Surveys in differential-algebraic equations III

CERN Document Server

Reis, Timo

2015-01-01

The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Flexibility of DAE formulations - Reachability analysis and deterministic global optimization - Numerical linear algebra methods - Boundary value problems The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.

Differential-algebraic solutions of the heat equation

OpenAIRE

Buchstaber, Victor M.; Netay, Elena Yu.

2014-01-01

In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.

Lie algebras and linear differential equations.

Science.gov (United States)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

Quadratic algebras applied to noncommutative integration of the Klein-Gordon equation: Four-dimensional quadratic algebras containing three-dimensional nilpotent lie algebras

International Nuclear Information System (INIS)

Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.

1995-01-01

The study is continued on noncommutative integration of linear partial differential equations in application to the exact integration of quantum-mechanical equations in a Riemann space. That method gives solutions to the Klein-Gordon equation when the set of noncommutative symmetry operations for that equation forms a quadratic algebra consisting of one second-order operator and of first-order operators forming a Lie algebra. The paper is a continuation of, where a single nontrivial example is used to demonstrate noncommutative integration of the Klein-Gordon equation in a Riemann space not permitting variable separation

Variational linear algebraic equations method

International Nuclear Information System (INIS)

Moiseiwitsch, B.L.

1982-01-01

A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)

An algebraic approach to the scattering equations

Energy Technology Data Exchange (ETDEWEB)

Huang, Rijun; Rao, Junjie [Zhejiang Institute of Modern Physics, Zhejiang University,Hangzhou, 310027 (China); Feng, Bo [Zhejiang Institute of Modern Physics, Zhejiang University,Hangzhou, 310027 (China); Center of Mathematical Science, Zhejiang University,Hangzhou, 310027 (China); He, Yang-Hui [School of Physics, NanKai University,Tianjin, 300071 (China); Department of Mathematics, City University,London, EC1V 0HB (United Kingdom); Merton College, University of Oxford,Oxford, OX14JD (United Kingdom)

2015-12-10

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

An algebraic approach to the scattering equations

International Nuclear Information System (INIS)

Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui

2015-01-01

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

Hecke symmetries and characteristic relations on reflection equation algebras

International Nuclear Information System (INIS)

Gurevich, D.I.; Pyatov, P.N.

1996-01-01

We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived. 10 refs

Properties of coupled-cluster equations originating in excitation sub-algebras

Science.gov (United States)

Kowalski, Karol

2018-03-01

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

ASYS: a computer algebra package for analysis of nonlinear algebraic equations systems

International Nuclear Information System (INIS)

Gerdt, V.P.; Khutornoj, N.V.

1992-01-01

A program package ASYS for analysis of nonlinear algebraic equations based on the Groebner basis technique is described. The package is written in REDUCE computer algebra language. It has special facilities to treat polynomial ideals of positive dimension, corresponding to algebraic systems with infinitely many solutions. Such systems can be transformed to an equivalent set of subsystems with reduced number of variables in completely automatic way. It often allows to construct the explicit form of a solution set in many problems of practical importance. Some examples and results of comparison with the standard Reduce package GROEBNER and special-purpose systems FELIX and A1PI are given. 21 refs.; 2 tabs

Exact solution of some linear matrix equations using algebraic methods

Science.gov (United States)

Djaferis, T. E.; Mitter, S. K.

1977-01-01

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Algebraic quantity equations before Fisher and Pigou

OpenAIRE

Thomas M. Humphrey

1984-01-01

Readers of this Review are doubtlessly familiar with the famous equation of exchange, MV=PQ, frequently employed to analyze the price level effects of monetary shocks. One might think the algebraic formulation of the equation is an outgrowth of the 20th century tendency toward mathematical modeling and statistical testing. Indeed, textbooks typically associate the transaction velocity version of the equation with Irving Fisher and the alternative Cambridge cash balance version with A. C. Pigo...

SUPPORTING STUDENTS’ UNDERSTANDING OF LINEAR EQUATIONS WITH ONE VARIABLE USING ALGEBRA TILES

Directory of Open Access Journals (Sweden)

Sari Saraswati

2016-01-01

Full Text Available This research aimed to describe how algebra tiles can support students’ understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students use the algebra tiles to find a method to solve linear equations with one variable. Design research was used as an approach in this study. It consists of three phases, namely preliminary design, teaching experiment and retrospective analysis. Video registrations, students’ written works, pre-test, post-test, field notes, and interview are technic to collect data. The data were analyzed by comparing the hypothetical learning trajectory (HLT and the actual learning process. The result shows that algebra tiles could supports students’ understanding to find the formal solution of linear equation with one variable.Keywords: linear equation with one variable, algebra tiles, design research, balancing method, HLT DOI: http://dx.doi.org/10.22342/jme.7.1.2814.19-30

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

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Rutwig Campoamor-Stursberg

2016-03-01

Full Text Available A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.

The algebraic structure of lax equations for infinite matrices

NARCIS (Netherlands)

Helminck, G.F.

2002-01-01

In this paper we discuss the algebraic structure of the tower of differential difference equations that one can associate with any commutative subalgebra of $M_k(\\mathbb{C})$. These equations can be formulated conveniently in so-called Lax equations for infinite upper- resp. lowertriangular matrices

Beltrami algebra and symmetry of Beltrami equation on Riemann surfaces

International Nuclear Information System (INIS)

Guo Hanying; Xu Kaiwen; Shen Jianmin; Wang Shikun

1989-12-01

It is shown that the Beltrami equation has an infinite dimensional symmetry, namely the Beltrami algebra, on its solution spaces. The Beltrami algebra with central extension and its supersymmetric version are explicitly found. (author). 12 refs

Langevin equation with the deterministic algebraically correlated noise

Energy Technology Data Exchange (ETDEWEB)

Ploszajczak, M. [Grand Accelerateur National d`Ions Lourds (GANIL), 14 - Caen (France); Srokowski, T. [Grand Accelerateur National d`Ions Lourds (GANIL), 14 - Caen (France)]|[Institute of Nuclear Physics, Cracow (Poland)

1995-12-31

Stochastic differential equations with the deterministic, algebraically correlated noise are solved for a few model problems. The chaotic force with both exponential and algebraic temporal correlations is generated by the adjoined extended Sinai billiard with periodic boundary conditions. The correspondence between the autocorrelation function for the chaotic force and both the survival probability and the asymptotic energy distribution of escaping particles is found. (author). 58 refs.

Langevin equation with the deterministic algebraically correlated noise

International Nuclear Information System (INIS)

Ploszajczak, M.; Srokowski, T.

1995-01-01

Stochastic differential equations with the deterministic, algebraically correlated noise are solved for a few model problems. The chaotic force with both exponential and algebraic temporal correlations is generated by the adjoined extended Sinai billiard with periodic boundary conditions. The correspondence between the autocorrelation function for the chaotic force and both the survival probability and the asymptotic energy distribution of escaping particles is found. (author)

SUPPORTING STUDENTS’ UNDERSTANDING OF LINEAR EQUATIONS WITH ONE VARIABLE USING ALGEBRA TILES

Directory of Open Access Journals (Sweden)

Sari Saraswati

2016-01-01

Full Text Available This research aimed to describe how algebra tiles can support students’ understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students use the algebra tiles to find a method to solve linear equations with one variable. Design research was used as an approach in this study. It consists of three phases, namely preliminary design, teaching experiment and retrospective analysis. Video registrations, students’ written works, pre-test, post-test, field notes, and interview are technic to collect data. The data were analyzed by comparing the hypothetical learning trajectory (HLT and the actual learning process. The result shows that algebra tiles could supports students’ understanding to find the formal solution of linear equation with one variable.

Searching dependency between algebraic equations: An algorithm applied to automated reasoning

International Nuclear Information System (INIS)

Yang Lu; Zhang Jingzhong

1990-01-01

An efficient computer algorithm is given to decide how many branches of the solution to a system of algebraic also solve another equation. As one of the applications, this can be used in practice to verify a conjecture with hypotheses and conclusion expressed by algebraic equations, despite the variety of reducible or irreducible. (author). 10 refs

Algebraic equations an introduction to the theories of Lagrange and Galois

CERN Document Server

Dehn, Edgar

2004-01-01

Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form. And they are given a more thorough exposition than is customary. For this reason, and also because the author concentrates on concrete applications of algebraic theory, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical ex

Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

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Zhaolin Jiang

2014-01-01

Full Text Available The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of n×n complex skew-circulant matrices are displayed in this paper.

On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations

International Nuclear Information System (INIS)

Zhang Yu-Feng; Tam, Honwah

2016-01-01

In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A_1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A_1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. (paper)

On the economical solution method for a system of linear algebraic equations

Directory of Open Access Journals (Sweden)

Jan Awrejcewicz

2004-01-01

Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22. The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

Solving differential–algebraic equation systems by means of index reduction methodology

DEFF Research Database (Denmark)

Sørensen, Kim; Houbak, Niels; Condra, Thomas

2006-01-01

of a number of differential equations and algebraic equations — a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately...... stiff ODEs and index 1 DAEs by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of ordinary differential equations — ODEs....

Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles

Science.gov (United States)

Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim

2016-01-01

This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…

Reduced minimax filtering by means of differential-algebraic equations

NARCIS (Netherlands)

V. Mallet; S. Zhuk (Sergiy)

2011-01-01

htmlabstractA reduced minimax state estimation approach is proposed for high-dimensional models. It is based on the reduction of the ordinary differential equation with high state space dimension to the low-dimensional Differential-Algebraic Equation (DAE) and on the subsequent application of the

On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds

International Nuclear Information System (INIS)

Kashaev, R.M.; Savel'ev, M.V.; Savel'eva, S.A.

1990-01-01

Nonlinear equations associated through a zero curvature type representation with Lie algebras S 0 Diff T 2 and of infinitesimal diffeomorphisms of (S 1 ) 2 , and also with a new infinite-dimensional Lie algebras. In particular, the general solution (in the sense of the Goursat problem) of the heavently equation which describes self-dual Einstein spaces with one rotational Killing symmetry is discussed, as well as the solutions to a generalized equation. The paper is supplied with Appendix containing the definition of the continuum graded Lie algebras and the general construction of the nonlinear equations associated with them. 11 refs

Solving differential-algebraic equation systems by means of index reduction methodology

DEFF Research Database (Denmark)

Sørensen, Kim; Houbak, Niels; Condra, Thomas Joseph

2006-01-01

of a number of differential equations and algebraic equations - a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately...... stiff ODE’s and index 1 DAE’s by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of Ordinary- Differential-Equations - ODE’s....

Solution of systems of linear algebraic equations by the method of summation of divergent series

International Nuclear Information System (INIS)

Kirichenko, G.A.; Korovin, Ya.S.; Khisamutdinov, M.V.; Shmojlov, V.I.

2015-01-01

A method for solving systems of linear algebraic equations has been proposed on the basis on the summation of the corresponding continued fractions. The proposed algorithm for solving systems of linear algebraic equations is classified as direct algorithms providing an exact solution in a finite number of operations. Examples of solving systems of linear algebraic equations have been presented and the effectiveness of the algorithm has been estimated [ru

Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation

Directory of Open Access Journals (Sweden)

Mitsuo Kato

2018-01-01

Full Text Available A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painlevé VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.

Stability criteria for neutral delay differential-algebraic equations

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FAN Ni

2013-10-01

Full Text Available The asymptotic stability of neutral delay differential-algebraic equations is studied in this paper.Two stability criteria described by evaluating a corresponding harmonic function on the boundary of a torus region are presented.

Linear algebra a first course with applications to differential equations

CERN Document Server

Apostol, Tom M

2014-01-01

Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.

Quadratic algebras in the noncommutative integration method of wave equation

International Nuclear Information System (INIS)

Varaksin, O.L.

1995-01-01

The paper deals with the investigation of applications of the method of noncommutative integration of linear differential equations by partial derivatives. Nontrivial example was taken for integration of three-dimensions wave equation with the use of non-Abelian quadratic algebras

Nonlinear elliptic equations and nonassociative algebras

CERN Document Server

Nadirashvili, Nikolai; Vlăduţ, Serge

2014-01-01

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions...

Surveys in differential-algebraic equations IV

CERN Document Server

Reis, Timo

2017-01-01

The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs) which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - History of DAEs - DAE aspects of mechanical multibody systems - Model reduction of DAEs - Observability for DAEs - Numerical Analysis for DAEs The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.

Equations of motion of higher-spin gauge fields as a free differential algebra

International Nuclear Information System (INIS)

Vasil'ev, M.A.

1988-01-01

It is shown that the introduction of auxiliary dynamical variables that generalize the gravitational Weyl tensor permits one to reduce the equations of motion of free massless fields of all spins in the anti-de Sitter O(3,2) space to a form characteristic of free differential algebras. The equations of motion of auxiliary gauge fields introduced previously are modified analogously. Arguments are presented to the effect that the equations of motion of interacting massless fields of all spins should be described in terms of a free differential algebra which is a deformation of a known free differential algebra generated by 1- and 0-forms in the adjoint representation of a nonabelian superalgebra of higher spins and auxiliary fields

The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders

International Nuclear Information System (INIS)

Gurau, Razvan

2012-01-01

Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson equations, generalizing the loop equations of matrix models, translate into constraints satisfied by the partition function. The constraints have been shown, in the large N limit, to close a Lie algebra indexed by colored rooted D-ary trees yielding a first generalization of the Virasoro algebra in arbitrary dimensions. In this paper we complete the Schwinger Dyson equations and the associated algebra at all orders in 1/N. The full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.

Quadratic PBW-Algebras, Yang-Baxter Equation and Artin-Schelter Regularity

International Nuclear Information System (INIS)

Gateva-Ivanova, Tatiana

2010-08-01

We study quadratic algebras over a field k. We show that an n-generated PBW-algebra A has finite global dimension and polynomial growth iff its Hilbert series is H A (z) = 1/(1-z) n . A surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relations are binomials xy - c xy zt, c xy is an element of k x , which are square-free and nondegenerate. We prove that in this case various good algebraic and homological properties are closely related. The main result shows that for an n-generated quantum binomial algebra A the following conditions are equivalent: (i) A is a PBW-algebra with finite global dimension; (ii) A is PBW and has polynomial growth; (iii) A is an Artin-Schelter regular PBW-algebra; (iv) A is a Yang-Baxter algebra; (v) H A (z) = 1/(1-z) n ; (vi) The dual A ! is a quantum Grassman algebra; (vii) A is a binomial skew polynomial ring. This implies that the problem of classification of Artin-Schelter regular PBW-algebras of global dimension n is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation (X,r), on sets X of order n.| (author)

Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations

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Kenichi Kondo

2013-11-01

Full Text Available Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.

A generalized variational algebra and conserved densities for linear evolution equations

International Nuclear Information System (INIS)

Abellanas, L.; Galindo, A.

1978-01-01

The symbolic algebra of Gel'fand and Dikii is generalized to the case of n variables. Using this algebraic approach a rigorous characterization of the polynomial kernel of the variational derivative is given. This is applied to classify all the conservation laws for linear polynomial evolution equations of arbitrary order. (Auth.)

Numerical algebra, matrix theory, differential-algebraic equations and control theory festschrift in honor of Volker Mehrmann

CERN Document Server

Bollhöfer, Matthias; Kressner, Daniel; Mehl, Christian; Stykel, Tatjana

2015-01-01

This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on ...

Closure of the gauge algebra, generalized Lie equations and Feynman rules

International Nuclear Information System (INIS)

Batalin, I.A.

1984-01-01

A method is given by which an open gauge algebra can always be closed and even made abelian. As a preliminary the generalized Lie equations for the open group are obtained. The Feynman rules for gauge theories with open algebras are derived by reducing the gauge theory to a non-gauge one. (orig.)

Using Computer Symbolic Algebra to Solve Differential Equations.

Science.gov (United States)

Mathews, John H.

1989-01-01

This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. (YP)

Numerical Solution of Differential Algebraic Equations and Applications

DEFF Research Database (Denmark)

Thomsen, Per Grove

2005-01-01

These lecture notes have been written as part of a special course on the numerical solution of Differential Algebraic Equations and applications . The course was held at IMM in the spring of 2005. The authors of the different chapters have all taken part in the course and the chapters are written...

The Max-Plus Algebra of the Natural Numbers has no Finite Equational Basis

DEFF Research Database (Denmark)

Aceto, Luca; Esik, Zoltan; Ingolfsdottir, Anna

2003-01-01

This paper shows that the collection of identities which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum is not finitely based. Moreover, it is proven that, for every n, the equations in at most n variables that hold in N do not form...... an equational basis. As a stepping stone in the proof of these facts, several results of independent interest are obtained. In particular, explicit descriptions of the free algebras in the variety generated by N are offered. Such descriptions are based upon a geometric characterization of the equations...

On solvability of some quadratic functional-integral equation in Banach algebra

International Nuclear Information System (INIS)

Darwish, M.A.

2007-08-01

Using the technique of a suitable measure of non-compactness in Banach algebra, we prove an existence theorem for some functional-integral equations which contain, as particular cases, a lot of integral and functional-integral equations that arise in many branches of nonlinear analysis and its applications. Also, the famous Chandrasekhar's integral equation is considered as a special case. (author)

Extended trigonometric Cherednik algebras and nonstationary Schrödinger equations with delta-potentials

International Nuclear Information System (INIS)

Hartwig, J. T.; Stokman, J. V.

2013-01-01

We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.

Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space

International Nuclear Information System (INIS)

Daszkiewicz, Marcin; Walczyk, Cezary J.

2008-01-01

The Newton equation describing particle motion in a constant external field force on canonical, Lie-algebraic, and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of the particle is generated. We also indicate that in the case of spatial coordinates commuting in a Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces

Stability of the Exponential Functional Equation in Riesz Algebras

Directory of Open Access Journals (Sweden)

Bogdan Batko

2014-01-01

Full Text Available We deal with the stability of the exponential Cauchy functional equation F(x+y=F(xF(y in the class of functions F:G→L mapping a group (G, + into a Riesz algebra L. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.

A novel algebraic procedure for solving non-linear evolution equations of higher order

International Nuclear Information System (INIS)

Huber, Alfred

2007-01-01

We report here a systematic approach that can easily be used for solving non-linear partial differential equations (nPDE), especially of higher order. We restrict the analysis to the so called evolution equations describing any wave propagation. The proposed new algebraic approach leads us to traveling wave solutions and moreover, new class of solution can be obtained. The crucial step of our method is the basic assumption that the solutions satisfy an ordinary differential equation (ODE) of first order that can be easily integrated. The validity and reliability of the method is tested by its application to some non-linear evolution equations. The important aspect of this paper however is the fact that we are able to calculate distinctive class of solutions which cannot be found in the current literature. In other words, using this new algebraic method the solution manifold is augmented to new class of solution functions. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDE does not exist. Otherwise, for practical use the algebraic construction of new class of solutions is of fundamental interest

Linear representation of algebras with non-associative operations which are satisfy in the balanced functional equations

International Nuclear Information System (INIS)

Ehsani, Amir

2015-01-01

Algebras with a pair of non-associative binary operations (f, g) which are satisfy in the balanced quadratic functional equations with four object variables considered. First, we obtain a linear representation for the operations, of this kind of binary algebras (A,f,g), over an abelian group (A, +) and then we generalize the linear representation of operations, to an algebra (A,F) with non-associative binary operations which are satisfy in the balanced quadratic functional equations with four object variables. (paper)

Truncatable bootstrap equations in algebraic form and critical surface exponents

Energy Technology Data Exchange (ETDEWEB)

Gliozzi, Ferdinando [Dipartimento di Fisica, Università di Torino andIstituto Nazionale di Fisica Nucleare - sezione di Torino,Via P. Giuria 1, Torino, I-10125 (Italy)

2016-10-10

We describe examples of drastic truncations of conformal bootstrap equations encoding much more information than that obtained by a direct numerical approach. A three-term truncation of the four point function of a free scalar in any space dimensions provides algebraic identities among conformal block derivatives which generate the exact spectrum of the infinitely many primary operators contributing to it. In boundary conformal field theories, we point out that the appearance of free parameters in the solutions of bootstrap equations is not an artifact of truncations, rather it reflects a physical property of permeable conformal interfaces which are described by the same equations. Surface transitions correspond to isolated points in the parameter space. We are able to locate them in the case of 3d Ising model, thanks to a useful algebraic form of 3d boundary bootstrap equations. It turns out that the low-lying spectra of the surface operators in the ordinary and the special transitions of 3d Ising model form two different solutions of the same polynomial equation. Their interplay yields an estimate of the surface renormalization group exponents, y{sub h}=0.72558(18) for the ordinary universality class and y{sub h}=1.646(2) for the special universality class, which compare well with the most recent Monte Carlo calculations. Estimates of other surface exponents as well as OPE coefficients are also obtained.

Sensitivity theory for general non-linear algebraic equations with constraints

International Nuclear Information System (INIS)

Oblow, E.M.

1977-04-01

Sensitivity theory has been developed to a high state of sophistication for applications involving solutions of the linear Boltzmann equation or approximations to it. The success of this theory in the field of radiation transport has prompted study of possible extensions of the method to more general systems of non-linear equations. Initial work in the U.S. and in Europe on the reactor fuel cycle shows that the sensitivity methodology works equally well for those non-linear problems studied to date. The general non-linear theory for algebraic equations is summarized and applied to a class of problems whose solutions are characterized by constrained extrema. Such equations form the basis of much work on energy systems modelling and the econometrics of power production and distribution. It is valuable to have a sensitivity theory available for these problem areas since it is difficult to repeatedly solve complex non-linear equations to find out the effects of alternative input assumptions or the uncertainties associated with predictions of system behavior. The sensitivity theory for a linear system of algebraic equations with constraints which can be solved using linear programming techniques is discussed. The role of the constraints in simplifying the problem so that sensitivity methodology can be applied is highlighted. The general non-linear method is summarized and applied to a non-linear programming problem in particular. Conclusions are drawn in about the applicability of the method for practical problems

Teaching materials of algebraic equation

Science.gov (United States)

Widodo, S. A.; Prahmana, R. C. I.; Purnami, A. S.; Turmudi

2017-12-01

The purpose of this paper is to know the effectiveness of teaching materials algebraic equation. This type of research used experimental method. The population in this study is all students of mathematics education who take numerical method in sarjanawiyata tamansiswa of university; the sample is taken using cluster random sampling. Instrument used in this research is test and questionnaire. The test is used to know the problem solving ability and achievement, while the questionnaire is used to know the student's response on the teaching materials. Data Analysis technique of quantitative used Wilcoxon test, while the qualitative data used grounded theory. Based on the results of the test can be concluded that the development of teaching materials can improve the ability to solve problems and achievement.

Algebraic solution for the vector potential in the Dirac equation

Energy Technology Data Exchange (ETDEWEB)

Booth, H.S. [School of Mathematics and Physics, University of Tasmania, Hobart Tas (Australia); Centre for Mathematics and its Applications, Australian National University (Australia)]. E-mail: hbooth@wintermute.anu.edu.au; Legg, G.; Jarvis, P.D. [School of Mathematics and Physics, University of Tasmania, Hobart Tas (Australia)

2001-07-20

The Dirac equation for an electron in an external electromagnetic field can be regarded as a singular set of linear equations for the vector potential. Radford's method of algebraically solving for the vector potential is reviewed, with attention to the additional constraints arising from non-maximality of the rank. The extension of the method to general spacetimes is illustrated by examples in diverse dimensions with both c- and a-number wavefunctions. (author)

Algebraic models for the hierarchy structure of evolution equations at small x

International Nuclear Information System (INIS)

Rembiesa, P.; Stasto, A.M.

2005-01-01

We explore several models of QCD evolution equations simplified by considering only the rapidity dependence of dipole scattering amplitudes, while provisionally neglecting their dependence on transverse coordinates. Our main focus is on the equations that include the processes of pomeron splittings. We examine the algebraic structures of the governing equation hierarchies, as well as the asymptotic behavior of their solutions in the large-rapidity limit

Classification of all solutions of the algebraic Riccati equations for infinite-dimensional systems

NARCIS (Netherlands)

Iftime, O; Curtain, R; Zwart, H

2003-01-01

We obtain a complete classification of all self-adjoint solution of the control algebraic Riccati equation for infinite-dimensional systems under the following assumptions: the system is output stabilizable, strongly detectable and the filter Riccati equation has an invertible self-adjoint

A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

International Nuclear Information System (INIS)

Zhang Yufeng; Xu Xixiang

2004-01-01

A subalgebra of loop algebra A-bar 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G-bar is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

Science.gov (United States)

Alim, Murad

2017-08-01

The tt * equations define a flat connection on the moduli spaces of {2d, \\mathcal{N}=2} quantum field theories. For conformal theories with c = 3 d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the tt * equations, restricted to the conformal directions, in the cases d = 1, 2, 3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with the generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of tt * and provides these with a finer and algebraic meaning. For sigma models into lattice polarized K3 manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for d = 1 and 3. The generators of the differential rings of special functions are given by quasi-modular forms for d = 1 and their generalizations in d = 2, 3. Some explicit examples are worked out including the case of the mirror of the quartic in {\\mathbbm{P}^3}, where due to further algebraic constraints, the differential ring coincides with quasi modular forms.

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

Science.gov (United States)

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

A direct algebraic method applied to obtain complex solutions of some nonlinear partial differential equations

International Nuclear Information System (INIS)

Zhang Huiqun

2009-01-01

By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.

Equations of motion for a spectrum-generating algebra: Lipkin-Meshkov-Glick model

International Nuclear Information System (INIS)

Rosensteel, G; Rowe, D J; Ho, S Y

2008-01-01

For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin-Meshkov-Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 10 6 , computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point

Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra

Directory of Open Access Journals (Sweden)

J. C. Ndogmo

2016-01-01

Full Text Available An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution.

Prolongation Loop Algebras for a Solitonic System of Equations

Directory of Open Access Journals (Sweden)

Maria A. Agrotis

2006-11-01

Full Text Available We consider an integrable system of reduced Maxwell-Bloch equations that describes the evolution of an electromagnetic field in a two-level medium that is inhomogeneously broadened. We prove that the relevant Bäcklund transformation preserves the reality of the n-soliton potentials and establish their pole structure with respect to the broadening parameter. The natural phase space of the model is embedded in an infinite dimensional loop algebra. The dynamical equations of the model are associated to an infinite family of higher order Hamiltonian systems that are in involution. We present the Hamiltonian functions and the Poisson brackets between the extended potentials.

Algorithm for solving polynomial algebraic Riccati equations and its application

Czech Academy of Sciences Publication Activity Database

Augusta, Petr; Augustová, Petra

2012-01-01

Roč. 1, č. 4 (2012), s. 237-242 ISSN 2223-7038 R&D Projects: GA ČR GPP103/12/P494 Institutional support: RVO:67985556 Keywords : Numerical algorithms * algebraic Riccati equation * spatially distributed systems * optimal control Subject RIV: BC - Control Systems Theory http://lib.physcon.ru/doc?id=8b4876d6a57d

An introduction to the history of algebra solving equations from Mesopotamian times to the Renaissance

CERN Document Server

Sesiano, Jacques

2009-01-01

This text should not be viewed as a comprehensive history of algebra before 1600, but as a basic introduction to the types of problems that illustrate the earliest forms of algebra. It would be particularly useful for an instructor who is looking for examples to help enliven a course on elementary algebra with problems drawn from actual historical texts. -Warren Van Egmond about the French edition for MathSciNet This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever

On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

International Nuclear Information System (INIS)

Man, Yiu-Kwong

2010-01-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided. (fast track communication)

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

International Nuclear Information System (INIS)

Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.; Shirokov, I.V.

1995-01-01

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented

On the structure of the commutative Z2 graded algebra valued integrable equations

International Nuclear Information System (INIS)

Konopelchenko, B.G.

1980-01-01

Partial differential equations integrable by the linear matrix spectral problem of arbitrary order are considered for the case that the 'potentials' take their values in the commutative infinte-dimensional Z 2 graded algebra (superalgebra). The general form of the integrable equations and their Baecklund transformations are found. The infinite sets of the integrals of the motion are constructed. The hamiltonian character of the integrable equations is proved. (orig.)

Existence and Uniqueness of Solution of Schrodinger equation in extended Colombeau algebra

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Fariba Fattahi

2014-09-01

Full Text Available In this paper, we establish the existence and uniquenessresult of the linear Schr¨odinger equation with Marchaudfractional derivative in Colombeau generalized algebra.The purpose of introducing Marchaud fractional derivativeis regularizing it in Colombeau sense.

A q-Schroedinger algebra, its lowest weight representations and generalized q-deformed heat equations

International Nuclear Information System (INIS)

Dobrev, V.K.; Doebner, H.D.; Mrugalla, C.

1995-12-01

We give a q-deformation S-perpendicular q of the centrally extended Schroedinger algebra. We construct the lowest weight representations of S-perpendicular q , starting from the Verma modules over S-perpendicular q , finding their singular vectors and factoring the Verma submodules built on the singular vectors. We also give a vector-field realization of S-perpendicular q which provides polynomial realization of the lowest weight representations and an infinite hierarchy of q-difference equations which may be called generalized q-deformed heat equations. We also apply our methods to the on-shell q-Schroedinger algebra proposed by Floreanini and Vinet. (author). 12 refs

Computer programs for the solution of systems of linear algebraic equations

Science.gov (United States)

Sequi, W. T.

1973-01-01

FORTRAN subprograms for the solution of systems of linear algebraic equations are described, listed, and evaluated in this report. Procedures considered are direct solution, iteration, and matrix inversion. Both incore methods and those which utilize auxiliary data storage devices are considered. Some of the subroutines evaluated require the entire coefficient matrix to be in core, whereas others account for banding or sparceness of the system. General recommendations relative to equation solving are made, and on the basis of tests, specific subprograms are recommended.

Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra

Science.gov (United States)

Awata, Hidetoshi; Kanno, Hiroaki; Mironov, Andrei; Morozov, Alexei; Morozov, Andrey; Ohkubo, Yusuke; Zenkevich, Yegor

2017-07-01

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki algebra Uq ,t(gl^ ^ 1) . We demonstrate that certain refined topological string amplitudes satisfy these equations and find that the braiding transformations are performed by the R matrix of Uq ,t(gl^ ^ 1) . The resulting system is the uplifting of the u^1 Wess-Zumino-Witten model. The solutions to the (q ,t ) KZE are identified with the (spectral dual of) building blocks of the Nekrasov partition function for five-dimensional linear quiver gauge theories. We also construct an elliptic version of the KZE and discuss its modular and monodromy properties, the latter being related to a dual version of the KZE.

A Novel Partial Differential Algebraic Equation (PDAE) Solver

DEFF Research Database (Denmark)

Lim, Young-il; Chang, Sin-Chung; Jørgensen, Sten Bay

2004-01-01

For solving partial differential algebraic equations (PDAEs), the space-time conservation element/solution element (CE/SE) method is addressed in this study. The method of lines (MOL) using an implicit time integrator is compared with the CE/SE method in terms of computational efficiency, solution...... or nonlinear adsorption isotherm are solved by the two methods. The CE/SE method enforces both local and global flux conservation in space and time, and uses a simple stencil structure (two points at the previous time level and one point at the present time level). Thus, accurate and computationally...

Expert Strategies in Solving Algebraic Structure Sense Problems: The Case of Quadratic Equations

Science.gov (United States)

Jupri, Al; Sispiyati, R.

2017-02-01

Structure sense, an intuitive ability towards symbolic expressions, including skills to interpret, to manipulate, and to perceive symbols in different roles, is considered as a key success in learning algebra. In this article, we report results of three phases of a case study on solving algebraic structure sense problems aiming at testing the appropriateness of algebraic structure sense tasks and at investigating expert strategies dealing with the tasks. First, we developed three tasks on quadratic equations based on the characteristics of structure sense for high school algebra. Next, we validated the tasks to seven experts. In the validation process, we requested these experts to solve each task using two different strategies. Finally, we analyzing expert solution strategies in the light of structure sense characteristics. We found that even if eventual expert strategies are in line with the characteristics of structure sense; some of their initial solution strategies used standard procedures which might pay less attention to algebraic structures. This finding suggests that experts have reconsidered their procedural work and have provided more efficient solution strategies. For further investigation, we consider to test the tasks to high school algebra students and to see whether they produce similar results as experts.

On Robust Stability of Differential-Algebraic Equations with Structured Uncertainty

Directory of Open Access Journals (Sweden)

A. Kononov

2018-03-01

Full Text Available We consider a linear time-invariant system of differential-algebraic equations (DAE, which can be written as a system of ordinary differential equations with non-invertible coefficients matrices. An important characteristic of DAE is the unsolvability index, which reflects the complexity of the internal structure of the system. The question of the asymptotic stability of DAE containing the uncertainty given by the matrix norm is investigated. We consider a perturbation in the structured uncertainty case. It is assumed that the initial nominal system is asymptotically stable. For the analysis, the original equation is reduced to the structural form, in which the differential and algebraic subsystems are separated. This structural form is equivalent to the input system in the sense of coincidence of sets of solutions, and the operator transforming the DAE into the structural form possesses the inverse operator. The conversion to structural form does not use a change of variables. Regularity of matrix pencil of the source equation is the necessary and sufficient condition of structural form existence. Sufficient conditions have been obtained that perturbations do not break the internal structure of the nominal system. Under these conditions robust stability of the DAE with structured uncertainty is investigated. Estimates for the stability radius of the perturbed DAE system are obtained. The text of the article is from the simpler case, in which the perturbation is present only for an unknown function, to a more complex one, under which the perturbation is also present in the derivative of the unknown function. We used values of the real and the complex stability radii of explicit ordinary differential equations for obtaining the results. We consider the example illustrating the obtained results.

First order linear ordinary differential equations in associative algebras

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Gordon Erlebacher

2004-01-01

Full Text Available In this paper, we study the linear differential equation $$ frac{dx}{dt}=sum_{i=1}^n a_i(t x b_i(t + f(t $$ in an associative but non-commutative algebra $mathcal{A}$, where the $b_i(t$ form a set of commuting $mathcal{A}$-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.

"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra

OpenAIRE

Gerbracht, Eberhard H. -A.

2008-01-01

In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit. In the linear case, we apply basic "symbolic" methods from linear algebra to the system of differential equations which is used to model the analog circuit. For nonlinear circuits and their corresponding nonlinear differential equations, we show how to employ computer algebra tools implemented in Maple, which are based on diff...

Algebraic treatment of second Poeschl-Teller, Morse-Rosen and Eckart equations

International Nuclear Information System (INIS)

Barut, A.O.; Inomata, A.; Wilson, R.

1987-01-01

The method of algebraic treatment is applied to the non-compact case to solve a family of second Poeschl-Teller, Morse-Rosen and Eckart equations with quantized coupling constants. Both discrete and continuous spectra, bound state and scattering wave functions (transmission coefficients) are found from the matrix elements of group representations. (author). 24 refs, 1 tab

Singular vectors and invariant equations for the Schroedinger algebra in n ≥ 3 space dimensions. The general case

International Nuclear Information System (INIS)

Dobrev, V. K.; Stoimenov, S.

2010-01-01

The singular vectors in Verma modules over the Schroedinger algebra s(n) in (n + 1)-dimensional space-time are found for the case of general representations. Using the singular vectors, hierarchies of equations invariant under Schroedinger algebras are constructed.

The Weyl approach to the representation theory of reflection equation algebra

International Nuclear Information System (INIS)

Saponov, P A

2004-01-01

The present paper deals with the representation theory of reflection equation algebra, connected to a Hecke type R-matrix. Up to some reasonable additional conditions, the R-matrix is arbitrary (not necessary originating from quantum groups). We suggest a universal method for constructing finite dimensional irreducible representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed. The modules are parametrized by Young diagrams. The spectrum of central elements s k Tr q L k is calculated in the single-row and single-column representations. A rule for the decomposition of the tensor product of modules into a direct sum of irreducible components is also suggested

Exploring Students' Understanding of Ordinary Differential Equations Using Computer Algebraic System (CAS)

Science.gov (United States)

Maat, Siti Mistima; Zakaria, Effandi

2011-01-01

Ordinary differential equations (ODEs) are one of the important topics in engineering mathematics that lead to the understanding of technical concepts among students. This study was conducted to explore the students' understanding of ODEs when they solve ODE questions using a traditional method as well as a computer algebraic system, particularly…

Analytic, Algebraic and Geometric Aspects of Differential Equations

CERN Document Server

Haraoka, Yoshishige; Michalik, Sławomir

2017-01-01

This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Będlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics. The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers. It will be of i...

Exact algebraization of the signal equation of spoiled gradient echo MRI

Energy Technology Data Exchange (ETDEWEB)

Dathe, Henning [Department of Orthodontics, Biomechanics Group, University Medical Centre, Goettingen (Germany); Helms, Gunther, E-mail: ghelms@gwdg.d [MR-Research in Neurology and Psychiatry, University Medical Centre, Goettingen (Germany)

2010-08-07

The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo acquisition in the steady state, created by a small flip angle {alpha} at a repetition time TR much shorter than the longitudinal relaxation time T{sub 1}. We describe two parameter transformations of {alpha} and TR/T{sub 1}, which render the Ernst equation as a low-order rational function. Computer algebra can be readily applied for analytically solving protocol optimization, as shown for the dual flip angle experiment. These transformations are based on the half-angle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small {alpha} and small TR/T{sub 1} with a third-order error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in {alpha} and TR/T{sub 1}. This reveals a fundamental relationship between the square of the flip angle and TR/T{sub 1} which characterizes the Ernst angle, constant degree of T{sub 1}-weighting and the influence of the local radio-frequency field.

Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays.

Science.gov (United States)

Sun, Leping

2016-01-01

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

Analytical Solutions for Systems of Singular Partial Differential-Algebraic Equations

Directory of Open Access Journals (Sweden)

U. Filobello-Nino

2015-01-01

Full Text Available This paper proposes power series method (PSM in order to find solutions for singular partial differential-algebraic equations (SPDAEs. We will solve three examples to show that PSM method can be used to search for analytical solutions of SPDAEs. What is more, we will see that, in some cases, Padé posttreatment, besides enlarging the domain of convergence, may be employed in order to get the exact solution from the truncated series solutions of PSM.

EXACT SOLITARY WAVE SOLUTIONS TO A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS USING DIRECT ALGEBRAIC METHOD

Institute of Scientific and Technical Information of China (English)

无

2008-01-01

Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.

Algebraic inversion of the Dirac equation for the vector potential in the non-Abelian case

International Nuclear Information System (INIS)

Inglis, S M; Jarvis, P D

2012-01-01

We study the Dirac equation for spinor wavefunctions minimally coupled to an external field, from the perspective of an algebraic system of linear equations for the vector potential. By analogy with the method in electromagnetism, which has been well-studied, and leads to classical solutions of the Maxwell–Dirac equations, we set up the formalism for non-Abelian gauge symmetry, with the SU(2) group and the case of four-spinor doublets. An extended isospin-charge conjugation operator is defined, enabling the hermiticity constraint on the gauge potential to be imposed in a covariant fashion, and rendering the algebraic system tractable. The outcome is an invertible linear equation for the non-Abelian vector potential in terms of bispinor current densities. We show that, via application of suitable extended Fierz identities, the solution of this system for the non-Abelian vector potential is a rational expression involving only Pauli scalar and Pauli triplet, Lorentz scalar, vector and axial vector current densities, albeit in the non-closed form of a Neumann series. (paper)

Existence Results for Some Nonlinear Functional-Integral Equations in Banach Algebra with Applications

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Lakshmi Narayan Mishra

2016-04-01

Full Text Available In the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. By utilizing the techniques of noncompactness measures, we operate the fixed point theorems such as Darbo's theorem in Banach algebra concerning the estimate on the solutions. The results obtained in this paper extend and improve essentially some known results in the recent literature. We also provide an example of nonlinear functional-integral equation to show the ability of our main result.

Head First Algebra A Learner's Guide to Algebra I

CERN Document Server

Pilone, Tracey

2008-01-01

Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. Does it make sense to buy two years of insurance on a car that depreciates as soon as you drive i

Lie Algebras and Integrable Systems

International Nuclear Information System (INIS)

Zhang Yufeng; Mei Jianqin

2012-01-01

A 3 × 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3 × 3 Lie subalgebra into a 2 × 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation. (general)

Symmetries of the Schrodinger Equation and Algebra/Superalgebra Duality

International Nuclear Information System (INIS)

Toppan, Francesco

2014-12-01

Some key features of the symmetries of the Schroedinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving rst and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation dependent notion of on-shell symmetry is introduced. The difference in associating the time derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric. (author)

Numerical Methods for a Class of Differential Algebraic Equations

Directory of Open Access Journals (Sweden)

Lei Ren

2017-01-01

Full Text Available This paper is devoted to the study of some efficient numerical methods for the differential algebraic equations (DAEs. At first, we propose a finite algorithm to compute the Drazin inverse of the time varying DAEs. Numerical experiments are presented by Drazin inverse and Radau IIA method, which illustrate that the precision of the Drazin inverse method is higher than the Radau IIA method. Then, Drazin inverse, Radau IIA, and Padé approximation are applied to the constant coefficient DAEs, respectively. Numerical results demonstrate that the Padé approximation is powerful for solving constant coefficient DAEs.

Multi-matrix loop equations: algebraic and differential structures and an approximation based on deformation quantization

International Nuclear Information System (INIS)

Krishnaswami, Govind S.

2006-01-01

Large-N multi-matrix loop equations are formulated as quadratic difference equations in concatenation of gluon correlations. Though non-linear, they involve highest rank correlations linearly. They are underdetermined in many cases. Additional linear equations for gluon correlations, associated to symmetries of action and measure are found. Loop equations aren't differential equations as they involve left annihilation, which doesn't satisfy the Leibnitz rule with concatenation. But left annihilation is a derivation of the commutative shuffle product. Moreover shuffle and concatenation combine to define a bialgebra. Motivated by deformation quantization, we expand concatenation around shuffle in powers of q, whose physical value is 1. At zeroth order the loop equations become quadratic PDEs in the shuffle algebra. If the variation of the action is linear in iterated commutators of left annihilations, these quadratic PDEs linearize by passage to shuffle reciprocal of correlations. Remarkably, this is true for regularized versions of the Yang-Mills, Chern-Simons and Gaussian actions. But the linear equations are underdetermined just as the loop equations were. For any particular solution, the shuffle reciprocal is explicitly inverted to get the zeroth order gluon correlations. To go beyond zeroth order, we find a Poisson bracket on the shuffle algebra and associative q-products interpolating between shuffle and concatenation. This method, and a complementary one of deforming annihilation rather than product are shown to give over and underestimates for correlations of a gaussian matrix model

Linear algebraic methods applied to intensity modulated radiation therapy.

Science.gov (United States)

Crooks, S M; Xing, L

2001-10-01

Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.

Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method

International Nuclear Information System (INIS)

Fan Engui

2002-01-01

A new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system. Compared with most of the existing tanh methods, the Jacobi elliptic function method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters. The solutions obtained in this paper include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions and (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper, KdV-MKdV, Ito's fifth MKdV, Hirota, Nizhnik-Novikov-Veselov, Broer-Kaup, generalized coupled Hirota-Satsuma, coupled Schroedinger-KdV, (2+1)-dimensional dispersive long wave, (2+1)-dimensional Davey-Stewartson equations. In addition, as an illustrative sample, the properties of the soliton solutions and Jacobi doubly periodic solutions for the Hirota equation are shown by some figures. The links among our proposed method, the tanh method, extended tanh method and the Jacobi elliptic function method are clarified generally. (author)

Algebra & trigonometry super review

CERN Document Server

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

Analytical solutions for systems of partial differential-algebraic equations.

Science.gov (United States)

Benhammouda, Brahim; Vazquez-Leal, Hector

2014-01-01

This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.

Equations of motion of interacting massless fields of all spins as a free differential algebra

Energy Technology Data Exchange (ETDEWEB)

Vasiliev, M A

1988-08-11

It is argued that the equations of motion of interacting massless fields of all spins s=0, 1, ..., infinity can naturally be formulated in terms of a free differential algebra (FDA) constructed from one-forms and zero-forms that belong both to the adjoint representation of the infinite-dimensional superalgebra of higher spins and auxiliary fields proposed previously. This FDA is found explicitly in the first non-trivial order in the zero-forms. Various properties of the proposed FDA are discussed including the ways for incorporating internal (Yang-Mills) gauge symmetries via associative algebras.

Strongly \\'etale difference algebras and Babbitt's decomposition

OpenAIRE

Tomašić, Ivan; Wibmer, Michael

2015-01-01

We introduce a class of strongly \\'{e}tale difference algebras, whose role in the study of difference equations is analogous to the role of \\'{e}tale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's decomposition theorem and we present applications to difference algebraic groups and the compatibility problem.

Algebra

CERN Document Server

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

Quantitative Algebraic Reasoning

DEFF Research Database (Denmark)

Mardare, Radu Iulian; Panangaden, Prakash; Plotkin, Gordon

2016-01-01

We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We define an equality relation indexed by rationals: a =ε b which we think of as saying that “a is approximately equal to b up to an error of ε”. We have 4 interesting examples where we have a quantitative...... equational theory whose free algebras correspond to well known structures. In each case we have finitary and continuous versions. The four cases are: Hausdorff metrics from quantitive semilattices; pWasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed...

Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrodynamical equations

Energy Technology Data Exchange (ETDEWEB)

Prykarpatsky, Anatoliy K [Department of Mining Geodesy, AGH University of Science and Technology, Cracow 30059 (Poland); Artemovych, Orest D [Department of Algebra and Topology, Faculty of Mathematics and Informatics of the Vasyl Stefanyk Pre-Carpathian National University, Ivano-Frankivsk (Ukraine); Popowicz, Ziemowit [Institute of Theoretical Physics, University of Wroclaw (Poland); Pavlov, Maxim V, E-mail: pryk.anat@ua.f, E-mail: artemo@usk.pk.edu.p, E-mail: ziemek@ift.uni.wroc.p, E-mail: M.V.Pavlov@lboro.ac.u [Department of Mathematical Physics, P.N. Lebedev Physical Institute, 53 Leninskij Prospekt, Moscow 119991 (Russian Federation)

2010-07-23

A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg-de Vries dynamical system.

Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrodynamical equations

International Nuclear Information System (INIS)

Prykarpatsky, Anatoliy K; Artemovych, Orest D; Popowicz, Ziemowit; Pavlov, Maxim V

2010-01-01

A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg-de Vries dynamical system.

The bubble algebra: structure of a two-colour Temperley-Lieb Algebra

International Nuclear Information System (INIS)

Grimm, Uwe; Martin, Paul P

2003-01-01

We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations

Intermediate algebra & analytic geometry

CERN Document Server

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

Analysis of the F. Calogero Type Projection-Algebraic Scheme for Differential Operator Equations

International Nuclear Information System (INIS)

Lustyk, Miroslaw; Bogolubov, Nikolai N. Jr.; Blackmore, Denis; Prykarpatsky, Anatoliy K.

2010-12-01

The existence, convergence, realizability and stability of solutions of differential operator equations obtained via a novel projection-algebraic scheme are analyzed in detail. This analysis is based upon classical discrete approximation techniques coupled with a recent generalization of the Leray-Schauder fixed point theorem. An example is included to illustrate the efficacy of the projection scheme and analysis strategy. (author)

"Real-Time Optical Laboratory Linear Algebra Solution Of Partial Differential Equations"

Science.gov (United States)

Casasent, David; Jackson, James

1986-03-01

A Space Integrating (SI) Optical Linear Algebra Processor (OLAP) employing space and frequency-multiplexing, new partitioning and data flow, and achieving high accuracy performance with a non base-2 number system is described. Laboratory data on the performance of this system and the solution of parabolic Partial Differential Equations (PDEs) is provided. A multi-processor OLAP system is also described for the first time. It use in the solution of multiple banded matrices that frequently arise is then discussed. The utility and flexibility of this processor compared to digital systolic architectures should be apparent.

Solving the generalized Langevin equation with the algebraically correlated noise

International Nuclear Information System (INIS)

Srokowski, T.; Ploszajczak, M.

1997-01-01

The Langevin equation with the memory kernel is solved. The stochastic force possesses algebraic correlations, proportional to 1/t. The velocity autocorrelation function and related quantities characterizing transport properties are calculated at the assumption that the system is in the thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble Levy walks with divergent moments of the velocity distribution. The motion of a Brownian particle is considered both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle. (author)

College algebra

CERN Document Server

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

Discrete integrable systems and deformations of associative algebras

International Nuclear Information System (INIS)

Konopelchenko, B G

2009-01-01

Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. Theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations. It is demonstrated that many discrete equations such as discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BKP equations, discrete Hirota-Miwa equations for KP and BKP hierarchies are particular realizations of the central system. An interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful. An interpretation of the Menelaus relation (discrete Schwarzian KP equation), discrete Hirota-Miwa equation for KP hierarchy, consistency around the cube as the associativity conditions and the concept of gauge equivalence, for instance, between the Menelaus and KP configurations are particular examples.

Basic linear algebra

CERN Document Server

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:...

Pre-Algebra Essentials For Dummies

CERN Document Server

Zegarelli, Mark

2010-01-01

Many students worry about starting algebra. Pre-Algebra Essentials For Dummies provides an overview of critical pre-algebra concepts to help new algebra students (and their parents) take the next step without fear. Free of ramp-up material, Pre-Algebra Essentials For Dummies contains content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical pre-algebra course, from fractions, decimals, and percents to scientific notation and simple variable equations. This guide is also a perfect reference for parents who need to review critical pre-algebra

A type of loop algebra and the associated loop algebras

International Nuclear Information System (INIS)

Tam Honwah; Zhang Yufeng

2008-01-01

A higher-dimensional twisted loop algebra is constructed. As its application, a new Lax pair is presented, whose compatibility gives rise to a Liouville integrable hierarchy of evolution equations by making use of Tu scheme. One of the reduction cases of the hierarchy is an analogous of the well-known AKNS system. Next, the twisted loop algebra, furthermore, is extended to another higher dimensional loop algebra, from which a hierarchy of evolution equations with 11-potential component functions is obtained, whose reduction is just standard AKNS system. Especially, we prove that an arbitrary linear combination of the four Hamiltonian operators directly obtained from the recurrence relations is still a Hamiltonian operator. Therefore, the hierarchy with 11-potential functions possesses 4-Hamiltonian structures. Finally, an integrable coupling of the hierarchy is worked out

Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory

KAUST Repository

Hall, Cameron L.; Chapman, S. Jonathan; Ockendon, John R.

2010-01-01

The system of algebraic equations given by σn j=0, j≠=i sgn(xi-xj )|xi-xj|a = 1, i = 1, 2, ⋯ , n, x0 = 0, appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole. We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n→∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment, but, up to corrections of logarithmic order, it also leads to a differential equation. The continuum approximation is valid only for i neither too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem. © 2010 Society for Industrial and Applied Mathematics.

A new generalized algebra method and its application in the (2 + 1) dimensional Boiti-Leon-Pempinelli equation

International Nuclear Information System (INIS)

Ren Yujie; Liu Shutian; Zhang Hongqing

2007-01-01

In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics

Colored Quantum Algebra and Its Bethe State

International Nuclear Information System (INIS)

Wang Jin-Zheng; Jia Xiao-Yu; Wang Shi-Kun

2014-01-01

We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation. (general)

Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model

International Nuclear Information System (INIS)

Li, Zi-Min; Batchelor, Murray T

2015-01-01

We obtain the exceptional part of the eigenspectrum of the generalized Rabi model, also known as the driven Rabi model, in terms of the roots of a set of algebraic equations. This approach provides a product form for the wavefunction components and allows an explicit connection with recent results obtained for the wavefunction in terms of truncated confluent Heun functions. Other approaches are also compared. For particular parameter values the exceptional part of the eigenspectrum consists of doubly degenerate crossing points. We give a proof for the number of roots of the constraint polynomials and discuss the number of crossing points. (paper)

Computer programs for nonlinear algebraic equations

International Nuclear Information System (INIS)

Asaoka, Takumi

1977-10-01

We have provided principal computer subroutines for obtaining numerical solutions of nonlinear algebraic equations through a review of the various methods. Benchmark tests were performed on these subroutines to grasp the characteristics of them compared to the existing subroutines. As computer programs based on the secant method, subroutines of the Muller's method using the Chambers' algorithm were newly developed, in addition to the equipment of subroutines of the Muller's method itself. The programs based on the Muller-Chambers' method are useful especially for low-order polynomials with complex coefficients except for the case of finding the triple roots, three close roots etc. In addition, we have equipped subroutines based on the Madsen's algorithm, a variant of the Newton's method. The subroutines have revealed themselves very useful as standard programs because all the roots are found accurately for every case though they take longer computing time than other subroutines for low-order polynomials. It is shown also that an existing subroutine of the Bairstow's method gives the fastest algorithm for polynomials with complex coefficients, except for the case of finding the triple roots etc. We have provided also subroutines to estimate error bounds for all the roots produced with the various algorithms. (auth.)

Study of some properties of partial differential equations by Lie algebra method

International Nuclear Information System (INIS)

Chongdar, A.K.; Ludu, A.

1990-05-01

In this note we present a system of optimal subalgebras of the Lie algebra obtained in course of investigating hypergeometric polynomial. In addition to this we have obtained some reduced equation and invariants of the P.D.E. obtained under certain transformation while studying hypergeometric polynomial by Weisner's method. Some topological properties of the solutions of P.D.E. are pointed out by using the extended jet bundle formalism. Some applications of our work on plasma physics and hydrodynamics are also cited. (author). 8 refs

Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras

International Nuclear Information System (INIS)

Khadzhiivanov, L.K.; Todorov, I.T.; Isaev, A.P.; Pyatov, P.N.; Ogievetskij, O.V.

1998-01-01

The quantum dynamical Yang-Baxter (or Gervais-Neveu-Felder) equation defines an R-matrix R cap (p), where p stands for a set of mutually commuting variables. A family of SL (n)-type solutions of this equation provides a new realization of the Hecke algebra. We define quantum antisymmetrizers, introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type R cap (p) a 1 a 2 = a 1 a 2 R cap. It is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the WZNW model

Space-time algebra for the generalization of gravitational field

Indian Academy of Sciences (India)

The Maxwell–Proca-like field equations of gravitolectromagnetism are formulated using space-time algebra (STA). The gravitational wave equation with massive gravitons and gravitomagnetic monopoles has been derived in terms of this algebra. Using space-time algebra, the most generalized form of ...

Helmholtz algebraic solitons

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.

Helmholtz algebraic solitons

International Nuclear Information System (INIS)

Christian, J M; McDonald, G S; Chamorro-Posada, P

2010-01-01

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.

Expansion of the Lie algebra and its applications

International Nuclear Information System (INIS)

Guo Fukui; Zhang Yufeng

2006-01-01

We take the Lie algebra A1 as an example to illustrate a detail approach for expanding a finite dimensional Lie algebra into a higher-dimensional one. By making use of the late and its resulting loop algebra, a few linear isospectral problems with multi-component potential functions are established. It follows from them that some new integrable hierarchies of soliton equations are worked out. In addition, various Lie algebras may be constructed for which the integrable couplings of soliton equations are obtained by employing the expanding technique of the the Lie algebras

Equivalent construction of the infinitesimal time translation operator in algebraic dynamics algorithm for partial differential evolution equation

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.

A linear algebraic approach to electron-molecule collisions

International Nuclear Information System (INIS)

Collins, L.A.; Schnieder, B.I.

1982-01-01

The linear algebraic approach to electron-molecule collisions is examined by firstly deriving the general set of coupled integrodifferential equations that describe electron collisional processes and then describing the linear algebraic approach for obtaining a solution to the coupled equations. Application of the linear algebraic method to static-exchange, separable exchange and effective optical potential, is examined. (U.K.)

Solution of the Schrodinger Equation for a Diatomic Oscillator Using Linear Algebra: An Undergraduate Computational Experiment

Science.gov (United States)

Gasyna, Zbigniew L.

2008-01-01

Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)

Advanced linear algebra for engineers with Matlab

CERN Document Server

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

Deformation of the exterior algebra and the GLq (r, included in) algebra

International Nuclear Information System (INIS)

El Hassouni, A.; Hassouni, Y.; Zakkari, M.

1993-06-01

The deformation of the associative algebra of exterior forms is performed. This operation leads to a Y.B. equation. Its relation with the braid group B n-1 is analyzed. The correspondence of this deformation with the GL q (r, included in) algebra is developed. (author). 9 refs

Algebra for Gifted Third Graders.

Science.gov (United States)

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)

Robust Algebraic Multilevel Methods and Algorithms

CERN Document Server

Kraus, Johannes

2009-01-01

This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. Provides a systematic presentation of the recent advances in robust algebraic multilevel methods. Can be used for advanced courses on the topic.

The algebraic geometry of multimonopoles

International Nuclear Information System (INIS)

Nahm, W.

1982-11-01

Multimonopole solutions of the Bogomolny equation are treated by a transform to an ordinary differential equation. The solution of this equation yields algebraic curves and holomorphic line bundles over them. (orig.)

Implementing the Standards: Teaching Informal Algebra.

Science.gov (United States)

Schultz, James E.

1991-01-01

Presents suggestions for developing algebraic concepts beginning in the early grades to develop a gradual building from informal to formal algebraic concepts that progresses over the K-12 curriculum. Includes suggestions for representing relationships, solving equations, employing meaningful applications of algebra, and using of technology. (MDH)

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper-and-Pencil

Science.gov (United States)

Fonger, Nicole L.; Davis, Jon D.; Rohwer, Mary Lou

2018-01-01

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is…

Ward identities and differential equations for supercharacters of N = 1 super-Kac-Moody algebras on supertorus

International Nuclear Information System (INIS)

Huang Chaoshang; Xu Kaiwen; Zhao Zhiyong.

1989-09-01

By using Bernard's method, the Ward identities for N = 1 super-Kac-Moody algebras on supertorus are completely given in the sense that any correlation function with currents inserted in it can be reduced from the correlation functions without insertion. The differential equations for the super-characters on supertorus are derived from the Ward identities. (author). 7 refs

Learning Activity Package, Algebra.

Science.gov (United States)

Evans, Diane

A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…

An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

KAUST Repository

Liu, Da-Yan; Tian, Yang; Boutat, Driss; Laleg-Kirati, Taous-Meriem

2015-01-01

This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

KAUST Repository

Liu, Da-Yan

2015-04-30

This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

Classical algebra its nature, origins, and uses

CERN Document Server

Cooke, Roger L

2008-01-01

This insightful book combines the history, pedagogy, and popularization of algebra to present a unified discussion of the subject. Classical Algebra provides a complete and contemporary perspective on classical polynomial algebra through the exploration of how it was developed and how it exists today. With a focus on prominent areas such as the numerical solutions of equations, the systematic study of equations, and Galois theory, this book facilitates a thorough understanding of algebra and illustrates how the concepts of modern algebra originally developed from classical algebraic precursors. This book successfully ties together the disconnect between classical and modern algebraand provides readers with answers to many fascinating questions that typically go unexamined, including: What is algebra about? How did it arise? What uses does it have? How did it develop? What problems and issues have occurred in its history? How were these problems and issues resolved? The author answers these questions and more,...

Three semi-direct sum Lie algebras and three discrete integrable couplings associated with the modified K dV lattice equation

International Nuclear Information System (INIS)

Yu Zhang; Zhang Yufeng

2009-01-01

Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified K dV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of soliton equations.

Quadratic algebra approach to relativistic quantum Smorodinsky-Winternitz systems

International Nuclear Information System (INIS)

Marquette, Ian

2011-01-01

There exists a relation between the Klein-Gordon and the Dirac equations with scalar and vector potentials of equal magnitude and the Schroedinger equation. We obtain the relativistic energy spectrum for the four relativistic quantum Smorodinsky-Winternitz systems from their quasi-Hamiltonian and the quadratic algebras studied by Daskaloyannis in the nonrelativistic context. We also apply the quadratic algebra approach directly to the initial Dirac equation for these four systems and show that the quadratic algebras obtained are the same than those obtained from the quasi-Hamiltonians. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras can be applied to the quantum relativistic case.

On W1+∞ 3-algebra and integrable system

Directory of Open Access Journals (Sweden)

Min-Ru Chen

2015-02-01

Full Text Available We construct the W1+∞ 3-algebra and investigate its connection with the integrable systems. Since the W1+∞ 3-algebra with a fixed generator W00 in the operator Nambu 3-bracket recovers the W1+∞ algebra, it is intrinsically related to the KP hierarchy. For the general case of the W1+∞ 3-algebra, we directly derive the KP and KdV equations from the Nambu–Poisson evolution equation with the different Hamiltonian pairs of the KP hierarchy. Due to the Nambu–Poisson evolution equation involves two Hamiltonians, the deep relationship between the Hamiltonian pairs of KP hierarchy is revealed. Furthermore we give a realization of the W1+∞ 3-algebra in terms of a complex bosonic field. Based on the Nambu 3-brackets of the complex bosonic field, we derive the (generalized nonlinear Schrödinger equation and give an application in optical soliton.

Templates for Linear Algebra Problems

NARCIS (Netherlands)

Bai, Z.; Day, D.; Demmel, J.; Dongarra, J.; Gu, M.; Ruhe, A.; Vorst, H.A. van der

1995-01-01

The increasing availability of advanced-architecture computers is having a very signicant eect on all spheres of scientic computation, including algorithm research and software development in numerical linear algebra. Linear algebra {in particular, the solution of linear systems of equations and

Exact WKB analysis and cluster algebras

International Nuclear Information System (INIS)

Iwaki, Kohei; Nakanishi, Tomoki

2014-01-01

We develop the mutation theory in the exact WKB analysis using the framework of cluster algebras. Under a continuous deformation of the potential of the Schrödinger equation on a compact Riemann surface, the Stokes graph may change the topology. We call this phenomenon the mutation of Stokes graphs. Along the mutation of Stokes graphs, the Voros symbols, which are monodromy data of the equation, also mutate due to the Stokes phenomenon. We show that the Voros symbols mutate as variables of a cluster algebra with surface realization. As an application, we obtain the identities of Stokes automorphisms associated with periods of cluster algebras. The paper also includes an extensive introduction of the exact WKB analysis and the surface realization of cluster algebras for nonexperts. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’. (paper)

Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra

Energy Technology Data Exchange (ETDEWEB)

Alvarez-Nodarse, R [Departamento de Analisis Matematico, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla (Spain); Atakishiyev, N M [Instituto de Matematicas, UNAM, Apartado Postal 273-3, CP 62210 Cuernavaca, Morelos, Mexico (Germany); Costas-Santos, R S [Departamento de Matematicas, EPS, Universidad Carlos III de Madrid, Ave. Universidad 30, E-28911, Leganes, Madrid (Spain)

2005-01-07

We argue that one can factorize the difference equation of hypergeometric type on non-uniform lattices in the general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues, this directly leads to the dynamical symmetry algebra su{sub q}(1, 1), whose generators are explicitly constructed in terms of the difference operators, obtained in the process of factorization. Thus all models with the q-linear spectrum (some of them, but not all, previously considered in a number of publications) can be treated in a unified form.

Groups of integral transforms generated by Lie algebras of second-and higher-order differential operators

International Nuclear Information System (INIS)

Steinberg, S.; Wolf, K.B.

1979-01-01

The authors study the construction and action of certain Lie algebras of second- and higher-order differential operators on spaces of solutions of well-known parabolic, hyperbolic and elliptic linear differential equations. The latter include the N-dimensional quadratic quantum Hamiltonian Schroedinger equations, the one-dimensional heat and wave equations and the two-dimensional Helmholtz equation. In one approach, the usual similarity first-order differential operator algebra of the equation is embedded in the larger one, which appears as a quantum-mechanical dynamic algebra. In a second approach, the new algebra is built as the time evolution of a finite-transformation algebra on the initial conditions. In a third approach, the algebra to inhomogeneous similarity algebra is deformed to a noncompact classical one. In every case, we can integrate the algebra to a Lie group of integral transforms acting effectively on the solution space of the differential equation. (author)

Distribution of the Discretization and Algebraic Error in Numerical Solution of Partial Differential Equations

Czech Academy of Sciences Publication Activity Database

Papež, Jan; Liesen, J.; Strakoš, Z.

2014-01-01

Roč. 449, 15 May (2014), s. 89-114 ISSN 0024-3795 R&D Projects: GA AV ČR IAA100300802; GA ČR GA201/09/0917 Grant - others:GA MŠk(CZ) LL1202; GA UK(CZ) 695612 Institutional support: RVO:67985807 Keywords : numerical solution of partial differential equations * finite element method * adaptivity * a posteriori error analysis * discretization error * algebra ic error * spatial distribution of the error Subject RIV: BA - General Mathematics Impact factor: 0.939, year: 2014

UCSMP Algebra. What Works Clearinghouse Intervention Report

Science.gov (United States)

What Works Clearinghouse, 2007

2007-01-01

"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…

New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation

Directory of Open Access Journals (Sweden)

Liu Jianzhou

2009-01-01

Full Text Available By using singular value decomposition and majorization inequalities, we propose new inequalities for the trace of the product of two arbitrary real square matrices. These bounds improve and extend the recent results. Further, we give their application in the algebraic Riccati equation. Finally, numerical examples have illustrated that our results are effective and superior.

The Universal C*-Algebra of the Electromagnetic Field

Science.gov (United States)

Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio

2016-02-01

A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of the field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.

Structure of Lie point and variational symmetry algebras for a class of odes

Science.gov (United States)

Ndogmo, J. C.

2018-04-01

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method

Directory of Open Access Journals (Sweden)

Murat Osmanoglu

2013-01-01

Full Text Available We have considered linear partial differential algebraic equations (LPDAEs of the form , which has at least one singular matrix of . We have first introduced a uniform differential time index and a differential space index. The initial conditions and boundary conditions of the given system cannot be prescribed for all components of the solution vector here. To overcome this, we introduced these indexes. Furthermore, differential transform method has been given to solve LPDAEs. We have applied this method to a test problem, and numerical solution of the problem has been compared with analytical solution.

Computer Algebra Systems in Undergraduate Instruction.

Science.gov (United States)

Small, Don; And Others

1986-01-01

Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)

Matrices and linear algebra

CERN Document Server

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

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)

Ebrahimi-Fard, K.

2006-06-01

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.)

INPUT-OUTPUT STRUCTURE OF LINEAR-DIFFERENTIAL ALGEBRAIC SYSTEMS

NARCIS (Netherlands)

KUIJPER, M; SCHUMACHER, JM

Systems of linear differential and algebraic equations occur in various ways, for instance, as a result of automated modeling procedures and in problems involving algebraic constraints, such as zero dynamics and exact model matching. Differential/algebraic systems may represent an input-output

Linear algebra

CERN Document Server

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

Einstein algebras and general relativity

International Nuclear Information System (INIS)

Heller, M.

1992-01-01

A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the space-time of a straight cosmic string with quasiregular singularity is constructed. 22 refs

Stability of Linear Equations--Algebraic Approach

Science.gov (United States)

Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.

2012-01-01

This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…

Galilean Duffin-Kemmer-Petiau algebra and symplectic structure

CERN Document Server

Fernandes, M C B; Vianna, J D M

2003-01-01

We develop the Duffin-Kemmer-Petiau (DKP) approach in the phase-space picture of quantum mechanics by considering DKP algebras in a Galilean covariant context. Specifically, we develop an algebraic calculus based on a tensor algebra defined on a five-dimensional space which plays the role of spacetime background of the non-relativistic DKP equation. The Liouville operator is determined and the Liouville-von Neumann equation is written in two situations: the free particle and a particle in an external electromagnetic field. A comparison between the non-relativistic and the relativistic cases is commented.

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

Science.gov (United States)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Traditional vectors as an introduction to geometric algebra

International Nuclear Information System (INIS)

Carroll, J E

2003-01-01

The 2002 Oersted Medal Lecture by David Hestenes concerns the many advantages for education in physics if geometric algebra were to replace standard vector algebra. However, such a change has difficulties for those who have been taught traditionally. A new way of introducing geometric algebra is presented here using a four-element array composed of traditional vector and scalar products. This leads to an explicit 4 x 4 matrix representation which contains key requirements for three-dimensional geometric algebra. The work can be extended to include Maxwell's equations where it is found that curl and divergence appear naturally together. However, to obtain an explicit representation of space-time algebra with the correct behaviour under Lorentz transformations, an 8 x 8 matrix representation has to be formed. This leads to a Dirac representation of Maxwell's equations showing that space-time algebra has hidden within its formalism the symmetry of 'parity, charge conjugation and time reversal'

Introduction to algebra and trigonometry

CERN Document Server

Kolman, Bernard

1981-01-01

Introduction to Algebra and Trigonometry provides a complete and self-contained presentation of the fundamentals of algebra and trigonometry.This book describes an axiomatic development of the foundations of algebra, defining complex numbers that are used to find the roots of any quadratic equation. Advanced concepts involving complex numbers are also elaborated, including the roots of polynomials, functions and function notation, and computations with logarithms. This text also discusses trigonometry from a functional standpoint. The angles, triangles, and applications involving triangles are

Semi-classical propagation of wavepackets for the phase space Schroedinger equation: interpretation in terms of the Feichtinger algebra

International Nuclear Information System (INIS)

Gosson, Maurice A de

2008-01-01

The nearby orbit method is a powerful tool for constructing semi-classical solutions of Schroedinger's equation when the initial datum is a coherent state. In this paper, we first extend this method to arbitrary squeezed states and thereafter apply our results to the Schroedinger equation in phase space. This adaptation requires the phase-space Weyl calculus developed in previous work of ours. We also study the regularity of the semi-classical solutions from the point of view of the Feichtinger algebra familiar from the theory of modulation spaces

Quasi exactly solvable operators and abstract associative algebras

International Nuclear Information System (INIS)

Brihaye, Y.; Kosinski, P.

1998-01-01

We consider the vector spaces consisting of direct sums of polynomials of given degrees and we show how to classify the linear differential operators preserving these spaces. The families of operators so obtained are identified as the envelopping algebras of particular abstract associative algebras. Some of these operators can be transformed into quasi exactly solvable Schroedinger operators which, having a hidden algebra, can be partially solved algebraically; we exhibit however a series of Schoedinger equations which, while completely solvable algebraically, do not possess a hidden algebra

From affine Hecke algebras to boundary symmetries

International Nuclear Information System (INIS)

Doikou, Anastasia

2005-01-01

Motivated by earlier works we employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the U q (gl n -bar ) case. The corresponding N site spin chain with open boundary conditions is then constructed and boundary non-local charges associated to the non-diagonal solutions of the reflection equation are derived, as coproduct realizations of the reflection algebra. With the help of linear intertwining relations involving the aforementioned solutions of the reflection equation, the symmetry of the open spin chain with the corresponding boundary conditions is exhibited, being essentially a remnant of the U q (gl n -bar ) algebra. More specifically, we show that representations of certain boundary non-local charges commute with the generators of the affine Hecke algebra and with the local Hamiltonian of the open spin chain for a particular choice of boundary conditions. Furthermore, we are able to show that the transfer matrix of the open spin chain commutes with a certain number of boundary non-local charges, depending on the choice of boundary conditions

Q-systems as cluster algebras

International Nuclear Information System (INIS)

Kedem, Rinat

2008-01-01

Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211-21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras g in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497-529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119-44)

Waterloo Workshop on Computer Algebra

CERN Document Server

Zima, Eugene; WWCA-2016; Advances in computer algebra : in honour of Sergei Abramov's' 70th birthday

2018-01-01

This book discusses the latest advances in algorithms for symbolic summation, factorization, symbolic-numeric linear algebra and linear functional equations. It presents a collection of papers on original research topics from the Waterloo Workshop on Computer Algebra (WWCA-2016), a satellite workshop of the International Symposium on Symbolic and Algebraic Computation (ISSAC’2016), which was held at Wilfrid Laurier University (Waterloo, Ontario, Canada) on July 23–24, 2016.   This workshop and the resulting book celebrate the 70th birthday of Sergei Abramov (Dorodnicyn Computing Centre of the Russian Academy of Sciences, Moscow), whose highly regarded and inspirational contributions to symbolic methods have become a crucial benchmark of computer algebra and have been broadly adopted by many Computer Algebra systems.

Introductory modern algebra a historical approach

CERN Document Server

Stahl, Saul

2013-01-01

Praise for the First Edition ""Stahl offers the solvability of equations from the historical point of view...one of the best books available to support a one-semester introduction to abstract algebra.""-CHOICE Introductory Modern Algebra: A Historical Approach, Second Edition presents the evolution of algebra and provides readers with the opportunity to view modern algebra as a consistent movement from concrete problems to abstract principles. With a few pertinent excerpts from the writings of some of the greatest mathematicians, the Second Edition uniquely facilitates the understanding of pi

{kappa}-deformed realization of D=4 conformal algebra

Energy Technology Data Exchange (ETDEWEB)

Klimek, M. [Technical Univ. of Czestochowa, Inst. of Mathematics and Computer Science, Czestochowa (Poland); Lukierski, J. [Universite de Geneve, Department de Physique Theorique, Geneve (Switzerland)

1995-07-01

We describe the generators of {kappa}-conformal transformations, leaving invariant the {kappa}-deformed d`Alembert equation. In such a way one obtains the conformal extension of-shell spin spin zero realization of {kappa}-deformed Poincare algebra. Finally the algebraic structure of {kappa}-deformed conformal algebra is discussed. (author). 23 refs.

Infinite-Dimensional Symmetry Algebras as a Help Toward Solutions of the Self-Dual Field Equations with One Killing Vector

Science.gov (United States)

Finley, Daniel; McIver, John K.

2002-12-01

The sDiff(2) Toda equation determines all self-dual, vacuum solutions of the Einstein field equations with one rotational Killing vector. Some history of the searches for non-trivial solutions is given, including those that begin with the limit as n → ∞ of the An Toda lattice equations. That approach is applied here to the known prolongation structure for the Toda lattice, hoping to use Bäcklund transformations to generate new solutions. Although this attempt has not yet succeeded, new faithful (tangent-vector) realizations of A∞ are described, and a direct approach via the continuum Lie algebras of Saveliev and Leznov is given.

The impact of fraction magnitude knowledge on algebra performance and learning.

Science.gov (United States)

Booth, Julie L; Newton, Kristie J; Twiss-Garrity, Laura K

2014-02-01

Knowledge of fractions is thought to be crucial for success with algebra, but empirical evidence supporting this conjecture is just beginning to emerge. In the current study, Algebra 1 students completed magnitude estimation tasks on three scales (0-1 [fractions], 0-1,000,000, and 0-62,571) just before beginning their unit on equation solving. Results indicated that fraction magnitude knowledge, and not whole number knowledge, was especially related to students' pretest knowledge of equation solving and encoding of equation features. Pretest fraction knowledge was also predictive of students' improvement in equation solving and equation encoding skills. Students' placement of unit fractions (e.g., those with a numerator of 1) was not especially useful for predicting algebra performance and learning in this population. Placement of non-unit fractions was more predictive, suggesting that proportional reasoning skills might be an important link between fraction knowledge and learning algebra. Copyright © 2013 Elsevier Inc. All rights reserved.

Higher spin fields and the Gelfand-Dickey algebra

International Nuclear Information System (INIS)

Bakas, I.

1989-01-01

We show that in 2-dimensional field theory, higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by Gelfand and Dickey to describe integrable nonlinear differential equations in Lax form. The spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined. This provides a conformal field theory approach to the representation theory of Gelfand-Dickey algebras. (orig.)

A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.

Science.gov (United States)

DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J

2016-12-01

Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions. Copyright © 2016 Elsevier Inc. All rights reserved.

Algebraic features of some generalizations of the Lotka-Volterra system

Science.gov (United States)

Bibik, Yu. V.; Sarancha, D. A.

2010-10-01

For generalizations of the Lotka-Volterra system, an integration method is proposed based on the nontrivial algebraic structure of these generalizations. The method makes use of an auxiliary first-order differential equation derived from the phase curve equation with the help of this algebraic structure. Based on this equation, a Hamiltonian approach can be developed and canonical variables (moreover, action-angle variables) can be constructed.

Topics in quaternion linear algebra

CERN Document Server

Rodman, Leiba

2014-01-01

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses...

Generalized Galilean algebras and Newtonian gravity

Science.gov (United States)

González, N.; Rubio, G.; Salgado, P.; Salgado, S.

2016-04-01

The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.

Isomorphism of Intransitive Linear Lie Equations

Directory of Open Access Journals (Sweden)

Jose Miguel Martins Veloso

2009-11-01

Full Text Available We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.

Mastering algebra retrains the visual system to perceive hierarchical structure in equations.

Science.gov (United States)

Marghetis, Tyler; Landy, David; Goldstone, Robert L

2016-01-01

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

Linear algebra

CERN Document Server

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...

Canonical formulation of the self-dual Yang-Mills system: Algebras and hierarchies

International Nuclear Information System (INIS)

Chau, L.; Yamanaka, I.

1992-01-01

We construct a canonical formulation of the self-dual Yang-Mills system formulated in the gauge-invariant group-valued J fields and derive their Hamiltonian and the quadratic algebras of the fundamental Dirac brackets. We also show that the quadratic algebras satisfy Jacobi identities and their structure matrices satisfy modified Yang-Baxter equations. From these quadratic algebras, we construct Kac-Moody-like and Virasoro-like algebras. We also discuss their related symmetries, involutive conserved quantities, and hierarchies of nonlinear and linear equations

Elementary matrix algebra

CERN Document Server

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

Contribution to the resolution of algebraic differential equations. Application to electronic circuits and nuclear reactors

International Nuclear Information System (INIS)

Monsef, Youssef.

1977-05-01

This note deals with the resolution of large algebraic differential systems involved in the physical sciences, with special reference to electronics and nuclear physics. The theoretical aspect of the algorithms established and developed for this purpose is discussed in detail. A decomposition algorithm based on the graph theory is developed in detail and the regressive analysis of the error involved in the decomposition is carried out. The specific application of these algorithms on the analyses of non-linear electronic circuits and to the integration of algebraic differential equations simulating the general operation of nuclear reactors coupled to heat exchangers is discussed in detail. To conclude, it is shown that the development of efficient digital resolution techniques dealing with the elements in order is sub-optimal for large systems and calls for the revision of conventional formulation methods. Thus for a high-order physical system, the larger, the number of auxiliary unknowns introduced, the easier the formulation and resolution, owing to the elimination of any form of complex matricial calculation such as those given by the state variables method [fr

Solving the Unknown with Algebra: Poster/Teaching Guide for Pre-Algebra Students. Expect the Unexpected with Math[R

Science.gov (United States)

Actuarial Foundation, 2013

2013-01-01

"Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…

Pre-Service Teachers' Perceptions and Beliefs of Technological Pedagogical Content Knowledge on Algebra

Science.gov (United States)

Lin, Cheng-Yao; Kuo, Yu-Chun; Ko, Yi-Yin

2015-01-01

The purpose of this study was to investigate elementary pre-service teachers' content knowledge in algebra (Linear Equation, Quadratic Equation, Functions, System Equations and Polynomials) as well as their technological pedagogical content knowledge (TPACK) in teaching algebra. Participants were 79 undergraduate pre-service teachers who were…

Handbook of algebra Vol. 1

CERN Document Server

1996-01-01

Handbook of Algebra defines algebra as consisting of many different ideas, concepts and results. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. Each chapter of the book combines some of the features of both a graduate-level textbook and a research-level survey. This book is divided into eight sections. Section 1A focuses on linear algebra and discusses such concepts as matrix functions and equations and random matrices. Section 1B cover linear d

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...

Lie groups and algebraic groups

Indian Academy of Sciences (India)

We give an exposition of certain topics in Lie groups and algebraic groups. This is not a complete ... of a polynomial equation is equivalent to the solva- bility of the equation ..... to a subgroup of the group of roots of unity in k (in particular, it is a ...

Certain algebraic structures and their applications to physics

International Nuclear Information System (INIS)

Salingaros, N.A.

1978-01-01

The aim of this thesis is to understand internal and external symmetries in Physics as arising from the same algebra by different processes, while the algebra itself arises out of the geometry of space-time. The result obtained is the Associative Generalized Algebra of Tensor Types. This algebra is constructed from the differential forms of spacetime, and is an algebra in the mathematical sense, describing all tensor types together. It is associative, and therefore very easy to use. A calculational formalism is developed that simplifies algebraic manipulations. The construction allows a classification of algebras that appear useful in Physics. The geometry excludes self-dual Minkowski bivector fields, but allows self-dual Euclidean bivector fields, a result, with important consequences in the theory of solutions of Yang-Mills gauge fields are demonstrated. There is only one bivector field, and every other bivector field, such as the electromagnetic field, is isomorphic to it. An exhaustive classification of the transformations of all fields in space-time yields the result that the only transformations of the electromagnetic field are the Lorentz transformations and the duality rotation. A fundamental asymmetry between the electric and magnetic fields are demonstrated. The derivative in the algebra is associative, and combines the Cartan exterior derivative with the coderivative of Hodge. The simplest derivative equations satisfied by a field in flat space-time are precisely the Maxwell equations

Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's equation over arbitrary domains

Directory of Open Access Journals (Sweden)

Tsugio Fukuchi

2014-06-01

Full Text Available The finite difference method (FDM based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.

Sugawara operators for classical Lie algebras

CERN Document Server

Molev, Alexander

2018-01-01

The celebrated Schur-Weyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody algebras. The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical \\mathcal{W}-algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the Harish-Chandra isomorphism connec...

College Algebra I.

Science.gov (United States)

Benjamin, Carl; And Others

Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra I. Topics covered include: sets; vocabulary; linear equations; inequalities; real numbers; operations; factoring; fractions; formulas; ratio, proportion, and…

Generalized NLS hierarchies from rational W algebras

International Nuclear Information System (INIS)

Toppan, F.

1993-11-01

Finite rational W algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. The problem of relating these algebras to integrable hierarchies of equations is studied by showing how to associate to a rational W algebra its corresponding hierarchy. Two examples are worked out, the sl(2)/U(1) coset, leading to the Non-Linear Schroedinger hierarchy, and the U(1) coset of the Polyakov-Bershadsky W algebra, leading to a 3-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies. (author). 19 refs

Elementary Algebra Connections to Precalculus

Science.gov (United States)

Lopez-Boada, Roberto; Daire, Sandra Arguelles

2013-01-01

This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…

ODE/IM correspondence and modified affine Toda field equations

Energy Technology Data Exchange (ETDEWEB)

Ito, Katsushi; Locke, Christopher

2014-08-15

We study the two-dimensional affine Toda field equations for affine Lie algebra g{sup ^} modified by a conformal transformation and the associated linear equations. In the conformal limit, the associated linear problem reduces to a (pseudo-)differential equation. For classical affine Lie algebra g{sup ^}, we obtain a (pseudo-)differential equation corresponding to the Bethe equations for the Langlands dual of the Lie algebra g, which were found by Dorey et al. in study of the ODE/IM correspondence.

Algebraic Systems and Pushdown Automata

Science.gov (United States)

Petre, Ion; Salomaa, Arto

We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.

Lie Algebras Associated with Group U(n)

International Nuclear Information System (INIS)

Zhang Yufeng; Dong Huanghe; Honwah Tam

2007-01-01

Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra A 1 are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.

Development of abstract mathematical reasoning: the case of algebra.

Science.gov (United States)

Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

2014-01-01

Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

A connection between the Einstein and Yang-Mills equations

International Nuclear Information System (INIS)

Mason, L.J.; Newman, E.T.

1989-01-01

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unified equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie-algebra as that of the volume preserving 3-dimensional diffeomorphisms). When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einstein vacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of an SO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations. (orig.)

On reduction and exact solutions of nonlinear many-dimensional Schroedinger equations

International Nuclear Information System (INIS)

Barannik, A.F.; Marchenko, V.A.; Fushchich, V.I.

1991-01-01

With the help of the canonical decomposition of an arbitrary subalgebra of the orthogonal algebra AO(n) the rank n and n-1 maximal subalgebras of the extended isochronous Galileo algebra, the rank n maximal subalgebras of the generalized extended classical Galileo algebra AG(a,n) the extended special Galileo algebra AG(2,n) and the extended whole Galileo algebra AG(3,n) are described. By using the rank n subalgebras, ansatze reducing the many dimensional Schroedinger equations to ordinary differential equations is found. With the help of the reduced equation solutions exact solutions of the Schroedinger equation are considered

Some quantum Lie algebras of type Dn positive

International Nuclear Information System (INIS)

Bautista, Cesar; Juarez-Ramirez, Maria Araceli

2003-01-01

A quantum Lie algebra is constructed within the positive part of the Drinfeld-Jimbo quantum group of type D n . Our quantum Lie algebra structure includes a generalized antisymmetry property and a generalized Jacobi identity closely related to the braid equation. A generalized universal enveloping algebra of our quantum Lie algebra of type D n positive is proved to be the Drinfeld-Jimbo quantum group of the same type. The existence of such a generalized Lie algebra is reduced to an integer programming problem. Moreover, when the integer programming problem is feasible we show, by means of the generalized Jacobi identity, that the Poincare-Birkhoff-Witt theorem (basis) is still true

Handbook of linear algebra

CERN Document Server

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

Riemann type algebraic structures and their differential-algebraic integrability analysis

Directory of Open Access Journals (Sweden)

Prykarpatsky A.K.

2010-06-01

Full Text Available The differential-algebraic approach to studying the Lax type integrability of generalized Riemann type equations is devised. The differentiations and the associated invariant differential ideals are analyzed in detail. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.

Commutator identities on associative algebras and integrability of nonlinear pde's

OpenAIRE

Pogrebkov, A. K.

2007-01-01

It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to associate to such commutator identity both nonlinear equation and its Lax pair. Thus problem of construction of new integrable pde's reduces to construction of commutator identities on associative algebras.

Towers of algebras in rational conformal field theories

International Nuclear Information System (INIS)

Gomez, C.; Sierra, G.

1991-01-01

This paper reports on Jones fundamental construction applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde's operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors

Division algebras and extended super KdVs

International Nuclear Information System (INIS)

Toppan, F.

2001-05-01

The division algebras R, C, H, O are used to construct and analyze the N = 1, 2, 4, 8 supersymmetric extensions of the KdV hamiltonian equation. In particular a global N = 8 super-KdV system is introduced and shown to admit a Poisson bracket structure given by the 'Non-Associate N = 8 Superconformal Algebra'. (author)

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

Science.gov (United States)

Liu, Jianzhou; Wang, Li; Zhang, Juan

2017-11-01

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.

Transitive Lie algebras of vector fields: an overview

NARCIS (Netherlands)

Draisma, J.

2011-01-01

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on

Applications of an anti-symmetry loop algebra and its expanding forms

International Nuclear Information System (INIS)

Zhang Yufeng; Yan Qingyou

2004-01-01

Constructing an anti-symmetry subalgebra A-tilde 2 of loop algebra A-tilde 2 gives the well-known Jaulent-Miodek (JM) hierarchy, the JM equation and its new Lax pair. Further, the Darboux transformation of the JM equation is deduced by anstaz method. By making use of a high-order loop algebra and Tu scheme, an expanding integrable model of the JM hierarchy is obtained. A direct expansion A-macron 2 * of loop algebra A-tilde 2 by considering the definition of Lie algebra is presented, which is used to establish two isospectral problems. It follows that corresponding two new integrable systems are engendered, which possess bi-Hamiltonian structures, respectively. Furthermore, a scalar transformation is applied to turn the loop algebra A-bar 2 * into its equivalent subalgebra A-tilde 1 of loop algebra A-tilde 1 . With the help of A-tilde 1 , another new high-order loop algebra G-bar is constructed, which is used to obtain an expanding integrable model of one of two integrable systems presented

Algebraic geometry and effective lagrangians

International Nuclear Information System (INIS)

Martinec, E.J.; Chicago Univ., IL

1989-01-01

N=2 supersymmetric Landau-Ginsburg fixed points describe nonlinear models whose target spaces are algebraic varieties in certain generalized projective spaces; the defining equation is precisely the zero set of the superpotential, considered as a condition in the projective space. The ADE classification of modular invariants arises as the classification of projective descriptions of P 1 ; in general, the hierarchy of fixed points is conjectured to be isomorphic to the classification of quasihomogeneous singularities. The condition of vanishing first Chern class is an integrality condition on the Virasoro central charge; the central charge is determined by the superpotential. The operator algebra is given by the algebra of Wick contractions of perturbations of the superpotential. (orig.)

Kinematic algebras, groups for elementary particles, and the geometry of momentum space

International Nuclear Information System (INIS)

Izmest'ev, A.A.

1986-01-01

It is shown that to each n-dimensional (n≥2) homogeneous isotropic Riemannian momentum (coordinate) space there corresponds a definite kinematic local algebra of operators N/sub a/, M/sub a//sub b/, P/sub a//sub ,/ ω(a,b = 1,2,...,n). In the three-dimensional case this gives the possibility of classifying particles in accordance with the algebras of the types of momentum space. The approach developed also makes it possible to obtain generalized equations describing particles of the different types. The operators under consideration satisfy not only the relevant algebra but also relations independent of the algebra that coincide in form with the Maxwell equations

Chemical Equation Balancing.

Science.gov (United States)

Blakley, G. R.

1982-01-01

Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)

On W algebras commuting with a set of screenings

Energy Technology Data Exchange (ETDEWEB)

Litvinov, Alexey [Landau Institute for Theoretical Physics,Akademika Semenova av., 1-A , Chernogolovka (Russian Federation); Kharkevich Institute for Information Transmission Problems,Bolshoy Karetny per. 19-1, Moscow, 127051 (Russian Federation); Spodyneiko, Lev [Landau Institute for Theoretical Physics,Akademika Semenova av., 1-A , Chernogolovka (Russian Federation); California Institute of Technology, Department of Physics,1200 East California Boulevard, Pasadena, 91125 (United States)

2016-11-22

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.

Clifford algebras, spinors, spin groups and covering groups

International Nuclear Information System (INIS)

Magneville, C.; Pansart, J.P.

1991-03-01

The Dirac equation uses matrices named Υ matrices which are representations of general algebraic structures associated with a metric space. These algebras are the Clifford algebras. In the first past, these algebras are studied. Then the notion of spinor is developed. It is shown that Majorana and Weyl spinors only exist for some particular metric space. In the second part, Clifford and spinor groups are studied. They may be interpreted as the extension of the notion of orthogonal group for Clifford algebras and their spaces for representation. The rotation of a spinor is computed. In the last part, the connexion between the spinor groups and the Universal Covering Groups is presented [fr

Titration Calculations with Computer Algebra Software

Science.gov (United States)

Lachance, Russ; Biaglow, Andrew

2012-01-01

This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…

Mathematical methods linear algebra normed spaces distributions integration

CERN Document Server

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

New solutions of the confluent Heun equation

Directory of Open Access Journals (Sweden)

Harold Exton

1998-05-01

Full Text Available New compact triple series solutions of the confluent Heun equation (CHE are obtained by the appropriate applications of the Laplace transform and its inverse to a suitably constructed system of soluble differential equations. The computer-algebra package MAPLE V is used to tackle an auxiliary system of non-linear algebraic equations. This study is partly motivated by the relationship between the CHE and certain Schrödininger equations.

An algorithm for analysis of the structure of finitely presented Lie algebras

Directory of Open Access Journals (Sweden)

Vladimir P. Gerdt

1997-12-01

Full Text Available We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

Basic math and pre-algebra for dummies

CERN Document Server

Zegarelli, Mark

2014-01-01

Tips for simplifying tricky basic math and pre-algebra operations Whether you're a student preparing to take algebra or a parent who wants or needs to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary math skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations. Explanations and practical examples that mirror today's teaching methodsRelevant cultural vernacular and referencesStandard For Dummies materials that

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

International Nuclear Information System (INIS)

Dong Huanhe; Yang Jiming; Wang Hui

2010-01-01

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM). (general)

Algebraic design theory

CERN Document Server

Launey, Warwick De

2011-01-01

Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book...

Algebraic aspects of the higher-spin problem

Energy Technology Data Exchange (ETDEWEB)

Vasiliev, M A [European Organization for Nuclear Research, Geneva (Switzerland)

1991-03-21

A general algebraic construction is established, which underlies the previously proposed consistent equations of interacting gauge fields of all spins in 3+1 dimensions. This construction makes a verification of the consistency (gauge invariance) of the higher-spin equations trivial and indicates how these equations can be generalized to higher dimensions and/or conformal-type higher-spin theories. (orig.).

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

Directory of Open Access Journals (Sweden)

Moysis Lazaros

2017-03-01

Full Text Available This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR representations A(σβ(k = 0, where σ denotes the shift forward operator and A(σ is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ. This work deals with the inverse problem of constructing a family of polynomial matrices A(σ such that the system A(σβ(k = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009 for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

Abstract numeric relations and the visual structure of algebra.

Science.gov (United States)

Landy, David; Brookes, David; Smout, Ryan

2014-09-01

Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic.

Numerical linear algebra theory and applications

CERN Document Server

Beilina, Larisa; Karchevskii, Mikhail

2017-01-01

This book combines a solid theoretical background in linear algebra with practical algorithms for numerical solution of linear algebra problems. Developed from a number of courses taught repeatedly by the authors, the material covers topics like matrix algebra, theory for linear systems of equations, spectral theory, vector and matrix norms combined with main direct and iterative numerical methods, least squares problems, and eigen problems. Numerical algorithms illustrated by computer programs written in MATLAB® are also provided as supplementary material on SpringerLink to give the reader a better understanding of professional numerical software for the solution of real-life problems. Perfect for a one- or two-semester course on numerical linear algebra, matrix computation, and large sparse matrices, this text will interest students at the advanced undergraduate or graduate level.

D=2 and D=4 realization of κ-conformal algebra

International Nuclear Information System (INIS)

Klimek, M.

1996-01-01

The generators of κ-conformal transformations leaving the κ-deformed d'Alembert equation invariant are described. The algebraic structure of the conformal extension of the off-shell spin zero realization of κ-Poincare algebra is discussed for D=4. The D=2 off-shell realization of κ-conformal algebra for an arbitrary spin and its commutation relations were studied. 14 refs

Algebraic relaxation of a time correlation function

International Nuclear Information System (INIS)

Srivastava, S.; Kumar, C.N.; Tankeshwar, K.

2004-06-01

A second order non-linear differential equation obtained from Mori's integro- differential equation is shown to transform to another form which provides algebraic decay to a time correlation function. Involved parameters in algebraic formula are related to exact properties of the corresponding correlation function. The model has been used to study a sol-gel system which is known, experimentally, to exhibit a power law decay to stress auto-correlation function. The expression obtained for the viscosity shows a logarithmic divergence at some critical value of the parameter. Some features of the model have also been tested using available information about Lennard-Jones fluids. (author)

Maxwell Equations and the Redundant Gauge Degree of Freedom

Science.gov (United States)

Wong, Chun Wa

2009-01-01

On transformation to the Fourier space (k,[omega]), the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the…

Solutions of the Yang-Baxter equation: Descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras

International Nuclear Information System (INIS)

Finch, P.E.; Dancer, K.A.; Isaac, P.S.; Links, J.

2011-01-01

The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the two-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are viewed as descendants of the six-vertex model case, are then obtained using tensor product graph methods which were originally formulated for quantum algebras. Connections with the Fateev-Zamolodchikov model are discussed.

Classical theory of algebraic numbers

CERN Document Server

Ribenboim, Paulo

2001-01-01

Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields Part One is devoted to residue classes and quadratic residues In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, iner...

A modified linear algebraic approach to electron scattering using cubic splines

International Nuclear Information System (INIS)

Kinney, R.A.

1986-01-01

A modified linear algebraic approach to the solution of the Schrodiner equation for low-energy electron scattering is presented. The method uses a piecewise cubic-spline approximation of the wavefunction. Results in the static-potential and the static-exchange approximations for e - +H s-wave scattering are compared with unmodified linear algebraic and variational linear algebraic methods. (author)

Semi-direct sums of Lie algebras and continuous integrable couplings

International Nuclear Information System (INIS)

Ma Wenxiu; Xu Xixiang; Zhang Yufeng

2006-01-01

A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems

Remarks on second-order quadratic systems in algebras

Directory of Open Access Journals (Sweden)

Art Sagle

2017-10-01

Full Text Available This paper is an addendum to our earlier paper [8], where a systematic study of quadratic systems of second order ordinary differential equations defined in commutative algebras was presented. Here we concentrate on special solutions and energy considerations of some quadratic systems defined in algebras which need not be commutative, however, we shall throughout assume the algebra to be associative. We here also give a positive answer to an open question, concerning periodic motions of such systems, posed in our earlier paper.

Vertex algebras and algebraic curves

CERN Document Server

Frenkel, Edward

2004-01-01

Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book co...

Some remarks on unilateral matrix equations

International Nuclear Information System (INIS)

Cerchiai, Bianca L.; Zumino, Bruno

2001-01-01

We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials

Algebraic partial Boolean algebras

International Nuclear Information System (INIS)

Smith, Derek

2003-01-01

Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E 8

Auxiliary equation method for solving nonlinear partial differential equations

International Nuclear Information System (INIS)

Sirendaoreji,; Jiong, Sun

2003-01-01

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation

Differential Equations Compatible with KZ Equations

International Nuclear Information System (INIS)

Felder, G.; Markov, Y.; Tarasov, V.; Varchenko, A.

2000-01-01

We define a system of 'dynamical' differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the 'dual' variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions

Five-dimensional Monopole Equation with Hedge-Hog Ansatz and Abel's Differential Equation

OpenAIRE

Kihara, Hironobu

2008-01-01

We review the generalized monopole in the five-dimensional Euclidean space. A numerical solution with the Hedge-Hog ansatz is studied. The Bogomol'nyi equation becomes a second order autonomous non-linear differential equation. The equation can be translated into the Abel's differential equation of the second kind and is an algebraic differential equation.

Some Evolution Hierarchies Derived from Self-dual Yang-Mills Equations

International Nuclear Information System (INIS)

Zhang Yufeng; Hon, Y.C.

2011-01-01

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra Ē of the Lie algebra E and the reduced self-dual Yang-Mills equations, we obtain an expanding integrable model of the Giachetti-Johnson (GJ) hierarchy whose Hamiltonian structure can also be derived by using the trace identity. This provides a much simpler construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra g N . As an application, we apply the loop algebra E-tilde of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup-Newell (KN) hierarchy which, consisting of two arbitrary parameters α and β, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra F of the Lie algebra F to obtain an expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R 3 based on the self-dual Yang-Mills equations, which include Poisson structures, irregular lines, and the reduced equations. (general)

The Virasoro algebra in integrable hierarchies and the method of matrix models

International Nuclear Information System (INIS)

Semikhatov, A.M.

1992-01-01

The action of the Virasoro algebra on hierarchies of nonlinear integrable equations, and also the structure and consequences of Virasoro constraints on these hierarchies, are studied. It is proposed that a broad class of hierarchies, restricted by Virasoro constraints, can be defined in terms of dressing operators hidden in the structure of integrable systems. The Virasoro-algebra representation constructed on the dressing operators displays a number of analogies with structures in conformal field theory. The formulation of the Virasoro constraints that stems from this representation makes it possible to translate into the language of integrable systems a number of concepts from the method of the 'matrix models' that describe nonperturbative quantum gravity, and, in particular, to realize a 'hierarchical' version of the double scaling limit. From the Virasoro constraints written in terms of the dressing operators generalized loop equations are derived, and this makes it possible to do calculations on a reconstruction of the field-theoretical description. The reduction of the Kadomtsev-Petviashvili (KP) hierarchy, subject to Virasoro constraints, to generalized Korteweg-deVries (KdV) hierarchies is implemented, and the corresponding representation of the Virasoro algebra on these hierarchies is found both in the language of scalar differential operators and in the matrix formalism of Drinfel'd and Sokolov. The string equation in the matrix formalism does not replicate the structure of the scalar string equation. The symmetry algebras of the KP and N-KdV hierarchies restricted by Virasoro constraints are calculated: A relationship is established with algebras from the family W ∞ (J) of infinite W-algebras

A new subalgebra of the Lie algebra A2 and two types of integrable Hamiltonian hierarchies, expanding integrable models

International Nuclear Information System (INIS)

Yan Qingyou; Zhang Yufeng; Wei Xiaopeng

2004-01-01

A new subalgebra G of the Lie algebra A 2 is first constructed. Then two loop algebra G-bar 1 , G-bar 2 are presented in terms of different definitions of gradations. Using G-bar 1 , G-bar 2 designs two isospectral problems, respectively. Again utilizing Tu-pattern obtains two types of various integrable Hamiltonian hierarchies of evolution equations. As reduction cases, the well-known Schroedinger equation and MKdV equation are obtained. At last, we turn the subalgebras G-bar 1 , G-bar 2 of the loop algebra A-bar 2 into equivalent subalgebras of the loop algebra A-bar 1 by making a suitable linear transformation so that the two types of 5-dimensional loop algebras are constructed. Two kinds of integrable couplings of the obtained hierarchies are showed. Specially, the integrable couplings of Schroedinger equation and MKdV equation are obtained, respectively

Galois theory of difference equations

CERN Document Server

Put, Marius

1997-01-01

This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.

Residues and duality for projective algebraic varieties

CERN Document Server

Kunz, Ernst; Dickenstein, Alicia

2008-01-01

This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of K�hler differential forms and their residues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial differential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text. The book is intended as an introduction to more advanced treatments and further applications of the subject, to which numerous bibliographical hints are given.

Spherical Hecke algebra in the Nekrasov-Shatashvili limit

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Bourgine, Jean-Emile [Asia Pacific Center for Theoretical Physics (APCTP),Pohang, Gyeongbuk 790-784 (Korea, Republic of)

2015-01-21

The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of N=2 gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter ϵ{sub 2}. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.

Learning Activity Package, Algebra 93-94, LAPs 12-22.

Science.gov (United States)

Evans, Diane

A set of 11 teacher-prepared Learning Activity Packages (LAPs) in beginning algebra, these units cover sets, properties of operations, operations over real numbers, open expressions, solution sets of equations and inequalities, equations and inequalities with two variables, solution sets of equations with two variables, exponents, factoring and…

Complex solutions for generalised fitzhughnagumo equation

International Nuclear Information System (INIS)

Neirameh, A.

2014-01-01

During present investigation, a direct algebraic method on complex solutions of nonlinear partial differential equation is developed and tested in the case of generalized Burgers-Huxley equation. The proposed scheme can be used in a wide class of nonlinear reaction-diffusion equations. These calculations demonstrate that the accuracy of the direct algebraic solutions is quite high even in the case of a small number of grid points. This method is a very reliable, simple, small computation costs, flexible, and convenient alternative method. (author)

The algebraic locus of Feynman integrals

OpenAIRE

Kol, Barak

2016-01-01

In the Symmetries of Feynman Integrals (SFI) approach, a diagram's parameter space is foliated by orbits of a Lie group associated with the diagram. SFI is related to the important methods of Integrations By Parts and of Differential Equations. It is shown that sometimes there exist a locus in parameter space where the set of SFI differential equations degenerates into an algebraic equation, thereby enabling a solution in terms of integrals associated with degenerations of the diagram. This i...

[Feature extraction for breast cancer data based on geometric algebra theory and feature selection using differential evolution].

Science.gov (United States)

Li, Jing; Hong, Wenxue

2014-12-01

The feature extraction and feature selection are the important issues in pattern recognition. Based on the geometric algebra representation of vector, a new feature extraction method using blade coefficient of geometric algebra was proposed in this study. At the same time, an improved differential evolution (DE) feature selection method was proposed to solve the elevated high dimension issue. The simple linear discriminant analysis was used as the classifier. The result of the 10-fold cross-validation (10 CV) classification of public breast cancer biomedical dataset was more than 96% and proved superior to that of the original features and traditional feature extraction method.

Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces

Science.gov (United States)

Ito, Kazufumi

1987-01-01

The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.

Linear Algebra and Smarandache Linear Algebra

OpenAIRE

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 vector spaces over finite p...

Methods of algebraic geometry in control theory

CERN Document Server

Falb, Peter

1999-01-01

"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is qui...

Extension of the constraint algebra for a closed string with a world surface of fixed topology

International Nuclear Information System (INIS)

Kashaev, R.M.; Osipov, A.A.

1989-01-01

The recently proposed choice of gauge in which the constraints and auxiliary conditions form a closed algebra is extended to the case of the Krichever--Novikov generalized graded algebras. It is shown that the central element of the extended algebra can be represented by an inexact form on a closed contour of the world surface of the string. A realization of the given algebra in terms of string variables is obtained. For this purpose, the classical dynamics of a closed bosonic string with a world surface of fixed genus is discussed. The dynamical variables are introduced in a covariant way and Hamilton equations are obtained in terms of them. These equations are equivalent to the Lagrange equations only in the case of a harmonic function of ''time.''

Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras

International Nuclear Information System (INIS)

Gebert, R.W.

1993-09-01

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ''physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathematics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. (orig.)

Chiral algebras in Landau-Ginzburg models

Science.gov (United States)

Dedushenko, Mykola

2018-03-01

Chiral algebras in the cohomology of the {\\overline{Q}}+ supercharge of two-dimensional N=(0,2) theories on flat spacetime are discussed. Using the supercurrent multiplet, we show that the answer is renormalization group invariant for theories with an R-symmetry. For N=(0,2) Landau-Ginzburg models, the chiral algebra is determined by the operator equations of motion, which preserve their classical form, and quantum renormalization of composite operators. We study these theories and then specialize to the N=(2,2) models and consider some examples.

Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra

Science.gov (United States)

Knight, D. G.

2006-01-01

This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,…

Zorn algebra in general relativity

International Nuclear Information System (INIS)

Oliveira, C.G.; Maia, M.D.

The covariant differential properties of the split Cayley subalgebra of local real quaternion tetrads is considered. Referred to this local quaternion tetrad several geometrical objects are given in terms of Zorn-Weyl matrices. Associated to a pair of real null vectors we define two-component spinor fields over the curved space and the associated Zorn-Weyl matrices which satisfy the Dirac equation written in terms of the Zorn algebra. The formalism is generalized by considering a field of complex tetrads defining a Hermitian second rank tensor. The real part of this tensor describes the gravitational potentials and the imaginary part the electromagnetic potentials in the Lorentz gauge. The motion of a charged spin zero test body is considered. The Zorn-Weyl algebra associated to this generalized formalism has elements belonging to the full octonion algebra [pt

Relativistic algebraic spinors and quantum motions in phase space

International Nuclear Information System (INIS)

Holland, P.R.

1986-01-01

Following suggestions of Schonberg and Bohm, we study the tensorial phase space representation of the Dirac and Feynman-Gell-Mann equations in terms of the complex Dirac algebra C 4 , a Jordan-Wigner algebra G 4 , and Wigner transformations. To do this we solve the problem of the conditions under which elements in C 4 generate minimal ideals, and extend this to G 4 . This yields the linear theory of Dirac spin spaces and tensor representations of Dirac spinors, and the spin-1/2 wave equations are represented through fermionic state vectors in a higher space as a set of interconnected tensor relations

New solutions of the generalized ellipsoidal wave equation

Directory of Open Access Journals (Sweden)

Harold Exton

1999-10-01

Full Text Available Certain aspects and a contribution to the theory of new forms of solutions of an algebraic form of the generalized ellipsoidal wave equation are deduced by considering the Laplace transform of a soluble system of linear differential equations. An ensuing system of non-linear algebraic equations is shown to be consistent and is numerically implemented by means of the computer algebra package MAPLE V. The main results are presented as series of hypergeometric type of there and four variables which readily lend themselves to numerical handling although this does not indicate all of the detailedanalytic properties of the solutions under consideration.

Classical algebraic chromodynamics

International Nuclear Information System (INIS)

Adler, S.L.

1978-01-01

I develop an extension of the usual equations of SU(n) chromodynamics which permits the consistent introduction of classical, noncommuting quark source charges. The extension involves adding a singlet gluon, giving a U(n) -based theory with outer product P/sup a/(u,v) = (1/2)(d/sup a/bc + if/sup a/bc)(u/sup b/v/sup c/ - v/sup b/u/sup c/) which obeys the Jacobi identity, inner product S (u,v) = (1/2)(u/sup a/v/sup a/ + v/sup a/u/sup a/), and with the n 2 gluon fields elevated to algebraic fields over the quark color charge C* algebra. I show that provided the color charge algebra satisfies the condition S (P (u,v),w) = S (u,P (v,w)) for all elements u,v,w of the algebra, all the standard derivations of Lagrangian chromodynamics continue to hold in the algebraic chromodynamics case. I analyze in detail the color charge algebra in the two-particle (qq, qq-bar, q-barq-bar) case and show that the above consistency condition is satisfied for the following unique (and, interestingly, asymmetric) choice of quark and antiquark charges: Q/sup a//sub q/ = xi/sup a/, Q/sup a//sub q/ = xi-bar/sup a/ + delta/sup a/0(n/2)/sup 3/2/1, with xi/sup a/xi/sup b/ = (1/2)(d/sup a/bc + if/sup a/bc) xi/sup c/, xi-bar/sup a/xi-bar/sup b/ = -(1/2)(d/sup a/bc - if/sup a/bc) xi-bar/sup c/. The algebraic structure of the two-particle U(n) force problem, when expressed on an appropriately diagonalized basis, leads for all n to a classical dynamics problem involving an ordinary SU(2) Yang-Mills field with uniquely specified classical source charges which are nonparallel in the color-singlet state. An explicit calculation shows that local algebraic U(n) gauge transformations lead only to a rigid global rotation of axes in the overlying classical SU(2) problem, which implies that the relative orientations of the classical source charges have physical significance

Some quantum Lie algebras of type D{sub n} positive

Energy Technology Data Exchange (ETDEWEB)

Bautista, Cesar [Facultad de Ciencias de la Computacion, Benemerita Universidad Autonoma de Puebla, Edif 135, 14 sur y Av San Claudio, Ciudad Universitaria, Puebla Pue. CP 72570 (Mexico); Juarez-Ramirez, Maria Araceli [Facultad de Ciencias Fisico-Matematicas, Benemerita Universidad Autonoma de Puebla, Edif 158 Av San Claudio y Rio Verde sn Ciudad Universitaria, Puebla Pue. CP 72570 (Mexico)

2003-03-07

A quantum Lie algebra is constructed within the positive part of the Drinfeld-Jimbo quantum group of type D{sub n}. Our quantum Lie algebra structure includes a generalized antisymmetry property and a generalized Jacobi identity closely related to the braid equation. A generalized universal enveloping algebra of our quantum Lie algebra of type D{sub n} positive is proved to be the Drinfeld-Jimbo quantum group of the same type. The existence of such a generalized Lie algebra is reduced to an integer programming problem. Moreover, when the integer programming problem is feasible we show, by means of the generalized Jacobi identity, that the Poincare-Birkhoff-Witt theorem (basis) is still true.

Applications of Lie algebras in the solution of dynamic problems

International Nuclear Information System (INIS)

Fellay, G.

1983-01-01

The purpose of this paper is to give some insight into the Lie-algebras and their applications. The first part introduces the elementary properties of such algebras, e.g. nilpotency, solvability, etc. The second part shows how to use the demonstrated theory for solving differential equations with time-dependent coefficients. (Auth.)

The MV formalism for IBL$_infty$- and BV$_infty$-algebras

Czech Academy of Sciences Publication Activity Database

Markl, Martin; Voronov, A.A.

2017-01-01

Roč. 107, č. 8 (2017), s. 1515-1543 ISSN 0377-9017 Institutional support: RVO:67985840 Keywords : MV-algebra * IBL$_infty$-algebra * master equation Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 1.671, year: 2016 https://link.springer.com/article/10.1007/s11005-017-0954-y

Balancing the Equation: Do Course Variations in Algebra 1 Provide Equal Student Outcomes?

Science.gov (United States)

Kenfield, Danielle M.

2013-01-01

Historically, algebra has served as a gatekeeper that divides students into academic programs with varying opportunities to learn and controls access to higher education and career opportunities. Successful completion of Algebra 1 demonstrates mathematical proficiency and allows access to a sequential and progressive path of advanced study that…

Ten-Year-Old Students Solving Linear Equations

Science.gov (United States)

Brizuela, Barbara; Schliemann, Analucia

2004-01-01

In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…

Classical Affine W-Algebras and the Associated Integrable Hamiltonian Hierarchies for Classical Lie Algebras

Science.gov (United States)

De Sole, Alberto; Kac, Victor G.; Valeri, Daniele

2018-06-01

We prove that any classical affine W-algebra W (g, f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.

Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

OpenAIRE

Kolev, Boris

2006-01-01

23 pages; International audience; This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures called affine or modified Lie-Poisson structures are involved in the integrability of certain Euler equations that arise as models of shallow water waves.

Riemann surfaces, Clifford algebras and infinite dimensional groups

International Nuclear Information System (INIS)

Carey, A.L.; Eastwood, M.G.; Hannabuss, K.C.

1990-01-01

We introduce of class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a 'gauge' group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces. (orig.)

Differential equations with applications in cancer diseases.

Science.gov (United States)

Ilea, M; Turnea, M; Rotariu, M

2013-01-01

Mathematical modeling is a process by which a real world problem is described by a mathematical formulation. The cancer modeling is a highly challenging problem at the frontier of applied mathematics. A variety of modeling strategies have been developed, each focusing on one or more aspects of cancer. The vast majority of mathematical models in cancer diseases biology are formulated in terms of differential equations. We propose an original mathematical model with small parameter for the interactions between these two cancer cell sub-populations and the mathematical model of a vascular tumor. We work on the assumption that, the quiescent cells' nutrient consumption is long. One the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. MATLAB simulations obtained for transition rate from the quiescent cells' nutrient consumption is long, we show a similar asymptotic behavior for two solutions of the perturbed problem. In this system, the small parameter is an asymptotic variable, different from the independent variable. The graphical output for a mathematical model of a vascular tumor shows the differences in the evolution of the tumor populations of proliferating, quiescent and necrotic cells. The nutrient concentration decreases sharply through the viable rim and tends to a constant level in the core due to the nearly complete necrosis in this region. Many mathematical models can be quantitatively characterized by ordinary differential equations or partial differential equations. The use of MATLAB in this article illustrates the important role of informatics in research in mathematical modeling. The study of avascular tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

Super Virasoro algebra and solvable supersymmetric quantum field theories

International Nuclear Information System (INIS)

Yamanaka, Itaru; Sasaki, Ryu.

1987-09-01

Interesting and deep relationships between super Virasoro algebras and super soliton systems (super KdV, super mKdV and super sine-Gordon equations) are investigated at both classical and quantum levels. An infinite set of conserved quantities responsible for solvability is characterized by super Virasoro algebras only. Several members of the infinite set of conserved quantities are derived explicitly. (author)

Bosonic Fradkin-Tseytlin equations unfolded

Energy Technology Data Exchange (ETDEWEB)

Shaynkman, O.V. [I.E.Tamm Theory Department, Lebedev Physical Institute,Leninski prospect 53, 119991, Moscow (Russian Federation)

2016-12-22

We test infinite-dimensional extension of algebra su(k,k) proposed by Fradkin and Linetsky as the candidate for conformal higher spin algebra. Adjoint and twisted-adjoint representations of su(k,k) on the space of this algebra are carefully explored. For k=2 corresponding unfolded system is analyzed and it is shown to encode Fradkin-Tseytlin equations for the set of all integer spins 1,2,… with infinite multiplicity.

Quantum W-algebras and elliptic algebras

International Nuclear Information System (INIS)

Feigin, B.; Kyoto Univ.; Frenkel, E.

1996-01-01

We define a quantum W-algebra associated to sl N as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary W-algebra of sl N , or the q-deformed classical W-algebra of sl N . We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U q (n). (orig.)

Benney's long wave equations

International Nuclear Information System (INIS)

Lebedev, D.R.

1979-01-01

Benney's equations of motion of incompressible nonviscous fluid with free surface in the approximation of long waves are analyzed. The connection between the Lie algebra of Hamilton plane vector fields and the Benney's momentum equations is shown

Soliton surfaces associated with sigma models: differential and algebraic aspects

International Nuclear Information System (INIS)

Goldstein, P P; Grundland, A M; Post, S

2012-01-01

In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the CP N-1 sigma model with finite action, defined in the Riemann sphere, are themselves solutions of the Euler–Lagrange equations for sigma models. On the other hand, we show that the Euler–Lagrange equations for surfaces immersed in the Lie algebra su(N), with conformal coordinates, that are extremals of the area functional, subject to a fixed polynomial identity, are exactly the Euler–Lagrange equations for sigma models. In addition to these differential constraints, the algebraic constraints, in the form of eigenvalues of the immersion functions, are systematically treated. The spectrum of the immersion functions, for different dimensions of the model, as well as its symmetry properties and its transformation under the action of the ladder operators are discussed. Another approach to the dynamics is given, i.e. description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model. (paper)

Virasoro algebra action on integrable hierarchies and Virasoro contraints in matrix models

International Nuclear Information System (INIS)

Semikhatov, A.M.

1991-01-01

The action of the Virasoro algebra on integrable hierarchies of non-linear equations and on related objects ('Schroedinger' differential operators) is investigated. The method consists in pushing forward the Virasoro action to the wave function of a hierarchy, and then reconstructing its action on the dressing and Lax operators. This formulation allows one to observe a number of suggestive similarities between the structures involved in the description of the Virasoro algebra on the hierarchies and the structure of conformal field theory on the world-sheet. This includes, in particular, an 'off-shell' hierarchy version of operator products and of the Cauchy kernel. In relation to matrix models, which have been observed to be effectively described by integrable hierarchies subjected to Virasoro constraints, I propose to define general Virasoro-constrained hierarchies also in terms of dressing operators, by certain equations which carry the information of the hierarchy and the Virasoro algebra simultaneously and which suggest an interpretation as operator versions of recursion/loop equations in topological theories. These same equations provide a relation with integrable hierarchies with quantized spectral parameter introduced recently. The formulation in terms of dressing operators allows a scaling (continuum limit) of discrete (i.e. lattice) hierarchies with the Virasoro constraints into 'continuous' Virasoro-constrained hierarchies. In particular, the KP hierarchy subjected to the Virasoro constraints is recovered as a scaling limit of the Virasoro-constrained Toda hierarchy. The dressing operator method also makes is straightforward to identify the full symmetry algebra of Virasoro-constrained hierarchies, which is related to the family of W ∞ (J) algebras introduced recently. (orig./HS)

Convergence of the standard RLS method and UDUT factorisation of covariance matrix for solving the algebraic Riccati equation of the DLQR via heuristic approximate dynamic programming

Science.gov (United States)

Moraes Rêgo, Patrícia Helena; Viana da Fonseca Neto, João; Ferreira, Ernesto M.

2015-08-01

The main focus of this article is to present a proposal to solve, via UDUT factorisation, the convergence and numerical stability problems that are related to the covariance matrix ill-conditioning of the recursive least squares (RLS) approach for online approximations of the algebraic Riccati equation (ARE) solution associated with the discrete linear quadratic regulator (DLQR) problem formulated in the actor-critic reinforcement learning and approximate dynamic programming context. The parameterisations of the Bellman equation, utility function and dynamic system as well as the algebra of Kronecker product assemble a framework for the solution of the DLQR problem. The condition number and the positivity parameter of the covariance matrix are associated with statistical metrics for evaluating the approximation performance of the ARE solution via RLS-based estimators. The performance of RLS approximators is also evaluated in terms of consistence and polarisation when associated with reinforcement learning methods. The used methodology contemplates realisations of online designs for DLQR controllers that is evaluated in a multivariable dynamic system model.

Nonholonomic deformation of generalized KdV-type equations

International Nuclear Information System (INIS)

Guha, Partha

2009-01-01

Karasu-Kalkani et al (2008 J. Math. Phys. 49 073516) recently derived a new sixth-order wave equation KdV6, which was shown by Kupershmidt (2008 Phys. Lett. 372A 2634) to have an infinite commuting hierarchy with a common infinite set of conserved densities. Incidentally, this equation was written for the first time by Calogero and is included in the book by Calogero and Degasperis (1982 Lecture Notes in Computer Science vol 144 (Amsterdam: North-Holland) p 516). In this paper, we give a geometric insight into the KdV6 equation. Using Kirillov's theory of coadjoint representation of the Virasoro algebra, we show how to obtain a large class of KdV6-type equations equivalent to the original equation. Using a semidirect product extension of the Virasoro algebra, we propose the nonholonomic deformation of the Ito equation. We also show that the Adler-Kostant-Symes scheme provides a geometrical method for constructing nonholonomic deformed integrable systems. Applying the Adler-Kostant-Symes scheme to loop algebra, we construct a new nonholonomic deformation of the coupled KdV equation.

Killing vectors in algebraically special space-times

International Nuclear Information System (INIS)

Torres del Castillo, G.F.

1984-01-01

The form of the isometric, homothetic, and conformal Killing vectors for algebraically special metrics which admit a shear-free congruence of null geodesics is obtained by considering their complexification, using the existence of a congruence of null strings. The Killing equations are partially integrated and the reasons which permit this reduction are exhibited. In the case where the congruence of null strings has a vanishing expansion, the Killing equations are reduced to a single master equation

Recoupling Lie algebra and universal ω-algebra

International Nuclear Information System (INIS)

Joyce, William P.

2004-01-01

We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an ω-algebra defined in this paper. ω-algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping ω-algebra of recoupling Lie algebras and prove a generalized Poincare-Birkhoff-Witt theorem. As an example we consider the algebras over an arbitrary recoupling of Z n graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure

Linear-algebraic approach to electron-molecule collisions: General formulation

International Nuclear Information System (INIS)

Collins, L.A.; Schneider, B.I.

1981-01-01

We present a linear-algebraic approach to electron-molecule collisions based on an integral equations form with either logarithmic or asymptotic boundary conditions. The introduction of exchange effects does not alter the basic form or order of the linear-algebraic equations for a local potential. In addition to the standard procedure of directly evaluating the exchange integrals by numerical quadrature, we also incorporate exchange effects through a separable-potential approximation. Efficient schemes are developed for reducing the number of points and channels that must be included. The method is applied at the static-exchange level to a number of molecular systems including H 2 , N 2 , LiH, and CO 2

Quantum ergodicity and a quantum measure algebra

International Nuclear Information System (INIS)

Stechel, E.B.

1985-01-01

A quantum ergodic theory for finite systems (such as isolated molecules) is developed by introducing the concept of a quantum measure algebra. The basic concept in classical ergodic theory is that of a measure space. A measure space is a set M, together with a specified sigma algebra of subsets in M and a measure defined on that algebra. A sigma algebra is closed under the formation of intersections and symmetric differences. A measure is a nonnegative and countably additive set function. For this to be further classified as a dynamical system, a measurable transformation is introduced. A measurable transformation is a mapping from a measure space into a measure space, such that the inverse image of every measurable set is measurable. In conservative dynamical systems, a measurable transformation is measure preserving, which is to say that the inverse image of every measurable set has the same measure as the original set. Once the measure space and the measurable transformation are defined, ergodic theory can be investigated on three levels: describable as analytic, geometric and algebraic. The analytic level studies linear operators induced by a transformation. The geometric level is concerned directly with transformations on a measure space and the algebraic treatments substitute a measure algebra for the measure space and basically equate sets that differ only by sets of measure zero. It is this latter approach that is most directly paralleled here. A measure algebra for a quantum dynamical system is defined within which stochastic concepts in quantum mechanics can be investigated. The quantum measure algebra differs from a normal measure algebra only in that multiplication is noncommutative and addition is nonassociative. Nonetheless, the quantum measure algebra preserves the essence of a normal measure algebra

Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

Science.gov (United States)

Morozov, Oleg I.

2018-06-01

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDE s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

An algebraic scheme associated with the non-commutative KP hierarchy and some of its extensions

International Nuclear Information System (INIS)

Dimakis, Aristophanes; Mueller-Hoissen, Folkert

2005-01-01

A well-known ansatz ('trace method') for soliton solutions turns the equations of the (non-commutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the non-commutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the non-commutative KP hierarchy. Relations with Rota-Baxter algebras are established

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.

Science.gov (United States)

Yu, Zhang; Zhang, Yufeng

2009-01-15

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.

Linear algebra

CERN Document Server

Said-Houari, Belkacem

2017-01-01

This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing t...

Generating higher-order Lie algebras by expanding Maurer-Cartan forms

International Nuclear Information System (INIS)

Caroca, R.; Merino, N.; Salgado, P.; Perez, A.

2009-01-01

By means of a generalization of the Maurer-Cartan expansion method, we construct a procedure to obtain expanded higher-order Lie algebras. The expanded higher-order Maurer-Cartan equations for the case G=V 0 +V 1 are found. A dual formulation for the S-expansion multialgebra procedure is also considered. The expanded higher-order Maurer-Cartan equations are recovered from S-expansion formalism by choosing a special semigroup. This dual method could be useful in finding a generalization to the case of a generalized free differential algebra, which may be relevant for physical applications in, e.g., higher-spin gauge theories.

The investigation of platonic solids symmetry operations with clifford algebra

International Nuclear Information System (INIS)

Kilic, A.

2005-01-01

The geometric algebra produces the new fields of view in the modern mathematical physics, definition of bodies and rearranging for equations of mathematics and physics. The new mathematical approaches play an important role in the progress of physics. After presenting Clifford algebra and quarantine's, the symmetry operations with Clifford algebra and quarantine's are defined. This symmetry operations are applied to a Platonic solids, which are called as tetrahedron, cube, octahedron, icosahedron and dodecahedron. Also, the vertices of Platonic solids presented in the Cartesian coordinates are calculated

Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra

NARCIS (Netherlands)

N.W. van den Hijligenberg; R. Martini

1995-01-01

textabstractWe discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra

Non-isospectral flows of noncommutative differential-difference KP equation

International Nuclear Information System (INIS)

Huang, Lin; Ilangovane, R.; Tamizhmani, K.M.; Zhang, Da-jun

2013-01-01

We present master symmetries of noncommutative differential-difference KP equation by considering Sato approach, where the field variables are defined over associative algebras. The Lie algebraic structures of generalized and master symmetries are given. They form a Virasoro Lie algebraic structure

Renormalization group flows and continual Lie algebras

International Nuclear Information System (INIS)

Bakas, Ioannis

2003-01-01

We study the renormalization group flows of two-dimensional metrics in sigma models using the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by (d/dt;1), with anti-symmetric Cartan kernel K(t,t') = δ'(t-t'); as such, it coincides with the Cartan matrix of the superalgebra sl(N vertical bar N+1) in the large-N limit. The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via Baecklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Z n to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra (d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown. (author)

The Yoneda algebra of a K2 algebra need not be another K2 algebra

OpenAIRE

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.

Algebraic Bethe ansatz for the Izergin-Korepin R matrix

International Nuclear Information System (INIS)

Tarasov, V.O.

1989-01-01

The authors propose a generalization of the algebraic Bethe ansatz for the Izergin-Korepin R matrix - the simplest unstudied odd-dimensional solution of the Yang-Baxter equation - and they discuss some related questions. The first section of the paper is an introduction. In the second they indicate a way of generalizing the algebraic Bethe ansatz to the case of the Izergin-Korepin R matrix. The simplest monodromy matrices (L operators) for this R matrix are described in the third section. The fourth section is devoted to the proof of the proposed generalization of the algebraic Bethe ansatz

Symmetric linear systems - An application of algebraic systems theory

Science.gov (United States)

Hazewinkel, M.; Martin, C.

1983-01-01

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Automatic Construction of Finite Algebras

Institute of Scientific and Technical Information of China (English)

张健

1995-01-01

This paper deals with model generation for equational theories,i.e.,automatically generating (finite)models of a given set of (logical) equations.Our method of finite model generation and a tool for automatic construction of finite algebras is described.Some examples are given to show the applications of our program.We argue that,the combination of model generators and theorem provers enables us to get a better understanding of logical theories.A brief comparison betwween our tool and other similar tools is also presented.

How to be Brilliant at Algebra

CERN Document Server

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.

Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking

Science.gov (United States)

Pearn, Catherine; Stephens, Max

2015-01-01

Many researchers argue that a deep understanding of fractions is important for a successful transition to algebra. Teaching, especially in the middle years, needs to focus specifically on those areas of fraction knowledge and operations that support subsequent solution processes for algebraic equations. This paper focuses on the results of Year 6…

Continual Lie algebras and noncommutative counterparts of exactly solvable models

Science.gov (United States)

Zuevsky, A.

2004-01-01

Noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of Saveliev-Vershik continual Lie algebras. Examples of noncommutative Liouville and sin/h-Gordon equations are given. The simplest soliton solution to the noncommutative sine-Gordon equation is found.

Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

Directory of Open Access Journals (Sweden)

K. S. Mahomed

2013-01-01

Full Text Available By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y′′′=0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.

Contribution to the minimization of time for the solution of algebraic differential equations system

International Nuclear Information System (INIS)

Michael, Samir.

1982-11-01

This note deals with the solution of large algebraic-differential systems involved in physical sciences specially in electronics and nuclear physics. The theoretical aspects for the stability of multistep methods is presented in detail. The stability condition is developed and we present our own conditions of stability. These conditions give rise to many new formulae that have very small truncation error. However for a real time simulation, it is necessary to obtain a very high computation speed. For this purpose, we have considered a multiprocessor machine and we have investigated the parallelization of the algorithm of generalized GEAR method. For a linear system, the method of GAUSS-JORDAN is used with some modifications. A new algorithm is presented for parallel matrix multiplication. This research work has been applied to the resolution of a system of equations corresponding to an experiment of gamma thermometry in a nuclear reactor (four thermometers in this case) [fr

The algebraic criteria for the stability of control systems

Science.gov (United States)

Cremer, H.; Effertz, F. H.

1986-01-01

This paper critically examines the standard algebraic criteria for the stability of linear control systems and their proofs, reveals important previously unnoticed connections, and presents new representations. Algebraic stability criteria have also acquired significance for stability studies of non-linear differential equation systems by the Krylov-Bogoljubov-Magnus Method, and allow realization conditions to be determined for classes of broken rational functions as frequency characteristics of electrical network.

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

International Nuclear Information System (INIS)

Yu Zhang; Zhang Yufeng

2009-01-01

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

Science.gov (United States)

Yu, Zhang; Zhang, Yufeng

2009-01-01

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092

The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

International Nuclear Information System (INIS)

Baseilhac, Pascal

2006-01-01

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite-dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered

Introduction to quantized LIE groups and algebras

International Nuclear Information System (INIS)

Tjin, T.

1992-01-01

In this paper, the authors give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups the authors study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then the authors explain in detail the concept of quantization for them. As an example the quantization of sl 2 is explicitly carried out. Next, the authors show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction, the authors explicitly construct the universal R matrix for the quantum sl 2 algebra. In the last section, the authors deduce all finite-dimensional irreducible representations for q a root of unity. The authors also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory

Cylindric-like algebras and algebraic logic

CERN Document Server

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.

Parametric Equations: Push 'Em Back, Push 'Em Back, Way Back!

Science.gov (United States)

Cieply, Joseph F.

1993-01-01

Stresses using the features of graphing calculators to teach parametric equations much earlier in the curriculum than is presently done. Examples using parametric equations to teach slopes and lines in beginning algebra, inverse functions in advanced algebra, the wrapping function, and simulations of physical phenomena are presented. (MAZ)

Properties of an associative algebra of tensor fields. Duality and Dirac identities

International Nuclear Information System (INIS)

Salingaros, N.; Dresden, M.

1979-01-01

An algebra of forms in Minkowski space has been constructed. A multiplication between forms is defined as an extension of the quaternionic multiplications. The algebra obtained is associative with respect to this multiplication of order 16. Duality is expressed as (new) multiplication by a basis element. Vector identities in the algebra lead to a number of new trace identities. A new derivative operator expresses the four Maxwell equations in an especially transparent form

Integrable N dimensional systems on the Hopf algebra and q deformations

International Nuclear Information System (INIS)

Lisitsyn, Ya.V.; Shapovalov, A.V.

2000-01-01

The class of integrable classic and quantum systems on the Hopf algebra, describing the n of interacting particles, is plotted. The general structure of the integrable Hamiltonian system for the Hopf algebra A(g) of the Lee simple algebra g is obtained, wherefrom it follows, that motion integrals depend on the linear combinations k of the phase space coordinates. The q-deformation standard procedure is carried out and the corresponding integrable system is obtained. The general scheme is illustrated by the examples of the sl(2), sl(3) and o(3, 1) algebras. The exact solution is achieved for the N-dimensional Hamiltonian system quantum analog on the Hopf algebra A (sl(2)) through the method of noncommutative integration of linear differential equations [ru

GPU TECHNOLOGIES EMBODIED IN PARALLEL SOLVERS OF LINEAR ALGEBRAIC EQUATION SYSTEMS

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Sidorov Alexander Vladimirovich

2012-10-01

Full Text Available The author reviews existing shareware solvers that are operated by graphical computer devices. The purpose of this review is to explore the opportunities and limitations of the above parallel solvers applicable for resolution of linear algebraic problems that arise at Research and Educational Centre of Computer Modeling at MSUCE, and Research and Engineering Centre STADYO. The author has explored new applications of the GPU in the PETSc suite and compared them with the results generated absent of the GPU. The research is performed within the CUSP library developed to resolve the problems of linear algebra through the application of GPU. The author has also reviewed the new MAGMA project which is analogous to LAPACK for the GPU.

Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra

NARCIS (Netherlands)

van den Hijligenberg, N.W.; van den Hijligenberg, N.W.; Martini, Ruud

1995-01-01

We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of

Generalization of the Knizhnik-Zamolodchikov-equations

International Nuclear Information System (INIS)

Alekseev, A.Yu.; Recknagel, A.; Schomerus, V.

1996-09-01

In this letter we introduce a generalization of the Knizhnik-Zamolodchikov equations from affine Lie algebras to a wide class of conformal field theories (not necessarily rational). The new equations describe correlation functions of primary fields and of a finite number of their descendents. Our proposal is based on Nahm's concept of small spaces which provide adequate substitutes for the lowest energy subspaces in modules of affine Lie algebras. We explain how to construct the first order differential equations and investigate properties of the associated connections, thereby preparing the grounds for an analysis of quantum symmetries. The general considerations are illustrated in examples of Virasoro minimal models. (orig.)

W-algebra for solving problems with fuzzy parameters

Science.gov (United States)

Shevlyakov, A. O.; Matveev, M. G.

2018-03-01

A method of solving the problems with fuzzy parameters by means of a special algebraic structure is proposed. The structure defines its operations through operations on real numbers, which simplifies its use. It avoids deficiencies limiting applicability of the other known structures. Examples for solution of a quadratic equation, a system of linear equations and a network planning problem are given.

Cluster algebras in mathematical physics

International Nuclear Information System (INIS)

Francesco, Philippe Di; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito

2014-01-01

This special issue of Journal of Physics A: Mathematical and Theoretical contains reviews and original research articles on cluster algebras and their applications to mathematical physics. Cluster algebras were introduced by S Fomin and A Zelevinsky around 2000 as a tool for studying total positivity and dual canonical bases in Lie theory. Since then the theory has found diverse applications in mathematics and mathematical physics. Cluster algebras are axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) subdivided into overlapping subsets (clusters) of the same cardinality subject to certain polynomial relations. A cluster algebra of rank n can be viewed as a subring of the field of rational functions in n variables. Rather than being presented, at the outset, by a complete set of generators and relations, it is constructed from the initial seed via an iterative procedure called mutation producing new seeds successively to generate the whole algebra. A seed consists of an n-tuple of rational functions called cluster variables and an exchange matrix controlling the mutation. Relations of cluster algebra type can be observed in many areas of mathematics (Plücker and Ptolemy relations, Stokes curves and wall-crossing phenomena, Feynman integrals, Somos sequences and Hirota equations to name just a few examples). The cluster variables enjoy a remarkable combinatorial pattern; in particular, they exhibit the Laurent phenomenon: they are expressed as Laurent polynomials rather than more general rational functions in terms of the cluster variables in any seed. These characteristic features are often referred to as the cluster algebra structure. In the last decade, it became apparent that cluster structures are ubiquitous in mathematical physics. Examples include supersymmetric gauge theories, Poisson geometry, integrable systems, statistical mechanics, fusion products in infinite dimensional algebras, dilogarithm

Quantum deformations of conformal algebras with mass-like deformation parameters

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Frydryszak, Andrzej; Lukierski, Jerzy; Mozrzymas, Marek; Minnaert, Pierre

1998-01-01

We recall the mathematical apparatus necessary for the quantum deformation of Lie algebras, namely the notions of coboundary Lie algebras, classical r-matrices, classical Yang-Baxter equations (CYBE), Froebenius algebras and parabolic subalgebras. Then we construct the quantum deformation of D=1, D=2 and D=3 conformal algebras, showing that this quantization introduce fundamental mass parameters. Finally we consider with more details the quantization of D=4 conformal algebra. We build three classes of sl(4,C) classical r-matrices, satisfying CYBE and depending respectively on 8, 10 and 12 generators of parabolic subalgebras. We show that only the 8-dimensional r-matrices allow to impose the D=4 conformal o(4,2)≅su(2,2) reality conditions. Weyl reflections and Dynkin diagram automorphisms for o(4,2) define the class of admissible bases for given classical r-matrices

Partially-massless higher-spin algebras and their finite-dimensional truncations

International Nuclear Information System (INIS)

Joung, Euihun; Mkrtchyan, Karapet

2016-01-01

The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS d+1 are studied. The algebras involving PM generators up to depth 2 (ℓ−1) are defined as the maximal symmetries of free conformal scalar field with 2 ℓ order wave equation in d dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of (A)dS d+1 isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of ℓ−d/2 , which coincides with the annihilator of the one-row ℓ-box Young diagram representation of so d+2 . Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.

Solving Langevin equation with the stochastic algebraically correlated noise

International Nuclear Information System (INIS)

Ploszajczak, M.; Srokowski, T.

1996-01-01

Long time tail in the velocity and force autocorrelation function has been found recently in the molecular dynamics simulations of the peripheral collisions of ions. Simulation of those slowly decaying correlations in the stochastic transport theory requires the development of new methods of generating stochastic force of arbitrarily long correlation times. The Markovian process and the multidimensional Kangaroo process which permit describing various algebraic correlated stochastic processes are proposed. (author)

Algebraic reasoning and bat-and-ball problem variants: Solving isomorphic algebra first facilitates problem solving later.

Science.gov (United States)

Hoover, Jerome D; Healy, Alice F

2017-12-01

The classic bat-and-ball problem is used widely to measure biased and correct reasoning in decision-making. University students overwhelmingly tend to provide the biased answer to this problem. To what extent might reasoners be led to modify their judgement, and, more specifically, is it possible to facilitate problem solution by prompting participants to consider the problem from an algebraic perspective? One hundred ninety-seven participants were recruited to investigate the effect of algebraic cueing as a debiasing strategy on variants of the bat-and-ball problem. Participants who were cued to consider the problem algebraically were significantly more likely to answer correctly relative to control participants. Most of this cueing effect was confined to a condition that required participants to solve isomorphic algebra equations corresponding to the structure of bat-and-ball question types. On a subsequent critical question with differing item and dollar amounts presented without a cue, participants were able to generalize the learned information to significantly reduce overall bias. Math anxiety was also found to be significantly related to bat-and-ball problem accuracy. These results suggest that, under specific conditions, algebraic reasoning is an effective debiasing strategy on bat-and-ball problem variants, and provide the first documented evidence for the influence of math anxiety on Cognitive Reflection Test performance.

Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

Science.gov (United States)

Jiang, Yunfeng; Zhang, Yang

2018-03-01

In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 super-Yang-Mills theory.

Algebraic theory of locally nilpotent derivations

CERN Document Server

Freudenburg, Gene

2017-01-01

This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves. More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. A lot of new material is included in this expanded second edition, such as canonical factoriza...

Bootstrapping non-commutative gauge theories from L∞ algebras

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Blumenhagen, Ralph; Brunner, Ilka; Kupriyanov, Vladislav; Lüst, Dieter

2018-05-01

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS5 sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L∞ algebra. The appearance of a non-trivial A∞ algebra is discussed, as well.

Introduction to relation algebras relation algebras

CERN Document Server

Givant, Steven

2017-01-01

The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field. The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly ...

From Rota-Baxter algebras to pre-Lie algebras

International Nuclear Information System (INIS)

An Huihui; Ba, Chengming

2008-01-01

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension ≤3 and their corresponding pre-Lie algebras

A Dirac algebraic approach to supersymmetry

International Nuclear Information System (INIS)

Guersey, F.

1984-01-01

The power of the Dirac algebra is illustrated through the Kaehler correspondence between a pair of Dirac spinors and a 16-component bosonic field. The SO(5,1) group acts on both the fermion and boson fields, leading to a supersymmetric equation of the Dirac type involving all these fields. (author)

Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

International Nuclear Information System (INIS)

Marquette, Ian

2013-01-01

We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently

Stability of a Generalized Euler-Lagrange Type Additive Mapping and Homomorphisms in C∗-Algebras

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Abbas Najati

2009-01-01

Full Text Available Let X,Y be Banach modules over a C∗-algebra and let r1,…,rn∈ℝ be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C∗-algebra: ∑j=1nf(−rjxj+∑1≤i≤n,i≠jrixi+2∑i=1nrif(xi=nf(∑i=1nrixi. We show that if ∑i=1nri≠0, ri,rj≠0 for some 1≤iequation mentioned above then the mapping f:X→Y is Cauchy additive. As an application, we investigate homomorphisms in unital C∗-algebras.

Yoneda algebras of almost Koszul algebras

Indian Academy of Sciences (India)

Abstract. Let k be an algebraically closed field, A a finite dimensional connected. (p,q)-Koszul self-injective algebra with p, q ≥ 2. In this paper, we prove that the. Yoneda algebra of A is isomorphic to a twisted polynomial algebra A![t; β] in one inde- terminate t of degree q +1 in which A! is the quadratic dual of A, β is an ...

General algebraic theory of identical particle scattering

International Nuclear Information System (INIS)

Bencze, G.; Redish, E.F.

1978-01-01

We consider the nonrelativistic N-body scattering problem for a system of particles in which some subsets of the particles are identical. We demonstrate how the particle identity can be included in a general class of linear integral equations for scattering operators or components of scattering operators. The Yakubovskii, Yakubovskii--Narodestkii, Rosenberg, and Bencze--Redish--Sloan equations are included in this class. Algebraic methods are used which rely on the properties of the symmetry group of the system. Operators depending only on physically distinguishable labels are introduced and linear integral equations for them are derived. This procedure maximally reduces the number of coupled equations while retaining the connectivity properties of the original equations

Quantum cluster algebras and quantum nilpotent algebras

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Goodearl, Kenneth R.; Yakimov, Milen T.

2014-01-01

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197

An introduction to algebraic geometry and algebraic groups

CERN Document Server

Geck, Meinolf

2003-01-01

An accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups of Lie type.The text covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups

Extreme compression behaviour of equations of state

International Nuclear Information System (INIS)

Shanker, J.; Dulari, P.; Singh, P.K.

2009-01-01

The extreme compression (P→∞) behaviour of various equations of state with K' ∞ >0 yields (P/K) ∞ =1/K' ∞ , an algebraic identity found by Stacey. Here P is the pressure, K the bulk modulus, K ' =dK/dP, and K' ∞ , the value of K ' at P→∞. We use this result to demonstrate further that there exists an algebraic identity also between the higher pressure derivatives of bulk modulus which is satisfied at extreme compression by different types of equations of state such as the Birch-Murnaghan equation, Poirier-Tarantola logarithmic equation, generalized Rydberg equation, Keane's equation and the Stacey reciprocal K-primed equation. The identity has been used to find a relationship between λ ∞ , the third-order Grueneisen parameter at P→∞, and pressure derivatives of bulk modulus with the help of the free-volume formulation without assuming any specific form of equation of state.

Infinite sets of conservation laws for linear and non-linear field equations

International Nuclear Information System (INIS)

Niederle, J.

1984-01-01

The work was motivated by a desire to understand group theoretically the existence of an infinite set of conservation laws for non-interacting fields and to carry over these conservation laws to the case of interacting fields. The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of its space-time symmetry group was established. It is shown that in the case of the Korteweg-de Vries (KdV) equation to each symmetry of the corresponding linear equation delta sub(o)uxxx=u sub() determined by an element of the enveloping algebra of the space translation algebra, there corresponds a symmetry of the full KdV equation

Commuting quantum traces: the case of reflection algebras

Energy Technology Data Exchange (ETDEWEB)

Avan, Jean [Laboratory of Theoretical Physics and Modelization, University of Cergy, 5 mail Gay-Lussac, Neuville-sur-Oise, F-95031, Cergy-Pontoise Cedex (France); Doikou, Anastasia [Theoretical Physics Laboratory of Annecy-Le-Vieux, LAPTH, BP 110, Annecy-Le-Vieux, F-74941 (France)

2004-02-06

We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures. Products of their respective solutions yield commuting quantum traces.

Computational algebraic geometry of epidemic models

Science.gov (United States)

Rodríguez Vega, Martín.

2014-06-01

Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.

Algebraic Side-Channel Attack on Twofish

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Chujiao Ma

2017-05-01

Full Text Available While algebraic side-channel attack (ASCA has been successful in breaking simple cryptographic algorithms, it has never been done on larger or more complex algorithms such as Twofish. Compared to other algorithms that ASCA has been used on, Twofish is more difficult to attack due to the key-dependent S-boxes as well as the complex key scheduling. In this paper, we propose the first algebraic side-channel attack on Twofish, and examine the importance of side-channel information in getting past the key-dependent S-boxes and the complex key scheduling. The cryptographic algorithm and side-channel information are both expressed as boolean equations and a SAT solver is used to recover the key. While algebraic attack by itself is not sufficient to break the algorithm, with the help of side-channel information such as Hamming weights, we are able to correctly solve for 96 bits of the 128 bits key in under 2 hours with known plaintext/ciphertext.

Algebraic computing program for studying the gauge theory

International Nuclear Information System (INIS)

Zet, G.

2005-01-01

An algebraic computing program running on Maple V platform is presented. The program is devoted to the study of the gauge theory with an internal Lie group as local symmetry. The physical quantities (gauge potentials, strength tensors, dual tensors etc.) are introduced either as equations in terms of previous defined quantities (tensors), or by manual entry of the component values. The components of the strength tensor and of its dual are obtained with respect to a given metric of the space-time used for describing the gauge theory. We choose a Minkowski space-time endowed with spherical symmetry and give some example of algebraic computing that are adequate for studying electroweak or gravitational interactions. The field equations are also obtained and their solutions are determined using the DEtools facilities of the Maple V computing program. (author)

Contact Geometry of Hyperbolic Equations of Generic Type

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Dennis The

2008-08-01

Full Text Available We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampère (class 6-6, Goursat (class 6-7 and generic (class 7-7 hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric generic hyperbolic equations are derived and explicit symmetry algebras are presented. Moreover, these maximally symmetric equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional generic hyperbolic structures is also given.

Closed string field theory: Quantum action and the Batalin-Vilkovsky master equation

International Nuclear Information System (INIS)

Zwiebach, B.

1993-01-01

The complete quantum theory of covariant closed strings is constructed in detail. The nonpolynomial action is defined by elementary vertices satisfying recursion relations that give rise to Jacobi-like identities for an infinite chain of string field products. The genus zero string field algebra is the homotopy Lie algebra L ∞ encoding the gauge symmetry of the classical theory. The higher genus algebraic structure implies the Batalin-Vilkovisky (BV) master equation and thus consistent BRST quantization of the quantum action. From the L ∞ algebra, and the BV equation on the off-shell state space we derive the L ∞ algebra, and the BV equation on physical states that were recently constructed in d=2 string theory. The string diagrams are surfaces with minimal area metrics, foliated by closed geodesics of length 2π. These metrics generalize quadratic differentials in that foliation bands can cross. The string vertices are succinctly characterized; they include the surfaces whose foliation bands are all of height smaller than 2π. (orig.)

An su(1, 1) algebraic approach for the relativistic Kepler-Coulomb problem

International Nuclear Information System (INIS)

Salazar-Ramirez, M; Granados, V D; MartInez, D; Mota, R D

2010-01-01

We apply the Schroedinger factorization method to the radial second-order equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the su(1, 1) Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.

Generation of exact solutions to the Einstein field equations for homogeneous space--time

International Nuclear Information System (INIS)

Hiromoto, R.E.

1978-01-01

A formalism is presented capable of finding all homogeneous solutions of the Einstein field equations with an arbitrary energy-stress tensor. Briefly the method involves the classification of the four-dimensional Lie algebra over the reals into nine different broad classes, using only the Lorentz group. Normally the classification of Lie algebras means that one finds all essentially different solutions of the Jacobi identities, i.e., there exists no nonsingular linear transformation which transforms two sets of structure constants into the other. This approach is to utilize the geometrical considerations of the homogeneous spacetime and field equations to be solved. Since the set of orthonormal basis vectors is not only endowed with a Minkowskian metric, but also constitutes the vector space of our four-dimensional Lie algebras, the Lie algebras are classified against the Lorentz group restricts the linear group of transformations, denoting the essentially different Lie algebras, into nine different broad classes. The classification of the four-dimensional Lie algebras represents the unification of various methods previously introduced by others. Where their methods found only specific solutions to the Einstein field equations, systematic application of the nine different classes of Lie algebras guarantees the extraction of all solutions. Therefore, the methods of others were extended, and their foundations of formalism which goes beyond the present literature of exact homogeneous solutions to the Einstein field equations is built upon

Differential Hopf algebra structures on the Universal Enveloping Algebra of a Lie Algebra

NARCIS (Netherlands)

van den Hijligenberg, N.W.; van den Hijligenberg, N.; Martini, Ruud

1995-01-01

We discuss a method to construct a De Rham complex (differential algebra) of Poincaré–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 algebrastructure of U(g).

An algebraic method to develop well-posed PML models Absorbing layers, perfectly matched layers, linearized Euler equations

International Nuclear Information System (INIS)

Rahmouni, Adib N.

2004-01-01

In 1994, Berenger [Journal of Computational Physics 114 (1994) 185] proposed a new layer method: perfectly matched layer, PML, for electromagnetism. This new method is based on the truncation of the computational domain by a layer which absorbs waves regardless of their frequency and angle of incidence. Unfortunately, the technique proposed by Berenger (loc. cit.) leads to a system which has lost the most important properties of the original one: strong hyperbolicity and symmetry. We present in this paper an algebraic technique leading to well-known PML model [IEEE Transactions on Antennas and Propagation 44 (1996) 1630] for the linearized Euler equations, strongly well-posed, preserving the advantages of the initial method, and retaining symmetry. The technique proposed in this paper can be extended to various hyperbolic problems

The relation between quantum W algebras and Lie algebras

International Nuclear Information System (INIS)

Boer, J. de; Tjin, T.

1994-01-01

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary sl 2 embeddings we show that a large set W of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set W contains many known W algebras such as W N and W 3 (2) . Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any W algebra in W can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore any realization of this semisimple affine Lie algebra leads to a realization of the W algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in W. Some examples are explicitly worked out. (orig.)

Gauge fields in algebraically special space-times

International Nuclear Information System (INIS)

Torres del Castillo, G.F.

1985-01-01

It is shown that in an algebraically special space-time which admits a congruence of null strings, a source-free gauge field aligned with the congruence is determined by a matrix potential which has to satisfy a second-order differential equation with quadratic nonlinearities. The Einstein--Yang--Mills equations are then reduced to a scalar and two matrix equations. In the case of self-dual gauge fields in a self-dual space-time, the existence of an infinite set of conservation laws, of an associated linear system, and of infinitesimal Baecklund transformations is demonstrated. All the results apply for an arbitrary gauge group

2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras

International Nuclear Information System (INIS)

Ayupov, Shavkat; Kudaybergenov, Karimbergen

2016-01-01

The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2 n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation. (paper)

Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems.

Science.gov (United States)

Geary, David C; Hoard, Mary K; Nugent, Lara; Rouder, Jeffrey N

2015-12-01

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial-numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities. Copyright © 2015 Elsevier Inc. All rights reserved.

Secondary School Pre-Service Mathematics Teachers' Content Knowledge of Algebraic Word Problem in Nigeria

Science.gov (United States)

Usman, Ahmed Ibrahim

2015-01-01

Knowledge and understanding of mathematical operations serves as a pre-reequisite for the successful translation of algebraic word problems. This study explored pre-service teachers' ability to recognize mathematical operations as well as use of those capabilities in constructing algebraic expressions, equations, and their solutions. The outcome…

Regular Riemann-Hilbert transforms, Baecklund transformations and hidden symmetry algebra for some linearization systems

International Nuclear Information System (INIS)

Chau Ling-Lie; Ge Mo-Lin; Teh, Rosy.

1984-09-01

The Baecklund Transformations and the hidden symmetry algebra for Self-Dual Yang-Mills Equations, Landau-Lifshitz equations and the Extended Super Yang-Mills fields (N>2) are discussed on the base of the Regular Riemann-Hilbert Transform and the linearization equations. (author)

Algebraic structure of open string interactions

International Nuclear Information System (INIS)

Ramond, P.; Rodgers, V.G.J.

1986-05-01

Starting from the gauge invariant equations of motion for the free open string we show how to generate interactions by analogy with Yang-Mills. We postulate Non-Abelian transformation laws acting on the fields of the gauge invariant free open string theory. By demanding algebraic closure we then derive a set of consistency requirements and show that they lead to the construction of the minimal interacting equations which contain no cubic terms away from the physical gauge. We present explicit solutions to lowest interacting order for both vertices and structure functions. 14 refs

Algebraic structure of open-string interactions

International Nuclear Information System (INIS)

Ramond, P.; Rodgers, V.G.J.

1986-01-01

Starting from the gauge-invariant equations of motion for the free open string we show how to generate interactions by analogy with the Yang-Mills system. We postulate non-Abelian transformation laws acting on the fields of the gauge-invariant free open-string theory. By demanding algebraic closure we then derive a set of consistency requirements and show that they lead to the construction of the minimal interacting equations which contain no cubic terms away from the physical gauge. We present explicit solutions to lowest interacting order for both vertices and structure functions

Symmetries of stochastic differential equations: A geometric approach

Energy Technology Data Exchange (ETDEWEB)

De Vecchi, Francesco C., E-mail: francesco.devecchi@unimi.it; Ugolini, Stefania, E-mail: stefania.ugolini@unimi.it [Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, Milano (Italy); Morando, Paola, E-mail: paola.morando@unimi.it [DISAA, Università degli Studi di Milano, via Celoria 2, Milano (Italy)

2016-06-15

A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.

Quantum cluster algebra structures on quantum nilpotent algebras

CERN Document Server

Goodearl, K R

2017-01-01

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts.

Tensor algebra over Hilbert space: Field theory in classical phase space

International Nuclear Information System (INIS)

Matos Neto, A.; Vianna, J.D.M.

1984-01-01

It is shown using tensor algebras, namely Symmetric and Grassmann algebras over Hilbert Space that it is possible to introduce field operators, associated to the Liouville equation of classical statistical mechanics, which are characterized by commutation (for Symmetric) and anticommutation (for Grassmann) rules. The procedure here presented shows by construction that many-particle classical systems admit an algebraic structure similar to that of quantum field theory. It is considered explicitly the case of n-particle systems interacting with an external potential. A new derivation of Schoenberg's result about the equivalence between his field theory in classical phase space and the usual classical statistical mechanics is obtained as a consequence of the algebraic structure of the theory as introduced by our method. (Author) [pt

Quadratic algebras

CERN Document Server

Polishchuk, Alexander

2005-01-01

Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various areas of mathematics. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, K-theory, number theory, and noncommutative linear algebra. The book offers a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincar�-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.

Regularity of C*-algebras and central sequence algebras

DEFF Research Database (Denmark)

Christensen, Martin S.

The main topic of this thesis is regularity properties of C*-algebras and how these regularity properties are re ected in their associated central sequence algebras. The thesis consists of an introduction followed by four papers [A], [B], [C], [D]. In [A], we show that for the class of simple...... Villadsen algebra of either the rst type with seed space a nite dimensional CW complex, or the second type, tensorial absorption of the Jiang-Su algebra is characterized by the absence of characters on the central sequence algebra. Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen...... algebra of the second type with innite stable rank fails the corona factorization property. In [B], we consider the class of separable C*-algebras which do not admit characters on their central sequence algebra, and show that it has nice permanence properties. We also introduce a new divisibility property...

Real division algebras and other algebras motivated by physics

International Nuclear Information System (INIS)

Benkart, G.; Osborn, J.M.

1981-01-01

In this survey we discuss several general techniques which have been productive in the study of real division algebras, flexible Lie-admissible algebras, and other nonassociative algebras, and we summarize results obtained using these methods. The principal method involved in this work is to view an algebra A as a module for a semisimple Lie algebra of derivations of A and to use representation theory to study products in A. In the case of real division algebras, we also discuss the use of isotopy and the use of a generalized Peirce decomposition. Most of the work summarized here has appeared in more detail in various other papers. The exceptions are results on a class of algebras of dimension 15, motivated by physics, which admit the Lie algebra sl(3) as an algebra of derivations

Linear algebra and analytic geometry for physical sciences

CERN Document Server

Landi, Giovanni

2018-01-01

A self-contained introduction to finite dimensional vector spaces, matrices, systems of linear equations, spectral analysis on euclidean and hermitian spaces, affine euclidean geometry, quadratic forms and conic sections. The mathematical formalism is motivated and introduced by problems from physics, notably mechanics (including celestial) and electro-magnetism, with more than two hundreds examples and solved exercises. Topics include: The group of orthogonal transformations on euclidean spaces, in particular rotations, with Euler angles and angular velocity. The rigid body with its inertia matrix. The unitary group. Lie algebras and exponential map. The Dirac’s bra-ket formalism. Spectral theory for self-adjoint endomorphisms on euclidean and hermitian spaces. The Minkowski spacetime from special relativity and the Maxwell equations. Conic sections with the use of eccentricity and Keplerian motions. An appendix collects basic algebraic notions like group, ring and field; and complex numbers and integers m...

Selfdual strings and loop space Nahm equations

International Nuclear Information System (INIS)

Gustavsson, Andreas

2008-01-01

We give two independent arguments why the classical membrane fields should be take values in a loop algebra. The first argument comes from how we may construct selfdual strings in the M5 brane from a loop space version of the Nahm equations. The second argument is that there appears to be no infinite set of finite-dimensional Lie algebras (such as su(N) for any N) that satisfies the algebraic structure of the membrane theory

An analytical solution of the Navier-Stokes equation for internal flows

International Nuclear Information System (INIS)

Lyberg, Mats D; Tryggeson, Henrik

2007-01-01

This paper derives a solution to the Navier-Stokes equation by considering vorticity generated at system boundaries. The result is an explicit expression for the velocity. The Navier-Stokes equation is reformulated as a divergence and integrated, giving a tensor equation that splits into a symmetric and a skew-symmetric part. One equation gives an algebraic system of quadratic equations involving velocity components. A system of nonlinear partial differential equations is reduced to algebra. The velocity is then explicitly calculated and shown to depend on boundary conditions only. This removes the need to solve the Navier-Stokes equation by a 3D numerical computation, replacing it by computation of 2D surface integrals over the boundary. (fast track communication)

Study of solving a Toda dynamic system with loop algebra

International Nuclear Information System (INIS)

Zhu Qiao; Yang Zhanying; Shi Kangjie; Wen Junqing

2006-01-01

The authors construct a Toda system with Loop algebra, and prove that the Lax equation L=[L,M] can be solved by means of solving a regular Riemann-Hilbert problem. In our system, M in Lax pair is an antisymmetrical matrix, while L=L + + M, and L + is a quasi-upper triangular matrix of loop algebra. In order to check our result, the authors exactly solve an R-H problem under a given initial condition as an example. (authors)

A new sub-equation method applied to obtain exact travelling wave solutions of some complex nonlinear equations

International Nuclear Information System (INIS)

Zhang Huiqun

2009-01-01

By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.

Hom-Novikov algebras

International Nuclear Information System (INIS)

Yau, Donald

2011-01-01

We study a twisted generalization of Novikov algebras, called Hom-Novikov algebras, in which the two defining identities are twisted by a linear map. It is shown that Hom-Novikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions 2 or 3 are computed, and their associated Hom-Novikov algebras are described explicitly. Another class of Hom-Novikov algebras is constructed from Hom-commutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel'fand. Two other classes of Hom-Novikov algebras are constructed from Hom-Lie algebras together with a suitable linear endomorphism, generalizing a construction due to Bai and Meng.

Systematic tools for chemical equation balancing

International Nuclear Information System (INIS)

Filby, E.E.; Idaho National Engineering Lab., Idaho Falls, ID; Idaho Univ., Idaho Falls, ID

1989-01-01

One of the most important skills that chemists and chemical engineers must develop is the ability to balance chemical equations. The normal first method taught is ''balancing by inspection'', which is sometimes explained as simply ''mental algebra.'' Every textbook surveyed for this paper presents equation balancing first as a matter of trial and error; this includes four very recently published books. Very little further guidance is provided until oxidation-reduction reactions must be balanced. The most commonly taught approaches for balancing, redox equations have been the oxidation state change and ion-electron methods. Unfortunately, redox reactions are usually treated as a new topic, and what the student has teamed about ''ordinary'' equations is of little or no help. All too often, these contradictions simply confuse and frustrate students, and equation balancing is relegated to the status of a black art. This is ironic because such,confusion and frustration is not necessary: Chemical equations can, in fact, be balanced by explicitly definable mathematical methods. The purpose of this paper is to outline the algebraic methods involved

Algebraic mesh generation for large scale viscous-compressible aerodynamic simulation

International Nuclear Information System (INIS)

Smith, R.E.

1984-01-01

Viscous-compressible aerodynamic simulation is the numerical solution of the compressible Navier-Stokes equations and associated boundary conditions. Boundary-fitted coordinate systems are well suited for the application of finite difference techniques to the Navier-Stokes equations. An algebraic approach to boundary-fitted coordinate systems is one where an explicit functional relation describes a mesh on which a solution is obtained. This approach has the advantage of rapid-precise mesh control. The basic mathematical structure of three algebraic mesh generation techniques is described. They are transfinite interpolation, the multi-surface method, and the two-boundary technique. The Navier-Stokes equations are transformed to a computational coordinate system where boundary-fitted coordinates can be applied. Large-scale computation implies that there is a large number of mesh points in the coordinate system. Computation of viscous compressible flow using boundary-fitted coordinate systems and the application of this computational philosophy on a vector computer are presented

Does Early Algebraic Reasoning Differ as a Function of Students' Difficulty with Calculations versus Word Problems?

Science.gov (United States)

Powell, Sarah R; Fuchs, Lynn S

2014-08-01

According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2 nd - grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.

Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

Science.gov (United States)

Powell, Sarah R.; Fuchs, Lynn S.

2014-01-01

According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2nd- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044

A generalized Zakharov-Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter

International Nuclear Information System (INIS)

Zhang Yufeng; Tam, Honwah; Feng Binlu

2011-01-01

Highlights: → A generalized Zakharov-Shabat equation is obtained. → The generalized AKNS vector fields are established. → The finite-band solution of the g-ZS equation is obtained. → By using a Lie algebra presented in the paper, a new soliton hierarchy with an arbitrary parameter is worked out. - Abstract: In this paper, a generalized Zakharov-Shabat equation (g-ZS equation), which is an isospectral problem, is introduced by using a loop algebra G ∼ . From the stationary zero curvature equation we define the Lenard gradients {g j } and the corresponding generalized AKNS (g-AKNS) vector fields {X j } and X k flows. Employing the nonlinearization method, we obtain the generalized Zhakharov-Shabat Bargmann (g-ZS-B) system and prove that it is Liouville integrable by introducing elliptic coordinates and evolution equations. The explicit relations of the X k flows and the polynomial integrals {H k } are established. Finally, we obtain the finite-band solutions of the g-ZS equation via the Abel-Jacobian coordinates. In addition, a soliton hierarchy and its Hamiltonian structure with an arbitrary parameter k are derived.

Linear algebra meets Lie algebra: the Kostant-Wallach theory

OpenAIRE

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.

On symmetry reduction and exact solutions of the linear one-dimensional Schroedinger equation

International Nuclear Information System (INIS)

Barannik, L.L.

1996-01-01

Symmetry reduction of the Schroedinger equation with potential is carried out on subalgebras of the Lie algebra which is the direct sum of the special Galilei algebra and one-dimensional algebra. Some new exact solutions are obtained

Perturbative quantum field theory via vertex algebras

International Nuclear Information System (INIS)

Hollands, Stefan; Olbermann, Heiner

2009-01-01

In this paper, we explain how perturbative quantum field theory can be formulated in terms of (a version of) vertex algebras. Our starting point is the Wilson-Zimmermann operator product expansion (OPE). Following ideas of a previous paper (S. Hollands, e-print arXiv:0802.2198), we consider a consistency (essentially associativity) condition satisfied by the coefficients in this expansion. We observe that the information in the OPE coefficients can be repackaged straightforwardly into 'vertex operators' and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras. We develop a general theory of perturbations of the algebras that we encounter, similar in nature to the Hochschild cohomology describing the deformation theory of ordinary algebras. The main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction Lagrangian (such as λφ 4 ) in practice, using the consistency condition and the corresponding nonlinear field equation. We derive graphical rules, which display the vertex operators (i.e., OPE coefficients) in terms of certain multiple series of hypergeometric type.

Geometry, algebra and applications from mechanics to cryptography

CERN Document Server

Encinas, Luis; Gadea, Pedro; María, Mª

2016-01-01

This volume collects contributions written by different experts in honor of Prof. Jaime Muñoz Masqué. It covers a wide variety of research topics, from differential geometry to algebra, but particularly focuses on the geometric formulation of variational calculus; geometric mechanics and field theories; symmetries and conservation laws of differential equations, and pseudo-Riemannian geometry of homogeneous spaces. It also discusses algebraic applications to cryptography and number theory. It offers state-of-the-art contributions in the context of current research trends. The final result is a challenging panoramic view of connecting problems that initially appear distant.

Essential linear algebra with applications a problem-solving approach

CERN Document Server

Andreescu, Titu

2014-01-01

This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject’s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality. Key features include: • a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory;  • over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance; • an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them.   Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course.    ...

Algorithmic Verification of Linearizability for Ordinary Differential Equations

KAUST Repository

Lyakhov, Dmitry A.

2017-07-19

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

Deformed Heisenberg algebra, fractional spin fields, and supersymmetry without fermions

International Nuclear Information System (INIS)

Plyushchay, M.S.

1996-01-01

Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a - ,a + ]=1+νK, involving the Klein operator K, {K,a ± }=0, K 2 =1. The connection of the minimal set of equations with the earlier proposed open-quote open-quote universal close-quote close-quote vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously broken N=2 supersymmetry allowing us to realize a Bose endash Fermi transformation and spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2 parallel 2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. The possibility of open-quote open-quote superimposing close-quote close-quote the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that the osp(2 parallel 2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model. Copyright copyright 1996 Academic Press, Inc

DDASAC, Double-Precision Differential or Algebraic Sensitivity Analysis

International Nuclear Information System (INIS)

Caracotsios, M.; Stewart, W.E.; Petzold, L.

1997-01-01

1 - Description of program or function: DDASAC solves nonlinear initial-value problems involving stiff implicit systems of ordinary differential and algebraic equations. Purely algebraic nonlinear systems can also be solved, given an initial guess within the region of attraction of a solution. Options include automatic reconciliation of inconsistent initial states and derivatives, automatic initial step selection, direct concurrent parametric sensitivity analysis, and stopping at a prescribed value of any user-defined functional of the current solution vector. Local error control (in the max-norm or the 2-norm) is provided for the state vector and can include the sensitivities on request. 2 - Method of solution: Reconciliation of initial conditions is done with a damped Newton algorithm adapted from Bain and Stewart (1991). Initial step selection is done by the first-order algorithm of Shampine (1987), extended here to differential-algebraic equation systems. The solution is continued with the DASSL predictor- corrector algorithm (Petzold 1983, Brenan et al. 1989) with the initial acceleration phase detected and with row scaling of the Jacobian added. The backward-difference formulas for the predictor and corrector are expressed in divide-difference form, and the fixed-leading-coefficient form of the corrector (Jackson and Sacks-Davis 1980, Brenan et al. 1989) is used. Weights for error tests are updated in each step with the user's tolerances at the predicted state. Sensitivity analysis is performed directly on the corrector equations as given by Catacotsios and Stewart (1985) and is extended here to the initialization when needed. 3 - Restrictions on the complexity of the problem: This algorithm, like DASSL, performs well on differential-algebraic systems of index 0 and 1 but not on higher-index systems; see Brenan et al. (1989). The user assigns the work array lengths and the output unit. The machine number range and precision are determined at run time by a

Hamiltonian structure of linearly extended Virasoro algebra

International Nuclear Information System (INIS)

Arakelyan, T.A.; Savvidi, G.K.

1991-01-01

The Hamiltonian structure of linearly extended Virasoro algebra which admits free bosonic field representation is described. An example of a non-trivial extension is found. The hierarchy of integrable non-linear equations corresponding to this Hamiltonian structure is constructed. This hierarchy admits the Lax representation by matrix Lax operator of second order

Algebra & trigonometry II essentials

CERN Document Server

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 II includes logarithms, sequences and series, permutations, combinations and probability, vectors, matrices, determinants and systems of equations, mathematica

Exact multi-line soliton solutions of noncommutative KP equation

International Nuclear Information System (INIS)

Wang, Ning; Wadati, Miki

2003-01-01

A method of solving noncommutative linear algebraic equations plays a key role in the extension of the ∂-bar -dressing on the noncommutative space-time manifold. In this paper, a solution-generating method of noncommutative linear algebraic equations is proposed. By use of the proposed method, a class of multi-line soliton solutions of noncommutative KP (ncKP) equation is constructed explicitly. The method is expected to be of use for constructions of noncommutative soliton equations. The significance of the noncommutativity of coordinates is investigated. It is found that the noncommutativity of the space-time coordinate has a role to split the spatial waveform of the classical multi-line solitons and reform it to a new configuration. (author)

A discrete variational identity on semi-direct sums of Lie algebras

Energy Technology Data Exchange (ETDEWEB)

M, Wenxiu [Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700 (United States)

2007-12-14

The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant {gamma} involved in the variational identity is determined through the corresponding solution to the stationary discrete zero-curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy.

A discrete variational identity on semi-direct sums of Lie algebras

International Nuclear Information System (INIS)

M, Wenxiu

2007-01-01

The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant γ involved in the variational identity is determined through the corresponding solution to the stationary discrete zero-curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy

Extended Virasoro algebra and algebra of area preserving diffeomorphisms

International Nuclear Information System (INIS)

Arakelyan, T.A.

1990-01-01

The algebra of area preserving diffeomorphism plays an important role in the theory of relativistic membranes. It is pointed out that the relation between this algebra and the extended Virasoro algebra associated with the generalized Kac-Moody algebras G(T 2 ). The highest weight representation of these infinite-dimensional algebras as well as of their subalgebras is studied. 5 refs

Solving algebraic computational problems in geodesy and geoinformatics the answer to modern challenges

CERN Document Server

Awange, Joseph L

2004-01-01

While preparing and teaching 'Introduction to Geodesy I and II' to - dergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taughtrequiredsomeskillsinalgebra,andinparticular,computeral- bra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we haveattemptedtoputtogetherbasicconceptsofabstractalgebra which underpin the techniques for solving algebraic problems. Algebraic c- putational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds,theconceptsand techniquespresented hereinarenonetheless- plicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robo...

An application of the division algebras, Jordan algebras and split composition algebras

International Nuclear Information System (INIS)

Foot, R.; Joshi, G.C.

1992-01-01

It has been established that the covering group of the Lorentz group in D = 3, 4, 6, 10 can be expressed in a unified way, based on the four composition division algebras R, C, Q and O. In this paper, the authors discuss, in this framework, the role of the complex numbers of quantum mechanics. A unified treatment of quantum-mechanical spinors is given. The authors provide an explicit demonstration that the vector and spinor transformations recently constructed from a subgroup of the reduced structure group of the Jordan algebras M n 3 are indeed the Lorentz transformations. The authors also show that if the division algebras in the construction of the covering groups of the Lorentz groups in D = 3, 4, 6, 10 are replaced by the split composition algebras, then the sequence of groups SO(2, 2), SO(3, 3) and SO(5, 5) result. The analysis is presumed to be self-contained as the relevant aspects of the division algebras and Jordan algebras are reviewed. Some applications to physical theory are indicated

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

Science.gov (United States)

Abdelaziz, Y.; Maillard, J.-M.

2017-05-01

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same {}_2F1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus {}_2F1 hypergeometric function example. We then focus on identities relating the same {}_2F1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to {}_3F2 , hypergeometric functions, and show that one just reduces to the previous {}_2F1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a _2F1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or {}_2F1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to

Chew-Low equations as Cremoma transformations

International Nuclear Information System (INIS)

Rerikh, K.V.

1982-01-01

The Chew-Low equations for the p-wave pion-nucleon scattering with the crossing-symmetry matrix (3x3) are investigated in their well-known formulation as a system of nonlinear difference equations. These equations interpreted as geometrical transformations are shown to be a special case of the Cremona transformaions. Using the properties of the Cremona transformations we obtain the general 3-parametric functional equation on invariant algebraic and nonalgebraic curves in the space solutions of the Chew- Low equations. It is proved that there exists only one invariant algebraic curve, the parabola corresponding to the well-known solution. Analysis of the general functional equation on invariant nonalgebraic curves makes it possible to select in addition to this parabola 3 invariant forms defining implicitly 3 nonalgebraic curves and to concretize for them the general equation by means of fixing the parameters. From the transformational properties of the invariant forms with respect to the Cremona transformations, there follows an important result that the ration of these forms in proper powers is the general integral of the nonlinear system of the Chew-Low equations, which is an even antiperiodic function. The structure of the second general integral is given and the functional equations which determinne this integral are presented [ru

Interactions Between Representation Ttheory, Algebraic Topology and Commutative Algebra

CERN Document Server

Pitsch, Wolfgang; Zarzuela, Santiago

2016-01-01

This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest dev...

The universal R-matrix and its associated quantum algebra as functionals of the classical r-matrix: the sl2 case

International Nuclear Information System (INIS)

Freidel, L.; Maillet, J.M.

1992-09-01

Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra U h (sl 2 ) and its universal quantum R-matrix are explicitly constructed as functionals of the associated classical r-matrix. In this framework, the quantum algebra U h (sl 2 ) is naturally imbedded in the universal enveloping algebra of the sl 2 current algebra. (author) 13 refs

BRST, generalized Maurer-Cartan equations and CFT

Energy Technology Data Exchange (ETDEWEB)

Zeitlin, Anton M. [Department of Mathematics, Yale University, 442 Dunham Lab, 10 Hillhouse Ave., New Haven, CT 06511 (United States); St. Petersburg Department of Steklov Mathematical Institute, Fontanka, 27, St. Petersburg 191023 (Russian Federation)]. E-mail: zam@math.ipme.ru

2006-12-25

The paper is devoted to the study of BRST charge in perturbed two-dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via the OPE. The corresponding equations are constructed and their symmetries are studied up to the second order in formal coupling constant. It appears that the obtained equations can be interpreted as generalized Maurer-Cartan ones. We study two concrete examples in detail: the bosonic nonlinear sigma model and perturbed first order theory. In particular, we show that the Einstein equations, which are the conformal invariance conditions for both these perturbed theories, expanded up to the second order, can be rewritten in such generalized Maurer-Cartan form.

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

Science.gov (United States)

Campoamor-Stursberg, R.

2018-03-01

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot-Guldberg-Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot-Guldberg-Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

A Continuous Formulation for Logical Decisions in Differential Algebraic Systems using Mathematical Programs with Complementarity Constraints

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Kody M. Powell

2016-03-01

Full Text Available This work presents a methodology to represent logical decisions in differential algebraic equation simulation and constrained optimization problems using a set of continuous algebraic equations. The formulations may be used when state variables trigger a change in process dynamics, and introduces a pseudo-binary decision variable, which is continuous, but should only have valid solutions at values of either zero or one within a finite time horizon. This formulation enables dynamic optimization problems with logical disjunctions to be solved by simultaneous solution methods without using methods such as mixed integer programming. Several case studies are given to illustrate the value of this methodology including nonlinear model predictive control of a chemical reactor using a surge tank with overflow to buffer disturbances in feed flow rate. Although this work contains novel methodologies for solving dynamic algebraic equation (DAE constrained problems where the system may experience an abrupt change in dynamics that may otherwise require a conditional statement, there remain substantial limitations to this methodology, including a limited domain where problems may converge and the possibility for ill-conditioning. Although the problems presented use only continuous algebraic equations, the formulation has inherent non-smoothness. Hence, these problems must be solved with care and only in select circumstances, such as in simulation or situations when the solution is expected to be near the solver’s initial point.

Solving Differential Equations in R: Package deSolve

Science.gov (United States)

In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...

Duncan F. Gregory, William Walton and the development of British algebra: 'algebraical geometry', 'geometrical algebra', abstraction.

Science.gov (United States)

Verburgt, Lukas M

2016-01-01

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.

Monomial algebras

CERN Document Server

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.

Algebra

CERN Document Server

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.

Some Aspects of Extended Kinetic Equation

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Dilip Kumar

2015-09-01

Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.

Iwahori-Hecke algebras and Schur algebras of the symmetric group

CERN Document Server

Mathas, Andrew

1999-01-01

This volume presents a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the q-Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems. The main results are proved by showing that the Iwahori-Hecke algebras and q-Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in Chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the q-Schur algebras. T...

New simple algebraic root locus method for design of feedback control systems

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Cingara Aleksandar M.

2008-01-01

Full Text Available New concept of algebraic characteristic equation decomposition method is presented to simplify the design of closed-loop systems for practical applications. The method consists of two decompositions. The first one, decomposition of the characteristic equation into two lower order equations, was performed in order to simplify the analysis and design of closed loop systems. The second is the decomposition of Laplace variable, s, into two variables, damping coefficient, ζ, and natural frequency, ω n. Those two decompositions reduce the design of any order feedback systems to setting of two complex dominant poles in the desired position. In the paper, we derived explicit equations for six cases: first, second and third order system with P and PI. We got the analytical solutions for the case of fourth and fifth order characteristic equations with the P and PI controller; one may obtain a complete analytical solution of controller gain as a function of the desired damping coefficient. The complete derivation is given for the third order equation with P and PI controller. We can extend the number of specified poles to the highest order of the characteristic equation working in a similar way, so we can specify the position of each pole. The concept is similar to the root locus but root locus is implicit, which makes it more complicated and this is simpler explicit root locus. Standard procedures, root locus and Bode diagrams or Nichol Charts, are neither algebraic nor explicit. We basically change controller parameters and observe the change of some function until we get the desired specifications. The derived method has three important advantage over the standard procedures. It is general, algebraic and explicit. Those are the best poles design results possible; it is not possible to get better controller design results.

Algebra of pseudo-differential operators over C*-algebra

International Nuclear Information System (INIS)

Mohammad, N.

1982-08-01

Algebras of pseudo-differential operators over C*-algebras are studied for the special case when in Hormander class Ssub(rho,delta)sup(m)(Ω) Ω = Rsup(n); rho = 1, delta = 0, m any real number, and the C*-algebra is infinite dimensional non-commutative. The space B, i.e. the set of A-valued C*-functions in Rsup(n) (or Rsup(n) x Rsup(n)) whose derivatives are all bounded, plays an important role. A denotes C*-algebra. First the operator class Ssub(phi,0)sup(m) is defined, and through it, the class Lsub(1,0)sup(m) of pseudo-differential operators. Then the basic asymptotic expansion theorems concerning adjoint and product of operators of class Ssub(1,0)sup(m) are stated. Finally, proofs are given of L 2 -continuity theorem and the main theorem, which states that algebra of all pseudo-differential operators over C*-algebras is itself C*-algebra

Commutative algebra with a view toward algebraic geometry

CERN Document Server

Eisenbud, David

1995-01-01

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algeb...

Diffeomorphism-type symmetries of the self-dual Yang-Mills equations

International Nuclear Information System (INIS)

Ivanova, T.A.

1998-01-01

The infinite-dimensional algebra of diffeomorphism-type symmetries of the self-dual Yang-Mills equations is described as the algebra of 0-cochains with values in a sheaf of germs of holomorphic sections of the (1,0) tangent bundle over the twistor space. It is shown that the extended conformal symmetries are obtained as particular cases of the aforementioned algebra

Jordan algebras versus C*- algebras

International Nuclear Information System (INIS)

Stormer, E.

1976-01-01

The axiomatic formulation of quantum mechanics and the problem of whether the observables form self-adjoint operators on a Hilbert space, are discussed. The relation between C*- algebras and Jordan algebras is studied using spectral theory. (P.D.)

Solutions to Arithmetic Convolution Equations

Czech Academy of Sciences Publication Activity Database

Glöckner, H.; Lucht, L.G.; Porubský, Štefan

2007-01-01

Roč. 135, č. 6 (2007), s. 1619-1629 ISSN 0002-9939 R&D Projects: GA ČR GA201/04/0381 Institutional research plan: CEZ:AV0Z10300504 Keywords : arithmetic functions * Dirichlet convolution * polynomial equations * analytic equations * topological algebras * holomorphic functional calculus Subject RIV: BA - General Mathematics Impact factor: 0.520, year: 2007

Solving Differential Equations in R: Package deSolve

NARCIS (Netherlands)

Soetaert, K.E.R.; Petzoldt, T.; Setzer, R.W.

2010-01-01

In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The

The lie-algebraic structures and integrability of differential and differential-difference nonlinear dynamical systems

International Nuclear Information System (INIS)

Prykarpatsky, A.K.; Blackmore, D.L.; Bogolubov, N.N. Jr.

2007-05-01

The infinite-dimensional operator Lie algebras of the related integrable nonlocal differential-difference dynamical systems are treated as their hidden symmetries. As a result of their dimerization the Lax type representations for both local differential-difference equations and nonlocal ones are obtained. An alternative approach to the Lie-algebraic interpretation of the integrable local differential-difference systems is also proposed. The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the centrally extended Lie algebra of integro-differential operators with matrix-valued coefficients coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is obtained by means of a specially constructed Baecklund transformation. The Hamiltonian description for the corresponding set of additional symmetry hierarchies is represented. The relation of these hierarchies with Lax type integrable (3+1)-dimensional nonlinear dynamical systems and their triple Lax type linearizations is analyzed. The Lie-algebraic structures, related with centrally extended current operator Lie algebras are discussed with respect to constructing new nonlinear integrable dynamical systems on functional manifolds and super-manifolds. Special Poisson structures and related with them factorized integrable operator dynamical systems having interesting applications in modern mathematical physics, quantum computing mathematics and other fields are constructed. The previous purely computational results are explained within the approach developed. (author)

Algebraic Reconstruction of Current Dipoles and Quadrupoles in Three-Dimensional Space

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Takaaki Nara

2013-01-01

Full Text Available This paper presents an algebraic method for an inverse source problem for the Poisson equation where the source consists of dipoles and quadrupoles. This source model is significant in the magnetoencephalography inverse problem. The proposed method identifies the source parameters directly and algebraically using data without requiring an initial parameter estimate or iterative computation of the forward solution. The obtained parameters could be used for the initial solution in an optimization-based algorithm for further refinement.

Implicative Algebras

African Journals Online (AJOL)

Tadesse

In this paper we introduce the concept of implicative algebras which is an equivalent definition of lattice implication algebra of Xu (1993) and further we prove that it is a regular Autometrized. Algebra. Further we remark that the binary operation → on lattice implicative algebra can never be associative. Key words: Implicative ...

Lie Algebras for Constructing Nonlinear Integrable Couplings

International Nuclear Information System (INIS)

Zhang Yufeng

2011-01-01

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their Hamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations. (general)

Open algebraic surfaces

CERN Document Server

Miyanishi, Masayoshi

2000-01-01

Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. There is a long history of research on projective algebraic surfaces, and there exists a beautiful Enriques-Kodaira classification of such surfaces. The research accumulated by Ramanujan, Abhyankar, Moh, and Nagata and others has established a classification theory of open algebraic surfaces comparable to the Enriques-Kodaira theory. This research provides powerful methods to study the geometry and topology of open algebraic surfaces. The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities. This book contains a comprehensive account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces. Prerequisite to understanding the text is a basic b...

Cognitive Load in Algebra: Element Interactivity in Solving Equations

Science.gov (United States)

Ngu, Bing Hiong; Chung, Siu Fung; Yeung, Alexander Seeshing

2015-01-01

Central to equation solving is the maintenance of equivalence on both sides of the equation. However, when the process involves an interaction of multiple elements, solving an equation can impose a high cognitive load. The balance method requires operations on both sides of the equation, whereas the inverse method involves operations on one side…

Separable algebras

CERN Document Server

Ford, Timothy J

2017-01-01

This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups. The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.

Lie symmetries and differential galois groups of linear equations

NARCIS (Netherlands)

Oudshoorn, W.R.; Put, M. van der

2002-01-01

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In

Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations

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M. Arshad

Full Text Available In this manuscript, we constructed different form of new exact solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations by utilizing the modified extended direct algebraic method. New exact traveling wave solutions for both equations are obtained in the form of soliton, periodic, bright, and dark solitary wave solutions. There are many applications of the present traveling wave solutions in physics and furthermore, a wide class of coupled nonlinear evolution equations can be solved by this method. Keywords: Traveling wave solutions, Elliptic solutions, Generalized coupled Zakharov–Kuznetsov equation, Dispersive long wave equation, Modified extended direct algebraic method

Flux algebra, Bianchi identities and Freed-Witten anomalies in F-theory compactifications

International Nuclear Information System (INIS)

Aldazabal, G.; Camara, P.G.; Rosabal, J.A.

2009-01-01

We discuss the structure of 4D gauged supergravity algebras corresponding to globally non-geometric compactifications of F-theory, admitting a local geometric description in terms of 10D supergravity. By starting with the well-known algebra of gauge generators associated to non-geometric type IIB fluxes, we derive a full algebra containing all, closed RR and NSNS, geometric and non-geometric dual fluxes. We achieve this generalization by a systematic application of SL(2,Z) duality transformations and by taking care of the spinorial structure of the fluxes. The resulting algebra encodes much information about the higher dimensional theory. In particular, tadpole equations and Bianchi identities are obtainable as Jacobi identities of the algebra. When a sector of magnetized (p,q) 7-branes is included, certain closed axions are gauged by the U(1) transformations on the branes. We indicate how the diagonal gauge generators of the branes can be incorporated into the full algebra, and show that Freed-Witten constraints and tadpole cancellation conditions for (p,q) 7-branes can be described as Jacobi identities satisfied by the algebra mixing bulk and brane gauge generators

Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

Science.gov (United States)

Hoque, Md. Fazlul; Marquette, Ian; Post, Sarah; Zhang, Yao-Zhong

2018-04-01

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

Generalized EMV-Effect Algebras

Science.gov (United States)

Borzooei, R. A.; Dvurečenskij, A.; Sharafi, A. H.

2018-04-01

Recently in Dvurečenskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.

The application of Max-Plus/Min-Plus algebra for an automated analysis; Einsatz der Max-Plus/Min-Plus Algebra zur automatisierten Sicherheitsanalyse

Energy Technology Data Exchange (ETDEWEB)

Laengst, W.; Lapp, A.; Stuebbe, K.; Schirmer, J.; Kraft, D. [Robert Bosch GmbH, Stuttgart (Germany). Zentralbereich Forschung und Vorausentwicklung

2003-07-01

The methodology introduced in this article is capable of describing a binary behavior of distributed discrete event systems. This is the basis for a safety analysis in early phases of system development and for an automated determination of failure dependencies. For this purpose a system equation similar to the state-space representation in system theory is used. The equation is evaluated applying the Max-Plus/Min-Plus algebra. The procedure is exemplified by a simplified braking system. (orig.) [German] In dem vorliegenden Artikel wird ein Verfahren zur binaeren Beschreibung von verteilten ereignisdiskreten Systemen vorgestellt. Dieses wird als Grundlage fuer eine Sicherheitsanalyse in einer fruehen Phase der Systementwicklung und zur automatisierten Ermittlung von Fehlerabhaengigkeiten eingesetzt. Hierzu wird eine Systemgleichung verwendet, die aehnlich zu der Zustandsraumdarstellung der konventionellen Systemtheorie ist. Dabei erfolgt die Auswertung der Systemgleichung durch Anwendung der Max-Plus/Min-Plus Algebra. Die Vorgehensweise wird anhand des Beispiels eines vereinfachten Bremssystems erlaeutert. (orig.)

Special set linear algebra and special set fuzzy linear algebra

OpenAIRE

Kandasamy, W. B. Vasantha; Smarandache, Florentin; Ilanthenral, K.

2009-01-01

The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analog...

ODE/IM correspondence and Bethe ansatz for affine Toda field equations

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Katsushi Ito

2015-07-01

Full Text Available We study the linear problem associated with modified affine Toda field equation for the Langlands dual gˆ∨, where gˆ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the Q-functions for g-type quantum integrable models. The ψ-system for the solutions associated with the fundamental representations of g leads to Bethe ansatz equations associated with the affine Lie algebra gˆ. We also study the A2r(2 affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T–Q relations.

On Approximate Solutions of Functional Equations in Vector Lattices

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Bogdan Batko

2014-01-01

Full Text Available We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra. The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x+y+F(x+F(y≠0⇒F(x+y=F(x+F(y in Riesz spaces, the Cauchy equation with squares F(x+y2=(F(x+F(y2 in f-algebras, and the quadratic functional equation F(x+y+F(x-y=2F(x+2F(y in Riesz spaces.

Banach Synaptic Algebras

Science.gov (United States)

Foulis, David J.; Pulmannov, Sylvia

2018-04-01

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions

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R. Naz

2015-01-01

Full Text Available We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density g(x, and the applied load denoted by f(u, a function of transverse displacement u(t,x. The complete Lie group classification is obtained for different forms of the variable lineal mass density g(x and applied load f(u. The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of g(x. For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when g(x is constant with variable applied load f(u. For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.

Grassmann algebras

International Nuclear Information System (INIS)

Garcia, R.L.

1983-11-01

The Grassmann algebra is presented briefly. Exponential and logarithm of matrices functions, whose elements belong to this algebra, are studied with the help of the SCHOONSCHIP and REDUCE 2 algebraic manipulators. (Author) [pt

Graded algebras of the second rank and integration of nonlinear equations Ysub(z)sub(z) = exp(2Y) - exp(-2Y), Ysub(z)sub(z) = 2 exp(Y) - exp(-2Y)

International Nuclear Information System (INIS)

Leznov, A.N.; Smirnov, V.G.

1981-01-01

In the terms of the notions of the theory of infinite-dimensional algebras of finite growth of the second rank, we have derived solutions to the equations Ysub(z)sub(z) = exp(2Y) - exp(-2Y); Ysub(z)sub(z) = 2 exp(Y) - exp(-2Y) dependent on two arbitrary functions. (orig.)

Algebraic geometry

CERN Document Server

Lefschetz, Solomon

2005-01-01

An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.

Recurrence approach and higher order polynomial algebras for superintegrable monopole systems

Science.gov (United States)

Hoque, Md Fazlul; Marquette, Ian; Zhang, Yao-Zhong

2018-05-01

We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in a generalized Taub-Newman-Unti-Tamburino (NUT) space. The Schrödinger equation of this model is solved in spherical coordinates in the framework of Stäckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show that the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.

Efficient simulation of gas-liquid pipe flows using a generalized population balance equation coupled with the algebraic slip model

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