Mathematics and statistics

Proof net

A proof net is a graph-based representation of a proof in linear logic, introduced by Jean-Yves Girard in 1987 to eliminate trivial rule permutations of sequent calculus derivations.

Proof that e is irrational

Proof that e is irrational: the demonstration, first given by Euler in the 1730s, that Euler's number e cannot be written as a ratio of two integers.

Proof that π is irrational

Proof that π is irrational, first given by Johann Heinrich Lambert in 1761, established that π cannot be written as a ratio of two integers.

Proof-theoretic semantics

Proof-theoretic semantics is an alternative to model-theoretic semantics that explains the meaning of logical constants in terms of the inference rules governing their behavior in proofs.

Proofs of convergence of random variables

Proofs of convergence of random variables presents the four modes of convergence in probability theory, the strict hierarchy among them, and proof tools such as Borel–Cantelli and uniform integrability.

Propagation of chaos

Propagation of chaos is a property of large systems of interacting random particles: as the number of particles tends to infinity, any fixed finite subset behaves as if independent.

Propensity score

A propensity score is the probability of receiving a treatment given observed covariates, used to reduce confounding in observational studies where treatment is not randomly assigned.

Propensity score matching

Propensity score matching, also known as propensity scoring, is a statistical technique that matches treated and untreated subjects with similar scores to mimic a randomized trial.

Propensity score weighting

Propensity score weighting is a statistical method for causal inference in observational studies that weights each subject by the inverse of the probability of the treatment received.

Proper forcing

Proper forcing is a property of forcing notions in set theory, introduced by Saharon Shelah in 1980, that preserves ω₁ and is preserved under countable-support iterations.

Proper forcing axiom

The Proper Forcing Axiom (PFA) is a strengthening of Martin's Axiom in set theory, formulated by James Baumgartner in the early 1980s using Shelah's notion of properness.

Proper morphism

In algebraic geometry, a proper morphism is a morphism of schemes that is separated, of finite type, and universally closed; projective space is proper, while affine space is not.

Proper orthogonal decomposition

The proper orthogonal decomposition (POD), also called the Karhunen–Loève decomposition, extracts orthogonal modes whose first few capture the most energy, a standard tool in fluid mechanics.

Property P (knot theory)

Property P is a property of knots in the 3-sphere, conjectured for every non-trivial knot and proved by Kronheimer and Mrowka in 2004 using gauge theory.

Property testing

Property testing is a subfield of algorithm theory in which a randomized procedure decides from a small sample whether an object has a property or is far from it.

Proportional hazards model

The proportional hazards model, usually called Cox regression or the Cox model, is a statistical regression model for time-to-event data proposed by David Cox in 1972.

Proportional meta-analysis

Proportional meta-analysis is a biostatistical evidence-synthesis method that pools a single proportion, such as disease prevalence or adverse event rates, from the proportions reported by multiple studies.

Proportionality (mathematics)

Proportionality is the mathematical relation between two varying quantities whose ratio or product is constant: a fixed ratio gives direct proportion and a fixed product gives inverse proportion.

Propositional calculus

A propositional calculus is a formal proof system for propositional logic, given by axioms and inference rules such as modus ponens; first axiomatized by Frege in 1879, it is sound and complete.

Propositional dynamic logic

Propositional dynamic logic (PDL) is a modal logic for reasoning about programs, in which [α]A means A holds after every execution of program α. Fischer and Ladner defined it in 1979.

Propositional logic

Propositional logic, also known as sentential logic or sentential calculus, studies reasoning with whole propositions that are true or false, joined by connectives like and, or, and if-then.

Propositional proof system

A propositional proof system is any polynomial-time computable function whose range is exactly the propositional tautologies, the abstract definition by Cook and Reckhow in 1979 that founded proof complexity.

Protein–protein interaction prediction

Protein–protein interaction prediction is the use of computational methods, from genomic context and co-evolution to deep learning, to infer whether two proteins bind or otherwise associate.

Provability logic

Provability logic is a modal logic, centered on the system GL, that studies what arithmetical theories like Peano Arithmetic can prove about their own provability.

Pseudo-observations

Pseudo-observations are rank-transformed values of a multivariate sample, scaled by 1/(n+1) so margins are approximately uniform, making them the standard input for semiparametric copula inference.

Pseudomathematics

Pseudomathematics, also called mathematical crankery or pseudo-maths, is purported mathematical work that fails accepted standards of logic and rigor, typically by amateurs targeting proved-impossible problems like squaring the circle.

Pseudometric space

A pseudometric space is a set with a distance function satisfying the metric axioms except that distinct points may lie at distance zero, as with functions equal almost everywhere.

Pseudorandomness

Pseudorandomness is the property of a deterministic process whose output appears random when its internal workings are hidden; cryptographic pseudorandomness means computationally indistinguishable from truly random output.

Pseudoscalar

A pseudoscalar in geometric algebra (Clifford algebra) is the product of all orthonormal basis vectors, representing a unit oriented volume and fixing the space's orientation.