6,052 articles
Ideal (ring theory)
An ideal in ring theory is a subset of a ring closed under addition and multiplication by any ring element, introduced by Richard Dedekind in 1871.
Ideal class group
The ideal class group of a Dedekind domain measures the failure of unique factorization; it is trivial exactly when the domain is a principal ideal domain.
Idele class group
The idele class group of a global field K is the quotient of the idele group by the diagonal K×, recovering ideal class groups and receiving the reciprocity map.
Idempotence
Idempotence is the property of an element e of a ring satisfying e² = e, equivalently e(1 − e) = 0; multiplication by e acts like a projection.
Identity (mathematics)
In mathematics, an identity is an element that leaves every other element unchanged when combined with it, or an equation, such as a + 0 = a, true for every value of its variables.
Identity matrix
The identity matrix, also called the unit matrix, is the square matrix with ones on the diagonal and zeros elsewhere that leaves any matrix unchanged under multiplication.
IEC 61508
IEC 61508 is an international standard for the functional safety of electrical, electronic, and programmable electronic safety-related systems, first published in 1998 and defining four safety integrity levels.
IEEE 754
IEEE 754, the IEEE Standard for Binary Floating-Point Arithmetic, is a technical standard first published in 1985 that defines floating-point formats, rounding rules, and exception handling in computing.
If and only if
"If and only if" (iff) is the connective of formal logic and mathematics asserting that two statements each imply the other, true exactly when their truth values match.
Image (mathematics)
In mathematics, the image of a function is the set of all outputs it produces, and the image of a set under a function collects those outputs.
Imaginary number
An imaginary number, also called a purely imaginary number, is a number whose square is negative, such as multiples of the imaginary unit i, where i squared equals −1.
Imaginary unit
The imaginary unit is the number i, also known as the square root of minus one, whose square is -1; it extends the real numbers to the complex numbers.
Immersed boundary method
The immersed boundary method is a numerical technique for simulating fluid–structure interaction on a fixed grid, introduced by Charles S. Peskin in 1972 for heart valve flow.
Immortal time bias
Immortal time bias is a systematic error in observational studies with time-to-event outcomes, arising when follow-up time during which the outcome cannot occur is misclassified or excluded, usually favoring the treatment.
Immune set
An immune set is an infinite set of natural numbers that contains no infinite recursively enumerable subset, so no algorithm can enumerate an infinite subcollection of it.
Imperialist competitive algorithm
The imperialist competitive algorithm is a population-based metaheuristic for optimization, introduced in 2007, in which candidate solutions called countries form empires that compete and assimilate until one remains.
Implementation and software for sequential Monte Carlo
Implementation and software for sequential Monte Carlo covers the dominant per-particle costs, GPU and CPU parallel resampling, adaptive resampling criteria, and libraries such as SMCTC, LibBi, and particles.
Implementation of mathematics in set theory
The implementation of mathematics in set theory defines every mathematical object, such as numbers, functions, and ordered pairs, as a set, giving ZFC its foundational role in mathematics.
Implicational propositional calculus
The implicational propositional calculus is the fragment of propositional logic in which the only connective is material implication, with theorems derived from axiom schemas by modus ponens.
Implicit differentiation
Implicit differentiation is a calculus technique for finding the derivative of a function defined implicitly by an equation, treating y as a function of x via the chain rule.
Implicit function
An implicit function, or implicitly defined function, is a function defined by an equation F(x, y) = 0 rather than an explicit formula y = f(x).
Implicit function theorem
The implicit function theorem is a calculus result stating that an equation can locally be solved for one variable as a differentiable function of the others.
Implicit surface
An implicit surface is a surface in space defined as the set of points where a scalar function f takes a fixed value, conventionally f(x, y, z) = 0.
Importance sampling
Importance sampling is a Monte Carlo variance-reduction technique that estimates expectations under a target distribution using samples from a proposal distribution, weighted to correct the change of measure.
Imprecise probability
Imprecise probability is a family of mathematical models that represent uncertainty with sets of probability measures, interval probabilities, or lower and upper bounds, named in Peter Walley's 1991 book.
Improper integral
An improper integral, or improper Riemann integral, extends the definite integral to unbounded intervals or unbounded integrands, defined as a limit that either converges or diverges.
Impulse response
The impulse response, also called the impulse response function, is the output of a system to a unit impulse; with convolution, it completely describes a linear time-invariant system.
Imputation (statistics)
Imputation, also called data imputation, is the statistical process of replacing missing values in a dataset with substituted values so analyses can proceed on complete records.
Inaccessible cardinal
An inaccessible cardinal is an uncountable cardinal that cannot be reached from smaller cardinals by successor, union, and power set operations, and whose existence ZFC cannot prove.
Incidence algebra
An incidence algebra is an associative algebra attached to a locally finite partially ordered set, introduced systematically by Gian-Carlo Rota in 1964 as the home of Möbius inversion.