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Inequality (mathematics)
An inequality, or mathematical inequality, is a mathematical relation stating that one quantity is greater than or less than another, written with the symbols , ≤, and ≥.
Inference
Inference, also known as logical inference, is the process of deriving a conclusion from premises with certainty, as in deduction, or to a lesser degree, as in induction and abduction.
Inference engine
An inference engine is a software module that applies the stored knowledge of an expert system to problem facts, deriving new facts and deciding which actions to take.
Infimum and supremum
The infimum and supremum, also called the greatest lower bound and least upper bound, generalize the minimum and maximum of a set of real numbers.
Infinitary logic
Infinitary logic, also called an infinitary language, is a family of extensions of first-order logic allowing infinitely long formulas; the best-studied case, L(ω₁,ω), permits countably infinite conjunctions and disjunctions.
Infinite divisibility (probability)
Infinite divisibility, also called an infinitely divisible distribution, is a probability distribution representable as a sum of n independent, identically distributed variables for every n.
Infinite mixture model
An infinite mixture model is a Bayesian mixture model with a prior over infinitely many components, so the number of clusters is inferred from data.
Infinite monkey theorem
The infinite monkey theorem states that a monkey typing random keys forever will almost surely type any text; a 2024 study showed real monkeys never could.
Infinite product
An infinite product is a product of infinitely many factors, defined as the limit of its partial products, and is the multiplicative analog of an infinite series.
Infinite set
An infinite set is a set that is not finite, so its elements cannot be counted with a natural number; Cantor showed in 1874 that infinities come in different sizes.
Infinitesimal
An infinitesimal, or infinitesimal number, is a nonzero quantity smaller in absolute value than every positive real number, realized rigorously in the hyperreal numbers introduced by Abraham Robinson in 1960.
Infinitesimal generator (stochastic processes)
The infinitesimal generator of a Markov process is the linear operator giving the instantaneous rate of change of expected values, and it determines the process via the Hille–Yosida theorem.
Infinity
Infinity is the condition of being without any finite bound in size, extent, or quantity, appearing in mathematics as both endless process and completed infinite objects.
Infinity symbol
The infinity symbol (∞), also known as the sideways eight or infinity sign, is a mathematical sign for infinite quantity, introduced by English mathematician John Wallis in 1655.
Inflection point
An inflection point, also written point of inflection or inflexion point, is a point on a function's graph where concavity changes, so the second derivative changes sign.
Influence function (statistics)
The influence function, also called the influence curve, is a statistic measuring a single outlier's effect on an estimator's asymptotic version, introduced by Frank Hampel in 1974.
Information
Information is a quantifiable measure of resolved uncertainty, formalized by Claude Shannon in his 1948 paper A Mathematical Theory of Communication, with the entropy formula and the bit as its unit.
Information theory
Information theory, also known as Shannon theory, is the mathematical study of the quantification, storage, and transmission of information, founded by Claude Shannon in his 1948 paper.
Inglehart–Welzel cultural map of the world
The Inglehart–Welzel cultural map of the world is a two-dimensional map of national values built from World Values Survey data by Ronald Inglehart and Christian Welzel.
Ingleton's inequality
Ingleton's inequality is a linear inequality on matroid ranks, proved by Aubrey Ingleton in 1971, that holds for every matroid representable over a field but is not implied by the rank axioms.
Injective dimension
Injective dimension, denoted id(M), is the least number of injective modules needed to resolve a module over a ring, measured via Ext; finiteness characterizes Gorenstein and regular rings.
Injective function
An injective function, also called a one-to-one function or injection, maps distinct inputs to distinct outputs, so no value in the codomain is hit more than once.
Injective hull
An injective hull, also called an injective envelope, is the smallest injective module containing a given module M, unique up to isomorphism; every module has one.
Injective module
An injective module is a module J over a ring R such that every homomorphism from a submodule into J extends to the whole module, the dual of projectivity.
Inner automorphism
An inner automorphism of a group is conjugation by a fixed element, sending x to gxg⁻¹; the inner automorphisms form a normal subgroup of the automorphism group.
Inner model
An inner model is a transitive class of sets containing all the ordinals and satisfying ZF, introduced by Kurt Gödel, whose constructible universe L gave the first relative consistency proofs in set theory.
Inner model theory
Inner model theory is the branch of set theory that constructs canonical inner models of ZFC satisfying large cardinal hypotheses, using fine structure theory, iteration trees, and extender models.
Inner product space
An inner product space, also called a pre-Hilbert space, is a vector space with an inner product that generalizes the dot product, giving lengths, angles, and orthogonality.
Innovation process
The innovation process is the sequence of one-step-ahead prediction errors of a stochastic process, forming the basis of linear time series models, the Kalman filter's correction, and model-fit diagnostics.
Innovations process
The innovations process is the part of a noisy observation record carrying new information about an unobserved signal, the basis of the innovations approach to filtering.