Mathematics and statistics

Stochastic matrix

A stochastic matrix, also called a Markov matrix or transition probability matrix, is a square matrix whose rows or columns sum to 1, so each is a probability distribution.

The stochastic maximum principle states necessary conditions for optimality of controls of systems governed by stochastic differential equations, generalizing Pontryagin's deterministic maximum principle to the stochastic setting.

Stochastic orders of dependence

Stochastic orders of dependence are mathematical orderings that rank probability distributions by dependence strength, with the concordance ordering, formalized by Harry Joe in 1990, as the central example.

Stochastic partial differential equation

A stochastic partial differential equation (SPDE) is a PDE perturbed by random forcing or coefficients, describing systems driven by noise, from growing interfaces and turbulent fluids to climate models.

Stochastic process

A stochastic process, also called a random process, is a family of random variables indexed by time that describes a system evolving under random influences.

Stochastic programming

Stochastic programming is a mathematical optimization framework for decision-making under uncertainty, optimizing an expected outcome or risk measure when some model parameters are random variables with known probability distributions.

Stochastic simulation

Stochastic simulation is a computational method that generates random sample paths of a system and estimates expectations from the outputs, used when quantities are mathematically intractable.

Stockwell transform

The Stockwell transform, or S-transform, is a time-frequency analysis method introduced in 1996 that applies a frequency-dependent Gaussian window, used widely in seismology and power systems.

Stokes solver

A Stokes solver is a numerical method for computing velocity and pressure fields in slow, viscous, incompressible flow, used in biological, suspension, and mantle convection simulations.

Stokes–Darcy model

The Stokes–Darcy model couples free-fluid flow, governed by the Stokes equations, with flow through a porous medium, governed by Darcy's law, linked by interface conditions on mass, forces, and tangential velocity.

Stokes' theorem

Stokes' theorem, also called the Kelvin–Stokes theorem, is a vector calculus result relating a line integral around a closed curve to the surface integral of the curl.

Stone duality

Stone duality is a correspondence in mathematics identifying every Boolean algebra with the clopen subsets of a compact, totally disconnected Stone space, established by Marshall H. Stone in the 1930s.

Stone space

A Stone space, also called a Boolean or profinite space, is a compact, Hausdorff, totally disconnected space dual to a Boolean algebra via Stone's 1936 theorem.

Stone–Čech compactification

The Stone–Čech compactification βX of a topological space X is the largest compact Hausdorff compactification, into which every bounded continuous real-valued function on X extends uniquely.

Stone–von Neumann theorem

The Stone–von Neumann theorem, or Weyl relations theorem, states that quantum canonical commutation relations have essentially one irreducible representation, proved by von Neumann in 1931.

Stone–Weierstrass theorem

The Stone–Weierstrass theorem is a result of real analysis, proved by Marshall Stone in 1937, generalizing Weierstrass's approximation theorem: any point-separating subalgebra of continuous functions on a compact space is dense.

Stone's representation theorem for Boolean algebras

Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the clopen subsets of its Stone space; Marshall Stone proved it in 1936.

Stopping time

A stopping time, also known as a Markov time or optional time, is a random time at which a stochastic process may be stopped, decidable from observed information alone.

Strange attractor

A strange attractor is an attractor of a dynamical system with a complicated, typically fractal structure rather than a point or closed curve, arising in chaotic systems whose long-term behavior appears unpredictable.

Strange loop

A strange loop is a hierarchical system that twists back on itself so moving through its levels returns to the start, coined by Douglas Hofstadter in 1979.

Strassen algorithm

The Strassen algorithm, or Strassen's method, is a recursive matrix multiplication algorithm Volker Strassen published in 1969 that uses seven multiplications of half-sized submatrices instead of eight, the first with sub-cubic complexity.

Strategy (game theory)

In game theory, a strategy is a complete plan of action specifying a player's move in every possible state, including states never reached in play.

Stratification score

The stratification score is the estimated probability of case status given confounders, used in retrospective studies to form strata balancing cases and controls, the case-control analog of the propensity score.

Stratified analysis (statistics)

Stratified analysis is a statistical method for controlling confounding by splitting data into subgroups sharing a confounder's level, then pooling stratum-specific estimates into an adjusted overall effect.

Stratified sampling

Stratified sampling, or stratified random sampling, is a probability sampling method that divides a population into homogeneous subgroups called strata, sampling independently within each to improve precision.

Stratonovich integral

The Stratonovich integral is a stochastic integral, introduced by R. L. Stratonovich in 1966, that evaluates the integrand at interval midpoints and obeys the ordinary chain rule of calculus.

Straw man

A straw man is an informal fallacy in which a speaker misrepresents an opponent's position and then refutes the distorted version instead of the real one.

Streaming model (algorithms)

The streaming model is a computational model where input arrives as a sequential stream and an algorithm with small memory processes it in one or a few passes.

Strichartz estimates

Strichartz estimates are space-time integral inequalities for dispersive partial differential equations, such as the Schrödinger and wave equations, introduced by Robert Strichartz in 1977 and now a fundamental tool for nonlinear dispersive equations.