In mathematics , multilinearalgebra extends the methods of linear algebra . Just as linear algebra is built on the concept of a vector space vector and develops the theory of vector space s, multilinearalgebra builds on the concepts of p vector s and multivector s with Grassmann algebra . Origin In a vector ... of the determinant is the most immediate application. Multilinearalgebra also has applications ... , and re published in 1862, his work was slow to find acceptance as ordinary linear algebra provided sufficient challenges to comprehension. The topic of multilinearalgebra is applied in some studies ... in multivariate calculus, and their manipulation is done with exterior algebra . After some preliminary work by Elwin Bruno Christoffel , a major advance in multilinearalgebra came in the work of Gregorio ... form of multilinearalgebra that Marcel Grossman and Michele Besso introduced to Albert Einstein ... of Mercury, established multilinearalgebra and tensor s as physically important mathematics ... more abstractly. The Nicolas Bourbaki Bourbaki group s treatise MultilinearAlgebra was especially influential &mdash in fact the term multilinearalgebra was probably coined there. Citation needed date April 2008 One reason at the time was a new area of application, homological algebra . The development ... is that by re expressing problems in terms of multilinearalgebra, there is a clear and well defined ... once more the content of old and new viewpoints. Topics in multilinearalgebra The subject matter of multilinearalgebra has evolved less than the presentation down the years. Here are further pages ... theory . From the point of view of applications Some of the ways in which multilinearalgebra concepts ... 1432 1807 doi 10.1007 BF01454201 Mathematics footer Category Multilinearalgebra ca lgebra multilineal .... This practical reference led to the use of the word tensor to describe the elements of the multilinear space. The extra structure in a multilinear space has led it to play an important role in various ... more details
In multilinearalgebra , a multilinear form is a Map mathematics map of the type math f V n to bold K , math where V is a vector space over the field mathematics field K , that is separately linear in each its n variables. ref MathWorld title Multilinear Form urlname MultilinearForm ref For n 2, i.e. only two variables, one calls a bilinear form . An important type of multilinear forms are alternating multilinear forms which have the additional property of changing their sign under exchange of two arguments. When K has characteristic algebra characteristic other than 2, this is equivalent to saying that math f dots,x, dots,x, dots 0 , math i.e. the form vanishes if supplied the same argument twice. The exceptional case of characteristic 2 requires more care. Special cases of these are determinant forms and differential form s. See also Homogeneous polynomial References Reflist DEFAULTSORT Multilinear Form Category Multilinearalgebra cs Multiline rn forma de Multilinearform es Forma multilineal fr Forme multilin aire pt Fun o n linear zh ... more details
In algebra, a multilinear polynomial is a polynomial that is linear in each of its variables. In other words, no variable occurs to a power of 2 or higher or alternatively, each monomial is a constant times a product of distinct variables. They are important in the study of polynomial identity testing , because if a multilinear polynomial is zero on a set of vectors that Linear span span the space, it will be zero everywhere. The Degree of a polynomial degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial. ref A. Giambruno, Mikhail Zaicev. Polynomial Identities and Asymptotic Methods. AMS Bookstore, 2005 ISBN 978 0 82183829 7. Section 1.3. ref References references Category Polynomials ... more details
In linear algebra , a multilinear map is a function mathematics function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function math f colon V ... function of math v i math . ref Lang. Algebra. Springer 3rd edition January 8, 2002 ref A multilinear map of two variables is a bilinear map . More generally, a multilinear map of k variables is called a k linear map . If the codomain of a multilinear map is the field of scalars, it is called a multilinear form . Multilinear maps and multilinear forms are fundamental objects of study in multilinearalgebra . If all variables belong to the same space, one can consider symmetric function symmetric ... Multilinear projection References references Category Multilinearalgebra de Multilineare Abbildung ... Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear ... is an antisymmetric multilinear function of the columns or rows of a square matrix . If math F colon ... m to mathbb R n math . The Multilinear subspace learning Tensor to vector projection .28TVP.29 tensor to vector projection in multilinear subspace learning is a multilinear map as well. Coordinate representation Let math f colon V 1 times cdots times V n to W text , math be a multilinear map between ... has dimension math d math . If we choose a basis linear algebra basis math textbf e i1 , ldots, textbf ... math completely determine the multilinear function math f math . In particular, if math textbf v i ... Relation to tensor products There is a natural one to one correspondence between multilinear maps ... by the formula math F v 1 otimes cdots otimes v n f v 1, ldots,v n . math Multilinear functions on n × n matrices One can consider multilinear functions, on an n × n matrix over a commutative ... A be such a matrix and math a i math , 1 i n be the rows of A . Then the multilinear function D can ... × 2 matrices math D A A 1,1 A 2,2 A 1,2 A 2,1 , math Properties A multilinear map has a value ... more details
and applications , SIAM Review 51 3 2009 455 500. ref . They both employ multilinearalgebra ... Machine learning Category Multilinearalgebra Category Tensors Compu AI stub ...Multilinear subspace learning MSL aims to learn a specific small part of a large space of multidimensional ... haiping Publication SurveyMSL PR2011.pdf title A Survey of Multilinear Subspace Learning for Tensor ... tensor space or vector space is named as Multilinear subspace learning Multilinear projection multilinear projection ref name MSLsurvey ref name mlprjICCV cite conference first M.A.O. last Vasilescu first2 D. last2 Terzopoulos title Multilinear Projection for Appearance Based Recognition in the Tensor ... 2007 doi 10.1109 ICCV.2007.4409067 ref . Multilinear subspace learning Algorithms MSL methods are higher ..., http dx.doi.org 10.1109 TNN.2007.901277 MPCA Multilinear principal component analysis of tensor ... reduction . Multilinear projection A multilinear subspace is defined through a multilinear .... Vandewalle, http portal.acm.org citation.cfm?id 354398 A multilinear singular value decomposition , SIAM ... from a tensor to a scalar is an elementary multilinear projection EMP . In EMP, a tensor is projected ... and have convergence problem. The solution obtained is local optimum . Algorithms Multilinear extension of principal component analysis PCA Multilinear principal component analysis Multilinear Principal Component Analysis MPCA ref name MPCA Uncorrelated Multilinear Principal Component Analysis UMPCA ... Uncorrelated multilinear principal component analysis for unsupervised multilinear subspace learning , IEEE Trans. Neural Netw., vol. 20, no. 11, pp. 1820 1836, Nov. 2009. ref Multilinear extension of linear ... Multilinear Discriminant Analysis UMLDA ref H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, http dx.doi.org 10.1109 TNN.2008.2004625 Uncorrelated multilinear discriminant analysis with regularization ... A survey of multilinear subspace learning for tensor data http www.dsp.utoronto.ca haiping ... more details
The term algebra is defined below after first defining a ring . ring In mathematics , a ring is an associative ring with a map A A which is an antiautomorphism and an Semigroup with involution involution ... over any ring. algebra A algebra A is a ring that is an associative algebra over a commutative ring ... , math x,y in A math . A homomorphism math f colon A to B math is algebra homomorphism that is compatible ... numbers. A operation on a algebra is an operation on an algebra over a ring that behaves similarly to taking ... of a algebra is the field of complex numbers C where is just complex conjugation . More generally, the conjugation involution in any Cayley Dickson algebra such as the complex numbers, quaternion s and octonion ... ring matrix algebra of n × n matrix mathematics matrices over C with given by the conjugate ... is also a star algebra. In Hecke algebra , an involution is important to the Kazhdan Lusztig polynomial ... curve becomes a algebra over the integers, where the involution is given by taking the dual ... notes on abelian varieties . Hopf algebra Examples Involutive Hopf algebras are important examples ... being The group Hopf algebra a group ring , with involution given by math g mapsto g 1 . math ... form a Jordan algebra The skew Hermitian elements form a Lie algebra If 2 is invertible, then math ... and anti symmetrizing , so the algebra decomposes as a direct sum of symmetric and anti symmetric Hermitian and skew Hermitian elements. This decomposition is as a vector space, not as an algebra, because the idempotents are operators, not elements of the algebra. Skew structures Given a ring, there is also the map math x mapsto x math . This is not a ring structure unless the characteristic algebra ... elements, and the imaginary numbers are the skew Hermitian. See also B algebra C algebra von Neumann algebra Baer ring operator algebra This article is no longer a stub, but there is more to be said about algebras which are not B or C algebras. DEFAULTSORT Algebra Category Algebras fr ... more details
Orphan date January 2012 A algebra or, more explicitly, a closed algebra is the name occasionally used in physics ref John A. Holbrook, David W. Kribs, and Raymond Laflamme. Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction. Quantum Information Processing . Volume 2, Number 5, p. 381&ndash 419. Oct 2003. ref for a finite dimensional C algebra . The dagger, , is used in the name because physicists typically use the symbol to denote a hermitian adjoint , and are often not worried about the subtleties associated with an infinite number of dimensions. Mathematicians usually use the asterisk, , to denote the hermitian adjoint. algebras feature prominently in quantum mechanics , and especially quantum information science . References references Category C algebras physics stub algebra stub ... more details
about the branch of mathematics pp move indef sprotect small yes Algebra from Arabic language Arabic al jebr meaning reunion of broken parts ref cite web title algebra work Online Etymology Dictionary ... , topology , combinatorics , and number theory , algebra is one of the main branches of pure mathematics . Elementary algebra , often part of the curriculum in secondary education , introduces ... be done for a variety of reasons, including equation solving . Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations ... algebra . History Main History of algebra Timeline of algebra File Image Al Kit b al mu ta ar f is b ... Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually ... s Algebra made use of lettered diagrams but all coefficients in the equations used in the Algebra are specific ... called the father of algebra , was an Alexandria n Greek mathematics Greek mathematician and the author ... 8 ref While the word algebra comes from the Arabic language lang ar transl ar al jabr restoration ... algebra as a mathematical discipline that is independent of geometry and arithmetic . ref citation title Al Khwarizmi The Beginnings of Algebra author Roshdi Rashed publisher Saqi Books date November 2009 isbn 0 86356 430 5 ref The roots of algebra can be traced to the ancient Babylonian mathematics ... Diophantus, Father of Algebra ref as well as Indian mathematics Indian mathematicians such as Brahmagupta ... s Brahmasphutasiddhanta are on a higher level. ref http www.algebra.com algebra about history History of Algebra ref For example, the first complete arithmetic solution including zero and negative ... of algebra but in more recent times there is much debate over whether al Khwarizmi, who founded the discipline ... Edition Wiley, 1991 , pages 178, 181 ref Those who support Diophantus point to the fact that the algebra found in Al Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica ... more details
Multilinear principal component analysis MPCA ref name MPCA H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, http www.dsp.utoronto.ca haiping Publication MPCA TNN08 rev2010.pdf MPCA Multilinear principal component analysis of tensor objects , IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18 39, Jan. 2008. ref is a mathematical procedure that uses multiple orthogonal transformations to convert a set of multidimensional objects into another set of multidimensional objects of lower dimensions. There is one orthogonal transformation for each dimension mode . This transformation aims to capture as high a variance as possible, accounting for as much of the variability in the data as possible, subject to the constraint of mode wise orthogonality. MPCA is a multilinear extension of principal component analysis PCA and it is a basic algorithm in multilinear subspace learning . Its origin can be traced back to the Tucker decomposition ref Cite journal author Ledyard R Tucker title Some mathematical notes on three mode factor analysis journal Psychometrika volume 31 issue 3 month September year ... citation.cfm?id 354398 A multilinear singular value decomposition , SIAM Journal of Matrix Analysis ... MPCA performs feature extraction by determining a Multilinear subspace learning Multilinear projection multilinear projection that captures most of the original tensorial input variations. As in PCA ... www.dsp.utoronto.ca haiping Publication UMPCA TNN09.pdf Uncorrelated multilinear principal component analysis for unsupervised multilinear subspace learning , IEEE Trans. Neural Netw., vol. 20, no. 11 ... uncorrelated multilinear features. Feature selection MPCA produces tensorial features. For conventional ... url http www.dsp.utoronto.ca haiping Publication SurveyMSL PR2011.pdf title A Survey of Multilinear ... . ref Non negative MPCA NMPCA ref Y. Panagakis, C. Kotropoulos, G. R. Arce, Non negative multilinear ... RMPCA ref K. Inoue, K. Hara, K. Urahama, Robust multilinear principal component analysis , Proc. IEEE ... more details
1989 isbn 3 540 64243 9 Category Algebras Category Multilinearalgebra Category Polynomials Category ...In mathematics , the symmetric algebra S V also denoted Sym V on a vector space V over a field mathematics field K is the Free object free commutative unital algebra unital associative algebra over K containing ... tensor s in V . A Frobenius algebra whose bilinear form is symmetric bilinear form symmetric is also called a symmetric algebra , but is not discussed here. Construction It turns out that S V is in effect ... this way has some advantage. It is possible to use the tensor algebra T V to describe the symmetric algebra S V . In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of T V by the ideal ring theory ideal ... algebra , into summands S sup k sup V which consist of the linear span of the monomial s in vectors ... square and denoted Sym sup 2 sup V . It has a universal property with respect to symmetric multilinear operators defined on V sup k sup . Distinction with symmetric tensors The symmetric algebra and symmetric tensor symmetric tensors are easily confused the symmetric algebra is a quotient of the tensor algebra, while the symmetric tensors are a subspace of the tensor algebra. The symmetric algebra must be a quotient to satisfy its universal property since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra . Conversely, symmetric tensors are defined as invariants given the natural action of the symmetric group on the tensor algebra, the symmetric tensors ... algebra can be identified. In any characteristic, there is a symmetrization map from the symmetric algebra to the symmetric tensors, given by math v 1 cdots v k mapsto sum sigma in S n v ... in the tensor algebra and the quotient to the symmetric algebra is multiplication by math k math ... more details
In abstract algebra , an alternative algebra is an algebra over a field algebra in which multiplication ... math math yx x y xx math for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly Algebra over a field Non associative algebras nonassociative algebras ... is alternating form alternating . The associator is a multilinear map trilinear map given by math x,y,z xy z x yz math By definition a multilinear map is alternating if it vanishes whenever two of it arguments are equal. The left and right alternative identities for an algebra are equivalent to math ... called flexible identity math xy x x yx . math The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of left alternative identity math x xy xx y math right .... The converse holds so long as the characteristic algebra characteristic of the base field is not 2. Properties Artin s theorem states that in an alternative algebra the subalgebra generated by any two elements is associative . Conversely, any algebra for which this is true is clearly alternative. It follows ... in an alternative algebra. A generalization of Artin s theorem states that whenever three elements math x,y,z math in an alternative algebra associate i.e. math x,y,z 0 math the subalgebra generated ... math a x ay axa y math math xa y a x aya math math ax ya a xy a math hold in any alternative algebra. In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover ... and forms a Moufang loop . This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra. Applications The projective plane over any alternative ... algebra s was given by Guy Roos in 2008 He shows page 162 the relation for an algebra A with unit ... algebra. See also Zorn ring Maltsev algebra References Guy Roos 2008 Exceptional symmetric domains ... more details
Category Algebras Category Multilinearalgebra ar de Tensoralgebra es lgebra tensorial ...In mathematics , the tensor algebra of a vector space V , denoted T V or T sup sup V , is the algebra over a field algebra of tensor s on V of any rank with multiplication being the tensor product . It is the free algebra on V , in the sense of being left adjoint to the forgetful functor from algebras to vector spaces it is the most general algebra containing V , in the sense of the corresponding universal property see below . The tensor algebra also has two Coalgebra structures coalgebra structures ... , and can be extended with an antipode to a Hopf algebra structure. Note In this article, all algebras are assumed to be unital algebra unital and associative algebra associative . Construction ... to all of T V . This multiplication rule implies that the tensor algebra T V is naturally a graded algebra with T sup k sup V serving as the grade k subspace. This grading can be extended to a Z ... in straightforward manner to the tensor algebra of any module mathematics module M over a commutative ... tensor products cannot be formed. Adjunction and universal property The tensor algebra T V is also called the free algebra on the vector space V , and is functorial. As with other free object ..., it s the functor which sends each K algebra to its underlying vector space. Explicitly, the tensor algebra satisfies the following universal property , which formally expresses the statement that it is the most general algebra containing V Any linear transformation f V &rarr A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T V to A as indicated by the following ... algebra Here i is the canonical inclusion of V into T V the unit of the adjunction . One can, in fact, define the tensor algebra T V as the unique algebra satisfying this property specifically, it is unique ... exists. The above universal property shows that the construction of the tensor algebra is functorial ... more details
Enveloping algebra in mathematics may refer to The universal enveloping algebra of a Lie algebra The enveloping algebra of a general non associative algebra disambig ... more details
Affine algebra may refer to affine Lie algebra , a type of Kac Moody algebras the Lie algebra of the affine group finitely generated algebra disambig ... more details
a basis, and not all subsets of a module that span the space contains a basis. In multilinearalgebra ... origin are vector subspaces in R sup 3 sup . Linear algebra is the branch of mathematics charged ... Algebra url http mathworld.wolfram.com LinearAlgebra.html work From MathWorld A Wolfram Web Resource. publisher Wolfram accessdate 16 April 2012 ref Linear algebra is central to both pure and applied mathematics. For instance abstract algebra arises by relaxing the axioms leading to a number of generalizations. However, linear algebra focuses more on finite dimensional spaces whereas functional ... sometimes be approximated by linear ones. History The study of linear algebra and matrices first ... Marie title A Brief History of Linear Algebra and Matrix Theory url http darkwing.uoregon.edu vitulli ... 01 24 2012 ref The study of matrix algebra first emerged in England in the mid 1800s. Sylvester ... as abstract algebra . The use of matrices in quantum mechanics , special relativity , and statistics did much to spread the subject of linear algebra beyond pure mathematics. The development of computers ..., and linear algebra became an essential tool for modelling and simulations. ref Vitulli, Marie ... elimination . Scope of study Vector spaces The main structures of linear algebra are vector space s. A vector ..., function mathematics functions , polynomial ring polynomials , vectors, or matrices. Linear algebra ... of other algebraic structures, linear algebra studies mappings between vector spaces ... structure, two isomorphic vector spaces are essentially the same from the linear algebra point of view. One essential question in linear algebra is whether a mapping is an isomorphism or not, and this question ..., linear algebra is interested in finding its Range mathematics range or image and the set of elements ..., span, and basis Again in analogue with theories of other algebraic objects, linear algebra is interested ... V , which we call a Basis linear algebra basis of V . Any set of vectors that spans V contains a basis ... more details
Matrix algebra may refer to Matrix theory , is the branch of mathematics that studies matrix mathematics matrices Matrix ring , thought of as an algebra over a field or a commutative ring disambig pl Algebra macierzy ... more details
Wiktionarypar algebra The word Algebra describes one of the main branches of mathematics. It can also ... al Khw rizm . As a branch of mathematics The term algebra may also refer to a more specialized branch of mathematics within the general field of Algebra Elementary algebra , i.e. high school algebra. Abstract algebra Linear algebra Relational algebra Universal algebra The term is also traditionally used for the field of Computer algebra , dealing with software systems for symbolic mathematical computation, which often offer capabilities beyond what is normally understood to be algebra . As a mathematical structure Several different classes of algebraic structures are known as Algebra ... include In ring theory and linear algebraAlgebra ring theory Algebra over a commutative ring a module equipped with a bilinear product Algebra over a field a vector space equipped with a bilinear vector product Associative algebra a module mathematics module equipped with an associative bilinear vector product Superalgebra a math mathbb Z 2 math graded algebra Lie algebra s, Poisson algebra s, and Jordan algebra s are important examples of potentially nonassociative algebras. In functional analysis Banach algebra an associative algebra A over the real number real or complex number complex numbers which at the same time is also a Banach space . Operator algebra continuous function topology .... algebra An algebra with a notion of adjoint of an operator adjoints . C algebra a Banach algebra equipped with a unary Involution mathematics involution operation. Von Neumann algebra or W algebra ... algebra structure Heyting algebra In measure theory Algebra over a set a collection of sets closed under finite unions and complementation Sigma algebra a collection of sets closed under countable unions and complementation The term algebra can also describe more general structures In category theory and computer science F algebra math F math algebra F coalgebra math F math coalgebra Other Algebra ... more details
In mathematics , an algebra bundle is a fiber bundle whose fiber s are algebra over a field algebra s and local trivialization s respect the algebra structure. It follows that the transition function s are algebra isomorphism s. Since algebras are also vector space s, every algebra bundle is a vector bundle . Examples include the tensor bundle , exterior bundle , and symmetric bundle associated to a given vector bundle , as well as the Clifford bundle associated to any Riemannian vector bundle. See also Lie algebra bundle References 1. W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and Cohomology, Vo. 2, Academic Press, New Yark, 1973 2. C. Chidambara and B.S. Kiranagi, On Cohomology of Associative algebra bundles, J. Ramanujan Math. Soc., Vol. 9 1 , 1994. pp. 1 12 3. B.S. Kiranagi and R. Rajendra, Revisiting Hochschild Cohomology for Algebra Bundles, Journal of Algebra and Its Applications Vol. 7, No. 6 2008 685 715. DEFAULTSORT Algebra Bundle Category Vector bundles topology stub algebra stub ... more details
A braid algebra can be A Gerstenhaber algebra also called an antibracket algebra . An algebra related to the braid group . disambig 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. ... more details
Journal of Algebra ISSN 0021 8693 is a leading international mathematical research journal in abstract algebraalgebra . An imprint of Academic Press , it is presently published by Elsevier . Journal of Algebra was founded by Graham Higman , who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor in chief. In 2004, Journal of Algebra announced vol. 276, no. 1 and 2 the creation of a new section on Computational Algebra, with a separate editorial board. The first issue completely devoted to Computational Algebra was vol. 292, no. 1 October 2005 . The current Editor in Chief of the Journal of Algebra is Michel Brou , Universit Paris Diderot , whereby Gerhard Hiss, Rheinische Westf lische Technische Hochschule Aachen RWTS is Editor of the Computational Algebra section. External links http www.sciencedirect.com science journal 00218693 Journal of Algebra at ScienceDirect sci journal stub Category Mathematics journals Category Publications established in 1964 nl Journal of Algebra ... more details
In mathematics , the Griess algebra is a commutative Algebra over a field Non associative algebras non associative algebra on a real number real vector space of dimension 196884 that has the Monster group M as its automorphism group . It is named after mathematician R. L. Griess , who constructed it in 1980 and subsequently used it in 1982 to construct M . The Monster fixes vectorwise a 1 space in this algebra and acts absolutely irreducibly on the 196883 dimensional orthogonal complement of this 1 space. The Monster preserves the standard inner product on the 196884 space. Griess s construction was later simplified by Jacques Tits and John H. Conway . The Griess algebra is the same as the degree 2 piece of the monster vertex algebra , and the Griess product is one of the vertex algebra products. References R. L. Griess, Jr, The Friendly Giant , Inventiones Mathematicae 69 1982 , 1 102 algebra stub Category Nonassociative algebras ... more details
In mathematics In abstract algebra and mathematical logic a derivative algebra abstract algebra derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topological space topology and which provides algebraic semantics for the modal logic wK3 . In differential geometry a derivative algebra is a vector space with a product operation that has similar behaviour to the standard cross product of 3 vector geometric vector s. Citation needed date July 2009 disambig ... more details
Difference algebra is analogous to differential algebra but concerned with difference equation s rather than differential equation s. References Alexander Levin 2008 , http books.google.co.uk books?id 15pgjT5PeY0C Difference algebra , Springer, ISBN 9781402069468 Richard M. Cohn 1979 , http books.google.co.uk books?id Fs8oAAAACAAJ& Difference algebra , R.E. Krieger Pub. Co., ISBN 9780882756516 algebra stub Category Algebras ... more details
Unreferenced date December 2009 In mathematics , a supercommutative algebra is a superalgebra i.e. a Z sub 2 sub graded algebra such that for any two homogeneous element s x , y we have math yx 1 x y xy. , math Equivalently, it is a superalgebra where the supercommutator math x,y xy 1 x y yx , math always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as skew commutative associative algebras to emphasize the anti commutation, or, to emphasize the grading, graded commutative or, if the supercommutativity is understood, simply commutative . Any commutative algebra is a supercommutative algebra if given the trivial gradation i.e. all elements are even . Grassmann algebra s also known as exterior algebra s are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra. The even subalgebra of a supercommutative algebra is always a commutative algebra . That is, even elements always commute. Odd elements, on the other hand, always anticommute. That is, math xy yx 0 , math for odd x and y . In particular, the square of any odd element x vanishes whenever 2 is invertible math x 2 0. , math Thus a commutative superalgebra with 2 invertible and nonzero degree one component always contains nilpotent elements. See also Commutative algebra Lie superalgebra DEFAULTSORT Supercommutative Algebra Category Algebras Category Super linear algebra it Algebra supercommutativa ... more details
Noref date November 2009 In mathematics , a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication math cdot A times A longrightarrow A math math a,b longmapsto a cdot b math that makes it an algebra over a field algebra over K . A unital associative algebra associative topological algebra is a topological ring . An example of a topological algebra is the algebra C 0,1 of continuous real valued functions on the closed unit interval 0,1 , or more generally any Banach algebra . The term was coined by David van Dantzig it appears in the title of his Thesis doctoral dissertation 1931 . The natural notion of subspace in a topological algebra is that of a topologically closed subalgebra . A topological algebra A is said to be generated by a subset S if A itself is the smallest closed subalgebra of A that contains S . For example by the Stone Weierstrass theorem , the set id sub 0,1 sub consisting only of the identity function id sub 0,1 sub is a generating set of the Banach algebra C 0,1 . Category Topological vector spaces Category Topological algebra Category Algebras topology stub pl Algebra topologiczna uk ... more details
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