1,733 articles
Picard group
The Picard group Pic(X) of a scheme or ringed space X is the group of isomorphism classes of line bundles on X, with tensor product as the group operation.
Pierre de Fermat
Pierre de Fermat (1601–1665) was a French lawyer in Toulouse who did mathematics as an amateur, founding modern number theory and probability, and stating Fermat's Last Theorem.
Pivot element
A pivot element, central to complete pivoting and other pivoting strategies, is the entry by which Gaussian elimination divides rows below it to create zero multipliers.
Planar algebra
A planar algebra is an algebraic structure of vector spaces acted on by planar diagrams, introduced by Vaughan Jones in 1999 to axiomatize subfactor invariants.
Plane partition
A plane partition is a partition of an integer n whose parts are arranged in a two-dimensional array, weakly decreasing across rows and down columns.
Plethysm
Plethysm is an operation on symmetric functions, written f[g], in which one symmetric function is substituted into another, introduced by D. E. Littlewood in 1936.
Plünnecke–Ruzsa inequality
The Plünnecke–Ruzsa inequality is a theorem in additive combinatorics stating that if |A+B| ≤ K|A|, then |kB − ℓB| ≤ K^(k+ℓ)|A|, proved by Plünnecke in 1970 and Ruzsa in 1989.
Plus and minus signs
The plus sign (+) and minus sign (−) are mathematical symbols for addition and subtraction, first printed in 1489 in Johannes Widmann's Leipzig arithmetic book.
Plus–minus sign
The plus or minus symbol (±) is a mathematical sign indicating a value may be either added or subtracted, also used to report measurement uncertainty.
Pochhammer symbol
The Pochhammer symbol, also called the shifted factorial, is the notation (a)n for the rising factorial a(a+1)(a+2)⋯(a+n-1); in combinatorics it can instead mean the falling factorial.
Poincaré algebra
The Poincaré algebra is the ten-dimensional Lie algebra of the Poincaré group, the symmetry group of flat spacetime, named for Henri Poincaré, who studied the group in 1905.
Point group
A point group is a group of symmetry operations, such as rotations and reflections, that all leave one common point fixed; crystallography has exactly 32 such groups.
Polar decomposition
The polar decomposition, or polar factorization, is a matrix factorization A = UH into a unitary or orthonormal-column factor U and a Hermitian positive semidefinite factor H.
Polignac's conjecture
Polignac's conjecture, stated by Alphonse de Polignac in 1849, claims every even number occurs infinitely often as a difference between consecutive primes; its gap-2 case is the twin prime conjecture.
Pollard's rho algorithm for logarithms
Pollard's rho algorithm for logarithms is a probabilistic method for solving the discrete logarithm problem, found by John Pollard in 1974 and published in 1978, running in expected O(√n) operations.
Pólya conjecture
The Pólya conjecture is a 1919 statement in number theory asserting the summatory Liouville function is never positive. Brian Haselgrove disproved it in 1958, and the smallest counterexample, 906,150,257, was found in 1980.
Polynomial
A polynomial is a mathematical expression built from powers of one or more variables multiplied by coefficients, forming the basic object of algebra and a key tool in cryptography and numerical analysis.
Polynomial long division
Polynomial long division, also known as the polynomial division algorithm, divides one polynomial by another to produce a quotient and a remainder, like arithmetic long division.
Polynomial ring
A polynomial ring, also called a polynomial algebra, is a ring whose elements are polynomials in one or more indeterminates, written R[x] over a base ring R.
Polynomial Szemerédi theorem
The polynomial Szemerédi theorem is a result in additive combinatorics, proved by Bergelson and Leibman in 1996, extending Szemerédi's theorem to polynomial patterns in dense integer sets.
Posit arithmetic
Posit arithmetic is a format for representing and computing with real numbers, designed by John Gustafson and Isaac Yonemoto in 2017 as a drop-in replacement for IEEE 754 floating point.
Positive definite matrix
A positive definite matrix is a Hermitian or real symmetric matrix for which the quadratic form is positive for every nonzero vector, equivalently all its eigenvalues are positive.
Power iteration
Power iteration, also called the power method or Von Mises iteration, is a numerical linear algebra algorithm that finds a square matrix's dominant eigenvalue and a corresponding eigenvector.
Power of 10
A power of 10 is a number like 100 or 1000, written 10 to the nth power, that underlies decimal notation and the SI metric prefixes.
Power of two
A power of two is a number written as 2 to a whole-number exponent, beginning 1, 2, 4, 8, 16, 32, and important in computing.
Power-associative algebra
A power-associative algebra is a not-necessarily-associative algebra in which powers of any single element are well defined, a concept introduced by A. A. Albert in his 1948 paper Power-Associative Rings.
Practical number
A practical number is a positive integer such that every integer up to n is a sum of distinct divisors of n; 12 is the classic example.
Preconditioner
A preconditioner is a matrix, or any linear operator, applied to a linear system to give it better spectral properties so iterative methods converge faster.
Predual and ultraweak topology of von Neumann algebras
The predual of a von Neumann algebra is the Banach space whose dual it is, unique up to isometric isomorphism; it determines the ultraweak topology and normal maps.
Presheaf with transfers
A presheaf with transfers is a contravariant additive functor from smooth schemes with finite correspondences to abelian groups, used by Voevodsky to build motivic cohomology.