euler characteristic

    Alternating cell or homology-rank counts define a topological invariant in suitable finite settings.

  • Exact sequences

    At each position, the incoming image equals the outgoing kernel.

  • Higher homotopy groups

    Based maps from spheres modulo homotopy measure higher-dimensional homotopy structure.

  • Homology groups

    Cycles modulo boundaries measure holes through a chain complex.

  • Homotopy equivalence

    Maps with inverse compositions homotopic to identities describe spaces with the same homotopy type.

  • Persistent homology

    Homological features are tracked across a nested family of spaces or data scales.

  • Simplicial complexes

    Simplices fitted together along faces provide combinatorial models of spaces.

Information and Coding Theory

  • Channel capacity

    Capacity bounds achievable reliable information rates for a specified channel model.

  • Conditional entropy

    Conditional entropy averages remaining uncertainty after another variable is observed.

  • Error correction bounds

    Minimum distance d permits correcting up to floor((d-1)/2) arbitrary symbol errors.

  • Hamming distance

    Hamming distance counts differing positions between equal-length strings.

  • Huffman coding

    A greedy merging procedure builds an optimal binary prefix code for given symbol probabilities.

  • Linear codes

    Codewords forming a finite-field vector subspace enable algebraic encoding and decoding.

  • Mutual information

    Mutual information measures statistical dependence as the reduction in uncertainty from observing another variable.

  • Parity checks

    A parity-check matrix produces a syndrome that diagnoses violations of code constraints.

  • Prefix codes

    No codeword is a prefix of another, enabling unambiguous immediate decoding.

  • Relative entropy

    KL divergence compares distributions and is generally asymmetric, with support conditions affecting finiteness.

  • Shannon entropy

    Entropy is expected negative log probability for a discrete distribution.

  • Source coding

    Source coding represents messages efficiently, with entropy constraining lossless average code length.

Representation Theory

  • Characters

    A finite-dimensional representation's character assigns the trace of each represented element.

  • Complete reducibility

    A representation is completely reducible when it is a direct sum of irreducible representations.

  • Group representations

    A group representation is a homomorphism into invertible linear transformations.

  • Induced representations

    A subgroup representation can be extended through an induction construction to the larger group.

  • Intertwining operators

    An intertwiner is a linear map commuting with the represented group actions.

  • Invariant subspaces

    An invariant subspace is preserved by every transformation in a representation.

  • Irreducible representations

    An irreducible representation has no nonzero proper invariant subspace.

  • Lie algebras

    A Lie algebra has a bilinear alternating bracket satisfying the Jacobi identity.

  • Maschke's theorem

    Finite-group representations are completely reducible when the field characteristic does not divide group order.

  • Schur's lemma

    Nonzero intertwiners between irreducible representations are isomorphisms; scalar conclusions need field hypotheses.

  • Tensor products of representations

    Actions on factor spaces induce a compatible action on their tensor product.

  • Weights and roots

    Weights and roots organize representations and structure relative to suitable commuting subalgebras.

Cryptography

  • Chinese remainder theorem

    Pairwise coprime moduli give a unique combined congruence class modulo their product.

  • Cryptographic hashes

    Cryptographic hashes aim at preimage and collision resistance; hashing alone is not encryption.

  • Cryptographic security models

    Security depends on defined adversary capabilities, assumptions and failure probability.

  • Diffie–Hellman key agreement

    Parties derive a shared secret through group exponentiation; authentication is needed against interception.

  • Digital signatures

    A private signing key and public verification key authenticate messages under security assumptions.

  • Elliptic curve cryptography

    Finite-field elliptic curve groups support cryptography based on hard discrete logarithms.

  • Modular arithmetic

    Congruence identifies integers with the same remainder modulo a positive integer.

  • Modular inverses

    An integer has an inverse modulo n exactly when it is coprime to n.

  • Public-key encryption

    Public encryption keys and private decryption keys have different roles.

  • RSA

    RSA uses modular exponentiation with related exponents; secure encryption requires appropriate padding and parameters.

  • Symmetric encryption

    A shared secret key supports encryption and decryption under a specified scheme.

  • Zero-knowledge proofs

    A proof can establish a statement without revealing extra witness information in a formal security model.

Discrete mathematics

  • Combinations

    Choose k distinct elements from an n-element set without order, for integers 0≤k≤n0\le k\le n. Each chosen subset corresponds to k!(n−kn-k)! full permutations. Repetition-allowed selections are a different counting problem and need their own assumptions.

  • Combinatorial counting

    Count finite outcomes with disjoint-case sums, successive-choice products and equal-fiber division or bijections. Check that cases are disjoint and multiplicities are constant. Multiplication of choice counts is not a probability-independence assumption.

  • Euclidean algorithm

    The Euclidean algorithm computes a greatest common divisor by repeated division with remainder. Replacing (a,b) by (b,r) preserves the common divisors, and the strictly smaller nonnegative remainders force termination.

  • Greatest common divisor

    For integers a and b that are not both zero, the greatest common divisor is the unique positive integer dividing both that is divisible by every other common divisor. Signs do not change its positive value.

  • Permutations

    Permutations record order. Distinct n objects have n! complete arrangements; indistinguishable repeated groups require dividing out their internal exchanges. Specify the multiplicities, whether all objects are used and whether rotations count as different arrangements.

Topology

  • Connectedness

    A connected space cannot be separated into two disjoint nonempty open pieces.

  • Fundamental group

    Based loops modulo endpoint-fixed homotopy form a group under concatenation.

  • Hausdorff spaces

    Distinct points admit disjoint neighborhoods in a Hausdorff space.

  • Homeomorphisms

    A homeomorphism is a bijection continuous in both directions.

  • Homotopy

    A homotopy continuously deforms one map into another.

  • Open and closed sets

    Closed sets have open complements; a set can be both open and closed.

  • Path connectedness

    Every pair of points in a path-connected space can be joined by a continuous path.

  • Product and quotient topologies

    Product and quotient constructions create topologies compatible with coordinate maps or identifications.

  • Topological continuity

    A map is continuous when inverse images of open sets are open.

  • Topological manifolds

    A manifold locally resembles Euclidean space with suitable separation and countability conditions.

  • Topological spaces

    A topology specifies open sets closed under arbitrary unions and finite intersections.

Optimization and Operations Research

  • Convex functions

    A convex function lies below chords connecting points on its graph.

  • Convex optimization

    Minimizing a convex function on a convex feasible set makes local minima global.

  • Convex sets

    Every segment joining two points of a convex set stays inside the set.

  • Gradient descent

    Iterates move opposite the gradient; step-size and smoothness conditions control convergence.

  • Newton optimization

    Gradient and Hessian information form a local quadratic step, often with globalization safeguards.

Graph Theory

  • Graph coloring

    Proper vertex coloring assigns different colors to adjacent vertices.

  • Graph connectivity

    Paths determine connected components; vertex and edge connectivity quantify robustness to removal.

  • Graph Laplacian

    For a simple undirected graph, degree matrix minus adjacency matrix encodes connectivity and diffusion.

  • Planar graphs

    A planar graph can be drawn without edge crossings except at shared endpoints.

  • Vertex degrees

    Degree counts incident edge ends; directed graphs distinguish incoming and outgoing degrees.