Mathematics

Policy iteration solves sequential decision problems by repeatedly evaluating a policy and improving its action choices until an optimal policy is obtained.

A polyhedron is a three-dimensional geometric object with polygonal faces; in convex mathematics, the term also denotes an intersection of finitely many half-spaces.

A polynomial is a finite sum of terms formed from coefficients and variables raised to nonnegative integer powers.

A symmetric or Hermitian matrix whose quadratic form is strictly positive for every nonzero vector.

A posterior distribution describes uncertainty about unknown quantities after observed data are incorporated through a probabilistic model.

Posterior predictive checking assesses a Bayesian model by comparing observed data with replicated data generated from its posterior predictive distribution.

A posterior predictive distribution describes unobserved outcomes by averaging their conditional distributions over Bayesian posterior uncertainty about model parameters.

A power law relates quantities through a constant exponent, producing scale-invariant relationships and, in probability, distributions with slowly decaying tails.

A power series is an infinite sum of nonnegative integer powers of a variable, used to represent functions and encode sequences.

The power set of a set is the collection of all its subsets, including the empty set and the original set.

A prime number is an integer greater than one whose only positive divisors are one and itself, providing the basic factors of all positive integers.

The prime number theorem states that the number of primes not exceeding x is asymptotic to x divided by the natural logarithm of x.

Principal component analysis transforms correlated variables into orthogonal components ordered by explained variance, enabling linear dimensionality reduction and data exploration.

A prior distribution represents uncertainty about an unknown quantity before the data used in a particular Bayesian analysis are incorporated.

A probabilistic graphical model represents a probability distribution through a graph and local functions, enabling structured reasoning about uncertainty.

Probability is the branch of mathematics that measures how likely uncertain events are, using numbers from 0 to 1.

A probability density function describes an absolutely continuous probability distribution, with probabilities obtained by integration over sets of possible values.

A probability distribution specifies how probability is assigned to the possible values of a random variable or to measurable sets of outcomes.

A probability mass function assigns probabilities to the individual values of a discrete random variable.

A probability space is a mathematical structure consisting of possible outcomes, measurable events, and a measure assigning probabilities to those events.

The product topology equips a Cartesian product of topological spaces with the coarsest topology making every coordinate projection continuous.

Projective geometry studies geometric properties preserved by projective transformations, emphasizing incidence, duality, and cross-ratio rather than distance and angle.

Proof by contradiction establishes a proposition by showing that its negation, together with accepted premises, entails a logical contradiction.

Proof theory studies formal proofs, their structure and transformations, and the strength and computational content of mathematical theories.

The Pythagorean theorem relates the side lengths of a right triangle and underlies Euclidean distance, coordinate geometry, and orthogonal decompositions.

The QR algorithm computes matrix eigenvalues and Schur decompositions through repeated orthogonal or unitary similarity transformations.

QR decomposition factors a matrix into orthogonal or unitary and triangular factors, supporting stable least-squares solutions and eigenvalue computations.

A quotient set is the set of equivalence classes determined by an equivalence relation on a given set.

A quotient space identifies equivalent points of a topological space and equips the resulting set with the finest topology making the canonical projection continuous.

A quotient vector space identifies vectors whose difference belongs to a specified subspace, inheriting addition and scalar multiplication from the original space.

A random variable is a measurable function that assigns a numerical value to each outcome in a probability space.

A random walk is a stochastic process whose position changes through successive randomly determined steps.

A theorem stating that a linear map’s rank plus its nullity equals the dimension of its finite-dimensional domain.

A rectified linear unit is a neural-network activation function that outputs zero for negative inputs and leaves positive inputs unchanged.

A recurrence relation specifies terms of a sequence through other terms, usually earlier ones, and initial conditions select a particular solution.

Regularization stabilizes mathematical estimation problems and controls model complexity through penalties, constraints, or changes to the learning process.

Representation theory studies algebraic structures through their actions on vector spaces, translating abstract symmetry into linear algebra.

Richard Bellman was an American applied mathematician who developed dynamic programming and advanced the mathematical study of sequential decisions and control.

The Riemann curvature tensor measures intrinsic curvature through the noncommutativity of covariant differentiation and governs parallel transport and geodesic deviation.

The Riemann hypothesis asserts that every nontrivial zero of the Riemann zeta function has real part one-half, governing fluctuations in the distribution of primes.

The Riemann integral defines accumulated quantities through limits of finite sums and exists for bounded functions whose discontinuities form a set of measure zero.

A Riemann sum approximates an integral by adding function values multiplied by the widths of corresponding subintervals.

The Riemann zeta function is a complex function whose analytic properties encode fundamental information about prime numbers and their distribution.

A ring is an algebraic structure with addition and multiplication satisfying laws that generalize integer arithmetic.

British statistician and geneticist who developed foundational methods of statistical inference, experimental design, and population genetics.

A rule of inference specifies a permissible step in a formal derivation, allowing a conclusion to be obtained from premises under stated conditions.

A saddle point is a stationary point with nearby values both above and below it, or a solution defined by opposing minimization and maximization conditions.

The sample mean is the arithmetic average of observed values, used to describe a dataset’s center and estimate a population mean.

A sample space is the set of all possible outcomes represented in a probability model, forming the underlying domain on which events and random variables are defined.

The probability distribution of a statistic across repeated samples drawn under a specified sampling process.

A factorization expressing any complex square matrix as a unitary similarity of an upper triangular matrix, with a real block-triangular counterpart.

Second-order logic extends first-order logic by allowing quantification over properties, relations, and functions, with expressive power and proof-theoretic behavior determined by its semantics.

A self-adjoint operator equals its adjoint, including its domain, and admits a real spectral representation fundamental to analysis and quantum mechanics.

Self-information is the negative logarithm of an event’s probability, measuring how surprising its occurrence is under a specified probability model.

A separable space is a topological space containing a countable dense subset, allowing its points to be approximated by a countable collection.

A family of formal proof systems that represents deductions as transformations of sequents, central to proof theory and automated reasoning.

A set partition divides a set into nonempty, mutually disjoint subsets whose union is the entire set.

A sigma-algebra is a collection of subsets closed under complements and countable unions, providing the framework for measurable sets and probabilistic events.

A significance level is a prespecified bound on the probability of rejecting a true null hypothesis in statistical hypothesis testing.

Singular value decomposition factors any real or complex matrix into orthogonal or unitary directions and nonnegative scaling factors, enabling rank analysis, approximation, and least-squares computation.