Mathematics and statistics

Perron method (verification)

The Perron method is an existence proof technique for viscosity solutions of Hamilton–Jacobi PDEs, constructing a value function that encodes the set of states from which a system can be driven to a target.

Perron–Frobenius theorem

The Perron–Frobenius theorem is a linear algebra result stating that a matrix with positive entries, or an irreducible nonnegative one, has a dominant positive eigenvalue with a positive eigenvector.

Persistence diagram

A persistence diagram is a multiset of points in the extended plane, each recording the birth and death scales of a topological feature in persistent homology.

Personal income in the United States

Personal income in the United States is a Bureau of Economic Analysis national accounts measure summing all income received by persons from all sources, published monthly.

Perturbation theory

Perturbation theory, also called perturbation analysis, is a group of methods for approximating hard mathematical and physical problems by starting from a related exactly solvable one.

Peter Bosler

Peter Bosler is an American applied mathematician at Sandia National Laboratories who works on Lagrangian numerical methods for climate modeling and won a 2025 PECASE award.

Peter–Weyl theorem

The Peter–Weyl theorem is a 1927 result of Fritz Peter and Hermann Weyl stating that matrix coefficients of irreducible representations form an orthonormal basis of L²(G) for compact groups.

Petersen graph

The Petersen graph is a cubic graph on 10 vertices and 15 edges, constructed by Julius Petersen in 1898 as the smallest snark in graph theory.

Petri net

A Petri net, or PT net, is a mathematical model of concurrent computation in which places and transitions exchange tokens, introduced by Carl Adam Petri in 1962.

Petrov–Galerkin method

The Petrov–Galerkin method is a finite element technique for solving partial differential equations in which trial and test functions come from different function spaces, improving stability.

PETSc

PETSc, the Portable, Extensible Toolkit for Scientific Computation, is a library for the scalable, parallel solution of scientific applications modeled by partial differential equations, running from laptops to GPU-based supercomputers.

Pfaffian

The Pfaffian is a polynomial in the entries of a skew-symmetric matrix whose square equals the determinant, defined for even-order matrices and named by Cayley in 1852.

Pharmacovigilance

Pharmacovigilance, also known as drug safety, is the science of detecting, assessing, understanding, and preventing adverse effects and other drug-related problems, as defined by the World Health Organization.

Phase reduction

Phase reduction is a mathematical technique that replaces a high-dimensional model of a weakly perturbed limit-cycle oscillator with a single phase variable, making synchronization of neural, chemical, and circadian oscillators tractable.

Phase space

Phase space is the space of all possible instantaneous states of a physical system, where a single point specifies the complete state; for n degrees of freedom it is 2n-dimensional.

Phase space reconstruction

Phase space reconstruction is a nonlinear time series analysis method that rebuilds a dynamical system's attractor from a single scalar signal using time-delayed coordinates, justified by Takens' embedding theorem.

Phase-type distribution

A phase-type distribution is the distribution of the time until absorption in a finite continuous-time Markov chain with transient phases, systematized by Marcel F. Neuts and applied in queueing, reliability, and insurance.

Phi coefficient

The phi coefficient is a measure of the association between two binary variables, computed from a 2 × 2 table and ranging from −1 to +1.

Philippine Statistics Authority

The Philippine Statistics Authority (PSA) is the central statistical agency of the Philippines, created in 2013 to merge four statistical agencies; it conducts national censuses, civil registration, and the PhilSys identification system.

Phragmén–Lindelöf principle

The Phragmén–Lindelöf principle, also known as the Phragmén–Lindelöf theorem, is a result in complex analysis extending the maximum modulus principle to unbounded domains, proved by Phragmén and Lindelöf in 1908.

Phrase structure rules

Phrase structure rules are rewrite rules in generative grammar, formalized by Noam Chomsky in the 1950s, that expand categories like S → NP VP to build sentence trees.

Pi

Pi, also known as Archimedes' constant, is the ratio of a circle's circumference to its diameter, an irrational number equal to about 3.14159 whose digits continue without end.

Pi calculus

The pi calculus is a process calculus for concurrent systems in which communication links are names that can be created dynamically and passed as messages, modeling changing structure.

Pi-system

A pi-system (Π-system) is a collection of subsets of a set closed under finite intersections; a workhorse of measure-theoretic probability, it powers Dynkin's pi-lambda theorem and uniqueness of measures.

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.

Picard iteration

Picard iteration, also called Picard–Lindelöf iteration, is a fixed-point method for solving initial value problems for ordinary differential equations, used mainly to prove that a unique solution exists.

Picard theorem

Picard theorem, a pair of results in complex analysis proved by Charles-Émile Picard in 1879: a non-constant entire function takes every complex value with at most one exception.

Picard–Lindelöf theorem

The Picard–Lindelöf theorem, also called the Cauchy–Lipschitz theorem or Picard's existence theorem, guarantees that an initial value problem for an ordinary differential equation whose right-hand side is Lipschitz continuous has a unique local solution.

PICO process

The PICO process is a framework that structures clinical questions into Population, Intervention, Comparison, and Outcome, proposed in 1995 and standard in evidence-based medicine and systematic reviews.