algebra , an algebraic closure of a field mathematics field K is an algebraic extension of K that is algebraically ... in mathematics. Using Zorn s lemma , it can be shown that every field has an algebraic closure, ref ... Company. pp. 11 12. ref and that the algebraic closure of a field K is unique up to an isomorphism ... speak of the algebraic closure of K , rather than an algebraic closure of K . The algebraic closure of a field K can be thought of as the largest algebraic extension of K . To see this, note that if L is any algebraic extension of K , then the algebraic closure of L is also an algebraic closure of K , and so L is contained within the algebraic closure of K . The algebraic closure of K is also ... containing K , then the elements of M which are algebraic extension algebraic over K form an algebraic closure of K . The algebraic closure of a field K has the same cardinal number cardinality as K ... states that the algebraic closure of the field of real number s is the field of complex number s. The algebraic closure of the field of rational number s is the field of algebraic number s. There are many ... the field of algebraic numbers these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q . For a finite field of prime number prime order p , the algebraic closure is a countably infinite field which contains a copy of the field of order ... expansion . Separable closure An algebraic closure K sup alg sup of K contains a unique separable extension K sup sep sup of K containing all algebraic separable extension s of K within K sup alg sup ... 1. Saying this another way, K is contained in a separably closed algebraic extension field. It is essentially unique up to isomorphism . The separable closure is the full algebraic closure if and only ... over K , math K X sqrt p X supset K X math is a non separable algebraic field extension. In general ... Algebraically closed field Algebraic extension Notes reflist DEFAULTSORT Algebraic Closure ... more details
unreferenced date June 2008 Algebraic enumeration is a subfield of enumeration that deals with counting algebra ic objects. Algebraic properties and symmetry symmetries of the objects being counted can be taken advantage of to simplify the counting process. Category Enumerative combinatorics combin stub ... more details
wiktionary functionFunction may refer to Diatonic function , a term in music theory Function E 40 song , a 2012 song by American rapper E 40 featuring YG rapper YG , iAmSu & Problem Function biology , explaining why a feature survived selection Function computer science , or subroutine, a portion of code within a larger program, performs a specific task Function engineering , related to the selected property of a system Function language , in linguistics, a way of achieving an aim using language Function mathematics , an abstract entity that associates an input to a corresponding output according to some rule Function model , a structured representation of the functions, activities or processes Function object , or functor or functionoid, a concept of object oriented programming Function Drinks , a beverage company based in Redondo Beach, California. An organised event such as a party or meeting See also Functionalism disambiguation Function hall Functional disambiguation Functionality in polymer chemistry see Structural unit Functor disambiguation bg bs Funkcija vor ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko id Fungsi it Funzione lt Funkcija lmo Funziun nl Functie ja no Funksjon nn Funksjon pl Funkcja ujednoznacznienie pt Fun o desambigua o ro Func ie dezambiguizare ru simple Function sk Funkcia sl Funkcija razlo itev sr sh Funkcija razvrstavanje sv Funktion olika betydelser th uk zh ... more details
Algebraic notation can mean In mathematics and computers, infix notation , the practice of representing a binary operator and operands with the operator between the two operands as in 2 2 Algebraic notation chess , one of the most popular systems for notating the placement and movement of pieces in a chess game In linguistics, recursive categorical syntax , also known as algebraic syntax , a theory of how natural languages are structured Mathematical notation for algebra disambig ... more details
In the mathematics mathematical field of knot theory , an algebraic link is a link knot theory link formed by taking the numerator closure of an tangle mathematics algebraic tangle . References Colin Adams, The Knot Book , American Mathematical Society, ISBN 0 8050 7380 9 Category Knot theory knottheory stub ... more details
In mathematics, S function may refer to sigmoid function Schur polynomials In physics, it may refer to Action physics action functional 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 mathematics , an algebraic expression is an Expression mathematics expression that contains Variable mathematics variables and a finite number of algebraic Operation mathematics operations addition , subtraction , multiplication , division and exponentiation to a rational exponent . A rational algebraic expression or rational expression is an algebraic expression that can be written as a quotient of polynomial s, such as math x 2 2x 4 math . An irrational algebraic expression is one that is not rational, such as math sqrt x 4 math . Some but not all polynomial equation s with Rational number rational coefficients have a solution that is an algebraic expression with a finite number of operations involving just those coefficients that is, can be Algebraic solution solved algebraically . This can be done for all such equations of degree one, two, three, or four but for given n 5 it can be done for some equations but Abel Ruffini theorem not for others . References cite book last1 Morris first1 Christopher G. title Academic Press dictionary of science and technology page 74 year 1992 url http books.google.co.uk books?id nauWlPTBcjIC&lpg PA74&dq algebraic 20expression 20over 20a 20field&pg PA74 v onepage&q&f false cite book last1 James first1 Robert Clarke last2 James first2 Glenn title Mathematics dictionary page 8 year 1992 url http books.google.co.uk books?id UyIfgBIwLMQC&lpg PA8&dq algebraic 20expression 20over 20a 20field&pg PA8 v onepage&q&f false External links MathWorld title Algebraic Expression id AlgebraicExpression Category Elementary algebra ... more details
In mathematics , an algebraic torus is a type of commutative affine algebraic group . These groups were named by analogy with the theory of tori in Lie group theory see maximal torus . The theory of tori is in some sense opposite to that of unipotent group s, because tori have rich arithmetic structure but no deformations. Definition Given a base Scheme mathematics scheme S , an algebraic torus over S is defined to be a group scheme over S that is flat topology fpqc locally isomorphic to a finite product of multiplicative groups. In other words, there exists a faithfully flat map X S such that any point in X has a quasi compact open neighborhood U whose image is an open affine in S , such that base change to U yields a finite product of copies of GL sub 1, U sub G sub m sub U . One particularly important case is when S is the spectrum of a field K , making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of G sub m sub L . In general, the multiplicity of this product i.e., the dimension of the scheme is called the Rank mathematics rank dn date July 2011 of the torus, and it is a locally constant function on S . If a torus is isomorphic to a product of multiplicative groups G sub m sub S , the torus is said to be split . All tori over separably closed fields are split, and any non separably closed field .... The weight lattice math X bullet T math is the group of algebraic homomorphisms T G sub m sub , and the coweight lattice math X bullet T math is the group of algebraic homomorphisms ... between the category of tori over K with algebraic homomorphisms and the category of finitely ... gives a canonical embedding of S into GL sub 2 sub , and composition with determinant gives an algebraic ... structure sheaf as a locally free math mathcal O S math module, and it is a locally constant function ... number of algebraic tori Annals of Mathematics 78 1 1963. Category Algebraic groups Category Article ... more details
Image VEST Core4 LowLevel.png thumbnail 320px right VEST 4 T function followed by a transposition layer In cryptography , a T function is a bijection bijective mapping that updates every bit of the state computer science state in a way that can be described as math x i x i f x 0, cdots, x i 1 math , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T function is called triangular . Thanks to their bijectivity no collisions, therefore no entropy loss regardless of the used Boolean function s and regardless of the selection of inputs as long as they all come from one side of the output bit , T functions are now widely used in cryptography to construct block cipher s, stream cipher s, PRNG s and cryptographic hash function hash functions . T functions were first proposed in 2002 by Alexander Klimov A. Klimov and Adi Shamir A. Shamir in their paper A New Class of Invertible Mappings . Ciphers such as TSC 1 , TSC 3 , TSC 4 , ABC stream cipher ABC , Mir 1 and VEST are built with different types of T functions. Because arithmetic operation s such as addition , subtraction and multiplication are also T functions triangular T functions , software efficient word based T functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T functions based on arithmetic operations is predictability of their period mathematics period , which is highly attractive to cryptographers. Although triangular T functions are naturally vulnerable to guess and determine attacks, well chosen bitwise transposition mathematics transposition ... bit. Subsequent transposition of the output bits and iteration of the T function also do not affect ... and losing the T function bias of depending only on the less significant bits of the state. References ... more details
Informally mathematical logic , an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variable s. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first order logic involving only algebraic sentences. Saying that a theory is algebraic is a stronger condition than saying it is elementary theory elementary . Informal Interpretation An algebraic theory consists of a set of n ary functional terms with additional rules axioms . E.g. a group theory is an algebraic theory because it has two functional terms, a binary operation a b a nullary operation 1 neutral element , and a unary operation x x sup 1 sup with the rules of associativity, neutrality and inversion. This is opposed to Geometric theory which involves partial functions or binary relationships or existential quantors see e.g. Eucledian geometry where the existence of points or lines is postulated. Category based Model Theoretical Interpretation See also Lawvere theory and Equational logic An Algebraic Theory T is a category theory category whose objects are natural numbers 0, 1, 2,..., and which, for each n, has an n tuple of morphism morphisms proj sub i sub n 1, i 1,..., n This allows interpreting n as a cartesian product of n copies of 1. Example. Let s define an algebraic theory T taking hom n, m m tuples of polynomials of n free variables X sub 1 sub ,..., X sub n sub with integer quotients and with substitution as composition ... ring commutative rings . In an algebraic theory, any morphism n m can be described as m morphisms ... 5 tr5abs.html Functorial Semantics of Algebraic Theories, Proceedings of the National Academy of Science ... bs.de adamek algebraic.theories.pdf Algebraic Theories. A Categorical Introduction To General Algebra ..., North Holland 1977 http ncatlab.org nlab show algebraic theory ncatlab, Algebraic Theory Related Algebraic sentence Algebraic definition Category Mathematical logic ... more details
Orphan date February 2009 Algebraic biology applies the algebraic methods of symbolic computation to the study of biological problems, especially in genomics , proteomics , analysis of molecular structure s and study of gene s. References Michael P Barnett , Symbolic calculation in the life sciences trends and prospects, Algebraic Biology 2005 &ndash Computer Algebra in Biology , edited by H. Anai, K. Horimoto, Universal Academy Press, Tokyo, 2006. on line http www.princeton.edu allengrp ms annobib mb.pdf .pdf format University of Oxford Research interests Centre of mathematical biology. Oxford. updated 12 01 10 cited 15 12 11 available from http www.maths.ox.ac.uk groups mathematical biology research Further reading cite journal date 2005 11 10 title Symbolic Calculation in the Life Sciences &mdash Some Trends and Prospects author Michael P. Barnett journal Algebraic Biology format PDF url http www.princeton.edu allengrp ms annobib mb.pdf publisher Universal Academy Press, Inc. External links http www.risc.uni linz.ac.at about conferences ab2007 Algebraic Biology 2007 Conference on algebraic biology Category Mathematical and theoretical biology Category Algebra Biology stub ar ... more details
In mathematics , if L is a field extension of K , then an element a of L is called an algebraic element over K , or just algebraic over K , if there exists some non zero polynomial g x with coefficient s in K such that g a 0. Elements of L which are not algebraic over K are called transcendental over K . These notions generalize the algebraic number s and the transcendental number s where the field extension is C Q , C being the field of complex number s and Q being the field of rational number s . Examples The square root of two 2 is algebraic over Q , since it is the root of the polynomial g x x sup 2 sup 2 whose coefficients are rational. Pi is transcendental over Q but algebraic over the field of real number s R it is the root of g x x , whose coefficients 1 and are both real, but not of any polynomial with only rational coefficients. The definition of the term transcendental number uses C Q , not C R . Properties The following conditions are equivalent for an element a of L a is algebraic over K the field extension K a K has finite degree, i.e. the dimension of a vector space dimension of K a as a K vector space is finite. Here K a denotes the smallest subfield of L containing K and a K a K a , where K a is the set of all elements of L that can be written in the form g a with a polynomial g whose coefficients lie in K . This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K . The set of all elements of L which are algebraic over K is a field that sits in between L and K . If a is algebraic over K , then there are many non zero polynomials g x with coefficients in K such that g a 0. However there is a single one with smallest degree and with leading coefficient 1. This is the minimal ... algebraic elements over them except their own elements are called algebraically closed field algebraically ... Abstract algebra ca Element algebraic de Algebraisches Element el es Elemento ... more details
Weisstein, Eric W. http mathworld.wolfram.com AlgebraicConnectivity.html Algebraic Connectivity . From ... graph has a traditional connectivity graph theory connectivity of 3, but an algebraic connectivity of only 0.243. The algebraic connectivity of a Graph mathematics graph G is greater than 0 if and only if G is a connected graph . Furthermore, the value of the algebraic connectivity is bounded ... of a connected graph is n and the Distance graph theory diameter is D , the algebraic connectivity ... 0.222 0.722 1, but for many large graphs the algebraic connectivity is much closer to the lower bound than the upper. Unlike the traditional connectivity, the algebraic connectivity ... graph s, the algebraic connectivity decreases with the number of vertices, and increases with the average ... on Complex Systems, 2006. ref The exact definition of the algebraic connectivity depends on the type ... model , the Laplacian matrix arises naturally, and so the algebraic connectivity gives an indication ... of a Connected Age , Vintage, 2003. ref and in fact the algebraic connectivity is closely related to the reciprocal of the average distance. ref name Mohar The algebraic connectivity also relates ... the algebraic connectivity. ref Norman Biggs. Algebraic Graph Theory , 2nd ed, Cambridge University Press, 1993, pp. 28 & 58. ref The Fiedler vector The original theory related to algebraic connectivity was produced by Miroslav Fiedler . ref M. Fiedler. Algebraic connectivity of Graphs , Czechoslovak Mathematical Journal 23 98 , 1973. ref ref M. Fiedler. Laplacian of graphs and algebraic connectivity ... with the algebraic connectivity has been named the Fiedler vector . The Fiedler vector can be used ... . See also Connectivity graph theory Graph property References reflist Category Algebraic graph theory ... more details
In mathematical logic , algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic igflff logic focuses on the identification and algebraic description of model theory models appropriate for the study of various logics in the form of classes of algebras that constitute the algebraic semantics for these deductive system s and connected ... algebraic logic. ref name review Works in the more recent abstract algebraic logic AAL focus ... the Leibniz operator . ref name review jstor 3094793 ref Algebras as models of logics Algebraic logic treats algebraic structure algebraic structures , often lattice order bounded lattices , as models interpretations of certain logic s, making logic a branch of the order theory . In algebraic logic ... or mathematical systems, and the algebraic structure which are its models are shown on the right ... nonclassical logic s are typically modeled by what are called Boolean algebras with operators. Algebraic ... , having the expressive power of set theory Relation algebra , arguably the paradigmatic algebraic ... br Polyadic algebra Predicate functor logic Set theory Combinatory logic Relation algebra Algebraic logic is mainly based on square roots. History Algebraic logic is, perhaps, the oldest approach ... all of Leibniz s known work on algebraic logic was published only in 1903 after Louis Couturat ... s volume into English. Brady 2000 discusses the rich historical connections between algebraic ... logicians in the algebraic tradition. Alfred Tarski , the founder of set theory set theoretic model ... as the starting point of abstract algebraic logic. Modern mathematical logic began in 1847, with two ... of some writings by Leopold Loewenheim and Thoralf Skolem , algebraic logic went into eclipse ... s 1940 re exposition of relation algebra. Leibniz had no influence on the rise of algebraic logic ... Dunn author2 Gary M. Hardegree title Algebraic methods in philosophical logic year 2001 publisher ... more details
split algebraic semantics computer science algebraic semantics mathematical logic date February 2012 discuss talk Algebraic semantics Unreferenced date December 2009 Expert subject mathematics date February 2012 In programming language theory , the algebraic semantics of a programming language is a form of axiomatic semantics based on algebra ic laws for describing and reasoning about program semantics in a formal methods formal manner. In mathematical logic , algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic . For example, the modal logic S4 is characterized by the class of interior algebra topological boolean algebras &mdash that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of Boolean algebra structure boolean algebras characterizes classical propositional logic, and the class of Heyting algebra s propositional intuitionistic logic . See also OBJ programming language Semantics of programming languages Axiomatic semantics Operational semantics Denotational semantics Further reading cite book author1 Josep Maria Font author2 Ram n Jansana title A general algebraic semantics for sentential logics year 1996 publisher Springer Verlag isbn 9783540616993 2nd published by Association for Symbolic Logic ASL in 2009 http projecteuclid.org euclid.lnl 1235416965 open access at Project Euclid cite book author1 W.J. Blok author2 Don Pigozzi title Algebraizable logics publisher American Mathematical Society year 1989 isbn 0821824597 cite book author Janusz Czelakowski title Protoalgebraic logics year 2001 publisher Springer isbn 9780792369400 cite book author1 J. Michael Dunn author2 Gary M. Hardegree title Algebraic methods in philosophical logic year ... chapter Algebraic Semantic cite book author1 Joseph Goguen author2 Grant Malcolm title Algebraic semantics of imperative programs year 1996 publisher MIT Press isbn 9780262071727 DEFAULTSORT Algebraic ... more details
Algebraic statistics is the use of algebra to advance statistics . Algebra has been useful for design ..., algebraic statistics has been associated with the design of experiments and multivariate analysis especially time series . In recent years, the term algebraic statistics has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics. The tradition of algebraic statistics In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics ... for experimental designs. Algebraic analysis and abstract statistical inference Haar measure Invariant ... statistics. Encompassing previous results on probability theory on algebraic structures, Ulf Grenander ... theory by Lucien Le Cam . Recent work using commutative algebra and algebraic geometry In recent years, the term algebraic statistics has been used more restrictively, to label the use of algebraic ... state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic variety algebraic ... an algebraic variety or surface in R sup 3 sup , and this variety, when intersected with the simplex given by math sum i 0 2 p i 1 quad mbox and quad 0 leq p i leq 1, math yields a piece of an algebraic ... www.math.harvard.edu seths assc.html Algebraic Statistics Short Course , lecture notes by Seth Sullivant L. Pachter and Bernd Sturmfels B. Sturmfels . Algebraic Statistics for Computational Biology. Cambridge University Press 2005. G. Pistone, E. Riccomango, H. P. Wynn. Algebraic Statistics. CRC Press, 2001. Drton, Mathias, Sturmfels, Bernd, Sullivant, Seth . Lectures on Algebraic Statistics , Springer 2009. Paolo Gibilisco, Eva Riccomagno, Maria Piera Rogantin, Henry Wynn Henry P. Wynn . Algebraic ... of Algebraic Statistics Journal of Algebraic Statistics DEFAULTSORT Algebraic Statistics Category ... more details
In algebraic geometry , algebraic stacks are generalizations of algebraic variety algebraic varieties , scheme mathematics schemes , and algebraic space s. They were originally introduced by harvs txt author1 link Pierre Deligne author1 Deligne author2 link David Mumford author2 Mumford year 1969 to define the fine moduli space of genus g curves their definition is currently referred to as Deligne Mumford stacks Deligne Mumford stacks . When viewed in this light, algebraic stacks are an algebraic analogue of orbifold s. They were generalized by harvs txt author link Michael Artin last Artin year 1974 to what is now called an Artin stacks Artin stack . The term algebraic stack is somewhat ambiguous it originally meant Deligne Mumford stack, but now usually means Artin stack. Motivation When ... category of schemes considered together with the tale Grothendieck topology . Technically an algebraic stack is a stack descent theory stack that can be suitably covered by algebraic space s with respect ... groups to be algebraic groups. Properties More generally a stack refers to any category mathematics ... as relatively fine moduli space s . Examples The moduli space of algebraic curves Deligne Mumford ... as a fine moduli space as an algebraic variety because in particular there are elliptic curves admitting nontrivial automorphisms though there is an algebraic variety forming a coarse moduli space . For elliptic ... Artin title Versal deformations and algebraic stacks doi 10.1007 BF01390174 id MR 0399094 year 1974 ... 75 109 Citation last1 G mez first1 Tom s L. title Algebraic stacks arxiv math 9911199 doi 10.1007 ... of algebraic stacks induce morphisms of lisse tale topoi. Some of these errors were fixed ... algebraic geometry stacks git Stacks Project http www.msri.org publications ln msri 2002 ... small start 56kps video link in Real Player to watch the lecture. DEFAULTSORT Algebraic Stack Category Algebraic geometry Category Category theory de Stack Kategorientheorie ... more details
In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi projective schemes over a field. It was introduced by harvs txt first1 Marc last1 Levine author1 link Marc Levine first2 Fabien last2 Morel author2 link Fabien Morel year1 2001 year2 2001b . An oriented cohomology theory on the category of smooth quasi projective schemes Sm over a field k consists of a contravariant functor A from Sm to commutative graded rings, together with push forward maps f sub sub whenever f Y X has relative dimension d for some d . These maps have to satisfy various conditions similar to those satisfied by complex cobordism. In particular they are oriented , which means roughly that they behave well on vector bundles this is closely related to the condition that a generalized cohomology theory has a complex orientation . Over a field of characteristic 0, algebraic cobordism is the universal oriented cohomology theory for smooth varieties. In other words there is a unique morphism of oriented cohomology theories from algebraic cobordism to any other oriented cohomology theory. harvtxt Levine 2002 and harvtxt Levine Morel 2007 give surveys of algebraic cobordism. References Citation last1 Levine first1 M editor1 last Li editor1 first Tatsien title Proceedings of the International Congress of Mathematicians, Vol. II Beijing, 2002 url http mathunion.org ICM ICM2002.2 publisher Higher Ed. Press location Beijing isbn 978 7 04 008690 4 mr 1957020 year 2002 chapter Algebraic cobordism pages 57 66 Citation last1 Levine first1 Marc last2 Morel first2 Fabien title Cobordisme alg brique. I doi 10.1016 S0764 4442 01 01832 8 mr 1843195 year 2001 journal Comptes Rendus de l Acad mie des Sciences. S rie I. Math matique issn 0764 4442 volume 332 issue 8 pages 723 728 Citation ... first2 Fabien title Algebraic cobordism publisher Springer Verlag location Berlin, New York series ... 8 mr 2286826 year 2007 Category Algebraic geometry ... more details
Algebraic holography , also sometimes called Rehren duality , is an attempt to understand the holographic principle of quantum gravity within the framework of algebraic quantum field theory , due to Karl Henning Rehren . It is sometimes described as an alternative formulation of the AdS CFT correspondence of string theory , but some string theorists reject this statement http golem.ph.utexas.edu distler blog archives 000987.html . The theories discussed in algebraic holography do not satisfy the usual holographic principle because their entropy follows a higher dimensional power law. Citation needed date June 2008 Rehren s duality The conformal boundary of an anti de Sitter space or its universal covering space is the conformal Minkowski space or its universal covering space with one fewer dimension. Let s work with the universal covering spaces. In algebraic quantum field theory AQFT , a QFT in the conformal space is given by a conformally covariant net of C algebras over the conformal space and the QFT in AdS is given a covariant net of C algebras over AdS. Any two distinct null geodesic hypersurfaces of codimension 1 which intersect at more than just a point in AdS divides AdS into four distinct regions, two of which are spacelike. Any of the two spacelike regions is called a wedge. It s a geometrical fact that the conformal boundary of a wedge is a double cone in the conformal boundary and that any double cone in the conformal boundary is associated with a unique wedge. In other words, we have a one to one correspondence between double cones in CFT and wedges in AdS. It s easy to check that any CFT defined in terms of algebras over the double cones which satisfy the Haag Kastler axiom s also gives rise to a net over AdS which satisfies these axioms if we assume that the algebra ... cone and vice versa. This correspondence between AQFTs on both sides is called algebraic holography ... Karl Henning Rehren , Algebraic Holography For a classical counterpart to Rehren duality see ... more details
Algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebra s that generalizes the Weyl character formula character of a finite dimensional representation and is analogous to the Harish Chandra character of the representations of semisimple Lie group s. Definition Let math mathfrak g math be a semisimple Lie algebra with a fixed Cartan subalgebra math mathfrak h , math and let the abelian group math A mathbb Z mathfrak h math consist of the possibly infinite formal integral linear combinations of math e mu math , where math mu in mathfrak h math , the complex vector space of weights. Suppose that math V math is a locally finite weight module . Then the algebraic character of math V math is an element of math A math defined by the formula math ch V sum mu dim V mu e mu , math where the sum is taken over all weight space s of the module math V. math Example The algebraic character of the Verma module math M lambda math with the highest weight math lambda math is given by the formula math ch M lambda frac e lambda prod alpha 0 1 e alpha , math with the product taken over the set of positive roots. Properties Algebraic characters are defined for locally finite weight module s and are additive , i.e. the character of a direct sum of modules is the sum of their characters. On the other hand, although one can define multiplication of the formal exponents by the formula math e mu cdot e nu e mu nu math and extend it to their finite linear combinations by linearity, this does not make math A math into a ring, because of the possibility of formal infinite sums. Thus the product of algebraic characters is well defined only in restricted situations for example, for the case of a highest weight module , or a finite dimensional module. In good situations, the algebraic character is multiplicative , i.e., the character of the tensor product of two weight modules is the product of their characters. Generalization Characters also ... more details
In functional analysis , a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the Interior topology interior . It is the subset of points contained in a given set that it is absorbing set absorbing with respect to, i.e. the radial set radial points of the set. ref name coherent cite journal title Coherent Risk Measures, Valuation Bounds, and math mu, rho math Portfolio Optimization first1 Stefan last1 Jaschke first2 Uwe last2 K uchler year 2000 ref Formally, if math X math is a linear space then the algebraic interior of math A subseteq X math is math operatorname core A left x 0 in A forall x in X, exists t x 0, forall t in 0,t x , x 0 tx in A right . math ref cite book author Nikola Kapitonovich Nikol ski title Functional analysis I linear functional analysis year 1992 publisher Springer isbn 9783540505846 ref Note that in general math text core A neq text core text core A math , but if math A math is a convex set then math text core A text core text core A math . In fact, if math A math is a convex set then if math x 0 in text core A , y in A, 0 lambda leq 1 math then math lambda x 0 1 lambda y in text core A math . Example If math A subset mathbb R 2 math such that math A x in mathbb R 2 x 2 geq x 1 2 text or x 2 leq 0 math then math 0 in text core A math , but math 0 not in text int A math and math 0 not in text core text core A math . Properties Let math A,B subset X math then math A math is Absorbing set absorbing if and only if math 0 in operatorname core A math . ref name coherent math A operatorname core B subset operatorname core A B math ref name Zalinescu cite book last Z linescu first C. title Convex analysis in general vector spaces publisher World Scientific Publishing Co., Inc River Edge, NJ, 2002 pages 2 3 isbn 981 238 067 1 mr 1921556 ref math A operatorname core B operatorname core A B math if math B operatorname core B math ref name Zalinescu See also Interior ... more details
Function field may refer to Function field of an algebraic variety Function field scheme theory Algebraicfunction field Function field sieve Function field analogy mathdab Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. de Funktionenk rper ... more details
Unreferenced date February 2007 orphan date November 2009 In mathematics , an algebraic vector bundle is a vector bundle for which all the transition map s are algebraicfunction s. All math SU 2 math instanton s over the sphere math S 4 math are algebraic vector bundles. DEFAULTSORT Algebraic Vector Bundle Category Vector bundles topology stub algebra stub ... more details
A transcendental function is a function mathematics function that does not satisfy a polynomial equation whose coefficient s are themselves polynomials, in contrast to an algebraicfunction , which does ... , the natural logarithm , is a transcendental function. Algebraic and transcendental functions details ... and the square root function. The operation of taking the indefinite integral of an algebraic ... meaningless results. Exceptional set If z is an algebraicfunction and is an algebraic number then ... function, z e sup z sup , then the only algebraic number where is also algebraic is 0, where 1. For a given transcendental function this set of algebraic numbers giving algebraic results is called the exceptional set of the function, ref D. Marques ..., and if is algebraic and irrational then sup sup is transcendental. Thus the function 2 sup ... function is not easy, it is known that given any subset of the algebraic numbers, say A , there is a transcendental ... if a function is transcendental just by looking at its values at algebraic numbers. In fact, Alex ... preprints, number 66, 1998. ref See also Algebraicfunction Analytic function Complex function Generalized ... . ref In other words, a transcendental function is a function that wiktionary transcend transcends algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations ... the exponential function , the logarithm , and the trigonometric function s. Formally, an analytic function z of the real or complex variables z sub 1 sub , , z sub n sub is transcendental if the n 1 functions z sub 1 sub , , z sub n sub , z are algebraic independence algebraically independent . ref M. Waldschmidt, Diophantine approximation on linear algebraic groups , Springer 2000 . ref That is, is Algebraic element transcendental over the field C z sub 1 sub , , z ... function base of the natural logarithm , then we get that math e x math is a transcendental function ... more details
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