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.
GOLDMAN ALGEBRA, OPERS AND THE SWAPPING ALGEBRA
Labourie, François
2012-01-01
We define a Poisson Algebra called the {\\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\\mathsf{SL}_n(\\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symple...
Operator algebras and topology
International Nuclear Information System (INIS)
These notes, based on three lectures on operator algebras and topology at the 'School on High Dimensional Manifold Theory' at the ICTP in Trieste, introduce a new set of tools to high dimensional manifold theory, namely techniques coming from the theory of operator algebras, in particular C*-algebras. These are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. A central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. This is an isomorphism conjecture, as discussed in the talks of Luck, but with a certain special flavor. Nevertheless, it has important direct applications to the topology of manifolds, it implies e.g. the Novikov conjecture. In the first chapter, the Baum-Connes conjecture will be explained and put into our context. Another application of the Baum-Connes conjecture is to the positive scalar curvature question. This will be discussed by Stephan Stolz. It implies the so-called 'stable Gromov-Lawson-Rosenberg conjecture'. The unstable version of this conjecture said that, given a closed spin manifold M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second chapter. The third chapter introduces another set of invariants, also using operator algebra techniques, namely L2-cohomology, L2-Betti numbers and other L2-invariants. These invariants, their basic properties, and the central questions about them, are introduced in the third chapter. (author)
Simple Algebras of Invariant Operators
Institute of Scientific and Technical Information of China (English)
Xiaorong Shen; J.D.H. Smith
2001-01-01
Comtrans algebras were introduced in as algebras with two trilinear operators, a commutator [x, y, z] and a translator , which satisfy certain identities. Previously known simple comtrans algebras arise from rectangular matrices, simple Lie algebras, spaces equipped with a bilinear form having trivial radical, spaces of hermitian operators over a field with a minimum polynomial x2+1. This paper is about generalizing the hermitian case to the so-called invariant case. The main result of this paper shows that the vector space of n-dimensional invariant operators furnishes some comtrans algebra structures, which are simple provided that certain Jordan and Lie algebras are simple.
Operator product expansion algebra
Energy Technology Data Exchange (ETDEWEB)
Holland, Jan [CPHT, Ecole Polytechnique, Paris-Palaiseau (France)
2014-07-01
The Operator Product Expansion (OPE) is a theoretical tool for studying the short distance behaviour of products of local quantum fields. Over the past 40 years, the OPE has not only found widespread computational application in high-energy physics, but, on a more conceptual level, it also encodes fundamental information on algebraic structures underlying quantum field theories. I review new insights into the status and properties of the OPE within Euclidean perturbation theory, addressing in particular the topics of convergence and ''factorisation'' of the expansion. Further, I present a formula for the ''deformation'' of the OPE algebra caused by a quartic interaction. This formula can be used to set up a novel iterative scheme for the perturbative computation of OPE coefficients, based solely on the zeroth order coefficients (and renormalisation conditions) as initial input.
Operator product expansion algebra
Energy Technology Data Exchange (ETDEWEB)
Holland, Jan [School of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG (United Kingdom); Hollands, Stefan [School of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG (United Kingdom); Institut für Theoretische Physik, Universität Leipzig, Brüderstr. 16, Leipzig, D-04103 (Germany)
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
Operation of Algebraic Fractions
Institute of Scientific and Technical Information of China (English)
2008-01-01
<正>The first step in factorizing algebraic expressions is to take out the common factors of all the terms of the expression.For example,2x~2+14x+24=2(x~2+7x+12)=2(x+3)(x+4) The three identities are also useful in factorizing some quadratic expressions:
A Jacobi identity for intertwining operator algebras
Huang, Y Z
1997-01-01
We find a Jacobi identity for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. We prove that intertwining operators for a suitable vertex operator algebra satisfy this Jacobi identity. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.
Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras
Ashihara, Takahiro; Miyamoto, Masahiko
2008-01-01
If a vertex operator algebra $V=\\oplus_{n=0}^{\\infty}V_n$ satisfies $\\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In t...
On Axiomatic Approaches to Intertwining Operator Algebras
Chen, Ling
2015-01-01
We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator algebras were given in [H9] but some of the proofs were postponed to an unpublished monograph. In this paper, we give the proofs of these results in [H9] and we formulate and prove results for general intertwining operator algebras without assuming that the modules involved are irreducible. In particular, we construct fusing and braiding isomorphisms for general intertwining operator algebras and prove that they satisfy the genus-zero Moore-Seiberg equations. We show that the Jacobi identity for intertwining operator algebras is equivalent to generalized rationality, commutativity and associativity properties of intertwining operator algebras. We introduce the locality for intertwining operator algebras and show that the Jacobi identity is equi...
Reflexive Operator Algebras on Banach Spaces
Merlevède, Florence; Peligrad, Costel; Peligrad, Magda
2012-01-01
In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their...
Spatial-Operator Algebra For Robotic Manipulators
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Exceptional Vertex Operator Algebras and the Virasoro Algebra
Tuite, Michael P.
2008-01-01
We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Exam...
Compact Weighted Composition Operators on Function Algebras
TAKAGI, Hiroyuki
1988-01-01
A weighted endomorphism of an algebra is an endomorphism followed by a multiplier. In [6] and [4], H. Kamowitz characterized compact weighted endomorphisms of $C(X)$ and the disc algebra. In this note we define a weighted composition operator on a function algebra as a generalization of a weighted endomorphism, and characterize compact weighted composition operators on a function algebra satisfying a certain condition [Theorem 2]. This theorem not only includes Kamowitz's results as ...
On ultraproducts of operator algebras
Institute of Scientific and Technical Information of China (English)
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.
Lectures on algebraic quantum field theory and operator algebras
Energy Technology Data Exchange (ETDEWEB)
Schroer, Bert [Berlin Univ. (Germany). Institut fuer Theoretische Physik. E-mail: schroer@cbpf.br
2001-04-01
In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as why mathematicians are/should be interested in algebraic quantum field theory would be equally fitting. besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to frontier research problems in mathematical. (author)
Niibori, Hidekazu; Sagaki, Daisuke
2009-01-01
Let $r \\in \\BC$ be a complex number, and $d \\in \\BZ_{\\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\\Vam$ is not simple), we give a gene...
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...
Operator algebras for multivariable dynamics
Davidson, Kenneth R.; Katsoulis, Elias G.
2007-01-01
Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\\tau_i:X \\to X$ for $1 \\le i \\le n$. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\\A(X, \\tau)$ and the semicrossed product $\\rC_0(X)\\times_\\tau\\Fn$. We introduce a concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise conjugacy class of the sy...
Weighted composition operators and locally convex algebras
Institute of Scientific and Technical Information of China (English)
Edoardo; Vesentini
2005-01-01
The Gleason-Kahane-Zelazko theorem characterizes the continuous homomorphism of an associative, locally multiplicatively convex, sequentially complete algebra A into the field C among all linear forms on A. This characterization will be applied along two different directions. In the case in which A is a commutative Banach algebra, the theorem yields the representation of some classes of continuous linear maps A: A → A as weighted composition operators, or as composition operators when A is a continuous algebra endomorphism. The theorem will then be applied to explore the behaviour of continuous linear forms on quasi-regular elements, when A is either the algebra of all Hilbert-Schmidt operators or a Hilbert algebra.
Homogeneous conformal averaging operators on semisimple Lie algebras
Kolesnikov, Pavel
2014-01-01
In this note we show a close relation between the following objects: Classical Yang---Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular part of a solution of CYBE (in the operator form) on a Lie algebra $\\mathfrak g$ determines an averaging operator on the corresponding current conformal algebra $\\mathrm{Cur} \\mathfrak g$. For a finite-dimensional semisimple Lie algebra $\\mathfrak g$, we desc...
Duality theories for Boolean algebras with operators
Givant, Steven
2014-01-01
In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as a hybrid of the first two. This text will be of interest to graduate students and researchers in the fields of mathematics, computer science, logic, and philosophy who are interested in exploring special or general classes of Boolean algebras with operators. Readers should be familiar with the basic arithmetic and theory of Boolean algebras, as well as the fundamentals of point-set topology.
Nijenhuis Operators on n-Lie Algebras
Jie-Feng, Liu; Yun-He, Sheng; Yan-Qiu, Zhou; Cheng-Ming, Bai
2016-06-01
In this paper, we study (n ‑ 1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples. Supported by National Natural Science Foundation of China under Grant Nos. 11471139, 11271202, 11221091, 11425104, Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022, and National Natural Science Foundation of Jilin Province under Grant No. 20140520054JH
Ahmed, Tarek Sayed
2013-01-01
We give some general theorems on free algebras of varieties of Boolean algebras with operators; a hitherto new result is obtained for Pinter's substitution algebras. For n\\geq 3, and m>1, there is a generating set of the free algebra freely generated by m elements, which is not a free set of generators.
C*-algebras and operator theory
Murphy, Gerald J
1990-01-01
This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.
Automorphism groups and derivation algebras of finitely generated vertex operator algebras
Dong, C.; Griess Jr., R. L.
2002-01-01
We investigate the general structure of the automorphism group and the Lie algebra of derivations of a finitely generated vertex operator algebra. The automorphism group is isomorphic to an algebraic group. Under natural assumptions, the derivation algebra has an invariant bilinear form and the ideal of inner derivations is nonsingular.
Almost-graded central extensions of Lax operator algebra
Schlichenmaier, Martin
2011-01-01
Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for $\\gl(n)$, with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. These results ...
Primitive parallel operations for computational linear algebra
Energy Technology Data Exchange (ETDEWEB)
Panetta, J.
1985-01-01
This work is a small step in the direction of code portability over parallel and vector machines. The proposal consists of a style of programming and a set of parallel operators built over abstract data types. Objects and operators are directed to the Computational Linear Algebra area, although the principles of the proposal can be applied to any other area. A subset of the operators was implemented on a 64-processor, distributed memory MIMD machine, and the results are that computationally intensive operators achieve asymptotically optimal speed-ups, but data movement operators are inefficient, some even intrinsically sequential.
Mappings of Multimode Bose Algebra Preserving Number Operators
Doresic, Miroslav; Meljanac, Stjepan; Milekovic, Marijan
1994-01-01
We define a class of deformed multimode oscillator algebras which possess number operators and can be mapped to multimode Bose algebra.We construct and discuss the states in the Fock space and the corresponding number operators.
Framed vertex operator algebras, codes and the moonshine module
Dong, C; Hoehn, G
1997-01-01
For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.
Framed vertex operator algebras, codes and the moonshine module
Dong, C.; Griess Jr., R. L.; Hoehn, G.
1997-01-01
For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator alg...
DERIVATIONS ON DIFFERENTIAL OPERATOR ALGEBRA AND WEYL ALGEBRA
Institute of Scientific and Technical Information of China (English)
CHENCAOYU
1996-01-01
Let L be an n-dimensional nilpotent Lie algebra with a basis{x1…,xn),and every xi acts as a locally nilpotent derivation on algebra A. This paper shows that there exists a set of derivations{y1,…,yn}on U(L) such that (A#U(L))#k{y,1,…,yn] is ismorphic to the Weyl algebra An(A).The author also uses the de4rivations to obtain a necessary and sufficient condition for a finite dimesional Lie algebra to be nilpotent.
Topological isomorphisms for some universal operator algebras
Hartz, Michael
2012-01-01
Let $I$ be a radical homogeneous ideal of complex polynomials in $d$ variables, and let $\\mathcal A_I$ be the norm-closed non-selfadjoint algebra generated by the compressions of the $d$-shift on Drury-Arveson space $H^2_d$ to the co-invariant subspace $H^2_d \\ominus I$. Then $\\mathcal A_I$ is the universal operator algebra for commuting row contractions subject to the relations in $I$. In this note, we study the question, under which conditions there are topological isomorphisms between two such algebras $\\mathcal A_I$ and $\\mathcal A_J$. We provide a positive answer to a conjecture of Davdison, Ramsey and Shalit: that $\\mathcal A_I$ and $\\mathcal A_J$ are topologically isomorphic if and only if there is an invertible linear map $A$ on $\\mathbb C^d$ which maps the vanishing locus of $J$ isometrically onto the vanishing locus of $I$. Most of the proof is devoted to showing that finite algebraic sums of full Fock spaces over subspaces of $\\mathbb C^d$ are closed. This allows us to show that the map $A$ induces...
Invariant subspaces of subgraded Lie algebras of compact operators
Kennedy, Matthew; Turovskii, Yuri
2008-01-01
We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the structure of such algebras. As an application, we prove a number of results on the existence of invariant subspaces for algebraic structures of compact operators. Along the way we obtain new criteria for the triangularizability of a Lie algebra of compact operators.
The radical of a vertex operator algebra
Dong, C.; Li, H.; Mason, G.; Montague, P
1996-01-01
The radical $J(V)$ of a vertex operator algebra $V$ is defined to be the subspace of $V$ consisting of vectors $v$ such that the zero mode $o(v)=0$ on $V$ where $o(v)=v_{wt v-1}$ if $v$ is homogeneous. We establish various facts about $o(v),$ including the determination of $J(V)$ which is shown to be essentially equal to $(L(0)+L(-1))V.$
Spatial Operator Algebra for multibody system dynamics
Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.
1992-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Bispectral algebras of commuting ordinary differential operators
Bakalov, B N; Yakimov, M T
1997-01-01
We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N. It enables us to obtain all previously known classes or examples of bispectral operators. The suggested method is completely algorithmic, which allows us to present explicitly new examples. We conjecture that the class built in the present paper exhausts all bispectral scalar operators. This paper is the third of a series of papers (hep-th/9510211, q-alg/9602010, q-alg/9602012) on the bispectral problem.
Rota-Baxter operators on Witt and Virasoro algebras
Gao, Xu; Liu, Ming; Bai, Chengming; Jing, Naihuan
2016-10-01
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
Algebras of unbounded operators and physical applications: a survey
Bagarello, Fabio
2009-01-01
After an historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance in physical applications.
On Monotone Product of Operator Algebras
Institute of Scientific and Technical Information of China (English)
Wen Ming WU; Li Guang WANG
2007-01-01
In this note, we give complete descriptions of the structure of the monotone product of two yon Neumann algebras and two C*-algebras. We show that the monotone product of two simple yon Neumann algebras and C*-algebras aren't simple again. We also show that the monotone product of two hyperfinite von Neumann algebras is again hyperfinite and determine the type of the monotone product of two factors.
International Nuclear Information System (INIS)
The graphic technique of 'trees' developed in the previous paper is used for the construction of the q-analogue of the tensor operator algebra. The adjoint action of the suq(2) generator on tensor operators is discussed and adjoint R-matrix is introduced. A set of formulae for the calculation of the matrix elements of tensor operators and their combinations is derived. As an application, the recurrent relations for the suq(2) Clebsh-Gordan and Racah coefficients are obtained
Spatial-Operator Algebra For Flexible-Link Manipulators
Jain, Abhinandan; Rodriguez, Guillermo
1994-01-01
Method of computing dynamics of multiple-flexible-link robotic manipulators based on spatial-operator algebra, which originally applied to rigid-link manipulators. Aspects of spatial-operator-algebra approach described in several previous articles in NASA Tech Briefs-most recently "Robot Control Based on Spatial-Operator Algebra" (NPO-17918). In extension of spatial-operator algebra to manipulators with flexible links, each link represented by finite-element model: mass of flexible link apportioned among smaller, lumped-mass rigid bodies, coupling of motions expressed in terms of vibrational modes. This leads to operator expression for modal-mass matrix of link.
A spatial operator algebra for manipulator modeling and control
Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.
1991-01-01
A recently developed spatial operator algebra for manipulator modeling, control, and trajectory design is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics.
Some operator algebraic techniques in loop quantum gravity
Energy Technology Data Exchange (ETDEWEB)
Kaminski, Diana
2011-08-19
The aim of the work is to give a mathematical description for a theory of Quantum Gravity. The following objects are studied (1) the two known algebras of Quantum Gravity in the Loop Quantum Gravity approach, which are the holonomy-flux {sup *}-algebra and the Weyl C{sup *}-algebra, (2) modifications of these algebras and new algebras of Loop Quantum Gravity,(3) states and representations of the algebras and (4) the concept of quantum constraints and KMS-Theory in Loop Quantum Gravity. In comparison to other theories of quantum physics it is obtained that, the two known algebras are not the only algebras in the Loop Quantum Gravity framework. Surprisingly, a huge amount of different algebras in Loop Quantum Gravity are presented in this dissertation. The idea of the construction of these algebras is to establish a finite set of operators, which generate (in the sense of Woronowicz, Schmuedgen and Inoue) the different O{sup *}- or C{sup *}-algebras of quantum gravity. In the Loop Quantum Gravity approach usually the basic classical variables are connections and electric fluxes. Studying the three constraints appearing in the canonical quantisation of classical general relativity in the ADM-formalism some other variables like curvature appear. Consequently, the main difficulty of the quantisation of gravity is to find a suitable replacement of the set of elementary classical variables and constraints. The important aim of this dissertation is to modify the holonomy-flux {sup *}-algebra and the Weyl C{sup *}-algebra in such a way that the set of constraints of Quantum Gravity in the formulation of the Ashtekar variables is a sub-algebra of the modified O{sup *}- or C{sup *}-algebra, which is generated by a set of the operators associated to holonomies, fluxes, diffeomorphisms and in some cases even the curvature. An exceptional algebra satisfying this property, is proposed.
An $E_8$-approach to the moonshine vertex operator algebra
Shimakura, Hiroki
2010-01-01
In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\\sqrt2E_8}^+$, and describe it by the quadratic space over $\\F_2$ associated to $V_{\\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex operator algebra analogue of "An $E_8$-approach to the Leech lattice and the Conway group" by Lepowsky and Meurman. Moreover, we find new analogies among the moonshine vertex operator algebra, the Leech lattice and the extended binary Golay code.
Campoamor-Stursberg, R
2008-01-01
Given a semidirect product $\\frak{g}=\\frak{s}\\uplus\\frak{r}$ of semisimple Lie algebras $\\frak{s}$ and solvable algebras $\\frak{r}$, we construct polynomial operators in the enveloping algebra $\\mathcal{U}(\\frak{g})$ of $\\frak{g}$ that commute with $\\frak{r}$ and transform like the generators of $\\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\\frak{r}$. Such operators are said to generate a virtual copy of $\\frak{s}$ in $\\mathcal{U}(\\frak{g})$, and allow to compute the Casimir operators of $\\frak{g}$ in closed form, using the classical formulae for the invariants of $\\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.
International Conference on Semigroups, Algebras and Operator Theory
Meakin, John; Rajan, A
2015-01-01
This book discusses recent developments in semigroup theory and its applications in areas such as operator algebras, operator approximations and category theory. All contributing authors are eminent researchers in their respective fields, from across the world. Their papers, presented at the 2014 International Conference on Semigroups, Algebras and Operator Theory in Cochin, India, focus on recent developments in semigroup theory and operator algebras. They highlight current research activities on the structure theory of semigroups as well as the role of semigroup theoretic approaches to other areas such as rings and algebras. The deliberations and discussions at the conference point to future research directions in these areas. This book presents 16 unpublished, high-quality and peer-reviewed research papers on areas such as structure theory of semigroups, decidability vs. undecidability of word problems, regular von Neumann algebras, operator theory and operator approximations. Interested researchers will f...
On the uniqueness of the moonshine vertex operator algebra
Dong, Chongying; Griess Jr., Robert L.; lam, Ching Hung
2005-01-01
It is proved that a vertex operator algebra is isomorphic to the moonshine VOA of Frenkel-Lepowsky-Meurman if it satisfies certain conditions. Our two main theorems establish a weak version of the FLM uniqueness conjecture for the moonshine vertex operator algebra. We believe that these are the first such results.
Norton's Trace Formulae for the Griess Algebra of a Vertex Operator Algebra with Larger Symmetry
Matsuo, Atsushi
2000-01-01
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module $V^\
Norton's trace formulae for the Griess algebra of a vertex operator algebra with larger symmetry
International Nuclear Information System (INIS)
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module of I. B. Frenkel et al. (1984), with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c=1/2, the W3 algebra at c=4/5 or the W4 algebra at c=1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster. (orig.)
Algebraic characterization of RNA operations for DNA-based computation
Institute of Scientific and Technical Information of China (English)
LI Shuchao
2004-01-01
Any RNA strand can be presented by a word in the language X*, where X={A,C,G,U}. By encoding A as 010, C as 000, G as 111, and U as 101, the RNA operations can be considered as the performance of concatenation, union, reverse, complement, in terms of the algebraic characterization. The concatenation and union play the roles of multiplication and addition over some algebraic structures, respectively. The rest of the operations turn out to be the homomorphisms or anti-homomorphisms of these algebraic structures. Using this technique, we find the relationship among these RNA operations.
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,...
Why Do the Quantum Observables Form a Jordan Operator Algebra?
Niestegge, Gerd
2010-01-01
The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.
Semantics for a Quantum Programming Language by Operator Algebras
Cho, K.
2014-01-01
This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite category of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger's first-order functional quantum programming la...
Algebraic Properties of Toeplitz Operators on the Polydisk
Directory of Open Access Journals (Sweden)
Bo Zhang
2011-01-01
Full Text Available We discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydisk Dn. Firstly, we introduce Toeplitz operators with quasihomogeneous symbols and property (P. Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite rank product problem for Toeplitz operators on the polydisk.
Operator algebra of free conformal currents via twistors
Energy Technology Data Exchange (ETDEWEB)
Gelfond, O.A. [Institute of System Research of Russian Academy of Sciences, Nakhimovsky prospect 36-1, 117218 Moscow (Russian Federation); Vasiliev, M.A., E-mail: vasiliev@lpi.ru [I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute, Leninsky prospect 53, 119991 Moscow (Russian Federation)
2013-11-21
Operator algebra of (not necessarily free) higher-spin conformal conserved currents in generalized matrix spaces, that include 3d Minkowski space–time as a particular case, is shown to be determined by an associative algebra M of functions on the twistor space. For free conserved currents, M is the universal enveloping algebra of the higher-spin algebra. Proposed construction greatly simplifies computation and analysis of correlators of conserved currents. Generating function for n-point functions of 3d (super)currents of all spins, built from N free constituent massless scalars and spinors, is obtained in a concise form of certain determinant. Our results agree with and extend earlier bulk computations in the HS AdS{sub 4}/CFT{sub 3} framework. Generating function for n-point functions of 4d conformal currents is also presented.
Jorgensen, PET
1987-01-01
Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas.This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly e
Expressing OLAP operators with the TAX XML algebra
Hachicha, Marouane; Darmont, Jérôme
2008-01-01
With the rise of XML as a standard for representing business data, XML data warehouses appear as suitable solutions for Web-based decision-support applications. In this context, it is necessary to allow OLAP analyses over XML data cubes (XOLAP). Thus, XQuery extensions are needed. To help define a formal framework and allow much-needed performance optimizations on analytical queries expressed in XQuery, having an algebra at one's disposal is desirable. However, XOLAP approaches and algebras from the literature still largely rely on the relational model and/or only feature a small number of OLAP operators. In opposition, we propose in this paper to express a broad set of OLAP operators with the TAX XML algebra.
BRST-operator for quantum Lie algebra and differential calculus on quantum groups
International Nuclear Information System (INIS)
For A Hopf algebra one determined structure of differential complex in two dual external Hopf algebras: A external expansion and in A* dual algebra external expansion. The Heisenberg double of these two Hopf algebras governs the differential algebra for the Cartan differential calculus on A algebra. The forst differential complex is the analog of the de Rame complex. The second complex coincide with the standard complex. Differential is realized as (anti)commutator with Q BRST-operator. Paper contains recursion relation that determines unequivocally Q operator. For Uq(gl(N)) Lie quantum algebra one constructed BRST- and anti-BRST-operators and formulated the theorem of the Hodge expansion
New modular form identities associated to generalized vertex operator algebras
Zuevsky, A.
2015-01-01
New identities appearing from consideration of higher genus characters for generalized vertex operator algebras with a formal VOSA parameter associated to a local coordinate on a self-sewn Riemann surface are considered. Genus two version of twisted product Jacobi identity is reviewed. Further generalization of classical number theory identities for modular forms are proposed.
and as Vertex Operator Extensionsof Dual Affine Algebras
Bowcock, P.; Feigin, B. L.; Semikhatov, A. M.; Taormina, A.
We discover a realisation of the affine Lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with ``dual'' levels (and an auxiliary level-1 algebra). The duality relation between the levels is . We construct the representation of on a sum of tensor products of , , and modules and decompose it into a direct sum over the spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs. Conversely, we show how the representations are constructed from representations.
Some topics pertaining to algebras of linear operators
Semmes, Stephen
2002-01-01
On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic geometry as on a graph, for instance. Of course this is a common theme which is considered in numerous settings. From an analysts' perspective, compact groups, their representations, and more general topological groups and their representations are basic object...
Spatial operator algebra for flexible multibody dynamics
Jain, A.; Rodriguez, G.
1993-01-01
This paper presents an approach to modeling the dynamics of flexible multibody systems such as flexible spacecraft and limber space robotic systems. A large number of degrees of freedom and complex dynamic interactions are typical in these systems. This paper uses spatial operators to develop efficient recursive algorithms for the dynamics of these systems. This approach very efficiently manages complexity by means of a hierarchy of mathematical operations.
Logic TK: Algebraic Notions from Tarski’s Consequence Operator
Directory of Open Access Journals (Sweden)
Hércules A. Feitosa
2010-04-01
Full Text Available Tarski presented his definition of consequence operator to explain the most important notions which any logical consequence concept must contemplate. A Tarski space is a pair constituted by a nonempty set and a consequence operator. This structure characterizes an almost topological space. This paper presents an algebraic view of the Tarski spaces and introduces a modal propositional logic which has as a model exactly the closed sets of a Tarski space.
Robot Control Based On Spatial-Operator Algebra
Rodriguez, Guillermo; Kreutz, Kenneth K.; Jain, Abhinandan
1992-01-01
Method for mathematical modeling and control of robotic manipulators based on spatial-operator algebra providing concise representation and simple, high-level theoretical frame-work for solution of kinematical and dynamical problems involving complicated temporal and spatial relationships. Recursive algorithms derived immediately from abstract spatial-operator expressions by inspection. Transition from abstract formulation through abstract solution to detailed implementation of specific algorithms to compute solution greatly simplified. Complicated dynamical problems like two cooperating robot arms solved more easily.
ALGEBRAIC METHODS IN PARTIAL DIFFERENTIAL OPERATORS
Institute of Scientific and Technical Information of China (English)
Djilali Behloul
2005-01-01
In this paper we build a class of partial differential operators L having the following property: if u is a meromorphic function in Cn and Lu is a rational function A/q, with q homogenous, then u is also a rational function.
Genus Two Zhu Theory for Vertex Operator Algebras
Gilroy, Thomas
2015-01-01
We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion expressing an arbitrary genus two $n$--point correlation function in terms of $(n-1)$--point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half plane for holomorphic $1$--forms, the normalised bidifferential of the second kind and the Heisenberg partition function. We also prove that the holomorphic mapping from the sewing parameter domain to the Siegel upper half plane is injective but not surjective.
Wilson operator algebras and ground states for coupled BF theories
Tiwari, Apoorv; Chen, Xiao; Ryu, Shinsei
2016-01-01
The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theori...
Algebraic Quantization, Good Operators and Fractional Quantum Numbers
Aldaya Valverde, Víctor; Calixto Molina, Manuel; Guerrero García, Julio
1995-01-01
The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the “failure” of the Ehrenfest theorem is clarified in terms of the already defined notion of good (and bad) operators. The analysis of “constrained” Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric ...
Spatial operator algebra framework for multibody system dynamics
Rodriguez, G.; Jain, Abhinandan; Kreutz, K.
1989-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Analysis on singular spaces: Lie manifolds and operator algebras
Nistor, Victor
2016-07-01
We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called "Lie manifolds" -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.
Exceptional Lie Algebra $E_{7(-25)}$ (Multiplets and Invariant Differential Operators)
Dobrev, V K
2008-01-01
In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra $E_{7(-25)}$. Our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of $n$-dimensional Minkowski space-time. This class of algebras is identified and summarized in a table. Another motivation is related to the AdS/CFT correspondence. We give the multiplets of indecomposable elementary representations, including the necessary data for all relevant invariant differential operators.
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.
Disjunctive normal forms for any class of Boolean algebras with operators
Khaled, Mohamed
2015-01-01
Disjunctive normal forms can provide elegant and constructive proofs of many standard results such as completeness, decidability and so on. They were also used to show non atomicity of some free algebras of specific Boolean algebras with operators. Here, we generalize the normal forms for any class of Boolean algebras with operators.
Institute of Scientific and Technical Information of China (English)
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.
Norton's trace formulae for the Griess algebra of a vertex operator algebra with larger symmetry
Energy Technology Data Exchange (ETDEWEB)
Matsuo, A. [Cambridge Univ. (United Kingdom). Dept. of Pure Mathematics and Mathematical Statistics
2001-12-01
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module of I. B. Frenkel et al. (1984), with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c=1/2, the W{sub 3} algebra at c=4/5 or the W{sub 4} algebra at c=1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster. (orig.)
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
Lie Subalgebras in a Certain Operator Lie Algebra with Involution
Institute of Scientific and Technical Information of China (English)
Shan Li SUN; Xue Feng MA
2011-01-01
We show in a certain Lie'-algebra,the connections between the Lie subalgebra G+:＝G+G*+[G,G*],generated by a Lie subalgebra G,and the properties of G.This allows us to investigate some useful information about the structure of such two Lie subalgebras.Some results on the relations between the two Lie subalgebras are obtained.As an application,we get the following conclusion:Let A (∪) B(X)be a space of self-adjoint operators and L:＝A ⊕ iA the corresponding complex Lie*-algebra.G+＝G+G*+[G,G*]and G are two LM-decomposable Lie subalgebras of,L with the decomposition G+＝R(G+)+S,G＝RG+SG,and RG (∪) R(C+).Then G+ is ideally finite iff RG+:＝RG+RG*+[RG,RG*]is a quasisolvable Lie subalgebra,SG+:＝SG+SG*+[SG,SG*]is an ideally finite semisimple Lie subalgebra,and [RG,SG]＝[RG*,SG]＝{0}.
S-numbers of elementary operators on C*-algebras
Anoussis, M; Todorov, I G
2008-01-01
We study the s-numbers of elementary operators acting on C*-algebras. The main results are the following: If $\\tau$ is any tensor norm and $a,b\\in B(H)$ are such that the sequences $s(a),s(b)$ of their singular numbers belong to a stable Calkin space $J$ then the sequence of approximation numbers of $a\\otimes_{\\tau} b$ belongs to $J$. If $A$ is a C*-algebra, $J$ is a stable Calkin space, $s$ is an s-number function, and $a_i, b_i \\in A,$ $i=1,...,m$ are such that $s(\\pi(a_i)), s(\\pi(b_i)) \\in J$, $i=1,...,m$ for some faithful representation $\\pi$ of $A$ then $s(\\sum_{i=1}^{m} M_{a_i,b_i})\\in J$. The converse implication holds if and only if the ideal of compact elements of $A$ has finite spectrum. We also prove a quantitative version of a result of Ylinen.
Algebra of 2D periodic operators with local and perpendicular defects
DEFF Research Database (Denmark)
Kutsenko, Anton
2016-01-01
We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm for finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations on some matrix-valued functions and a few number of int...
-Orthomorphisms and -Linear Operators on the Order Dual of an -Algebra
Directory of Open Access Journals (Sweden)
Ying Feng
2012-01-01
Full Text Available We consider the -orthomorphisms and -linear operators on the order dual of an -algebra. In particular, when the -algebra has the factorization property (not necessarily unital, we prove that the orthomorphisms, -orthomorphisms, and -linear operators on the order dual are precisely the same class of operators.
Mikusi\\'nski's Operational Calculus with Algebraic Foundations and Applications to Bessel Functions
Bengochea, Gabriel; G, Gabriel López
2013-01-01
We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde. With this operational calculus we study the solution of certain Bessel type equations.
Wilson operator algebras and ground states for coupled BF theories
Tiwari, Apoorv; Ryu, Shinsei
2016-01-01
The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theories on the three-torus, we explicitly calculate the $\\mathcal{S}$- and $\\mathcal{T}$-matrices, which encode fractional braiding statistics and topological spin of loop-like excitations, respectively. In the coupled $BF$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with point-like excitations, form composite particles.
Structure of associative subalgebras of Jordan operator algebras
Hamhalter, J
2011-01-01
We show that any order isomorphism between ordered structures of associative unital JB-subalgebras of JBW algebras is implemented naturally by a Jordan isomorphism. Consequently, JBW algebras are determined by the structure of their associative unital JB subalgebras. Further we show that in a similar way it is possible to reconstruct Jordan structure from the order structure of associative subalgebras endowed with an orthogonality relation. In case of abelian subalgebras of von Neuman algebra it is we shown that order isomorphisms of the structure of abelian C*-subalgebras that are well behaved with respect to the structure of two by two matrices over original algebra are implemented by *-isomorphisms.
Algebraic models of deviant modal operators based on de Morgan and Kleene lattices
Cattaneo, G.; Ciucci, DE; Dubois, D.
2011-01-01
An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and...
Structure and properties of the algebra of partially transposed permutation operators
Energy Technology Data Exchange (ETDEWEB)
Mozrzymas, Marek [Institute for Theoretical Physics, University of Wrocław, 50-204 Wrocław (Poland); Horodecki, Michał; Studziński, Michał [Institute for Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk (Poland); National Quantum Information Centre of Gdańsk, 81-824 Sopot (Poland)
2014-03-15
We consider the structure of algebra of operators, acting in n-fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S(n − 1) induced by irreducible representations of the group S(n − 2). The second kind is structurally connected with irreducible representations of the group S(n − 1)
Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach
Alekseev, Vadim
2011-01-01
We recast the Foelner condition in an operator algebraic setting and prove that it implies a certain dimension flatness property. Furthermore, it is proven that the Foelner condition generalizes the existing notions of amenability and that the enveloping von Neumann algebra arising from a Foelner algebra is automatically injective. As an application we show how our techniques unify the previously known results concerning vanishing of L^2-Betti numbers for amenable groups, groupoids and quantum groups and moreover provides a large class of new examples of algebras with vanishing L^2-Betti numbers.
First order differential operator associated to the Cauchy-Riemann operator in a Clifford algebra
International Nuclear Information System (INIS)
The complex differentiation transforms holomorphic functions into holomorphic functions. Analogously, the conjugate Cauchy-Riemann operator of the Clifford algebra transforms regular functions into regular functions. This paper determines more general first order operator L (matrix-type) for which Lu is regular provided u is regular. For such operator L, the initial value problem ∂u / ∂t = L (t, x, u, ∂u / ∂x) (1) u(0, x) = φ(x) (2) is solvable for an arbitrary regular function φ and the solution is regular in x for each t. (author)
Spatiality of Derivations of Operator Algebras in Banach Spaces
Directory of Open Access Journals (Sweden)
Quanyuan Chen
2011-01-01
Full Text Available Suppose that A is a transitive subalgebra of B(X and its norm closure A¯ contains a nonzero minimal left ideal I. It is shown that if δ is a bounded reflexive transitive derivation from A into B(X, then δ is spatial and implemented uniquely; that is, there exists T∈B(X such that δ(A=TA−AT for each A∈A, and the implementation T of δ is unique only up to an additive constant. This extends a result of E. Kissin that “if A¯ contains the ideal C(H of all compact operators in B(H, then a bounded reflexive transitive derivation from A into B(H is spatial and implemented uniquely.” in an algebraic direction and provides an alternative proof of it. It is also shown that a bounded reflexive transitive derivation from A into B(X is spatial and implemented uniquely, if X is a reflexive Banach space and A¯ contains a nonzero minimal right ideal I.
Nonmeromorphic operator product expansion and C{sub 2}-cofiniteness for a family of W-algebras
Energy Technology Data Exchange (ETDEWEB)
Carqueville, Nils; Flohr, Michael [Physikalisches Institut, University of Bonn, Nussallee 12, 53115 Bonn (Germany)
2006-01-27
We prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras, the triplet W-algebras, using results from P(z)-tensor product theory. While doing this, we also show that all these vertex operator algebras are C{sub 2}-cofinite.
Path operator algebras in conformal quantum field theories
International Nuclear Information System (INIS)
Two different kinds of path algebras and methods from noncommutative geometry are applied to conformal field theory: Fusion rings and modular invariants of extended chiral algebras are analyzed in terms of essential paths which are a path description of intertwiners. As an example, the ADE classification of modular invariants for minimal models is reproduced. The analysis of two-step extensions is included. Path algebras based on a path space interpretation of character identities can be applied to the analysis of fusion rings as well. In particular, factorization properties of character identities and therefore of the corresponding path spaces are - by means of K-theory - related to the factorization of the fusion ring of Virasoro- and W-algebras. Examples from nonsupersymmetric as well as N=2 supersymmetric minimal models are discussed. (orig.)
A C-ALGEBRA APPROACH TO THE IRREDUCIBILITY OFCOWEN-DOUGLAS OPERATORS
Institute of Scientific and Technical Information of China (English)
XUEYIFENG; WANGZONGYAO
1999-01-01
The authors consider the irreducibility of the Cowen-Douglas operator T. It is proved that Tis irreducible Lff the unital CI-algebra generated by some non-zero blocks in the decomposition of T with respect to
GLAME@lab: An M-script API for Linear Algebra Operations on Graphics Processors
Barrachina Mir, Sergio; Castillo Catalán, Maribel; Igual Peña, Francisco Daniel; Mayo, Rafael; Quintana Ortí, Enrique S.
2008-01-01
We propose two high-level application programming interfaces (APIs) to use a graphics processing unit (GPU) as a coprocessor for dense linear algebra operations. Combined with an extension of the FLAME API and an implementation on top of NVIDIA CUBLAS, the result is an efficient and user-friendly tool to design, implement, and execute dense linear algebra operations on the current generation of NVIDIA graphics processors, of wide-appeal to scientists and engineers. As an applicati...
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.
Norton's Trace Formulae for the Griess Algebraof a Vertex Operator Algebra with Larger Symmetry
Matsuo, Atsushi
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module V , with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c= 1/2, the W3 algebra at c= 4/5 or the W4 algebra at c= 1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster.
Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches
Skrypnyk, T.
2013-10-01
For each finite-dimensional simple Lie algebra {g}, starting from a general {g}⊗ {g}-valued solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras widetilde{{g}}-_r of {g}-valued meromorphic functions. We outline two ways of embedding of the Lie algebra widetilde{{g}}-_r into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of widetilde{{g}}-_r into Lie algebra widetilde{{g}}(u^{-1},u)) of formal Laurent power series. The second is an embedding of widetilde{{g}}-_r as a quasigraded Lie subalgebra into a quasigraded Lie algebra widetilde{{g}}_r: widetilde{{g}}_r=widetilde{{g}}-_r+widetilde{{g}}+_r, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces widetilde{{g}}^*_r, (widetilde{{g}}^{± }_r)^* and explicit form of the Lax operators L(u), L±(u) as elements of these spaces. We develop a theory of integrable finite-dimensional hamiltonian systems and soliton hierarchies based on Lie algebras widetilde{{g}}_r, widetilde{{g}}^{± }_r. We consider examples of such systems and soliton equations and obtain the most general form of integrable tops, Kirchhoff-type integrable systems, and integrable Landau-Lifshitz-type equations corresponding to the Lie algebra {g}.
Some G-M-type Banach spaces and K-groups of operator algebras on them
Institute of Scientific and Technical Information of China (English)
ZHONG Huaijie; CHEN Dongxiao; CHEN Jianlan
2004-01-01
By providing several new varieties of G-M-type Banachspaces according to decomposable and compoundable properties, this paper discusses the operator structures of thesespaces and the K-theory of the algebra of the operators on these G-M-type Banach spaces throughcalculation of the K-groups of the operator ideals contained in the class of Riesz operators.
The IBM RISC System/6000 and linear algebra operations
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. (Tennessee Univ., Knoxville, TN (USA). Dept. of Computer Science Oak Ridge National Lab., TN (USA)); Mays, P. (Numerical Algorithms Group Ltd., Oxford (UK)); Radicati di Brozolo, G. (IBM European Center for Scientific and Engineering Computing, Rome (Italy))
1991-01-01
This paper discusses the IBM RISC System/6000 workstation and a set of experiments with blocked algorithms commonly used in solving problems in numerical linear algebra. We describe the performance of these algorithms and discuss the techniques used in achieving high performance on such an architecture. 10 refs., 11 figs., 6 tabs.
International Nuclear Information System (INIS)
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 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 E7(7) which is parabolically related to the CLA E7(−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 so(p − 1, q − 1). Further we consider the algebras sl(2n, R) and for n = 2k the algebras su* (4k) which are parabolically related to the CLA su(n,n). Further we consider the algebras sp(r,r) which are parabolically related to the CLA sp(2r, R). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14).
Quantum Measurement Problem and Systems Selfdescription in Operators Algebras Formalism
Mayburov, S.
2002-01-01
Quantum Measurement problem studied in Information Theory approach of systems selfdescription which exploits the information acquisition incompleteness for the arbitrary information system. The studied model of measuring system (MS) consist of measured state S environment E and observer $O$ processing input S signal. $O$ considered as the quantum object which interaction with S,E obeys to Schrodinger equation (SE). MS incomplete or restricted states for $O$ derived by the algebraic QM formali...
6-transposition property of $\\tau$-involutions of vertex operator algebras
Sakuma, Shinya
2006-01-01
In this paper, we study the subalgebra generated by two Ising vectors in the Griess algebra of a vertex operator algebra. We show that the structure of it is uniquely determined by some inner products of Ising vectors. We prove that the order of the product of two $\\tau$-involutions is less than or equal to 6 and we determine the inner product of two Ising vectors.
Boolean Functions, Quantum Gates, Hamilton Operators, Spin Systems and Computer Algebra
Hardy, Yorick; Steeb, Willi-Hans
2014-01-01
We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.
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 ...
On the Action of Steenrod Operations on Polynomial Algebras
KARACA, İsmet
1998-01-01
Let \\( \\bba \\) be the mod-\\( p \\) Steenrod Algebra. Let \\( p \\) be an odd prime number and \\( Zp = Z/pZ \\). Let \\( Ps = Zp [x1,x2,\\ldots,xs]. \\) A polynomial \\( N \\in Ps \\) is said to be hit if it is in the image of the action \\( A \\otimes Ps \\ra Ps. \\) In [10] for \\( p=2, \\) Wood showed that if \\( \\a(d+s) > s \\) then every polynomial of degree \\( d \\) in \\( Ps \\) is hit where \\( \\a(d+s) \\) denotes the number of ones in the binary expansion of \\( d+s \\). Latter in [6] Monks extended a resu...
Matrix preconditioning: a robust operation for optical linear algebra processors.
Ghosh, A; Paparao, P
1987-07-15
Analog electrooptical processors are best suited for applications demanding high computational throughput with tolerance for inaccuracies. Matrix preconditioning is one such application. Matrix preconditioning is a preprocessing step for reducing the condition number of a matrix and is used extensively with gradient algorithms for increasing the rate of convergence and improving the accuracy of the solution. In this paper, we describe a simple parallel algorithm for matrix preconditioning, which can be implemented efficiently on a pipelined optical linear algebra processor. From the results of our numerical experiments we show that the efficacy of the preconditioning algorithm is affected very little by the errors of the optical system.
Operator Algebra Quantum Homogeneous Spaces of Universal Gauge Groups
Mahanta, Snigdhayan; Mathai, Varghese
2011-09-01
In this paper, we quantize universal gauge groups such as SU(∞), as well as their homogeneous spaces, in the σ- C*-algebra setting. More precisely, we propose concise definitions of σ- C*-quantum groups and σ- C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute these groups for the quantum homogeneous spaces associated to the quantum version of the universal gauge group SU(∞).
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)
BRST and anti-BRST operators for quantum linear algebra Uq(gl(N))
International Nuclear Information System (INIS)
For a quantum Lie algebra Uq(gl(N)) we construct BRST, anti-BRST and Laplace operators. The (anti)commutator with the BRST operator defines the differential on the de Rham complex over the quantum group GLq(N). The Hodge decomposition theorem for this complex is formulated
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
Mason, Geoffrey
2009-01-01
We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.
On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions
Creutzig, Thomas; Wood, Simon
2014-01-01
By working in the generality of the singlet vertex operator algebras we connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter $\\epsilon$. We then apply modular transformation properties of regularised partial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of $\\epsilon$. The map from the space of characters equipped with the Ver...
Which multiplier algebras are $W^*$-algebras?
Akemann, Charles A.; Amini, Massoud; Asadi, Mohammad B.
2013-01-01
We consider the question of when the multiplier algebra $M(\\mathcal{A})$ of a $C^*$-algebra $\\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebr...
Algebraic Properties of Dual Toeplitz Operators on the Orthogonal Complement of the Dirichlet Space
Institute of Scientific and Technical Information of China (English)
Tao YU; Shi Yue WU
2008-01-01
In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space.We completely characterize commuting dual Toeplitz operators with harmonic symbols,and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic.We also obtain the sufficient and necessary conditions on the harmonic symbols for SψSψ=Sψψ.
Construction of the Model of the Lambda Calculus System with Algebraic Operators
Institute of Scientific and Technical Information of China (English)
陆汝占; 张政; 等
1991-01-01
A lambda system with algebraic operators,Lambda-plus system,is introduced.After giving the definitions of the system,we present a sufficient condition for formulating a model of the system.Finally,a model of such system is constructed.
A chain morphism for Adams operations on rational algebraic K-theory
DEFF Research Database (Denmark)
Feliu, Elisenda
2010-01-01
For any regular noetherian scheme X and every k>0, we define a chain morphism between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by Q. These morphisms induce in homology the Adams operations defined by Gillet and Soulé...
Integral forms in vertex operator algebras which are invariant under finite groups
Griess, Robert L
2012-01-01
For certain vertex operator algebras (e.g., lattice type) and given finite group of automorphisms, we prove existence of a positive definite integral form invariant under the group. Applications include an integral form in the Moonshine VOA which is invariant under the Monster, and examples in other lattice type VOAs.
Hypercyclic operators on algebra of symmetric snalytic functions on $\\ell_p$
Directory of Open Access Journals (Sweden)
Z. H. Mozhyrovska
2016-06-01
Full Text Available In the paper, it is proposed a method of construction of hypercyclic composition operators on $H(\\mathbb{C}^n$ using polynomial automorphisms of $\\mathbb{C}^n$ and symmetric analytic functions on $\\ell_p.$ In particular, we show that an ``symmetric translation'' operator is hypercyclic on a Frechet algebra of symmetric entire functions on $\\ell_p$ which are bounded on bounded subsets.
An $S_3$-symmetry of the Jacobi Identity for Intertwining Operator Algebras
Chen, Ling
2015-01-01
We prove an $S_{3}$-symmetry of the Jacobi identity for intertwining operator algebras. Since this Jacobi identity involves the braiding and fusing isomorphisms satisfying the genus-zero Moore-Seiberg equations, our proof uses not only the basic properties of intertwining operators, but also the properties of braiding and fusing isomorphisms and the genus-zero Moore-Seiberg equations. Our proof depends heavily on the theory of multivalued analytic functions of several variables, especially th...
An $S_3$-symmetry of the Jacobi Identity for Intertwining Operator Algebras
Chen, Ling
2015-01-01
We prove an $S_{3}$-symmetry of the Jacobi identity for intertwining operator algebras. Since this Jacobi identity involves the braiding and fusing isomorphisms satisfying the genus-zero Moore-Seiberg equations, our proof uses not only the basic properties of intertwining operators, but also the properties of braiding and fusing isomorphisms and the genus-zero Moore Seiberg equations. Our proof depends heavily on the theory of multivalued analytic functions of several variables, especially the theory of analytic extensions.
Unitary operator bases and q-deformed algebras
Energy Technology Data Exchange (ETDEWEB)
Galleti, D.; Lunardi, J.T.; Pimentel, B.M. [Instituto de Fisica Teorica (IFT), Sao Paulo, SP (Brazil); Lima, C.L. [Sao Paulo Univ., SP (Brazil). Inst. de Fisica
1995-11-01
Starting from the Schwinger unitary operator bases formalism constructed out of a finite dimensional state space, the well-know q-deformed communication relation is shown to emergence in a natural way, when the deformation parameter is a root of unity. (author). 14 refs.
Unitary operator bases and Q-deformed algebras
Energy Technology Data Exchange (ETDEWEB)
Galetti, D.; Pimentel, B.M. [Instituto de Fisica Teorica (IFT), Sao Paulo, SP (Brazil); Lima, C.L. [Sao Paulo Univ., SP (Brazil). Inst. de Fisica. Grupo de Fisica Nuclear e Teorica e Fenomenologia de Particulas Elementares; Lunardi, J.T. [Universidade Estadual de Ponta Grossa, PR (Brazil). Dept. de Matematica e Estatistica
1998-03-01
Starting from the Schwinger unitary operator bases formalism constructed out of a finite dimensional state space, the well-know q-deformed commutation relation is shown to emerge in a natural way, when the deformation parameter is a root of unity. (author)
Quantum exchange algebra and exact operator solution of A sub 2 -Toda field theory
Takimoto, Y; Kurokawa, H; Fujiwara, T
1999-01-01
Locality is analyzed for Toda field theories by noting novel chiral description in the conventional non-chiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to non-trivial relations among the R-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for a A sub 2 -system by extending the algebraic method developed for the Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.
Quantum Exchange Algebra and Exact Operator Solution of $A_{2}$-Toda Field Theory
Takimoto, Y; Kurokawa, H; Fujiwara, T
1999-01-01
Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to nontrivial relations among the ${\\cal R}$-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for $A_2$-system by extending the algebraic method developed for Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.
Korf, Lisa A.; Schroeck, Franklin E.
2015-12-01
We consider an effect algebra of phase space localization operators for a quantum mechanical Hilbert space that contains no non-trivial projections, and the C*-algebra generated by it. This C∗-algebra forms an informationally complete set in the original Hilbert space. Its elements are shown to have singular-value-based decompositions that permit their characterization in terms of limits of linear combinations of products of pairs of the phase space fuzzy localization operators. Through these results, it is shown that the informational completeness of the C*-algebra can be greatly reduced to the informational completeness of the set of products of pairs formed from the elements of the effect algebra.
The origin of the algebra of quantum operators in the stochastic formulation of quantum mechanics
Davidson, Mark P.
2001-01-01
The origin of the algebra of the non-commuting operators of quantum mechanics is explained in the general Fenyes-Nelson stochastic models in which the diffusion constant is a free parameter. This is achieved by continuing the diffusion constant to imaginary values, a continuation which destroys the physical interpretation, but does not affect experimental predictions. This continuation leads to great mathematical simplification in the stochastic theory, and to an understanding of the entire m...
Finite Blaschke product and the multiplication operators on Sobolev disk algebra
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Let R(D) be the algebra generated in Sobolev space W22(D) by the rational functions with poles outside the unit disk D. In this paper the multiplication operators Mg on R(D) is studied and it is proved that Mg ～ Mzn if and only if g is an n-Blaschke product. Furthermore, if g is an n-Blaschke product, then Mg has uncountably many Banach reducing subspaces if and only if n > 1.
Tensor operators of the Lie algebras U(n) and o(ν)
International Nuclear Information System (INIS)
Formulas for matrix elements of tensor operators of the Lie algebras U(n) and o(ν) are derived in bases of irreducible representations. The formulas are simple for applications in calculations and are of the form of the Bell polynomials in sums of powers of the angular lengths. Recurrent formulas and generating functions are also obtained. An example of using the formulas obtained in nuclear physics is presented
Lattice-integrality of certain group-invariant integral forms in vertex operator algebras
Dong, Chongying; Griess Jr., Robert L.
2014-01-01
Certain vertex operator algebras have integral forms (integral spans of bases which are closed under the countable set of products). It is unclear when they (or integral multiples of them) are integral as lattices under the natural bilinear form on the VOA. We show that lattice-integrality may be arranged under some hypotheses, including cases of integral forms invariant by finite groups. In particular, there exists a lattice-integral Monster-invariant integral form in the Moonshine VOA.
Kleyn, Aleks
2007-01-01
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
Hasse-Schmidt derivations on Grassmann algebras with applications to vertex operators
Gatto, Letterio
2016-01-01
This book provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra (an analogue of the Taylor expansion for real-valued functions), and shows how this notion provides a natural framework for many ostensibly unrelated subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (Putzer's method revisited), universal decomposition of a polynomial in the product of two monic polynomials of fixed smaller degree, Schubert calculus for Grassmannian varieties, and vertex operators obtained with the help of Schubert calculus tools (Giambelli's formula). Significant emphasis is placed on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank, which then leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP) hierarchy describing the Plücker embedding of ...
Jeribi, Aref
2015-01-01
Uncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and Banach algebras. The book provides researchers and graduate students with a unified survey of the fundamental principles of fixed point theory in Banach spaces and algebras. The authors present several exten
Compactness characterization of operators in the Toeplitz algebra of the Fock space $F_\\alpha ^p$
Bauer, Wolfram
2011-01-01
Let BT be the class of functions $f$ on $\\mathbb{C}^n$ where the Berezin transform $B_\\alpha (|f|)$ associated to the standard weighted Fock space $F_\\alpha ^2$ is bounded, and for $1 < p < \\infty$ let $\\mathcal{T}_p$ be the norm closure of the algebra generated by Toeplitz operators with BT symbols acting on $F_\\alpha ^p$. In this paper, we will show that an operator $A$ is compact on $F_\\alpha ^p$ if and only if $A \\in \\mathcal{T}_p$ and the Berezin transform $B_\\alpha (A)$ of $A$ vanishes at infinity.
Chen, Hsian-Yang; Lam, Ching Hung
2014-06-01
In this paper, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors I such that (1) for any e ≠ f ∈ I, the subVOA VOA(e, f) generated by e and f is isomorphic to either U2B or U3C; and (2) the subgroup generated by the corresponding Miyamoto involutions {τe | e ∈ I} is isomorphic to the Weyl group of a root system of type An, Dn, E6, E7 or E8. The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of these vertex operator algebras are determined.
Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra
Griess Jr., Robert L.; Hoehn, Gerald
2001-01-01
The concept of a framed vertex operator algebra was studied in [DGH] (q-alg/9707008). This article is an analysis of all Virasoro frame stabilizers of the lattice VOA V for the E_8 root lattice, which is isomorphic to the E_8-level 1 affine Kac-Moody VOA V. We analyze the frame stabilizers, both as abstract groups and as subgroups of the Lie group Aut(V) = E_8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V). In particu...
Rigged modules I: modules over dual operator algebras and the Picard group
Blecher, David P.; Kashyap, Upasana
2016-01-01
In a previous paper we generalized the theory of W*-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel "Rigged modules II", giving many new results about weak*-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative a...
Clifford algebra, geometric algebra, and applications
Lundholm, Douglas
2009-01-01
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II
Mason, Geoffrey
2011-01-01
We continue our program to define and study $n$-point correlation functions for a vertex operator algebra $V$ on a higher genus compact Riemann surface obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras $V_L$. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. We also compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity. We compare our results with those obtained in the companion paper, when a pair of tori are sewn together, and show that the partition functions are not compatible in the neighborhood of a two-tori degeneration point. The \\emph{normalized} partition functions of a lattice theory $V_L$ \\emph{are} compatible, e...
Derivations on the Algebra of Operators in Hilbert C*-Modules
Institute of Scientific and Technical Information of China (English)
Peng Tong LI; De Guang HAN; Wai Shing TANG
2012-01-01
Let (M) be a full Hilbert C*-module over a C*-algebra (A),and let End*(A)((M)) be the algebra of adjointable operators on (M).We show that if (A) is unital and commutative,then every derivation of End*(A)((M)) is an inner derivation,and that if (A) is σ-unital and commutative,then innerness of derivations on “compact” operators completely decides innerness of derivations on End*(A)((M)).If (A) is unital (no commutativity is assumed) such that every derivation of (A) is inner,then it is proved that every derivation of End*(A)(Ln((A))) is also inner,where Ln((A)) denotes the direct sum of n copies of (A).In addition,in case (A) is unital,commutative and there exist x0,y0 ∈(M) such that〈x0,y0〉＝1,we characterize the linear (A)-module homomorphisms on End*(A)((M)) which behave like derivations when acting on zero products.
BRST and anti-BRST operators for quantum linear algebra U{sub q}(gl(N))
Energy Technology Data Exchange (ETDEWEB)
Isaev, A.P. E-mail: isaevap@thsun1.jinr.ru; Ogievetsky, O.V. E-mail: isaevap@thsun1.jinr.ru
2001-09-01
For a quantum Lie algebra U{sub q}(gl(N)) we construct BRST, anti-BRST and Laplace operators. The (anti)commutator with the BRST operator defines the differential on the de Rham complex over the quantum group GL{sub q}(N). The Hodge decomposition theorem for this complex is formulated.
Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph
Pineda, Jesús A; Caicedo, Mario I
2015-01-01
In this work we discuss the elements required for the construction of the operator algebra for the space of paths over a simply laced $SU(3)$ graph. These operators are an important step in the construction of the bialgebra required to find the partition functions of some modular invariant CFTs. We define the cup and cap operators associated with back-and-forth sequences and add them to the creation and annihilation operators in the operator algebra as they are required for the calculation of the full space of essential paths prescribed by the fusion algebra. These operators require collapsed triangular cells that had not been found in previous works; here we provide explicit values for these cells and show their importance in order for the cell system to fulfill the Kuperberg relations for $SU(3)$ tangles. We also find that demanding that our operators satisfy the Temperley-Lieb algebra leads one naturally to consider operators that create and annihilate closed triangular sequences, which in turn provides an...
Institute of Scientific and Technical Information of China (English)
An Hui-hui; Wang Zhi-chun
2016-01-01
L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.
Omar, Mohamed A
2014-01-01
Initial transient oscillations inhibited in the dynamic simulations responses of multibody systems can lead to inaccurate results, unrealistic load prediction, or simulation failure. These transients could result from incompatible initial conditions, initial constraints violation, and inadequate kinematic assembly. Performing static equilibrium analysis before the dynamic simulation can eliminate these transients and lead to stable simulation. Most exiting multibody formulations determine the static equilibrium position by minimizing the system potential energy. This paper presents a new general purpose approach for solving the static equilibrium in large-scale articulated multibody. The proposed approach introduces an energy drainage mechanism based on Baumgarte constraint stabilization approach to determine the static equilibrium position. The spatial algebra operator is used to express the kinematic and dynamic equations of the closed-loop multibody system. The proposed multibody system formulation utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are formulated using the Cartesian coordinates and the joint coordinates to form an augmented set of differential algebraic equations. Then system connectivity matrix is derived from the system topological relations and used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations.
Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C.
2016-05-01
Three loop ladder and V-topology diagrams contributing to the massive operator matrix element AQg are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.
Hoehn, Gerald
1996-01-01
We investigate self-dual vertex operator algebras (VOAs) and super algebras (SVOAs). Using the genus one correlation functions, it is shown that self-dual SVOAs exist only for half-integral central charges. It is described how self-dual SVOAs can be constructed from self-dual VOAs of larger central charge. The analogy with integral lattices and binary codes is emphasized. One main result is the construction of the shorter Moonshine module, a self-dual SVOA of central charge 23.5 on which the Baby monster - the second largest sporadic simple group - acts by automorphisms. The shorter Moonshine module has the character q^(-47/48)*(1+ 4371q^(3/2)+ 96256q^2+ 1143745q^(5/2) +...) and is the "shorter cousin" of the Moonshine module. Its lattice and code analog are the shorter Leech lattice and shorter Golay code. We conjecture that the shorter Moonshine module is the unique SVOA with this character. The final chapter introduces the notion of extremal VOAs and SVOAs. These are self-dual (S)VOAs with character having...
关于幂等算子代数的注记%A note on idempotent operator algebras
Institute of Scientific and Technical Information of China (English)
郅伟萍
2008-01-01
A characterization for an idempotent operator algebra is established.Idempotent operator algebras acting on a Hilbert space H are defined. Let Ω be a subalgebra of B(H). The powers of Ω are defined as follows:Ω1=Ω,Ωn=Ω (n-1)Ω +Ω (n-2)Ω 2+...+ΩΩ (n-1),n=1,2,.... Then Ω is idempotent when Ω2=Ω. Some important properties of idempotent operator algebras are presented.Some sufficient conditions for a operator algebra to be an idempotent operator algebra are discussed. If Ω is not unital and for each element A in Ω, that exists a nonzero number λA such that R(A)N(A-λA), then Ω is an idempotent operator algebra.%给出幂等算子代数的一个刻画.定义了希尔波特空间H的幂等算子代数.设Ω是B(H)上的一个子代数,且满足Ω1=Ω,Ωn=Ω (n-1)Ω +Ω (n-2)Ω 2+...+ΩΩ (n-1),n=1,2,...,当Ω 2=Ω时,Ω是幂等的.经过研究,得出了幂等算子代数的一些重要性质.从而,进一步得到一个算子代数是幂等算子代数的充分条件.如果Ω不含单位元,对Ω中的每一个元A,都存在一个非零复数λA,使得R(A)N(A-λA),那么,Ω就是幂等算子代数.
Directory of Open Access Journals (Sweden)
M. Heydari
2013-05-01
Full Text Available A new and effective direct method to determine the numerical solution of linear and nonlinear differential-algebraic equations (DAEs is proposed. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a differentialalgebraic equation can be transformed to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique
Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra $A(V_{L})$
Dong, Chongying; Li, Haisheng; Mason, Geoffrey
1996-01-01
It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.
Factorizations of invertible operators and $K$-theory of $C^*$-algebras
Zhang, Shuang
1992-01-01
Let $\\Scr A$ be a unital C*-algebra. We describe \\it K-skeleton factorizations \\rm of all invertible operators on a Hilbert C*-module $\\Scr H_{\\Scr A}$, in particular on $\\Scr H=l^2$, with the Fredholm index as an invariant. We then outline the isomorphisms $K_0(\\Scr A) \\cong \\pi _{2k}([p]_0)\\cong \\pi _{2k} ({GL}^p_r(\\Scr A))$ and $K_1(\\Scr A)\\cong \\pi _{2k+1}([p]_0)\\cong \\pi _{2k+1}(GL^p_r(\\Scr A))$ for $k\\ge 0 $, where $[p]_0$ denotes the class of all compact perturbations of a projection $...
Linear algebra and linear operators in engineering with applications in Mathematica
Davis, H Ted
2000-01-01
Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical ...
Ablinger, J; Blümlein, J; De Freitas, A; von Manteuffel, A; Schneider, C
2015-01-01
Three loop ladder and $V$-topology diagrams contributing to the massive operator matrix element $A_{Qg}$ are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable $N$ and the dimensional parameter $\\varepsilon$. Given these representations, the desired Laurent series expansions in $\\varepsilon$ can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural ...
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
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
Universal Algebras of Hurwitz Numbers
A. Mironov; Morozov, A; Natanzon, S.
2009-01-01
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
Energy Technology Data Exchange (ETDEWEB)
Ablinger, J.; Schneider, C. [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation; Behring, A.; Bluemlein, J.; Freitas, A. de [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Manteuffel, A. von [Mainz Univ. (Germany). Inst. fuer Physik
2015-09-15
Three loop ladder and V-topology diagrams contributing to the massive operator matrix element A{sub Qg} are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.
Spectral theory of linear operators and spectral systems in Banach algebras
Müller, Vladimir
2003-01-01
This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg...
Alphan, Hakan
2011-11-01
The aim of this study is to compare various image algebra procedures for their efficiency in locating and identifying different types of landscape changes on the margin of a Mediterranean coastal plain, Cukurova, Turkey. Image differencing and ratioing were applied to the reflective bands of Landsat TM datasets acquired in 1984 and 2006. Normalized Difference Vegetation index (NDVI) and Principal Component Analysis (PCA) differencing were also applied. The resulting images were tested for their capacity to detect nine change phenomena, which were a priori defined in a three-level classification scheme. These change phenomena included agricultural encroachment, sand dune afforestation, coastline changes and removal/expansion of reed beds. The percentage overall accuracies of different algebra products for each phenomenon were calculated and compared. The results showed that some of the changes such as sand dune afforestation and reed bed expansion were detected with accuracies varying between 85 and 97% by the majority of the algebra operations, while some other changes such as logging could only be detected by mid-infrared (MIR) ratioing. For optimizing change detection in similar coastal landscapes, underlying causes of these changes were discussed and the guidelines for selecting band and algebra operations were provided.
Real Operator Algebras on a Complex Hilbert Space%复Hilbert空间上的实算子代数
Institute of Scientific and Technical Information of China (English)
李炳仁
2009-01-01
We study real operator algebras on a complex Hilbert space H. From H, we can get a real Hilbert space H_r. Further, we have a complex Hilbert space H_(rc)= H_r+iH_r. Through this process, we prove the following. If A and M are uniformly closed and weakly closed real * operator algebras on H respectively, then their complex span A + iA and M + iM are (complex) C*-algebra and (complex) von Neumann algebra on H, respectively. Here, we don't need the condition: A∩iA = {0}, M∩iM = {0}. So our result is a generalization of Stormer's result.%本文研究在一个复Hilbert空间H上的实算子代数.从H可以得到一个实Hilbert空间Hr珥,进而又有一个复Hilbert空间H_(rc)=H_r+iH_r.通过这个过程,证明了如下结果.如果A,M分别是H上一致闭的,弱闭的实*算子代数,则它们的复扩张A+iA,M4-iM分别是日上的(复)C~*代数,(复)von Neumann代数.这里,不需要条件A∩iA={0},MniM={0}.因此,我们的结果是Stormer结果的推广.
Relations Between BZMVdM-Algebra and Other Algebras
Institute of Scientific and Technical Information of China (English)
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.
Scalar product for the tensor operators of the quantum algebra Ŭq(su2) by the Wigner-Eckart theorem
Fakhri, H.; Nouraddini, M.
2015-07-01
Tensor operators as the irreducible submodules corresponding to the adjoint representation of the quantum algebra Ŭq(su2) are equipped with q-analogue of the Hilbert-Schmidt scalar product by using the Wigner-Eckart theorem. Then, it is used to show that the adjoint representation of the quantum algebra Ŭq(su2) is a *-representation.
Wassermann, Antony
1998-01-01
Fusion of positive energy representations is defined using Connes' tensor product for bimodules over a von Neumann algebra. Fusion is computed using the analytic theory of primary fields and explicit solutions of the Knizhnik-Zamolodchikov equation.
Chappell, Isaac
2009-01-01
Using the previous construction of the geometrical representation (GR) of the centerless 1D, N = 4 extended Super Virasoro algebra, we construct the corresponding Short Distance Operation Product Expansions for the deformed version of the algebra. This algebra differs from the regular algebra by the addition of terms containing the Levi-Civita tensor. How this addition changes the super-commutation relations and affects the Short Distance Operation Product Expansions (OPEs) of the associated fields is investigated. The Method of Coadjoint Orbits, which removes the need first to find Lagrangians invariant under the action of the symmetries, is used to calculate the expansions. Finally, an alternative method involving Clifford algebras is investigated for comparison.
Deskins, W E
1996-01-01
This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diop
lam, Ching Hung; Chen, Hsian-Yang
2013-01-01
In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside th...
Matrix Operations for Engineers and Scientists An Essential Guide in Linear Algebra
Jeffrey, Alan
2010-01-01
Engineers and scientists need to have an introduction to the basics of linear algebra in a context they understand. Computer algebra systems make the manipulation of matrices and the determination of their properties a simple matter, and in practical applications such software is often essential. However, using this tool when learning about matrices, without first gaining a proper understanding of the underlying theory, limits the ability to use matrices and to apply them to new problems. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations. These students require a straightforward introduction to linear algebra illustrated by applications to which they can relate. It caters of the needs of undergraduate engineers in all disciplines, and provides considerable detail where it is likely to be helpful. According to the author the best way to understand the theory of matrices is by working simple exercises designe...
Left Artinian Algebraic Algebras
Institute of Scientific and Technical Information of China (English)
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.
Tree technique and irreducible tensor operators for SUq(2) quantum algebra, 9j-symbols
International Nuclear Information System (INIS)
The graphic technique of Kuznetsov-Smorodinov for the SUq(2) quantum algebra is discussed. The transformation of trees including the braiding of branches is considered. Using the universal R-matrix the q-analog of 9j-symbol is introduced and its symmetry are examined
Computer Program For Linear Algebra
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
International Nuclear Information System (INIS)
This proceedings paper aims to show the efficiency of an expectation value identity for a given algebraic function operator which is assumed to be depending pn only position operator. We show that this expectation value formula becomes enabled to determine the eigenstates of the quantum system Hamiltonian as long as it is autonomous and an appropriate basis set in position operator is used. This approach produces a denumerable infinite recursion which may be considered as revisited but at the same time generalized form of the recursions over the natural number powers of the position operator. The content of this short paper is devoted not only to the formulation of the new method but also to show that this novel approach is capable of catching the eigenvalues and eigenfunctions for Hydrogen-like systems, beyond that, it can give a hand to us to reveal the wavefunction structure. So it has also somehow a confirmative nature
Energy Technology Data Exchange (ETDEWEB)
Kalay, Berfin; Demiralp, Metin [İstanbul Technical University, Informatics Institute, Maslak, 34469, İstanbul (Turkey)
2015-12-31
This proceedings paper aims to show the efficiency of an expectation value identity for a given algebraic function operator which is assumed to be depending pn only position operator. We show that this expectation value formula becomes enabled to determine the eigenstates of the quantum system Hamiltonian as long as it is autonomous and an appropriate basis set in position operator is used. This approach produces a denumerable infinite recursion which may be considered as revisited but at the same time generalized form of the recursions over the natural number powers of the position operator. The content of this short paper is devoted not only to the formulation of the new method but also to show that this novel approach is capable of catching the eigenvalues and eigenfunctions for Hydrogen-like systems, beyond that, it can give a hand to us to reveal the wavefunction structure. So it has also somehow a confirmative nature.
Quantum double actions on operator algebras and orbifold quantum field theories
International Nuclear Information System (INIS)
Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1 dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitary locally compact groups and our methods are adapted to chiral theories on the circle. (orig.)
Shimakura, Hiroki
2013-01-01
In this article, we prove that the full automorphism group of the Z_2-orbifold of the Barnes-Wall lattice vertex operator algebra of central charge 32 has the shape $2^{27}.E_6(2)$. In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated to $E_6(2)$.
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.
Lefschetz, Solomon
2012-01-01
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.
Topological ∗-algebras with *-enveloping Algebras II
Indian Academy of Sciences (India)
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.
Algebraic functions of complexity one, a Weierstrass theorem, and three arithmetic operations
Beloshapka, V. K.
2016-07-01
The Weierstrass theorem concerning functions admitting an algebraic addition theorem enables us to give an explicit description of algebraic functions of two variables of analytical complexity one. Their description is divided into three cases: the general case, which is elliptic, and two special ones, a multiplicative and an additive one. All cases have a unified description; they are the orbits of an action of the gauge pseudogroup. The first case is a 1-parameter family of orbits of "elliptic addition," the second is the orbit of multiplication, and the third of addition. Here the multiplication and addition can be derived from the "elliptic addition" by passages to a limit. On the other hand, the elliptic orbits correspond to complex structures on the torus, the multiplicative orbit corresponds to the complex structure on the cylinder, and the additive one to that on the complex plane. This work was financially supported by the Russian Foundation for Basic Research under grants nos. 14-00709-a and 13-01-12417-ofi-m2.
Symmetric Extended Ockham Algebras
Institute of Scientific and Technical Information of China (English)
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
Brouder, Christian
2002-01-01
The Laplace Hopf algebra created by Rota and coll. is generalized to provide an algebraic tool for combinatorial problems of quantum field theory. This framework encompasses commutation relations, normal products, time-ordered products and renormalisation. It considers the operator product and the time-ordered product as deformations of the normal product. In particular, it gives an algebraic meaning to Wick's theorem and it extends the concept of Laplace pairing to prove that the renormalise...
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
We give an equivalent construction of the infinitesimal time translation operator for partial differential evolution equation in the algebraic dynamics algorithm proposed by Shun-Jin Wang and his students. Our construction involves only simple partial differentials and avoids the derivative terms of δ function which appear in the course of computation by means of Wang-Zhang operator. We prove Wang’s equivalent theorem which says that our construction and Wang-Zhang’s are equivalent. We use our construction to deal with several typical equations such as nonlinear advection equation, Burgers equation, nonlinear Schrodinger equation, KdV equation and sine-Gordon equation, and obtain at least second order approximate solutions to them. These equations include the cases of real and complex field variables and the cases of the first and the second order time derivatives.
Stability of functional equations in Banach algebras
Cho, Yeol Je; Rassias, Themistocles M; Saadati, Reza
2015-01-01
Some of the most recent and significant results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C*-algebras, C*-ternary algebras, non-Archimedean Banach algebras and multi-normed algebras are presented in this book. A brief introduction for functional equations and their stability is provided with historical remarks. Since the homomorphisms and derivations in Banach algebras are additive and R-linear or C-linear, the stability problems for additive functional equations and additive mappings are studied in detail. The latest results are discussed and examined in stability theory for new functional equations and functional inequalities in Banach algebras and C*-algebras, non-Archimedean Banach algebras, non-Archimedean C*-algebras, multi-Banach algebras and multi-C*-algebras. Graduate students with an understanding of operator theory, functional analysis, functional equations and analytic inequalities will find this book useful for furthering their understanding and discovering the l...
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
Developable algebraic surfaces
Institute of Scientific and Technical Information of China (English)
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.
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...
Algebra Operations on Counting Bloom Filters%计数布鲁姆过滤器代数运算
Institute of Scientific and Technical Information of China (English)
田小梅; 张大方; 谢鲲; 史长琼; 杨晓波
2012-01-01
文中探讨计数布鲁姆过滤器的代数运算和集合运算的一致性关系,研究使用计数布鲁姆过滤器代数运算进行集合成员查询的性能.理论分析和实验结果表明,计数布鲁姆过滤器的并、交、补、减、异或运算产生的新过滤器依然保持计数布鲁姆过滤器的特征,支持元素的删除操作,不会出现假阴性,能用于集合并集、交集、补集、差集及对称差的成员查询；当使用两个原始的计数布鲁姆过滤器查询补集、差集及对称差元素时,会存在部分本来属于补集、差集或对称差的元素被判为不属于补集、差集或对称差的问题,而使用计数布鲁姆过滤器代数运算后的过滤器进行补集、差集及对称差成员查询,则不存在上述问题,空间效率能提高一倍,时间效率亦能显著地得到改善.计数布鲁姆过滤器代数运算的使用有利于进一步扩展计数布鲁姆过滤器的应用范围.譬如计数布鲁姆过滤器减运算可用作一种新的集合调和方法,用于分布式系统中大型文件的分发.%This paper examines the consistence between algebra operations on counting Bloom filters and algebra operations on data sets, and studies the membership query performances of algebra operations on counting Bloom filters. Theoretical analyses and simulations show that the counting Bloom filter which is Ored(ANDed, COMPLEMENTed, SUBTRACTed, XORed) from the original counting Bloom filters can support membership query on data set Ored (ANDed, COMPLEMENTed, SUBTRACTed, XORed) from the original data sets. When using the two original counting Bloom filter to query elements belonged to complementary set, differences or symmetric differences of the two sets, some complementary set elements, differences or symmetric differences of the sets will be misjudged, while the query method using algebra operations on counting Bloom filters has no false negatives and gain a remarkable improvement in
Quantum algebra of $N$ superspace
Hatcher, N; Stephany, J
2006-01-01
We identify the quantum algebra of position and momentum operators for a quantum system in superspace bearing an irreducible representation of the super Poinca\\'e algebra. This algebra is noncommutative for the position operators. We use the properties of superprojectors in D=4 $N$ superspace to construct explicit position and momentum operators satisfying the algebra. They act on wave functions corresponding to different supermultiplets classified by its superspin. We show that the quantum algebra associated to the massive superparticle is a particular case described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently.
Villarreal, Rafael
2015-01-01
The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.
Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations
Institute of Scientific and Technical Information of China (English)
Ling CHEN
2011-01-01
We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.
On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_{2}
Directory of Open Access Journals (Sweden)
Sergii Kuzhel
2012-01-01
\\(\\Sigma_{J_{\\vec{\\beta}}}\\ are unitarily equivalent for different \\(\\vec{\\alpha}, \\vec{\\beta} \\in \\mathbb{S}^2\\ and describe in detail the structure of operators \\(A \\in \\Sigma_{J_{\\vec{\\alpha}}}\\ with empty resolvent set.
Extended finite operator calculus as an example of algebraization of analysis
Kwasniewski, A. K.
2008-01-01
A calculus of sequences started by professor morgan ward constitutes the general scheme for extensions of classical operator calculus of the distinguished gian carlo rota considered by many afterwards and after ward morgan. Because of the historically now established notation we call the wardian calculus of sequences in its afterwards elaborated form a psi calculus. The psi calculus in parts appears to be almost automatic, natural extension of classical operator calculus or equivalently of um...
Izhakian, Zur; Rowen, Louis
2008-01-01
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...
Quantum computation using geometric algebra
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
Jordan Elementary Map on Operator Algebras%算子代数上的Jordan初等映射
Institute of Scientific and Technical Information of China (English)
安润玲; 侯晋川
2012-01-01
Let R and R′be two rings.Under some mild assumptions on R,we show that, if maps M：R→R＇ and M~＊：R＇→R are surjective and satisfy M（AM＊（B）C＋CM＊（B）A）=M（A）BM（C）＋M（C）BM（A）,M＊（BM（A）D＋DM（A）B）=M＊（B）AM＊（D）＋M＊（D）AM＊（B） for all A,C∈R and B,D∈R＇,then both M and M~＊ are additive.And if R and R＇ have identity I and I＇,respectively,and M（I）,M~＊（I＇） are invertible,then there is a Jordan ring isomorphism N such that M（A） = N（A）M（I） and M~＊（B） = N~（-1）（BM（I））.In particular,if R = R＇ is a standard operator algebra,or a nest algebra on a Hilbert space,then we can conclude that both M and M~＊ are additive,and M（A） = SAT,M~＊（B） = TBS or M（A） = TA~＊S,M~＊（B） =（SBT）~＊,where S and T are bounded invertible linear or conjugate linear operators.%给定两个环R,R＇.对于满足一定条件的环R,本文证明了若M：R→R＇,M＊：R＇→R为满射且对A,C∈R和B,D∈R＇满足M（AM＊（B）C＋CM＊（B）A）=M（A）BM（C）＋M（C）BM（A）,M＊（BM（A）D＋DM（A）B）=M＊（B）AM＊（D）＋M＊（D）AM＊（B）则M和M＊是可加的;若R和R＇分别包含单位I和I＇,M（I）,M＊（I＇）可逆,则存在环同构N使得M（A）=N（A）M（I）,M＊（B）=N~（-1）（BM（I））.特别地,若R=R＇为标准算子代数或Hilbert空间套代数,则M和M＊可加且存在有界可逆的线性或共轭线性算子S和T使得M（A）=SAT,M＊（B）=TBS或M（A）=TA＊S,M＊（B）=（SBT）＊对任意的A,B∈R成立.
On Dunkl angular momenta algebra
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
Institute of Scientific and Technical Information of China (English)
FAN Hong-Yi; WANG Yong
2006-01-01
With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.
Operator algebra from fusion rules: The infinite number of Ising theories
Energy Technology Data Exchange (ETDEWEB)
Fuchs, J. (Nationaal Inst. voor Kernfysica en Hoge-Energiefysica (NIKHEF), Amsterdam (Netherlands). Sectie H)
1989-12-25
It is described how the fusion rules of a conformal field theory can be employed to derive differential equations for the four-point functions of the theory, and thus to determine eventually the operator product coefficients for primary fields. The results are applied to the Ising fusion rules. A set of theories possessing these fusion rules is found which is labelled by two discrete parameters. For a specific value of one of the parameters, these are the level one Spin(2m+1) Wess-Zumino-Witten theories; it is shown that they represent an infinite number of inequivalent theories. (orig.).
Baykara, N. A.
2015-12-01
Recent studies on quantum evolutionary problems in Demiralp's group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one.
International Nuclear Information System (INIS)
Recent studies on quantum evolutionary problems in Demiralp’s group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one
Energy Technology Data Exchange (ETDEWEB)
Baykara, N. A. [Marmara University, Faculty of Sciences and Letters, Mathematics Department, Göztepe Campus, 34730, Istanbul (Turkey)
2015-12-31
Recent studies on quantum evolutionary problems in Demiralp’s group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one.
Optimizing relational algebra operations using discrimination-based joins and lazy products
DEFF Research Database (Denmark)
Henglein, Fritz
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......- tically efficient. In contrast to hashing, discrimination is fully ab- stract (only depends on which equivalences hold on its inputs), and in contrast to comparison-based sorting, it does not require an or- dering relation on its inputs. In particular, it is applicable to ref- erences (pointers......). Furthermore, it has better asymptotic computa- tional complexity than both sorting and hashing. We represent cross-products and unions lazily (symbolically) as formal products of the argument sets (relations). This allows the selection operation to recognize on the fly whenever it is applied to a cross...
Institute of Scientific and Technical Information of China (English)
AN RUNLING; HOU JINCHUAN
2005-01-01
Let H and K be indefinite inner product spaces. This paper shows that a bijective map Φ: B(H) → B(K) satisfies Φ(AB+ + B+A) = Φ(A)Φ(B)+ + Φ(B)+Φ(A) for every pair A,B ∈ B(H) if and only if either Φ(A) = cUAU+ for all A or Φ(A) = cUA+U+ for all A; Φ satisfies Φ(AB+A) = Φ(A)Φ(B)+Φ(A) for every pair A, B ∈ B(H) if and only if either Φ(A) = UAV for all A or Φ(A) = UA+V for all A, where A+ denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U+U = c-1I and V+V = cI for some nonzero real number c.
Intermediate algebra a textworkbook
McKeague, Charles P
1985-01-01
Intermediate Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in intermediate algebra. The publication first takes a look at basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on properties of exponents, polynomials, sums, and differences, multiplication of polynomials, inequalities involving absolute value, word problems, first-degree inequalities, real numbers, opposites, reciprocals, and absolute value, and addition and subtraction of real numbers. The text then ex
Beginning algebra a textworkbook
McKeague, Charles P
1985-01-01
Beginning Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in algebra. The publication first elaborates on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on solving linear systems by graphing, elimination method, graphing ordered pairs and straight lines, linear and compound inequalities, addition and subtraction of real numbers, and properties of real numbers. The text then examines exponents and polynomials, factoring, and rational expressions. Topics include multiplication and division
Algebraic solution of master equations
R. Rangel; L. Carvalho
2003-01-01
We present a simple analytical method to solve master equations for finite temperatures and any initial conditions, which consists in the expansion of the density operator into normal modes. These modes and the expansion coefficients are obtained algebraically by using ladder superoperators. This algebraic technique is successful in cases in which the Liouville superoperator is quadratic in the creation and annihilation operators.
Energy Technology Data Exchange (ETDEWEB)
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.
Directory of Open Access Journals (Sweden)
A. V. Pavlov
2014-01-01
Full Text Available A mechanism of cognitive dissonance reducing is demonstrated with approach for non-monotonic fuzzy-valued logics by Fourier-holography technique implementation developing. Cognitive dissonance occurs under perceiving of new information that contradicts to the existing subjective pattern of the outside world, represented by double Fourier-transform cascade with a hologram – neural layers interconnections matrix of inner information representation and logical conclusion. The hologram implements monotonic logic according to “General Modus Ponens” rule. New information is represented by a hologram of exclusion that implements interconnections of logical conclusion and exclusion for neural layers. The latter are linked by Fourier transform that determines duality of the algebra forming operations of conjunction and disjunction. Hologram of exclusion forms conclusion that is dual to the “General Modus Ponens” conclusion. It is shown, that trained for the main rule and exclusion system can be represented by two-layered neural network with separate interconnection matrixes for direct and inverse iterations. The network energy function is involved determining the cyclic dynamics character; dissipative factor causing convergence type of the dynamics is analyzed. Both “General Modus Ponens” and exclusion holograms recording conditions on the dynamics and convergence of the system are demonstrated. The system converges to a stable status, in which logical conclusion doesn’t depend on the inner information. Such kind of dynamics, leading to tolerance forming, is typical for ordinary kind of thinking, aimed at inner pattern of outside world stability. For scientific kind of thinking, aimed at adequacy of the inner pattern of the world, a mechanism is needed to stop the net relaxation; the mechanism has to be external relative to the model of logic. Computer simulation results for the learning conditions adequate to real holograms recording are
Axler, Sheldon
2015-01-01
This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the ...
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...
Algebra-Geometry of Piecewise Algebraic Varieties
Institute of Scientific and Technical Information of China (English)
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.
The algebraic structure of the Onsager algebra
DATE, ETSURO; Roan, Shi-shyr
2000-01-01
We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.
Some C∗-algebras which are coronas of non-C∗-Banach algebras
Voiculescu, Dan-Virgil
2016-07-01
We present results and motivating problems in the study of commutants of hermitian n-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗-algebras which arise in this context as coronas of non-C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
Vertex Algebras, Kac-Moody Algebras, and the Monster
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
Semigroups and computer algebra in algebraic structures
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Congruence Kernels of Orthoimplication Algebras
Directory of Open Access Journals (Sweden)
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.
Hochschild homology of structured algebras
DEFF Research Database (Denmark)
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 moduli space action on the Hochschild complex of open TCFTs, the Tradler–Zeinalian and Kaufmann actions of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex....
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...
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.
Issa, A. Nourou
2010-01-01
Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative 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. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-M...
Cooperstein, Bruce
2010-01-01
Vector SpacesFieldsThe Space FnVector Spaces over an Arbitrary Field Subspaces of Vector SpacesSpan and IndependenceBases and Finite Dimensional Vector SpacesBases and Infinite Dimensional Vector SpacesCoordinate VectorsLinear TransformationsIntroduction to Linear TransformationsThe Range and Kernel of a Linear TransformationThe Correspondence and Isomorphism TheoremsMatrix of a Linear TransformationThe Algebra of L(V, W) and Mmn(F)Invertible Transformations and MatricesPolynomialsThe Algebra of PolynomialsRoots of PolynomialsTheory of a Single Linear OperatorInvariant Subspaces of an Operator
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
Algebra & trigonometry I essentials
REA, Editors of
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. Algebra & Trigonometry I includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, eq
On integral forms for vertex algebras associated with affine Lie algebras and lattices
McRae, Robert
2014-01-01
We revisit the construction of integral forms for vertex (operator) algebras $V_L$ based on even lattices $L$ using generators instead of bases, and we construct integral forms for $V_L$-modules. We construct integral forms for vertex (operator) algebras based on highest-weight modules for affine Lie algebras and we exhibit natural generating sets. For vertex operator algebras in general, we give conditions showing when an integral form contains the standard conformal vector generating the Vi...
Congruences on Balanced Pseudocomplemented Ockham Algebras
Institute of Scientific and Technical Information of China (English)
Jie FANG
2009-01-01
The variety bpO consists of those algebras (L;∧,∨, f,* ) of type where (L; ∧, ∨, f, 0, 1) is an Ockham algebra, (L; ∧, ∨, *, 0, 1) is a p-algebra, and the operations x→f(x) and x →x* satisfy the identities f(x*) = x** and [f(x)]* = f2(x). In this note, we show that the compact congruences on a bpO-algebra form a dual Stone lattice. Using this, we characterize the algebras in which every principal congruence is complemented. We also give a description of congruence coherent bpO-algebras.
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
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
Holtz, Olga; Ron, Amos
2007-01-01
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\\cal H}(X)$. This well-known line of study is particularly interesting in case $n\\eqbd\\rank X \\ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\\it external, central, and internal.} Each algebraic structure is ...
Toeplitz Algebras on Dirichlet Spaces
Institute of Scientific and Technical Information of China (English)
TAN Yan-hua; WANG Xiao-feng
2001-01-01
In the present paper, some properties of Toeplitz algebras on Dirichlet spaces for several complex variables are discussed; in particular, the automorphism group of the Toeplitz C* -algebra, (C1), generated by Toeplitz operators with C1-symbols is discussed. In addition, the first cohomology group of (C1) is computed.
基于8种常用蕴涵算子上的模糊布尔代数%Fuzzy Boolean Algebras Based on Eight Familiar Kinds of Implication Operator
Institute of Scientific and Technical Information of China (English)
陈华新
2012-01-01
Based on the work of "Fuzzy Boolean Algebras Based on Implication Operator", present work gives eight kinds of equivalent forms of fuzzy Boolean algebras based on familiar eight kinds of implication operator by using inequalities characterizatian method. This paper promotes the results of the corresponding fuzzy algebra, and enriches riches theoretical results of fuzzy algebra.%在文献[1]的基础上,利用不等式的刻画方法,给出8种常用的R-蕴涵算子下的R-模糊布尔代数的8种等价形式,推广了现有相应模糊代数的结果,丰富了模糊代数的理论成果.
Das, Tapas
2015-01-01
The second order $N$-dimensional Schr\\"odinger equation with pseudoharmonic potential is reduced to a first order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Our results generalize all other previous works that done for various potential combinations in the case of lower dimensions.The Ladder operators are also constructed for the pseudoharmonic potential in $N$-dimensions.Lie algebra associated with these operators are studied and found that they satisfy the commutation relations for the SU(1,1) group. Matrix elements of different operators such as $z$, $z\\frac{d}{dz}$ are derived and finally the Casimir operator is discussed briefly.
Monakhov, Vadim V
2016-01-01
We introduced fermionic variables in complex modules over real Clifford algebras of even dimension which are analog of the Witt basis. We built primitive idempotents which are a set of equivalent Clifford vacuums. It is shown that the modules are decomposed into direct sum of minimal left ideals generated by these idempotents and that the fermionic variables can be considered as more fundamental mathematical objects than spinors.
Miki, Kei
2016-07-01
Highest weight modules for U q ( g l 2 ̂ ) are endowed with a structure of modules for the quantum toriodal algebra U κ of type sl2. Using this, we define U κ actions on the space of vertex operators for irreducible highest weight U q ( g l 2 ̂ ) modules. Highest or lowest weight vectors of the thus obtained U κ modules are expressed in terms of an intertwiner for U q ( s l 2 ̂ ) modules and an extra boson. The submodules generated by these vectors are investigated.
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...
Semiprojectivity of universal -algebras generated by algebraic elements
DEFF Research Database (Denmark)
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....
Algebraic totality, towards completeness
Tasson, Christine
2009-01-01
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${\\mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${\\mathcal{B}}^n\\to{\\mathcal{B}}$ for every n by an algebraic metho...
Institute of Scientific and Technical Information of China (English)
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.
GCD, LCM, and Boolean Algebra?
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Hazewinkel, Michiel
2004-01-01
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generaliz...
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...
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
Associative subalgebras of the Griess algebra and related topics
Dong, C; Mason, G; Norton, S P
1996-01-01
It is shown how certain idempotents in the Griess algebra generate the discrete series representations for the Virasoro algebra inside the Frenkel-Lepowsky-Meurman's moonshine module vertex operator algebra. It is also shown that each Niemeier lattice determines (in many ways) certain maximal associative subalgebras of the Griess algebra.
Directory of Open Access Journals (Sweden)
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.
Properties Of Operators On Linear De.Morgan Algebra(II)%线性De.Morgan代数上算子的性质(Ⅱ)
Institute of Scientific and Technical Information of China (English)
成央金
2000-01-01
At first,we propose the θ operator and θ* operator on De.Morgan algebra.Next investigate their properties and apply them to solve inqualities in one unknown.especially obtain the result:aθ*b=((-a)+e)(aθb)%讨论了θ算子与θ*算子,建立了θ*算子的类似于剩余算子的性质,由此可以看出可把θ*算子作为一种新的蕴涵算子.研究了θ算子与θ*算子的关系获得了aθ*b=(e+)(aθb),给出了含一个变元的不等式的求解.
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
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
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
Quantum algebra of N superspace
International Nuclear Information System (INIS)
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the κ-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra
Energy Technology Data Exchange (ETDEWEB)
Bowcock, P.; Taormina, A. [Durham Univ. (United Kingdom). Dept. of Mathematics; Feigin, B.L. [Landau Inst. of Theoretical Physics, Moscow (Russian Federation); Semikhatov, A.M. [Rossijskaya Akademiya Nauk, Moscow (Russian Federation). Fizicheskij Institut
2000-11-01
We discover a realisation of the affine Lie superalgebra sl(2 vertical stroke 1) and of the exceptional affine superalgebra D(2 vertical stroke 1;{alpha}) as vertex operator extensions of two sl(2) algebras with ''dual'' levels (and an auxiliary level-1 sl(2) algebra). The duality relation between the levels is (k{sub 1}+1)(k{sub 2}+1)=1. We construct the representation of sl(2 vertical stroke 1){sub k{sub 1}} on a sum of tensor products of sl(2){sub k{sub 1}}, sl(2){sub k{sub 2}}, and sl(2){sub 1} modules and decompose it into a direct sum over the sl(2 vertical stroke 1){sub k{sub 1}} spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to D(2 vertical stroke 1;k{sub 2}){sub k{sub 1}} is traced to the properties of sl(2)+ sl(2)+ sl(2) embeddings into D(2 vertical stroke 1;{alpha}) and their relation with the dual sl(2) pairs. Conversely, we show how the sl (2){sub k{sub 2}} representations are constructed from sl(2 vertical stroke 1){sub k{sub 1}} representations. (orig.)
Bowcock, P; Semikhatov, A M; Taormina, A
2000-01-01
We discover a realisation of the affine Lie superalgebra sl(2|1) and of the exceptional affine superalgebra D(2|1;alpha) as vertex operator extensions of two affine sl(2) algebras with dual levels (and an auxiliary level 1 sl(2) algebra). The duality relation between the levels is (k+1)(k'+1)=1. We construct the representation of sl(2|1) at level k' on a sum of tensor products of sl(2) at level k, sl(2) at level k' and sl(2) at level 1 modules and decompose it into a direct sum over the sl(2|1) spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to the affine D(2|1;k') at level k is traced to properties of sl(2)+sl(2)+sl(2) embeddings into D(2|1;alpha) and their relation with the dual sl(2) pairs. Conversely, we show how the level k' sl(2) representations are constructed from level k sl(2|1) representations.
On Nilpotent Extensions of Algebras
Institute of Scientific and Technical Information of China (English)
Adam W. Marczak; Jerzy Plonka
2007-01-01
In this paper, we investigate essentially n-ary term operations of nilpotent extensions of algebras. We detect the connection between term operations of an original algebra and its nilpotent extensions. This structural point of view easily leads to the conclusion that the number of distinct essentially n-ary term operations of a proper algebraic nilpotent extension (ひ) of an algebra (ワ) is given by the formula pn(ひ)={pn(ワ)+1 for n=1,{pn(ワ) otherwise. We show that in general the converse theorem is not true. However, we suppose that if a variety V is uniquely determined by its pn-sequences, the converse theorem is also satisfied. In the second part of the paper, we characterize generics of nilpotent shifts of varieties and describe cardinalities of minimal generics. We give a number of examples and pose some problems.
Indian Academy of Sciences (India)
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.
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.
Rota-Baxter algebras and the Hopf algebra of renormalization
Energy Technology Data Exchange (ETDEWEB)
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.)
Rota-Baxter algebras and the Hopf algebra of renormalization
International Nuclear Information System (INIS)
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.)
Fuzzy Boolean Algebras Based on Implication Operator%基于蕴涵算子上的模糊布尔代数
Institute of Scientific and Technical Information of China (English)
陈华新
2011-01-01
文中给出R-模糊布尔代数的定义,讨论了其与模糊布尔代数的关系,证明在一定的条件下,有限个R-模糊布尔代数的交(并)还是R-模糊布尔代数,R-模糊布尔代数的同态像(原像)仍是R-模糊布尔代数.%In this paper ,we introduce the definition of fuzzy Boolean algebra. Based on that, the differences and connection between R-fuzzy Boolean algebra and fuzzy Boolean algebra are discussed. Furhtermore, it is proved that the finite intersection (union) of R-fuzzy Boolean algebra is still R-fuzzy Boolean algebra , and the homomorphic image (preimage) of R-fuzzy Boolean algebra is still R-fuzzy Boolean algebra.
Literal algebra for satellite dynamics. [perturbation analysis
Gaposchkin, E. M.
1975-01-01
A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.
Central simple Poisson algebras
Institute of Scientific and Technical Information of China (English)
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.
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. U...
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-12-15
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
Chirvasitu, Alex; Smith, S. Paul
2015-01-01
This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "...
Simplicities and Automorphisms of a Sp ecial Infinite Dimensional Lie Algebra
Institute of Scientific and Technical Information of China (English)
YU De-min; LI Ai-hua
2013-01-01
In this paper, a special infinite dimensional Lie algebra is studied. The infinite dimensional Lie algebra appears in the fields of conformal theory, mathematical physics, statistic mechanics and Hamilton operator. The infinite dimensional Lie algebras is pop-ularized Virasoro-like Lie algebra. Isomorphisms, homomorphisms, ideals of the infinite dimensional Lie algebra are studied.
Nonmonotonic logics and algebras
Institute of Scientific and Technical Information of China (English)
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.
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.
Energy Technology Data Exchange (ETDEWEB)
Casasent, D.; Ghosh, A.
1983-01-01
Many of the linear algebra operations and algorithms possible on optical matrix-vector processors are reviewed. Emphasis is given to the use of direct solutions and their realization on systolic optical processors. As an example, implicit and explicit solutions to partial differential equations are considered. The matrix-decomposition required is found to be the major operation recommended for optical realization. The pipelining and flow of data and operations are noted to be key issues in the realization of any algorithm on an optical systolic array processor. A realization of the direct solution by householder qr decomposition is provided as a specific case study. 19 references.
Perturbations of C*-algebraic Invariants
DEFF Research Database (Denmark)
Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.;
2010-01-01
The setting of the article is the so-called theory of perturbations of algebras of operators. It is shown that several of the properties a C*-algebra may have are preseved under pertubations. The main result states that Pisier's concept finite length is a stasble property.......The setting of the article is the so-called theory of perturbations of algebras of operators. It is shown that several of the properties a C*-algebra may have are preseved under pertubations. The main result states that Pisier's concept finite length is a stasble property....
Parsing with Regular Expressions & Extensions to Kleene Algebra
DEFF Research Database (Denmark)
Grathwohl, Niels Bjørn Bugge
. In the second part of this thesis, we study two extensions to Kleene algebra. Chomsky algebra is an algebra with a structure similar to Kleene algebra, but with a generalized mu-operator for recursion instead of the Kleene star. We show that the axioms of idempotent semirings along with continuity of the mu......-operator completely axiomatize the equational theory of the context-free languages. KAT+B! is an extension to Kleene algebra with tests (KAT) that adds mutable state. We describe a test algebra B! for mutable tests and give a commutative coproduct between KATs. Combining the axioms of B! with those of KAT and some...
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...
李代数D8到李代数E8嵌入关系的顶点算子代数类似%Vertex operator algebra analogue of the embedding D8 into E8
Institute of Scientific and Technical Information of China (English)
楚彦军; 程俊芳; 郑驻军
2012-01-01
Frenkel I,Lepowsky J,MeurmanA利用E8-格的方法构造月光顶点算子代数.由此过程可知,D8格顶点算子代数到E8格顶点算子代数的嵌入关系是不平凡的,而且这种嵌入关系应用到共形场论中有困难.结合一些新发展的顶点代数理论,给出了顶点算子代数LD8(1,0)到顶点算子代数LE8(1,0)嵌入关系的一种实现.这也表明LE8(1,0)作为LD8(1,0)模,同构于LE8(1,0)由其单模LD8(1,(ω)8)的扩张.在此基础上,得到LD8(1,0)在LE8(1,0)中的commutant子代数是由真空向量生成的一维平凡子代数.我们希望这样的嵌入关系对理解与月光顶点算子代数的构造相关的嵌入关系有较大帮助.%From Frenkel-Lepowsky-Meurman' s construction of the moonshine vertex operator algebra with the methods of Es - lattice originally, there is a nontrivial embedding of the Dg - lattice vertex operator algebra into the ￡g - lattice vertex operator algebra , and this embedding relation is difficult to be used into conformal field theories. Associating to the recent vertex algebra theory, we give a realization of the embedding of the vertex operator algebra L^ (1,0) into the vertex operator algebra LE% (1,0) and show that as an L^ (1,0 ) - module, LE% (1,0) is isomorphic to the extension of L^ (1,0) by its simple module Z,^ (1 ,a>g ). We also get the commutant of L^ (1,0) in Z,￡g (1,0) , which is trivial. We expect it is helpful to study the embedding relations associated to the construction of the moonshine vertex operator algebra.
Algebras, dialgebras, and polynomial identities
Bremner, Murray R
2012-01-01
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.
Solvable quadratic Lie algebras
Institute of Scientific and Technical Information of China (English)
无
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.
Grabowski, Jan
2015-01-01
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating ...
Algebras with actions and automata
Directory of Open Access Journals (Sweden)
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].
Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras
Konig, Martinvaldo
2014-10-01
Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.
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.
Institute of Scientific and Technical Information of China (English)
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.
Li, Haisheng; Tan, Shaobin; Wang, Qing
2012-01-01
In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra with...
Institute of Scientific and Technical Information of China (English)
Hoger GHAHRAMANI
2014-01-01
Let A be a subalgebra of B(X ) containing the identity operator I and an idem-potent P . Suppose that α,β :A→A are ring epimorphisms and there exists some nest N on X such thatα(P )(X ) andβ(P )(X ) are non-trivial elements of N . Let A contain all rank one operators in AlgN and δ :A→B(X ) be an additive mapping. It is shown that, if δ is (α,β)-derivable at zero point, then there exists an additive (α,β)-derivation τ : A→ B(X ) such that δ(A) = τ(A)+α(A)δ(I ) for all A ∈ A. It is also shown that if δ is generalized (α,β)-derivable at zero point, then δ is an additive generalized (α,β)-derivation. Moreover, by use of this result, the additive maps (generalized) (α,β)-derivable at zero point on several nest algebras, are also characterized.
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...
Algebras of Measurements: the logical structure of Quantum Mechanics
Lehmann, D; Gabbay, D M; Engesser, Kurt; Gabbay, Dov M.; Lehmann, Daniel
2005-01-01
In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute. Keywords: Quantum measurements, Measurement algebras, Quantum Logic. PACS: 02.10.-v.
Yoneda algebras of almost Koszul algebras
Indian Academy of Sciences (India)
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].
Kinds of Knowledge in Algebra.
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY
Institute of Scientific and Technical Information of China (English)
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.
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.
Alternative algebraic approaches in quantum chemistry
Energy Technology Data Exchange (ETDEWEB)
Mezey, Paul G., E-mail: paul.mezey@gmail.com [Canada Research Chair in Scientific Modeling and Simulation, Department of Chemistry and Department of Physics and Physical Oceanography, Memorial University of Newfoundland, 283 Prince Philip Drive, St. John' s, NL A1B 3X7 (Canada)
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
The algebras of large N matrix mechanics
Energy Technology Data Exchange (ETDEWEB)
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 Yoneda algebra of a K2 algebra need not be another K2 algebra
Cassidy, T.; Phan, C.; Shelton, B.
2010-01-01
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.
Enveloping algebras of some quantum Lie algebras
Pourkia, Arash
2014-01-01
We define a family of Hopf algebra objects, $H$, in the braided category of $\\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\\psi^2_{H\\otimes H}=id_{H\\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $\\psi^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analog...
Regular algebra and finite machines
Conway, John Horton
2012-01-01
World-famous mathematician John H. Conway based this classic text on a 1966 course he taught at Cambridge University. Geared toward graduate students of mathematics, it will also prove a valuable guide to researchers and professional mathematicians.His topics cover Moore's theory of experiments, Kleene's theory of regular events and expressions, Kleene algebras, the differential calculus of events, factors and the factor matrix, and the theory of operators. Additional subjects include event classes and operator classes, some regulator algebras, context-free languages, communicative regular alg
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Energy Technology Data Exchange (ETDEWEB)
Krishnaswami, Govind S [Institute for Theoretical Physics and Spinoza Institute, Utrecht University, Postbus 80.195, 3508 TD, Utrecht (Netherlands)], E-mail: govind.krishnaswami@durham.ac.uk
2008-04-11
We consider large-N multi-matrix models whose action closely mimics that of Yang-Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger-Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G({xi}), are quadratic equations S{sup i}G=G{xi}{sup i}G in concatenation of correlations. The Schwinger-Dyson operator S{sup i} is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to concatenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost correlations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger-Dyson operator as a whole is also a derivation of the graded shuffle product. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme.
Dzhumadil'daev, A. S.
2002-01-01
Algebras with identity $(a\\star b)\\star (c\\star d) -(a\\star d)\\star(c\\star b)$ $=(a,b,c)\\star d-(a,d,c)\\star b$ are studied. Novikov algebras under Jordan multiplication and Leibniz dual algebras satisfy this identity. If algebra with such identity has unit, then it is associative and commutative.
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
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
Directory of Open Access Journals (Sweden)
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.
包含紧算子理想的Toeplitz算子代数的刻画%A Character of Toeplitz Algebras Containing the Ideal of Compact Operators
Institute of Scientific and Technical Information of China (English)
许庆祥
2001-01-01
设G为一个torsion一free的离散群， (G，G+)为一个拟序群。记TG+(G)为相应的T0eplitz算子代数，K( 2(G+))为 2(G+)上的紧算子全体。本文证明了K( 2(G+)) TG+(G)当且仅当下列两个条件同时满足： (1)(G，G+)为一个序群； (2)G中存在一个最小的正元。%Let G be a discrete torsion-free group and (G, G+) a qua si-ordered group. Let TG+ (G) be the corresponding Toeplitz algebra. In this note, we show that TG+ (G) contains the ideal of compact operators on 2(G+) if and only if the following two conditions are satisfied: (1) (G, G+)is an ordered group; (2) G admits a least positive element.
RICCATI EQUATIONS ON NONCOMMUTATIVE BANACH ALGEBRAS
Curtain, Ruth
2011-01-01
Conditions for the existence of a solution of a Riccati equation to be in some prescribed noncommutative involutive Banach algebras are given. The Banach algebras are inverse-closed subalgebras of the space of bounded linear operators on some Hilbert space, and the Riccati equation has an exponentia
Relational algebra as formalism for hardware design
Berg, ten A.J.W.M.; Huijs, C.; Krol, Th.
1993-01-01
This paper introduces relational algebra as an elegant formalism to describe hardware behaviour. Hardware behaviour is modelled by functions that are represented by sets of tables. Relational algebra, developed for designing large and consistent databases is capable to operate on sets of tables and
Generalized Quantum Current Algebras
Institute of Scientific and Technical Information of China (English)
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.
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.
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...
Multiparameter Twisted Weyl Algebras
Futorny, Vyacheslav; Hartwig, Jonas T.
2011-01-01
We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs...
Algebraic Proofs over Noncommutative Formulas
Tzameret, Iddo
2010-01-01
We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege---yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in [BIKPRS96,GH03]. We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas ("PC over ordered formulas", for short). This is an algebraic propositional proof system that operates with noncommutative polynomials in which the order of products in all monomials respects a fixed linear order on the variables, and where proof-lines are written as noncommutative formulas. We show that the latter proof system is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR) and admits polynomial-size refutations for the pigeonhole principle and the Tseitin's formulas. We...
Bonatsos, D; Raychev, P P; Terziev, P A; Bonatsos, Dennis
2003-01-01
The rotational invariance under the usual physical angular momentum of the SUq(2) Hamiltonian for the description of rotational molecular spectra is explicitly proved and a connection of this Hamiltonian to the formalism of Amal'sky is provided. In addition, a new Hamiltonian for rotational spectra is introduced, based on the construction of irreducible tensor operators (ITOs) under SUq(2) and use of q-deformed tensor products and q-deformed Clebsch-Gordan coefficients. The rotational invariance of this SUq(2) ITO Hamiltonian under the usual physical angular momentum is explicitly proved and a simple closed expression for its energy spectrum (the ``hyperbolic tangent formula'') is introduced. Numerical tests against an experimental rotational band of HF are provided.
On q-deformed infinite-dimensional n-algebra
Directory of Open Access Journals (Sweden)
Lu Ding
2016-03-01
Full Text Available The q-deformation of the infinite-dimensional n-algebras is investigated. Based on the structure of the q-deformed Virasoro–Witt algebra, we derive a nontrivial q-deformed Virasoro–Witt n-algebra which is nothing but a sh-n-Lie algebra. Furthermore in terms of the pseud-differential operators, we construct the (cosine n-algebra and the q-deformed SDiff(T2 n-algebra. We find that they are the sh-n-Lie algebras for the n even case. In terms of the magnetic translation operators, an explicit physical realization of the (cosine n-algebra is given.
NORMAL-FORM IN A DELAY-INSENSITIVE ALGEBRA
GROENBOOM, R; JOSEPHS, MB; LUCASSEN, PG; UDDING, JT; Furber, S; Edwards, M
1993-01-01
A process algebra is given for specifying delay-insensitive processes. We show in two steps that expressions in this algebra have a normal form, as a consequence of which the algebra is complete. First, the number of operators in process expressions is reduced by a set of rewrite laws. The resulting
Institute of Scientific and Technical Information of China (English)
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
Maps from the enveloping algebra of the positive Witt algebra to regular algebras
Sierra, Susan J.; Walton, Chelsea
2015-01-01
We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.
Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
Directory of Open Access Journals (Sweden)
Kenny De Commer
2013-12-01
Full Text Available Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to g by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping ∗-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.
Graph model of the Heisenberg-Weyl algebra
Blasiak, P.; Horzela, A.; Duchamp, G. H. E.; Penson, K. A.; Solomon, A. I.
2007-01-01
We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple interpretation involving, for example, the natural composition of graphs. This provides a deeper insight into the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorial underpinning of its abstract formalism.
A Unified Algebraic Approach to Classical Yang-Baxter Equation
Bai, Chengming
2007-01-01
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical ...
Linear algebra algorithms for divisors on an algebraic curve
Khuri-Makdisi, Kamal
2001-01-01
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point o...
Spacetime algebra and electron physics
Doran, C J L; Gull, S F; Somaroo, S; Challinor, A D
1996-01-01
This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include non-relativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, propagators and scattering theory, spin precession, tunnelling times, spin measurement, multiparticle quantum mechanics, relativistic multiparticle wave equations, and semiclassical mechanics.
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
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.
The quantum algebra of superspace
Hatcher, N; Stephany, J
2006-01-01
We present the complete set of N=1, D=4 quantum algebras associated to massive superparticles. We obtain the explicit solution of these algebras realized in terms of unconstrained operators acting on the Hilbert space of superfields. These solutions are expressed in terms of the chiral, anti-chiral and tensorial projectors which define the three irreducible representations of the supersymmetry on the superfields. In each case the space-time variables are non-commuting and their commutators are proportional to the internal angular momentum of the representation. The quantum algebras associated to the chiral or anti-chiral projectors is the one obtained by the quantization of the Casalbuoni-Brink-Schwarz massive superparticle. We present a new action for the tensorial case and show that their wave functions are restricted to be tensorial superfields.
Algebraic realization of rotational dynamics
International Nuclear Information System (INIS)
It is shown that the dynamics of a quantum rotor can be realized in terms of the SU(3)→SO(3) group algebra. Specifically, an analytic result is given for mapping from the hamiltonian of a trixial rotor to its algebraic image. Under the mapping invariants of the rotor are carried into Casimir invariants of the algebraic theory. Results for spectra and transition rates and various sums are given to demonstrate the effectiveness of the mapping. The theory gives physical significance to operators that were first introduced by Racah as a means for resolving the SU(3)→SO(3) state labelling problem. As the SU(3)→SO(3) structure is common to the rotational limit of several nuclear models, the theory also offers an opportunity to explore in a new way the microscopic underpinnings of rotational phenomena in nuclei. (orig.)
A Metric Conceptual Space Algebra
Adams, Benjamin; Raubal, Martin
The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.
Representations of twisted current algebras
Lau, Michael
2013-01-01
We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.
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.
Process algebra with four-valued logic
Bergstra, J.A.; Ponse, A.
2000-01-01
We propose a combination of a fragment of four-valued logic and process algebra. We present an operational semantics in SOS-style, and a completeness result for ACP with conditionals and four-valued logic.
The Lie algebra of the N=2-string
Energy Technology Data Exchange (ETDEWEB)
Kugel, K.
2006-07-01
The theory of generalized Kac-Moody algebras is a generalization of the theory of finite dimensional simple Lie algebras. The physical states of some compactified strings give realizations of generalized Kac-Moody algebras. For example the physical states of a bosonic string moving on a 26 dimensional torus form a generalized Kac-Moody algebra and the physical states of a N=1 string moving on a 10 dimensional torus form a generalized Kac-Moody superalgebra. A natural question is whether the physical states of the compactified N=2-string also realize such an algebra. In this thesis we construct the Lie algebra of the compactified N=2-string, study its properties and show that it is not a generalized Kac-Moody algebra. The Fock space of a N=2-string moving on a 4 dimensional torus can be described by a vertex algebra constructed from a rational lattice of signature (8,4). Here 6 coordinates with signature (4,2) come from the matter part and 6 coordinates with signature (4,2) come from the ghost part. The physical states are represented by the cohomology of the BRST-operator. The vertex algebra induces a product on the vector space of physical states that defines the structure of a Lie algebra on this space. This Lie algebra shares many properties with generalized Kac-Moody algebra but we will show that it is not a generalized Kac-Moody algebra. (orig.)
Bounded Algebra and Current-Mode Digital Circuits
Institute of Scientific and Technical Information of China (English)
WU Xunwei; Massoud Pedram
1999-01-01
This paper proposes two boundedarithmetic operations, which are easily realized with current signals.Based on these two operations, a bounded algebra system suitable fordescribing current-mode digital circuits is developed and itsrelationship with the Boolean algebra, which is suitable for representingvoltage-mode digital circuits, is investigated. Design procedure forcurrent-mode circuits using the proposed algebra system is demonstratedon a number of common circuit elements which are used to realizearithmetic operations, such as adders and multipliers.
Derivations of certain operator algebras
Directory of Open Access Journals (Sweden)
Jiankui Li
2000-01-01
conditions under which any derivation δ from into L(H must be inner. The conditions include (1 H−≠H, (2 0+≠0, (3 there is a nontrivial projection in which is in , and (4 δ is norm continuous. We also give some pplications.
Homogeneous Construction of the Toroidal Lie Algebra of Type A1
Institute of Scientific and Technical Information of China (English)
Haifeng Lian; Cui Chen; Qinzhu Wen
2007-01-01
In this paper,we consider an analogue of the level two homogeneous construc-tion of the affine Kac-Moody algebra A1(1) by vertex operators.We construct modules for the toroidal Lie algebra and the extended toroidal Lie algebra of type A1.We also prove that the module is completely reducible for the extended toroidal Lie algebra.
Accelerating Dense Linear Algebra on the GPU
DEFF Research Database (Denmark)
Sørensen, Hans Henrik Brandenborg
and matrix-vector operations on GPUs. Such operations form the backbone of level 1 and level 2 routines in the Basic Linear Algebra Subroutines (BLAS) library and are therefore of great importance in many scientific applications. The target hardware is the most recent NVIDIA Tesla 20-series (Fermi...... architecture). Most of the techniques I discuss for accelerating dense linear algebra are applicable to memory-bound GPU algorithms in general....
Advanced linear algebra for engineers with Matlab
Dianat, Sohail A
2009-01-01
Matrices, Matrix Algebra, and Elementary Matrix OperationsBasic Concepts and NotationMatrix AlgebraElementary Row OperationsSolution of System of Linear EquationsMatrix PartitionsBlock MultiplicationInner, Outer, and Kronecker ProductsDeterminants, Matrix Inversion and Solutions to Systems of Linear EquationsDeterminant of a MatrixMatrix InversionSolution of Simultaneous Linear EquationsApplications: Circuit AnalysisHomogeneous Coordinates SystemRank, Nu
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.
Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra
Lagraa, M.; Touhami, N.
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.
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.
Planar Algebra of the Subgroup-Subfactor
Gupta, Ved Prakash
2008-01-01
We give an identification between the planar algebra of the subgroup-subfactor $R \\rtimes H \\subset R \\rtimes G$ and the $G$-invariant planar subalgebra of the planar algebra of the bipartite graph $\\star_n$, where $n = [G : H]$. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant, of $R \\rtimes H \\subset R \\rtimes G$ in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor $R^G \\subset R^H$ and the $G$-invariant planar subalgebra of the planar algebra of the `flip' of $\\star_n $.
Planar Algebra of the Subgroup-Subfactor
Indian Academy of Sciences (India)
Ved Prakash Gupta
2008-11-01
We give an identification between the planar algebra of the subgroup-subfactor $R \\rtimes H \\subset R \\rtimes G$ and the -invariant planar subalgebra of the planar algebra of the bipartite graph $\\star_n$, where $n=[G:H]$. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of $R \\rtimes H \\subset R \\rtimes G$ in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor $R^G \\subset R^H$ and the -invariant planar subalgebra of the planar algebra of the `flip’ of $\\star_n$.
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.
REAL PIECEWISE ALGEBRAIC VARIETY
Institute of Scientific and Technical Information of China (English)
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.
Deficiently Extremal Gorenstein Algebras
Indian Academy of Sciences (India)
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.
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.
Jespers, Eric; Riley, David; Siciliano, Salvatore
2007-01-01
An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically generated GI-algebra locally finite?
Ellipticity in Pseudodifferential Algebras of Toeplitz Type
Seiler, Jörg
2011-01-01
Let L^\\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\\star be a subalgebra of operators of the form P_1AP_0, where P_0 and P_1 are two projections. The elements of L^\\star act as linear continuous operators in certain scales of abstract Sobolev spaces, the elements of the subalgebra in the corresponding subspaces determined by the projections. We study how the ellipticity in L^\\star descends to T^\\star, focusing on parametrix construction, Fredholm property, and homogeneous principal symbols. Applications concern SG-pseudodifferential operators, pseudodifferential operators on manifolds with conical singularities, and Boutet de Monvel's algebra for boundary value problems. In particular, we derive invertibilty of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvel's algebra. We indicate how the concept generalizes to parameter-dependent operators.
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
On the concept of universal W-algebra
International Nuclear Information System (INIS)
Ordinary quadratic WN-algebras are in fact particular subsets of naturally closed algebras of Casimir-like operators TrJn, formed from generators J of Sl(N) Kac-Moody algebra. These subsets arise after Hamiltonian reduction and some additional less universal restrictions, specifying non-reducible closed subalgebra and ignored in the text. In the limit of N=∞ a structure of linear Lie algebra is restored and a new object arises, naturally interpreted as universal W-algebra. 14 refs
Indian Academy of Sciences (India)
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.
Taghavi, Ali
2013-01-01
We study some properies of $Z^{*}$ algebras, thos C^* algebra which all positive elements are zero divisors. We show by means of an example that an extension of a Z* algebra by a Z* algebra is not necessarily Z* algebra. However we prove that an extension of a non Z* algebra by a non Z* algebra is again a Z^* algebra. As an application of our methods, we prove that evey compact subset of the positive cones of a C* algebra has an upper bound in the algebra.
Shilov, Georgi E
1977-01-01
Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.
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.
The Boolean algebra and central Galois algebras
Directory of Open Access Journals (Sweden)
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.
Poincare invariant algebra from instant to light-front quantization
International Nuclear Information System (INIS)
We present the Poincare algebra interpolating between instant and light-front time quantizations. The angular momentum operators satisfying SU(2) algebra are constructed in an arbitrary interpolation angle and shown to be identical to the ordinary angular momentum and Leutwyler-Stern angular momentum in the instant and light-front quantization limits, respectively. The exchange of the dynamical role between the transverse angular mometum and the boost operators is manifest in our newly constructed algebra
Algebras of Measurements: The Logical Structure of Quantum Mechanics
Lehmann, Daniel; Engesser, Kurt; Gabbay, Dov M.
2006-04-01
In quantum physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute.
On Derivations Of Genetic Algebras
International Nuclear Information System (INIS)
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
Algebra a complete introduction : teach yourself
Neill, Hugh
2013-01-01
Algebra: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using Algebra. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all the key areas of algebra including elementary operations, linear equations, formulae, simultaneous equations, quadratic equations, logarithms, variation, laws and sequences. Everything you will need is here in this one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.
Higher Dimensional Classical W-Algebras
Martínez-Moras, F; Martinez-Moras, Fernando; Ramos, Eduardo
1993-01-01
Classical $W$-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. These $W$-algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of Diff$S(1)$ is taken by the algebra of generators of local diffeomorphisms in two dimensions.
Pérez López, César
2014-01-01
MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Matrix Algebra introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. Starting with a look at symbolic and numeric variables, with an emphasis on vector and matrix variables, you will go on to examine functions and operations that support vectors and matrices as arguments, including those based on analytic parent functions. Computational methods for finding eigenvalues and eigenvectors of matrices are detailed, leading to various matrix decompositions. Applications such as change of bases, the classification of quadratic forms and ...
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.
Stable endomorphism algebras of modules over special biserial algebras
Schröer, Jan; Zimmermann, Alexander
2002-01-01
We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle.
$L_{\\infty}$ algebra structures of Lie algebra deformations
Gao, Jining
2004-01-01
In this paper,we will show how to kill the obstructions to Lie algebra deformations via a method which essentially embeds a Lie algebra into Strong homotopy Lie algebra or $L_{\\infty}$ algebra. All such obstructions have been transfered to the revelvant $L_{\\infty}$ algebras which contain only three terms
Omni-Lie Color Algebras and Lie Color 2-Algebras
Zhang, Tao
2013-01-01
Omni-Lie color algebras over an abelian group with a bicharacter are studied. The notions of 2-term color $L_{\\infty}$-algebras and Lie color 2-algebras are introduced. It is proved that there is a one-to-one correspondence between Lie color 2-algebras and 2-term color $L_{\\infty}$-algebras.
Deformations of Quasicoherent Sheaves of Algebras
Lunts, Valery A.
2001-01-01
Gerstenhaber and Schack ([GS]) developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherent sheaves of algebras on a quasiprojective scheme $X$ in terms of sheaf cohomology on $X$ and $X\\times X$. These results are applied to the study of deformations of the sheaf $D_X$ of differential operators on $X$. In particular, in case $X$ is...
Generalized join-hemimorphisms on Boolean algebras
Sergio Celani
2003-01-01
We introduce the notions of generalized join-hemimorphism and generalized Boolean relation as an extension of the notions of join-hemimorphism and Boolean relation, respectively. We prove a duality between these two notions. We will also define a generalization of the notion of Boolean algebra with operators by considering a finite family of Boolean algebras endowed with a generalized join-hemimorphism. Finally, we define suitable notions of subalgebra, congruences, Boole...
Linear Maps Preserving Idempotence on Nest Algebras
Institute of Scientific and Technical Information of China (English)
Jian Lian CUI; Jin Chuan HOU
2004-01-01
In this paper, we discuss the rank-1-preserving linear maps on nest algebras of Hilbertspace operators. We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear bijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism.
Algebraic Curves for Factorized String Solutions
Dekel, Amit
2013-01-01
We show how to construct an algebraic curve for factorized string solution in the context of the AdS/CFT correspondence. We define factorized solutions to be solutions where the flat-connection becomes independent of one of the worldsheet variables by a similarity transformation with a matrix $S$ satisfying $S^{-1}d S=const$. Using the factorization property we construct a well defined Lax operator and an associated algebraic curve. The construction procedure is local and does not require the...
An elliptic quantum algebra for sl$_{2}$
Foda, O E; Jimbo, M; Kedem, R; Miwa, T; Yan, H
1994-01-01
An elliptic deformation of \\widehat{sl}_2 is proposed. Our presentation of the algebra is based on the relation RLL=LLR^*, where R and R^* are eight-vertex R-matrices with the elliptic moduli chosen differently. In the trigonometric limit, this algebra reduces to a quotient of that proposed by Reshetikhin and Semenov-Tian-Shansky. Conjectures concerning highest weight modules and vertex operators are formulated, and the physical interpretation of R^* is discussed.
Combinatorial Algebra for second-quantized Quantum Theory
Blasiak, P; Solomon, A I; Horzela, A; Penson, K A
2010-01-01
We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H), the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.
Sahai, Vivek
2013-01-01
Beginning with the basic concepts of vector spaces such as linear independence, basis and dimension, quotient space, linear transformation and duality with an exposition of the theory of linear operators on a finite dimensional vector space, this book includes the concept of eigenvalues and eigenvectors, diagonalization, triangulation and Jordan and rational canonical forms. Inner product spaces which cover finite dimensional spectral theory and an elementary theory of bilinear forms are also discussed. This new edition of the book incorporates the rich feedback of its readers. We have added new subject matter in the text to make the book more comprehensive. Many new examples have been discussed to illustrate the text. More exercises have been included. We have taken care to arrange the exercises in increasing order of difficulty. There is now a new section of hints for almost all exercises, except those which are straightforward, to enhance their importance for individual study and for classroom use.
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.
Finite-dimensional (*)-serial algebras
Institute of Scientific and Technical Information of China (English)
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.
Directory of Open Access Journals (Sweden)
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.
On the Toroidal Leibniz Algebras
Institute of Scientific and Technical Information of China (English)
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.
Salaün, Gwen; Serwe, Wendelin
2005-01-01
A natural approach for the description of asynchronous hardware designs are hardware process algebras, such as Martin's CHP (Communicating Hardware Processes), Tangram, or BALSA, which are extensions of standard process algebras with particular operators exploiting the implementation of synchronisation using handshake protocols. In this research report, we give a structural operational semantics for value-passing CHP. Compared to existing semantics of CHP defined by translation into Petri net...
On W algebras commuting with a set of screenings
Litvinov, Alexey
2016-01-01
We consider the problem of classification of all W algebras which commute with a set of exponential screening operators. Assuming that the W algebra has a nontrivial current of spin 3, we find equations satisfied by the screening operators and classify their solutions.
Algebraic nonlinear collective motion
Troupe, J.; Rosensteel, G.
1999-01-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real number $\\Lambda$. The $\\Lambda=0$ solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear g...
Geometric Algebras and Extensors
Fernandez, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2007-01-01
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to met...
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
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
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
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...
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
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
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.
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
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.
Indian Academy of Sciences (India)
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.
On the number of finite algebraic structures
Aichinger, Erhard; McKenzie, Ralph
2011-01-01
We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related -- i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.
(Quasi-)Poisson enveloping algebras
Yang, Yan-Hong; Yuan YAO; Ye, Yu
2010-01-01
We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.
On algebraic volume density property
Kaliman, Shulim; Kutzschebauch, Frank
2012-01-01
A smooth affine algebraic variety $X$ equipped with an algebraic volume form $\\omega$ has the algebraic volume density property (AVDP) if the Lie algebra generated by completely integrable algebraic vector fields of $\\omega$-divergence zero coincides with the space of all algebraic vector fields of $\\omega$-divergence zero. We develop an effective criterion of verifying whether a given $X$ has AVDP. As an application of this method we establish AVDP for any homogeneous space $X=G/R$ that admi...
Some Properties of Intuitionistic Fuzzy Lie Algebras over a Fuzzy Field
Antony, P. L.; Lilly, P. L.
2011-01-01
The concept of intuitionistic fuzzy Lie algebra over a fuzzy field is introduced. We study the "necessity" and "possibility" operators on intuitionistic fuzzy Lie algebra over a fuzzy field and give some properties of homomorphic images.
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
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
Institute of Scientific and Technical Information of China (English)
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).
An Invitation to Algebraic Statistics: New Outlook and Opportunities
Çetin, Eyüp
2012-01-01
Algebra, a branch of pure mathematics, now advances statistics and operations research of applied mathematics. This synergy is called algebraic statistics as a new discipline. Algebraic statistics offers statisticians, management scientists, business researchers, econometricians and algebraists new opportunities, horizons and connections to advance their fields and related application areas. In this effort, this young, vibrant, quickly growing, and active discipline is briefly discussed and s...
Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers
Zheng, Quan
2016-05-01
We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.
Mathematical methods linear algebra normed spaces distributions integration
Korevaar, Jacob
1968-01-01
Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector
Instantaneous Point, Line, and Plane Motions Using a Clifford Algebra
Institute of Scientific and Technical Information of China (English)
Kwun-Lon Ting; Yi Zhang
2004-01-01
The motions of points, lines, and planes, embedded in a rigid body are expressed in a unified algebraic framework using a Clifford algebra. A Clifford algebra based displacement operator is addressed and its higher derivatives from which the coordinate-independent characteristic numbers with simple geometric meaning are defined. Because of the coordinate independent feature, no tedious coordinate transformation typically found in the conventional instantaneous invariants methods is needed.
Random walks on the BMW monoid: an algebraic approach
Wolff, Sarah
2016-01-01
We consider Metropolis-based systematic scan algorithms for generating Birman-Murakami-Wenzl (BMW) monoid basis elements of the BMW algebra. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We translate these walks into left multiplication operators in the corresponding BMW algebra. Taking this algebraic perspective enables the use of tools from representation theory to analyze the walks; in particular, we develop ...
Boolean Differential Operators
Catumba, Jorge; Diaz, Rafael
2012-01-01
We consider four combinatorial interpretations for the algebra of Boolean differential operators. We show that each interpretation yields an explicit matrix representation for Boolean differential operators.
Symplectic maps from cluster algebras
Fordy, Allan
2011-01-01
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding %associated quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a % symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The deg...
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…
Directory of Open Access Journals (Sweden)
Carlos C. Peña
2000-05-01
Full Text Available Topological algebras of sequences of complex numbers are introduced, endowed with a Hadamard product type. The complex homomorphisms on these algebras are characterized, and units, prime cyclic ideals, prime closed ideals, and prime minimal ideals, discussed. Existence of closed and maximal ideals are investigated, and it is shown that the Jacobson and nilradicals are both trivial.
Heinicke, C; Heinicke, Christian; Hehl, Friedrich W.
2001-01-01
We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some computer algebra systems and packages relevant to applications in gravitational physics. We conclude by presenting a couple of typical examples.
Introduction to noncommutative algebra
Brešar, Matej
2014-01-01
Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
Elements of mathematics algebra
Bourbaki, Nicolas
2003-01-01
This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...
A Representation of Quantum Measurement in Nonassociative Algebras
Niestegge, Gerd
2010-01-01
Starting from an abstract setting for the Lueders - von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for i...
Interpolation Problems for Nest Algebra Modules
Institute of Scientific and Technical Information of China (English)
李鹏同; 鲁世杰
2002-01-01
Let U be a weakly closed nest algebra module acting on a Hilbert space H.Given two operators X and Y in B(H),a necessary and sufficient condition for the existence of an operator T in U satisfying TX=Y is provided.
The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
Directory of Open Access Journals (Sweden)
Karl Hallowell
2007-09-01
Full Text Available Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R. These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.
Optical systolic solutions of linear algebraic equations
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
Linear algebra algorithms for divisors on an algebraic curve
Khuri-Makdisi, Kamal
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
Algebraic Properties of Toeplitz Operators on Discrete Commutative Groups%离散交换群上Toeplitz算子的代数性质
Institute of Scientific and Technical Information of China (English)
郭训香
2008-01-01
In this Paper,a generalized Toeplitz operator is defined and some of results about the classical Toeplitz operator are generalized.In particular,we obtain the necessary and sufficient condition for the product of two such Toeplitz operators to still be Toeplitz operator and the necessary and sufficient condition for such Toeplitz operator to be normal operator.Finally,a necessary condition for two such Toeplitz operators to be commutative is established.
The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra
Indian Academy of Sciences (India)
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.
Graded Lie Algebra Generating of Parastatistical Algebraic Relations
Institute of Scientific and Technical Information of China (English)
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.
The role of C*-algebras in infinite dimensional numerical linear algebra
Arveson, W
1993-01-01
This is a survey of four recent papers which deal with the relationship of simple C*-algebras to the problem of computing the spectra of self-adjoint operators in the general case, especially when the spectrum is not discrete. It is an expanded version of a talk presented at the 50 year C*-algebra celebration, held at the annual meeting of the AMS in San Antonio during January, 1993.
Quasi-lisse vertex algebras and modular linear differential equations
Arakawa, Tomoyuki
2016-01-01
We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level $-h^{\\vee}/6-1$, which expresses the homogeneous Schur limit of the superconformal index of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
Directory of Open Access Journals (Sweden)
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.
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
Kandasamy, W B Vasantha
2008-01-01
In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized world. Some other structures ought to replace linear algebras which are over a century old. Super linear algebras that use super matrices can store data not only in a block but in multiple blocks so it is certainly more powerful than the usual matrices. This book has 3 chapters. Chapter one introduces the notion of super vector spaces and enumerates a number of properties. Chapter two defines the notion of sup...
Permutation Centralizer Algebras and Multi-Matrix Invariants
Mattioli, Paolo
2016-01-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multi-matrix gauge invariant observables. One family of such non-commutative algebras is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of 2-matrix models. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The centre of the algebra allows efficient computation of a sector of multi-matrix correlator...
Monotone complete C*-algebras and generic dynamics
Saitô, Kazuyuki
2015-01-01
This monograph is about monotone complete C*-algebras, their properties and the new classification theory. A self-contained introduction to generic dynamics is also included because of its important connections to these algebras. Our knowledge and understanding of monotone complete C*-algebras has been transformed in recent years. This is a very exciting stage in their development, with much discovered but with many mysteries to unravel. This book is intended to encourage graduate students and working mathematicians to attack some of these difficult questions. Each bounded, upward directed net of real numbers has a limit. Monotone complete algebras of operators have a similar property. In particular, every von Neumann algebra is monotone complete but the converse is false. Written by major contributors to this field, Monotone Complete C*-algebras and Generic Dynamics takes readers from the basics to recent advances. The prerequisites are a grounding in functional analysis, some point set topology and an eleme...
Two types of q-deformed Wigner algebra
Energy Technology Data Exchange (ETDEWEB)
Chung, Won Sang [Department of Physics and Research Institute of Natural Science, College of Natural Science, Gyeongsang National University, Jinju (Korea, Republic of)
2014-07-01
In this paper we introduce two types of q-deformed Wigner algebras. One is q-deformation of Wigner algebra with q-reflection symmetry and another is q-deformation of Wigner algebra with reflection symmetry. For two types of q-deformed Wigner algebras, we investigate the representation and the eigenvalue of the position operator. Like q-calculus, we introduce the (q; ν)- numbers, (q; ν)-derivatives and (q; ν)-Hermite polynomials for two algebras. For the deformation parameter q = 1 - ε with small ε, we discuss the thermodynamics of the particle obeying the q-deformed Wigner algebra. (Copyright copyright 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
International Nuclear Information System (INIS)
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras
Introduction to Image Algebra Ada
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Equational axioms of test algebra
Hollenberg, M.
2008-01-01
We present a complete axiomatization of test algebra ([24,18,29]), the two-sorted algebraic variant of Propositional Dynamic Logic (PDL,[21,7]). The axiomatization consists of adding a finite number of equations to any axiomatization of Kleene algebra ([15,26,17,4]) and algebraic translations of the
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
Four Lie algebras associated with R6 and their applications
Zhang, Yufeng; Tam, Honwah
2010-09-01
The first part in the paper reads that a three-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which isospectral problems are established. By employing zero curvature equations, a modified Kaup-Newell (mKN) soliton hierarchy of evolution equations is obtained. The corresponding hereditary operator and Hamiltonian structure are worked out, respectively. Then two types of enlarging semisimple Lie algebras isomorphic to the linear space R6 are followed to construct, one of them is a complex Lie algebra. Their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mKN hierarchy. The hereditary operators, Hamiltonian structures of the hierarchies are produced again, respectively. The exact computing formulas of the constant γ appearing in the trace identity and the variational identity are derived under the semisimple algebras. The second part of this paper is devoted to constructing two kinds of Lie algebras by using product of complex vectors, which are also isomorphic to the linear space R6. Then we make use of the corresponding loop algebras to produce two integrable hierarchies along with bi-Hamiltonian structures. From various aspects, we give some ways for constructing Lie algebras which have extensive applications in generating integrable Hamiltonian systems.
Symplectic algebraic dynamics algorithm
Institute of Scientific and Technical Information of China (English)
2007-01-01
Based on the algebraic dynamics solution of ordinary differential equations andintegration of ,the symplectic algebraic dynamics algorithm sn is designed,which preserves the local symplectic geometric structure of a Hamiltonian systemand possesses the same precision of the na ve algebraic dynamics algorithm n.Computer experiments for the 4th order algorithms are made for five test modelsand the numerical results are compared with the conventional symplectic geometric algorithm,indicating that sn has higher precision,the algorithm-inducedphase shift of the conventional symplectic geometric algorithm can be reduced,and the dynamical fidelity can be improved by one order of magnitude.
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
Directory of Open Access Journals (Sweden)
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.
Michael Roitman
2003-01-01
In this paper we prove that for any commutative (but in general non-associative) algebra $A$ with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra $V = V_0 \\oplus V_2 \\oplus V_3\\oplus ...$, such that $\\dim V_0 = 1$ and $V_2$ contains $A$. We can choose $V$ so that if $A$ has a unit $e$, then $2e$ is the Virasoro element of $V$, and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as well. In addition, the algebra $V$ can be chosen with...
Symmetries of faces models and the double triangle algebra
Trinchero, R
2005-01-01
Symmetries of trigonometric integrable two dimensional statistical face models are considered. The corresponding symmetry operators on the Hilbert space of states of the quantum version of these models define a weak *-Hopf algebra isomorphic to the Ocneanu double triangle algebra(DTA).
Introduction to Matrix Algebra, Student's Text, Unit 23.
Allen, Frank B.; And Others
Unit 23 in the SMSG secondary school mathematics series is a student text covering the following topics in matrix algebra: matrix operations, the algebra of 2 X 2 matrices, matrices and linear systems, representation of column matrices as geometric vectors, and transformations of the plane. Listed in the appendix are four research exercises in…
Relating Kac-Moody, Virasoro and Krichever-Novikov algebras
Energy Technology Data Exchange (ETDEWEB)
Alberty, J.; Taormina, A.; Van Baal, P.
1988-12-01
We demonstrate that the Kac-Moody and Virasoro-like algebras on Riemann surfaces of arbitrary genus with two punctures introduced by Krichever and Novikov are in two ways linearly related to Kac-Moody and Virasoro algebras on S/sup 1/. The two relations differ by a Bogoliubov transformation, and we discuss the connection with the operator formalism.
Relating Kac-Moody, Virasoro and Krichever-Novikov algebras
Alberty, José; Taormina, Anne; van Baal, Pierre
1988-06-01
We demonstrate that the Kac-Moody and Virasoro-like algebras on Riemann surfaces of arbitrary genus with two punctures introduced by Krichever and Novikov are in two ways linearly related to Kac-Moody and Virasoro algebras on S 1. The two relations differ by a Bogoliubov transformation, and we discuss the connection with the operator formalism.
The Virasoro Algebra and Some Exceptional Lie and Finite Groups
Directory of Open Access Journals (Sweden)
Michael P. Tuite
2007-01-01
Full Text Available We describe a number of relationships between properties of the vacuum Verma module of a Virasoro algebra and the automorphism group of certain vertex operator algebras. These groups include the Deligne exceptional series of simple Lie groups and some exceptional finite simple groups including the Monster and Baby Monster.
Invariant differential operators
Dobrev, Vladimir K
2016-01-01
With applications in quantum field theory, elementary particle physics and general relativity, this two-volume work studies invariance of differential operators under Lie algebras, quantum groups, superalgebras including infinite-dimensional cases, Schrödinger algebras, applications to holography. This first volume covers the general aspects of Lie algebras and group theory.
Meadow enriched ACP process algebras
J.A. Bergstra; Middelburg, C.A.
2009-01-01
We introduce the notion of an ACP process algebra. The models of the axiom system ACP are the origin of this notion. ACP process algebras have to do with processes in which no data are involved. We also introduce the notion of a meadow enriched ACP process algebra, which is a simple generalization of the notion of an ACP process algebra to processes in which data are involved. In meadow enriched ACP process algebras, the mathematical structure for data is a meadow.
Algebraic Properties of Propositional Calculus
Schuh, Bernd R.
2009-01-01
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to subs...
Hom-power associative algebras
Yau, Donald
2010-01-01
A generalization of power associative algebra, called Hom-power associative algebra, is studied. The main result says that a multiplicative Hom-algebra is Hom-power associative if and only if it satisfies two identities of degrees three and four. It generalizes Albert's result that power associativity is equivalent to third and fourth power associativity. In particular, multiplicative right Hom-alternative algebras and non-commutative Hom-Jordan algebras are Hom-power associative.
Arrangement Computation for Planar Algebraic Curves
Berberich, Eric; Kobel, Alexander; Sagraloff, Michael
2011-01-01
We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system. We implemented our algorithm as a prototypical package of the C++-library CGAL. Our implementation exploits graphics hardware to expedite the resultant and gcd...
Supersymmetric extension of the Snyder algebra
Energy Technology Data Exchange (ETDEWEB)
Gouba, L., E-mail: lgouba@ictp.it [Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste (Italy); Stern, A., E-mail: astern@bama.ua.edu [Dept. of Physics and Astronomy, Univ. of Alabama, Tuscaloosa, Al 35487 (United States)
2012-04-11
We obtain a minimal supersymmetric extension of the Snyder algebra and study its representations. The construction differs from the general approach given in Hatsuda and Siegel ( (arXiv:hep-th/0311002)) and does not utilize super-de Sitter groups. The spectra of the position operators are discrete, implying a lattice description of space, and the lattice is compatible with supersymmetry transformations. -- Highlights: Black-Right-Pointing-Pointer A new supersymmetric extension of the Snyder algebra is constructed. Black-Right-Pointing-Pointer The extension is minimal and the construction does not involve supersymmetric de Sitter algebras. Black-Right-Pointing-Pointer An involution is defined for the system and discrete representations are constructed. Black-Right-Pointing-Pointer The representations imply a spatial lattice and the lattice spacing is half that of the bosonic case. Black-Right-Pointing-Pointer A differential operator representation is given for fields on super-momentum space.
An algebra of discrete event processes
Heymann, Michael; Meyer, George
1991-01-01
This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.
The Algebraic Properties of Concept Lattice
Institute of Scientific and Technical Information of China (English)
KaisheQu; JiyeLiang; JunhongWang; ZhongzhiShi
2004-01-01
Concept lattice is a powerful tool for data analysis. It has been applied widely to machine learning, knowledge discovery and software engineering and so on. Some aspects of concept lattice have been studied widely such as building lattice and rules extraction, as for its algebraic properties, there has not been discussed systematically. The paper suggests a binary operation between the elements for the set of all concepts in formal context. This turns the concept lattice in general significance into those with operators. We also proved that the concept lattice is a lattice in algebraic significance and studied its algebraic properties.These results provided theoretical foundation and a new method for further study of concept lattice.
On isomorphisms of integral table algebras
Institute of Scientific and Technical Information of China (English)
FAN; Yun(樊恽); SUN; Daying(孙大英)
2002-01-01
For integral table algebras with integral table basis T, we can consider integral R-algebra RT over a subring R of the ring of the algebraic integers. It is proved that an R-algebra isomorphism between two integral table algebras must be an integral table algebra isomorphism if it is compatible with the so-called normalizings of the integral table algebras.
The quantum holonomy-diffeomorphism algebra and quantum gravity
Aastrup, Johannes; Grimstrup, Jesper Møller
2016-03-01
We introduce the quantum holonomy-diffeomorphism ∗-algebra, which is generated by holonomy-diffeomorphisms on a three-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac-type operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial three-dimensional Dirac operator and Dirac-Hamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a Yang-Mills-type operator over the space of SU(2)-connections.
Cameron, Peter J
2007-01-01
This Second Edition of a classic algebra text includes updated and comprehensive introductory chapters,. new material on axiom of Choice, p-groups and local rings, discussion of theory and applications, and over 300 exercises. It is an ideal introductory text for all Year 1 and 2 undergraduate students in mathematics. - ;Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with. applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the th...
Indian Academy of Sciences (India)
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).
Introduction to abstract algebra
Nicholson, W Keith
2012-01-01
Praise for the Third Edition ". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."-Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately be
Institute of Scientific and Technical Information of China (English)
Antonio AIZPURU; Antonio GUTI(E)RREZ-D(A)VILA
2004-01-01
In this paper we will study some families and subalgebras ( ) of ( )(N) that let us characterize the unconditional convergence of series through the weak convergence of subseries ∑i∈A xi, A ∈ ( ).As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.
Geometric Algebra for Physicists
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
Andrilli, Stephen
2010-01-01
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, expl
Intermediate algebra & analytic geometry
Gondin, William R
1967-01-01
Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system
Law, Shirley
2014-01-01
International audience A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The...
Holomorphically Equivalent Algebraic Embeddings
Feller, Peter; Stampfli, Immanuel
2014-01-01
We prove that two algebraic embeddings of a smooth variety $X$ in $\\mathbb{C}^m$ are the same up to a holomorphic coordinate change, provided that $2 \\dim X + 1$ is smaller than or equal to $m$. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of $\\mathbb{C}^m$.
A Canonical Laplacian on the Algebra of Densities on a Projectively Connected Manifold
George, Jacob
2009-01-01
On a manifold with a projective connection we canonically assign a second order differential operator acting on the algebra of all densities to any tensor density $S^{ij}$ of fixed weight $\\lambda$. In particular, this implies that on any projectively connected manifold, a `bracket' (symmetric biderivation) on the algebra of functions extends canonically to the algebra of densities.
Pawlak algebra and approximate structure on fuzzy lattice.
Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai
2014-01-01
The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.
Rao, Shrisha
2009-01-01
Every system of any significant size is created by composition from smaller sub-systems or components. It is thus fruitful to analyze the fault-tolerance of a system as a function of its composition. In this paper, two basic types of system composition are described, and an algebra to describe fault tolerance of composed systems is derived. The set of systems forms monoids under the two composition operators, and a semiring when both are concerned. A partial ordering relation between systems is used to compare their fault-tolerance behaviors.
Linear algebra, geometry and transformation
Solomon, Bruce
2014-01-01
Vectors, Mappings and Linearity Numeric Vectors Functions Mappings and Transformations Linearity The Matrix of a Linear Transformation Solving Linear Systems The Linear SystemThe Augmented Matrix and RRE Form Homogeneous Systems in RRE Form Inhomogeneous Systems in RRE Form The Gauss-Jordan Algorithm Two Mapping Answers Linear Geometry Geometric Vectors Geometric/Numeric Duality Dot-Product Geometry Lines, Planes, and Hyperplanes System Geometry and Row/Column Duality The Algebra of Matrices Matrix Operations Special Matrices Matrix Inversion A Logical Digression The Logic of the Inversion Alg
Topological gravity with exchange algebra
Aoyama, S.
1993-01-01
A topological gravity is obtained by twisting the effective $(2,0)$ super\\-gravity. We show that this topological gravity has an infinite number of BRST invariant quantities with conformal weight $0$. They are a tower of OSp$(2,2)$ multiplets and satisfy the classical exchange algebra of OSp$(2,2)$. We argue that these BRST invariant quantities become physical operators in the quantum theory and their correlation functions are braided according to the quantum OSp$(2,2)$ group. These propertie...
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
Hijligenberg, van den, N.W.; 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 $U(g)$. The construction of such differential structures is interpreted in terms of colour Lie superalgebras.
Axis Problem of Rough 3-Valued Algebras
Institute of Scientific and Technical Information of China (English)
Jianhua Dai; Weidong Chen; Yunhe Pan
2006-01-01
The collection of all the rough sets of an approximation space has been given several algebraic interpretations, including Stone algebras, regular double Stone algebras, semi-simple Nelson algebras, pre-rough algebras and 3-valued Lukasiewicz algebras. A 3-valued Lukasiewicz algebra is a Stone algebra, a regular double Stone algebra, a semi-simple Nelson algebra, a pre-rough algebra. Thus, we call the algebra constructed by the collection of rough sets of an approximation space a rough 3-valued Lukasiewicz algebra. In this paper,the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras, are studied. Whether the rough 3-valued Lukasiewicz algebra is a axled 3-valued Lukasiewicz algebra is examined.
Algebraic structures and eigenstates for integrable collective field theories
International Nuclear Information System (INIS)
Conditions for the construction of polynomial eigen-operators for the Hamiltonian of collective string field theories are explored. Such eigen-operators arise for only one monomial potential v(x)=μx2 in the collective field theory. They form a w∞-algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non-zero-energy polynomial eigen-operators. This analysis leads us to consider a particular potential ν(x)=μx2+g/x2. A Lie algebra of polynomial eigen-operators is then constructed for this potential. It is a symmetric 2-index Lie algebra, also represented as a subalgebra of U(sl(2)). (orig.)
Noise limitations in optical linear algebra processors.
Batsell, S G; Jong, T L; Walkup, J F; Krile, T F
1990-05-10
A general statistical noise model is presented for optical linear algebra processors. A statistical analysis which includes device noise, the multiplication process, and the addition operation is undertaken. We focus on those processes which are architecturally independent. Finally, experimental results which verify the analytical predictions are also presented.
Quasi-algebras and general Weyl quantization
International Nuclear Information System (INIS)
In this paper we show how the systematic use of the topological properties of the quasi-sup(*)-algebra L(S,S') leads to a systematization of the quantization procedure. With that as background, the multiplication of certain classes of pairs of operators of L(S,S') and the corresponding twisted product of their sybmols are defined. (orig./HSI)
Learning Activity Package, Pre-Algebra.
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) for individualized instruction in topics in pre-algebra, the units cover the decimal numeration system; number theory; fractions and decimals; ratio, proportion, and percent; sets; properties of operations; rational numbers; real numbers; open expressions; and open rational…
Algebra 1Q, Mathematics: 5215.12.
Hirigoyen, Hector
This is the second of the six guidebooks on minimum course content for first-year algebra; it includes the ordered field properties of the real number system, solution of linear equations and inequalities, verbal problems, exponents and operations with polynomials. Overall goals for the course are stated; performance objectives for each unit, a…
Algebra 1p, Mathematics: 5215.11.
Strachan, Florence; Hirigoyen, Hector
This is the first of six guidebooks on minimum course content for first-year algebra; it introduces the language of sets, the fundamental operations and properties of the real number system, the use of variables, and the solution of simple linear equations and inequalities. Overall goals for the course are stated; then performance objectives, a…
How to be Brilliant at Algebra
Webber, Beryl
2010-01-01
How to be Brilliant at Algebra is contains 40 photocopiable worksheets designed to improve students' understanding of number relationships and patterns. They will learn about: odds and evens; patterns; inverse operations; variables; calendars; equations; pyramid numbers; digital root patterns; prime numbers; Fibonacci numbers; Pascal's triangle.
Schematic limits of rank 4 Azuyama bundles are the locally-Witt algebras
Venkata-Balaji, T E
2002-01-01
It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction basechanges well. This generalises the main theorem of Part I of an earlier work and clarifies it by showing that the algebraic operation of forming the even Clifford algebra (=Witt algebra) of a rank 3 quadratic module essentially translates to performing the geometric operation of taking the schematic image of the scheme of Azumaya algebra structures.
SLAPP: A systolic linear algebra parallel processor
Energy Technology Data Exchange (ETDEWEB)
Drake, B.L.; Luk, F.T.; Speiser, J.M.; Symanski, J.J. (Naval Ocean Systems Center and Cornell Univ.)
1987-07-01
Systolic array computer architectures provide a means for fast computation of the linear algebra algorithms that form the building blocks of many signal-processing algorithms, facilitating their real-time computation. For applications to signal processing, the systolic array operates on matrices, an inherently parallel view of the data, using numerical linear algebra algorithms that have been suitably parallelized to efficiently utilize the available hardware. This article describes work currently underway at the Naval Ocean Systems Center, San Diego, California, to build a two-dimensional systolic array, SLAPP, demonstrating efficient and modular parallelization of key matric computations for real-time signal- and image-processing problems.
An Algebraic Hardware/Software Partitioning Algorithm
Institute of Scientific and Technical Information of China (English)
秦胜潮; 何积丰; 裘宗燕; 张乃孝
2002-01-01
Hardware and software co-design is a design technique which delivers computer systems comprising hardware and software components. A critical phase of the co-design process is to decompose a program into hardware and software. This paper proposes an algebraic partitioning algorithm whose correctness is verified in program algebra. The authors introduce a program analysis phase before program partitioning and develop a collection of syntax-based splitting rules. The former provides the information for moving operations from software to hardware and reducing the interaction between components, and the latter supports a compositional approach to program partitioning.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Over a field F of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector space A［D]=A［D] from any pair of a commutative associative algebra A with an identity element and the polynomial algebra ［D] of a commutative derivation subalgebra D of A. We prove that A[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only if A is D－simple and A［D] acts faithfully on A. Thus we obtain a lot of simple algebras.
Algebra, Logic and Qubits Quantum Abacus
Vlasov, A Yu
2000-01-01
The canonical anticommutation relations (CAR) for fermion systems can be represented by finite-dimensional matrix algebra, but it is impossible for canonical commutation relations (CCR) for bosons. After description of more simple case with representation CAR and (bounded) quantum computational networks via Clifford algebras in the paper are discussed CCR. For representation of the algebra it is not enough to use quantum networks with fixed number of qubits and it is more convenient to consider Turing machine with essential operation of appending new cells for description of infinite tape in finite terms --- it has straightforward generalization for quantum case, but for CCR it is necessary to work with symmetrized version of the quantum Turing machine. The system is called here quantum abacus due to understanding analogy with the ancient counting devices (abacus).
KWIC Index for Numerical Linear Algebra
Energy Technology Data Exchange (ETDEWEB)
Carpenter, J.A.
1983-07-01
This report is a sequel to ORNL/CSD-106 in the ongoing supplements to Professor A.S. Householder's KWIC Index for Numerical Algebra. Beginning with the previous supplement, the subject has been restricted to Numerical Linear Algebra, roughly characterized by the American Mathematical Society's classification sections 15 and 65F but with little coverage of infinite matrices, matrices over fields of characteristics other than zero, operator theory, optimization and those parts of matrix theory primarily combinatorial in nature. Some consideration is given to the uses of graph theory in Numerical Linear Algebra, particularly with respect to algorithms for sparse matrix computations. The period covered by this report is roughly the calendar year 1982 as measured by the appearance of the articles in the American Mathematical Society's Contents of Mathematical Publications lagging actual appearance dates by up to nearly half a year. The review citations are limited to the Mathematical Reviews (MR).
Pentagon Relations in Direct Sums and Grassmann Algebras
Korepanov, Igor G.; Sadykov, Nurlan M.
2013-01-01
We construct vast families of orthogonal operators obeying pentagon relation in a direct sum of three n-dimensional vector spaces. As a consequence, we obtain pentagon relations in Grassmann algebras, making a far reaching generalization of exotic Reidemeister torsions.
Linear algebra on a Cray X-MP. Technical document
Energy Technology Data Exchange (ETDEWEB)
Freund, R.F.
1990-04-01
This paper discusses basic issues of vectorization as well as memory organization and contention for vector machines. There is an analysis of the implications of these issues for the performance of basic linear algebra operations, SAXPY and SDOT.
Institute of Scientific and Technical Information of China (English)
WANG Shundin; ZHANG Hua
2008-01-01
Using functional derivative technique In quantum field theory,the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations.The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by Introducing the time translation operator.The functional partial differential evolution equations were solved by algebraic dynam-ics.The algebraic dynamics solutions are analytical In Taylor series In terms of both initial functions and time.Based on the exact analytical solutions,a new nu-merical algorithm-algebraic dynamics algorithm was proposed for partial differ-ential evolution equations.The difficulty of and the way out for the algorithm were discussed.The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Institute of Scientific and Technical Information of China (English)
2008-01-01
Using functional derivative technique in quantum field theory, the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations. The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynam-ics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new nu-merical algorithm—algebraic dynamics algorithm was proposed for partial differ-ential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Clone Networks, Clone Extensions and Biregularizations of Varieties of Algebras
Institute of Scientific and Technical Information of China (English)
J. Plonka
2001-01-01
We consider algebras of type τ- without nullary operations. An identity ψ≈ψ of type τ is clone compatible if ψ and ψ are the same variable or the sets of fundamental operation symbols in ψ and ψ are non-empty and identical. For a variety V, we denote by Vc the variety defined by all clone compatible identities from Id(V). In this paper, we give a construction of algebras called a clone network. Under some assumptions, we describe algebras from Vc by means of this construction. We find some properties of Vc and applications.
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.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Rings of quotients of incidence algebras and path algebras
DEFF Research Database (Denmark)
Esparza, Eduardo Ortega
2006-01-01
We compute the maximal right/left/symmetric rings of quotients of finite dimensional incidence and graph algebras. We show that these rings of quotients are Morita equivalent to incidence algebras and path algebras respectively, with respect to simpler, well determined partially ordered sets...
The Planar Algebra Associated to a Kac Algebra
Indian Academy of Sciences (India)
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'.
Structure of Solvable Quadratic Lie Algebras
Institute of Scientific and Technical Information of China (English)
ZHU Lin-sheng
2005-01-01
@@ Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics[10,12,13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras.
Blyth, T S
2002-01-01
Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers:...
Algebraic quantum field theory
International Nuclear Information System (INIS)
The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)
Jarvis, Frazer
2014-01-01
The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the fi...
Directory of Open Access Journals (Sweden)
Michael Roitman
2008-08-01
Full Text Available In this paper we prove that for any commutative (but in general non-associative algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V_0 oplus V2 oplus V3 oplus ..., such that dim V_0 = 1 and V_2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Energy Technology Data Exchange (ETDEWEB)
Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado
1997-10-01
The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory.
Algebra for Gifted Third Graders.
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Indian Academy of Sciences (India)
Anil K Karn
2003-02-01
Order unit property of a positive element in a *-algebra is defined. It is proved that precisely projections satisfy this order theoretic property. This way, unital hereditary *-subalgebras of a *-algebra are characterized.
Extended Griess algebras and Matsuo-Norton trace formulae
Yamauchi, Hiroshi
2012-01-01
We introduce the Z_2-extended Griess algebra of a vertex operator superalgebra with an involution and derive the Matsuo-Norton trace formulae for the extended Griess algebra based on conformal design structure. We illustrate an application of our formulae by reformulating the one-to-one correspondence between 2A-elements of the Baby-monster simple group and N=1 c=7/10 Virasoro subalgebras inside the Baby-monster vertex operator superalgebra.
Principles of algebraic geometry
Griffiths, Phillip A
1994-01-01
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special top
Reed, Nat
2011-01-01
For grades 3-5, our State Standards-based combined resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to review the concepts in unique ways. The task sheets introduce the mathematical concepts to the students around a central problem taken from real-life experiences, while the drill sheets provide warm-up and timed practice questions for the students to strengthen their procedural proficiency skills. Included are opportunities for problem-solving, patterning, algebraic graphing, equations and determining averages. The combined task & drill sheets
Reed, Nat
2011-01-01
For grades 6-8, our State Standards-based combined resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to review the concepts in unique ways. The task sheets introduce the mathematical concepts to the students around a central problem taken from real-life experiences, while the drill sheets provide warm-up and timed practice questions for the students to strengthen their procedural proficiency skills. Included are opportunities for problem-solving, patterning, algebraic graphing, equations and determining averages. The combined task & drill sheets
Recollements of extension algebras
Institute of Scientific and Technical Information of China (English)
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)).
Hogben, Leslie
2013-01-01
With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.New to the Second EditionSeparate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of
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
Kendig, Keith
2015-01-01
Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring th
Hohn, Franz E
2012-01-01
This complete and coherent exposition, complemented by numerous illustrative examples, offers readers a text that can teach by itself. Fully rigorous in its treatment, it offers a mathematically sound sequencing of topics. The work starts with the most basic laws of matrix algebra and progresses to the sweep-out process for obtaining the complete solution of any given system of linear equations - homogeneous or nonhomogeneous - and the role of matrix algebra in the presentation of useful geometric ideas, techniques, and terminology.Other subjects include the complete treatment of the structur
International Nuclear Information System (INIS)
In this paper, we begin the study of zero-dimensional field theories with fields taking values in a semistrict Lie 2-algebra. These theories contain the IKKT matrix model and various M-brane related models as special cases. They feature solutions that can be interpreted as quantized 2-plectic manifolds. In particular, we find solutions corresponding to quantizations of ℝ3, S3 and a five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie 2-algebra models around the solution corresponding to quantized ℝ3, we obtain higher BF-theory on this quantized space
Partially ordered algebraic systems
Fuchs, Laszlo
2011-01-01
Originally published in an important series of books on pure and applied mathematics, this monograph by a distinguished mathematician explores a high-level area in algebra. It constitutes the first systematic summary of research concerning partially ordered groups, semigroups, rings, and fields. The self-contained treatment features numerous problems, complete proofs, a detailed bibliography, and indexes. It presumes some knowledge of abstract algebra, providing necessary background and references where appropriate. This inexpensive edition of a hard-to-find systematic survey will fill a gap i
Energy Technology Data Exchange (ETDEWEB)
Christian, J M; McDonald, G S [Joule Physics Laboratory, School of Computing, Science and Engineering, Materials and Physics Research Centre, University of Salford, Salford M5 4WT (United Kingdom); Chamorro-Posada, P, E-mail: j.christian@salford.ac.u [Departamento de Teoria de la Senal y Comunicaciones e Ingenieria Telematica, Universidad de Valladolid, ETSI Telecomunicacion, Campus Miguel Delibes s/n, 47011 Valladolid (Spain)
2010-02-26
We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons.
Endomorphisms of graph algebras
DEFF Research Database (Denmark)
Conti, Roberto; Hong, Jeong Hee; Szymanski, Wojciech
2012-01-01
We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D...... that the restriction to the diagonal MASA of an automorphism which globally preserves both D_E and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C...
Automorphism groups of pointed Hopf algebras
Institute of Scientific and Technical Information of China (English)
YANG Shilin
2007-01-01
The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.
Derivations of generalized Weyl algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai(苏育才)
2003-01-01
A class of the associative and Lie algebras A[D] = A × F[D] of Weyl type are studied, where Ais a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] isthe polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A suchthat A is D-simple. The derivations of these associative and Lie algebras are precisely determined.
The theory of algebraic numbers
Pollard, Harry
1998-01-01
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Optimal Algorithm for Algebraic Factoring
Institute of Scientific and Technical Information of China (English)
支丽红
1997-01-01
This paper presents on optimized method for factoring multivariate polynomials over algebraic extension fields defined by an irreducible ascending set. The basic idea is to convert multivariate polynomials to univariate polynomials and algebraic extension fields to algebraic number fields by suitable integer substituteions.Then factorize the univariate polynomials over the algebraic number fields.Finally,construct mulativariate factors of the original polynomial by Hensel lemma and TRUEFACTOR test.Some examples with timing are included.
Vibrational spectra of nickel metalloporphyrins: An algebraic approach
Indian Academy of Sciences (India)
Srinivasa Rao Karumuri; Joydeep Choudhury; Nirmal Kumar Sarkar; Ramendu Bhattacharjee
2009-03-01
One of the most interesting areas of current research in molecular physics is the study of the vibrationally excitated states of medium and large molecules. In view of the considerable amount of experimental activity in this area, one needs theoretical models within which to interpret experimental data. Using Lie algebraic method, the vibrational energy levels of nickel metalloporphyrins like Ni(OEP), Ni porphyrin and Ni(TPP) are calculated for 16 vibrational modes. The algebraic Hamiltonian $$H = E_{0} + \\sum_{i=1}^{n} A_{i}C_{i} + \\sum_{i < j} A_{ij}C_{ij} + \\sum_{i < j}^{n} _{ij}M_{ij}$,$$ where , and are the algebraic parameters which vary from molecule to molecule and , and are algebraic operators. The vibrational energy levels are calculated using algebraic model Hamiltonian and the results are compared with the experimental values. The results obtained by this model are very accurate.