6,052 articles
Sphere packing in dimensions 8 and 24
Sphere packing in dimensions 8 and 24 was solved in 2016 by Maryna Viazovska and colleagues, who proved the E₈ and Leech lattices are densest.
Spherical coordinate system
The spherical coordinate system, also called spherical coordinates, is a three-dimensional coordinate system that locates a point by its distance from an origin and two angles.
Spherical harmonics
Spherical harmonics, also written Ylm, are eigenfunctions of the Laplace-Beltrami operator on the sphere, introduced by Laplace in 1782 as a spherical analog of Fourier series.
Spherical measure
Spherical measure is a mathematical term for either the surface measure on the unit sphere, used for spherical averages, or a ball-covering outer measure close to Hausdorff measure.
Spherical trigonometry
Spherical trigonometry, or spherical trig, is the branch of trigonometry that deals with triangles drawn on a sphere, the basis of great-circle and celestial navigation.
Spheroid
A spheroid, also called an ellipsoid of revolution, is a quadric surface made by rotating an ellipse, giving oblate or prolate shapes like Earth or a rugby ball.
Spheroidal wave functions
Spheroidal wave functions are the special functions that arise when the wave equation is separated in spheroidal coordinates, reducing to Legendre functions when the spheroidal parameter vanishes.
Spin structure
A spin structure is a lift of an oriented Riemannian manifold's frame bundle to the group Spin(n), the data needed to define spinor fields and Dirac operators on curved spaces.
Spinor representation
The spinor representation, also called the spin representation, is the simplest faithful linear representation of the spinor group, equivalently of an even Clifford algebra; in even dimension it splits into two half-spinor representations.
Spiral
A spiral, in mathematics, is a curve whose radius changes as it revolves about a fixed point, winding ever outward or inward; Archimedes studied it about 225 BC.
Spline (mathematics)
A spline, also called a spline curve or spline function, is a piecewise polynomial function whose pieces join smoothly at breakpoints, used to interpolate data.
Spline regression
Spline regression fits piecewise polynomials, typically cubic, constrained to join smoothly at knots to model nonlinear relationships; variants include smoothing splines, P-splines, natural splines, and Friedman's MARS.
Split graph
A split graph is a graph whose vertex set partitions into a clique and an independent set, defined by Földes and Hammer in 1977 and recognizable in linear time.
Split-complex number
The split-complex number, also called a Lorentz or double number, is a number x + yj where j² = 1, unlike the complex unit i² = −1.
Split-plot design
A split-plot design is an experimental design applying hard-to-change factors to large whole plots and easy-to-change factors to nested subplots, giving two randomizations and two error terms.
Split-quaternion
The split-quaternion, also called a coquaternion or para-quaternion, is a four-dimensional non-commutative algebra over the real numbers, defined by James Cockle in 1849, containing zero divisors and isomorphic to 2 × 2 real matrices.
Split-step Fourier method
The split-step Fourier method (SSFM) is a numerical technique for integrating evolution equations, most prominently the nonlinear Schrödinger equation, by alternating linear substeps in Fourier space with nonlinear substeps in physical space.
Splitting field
A splitting field of a polynomial over a field F is the smallest field extension of F in which the polynomial factors completely into linear factors.
Sporadic group
A sporadic group is one of the 26 finite simple groups outside the infinite families, ranging from the Mathieu group M11 to the monster group.
SPSS
SPSS, short for Statistical Package for the Social Sciences, is commercial statistical analysis software developed at Stanford, first released in 1968, and acquired by IBM in 2009.
Spurious relationship
A spurious relationship, or spurious correlation, is a statistical association between two variables without a direct causal link, produced by chance, a third factor, or data selection.
Square
A square, also called a regular quadrilateral, is a plane figure with four equal sides and four right angles, its area being the side length squared.
Square (algebra)
In algebra, the square of a number or expression is the result of multiplying it by itself, written x², so named because a square's area is its side squared.
Square matrix
A square matrix is a matrix with the same number of rows and columns, an n × n grid. It is the setting for determinants, trace, eigenvalues, and inverses.
Square number
A square number, or perfect square, is an integer that is the product of an integer with itself, such as 0, 1, 4, 9, 16, and 25.
Square root
The square root, also called the second root, of a number x is a number r satisfying r² = x; the nonnegative root is the principal square root, written √x.
Square root of 2
The square root of 2, sometimes called Pythagoras's constant, is the positive real number that multiplies by itself to give 2, about 1.4142135, and cannot be written as a ratio of two integers.
Square root of 3
The square root of 3, also called Theodorus's constant, is the irrational number whose square is 3, approximately 1.7320508, giving side ratios in the 30-60-90 triangle.
Square root of 5
The square root of 5 is the positive real number whose square is 5, an irrational number, and it links the regular pentagon to the golden ratio.
Square root of a matrix
A square root of a matrix is a matrix B whose square B² equals the given matrix A. Roots may not exist and are generally not unique.