Mathematics and statistics

Partial differential equation

A partial differential equation (PDE) is an equation for a function of several variables involving its partial derivatives, describing how quantities vary in space and time.

Partial differential sign

∂, called the partial differential sign or "curly d", is the mathematical symbol for partial derivatives, which differentiate one variable while the others are held constant.

Partial fraction decomposition

Partial fraction decomposition, or partial fraction expansion, rewrites a rational function as a polynomial plus simpler fractions, the standard first step for integrating rational functions.

Partial function

A partial function from X to Y assigns at most one value to each element of X but need not be defined everywhere; total functions are the special case.

Partial information decomposition

Partial information decomposition is an information theory framework that splits what multiple sources reveal about a target into redundant, unique, and synergistic components, introduced by Williams and Beer in 2010.

Partial least squares discriminant analysis

Partial least squares discriminant analysis (PLS-DA) is a supervised classification method that runs PLS regression against dummy-coded class labels, widely used in metabolomics and spectroscopy but prone to overfitting.

Partial least squares path modeling

Partial least squares path modeling (PLS-SEM, also called PLS-PM) is a variance-based structural equation modeling method estimating latent variables as weighted composites, suited to small samples.

Partial least squares regression

Partial least squares regression (PLS) projects many correlated predictors onto a few latent components and regresses on them; proposed by Herman Wold in the 1960s.

Partial order reduction

Partial order reduction is a model checking technique that combats state explosion in concurrent systems by exploiting independence between transitions, preserving properties like LTL without the next operator.

Partial word

A partial word is a finite or infinite sequence over a finite alphabet in which some positions, called holes, are undefined; it extends ordinary words in combinatorics on words.

Partially observable Markov decision process

A partially observable Markov decision process (POMDP) is a mathematical model for sequential decision-making under uncertainty, where an agent acts on noisy observations and a belief state.

Partially ordered set

A partially ordered set, or poset, is a set with a reflexive, antisymmetric, transitive relation, so some elements rank relative to each other while others are unrelated.

Particle filter

Particle filter, also known as Sequential Monte Carlo, is an algorithm that estimates the hidden state of a dynamic system using weighted random samples, introduced in 1993.

Particle filtering

Particle filtering is a sequential Monte Carlo method that estimates the hidden state of a nonlinear, non-Gaussian system from noisy observations, using weighted sample particles; the modern bootstrap filter dates to 1993.

Particle MCMC

Particle MCMC (PMCMC) is a family of Monte Carlo algorithms combining particle filters with MCMC for Bayesian inference on non-linear state-space models, introduced by Andrieu, Doucet, and Holenstein in 2010.

Particle physics and representation theory

Particle physics and representation theory describe elementary particles as irreducible representations of the Poincaré group, classified by mass and spin, as Wigner showed in 1939.

Particle smoothing

Particle smoothing is a family of sequential Monte Carlo methods that approximates the distributions of a state-space model's past states conditioned on all observations, in nonlinear and non-Gaussian settings.

Particular values of the Riemann zeta function

The particular values of the Riemann zeta function, also called zeta constants, are its values at integer arguments, such as ζ(2) = π²/6 and ζ(−1) = −1/12.

Partition (number theory)

In number theory, a partition of a nonnegative integer n is an unordered collection of positive integers summing to n, counted by the partition function p(n).

Partition function (number theory)

The partition function p(n) counts the ways a nonnegative integer can be written as a sum of positive integers, ignoring order; for example, p(4) = 5.

Partition lattice

The partition lattice is the partially ordered set of all set partitions of a finite set, ordered by refinement, written Πn and counted by Bell numbers.

Partition matroid

A partition matroid is a standard matroid example whose ground set is divided into disjoint parts, with a set independent when it contains at most one element from each part.

Partition numbers

The partition numbers p(n) count the ways a nonnegative integer n can be written as a sum of positive integers, ignoring order; the sequence begins 1, 1, 2, 3, 5, 7.

Partition of a set

A partition of a set, or set partition, is a collection of nonempty, pairwise disjoint subsets, called blocks, whose union is the whole set, with every element in exactly one block.

Partition of unity

A partition of unity is a collection of functions on a space, valued between 0 and 1, that sum to 1 at every point and exist when the space is paracompact.

Pascal's pyramid

Pascal's pyramid, also called Pascal's tetrahedron, is a three-dimensional arrangement of the trinomial coefficients of (x+y+z)^n, the analog of Pascal's triangle, with each entry the sum of the three above it.

Pascal's theorem

Pascal's theorem, also called the mystic hexagram theorem, is a projective geometry result: opposite sides of a hexagon inscribed in a conic meet in three collinear points.

Pascal's triangle

Pascal's triangle, also called the Khayyam or Yang Hui triangle, is a triangular array of binomial coefficients where each number is the sum of the two above it.

Path (graph theory)

In graph theory, a path is a walk in a graph that repeats no vertex and therefore no edge; a graph is connected when every pair of vertices has a path between them.