Category:Linear algebra

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Topic: Category:Linear algebra

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	Concurrency Abstracts
		Algebraic operations on schedules can then be defined as constructions in the category of schedules.
		Its algebraic structure is essentially that of linear logic, with its morphisms being consequence-preserving renamings of propositions, and with its operations forming the core of a natural concurrent programming language.
		We develop a primitive algebraic model of this duality of time and information for rigid local computation, or straightline code, in the absence of choice and concurrency, where time and information are linearly ordered.
	boole.stanford.edu /abstracts.html

	Dual space : Duality (linear algebra)
		In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.
		f produces an injective homomorphism between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphism iff W is finite-dimensional.
		If V consists of the space of geometrical vectors (arrows) in the plane, then the elements of the dual V* can be intuitively represented as collections of parallel lines.
	www.eurofreehost.com /du/Duality_(linear_algebra)_2.html (408 words)

	Articles - Kernel (algebra)
		The notion of kernel in category theory is a generalisation of the kernels of abelian algebras; see Kernel (category theory).
		This is because monoids are not Mal'cev algebras.
		For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings).
	www.kamero.net /articles/Kernel_of_a_homomorphism (408 words)

	Courses—Rensselaer Catalog 9899
		The mathematical topics covered are selected from calculus, linear algebra, differential equations, numerical methods, and Fourier analysis.
		Functions of several variables, introductory linear algebra, and other analytical techniques needed for further study in probability, statistics, and operations research.
		Additional topics may be chosen from Mobius strips, Klein bottles, identification spaces, homotopy, the fundamental group of a surface, sequences in topological spaces, pseudo-metric spaces, completeness, Baire category, space-filling curves, weak topologies, quotient spaces, strong topologies, hyperspaces, the Hausdorff metric, and topological dimension.
	www.rpi.edu /dept/catalog/98-99/Courses/MATH.html (408 words)

	plotkin.htm
		Definition of a free algebra Semigroups Groups Rings Linear spaces and modules Linear algebras Some many-sorted structures Representations of groups and semigroups Linear representations Automata Affine spaces and affine automata CATEGORIES General information and examples Definition of a category Examples Subcategories Monomorphisms, epimorphisms and isomorphisms Duality Functors Natural transformations of functors.
		The origins Linear spaces and modules Associative linear algebras Group algebras and semigroup algebras Other structures Homomorphisms.
		Categories of functors Equivalence of categories Some technical notions Universal objects Direct and free products (products and coproducts) Other examples of universal objects Tensor products of modules Adjoint functors Cones, equalizers, and limits THE CATEGORY OF SETS, TOPOI.
	www.mmsysgrp.com /plotkin.htm (408 words)

	plotkin.htm
		Definition of a free algebra Semigroups Groups Rings Linear spaces and modules Linear algebras Some many-sorted structures Representations of groups and semigroups Linear representations Automata Affine spaces and affine automata CATEGORIES General information and examples Definition of a category Examples Subcategories Monomorphisms, epimorphisms and isomorphisms Duality Functors Natural transformations of functors.
		The origins Linear spaces and modules Associative linear algebras Group algebras and semigroup algebras Other structures Homomorphisms.
		Equality in Halmos algebras Cylindric algebras Homorphisms and structure of Halmos algebras.
	www.mmsysgrp.com /plotkin.htm (408 words)

	plotkin.htm
		Definition of a free algebra Semigroups Groups Rings Linear spaces and modules Linear algebras Some many-sorted structures Representations of groups and semigroups Linear representations Automata Affine spaces and affine automata CATEGORIES General information and examples Definition of a category Examples Subcategories Monomorphisms, epimorphisms and isomorphisms Duality Functors Natural transformations of functors.
		The origins Linear spaces and modules Associative linear algebras Group algebras and semigroup algebras Other structures Homomorphisms.
		Categories of functors Equivalence of categories Some technical notions Universal objects Direct and free products (products and coproducts) Other examples of universal objects Tensor products of modules Adjoint functors Cones, equalizers, and limits THE CATEGORY OF SETS, TOPOI.
	www.mmsysgrp.com /plotkin.htm (201 words)

	week40
		However, to some of us monoidal closed categories are rather familiar things, and in fact anyone who knows about vector spaces, linear maps, and the vector spaces Hom(V,W) and V tensor W knows a really good example of a monoidal closed category.
		Thus monoidal closed categories can be viewed as an abstraction of linear algebra, and indeed this is how "linear logic" got its name.
		It seems that I should read the following papers, too, before I really understand the connection between linear logic and category theory:
	math.ucr.edu /home/baez/week40.html (1002 words)

	Algebraic K-theory, groups and categories
		These include a solution of the long standing conjecture that the cohomology algebra of a finite group scheme is finitely generated and a further development of methods here to achieve a complete determination of all Ext-groups between classical functors in the category of strict polynomial functors of finite degree.
		The main results in category theory continue the programme of Categorical Galois Theory, which is shown to have a wide range of applications, involving Descent Theory, internal groupoids and commutator theory, and to link with the homotopical algebra methods developed by the Bangor group.
		And there are several important levels of generality, where it is desirable to have certain simplified descriptions of internal groupoids (for example it is well known that in the category of groups they are precisely crossed modules).
	www.bangor.ac.uk /~mas010/intasrep.html (1002 words)

	Concurrency Abstracts
		Algebraic operations on schedules can then be defined as constructions in the category of schedules.
		Its algebraic structure is essentially that of linear logic, with its morphisms being consequence-preserving renamings of propositions, and with its operations forming the core of a natural concurrent programming language.
		We prove full completeness for a fragment of the linear logic of the self-dual monoidal category of Chu spaces over 2, namely that the proofs between semisimple (conjunctive normal form) formulas of multiplicative linear logic without constants having two occurrences of each variable are in bijection with the dinatural transformations between the corresponding functors.
	boole.stanford.edu /abstracts.html (1002 words)

	MathGuide - Simple Search
		Field theory and polynomials; Commutative rings and algebras; Algebraic geometry; Linear and multilinear algebra, matrix theory; Associative rings and algebras; Nonassociative rings and algebras; Category theory, homological algebra
		Linear and multilinear algebra, matrix theory; Partial differential equations; Approximations and expansions; Integral equations; Functional analysis; Calculus of variations and optimal control, optimization; Numerical analysis
		Numerical analysis; Linear and multilinear algebra, matrix theory; Ordinary differential equations; Approximations and expansions; Calculus of variations and optimal control, optimization
	www.mathguide.de /cgi-bin/ssgfi/suche.pl?db=math&tag=SUC&words=15-XX&sort=&dsp=minitemp&COL=SUB (178 words)

	PREFACE
		The major areas represented in Magma V2.12 include group theory, ring theory, commutative algebra, arithmetic fields and their completions, module theory and lattice theory, finite dimensional algebras, Lie theory, representation theory, the elements of homological algebra, general schemes and curve schemes, modular forms and modular curves, finite incidence structures, linear codes and much else.
		For example, within the variety of algebras, the family of finitely presented algebras constitutes an abstract category, while the family of matrix algebras constitutes a concrete category.
		However, categories based on a concrete representation are as least as important as the abstract category in most varieties.
	www.math.lsu.edu /magma/preface.htm (713 words)

	Paul Mitchener's Homepage
		A C-category* can be defined to be a closed subcategory of a category consisting of a collection of Hilbert spaces and bounded linear operators between them.
		A C-algebra* is a closed algebra of bounded linear operators from a given Hilbert space to itself.
		In a recent article, John Roe proved by a direct computation that the Baum-Connes assembly map can be expressed as a boundary map associated to a certain short exact sequence of C*-algebras.
	www.uni-math.gwdg.de /mitch/research.html (713 words)

	ACT's CAAP Tests: Mathematics Test
		Items in this category are based on advanced algebra concepts including rational exponents, exponential and logarithmic functions, complex numbers, matrices, inverses of functions, and domains and ranges.
		Knowledge and skills assessed in this category may include graphing in the standard coordinate plane or the real number line, graphing conics, linear equations in two variables, graphing systems of equations, and similar types of skills.
		The College Algebra subscore is composed of test questions from the College Algebra and Trigonometry content areas.
	www.act.org /caap/tests/math.html (348 words)

	PREFACE
		Algebraic Design Philosophy: The design principles underpinning both the user language and system architecture are based on ideas from universal algebra and category theory.
		For example, within the variety of algebras, the family of finitely presented algebras constitutes an abstract category, while the family of matrix algebras constitutes a concrete category.
		The major areas represented in Magma V2.12 include group theory, ring theory, commutative algebra, arithmetic fields and their completions, module theory and lattice theory, finite dimensional algebras, Lie theory, representation theory, the elements of homological algebra, general schemes and curve schemes, modular forms and modular curves, finite incidence structures, linear codes and much else.
	www.math.lsu.edu /magma/preface.htm (713 words)

	h1126
		We compute the mod 2 homology of the general linear group GL(F) as a Hopf algebra over the Steenrod algebra.
		We show that LG is a Frobenius algebra object in the K(n)-local stable category, and we recall the connection between Frobenius algebras and topological quantum field theories to help analyse this structure.
		http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localhom Author: Stephen A. Mitchell Title: The mod 2 homology of the general linear group of a 2-adic local field e-mail: mitchell@math.washington.edu Let F be a finite extension of the 2-adic rational numbers.
	www.lehigh.edu /~dmd1/h1126 (713 words)

	Cohomology and the Bar Construction
		Given a linear operad O, construct a simplicial linear operad O_infinity whose algebras in the category of simplicial vector spaces are "weak O-algebras".
		Given a linear operad O over a commutative ring k, we can speak of its algebras over the commutative ring k[[x]], or indeed any commutative ring containing k (???).
		Show that "two-layer" semistrict (?) simplicial Lie algebras nontrivial only in dimensions 0 and n are classified by Lie algebra cohomology H^{n+2}(G,A).
	math.ucr.edu /home/baez/hda/bar.html (713 words)

	PlanetMath: isomorphism
		In the category of vector spaces and linear transformations, a linear transformation is an isomorphism if and only if it is an invertible linear transformation.
		(Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
		In the category of topological spaces and continuous maps, a continuous map is an isomorphism if and only if it is a homeomorphism.
	planetmath.org /encyclopedia/Isomorphism2.html (713 words)

	15: Linear and multilinear algebra; matrix theory
		Classic topics in linear algebra and matrix theory are at the center of the diagram: 15A03: Vector spaces, 15A04: Linear transformations, 15A15: Determinants, and 15A21: Canonical forms (e.g.
		Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure.
		In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis.
	www.math.niu.edu /~rusin/known-math/index/15-XX.html (713 words)

	Information and Computation Bibliography
		In particular, \Pi(SUP), the full subcategory of BC with all prime-algebraic lattices as objects, is such a categorical semantics.
		A maximal monoidal closed category of distributive algebraic domains.
		We study the category BC of bounded complete dcpos and maps preserving all suprema (linear maps).
	theory.lcs.mit.edu /~iandc/References/huth1995:10.html (351 words)

	HJM, Vol. 29, No. 1, 2003
		In the present paper, we further show that the hypercyclic operators on a separable infinite dimensional Frechet space form a dense subset of the algebra of continuous linear operators in the strong operator topology.
		In this paper, considering a lattice-triangular category C, we study the basic properties of the "category of fractions" I(C) associated to C (the universal property and the representation theorem for I(C) are also presented).
		This question was answered recently in the positive, and the result was generalized to the Frechet space case, in papers of Ansari, Bernal-Gonzalez, and Bonet and Peris.
	www.math.uh.edu /~hjm/Vol29-1.html (1785 words)

	The Future of Mathematical Text: Mayans: JoDI
		Expositions of basic calculus, linear algebra, and ordinary differential equations are common, and there is no need to add to this work here.
		The text is on linear algebra, a subject with connections to many parts of mathematics, to numerical work, and to a wide range of applications.
		For example, any book covering derivatives of a real variable will have L'Hôpital's rule, but most will not cover Baire's theorem on the continuous points of a derivative, because Baire category is a little more subtle.
	jodi.ecs.soton.ac.uk /Articles/v05/i01/Mayans (1785 words)

	Free Module, Basis
		These module homomorphisms are called linear transformations in the world of linear algebra, and they can be represented by matrices, where matrix multiplication implements function composition.
		For now, we'll call this the definition of a "free module", but if you study category theory, you'll find that a free object has a more general definition, and when that definition is applied to the category of unitary modules, the "direct sum" definition falls out.
		The module automorphisms correspond to the nonsingular matrices.
	www.mathreference.com /mod,basis.html (512 words)

	Best Book Buys - Normed linear spaces Books
		Subject Category > Mathematics > Algebra / Linear > Normed linear spaces
		Books > Browse > Subject Category > Mathematics > Algebra / Linear > Normed linear spaces
		Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras
	www.bestwebbuys.com /Linear_Algebra-N_10020476-books.html (512 words)

	Category theory preprints 2002
		These latter objects arise in problems within algebraic K-theory, an area of algebra linked to the study of decomposition and normal form theorems in linear algebra.
		The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.
		We extend Shrimpton's investigations on the morphism-digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal.
	www.informatics.bangor.ac.uk /public/mathematics/research/preprints/02/cathom02.html (512 words)

	Category theory preprints 2002
		These latter objects arise in problems within algebraic K-theory, an area of algebra linked to the study of decomposition and normal form theorems in linear algebra.
		The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.
		We extend Shrimpton's investigations on the morphism-digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal.
	www.informatics.bangor.ac.uk /public/mathematics/research/preprints/02/cathom02.html (838 words)

	Entropic Hopf algebras and models of non-commutative logic
		We show that the category of modules over an entropic Hopf algebra is an entropic category (possibly after application of the Chu construction).
		It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic.
		Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category.
	www.emis.ams.org /journals/TAC/volumes/10/17/10-17abs.html (838 words)

	Graduate Programs in Mathematics
		Submanifolds, fundamental theorem for hypersurfaces, variations of the length integral, Jacobi fields, comparison theorem, Morse index theorem, almost complex and complex manifolds, Hermitian and Kaehlerian metrics, homogeneous spaces, symmetric spaces and symmetric Lie algebra, characteristic classes.
		Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, category and uniform boundedness theorem, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem.
		Hölder spaces, Sobolev spaces, Sobolev embedding theorems, existence and regularity for second-order elliptic equations, maximum principles, second-order linear parabolic and hyperbolic equations, methods for non-linear PDE's, variational methods, fixed point theorems of Banach and Schauder.
	www.boun.edu.tr /graduate/sciences_and_engineering/mathematics.html (1655 words)

	Category theory
		Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
		One may check that the map from the category of topological spaces with a distinguished point to the category of groups is functorial: a topological (homo/iso)morphism will naturally correspond to a group (homo/iso)morphism.
		These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor.
	www.sciencedaily.com /encyclopedia/category_theory (1655 words)

	Glossary
		In mathematics, usually means basis in the sense of linear algebra; a minimal set of vectors that spans a vector space.
		The study of abstracted collections of mathematical objects, such as the category of sets or the category of vector spaces, together with abstracted operations sending one object to another, such as the collection of functions from one set to another or linear transformations from one vector space to another.
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
	www.math.ucdavis.edu /profiles/glossary.html (1655 words)

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In the News (Wed 30 May 12)