Mathematics and statistics

Shannon–Hartley theorem

The Shannon–Hartley theorem, or Shannon–Hartley law, gives the maximum error-free data rate of a noisy channel, C = W log2(1 + S/N), stated by Claude Shannon in 1948.

Shape

Shape is the form of an object, defined as everything about a figure that remains after removing position, orientation, and scale, studied across geometry, statistics, and computing.

Shape derivative

The shape derivative measures how a shape functional, a function assigning a number to a domain, changes when the domain is smoothly deformed, and is central to shape optimization.

A shape grammar is a rule-based production system that generates shapes by recursively applying shape-replacement rules, introduced by George Stiny and James Gips in 1971 and widely used in architectural design.

Shapiro–Wilk test

The Shapiro–Wilk test is a statistical test of whether a data sample was drawn from a normal distribution, proposed by Samuel Shapiro and Martin Wilk in 1965.

Shapley value

The Shapley value is a cooperative game theory solution concept giving each player a fair payoff, introduced by Lloyd S. Shapley in 1953 and uniquely determined by fairness axioms.

Sheaf (mathematics)

A sheaf (mathematics) assigns data to the open sets of a topological space so that compatible local data glue uniquely; introduced by Jean Leray in the 1940s.

Sheaf cohomology

Sheaf cohomology is the cohomology theory attached to a sheaf on a topological space or scheme, measuring the failure of local sections to glue into global ones.

Sheafification

Sheafification is the construction that turns a presheaf into a sheaf, its best approximation, and the standard way to build quotient sheaves, image sheaves, and inverse image sheaves.

Shear locking

Shear locking is a numerical artifact of the finite element method in which thin beam, plate, or shell elements become artificially stiff, giving deflections far too small.

Shear mapping

Shear mapping, or shear transformation, is a linear map that shifts each point parallel to a fixed line by an amount proportional to its distance from it, preserving area.

Shearlet transform

The shearlet transform is a multiscale mathematical transform representing images with scaled, sheared, and translated wavelet-like functions, introduced in 2005 and used in denoising, compression, and tomography.

Sheffer sequence

A Sheffer sequence, also called a Sheffer polynomial or poweroid, is a polynomial sequence in umbral calculus that includes the Hermite, Laguerre, and Charlier polynomials.

Sheffer stroke

The Sheffer stroke, or NAND, is a binary logical connective false only when both inputs are true, named for Henry M. Sheffer, who published it in 1913.

Shelah cardinal

A Shelah cardinal is a large cardinal in set theory, introduced by Saharon Shelah and Hugh Woodin in 1990, lying between Woodin and superstrong cardinals in consistency strength.

Shift–Or algorithm

The Shift–Or algorithm is a bit-parallel string matching method that finds all occurrences of a pattern in a text using only shifts and bitwise logical operations, scanning in O(n) time.

Shimura variety

A Shimura variety is a higher-dimensional analog of a modular curve, introduced by Goro Shimura in the 1960s; examples include modular, Siegel, and Hilbert modular varieties.

Shinichi Mochizuki (望月新一)

Shinichi Mochizuki (望月新一) is a Japanese mathematician at Kyoto University's RIMS, known for anabelian geometry and for his disputed claimed proof of the abc conjecture.

Shock-capturing scheme

A shock-capturing scheme is a numerical method for hyperbolic conservation laws that resolves shock waves and other discontinuities automatically, smearing them over grid cells instead of tracking the front explicitly.

Shoelace formula

The shoelace formula, also known as Gauss's area formula or the surveyor's formula, is a mathematical algorithm that computes a simple polygon's area from its vertices' coordinates.

Shooting method

The shooting method is a numerical technique for solving boundary value problems of ordinary differential equations, adjusting guessed initial conditions until the boundary conditions hold.

Shortest path algorithm

Shortest path algorithms find minimum cost routes through weighted graphs, applying to network protocols and traffic planning. They are typically organized into single-source (SSSP) or all-pairs (APSP) problems.

Shortest path problem

The shortest path problem, also called the single-pair shortest-path problem, asks for a minimum-weight path between two vertices in a weighted graph, underlying routing and pathfinding systems.

Shortest vector problem

The shortest vector problem (SVP) is the computational problem of finding the shortest nonzero vector in a lattice, a problem NP-hard to approximate and central to post-quantum cryptography.

Shrinkage estimation

Shrinkage estimation is a statistical technique that biases estimates toward zero or a target value, reducing variance and lowering overall mean squared error, as in the James–Stein estimator.

Shrinkage estimator

A shrinkage estimator is a statistical estimator that deliberately pulls an ordinary estimate toward a central value such as zero, trading added bias for lower variance.

Shuffled frog leaping algorithm

The shuffled frog leaping algorithm (SFLA) is a population-based memetic metaheuristic for discrete and combinatorial optimization, introduced by Eusuff and Lansey in 2003 for water distribution network design.

Sidon sequence

A Sidon sequence, also called a B₂-sequence or Sidon set, is a sequence of integers in which all pairwise sums are distinct, introduced by Simon Sidon in 1932.

Siegel zero

A Siegel zero, or Landau–Siegel zero, is a hypothetical real zero very close to 1 of a Dirichlet L-function, a central open problem in analytic number theory.