In mathematics , a finiteset is a Set mathematics set that has a finite number of element mathematics elements . For example, math 2,4,6,8,10 , math is a finiteset with five elements. The number of elements of a finiteset is a natural number non negative integer , and is called the cardinality of the set. A set that is not finite is called infinite . For example, the set of all positive integers ... , which states that there cannot exist an injective function from a larger finiteset to a smaller finiteset. Definition and terminology Formally, a set S is called finite if there exists a bijection ... of the set, and is denoted S . Note that the empty set is considered finite, with cardinality zero. If a set is finite, its elements may be written as a sequence math S x 1,x 2, ldots,x n . math In combinatorics , a finiteset with n elements is sometimes called an n set and a subset with k elements is called a k subset . For example, the set 5,6,7 is a 3 set, a finiteset with three elements, and 6,7 is a 2 subset of it. Basic properties Any proper subset of a finiteset S is finite ... set S and a proper subset of S . Any set with this property is called Dedekind finite . Using the standard Zermelo Fraenkel set theory ZFC axioms for set theory , every Dedekind finiteset is also .... The union set theory union of two finite sets is finite, with math S cup T le S T . math In fact ..., the Cartesian product of finitely many finite sets is finite. A finiteset with n elements has 2 sup n sup distinct subsets. That is, the power set of a finiteset is finite, with cardinality 2 sup n sup . Any subset of a finiteset is finite. The set of values of a function when applied to elements of a finiteset is finite. All finite sets are countable , but not all countable sets are finite ... sets to be countable. The free semilattice over a finiteset is the set of its non empty subsets ... date October 2009 S is a finiteset. That is, S can be placed into a one to one correspondence with the set ... more details
wiktionary Finite is the opposite of infinite . It may refer to A finite number or value that is, a real number or complex with finite modulus?? Finiteset , having a number of elements given by some natural number Finite verb , being inflected for person and for tense disambig de Endlichkeit fr Fini ko it Finito simple Finite ... more details
In mathematics, a family of sets family math mathcal F math of Set mathematics sets is of finite character provided it has the following properties For each math A in mathcal F math , every finitesetfinite subset of math A math belongs to math mathcal F math . If every finite subset of a given set math A math belongs to math mathcal F math , then math A math belongs to math mathcal F math . Properties A family math mathcal F math of sets of finite character enjoys the following properties For each math A in mathcal F math , every finite or infinite subset of math A math belongs to math mathcal F math . Tukey s lemma In math mathcal F math , partial order partially ordered by inclusion, the Union set theory union of every Total order Chains chain of elements of math mathcal F math also belong to math mathcal F math , therefore, by Zorn Lemma Zorn s lemma , math mathcal F math contains at least one maximal element. Example Let V be a vector space , and let F be the family of linearly independent subsets of V . Then F is a family of finite character because a subset X &sube V is linearly dependent iff X has a finite subset which is linearly dependent . Therefore, in every vector space , there exists a maximal family of linearly independent elements. As a maximal family is a vector basis , every vector space has a possibly infinite vector basis. PlanetMath attribution id 3692 title finite character Mathlogic stub Category Set families eo Ara sistemo de finia speco fr Ensemble de parties de caract re fini ... more details
of finite type There is another finiteness condition on morphisms of schemes, morphisms of finite type , which is much weaker than being finite. Morally, a morphism of finite type corresponds to a set ... scheme s is a finite morphism , if math Y math has an open cover by affine schemes math V i mbox ... Z rightarrow mathbb Z x, y, z langle y 3 x 4 z rangle math . This is an example of a morphism of finite ... of rings math B i rightarrow A ij math . The morphism f is called locally of finite type , if math ... the open cover math f 1 V i bigcup j U ij math can be chosen to be finite, then f is called of finite type . For example, if math k math is a field mathematics field , the scheme math mathbb ... rings math k to k X 1, ldots,X n . math This is a morphism of finite type, but if math n 0 math then it is not a finite morphism. On the other hand, if we take the affine scheme math mbox Spec ... homomorphism math k X to k X,Y langle Y 2 X 3 X rangle. math Then this morphism is a finite morphism. Properties of finite morphisms In the following, f X Y denotes a finite morphism. The composition of two finite maps is finite. Any Grothendieck s relative point of view base change of a finite morphism is finite, i.e. if math g Z rightarrow Y math is another arbitrary morphism, then the canonical morphism math X times Y Z rightarrow Z math is finite. This corresponds to the following algebraic ... module. Closed immersion s are finite, as they are locally given by math A rightarrow A I math , where I is the Ideal ring theory ideal corresponding to the closed subscheme. Finite morphisms are closed ... of Cohen Seidenberg. Finite morphisms have finite fibres i.e. they are quasi finite morphism quasi finite . This follows from the fact that any finite k algebra, for any field k is an Artinian ring . Slightly more generally, for a finite surjective morphism f , one has dim X dim Y . Conversely, proper, quasi finite maps are finite. This is a consequence of the Stein factorization . Finite morphisms ... more details
The term locally finite has a number of different meanings in mathematics Locally finite collection of sets in a topological space Locally finite group Locally finite measure Locally finite poset Locally finite variety in the sense of universal algebra disambig ... more details
cleanup rewrite date September 2011 Groups In mathematics and abstract algebra , a finite group is a group mathematics group whose underlying set mathematics set G has finitesetfinite ly many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local analysis local theory of finite groups, and the theory of solvable group s and nilpotent group s. A complete determination of the structure of all finite groups is too much to hope for the number of possible structures soon becomes overwhelming. However, the complete classification of finite simple groups classification of the finite simple groups was achieved, meaning that the building blocks from which all finite groups can be built are now known, as each finite group has a composition series . During the second half of the twentieth century, mathematicians such as Claude Chevalley Chevalley and Robert Steinberg Steinberg also increased our understanding of finite analogs of classical groups , and other related groups. One such family of groups is the family of general linear group s over finite field s. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure ... symmetry , is strongly influenced by the associated Weyl group s. These are finite groups generated by reflections which act on a finite dimensional Euclidean space . The properties of finite ... orders uses the classification of finite simple groups . For any positive integer n there are at most ... of finite simple groups List of finite simple groups Lagrange s theorem group theory Lagrange ... groups Representation theory of finite groups Modular representation theory Monstrous moonshine Profinite group Infinite group theory Finite ring Div col end Notes references External references Number ... for groups of small order Category Finite groups Category Properties of groups ar ca Grup ... more details
A finite geometry is any geometry geometric system that has only a finitesetfinite number of point geometry points . Euclidean geometry , for example, is not finite, because a Euclidean line contains ... . A finite geometry can have any finite number of dimensions. Finite geometries may be constructed via linear algebra , as vector space s over a finite field , and called Galois geometry Galois geometries , or can be defined purely combinatorially. Many, but not all, finite geometries are Galois geometries for example, any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field the projectivization of a vector space over a finite field , so ... planes which are not isomorphic to projective spaces over finite fields, namely the non Desarguesian plane s, so in this case there is a distinction. Finite planes The following remarks apply only to finite planes . There are two kinds of finite plane geometry affine geometry affine and projective ... parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axiom s. An affine plane geometry is a nonempty set math X math whose ... ell cap ell varnothing. math There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial either empty set empty or too ... Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are parallel ..., a finite affine plane of order math n math has math n 2 math points and math n 2 n math lines each ... thumb 200px right Finite affine plane of order 3, containing 9 points and 12 lines. A projective plane geometry is a nonempty set math X math whose elements are called points , along with a nonempty ... of any two distinct lines contains exactly one point. There exists a set of four points, no three of which ... A finite plane of order n is one such that each line has n points for an affine plane , or such that each ... more details
A finite map can be one of the following In computer science , finite map is a synonym for an associative array . A finite map algebraic geometry finite map in algebraic geometry is a surjective regular map algebraic geometry regular map with zero dimensional fibers. disambig ... more details
representation theory , and the groups of Lie type . Finite fields are an active area of research, including recent results on the Kakeya set Kakeya sets in vector spaces over finite fields Kakeya ... of primitive roots primitive root . Finite fields appear in the following chain of subclass set theory ...In abstract algebra , a finite field or Galois field so named in honor of variste Galois is a field mathematics field that contains a finite number of elements. Finite fields are important in number theory ... arXiv quant ph 9605005 ref The finite fields are classified by size there is exactly one finite field up to isomorphism of size p sup k sup for each prime p and positive integer k . Each finite field ... group of the field is a cyclic group . Wedderburn s little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field. Finite fields have ... domain s principal ideal domain s Euclidean domain s field mathematics field s finite fields . Classification The finite fields are classified as follows Harv Jacobson 2009 loc 4.13,p. 287 The order , or number of elements, of a finite field is of the form p sup n sup , where p is a prime number ... prime number p and positive integer n , there exists a finite field with p sup n sup elements. Any two finite fields with the same number of elements are isomorphic . That is, under some renaming ... tables of the other one. This classification justifies using a naming scheme for finite fields that specifies only the order of the field. One notation for a finite field is F sub p ... algebra Case of fields Prime field . An example of such a finite field is the Ring mathematics ring Z p Z . It is a finite field with p elements, usually labelled 0, 1, 2, ..., p 1, where ... up to 1 with coefficients in Z 2 Z , i.e. the set 0, 1, T , 1 T see below for more details . Notice ... on a set of 6 elements fails to satisfy the mathematical definition of a field mathematics field ... more details
Use mdy dates date September 2011 A finite difference is a mathematical expression of the form f x b &minus f x a . If a finite difference is divided by b &minus a , one gets a difference quotient . The approximation of derivatives by finite differences plays a central role in finite difference method s for the numerical analysis numerical solution of differential ... equations by replacing iteration notation with finite differences. Forward, backward, and central ... this section is linked to further down in the article In an analogous way one can obtain finite ... of finite differences, explained below. If necessary, the finite difference can be centered ... n h fg, x sum limits k 0 n binom n k Delta k h f, x Delta n k h g, x kh . math Finite difference methods An important application of finite differences is in numerical analysis , especially in numerical ... is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference method s. Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal ... 0 frac x a h choose k sum j 0 k 1 k j k choose j f a j h . math Calculus of finite differences The forward ... Boole, George , 1872 . A Treatise On The Calculus of Finite Differences , 2nd ed., Macmillan and Company ... calculus of finite differences&ots vOzf92u Ak&sig b vtnZz9U1zQi3hLmq6Rjk6mC8o v onepage Also, Dover ... , and I is the identity operator . The finite difference of higher orders can be defined in recursive ... arsinh tfrac12 delta h . math The calculus of finite differences is related to the umbral calculus ... f x thus map systematically to umbral finite difference analogs involving f xT sub h sub sup 1 ... integral, is the indefinite sum or antidifference operator. Rules for calculus of finite difference ... nabla f n f b f a 1 math br ref cite book last Levy first H. coauthor Lessman, F. title Finite Difference ... more details
Noref date May 2010 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 circle group T . From an abstract point of view, the characterization of trigonometric polynomials amongst other functions F , in the harmonic analysis of the circle, is that for functions F in any of the typical function space s, F is a trigonometric polynomial if and only if its Fourier coefficient s a sub n sub vanish for n large enough, and that this in turn is equivalent to the statement that all the translates F t by a fixed angle lie in a finite dimensional subspace. One implication here is trivial, and the other, starting from a finite dimensional invariant subspace , follows from complete reducibility of representations of T . From this formulation, the general definition can be seen for a representation of K on a vector space V , a K finite vector v in V is one for which the k . v for k in K span a finite dimensional subspace. The union of all finite dimension K invariant subspaces is itself a subspace, and K invariant, and consists of all the K finite vectors. When all v are K finite, the representation itself is called K finite. Category Representation theory of groups Reference Lectures on Lie Groups ans Lie Algebras by Roger Carter, Graeme Segal and Ian Macdonald ... more details
A finite verb is a verb that is Inflection inflected for grammatical person person and for grammatical tense tense according to the rules and categories of the languages in which it occurs. Finite verbs can form independent clause s, which can stand on their own as complete Sentence linguistics sentence s. The finite forms of a verb are the forms where the verb shows tense, person or number. Non finite verb forms have no person or number, but some types can show tense. Finite verb forms include I go, she goes, he went Non finite verb forms include to go, going, gone Indo European languages In the Indo European language s such as English , only verbs in certain grammatical mood moods are finite. These include the indicative mood expressing a state of affairs e.g., The bulldozer demolished the restaurant, The leaves were yellow and stiff. the Imperative mood imperative mood giving a command e. g., Come here , Be a good boy the subjunctive mood typically used in dependent clauses e. g., It is required ... expressing a wish or hope . Non existent as a mood in English. Verb forms that are non finite verb not finite include the infinitive the participle s e. g., The broken window... , The wheezing gentleman ... complete sentence or clause must contain a finite verb. However, sentences lacking a finite ... where it could have a subject grammar subject and a finite verb form compare I appreciate your help . Finite verbs in syntax Finite verbs play a particularly important role in syntactic analyses of sentence structure. In phrase structure grammar phrase structure grammars , the finite verb is the head of a finite verb phrase VP and as such, it is the head of the entire sentence, and in dependency grammar dependency grammars , the finite verb is the root of the entire clause and is thus the most prominent structural unit in the clause. See also Non finite verb Balancing and deranking Grammatical ... da Finit verbum de Finite Verbform es conjugaci n Formas personales del verbo li Persoensv rm nl ... more details
Finite part may refer to Cauchy principal value Hadamard finite part mathdab Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
In formal language formal language theory , a class of languages math mathcal L math has finite thickness if for every string s , there are only finitely many consistent languages in math mathcal L math . This condition was introduced by Dana Angluin in connection with learning, as a sufficient condition for language identification in the limit . The related notion of M finite thickness We say that math mathcal L math satisfies the MEF condition if for each string s and each consistent language L in the class, there is a minimal consistent language in math mathcal L math , which is a sublanguage of L. Symmetrically, we say that math mathcal L math satisfies the MFF condition if for every string s there are only finitely many minimal consistent languages in math mathcal L math . Finally, math mathcal L math is said to have M finite thickness if it satisfies both the MEF and MFF conditions. Finite thickness implies M finite thickness. However, there are classes that are of M finite thickness but not of finite thickness for example, let math L n math be a class of languages such that math L 0 subseteq L 1 subseteq ldots math . comp sci theory stub Category Formal languages ... more details
In mathematics , more specifically abstract algebra , a finite ring is a ring mathematics ring not necessarily with a multiplicative identity 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 is an example of an abelian group abelian finite group , but the concept of finite rings in their own right has a more recent history ... notes see reference . The occasion of non commutativity in finite rings was described in 1968 in the same journal 75 512&ndash 14 by K. Eldrige in two theorems If the order m of a finite ring with 1 ... non commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper ... references in the topic of finite rings, such as Robert Ballieu 1947 Anneaux finis in Ann ... review MR0022841 of Ballieu. These are a few of the facts that are known about the number of finite rings of a given order suppose p and q represent distinct prime numbers There are two finite rings of order p . There are four finite rings of order pq . There are eleven finite rings of order p sup 2 sup . There are twenty two finite rings of order p sup 2 sup q . There are fifty two finite rings of order eight. There are 3 p 50 finite rings of order p sup 3 sup , p 2 ... s little theorem Artin Wedderburn theorem There are other deep aspects to the theory of finite rings ... s little theorem , asserts that any finite division ring is necessarily commutative and therefore a finite field . Nathan Jacobson later discovered yet another condition which guarantees commutativity ... of finite simple ring s is relatively straightforward in nature. More specifically, any finite simple ring is isomorphic to the ring math M n mathbb F q math of n by n matrices over a finite field ... is Wedderburn s little theorem . On the other hand, the classification of finite simple groups ... of journal pages. Therefore, in some respects, the theory of finite rings is simpler than that of finite ... more details
The term finite mathematics refers either to discrete mathematics , or to a course conventionally required of business students, in which the curriculum brings together several mathematical topics, including basic probability theory , an introduction to linear programming , some theory of matrix mathematics matrices and determinants, and sometimes an abbreviated account of calculus . disambig Category Mathematical disambiguation ... more details
orphan date August 2010 refimprove date March 2009 A finite wing is an wing aerodynamic wing with tips that result in trailing vortices . ref http www.aerospaceweb.org question aerodynamics q0167.shtml ref This is in contrast to an infinite wing . References reflist Category Aircraft wing design aviation stub ... more details
wiktionarypar set TOCright Set may refer to Mathematics and programming Set mathematics , A collection of well defined and distinct objects Set theory Category of sets Set computer science , a data type that is a collection of unique values Set C , an implementation of Set computer science in the C language Standard Template Library STL Chemistry SET, single electron transfer Psychology Set psychology , a set of expectations which shapes perception or thought Technology Set, to make become solid see Solidification A television setSET, Secure Electronic Transaction , a standard protocol for securing credit card transactions over insecure networks SET, Single Electron Transistor , a device to amplify currents in nanoelectronics SET, single ended triode , a type of electronic amplifier SET Awards, The Science, Engineering & Technology Student of the Year Awards Set command , a command for manipulating ..., a Quality Assurance job title in some software companies Saw set , the process of setting the teeth of a saw so each tooth protrudes to the side of the blade Arts Theatre Theatrical scenery Set construction , construction of scenery for theatrical, movie and video production Dancing Set, the basic square formation in Square dance square dancing Music DJ set or DJ mix , a musical performance by a DJ Set theory music , dealing with concepts for categorizing musical objects and describing their relationships Set music , a collection of discrete entities, for example pitch sets, duration sets, and timbre sets Set Thompson Twins album Set Thompson Twins album Set Alex Chilton album Set Alex Chilton album Set list , a list of songs to be performed at a concert In the professional jargon of music production and critique, a set refers to an album of songs. Fiction Set or Father Set , a fictional ... and games A chess setSet, a signal used in American football Set darts , when one player wins three legs Set game , a card game Set, a unit of play in tennis Set, a term for three of a kind in poker ... more details
dimension . A collection of subsets of a topological space X is said to be locally finite , if each ... of the sets in the collection. Note that the term Locally finite disambiguation locally finite has different meanings in other mathematical fields. Examples and properties A finitesetfinite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite ... n . A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of R of the form &minus n , n with integer n . If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. The converse, however, can fail if the closures of the sets are not distinct. For example, in the finite complement topology on R the collection of all open sets is not locally finite, but the collection of all closures of these sets is locally finite since the only closures are R and the empty set . Compact spaces No infinite collection of a compact space can be locally finite. Indeed, let G sub a sub be an infinite family of subsets of a space and suppose this collection is locally finite. For each point x of this space ... has a finite subcover, U sub a sub 1 sub sub ...... U sub a sub n sub sub . Since each U sub ... every open cover admits a locally finite open refinement topology refinement is called paracompact space paracompact . Every locally finite collection of subsets of a topological space X is also Point finite collection point finite . A topological space in which every open cover admits a point finite ... cover topology cover of a Lindel f space space can be locally finite, by essentially the same argument ... is locally finite. Closed sets It is clear from the definition of a topology that a finite union of closed .... However, if we consider a locally finite collection of closed sets, the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closed ... more details
In mathematics , a locally finite measure is a Measure mathematics measure for which every point of the measure space has a Neighbourhood mathematics neighbourhood of Wikt finitefinite measure. Definition Let X , T be a Hausdorff space Hausdorff topological space and let &Sigma be a sigma algebra &sigma algebra on X that contains the topology T so that every open set is a measurable set , and &Sigma is at least as fine as the Borel sigma algebra Borel &sigma algebra on X . A measure signed measure complex measure &mu defined on &Sigma is called locally finite if, for every point p of the space X , there is an open neighbourhood N sub p sub of p such that the &mu measure of N sub p sub is finite. In more condensed notation, &mu is locally finite if and only if math forall p in X, exists N p in T mbox s.t. p in N p mbox and left mu N p right infty. math Examples Any probability measure on X is locally finite, since it assigns unit measure the whole space. Similarly, any measure that assigns finite measure to the whole space is locally finite. Lebesgue measure on Euclidean space is locally finite. By definition, any Radon measure is locally finite. Counting measure is sometimes locally finite and sometimes not counting measure on the integer s with their usual discrete topology is locally finite, but counting measure on the real line with its usual Borel topology is not. See also Inner regular measure Strictly positive measure References Unreferenced date December 2006 Category Measures measure theory fr Mesure localement finie nl Lokaal eindige maat pl Miara lokalnie sko czona ... more details
of a Set mathematics set X is called finite if X is a finite real number rather than . The measure is called finite if X is the countable set countable Union set theory union of measurable sets of finite measure. A set in a measure space is said to have finite measure if it is a countable union of sets with finite measure. Examples Lebesgue measure For example, Lebesgue measure on the real number s is not finite, but it is finite. Indeed, consider the Interval mathematics closed interval ... numbers with the counting measure the measure of any finiteset is the number of elements in the set, and the measure of any infinite set is infinity. This measure is not finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. But, the set of natural numbers math mathbb N math with the counting measure is finite. Locally compact groups Locally compact group s which are compact space compact are finite under Haar measure . For example, all connected space connected , locally compact ...lowercase finite measure redirect Refimprove date March 2009 In mathematics , a positive or signed ... of G , must be G itself. Thus all connected Lie group s are finite under Haar measure ... finite. One example in math scriptstyle R math is for all math scriptstyle A subset R math , math scriptstyle ..., both are translation invariant. Properties The class of finite measures has some very convenient ... similar on the measures involved. Though measures which are not finite are sometimes regarded ... dimension r , then all lower dimensional Hausdorff measure s are non finite if considered as measures on X . Equivalence to a probability measure Any finite measure on a space X is Equivalence ... , n N , be a covering of X by pairwise disjoint measurable sets of finite measure, and let ... math is then a probability measure on X with precisely the same null set s as . DEFAULTSORT ... more details
In mathematics, a locally finite poset is a partially ordered set P such that for all x , y &isin P , the poset Interval interval x , y consists of finiteset finitely many elements. Given a locally finite poset P we can define its incidence algebra . Elements of the incidence algebra are functions that assign to each interval x , y of P a real number x , y . These functions form an associative algebra with a product defined by math f g x,y sum x leq z leq y f x,z g z,y . math See incidence algebra for more details In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime . References Stanley, Richard P. Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113 116. DEFAULTSORT Locally Finite Poset Category Order theory algebra stub ... more details
sup &minus 1 sup f x . In other words, every fiber is a discrete hence finiteset. For every point x of X , the scheme nowrap f sup &minus 1 sup f x X × sub Y sub Spec &kappa f x is a finite &kappa ... is quasi finite if it is Glossary of scheme theory finite type and satisfies any of the following ... O X,x otimes kappa f x math is finitely generated over math kappa f x math . Quasi finite morphisms ... not include the finite type hypothesis. This hypothesis was added to the definition in l ments ... x in X , f is said to be quasi finite at x if there exist open affine neighborhoods U of x ... finite. f is locally quasi finite if it is quasi finite at every point in X . ref EGA III, Err sub III sub , 20. ref A quasi compact locally quasi finite morphism is quasi finite. Properties For a morphism f , the following properties are true. ref EGA II, Proposition 6.2.4. ref If f is quasi finite, then the induced map f sub red sub between reduced scheme s is quasi finite. If f is a closed immersion, then f is quasi finite. If X is noetherian and f is an immersion, then f is quasi finite. If nowrap g Y &rarr Z , and if nowrap g f is quasi finite, then f is quasi finite if any of the following ... finiteness is preserved by base change. The composite and fiber product of quasi finite morphisms is quasi finite. ref EGA II, Proposition 6.2.4. ref If f is Glossary of scheme theory Unramified and tale morphisms unramified at a point x , then f is quasi finite at x . Conversely, if f is quasi finite at x , and if also math mathcal O f 1 f x ,x math , the local ring of x in the fiber f sup &minus 1 sup f x , is a field and a finite separable extension of &kappa f x , then f is unramified at x . ref EGA IV sub 4 sub , Th or me 17.4.1. ref Finite morphism s are quasi finite. ref EGA II, Corollaire 6.1.7. ref A quasi finite proper morphism locally of finite presentation is finite. ref EGA ... compact and quasi separated. Let f be quasi finite, separated and of finite presentation. Then f ... more details
The fuzzy finite element method combines the well established finite element method with the concept of fuzzy number s, the latter being a special case of a fuzzy set ref Michael Hanss , 2005. Applied Fuzzy Arithmetic, An Introduction with Engineering Applications . Springer, ISBN 3 540 24201 5 ref . The advantage of using fuzzy number s instead of real numbers lies in the incorporation of uncertainty on material properties, parameters, geometry, initial conditions, etc. in the finite element analysis. One way to establish a fuzzy finite element FE analysis is to use existing FE software in house or commercial as an inner level module to compute a deterministic result, and to add an outer level loop to handle the fuzziness uncertainty . This outer level loop comes down to solving an Optimization mathematics optimization problem. If the inner level deterministic module produces monotonic behavior with respect to the input variables, then the outer level optimization problem is greatly simplified, since in this case the extrema will be located at the vertex geometry vertices of the domain mathematics domain . See also div style moz column count 2 column count 2 Finite element method Fuzzy number Fuzzy set Uncertainty div References reflist logic DEFAULTSORT Fuzzy Finite Element Category Fuzzy logic fa pt Elemento infinito difuso ... more details
In the theory of computation , a generalized nondeterministic finite automaton GNFA , also known as expression automaton or generalized nondeterministic finite state machine is a variation of nondeterministic finite automaton NFA where each transition is labeled with any regular expression . The GNFA reads blocks of symbols from the input which constitute a string as defined by the regular expression on the transition. There are several differences between a standard finite state machine and a generalized nondeterministic finite state machine. A gNFA must have only one start state and one accept state, and these cannot be the same state, whereas a NFA or DFA both may have several accept states, and the start state can be an accept state. A gNFA must have only one transition between any two states, whereas a NFA or DFA both allow for numerous transitions between states. In a gNFA, a state has a single transition to every state in the machine, although often it is a convention to ignore the transitions that are labelled with the empty set when drawing generalized nondeterministic finite state machines. Formal definition A GNFA can be defined as a n tuple 5 tuple , S , , T , s , a , consisting of a finiteset of states S a finiteset called the alphabet a transition function mathematics function T S & x2216 a × S & x2216 s R a start state s S an accept state a S where R is the collection of all regular expressions over the alphabet . The transition function takes as its ... from other finite state machines, which take as input a single state and an input from the alphabet or the empty string in the case of nondeterministic finite state machines and outputs the next state or the set of possible states in the case of nondeterministic finite state machines . A deterministic finite automaton DFA or nondeterministic finite automaton NFA can easily be converted into a GNFA ... . This shows that GNFAs recognize the same set of formal language s as DFAs and NFAs. References ... more details
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