Mathematics and statistics

Scalar curvature

Scalar curvature, also called the Ricci scalar, is a single number at each point of a Riemannian manifold measuring its curvature, and it appears in general relativity.

Scalar triple product

The scalar triple product a·(b×c) is the dot product of one vector with the cross product of the other two, whose absolute value equals the parallelepiped's volume.

Scale (descriptive set theory)

In descriptive set theory, a scale, or scale property, is a structured family of norms on a pointset that selects canonical elements and uniformizes relations, first isolated by Yiannis Moschovakis.

Scale-free network

A scale-free network is a network whose degree distribution follows a power law, giving a few highly connected hubs, robustness to random failure, and fragility to hub attacks.

Scaled boundary finite element method

The scaled boundary finite element method (SBFEM) is a semi-analytical numerical method that solves partial differential equations by discretizing only a domain's boundary, reducing the problem dimension by one.

Scaling (geometry)

Scaling, also known as dilation or homothety, is a geometric transformation that multiplies distances from a fixed center by a scale factor, enlarging or shrinking figures.

Scaling function (wavelets)

A scaling function in wavelet theory is a function whose dilated and translated copies generate the nested approximation spaces of a multiresolution analysis, acting as a low-resolution averaging kernel.

Scan statistic

The scan statistic is a statistical method that detects clusters in spatial or temporal point data by moving a window of varying size across the study region.

Scatter plot

A scatter plot, also called a scatter diagram, is a graph of points showing two variables, one on each axis, to reveal relationships between them.

Scenario optimization

Scenario optimization is a method in stochastic optimization that solves chance-constrained problems by drawing random constraint scenarios and optimizing over them, with a certified violation guarantee.

Schauder fixed-point theorem

The Schauder fixed-point theorem, proved by Juliusz Schauder in 1930, extends the Brouwer fixed-point theorem to Banach spaces and underlies existence proofs for differential and integral equations.

Scheffé test

The Scheffé test is a post-hoc procedure used after ANOVA to test any contrast among group means, pairwise or complex, keeping the family-wise error rate at α.

Scheme (mathematics)

A scheme is the basic geometric object of modern algebraic geometry, a locally ringed space built from the spectra of commutative rings, introduced by Alexander Grothendieck in the late 1950s.

Scheme-theoretic image

The scheme-theoretic image of a morphism f: X → Y is the smallest closed subscheme of Y through which f factors, cut out by the kernel of O_Y → f*O_X.

Schinzel's hypothesis H

Schinzel's hypothesis H is a 1958 conjecture in number theory stating that irreducible integer polynomials take prime values simultaneously for infinitely many integers, implying the twin prime conjecture.

Schnirelmann density

Schnirelmann density is a density for sets of integers, introduced by Lev Schnirelmann in 1930, defined as the infimum of the proportion of the first n positive integers in the set.

Schubert calculus

Schubert calculus is the branch of mathematics studying the intersection theory of Schubert varieties on flag manifolds and Grassmannians, translating enumerative geometry problems into cohomology ring computations; its rigorous foundation was Hilbert's 15th problem.

Schur complement

The Schur complement is a matrix constructed from a block matrix, equal to A22 − A21A11⁻¹A12, arising when one group of variables is eliminated from a linear system.

Schur decomposition

The Schur decomposition writes a square matrix A as A = UTU, with U unitary and T upper triangular; it exists for every square matrix and underlies the QR algorithm.

Schur functions

Schur functions are the holomorphic functions on the open unit disk bounded by one in modulus; Schur's 1917 paper associated each with a sequence of parameters.

Schur functor

A Schur functor is a polynomial construction on vector spaces indexed by a Young diagram, generalizing symmetric and exterior powers; it arose in Issai Schur's 1901 thesis.

Schur multiplier

The Schur multiplier of a group G is the abelian group H₂(G, ℤ), introduced by Issai Schur in 1904; it classifies central extensions of G and the obstruction to lifting projective representations.

Schur polynomial

A Schur polynomial is a symmetric homogeneous polynomial indexed by a partition, forming a basis of symmetric functions and serving as characters of irreducible representations of GL_n.

Schur–Weyl duality

Schur–Weyl duality, also called Frobenius–Schur duality, is a theorem in representation theory describing how the general linear group and the symmetric group act on tensor powers of a vector space.

Schur's lemma

Schur's lemma is a fundamental result of representation theory, proved by Issai Schur in 1907, stating that a homomorphism between irreducible representations is either zero or an isomorphism.

Schur's theorem (Ramsey theory)

Schur's theorem is a 1916 result in Ramsey theory: every k-coloring of 1 through n contains same-colored x, y, z with x + y = z.

Schwarz alternating method

The Schwarz alternating method is an iterative domain decomposition technique for solving partial differential equations on overlapping subdomains, devised by Hermann Schwarz in 1869 and now used mainly as a parallel preconditioner.

Scikit-learn

Scikit-learn, also known as Sklearn, is an open source Python machine learning library built on SciPy, started as a 2007 Google Summer of Code project and first released in 2010.

Scilab

Scilab is a free and open-source platform for numerical computation, providing a high-level interpreted programming language, hundreds of mathematical functions, 2-D and 3-D graphics, and the Xcos graphical modeler and simulator.