Mathematics

The Cauchy–Schwarz inequality bounds the absolute inner product of two vectors by the product of their norms, with equality precisely when the vectors are linearly dependent.

A family of probability theorems describing when suitably standardized sums of random variables converge in distribution to a normal distribution.

The chain rule expresses the derivative of a composite function in terms of the derivatives of its component functions.

Channel capacity is the supremum of information rates achievable over a specified communication channel with decoding error probability tending to zero.

The Chapman–Kolmogorov equations express how Markov transition probabilities compose across successive time intervals by summing or integrating over intermediate states.

A polynomial associated with a square matrix or finite-dimensional linear operator whose roots are its eigenvalues, counted with algebraic multiplicity.

Chebyshev’s inequality bounds the probability that a random variable deviates from its mean using only its variance.

Cholesky decomposition factors a positive-definite matrix into a triangular matrix and its conjugate transpose, enabling efficient linear-system solutions and statistical computations.

Claude Shannon was an American mathematician and engineer who founded information theory and established key principles of digital circuit design.

A closed set is a subset of a topological space whose complement is open; in metric spaces, it contains every limit of its convergent sequences.

The closure of a subset of a topological space is the smallest closed set containing it, equivalently the set of points whose every neighborhood meets that subset.

Cluster analysis groups observations according to similarity or statistical structure, revealing patterns without requiring predefined class labels.

The codomain is the specified target set of a function, containing every output but not necessarily consisting only of outputs actually attained.

The coefficient of determination, usually denoted R², measures a model’s reduction in squared prediction error relative to a specified baseline.

Combinatorics studies discrete structures, their enumeration, existence, properties, and optimal arrangements.

A compact space is a topological space in which every open cover contains a finite subcover, generalizing key properties of closed, bounded subsets of Euclidean space.

The compactness theorem states that a set of first-order sentences has a model exactly when every finite subset has a model.

A complete metric space is a metric space in which every Cauchy sequence converges to a point belonging to the space.

Complex analysis studies functions of complex variables, especially holomorphic functions, their integrals, singularities, and geometric properties.

A condition number measures how strongly small changes in a problem’s input can affect its solution.

Conditional entropy measures the average uncertainty remaining about a random variable when another variable is known.

Conditional expectation is the mean of a random variable given specified information, defined rigorously by measurability and integral-preservation conditions.

Conditional independence means that, given specified information, learning one random variable provides no additional information about another.

Conditional probability measures the probability of an event when specified information or another event is taken as given.

A confidence interval estimates an unknown parameter using a procedure calibrated to cover its true value at a specified long-run frequency.

A conic section is a plane curve obtained by intersecting a cone with a plane, encompassing circles, ellipses, parabolas, and hyperbolas.

A conjugate prior is a prior distribution whose family is preserved when it is updated by a specified likelihood.

The conjugate transpose of a complex matrix exchanges its rows and columns and replaces each entry by its complex conjugate.

A consistent estimator converges in probability to its target parameter as the sample size tends to infinity.

A continuous function preserves limits: sufficiently small changes in its input produce arbitrarily small changes in its output.

A stochastic process on a finite or countable state space that evolves in continuous time and satisfies the Markov property.

The continuum hypothesis asserts that no infinite cardinality lies strictly between those of the natural numbers and the real numbers.

A contraction mapping uniformly reduces distances by a factor less than one, guaranteeing a unique fixed point when it maps a nonempty complete metric space into itself.

Convergence in probability means that the probability of any fixed positive deviation from a limiting random variable tends to zero.

A convex combination is a weighted sum of points with nonnegative real coefficients whose sum is one.

A convex function assigns to a weighted average of inputs a value no greater than the corresponding weighted average of its values.

The convex hull of a set is the smallest convex set containing it, equivalently the set of all finite convex combinations of its points.

Convex optimization studies the minimization of convex functions over convex feasible sets, combining global optimality guarantees with structured numerical methods.

A convex set contains the entire line segment joining any two of its points, making it a fundamental object in geometry, analysis, and optimization.

Convolution combines functions or sequences through shifted products, linking mathematical analysis, probability, signal processing, and neural networks.

A coset is a translate of a subgroup, used to partition groups, count subgroup indices, and construct quotient structures.

Cosine similarity measures the directional alignment of two nonzero vectors using their normalized inner product, independently of their magnitudes.

A countable set is a finite set or an infinite set whose elements can be put in one-to-one correspondence with the natural numbers.

Covariance measures how two random variables vary together through the expected product of their deviations from their means.

A covariance matrix records the variances and pairwise covariances of a random vector, describing its second-order variability and linear dependence.

A covariant derivative differentiates vector and tensor fields using a connection, making the result independent of the chosen coordinates or local frame.

A credible interval is a range containing a specified posterior probability for an unknown quantity under a Bayesian statistical model.

Cross-entropy measures the expected logarithmic loss incurred when one probability distribution is used to describe outcomes generated by another.

Cross-validation estimates predictive performance by repeatedly fitting models on subsets of data and evaluating them on held-out observations.

A cumulative distribution function gives the probability that a real-valued random variable is less than or equal to a specified threshold, uniquely characterizing its distribution.

A structural relationship between logical propositions and types, proofs and programs, and proof simplification and computation.

The curse of dimensionality describes geometric, statistical, and computational difficulties that arise as the number of dimensions in a problem increases.

A graphical criterion that identifies conditional independence relations implied by a directed acyclic graph.

De Morgan’s laws describe how negation interchanges conjunction and disjunction, with corresponding identities for sets and quantified statements.

Decision theory studies how choices can be evaluated using preferences, uncertainty, consequences, and formal criteria of rationality.

Degrees of freedom describe independent variation remaining after constraints, with applications in mathematics, statistical inference, and mechanics.

A dense set is a subset whose closure is the entire ambient space, so that every nonempty open region contains one of its points.

A derivative measures a function’s instantaneous rate of change and describes its local linear behavior.

Detailed balance is the condition that every transition between two states is exactly balanced by its reverse under a specified stationary distribution.

A determinant is a scalar associated with a square matrix that characterizes invertibility and, over the real numbers, signed volume scaling.