Relative Homological Algebra Volume 1
2011-01-01
This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists. The book is also suitable for an introductory course in commutative and ordinary homological algebra.
Shafarevich, I
1994-01-01
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
Homology theory on algebraic varieties
Wallace, Andrew H
1958-01-01
Homology Theory on Algebraic Varieties, Volume 6 deals with the principles of homology theory in algebraic geometry and includes the main theorems first formulated by Lefschetz, one of which is interpreted in terms of relative homology and another concerns the Poincaré formula. The actual details of the proofs of these theorems are introduced by geometrical descriptions, sometimes aided with diagrams. This book is comprised of eight chapters and begins with a discussion on linear sections of an algebraic variety, with emphasis on the fibring of a variety defined over the complex numbers. The n
Homology and cohomology of Rees semigroup algebras
Grønbæk, Niels; Gourdeau, Frédéric; White, Michael C.
2011-01-01
Let S by a Rees semigroup, and let 1¹(S) be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of l¹(S) is isomorphic to those of the underlying discrete group algebra.......Let S by a Rees semigroup, and let 1¹(S) be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of l¹(S) is isomorphic to those of the underlying discrete group algebra....
Threading homology through algebra selected patterns
Boffi, Giandomenico
2006-01-01
Aimed at graduate students and researchers in mathematics, this book takes homological themes, such as Koszul complexes and their generalizations, and shows how these can be used to clarify certain problems in selected parts of algebra, as well as their success in solving a number of them. - ;Threading Homology through Algebra takes homological themes (Koszul complexes and their variations, resolutions in general) and shows how these affect the perception of certain problems in selected parts of algebra, as well as their success in solving a number of them. The text deals with regular local ri
Hochschild homology of structured algebras
Wahl, Nathalie; Westerland, Craig Christopher
2016-01-01
We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication—we think of such algebras as A∞-algebras “with extra structure”. As applications, we obtain an integral version of the Costello–Kontsevich–Soibelman...
Ganea Term for Homology of Leibniz n-Algebras
J.M. Casas
2005-01-01
We extend the five-term exact sequence of homology with trivial coefficients of Leibniz n-algebras nH L1 ( K ) → nH L1 (L) → M → nH L0( K ) → nH L0( L ) → 0 associated to a central extension of Leibniz n-algebras 0 → M →K → L → 0 by means of a sixth term which is a generalization of the Ganea term for homology of Leibniz algebras. We use this sequence in order to analyze several questions related with the centre and central extensions of a Leibniz n-algebra.
Homological Algebra of Semimodules and Semicontramodules
Positselski, Leonid
2010-01-01
This is a monograph in semi-infinite homological algebra, concentrated mostly on the semi-infinite theory of associative algebraic structures, but including also some material on the semi-infinite homology and cohomology of Lie algebras and topological groups. The main objects of study are the double-sided derived functors SemiExt and SemiTor, and the phenomenon of comodule-contramodule correspondence, connecting them with the more conventional, one-sided Ext and CtrTor. Contramodules, introduced originally by Eilenberg and Moore in 1960's but almost forgotten for four decades, play a very pro
Homological Dimensions of the Extension Algebras of Monomial Algebras
Hong Bo SHI
2015-01-01
The main objective of this paper is to study the dimension trees and further the homo-logical dimensions of the extension algebras — dual and trivially twisted extensions — with a unified combinatorial approach using the two combinatorial algorithms — Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden sub-tle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more eﬃ cient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.
Homology theory on algebraic varieties
Wallace, Andrew H
2014-01-01
Concise and authoritative, this monograph is geared toward advanced undergraduate and graduate students. The main theorems whose proofs are given here were first formulated by Lefschetz and have since turned out to be of fundamental importance in the topological aspects of algebraic geometry. The proofs are fairly elaborate and involve a considerable amount of detail; therefore, some appear in separate chapters that include geometrical descriptions and diagrams.The treatment begins with a brief introduction and considerations of linear sections of an algebraic variety as well as singular and h
Threading homology through algebra selected patterns
Boffi, Giandomenico
2006-01-01
Aimed at graduate students and researchers in mathematics, this book takes homological themes, such as Koszul complexes and their generalizations, and shows how these can be used to clarify certain problems in selected parts of algebra, as well as their success in solving a number of them.
Periodic cyclic homology of affine Hecke algebras
Solleveld, Maarten
2009-01-01
This is the author's PhD-thesis, which was written in 2006. The version posted here is identical to the printed one. Instead of an abstract, the short list of contents: Preface 5 1 Introduction 9 2 K-theory and cyclic type homology theories 13 3 Affine Hecke algebras 61 4 Reductive p-adic groups 103 5 Parameter deformations in affine Hecke algebras 129 6 Examples and calculations 169 A Crossed products 223 Bibliography 227 Index 237 Samenvatting 245 Curriculum vitae 253
Introduction to relation algebras relation algebras
Givant, Steven
2017-01-01
The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field. The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly ...
Relative Derived Equivalences and Relative Homological Dimensions
Sheng Yong PAN
2016-01-01
Let A be a small abelian category. For a closed subbifunctor F of Ext1A (−,−), Buan has generalized the construction of Verdier’s quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.
Cyclic structures in algebraic (co)homology theories
Kowalzig, Niels
2010-01-01
This note discusses the cyclic cohomology of a left Hopf algebroid ($\\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd modules.
Homology of Lie algebra of supersymmetries and of super Poincare Lie algebra
Movshev, M.V. [Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651 (United States); Schwarz, A., E-mail: schwarz@math.ucdavis.edu [Department of Mathematics, University of California, Davis, CA 95616 (United States); Xu, Renjun [Department of Physics, University of California, Davis, CA 95616 (United States)
2012-01-11
We study the homology and cohomology groups of super Lie algebras of supersymmetries and of super Poincare Lie algebras in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions {<=}11. For dimensions D=10,11 we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry Lie algebras.
Homological unimodularity and Calabi-Yau condition for Poisson algebras
Lü, Jiafeng; Wang, Xingting; Zhuang, Guangbin
2017-09-01
In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi-Yau algebra if the Poisson structure is unimodular.
Vijay Kodiyalam; R Srinivasan; V S Sunder
2000-08-01
In this paper, we study a tower $\\{A^G_n(d):n≥ 1\\}$ of finite-dimensional algebras; here, represents an arbitrary finite group, denotes a complex parameter, and the algebra $A^G_n(d)$ has a basis indexed by `-stable equivalence relations' on a set where acts freely and has 2 orbits. We show that the algebra $A^G_n(d)$ is semi-simple for all but a finite set of values of , and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the `generic case'. Finally we determine the Bratteli diagram of the tower $\\{A^G_n(d): n≥ 1\\}$ (in the generic case).
Relative K-homology and normal operators
Manuilov, Vladimir; Thomsen, Klaus
2009-01-01
Let $A$ be a $C^*$-algebra, $J \\subset A$ a $C^*$-subalgebra, and let $B$ be a stable $C^*$-algebra. Under modest assumptions we organize invertible $C^*$-extensions of $A$ by $B$ that are trivial when restricted onto $J$ to become a group $\\mathrm{Ext}_J^{-1}(A,B)$, which can be computed by a six......-term exact sequence which generalizes the excision six-term exact sequence in the first variable of KK-theory. Subsequently we investigate the relative K-homology which arises from the group of relative extensions by specializing to abelian $C^*$-algebras. It turns out that this relative K-homology carries...... substantial information also in the operator theoretic setting from which the BDF theory was developed and we conclude the paper by extracting some of this information on approximation of normal operators....
Relations Between BZMVdM-Algebra and Other Algebras
高淑萍; 邓方安; 刘三阳
2003-01-01
Some properties of BZMVdM-algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMVdM-algebra can produce a quasi-lattice implication algebra. The relations between BZMVdM-algebra and other algebras are discussed in detail. A pseudo-distance function is defined in linear BZMVdM-algebra, and its properties are derived.
A definition of graph homology and graph K-theory of algebras
Movshev, M. V.
1999-01-01
We introduce and study elementary properties of graph homology of algebras. This new homology theory shares many features of cyclic and Hochschild homology. We also define a graph K-theory together with an analog of Chern character.
Algebras related to posets of hyperplanes
Jeurnink, G.A.M.
2000-01-01
We compare two noncommutative algebras which are related to arrangements of hyperplanes. For three special arrangements the induced approximately finite dimensional $C^*$-algebra and the graded Orlik-Solomon-algebra are investigated.
(Co)Homology and Universal Central Extension of Hom-Leibniz Algebras
Yong Sheng CHENG; Yu Cai SU
2011-01-01
Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Horn-Lie alge- bras. In this paper, we develop some structure theory (such as (co)homology groups, universal central extensions) of Hom-Leibniz algebras based on some works of Loday and Pirashvili.
Equivalence Relations of -Algebra Extensions
Changguo Wei
2010-04-01
In this paper, we consider equivalence relations of *-algebra extensions and describe the relationship between the isomorphism equivalence and the unitary equivalence. We also show that a certain group homomorphism is the obstruction for these equivalence relations to be the same.
Semi-algebraic partition and basis of Borel-Moore homology of hyperplane arrangements
Ito, Ko-Ki
2011-01-01
We describe an explicit semi-algebraic partition for the complement of the hyperplane arrangement such that each piece is contractible and forms a basis of Borel-Moore homology. We also give explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
Homology for higher-rank graphs and twisted C*-algebras
Kumjian, Alex; Sims, Aidan
2011-01-01
We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set \\tilde{Q}(\\Lambda) from a k-graph {\\Lambda} and demonstrate that the homology and topological realisation of {\\Lambda} coincide with those of \\tilde{Q}(\\Lambda) as defined by Grandis.
A New Formalism for Relational Algebra
Schwartzbach, Michael Ignatieff; Larsen, Kim Skak; Schmidt, Erik Meineche
1992-01-01
We present a new formalism for relational algebra, the FC language, which is based on a novel factorization of relations. The acronym stands for factorize and combine. A pure version of this language is equivalent to relational algebra in the sense that semantics preserving translations exist...
Relational Representations of Strongly Algebraic Lattices
XU Guang-hong; RAO San-ping
2012-01-01
In this paper,we introduce and investigate the strongly regular relation.Then we give the relational representations and an intrinsic characterization of strongly algebraic lattices via mapping relation and strongly regular relation.
Relational Algebra and SQL: Better Together
McMaster, Kirby; Sambasivam, Samuel; Hadfield, Steven; Wolthuis, Stuart
2013-01-01
In this paper, we describe how database instructors can teach Relational Algebra and Structured Query Language together through programming. Students write query programs consisting of sequences of Relational Algebra operations vs. Structured Query Language SELECT statements. The query programs can then be run interactively, allowing students to…
The graded Jacobi algebras and (co)homology
Grabowski, Janusz; Marmo, Giuseppe
2002-01-01
Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This...
A Relational Algebra Query Language for Programming Relational Databases
McMaster, Kirby; Sambasivam, Samuel; Anderson, Nicole
2011-01-01
In this paper, we describe a Relational Algebra Query Language (RAQL) and Relational Algebra Query (RAQ) software product we have developed that allows database instructors to teach relational algebra through programming. Instead of defining query operations using mathematical notation (the approach commonly taken in database textbooks), students…
Graded Lie Algebra Generating of Parastatistical Algebraic Relations
JING Si-Cong; YANG Wei-Min; LI Ping
2001-01-01
A new kind of graded Lie algebra (We call it Z2,2 graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the Z2,2 graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.
Hopf C~* -algebras related to the Latin square
郭懋正; 蒋立宁; 钱敏
2000-01-01
A sufficient condition is given for the multiparametric Hopf algebras to be Hopf * -algebras. Then a special subclass of the * -algebra related to a Latin square is given. After being completed, its generators are all of norm one.
Hopf C*-algebras related to the Latin square
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2000-01-01
A sufficient condition is given for the multiparametric Hopf algebras to be Hopf*-algebras. Then a special subclass of the *-algebra related to a Latin square is given. After being completed, its generators are all of norm one.
SWITCHING-ALGEBRAIC ANALYSIS OF RELATIONAL DATABASES
Ali Muhammad Ali Rushdi
2014-01-01
Full Text Available There is an established equivalence between relational database Functional Dependencies (FDs and a fragment of switching algebra that is built typically of Horn clauses. This equivalence pertains to both concepts and procedures of the FD relational database domain and the switching algebraic domain. This study is an exposition of the use of switching-algebraic tools in solving problems typically encountered in the analysis and design of relational databases. The switching-algebraic tools utilized include purely-algebraic techniques, purely-visual techniques employing the Karnaugh map and intermediary techniques employing the variable-entered Karnaugh map. The problems handled include; (a the derivation of the closure of a Dependency Set (DS, (b the derivation of the closure of a set of attributes, (c the determination of all candidate keys and (d the derivation of irredundant dependency sets equivalent to a given DS and consequently the determination of the minimal cover of such a set. A relatively large example illustrates the switching-algebraic approach and demonstrates its pedagogical and computational merits over the traditional approach.
Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras
Victor Nistor
2001-01-01
Full Text Available We give a detailed calculation of the Hochschild and cyclic homology of the algebra 𝒞c∞(G of locally constant, compactly supported functions on a reductive p-adic group G. We use these calculations to extend to arbitrary elements the definition of the higher orbital integrals introduced by Blanc and Brylinski (1992 for regular semi-simple elements. Then we extend to higher orbital integrals some results of Shalika (1972. We also investigate the effect of the induction morphism on Hochschild homology.
Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies
Bonora, L., E-mail: bonora@sissa.it [International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste (Italy); INFN, Sezione di Trieste (Italy); Bytsenko, A.A., E-mail: abyts@uel.br [Departamento de Fisica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina (Brazil)
2011-11-11
There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS{sub 3}, which also describe the three-dimensional Euclidean black holes, the pure N=1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2,Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.
Aisaka, Y; Aisaka, Yuri; Kazama, Yoichi
2003-01-01
Based on a novel first class algebra, we develop an extension of the pure spinor (PS) formalism of Berkovits, in which the PS constraints are removed. By using the homological perturbation theory in an essential way, the BRST-like charge $Q$ of the conventional PS formalism is promoted to a bona fide nilpotent charge $\\hat{Q}$, the cohomology of which is equivalent to the constrained cohomology of $Q$. This construction requires only a minimum number (five) of additional fermionic ghost-antighost pairs and the vertex operators for the massless modes of open string are obtained in a systematic way. Furthermore, we present a simple composite "$b$-ghost" field $B(z)$ which realizes the important relation $T(z) = \\{\\hat{Q}, B(z)\\} $, with $T(z)$ the Virasoro operator, and apply it to facilitate the construction of the integrated vertex. The present formalism utilizes U(5) parametrization and the manifest Lorentz covariance is yet to be achieved.
A Relational Localisation Theory for Topological Algebras
2012-01-01
In this thesis, we develop a relational localisation theory for topological algebras, i.e., a theory that studies local approximations of a topological algebra’s relational counterpart. In order to provide an appropriate framework for our considerations, we first introduce a general Galois theory between continuous functions and closed relations on an arbitrary topological space. Subsequently to this rather foundational discussion, we establish the desired localisation theory comprising the i...
Relational Lattice Foundation For Algebraic Logic
Tropashko, Vadim
2009-01-01
Relational Lattice is a succinct mathematical model for Relational Algebra. It reduces the set of six classic relational algebra operators to two: natural join and inner union. In this paper we push relational lattice theory in two directions. First, we uncover a pair of complementary lattice operators, and organize the model into a bilattice of four operations and four distinguished constants. We take a notice a peculiar way bilattice symmetry is broken. Then, we give axiomatic introduction of unary negation operation and prove several laws, including double negation and De Morgan. Next we reduce the model back to two basic binary operations and twelve axioms, and exhibit a convincing argument that the resulting system is complete in model-theoretic sense. The final part of the paper casts relational lattice perspective onto database dependency theory.
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
Barakat, Mohamed
2010-01-01
In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We follow this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R. As expected, solving one-sided (in)homogenous linear systems over R is the key algorithm. For a finitely generated maximal ideal m in a commutative ring R we show how solving (in)homogenous linear systems over R_m can be reduced to solving associated systems over R, yielding, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of local polynomial rings we compare an existing implementation of Mora's algorithm with the implementation of our homologically motivated alternative approach.
Translating cosmological special relativity into geometric algebra
Horn, Martin Erik
2012-11-01
Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.
Abstract Numeric Relations and the Visual Structure of Algebra
Landy, David; Brookes, David; Smout, Ryan
2014-01-01
Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition,…
Hopf代数交叉积的同调维数%Homological Dimension of Crossed Products of Hopf Algebras
WANG Zhi-xi; WU Yan-hui
2004-01-01
Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra. In this paper, we characterize the projectivity (injectivity) of M as a left A#σH-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σH, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.
Factoring Algebraic Error for Relative Pose Estimation
Lindstrom, P; Duchaineau, M
2009-03-09
We address the problem of estimating the relative pose, i.e. translation and rotation, of two calibrated cameras from image point correspondences. Our approach is to factor the nonlinear algebraic pose error functional into translational and rotational components, and to optimize translation and rotation independently. This factorization admits subproblems that can be solved using direct methods with practical guarantees on global optimality. That is, for a given translation, the corresponding optimal rotation can directly be determined, and vice versa. We show that these subproblems are equivalent to computing the least eigenvector of second- and fourth-order symmetric tensors. When neither translation or rotation is known, alternating translation and rotation optimization leads to a simple, efficient, and robust algorithm for pose estimation that improves on the well-known 5- and 8-point methods.
Finite dimensional semigroup quadratic algebras with minimal number of relations
Iyudu, Natalia
2011-01-01
A quadratic semigroup algebra is an algebra over a field given by the generators $x_1,...,x_n$ and a finite set of quadratic relations each of which either has the shape $x_jx_k=0$ or the shape $x_jx_k=x_lx_m$. We prove that a quadratic semigroup algebra given by $n$ generators and $d\\leq \\frac{n^2+n}{4}$ relations is always infinite dimensional. This strengthens the Golod--Shafarevich estimate for the above class of algebras. Our main result however is that for every $n$, there is a finite dimensional quadratic semigroup algebra with $n$ generators and $\\delta_n$ generators, where $\\delta_n$ is the first integer greater than $\\frac{n^2+n}{4}$. This shows that the above Golod-Shafarevich type estimate for semigroup algebras is sharp.
Generalized Path Algebras and Pointed Hopf Algebras%广义路代数与点Hopf代数
张寿传; 张耀中; 郭夕敬
2009-01-01
Most of pointed Hopf algebras of dimension pm with large coradical are shown to be generalized path algebras. By the theory of generalized path algebras, the representations, homological dimensions and radicals of these Hopf algebras are obtained. The relations between the radicals of path algebras and connectivity of directed graphs are given.
Dobrev, V K
2013-01-01
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of {\\it parabolic relation} between two non-compact semisimple Lie algebras g and g' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E_{7(7)} which is parabolically related to the CLA E_{7(-25)}, the parabolic subalgebras including E_{6(6)} and E_{6(-6)} . Other interesting examples are the orthogonal algebras so(p,q) all of which are parabolically related to the conformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including the Lorentz subalgebra so(n-1,1) and its analogs so(p-1,...
Relativity symmetries and Lie algebra contractions
Cho, Dai-Ning; Kong, Otto C.W., E-mail: otto@phy.ncu.edu.tw
2014-12-15
We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics.
姚海楼; 平艳茹
2008-01-01
In this paper,let A be a finite dimensional associative algebra over an algebraically closed field k,modA be the category of finite dimensional left A-module and X1,X2,...,Xn,in modA be a complete exceptional sequence,and let E be the endomorphism algebra of X1,X2,...,Xn.We study the global dimension of E,and calculate the Hochschild cohomology and homology groups of E.
Traces of differential forms and Hochschild homology
Hübl, Reinhold
1989-01-01
This monograph provides an introduction to, as well as a unification and extension of the published work and some unpublished ideas of J. Lipman and E. Kunz about traces of differential forms and their relations to duality theory for projective morphisms. The approach uses Hochschild-homology, the definition of which is extended to the category of topological algebras. Many results for Hochschild-homology of commutative algebras also hold for Hochschild-homology of topological algebras. In particular, after introducing an appropriate notion of completion of differential algebras, one gets a natural transformation between differential forms and Hochschild-homology of topological algebras. Traces of differential forms are of interest to everyone working with duality theory and residue symbols. Hochschild-homology is a useful tool in many areas of k-theory. The treatment is fairly elementary and requires only little knowledge in commutative algebra and algebraic geometry.
Relative local control and the block source algebras
樊恽
1997-01-01
The local control of pointed groups is generalized to the concept of relative local control,and it is proved that there exists a lifting for a covering of a block source algebra if the relative local control holds.As an application,a result is proved on the source algebras of blocks,whose defect groups are direct products of a normal subgroup and a subgroup that gives a relative local control.
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
Constitutive relations in optics in terms of geometric algebra
Dargys, A.
2015-11-01
To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.
The q-AGT-W relations via shuffle algebras
Neguţ, Andrei
2016-01-01
We construct the action of the q-deformed W-algebra on its level r representation geometrically, using the moduli space of U(r) instantons on the plane and the double shuffle algebra. We give explicit formulas for the action of W-currents in the fixed point basis of the level r representation, and prove a relation between the Carlsson-Okounkov Ext operator and vertex operators for the deformed W-algebra. We interpret this result as a q-deformed version of the AGT-W relations.
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
f-Deformed Boson Algebra Related to Gentile Statistics
Chung, Won Sang; Hassanabadi, Hassan
2017-06-01
In this paper the deformed boson algebra giving the Gentile distribution function is constructed by using the model of ideal gas of deformed bosons and some properties of a root of unity. As an example we discuss the quantum optical problem related to the Gentile (or f-deformed) boson algebra with large but finite M. For this algebra we construct the Gentile (or f-deformed) coherent state and discuss its nonclassical properties such as sub-Poissonian statistics and anti-bunching effect.
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
2013-01-01
The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.
The graded Lie algebra of general relativity
Reiterer, Michael
2014-01-01
We construct a graded Lie algebra in which a solution to the vacuum Einstein equations is any element of degree 1 whose bracket with itself is zero. Each solution generates a cochain complex, whose first cohomology is linearized gravity about that solution. We gauge-fix to get a smaller cochain complex with the same cohomologies (deformation retraction). The new complex is much smaller, it consists of the solution spaces of linear homogeneous wave equations (symmetric hyperbolic equations). The algorithm that produces these gauges and wave equations is both for linearized gravity and the full Einstein equations. The gauge groupoid is the groupoid of rank 2 complex vector bundles.
Green's-Like Relations on Algebras and Varieties
K. Denecke
2008-01-01
Full Text Available There are five equivalence relations known as Green's relations definable on any semigroup or monoid, that is, on any algebra with a binary operation which is associative. In this paper, we examine whether Green's relations can be defined on algebras of any type τ. Some sort of (super-associativity is needed for such definitions to work, and we consider algebras which are clones of terms of type τ, where the clone axioms including superassociativity hold. This allows us to define for any variety V of type τ two Green's-like relations ℒV and ℛV on the term clone of type τ. We prove a number of properties of these two relations, and describe their behaviour when V is a variety of semigroups.
Persistent Homology for Random Fields and Complexes
Adler, Robert J; Borman, Matthew S; Subag, Eliran; Weinberger, Shmuel
2010-01-01
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.
Recursion relations and branching rules for simple Lie algebras
Lyakhovsky, V D
1995-01-01
The branching rules between simple Lie algebras and its regular (maximal) simple subalgebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights \\Gamma specific for each injection. The structure of \\Gamma is easily deduced from the correspondence between the root systems of algebra and subalgebra. The recursion relations thus obtained give rise to simple and effective algorithm for branching rules. The details are exposed by performing the explicit decomposition procedure for A_{3} \\oplus u(1) \\rightarrow B_{4} injection.
Sepanski, Mark R
2010-01-01
Mark Sepanski's Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems
Noncommutative Poisson brackets on Loday algebras and related deformation quantization
UCHINO, Kyousuke
2010-01-01
We introduce a new type of algebra which is called a Loday-Poisson algebra. The class of the Loday-Poisson algebras forms a special subclass of Aguiar's dual-prePoisson algebas (\\cite{A}). We will prove that there exists a unique Loday-Poisson algebra over a Loday algebra, like the Lie-Poisson algebra over a Lie algebra. Thus, Loday-Poisson algebras are regarded as noncommutative analogues of Lie-Poisson algebras. We will show that the polinomial Loday-Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday's associative dialgebra.
Relative rates of homologous and nonhomologous recombination in transfected DNA.
Roth, D B; Wilson, J H
1985-05-01
Both homologous and nonhomologous recombination events occur at high efficiency in DNA molecules transfected into mammalian cells. Both types of recombination occur with similar overall efficiencies, as measured by an endpoint assay, but their relative rates are unknown. In this communication, we measure the relative rates of homologous and nonhomologous recombination in DNA transfected into monkey cells. This measurement is made by using a linear simian virus 40 genome that contains a 131-base-pair duplication at its termini. Once inside the cell, this molecule must circularize to initiate lytic infection. Circularization can occur either by direct, nonhomologous end-joining or by homologous recombination within the duplicated region. Although the products of the two recombination pathways are different, they are equally infectious. Since homologous and nonhomologous recombination processes are competing for the same substrate, the relative amounts of the products of each pathway should reflect the relative rates of homologous and nonhomologous recombination. Analysis of individual recombinant genomes from 164 plaques indicates that the rate of circularization by nonhomologous recombination is 2- to 3-fold higher than the rate of homologous recombination. The assay system described here may prove to be useful for testing procedures designed to influence the relative rates of homologous and nonhomologous recombination.
Maximal-acceleration phase space relativity from Clifford algebras
Castro, C
2002-01-01
We present a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). Crucial in order to establish this link is the use of Clifford algebras in phase spaces. The maximal proper-acceleration bound is a = c^2/ \\Lambda in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. We present the reasons why an Extended Scale Relativity based on Clifford spaces is physically more appealing than those based on kappa-deformed Poincare algebras and the inhomogeneous quantum groups operating in quantum Minkowski spacetimes. The main reason being that the Planck scale should not be taken as a deformation parameter to construct quantum algebras but should exist already as the minimum scale in Clifford spaces.
Lagrangian Quantum Homology for Lagrangian cobordism
Singer, Berit
2015-01-01
We extend the definition of Lagrangian quantum homology to monotone Lagrangian cobordism and establish its general algebraic properties. In particular we develop a relative version of Lagrangian quantum homology associated to a cobordism relative to a part of its boundary and study relations of this invariant to the ambient quantum homology.
Relative rates of homologous and nonhomologous recombination in transfected DNA.
Roth, D B; Wilson, J H
1985-01-01
Both homologous and nonhomologous recombination events occur at high efficiency in DNA molecules transfected into mammalian cells. Both types of recombination occur with similar overall efficiencies, as measured by an endpoint assay, but their relative rates are unknown. In this communication, we measure the relative rates of homologous and nonhomologous recombination in DNA transfected into monkey cells. This measurement is made by using a linear simian virus 40 genome that contains a 131-ba...
A Voronoi-based spatial algebra for spatial relations
无
2002-01-01
Spatial relation between spatial objects is a very important topic for spatial reasoning, query and analysis in geographical information systems (GIS). The most popular models in current use have fundamental deficiencies in theory. In this paper, a generic algebra for spatial relations is presented, in which (i) appropriate operators from set operators (i.e. union, intersection, difference, difference by, symmetric difference, etc.) are utilized to distinguish the spatial relations between neighboring spatial objects; (ii) three types of values are used for the computational results of set operations-content, dimension and number of connected components; and (iii) a spatial object is treated as a whole but the Voronoi region of an object is employed to enhance its interaction with its neighbours. This algebra overcomes the shortcomings of the existing models and it can effectively describe the relations of spatial objects.
Canonical Noncommutativity Algebra for the Tetrad Field in General Relativity
Kober, Martin
2011-01-01
General relativity under the assumption of noncommuting components of the tetrad field is considered in this paper. Since the algebraic properties of the tetrad field representing the gravitational field are assumed to correspond to the noncommutativity algebra of the coordinates in the canonical case of noncommutative geometry, this idea is closely related to noncommutative geometry as well as to canonical quantization of gravity. According to this presupposition there are derived generalized field equations for general relativity which are obtained by replacing the usual tetrad field by the tetrad field operator within the actions and then building expectation values of the corresponding field equations between coherent states. These coherent states refer to creation and annihilation operators created from the components of the tetrad field operator. In this sense the obtained theory could be regarded as a kind of semiclassical approximation of a complete quantum description of gravity. The consideration pr...
Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras
Ondrus, Matthew; Wiesner, Emilie
2016-07-01
This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
[Relations between biomedical variables: mathematical analysis or linear algebra?].
Hucher, M; Berlie, J; Brunet, M
1977-01-01
The authors, after a short reminder of one pattern's structure, stress on the possible double approach of relations uniting the variables of this pattern: use of fonctions, what is within the mathematical analysis sphere, use of linear algebra profiting by matricial calculation's development and automatiosation. They precise the respective interests on these methods, their bounds and the imperatives for utilization, according to the kind of variables, of data, and the objective for work, understanding phenomenons or helping towards decision.
Relational Algebra in Spatial Decision Support Systems Ontologies.
Diomidous, Marianna; Chardalias, Kostis; Koutonias, Panagiotis; Magnita, Adrianna; Andrianopoulos, Charalampos; Zimeras, Stelios; Mechili, Enkeleint Aggelos
2017-01-01
Decision Support Systems (DSS) is a powerful tool, for facilitates researchers to choose the correct decision based on their final results. Especially in medical cases where doctors could use these systems, to overcome the problem with the clinical misunderstanding. Based on these systems, queries must be constructed based on the particular questions that doctors must answer. In this work, combination between questions and queries would be presented via relational algebra.
Representations of a Noncommutative Associative Algebra Related to Quantum Torus of Rank Three
Shang Yuan LIN; Bin XIN
2005-01-01
In this paper, we present some modules over the rank-three quantized Weyl algebra, which are closely related to modules over some vertex algebras. The isomorphism classes among these modules are also determined.
Abstract numeric relations and the visual structure of algebra.
Landy, David; Brookes, David; Smout, Ryan
2014-09-01
Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic.
RELATIONS OF DERIVATIVE ALGEBRA AND RING OF DIFFERENTIAL OPERATORS IN CHARACTERISTIC p>0
张江峰
2002-01-01
Let K be a field of characteristic p>0. We prove that the derivative algebra of K[x1,…,xn] is a proer subring of the ring of differential operators of K[x1,…,xn]. A concrete example is given to show that there is a differential operator of order p that does not belong to the derivative algebra. By these results, is follows that the derivative algebra is Morita equivalent to K[xp1,…,xpn], and hence its global homological dimension, Krull dimension, K0 group and some other properties are got.
Gorenstein homological dimensions
Holm, Henrik Granau
2004-01-01
In basic homological algebra, the projective, injective and 2at dimensions of modules play an important and fundamental role. In this paper, the closely related Gorenstein projective, Gorenstein injective and Gorenstein 2at dimensions are studied. There is a variety of nice results about Gorenstein...
Gorenstein homological dimensions
Holm, Henrik Granau
2004-01-01
In basic homological algebra, the projective, injective and 2at dimensions of modules play an important and fundamental role. In this paper, the closely related Gorenstein projective, Gorenstein injective and Gorenstein 2at dimensions are studied. There is a variety of nice results about Gorenstein...
Fahrenberg, Uli
2004-01-01
We introduce a new notion of directed homology for semicubical sets. We show that it respects directed homotopy and is functorial, and that it appears to enjoy some good algebraic properties. Our work has applications to higher-dimensional automata.......We introduce a new notion of directed homology for semicubical sets. We show that it respects directed homotopy and is functorial, and that it appears to enjoy some good algebraic properties. Our work has applications to higher-dimensional automata....
Relating Reasoning Methodologies in Linear Logic and Process Algebra
Yuxin Deng
2012-11-01
Full Text Available We show that the proof-theoretic notion of logical preorder coincides with the process-theoretic notion of contextual preorder for a CCS-like calculus obtained from the formula-as-process interpretation of a fragment of linear logic. The argument makes use of other standard notions in process algebra, namely a labeled transition system and a coinductively defined simulation relation. This result establishes a connection between an approach to reason about process specifications and a method to reason about logic specifications.
Matrix formulae and skein relations for cluster algebras from surfaces
Musiker, Gregg
2011-01-01
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fraction field of) A(S,M), using products of elements of PSL_2(R). We give a direct proof that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [MSW, MSW2]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [FG1, FG2, FG3], who worked in the coefficient-free case. The results of this paper will be used in [MSW2] in order to show that certain collections of arcs and loops comprise a vector-space basis for A(S,M).
Optimal uncertainty relations in a modified Heisenberg algebra
Abdelkhalek, Kais; Fiedler, Leander; Mangano, Gianpiero; Schwonnek, René
2016-01-01
Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations which are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min- and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min-entropy is exactly one bit.
Optimal uncertainty relations in a modified Heisenberg algebra
Abdelkhalek, Kais; Chemissany, Wissam; Fiedler, Leander; Mangano, Gianpiero; Schwonnek, René
2016-12-01
Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations that are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows us to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min entropy is exactly 1 bit.
Dolotin, V.; Morozov, A.
2014-01-01
We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful.
Dolotin, V
2014-01-01
We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful.
Dolotin, V.; Morozov, A.
2014-01-15
We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N−1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov–Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful.
Quantum walks, deformed relativity and Hopf algebra symmetries.
Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo
2016-05-28
We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras.
A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.
DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J
2016-12-01
Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions.
Relational Thinking: Learning Arithmetic in Order to Promote Algebraic Thinking
Napaphun, Vishnu
2012-01-01
Trends in the curriculum reform propose that algebra should be taught throughout the grades, starting in elementary school. The aim should be to decrease the discontinuity between the arithmetic in elementary school and the algebra in upper grades. This study was conducted to investigate and characterise upper elementary school students…
Composing Cardinal Direction Relations Basing on Interval Algebra
Chen, Juan; Jia, Haiyang; Liu, Dayou; Zhang, Changhai
Direction relations between extended spatial objects are important commonsense knowledge. Skiadopoulos proposed a formal model for representing direction relations between compound regions (the finite union of simple regions), known as SK-model. It perhaps is currently one of most cognitive plausible models for qualitative direction information, and has attracted interests from artificial intelligence and geographic information system. Originating from Allen first using composition table to process time interval constraints; composing has become the key technique in qualitative spatial reasoning to check the consistency. Due to the massive number of basic directions in SK-model, its composition becomes extraordinary complex. This paper proposed a novel algorithm for the composition. Basing the concepts of smallest rectangular directions and its original directions, it transforms the composition of basic cardinal direction relations into the composition of interval relations corresponding to Allen's interval algebra. Comparing with existing methods, this algorithm has quite good dimensional extendibility, that is, it can be easily transferred to the tridimensional space with a few modifications.
Categories and Commutative Algebra
Salmon, P
2011-01-01
L. Badescu: Sur certaines singularites des varietes algebriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algebriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de series formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all'algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.- C.B. Winters: An elementary lecture on algebraic spaces.
W(1+infinity) algebra as a symmetry behind AGT relation
Kanno, Shoichi; Shiba, Shotaro
2011-01-01
We give some evidences which imply that W(1+infinity) algebra describes the symmetry behind AGT(-W) conjecture: a correspondence between the partition function of N=2 supersymmetric quiver gauge theories and the correlators of Liouville (Toda) field theory.
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
Aguiar, Marcelo; Benedetti, Carolina; Bergeron, Nantel; Chen, Zhi; Diaconis, Persi; Hendrickson, Anders; Hsiao, Samuel; Isaacs, I Martin; Jedwab, Andrea; Johnson, Kenneth; Karaali, Gizem; Lauve, Aaron; Le, Tung; Lewis, Stephen; Li, Huilan; Magaard, Kay; Marberg, Eric; Novelli, Jean-Christophe; Pang, Amy; Saliola, Franco; Tevlin, Lenny; Thibon, Jean-Yves; Thiem, Nathaniel; Venkateswaran, Vidya; Vinroot, C Ryan; Yan, Ning; Zabrocki, Mike
2010-01-01
We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.
Exact Solutions of Two Coupled Harmonic Oscillators Related to the Sp(4, R) Lie Algebra
PAN Feng; DAI LianRong
2001-01-01
Exact solutions of the eigenvalue problem of two coupled harmonic oscillators related to the Sp(4, R) Lie algebra are derived by using an algebraic method. It is found that the energy spectrum of the system is determined by one-boson excitation energies built on a vector coherent state of Sp(4, R) U(2).``
DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J
2015-05-01
To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions. Copyright © 2015 Elsevier Inc. All rights reserved.
Hegedus, Stephen J.; Tapper, John; Dalton, Sara
2016-01-01
In this study, we examine the relationship between contextual variables related to teachers and student performance in Advanced Algebra classrooms in the USA. The data were gathered from a cluster-randomized study on the effects of SimCalc MathWorlds®, a curricular and technological intervention as a replacement for Algebra 2 curriculum, on…
Hegedus, Stephen J.; Tapper, John; Dalton, Sara
2016-01-01
In this study, we examine the relationship between contextual variables related to teachers and student performance in Advanced Algebra classrooms in the USA. The data were gathered from a cluster-randomized study on the effects of SimCalc MathWorlds®, a curricular and technological intervention as a replacement for Algebra 2 curriculum, on…
Atnafu, Mulugeta
2010-01-01
The purpose of this study was to examine the relation between the attitudes and components of attitude of the students towards algebra with their algebra achievements. The population for this study consists of all government tenth grade students and their mathematics teachers in Addis Ababa city administration. Sixteen tenth grade sections were…
Revisiting special relativity: a natural algebraic alternative to Minkowski spacetime.
James M Chappell
Full Text Available Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension [Formula: see text], with the unit imaginary producing the correct spacetime distance [Formula: see text], and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary [Formula: see text], with the Clifford bivector [Formula: see text] for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis [Formula: see text] and [Formula: see text]. We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.
Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime
Chappell, James M.; Iqbal, Azhar; Iannella, Nicolangelo; Abbott, Derek
2012-01-01
Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension , with the unit imaginary producing the correct spacetime distance , and the results of Einstein’s then recently developed theory of special relativity, thus providing an explanation for Einstein’s theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary , with the Clifford bivector for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis and . We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton’s scattering formula, and a simple formulation of Dirac’s and Maxwell’s equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane. PMID:23300566
Revisiting special relativity: a natural algebraic alternative to Minkowski spacetime.
Chappell, James M; Iqbal, Azhar; Iannella, Nicolangelo; Abbott, Derek
2012-01-01
Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension [Formula: see text], with the unit imaginary producing the correct spacetime distance [Formula: see text], and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary [Formula: see text], with the Clifford bivector [Formula: see text] for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis [Formula: see text] and [Formula: see text]. We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.
Dobrev, V K
2013-01-01
In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of {\\it parabolic relation} between two non-compact semisimple Lie algebras $\\cal G$ and $\\cal G'$ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra $E_{7(7)}$ which is parabolically related to the CLA $E_{7(-25)}$. Other interesting examples are the orthogonal algebras $so(p,q)$ all of which are parabolically related to the conformal algebra $so(n,2)$ with $p+q=n+2$, the parabolic subalgebras including the Lorentz subalgebra $so(n-1,1)$ and its analogs ...
An Improved Algorithm for Generating Database Transactions from Relational Algebra Specifications
Daniel J. Dougherty
2010-03-01
Full Text Available Alloy is a lightweight modeling formalism based on relational algebra. In prior work with Fisler, Giannakopoulos, Krishnamurthi, and Yoo, we have presented a tool, Alchemy, that compiles Alloy specifications into implementations that execute against persistent databases. The foundation of Alchemy is an algorithm for rewriting relational algebra formulas into code for database transactions. In this paper we report on recent progress in improving the robustness and efficiency of this transformation.
Deficiently Extremal Gorenstein Algebras
Pavinder Singh
2011-08-01
The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if / is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of / must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained.
Lloris Ruiz, Antonio; Parrilla Roure, Luis; García Ríos, Antonio
2014-01-01
This book presents a complete and accurate study of algebraic circuits, digital circuits whose performance can be associated with any algebraic structure. The authors distinguish between basic algebraic circuits, such as Linear Feedback Shift Registers (LFSRs) and cellular automata, and algebraic circuits, such as finite fields or Galois fields. The book includes a comprehensive review of representation systems, of arithmetic circuits implementing basic and more complex operations, and of the residue number systems (RNS). It presents a study of basic algebraic circuits such as LFSRs and cellular automata as well as a study of circuits related to Galois fields, including two real cryptographic applications of Galois fields.
Basic algebraic topology and its applications
Adhikari, Mahima Ranjan
2016-01-01
This book provides an accessible introduction to algebraic topology, a ﬁeld at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book oﬀers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. T...
Realizations of $\\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras
Juric, Tajron; Pikutic, Danijel
2015-01-01
Realizations of $\\kappa$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist satisfying normalization and cocycle conditions. $\\kappa$-deformed $\\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The $\\kappa$-Poincar\\'e-Weyl and $\\kappa$-Poincar\\'e-Hopf algebras are discussed. Left-right dual $\\kappa$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All known Drinfeld twists related to $\\kappa$-Minkowski space are obtained...
An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications
Tongsong Jiang
2014-01-01
Full Text Available This paper, by means of complex representation of a quaternion matrix, discusses the consimilarity of quaternion matrices, and obtains a relation between consimilarity and similarity of quaternion matrices. It sets up an algebraic bridge between consimilarity and similarity, and turns the theory of consimilarity of quaternion matrices into that of ordinary similarity of complex matrices. This paper also gives algebraic methods for finding coneigenvalues and coneigenvectors of quaternion matrices by means of complex representation of a quaternion matrix.
Central simple Poisson algebras
SU Yucai; XU Xiaoping
2004-01-01
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
Geary, David C.; Hoard, Mary K.; Nugent, Lara; Rouder, Jeffrey N.
2015-01-01
The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 (92 girls) 9th graders, controlling parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation, but not schema memory. Frequency of fact-retrieval errors was related to schema memory but not coordinate plane or expression evaluation accuracy. The results suggest the ANS may contribute to or is influenced by spatial-numerical and numerical only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest different brain and cognitive systems are engaged during the learning of different components of algebraic competence, controlling demographic and domain general abilities. PMID:26255604
An interpretation of E_n-homology as functor homology
Livernet, Muriel; Richter, Birgit
2009-01-01
We prove that E_n-homology of non-unital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by natural maps, ranging from Hochschild homology and its higher order versions to Gamma homology.
Relations between elliptic multiple zeta values and a special derivation algebra
Broedel, Johannes; Matthes, Nils; Schlotterer, Oliver
2016-04-01
We investigate relations between elliptic multiple zeta values (eMZVs) and describe a method to derive the number of indecomposable elements of given weight and length. Our method is based on representing eMZVs as iterated integrals over Eisenstein series and exploiting the connection with a special derivation algebra. Its commutator relations give rise to constraints on the iterated integrals over Eisenstein series relevant for eMZVs and thereby allow to count the indecomposable representatives. Conversely, the above connection suggests apparently new relations in the derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations for eMZVs over a wide range of weights and lengths.
Jia-feng; Lü
2007-01-01
[1]Priddy S.Koszul resolutions.Trans Amer Math Soc,152:39-60 (1970)[2]Beilinson A,Ginszburg V,Soergel W.Koszul duality patterns in representation theory.J Amer Math Soc,9:473-525 (1996)[3]Aquino R M,Green E L.On modules with linear presentations over Koszul algebras.Comm Algebra,33:19-36 (2005)[4]Green E L,Martinez-Villa R.Koszul and Yoneda algebras.Representation theory of algebras (Cocoyoc,1994).In:CMS Conference Proceedings,Vol 18.Providence,RI:American Mathematical Society,1996,247-297[5]Berger R.Koszulity for nonquadratic algebras.J Algebra,239:705-734 (2001)[6]Green E L,Marcos E N,Martinez-Villa R,et al.D-Koszul algebras.J Pure Appl Algebra,193:141-162(2004)[7]He J W,Lu D M.Higher Koszul Algebras and A-infinity Algebras.J Algebra,293:335-362 (2005)[8]Green E L,Marcos E N.δ-Koszul algebras.Comm Algebra,33(6):1753-1764 (2005)[9]Keller B.Introduction to A-infinity algebras and modules.Homology Homotopy Appl,3:1-35 (2001)[10]Green E L,Martinez-Villa R,Reiten I,et al.On modules with linear presentations.J Algebra,205(2):578-604 (1998)[11]Keller B.A-infinity algebras in representation theory.Contribution to the Proceedings of ICRA Ⅸ.Beijing:Peking University Press,2000[12]Lu D M,Palmieri J H,Wu Q S,et al.A∞-algebras for ring theorists.Algebra Colloq,11:91-128 (2004)[13]Weibel C A.An Introduction to homological algebra.Cambridge Studies in Avanced Mathematics,Vol 38.Cambridge:Cambridge University Press,1995
The Virasoro vertex algebra and factorization algebras on Riemann surfaces
Williams, Brian
2017-08-01
This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.
Relative Binding Free Energy Calculations Applied to Protein Homology Models.
Cappel, Daniel; Hall, Michelle Lynn; Lenselink, Eelke B; Beuming, Thijs; Qi, Jun; Bradner, James; Sherman, Woody
2016-12-27
A significant challenge and potential high-value application of computer-aided drug design is the accurate prediction of protein-ligand binding affinities. Free energy perturbation (FEP) using molecular dynamics (MD) sampling is among the most suitable approaches to achieve accurate binding free energy predictions, due to the rigorous statistical framework of the methodology, correct representation of the energetics, and thorough treatment of the important degrees of freedom in the system (including explicit waters). Recent advances in sampling methods and force fields coupled with vast increases in computational resources have made FEP a viable technology to drive hit-to-lead and lead optimization, allowing for more efficient cycles of medicinal chemistry and the possibility to explore much larger chemical spaces. However, previous FEP applications have focused on systems with high-resolution crystal structures of the target as starting points-something that is not always available in drug discovery projects. As such, the ability to apply FEP on homology models would greatly expand the domain of applicability of FEP in drug discovery. In this work we apply a particular implementation of FEP, called FEP+, on congeneric ligand series binding to four diverse targets: a kinase (Tyk2), an epigenetic bromodomain (BRD4), a transmembrane GPCR (A2A), and a protein-protein interaction interface (BCL-2 family protein MCL-1). We apply FEP+ using both crystal structures and homology models as starting points and find that the performance using homology models is generally on a par with the results when using crystal structures. The robustness of the calculations to structural variations in the input models can likely be attributed to the conformational sampling in the molecular dynamics simulations, which allows the modeled receptor to adapt to the "real" conformation for each ligand in the series. This work exemplifies the advantages of using all-atom simulation methods with
Odesskii, A V [L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow (Russian Federation)
2002-12-31
This survey is devoted to associative Z{sub {>=}}{sub 0}-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). Examples are considered of such algebras, depending on two continuous parameters (namely, on an elliptic curve and a point on it), that are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces, and other directions of modern investigations.
Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations
Seo, Jae Hong
Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.
无
2007-01-01
It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are,in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A), and show an A∞-structure on E(A). Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos.
Interactions Between Representation Ttheory, Algebraic Topology and Commutative Algebra
Pitsch, Wolfgang; Zarzuela, Santiago
2016-01-01
This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest dev...
A ALINEJAD; A GHAFFARI
2017-09-01
We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $\\ast$-semigroup $S$ when it is infinite non-discrete cancellative, $M_{a}(S)^{\\ast\\ast}$ does not admit an involution, and $M_{a}(S)^{\\ast\\ast}$ has atrivolution with range $M_{a}(S)$ if and only if $S$ is discrete. We also show that when $G$ isan amenable group, the second dual of the Fourier algebra of $G$ admits an involutionextending one of the natural involutions of $A(G)$ if and only if $G$ is finite. However,$A(G)^{\\ast\\ast}$ always admits trivolution.
Differential forms on singular varieties and cyclic homology
Brasselet, J P; Brasselet, Jean-Paul; Legrand, André
1996-01-01
A classical result of A. Connes asserts that the Frechet algebra of smooth functions on a smooth compact manifold X provides, by a purely algebraic procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild and cyclic homology of this algebra. In the situation of a Thom-Mather stratified variety, we construct a Frechet algebra of functions on the regular part and a module of poles along the singular part. We associate to these objects a complex of differential forms and an Hochschild complex, on the regular part, both with poles along the singular part. The de Rham cohomology of the first complex and the cylic homology of the second one are related to the intersection homology of the variety, the corresponding perversity is determined by the orders of poles.
Deformations of the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity.
Krasnov, Kirill
2008-02-29
We show that the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity can be nontrivially deformed by allowing the cosmological constant to become an arbitrary function of the (Weyl) curvature. Our result implies that there is not one but infinitely many (parametrized by an arbitrary function) four-dimensional generally covariant local gravity theories propagating 2 degrees of freedom.
van den Berg, J. B.; Ghrist, R.; Vandervorst, R. C.; Wójcik, W.
2015-09-01
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on R / Z ×D2. The periodic flow-lines define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a new invariant for such classes, the BRAID FLOER HOMOLOGY. This refinement of Floer homology, originally used for the Arnol'd Conjecture, yields a Morse-type forcing theory for periodic points of area-preserving diffeomorphisms of the 2-disc based on braiding. Contributions of this paper include (1) a monotonicity lemma for the behavior of the nonlinear Cauchy-Riemann equations with respect to algebraic lengths of braids; (2) establishment of the topological invariance of the resulting braid Floer homology; (3) a shift theorem describing the effect of twisting braids in terms of shifting the braid Floer homology; (4) computation of examples; and (5) a forcing theorem for the dynamics of Hamiltonian disc maps based on braid Floer homology.
Realizations of κ-Minkowski space, Drinfeld twists, and related symmetry algebras
Juric, Tajron; Meljanac, Stjepan; Pikutic, Danijel [Ruder Boskovic Institute, Theoretical Physics Division, Zagreb (Croatia)
2015-11-15
Realizations of κ-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the gl(n) generators. There are three one-parameter families of linear realizations for timelike and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. κ-Deformed igl(n)-Hopf algebras are presented for all cases. The κ-Poincare-Weyl and κ-Poincare-Hopf algebras are discussed. The left-right dual κ-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to κ-Minkowski space are obtained from our construction. Finally, some physical applications are discussed. (orig.)
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY
TaoChangli; LuShijie; ChenPeixin
2002-01-01
Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.
Nader Ali Makboul Hassan
2014-01-01
Full Text Available This paper is an attempt to stress the usefulness of the multi-variable special functions. In this paper, we derive certain generating relations involving 2-indices 5-variables 5-parameters Tricomi functions (2I5V5PTF by using a Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.
Extension of relational event algebra to a general decision making setting
Goodman, I.R.; Kramer, G.F.
1996-12-31
Relational Event Algebra (REA) is a new mathematical tool which provides an explicit algebraic reconstruction of events (appropriately designated as relational events) when initially only the formal probability values of such events are given as functions of known contributing event probabilities. In turn, once such relational events are obtained, one can then determine the probability of any finite logical combination, and in particular, various probabilistic distance measures among the events. A basic application of REA is to test hypotheses for the similarity of distinct models attempting to describe the same events such as in data fusion and combination of evidence. This paper considers new motivation for the use of REA, as well as a more general decision-making framework where system performance and redundancy / consistency tradeoffs are considered.
Fontana, Marco; Olberding, Bruce; Swanson, Irena
2011-01-01
Commutative algebra is a rapidly growing subject that is developing in many different directions. This volume presents several of the most recent results from various areas related to both Noetherian and non-Noetherian commutative algebra. This volume contains a collection of invited survey articles by some of the leading experts in the field. The authors of these chapters have been carefully selected for their important contributions to an area of commutative-algebraic research. Some topics presented in the volume include: generalizations of cyclic modules, zero divisor graphs, class semigrou
El-Chaar, Caroline
2012-01-01
In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. Using this fourth realization, we explicitly describe all its ideals.
Optimizing relational algebra operations using discrimination-based joins and lazy products
Henglein, Fritz
We show how to implement in-memory execution of the core re- lational algebra operations of projection, selection and cross-product eciently, using discrimination-based joins and lazy products. We introduce the notion of (partitioning) discriminator, which par- titions a list of values according...... to a specied equivalence relation on keys the values are associated with. We show how discriminators can be dened generically, purely functionally, and eciently (worst-case linear time) on top of the array-based basic multiset discrimination algorithm of Cai and Paige (1995). Discriminators provide the basis...... the selection operation to recognize on the y whenever it is applied to a cross-product, in which case it can choose an ecient discrimination-based equijoin implementation. The techniques subsume most of the optimization techniques based on relational algebra equalities, without need for a query preprocessing...
Optimizing relational algebra operations using discrimination-based joins and lazy products
Henglein, Fritz
We show how to implement in-memory execution of the core re- lational algebra operations of projection, selection and cross-product eciently, using discrimination-based joins and lazy products. We introduce the notion of (partitioning) discriminator, which par- titions a list of values according...... to a specied equivalence relation on keys the values are associated with. We show how discriminators can be dened generically, purely functionally, and eciently (worst-case linear time) on top of the array-based basic multiset discrimination algorithm of Cai and Paige (1995). Discriminators provide the basis...... the selection operation to recognize on the y whenever it is applied to a cross-product, in which case it can choose an ecient discrimination-based equijoin implementation. The techniques subsume most of the optimization techniques based on relational algebra equalities, without need for a query preprocessing...
Constraint algebra of general relativity from a formal continuum limit of canonical tensor model
Sasakura, Naoki [Yukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502 (Japan); Sato, Yuki [National Institute for Theoretical Physics, School of Physics andMandelstam Institute for Theoretical Physics, University of the Witwatersrand,Wits 2050 (South Africa)
2015-10-16
Canonical tensor model (CTM for short below) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. In the classical case, the constraints form a first-class constraint Poisson algebra with structures similar to that of the ADM formalism of general relativity, qualifying CTM as a possible discrete formalism for quantum gravity. In this paper, we show that, in a formal continuum limit, the constraint Poisson algebra of CTM with no cosmological constant exactly reproduces that of the ADM formalism. To this end, we obtain the expression of the metric tensor field in general relativity in terms of one of the dynamical rank-three tensors in CTM, and determine the correspondence between the constraints of CTM and those of the ADM formalism. On the other hand, the cosmological constant term of CTM seems to induce non-local dynamics, and is inconsistent with an assumption about locality of the continuum limit.
Bitopological spaces theory, relations with generalized algebraic structures and applications
Dvalishvili, Badri
2005-01-01
This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, a
On boundary fusion and functional relations in the Baxterized affine Hecke algebra
Babichenko, A., E-mail: babichen@weizmann.ac.il [Department of Mathematics, University of York, York YO10 5DD (United Kingdom); Regelskis, V., E-mail: v.regelskis@surrey.ac.uk [Department of Mathematics, University of Surrey, Guildford GU2 7XH (United Kingdom)
2014-04-15
We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations
Lytvynov, Eugene
2017-04-01
Let X=R^2 and let qin{C}, |q|=1. For x=(x^1,x^2) and y=(y^1,y^2) from X^2, we define a function Q(x,y) to be equal to q if x1 y1, and to Rq if x^1=y^1. Let partial_x^+, partial_x^- (xin X) be operator-valued distributions such that partial_x^+ is the adjoint of partial_x^-. We say that partial_x^+, partial_x^- satisfy the anyon commutation relations (ACR) if partial^+_xpartial_y^+=Q(y,x)partial_y^+partial_x^+ for xne y and partial^-_xpartial_y^+=δ(x-y)+Q(x,y)partial_y^+partial^-_x for (x,y)in X^2. In particular, for q = 1, the ACR become the canonical commutation relations and for q = -1, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of partial_x^+, partial_x^-. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator T on the real space L 2( X, dx), which commutes with any operator of multiplication by a bounded function ψ(x^1). In the case Rq 0), we discuss the corresponding particle density ρ(x):=partial_x^+partial_x^-. For Re qin(0,1], using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of ρ(x). This state is given by a negative binomial point process. A scaling limit of these states as κto∞ gives the gamma random measure, depending on parameter {R}q.
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations
Lytvynov, Eugene
2016-10-01
Let {X={R}^2} and let {qin{C}} , {|q|=1} . For {x=(x^1,x^2)} and {y=(y^1,y^2)} from {X^2} , we define a function {Q(x,y)} to be equal to q if {x1 y1} , and to {R}q if {x^1=y^1} . Let {partial_x^+} , {partial_x^-} ({xin X} ) be operator-valued distributions such that {partial_x^+} is the adjoint of {partial_x^-} . We say that {partial_x^+} , {partial_x^-} satisfy the anyon commutation relations (ACR) if {partial^+_xpartial_y^+=Q(y,x)partial_y^+partial_x^+} for {x≠ y} and {partial^-_xpartial_y^+=δ(x-y)+Q(x,y)partial_y^+partial^-_x} for {(x,y)in X^2} . In particular, for q = 1, the ACR become the canonical commutation relations and for q = -1, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of {partial_x^+} , {partial_x^-} . We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator T on the real space L 2(X, dx), which commutes with any operator of multiplication by a bounded function {ψ(x^1)} . In the case {R}q 0} ), we discuss the corresponding particle density {ρ(x):=partial_x^+partial_x^-} . For {Re qin(0,1]} , using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of {ρ(x)} . This state is given by a negative binomial point process. A scaling limit of these states as {κto∞} gives the gamma random measure, depending on parameter {R}q.
Nawata, Satoshi
2015-01-01
We provide various formulations of knot homology that are predicted by string dualities. In addition, we also explain the rich algebraic structure of knot homology which can be understood in terms of geometric representation theory in these formulations. These notes are based on lectures in the workshop "Physics and Mathematics of Link Homology" at Centre de Recherches Math\\'ematiques, Universit\\'e de Montr\\'eal.
Relations in Grassmann Algebra Corresponding to Three- and Four-Dimensional Pachner Moves
Igor G. Korepanov
2011-12-01
Full Text Available New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally – using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated boundary, and present example calculations confirming its nontriviality.
Optimizing relational algebra operations using generic equivalence discriminators and lazy products
Henglein, Fritz
2010-01-01
We show how to efficiently evaluate generic map-filter-product queries, generalizations of select-project-join (SPJ) queries in re- lational algebra, based on a combination of two novel techniques: generic discrimination-based joins and lazy (formal) products. Discrimination-based joins are based...... that discriminators can be constructed generically (by structural recursion on equivalence expressions), purely func- tionally, and efficiently (worst-case linear time). The array-based basic multiset discrimination algorithm of Cai and Paige (1995) provides a base discriminator that is both asymptotically and prac...... on relational algebra equalities, without need for a query preprocessing phase. They require no indexes and behave purely functionally. They can be considered a form of symbolic execution of set expressions that automate and encapsulate dynamic program transformation of such expressions and lead to asymptotic...
Matheus, Carlos
2009-01-01
We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus 3) and Forni-Matheus (in genus 4). We show that, in both cases, the action on the non trivial part of the homology is through finite groups. In particular, the action on some 4-dimensional invariant subspace of the homology leaves invariant a root system of $D_4$ type. This provides as a by-product a new proof of (slightly stronger versions of) the results of Forni and Matheus: the non trivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmuller disks of these two origamis are equal to zero.
2007-01-01
The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers were Michel Brion, Jens Carsten Jantzen, and Raphaël Rouquier. During the last years, the subject...... of algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a "general audience" of algebraic group......-theorists, and to stimulate contacts between participants. Each of the first four days was dedicated to one area of research that has recently seen decisive progress: \\begin{itemize} \\item structure and classification of wonderful varieties, \\item finite reductive groups and character sheaves, \\item quantum cohomology...
On the digital homology groups of digital images
Lee, Dae-Woong
2011-01-01
In this article we study the digital homology groups of digital images which are based on the singular homology groups of topological spaces in algebraic topology. Specifically, we define a digitally standard $n$-simplex, a digitally singular $n$-simplex, and the digital homology groups of digital images with $k$-adjacency relations. We then construct a covariant functor from a category of digital images and digitally continuous functions to the one of abelian groups and group homomorphisms, and investigate some fundamental and interesting properties of digital homology groups of digital images, such as the digital version of the dimension axiom which is one of the Eilenberg-Steenrod axioms.
PENG Jia-yin
2011-01-01
The notions of norm and distance in BCI-algebras are introduced,and some basic properties in normed BCI-algebras are given.It is obtained that the isomorphic(homomorphic)image and inverse image of a normed BCI-algebra are still normed BCI-algebras.The relations of normaled properties between BCI-algebra and Cartesian product of BCIalgebras are investigated.The limit notion of sequence of points in normed BCI-algebras is introduced,and its related properties are investigated.
SI JunRu
2009-01-01
The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called (s, t, d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An (s, t, d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an (s, t, d)-bi-Koszul algebra is discussed. Based on it, the notion of strongly (s, t, d)-bi-Koszul algebras is raised and their homological properties are further discussed.
无
2009-01-01
The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.
PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
Guenther, Uwe
2010-01-01
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.
Samuel, Pierre
2008-01-01
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal
Walker, Christine
2008-01-01
The purpose of this grounded theory study was to discover the factors that contribute to the success or failure of college algebra for students taking college algebra by distance education Internet, and then generate a theory of success or failure of the group of College Algebra Internet students at one Utah college. Qualitative data were collected and analyzed on students’ perceptions and perspectives of a College Algebra Internet course that they took during the spring or summer 2006 semest...
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
Vladimir S. Gerdjikov
2006-02-01
Full Text Available The construction of a family of real Hamiltonian forms (RHF for the special class of affine 1+1-dimensional Toda field theories (ATFT is reported. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. The construction method is illustrated on the explicit nontrivial example of RHF of ATFT related to the exceptional algebras E_6 and E_7. The involutions of the local integrals of motion are proved by means of the classical R-matrix approach.
Equivariant ordinary homology and cohomology
Costenoble, Steven R
2016-01-01
Filling a gap in the literature, this book takes the reader to the frontiers of equivariant topology, the study of objects with specified symmetries. The discussion is motivated by reference to a list of instructive “toy” examples and calculations in what is a relatively unexplored field. The authors also provide a reading path for the first-time reader less interested in working through sophisticated machinery but still desiring a rigorous understanding of the main concepts. The subject’s classical counterparts, ordinary homology and cohomology, dating back to the work of Henri Poincaré in topology, are calculational and theoretical tools which are important in many parts of mathematics and theoretical physics, particularly in the study of manifolds. Similarly powerful tools have been lacking, however, in the context of equivariant topology. Aimed at advanced graduate students and researchers in algebraic topology and related fields, the book assumes knowledge of basic algebraic topology and group act...
Representations of a Class of Associative Algebras Related to the Quantum Torus
叶从峰
2003-01-01
@@ 1 Introduction The motivation of this paper comes from the work of [5]. We know that vertex algebra theory is one of the important parts in the study of infinite dimensional Lie theory, while the lattice vertex algebras[2,7] form one of the most important and fundamental classes of vertex algebras.
Left Artinian Algebraic Algebras
S. Akbari; M. Arian-Nejad
2001-01-01
Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.
Algebraic partial Boolean algebras
Smith, Derek [Math Department, Lafayette College, Easton, PA 18042 (United States)
2003-04-04
Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A{sub 5} sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E{sub 8}.
Hepatoma-derived growth factor-related protein 2 promotes DNA repair by homologous recombination
Baude, Annika; Aaes, Tania Løve; Zhai, Beibei; Al-Nakouzi, Nader; Oo, Htoo Zarni; Daugaard, Mads; Rohde, Mikkel; Jäättelä, Marja
2016-01-01
We have recently identified lens epithelium-derived growth factor (LEDGF/p75, also known as PSIP1) as a component of the homologous recombination DNA repair machinery. Through its Pro-Trp-Trp-Pro (PWWP) domain, LEDGF/p75 binds to histone marks associated with active transcription and promotes DNA end resection by recruiting DNA endonuclease retinoblastoma-binding protein 8 (RBBP8/CtIP) to broken DNA ends. Here we show that the structurally related PWWP domain-containing protein, hepatoma-derived growth factor-related protein 2 (HDGFRP2), serves a similar function in homologous recombination repair. Its depletion compromises the survival of human U2OS osteosarcoma and HeLa cervix carcinoma cells and impairs the DNA damage-induced phosphorylation of replication protein A2 (RPA2) and the recruitment of DNA endonuclease RBBP8/CtIP to DNA double strand breaks. In contrast to LEDGF/p75, HDGFRP2 binds preferentially to histone marks characteristic for transcriptionally silent chromatin. Accordingly, HDGFRP2 is found in complex with the heterochromatin-binding chromobox homologue 1 (CBX1) and Pogo transposable element with ZNF domain (POGZ). Supporting the functionality of this complex, POGZ-depleted cells show a similar defect in DNA damage-induced RPA2 phosphorylation as HDGFRP2-depleted cells. These data suggest that HDGFRP2, possibly in complex with POGZ, recruits homologous recombination repair machinery to damaged silent genes or to active genes silenced upon DNA damage. PMID:26721387
Fundamentals of algebraic topology
Weintraub, Steven H
2014-01-01
This rapid and concise presentation of the essential ideas and results of algebraic topology follows the axiomatic foundations pioneered by Eilenberg and Steenrod. The approach of the book is pragmatic: while most proofs are given, those that are particularly long or technical are omitted, and results are stated in a form that emphasizes practical use over maximal generality. Moreover, to better reveal the logical structure of the subject, the separate roles of algebra and topology are illuminated. Assuming a background in point-set topology, Fundamentals of Algebraic Topology covers the canon of a first-year graduate course in algebraic topology: the fundamental group and covering spaces, homology and cohomology, CW complexes and manifolds, and a short introduction to homotopy theory. Readers wishing to deepen their knowledge of algebraic topology beyond the fundamentals are guided by a short but carefully annotated bibliography.
Homology of classical groups and K-theory
Mirzaii, B.
2004-01-01
The study of the homology groups of classical group over a ring R with coefficient A, where A is a commutative ring with trivial group action, seems important, notably because of their close relation to algebraic and Hermitian Ktheory and their appearance in the study of scissors congruence of polyh
Homology of classical groups and K-theory
Mirzaii, B.
2004-01-01
The study of the homology groups of classical group over a ring R with coefficient A, where A is a commutative ring with trivial group action, seems important, notably because of their close relation to algebraic and Hermitian Ktheory and their appearance in the study of scissors congruence of
Algebraic Topology, Rational Homotopy
1988-01-01
This proceedings volume centers on new developments in rational homotopy and on their influence on algebra and algebraic topology. Most of the papers are original research papers dealing with rational homotopy and tame homotopy, cyclic homology, Moore conjectures on the exponents of the homotopy groups of a finite CW-c-complex and homology of loop spaces. Of particular interest for specialists are papers on construction of the minimal model in tame theory and computation of the Lusternik-Schnirelmann category by means articles on Moore conjectures, on tame homotopy and on the properties of Poincaré series of loop spaces.
Homological Methods in Equations of Mathematical Physics
Krasil'shchik, Joseph; Verbovetsky, Alexander
1998-01-01
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invariants of differential equations are discussed. The lectures contain necessary introductory material on the geometric theory of differential equations and homological algebra.
Norén, Patrik
2013-01-01
Algebraic statistics brings together ideas from algebraic geometry, commutative algebra, and combinatorics to address problems in statistics and its applications. Computer algebra provides powerful tools for the study of algorithms and software. However, these tools are rarely prepared to address statistical challenges and therefore new algebraic results need often be developed. This way of interplay between algebra and statistics fertilizes both disciplines. Algebraic statistics is a relativ...
The Green formula and heredity of algebras
无
2005-01-01
[1]Green, J. A., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 1995, 120: 361-377.[2]Ringel, C. M., Green's theorem on Hall algebras, in Representations of Algebras and Related Topics, CMS Conference Proceedings 19, Providence, 1996, 185-245.[3]Xiao J., Drinfeld double and Ringel-Green theory of Hall Algebras, J. Algebra, 1997, 190: 100-144.[4]Sevenhant, B., Van den Bergh, M., A relation between a conjecture of Kac and the structure of the Hall algebra,J. Pure Appl. Algebra, 2001, 160: 319-332.[5]Deng B., Xiao, J., On double Ringel-Hall algebras, J. Algebra, 2002, 251: 110-149.
Anomaly in RTT relation for DIM algebra and network matrix models
Awata, H; Mironov, A; Morozov, A; Morozov, An; Ohkubo, Y; Zenkevich, Y
2016-01-01
We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to cancel the anomaly using a construction somewhat similar to the anomaly matching condition in gauge theory. We also describe the singular limit to the affine Yangian (4d Nekrasov functions), which breaks the spectral duality.
Differential Privacy for Relational Algebra: Improving the Sensitivity Bounds via Constraint Systems
Catuscia Palamidessi
2012-07-01
Full Text Available Differential privacy is a modern approach in privacy-preserving data analysis to control the amount of information that can be inferred about an individual by querying a database. The most common techniques are based on the introduction of probabilistic noise, often defined as a Laplacian parametric on the sensitivity of the query. In order to maximize the utility of the query, it is crucial to estimate the sensitivity as precisely as possible. In this paper we consider relational algebra, the classical language for queries in relational databases, and we propose a method for computing a bound on the sensitivity of queries in an intuitive and compositional way. We use constraint-based techniques to accumulate the information on the possible values for attributes provided by the various components of the query, thus making it possible to compute tight bounds on the sensitivity.
广义（p，λ）-Koszul 代数（模）%Some related studies on the generalized(p,λ)-Koszul algebras(modules)
王莉萍; 吴学超; 陈淼森
2014-01-01
It was mainly studied the(p,λ)-Koszul algebras with finite jump-degree. A notion of generalized (p,λ)-Koszul algebras(modules)was introduced,the basic homological properties and criterias of the gener-alized(p,λ)-Koszul algebras(modules)and some basic properties of the generalized(p,λ)-Koszul algebras (modules)were presented.%研究了有任意有限多个跳跃度的（p，λ）-Koszul 代数，引入了广义（p，λ）-Koszul 代数（模），给出了广义（p，λ）-Koszul 代数的基本同调性质、判定准则及广义（p，λ）-Koszul 模的一些基本性质。
Algebraic Geometry of Topological Spaces I
Cortiñas, Guillermo
2009-01-01
We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, seminormal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case when M=N^n gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case when M=Z^n. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C*-algebras, and for a homology the...
The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra
Vijay Kodiyalam; V S Sunder
2006-11-01
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.
2 Algebra and two-dimensional quasiexactly solvable Hamiltonian related to Poschl–Teller potential
H Panahi; H Rahmati
2014-07-01
In this article, we write the general form of the quasiexactly solvable Hamiltonian of 2 algebra via one special representation in the – two-dimensional space. Then, by choosing an appropriate set of coefficients and making a gauge rotation, we show that this Hamiltonian leads to the solvable Poschl–Teller model on an open infinite strip. Finally, we assign 2 hidden algebra to the Poschl–Teller potential and obtain its spectrum by using the representation space of 2 algebra.
DeWolf, M; Bassok, M; Holyoak, KJ
2015-01-01
© 2015 Elsevier Inc. To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuri...
Touzé, Antoine
2015-01-01
This book features a series of lectures that explores three different fields in which functor homology (short for homological algebra in functor categories) has recently played a significant role. For each of these applications, the functor viewpoint provides both essential insights and new methods for tackling difficult mathematical problems. In the lectures by Aurélien Djament, polynomial functors appear as coefficients in the homology of infinite families of classical groups, e.g. general linear groups or symplectic groups, and their stabilization. Djament’s theorem states that this stable homology can be computed using only the homology with trivial coefficients and the manageable functor homology. The series includes an intriguing development of Scorichenko’s unpublished results. The lectures by Wilberd van der Kallen lead to the solution of the general cohomological finite generation problem, extending Hilbert’s fourteenth problem and its solution to the context of cohomology. The focus here is o...
Homological Perturbation Theory and Mirror Symmetry
Jian ZHOU
2003-01-01
We explain how deformation theories of geometric objects such as complex structures,Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson al-gebras. We use homological perturbation theory to construct A∞ algebra structures on the cohomology,and their canonically defined deformations. Such constructions are used to formulate a version of A∞algebraic mirror symmetry.
Process algebra for Hybrid systems
Bergstra, J.A.; Middelburg, C.A.
2008-01-01
We propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from Baeten and Middelburg [Process Algebra with Timing, Springer, Chap. 4, 2002] and the process algebra with propositional signals from Baeten and Bergstra [Theoretical Computer
Process algebra for hybrid systems
Bergstra, J.A.; Middelburg, C.A.
2005-01-01
We propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from Baeten and Middelburg (Process Algebra with Timing, Springer,Berlin, 2002, Chapter 4), and the process algebra with propositional signals from Baeten and Bergstra(Theoret. Com
Cohen, A.M.; Liu, S.
2015-01-01
For each n ≥ 1, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular struc
Commutative algebra with a view toward algebraic geometry
Eisenbud, David
1995-01-01
Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algeb...
The newest release of the Ortocartan set of programs for algebraic calculations in relativity
Krasinski, A
2001-01-01
The program Ortocartan for algebraic calculations in relativity has just been implemented in the Codemist Standard Lisp and can now be used under the Windows 98 and Linux operating systems. The paper describes the new facilities and subprograms that have been implemented since the previous release in 1992. These are: the possibility to write the output as Latex input code and as Ortocartan's input code, the calculation of the Ellis evolution equations for the kinematic tensors of flow, the calculation of the curvature tensors from given (torsion-free) connection coefficients in a manifold of arbitrary dimension, the calculation of the lagrangian from a given metric by the Landau-Lifshitz method, the calculation of the Euler-Lagrange equations from a given lagrangian (only for sets of ordinary differential equations) and the calculation of first integrals of sets of ordinary differential equations of second order (the first integrals are assumed to be polynomials of second degree in the first derivatives of th...
Yantz. Jennifer
2013-01-01
The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting students' postsecondary success as STEM majors. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. In the present study, the connections…
Lefschetz, Solomon
2005-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
On String Topology Operations and Algebraic Structures on Hochschild Complexes
Rivera, Manuel
The field of string topology is concerned with the algebraic structure of spaces of paths and loops on a manifold. It was born with Chas and Sullivan's observation of the fact that the intersection product on the homology of a smooth manifold M can be combined with the concatenation product on the homology of the based loop space on M to obtain a new product on the homology of LM, the space of free loops on M. Since then, a vast family of operations on the homology of LM have been discovered. In this thesis we focus our attention on a non trivial coproduct of degree 1--dim(M) on the homology of LM modulo constant loops. This coproduct was described by Sullivan on chains on general position and by Goresky and Hingston in a Morse theory context. We give a Thom-Pontryagin type description for the coproduct. Using this description we show that the resulting coalgebra is an invariant on the oriented homotopy type of the underlying manifold. The coproduct together with the loop product induce an involutive Lie bialgebra structure on the S 1-equivariant homology of LM modulo constant loops. It follows from our argument that this structure is an oriented homotopy invariant as well. There is also an algebraic theory of string topology which is concerned with the structure of Hochschild complexes of DG Frobenius algebras and their homotopy versions. We make several observations about the algebraic theory around products, coproducts and their compatibilities. In particular, we describe a BV-coalgebra structure on the coHochschild complex of a DG cocommutative Frobenius coalgebra. Some conjectures and partial results regarding homotopy versions of this structure are discussed. Finally, we explain how Poincare duality may be incorporated into Chen's theory of iterated integrals to relate the geometrically constructed string topology operations to algebraic structures on Hochschild complexes.
Nasser, Ramzi; Carifio, James
The purpose of this study was to find out whether students perform differently on algebra word problems that have certain key context features and entail proportional reasoning, relative to their level of logical reasoning and their degree of field dependence/independence. Field-independent students tend to restructure and break stimuli into parts…
Maria Joita
2007-12-01
Full Text Available In this paper we characterize the order relation on the set of all nondegenerate completely n-positive linear maps between C*-algebras in terms of a self-dual Hilbert module induced by each completely n-positive linear map.
Lectures on algebraic statistics
Drton, Mathias; Sullivant, Seth
2009-01-01
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
Beilinson, Alexander
2004-01-01
Chiral algebras form the primary algebraic structure of modern conformal field theory. Each chiral algebra lives on an algebraic curve, and in the special case where this curve is the affine line, chiral algebras invariant under translations are the same as well-known and widely used vertex algebras. The exposition of this book covers the following topics: the "classical" counterpart of the theory, which is an algebraic theory of non-linear differential equations and their symmetries; the local aspects of the theory of chiral algebras, including the study of some basic examples, such as the ch
On dibaric and evolution algebras
Ladra, M; Rozikov, U A
2011-01-01
We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps $f, g$ of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. Using the pair $(f,g)$ we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a ...
Purely infinite simple reduced C*-algebras of one-relator separated graphs
Ara, Pere
2012-01-01
Given a separated graph $(E,C)$, there are two different C*-algebras associated to it, the full graph C*-algebra $C^*(E,C)$, and the reduced one $C^*_{\\text{red}} (E,C)$. For a large class of separated graphs $(E,C)$, we prove that $C^*_{\\text{red}} (E,C)$ either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C*-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.
Cylindric-like algebras and algebraic logic
Ferenczi, Miklós; Németi, István
2013-01-01
Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.
From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras
Leon, Zen-chen
2016-01-01
Starting from our revisit to Special Relativity here, we provide a reliable characterization of the entire 4-dimensional fundamental structures in our reality where the frame of discrete tangent space of $F^{1,3}$ is quantized to massless, zero-momentum particles distributing on a 4-dimensional regular base $\\{\\mathbb{N}\\cdot c_\\alpha\\}$ with metric $B(c_\\alpha,c_\\beta)=\\delta_{\\alpha\\beta}=\\text{diag}(+,+,+,+)$, determining the constant $c$ locally, as well as instant characterizations on all particles moving along the proper time of $\\tau\\in\\mathbb{N}$. Together with $\\phi_\\text{IV}$ on 1-dimensional space $\\{\\mathbb{N}\\cdot c_4\\}$ of $B(c_\\alpha,c_4)=0$, the quantized particles of tangent frame are split anti-symmetrically from roots $\\gamma_{\\alpha4}$ in a rank-4 Lie algebra with exactly $c_\\alpha$ the generators of its Cartan subalgebra. The same as the combined frame-Higgs valued in $\\gamma_{\\alpha4}$, every massive particles are related to some split frame and Higgs to obtain its unique 3-velocity, 4-v...
Kurosh, A G; Stark, M; Ulam, S
1965-01-01
Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. The book examines the parallelism between the theory of groups and the theory of rings; such examinations show the
Hecke algebras with unequal parameters
Lusztig, G
2003-01-01
Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over p-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives rese...
Algebraic statistics computational commutative algebra in statistics
Pistone, Giovanni; Wynn, Henry P
2000-01-01
Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.
Prediction of Algebraic Instabilities
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
A Specialization of Prinjective Ringel-Hall Algebra and the associated Lie algebra
Justyna KOSAKOWSKA
2008-01-01
In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type,by generators and relations.This gives us a generalisation of Serre relations for semisimple Lie algebras.Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.
The Planar Algebra Associated to a Kac Algebra
Vijay Kodiyalam; Zeph Landau; V S Sunder
2003-02-01
We obtain (two equivalent) presentations – in terms of generators and relations-of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the `rotation on 2-boxes'.
On Yang's Noncommutative Space Time Algebra, Holography, Area Quantization and C-space Relativity
Castro, C
2004-01-01
An isomorphism between Yang's Noncommutative space-time algebra (involving two length scales) and the holographic-area-coordinates algebra of C-spaces (Clifford spaces) is constructed via an AdS_5 space-time which is instrumental in explaining the origins of an extra (infrared) scale R in conjunction to the (ultraviolet) Planck scale lambda characteristic of C-spaces. Yang's space-time algebra allowed Tanaka to explain the origins behind the discrete nature of the spectrum for the spatial coordinates and spatial momenta which yields a minimum length-scale lambda (ultraviolet cutoff) and a minimum momentum p = (\\hbar / R) (maximal length R, infrared cutoff). The double-scaling limit of Yang's algebra : lambda goes to 0, and R goes to infinity, in conjunction with the large n infinity limit, leads naturally to the area quantization condition : lambda R = L^2 = n lambda^2 (in Planck area units) given in terms of the discrete angular-momentum eigenvalues n . The generalized Weyl-Heisenberg algebra in C-spaces is ...
Sutures and contact homology I
Colin, Vincent; Honda, Ko; Hutchings, Michael
2010-01-01
We define a relative version of contact homology for contact manifolds with convex boundary, and prove basic properties of this relative contact homology. Similar considerations also hold for embedded contact homology.
Linear Mappings of Quaternion Algebra
Kleyn, Aleks
2011-01-01
In the paper I considered linear and antilinear automorphisms of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of linear and antilinear automorphisms.
Frank Roumen
2017-01-01
Full Text Available We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.
Topological ∗-algebras with *-enveloping Algebras II
S J Bhatt
2001-02-01
Universal *-algebras *() exist for certain topological ∗-algebras called algebras with a *-enveloping algebra. A Frechet ∗-algebra has a *-enveloping algebra if and only if every operator representation of maps into bounded operators. This is proved by showing that every unbounded operator representation , continuous in the uniform topology, of a topological ∗-algebra , which is an inverse limit of Banach ∗-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-* algebra () of . Given a *-dynamical system (, , ), any topological ∗-algebra containing (, ) as a dense ∗-subalgebra and contained in the crossed product *-algebra *(, , ) satisfies ()=*(, , ). If $G = \\mathbb{R}$, if is an -invariant dense Frechet ∗-subalgebra of such that () = , and if the action on is -tempered, smooth and by continuous ∗-automorphisms: then the smooth Schwartz crossed product $S(\\mathbb{R}, B, )$ satisfies $E(S(\\mathbb{R}, B, )) = C^*(\\mathbb{R}, A, )$. When is a Lie group, the ∞-elements ∞(), the analytic elements () as well as the entire analytic elements () carry natural topologies making them algebras with a *-enveloping algebra. Given a non-unital *-algebra , an inductive system of ideals is constructed satisfying $A = C^*-\\mathrm{ind} \\lim I_$; and the locally convex inductive limit $\\mathrm{ind}\\lim I_$ is an -convex algebra with the *-enveloping algebra and containing the Pedersen ideal of . Given generators with weakly Banach admissible relations , we construct universal topological ∗-algebra (, ) and show that it has a *-enveloping algebra if and only if (, ) is *-admissible.
夏铁成; 李季
2008-01-01
Based on the generalization of Lie algebra An-1,two types of new Lie algebras were worked out and the integrability of the related hierarchies of evolution equations were proved in the sense of Liouville.
Point value characterizations and related results in the full Colombeau algebras G^e and G^d
Nigsch, Eduard
2011-01-01
We present a point value characterization for elements of the elementary full Colombeau algebra G^e and the diffeomorphism invariant full Colombeau algebra G^d. Moreover, several results from the special algebra G^s about generalized numbers and invertibility are extended to the elementary full algebra.
The Boolean algebra and central Galois algebras
George Szeto
2001-01-01
Full Text Available Let B be a Galois algebra with Galois group G, Jg={b∈B∣bx=g(xb for all x∈B} for g∈G, and BJg=Beg for a central idempotent eg. Then a relation is given between the set of elements in the Boolean algebra (Ba,≤ generated by {0,eg∣g∈G} and a set of subgroups of G, and a central Galois algebra Be with a Galois subgroup of G is characterized for an e∈Ba.
W1+∞ algebra as a symmetry behind the Alday-Gaiotto-Tachikawa relation
Kanno, Shoichi; Matsuo, Yutaka; Shiba, Shotaro
2011-07-01
We give some evidences which imply that W1+∞ algebra describes the symmetry behind the Alday-Gaiotto-Tachikawa(-Wyllard) conjecture: a correspondence between the partition function of N=2 supersymmetric quiver gauge theories and the correlators of Liouville (Toda) field theory.
Fong, Anthony; Jaquet, Karina; Finkelstein, Neal
2016-01-01
The information provided in this report shows how students perform when they repeat algebra I and how the level of improvement varies depending on initial course performance and the academic measure (course grades or CST scores). This information can help inform decisions and policies regarding whether and under what circumstances students should…
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
Warner, Seth
1990-01-01
Standard text provides an exceptionally comprehensive treatment of every aspect of modern algebra. Explores algebraic structures, rings and fields, vector spaces, polynomials, linear operators, much more. Over 1,300 exercises. 1965 edition.
Goodstein, R L
2007-01-01
This elementary treatment by a distinguished mathematician employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. Numerous examples appear throughout the text, plus full solutions.
Approximate Preservers on Banach Algebras and C*-Algebras
M. Burgos
2013-01-01
Full Text Available The aim of the present paper is to give approximate versions of Hua’s theorem and other related results for Banach algebras and C*-algebras. We also study linear maps approximately preserving the conorm between unital C*-algebras.
Kollár, János
1997-01-01
This volume contains the lectures presented at the third Regional Geometry Institute at Park City in 1993. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second- and third-year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics. Topics covered include the theory of surfaces from the viewpoint of recent higher-dimensional developments, providing an excellent introduction to more advanced topics such as the minimal model program. Also included is an introduction to Hodge theory and intersection homology based on the simple topological ideas of Lefschetz and an overview of the recent interactions between algebraic geometry and theoretical physics, which involve mirror symmetry and string theory.
Process Algebra and Markov Chains
Brinksma, Hendrik; Hermanns, H.; Brinksma, Hendrik; Hermanns, H.; Katoen, Joost P.
This paper surveys and relates the basic concepts of process algebra and the modelling of continuous time Markov chains. It provides basic introductions to both fields, where we also study the Markov chains from an algebraic perspective, viz. that of Markov chain algebra. We then proceed to study
Using homology relations within a database markedly boosts protein sequence similarity search.
Tong, Jing; Sadreyev, Ruslan I; Pei, Jimin; Kinch, Lisa N; Grishin, Nick V
2015-06-02
Inference of homology from protein sequences provides an essential tool for analyzing protein structure, function, and evolution. Current sequence-based homology search methods are still unable to detect many similarities evident from protein spatial structures. In computer science a search engine can be improved by considering networks of known relationships within the search database. Here, we apply this idea to protein-sequence-based homology search and show that it dramatically enhances the search accuracy. Our new method, COMPADRE (COmparison of Multiple Protein sequence Alignments using Database RElationships) assesses the relationship between the query sequence and a hit in the database by considering the similarity between the query and hit's known homologs. This approach increases detection quality, boosting the precision rate from 18% to 83% at half-coverage of all database homologs. The increased precision rate allows detection of a large fraction of protein structural relationships, thus providing structure and function predictions for previously uncharacterized proteins. Our results suggest that this general approach is applicable to a wide variety of methods for detection of biological similarities. The web server is available at prodata.swmed.edu/compadre.
Lectures on Algebraic Geometry I
Harder, Gunter
2012-01-01
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own. In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern metho
曹晨忠
1995-01-01
The ionization potential of organic homologs can be expressed as I_p=(∑X_i)/(a+bn).Here,X_i is the electronegativity(the average energy of valence electrons in a ground-state free atom)of the ith atomin an organic homologous molecule;n,the number of repeating units in the molecule;and(a+bn),the electronmoving range in the molecule orbit.The results of linear regression analysis show that the correlationcoefficients r are all "excellent"(r>0.990)for the 146 sets of photo electron spectroscopy data of 42 organichomologous series.
The first cohomology group of trivial extensions of special biserial algebras
无
2004-01-01
Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A × DA in terms of the combinatorics of the quiver and relations is determined.
Kojima, T
2007-01-01
We explicitly construct two classes of infinitly many commutative operators in terms of the deformed W-algebra $W_{qt}(sl_N^)$, and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal one, since they can be regarded as elliptic deformation of local and nonlocal integrals of motion for the $W_N$ algebra.
Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.
2016-08-01
We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.
J. W. Kitchen
1994-01-01
Full Text Available We study bundles of Banach algebras π:A→X, where each fiber Ax=π−1({x} is a Banach algebra and X is a compact Hausdorff space. In the case where all fibers are commutative, we investigate how the Gelfand representation of the section space algebra Γ(π relates to the Gelfand representation of the fibers. In the general case, we investigate how adjoining an identity to the bundle π:A→X relates to the standard adjunction of identities to the fibers.
Located Actions in Process Algebra with Timing
Bergstra, J.A.; Middelburg, C.A.
2004-01-01
We propose a process algebra obtained by adapting the process algebra with continuous relative timing from Baeten and Middelburg [Process Algebra with Timing, Springer, 2002, Chap. 4] to spatially located actions. This process algebra makes it possible to deal with the behaviour of systems with a kn
The 3-kind ideals relation and properties of BR0-algebras%BR0-代数三种理想的关系和性质
王娜; 吴洪博
2013-01-01
为了进一步研究BR0-代数的结构.首先在BR0-代数中给出了蕴涵理想,⊙-理想和V-理想的定义；其次,讨论了BR0-代数中这三种理想的关系；最后,研究了BR0-代数中理想的一些性质,并证明了极大理想存在定理.%In order to study the structure of BR0-algebras in depth,firstly,the definitions of implicative ideal,⊙-ideal and V-ideal are given on the BR0-algebras; Secondly,the relation of 3-kind ideals is discussed in the BR0-algebras; Finally,some properties of ideal are researched in the BR0-algebras and maximal ideal existence theorem is proved.
ALGEBRAIC OPERATION OF SPECIAL MATRICES RELATED TO METHOD OF LEAST SQUARES
XuFuhua
2003-01-01
The follwing situation in using the method of least squares to solve problems often occurs.After m experiments completed and a solution of least squares obtained,the(m+1)-th experiment is made further in order to improve the results.A method of algebraic operation of special matrices involed in the problem is given is this paper for obtaining a new solution for the m+1 experiments based upon the old solution for the primary m experiments .This method is valid for more general matrices.
Planar Para Algebras, Reflection Positivity
Jaffe, Arthur; Liu, Zhengwei
2017-05-01
We define a planar para algebra, which arises naturally from combining planar algebras with the idea of ZN para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects that are invariant under para isotopy. For each ZN, we construct a family of subfactor planar para algebras that play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras, which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), which we use on the matrix algebra generated by the Pauli matrices. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivity by relating the two reflections through the string Fourier transform.
On ultraproducts of operator algebras
LI; Weihua
2005-01-01
Some basic questions on ultraproducts of C*-algebras and yon Neumann algebras, including the relation to K-theory of C*-algebras are considered. More specifically,we prove that under certain conditions, the K-groups of ultraproduct of C*-algebras are isomorphic to the ultraproduct of respective K-groups of C*-algebras. We also show that the ultraproducts of factors of type Ⅱ1 are prime, i.e. not isomorphic to any non-trivial tensor product.
Issa, A Nourou
2010-01-01
Non-Hom-associative algebras and Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms.
Santhi Lasya
2011-09-01
Full Text Available Even in this World Wide Web era where there is unrestricted access to a lot of articles and books at a mouses click, the role of an organized library is immense. It is vital to have effective software to manage various functions in a library and the fundamental for effective software is the underlying database access and the queries used. And hence library databases become our use-case for this study. This paper starts off with considering a basic ER model of a typical library relational database. We would also list all the basic use-cases in a library management system. The next part of the paper deals with the sql queries used for performing certain functions in a library database management system. Along with the queries, we would generate reports for some of the use cases. The final section of the paper forms the crux of this library database study, wherein we would dwell on the concepts of query processing and query optimization in the relational database domain. We would analyze the above mentioned queries, by translating the query into a relational algebra expression and generating a query tree for the same. By converting algebra, we look at optimizing the query, and by generating a query tree, we would come up a cheapest cost plan.
Dargys, A
2016-01-01
To have a closed system, the Maxwell equations should be supplemented by constitutive relations which connect the primary electromagnetic fields $(\\bE,\\bB)$ with the secondary ones $(\\bD,\\bH)$ induced in a medium. Recently [Opt. Commun. \\textbf{354}, 259 (2015)] the allowed shapes of the constitutive relations that follow from the relativistic Maxwell equations formulated in terms of geometric algebra were constructed by author. In this paper the obtained general relativistic relations between $(\\bD,\\bH)$ and $(\\bE,\\bB)$ fields are transformed to four $6\\times 6$ matrices that are universal in constructing various combinations of constitutive relations in terms of more popular Gibbs-Heaviside vectorial calculus frequently used to investigate the electromagnetic wave propagation in anisotropic, birefringent, bianisotropic, chiral etc media.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
Quantitative Algebraic Reasoning
Mardare, Radu Iulian; Panangaden, Prakash; Plotkin, Gordon
2016-01-01
We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We deﬁne an equality relation indexed by rationals: a =ε b which we think of as saying that “a is approximately equal to b up to an error of ε”. We have 4 interesting examples where we have a quantitative...... equational theory whose free algebras correspond to well known structures. In each case we have ﬁnitary and continuous versions. The four cases are: Hausdorﬀ metrics from quantitive semilattices; pWasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed...
Hopf algebras in noncommutative geometry
Varilly, J C
2001-01-01
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups.
Twin TQFTs and Frobenius Algebras
Carmen Caprau
2013-01-01
Full Text Available We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C,W,z,z∗ consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism z:C→W with dual z∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Representations of fundamental groups of algebraic varieties
Zuo, Kang
1999-01-01
Using harmonic maps, non-linear PDE and techniques from algebraic geometry this book enables the reader to study the relation between fundamental groups and algebraic geometry invariants of algebraic varieties. The reader should have a basic knowledge of algebraic geometry and non-linear analysis. This book can form the basis for graduate level seminars in the area of topology of algebraic varieties. It also contains present new techniques for researchers working in this area.
Asymptotic aspect of derivations in Banach algebras.
Roh, Jaiok; Chang, Ick-Soon
2017-01-01
We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.
Sinan AYDIN
2009-04-01
Full Text Available Linear algebra is a basic course followed in mathematics, science, and engineering university departments.Generally, this course is taken in either the first or second year but there have been difficulties in teachingand learning. This type of active algebra has resulted in an increase in research by mathematics educationresearchers. But there is insufficient information on this subject in Turkish and therefore it has not beengiven any educational status. This paper aims to give a general overview of this subject in teaching andlearning. These education studies can be considered quadruple: a the history of linear algebra, b formalismobstacles of linear algebra and cognitive flexibility to improve teaching and learning, c the relation betweenlinear algebra and geometry, d using technology in the teaching and learning linear algebra.Mathematicseducation researchers cannot provide an absolute solution to overcome the teaching and learning difficultiesof linear algebra. Epistemological analyses and experimental teaching have shown the learning difficulties.Given these results, further advice and assistance can be offered locally.
A Gregory Bruce
Full Text Available Two gammaherpesviruses, Epstein-Barr virus (EBV (Lymphocryptovirus genus and Kaposi's sarcoma-associated herpesvirus (KSHV (Rhadinovirus genus have been implicated in the etiology of AIDS-associated lymphomas. Homologs of these viruses have been identified in macaques and other non-human primates. In order to assess the association of these viruses with non-human primate disease, archived lymphoma samples were screened for the presence of macaque lymphocryptovirus (LCV homologs of EBV, and macaque rhadinoviruses belonging to the RV1 lineage of KSHV homologs or the more distant RV2 lineage of Old World primate rhadinoviruses. Viral loads were determined by QPCR and infected cells were identified by immunolabeling for different viral proteins. The lymphomas segregated into three groups. The first group (n = 6 was associated with SIV/SHIV infections, contained high levels of LCV (1-25 genomes/cell and expressed the B-cell antigens CD20 or BLA.36. A strong EBNA-2 signal was detected in the nuclei of the neoplastic cells in one of the LCV-high lymphomas, indicative of a type III latency stage. None of the lymphomas in this group stained for the LCV viral capsid antigen (VCA lytic marker. The second group (n = 5 was associated with D-type simian retrovirus-2 (SRV-2 infections, contained high levels of RV2 rhadinovirus (9-790 genomes/cell and expressed the CD3 T-cell marker. The third group (n = 3 was associated with SIV/SHIV infections, contained high levels of RV2 rhadinovirus (2-260 genomes/cell and was negative for both CD20 and CD3. In both the CD3-positive and CD3/CD20-negative lymphomas, the neoplastic cells stained strongly for markers of RV2 lytic replication. None of the lymphomas had detectable levels of retroperitoneal fibromatosis herpesvirus (RFHV, the macaque RV1 homolog of KSHV. Our data suggest etiological roles for both lymphocryptoviruses and RV2 rhadinoviruses in the development of simian AIDS-associated lymphomas and indicate that
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Algebra-Geometry of Piecewise Algebraic Varieties
Chun Gang ZHU; Ren Hong WANG
2012-01-01
Algebraic variety is the most important subject in classical algebraic geometry.As the zero set of multivariate splines,the piecewise algebraic variety is a kind generalization of the classical algebraic variety.This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.
A Web-Based Authoring Tool for Algebra-Related Intelligent Tutoring Systems
Maria Virvou
2000-01-01
Full Text Available This paper describes the development of a web-based authoring tool for Intelligent Tutoring Systems. The tool aims to be useful to teachers and students of domains that make use of algebraic equations. The initial input to the tool is a "description" of a specific domain given by a human teacher. In return the tool provides assistance at the construction of exercises by the human teacher and then monitors the students while they are solving the exercises and provides appropriate feedback. The tool incorporates intelligence in its diagnostic component, which performs error diagnosis to students errors. It also handles the teaching material in a flexible and individualised way.
Constructions of Lie algebras and their modules
Seligman, George B
1988-01-01
This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. T...
Factorization algebras in quantum field theory
Costello, Kevin
2017-01-01
Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics.
Homotopy Theory of C*-Algebras
Ostvaer, Paul Arne
2010-01-01
Homotopy theory and C* algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C*-algebras. One basic idea of the setup is to merge C*-algebras and spaces studied in algebraic topology into one category comprising C*-spaces. These objects are suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures. With the foundations in place one is led to natural definitions of invariants for C*-spaces such as homology and cohomology theories, K-theory and zeta-functions. The text is largely self-contained. It
Remarks on Cyclotomic and Degenerate Cyclotomic BMW Algebras
Goodman, Frederick M
2010-01-01
We relate the structure of cyclotomic and degenerate cyclotomic BMW algebras, for arbitrary parameter values, to that for admissible parameter values. In particular, we show that these algebras are cellular. We characterize those parameter sets for affine BMW algebras over an algebraically closed field that permit the algebras to have non--trivial cyclotomic quotients.
Chisolm, Eric
2012-01-01
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric al...
2013-01-01
SUMMARY Homologous recombination is a universal process, conserved from bacteriophage to human, which is important for the repair of double-strand DNA breaks. Recombination in mitochondrial DNA (mtDNA) was documented more than 4 decades ago, but the underlying molecular mechanism has remained elusive. Recent studies have revealed the presence of a Rad52-type recombination system of bacteriophage origin in mitochondria, which operates by a single-strand annealing mechanism independent of the canonical RecA/Rad51-type recombinases. Increasing evidence supports the notion that, like in bacteriophages, mtDNA inheritance is a coordinated interplay between recombination, repair, and replication. These findings could have profound implications for understanding the mechanism of mtDNA inheritance and the generation of mtDNA deletions in aging cells. PMID:24006472
Kolman, Bernard
1985-01-01
College Algebra, Second Edition is a comprehensive presentation of the fundamental concepts and techniques of algebra. The book incorporates some improvements from the previous edition to provide a better learning experience. It provides sufficient materials for use in the study of college algebra. It contains chapters that are devoted to various mathematical concepts, such as the real number system, the theory of polynomial equations, exponential and logarithmic functions, and the geometric definition of each conic section. Progress checks, warnings, and features are inserted. Every chapter c
Garrett, Paul B
2007-01-01
Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal
An algebraic proof for the identities for the degree of syzygies in numerical semigroup
Neeraj Kumar
2012-05-01
Full Text Available In the article [4] two new identities for the degree of syzygies are given. We present an algebraic proof of them, using only basic homological algebra tools. We also extend these results.
Trading GRH for algebra: algorithms for factoring polynomials and related structures
Ivanyos, Gábor; Rónyai, Lajos; Saxena, Nitin
2008-01-01
In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n^{\\log n},\\log |k|) time "either" a nontrivial factor of f(x) "or" a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRH-free results, most striking of which are: (1) Given a noncommutative algebra over a finite field, we can find a zero divisor in deterministic subexponential time. (2) Given a positive integer r>4 such that either 4|r or r has two distinct prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of the r-th cyclotomic polynomial over a finite field. In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois...
On the Product and Factorization of Lattice Implication Algebras
秦克云; 宋振明; 等
1993-01-01
In this paper,the concepts of product and factorization of lattice implication algebra are proposed,the relation between lattice implication product algebra and its factors and some properties of lattice implication product algebras are discussed.
Congruence Kernels of Orthoimplication Algebras
I. Chajda
2007-10-01
Full Text Available Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices. Finally, it is proved that the lattice of all congruence kernels of an orthoimplication algebra is relatively pseudocomplemented and a simple description of the relative pseudocomplement is given.
McKeague, Charles P
1981-01-01
Elementary Algebra 2e, Second Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first tackles the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the substitution method, solving linear systems by graphing, solutions to linear equations in two variables, multiplication property of equality, word problems, addition property of equality, and subtraction, addition, multiplication, and division of real numbers. The manuscript then examines exponents and polynomials, factoring, and rational e
McKeague, Charles P
1986-01-01
Elementary Algebra, Third Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first ponders on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the elimination method, solving linear systems by graphing, word problems, addition property of equality, solving linear equations, linear inequalities, addition and subtraction of real numbers, and properties of real numbers. The text then takes a look at exponents and polynomials, factoring, and rational expressions. Topics include reducing ra
Reversed-phase thin-layer chromatography of homologs of Antimycin-A and related derivatives
Abidi, Sharon L.
1989-01-01
Using a reversed-phase high-performance liquid chromatographic (HPLC) technique, a mixture of antimycins A was separated into eight hitherto unreported subcomponents, Ala, Alb, A2a, A2b, A3a, A3b, A4a, and A4b. Although a base-line resolution of the known four major antimycins Al, A2, A3, and A4 was readily achieved with mobile phases containing acetate buffers, the separation of the new antibiotic subcomponents was highly sensitive to variation in mobile phase conditions. The type and composition of organic modifiers, the nature of buffer salts, and the concentration of added electrolytes had profound effects on capacity factors, separation factors, and peak resolution values. Of the numerous chromatographic systems examined, a mobile phase consisting of methanol-water (70:30) and 0.005 M tetrabutylammonium phosphate at pH 3.0 yielded the most satisfactory results for the separation of the subcomponents. Reversed-phase gradient HPLC separation of the dansylated or methylated antibiotic compounds produced superior chromatographic characteristics and the presence of added electrolytes was not a critical factor for achieving separation. Differences in the chromatographic outcome between homologous and structural isomers were interpretated based on a differential solvophobic interaction rationale. Preparative reversed-phase HPLC under optimal conditions enabled isolation of pure samples of the methylated antimycin subcomponents for use in structural studies.
张万里; 林安
2014-01-01
In this paper, the Assumption B1 and B2 are proposed basing on the idea of Flores-Baz´an et al. The relative algebraic interior of the sum for two sets is equal to the sum of the relative algebraic interior for these sets, the sum of the algebraic closure of a set and the relative algebraic interior of a set is equal to the sum of the relative algebraic interior for the two sets, the relative topological interior of the sum for two sets is equal to the sum of the relative topological interior for these sets, the sum of topological closure of set and the relative topological interior of set is equal to the sum of the relative topological interior for the two sets are proved. Furthermore, the equivalent relations between equality of the algebraic closure and the equality of algebraic interior are established. We also obtain the similar equivalent relations for the topological closure and the relative topological interior.%基于 Flores-Baz´an 等人的思想,提出了假设 B1和假设 B2,证明了集合和的相对代数内部等于相对代数内部的和；集合代数闭包与相对代数内部的和等于和的相对代数内部；集合和的相对拓扑内部等于相对拓扑内部的和；集合拓扑闭包与相对拓扑内部的和等于和的相对拓扑内部,建立了集合代数闭包相等与代数内部相等,拓扑闭包相等与拓扑内部相等之间的一些等价关系。
Høyrup, Jens
From the early fourteenth century onward, some Italian Abbacus manuscripts begin to use particular abbreviations for algebraic operations and objects and, to be distinguished from that, examples of symbolic operation. The algebraic abbreviations and symbolic operations we find in German Rechenmei...
李宝; 周林芳; 肖国镇
1999-01-01
For a class of algebraic-geometric codes, a type of recurring relation is introduced on the syndrome sequence of an error vector. Then, a new majority yoting scheme is developed. By applying the generalized Berlekamp-Massey algorithm, and incorporating the majority voting scheme, an efficient decoding algorithm up to half the Feng-Rao bound is developed for a class of algebraic-geometric codes, the complexity of which is O （ γo1n2）, where n is the code length, and γ is the genus of curve. On different algebraic curves, the complexity of the algorithm can be lowered by choosing base functions suitably. For example, on Hermitian curves the complexity is O(n7/3.
Algebraic Quantum Mechanics and Pregeometry
Hiley, D J B P G D B J
2006-01-01
We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford Algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra in a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra.
Calculations on Lie Algebra of the Group of Affine Symplectomorphisms
Zuhier Altawallbeh
2017-01-01
Full Text Available We find the image of the affine symplectic Lie algebra gn from the Leibniz homology HL⁎(gn to the Lie algebra homology H⁎Lie(gn. The result shows that the image is the exterior algebra ∧⁎(wn generated by the forms wn=∑i=1n(∂/∂xi∧∂/∂yi. Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology HH⁎-1(U(gn is isomorphic to H⁎-1Lie(gn,U(gnad. Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.
Beck, Carole; Boehler, Christian; Guirouilh Barbat, Josée; Bonnet, Marie-Elise; Illuzzi, Giuditta; Ronde, Philippe; Gauthier, Laurent R; Magroun, Najat; Rajendran, Anbazhagan; Lopez, Bernard S; Scully, Ralph; Boussin, François D; Schreiber, Valérie; Dantzer, Françoise
2014-05-01
The repair of toxic double-strand breaks (DSB) is critical for the maintenance of genome integrity. The major mechanisms that cope with DSB are: homologous recombination (HR) and classical or alternative nonhomologous end joining (C-NHEJ versus A-EJ). Because these pathways compete for the repair of DSB, the choice of the appropriate repair pathway is pivotal. Among the mechanisms that influence this choice, deoxyribonucleic acid (DNA) end resection plays a critical role by driving cells to HR, while accurate C-NHEJ is suppressed. Furthermore, end resection promotes error-prone A-EJ. Increasing evidence define Poly(ADP-ribose) polymerase 3 (PARP3, also known as ARTD3) as an important player in cellular response to DSB. In this work, we reveal a specific feature of PARP3 that together with Ku80 limits DNA end resection and thereby helps in making the choice between HR and NHEJ pathways. PARP3 interacts with and PARylates Ku70/Ku80. The depletion of PARP3 impairs the recruitment of YFP-Ku80 to laser-induced DNA damage sites and induces an imbalance between BRCA1 and 53BP1. Both events result in compromised accurate C-NHEJ and a concomitant increase in DNA end resection. Nevertheless, HR is significantly reduced upon PARP3 silencing while the enhanced end resection causes mutagenic deletions during A-EJ. As a result, the absence of PARP3 confers hypersensitivity to anti-tumoral drugs generating DSB. © The Author(s) 2014. Published by Oxford University Press.
Quadratic and 2-Crossed Modules of Algebras
Z. Arvasi; E. Ulualan
2007-01-01
In this work, we define the quadratic modules for commutative algebras and give relations among 2-crossed modules, crossed squares, quadratic modules and simplicial commutative algebras with Moore complex of length 2.
Planar Para Algebras, Reflection Positivity
Jaffe, Arthur
2016-01-01
We define the notion of a planar para algebra, which arises naturally from combining planar algebras with the idea of $\\Z_{N}$ para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under isotopy. For each $\\Z_{N}$, we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra. Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras which one can use in the study of quantum information. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivi...
Harmonic functions on groups and Fourier algebras
Chu, Cho-Ho
2002-01-01
This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals.
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.
WANG Renhong; ZHU Chungang
2004-01-01
The piecewise algebraic variety is a generalization of the classical algebraic variety. This paper discusses some properties of piecewise algebraic varieties and their coordinate rings based on the knowledge of algebraic geometry.
Marchuk, Nikolay
2011-01-01
Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann alg...
Edwards, Harold M
1995-01-01
In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject
Liesen, Jörg
2015-01-01
This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...
Recollements of extension algebras
CHEN; Qinghua(陈清华); LIN; Yanan(林亚南)
2003-01-01
Let A be a finite-dimensional algebra over arbitrary base field k. We prove: if the unbounded derived module category D-(Mod-A) admits symmetric recollement relative to unbounded derived module categories of two finite-dimensional k-algebras B and C:D-(Mod- B) ( ) D-(Mod- A) ( ) D-(Mod- C),then the unbounded derived module category D-(Mod - T(A)) admits symmetric recollement relative to the unbounded derived module categories of T(B) and T(C):D-(Mod - T(B)) ( ) D-(Mod - T(A)) ( ) D-(Mod - T(C)).
G.C. Rao
2012-11-01
Full Text Available A C- algebra is the algebraic form of the 3-valued conditional logic, which was introduced by F. Guzman and C. C. Squier in 1990. In this paper, some equivalent conditions for a C- algebra to become a boolean algebra in terms of congruences are given. It is proved that the set of all central elements B(A is isomorphic to the Boolean algebra of all C-algebras Sa, where a B(A. It is also proved that B(A is isomorphic to the Boolean algebra of all C-algebras Aa, where a B(A.
Stoll, R R
1968-01-01
Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand
Jacobson, Nathan
1979-01-01
Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its
Jacobson, Nathan
2009-01-01
A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as L
Allenby, Reg
1995-01-01
As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.Solutions to the exercises are available onlin
A question on the spectra of algebras of symmetric functions on L_∞ related to the moment problem
T. V. Vasylyshyn
2012-11-01
Full Text Available We consider a question on description of the set of characters of the algebra of bounded type symmetric analytic functions on L ∞ [0;1] and establish some connection with the trigonometric moment problem.
A question on the spectra of algebras of symmetric functions on L_∞ related to the moment problem
T. V. Vasylyshyn; A. V. Zagorodnyuk
2013-01-01
We consider a question on description of the set of characters of the algebra of bounded type symmetric analytic functions on L ∞ [0;1] and establish some connection with the trigonometric moment problem.
Parafermion Fields Constructed by Current Algebra
YANGZhan-Ying; SHIKang-Jie; WANGPei; ZHAOLiu
2004-01-01
In this letter, the parafermion fields constructed by current algebra are considered. It is proved that there must be a parafermion field with respect to each form of current algebra. We also obtain the corresponding representation and unitary relation of the parafermion field from any current algebra.
A Note on Solvable Polynomial Algebras
Huishi Li
2014-03-01
Full Text Available In terms of their defining relations, solvable polynomial algebras introduced by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990] are characterized by employing Gr\\"obner bases of ideals in free algebras, thereby solvable polynomial algebras are completely determinable and constructible in a computational way.
DERIVATIONS AND EXTENSIONS OF LIE COLOR ALGEBRA
Zhang Qingcheng; Zhang Yongzheng
2008-01-01
In this article, the authors obtain some results concerning derivations of fi-nitely generated Lie color algebras and discuss the relation between skew derivation space SkDer(L) and central extension H2(L, F) on some Lie color algebras. Meanwhile, they generalize the notion of double extension to quadratic Lie color algebras, a sufficient con-dition for a quadratic Lie color algebra to be a double extension and further properties are given.
Oliver, Bob; Pawałowski, Krzystof
1991-01-01
As part of the scientific activity in connection with the 70th birthday of the Adam Mickiewicz University in Poznan, an international conference on algebraic topology was held. In the resulting proceedings volume, the emphasis is on substantial survey papers, some presented at the conference, some written subsequently.
Tomás L Gómez
2001-02-01
This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.
A multi-set extended relational algebra: a formal approach to a practical issue
Grefen, Paul W.P.J.; By, de Rolf A.
1994-01-01
The relational data model is based on sets of tuples, i.e. it does not allow duplicate tuples an a relation. Many database languages and systems do require multi-set semantics though, either because of functional requirements or because of the high costs of duplicate removal in database operations.
Algebraically special space-time in relativity, black holes, and pulsar models
Adler, R. J.; Sheffield, C.
1973-01-01
The entire field of astronomy is in very rapid flux, and at the center of interest are problems relating to the very dense, rotating, neutron stars observed as pulsars. the hypothesized collapsed remains of stars known as black holes, and quasars. Degenerate metric form, or Kerr-Schild metric form, was used to study several problems related to intense gravitational fields.
Hochschild homology, global dimension, and truncated oriented cycles
Han, Yang
2010-01-01
It is shown that a bounded quiver algebra having a 2-truncated oriented cycle is of infinite Hochschild homology dimension and global dimension, which generalizes a result of Solotar and Vigu\\'{e}-Poirrier to nonlocal ungraded algebras having a 2-truncated oriented cycle of arbitrary length. Therefore, a bounded quiver algebra of finite global dimension has no 2-truncated oriented cycles. Note that the well-known "no loops conjecture", which has been proved to be true already, says that a bounded quiver algebra of finite global dimension has no loops, i.e., truncated oriented cycles of length 1. Moreover, it is shown that a monomial algebra having a truncated oriented cycle is of infinite Hochschild homology dimension and global dimension. Consequently, a monomial algebra of finite global dimension has no truncated oriented cycles.
Nearly projective Boolean algebras
Heindorf, Lutz; Shapiro, Leonid B
1994-01-01
The book is a fairly complete and up-to-date survey of projectivity and its generalizations in the class of Boolean algebras. Although algebra adds its own methods and questions, many of the results presented were first proved by topologists in the more general setting of (not necessarily zero-dimensional) compact spaces. An appendix demonstrates the application of advanced set-theoretic methods to the field. The intended readers are Boolean and universal algebraists. The book will also be useful for general topologists wanting to learn about kappa-metrizable spaces and related classes. The text is practically self-contained but assumes experience with the basic concepts and techniques of Boolean algebras.
Durka, R
2016-01-01
We explore the $S$-expansion framework to analyze freedom in closing the multiplication tables for the abelian semigroups. Including possibility of the zero element in the resonant decomposition and relating the Lorentz generator with the semigroup identity element leads to the wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results we find not only all the Maxwell algebras of type $\\mathfrak{B}_m$, $\\mathfrak{C}_m$, and recently introduced $\\mathfrak{D}_m$, but we also produce new examples. We discuss some prospects concerning further enlarging the algebras and provide all necessary constituents for constructing the gravity actions based on the obtained results.
Wadsworth, A R
2017-01-01
This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.
An introduction to algebraic topology
Rotman, Joseph J
1988-01-01
There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim ...
Division algebras and supersymmetry
Baez, John C
2009-01-01
Supersymmetry is deeply related to division algebras. Nonabelian Yang--Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green--Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this, in turn, is the existence of normed division algebras in dimensions 1, 2, 4 and 8: the real numbers, complex numbers, quaternions and octonions. Here we provide a self-contained account of how this works.
Weiss, Edwin
1998-01-01
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te
Algebra & trigonometry super review
2012-01-01
Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Algebra and Trigonometry Super Review includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, equations, linear equations and systems of linear equations, inequalities, relations and functions, quadratic equations, equations of higher order, ratios, proportions, and variations. Take the Super Review quizzes to see how much y
Uniform Algebras Over Complete Valued Fields
Mason, Jonathan W
2012-01-01
UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete valued field other than the complex numbers is used as the underlying field of the algebra. In the Archimedean setting, this generalisation is given by the theory of real function algebras introduced by S. H. Kulkarni and B. V. Limaye in the 1980s. This thesis establishes a broader theory accommodating any complete valued field as the underlying field by involving Galois automorphisms and using non-Archimedean analysis. The approach taken keeps close to the original definitions from the Archimedean setting. Basic function algebras are defined and generalise real function algebras to all complete valued fields. Several examples are provided. Each basic function algebra is shown to have a lattice of basic extensions related to the field structure. In the non-Archimedean settin...
Harvey, J A; Harvey, Jeffrey A.; Moore, Gregory
1998-01-01
We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko \\& Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any CY 3-fold there are two canonically associated algebras exchanged by mirror symmetry.
Harvey, Jeffrey A.; Moore, Gregory
We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi-Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry.
Generalized NLS Hierarchies from Rational $W$ Algebras
Toppan, F
1994-01-01
Finite rational $\\cw$ algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of equations, by showing how to associate to a rational $\\cw$ algebra its corresponding hierarchy. We work out two examples: the $sl(2)/U(1)$ coset, leading to the Non-Linear Schr\\"{o}dinger hierarchy, and the $U(1)$ coset of the Polyakov-Bershadsky $\\cw$ algebra, leading to a $3$-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies.
Physical Identifications for the Algebraic Quantities of Five-Dimensional Relativity
Wesson, Paul S
2010-01-01
When four-dimensional general relativity is embedded in an unconstrained man-ner in a fifth dimension, the physical quantities of spacetime can be interpreted as geometrical properties related to the extra dimension. It has become widespread to view the ten Einstein equations and the source terms of the energy-momentum tensor in this way. We now assign physical meanings to the other five equations involved. The scalar field acts like gravity, but concerns inertial as opposed to gravitational mass. The other four equations are conservation laws for 4D dynamics, but where the mass of a test particle is related to a local value of the cosmological 'constant'. Ways of testing these identifications are suggested.
Homological descent for motivic homology theories
Geisser, Thomas
2014-01-01
The purpose of this paper is to give homological descent theorems for motivic homology theories (for example, Suslin homology) and motivic Borel-Moore homology theories (for example, higher Chow groups) for certain hypercoverings.
Algebraic Thinking through Origami.
Higginson, William; Colgan, Lynda
2001-01-01
Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)
On the QFT relation between Donaldson-Witten invariants and Floer homology theory
Gianvittorio, R
1998-01-01
A TQFT in terms of general gauge fixing functions is discussed. In a covariant gauge it yields the Donaldson-Witten TQFT. The theory is formulated on a generalized phase space where a simplectic structure is introduced. The Hamiltonian is expressed as the anticommutator of off-shell nilpotent BRST and anti-BRST charges. Following original ideas of Witten a time reversal operation is introduced and an inner product is defined in terms of it. A non-covariant gauge fixing is presented giving rise to a manifestly time reversal invariant Lagrangean and a positive definite Hamiltonian, with the inner product previously introduced. As a consequence, the indefiniteness problem of some of the kinetic terms of the Witten's action is resolved. The construction allows then a consistent interpretation of Floer groups in terms of the cohomology of the BRST charge which is explicitly independent of the background metric. The relation between the BRST cohomology and the ground states of the Hamiltonian is then completely sta...
Very true operators on MTL-algebras
Wang Jun Tao
2016-01-01
Full Text Available The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.
Elements of algebraic coding systems
Cardoso da Rocha, Jr, Valdemar
2014-01-01
Elements of Algebraic Coding Systems is an introductory textto algebraic coding theory. In the first chapter, you'll gain insideknowledge of coding fundamentals, which is essential for a deeperunderstanding of state-of-the-art coding systems.This book is a quick reference for those who are unfamiliar withthis topic, as well as for use with specific applications such as cryptographyand communication. Linear error-correcting block codesthrough elementary principles span eleven chapters of the text.Cyclic codes, some finite field algebra, Goppa codes, algebraic decodingalgorithms, and applications in public-key cryptography andsecret-key cryptography are discussed, including problems and solutionsat the end of each chapter. Three appendices cover the Gilbertbound and some related derivations, a derivation of the Mac-Williams' identities based on the probability of undetected error,and two important tools for algebraic decoding-namely, the finitefield Fourier transform and the Euclidean algorithm for polynomials.
Brasseur Robert
2008-01-01
subset necessary to confer the lipocalin fold. This information has been used to assign LIR2 to lipocalins and to investigate its structure/function relationship. This study could be applied to other protein families with low pairwise similarity, such as the structurally related fatty acid binding proteins or avidins.
Homology of locally semialgebraic spaces
Delfs, Hans
1991-01-01
Locally semialgebraic spaces serve as an appropriate framework for studying the topological properties of varieties and semialgebraic sets over a real closed field. This book contributes to the fundamental theory of semialgebraic topology and falls into two main parts. The first dealswith sheaves and their cohomology on spaces which locally look like a constructible subset of a real spectrum. Topics like families of support, homotopy, acyclic sheaves, base-change theorems and cohomological dimension are considered. In the second part a homology theory for locally complete locally semialgebraic spaces over a real closed field is developed, the semialgebraic analogue of classical Bore-Moore-homology. Topics include fundamental classes of manifolds and varieties, Poincare duality, extensions of the base field and a comparison with the classical theory. Applying semialgebraic Borel-Moore-homology, a semialgebraic ("topological") approach to intersection theory on varieties over an algebraically closed field of ch...
Homology, convergence and parallelism.
Ghiselin, Michael T
2016-01-05
Homology is a relation of correspondence between parts of parts of larger wholes. It is used when tracking objects of interest through space and time and in the context of explanatory historical narratives. Homologues can be traced through a genealogical nexus back to a common ancestral precursor. Homology being a transitive relation, homologues remain homologous however much they may come to differ. Analogy is a relationship of correspondence between parts of members of classes having no relationship of common ancestry. Although homology is often treated as an alternative to convergence, the latter is not a kind of correspondence: rather, it is one of a class of processes that also includes divergence and parallelism. These often give rise to misleading appearances (homoplasies). Parallelism can be particularly hard to detect, especially when not accompanied by divergences in some parts of the body. © 2015 The Author(s).
Multi-mode q-oscillator algebras with q2(k+1)= 1 and related thermo field dynamics
高亚军
2002-01-01
A type of multi-mode q-oscillator algebra with q2(k+1) = 1 is set up and the associated qk-thermo field dynamics is constructed for all k = 1, 2, …, ∞ in a unified form. It is demonstrated that these qk-thermo field dynamics can all be nicely fitted into the algebraic formulation of statistical mechanics (axiomatized form for statistical physics). This means that we obtain infinitely many realizations of the algebraic scheme, which extend the consideration of Ojima[1981 Ann. Phys. 137 1] and contain the usual thermo field dynamics for the fermionic (k = 1) and bosonic (k = ∞)systems as special cases. As simple applications, the qk-statistical average of some operators are given.
Kouri, Donald J; Markovich, Thomas; Maxwell, Nicholas; Bodmann, Bernhard G
2009-07-02
We discuss a periodic variant of the Heisenberg-Weyl algebra, associated with the group of translations and modulations on the circle. Our study of uncertainty minimizers leads to a periodic version of canonical coherent states. Unlike the canonical, Cartesian case, there are states for which the uncertainty product associated with the generators of the algebra vanishes. Next, we explore the supersymmetric (SUSY) quantum mechanical setting for the uncertainty-minimizing states and interpret them as leading to a family of "hindered rotors". Finally, we present a standard quantum mechanical treatment of one of these hindered rotor systems, including numerically generated eigenstates and energies.
Quantum Lie algebras of type A$_{n}$
Sudbery, A I
1995-01-01
It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's. This gives rise to Lie algebra-like generators and relations for the locally finite part of the quantised enveloping algebra, and suggests a canonical Poincare-Birkhoff-Witt basis.
Results of Associated Implication Algebra on a Partial Ordered Set
无
2007-01-01
Some sufficient and necessary conditions that implication algebra on a partial ordered set is associated implication algebra are obtained, and the relation between lattice H implication algebra and associated implication algebra is discussed. Also, the concept of filter is proposed with some basic properties being studied.
Langlois, Michel
2014-01-01
In order to generalize the relativistic notion of boost to the case of non inertial particles and to general relativity, we come back to the definition of Lie group of Lorentz matrices and its Lie algebra and we study how this group acts on the Minskowski space. We thus define the notion of tangent boost along a worldline. This notion very general notion gives a useful tool both in special relativity (for non inertial particles or/and for non rectilinear coordinates) and in general relativity. We also introduce a matrix of the Lie algebra which, together with the tangent boost, gives the whole dynamical description of the considered system (acceleration and Thomas rotation). After studying the properties of Lie algebra matrices and of their reduced forms, we show that the Lie group of special Lorentz matrices has four one-parameter subgroups. These tools lead us to introduce the Thomas rotation in a quite general way. At the end of the paper, we present some examples using these tools and we consider the case...
Iachello, F
1995-01-01
1. The Wave Mechanics of Diatomic Molecules. 2. Summary of Elements of Algebraic Theory. 3. Mechanics of Molecules. 4. Three-Body Algebraic Theory. 5. Four-Body Algebraic Theory. 6. Classical Limit and Coordinate Representation. 8. Prologue to the Future. Appendices. Properties of Lie Algebras; Coupling of Algebras; Hamiltonian Parameters
Nonmonotonic logics and algebras
CHAKRABORTY Mihir Kr; GHOSH Sujata
2008-01-01
Several nonmonotonie logic systems together with their algebraic semantics are discussed. NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.
Semiprojectivity of universal -algebras generated by algebraic elements
Shulman, Tatiana
2012-01-01
Let be a polynomial in one variable whose roots all have multiplicity more than 1. It is shown that the universal -algebra of a relation , , is semiprojective and residually finite-dimensional. Applications to polynomially compact operators are given.......Let be a polynomial in one variable whose roots all have multiplicity more than 1. It is shown that the universal -algebra of a relation , , is semiprojective and residually finite-dimensional. Applications to polynomially compact operators are given....
Ito, Toshihiro; Itakura, Junya; Takahashi, Sakuma; Sato, Miwa; Mino, Megumi; Fushimi, Soichiro; Yamada, Masao; Morishima, Tuneo; Kunkel, Steven L; Matsukawa, Akihiro
2016-07-01
Influenza A virus causes acute respiratory infections that induce annual epidemics and occasional pandemics. Although a number of studies indicated that the virus-induced intracellular signaling events are important in combating influenza virus infection, the mechanism how specific molecule plays a critical role among various intracellular signaling events remains unknown. Raf/MEK/extracellular signal-regulated kinase cascade is one of the key signaling pathways during influenza virus infection, and the Sprouty-related Ena/vasodilator-stimulated phosphoprotein homology 1-domain-containing protein has recently been identified as a negative regulator of Raf-dependent extracellular signal-regulated kinase activation. Here, we examined the role of Raf/MEK/extracellular signal-regulated kinase cascade through sprouty-related Ena/vasodilator-stimulated phosphoprotein homology 1-domain-containing protein in influenza A viral infection because the expression of sprouty-related Ena/vasodilator-stimulated phosphoprotein homology 1-domain-containing protein was significantly enhanced in human influenza viral-induced pneumonia autopsy samples. Prospective animal trial. Research laboratory. Wild-type and sprouty-related Ena/vasodilator-stimulated phosphoprotein homology 1-domain-containing protein-2 knockout mice inoculated with influenza A. Wild-type or sprouty-related Ena/vasodilator-stimulated phosphoprotein homology 1-domain-containing protein-2 knockout mice were infected by intranasal inoculation of influenza A (A/PR/8). An equal volume of phosphate-buffered saline was inoculated intranasally into mock-infected mice. Influenza A infection of sprouty-related Ena/vasodilator-stimulated phosphoprotein homology 1-domain-containing protein-2 knockout mice led to higher mortality with greater viral load, excessive inflammation, and enhanced cytokine production than wild-type mice. Administration of MEK inhibitor, U0126, improved mortality and reduced both viral load and
Dimitrov, Bogdan G
2009-01-01
On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and found as a solution of quasilinear differential equations in partial derivatives for two different cases of "compactification + rescaling" and "rescaling + compactification". New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case $l=1$ of the standard gravitati...
Quaternionen and Geometric Algebra (Quaternionen und Geometrische Algebra)
Horn, Martin Erik
2007-01-01
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same time, Hestenes geometric algebra provides a didactically promising instrument to model phenomena in physics mathematically and in a tangible manner. Quaternions particularly have a catchy interpretation in the context of geometric algebra which can be used didactically. The relation between quaternions and geometric algebra is presented with a view to analysing its didactical possibilities.
Sun, Qiao; Nechitailo, Galina S.; Lu, Jinying; Liu, Min; Li, Huasheng
Usually, phenotype changes of plants were used to analayze the responding genetic damages. However, this method is time-consuming, laborious and needs a long period. Here, we developed an Arabidopsis thaliana homologous recombination reporter system, in which HR frequency and HR-related AtRAD54 gene expression level were used as mutagenic end points. Based on the system, effect of DNA damage by space-flight during the Shenzhou-9 mission was investigated. In this study, an Arabidopsis thaliana-line transgenic for GUS recombination substrates (R3L66, AtRAD54promoter:: GFP + GUS) was used to study the mutagenicity of space-flight, and the results showed that 13 days space-flight exposure of seedlings induced a significant increase in HRF compared with its ground-base three-dimensional clinostat (generally called a random positioning machine or RPM, an effective simulator of microgravity) controls and ground 1g controls. We also observed three-dimensional clinostat induced a significant increase in HRF and HR-related AtRAD54 gene expression level compared with ground 1g controls. Treatment with the ROS scavenger DMSO dramatically reduced the effects of simulated microgravity on the induction of HR and expression of the AtRAD54 gene, suggesting that ROS play a critical role in mediating the simulated microgravity mutagenic effects in plants. In order to understand the combined effects of radiation and microgravity (the main factors in space environment) on DNA damage, we further investigated the effects of modeled microgravity on radiation-induced bystander effects (RIBE) n vivo in A. thaliana plants using the expression level of the HR-related AtRAD54 gene as mutagenic end points. The results showed that the modeled microgravity significantly inhibited the up-regulated expression of the AtRAD54 gene in bystander aerial plants after root irradiation, suggesting a repressive effect of microgravity on RIBE.
Mahé, Louis; Roy, Marie-Françoise
1992-01-01
Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...
Rota-Baxter algebras and the Hopf algebra of renormalization
Ebrahimi-Fard, K.
2006-06-15
Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. Hereby the notion of Rota-Baxter algebras enters the scene. In this work we develop in detail several mathematical aspects of Rota-Baxter algebras as they appear also in other sectors closely related to perturbative renormalization, to wit, for instance multiple-zeta-values and matrix differential equations. The Rota-Baxter picture enables us to present the algebraic underpinning for the Connes-Kreimer Birkhoff decomposition in a concise way. This is achieved by establishing a general factorization theorem for filtered algebras. Which in turn follows from a new recursion formula based on the Baker-Campbell-Hausdorff formula. This allows us to generalize a classical result due to Spitzer to non-commutative Rota-Baxter algebras. The Baker-Campbell-Hausdorff based recursion turns out to be a generalization of Magnus' expansion in numerical analysis to generalized integration operators. We will exemplify these general results by establishing a simple representation of the combinatorics of renormalization in terms of triangular matrices. We thereby recover in the presence of a Rota-Baxter operator the matrix representation of the Birkhoff decomposition of Connes and Kreimer. (orig.)
Hidden torsion, 3-manifolds, and homology cobordism
Cha, Jae Choon
2011-01-01
This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pairwise homology cobordant, yet remain indistinguishable by any prior known homology cobordism invariants.
Introduction to applied algebraic systems
Reilly, Norman R
2009-01-01
This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as
Solvable quadratic Lie algebras
ZHU; Linsheng
2006-01-01
A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.
DÍaz, R.; Rivas, M.
2010-01-01
In order to study Boolean algebras in the category of vector spaces we introduce a prop whose algebras in set are Boolean algebras. A probabilistic logical interpretation for linear Boolean algebras is provided. An advantage of defining Boolean algebras in the linear category is that we are able to study its symmetric powers. We give explicit formulae for products in symmetric and cyclic Boolean algebras of various dimensions and formulate symmetric forms of the inclusion-exclusion principle.
Bliss, Gilbert Ames
1933-01-01
This book, immediately striking for its conciseness, is one of the most remarkable works ever produced on the subject of algebraic functions and their integrals. The distinguishing feature of the book is its third chapter, on rational functions, which gives an extremely brief and clear account of the theory of divisors.... A very readable account is given of the topology of Riemann surfaces and of the general properties of abelian integrals. Abel's theorem is presented, with some simple applications. The inversion problem is studied for the cases of genus zero and genus unity. The chapter on t
Bounds for Bilinear Complexity of Noncommutative Group Algebras
Pospelov, Alexey
2010-01-01
We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show nontrivial lower bounds for the rest of the group algebras. These lower bounds are built on the top of Bl\\"aser's results for semisimple algebras and algebras with large radical and the lower bound for arbitrary associative algebras due to Alder and Strassen. We also show subquadratic upper bounds for all group algebras turning into "almost linear" provided the exponent of matrix multiplication equals 2.
$W_{\\infty}$ algebra in the integer quantum Hall effects
Azuma, Hiroo
1994-01-01
We investigate the $W_{\\infty}$ algebra in the integer quantum Hall effects. Defining the simplest vacuum, the Dirac sea, we evaluate the central extension for this algebra. A new algebra which contains the central extension is called the $W_{1+\\infty}$ algebra. We show that this $W_{1+\\infty}$ algebra is an origin of the Kac-Moody algebra which determines the behavior of edge states of the system. We discuss the relation between the $W_{1+\\infty}$ algebra and the incompressibility of the int...
Operator algebras for analytic varieties
Davidson, Kenneth R; Shalit, Orr Moshe
2012-01-01
We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $\\cM_V$ of the multiplier algebra $\\cM$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball. The related algebras of continuous multipliers are also considered. We find that $\\cM_V$ is completely isometrically isomorphic to $\\cM_W$ if and only if $W$ is the image of $V$ under a biholomorphic automorphism of the ball. A similar condition characterizes when there exists a unital completely contractive homomorphism from $\\cM_V$ to $\\cM_W$. If one of the varieties is a homogeneous algebraic variety, then isometric isomorphism is shown to imply completely isometric isomorphism of the algebras. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. It is shown that if there is an isomorphism between $\\cM_V$ and $\\cM_W$, then there is a biholomorphism (with multiplier coordinates) between the varieties. We present a n...
Grätzer, George
1979-01-01
Universal Algebra, heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .," has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well-selected additional bibliography of over 1250 papers and books which makes this a fine work for students, instructors, and researchers in the field. "This book will certainly be, in the years to come, the basic reference to the subject." --- The American Mathematical Monthly (First Edition) "In this reviewer's opinion [the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially sui...
Equivalency of two-dimensional algebras
Santos, Gildemar Carneiro dos; Pomponet Filho, Balbino Jose S. [Universidade Federal da Bahia (UFBA), BA (Brazil). Inst. de Fisica
2011-07-01
Full text: Let us consider a vector z = xi + yj over the field of real numbers, whose basis (i,j) satisfy a given algebra. Any property of this algebra will be reflected in any function of z, so we can state that the knowledge of the properties of an algebra leads to more general conclusions than the knowledge of the properties of a function. However structural properties of an algebra do not change when this algebra suffers a linear transformation, though the structural constants defining this algebra do change. We say that two algebras are equivalent to each other whenever they are related by a linear transformation. In this case, we have found that some relations between the structural constants are sufficient to recognize whether or not an algebra is equivalent to another. In spite that the basis transform linearly, the structural constants change like a third order tensor, but some combinations of these tensors result in a linear transformation, allowing to write the entries of the transformation matrix as function of the structural constants. Eventually, a systematic way to find the transformation matrix between these equivalent algebras is obtained. In this sense, we have performed the thorough classification of associative commutative two-dimensional algebras, and find that even non-division algebra may be helpful in solving non-linear dynamic systems. The Mandelbrot set was used to have a pictorial view of each algebra, since equivalent algebras result in the same pattern. Presently we have succeeded in classifying some non-associative two-dimensional algebras, a task more difficult than for associative one. (author)
Yoneda algebras of almost Koszul algebras
Zheng Lijing
2015-11-01
Let be an algebraically closed field, a finite dimensional connected (, )-Koszul self-injective algebra with , ≥ 2. In this paper, we prove that the Yoneda algebra of is isomorphic to a twisted polynomial algebra $A^!$ [ ; ] in one indeterminate of degree +1 in which $A^!$ is the quadratic dual of , is an automorphism of $A^!$, and = () for each $t \\in A^!$. As a corollary, we recover Theorem 5.3 of [2].
Miyanishi, Masayoshi
2000-01-01
Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. There is a long history of research on projective algebraic surfaces, and there exists a beautiful Enriques-Kodaira classification of such surfaces. The research accumulated by Ramanujan, Abhyankar, Moh, and Nagata and others has established a classification theory of open algebraic surfaces comparable to the Enriques-Kodaira theory. This research provides powerful methods to study the geometry and topology of open algebraic surfaces. The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities. This book contains a comprehensive account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces. Prerequisite to understanding the text is a basic b...
Marchuk, Nikolay
2011-01-01
Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann algebra and Clifford algebra. Specialists consider models of gravity that based on a mathematical formalism with two metric tensors. We hope that the considered in this paper 2-metric exterior algebra can be useful for development of this model in gravitation theory. Especially in description of fermions in presence of a gravity field.
Noncommutative correspondence categories, simplicial sets and pro $C^*$-algebras
Mahanta, Snigdhayan
2009-01-01
We show that a $KK$-equivalence between two unital $C^*$-algebras produces a correspondence between their DG categories of finitely generated projective modules which is a $\\mathbf{K}_*$-equivalence, where $\\mathbf{K}_*$ is Waldhausen's $K$-theory. We discuss some connections with strong deformations of $C^*$-algebras and homological dualities. Motivated by a construction of Cuntz we associate a pro $C^*$-algebra to any simplicial set. We show that this construction is functorial with respect to proper maps of simplicial sets, that we define, and also respects proper homotopy equivalences. We propose to develop a noncommutative proper homotopy theory in the context of topological algebras.
Alfredo De Biasio
Full Text Available Proliferating Cell Nuclear Antigen (PCNA is an essential factor for DNA replication and repair. PCNA forms a toroidal, ring shaped structure of 90 kDa by the symmetric association of three identical monomers. The ring encircles the DNA and acts as a platform where polymerases and other proteins dock to carry out different DNA metabolic processes. The amino acid sequence of human PCNA is 35% identical to the yeast homolog, and the two proteins have the same 3D crystal structure. In this report, we give evidence that the budding yeast (sc and human (h PCNAs have highly similar structures in solution but differ substantially in their stability and dynamics. hPCNA is less resistant to chemical and thermal denaturation and displays lower cooperativity of unfolding as compared to scPCNA. Solvent exchange rates measurements show that the slowest exchanging backbone amides are at the β-sheet, in the structure core, and not at the helices, which line the central channel. However, all the backbone amides of hPCNA exchange fast, becoming undetectable within hours, while the signals from the core amides of scPCNA persist for longer times. The high dynamics of the α-helices, which face the DNA in the PCNA-loaded form, is likely to have functional implications for the sliding of the PCNA ring on the DNA since a large hole with a flexible wall facilitates the establishment of protein-DNA interactions that are transient and easily broken. The increased dynamics of hPCNA relative to scPCNA may allow it to acquire multiple induced conformations upon binding to its substrates enlarging its binding diversity.
Rigidification of algebras over essentially algebraic theories
Rosicky, J
2012-01-01
Badzioch and Bergner proved a rigidification theorem saying that each homotopy simplicial algebra is weakly equivalent to a simplicial algebra. The question is whether this result can be extended from algebraic theories to finite limit theories and from simplicial sets to more general monoidal model categories. We will present some answers to this question.
Distribution theory of algebraic numbers
Yang, Chung-Chun
2008-01-01
The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth''s theorem Subspace theorems Vojta''s conjectures L-functions.
The complex of looped diagrams and natural operations on Hochschild homology
Klamt, Angela
In this thesis natural operations on the (higher) Hochschild complex of a given family of algebras are investigated. We give a description of all formal operations (in the sense of Wahl) for the class of commutative algebras using Loday's lambda operation, Connes' boundary operator and shue produ...... of formal operations on Hochschild homology to higher Hochschild homology. We also generalize statements about the formal operations and give smaller models for the formal operations on higher Hochschild homology in certain cases....
Combinatorics and commutative algebra
Stanley, Richard P
1996-01-01
Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists. New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Included in this chapter is an outline of the proof of McMullen's g-conjecture for simplicial polytopes based on toric varieties, as well as a discussion of the face rings of such special ...
Fuzzy Dot Structure of BG-algebras
Tapan Senapati
2014-09-01
Full Text Available In this paper, the notions of fuzzy dot subalgebras is introduced together with fuzzy normal dot subalgebras and fuzzy dot ideals of BG-algebras. The homomorphic image and inverse image are investigated in fuzzy dot subalgebras and fuzzy dot ideals of BG-algebras. Also, the notion of fuzzy relations on the family of fuzzy dot subalgebras and fuzzy dot ideals of BG-algebras are introduced with some related properties.
Algebraic Systems and Pushdown Automata
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
Dong, Xiaotian; Su, Xiaoru; Yu, Jiong; Liu, Jingqi; Shi, Xiaowei; Pan, Qiaoling; Yang, Jinfeng; Chen, Jiajia; Li, Lanjuan; Cao, Hongcui
2017-01-01
Hypoxia-inducible factor 2 alpha (HIF2α), prolyl hydroxylase domain protein 2 (PHD2), and the von Hippel Lindau tumor suppressor protein (pVHL) are three principal proteins in the oxygen-sensing pathway. Under normoxic conditions, a conserved proline in HIF2α is hydroxylated by PHD2 in an oxygen-dependent manner, and then pVHL binds and promotes the degradation of HIF2α. However, the crystal structure of the HIF2α-pVHL complex has not yet been established, and this has limited research on the interaction between HIF and pVHL. Here, we constructed a structural model of a 23-residue HIF2α peptide (528-550)-pVHL-ElonginB-ElonginC complex by using homology modeling and molecular dynamics simulations. We also applied these methods to HIF2α mutants (HYP531PRO, F540L, A530 V, A530T, and G537R) to reveal structural defects that explain how these mutations weaken the interaction with pVHL. Homology modeling and molecular dynamics simulations were used to construct a three-dimensional (3D) structural model of the HIF2α-VHL complex. Subsequently, MolProbity, an active validation tool, was used to analyze the reliability of the model. Molecular mechanics energies combined with the generalized Born and surface area continuum solvation (MM-GBSA) and solvated interaction energy (SIE) methods were used to calculate the binding free energy between HIF2a and pVHL, and the stability of the simulation system was evaluated by using root mean square deviation (RMSD) analysis. We also determined the secondary structure of the system by using the definition of secondary structure of proteins (DSSP) algorithm. Finally, we investigated the structural significance of specific point mutations known to have clinical implications. We established a reliable structural model of the HIF2α-pVHL complex, which is similar to the crystal structure of HIF1α in 1LQB. Furthermore, we compared the structural model of the HIF2α-pVHL complex and the HIF2α (HYP531P, F540L, A530V, A530T, and G537
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
Lutfiyya, Lutfi A
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Modern Algebra includes set theory, operations, relations, basic properties of the integers, group theory, and ring theory.
The Yoneda algebra of a K_2 algebra need not be another K_2 algebra
Cassidy, T.; Phan, Van C.; Shelton, B.
2008-01-01
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
The Yoneda algebra of a K_2 algebra need not be another K_2 algebra
Cassidy, T; Phan, Van C.; Shelton, B.
2008-01-01
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
The isomorphism problem for some universal operator algebras
Davidson, Kenneth R; Shalit, Orr Moshe
2010-01-01
This paper addresses the isomorphism problem for the universal operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geo...
Algebraic cobordism theory attached to algebraic equivalence
Krishna, Amalendu
2012-01-01
After the algebraic cobordism theory of Levine-Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the zero-th semi-topological K-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional or special types of varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.
Workshop on Commutative Algebra
Simis, Aron
1990-01-01
The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results.
Probabilistic Concurrent Kleene Algebra
Annabelle McIver
2013-06-01
Full Text Available We provide an extension of concurrent Kleene algebras to account for probabilistic properties. The algebra yields a unified framework containing nondeterminism, concurrency and probability and is sound with respect to the set of probabilistic automata modulo probabilistic simulation. We use the resulting algebra to generalise the algebraic formulation of a variant of Jones' rely/guarantee calculus.
Generalized Quantum Current Algebras
ZHAO Liu
2001-01-01
Two general families of new quantum-deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enables one to define "tensor products" of these algebras. The standard quantum affine algebras turn out to be a very special case of the two algebra families, in which case the infinite Hopf family structure degenerates into a standard Hopf algebra. The relationship between the two algebraic families as well as thefr various special examples are discussed, and the free boson representation is also considered.
Perturbations of planar algebras
Das, Paramita; Gupta, Ved Prakash
2010-01-01
We introduce the concept of {\\em weight} of a planar algebra $P$ and construct a new planar algebra referred as the {\\em perturbation of $P$} by the weight. We establish a one-to-one correspondence between pivotal structures on 2-categories and perturbations of planar algebras by weights. To each bifinite bimodule over $II_1$-factors, we associate a {\\em bimodule planar algebra} bimodule corresponds naturally with sphericality of the bimodule planar algebra. As a consequence of this, we reproduce an extension of Jones' theorem (of associating 'subfactor planar algebras' to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. We show that the perturbation class of a bimodule planar algebra contains a unique spherical unimodular bimodule planar algeb...
Modular framization of the BMW algebra
Juyumaya, Jesus
2010-01-01
In this work we introduce the concept of Modular Framization or simply Framization. We construct a framization $F_{d,n}$ of the Birman--Wenzl--Murakami algebra, also known as BMW algebra, and start a systematic study of this framization. We show that $F_{d,n}$ is finite dimensional and the \\lq braid generators\\rq\\ of this algebra satisfy a quartic relation which is of minimal degree not containing the generators $t_i$. They also satisfy a quintic relation, as the smallest closed relation. We conjecture that the algebras $F_{d,n}$ support a Markov trace which allow to define polynomial invariants for unoriented knots in an analogous way that the Kauffman polynomial is derived from the BMW algebra. The idea originates from the Yokonuma--Hecke algebra, built from the classical Hecke algebra by adding framing generators and changing the Hecke algebra quadratic relation by a new quadratic relation which involves the framing generators. Using the Yokonuma--Hecke algebras and a Markov trace constructed on them\\cite{...
Minimax Rates for Homology Inference
Balakrishnan, Sivaraman; Sheehy, Don; Singh, Aarti; Wasserman, Larry
2011-01-01
Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In this paper, we consider the statistical problem of estimating the homology of a manifold from noisy samples under several different noise models. We derive upper and lower bounds on the minimax risk for this problem. Our upper bounds are based on estimators which are constructed from a union of balls of appropriate radius around carefully selected points. In each case we establish complementary lower bounds using Le Cam's lemma.
Topological convolution algebras
Alpay, Daniel
2012-01-01
In this paper we introduce a new family of topological convolution algebras of the form $\\bigcup_{p\\in\\mathbb N} L_2(S,\\mu_p)$, where $S$ is a Borel semi-group in a locally compact group $G$, which carries an inequality of the type $\\|f*g\\|_p\\le A_{p,q}\\|f\\|_q\\|g\\|_p$ for $p > q+d$ where $d$ pre-assigned, and $A_{p,q}$ is a constant. We give a sufficient condition on the measures $\\mu_p$ for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.
Testing algebraic geometric codes
CHEN Hao
2009-01-01
Property testing was initially studied from various motivations in 1990's.A code C (∩)GF(r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector's coordinates.The problem of testing codes was firstly studied by Blum,Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs).How to characterize locally testable codes is a complex and challenge problem.The local tests have been studied for Reed-Solomon (RS),Reed-Muller (RM),cyclic,dual of BCH and the trace subcode of algebraicgeometric codes.In this paper we give testers for algebraic geometric codes with linear parameters (as functions of dimensions).We also give a moderate condition under which the family of algebraic geometric codes cannot be locally testable.
Testing algebraic geometric codes
无
2009-01-01
Property testing was initially studied from various motivations in 1990’s. A code C GF (r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector’s coordinates. The problem of testing codes was firstly studied by Blum, Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs). How to characterize locally testable codes is a complex and challenge problem. The local tests have been studied for Reed-Solomon (RS), Reed-Muller (RM), cyclic, dual of BCH and the trace subcode of algebraicgeometric codes. In this paper we give testers for algebraic geometric codes with linear parameters (as functions of dimensions). We also give a moderate condition under which the family of algebraic geometric codes cannot be locally testable.
Palmkvist, Jakob, E-mail: palmkvist@ihes.fr [Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, FR-91440 Bures-sur-Yvette (France)
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
International Conference on Algebraic Topology
Cohen, Ralph; Miller, Haynes; Ravenel, Douglas
1989-01-01
These are proceedings of an International Conference on Algebraic Topology, held 28 July through 1 August, 1986, at Arcata, California. The conference served in part to mark the 25th anniversary of the journal Topology and 60th birthday of Edgar H. Brown. It preceded ICM 86 in Berkeley, and was conceived as a successor to the Aarhus conferences of 1978 and 1982. Some thirty papers are included in this volume, mostly at a research level. Subjects include cyclic homology, H-spaces, transformation groups, real and rational homotopy theory, acyclic manifolds, the homotopy theory of classifying spaces, instantons and loop spaces, and complex bordism.
Yangians and transvector algebras
Molev, A. I.
1998-01-01
Olshanski's centralizer construction provides a realization of the Yangian for the Lie algebra gl(n) as a subalgebra in the projective limit of a chain of centralizers in the universal enveloping algebras. We give a modified version of this construction based on a quantum analog of Sylvester's theorem. We then use it to get an algebra homomorphism from the Yangian to the transvector algebra associated with the general linear Lie algebras. The results are applied to identify the elementary rep...
Le Henanff, Gaëlle; Heitz, Thierry; Mestre, Pere; Mutterer, Jerôme; Walter, Bernard; Chong, Julie
2009-01-01
Background Grapevine protection against diseases needs alternative strategies to the use of phytochemicals, implying a thorough knowledge of innate defense mechanisms. However, signalling pathways and regulatory elements leading to induction of defense responses have yet to be characterized in this species. In order to study defense response signalling to pathogens in Vitis vinifera, we took advantage of its recently completed genome sequence to characterize two putative orthologs of NPR1, a key player in salicylic acid (SA)-mediated resistance to biotrophic pathogens in Arabidopsis thaliana. Results Two cDNAs named VvNPR1.1 and VvNPR1.2 were isolated from Vitis vinifera cv Chardonnay, encoding proteins showing 55% and 40% identity to Arabidopsis NPR1 respectively. Constitutive expression of VvNPR1.1 and VvNPR1.2 monitored in leaves of V. vinifera cv Chardonnay was found to be enhanced by treatment with benzothiadiazole, a SA analog. In contrast, VvNPR1.1 and VvNPR1.2 transcript levels were not affected during infection of resistant Vitis riparia or susceptible V. vinifera with Plasmopara viticola, the causal agent of downy mildew, suggesting regulation of VvNPR1 activity at the protein level. VvNPR1.1-GFP and VvNPR1.2-GFP fusion proteins were transiently expressed by agroinfiltration in Nicotiana benthamiana leaves, where they localized predominantly to the nucleus. In this system, VvNPR1.1 and VvNPR1.2 expression was sufficient to trigger the accumulation of acidic SA-dependent Pathogenesis-Related proteins PR1 and PR2, but not of basic chitinases (PR3) in the absence of pathogen infection. Interestingly, when VvNPR1.1 or AtNPR1 were transiently overexpressed in Vitis vinifera leaves, the induction of grapevine PR1 was significantly enhanced in response to P. viticola. Conclusion In conclusion, our data identified grapevine homologs of NPR1, and their functional analysis showed that VvNPR1.1 and VvNPR1.2 likely control the expression of SA-dependent defense genes
Homology group on manifolds and their foldings
M. Abu-Saleem
2010-03-01
Full Text Available In this paper, we introduce the definition of the induced unfolding on the homology group. Some types of conditional foldings restricted on the elements of the homology groups are deduced. The effect of retraction on the homology group of a manifold is dicussed. The unfolding of variation curvature of manifolds on their homology group are represented. The relations between homology group of the manifold and its folding are deduced.
Affine transformation crossed product like algebras and noncommutative surfaces
Arnlind, Joakim
2009-01-01
Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surfaces being inverse images of fourth order polynomials (in R^3) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Three-manifolds class field theory (Homology of coverings for a non-virtually Haken manifold)
Reznikov, A G
1996-01-01
This is a first in a series of papers, devoted to the relation betwwen three-manifolds and number fields. The present paper studies first homology of finite coverings of a three-manifold with primary interest in the Thurston $b_1$ conjecture.The main result reads: if $M$ does not yield the Thurston conjecture, then the pro-p completion of its fundamental group is a Poincaré duality pro-p group. Conceptually, it means that we have a ``p-adic'' three-manifold. We develop several algebraic techniques, including a new powerful specral seguence, to actually compute homology of coverings, assumong only information on homology of $M$, a thing never done before.A number of applications to the structure of finite group cohomology rings is also given.
Cellularity of certain quantum endomorphism algebras
Andersen, Henning Haahr; Lehrer, Gus; Zhang, Ruibin
2015-01-01
structure are described in terms of certain Temperley–Lieb-like diagrams. We also prove general results that relate endomorphism algebras of specialisations to specialisations of the endomorphism algebras. When ζ is a root of unity of order bigger than d we consider the Uζ-module structure...... we independently recover the weight multiplicities of indecomposable tilting modules for Uζ(sl2) from the decomposition numbers of the endomorphism algebras, which are known through cellular theory....
A Hopf algebra deformation approach to renormalization
Ionescu, L M; Ionescu, Lucian M.; Marsalli, Michael
2003-01-01
We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices. It is suggested that the QFTs obtained via deformation quantization and renormalization correspond to each other in the sense of Kontsevich/Cattaneo-Felder.
Algebras with actions and automata
W. Kühnel
1982-01-01
Full Text Available In the present paper we want to give a common structure theory of left action, group operations, R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces. The first section gives an axiomatic approach to algebraic structures relative to a base category B, slightly more powerful than that of monadic (tripleable functors. In section 2 we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories. In section 3 we treat the structures mentioned in the beginning as many-sorted algebras with fixed scalar or input object and show that they still have an algebraic (or monadic forgetful functor (theorem 3.3 and hence the general theory of algebraic structures applies. These structures were usually treated as one-sorted in the Lawvere-setting, the action being expressed by a family of unary operations indexed over the scalars. But this approach cannot, as the one developed here, describe continuity of the action (more general: the action to be a B-morphism, which is essential for the structures mentioned above, e.g. modules for a sheaf of rings or topological automata. Finally we discuss consequences of theorem 3.3 for the structure theory of various types of automata. The particular case of algebras with fixed natural numbers object has been studied by the authors in [23].
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
On indecomposable modules over the Virasoro algebra
sU; Yucai(
2001-01-01
［1］Chari, V. , Pressley, A., Unitary representations of the Virasoro algebra and a conjecture of Kac, Compositio Math, 1988,67: 315-342.［2］Feign, B. L. , Fuchs, D. B., Verma modules over the Virasoro algebra, Lecture Notes in Math, 1984, 1060: 230-245.［3］Kac, V. G., Some problems on infinite-dimensional Lie algebras and their representations, Lie algebras and related topics,Lecture Notes in Math., 1982, 933: 117-126.［4］Kac, V. G., Infinite Dimensional Lie Algebras, 2nd ed., Boston, Cambridge: Birkhauser, 1985.［5］Kaplansky, I., Santharoubane, L. J., Harish-Chandra modules over the Virasoro algebra, Infinite-dimensional groups with application, Math. Sci. Res. Inst. Pub., 1985, 4: 217-231.［6］Langlands, R., On unitary representations of the Virasoro algebra, Infinite-Dimensional Lie Algebras and Their Application,Singapore: World Scientific, 1986, 141-159.［7］Martin. C. , Piard, A., Indecomposable modules over the Virasoro Lie algebra and a conjecture of V Kac, Comm. Math.Phys., 1991, 137: 109-132.［8］Mathieu, O. , Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent Math., 1992, 107: 225-234.［9］Su, Y., A classification of indecomposable sl2(2)-modules and a conjecture of Kac on irreducible modules over the Virasoro algebra, J. Alg., 1993, 161: 33-46.［10］Su, Y. , Classification of Harish-Chandra modules over the super-Virasoro algebras, Comm. Alg., 1995, 23: 3653-3675.［11］Su, Y. , Simple modules over the high rank Virasoro algebras, Comm. Alg., 2001, in press.
Blyth, T S
2002-01-01
Most of the introductory courses on linear algebra develop the basic theory of finite dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num ber of illustrative and worked examples, as well as many exercises that are strategi cally placed throughout the text. Solutions to the ex...
Rhythmical bimanual force production: homologous and non-homologous muscles.
Kennedy, Deanna M; Boyle, Jason B; Rhee, Joohyun; Shea, Charles H
2015-01-01
The experiment was designed to determine participants' ability to coordinate a bimanual multifrequency pattern of isometric forces using homologous or non-homologous muscles. Lissajous feedback was provided to reduce perceptual and attentional constraints. The primary purpose was to determine whether the activation of homologous and non-homologous muscles resulted in different patterns of distortions in the left limb forces that are related to the forces produced by the right limb. The task was to rhythmically produce a 1:2 pattern of isometric forces by exerting isometric forces on the left side force transducer with the left arm that was coordinated with the pattern of isometric forces produced on the right side force transducer with the right arm. The results indicated that participants were able to 'tune-in' a 1:2 coordination patterns using homologous (triceps muscles of the left and right limbs) and using non-homologous muscles (biceps left limb and triceps right limb) when provided Lissajous feedback. However, distinct but consistent and identifiable distortions in the left limb force traces were observed for both the homologous and non-homologous tasks. For the homologous task, the interference occurred in the left limb when the right limb was initiating and releasing force. For the non-homologous task, the interference in the left limb force occurred only when the right limb was releasing force. In both conditions, the interference appeared to continue from the point of force initiation and/or release to peak force velocity. The overall results are consistent with the notion that neural crosstalk manifests differently during the coordination of the limbs depending upon whether homologous or non-homologous muscles are activated.
An introduction to Clifford algebras and spinors
Vaz, Jayme
2016-01-01
This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians. Among the existing approaches to Clifford algebras and spinors this book is unique in that it provides a didactical presentation of the topic and i...
Neutrosophic Bilinear Algebras and their Generalizations
Kandasamy, W B Vasantha
2010-01-01
This book introduces the concept of neutrosophic bilinear algebras and their generalizations to n-linear algebras, n>2. This book has five chapters. The first chapter is introductory in nature and gives a few essential definitions and references for the reader to make use of the literature in case the reader is not thorough with the basics. The second chapter deals with different types of neutrosophic bilinear algebras and bivector spaces and proves several results analogous to linear bialgebra. In chapter three the authors introduce the notion of n-linear algebras and prove several theorems related to them. Many of the classical theorems for neutrosophic algebras are proved with appropriate modifications. Chapter four indicates the probable applications of these algebraic structures. The final chapter suggests about 80 innovative problems for the reader to solve.
Goldmann, H
1990-01-01
The first part of this monograph is an elementary introduction to the theory of Fréchet algebras. Important examples of Fréchet algebras, which are among those considered, are the algebra of all holomorphic functions on a (hemicompact) reduced complex space, and the algebra of all continuous functions on a suitable topological space.The problem of finding analytic structure in the spectrum of a Fréchet algebra is the subject of the second part of the book. In particular, the author pays attention to function algebraic characterizations of certain Stein algebras (= algebras of holomorphic functions on Stein spaces) within the class of Fréchet algebras.
Jardino, Sergio
2010-01-01
We extend the concept of a generalized Lie 3-algebra, known to octonions $\\mathbb{O}$, to split-octonions $\\mathbb{SO}$. In order to do that, we introduce a notational device that unifies the two elements product of both of the algebras. We have also proved that $\\mathbb{SO}$ is a Malcev algebra and have recalculated known relations for the structure constants in terms of the introduced structure tensor. An application of the split Lie $3-$algebra to a Bagger and Lambert gauge theory is also discussed.
Dispersion Operators Algebra and Linear Canonical Transformations
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-02-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
A Construction of the "2221" Planar Algebra
Han, Richard
2011-01-01
In this paper, we construct the "2221" subfactor planar algebra by finding it as a subalgebra of the graph planar algebra of its principal graph. In particular, we give a presentation of the "2221" subfactor planar algebra consisting of generators and relations. As a corollary, we have a planar algebra proof of the existence of a subfactor with principal graph "2221". To show the subfactor property, we use the jellyfish algorithm for evaluating closed diagrams. Lastly, we show uniqueness up to conjugation of "2221".
Dispersion Operators Algebra and Linear Canonical Transformations
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-04-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Deformed Virasoro Algebras from Elliptic Quantum Algebras
Avan, J.; Frappat, L.; Ragoucy, E.
2017-09-01
We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 1990s. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the gl(2) case) a Liouville formula.
Clifford Algebra with Mathematica
Aragon-Camarasa, G; Aragon, J L; Rodriguez-Andrade, M A
2008-01-01
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, a package for Clifford algebra calculations for the computer algebra program Mathematica is introduced through a presentation of the main ideas of Clifford algebras and illustrative examples. This package can be a useful computational tool since allows the manipulation of all these mathematical objects. It also includes the possibility of visualize elements of a Clifford algebra in the 3-dimensional space.
Boicescu, V; Georgescu, G; Rudeanu, S
1991-01-01
The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.
Abian, Alexander
1973-01-01
Linear Associative Algebras focuses on finite dimensional linear associative algebras and the Wedderburn structure theorems.The publication first elaborates on semigroups and groups, rings and fields, direct sum and tensor product of rings, and polynomial and matrix rings. The text then ponders on vector spaces, including finite dimensional vector spaces and matrix representation of vectors. The book takes a look at linear associative algebras, as well as the idempotent and nilpotent elements of an algebra, ideals of an algebra, total matrix algebras and the canonical forms of matrices, matrix
Kobayashi, Hiroki; Komatsu, Shuhei; Ichikawa, Daisuke; Kawaguchi, Tsutomu; Hirajima, Shoji; Miyamae, Mahito; Okajima, Wataru; Ohashi, Takuma; Kosuga, Toshiyuki; Konishi, Hirotaka; Shiozaki, Atsushi; FUJIWARA, Hitoshi; Okamoto, Kazuma; Tsuda, Hitoshi; Otsuji, Eigo
2015-01-01
Background Denticleless E3 ubiquitin protein ligase homolog (DTL) has been identified in amplified region (1q32) of several cancers and has an oncogenic function. In this study, we tested whether DTL acts as a cancer-promoting gene through its activation/overexpression in gastric cancer (GC). Methods We analyzed 7 GC cell lines and 100 primary tumors that were curatively resected in our hospital between 2001 and 2003. Results Overexpression of the DTL protein was detected in GC cell lines (4/...
Probability on real Lie algebras
Franz, Uwe
2016-01-01
This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.
Algebraic topology a first course
Fulton, William
1995-01-01
To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: ...
Cellularity of diagram algebras as twisted semigroup algebras
Wilcox, Stewart
2010-01-01
The Temperley-Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups. This theorem, which generalises a recent result of East about semigroup algebras of inverse semigroups, allows us to easily reproduce the cellularity of these algebras.
Hom-alternative algebras and Hom-Jordan algebras
Makhlouf, Abdenacer
2009-01-01
The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.
Alternative description of three dimensional complex diassociative algebras with some constraints
Rikhsiboev, Ikrom M.; Venkatesan, Yuvendra Rao
2014-07-01
Considering significant of classification problems in modern algebra, especially in the algebras which related to Lie algebras, current research pursue investigation on structure theory of low dimensional diassociative algebras. Note that the classification of complex diassociative algebras in low dimensions have been presented in our recent studies, however this paper deals to provide description of such algebras with some constrains in dimension three, applying notion of annihilator.
Asveld, P.R.J.
1976-01-01
Operaties op formele talen geven aanleiding tot bijbehorende operatoren op families talen. Bepaalde onderwerpen uit de algebra (universele algebra, tralies, partieel geordende monoiden) kunnen behulpzaam zijn in de studie van verzamelingen van dergelijke operatoren.
A geometric model for Hochschild homology of Soergel bimodules
Webster, Ben; Williamson, Geordie
2008-01-01
An important step in the calculation of the triply graded link homology of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as B–equivariant intersection cohomology...... of B×B–orbit closures in G. We show that, in type A, these orbit closures are equivariantly formal for the conjugation B–action. We use this fact to show that, in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring...
Modular invariance and the fusion algebra
Dijkgraaf, Robbert; Verlinde, Erik
1988-12-01
We discuss the relation between modular transformations and the fusion algebra, and explain its proof. It is shown that the existence of off-diagonal modular invariant partition functions imply the existence of a non-trivial automorphism of the fusion algebra. This is illustrated using the SU(2) affine models.
Diassociative algebras and Milnor's invariants for tangles
Kravchenko, Olga
2010-01-01
We extend Milnor's mu-invariants of link homotopy to ordered tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.
Advanced Lukasiewicz calculus and MV-algebras
Mundici, Daniele
2011-01-01
This is a continuation of Vol. 7 of Trends in Logic. It wil cover the wealth of recent developments of Lukasiewicz Logic and their algebras (Chang MV-algebras), with particular reference to (de Finetti) coherent evaluation of continuously valued events, (Renyi) conditionals for such events, related algorithms.
FUZZY ALGEBRA IN TRIANGULAR NORM SYSTEM
宋晓秋; 潘志
1994-01-01
Triangular norm is a powerful tool in the theory research and application development of fuzzy sets. In this paper, using the triangular norm, we introduce some concepts such as fuzzy algebra, fuzzy o algebra and fuzzy monotone class, and discuss the relations among them, obtaining the following main conclusions.
Matrix Lie Algebras and Integrable Couplings
ZHANG Yu-Feng; GUO Fu-Kui
2006-01-01
Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.
Lie groups and Lie algebras for physicists
Das, Ashok
2015-01-01
The book is intended for graduate students of theoretical physics (with a background in quantum mechanics) as well as researchers interested in applications of Lie group theory and Lie algebras in physics. The emphasis is on the inter-relations of representation theories of Lie groups and the corresponding Lie algebras.
Mattson Solomon transform and algebra codes
Martínez-Moro, E.; Benito, Diego Ruano
2009-01-01
In this note we review some results of the first author on the structure of codes defined as subalgebras of a commutative semisimple algebra over a finite field (see Martínez-Moro in Algebra Discrete Math. 3:99-112, 2007). Generator theory and those aspects related to the theory of Gröbner bases...
Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra
Lagraa, M
1997-01-01
We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.
Lie algebraic Noncommutative Gravity
Banerjee, R; Samanta, S; Banerjee, Rabin; Mukherjee, Pradip; Samanta, Saurav
2007-01-01
The minimal (unimodular) formulation of noncommutative general relativity, based on gauging the Poincare group, is extended to a general Lie algebra valued noncommutative structure. We exploit the Seiberg -- Witten map technique to formulate the theory as a perturbative Lagrangian theory. Detailed expressions of the Seiberg -- Witten maps for the gauge parameters, gauge potentials and the field strengths have been worked out. Our results demonstrate that notwithstanding the introduction of more general noncommutative structure there is no first order correction, exactly as happens for a canonical (i.e. constant) noncommutativity.
Statecharts Via Process Algebra
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
刘卫锋
2013-01-01
将软集理论应用到布尔代数中，提出了软布尔代数、软布尔子代数、软布尔代数的软理想、软理想布尔代数等概念，研究了它们的相关性质，并初步讨论了软布尔代数与几类布尔代数的模糊子代数的关系。%The soft set theory is applied to the Boolean algebra.The concepts of soft Boolean algebra, soft Boolean sub-algebra, soft ideal of soft Boolean algebra and idealistic soft Boolean algebra are presented and some related algebraic properties are discussed.The relations between soft Boolean algebra and several kinds of fuzzy subalgebras of Boolean algebra are preliminarily investigated.
Polynomial algebras Poisson with regular structure of simplectical leaf
Odesskij, A V
2002-01-01
The Poisson polynomial algebras with certain regularity conditions are studied. The linear structure on the dual spaces of the semisimple Lie algebras, the Sklyanin quadratic elliptical algebras as well as the polynomial algebras constitute, in particular, the algebras of this class. Simple determinate relations between the brackets and Kazimir operators are established. These relations determine, in particular, that the sum of the Kazimir operators grades coincides with the dimensionality of the algebra for the Sklyanin elliptical algebras. The examples of such algebras are presented and it is shown, that some of them are naturally generated in the Hamiltonian integrated systems. The new class of the two-particle integrable systems, depending elliptically both on the coordinates and pulses, is among these examples
The use of ultraproducts in commutative algebra
Schoutens, Hans
2010-01-01
In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.
On the Algebraic Classification of Module Spectra
Patchkoria, Irakli
2011-01-01
Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $\\pi_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \\geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $\\pi_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. The main examples of ring spectra to which our result applies are the $p$-local real connective $K$-theory spectrum $ko_{(p)}$, the Johnson-Wilson spectrum E(2), and the truncated Brown-Peterson spectrum $BP$, for an odd prime $p$.
Equivariant characteristic classes of complex algebraic varieties
Cappell, Sylvain E; Schuermann, Joerg; Shaneson, Julius L
2010-01-01
Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global orbifolds. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold.
Connecting Arithmetic to Algebra
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Arithmetic: Prerequisite to Algebra?
Rotman, Jack W.
Drawing from research and observations at Lansing Community College (Michigan) (LCC), this paper argues that typical arithmetic courses do little to prepare students to master algebra, and proposes an alternative set of arithmetic skills as actual prerequisites to algebra. The first section offers a description of the algebra sequence at LCC,…
Bergstra, J.A.; Fokkink, W.J.; Middelburg, C.A.
2008-01-01
Timed frames are introduced as objects that can form a basis of a model theory for discrete time process algebra. An algebraic setting for timed frames is proposed and results concerning its connection with discrete time process algebra are given. The presented theory of timed frames captures the ba
Foundations of algebraic geometry
Weil, A
1946-01-01
This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.
REAL PIECEWISE ALGEBRAIC VARIETY
Ren-hong Wang; Yi-sheng Lai
2003-01-01
We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set, etc., we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in Cμ spline ring.
Equivariant Algebraic Cobordism
Heller, Jeremiah
2010-01-01
We define equivariant algebraic cobordism for a connected linear algebraic group $G$ over a field of characteristic zero. The construction is based on Totaro's idea of using algebraic approximations for $BG$. We establish the analogous of the properties of an oriented cohomology theory, prove some of the expected properties from an equivariant theory, and make a few computations.
Deep homology: a view from systematics.
Scotland, Robert W
2010-05-01
Over the past decade, it has been discovered that disparate aspects of morphology - often of distantly related groups of organisms - are regulated by the same genetic regulatory mechanisms. Those discoveries provide a new perspective on morphological evolutionary change. A conceptual framework for exploring these research findings is termed 'deep homology'. A comparative framework for morphological relations of homology is provided that distinguishes analogy, homoplasy, plesiomorphy and synapomorphy. Four examples - three from plants and one from animals - demonstrate that homologous developmental mechanisms can regulate a range of morphological relations including analogy, homoplasy and examples of uncertain homology. Deep homology is part of a much wider range of phenomena in which biological (genes, regulatory mechanisms, morphological traits) and phylogenetic levels of homology can both be disassociated. Therefore, to understand homology, precise, comparative, independent statements of both biological and phylogenetic levels of homology are necessary.
Bases of Schur algebras associated to cellularly stratified diagram algebras
Bowman, C
2011-01-01
We examine homomorphisms between induced modules for a certain class of cellularly stratified diagram algebras, including the BMW algebra, Temperley-Lieb algebra, Brauer algebra, and (quantum) walled Brauer algebra. We define the `permutation' modules for these algebras, these are one-sided ideals which allow us to study the diagrammatic Schur algebras of Hartmann, Henke, Koenig and Paget. We construct bases of these Schur algebras in terms of modified tableaux. On the way we prove that the (quantum) walled Brauer algebra and the Temperley-Lieb algebra are both cellularly stratified and therefore have well-defined Specht filtrations.
INPUT-OUTPUT STRUCTURE OF LINEAR-DIFFERENTIAL ALGEBRAIC SYSTEMS
KUIJPER, M; SCHUMACHER, JM
1993-01-01
Systems of linear differential and algebraic equations occur in various ways, for instance, as a result of automated modeling procedures and in problems involving algebraic constraints, such as zero dynamics and exact model matching. Differential/algebraic systems may represent an input-output relat
Changing Pre-Service Elementary Teachers' Attitudes to Algebra.
McGowen, Mercedes A.; Davis, Gary E.
This article addresses the question: "What are the implications for the preparation of prospective elementary teachers of 'early algebra' in the elementary grades curriculum?" Part of the answer involves language aspects of algebra: in particular, how a change in pre-service teachers' attitudes to algebra, from instrumental to relational, is…
Reflection equation algebras, coideal subalgebras, and their centres
Kolb, S.; Stokman, J.V.
2009-01-01
Reflection equation algebras and related U-q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so c
An improved Multiplicity Conjecture for codimension three Gorenstein algebras
2006-01-01
The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.
Deficiently Extremal Cohen-Macaulay Algebras
Chanchal Kumar; Pavinder Singh
2010-04-01
The aim of this paper is to study homological properties of deficiently extremal Cohen–Macaulay algebras. Eagon–Reiner showed that the Stanley–Reisner ring of a simplicial complex has a linear resolution if and only if the Alexander dual of the simplicial complex is Cohen–Macaulay. An extension of a special case of Eagon–Reiner theorem is obtained for deficiently extremal Cohen–Macaulay Stanley–Reisner rings.
Sugawara construction and Casimir operators for Krichever-Novikov algebras
Schlichenmaier, M; Schlichenmaier, Martin; Sheinman, Oleg K
1995-01-01
We show how to obtain from highest weight representations of Krichever-Novikov algebras of affine type (also called higher genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations are given and Casimir operators are constructed. In an appendix the Sugawara construction for the multi-point situation is done.
Van Valen, L M
1982-09-01
Homology is resemblance caused by a continuity of information. In biology it is a unified developmental phenomenon. Homologies among and within individuals intergrade in several ways, so historical homology cannot be separated sharply from repetitive homology. Nevertheless, the consequences of historical and repetitive homologies can be mutually contradictory. A detailed discussion of the rise and fall of the "premolar-analogy" theory of homologies of mammalian molar-tooth cusps exemplifies such a contradiction. All other hypotheses of historical homology which are based on repetitive homology, such as the foliar theory of the flower considered phyletically, are suspect.
基于C++的关系代数产生的安全SQL查询%Safe SQL Queries Generated by Relational Algebra Based on C++ Language
顾坤鹏; 宋顺林
2011-01-01
在使用C++开发数据库相关的应用程序时,SQL语句的产生在程序编译期间并不会进行必要的检查.本文研究在编译期间使用C++编译器对关系代数运算作检查,由关系代数生成正确的SQL查询,将运行期SQL查询的部分检查工作提前到程序的编译期间处理.%When developing applications based on a database with C++ , the SQL queries are not necessarily checked for correctness at compile-time but only at runtime. The paper studies on using C++ compiler to check the relational algebra, which generates correct SQL queries, that brings forward some part of runtime work to be done at compile-time.
Algorithms in Algebraic Geometry
Dickenstein, Alicia; Sommese, Andrew J
2008-01-01
In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its
Computer algebra and operators
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Antonio J Calderón Martín; Manuel Forero Piulestán; José M Sánchez Delgado
2012-05-01
We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras is of the form $M=\\mathcal{U}+\\sum_jI_j$ with $\\mathcal{U}$ a subspace of the abelian Malcev subalgebra and any $I_j$ a well described ideal of satisfying $[I_j, I_k]=0$ if ≠ . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.
The Necessary Fundamental Algebraic Competence in the Age of Computer Algebra Systems.
Heugl, Helmut
This lecture addresses the exploration of algebraic fundamental competence by examining the Austrian Computer Algebraic Systems (CAS). Data are used to support answers and conclusions related to two questions that explore the role that instrumental understanding plays in supporting a high level of relational understanding and the idea that…
Holly, Keith Allen
The comparative validities of four composites of variables for predicting grade point average in ninth grade modern algebra and performance on the "Cooperative Mathematics Test" were studied. A sample of 177 high school students was used. Findings showed that certain structure-of-intellect factors were useful in predicting algebra success,…
Fong, Anthony B.; Jaquet, Karina; Finkelstein, Neal
2014-01-01
This REL West study explores the prevalence of students repeating Algebra I, who is most likely to repeat the course, and the level of improvement for students who repeat. Using six years of data from a cohort of 3,400 first-time seventh grade students in a California school district, authors found that 44 percent of students repeated algebra I.…
Basheer, Maamoun; Schwalb, Herzl; Nesher, Maoz; Gilon, Dan; Shefler, Irit; Mekori, Yoseph A; Shapira, Oz M; Gorodetsky, Raphael
2010-11-01
Haptides are a family of short peptides homologous to C-termini sequences of fibrinogen chains β and γ (haptides Cβ and preCγ, respectively) which were previously shown to penetrate and bind cells. This work investigates the systemic effect of the haptides with possible clinical implications. Intra-arterial monitoring in rats recorded the haptides' effects on systemic blood pressure. In parallel, their effect was also tested in vitro on isolated rat peritoneal mast cells and on human mast cells. Intra-arterial monitoring in rats showed that intravenous administration of low haptides concentrations (35-560 μg/kg rat) caused a shocklike behavior with transient decrease in the systolic and diastolic blood pressure by up to 55% (P caused degranulation of the mast cells. We found that the haptides Cβ and preCγ activated mast cells causing histamine release, resulting in a steep decrease in blood pressure, comparable to anaphylactic shock. In treating vascular occlusive diseases, massive fibrinolysis is induced, and haptide-containing sequences are released. We suggest that treatment with histamine receptor blockers or with mast cell stabilizing agents in such pathological conditions may overcome this effect. Copyright © 2010 American Academy of Allergy, Asthma & Immunology. Published by Mosby, Inc. All rights reserved.
代业明
2011-01-01
From the current status quo of university teaching linear algebra,taking a mathematical methodology and epistemology as a guide,the article explore the relations between the linear algebra and the Mathematics of Middle School on mathematical knowledges,mathematical thoughts and methods,mathematical senses,so we can help students better to have transition from the Mathematics of middle school to learning of the linear algebra in university.%从当下大学线性代数教学现状出发,以数学方法论和认识论为指导,探究线性代数与中学数学在数学知识、数学思想方法、数学观念等方面的联系,以期帮助大学生顺利地从中学过渡到大学线性代数的学习。
Modern Algebra and Information Science.
Diodato, Virgil
1983-01-01
Mathematical concepts used in modern algebra are shown to be useful in modeling syndetic structure employed in information retrieval systems. The equivalence relation can represent "see also" components of traditional library catalogs; partial orderings and mappings assist in modeling syndetics in sophisticated systems such as thesauri.…
Reflection algebra and functional equations
Galleas, W.; Lamers, J.
2014-01-01
In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partitio
On Derivations Of Genetic Algebras
Mukhamedov, Farrukh; Qaralleh, Izzat
2014-11-01
A genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. In application of genetics this algebra often has a basis corresponding to genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. In this paper, we find the connection between the genetic algebras and evolution algebras. Moreover, we prove the existence of nontrivial derivations of genetic algebras in dimension two.
Rational equivariant K-homology of low dimensional groups
Lafont, Jean-François; Sánchez-García, Rubén J
2011-01-01
We consider groups G which have a cocompact, 3-manifold model for the classifying space \\underline{E}G. We provide an algorithm for computing the rationalized equivariant K-homology of \\underline{E}G. Under the additional hypothesis that the quotient 3-orbifold \\underline{E}G/G is geometrizable, the rationalized K-homology groups coincide with the rationalized K-theory of the reduced C*-algebra of G. We illustrate our algorithm on some concrete examples.
Cellularity of certain quantum endomorphism algebras
Andersen, Henning Haahr; Lehrer, G. I.; Zhang, R.
Let $\\tA=\\Z[q^{\\pm \\frac{1}{2}}][([d]!)\\inv]$ and let $\\Delta_{\\tA}(d)$ be an integral form of the Weyl module of highest weight $d \\in \\N$ of the quantised enveloping algebra $\\U_{\\tA}$ of $\\fsl_2$. We exhibit for all positive integers $r$ an explicit cellular structure for $\\End...... of endomorphism algebras, and another which relates the multiplicities of indecomposable summands to the dimensions of simple modules for an endomorphism algebra. Our cellularity result then allows us to prove that knowledge of the dimensions of the simple modules of the specialised cellular algebra above...... is equivalent to knowledge of the weight multiplicities of the tilting modules for $\\U_{\\zeta}(\\fsl_2)$. In the final section we independently determine the weight multiplicities of indecomposable tilting modules for $U_\\zeta(\\fsl_2)$ and the decomposition numbers of the endomorphism algebras. We indicate how...
Homology, Analogy, and Ethology.
Beer, Colin G.
1984-01-01
Because the main criterion of structural homology (the principle of connections) does not exist for behavioral homology, the utility of the ethological concept of homology has been questioned. The confidence with which behavioral homologies can be claimed varies inversely with taxonomic distance. Thus, conjectures about long-range phylogenetic…
Chen, Peiying; Gao, Rongsui; Chen, Shaochun; Pu, Li; Li, Pin; Huang, Ying; Lu, Ling
2012-12-01
Pericentrin is a large coiled-coil protein in mammalian centrosomes that serves as a multifunctional scaffold for anchoring numerous proteins. Recent studies have linked numerous human disorders with mutated or elevated levels of pericentrin, suggesting unrecognized contributions of pericentrin-related proteins to the development of these disorders. In this study, we characterized AnPcpA, a putative homolog of pericentrin-related protein in the model filamentous fungus Aspergillus nidulans, and found that it is essential for conidial germination and hyphal development. Compared to the hyphal apex localization pattern of calmodulin (CaM), which has been identified as an interactive partner of the pericentrin homolog, GFP-AnPcpA fluorescence dots are associated mainly with nuclei, while the accumulation of CaM at the hyphal apex depends on the function of AnPcpA. In addition, the depletion of AnPcpA by an inducible alcA promoter repression results in severe growth defects and abnormal nuclear segregation. Most interestingly, in mature hyphal cells, knockdown of pericentrin was able to significantly induce changes in cell shape and cytoskeletal remodeling; it resulted in some enlarged compartments with condensed nuclei and anucleate small compartments as well. Moreover, defects in AnPcpA significantly disrupted the microtubule organization and nucleation, suggesting that AnPcpA may affect nucleus positioning by influencing microtubule organization.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
W~t -approximation representations over quasi k-Gorenstein algebras
黄兆泳
1999-01-01
The notions of quasi k-Gorenstein algebras and W~t-approximation representations are introduced. The existence and uniqueness (up to projective equivalences) of W~t-approximation representations over quasi k-Gorenstein algebras are established. Some applications of W~t-approximation representations to homologically finite subcategories are given.
The Hall Algebra of Cyclic Serial Algebra
郭晋云
1994-01-01
In this paper, orders <1 and <2 on ((Z)+)nm are introduced and also regarded as orders on the isomorphism classes of finite modules of finite .cyclic algebra R with n simple modules and all the indecomposable projective modules have length m through a one-to-one correspondence between ((Z)+)nm and the isomorphism classes of finite R modules. Using this we prove that the Hall algebra of a cyclic serial algebra is identified with its Loewy subalgebra, and its rational extension has a basis of BPW type, and is a (((Z)+)nm, <2) filtered ring with the associated graded ring as an iterated skew polynomial ring. These results are also generalized to the Hall algebra of a tube over a finite field.
Definition and identification of homology domains.
Lawrence, C B; Goldman, D A
1988-03-01
A method is described for identifying and evaluating regions of significant similarity between two sequences. The notion of a 'homology domain' is employed which defines the boundaries of a region of sequence homology containing no insertions or deletions. The relative significance of different potential homology domains is evaluated using a non-linear similarity score related to the probability of finding the observed level of similarity in the region by chance. The sensitivity of the method is demonstrated by simulating the evolution of homology domains and applying the method to their detection. Several examples of the use of homology domain identification are given.
Structure of classical affine and classical affine fractional W-algebras
Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr [Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul 151-747 (Korea, Republic of)
2015-01-15
We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms of free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.
Evolution algebras and their applications
Tian, Jianjun Paul
2008-01-01
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
Algebra and geometry of Hamilton's quaternions
Krishnaswami, Govind S
2016-01-01
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.
Loop-deformed Poincar\\'e algebra
Mielczarek, Jakub
2013-01-01
In this essay we present evidence suggesting that loop quantum gravity leads to deformation of the local Poincar\\'e algebra within the limit of high energies. This deformation is a consequence of quantum modification of effective off-shell hypersurface deformation algebra. Surprisingly, the form of deformation suggests that the signature of space-time changes from Lorentzian to Euclidean at large curvatures. We construct particular realization of the loop-deformed Poincar\\'e algebra and find that it can be related to curved momentum space, which indicates the relationship with recently introduced notion of relative locality. The presented findings open a new way of testing loop quantum gravity effects.
Elementary n-Lie Algebras%基本n-Lie代数
白瑞蒲; 张艳艳
2007-01-01
In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras. We also give the relations between elementary n-algebras and E-algebras.
The semaphorontic view of homology.
Havstad, Joyce C; Assis, Leandro C S; Rieppel, Olivier
2015-11-01
The relation of homology is generally characterized as an identity relation, or alternatively as a correspondence relation, both of which are transitive. We use the example of the ontogenetic development and evolutionary origin of the gnathostome jaw to discuss identity and transitivity of the homology relation under the transformationist and emergentist paradigms respectively. Token identity and consequent transitivity of homology relations are shown to be requirements that are too strong to allow the origin of genuine evolutionary novelties. We consequently introduce the concept of compositional identity that is grounded in relations prevailing between parts (organs and organ systems) of a whole (organism). We recognize an ontogenetic identity of parts within a whole throughout the sequence of successive developmental stages of those parts: this is an intra-organismal character identity maintained throughout developmental trajectory. Correspondingly, we recognize a phylogenetic identity of homologous parts within two or more organisms of different species: this is an inter-species character identity maintained throughout evolutionary trajectory. These different dimensions of character identity--ontogenetic (through development) and phylogenetic (via shared evolutionary history)--break the transitivity of homology relations. Under the transformationist paradigm, the relation of homology reigns over the entire character (-state) transformation series, and thus encompasses the plesiomorphic as well as the apomorphic condition of form. In contrast, genuine evolutionary novelties originate not through transformation of ancestral characters (-states), but instead through deviating developmental trajectories that result in alternate characters. Under the emergentist paradigm, homology is thus synonymous with synapomorphy. © 2015 The Authors. Journal of Experimental Zoology Part B: Molecular and Developmental Evolution Published by Wiley Periodicals, Inc.
Finite-dimensional (*)-serial algebras
无
2010-01-01
Let A be a finite-dimensional associative algebra with identity over a field k. In this paper we introduce the concept of (*)-serial algebras which is a generalization of serial algebras. We investigate the properties of (*)-serial algebras, and we obtain suficient and necessary conditions for an associative algebra to be (*)-serial.
On the Toroidal Leibniz Algebras
Dong LIU; Lei LIN
2008-01-01
Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras gOC[t±11 ,...,t±v1] in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.
R. A. Borzooei
2006-01-01
Full Text Available We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
Borzooei, R. A.; Dudek, W. A.; Koohestani, N.
2006-01-01
We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Developable algebraic surfaces
CHEN Dongren; WANG Guojin
2004-01-01
An algebraic surface can be defined by an implicit polynomial equation F(x,y,z)=0. In this paper, general characterizations of developable algebraic surfaces of arbitrary degree are presented. Using the shift operators of the subscripts of Bézier ordinates, the uniform apparent discriminants of developable algebraic surfaces to their Bézier ordinates are given directly. To degree 2 algebraic surfaces, which are widely used in computer aided geometric design and graphics, all possible developable surface types are obtained. For more conveniently applying algebraic surfaces of high degree to computer aided geometric design, the notion of ε-quasi-developable surfaces is introduced, and an example of using a quasi-developable algebraic surface of degree 3 to interpolate three curves of degree 2 is given.
The algebra of q-pseudodifferential symbols and the q-W$_{KP}^{n}$ algebra
Mas, J; Mas, Javier; Seco, Marcos
1995-01-01
We construct q-deformations of the W_KP and related W-type algebras. These algebras arise as Gelfand-Dickey brackets on the space of q-pseudodifferential symbols. The construction establishes a continuous correspondence between the Toda lattice and the KP hierarchy at the level of their hamiltonian structures
Algebra Automorphisms of Quantized Enveloping Algebras Uq(■)
查建国
1994-01-01
The algebra automorphisms of the quantized enveloping algebra Uq(g) are discussed, where q is generic. To some extent, all quantum deformations of automorphisms of the simple Lie algebra g have been determined.
Cayley-Dickson and Clifford Algebras as Twisted Group Algebras
Bales, John W
2011-01-01
The effect of some properties of twisted groups on the associated algebras, particularly Cayley-Dickson and Clifford algebras. It is conjectured that the Hilbert space of square-summable sequences is a Cayley-Dickson algebra.
Frobenius morphisms and stable module categories of repetitive algebras
2008-01-01
Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.
Krichever-Novikov type algebras theory and applications
Schlichenmaier, Martin
2014-01-01
Krichever and Novikov introduced certain classes of infinite dimensionalLie algebrasto extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them toa more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are
Construction of the elliptic Gaudin system based on Lie algebra
CAO Li-ke; LIANG Hong; PENG Dan-tao; YANG Tao; YUE Rui-hong
2007-01-01
Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics.The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra.Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, An, Bn, Cn, Dn, and we calculate a classical r-matrix for the elliptic Gaudin system with spin.
Symmetric Extended Ockham Algebras
T.S. Blyth; Jie Fang
2003-01-01
The variety eO of extended Ockham algebras consists of those algealgebra with an additional endomorphism k such that the unary operations f and k commute. Here, we consider the cO-algebras which have a property of symmetry. We show that there are thirty two non-isomorphic subdirectly irreducible symmetric extended MS-algebras and give a complete description of them.2000 Mathematics Subject Classification: 06D15, 06D30
Krichever, Igor M.; Sheinman, Oleg K.
2007-01-01
In this paper we develop a general concept of Lax operators on algebraic curves introduced in [1]. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct the orthogonal and symplectic analogs of Lax operators, prove that they constitute almost graded Lie algebras and construct local central extensions of those Lie algebras.
Albert, A A
1939-01-01
The first three chapters of this work contain an exposition of the Wedderburn structure theorems. Chapter IV contains the theory of the commutator subalgebra of a simple subalgebra of a normal simple algebra, the study of automorphisms of a simple algebra, splitting fields, and the index reduction factor theory. The fifth chapter contains the foundation of the theory of crossed products and of their special case, cyclic algebras. The theory of exponents is derived there as well as the consequent factorization of normal division algebras into direct factors of prime-power degree. Chapter VI con
Algebraic extensions of fields
McCarthy, Paul J
1991-01-01
""...clear, unsophisticated and direct..."" - MathThis textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamenta
Balan, Adriana
2010-01-01
We extend Barr's well-known characterization of the final coalgebra of a $Set$-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a $Set$-monad $\\mathbf{M}$ for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad $\\mathbf{M}$ and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.
Underwood, Robert G
2015-01-01
This text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras, and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields, and linearly recursive sequences. The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforw...
Solomon, Alan D
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
Hijligenberg, N.W. van den; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
N.W. van den Hijligenberg; R. Martini
1995-01-01
textabstractWe discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra
Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra
van den Hijligenberg, N.W.; van den Hijligenberg, N.W.; Martini, Ruud
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of
Fredslund, Jakob; Schauser, Leif; Madsen, Lene Heegaard
2005-01-01
phylogenetically related species and outputs a list of possibly degenerate primer pairs fulfilling a number of criteria, such that the primers have a maximal probability of amplifying orthologous sequences in other phylogenetically related species. Operating on a genome-wide scale, PriFi automates the first steps...
Tubular algebras and affine Kac-Moody algebras
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.
Tubular algebras and affine Kac-Moody algebras
Zheng-xin CHEN; Ya-nan LIN
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)C1/I(A) of complex degenerate composition Lie algebras L(A)C1 by some ideals, where L(A)C1 is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)C1/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)C1 generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)C1 generated by simple A-modules.
Cătălin Ciupală
2005-02-01
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: -algebras. We also define the Frölicher–Nijenhuis bracket in the non-commutative geometry on -algebras.
Language Approaches to Beginning Algebra.
Rotman, Jack W.
1990-01-01
Ideas which apply language concepts to the study of algebra are presented. Discussed are algebraic notation, vocabulary, vocalization, and written assignments. The careful use of algebraic language in mathematics classes is emphasized. (CW)
GENERALIZED FUZZY FILTERS OF BL-ALGEBRAS
无
2007-01-01
The concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set is considered. In fact, this is a generalization of quasi-coincidence of a fuzzy point with a fuzzy set. By using this new idea, the notion of interval valued (∈, ∈∨q)-fuzzy filters in BL-algebras which is a generalization of fuzzy filters of BL-algebras, is defined, and related properties are investigated. In particular, the concept of a fuzzy subgroup with thresholds is extended to the concept of an interval valued fuzzy filter with thresholds in BL-algebras.
Algebraic Geometry and Number Theory Summer School
Sarıoğlu, Celal; Soulé, Christophe; Zeytin, Ayberk
2017-01-01
This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.
Orbifold Riemann surfaces and geodesic algebras
Chekhov, L O [Steklov Mathematical Institute, Moscow (Russian Federation)], E-mail: chekhov@mi.ras.ru
2009-07-31
We study the Teichmueller theory of Riemann surfaces with orbifold points of order 2 using the fat graph technique. The previously developed technique of quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebras of geodesic functions is applicable to the surfaces with orbifold points. We describe classical and quantum braid group relations for particular sets of geodesic functions corresponding to A{sub n} and D{sub n} algebras and describe their central elements for the Poisson and quantum algebras.
Adinkras for Clifford Algebras, and Worldline Supermultiplets
Doran, C F; Gates, S J; Hübsch, T; Iga, K M; Landweber, G D; Miller, R L
2008-01-01
Adinkras are a graphical depiction of representations of the N-extended supersymmetry algebra in one dimension, on the worldline. These diagrams represent the component fields in a supermultiplet as vertices, and the action of the supersymmetry generators as edges. In a previous work, we showed that the chromotopology (topology with colors) of an Adinkra must come from a doubly even binary linear code. Herein, we relate Adinkras to Clifford algebras, and use this to construct, for every such code, a supermultiplet corresponding to that code. In this way, we correlate the well-known classification of representations of Clifford algebras to the classification of Adinkra chromotopologies.
K-Theory for group C^*-algebras
Baum, Paul F
2009-01-01
These notes are based on a lecture course given by the first author in the Sedano Winter School on K-theory held in Sedano, Spain, on January 22-27th of 2007. They aim at introducing K-theory of C^*-algebras, equivariant K-homology and KK-theory in the context of the Baum-Connes conjecture.
Hestenes, David
2015-01-01
This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas. These same techniques, in the form of the ‘Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, a...
Algebraic topology and concurrency
Fajstrup, Lisbeth; Raussen, Martin; Goubault, Eric
2006-01-01
We show in this article that some concepts from homotopy theory, in algebraic topology,are relevant for studying concurrent programs. We exhibit a natural semantics of semaphore programs, based on partially ordered topological spaces, which are studied up to “elastic deformation” or homotopy......, giving information about important properties of the program, such as deadlocks, unreachables, serializability, essential schedules, etc. In fact, it is not quite ordinary homotopy that has to be used, but rather a “directed homotopy” that does not reverse the flow of time. We show some of the essential...... differences between ordinary and directed homotopy through examples. We also relate the topological view to a combinatorial view of concurrent programs closer to transition systems, through the notion of a cubical set. Finally we apply some of these concepts to the proof of the safeness of a two...
Uniform Exponential Growth in Algebras /
Briggs, Christopher Alan
2013-01-01
We consider uniform exponential growth in algebras. We give conditions for the uniform exponential growth of descending-filtered algebras and prove that an N-graded algebra has uniform exponential growth if it has exponential growth. We use this to prove that Golod- Shafarevich algebras and group algebras of Golod- Shafarevich groups have uniform exponential growth. We prove that the twisted Laurent extension of a free commutative polynomial algebra with respect to an endomorphism with some e...
Xu-Ping Yu; Jun-Li Zhu; Xue-Ping Yao; Shi-Cheng He; Hai-Ning Huang; Wei-Liang Chen; Yong-Hao Hu; De-Bao Li
2005-01-01
AIM: To identify the gene (s) related to the antagonistic activity of Enterobacter cloacae B8 and to elucidate its antagonistic mechanism.METHODS: Transposon-mediated mutagenesis and tagging method and cassette PCR-based chromosomal walking method were adopted to isolate the mutant strain(s) of B8 that lost the antagonistic activity and to clone DNA fragments around Tn5 insertion site. Sequence compiling and open reading frame (ORF) finding were done with DNAStar program and homologous sequence and conserved domain searches were performed with BlastN or BlastP programs at www. ncbi.nlm.nih.gov. To verify the gene involved in the antagonistic activity, complementation of a full-length clone of the anrFgene to the mutant B8F strain was used.RESULTS: A 3 321 bp contig around the Tn5 insertion site was obtained and an ORF of 2 634 bp in length designated as anrFgene encoding for a 877 aa polyketide synthase-like protein was identified. It had a homology of 83% at the nucleotide level and 79% ID/87% SIM at the protein level, to the admM gene of Pantoea agglornerans andrimid biosynthetic gene cluster (AY192157). The Tn5was inserted at 2 420 bp of the gene corresponding to the COG3319 (the thioesterase domain of type Ⅰ polyketide synthase) coding region on B8F. The antagonistic activity against Xanthomonas oryzae pv. oryzae was resumed with complementation of the full-length anrFgene to the mutant B8F.CONCLUSION: The anrFgene obtained is related to the antagonistic activity of B8, and the antagonistic substances produced by B8 are andrimid and/or its analogs.
From the Virasoro Algebra to Krichever--Novikov Type Algebras and Beyond
Schlichenmaier, Martin
2013-01-01
Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the two-point case. Schlichenmaier studied the most general multi-point situation with arbitrary splittings. Here we will review the path of developments from the Virasoro algebra to its higher genus and multi-point analogs. The starting point will be a Poisson algebra structure on the space of meromorphic forms of all weights. As sub-structures the vector field algebras, function algebras, Lie superalgebras and the related current algebras show up. All these algebras will be almost-graded...
Generalized conformal realizations of Kac-Moody algebras
Palmkvist, Jakob
2009-01-01
We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n =1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3×3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n =2. Moreover, we obtain their infinite-dimensional extensions for n ≥3. In the case of 2×2 matrices, the resulting Lie algebras are of the form so(p +n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).
Diversity-multiplexing Gain Tradeoff: a Tool in Algebra?
Vehkalahti, Roope
2011-01-01
Since the invention of space-time coding numerous algebraic methods have been applied in code design. In particular algebraic number theory and central simple algebras have been on the forefront of the research. In this paper we are turning the table and asking whether information theory can be used as a tool in algebra. We will first derive some corollaries from diversity-multiplexing gain (DMT) bounds by Zheng and Tse and later show how these results can be used to analyze the unit group of orders of certain division algebras. The authors do not claim that the algebraic results are new, but we do find that this interesting relation between algebra and information theory is quite surprising and worth pointing out.
de Silva, Vin; Salamon, Dietmar
2012-01-01
We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology.
Automorphism groups of some algebras
PARK; Hong; Goo; LEE; Jeongsig; CHOI; Seul; Hee; NAM; Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n).
Automorphism groups of some algebras
PARK Hong Goo; LEE Jeongsig; CHOI Seul Hee; CHEN XueQing; NAM Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m, m+n).
Non-solvable contractions of semisimple Lie algebras in low dimension
Campoamor-Stursberg, R [Dpto. GeometrIa y TopologIa, Fac. CC. Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, E-28040 Madrid (Spain)
2007-05-18
The problem of non-solvable contractions of Lie algebras is analysed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras of dimension n {<=} 8, and obtain the non-solvable contractions of the latter class of algebras.
Crossed products of Banach algebras. I
Dirksen, Sjoerd; Wortel, Marten
2011-01-01
We construct a crossed product Banach algebra from a Banach algebra dynamical system $(A,G,\\alpha)$ and a given uniformly bounded class $R$ of continuous covariant Banach space representations of that system. If $A$ has a bounded left approximate identity, and $R$ consists of non-degenerate continuous covariant representations only, then the non-degenerate bounded representations of the crossed product are in bijection with the non-degenerate $R$-continuous covariant representations of the system. This bijection, which is the main result of the paper, is also established for involutive Banach algebra dynamical systems and then yields the well-known representation theoretical correspondence for the crossed product $C^*$-algebra as commonly associated with a $C^*$-algebra dynamical system as a special case. Taking the algebra $A$ to be the base field, the crossed product construction provides, for a given non-empty class of Banach spaces, a Banach algebra with a relatively simple structure and with the property...
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY%弱代数性自反与强代数性自反
陶常利; 鲁世杰; 陈培鑫
2002-01-01
Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Derived equivalence of algebras
杜先能
1997-01-01
The derived equivalence and stable equivalence of algebras RmA and RmB are studied It is proved, using the tilting complex, that RmA and RmB are derived-equivalent whenever algebras A and B are derived-equivalent
Herriott, Scott R.; Dunbar, Steven R.
2009-01-01
The common understanding within the mathematics community is that the role of the college algebra course is to prepare students for calculus. Though exceptions are emerging, the curriculum of most college algebra courses and the content of most textbooks on the market both reflect that assumption. This article calls that assumption into question…
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…