Mathematics and statistics

Poincaré map

The Poincaré map, or first-return map, records where a continuous dynamical system's trajectories cross a transversal surface, reducing the flow to a lower-dimensional discrete map.

Poincaré plot

The Poincaré plot, also called a Lorenz plot, is a scatter plot of each value of a time series against the next, used in heart rate variability analysis.

Poincaré recurrence theorem

The Poincaré recurrence theorem, also called the Poincaré return theorem, states that in a finite measure-preserving dynamical system almost every point returns to its starting set infinitely often.

Point (geometry)

A point in geometry is an object that marks a location and has no size, one of the primitive objects from which lines and figures are built.

Point at infinity

A point at infinity, also called an ideal point, is a point added to each family of parallel lines so that any two distinct lines meet in exactly one point.

Point counting (stereology)

Point counting is a stereological method that estimates the volume or volume fraction of a structure by counting how many points of a regularly spaced grid fall on it in sections.

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.

Point location is a computational geometry method that preprocesses a subdivision of space so repeated queries quickly report the region containing a query point, with optimal structures achieving logarithmic query time in linear space.

Point pattern analysis

Point pattern analysis is the branch of statistics that tests whether the locations of events in space are clustered, dispersed, or random, fitting point process models.

Point process

A point process, sometimes studied as point process theory, is a stochastic model for randomly occurring events in time or space, such as customer arrivals or earthquakes.

Point spectrum

The point spectrum of a linear operator is the set of its eigenvalues, one of the three standard parts of the spectrum in functional analysis.

Pointclass

A pointclass is a collection of pointsets, subsets of Polish spaces, organizing definable sets of reals into hierarchies, chiefly the Borel and projective pointclasses of descriptive set theory.

Pointless topology

Pointless topology, also called point-free topology or locale theory, is an approach to topology that studies lattices of open sets called locales instead of sets of points.

Pointwise convergence

Pointwise convergence is a mode of convergence for sequences of functions where each point's values converge, without requiring the same rate everywhere; it is weaker than uniform convergence.

Pointwise mutual information

Pointwise mutual information, introduced by Robert Fano in 1961, is a measure of how much more often two outcomes co-occur than independence would predict, with wide use in linguistics and NLP.

Poisson approximation

Poisson approximation, known as the law of rare events or law of small numbers, approximates the distribution of many rare random events by a Poisson distribution with the same mean.

Poisson distribution

The Poisson distribution is a discrete probability distribution giving the probability that a number of events occurs in a fixed interval at a constant average rate, with mean and variance equal to the single parameter λ.

Poisson limit theorem

The Poisson limit theorem, or law of rare events, states that a binomial distribution with many trials and small success probability converges to a Poisson distribution.

Poisson point process

The Poisson point process, also known as the Poisson random point field, is a mathematical model for events scattered without interaction, with independent counts in disjoint regions.

Poisson regression

Poisson regression is a generalized linear model used to predict counts of events, such as hospital admissions, from explanatory variables, assuming the response follows a Poisson distribution.

Poisson sampling

Poisson sampling is a survey sampling design in which each population unit is selected independently with its own inclusion probability, giving a random sample size.

Poisson's equation

Poisson's equation, also known as the Poisson problem, is a second-order elliptic partial differential equation relating a potential's curvature to source density like charge or mass.

Polar coordinate system

The polar coordinate system, also known as polar coordinates, is a two-dimensional coordinate system describing each point by its distance and angle from a fixed pole.

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.

Polar plot

A polar plot is a graph of a curve on a polar coordinate system, plotting points at a radius and angle, often circles, cardioids, roses, and spirals.

Polar space

A polar space is a point-line incidence geometry satisfying the one-or-all axiom, introduced by F. D. Veldkamp in 1959 and tied to the classical groups of Lie type.

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.

Polish notation

Polish notation, also called prefix notation or Łukasiewicz notation, is a mathematical notation, invented by Jan Łukasiewicz in 1924, in which operators precede their operands.

Polish space

A Polish space, or Polish metric space, is a separable, completely metrizable topological space; examples include ℝⁿ, and the concept is central to descriptive set theory.