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Automata theory
Automata theory, also called theory of automata, is the branch of theoretical computer science that studies abstract machines called automata and the languages they recognize.
Automatic differentiation
Automatic differentiation, also called algorithmic differentiation or autodiff, computes derivatives of functions expressed as computer programs via the chain rule; its reverse mode underlies backpropagation in machine learning.
Automatic sequence
An automatic sequence is an infinite sequence whose nth term is produced by a finite automaton with output from the base-k digits of n, including the Thue–Morse sequence.
Automorphic form
An automorphic form is a function behaving in a controlled way under a discrete group of symmetries, generalizing modular forms and central to number theory and the Langlands program.
Automorphic representation
An automorphic representation is an irreducible representation occurring in the space of automorphic forms, packaging modular forms and their L-functions into representation theory for the Langlands program.
Automorphism group
An automorphism group is the group formed by all automorphisms of a mathematical object, such as a group, field, or graph, with composition as the operation.
Automotive Safety Integrity Level
Automotive Safety Integrity Level (ASIL) is a risk classification scheme in ISO 26262, grading vehicle functions ASIL A to ASIL D based on severity, exposure, and controllability.
Autoregression (statistics)
An autoregressive model is a time series model regressing a variable on its own past values plus white noise; proposed by Yule in 1927, it underlies ARMA and ARIMA models.
Autoregressive distributed lag model
The autoregressive distributed lag (ARDL) model is an econometric time-series regression explaining a variable by its own lags and current and lagged explanatory variables, used with the Pesaran bounds test for cointegration.
Autoregressive moving-average model
The autoregressive moving-average model, or ARMA model, is a statistical model for stationary time series that combines autoregressive and moving average terms; it underlies the ARIMA family.
Auxiliary particle filter
The auxiliary particle filter (APF) is a sequential Monte Carlo method introduced by Pitt and Shephard in 1999 that reweights particles using the next observation's likelihood before propagation.
Average
An average, also called a statistical average, is a single value chosen to represent the typical level of a set of numbers, usually the arithmetic mean.
Average absolute deviation
The average absolute deviation (AAD), also called the mean absolute deviation about the mean, measures dispersion as the average absolute distance of observations from the mean.
Average linkage clustering
Average linkage clustering, also known as UPGMA, is an agglomerative hierarchical clustering method that merges clusters with the smallest average pairwise distance, producing a dendrogram.
Average path length
Average path length is the mean of the shortest-path distances between all pairs of nodes in a network, counting the average number of steps needed to get from one node to another.
Average treatment effect
The average treatment effect (ATE) is a causal estimand defined as the expected difference in outcomes had everyone in a population been treated versus had no one been, widely used in epidemiology and the social sciences.
Axiom
An axiom is a proposition accepted without proof from which the theorems of a theory are derived, a method beginning with Euclid's Elements and formalized by Hilbert in 1899.
Axiom of choice
The axiom of choice is the principle of set theory that every collection of nonempty sets has a function selecting one element from each set, formulated by Zermelo in 1904.
Axiom of countable choice
The axiom of countable choice, usually denoted ACω, states that every countable family of nonempty sets has a choice function and is weaker than the full axiom of choice.
Axiom of dependent choice
The axiom of dependent choice (DC), also called the principle of dependent choices, is a weak form of the axiom of choice formulated by Paul Bernays in 1942.
Axiom of determinacy
The axiom of determinacy, introduced by Mycielski and Steinhaus in 1962 as an alternative to the axiom of choice, asserts that every infinite two-player game is determined.
Axiom of empty set
The axiom of empty set, part of Zermelo's 1908 axioms, asserts that a set with no elements exists; with extensionality it is unique and serves as zero in the von Neumann natural numbers.
Axiom of extensionality
The axiom of extensionality is the axiom of set theory asserting that two sets are equal exactly when they share the same elements, first axiomatized by Ernst Zermelo in 1908.
Axiom of global choice
The axiom of global choice, or global choice, strengthens the axiom of choice by asserting one class function that selects an element from every nonempty set.
Axiom of infinity
The axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory, asserting the existence of an inductive set and hence of infinite sets like the natural numbers.
Axiom of pairing
The axiom of pairing, also called the axiom of the unordered pair, is a Zermelo–Fraenkel axiom: for any two sets x and y, the set {x, y} exists.
Axiom of power set
The axiom of power set, part of Zermelo–Fraenkel set theory, asserts that every set has a power set containing all its subsets, grounding Cantor's theorem.
Axiom of projective determinacy
The axiom of projective determinacy (PD) is the set-theoretic statement that every projective set of reals is determined; it is consistent with, but independent of, ZFC.
Axiom of regularity
The axiom of regularity, also called the axiom of foundation, is a Zermelo–Fraenkel set theory axiom requiring every non-empty set to have an element disjoint from it.
Axiom of union
The axiom of union, or sum-set axiom, is one of the Zermelo–Fraenkel axioms: for any set x, there is a set containing exactly the members of x's members.