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Finiteness, finitude, or being finite, is the state of being limited or having an end, and is a counter to the concept of infinity. Humans are considered...

finitely presented may refer to: finitely presented group finitely presented monoid finitely presented module finitely presented algebra finitely presented...

example of a countable group that is not finitely generated. Every quotient of a finitely generated group G is finitely generated; the quotient group is generated...

R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related...

abelian group ( G , + ) {\displaystyle (G,+)} is called finitely generated if there exist finitely many elements x 1 , … , x s {\displaystyle x_{1},\dots...

categories are finitely complete and finitely cocomplete but neither complete nor cocomplete: The category of finite sets The category of finite abelian groups...

is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal...

geometrically finite polyhedron with an infinite number of sides. A geometrically finite polyhedron has only a finite number of cusps, and all but finitely many...

mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R...

a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field, a finite field...

an infinite finitely presented group with no nontrivial finite quotients. The quotient by the maximal proper normal subgroup is a finitely generated infinite...

free on a finite set. For instance, one way of defining a finitely generated group is that it is the image of a group homomorphism from a finitely generated...

of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem...

asserts that every fusion ring has finitely many categorifications. Stated more directly, this means that are only finitely many equivalence classes of fusion...

space X {\displaystyle X} is said to be point-finite if every point of X {\displaystyle X} lies in only finitely many members of U . {\displaystyle {\mathcal...

(respectively finitely related, finitely presented) if it has a presentation that is finitely generated (respectively finitely related, a finite presentation)...

Finite type refers to several related concepts in mathematics: Algebra of finite type, an associative algebra with finitely many generators Morphism of...

schemes can be found in the article on finite morphisms. Finite morphism Finitely generated algebra Finitely generated module Atiyah, Michael Francis;...

makes Ai a finitely generated module over Bi (in other words, a finite Bi-algebra). One also says that X is finite over Y. In fact, f is finite if and only...

A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line...

residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group, such...

In mathematics, a finite set is a collection of finitely many different things; the things are called elements or members of the set and are typically...

is one winner). Every play ends after finitely many moves. At any point in a play of G, there are but finitely many legal possibilities for the next move...

the field Q {\displaystyle \mathbb {Q} } of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and...

A finite difference is a mathematical expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often...

In mathematics education, Finite Math is a syllabus in college and university mathematics that is independent of calculus. A course in precalculus may...

is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces. If A {\displaystyle A} is a finitely-generated R {\displaystyle...

learning theory, a class C of languages has finite thickness if every string is contained in at most finitely many languages in C. This condition was introduced...

{\displaystyle X} is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection...

localization of the ring). The converse is true for finitely generated modules over Noetherian rings: a finitely generated module over a commutative Noetherian...

theoretical computer science, a finite-state machine (FSM) or finite-state automaton (FSA, plural: automata), finite automaton, or simply a state machine...

{\displaystyle B} is called an A {\displaystyle A} -algebra of finite type if B {\displaystyle B} can be finitely generated as an A {\displaystyle A} -algebra. It is...

English imperative). A finite transitive verb or a finite intransitive verb can function as the root of an independent clause. Finite verbs are distinguished...

a finite set. In other words, A {\displaystyle A} contains all but finitely many elements of X . {\displaystyle X.} If the complement is not finite, but...

Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical...

mathematics, the obstruction to a finitely dominated space X being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction w(X) which is...

ZFC by replacing them with finitely many axioms in the same language. A theory is finitely axiomatizable if there is a finite set of sentences whose deductive...

infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely presented groups. Given an integer n ≥ 1, a group...

finitely generated group is finitely generated (the images of the generators in the quotient give a finite generating set), a subgroup of a finitely generated...

structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian...

solid whose boundary can be covered by finitely many planes or that it is a solid formed as the union of finitely many convex polyhedra. Natural refinements...

condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational...

L-theorem if and only if A is a theorem of the theory of finite models of L. If L is finitely axiomatizable (and has a recursive set of inference rules)...

taking subgroups, extensions by amenable groups, and free products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups...

topologically finitely generated then it is in addition equal to its own profinite completion. In practice, the inverse system of finite groups is almost...

power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets). A topology on a finite set can also...

which turn out to be finitely presented, include the infinite Thompson groups T {\displaystyle T} and V {\displaystyle V} . Finitely presented torsion-free...

finitely generated abelian groups: any group which is quasi-isometric to a finitely generated abelian group contains a free abelian group of finite index...

infima). Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets)...

theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is...

additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have...

Look up finite, finitely, or finiteness in Wiktionary, the free dictionary. Finite, or variants, may refer to: Finite set, a set whose cardinality (number...

group cannot be finitely generated. General Burnside problem. If G is a finitely generated, periodic group, then is G necessarily finite? This question...

variables, as a special case. Precisely, it states: Given a normal domain A, finitely generated as an algebra over a field k, if L is a subfield of the field...

proof. Certainly the finiteness of this group is a necessary condition for E ( Q ) {\displaystyle E(\mathbb {Q} )} to be finitely generated; and it shows...

coherent A left coherent ring is a ring such that every finitely generated left ideal of it is a finitely presented module; in other words, it is coherent as...

{\displaystyle f:B\to F} such that f ( b ) = 0 {\displaystyle f(b)=0} for all but finitely many b {\displaystyle b} in B . {\displaystyle B.} These functions can...

abelian subgroup of finite index); being virtually nilpotent; being virtually free; being finitely presentable; being a finitely presentable group with...

which every finitely generated subgroup is finite. Since the cyclic subgroups of a locally finite group are finitely generated hence finite, every element...

abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. Every cyclic...

processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because...

conjecture could be carried over to the Euclidean case. Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ Fn be a Kakeya set, i.e. for each vector...

linear. Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above. Since any finitely linear group...

C} is polyhedral if it is the conical hull of finitely many vectors (this property is also called finitely-generated). I.e., there is a set of vectors {...

a starting point in Dunwoody's theory of accessibility for finitely generated and finitely presented groups. Since the ranks of the free factors are smaller...

finitely generated, but not finitely presented. However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely...

H_{3},} and H 4 {\displaystyle H_{4}} ). Every finite Coxeter group is the direct product of finitely many of these irreducible groups. Many, but not...

for the algorithm to run and how (finitely) much memory is available. In fact the problem of deciding whether a finitely presented group is trivial is undecidable...

functions is locally finitely presentable, since every set is the direct limit of its finite subsets, and finite sets are finitely presentable. The category...

set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends...

a locally finite space is a topological space in which every point has a finite neighborhood, that is, a neighborhood consisting of finitely many elements...

pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect...

integral points states that a curve of genus greater than zero has only finitely many integral points over any given number field. The theorem was first...

that certain ascending or descending sequences of subobjects must have finite length. Noetherian objects are named after Emmy Noether, who was the first...

corresponding theory for finitely generated modules over a Dedekind domain. Let us briefly recall the structure theory in the case of a finitely generated module...

to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated. The integers, considered...

elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of...

points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. A set of points...

used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences...

finite groups, when only finitely generated groups are considered: Does specifying an exponent force finiteness? The existence of infinite, finitely generated...

can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deal with finite sets, particularly...

finite graph Locally finite group Locally finite measure Locally finite operator in linear algebra Locally finite poset Locally finite space, a topological...

Finite number may refer to: Natural number, a countable number less than infinity, being the cardinality of a finite set Real number, such as may result...

Principal ideal: An ideal generated by one element. Finitely generated ideal: This type of ideal is finitely generated as a module. Primitive ideal: A left...

One first observes that for every finitely generated subgroup H of G is finitely presentable and that for every finite marked generating set Y of H the...

Liouvillian function is a function of one variable which is the composition of a finite number of algebraic operations ( addition, subtraction, multiplication,...

+ sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be...

Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge...

independent. Knight, Vince. "Finitely Repeated Games". Game Theory. Retrieved 6 December 2017. Benoit, J.P. & Krishna, V. (1985). "Finitely Repeated Games". Econometrica...

branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability...

finite type over A and if C is finite over B, then B is of finite type over A. (Here, "of finite type" means "finitely generated algebra" and "finite"...

Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part...

In logic, a finite-valued logic (also finitely many-valued logic) is a propositional calculus in which truth values are discrete. Traditionally, in Aristotle's...

mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable...

on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable. A simplified proof is given in (Rubinstein...

K(\{x_{1},\ldots ,x_{n}\}),} and one says that K ( S ) {\displaystyle K(S)} is finitely generated over K {\displaystyle K} . If S {\displaystyle S} consists of...

speaking, a step function is a piecewise constant function having only finitely many pieces. A function f : R → R {\displaystyle f\colon \mathbb {R} \rightarrow...

{O}}_{X,x}\otimes \kappa (f(x))} is finitely generated over κ ( f ( x ) ) {\displaystyle \kappa (f(x))} . Quasi-finite morphisms were originally defined...

can have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated...

\operatorname {Ass} (M)} is a finite set when M {\displaystyle M} is finitely generated. Let M {\displaystyle M} be a finitely generated module over a Noetherian...

any finitely generated group G {\displaystyle G} we have e ( G ) ∈ { 0 , 1 , 2 , ∞ } {\displaystyle e(G)\in \{0,1,2,\infty \}} . For any finitely generated...

cyclic group is a torsion abelian group. A finitely generated abelian group is torsion iff it is finite, see Finitely generated abelian group#Classification...

of finite character if for each A {\displaystyle A} , A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} if and only if every finite subset...

}}B)} is covered by finitely many affine open sets Spec  A {\displaystyle {\text{Spec }}A} where each A {\displaystyle A} is finitely generated as a B {\displaystyle...

In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying...

In mathematics, there are several finiteness theorems. Ahlfors finiteness theorem Finiteness theorem for a proper morphism Compactness theorem, in mathematical...

any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian...

submonoid of M containing S is M. If there is a finite set that generates M, then M is said to be a finitely generated monoid. A monoid whose operation is...

In linguistics, a non-finite clause is a dependent or embedded clause that represents a state or event in the same way no matter whether it takes place...

deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state...

the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups. The proof of...

structure of finitely generated linear groups. The theorem, proven by Tits, is stated as follows. Theorem— Let G {\displaystyle G} be a finitely generated...

correctness. The name "finite bypass" means that any process (concurrent part) of the algorithm is bypassed at most a finite number times before being...

Finite topology is a mathematical concept which has several different meanings. A finite topological space is a topological space, the underlying set of...

referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition. The...

have finite activity (only finitely many jumps on any finite time interval), finite variation (the sum of the absolute value of all jumps is finite on every...

Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate...

[\alpha ]} is a finitely generated Z {\displaystyle \mathbb {Z} } -module. α ∈ K is an algebraic integer if there exists a non-zero finitely generated Z {\displaystyle...

analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences...

of finitely presentable groups is one for which: P is an abstract property, that is, P is preserved under group isomorphism. There exists a finitely presentable...

being a Prüfer domain, i.e. every finitely generated ideal of R is projective: Ideal arithmetic Every non-zero finitely generated ideal I of R is invertible:...

of n {\displaystyle n} variables over a field k {\displaystyle k} is a finitely generated field extension K / k {\displaystyle K/k} which has transcendence...

while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups...

between finite and non-finite clauses. A finite clause contains a structurally central finite verb, whereas the structurally central word of a non-finite clause...

by R-Mod (see category of modules). Finitely generated An R-module M is finitely generated if there exist finitely many elements x1, ..., xn in M such...

product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial). Given X {\displaystyle...

locally finite if every finitely generated algebra has finite cardinality, or equivalently, if every finitely generated free algebra has finite cardinality...

idempotents commute. Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Zd. These semigroups have applications to commutative...

topological space is called countably compact if every countable open cover has a finite subcover. A topological space X is called countably compact if it satisfies...

infinity. This measure is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets...

Geometry in affine spaces over a finite fields has also been widely studied. For example, elliptic curves over finite fields are widely used in cryptography...

field K is finitely generated as an associative algebra over another field k, then K is a finite field extension of k; that is, K is finitely generated...

commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a...

property if all finitely generated free modules over R have a well-defined rank. In the case of fields, the IBN property is the fact that finite-dimensional...

In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if each of its transitions is uniquely determined by its...

In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950...

to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward. This table contains...

spaces or finitely generated spaces. The latter name stems from the fact that their topology is uniquely determined by the family of all finite subspaces...

Steinitz's theorem states that a finite extension of fields L / K {\displaystyle L/K} is simple if and only if there are only finitely many intermediate fields...

the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements...

reciprocals of the twin primes, of which there may be finitely many or infinitely many, is known to be finite and is called Brun's constant, approximately 1...

Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter...

be defined as a language recognised by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after...

computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. A CW complex is constructed by taking the...

number has only finitely many prime factors in its denominator, so a ∈ Z p {\displaystyle a\in \mathbb {Z} _{p}} for all but finitely many primes p {\displaystyle...

Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that k is a field and let...

The fuzzy finite element method combines the well-established finite element method with the concept of fuzzy numbers, the latter being a special case...

book, Finite and Infinite Games to topics of business and leadership. The book is based on Carse's distinction between two types of games: finite games...

{\displaystyle x\,} meeting only finitely many sets in O {\displaystyle {\mathcal {O}}\,} ; thus x {\displaystyle x\,} belongs to only finitely many sets in O {\displaystyle...

Hilbert's basis theorem the ideal I is finitely generated (as an ideal). Hence, I is finitely generated by finitely many invariants of G (because if we are...

there are only finitely many possibilities for the second sum (the topological reason). Hence, S σ {\displaystyle S_{\sigma }} is finitely generated. The...

generalized notions of volumes, an s-finite measure is a special type of measure. An s-finite measure is more general than a finite measure, but allows one to generalize...

known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two finitely generated subgroups...

is a finite extension of K, then the integral closure A ′ {\displaystyle A'} of A in L is a finitely generated A-module and is also a finitely generated...

of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators are...

Moore matrix, introduced by E. H. Moore (1896), is a matrix defined over a finite field. When it is a square matrix, its determinant is called a Moore determinant...

input. A common deterministic automaton is a deterministic finite automaton (DFA) which is a finite state machine, where for each pair of state and input symbol...

elements of Sub(A) are exactly the finitely generated substructures of A. Every substructure is the union of its finitely generated substructures; hence Sub(A)...

this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, and in general for discretization...

Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with...

then it is finitely generated, and any Cayley graph for G {\displaystyle G} is quasi-isometric to Y {\displaystyle Y} . Thus a group is (finitely generated...

Sequences can be finite, as in these examples, or infinite, such as the sequence of positive even integers (2, 4, 6, 8, ...). The length of a finite sequence...

is diffeomorphic to a sphere. Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism) compact n-dimensional...

finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements. Given a locally finite poset...

should not be confused with a representation. The relations are given as a (finite) binary relation R on Σ∗. To form the quotient monoid, these relations are...

quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension...

notions of groups and partially ordered sets. If the given category is finite (has finitely many objects and morphisms), then the following two definitions of...

quotient field is a finitely generated A {\displaystyle A} -module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L {\displaystyle...

group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or...

there exist only finitely many pro-p groups of coclass r. This finiteness result is fundamental for the classification of finite p-groups by means of...

which every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every locally cyclic group is abelian. Every finitely-generated...

groups are essentially different: one is finitely generated and countable, while the other is not finitely generated and has the cardinality of the continuum...

I}V_{i}} ), where only tuples with finitely many nonzero vectors are allowed. If the index set I {\displaystyle I} is finite, the two constructions agree,...

limited series differs from an ongoing series in that the number of issues is finite and determined before production, and it differs from a one shot in that...

all (finitely generated) submodules of projective left R-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated)...

particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries...

In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic...

then K [ V ] {\displaystyle K[V]} is finitely-generated. Another important result, due to Noether, is that any finitely-generated graded algebra R {\displaystyle...

Unsolved problem in mathematics Does each positive integer occur only finitely many times as a difference of perfect powers? More unsolved problems in...

approximations of X {\displaystyle X} which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values, the...

all but finitely many  i } {\displaystyle \left\{x=\left(x_{i}\right)_{i\in I}\in X\mathbin {\big \vert } x_{i}=z_{i}{\text{ for all but finitely many }}i\right\}}...

Wallace's theorem, is a useful tool in order to prove that the product of finitely many compact spaces is compact. The lemma uses the following terminology:...

construct a nondeterministic finite automaton (NFA), which is then made deterministic and the resulting deterministic finite automaton (DFA) is run on the...

algebraic variety with certain special varieties are accounted for by finitely many special varieties. The intersection of two algebraic varieties is...

7th degree by means of functions of only two arguments. Proof of the finiteness of certain complete systems of functions. Rigorous foundation of Schubert's...

well-defined and is additive for short exact sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module...

Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group...

In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product...

theorem states that for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that...

are finitely generated subgroups of a free group F(X) such that the intersection L = H ∩ K {\displaystyle L=H\cap K} is nontrivial, then L is finitely generated...

LZFSE (Lempel–Ziv Finite State Entropy) is an open source lossless data compression algorithm created by Apple Inc. It was released with a simpler algorithm...

Gelfand together with Mark Naimark in the same year. The full theory of the finite-dimensional representations of the Lie algebra of the Lorentz group is deduced...

that Bézout's identity holds for every pair of elements, and that every finitely generated ideal is principal. Bézout domains are a form of Prüfer domain...

Shannon entropy H, that can be contained within a given finite region of space which has a finite amount of energy—or equivalently, the maximum amount of...

looking for evidence to support the proposal that our universe might be finite in size due to its having a nontrivial topology. Other work includes black...

finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely...

{\displaystyle \emptyset } ⁠) and the latter has no elements at all. A set is finite if there exists a natural number ⁠ n {\displaystyle n} ⁠ such that the first...

multiply connected space like a 3 torus has everywhere zero curvature but is finite in extent, whereas a flat simply connected space is infinite in extent (such...

for every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module...

particular, if J is finite, then M is a finitely generated module. If I and J are finite sets, then the presentation is called a finite presentation; a module...

rather than a static product. The theory of finite semiotics explains semiosis as an effect of the finite nature of the human mind that occurs as an individual...

the opposite of finite a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause a non-finite clause is a clause...

hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of...

prime ideal of the ring is finitely generated. However, it is not enough to ask that all the maximal ideals are finitely generated, as there is a non-Noetherian...

additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected...

In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process...

or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature...

with a finite number of states is called a finite automaton (FA) or finite-state machine (FSM). The figure on the right illustrates a finite-state machine...

be generated by the conjugates of finitely many elements of F {\displaystyle F} , then G {\displaystyle G} is finitely presented. If S {\displaystyle S}...

equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable...

→ R {\displaystyle f\colon G\to R} of finite support ( f ( g ) {\displaystyle f(g)} is nonzero for only finitely many elements g {\displaystyle g} ), where...

observable subgroup for which the ring of functions on the quotient variety is finitely generated. Dolgachev, Igor (2003), Lectures on Invariant Theory, London...

has finite length as A-module. In particular, the image of I {\displaystyle I} there is finitely generated and so I {\displaystyle I} is finitely generated...

In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. Let ( X , T ) {\displaystyle...

\\|\\F_{2}\\|\\F_{1}\\|\\\ F_{0}.\end{array}}} A tower of fields may be finite or infinite. Q ⊆ R ⊆ C is a finite tower with rational, real and complex numbers. The sequence...

torsion-free subgroup of finite index. Every one-relator group is coherent. That is, every finitely generated subgroup is finitely presented. A one-relator...

the finite field Fp introduced below. Otherwise the prime field is isomorphic to Q. Finite fields (also called Galois fields) are fields with finitely many...

A (simple) arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of curved figures in continuous geometries...

says that the category of finite topological coverings of a complex algebraic variety is equivalent to the category of finite étale coverings of the variety...

p-elementary group is a direct product of a finite cyclic group of order relatively prime to p and a p-group. A finite group is an elementary group if it is...

be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements)...

Several methods of discretization can be applied: Finite volume method Finite elements method Finite difference method We begin with the incompressible...

regular examples of directed graphs are given by the Cayley graphs of finitely-generated groups, as well as Schreier coset graphs In category theory,...

effective) method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a...

has only finitely many solutions, for any given integer A {\displaystyle A} , and that n ! = P ( x ) {\displaystyle n!=P(x)} has only finitely many integer...

digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any...

this dependence when calculating the sample variance, a finite population correction (or finite population multiplier) of N − n N − 1 {\displaystyle {\frac...

Noether in 1926. It states that for any field k {\displaystyle k} , and any finitely generated commutative k-algebra A {\displaystyle A} , there exist elements...

mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan. In...

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite...

finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite)...

these counterexamples are finitely presented, and for some years it was considered possible that the conjecture held for finitely presented groups. However...

field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. An étale algebra is a special sort...

supercomputers. The simplest and the original implementation, later formalized as Finite Field Diffie–Hellman in RFC 7919, of the protocol uses the multiplicative...

Ado's theorem is a theorem characterizing finite-dimensional Lie algebras. Ado's theorem states that every finite-dimensional Lie algebra L over a field...

which have been featured in internet memes. He has released the albums Finite Form (2013), Viator (2015), Reviator (2017) Pop Food (2017), HiLo (2018)...

vertex groups Av are finitely generated then G is finitely generated. If A is finite and all the vertex groups Av are finitely presented and all the...

infinite index sets I {\displaystyle I} because only finitely many of the terms are non-zero. If finitely many Hilbert spaces H 1 , … , H n {\displaystyle...

lens, so the plane at which the image sensor is placed is said to be a finite conjugate. Within a system with relay lenses or eyepieces, there may be...

lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric...

In mathematics, a K-finite function is a type of generalized trigonometric polynomial. Here K is some compact group, and the generalization is from the...

modules that are not finitely generated. A finitely presented module (that is the quotient of a finitely generated free module by a finitely generated submodule)...

theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for finite rings, there is no distinction...

Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. Bers (1970) suggested...

In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric...

sponge function or sponge construction is any of a class of algorithms with finite internal state that take an input bit stream of any length and produce an...

numbers. (These are simply formal power series in which we also allow finitely many terms of negative degree.) Then K has a unique cyclic extension K...

by groups of finite Morley rank (see below). A group of finite Morley rank is an abstract group G such that the formula x = x has finite Morley rank for...

of finitely generated abelian groups is that every such torsion-free group is isomorphic to a Z r {\displaystyle \mathbb {Z} ^{r}} . A non-finitely generated...

In mathematics, subshifts of finite type are shift spaces defined by a finite set of forbidden words. They are used to model dynamical systems, and in...

a (countable) structure by its finitely generated substructures. Given a class K {\displaystyle \mathbf {K} } of finite relational structures, if K {\displaystyle...

Thue equations (bivariate homogeneous of degree at least 3) possess only finitely many integer solutions. Roth's theorem states that every irrational algebraic...

groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose...

Any two vertices in G can be connected by a unique simple path. If G has finitely many vertices, say n of them, then the above statements are also equivalent...

finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable. If we restrict ourselves to finitely...

According to this theorem, whenever the natural numbers are partitioned into finitely many subsets, there exist arbitrarily large sets of numbers all of whose...

pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by Tate (1958, 1963)...

membership required.) Webb, Ursula Hilda Mary (1978). Automorphisms of finitely generated nilpotent groups. warwick.ac.uk (PhD thesis). University of Warwick...

complement automaton. For instance, if A is a finite automaton, then a complement automaton for A as a finite automaton always exists, because the regular...

example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside's problem, conversely, asks whether a finitely generated periodic...

if and only if it is regular and has a countably locally finite (that is, 𝜎-locally finite) basis. A topological space X {\displaystyle X} is called...

by construction there are, for every given x , {\displaystyle x,} only finitely many nonzero terms in the sum, so issues regarding convergence of such...

A finite wing is an aerodynamic wing with tips that result in trailing vortices. This is in contrast to an infinite wing. According to John D. Anderson...

a specific procedure to make these divergent integrals finite and obtain (and predict) finite values for physically measurable quantities. The Bogoliubov–Parasyuk...

Charles Babbage. The name difference engine is derived from the method of finite differences, a way to interpolate or tabulate functions by using a small...

substantial power savings. A finite-state machine with data path (FSMD) is a mathematical abstraction which combines a finite-state machine, which controls...

locally finite, infinite graph. This means that every two vertices can be connected by a finite path, each vertex is adjacent to only finitely many other...

finitely many points removed, and the covering is ramified at finitely many points. A Kleinian group is called finitely generated if it has a finite number...

are given by a stationary finite Markov chain. An information source is a sequence of random variables ranging over a finite alphabet Γ {\displaystyle...

bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed...

theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable...

approximate identity, whose approximately unital modules are the same as K-finite representations of the pairs (g, K). The Hecke algebra of a pair is a generalization...

Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin. Every Artin algebra...

theory) for basic operations such as sum or products of ideals. Ideals in a finitely generated algebra over a field (that is, a quotient of a polynomial ring...

xn+1. An aperiodic monoid is an aperiodic semigroup which is a monoid. A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups...

Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point...

posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area...

infinity if, for every compact set S ⊆ X {\displaystyle S\subseteq X} only finitely many points p i {\displaystyle p_{i}} are in S . {\displaystyle S.} Then...

In mathematical physics, a caloron is the finite temperature generalization of an instanton. At zero temperature, instantons are the name given to solutions...

Banach–Tarski paradox, demonstrating that one can cut one sphere into finitely many pieces, smoothly rearrange them, and end with two solid spheres, which...

as a small skeletal thin finitely cocomplete cartesian closed category, and a Boolean algebra as a small skeletal thin finitely cocomplete *-autonomous...

the group of units of O, denoted O×, is a finitely generated abelian group. The fundamental theorem of finitely generated abelian groups therefore implies...

which every ideal is finitely generated; that is, all elements of any ideal can be written as a linear combinations of a finite set of elements, with...

games: finite and infinite. A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play. Finite games are...

of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite measures...

compact-finite open refinement. That is, given any open cover, we can find an open refinement with the property that every compact set meets only finitely many...

Unsolved problem in mathematics Do the Henson graphs have the finite model property? More unsolved problems in mathematics In graph theory, the Henson...

space. The integers modulo 2 form the finite field F 2 {\displaystyle \mathbb {F} _{2}} , the smallest finite field. It has two elements, usually denoted...

graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many...

countable set of all integer sequences that have only finitely many nonzero entries. Functions of finite support are used in defining algebraic structures...

computer proof, with Larry Thiel and S. Swiercz, of the nonexistence of a finite projective plane of order 10. Lam earned his PhD in 1974 under Herbert Ryser...

FEBio(Finite Elements for Biomechanics) is a software package for finite element analysis and was specifically designed for applications in biomechanics...

the Bass conjecture. For any finitely generated Z-algebra A, the groups K'n(A) are finitely generated (K-theory of finitely generated A-modules, also known...

category of finite S {\displaystyle \mathbb {S} } -sets (roughly because the permutation category is equivalent to the category of finite sets and bijections...

theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. The two-component unlink, consisting...

In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of...

Unsolvability of the Word Problem for Finitely Presented Groups" (PDF). Retrieved 6 December 2021. "Subgroups of finitely presented groups". Mathematics of...

enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically...

\sum _{i=m}^{n}\sum _{j=m}^{n}a_{i,j}=\sum _{i,j=m}^{n}a_{i,j}} . The term finite series is sometimes used when discussing the summation presented above....

groups Finitely generated free abelian groups Polycyclic groups Finitely generated recursively absolutely presented groups, including: Finitely presented...

sentence φ of vocabulary τ is finitely satisfiable. Proof According to the previous lemma, we can in fact use finitely many binary relation symbols. The...

over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Élie Cartan (1909), and its analogues over finite fields...

to the familiar notion of finitely generated modules.) Epimorphic images of finitely generated objects are again finitely generated. If U ⊆ X {\displaystyle...

neighborhood" is taken to be a finitely generated subgroup. An infinite group is said to be locally P if every finitely generated subgroup is P. For instance...

difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings. The word 'catenary' is derived...

a number field K; or more generally (for global fields or more general finitely-generated rings or fields). There is some tension here between concepts:...

used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set called an alphabet. A primary...

In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers...

IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set D of natural numbers are all those numbers...

equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite...

structure theorem for finitely generated modules over a principal ideal domain, which includes the fundamental theorem of finitely generated abelian groups...

some nonzero m ∈ M implies xnM = 0 for some positive integer n. A nonzero finitely generated module M over a commutative Noetherian ring is coprimary if and...

group theory, k-theory, and knot theory. Let F n {\textstyle F_{n}} be a finitely generated free group of rank n {\textstyle n} . An elementary Nielsen transformation...

mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and...

a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not within the graph. In the special case of a finite simple...

one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set. The notation for...

l collinear to p is either a singleton or the whole l. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects. A polar...

polynomials. If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements. The Racah...

In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups...

Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). More narrowly...

statistically independent samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges...

is locally cyclic if every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group...

the term finite field. finite type An algebra over a ring is said to be of finite type if it is finitely generated as an algebra. finitely generated...

definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals". Some literature refers to a commutative semi-local...

differential equations. The duality argument for estimating the error of the finite element method and a scheme for the weak enforcement of Dirichlet boundary...

In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph...

\right\}} is finite. A cover of X {\displaystyle X} is said to be point finite if every point of X {\displaystyle X} is contained in only finitely many sets...

theorem, which asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective...

convergence. Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles...

it is often used for finite populations when people are interested in measuring the process that created the existing finite population (this is called...

theorem), and G {\displaystyle G} is finitely generated (see Švarc-Milnor lemma). In particular, CAT(0) groups are finitely generated, and the space X {\displaystyle...

a commutative Noetherian local ring and M is a non-zero finitely generated R-module of finite projective dimension, then: p d R ( M ) + d e p t h ( M...

of equations is finite. A free monoid on a finite alphabet is compact. A free monoid on a countable alphabet is compact. A finitely generated free group...

equivalently a group whose derived series reaches the trivial group in finitely many steps Solvable extension, a field extension whose Galois group is...

computer hardware and software In particular, a finite-state automaton, an automaton limited to a finite state space Automatons (film), a 2006 film Automaton...

into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Class field theory provides detailed...

after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn,...

eRe} is also Dedekind-finite. A ring with finitely many nilpotents is Dedekind-finite. A unit-regular ring is Dedekind-finite. A counter-example can...

ring is finitely generated Hilbert's finiteness theorem, in invariant theory, stating that the ring of invariants of a reductive group is finitely generated...

the Banach–Tarski paradox demonstrates that it is impossible to find a finitely additive measure (or a Banach measure) defined on all subsets of a Euclidean...

left-noetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian...

to any topological space which has a finitely generated homology. Given a topological space which has a finitely generated homology, the Poincaré polynomial...

consistent for the variance estimate of the residuals when regressors have finite fourth moments and—by the Gauss–Markov theorem—optimal in the class of linear...

contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement...

collection of finitely generated subgroups H i {\displaystyle H_{i}} of a given group G {\displaystyle G} can be partially ordered by inclusion. Finite sets of...

In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field...

{\mathcal {E}}} such that: E {\displaystyle {\mathcal {E}}} is finitely complete (it has all finite limits). Equivalently, it has a terminal object, binary products...

is an American mathematician known for work in the theory of semigroups, finite-state automata, and algebraic approaches to differential equations. Rhodes...

(repetitive process) terminates with a simple fraction, the result is a finite continued fraction; if it continues indefinitely, the result is an infinite...

consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules...

approach to minimizing functions that can be decomposed into finite sums. By exploiting the finite sum structure, variance reduction techniques are able to...

permutable subgroup is normal. (The finiteness is used crucially in the proofs.) In summary, a subgroup H of a finite group G is permutable in G if and...

to the vertical at a given point. By extension, the concept applies to finite objects contained by a line or a plane, such as line segments, plane regions...

homomorphism (which is easily seen to be well-defined). If A {\displaystyle A} is finitely generated and abelian, then it can be written as the direct sum of its...

Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial...

curves over the rationals have finite Tate–Shafarevich groups. It is widely believed that these groups are always finite. Rubin graduated from Princeton...

Finite risk insurance is the term applied within the insurance industry to describe an alternative risk transfer product that is typically a multi-year...

finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, then Ker(X/K) is finite...

ANSA is a computer-aided engineering tool for finite element analysis and computational fluid dynamics analysis widely used in the automotive industry...

In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the...

Deeper: The D:finitive Worship Experience is a double compilation album by the band Delirious?. "Did You Feel the Mountains Tremble? (New Version)" - 9:01...

In mathematics the finite Fourier transform may refer to either another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g...

In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The...

in New York City, New York) is an American mathematician specializing in finite group theory. Lyons received his PhD in 1970 at the University of Chicago...

mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first...

) d . {\displaystyle a_{n}=a_{1}+(n-1)d.} A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called...

{\displaystyle N/aM} is finitely generated. Since a M {\displaystyle aM} is also finitely generated, it follows that N {\displaystyle N} is finitely generated; i...

is finitely complemented in X {\displaystyle X} , then so are its supersets in X {\displaystyle X} . If the base set X {\displaystyle X} is finite, then...

is finite. If μ {\displaystyle \mu } is a finitely additive, monotone, and locally finite then μ ( K ) {\displaystyle \mu (K)} is necessarily finite for...

p-group of finite coclass is solvable. Conjecture D: There are only finitely many pro p-groups of given coclass. Conjecture E: There are only finitely many...

L} preserving finite products. A model of a Lawvere theory in a category C {\displaystyle C} with finite products is a finite-product preserving...

number variables to help distinguish them from natural number variables. A finite set can be enumerated by successively labeling each element with the least...

invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting...

A finite map can be one of the following: In computer science, finite map is a synonym for an associative array. A finite map in algebraic geometry is...

Gelfand–Graev representation is a representation of a reductive group over a finite field introduced by Gelfand & Graev (1962), induced from a non-degenerate...

quadrilateral element, also known as the Q8 element is a type of element used in finite element analysis which is used to approximate in a 2D domain the exact solution...

that every finite separable extension is a simple extension. For a field of characteristic 0 such as the rationals, this means all finite extensions are...

dimension is infinite. Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules...

is said to be finitely variable, or to have the finite variability property, if during each bounded interval, the letter change a finite number of times...

F_{i}} are finitely generated free modules. 2.  A finitely presented module is a module that admits a finite free presentation. finitely generated A...

The theorem states that every finitely generated projective module over a polynomial ring is free. Geometrically, finitely generated projective modules...

finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring...

automaton in the same way that an alternating finite automaton is a generalisation of a nondeterministic finite automaton (NFA). The emptiness problem (deciding...

analysis of partial differential equations in general and finite difference method and finite element method in particular. Mitchell has authored several...

and those terms are finitely generated; thus, B I M {\displaystyle B_{I}M} is finitely generated. Conversely, if it is finitely generated, then it is...

Mod-R and is a left Kasch ring. A ring R is right finitely pseudo-Frobenius if and only if every finitely generated faithful right R module is a generator...

n) is not residually finite (i.e. if it is not the case that |m| = 1, |n| = 1, or |m| = |n|) it cannot be isomorphic to a finitely generated linear group...

over a finite field, using only multiplication of the matrix by long, thin matrices. Such matrices are considered as vectors of tuples of finite-field...

of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks...

different secondary characters and a basic theme. Series can either have a finite number of episodes like, for example, miniseries, a definite end, or be...

dimensional analogues of the situation described above; in the case of finite-dimensional Morse theory, invariance may be proved by proving that Morse...

the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices...

It is a kind of dictionary-matching algorithm that locates elements of a finite set of strings (the "dictionary") within an input text. It matches all strings...

In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ ) is a finite sequence of mathematically rigorous instructions, typically used to solve...

{\displaystyle N} are finitely generated R {\displaystyle R} -modules. The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension...

distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations...

measure: while the latter must be countably additive, the former must only be finitely additive. In many important applications the A {\displaystyle {\mathcal...

graphs have the finite model property? Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts? Does...

states that, when there exists a decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes...

different kinds of finite automata. The classical result in the area is that simulating an n {\displaystyle n} -state nondeterministic finite automaton by a...

{\displaystyle d} -dimensional Euclidean space, every convex combination of finitely many points from X {\displaystyle X} is also a convex combination of at...

FEFLOW (Finite Element subsurface FLOW system) is a computer program for simulating groundwater flow, mass transfer and heat transfer in porous media and...

A finite-state transducer (FST) is a finite-state machine with two memory tapes, following the terminology for Turing machines: an input tape and an output...

j ) {\displaystyle (x_{n_{j}})} . But suppose now that there are only finitely many peaks, let N {\displaystyle N} be the final peak if one exists (let...

In mathematics, a uniform tree is a locally finite tree which is the universal cover of a finite graph. Equivalently, the full automorphism group G = A...

group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), then there is a basis e 1...

Trefftz (1888–1937). It falls within the class of finite element methods. The hybrid Trefftz finite-element method has been considerably advanced since...

M {\displaystyle M} is a finitely cogenerated module if and only if s o c ( M ) {\displaystyle \mathrm {soc} (M)} is finitely generated and s o c ( M )...

trumpet) is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian idea that the archangel Gabriel...

algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension...

and only if it is a field. A ring with finitely many, say left, ideals is left Artinian. In particular, a finite ring (e.g., Z / n Z {\displaystyle \mathbb...

closure is compact Totally bounded set, a subset that can be covered by finitely many subsets of any fixed size This disambiguation page lists articles...

irreducible representation of the finite group Sp4(Fq), where Fq is a finite field with q elements. In this finite-field case, for q odd, θ10 has dimension...

characters. Common behavior selection algorithms include: Finite-state machines Hierarchical finite-state machines Decision trees Behavior trees Hierarchical...

group is a countable locally finite group, say U, which is uniquely characterized by the following properties. Every finite group G admits a monomorphism...

a finite field. It was proposed by Michael O. Rabin. Given an n-bit message m0,...,mn-1, we view it as a polynomial of degree n-1 over the finite field...

is finitely generated, which is stronger than the group itself being finitely generated: the free group on 2 or finitely more generators is finitely generated...

Python for specifying finite element discretizations of differential equations in terms of finite element variational forms; FIAT (finite element automatic...

the result above, the category of real finite-rank vector bundles on X is equivalent to the category of finitely generated projective modules over C(X)...

said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended...

finitely-generated group with polynomial growth on a standard Borel space induces a hyperfinite orbit equivalence relation. The action of a finitely-generated...

of finitely generated abelian groups Fundamental theorem of finitely generated modules over a principal ideal domain Fundamental theorem of finite distributive...

equilibrium through backwards induction is (UA, X) with the payoff (3, 4). For finitely repeated games, if a stage game has only one unique Nash equilibrium, the...

distribution is a symmetric probability distribution wherein each of some finite whole number n of outcome values are equally likely to be observed. Thus...

isomorphic; these include many finite stationary stochastic processes, including Markov chains and subshifts of finite type, Anosov flows and Sinai's...

theorem which states that every finitely generated recursively presented group can be homomorphically embedded in a finitely presented group. Most modern...

February 2019) was a mathematician in Cambridge, England, who worked on finite simple groups. Simon Norton was born into a Sephardi family of Iraqi descent...

weighted finite-state machine is a generalization of a finite-state machine in which the edges have weights, for example real numbers or integers. Finite-state...

where he was advised by Juris Hartmanis on the thesis, Complexity of Finitely Presented Algebras. He is known for his work at the intersection of logic...

particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the...

group of finite rank. They were introduced by M. C. R. Butler (1965). Butler, M. C. R. (1965), "A class of torsion-free abelian groups of finite rank",...

the Finnish mathematician Bertil Qvist [de], is a statement on ovals in finite projective planes. Standard examples of ovals are non-degenerate (projective)...

economics, scarcity refers to the basic fact of life that there exists only a finite amount of human and nonhuman resources which the best technical knowledge...

expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and...

intersection point. Let R be a Noetherian ring, I an ideal in R, and M a finitely generated R-module. The depth of I on M, written depthR(I, M) or just depth(I...

The intersection non-emptiness problem, also known as finite automaton intersection problem or the non-emptiness of intersection problem, is a PSPACE-complete...

Noetherian, a module over A is perfect if and only if it is finitely generated and of finite projective dimension. Perfect complexes are precisely the compact...

the number of occurrences. Naturally, the patterns can not be enumerated finitely in this case. They are represented usually by a regular grammar or regular...

symbols [ A ] {\displaystyle [A]} for any finitely generated abelian groups A. One first notes that any finite abelian group G satisfies that [ G ] = 0...

{\displaystyle \Sigma } is the Banach space consisting of all bounded and finitely additive signed measures on Σ {\displaystyle \Sigma } . The norm is defined...

Theistic finitism, also known as finitistic theism or finite godism, is the belief in a deity that is limited. It has been proposed by some philosophers...

of generating sets of L over K. In particular, a finitely generated field extension admits a finite transcendence basis. If no field K is specified, the...

e_{F}:=\sum _{v\in F}1_{v}} where F ranges over finite subsets of the vertex set of Γ. If the quiver has finitely many vertices and arrows, and the end vertex...

separable quantum states and mixed entangled states now known as Werner state, finitely correlated states aka matrix product states, mean field theory and entanglement...

peripheral subgroups are finitely generated nilpotent (Dahmani, Touikan) Any hyperbolic group, such as a free group of finite rank or the fundamental group...

in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are different from unique...

existence of a (finite) basis of V, and can also be phrased as saying that any linear map V → V can be written as the sum of (finitely many) rank-one linear...

modular representation theory of finite groups and related questions of algebraic topology, representation theory of finite algebras and homological algebra...

Kulikov. Let A be an abelian group. If A is finitely generated then by the fundamental theorem of finitely generated abelian groups, A is decomposable...

December 1948) is an American mathematician specializing in the theory of finite groups. Solomon studied as an undergraduate at Queens College and received...

form, proposed by Thomas Baldwin[citation needed]. There could have been finitely many things. For each thing, that thing might not have existed. The removal...

mechanics. Bathe is considered to be one of the pioneers in the field of finite element analysis and its applications. He was born in Berlin as a second...

finite approximations of the limit figure. This is done using the method which we will call box self-similarity where measurements are made on finite...

does: injective, projective, flat, faithful, simple, semisimple, finitely generated, finitely presented, Artinian, and Noetherian. Examples of properties not...

method of Lagrange multipliers as used to find extrema of a function of finitely many variables. Let X and Y be real Banach spaces. Let U be an open subset...

series with coefficients in the valuation ring of the field have only finitely many zeroes. It was introduced by Reinhold Strassman. Let K {\displaystyle...

objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and...

geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain...

{\displaystyle r} be the common ratio of a geometric progression. For a finite number of terms, the geometric sequence consists of the elements up to n...

{\displaystyle f} ) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of f {\displaystyle f} ....

classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each...

Specifically, it states that: For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such...

classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either...

two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique...

Since only finitely many coefficients a i {\displaystyle a_{i}} and b j {\displaystyle b_{j}} are non-zero, all sums in effect have only finitely many terms...

Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of...

then there exists such a line for all the animals. More generally, for finitely many red and blue points in d {\displaystyle d} -dimensional Euclidean...

the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every...

as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and...

Gábor Korchmáros (born 1948) is a Hungarian mathematician, who works on finite geometry. Korchmáros received in 1972 from the University of Budapest a...

states that every set of n {\displaystyle n} -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed...

(BTP) is a procedure for envy-free cake-cutting. It explicated the first finite procedure to produce an envy-free division of a cake among any positive...

A state register that stores the state of the Turing machine, one of finitely many. Among these is the special start state with which the state register...

(that is, intersects non-trivially) only finitely many cells of the complex. Thus f(en) meets at most finitely many cells of Y, so we can take e k ⊆ Y...

pointwise convergence, in which all but finitely many of the functions lie in a thin band at each point, but the finite set of functions which must be excluded...

defining a probability distribution over sparse binary matrices with a finite number of rows and an infinite number of columns. This distribution is suitable...

coupled gyrokinetic-Maxwell equations using a combination of finite difference, finite element and spectral methods. Given plasma equilibrium data, GYRO...

contributions on the finite element method, particularly the conception and implementation of the p- and hp-versions of the finite element method. He is...

In particular, it follows that on S 2 {\displaystyle S^{2}} there is no finitely additive measure defined on all subsets such that the measure of congruent...

indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic...

following sense. Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that...

approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method. The numerical solution u h...

all finite groups by the symmetries of its graphs. László Babai conjectures, more strongly, that each minor-closed family can represent only finitely many...

in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class...

subset is rational. For finite groups the following result of A. Anissimov and A. W. Seifert is well known: a subgroup H of a finitely generated group G is...

regular local rings For a commutative Noetherian local ring R, a finite (i.e. finitely generated) R-module M ≠ 0 {\displaystyle M\neq 0} is a Cohen–Macaulay...

collection of finitely generated subgroups H i {\displaystyle H_{i}} of a given group G {\displaystyle G} can be partially ordered by inclusion. Finite sets of...

functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially on vector spaces. For...

classified finite-dimensional simple and also semisimple algebras over fields. Simple algebras are building blocks of semisimple algebras: any finite-dimensional...

manifold is finitely generated if its Ricci curvature stays nonnegative. In an oversimplified interpretation, such a manifold has a finite number of "holes"...

xj, and ti is not equal to xj, choose δ smaller. Since there are only finitely many ti and xj, we can always choose δ sufficiently small. Now we add two...

showing what state (or states in the case of a nondeterministic finite automaton) a finite-state machine will move to, based on the current state and other...

projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective...

set A {\displaystyle A} of finite measure, E ∩ A {\displaystyle E\cap A} is measurable. σ {\displaystyle \sigma } -finite measures and measures arising...

programming a finite-state machine, and experiment with proofs, such as converting a nondeterministic finite automaton (NFA) to a deterministic finite automaton...

context of finite models. Other widely used techniques for proving inexpressibility results, such as the compactness theorem, do not work in finite models...

where the total size reading population has been normalized to 1. A typical finite-dimensional mixture model is a hierarchical model consisting of the following...

points for a global field k is finitely generated by the Mordell-Weil theorem. Hence, by the structure theorem for finitely generated abelian groups, it...

belong to D ( A ) {\displaystyle {\mathfrak {D}}(A)} , and that after finitely many steps, we end up with zero: ( A − λ I B ) k x = 0 {\displaystyle (A-\lambda...

the remainder of this article presumes that computable functions take finitely many natural numbers as arguments and produce a value that is a single...