A simple algebraic cancer equation: calculating how cancers may arise with normal mutation rates
Directory of Open Access Journals (Sweden)
Shibata Darryl
2010-01-01
Full Text Available Abstract Background The purpose of this article is to present a relatively easy to understand cancer model where transformation occurs when the first cell, among many at risk within a colon, accumulates a set of driver mutations. The analysis of this model yields a simple algebraic equation, which takes as inputs the number of stem cells, mutation and division rates, and the number of driver mutations, and makes predictions about cancer epidemiology. Methods The equation [p = 1 - (1 - (1 - (1 - udkNm ] calculates the probability of cancer (p and contains five parameters: the number of divisions (d, the number of stem cells (N × m, the number of critical rate-limiting pathway driver mutations (k, and the mutation rate (u. In this model progression to cancer "starts" at conception and mutations accumulate with cell division. Transformation occurs when a critical number of rate-limiting pathway mutations first accumulates within a single stem cell. Results When applied to several colorectal cancer data sets, parameter values consistent with crypt stem cell biology and normal mutation rates were able to match the increase in cancer with aging, and the mutation frequencies found in cancer genomes. The equation can help explain how cancer risks may vary with age, height, germline mutations, and aspirin use. APC mutations may shorten pathways to cancer by effectively increasing the numbers of stem cells at risk. Conclusions The equation illustrates that age-related increases in cancer frequencies may result from relatively normal division and mutation rates. Although this equation does not encompass all of the known complexity of cancer, it may be useful, especially in a teaching setting, to help illustrate relationships between small and large cancer features.
Stability of functional equations in Banach algebras
Cho, Yeol Je; Rassias, Themistocles M; Saadati, Reza
2015-01-01
Some of the most recent and significant results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C*-algebras, C*-ternary algebras, non-Archimedean Banach algebras and multi-normed algebras are presented in this book. A brief introduction for functional equations and their stability is provided with historical remarks. Since the homomorphisms and derivations in Banach algebras are additive and R-linear or C-linear, the stability problems for additive functional equations and additive mappings are studied in detail. The latest results are discussed and examined in stability theory for new functional equations and functional inequalities in Banach algebras and C*-algebras, non-Archimedean Banach algebras, non-Archimedean C*-algebras, multi-Banach algebras and multi-C*-algebras. Graduate students with an understanding of operator theory, functional analysis, functional equations and analytic inequalities will find this book useful for furthering their understanding and discovering the l...
Differential Equations for Algebraic Functions
Bostan, Alin; Chyzak, Frédéric; Salvy, Bruno; Lecerf, Grégoire; Schost, Éric
2007-01-01
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove ...
Hopf algebras and Dyson-Schwinger equations
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Algebraic dynamics solution and algebraic dynamics algorithm of Burgers equations
Institute of Scientific and Technical Information of China (English)
2008-01-01
Algebraic dynamics solution and algebraic dynamics algorithm of nonlinear partial differential evolution equations in the functional space are applied to Burgers equation. The results indicate that the approach is effective for analytical solutions to Burgers equation, and the algorithm for numerical solutions of Burgers equation is more stable, with higher precision than other existing finite difference algo-rithms.
Solving Absolute Value Equations Algebraically and Geometrically
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
Asymptotic-group analysis of algebraic equations
Shamrovskii, A. D.; I. V. Andrianov; J. Awrejcewicz
2004-01-01
Both the method of asymptotic analysis and the theory of extension group are applied to study the Descates equation. The proposed algorithm allows to obtain various variants of simplification and can be easily generalized to their algebraic and differential equations.
Algebraic entropy for differential-delay equations
Viallet, Claude M.
2014-01-01
We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.
An Algebraic Approach to the Scattering Equations
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.
Lie algebras and linear differential equations.
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.
Algebraic differential equations associated to some polynomials
Barlet, Daniel
2013-01-01
We compute the Gauss-Manin differential equation for any period of a polynomial in \\ $\\C[x_{0},\\dots, x_{n}]$ \\ with \\ $(n+2)$ \\ monomials. We give two general factorizations theorem in the algebra \\ $\\C$ \\ for such a differential equations.
The kinematic algebras from the scattering equations
International Nuclear Information System (INIS)
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant
Surveys in differential-algebraic equations III
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.
Surveys in differential-algebraic equations II
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 - Observers for DAEs - DAEs in chemical processes - Optimal control of DAEs - DAEs from a functional-analytic viewpoint - Algebraic methods 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.
On solving equations of algebraic sum of equal powers
Institute of Scientific and Technical Information of China (English)
WANG; Xinghua; YANG; Shijun
2006-01-01
It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics,it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.
Optical systolic solutions of linear algebraic equations
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
Algebras with Parastrophically Uncancellable Quasigroup Equations
Directory of Open Access Journals (Sweden)
Amir Ehsani
2016-07-01
Full Text Available We consider 48 parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non-Belousov (consists of 16 equations and non-balanced non-gemini (consists of 32 equations. A linear representation of a group (Abelian group for a pair of quasigroup operations satisfying one of these parastrophically uncancellable quadratic equations is obtained. As a consequence of these results, a linear representation for every operation of a binary algebra satisfying one of these hyperidentities is obtained.
Characteristic Algebras of Fully Discrete Hyperbolic Type Equations
Habibullin, Ismagil T.
2005-01-01
The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.
Wilsonian renormalization, differential equations and Hopf algebras
Thomas, Krajewski
2008-01-01
In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Finally, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally...
Upper bounds on the solution of coupled algebraic riccati equation
Czornik Adam; Świerniak Andrzej
2001-01-01
Upper bounds for eigenvalues of a solution to continuous time coupled algebraic Riccati equation (CCARE) and discrete time coupled algebraic Riccati equation (DCARE) are developed as special cases of bounds for the unified coupled algebraic Riccati equation (UCARE). They include bounds of the maximal eigenvalues, the sums of the eigenvalues and the trace.
Upper bounds on the solution of coupled algebraic riccati equation
Directory of Open Access Journals (Sweden)
Czornik Adam
2001-01-01
Full Text Available Upper bounds for eigenvalues of a solution to continuous time coupled algebraic Riccati equation (CCARE and discrete time coupled algebraic Riccati equation (DCARE are developed as special cases of bounds for the unified coupled algebraic Riccati equation (UCARE. They include bounds of the maximal eigenvalues, the sums of the eigenvalues and the trace.
Laurent phenomenon algebras and the discrete BKP equation
Okubo, Naoto
2016-09-01
We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada–Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.
A Unified Algebraic Approach to Classical Yang-Baxter Equation
Bai, Chengming
2007-01-01
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical ...
Algebraic dynamics solution to and algebraic dynamics algorithm for nonlinear advection equation
Institute of Scientific and Technical Information of China (English)
2008-01-01
Algebraic dynamics approach and algebraic dynamics algorithm for the solution of nonlinear partial differential equations are applied to the nonlinear advection equa-tion. The results show that the approach is effective for the exact analytical solu-tion and the algorithm has higher precision than other existing algorithms in nu-merical computation for the nonlinear advection equation.
Lie-algebraic symmetries of generalized Davey-Stewartson equations
International Nuclear Information System (INIS)
We identify the full Lie-algebraic structure of the generalized Davey-Stewartson (GDS) system of equations with symmetries of a specific of continual Lie algebras. In particular, we show that they are related to two copies of the Poisson bracket continual Lie algebra.
Inhomogeneous linear equation in Rota-Baxter algebra
Pietrzkowski, Gabriel
2014-01-01
We consider a complete filtered Rota-Baxter algebra of weight $\\lambda$ over a commutative ring. Finding the unique solution of a non-homogeneous linear algebraic equation in this algebra, we generalize Spitzer's identity in both commutative and non-commutative cases. As an application, considering the Rota-Baxter algebra of power series in one variable with q-integral as the Rota-Baxter operator, we show certain Eulerian identities.
Quantum Einstein’s equations and constraints algebra
Indian Academy of Sciences (India)
Fatimah Shojai; Ali Shojai
2002-01-01
In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the deBroglie–Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.
Deformed algebras: examples and application to Lax equations
Magnot, Jean-Pierre
2014-01-01
We develop here a concept of deformed algebras through three examples and an application. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \\cite{Ma2013} is only an example among others, and how they often give rise to regular Fr\\"olicher Lie groups. Then, we show how such deformed algebras arise in a formal integration of Lax equations through time scaling. The infinite dime...
ON THE MINIMAL NONNEGATIVE SOLUTION OF NONSYMMETRIC ALGEBRAIC RICCATI EQUATION
Institute of Scientific and Technical Information of China (English)
Xiao-xia Guo; Zhong-zhi Bai
2005-01-01
We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X2 - EX - F -= 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.
Algebraic Solutions of the Lam\\'e Equation, Revisited
Maier, Robert S.
2002-01-01
A minor error in the necessary conditions for the algebraic form of the Lam\\'e equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lam\\'e's differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was misse...
Difference equations and cluster algebras I: Poisson bracket for integrable difference equations
Inoue, Rei
2010-01-01
We introduce the cluster algebraic formulation of the integrable difference equations, the discrete Lotka-Volterra equation and the discrete Liouville equation, from the view point of the general T-system and Y-system. We also study the Poisson structure for the cluster algebra, and give the associated Poisson bracket for the two difference equations.
Clifford algebra and solution of Bargmann-Michel-Telegdi equation
International Nuclear Information System (INIS)
The Clifford algebra structure of the Minkowski space is presented in the article. The method of solving motion equations within the frames of formalism of this algebra is described. The solution of the spin motion equation (the Bargmann-Michel-Telegdi equations) is obtained by means of the plotted algorithm for the case of the magnetic gyroscope motion in the constant and homogenous electromagnetic field
Counting equations in algebraic attacks on block ciphers
DEFF Research Database (Denmark)
Knudsen, Lars Ramkilde; Miolane, Charlotte Vikkelsø
2010-01-01
This paper is about counting linearly independent equations for so-called algebraic attacks on block ciphers. The basic idea behind many of these approaches, e.g., XL, is to generate a large set of equations from an initial set of equations by multiplication of existing equations by the variables...
Affine Vertex Operator Algebras and Modular Linear Differential Equations
Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi
2016-05-01
In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.
Affine Vertex Operator Algebras and Modular Linear Differential Equations
Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi
2016-04-01
In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.
Algebraic methods for the solution of some linear matrix equations
Djaferis, T. E.; Mitter, S. K.
1979-01-01
The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
Institute of Scientific and Technical Information of China (English)
WANG Shundin; ZHANG Hua
2008-01-01
Using functional derivative technique In quantum field theory,the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations.The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by Introducing the time translation operator.The functional partial differential evolution equations were solved by algebraic dynam-ics.The algebraic dynamics solutions are analytical In Taylor series In terms of both initial functions and time.Based on the exact analytical solutions,a new nu-merical algorithm-algebraic dynamics algorithm was proposed for partial differ-ential evolution equations.The difficulty of and the way out for the algorithm were discussed.The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Institute of Scientific and Technical Information of China (English)
2008-01-01
Using functional derivative technique in quantum field theory, the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations. The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynam-ics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new nu-merical algorithm—algebraic dynamics algorithm was proposed for partial differ-ential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Linear algebra a first course with applications to differential equations
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.
Algebraic properties of the Dirac equation in three dimensions
International Nuclear Information System (INIS)
It is argued that, in three dimensions, spinors should have four components as a consequence of the algebraic structure realised from the Clifford algebra related to the Dirac equations. As an example, it is shown then that no induced mass appears in vacuum polarisation at 1-loop in 3 D quantum electrodynamics. (author)
Clifford algebras and the classical dynamical Yang-Baxter equation
Alekseev, Anton; Meinrenken, E.
2003-01-01
We describe a relationship of the classical dynamical Yang-Baxter equation with the following elementary problem for Clifford algebras: Given a vector space $V$ with quadratic form $Q_V$, how is the exponential of an element in $\\wedge^2(V)$ under exterior algebra multiplication related to its exponential under Clifford multiplication?
Hidden $sl_2$-algebra of finite-difference equations
Smirnov, Yuri; Turbiner, Alexander
1995-01-01
The connection between polynomial solutions of finite-difference equations and finite-dimensional representations of the $sl_2$-algebra is established (the talk given at the Wigner Symposium, Guadalajara, Mexico, August 1995, to be published in the Proceedings)
Linear boundary value problems for differential algebraic equations
Balla, Katalin; März, Roswitha
2003-01-01
By the use of the corresponding shift matrix, the paper gives a criterion for the unique solvability of linear boundary value problems posed for linear differential algebraic equations up to index 2 with well-matched leading coefficients. The solution is constructed by a proper Green function. Another characterization of the solutions is based upon the description of arbitrary affine linear subspaces of solutions to linear differential algebraic equations in terms of solutions to the adjoint ...
Difference algebraic relations among solutions of linear differential equations
Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael
2013-01-01
We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrab...
Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra
International Nuclear Information System (INIS)
In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs
Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations
International Nuclear Information System (INIS)
The known (2+1)-dimensional breaking soliton equation, the coupled KP equation with three potentials and a new (3+1)-dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the (1+1)-dimensional AKNS equations. The Abel-Jacobi coordinates are introduced to straighten out the associated flows, from which algebraic-geometrical solutions of the (2+1)-dimensional breaking soliton equation, the coupled KP equation and the (3+1)-dimensional evolution equation are explicitly given in terms of the Riemann theta functions
Institute of Scientific and Technical Information of China (English)
WANG; Shunjin; ZHANG; Hua
2006-01-01
The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system.The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics,and a new algorithm-algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method.In the new algorithm,the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator.The exact analytical piece-like solution of the ordinary differential equations is expressd in terms of Taylor series with a local convergent radius,and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.
Stability of Linear Equations--Algebraic Approach
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…
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1979-01-01
Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation 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 paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.
On algebraic relations between solutions of a generic Painleve equation
Nagloo, Ronnie
2011-01-01
We prove that if y" = f(y,y',t,\\alpha, \\beta,..) is a generic Painleve equation (i.e. an equation in one of the families PI-PVI but with the complex parameters \\alpha, \\beta,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school on "irreducibility" of the Painleve equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.
Linear spaces for index 2 differential-algebraic equations
Petry, Thomas
1995-01-01
In this paper we consider solution spaces of linear index-2-tractable differential algebraic equations. Relations between the solutions of the adjoint equations and the corresponding solution spaces are derived and, thus, a simple method for computing consistent initial values is provided.
Zeta functional equation on Jordan algebras of type II
Kayoya, J. B.
2005-02-01
Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation by using methods based on the algebraic properties of regular graded algebras which are in one-to-one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in this paper is a direct application of specific properties of Jordan algebras of type II.
Algebraic Integrability of Lotka-Volterra equations in three dimensions
Constandinides, Kyriacos
2009-01-01
We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions for the algebraic integrability of the corresponding systems. We also show that the conditions are sufficient.
Algebraic Approach to the Computation of the Defining Polynomial of the Algebraic Riccati Equation
Kitamoto, Takuya
The algebraic Riccati equation, which we denote by ’ARE’ in the rest of the paper, is one of the most important equations of the post modern control theory. It plays important role for solving H 2 and H ∞ optimal control problems.
Falcon, Raymond
2009-01-01
Teachers use action research in order to improve their teaching and student learning. This action research will analyze students' algebraic reasoning in finding values of variables in systems of equations pictorially and algebraically. This research will look at students solving linear systems of equations without knowing the algebraic algorithms.…
Introduction to linear algebra and differential equations
Dettman, John W
1986-01-01
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
A Novel Partial Differential Algebraic Equation (PDAE) Solver
DEFF Research Database (Denmark)
Lim, Young-il; Chang, Sin-Chung; Jørgensen, Sten Bay
2004-01-01
accuracy and stability. The space-time CE/SE method is successfully implemented to solve PDAE systems through combining an iteration procedure for nonlinear algebraic equations. For illustration, chromatographic adsorption problems including convection, diffusion and reaction terms with a linear or......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...
On systems of algebraic equations with parametric exponents
Yokoyama, Kazuhiro
2004-01-01
We deal with systems of algebraic equations with parametric exponents. As the first step for solving such systems, we introduce a simple formulation and basic notions in ideal theory. Then we give a concrete method for most simple cases, univariate case and 0-dimensional case.
Matrix solutions of wave equations and Clifford algebras
International Nuclear Information System (INIS)
We are extending the formation of matrix solutions un for linear and nonlinear wave equations by construction of unitary anti-Hermitian-anti-commuting matrices up to the eighth order. We use Clifford algebras C(0,n) with periodicity in modulo 8 to construct coupled matrix solutions. We also propose to use the matrix solutions for describing the intrinsic rotations of particles. (author)
The equationally-defined commutator a study in equational logic and algebra
Czelakowski, Janusz
2015-01-01
This monograph introduces and explores the notions of a commutator equation and the equationally-defined commutator from the perspective of abstract algebraic logic. An account of the commutator operation associated with equational deductive systems is presented, with an emphasis placed on logical aspects of the commutator for equational systems determined by quasivarieties of algebras. The author discusses the general properties of the equationally-defined commutator, various centralization relations for relative congruences, the additivity and correspondence properties of the equationally-defined commutator, and its behavior in finitely generated quasivarieties. Presenting new and original research not yet considered in the mathematical literature, The Equationally-Defined Commutator will be of interest to professional algebraists and logicians, as well as graduate students and other researchers interested in problems of modern algebraic logic.
Algebraic and transcendental solutions of some exponential equations
Sondow, Jonathan
2011-01-01
We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$, $x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential functions are given. The main tools used are classical theorems of Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential Diophantine equations.
Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras
Cherednik, Ivan
1998-01-01
This paper is the course of lectures delivered by the first author in Kyoto in 1996-97 and recorded by the others. We tried to follow closely the notes of the lectures not yielding to the temptation of giving more examples and names. The focus is on the relations of the Knizhnik-Zamolodchikov equations and Kac-Moody algebras to a new theory of spherical and hypergeometric functions based on affine and double affine Hecke algebras. Here mathematics and physics are closer than...
Dirac's equation in half of his algebra
International Nuclear Information System (INIS)
Relativistic quantum physics has, since 1928, been introduced in textbooks by using 4x4 complex number matrices. Matrices are actually not needed at all. If used, then only the block diagonal 4x4 matrices are really needed for both the momentum operator and for the wavefunction. This new approach should 'simplify' the mathematics (no matrices), so that this wave-equation material can finally be pushed down into the Junior/Senior-level Modern Physics course. (author)
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
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
On Detectability of Switched Linear Differential-Algebraic Equations
Tanwani, Aneel; Trenn, Stephan
2015-01-01
This paper addresses the notion of detectability for continuous-time switched systems comprising linear differential-algebraic equations (DAEs). It relates to studying asymptotic stability of the set of state trajectories corresponding to zero input and zero output. Due to the nature of solutions of switched DAEs, the problem reduces to stability of the trajectories emanating from a non-vanishing unobservable subspace, for which we first derive a geometric expression. The stability of state t...
SPR0 SUBSTITUTIONS AND FAMILIES OF ALGEBRAIC RICCATI EQUATIONS
Czech Academy of Sciences Publication Activity Database
Fernández-Anaya, G.; Martínez-García, J.C.; Kučera, Vladimír; Aguilar-George, D.
Roč. 42, č. 5 ( 2006 ), s. 605-616. ISSN 0023-5954 R&D Projects: GA MŠk(CZ) 1M0567 Institutional research plan: CEZ:AV0Z10750506 Keywords : linear time invariant systems * positive real substitutions * properties preservation * algebraic Riccati equations * H-infinity-norm bounded systems Subject RIV: BC - Control Systems Theory Impact factor: 0.293, year: 2006
An Algorithm to Solve algebraic Riccati Equations with Polynomials
Czech Academy of Sciences Publication Activity Database
Augusta, Petr
Szczecin: West Pomeranian University of Technology, 2012, s. 409-414. ISBN 978-1-4673-2123-5. [The 17th International Conference on Methods and Models in Automation and Robotics . Międzyzdroje (PL), 27.08.2012-30.08.2012] R&D Projects: GA ČR GPP103/12/P494 Institutional support: RVO:67985556 Keywords : algebraic Riccati equations * spatially distributed systems * Fourier transforms Subject RIV: BC - Control Systems Theory
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.
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.
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Truncatable bootstrap equations in algebraic form and critical surface exponents
Gliozzi, Ferdinando
2016-01-01
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...
Validated Simulation of Differential Algebraic Equations with Runge-Kutta Methods
Alexandre Dit Sandretto, Julien; Chapoutot, Alexandre
2015-01-01
Differential Algebraic Equations (DAEs) are a general and implicit form of differential equations. This mathematical object is often used to represent physical systems such as dynamics of solid or chemical interactions. These equations are different from Ordinary Differential Equations (ODEs) in sense that some of the dependent variables occur without their derivatives. These variables are called ``algebraic variables'', which means free of derivatives and not with respect to abstract algebra...
First order linear ordinary differential equations in associative algebras
Directory of Open Access Journals (Sweden)
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.
International Nuclear Information System (INIS)
We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlevé equations. These cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation-period property. We will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation. (paper)
Solutions of Riccati-Abel equation in terms of characteristics of general complex algebra
International Nuclear Information System (INIS)
The Riccati-Abel differential equation defined as an equation between the first order derivative and the cubic polynomial is explored. In the case of constant coefficients this equation is reduced into an algebraic equation. A method of derivation of a summation formula for solutions of the Riccati-Abel equation is elaborated. The solutions of the Riccati-Abel equation are expressed in terms of the characteristic functions of general complex algebra of the third order
SUPPORTING STUDENTS’ UNDERSTANDING OF LINEAR EQUATIONS WITH ONE VARIABLE USING ALGEBRA TILES
Sari Saraswati; Ratu Ilma Indra Putri; 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 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 ...
Solution of systems of linear algebraic equations by the method of summation of divergent series
International Nuclear Information System (INIS)
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
Space-time algebra for the generalization of gravitational field equations
Indian Academy of Sciences (India)
Süleyman Demir
2013-05-01
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 gravitoelectromagnetic Klein–Gordon equation has been obtained. Finally, the analogy in formulation between massive gravitational theory and electromagnetism has been discussed.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
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 ...
Projective BGG equations, algebraic sets, and compactifications of Einstein geometries
Cap, A; Hammerl, M
2010-01-01
For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivialisations arising from the special frames, normal solutions of classes of natural linear PDE (so-called first BGG equations) are shown to be necessarily polynomial in the generalised homogeneous coordinates; the polynomial system is the pull back of a polynomial system that solves the corresponding problem on the model. Thus questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects a...
Aghaei, S.; Chenaghlou, A.
2014-02-01
The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.
Algebraization of difference eigenvalue equations related to $U_q(sl_2)$
Wiegmann, P. B.; Zabrodin, A. V.
1995-01-01
A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polinomials (discrete polinomials). Such difference operators can be constructed by means of $U_q(sl_2)$, the quantum deformation of the $sl_2$ algebra. The roots of polinomials determine the spectrum and obey the Bethe Ansatz equations. A particular case of difference equations for $q$-hypergeometric and Askey-Wilso...
Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles
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…
A Direct Algebraic Method in Finding Particular Solutions to Some Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
LIUChun-Ping; CHENJian-Kang; CAIFan
2004-01-01
Firstly, a direct algebraic method and a routine way in finding traveling wave solutions to nonlinear evolution equations are explained. And then some new exact solutions for some evolution equations are obtained by using the method.
Institute of Scientific and Technical Information of China (English)
BAO Yuan-Peng; RUAN Hang-Yu; XIE Wen-Fang; LI Zhi-Fang
2008-01-01
Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for (2+1)-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton: 1) the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2) the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.
Non-associative algebras, Yang-Baxter equations, and quantum computers
International Nuclear Information System (INIS)
Non-associative algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras. The (quantum) Yang-Baxter equation and related structures are interesting topics, because they have applications in many areas of physics, mathematics, and computer science. Several new interpretations and results are presented in this paper
International Nuclear Information System (INIS)
Nonlinear equations associated through a zero curvature type representation with Lie algebras S0Diff T2 and of infinitesimal diffeomorphisms of (S1)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
Directory of Open Access Journals (Sweden)
Alvaro Alvarez-Parrilla
2012-01-01
Full Text Available A new technique for solving a certain class of systems of autonomous ordinary differential equations over n is introduced ( being the real or complex field. The technique is based on two observations: (1, if n has the structure of certain normed, associative, commutative, and with a unit, algebras over , then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2 a technique, previously introduced for solving differential equations over ℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.
Bracken, Paul
2015-01-01
Prolongation algebras which are determined by applying a version of the Wahlquist-Estabrook method to three different nonlinear partial differential equations can be employed to obtain not only Lax pairs but B\\"acklund transformations as well. By solving Maurer-Cartan equations for the related group specified by the prolongation algebra, a set of differential forms is obtained which can lead directly to these kinds of results. Although specific equations are studied, the approach should be ap...
Index-aware model order reduction methods applications to differential-algebraic equations
Banagaaya, N; Schilders, W H A
2016-01-01
The main aim of this book is to discuss model order reduction (MOR) methods for differential-algebraic equations (DAEs) with linear coefficients that make use of splitting techniques before applying model order reduction. The splitting produces a system of ordinary differential equations (ODE) and a system of algebraic equations, which are then reduced separately. For the reduction of the ODE system, conventional MOR methods can be used, whereas for the reduction of the algebraic systems new methods are discussed. The discussion focuses on the index-aware model order reduction method (IMOR) and its variations, methods for which the so-called index of the original model is automatically preserved after reduction.
Equations of motion for a time-dependent open system: An algebraic approach
International Nuclear Information System (INIS)
Highlights: ► Based on the concept of quantum densities an algebraic equation is introduced. ► Heisenberg equation and hypervirial theorem are derived based on the algebraic equation. ► Quantum Navier–Stokes equation is derived based on the algebraic equation. ► Differential form of the force law and local virial theorem are extracted. ► Central equations in QTAIM are extracted without referring to quantum stationary action. - Abstract: An algebraic approach based on the concept of local densities is introduced in order to provide an alternative derivation of several equations central to the hydrodynamical formulation of quantum mechanics. The origin of this work lays in an algebraic equation which is built based on the concept of quantum densities. This enables us to derive the regional and local forms of several significant quantum laws and equations, namely Heisenberg equation of motion, hypervirial theory and quantum Navier–Stokes equation. In particular, atomic force law and local virial theorem for a time-dependent open system are extracted without referring to rigorous Schwinger’s principle of stationary action
Equations of motion for a time-dependent open system: An algebraic approach
Energy Technology Data Exchange (ETDEWEB)
Nasertayoob, Payam [Department of Chemistry, Amirkabir University of Technology (Polytechnic), Tehran (Iran, Islamic Republic of); Department of Mathematics, Amirkabir University of Technology (Polytechnic), Tehran (Iran, Islamic Republic of); Sabbaghan, Masoud, E-mail: sabbagh@khayam.ut.ac.ir [Department of Mathematics, I.A.U. Lahijan Branch, Lahijan (Iran, Islamic Republic of)
2013-02-01
Highlights: ► Based on the concept of quantum densities an algebraic equation is introduced. ► Heisenberg equation and hypervirial theorem are derived based on the algebraic equation. ► Quantum Navier–Stokes equation is derived based on the algebraic equation. ► Differential form of the force law and local virial theorem are extracted. ► Central equations in QTAIM are extracted without referring to quantum stationary action. - Abstract: An algebraic approach based on the concept of local densities is introduced in order to provide an alternative derivation of several equations central to the hydrodynamical formulation of quantum mechanics. The origin of this work lays in an algebraic equation which is built based on the concept of quantum densities. This enables us to derive the regional and local forms of several significant quantum laws and equations, namely Heisenberg equation of motion, hypervirial theory and quantum Navier–Stokes equation. In particular, atomic force law and local virial theorem for a time-dependent open system are extracted without referring to rigorous Schwinger’s principle of stationary action.
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.
Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations
International Nuclear Information System (INIS)
The ℓ-conformal Galilei algebra, denoted by gl(d), is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by gl(d) with a central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of gl(d) and vector field representations for d = 1, 2. (paper)
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.
Computer algebra tools for higher symmetry analysis of nonlinear evolution equations
International Nuclear Information System (INIS)
Paper presents a computer-aided approach and a software package for symbolic algebraic computation to solve the problem of verifying the existence of the canonical Lie-Backlund symmetries for multicomponent quasilinear evolution equations with polynomial-nonlinearity and computing a given order symmetry. In the presence of arbitrary numerical parameters the problem is reduced to investigation and solving of nonlinear algebraic equations in those parameters. 16 refs.; 2 tabs
International Nuclear Information System (INIS)
The reflection equation algebra of Sklyanin is extended to the supersymmetric case. A graded reflection equation algebra is proposed and the corresponding graded (supersymmetric) boundary quantum inverse scattering method (QISM) is formulated. As an application, integrable open-boundary conditions for the doped spin-1 chain of the supersymmetric t-J model are studied in the framework of the boundary QISM. Diagonal boundary K-matrices are found and four classes of integrable boundary terms are determined. (author)
Representations of ℓ-conformai Galilei algebra and hierarchy of invariant equation
International Nuclear Information System (INIS)
The ℓ-conformai Galilei algebra, denoted by gℓ(d), is a particular non-semisimple Lie algebra specified by a positive integer d and a spin value ℓ. The algebra gℓ(d) admits central extensions. We study lowest weight representations, in particular Verma modules, of gℓ(d) with the central extensions for d = 1,2. We give a classification of irreducible modules over d = 1 algebras and a condition of the Verma modules over d = 2 algebras being reducible. As an application of the representation theory, hierarchies of differential equations are derived. The Lie group generated by gℓ(d) with the central extension is a kinematical symmetry of the differential equations.
On solvability of some quadratic functional-integral equation in Banach algebra
International Nuclear Information System (INIS)
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)
Algebraic equations an introduction to the theories of Lagrange and Galois
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
Directory of Open Access Journals (Sweden)
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.
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
Directory of Open Access Journals (Sweden)
I.Yu. Krivsky
2010-01-01
Full Text Available We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration is proved. In order to provide the corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.
Hydra: Automatic algorithm exploration from linear algebra equations
Duchâteau, Alexandre; Padua, David; Barthou, Denis
2013-01-01
International audience Hydra accepts an equation written in terms of operations on matrices and automatically produces highly efficient code to solve these equations. Processing of the equation starts by tiling the matrices. This transforms the equation into either a single new equation containing terms involving tiles or into multiple equations some of which can be solved in parallel with each other. Hydra continues transforming the equations using tiling and seeking terms that Hydra know...
The Hamiltonian structure of soliton equations and deformed scr(W)-algebras
International Nuclear Information System (INIS)
The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties and their relation to scr(W)-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel close-quote d endash Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand endash Dickey and the constrained KP hierarchies. Therefore, our results provide a unified description of the relation between the Hamiltonian structure of soliton equations and scr(W)-algebras, and it comprises almost all the results formerly obtained by other authors. The main result of this paper is an explicit general equation showing that the second Poisson bracket algebra is a deformation of the Dirac bracket algebra corresponding to the scr(W)-algebras obtained through Hamiltonian reduction. copyright 1997 Academic Press, Inc
The Algebra Solution to Mathematics Reform: Completing the Equation
Spielhagen, Frances R.
2011-01-01
How can we increase mathematics achievement among all students? This book provides a straightforward explanation of how changing mathematics tracking policies to provide algebra instruction to all students by at least eighth grade can bring about changes in both student achievement and teacher performance. Spielhagen chronicles the success of a…
New method for solving algebraic equations without the use of imaginary numbers
International Nuclear Information System (INIS)
A new method for solving algebraic equations without the use of imaginary numbers has been proposed. Complex-conjugate roots can be expressed in terms of their absolute value and real part. This method simplifies the process of solving quadratic equations and provides simple approximate but sufficiently accurate formulas for the roots of cubic equations. This method can also be useful for solving equations of higher orders
International Nuclear Information System (INIS)
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.
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations
Directory of Open Access Journals (Sweden)
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.
Stability of a Bi-Additive Functional Equation in Banach Modules Over a C⋆-Algebra
Directory of Open Access Journals (Sweden)
Won-Gil Park
2012-01-01
Full Text Available We solve the bi-additive functional equation f(x+y,z−w+f(x−y,z+w=2f(x,z−2f(y,w and prove that every bi-additive Borel function is bilinear. And we investigate the stability of a bi-additive functional equation in Banach modules over a unital C⋆-algebra.
Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1) Dimensions
Institute of Scientific and Technical Information of China (English)
ZHAO Xue-Qing; L(U)Jing-Fa
2004-01-01
The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.
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.
On the algebraic Bethe ansatz for the XXX spin chain: creation operators 'beyond the equator'
International Nuclear Information System (INIS)
Considering the XXX spin-1/2 chain in the framework of the algebraic Bethe ansatz, we make the following short comment: the product of the creation operators corresponding to the recently found solution of the Bethe equations 'on the wrong side of the equator' is just zero (not only its action on the pseudovacuum). (author). Letter-to-the-editor
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.
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
Energy Technology Data Exchange (ETDEWEB)
Man, Yiu-Kwong, E-mail: ykman@ied.edu.h [Department of Mathematics and Information Technology, Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories (Hong Kong)
2010-10-15
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)
International Nuclear Information System (INIS)
We give a q-deformation S-perpendicularq of the centrally extended Schroedinger algebra. We construct the lowest weight representations of S-perpendicularq, starting from the Verma modules over S-perpendicularq, finding their singular vectors and factoring the Verma submodules built on the singular vectors. We also give a vector-field realization of S-perpendicularq 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
Lie-algebraic approach for the Fokker-Planck equation with a nonlinear drift force
International Nuclear Information System (INIS)
In this paper we have investigated the algebraic structure of the Fokker-Planck equation with a nonlinear drift force. Using the Lie-algebraic approach we have obtained an exact form of the time evolution operator which, in turn, enables us to derive the propagator readily. Since the propagator is for a very general Fokker-Planck equation with a nonlinear drift force, results for any special case can be easily deduced from it. (Copyright (c) 1999 Elsevier Science B.V., Amsterdam. All rights reserved.)
Gauss对解代数方程的贡献%Gauss' contribution to solving algebraic equation
Institute of Scientific and Technical Information of China (English)
王宵瑜
2011-01-01
目的 分析Gauss(1777-1855)解方程的主要思想和具体步骤,研究其对解代数方程的贡献.方法 从原始文献出发,寻求求解代数方程的内涵.结果 Gauss证明了分圆方程根式可解；通过对Lagrange(1736-1813)与Gauss关于解代数方程方法的比较,得出Gauss解方程的方法是对Lagrange解方程方法的应用；最后Gauss在解方程的过程中得出一个根式扩张塔.结论 Gauss不仅证明了分圆方程是根式可解的,且在解方程的过程中得到根式扩张塔,改变了方程可解的定义.%Aim To investigate Gauss'(1777-1855) main idea and procedures of solving the cyclotomic equations, and then establish Gauss' contribution to solving algebraic equation. Methods Search for the inner meaning of solving algebraic equations from the original literature. Results Gauss proved the algebraic solvability of cyclotomic equation; Lagrange's and Gauss' methods of solving equation being compared, it shows that Gauss' method is the application of Lagrange's method. Gauss obtained a tower of radical field extension in solving algebraic e-quation. Conclusion Gauss proved the algebraic solvability of cyclotomic equations and got a tower of radical field extension, which changed the definition of radical solvability of the equation.
Differential-algebraic equations and applications in circuit theory
Reich, Sebastian
2010-01-01
Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this p...
Quadratic algebras and position-dependent mass Schr\\"odinger equations
Quesne, C
2007-01-01
During recent years, exact solutions of position-dependent mass Schr\\"odinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of $d$-dimensional radial harmonic oscillators with $d\\ge2$ (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice. This provides us with a counterpart of the well-known su(1,1) Lie algebraic approach to the constant-mass oscillators.
Flipping an Algebra Classroom: Analyzing, Modeling, and Solving Systems of Linear Equations
Kirvan, Rebecca; Rakes, Christopher R.; Zamora, Regie
2015-01-01
The present study investigated whether flipping an algebra classroom led to a stronger focus on conceptual understanding and improved learning of systems of linear equations for 54 seventh- and eighth-grade students using teacher journal data and district-mandated unit exam items. Multivariate analysis of covariance was used to compare scores on…
Numerical integration of differential-algebraic equations with harmless critical points
Dokchan, Rakporn
2011-01-01
Algebro-Differentialgleichungen (engl. differential-algebraic equations - DAEs) sind implizite singuläre gewöhnliche Differentialgleichungen, die restringierte dynamische Prozesse beschreiben. Sie unterscheiden sich von expliziten gewöhnlichen Differentialgleichungen dahingehend, dass Anfangswerte nicht beliebig vorgegeben werden können. Weiterhin sind in einer DAE neben Integrations- auch Differentiationsaufgaben involviert. Der Differentiationsindex gibt an, wieviele Differentiationen zur L...
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...
Alternative Representations for Algebraic Problem Solving: When Are Graphs Better than Equations?
Mielicki, Marta K.; Wiley, Jennifer
2016-01-01
Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation…
Energy Technology Data Exchange (ETDEWEB)
Vengerskii, P.S.; Kardash, A.I.; Sen`o, P.S. [L`vov State Univ. (Russian Federation)
1994-06-05
We consider the problem of finding real solutions of a system of nonlinear algebraic equations using interval analysis. Several versions of Newton and Runge interval iteration methods are presented. The computational aspects of their application are explained. 6 refs., 2 tabs.
New solutions of reflection equation derived from type B BMW algebras
Häring-Oldenburg, Reinhard
1996-09-01
We use B-type knot theory to find new solutions of Sklyanin's reflection equation in a systematic way. This generalizes the well known Baxterization of Birman - Wenzl algebras and should describe integrable systems which are restricted to a half plane.
A new algorithm for differential-algebraic equations based on HIDM
International Nuclear Information System (INIS)
A new algorithm is proposed to solve differential-algebraic equations. The algorithm is an extension of the algorithm of general purpose HIDM (higher order implicit difference method). A computer program named HDMTDV and based on the new algorithm is constructed and its high performance is proved numerically through several numerical computations, including index-2 problem of differential-algebraic equations and connected rigid pendulum equations. The new algorithm is also secular error free when applied to dissipationless dynamical systems. This nature is demonstrated numerically by computation of the Kepler motion. The new code can solve the initial value problem O = L { φ(x), [dφ(x)]/dx, [d2φ(x)]/dx2, x }, where L and φ are vectors of length N. The values of first or second derivatives of φ(x) are not always necessary in the equations. (author)
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
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 A1 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 A1, 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. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
I.Yu. Krivsky; Simulik, V. M.
2010-01-01
We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance) the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration ...
International Nuclear Information System (INIS)
Investigation into the method of non-commutative integration of linear differential equations in particular derivatives is continued in the work. Solutions of the Klein-Gordon equation in the case when lots of non-commutating operators of the Klein-Gordon equation symmetry form square-law algebra comprising a single second order operator and the first order operators forming the Lie algebra, are obtained using the given method. 10 refs
Multiple Representations for Systems of Linear Equations Via the Computer Algebra System Maple
Directory of Open Access Journals (Sweden)
Dann G. Mallet
2007-02-01
Full Text Available A number of different representational methods exist for presenting the theory of linear equations and associated solution spaces. Discussed in this paper are the findings of a case study where first year undergraduate students were exposed to a new (to the department method of teaching linear systems which used visual, algebraic and data-based representations constructed using the computer algebra system Maple. Positive and negative impacts on the students are discussed as they apply to representational translation and perceived learning.
Institute of Scientific and Technical Information of China (English)
Abbas NAJATI
2009-01-01
In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation f(x+y/2+z)-g(x-y/2+z)=h(y).This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soe., 72, 297-300 (1978).
Statistical and algebraic analysis of a family of random Boolean equations
International Nuclear Information System (INIS)
The statistical and algebraic properties of a family of random Boolean equations are studied in this paper. Based on studying the mechanism of influence propagation from some variable fixed to 1, the giant strongly connected component and magnetization of solutions are investigated, which exhibit the splitting phenomenon of solution space and a ferromagnetic transition. Furthermore, by analyzing the semi-group property of the solution space and the influence of propagation from some variable fixed to 0, the scale of generating elements is calculated, which undergoes linear, polynomial and exponential phases. Compared with the analysis by statistical mechanics, it is suggested that these different phases of algebraic complexity correspond to the structural complexity of the solution space as replica symmetry, one-step replica symmetry breaking and further-step replica symmetry breaking phases. It is supposed that the structural complexity of solution space can be interpreted from the viewpoint of algebra
International Nuclear Information System (INIS)
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
Directory of Open Access Journals (Sweden)
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.
Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation
International Nuclear Information System (INIS)
So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie-Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization. -- Highlights: → Generalized Hamiltonian fluid mechanics for the two-dimensional vorticity equation. → Enstrophy as Casimir of the Zeitlin (sine-bracket) truncation. → Algebraic construction of Nambu bracket by Bialynicki-Birula and Morrison method. → Infinite-mode limit yields continuum Nambu bracket. → Discussion of generalized Jacobi identity.
International Nuclear Information System (INIS)
It is shown that Heisenberg's commutation rule between the position co-ordinate and the corresponding canonically conjugate momentum may be interpreted by noncommuting geometrical structures. As in the absence of a magnetic field the Euclidean norm of the momentum space directly enters the kinetic energy, the momentum space can be mapped onto the quaternion field U2. Such a mapping preserves the norm of the momentum space. By that, the geometric and algebraic structure of the Pauli equation can be obtained and the relationship between the Pauli and the Dirac equation can be made apparent by noncommuting algebraic structures. In an appendix it will also be shown that the extension of the procedure to vector spaces equipped with Riemannian geometry makes no difficulties and a covariant quantization procedure can be formulated. (author)
On the vibrations of lumped parameter systems governed by differential-algebraic equations
Darbha, S.; Nakshatrala, K. B.; K.R. Rajagopal
2009-01-01
In this paper, we consider the vibratory motions of lumped parameter systems wherein the components of the system cannot be described by constitutive expressions for the force in terms of appropriate kinematical quantities. Such physical systems reduce to a system of differential-algebraic equations, which invariably need to be solved numerically. To illustrate the issues with clarity, we consider a simple system in which the dashpot is assumed to contain a "Bingham" fluid for which one canno...
Projected Runge-Kutta methods for differential algebraic equations of index 3
Schropp, Johannes
2003-01-01
In the present paper we introduce a new class of methods, Projected Runge-Kutta methods, for the solution of index 3 differential algebraic equations (DAEs) in Hessenberg form. The methods admit the integration of index 3 DAEs without any drift effects. This makes them particularly well suited for long term integration. Finally, implemented on the basis of the Radau5 code, the projected Runge-Kutta method admits larger step sizes for a prescribed tolerance than the corresponding classical sc...
Parabolic orbit determination. Comparison of the Olbers method and algebraic equations
Kuznetsov, V. B.
2016-05-01
In this paper, the Olbers method for the preliminary parabolic orbit determination (in the Lagrange-Subbotin modification) and the method based on systems of algebraic equations for two or three variables proposed by the author are compared. The maximum number of possible solutions is estimated. The problem of selection of the true solution from the set of solutions obtained both using additional equations and by the problem reduction to finding the objective function minimum is considered. The results of orbit determination of the comets 153P/Ikeya-Zhang and 2007 N3 Lulin are cited as examples.
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
International Nuclear Information System (INIS)
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
Energy Technology Data Exchange (ETDEWEB)
Grahovski, Georgi G., E-mail: G.Grahovski@leeds.ac.uk [Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT (United Kingdom); Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chausee, Sofia 1784 (Bulgaria); Mikhailov, Alexander V., E-mail: A.V.Mikhailov@leeds.ac.uk [Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT (United Kingdom)
2013-12-17
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
A new linear Dirac-like spin-3/2 wave equation using Clifford algebra
International Nuclear Information System (INIS)
A new linear Dirac-like wave equation for spin-3/2 is derived, employing four of the seven irreducible eight-dimensional matrices obeying the Clifford algebra C7 with the wave function having the needed eight components only. Though this wave equation is not manifestly covariant and the wave function employed is not locally covariant, it is relativistically invariant and by its very derivation is connected to the Weaver, Hammer and Good (Phys. Rev.; 135: B241 (1964)) formalism for spin-3/2 by a chain of transformations which can be arbitrarily chosen to be either unitary or non-unitary. (author)
Sheinman, O. K.
2015-12-01
Based on ℤ-gradings of semisimple Lie algebras and invariant polynomials on them, we construct hierarchies of Lax equations with a spectral parameter on a Riemann surface and prove the commutativity of the corresponding flows.
International Nuclear Information System (INIS)
The exact solutions of the Cauchy problem for two equations which are slight generalization of the so-called linear vectorial Fokker-Planck equation are found using the disentangling techniques of Feynman and algebraic (operational) methods. This approach may be considered as a generalization of the Suzuki's method for solving the one-dimensional linear Pokker-Planck equation
The Lie algebra of invariant group of the KdV, MKdV, or Burgers equation
International Nuclear Information System (INIS)
In this paper it is proved that all generators of the invariant group of the KdV, MKdV or Burgers equation form a commutative Lie algebra, from which it follows that for any symmetry of these three equations, the evolution equation possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations). The authors conjecture that this result holds for a yet wider class of equations. (Auth.)
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.
Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method
International Nuclear Information System (INIS)
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)
Bolaños, Marduk; Barberis-Blostein, Pablo
2015-11-01
We consider the Lindblad equation for a collection of multilevel systems coupled to independent environments. The equation is symmetric under the exchange of the labels associated with each system and thus the open-system dynamics takes place in the permutation-symmetric subspace of the operator space. The dimension of this space grows polynomially with the number of systems. We construct a basis of this space and a set of superoperators whose action on this basis is easily specified. For a given number of levels, M, these superoperators are written in terms of a bosonic realization of the generators of the Lie algebra {sl}({M}2). In some cases, these results enable finding an analytic solution of the master equation using known Lie-algebraic methods. To demonstrate this, we obtain an analytic expression for the state operator of a collection of three-level atoms coupled to independent radiation baths. When analytic solutions are difficult to find, the basis and the superoperators can be used to considerably reduce the computational resources required for simulations.
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.
A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra
Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei
2013-01-01
The analysis of the $\\mathbb{Z}_2^{3}$ Laplace-Dunkl equation on the $2$-sphere is cast in the framework of the Racah problem for the Hopf algebra $sl_{-1}(2)$. The related Dunkl-Laplace operator is shown to correspond to a quadratic expression in the total Casimir operator of the tensor product of three irreducible $sl_{-1}(2)$-modules. The operators commuting with the Dunkl Laplacian are seen to coincide with the intermediate Casimir operators and to realize a central extension of the Banna...
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 ostatní: 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 * algebraic error * spatial distribution of the error Subject RIV: BA - General Mathematics Impact factor: 0.939, year: 2014
Symmetries of the Schr\\"odinger Equation and Algebra/Superalgebra Duality
Toppan, Francesco
2014-01-01
Some key features of the symmetries of the Schr\\"odinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first 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.
Symmetries of the Schrödinger equation and algebra/superalgebra duality
Toppan, Francesco
2015-04-01
Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first 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.
Study of some properties of partial differential equations by Lie algebra method
International Nuclear Information System (INIS)
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
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.
Modeling boyciana-fish-human interaction with partial differential algebraic equations.
Jiang, Yushan; Zhang, Qingling; Wang, Haiyan
2016-07-01
Under the influence of human population distribution, the boyciana-fish ecological system is considered. First, the system can be described as a nonlinear partial differential algebraic equations system (PDAEs) with Neumann boundary conditions and ratio-dependent functional response. Second, we examine the system's persistence properties: the loacl stabilities of positive steady states, the absorbtion region and the global stability. And the proposed approach is illustrated by numerical simulation. Finally, by using the realistic data collected in the past fourteen years, the PDAEs parameter optimization model is built to predict the boyciana population. PMID:27155570
Graph theory, irreducibility, and structural analysis of differential-algebraic equation systems
Pryce, John D.; Nedialkov, Nedialko S.; Tan, Guangning
2014-01-01
The $\\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the DAE that can be exploited to speed up numerical solution. The paper compares this fine BTF with the usually coarser BTF derived from the sparsity pattern of the \\sigmx. It defines a Fine-Block Graph with weighted edges, which gives insight into the relatio...
Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model
International Nuclear Information System (INIS)
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)
Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory
Hall, Cameron L.
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.
Directory of Open Access Journals (Sweden)
M. Heydari
2013-05-01
Full Text Available A new and effective direct method to determine the numerical solution of linear and nonlinear differential-algebraic equations (DAEs is proposed. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a differentialalgebraic equation can be transformed to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique
Lie Algebraic Structures and Integrability of Long-Short Wave Equation in （2＋1）Dimensions
Institute of Scientific and Technical Information of China (English)
ZHAOXue-Qing; LüJing-Fa
2004-01-01
The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.
International Nuclear Information System (INIS)
A 64-dimensional extended real Clifford-Dirac algebra is introduced. On its basis, new pure matrix symmetries of the Dirac equation in the Foldy-Wouthuysen representation was found. Finally, spin 1 Poincare symmetries both for the Foldy-Wouthuysen and standard Dirac equations with nonzero mass are found.
International Nuclear Information System (INIS)
A generalized variable-coefficient algebraic method is applied to construct several new families of exact solutions of physical interest for (3+1)-dimensional Kadomtsev-Petviashvilli (KP) equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.
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.
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. PMID:27330880
International Nuclear Information System (INIS)
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
New solutions from algebraic equations for the Skyrme model coupled to a scalar meson
Energy Technology Data Exchange (ETDEWEB)
Braghin, Fabio L. [Universidade Federal do Rio Grande do Norte (IIF/UFRN), Natal, RN (Brazil). Inst. Internacional de Fisica
2010-07-01
Full text: In this work a modified Skyrme model is considered such as to incorporate the interaction of the hedgehog with a scalar field, based on a previous work. The Skyrme model is a model of the nucleon in which the baryon emerges as a topological soliton and its coupling to a scalar field can either correspond to the coupling to the lightest scalar isoscalar meson sigma and also to implement the spontaneous breakdown of chiral symmetry in a consistent way. Therefore it can be related to modifications of a dense interacting medium and it becomes suitable for investigating the role of the symmetry breaking and its restoration. A transcendental algebraic equation is found to be enough to extract a new class of profile solutions of the skyrmion in a constant background. The mass of the corresponding topological soliton was found to decrease considerably in the case small masses are associated to the scalar field. (author)
FAST SOLUTION FOR LARGE SCALE LINEAR ALGEBRAIC EQUATIONS IN FINITE ELEMENT ANALYSIS
Institute of Scientific and Technical Information of China (English)
Qi Zhaohui; Liu Yuqi; Hu Ping
2001-01-01
The computational efficiency of numerical solution of linear algebraic equations in finite elements can be improved in tow wqys. One is to decrease the fill-in numbers, which are new non-ze-ro numbers in the matrix of global stiffness generated during the process of elimination.The other is to reduce the computational operation of multiplying a real number by zero.Based on the fact that the order of elimination can determine how many fill-in numbers should be generated, we present a new method for optimization of numbering nodes. This method is quite different from bandwidth optimization. Fill-in numbers can be decreased in a large scale by the use of this method. The bi-factorization method is adoted to avoid multiplying real numbers by zero.For large scale finite element analysis, the method presented in this paper is more efficient than the traditional LDLT method.
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.
Directory of Open Access Journals (Sweden)
Christiane Quesne
2007-05-01
Full Text Available An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrödinger equations.
Haiyan Yuan; Jihong Shen
2014-01-01
This paper studies the asymptotic stability of the two-step Runge-Kutta methods for neutral delay integro differential-algebraic equations with many delays. It proves that A-stable two-step Runge-Kutta methods are asymptotically stable for neutral delay integro differential-algebraic equations with many delays.
International Nuclear Information System (INIS)
In this paper, an extended algebraic method with symbolic computation is applied to construct a series of travelling wave solutions of the one-dimensional generalized BBM equation of any order with positive and negative exponents. As a result, the proposed method gives many explicit exact solutions such as solitary wave solutions, periodic solutions, solitary patterns solutions and compacton solutions
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.
New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy
International Nuclear Information System (INIS)
A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-tilde2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.
Institute of Scientific and Technical Information of China (English)
张春蕊; 郑宝东
2003-01-01
Runge-Kutta方法对微分代数方程是正则的,是指数值解的有限渐进值与方程本身的渐进值是相等的.给出了保证Runge-Kutta方法对微分代数方程是正则的条件,证明了Runge-Kutta方法是正则的充要条件是折叠方法是正则的.%The numerical method for differential-algebraic equation is regular if it has the same set offinite asymptotic values as the underlying differential-algebraic system. The conditions that guaranteeregular properties of Runge-Kutta method for differential-algebraic equation are considered. It isproved that the Runge-Kutta method for differential-algebraic equation is regular if and only if thefold method is regular for differential-algebraic equation.
Directory of Open Access Journals (Sweden)
Adriana NASTASE
2015-12-01
Full Text Available A comparison of the behaviours of the elliptic with those of hyperbolic quadratic algebraic equations (QAEs with free and linear variable coefficients, in vicinity of their critical surfaces is made. The critic values of the elliptic and hyperbolic QAEs with variables coefficients are obtained by can-celling their great determinant. If only the free term of a QAE is variable from -∞ to + ∞ and the QAE are two-dimensional, an elliptic QAE is represented by coaxial ellipses, which decrease in size and collapse in their common centre. A hyperbolic QAE is represented by coaxial hyperbolas, which approach their asymptotes, degenerate in them, jump over them and go away from them. The real solutions of hyperbolic QAEs exist for all the values of free term and for elliptic QAE, if the value of the free term is greater than the critical one, the real solutions of elliptic QAEs do no longer exist. If, additionally, also the free term is variable, critical parabolas occur, if a plane of coefficients is used. The real solutions for elliptic QAE collapse along their critical parabola and do not exist inside of it. The hyperbolic QAE is represented by coaxial hyperbolas which degenerate in their asymptotes and jump over them along their critical parabola.
International Nuclear Information System (INIS)
In this paper, we study the Dirac equation with spin and pseudospin symmetry by the quadratic algebra approach for the 4-dimensional harmonic oscillator. By realization of the quadratic algebras in the deformed oscillator algebra, we obtain the relativistic energy spectrum. Also, by regarding the generalized Kustaanheimo–Stiefel transformation, we obtain the relativistic energy spectrum for the charge-dyon system with the U(1) monopole. (author)
Aghaei, S.; Chenaghlou, A.
2015-01-01
In this paper, we study the Dirac equation with spin and pseudospin symmetry by the quadratic algebra approach for the 4-dimensional harmonic oscillator. By realization of the quadratic algebras in the deformed oscillator algebra, we obtain the relativistic energy spectrum. Also, by regarding the generalized Kustaanheimo-Stiefel transformation, we obtain the relativistic energy spectrum for the charge-dyon system with the U(1) monopole.
Hewitt, Dave
2012-01-01
This study looks at a mixed ability group of 21 Year 5 primary students (aged 9-10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software "Grid Algebra"…
Balancing the Equation: Do Course Variations in Algebra 1 Provide Equal Student Outcomes?
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…
Energy Technology Data Exchange (ETDEWEB)
Ulmer, W.; Hartmann, H. (Frankfurt Univ. (Germany, F.R.). Inst. fuer Physikalische und Theoretische Chemie)
1978-09-01
It is shown that Heisenberg's commutation rule between the position co-ordinate and the corresponding canonically conjugate momentum may be interpreted by noncommuting geometrical structures. As in the absence of a magnetic field the Euclidean norm of the momentum space directly enters the kinetic energy, the momentum space can be mapped onto the quaternion field U/sub 2/. Such a mapping preserves the norm of the momentum space. By that, the geometric and algebraic structure of the Pauli equation can be obtained and the relationship between the Pauli and the Dirac equation can be made apparent by noncommuting algebraic structures. In an appendix it will also be shown that the extension of the procedure to vector spaces equipped with Riemannian geometry makes no difficulties and a covariant quantization procedure can be formulated.
Akbari, M. R.; Ganji, D. D.; Ahmadi, A. R.; Kachapi, Sayyid H. Hashemi
2014-03-01
In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.
A Dirac-Dunkl Equation on S 2 and the Bannai-Ito Algebra
De Bie, Hendrik; Genest, Vincent X.; Vinet, Luc
2016-05-01
The Dirac-Dunkl operator on the two-sphere associated to the Z23 reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.
代数Riccati方程可稳解的条件数%CONDITION NUMBERS OF THE STABILIZING SOLUTIONS OF THE ALGEBRAIC RICCATI EQUATIONS
Institute of Scientific and Technical Information of China (English)
刘新国
2001-01-01
The Condition numbers are defined for the stabilizing Solutionsof Continuoustime, discrete-time and the reverse discrete-time algebraic Riccati equations. The first-order perturbation expansions for the stabilizing Solutions are also obtained.
Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys–Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients
Energy Technology Data Exchange (ETDEWEB)
Kalchev, D
2012-04-02
This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the
Shape Optimization for Navier-Stokes Equations with Algebraic Turbulence Model: Existence Analysis
Czech Academy of Sciences Publication Activity Database
Bulíček, M.; Haslinger, J.; Málek, J.; Stebel, Jan
2009-01-01
Roč. 60, č. 2 (2009), s. 185-212. ISSN 0095-4616 R&D Projects: GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : optimal shape design * paper machine headbox * incompressible non-Newtonian fluid * algebraic turbulence model * outflow boundary condition Subject RIV: BA - General Mathematics Impact factor: 0.757, year: 2009
Mohammad Shahzad
2016-01-01
This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM) by State Space Exact Linearization (SSEL) technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustne...
Folly-Gbetoula, Mensah; Kara, A. H.
2015-04-01
Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the first-order source equation are obtained. Based on certain properties of iterative equations, finding the solutions is reduced to finding solutions of the second-order source equation. We have therefore found classes of solutions to the source equations by letting the parameters of the source equation be functions of a specific type such as monomials, functions of exponential and logarithmic type.
The exploration of Galois' algebraic equation theory%伽罗瓦的代数方程思想
Institute of Scientific and Technical Information of China (English)
赵晔; 王昌; 周畅
2011-01-01
目的 探讨在代数方程根式可解性理论的发展中,伽罗瓦(Evariste Galois,1811-1832)的代数方程理论思想发展过程.方法 采用历史考察与数理分析法.结果 伽罗瓦是通过引进"伽罗瓦群"、"正规子群"、"置换群"等概念开始建立他的理论,并且找出了根式扩张塔和可解群之间的对应关系,利用这种对应关系最终解决了代数方程根式可解性理论这一难题.结论 伽罗瓦继承了拉格朗日(J.L.Lagrange,1736-1813)问题转化的思想,并且把这一思想进行发展,使得人们对方程根式解问题的研究进入到对"结构"观念的研究,导致了抽象代数学科的诞生;伽罗瓦的研究思路是通过继承和发展前人的思想成果得出来的.%Aim To explore the theory of Galois' algebraic equation through the depth study on the radical solution theory. Methods Historical investigation and mathematical analysis. Results Galois established his theory by introducing some concepts such as "Galois group", "normal subgroup", "permutation group" etc, and identifying the corresponding relation between radical expansion tower and solvable group. He finally solved the problem of algebraic equation radical solution theory. Conclusion Galois inherited the thoughts of Lagrange's problem transforming, and he developed this thoughts so that people turned the research of algebraic equation radical solution theory into the study of structure concept. It caused the establishment of abstract algebra discipline. His research idea originated from inheriting and developing predecessors' achievements.
Desingularizing of Differential-Algebraic Equations%微分代数方程去奇异化分析
Institute of Scientific and Technical Information of China (English)
邹杰涛; 王瑞瑞; 杨志辉
2011-01-01
利用去奇异化方法讨论了拟线性微分代数方程在奇点邻域内光滑解的性质.通过尺度参数的微分同胚变换,将拟线性微分代数方程转化为相应的常微分方程,从而构造出在孤立奇点邻域内的初始微分代数方程的光滑解,给出解存在的充分条件,并进一步讨论了解的性质.%In this paper we use the desingularizing method to discuss smooth solutions of quasi-linear differential-algebraic equations (DAEs) in the neighborhood of singularities. Through a diffeomorphic transformation of the scale parameter, the original quasi-linear DAEs can be changed into a corresponding ODE from which we can construct smooth solutions of DAEs near the isolate singularities. Furthermore, a sufficient condition for the existence of smooth solutions of quasi-linear differential-algebraic equations is presented.
Daviau, Claude
2014-01-01
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.
Daviau, Claude; Bertrand, Jacques
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.
Czech Academy of Sciences Publication Activity Database
Haslinger, J.; Stebel, Jan
2011-01-01
Roč. 63, č. 2 (2011), s. 277-308. ISSN 0095-4616 R&D Projects: GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : optimal shape design * paper machine headbox * incompressible non-Newtonian fluid * algebraic turbulence model Subject RIV: BA - General Mathematics Impact factor: 0.952, year: 2011 http://link.springer.com/article/10.1007%2Fs00245-010-9121-x
On Lagrange＇s Algebraic Equation Theory and Its Influence%拉格朗日的代数方程求解理论及其影响
Institute of Scientific and Technical Information of China (English)
赵增逊
2012-01-01
Lagrange＇s theory of algebraic equations is an integral part of the resolution history of algebraic equations,and the theory has an important impact on later algebraist.To reveal the content of Lagrange＇s algebraic equations theory,express its far-reaching implications,the article,based on original literature, concisely describes the content of Lagrange＇s algebraic equations theory and its influence.Therefore, understanding Lagrange＇s theory of algebraic equations is not only conducive to the understanding of the auxiliary equation theory,the connotation of permutation theory,but also is more helpful to find out the entire resolution history of algebraic equation.%拉格朗日的代数方程求解理论是整个代数方程求解史中不可或缺的一部分,并且该理论对以后的代数学家产生了重要的影响。为展示拉格朗日代数方程求解理论的内容,说明该理论产生的深远影响,从原始文献出发,叙述了拉格朗日的代数方程求解理论的内容,重点阐述了该理论产生的重要影响。因此,清楚拉格朗日的代数方程求解理论不仅有利于了解辅助方程理论、置换思想的内涵,更有利于清楚整个代数方程的求解历史。
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
Ferrante, Augusto; Ntogramatzidis, Lorenzo
2013-01-01
The purpose of this paper is to investigate the role that the continuous-time generalised Riccati equation plays within the context of singular linear-quadratic optimal control. This equation has been defined following the analogy with the discrete-time generalised Riccati equation, but, differently from the discrete case, to date the importance of this equation in the context of optimal control is yet to be understood. This note addresses this point. We show in particular that when the conti...
Generalized Flip-Flop Input Equations Based on a Four-Valued Boolean Algebra
Tucker, Jerry H.; Tapia, Moiez A.
1996-01-01
A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.
Algebraic Equations and Inequalities: Issues for Research and Teaching. Research Forum
Bazzini, Luciana; Tsamir, Pessia
2004-01-01
The presentations address a variety of difficulties occurring in students' solutions of equations and inequalities, and suggest different reasons for these difficulties. When analyzing students' performances, [BB] and [TTT] mention students' tendencies to make irrelevant connections between equations and inequalities as a problematic phenomenon.…
Institute of Scientific and Technical Information of China (English)
FAN Hong-Yi; WANG Yong
2006-01-01
With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.
Electromagnetic energy-momentum equation without tensors: a geometric algebra approach
Sugon Jr., Quirino M.; McNamara, Daniel J.
2008-01-01
In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is ...
Institute of Scientific and Technical Information of China (English)
An Hui-hui; Wang Zhi-chun
2016-01-01
L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.
International Nuclear Information System (INIS)
Based on a new general ansaetz, a new general algebraic method named improved Riccati equation rational expansion method is devised for constructing multiple nontravelling wave solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. With the aid of symbolic computation, we choose the (2 + 1)-dimensional cubic nonlinear Schroedinger equation to illustrate the method. As a result, six families of new exact analytical solutions for this equation are found, which include some new and more general exact rational form soliton-like solutions and triangular periodic-like solutions
Norman, Joseph W
2011-01-01
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to G\\"odel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms is computed from the solution to the corresponding system of polynomial equations. A reference by a formula to its own provability is shown to be a recurrence relation, which can be either interpreted as such to generate a discrete dynamical system, or interpreted in a static way to create an additional simultaneous equation. In this framework the truth values of logical formulas and other polynomial objectives have complex data structures: sets of elementary values, or dynamical systems that generate sets of infinite sequences of such solution-value sets. Besides the routine result that a formula has a definite elementary value, these data structures encode several exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or contingent. These exceptions represent ...
Icardi, Matteo
2014-09-01
The inhomogeneous generalized population balance equation, which is discretized with the direct quadrature method of moment (DQMOM), is solved to predict the bubble size distribution (BSD) in a vertical pipe flow. The proposed model is compared with a more classical approach where bubbles are characterized with a constant mean size. The turbulent two-phase flow field, which is modeled using a Reynolds-Averaged Navier-Stokes equation approach, is assumed to be in local equilibrium, thus the relative gas and liquid (slip) velocities can be calculated with the algebraic slip model, thereby accounting for the drag, lift, and lubrication forces. The complex relationship between the bubble size distribution and the resulting forces is described accurately by the DQMOM. Each quadrature node and weight represents a class of bubbles with characteristic size and number density, which change dynamically in time and space to preserve the first moments of the BSD. The predictions obtained are validated against previously published experimental data, thereby demonstrating the advantages of this approach for large-scale systems as well as suggesting future extensions to long piping systems and more complex geometries. © 2014 Elsevier Inc.
Computer subroutine ISUDS accurately solves large system of simultaneous linear algebraic equations
Collier, G.
1967-01-01
Computer program, an Iterative Scheme Using a Direct Solution, obtains double precision accuracy using a single-precision coefficient matrix. ISUDS solves a system of equations written in matrix form as AX equals B, where A is a square non-singular coefficient matrix, X is a vector, and B is a vector.
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2013-01-01
Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…
Kazeykina, Anna
2012-01-01
We show that the Novikov--Veselov equation (an analog of KdV in dimension 2 + 1) at positive and negative energies does not have solitons with the space localization stronger than O(|x|^{-3}) as |x| \\to \\infty.
Realizations of Galilei algebras
International Nuclear Information System (INIS)
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations. (paper)
Picard-Fuchs equations, Integrable Systems, and higher Algebraic K-theory
del Angel, Pedro L.; Müller-Stach, Stefan
2002-01-01
This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman to our setup.
On the algebraic classification of perfect fluid solutions of Einstein's equations
International Nuclear Information System (INIS)
This thesis concerns the Einstein-Petrov problem for perfect fluid space-times. This is the problem of finding exact solutions of Einstein's perfect fluid field equations for which the Weyl tensor is of a specific Petrov type. No examples of exact perfect fluid space-times having a Weyl tensor of Petrov type III have ever appeared in the literature. This is the only Petrov Class for which this is so. Consequently an example of such a solution is sought. Using the Newman-Penrose formalism a particular solution is obtained and its properties are discussed. (author)
Free Markov processes and stochastic differential equations in von Neumann algebras
Gao, Mingchu
2008-01-01
Free Markov processes are investigated in Voiculescu’s free probability theory. We show that Voiculescu’s free Markov property implies a property called “weak Markov property”, which is the classical Markov property in the commutative case; while, in the general case, the “weak Markov property” is the same as the Markov property defined by Bozejko, Kummer, and Speicher. We also show that a kind of stochastic differential equations driven by free Levy processes has solutions. The solutions are...
Allenby, Reg
1995-01-01
As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.Solutions to the exercises are available onlin
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
Set-membership state estimation framework for uncertain linear differential-algebraic equations
Zhuk, Serhiy
2008-01-01
We investigate a problem of state estimation for the dynamical system described by the linear operator equation with unknown parameters in Hilbert space. We present explicit expressions for linear minimax estimation and error provided that any pair of uncertain parameters belongs to the quadratic bounding set. As an application of the introduced approach we introduce a notion of minimax directional observability and index of non-causality for linear noncausal DAEs. Application of these notions to the problem of state estimation for the linear uncertain noncausal DAEs allows to construct the state estimation in the form of the recursive minimax filter. A numerical example of the state estimation for 3D non-causal descriptor system is presented.
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...
Phantom energy from graded algebras
Chaves, Max; Singleton, Douglas
2006-01-01
We construct a model of phantom energy using the graded Lie algebra SU(2/1). The negative kinetic energy of the phantom field emerges naturally from the graded Lie algebra, resulting in an equation of state with w
Intermediate algebra & analytic geometry
Gondin, William R
1967-01-01
Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system
Left Artinian Algebraic Algebras
Institute of Scientific and Technical Information of China (English)
S. Akbari; M. Arian-Nejad
2001-01-01
Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.
Colored Quantum Algebra and Its Bethe State
International Nuclear Information System (INIS)
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)
Murthy, S; Arunkumar, M; V. Govindan
2015-01-01
In this paper, the authors introduce and investigate the general solution and generalized Ulam-Hyers stability of a generalized n-type additive-quadratic functional equation. g(x + 2y; u + 2v) + g(x 2y; u 2v) = 4[g(x + y; u + v) + g(x y; u v)] 6g(x; u) + g(2y; 2v) + g(2y;2v) 4g(y; v) 4g(y;v) Where is a positive integer with , in Banach Space and Banach Algebras using direct and fixed point methods.
Algebra & trigonometry super review
2012-01-01
Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Algebra and Trigonometry Super Review includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, equations, linear equations and systems of linear equations, inequalities, relations and functions, quadratic equations, equations of higher order, ratios, proportions, and variations. Take the Super Review quizzes to see how much y
Lefschetz, Solomon
2012-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
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.
Das, Tapas
2015-01-01
The second order $N$-dimensional Schr\\"odinger equation with pseudoharmonic potential is reduced to a first order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Our results generalize all other previous works that done for various potential combinations in the case of lower dimensions.The Ladder operators are also constructed for the pseudoharmonic potential in $N$-dimensions.Lie algebra associated with these operators are studied and found that they satisfy the commutation relations for the SU(1,1) group. Matrix elements of different operators such as $z$, $z\\frac{d}{dz}$ are derived and finally the Casimir operator is discussed briefly.
Yang-Mills gauge fields conserving the symmetry algebra of the Dirac equation in a homogeneous space
International Nuclear Information System (INIS)
We consider the Dirac equation with an external Yang-Mills gauge field in a homogeneous space with an invariant metric. The Yang-Mills fields for which the motion group of the space serves as the symmetry group for the Dirac equation are found by comparison of the Dirac equation with an invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The eigenfunctions and the corresponding eigenvalues for the Dirac equation are obtained in the space R2 × S2 by a noncommutative integration method
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
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.
A Deductive Approach towards Reasoning about Algebraic Transition Systems
Jun Fu; Jinzhao Wu; Hongyan Tan
2015-01-01
Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semanti...
Symplectic algebraic dynamics algorithm
Institute of Scientific and Technical Information of China (English)
2007-01-01
Based on the algebraic dynamics solution of ordinary differential equations andintegration of ,the symplectic algebraic dynamics algorithm sn is designed,which preserves the local symplectic geometric structure of a Hamiltonian systemand possesses the same precision of the na ve algebraic dynamics algorithm n.Computer experiments for the 4th order algorithms are made for five test modelsand the numerical results are compared with the conventional symplectic geometric algorithm,indicating that sn has higher precision,the algorithm-inducedphase shift of the conventional symplectic geometric algorithm can be reduced,and the dynamical fidelity can be improved by one order of magnitude.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
Intermediate algebra a textworkbook
McKeague, Charles P
1985-01-01
Intermediate Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in intermediate algebra. The publication first takes a look at basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on properties of exponents, polynomials, sums, and differences, multiplication of polynomials, inequalities involving absolute value, word problems, first-degree inequalities, real numbers, opposites, reciprocals, and absolute value, and addition and subtraction of real numbers. The text then ex
Pre-Algebra Essentials For Dummies
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
Exceptional Vertex Operator Algebras and the Virasoro Algebra
Tuite, Michael P.
2008-01-01
We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Exam...
Proposition Algebra with Projective Limits
Bergstra, J A
2008-01-01
Sequential logic deviates from propositional logic by taking into account that atomic propositions yield different Boolean values at different times during the sequential evaluation of a single proposition. Reactive valuations capture this dynamics of a proposition's environment. This logic is phrased as an equationally specified algebra rather than in the form of proof rules. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The proposition algebra PRA is developed in a fashion similar to the process algebra ACP and the program algebra PGA via an algebraic specification which has a meaningful initial algebra for which a range of courser congruences are considered important as well. In addition infinite objects (that is propositions, processes and programs respectively) are preferably dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts ...
Quantum computation using geometric algebra
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
Solitons as purely algebraic construction
International Nuclear Information System (INIS)
A new purely algebraic method for finding soliton solutions for nonlinear equations Without using the inverse scattering method is elaborated. As the examples the soliton solutions are given explicitly for both the well known nonlinear equations and equations which have not been discussed earler. The symmetry basis of the method is connected with the infinite-dimensional internal symmetry Lie algebra of the system under consideration
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:...
Izhakian, Zur; Rowen, Louis
2008-01-01
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...
International Nuclear Information System (INIS)
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.
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.
Czech Academy of Sciences Publication Activity Database
Dobeš, J.; Fořt, J.; Příhoda, Jaromír
Praha: Institute of Thermomechanics AS CR, v. v. i., 2007 - (Jonáš, P.; Uruba, V.), s. 13-14 ISBN 978-80-87012-07-9. [Colloquium FLUID DYNAMICS 2007. Prague (CZ), 24.10.2007-26.10.2007] R&D Projects: GA ČR GA101/07/1508 Institutional research plan: CEZ:AV0Z20760514 Keywords : bypass transition modelling * two-equation turbulence model * finite volume method solver Subject RIV: BK - Fluid Dynamics
International Nuclear Information System (INIS)
The geometrical properties of a flat tangent space-time local to the manifold of the Einstein--Schroedinger nonsymmetric theory to which an internal octonionic space is attached, is developed here. As an application of the theory, an octonionic Dirac equation for a spin-1/2 particle is also obtained, where is now used an octonionic-like gauge field. It is shown that the (quaternionic) nonsymmetric Yang--Mills theory can be easily recovered and from there, the usual gauge theory on a curved space
Semigroups and computer algebra in algebraic structures
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Algebra I Essentials For Dummies
Sterling, Mary Jane
2010-01-01
With its use of multiple variables, functions, and formulas algebra can be confusing and overwhelming to learn and easy to forget. Perfect for students who need to review or reference critical concepts, Algebra I Essentials For Dummies provides content focused on key topics only, with discrete explanations of critical concepts taught in a typical Algebra I course, from functions and FOILs to quadratic and linear equations. This guide is also a perfect reference for parents who need to review critical algebra concepts as they help students with homework assignments, as well as for adult learner
Homogeneous conformal averaging operators on semisimple Lie algebras
Kolesnikov, Pavel
2014-01-01
In this note we show a close relation between the following objects: Classical Yang---Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular part of a solution of CYBE (in the operator form) on a Lie algebra $\\mathfrak g$ determines an averaging operator on the corresponding current conformal algebra $\\mathrm{Cur} \\mathfrak g$. For a finite-dimensional semisimple Lie algebra $\\mathfrak g$, we desc...
An infinite algebra of quantum Dirac brackets
International Nuclear Information System (INIS)
A new algebraic approach to the theory with second-class constraints is proposed. The operator equations that generate automatically the infinite algebra of quantum Dirac brackets are formulated. First-class constraints are naturally involved into the new algebraic scheme. (orig.)
Carlen, E. A.; Carvalho, M. C.; Orlandi, E.
1999-06-01
This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v 0 of a front that is sufficiently small in L 2 norm, and sufficiently localized that ∫ x 2 v 0( x)2 dx<∞, yields a solution that relaxes to another front, selected by a conservation law, in the L 1 norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L 2 norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.
Directory of Open Access Journals (Sweden)
Kumare Vinodh
2016-03-01
Full Text Available This paper presents an analytical approach for solving the weighting matrices selection problem of a linear quadratic regulator (LQR for the trajectory tracking application of a magnetic levitation system. One of the challenging problems in the design of LQR for tracking applications is the choice of Q and R matrices. Conventionally, the weights of a LQR controller are chosen based on a trial and error approach to determine the optimum state feedback controller gains. However, it is often time consuming and tedious to tune the controller gains via a trial and error method. To address this problem, by utilizing the relation between the algebraic Riccati equation (ARE and the Lagrangian optimization principle, an analytical methodology for selecting the elements of Q and R matrices has been formulated. The novelty of the methodology is the emphasis on the synthesis of time domain design specifications for the formulation of the cost function of LQR, which directly translates the system requirement into a cost function so that the optimal performance can be obtained via a systematic approach. The efficacy of the proposed methodology is tested on the benchmark Quanser magnetic levitation system and a detailed simulation and experimental results are presented. Experimental results prove that the proposed methodology not only provides a systematic way of selecting the weighting matrices but also significantly improves the tracking performance of the system.
An "Arithmetic" Thinker Tackles Algebra
Armstrong, Alayne C.
2006-01-01
Working from Carolyn Kieran's categorization of "arithmetic" and "algebraic" thinkers, the article describes one eighth-grade "arithmetic" thinker's progress as she attempts to solve one- and two-step equations.
Picard–Vessiot extensions of artinian simple module algebras
Amano, Katsutoshi; Masuoka, Akira
2004-01-01
This paper pursues Takeuchi's Hopf algebraic approach [M. Takeuchi, A Hopf algebraic approach to the Picard–Vessiot theory, J. Algebra 122 (1989) 481–509] to the Picard–Vessiot (PV) theory for differential equations, to involve the PV extensions of difference equations. Differential fields and total difference rings in the standard PV theory are unified here by artinian simple (AS) module algebras over a cocommutative, pointed smooth Hopf algebra.
Questions on Algebraic Varieties
Marchionna, E
2011-01-01
P. Dolbeault: Residus et courants.- D. Mumford: Varieties defined by quadratic equations.- A. Neron: Hauteurs et theorie des intersections.- A. Seidenberg: Report on analytic product.- C.S. Seshadri: Moduli of p-vector bundles over an algebraic curve.- O. Zariski: Contributions to the problem of equi-singularity.
Head First Algebra A Learner's Guide to Algebra I
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
DEFF Research Database (Denmark)
2007-01-01
The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers were Michel Brion, Jens Carsten Jantzen, and Raphaël Rouquier. During the last years, the subject of...... algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a "general audience" of algebraic group-theorists, and to...
Liouville Extensions of Artinian Simple Module Algebras
Amano, Katsutoshi
2006-01-01
In a previous article (Amano and Masuoka, 2005), the author and Masuoka developed a Picard-Vessiot theory for module algebras over a cocommutative pointed smooth Hopf algebra D. By using the notion of Artinian simple (AS)D-module algebras, it generalizes and unifies the standard Picard-Vessiot theories for linear differential and difference equations. The purpose of this article is to define the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding Picar...
Method of averaging in Clifford algebras
Shirokov, D. S.
2014-01-01
In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros' vee group. This operator average" an action of Salingaros' vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using method of averaging we present solutions of system of commutator equations.
A Deductive Approach towards Reasoning about Algebraic Transition Systems
Directory of Open Access Journals (Sweden)
Jun Fu
2015-01-01
Full Text Available Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semantics of modalities and formulas is defined with solutions of algebraic equations. A proof system for this logic is constructed to verify properties of algebraic transition systems. The proof system combines with inference rules decision procedures on the theory of polynomial ideals to reduce a proof-search problem to an algebraic computation problem. The proof system proves to be sound but inherently incomplete. Finally, a typical example illustrates that reasoning about algebraic transition systems with our approach is feasible.
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Reed, Nat
2011-01-01
For grades 3-5, our State Standards-based combined resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to review the concepts in unique ways. The task sheets introduce the mathematical concepts to the students around a central problem taken from real-life experiences, while the drill sheets provide warm-up and timed practice questions for the students to strengthen their procedural proficiency skills. Included are opportunities for problem-solving, patterning, algebraic graphing, equations and determining averages. The combined task & drill sheets
Reed, Nat
2011-01-01
For grades 6-8, our State Standards-based combined resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to review the concepts in unique ways. The task sheets introduce the mathematical concepts to the students around a central problem taken from real-life experiences, while the drill sheets provide warm-up and timed practice questions for the students to strengthen their procedural proficiency skills. Included are opportunities for problem-solving, patterning, algebraic graphing, equations and determining averages. The combined task & drill sheets
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
International Nuclear Information System (INIS)
We calculate the cohomology of the BRS operator s modulo an auxiliary differential operator t where both operators act on invariant polynomials in anticommuting variables Ci and commuting variables Xi. Ci and Xi transform according to the adjoint representation of the Lie algebra of a compact Lie group. The cohomology classes of s modulo t are related to the solutions of the consistency equations which have to be satisfied by anomalies of Yang-Mills theories. The present investigation completes the proof of the completeness and nontriviality of these solutions and, as a by-product, determines the cohomology of the underlying Lie algebra. (orig.)
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
Segal algebras in commutative Banach algebras
INOUE, Jyunji; TAKAHASI, Sin-Ei
2014-01-01
The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.
Algebra-Geometry of Piecewise Algebraic Varieties
Institute of Scientific and Technical Information of China (English)
Chun Gang ZHU; Ren Hong WANG
2012-01-01
Algebraic variety is the most important subject in classical algebraic geometry.As the zero set of multivariate splines,the piecewise algebraic variety is a kind generalization of the classical algebraic variety.This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.
The algebraic structure of the Onsager algebra
DATE, ETSURO; Roan, Shi-shyr
2000-01-01
We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.
Explicit Traveling Waves and Invariant Algebraic Curves
Gasull, Armengol; Giacomini, Hector
2013-01-01
In this paper we introduce a precise definition of algebraic traveling wave solution for general n-th order partial differential equations. All examples of explicit traveling waves known by the authors fall in this category. Our main result proves that algebraic traveling waves exist if and only if an associated n- dimensional first order ordinary differential system has some invariant algebraic curve. As a paradigmatic application we prove that, for the celebrated Fisher- Kolmogorov equation...
Classical algebra its nature, origins, and uses
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,...
Holtz, Olga; Ron, Amos
2007-01-01
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\\cal H}(X)$. This well-known line of study is particularly interesting in case $n\\eqbd\\rank X \\ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\\it external, central, and internal.} Each algebraic structure is ...
Issa, A. Nourou
2010-01-01
Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-M...
On Axiomatic Approaches to Intertwining Operator Algebras
Chen, Ling
2015-01-01
We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator algebras were given in [H9] but some of the proofs were postponed to an unpublished monograph. In this paper, we give the proofs of these results in [H9] and we formulate and prove results for general intertwining operator algebras without assuming that the modules involved are irreducible. In particular, we construct fusing and braiding isomorphisms for general intertwining operator algebras and prove that they satisfy the genus-zero Moore-Seiberg equations. We show that the Jacobi identity for intertwining operator algebras is equivalent to generalized rationality, commutativity and associativity properties of intertwining operator algebras. We introduce the locality for intertwining operator algebras and show that the Jacobi identity is equi...
Generalized Galilean algebras and Newtonian gravity
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.
Classical particle with spin and Clifford algebra
International Nuclear Information System (INIS)
Equations of motion of classical particle with spin in electromagnetic field are derived in terms of the Clifford algebra of the Minkowsky space. The use of the Clifford algebra simplifies the derivation of these equations as well as their form and process of their solving. The equations also get an evident geometric interpretation. The perturbation theory for these equations is formulated which allows to analyze the motion and the polarization of particles in various electromagnetic fields
Quasi-projective relation algebras and directed cylindric algebras are categorially equivalent
Ahmed, Tarek Sayed
2013-01-01
We show that quasi-projective relation algebras and directed cylindric algebras are equivalent categorialy. We work out a Godels second incompleteness theorem for finite varibale fragments of first order logic. We show that distinct set theories (like one with CH, and another with its negation) give rise to equationally distinct simple directed cylindric algebras. This correspondance was worked out for quasi projective relation algebras (with a distinguished element corresponding to membershi...
Levandovskyy, Viktor
2005-01-01
Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations. In the thesis, we have developed a framework, unifying many important algebras in the classes of \\(G\\)- and \\(GR\\)-algebras and studied their ring-theoretic properties. Let \\(A\\) be a \\(G\\)-algebra in \\(n\\) variables. We establish necessary and sufficient conditions for \\(A\\) to have a Poincar'e-Birkhoff-Witt (PBW) basis. Furt...
International Nuclear Information System (INIS)
Since the works of Gelfand, Harish-Chandra, Kostant and Duflo, a new theory has earned its place in the field of mathematics, due to the abundance of its results and the coherence of its methods: the theory of enveloping algebras. This study is the first to present the whole subject in textbook form. The most recent results are included, as well as complete proofs, starting from the elementary theory of Lie algebras. (Auth.)
Spacetime algebra and electron physics
Doran, C J L; Gull, S F; Somaroo, S; Challinor, A D
1996-01-01
This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include non-relativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, propagators and scattering theory, spin precession, tunnelling times, spin measurement, multiparticle quantum mechanics, relativistic multiparticle wave equations, and semiclassical mechanics.
Which multiplier algebras are $W^*$-algebras?
Akemann, Charles A.; Amini, Massoud; Asadi, Mohammad B.
2013-01-01
We consider the question of when the multiplier algebra $M(\\mathcal{A})$ of a $C^*$-algebra $\\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebr...
Energy Technology Data Exchange (ETDEWEB)
Sternberg, K.
2007-02-08
Molten carbonate fuel cells (MCFCs) allow an efficient and environmentally friendly energy production by converting the chemical energy contained in the fuel gas in virtue of electro-chemical reactions. In order to predict the effect of the electro-chemical reactions and to control the dynamical behavior of the fuel cell a mathematical model has to be found. The molten carbonate fuel cell (MCFC) can indeed be described by a highly complex,large scale, semi-linear system of partial differential algebraic equations. This system includes a reaction-diffusion-equation of parabolic type, several reaction-transport-equations of hyperbolic type, several ordinary differential equations and finally a system of integro-differential algebraic equations which describes the nonlinear non-standard boundary conditions for the entire partial differential algebraic equation system (PDAE-system). The existence of an analytical or the computability of a numerical solution for this high-dimensional PDAE-system depends on the kind of the differential equations and their special characteristics. Apart from theoretical investigations, the real process has to be controlled, more precisely optimally controlled. Hence, on the basis of the PDAE-system an optimal control problem is set up, whose analytical and numerical solvability is closely linked to the solvability of the PDAE-system. Moreover the solution of that optimal control problem is made more difficult by inaccuracies in the underlying database, which does not supply sufficiently accurate values for the model parameters. Therefore the optimal control problem must also be investigated with respect to small disturbances of model parameters. The aim of this work is to analyze the relevant dynamic behavior of MCFCs and to develop concepts for their optimal process control. Therefore this work is concerned with the simulation, the optimal control and the sensitivity analysis of a mathematical model for MCDCs, which can be characterized
Introduction to algebra and trigonometry
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
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
Algebraic structures in quantum gravity
Tanasa, Adrian
2009-01-01
Starting from a recently-introduced algebraic structure on spin foam models, we define a Hopf algebra by dividing with an appropriate quotient. The structure, thus defined, naturally allows for a mirror analysis of spin foam models with quantum field theory, from a combinatorial point of view. A grafting operator is introduced allowing for the equivalent of a Dyson-Schwinger equation to be written. Non-trivial examples are explicitly worked out. Finally, the physical significance of the results is discussed.
Process algebra for synchronous communication
Bergstra, J. A.; Klop, Jan Willem
1984-01-01
Within the context of an algebraic theory of processes, an equational specification of process cooperation is provided. Four cases are considered: free merge or interleaving, merging with communication, merging with mutual exclusion of tight regions, and synchronous process cooperation. The rewrite system behind the communication algebra is shown to be confluent and terminating (modulo its permutative reductions). Further, some relationships are shown to hold between the four concepts of merg...
Algebraic entropy for algebraic maps
International Nuclear Information System (INIS)
We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Bäcklund transformations. (letter)
Homotopy DG algebras induce homotopy BV algebras
Terilla, John; Tradler, Thomas; Wilson, Scott O.
2011-01-01
Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential Batalin-Vilkovisky algebra. Moreover, if A is an A-infinity algebra, then TA is a commutative BV-infinity algebra.
Algebra & trigonometry II essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Algebra & Trigonometry II includes logarithms, sequences and series, permutations, combinations and probability, vectors, matrices, determinants and systems of equations, mathematica
Institute of Scientific and Technical Information of China (English)
WANG Renhong; ZHU Chungang
2004-01-01
The piecewise algebraic variety is a generalization of the classical algebraic variety. This paper discusses some properties of piecewise algebraic varieties and their coordinate rings based on the knowledge of algebraic geometry.
Edwards, Harold M
1995-01-01
In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject
Clifford algebra, geometric algebra, and applications
Lundholm, Douglas; Svensson, Lars
2009-01-01
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The v...
Cofree Hopf algebras on Hopf bimodule algebras
Fang, Xin; Jian, Run-Qiang
2013-01-01
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.
On uniform topological algebras
Azhari, M. El
2013-01-01
The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M (A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.
Marchuk, Nikolay
2011-01-01
Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann alg...
Hazewinkel, Michiel
2004-01-01
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generaliz...
Jacobson, Nathan
1979-01-01
Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its
Deskins, W E
1996-01-01
This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diop
Advanced linear algebra for engineers with Matlab
Dianat, Sohail A
2009-01-01
Matrices, Matrix Algebra, and Elementary Matrix OperationsBasic Concepts and NotationMatrix AlgebraElementary Row OperationsSolution of System of Linear EquationsMatrix PartitionsBlock MultiplicationInner, Outer, and Kronecker ProductsDeterminants, Matrix Inversion and Solutions to Systems of Linear EquationsDeterminant of a MatrixMatrix InversionSolution of Simultaneous Linear EquationsApplications: Circuit AnalysisHomogeneous Coordinates SystemRank, Nu
GOLDMAN ALGEBRA, OPERS AND THE SWAPPING ALGEBRA
Labourie, François
2012-01-01
We define a Poisson Algebra called the {\\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\\mathsf{SL}_n(\\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symple...
Rota-Baxter algebras and the Hopf algebra of renormalization
International Nuclear Information System (INIS)
Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. Hereby the notion of Rota-Baxter algebras enters the scene. In this work we develop in detail several mathematical aspects of Rota-Baxter algebras as they appear also in other sectors closely related to perturbative renormalization, to wit, for instance multiple-zeta-values and matrix differential equations. The Rota-Baxter picture enables us to present the algebraic underpinning for the Connes-Kreimer Birkhoff decomposition in a concise way. This is achieved by establishing a general factorization theorem for filtered algebras. Which in turn follows from a new recursion formula based on the Baker-Campbell-Hausdorff formula. This allows us to generalize a classical result due to Spitzer to non-commutative Rota-Baxter algebras. The Baker-Campbell-Hausdorff based recursion turns out to be a generalization of Magnus' expansion in numerical analysis to generalized integration operators. We will exemplify these general results by establishing a simple representation of the combinatorics of renormalization in terms of triangular matrices. We thereby recover in the presence of a Rota-Baxter operator the matrix representation of the Birkhoff decomposition of Connes and Kreimer. (orig.)
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.)
Smarandache Jordan Algebras - abstract
Vasantha Kandasamy, W. B.; Christopher, S.; A. Victor Devadoss
2004-01-01
We prove a S-commutative Jordan Algebra is a S-weakly commutative Jordan algebra. We define a S-Jordan algebra to be S-simple Jordan algebras if the S-Jordan algebra has no S-Jordan ideals. We obtain several other interesting notions and results on S-Jordan algebras.
Lin, I-hsiung
2005-01-01
This accessible book for beginners uses intuitive geometric concepts to create abstract algebraic theory with a special emphasis on geometric characterizations. The book applies known results to describe various geometries and their invariants, and presents problems concerned with linear algebra, such as in real and complex analysis, differential equations, differentiable manifolds, differential geometry, Markov chains and transformation groups. The clear and inductive approach makes this book unique among existing books on linear algebra both in presentation and in content.
Assessing non-uniqueness: An algebraic approach
Energy Technology Data Exchange (ETDEWEB)
Vasco, Don W.
2002-09-16
Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.
Algebraic analysis approach for multibody problems
International Nuclear Information System (INIS)
Here we propose an algebraic analysis approach for multibody Coulomb interactions. The momentum transfer cross section calculated by the algebraic approximation is close to the exact one. The CPU time required for the algebraic approximation is only about 20 min using a personal computer, whereas the exact analysis requires 15 h to integrate the entire set of multibody equations of motion, in which all the field particles are at rest. (author)
Indian Academy of Sciences (India)
Tomás L Gómez
2001-02-01
This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.
Oliver, Bob; Pawałowski, Krzystof
1991-01-01
As part of the scientific activity in connection with the 70th birthday of the Adam Mickiewicz University in Poznan, an international conference on algebraic topology was held. In the resulting proceedings volume, the emphasis is on substantial survey papers, some presented at the conference, some written subsequently.
Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series
Mainiero, Tom
2016-01-01
It is an observation of Kontsevich and Soibelman that generating series that produce certain (generalized) Donaldson Thomas invariants are secretly algebraic functions over the rationals. From a physical perspective this observation arises naturally for DT invariants that appear as BPS indices in theories of class S[A]: explicit algebraic equations (that completely determine these series) can be derived using (degenerate) spectral networks. In this paper, we conjecture an algebraic equation associated to DT invariants for the Kronecker 3-quiver with dimension vectors (3n,2n), n>0 in the non-trivial region of its stability parameter space. Using a functional equation due to Reineke, we show algebraicity of generating series for Euler characteristics of stable moduli for the Kronecker m-quiver assuming algebraicity of generating series for DT invariants. In the latter part of the paper we deduce very explicit results on the asymptotics of DT invariants/Euler characteristics under the assumption of algebraicity ...
Algebraic structure and Poisson integrals of a rotational relativistic Birkhoff system
Institute of Scientific and Technical Information of China (English)
罗绍凯; 陈向炜; 郭永新
2002-01-01
We have studied the algebraic structure of the dynamical equations of a rotational relativistic Birkhoff system. It is proven that autonomous and semi-autonomous rotational relativistic Birkhoff equations possess consistent algebraic structure and Lie algebraic structure. In general, non-autonomous rotational relativistic Birkhoff equations possess no algebraic structure, but a type of special non-autonomous rotational relativistic Birkhoff equation possesses consistent algebraic structure and consistent Lie algebraic structure. Then, we obtain the Poisson integrals of the dynamical equations of the rotational relativistic Birkhoff system. Finally, we give an example to illustrate the application of the results.
Algebra I Workbook For Dummies
Sterling, Mary Jane
2011-01-01
From signed numbers to story problems - calculate equations with ease Practice is the key to improving your algebra skills, and that's what this workbook is all about. This hands-on guide focuses on helping you solve the many types of algebra problems you'll encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, this workbook shows you how to work with fractions, exponents, factoring, linear and quadratic equations, inequalities, graphs, and more! 100s of problems! Hundreds of practice exercises and helpful explanations Explanations mi
Algebra a complete introduction : teach yourself
Neill, Hugh
2013-01-01
Algebra: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using Algebra. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all the key areas of algebra including elementary operations, linear equations, formulae, simultaneous equations, quadratic equations, logarithms, variation, laws and sequences. Everything you will need is here in this one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.
Classical theory of algebraic numbers
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...
Combinatorics and commutative algebra
Stanley, Richard P
1996-01-01
Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists. New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Included in this chapter is an outline of the proof of McMullen's g-conjecture for simplicial polytopes based on toric varieties, as well as a discussion of the face rings of such special ...
On deformations of Yang-Mills algebras
Movshev, M.
2005-01-01
This is a next paper from a sequel devoted to algebraic aspects of Yang-Mills theory. We undertake a study of deformation theory of Yang-Mills algebra YM - a ``universal solution'' of Yang-Mills equation. We compute (cyclic) (co)homology of YM.
The Algebra of Directed Acyclic Graphs
Fiore, Marcelo; Campos, Marco Devesas
2013-01-01
We give an algebraic presentation of directed acyclic graph structure, introducing a symmetric monoidal equational theory whose free PROP we characterise as that of finite abstract dags with input/output interfaces. Our development provides an initial-algebra semantics for dag structure.
Difference Galois theory of linear differential equations
Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael
2013-01-01
We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.
International Nuclear Information System (INIS)
A non-linear associative algebra is realized in terms of translation and dilation operators, and a wavelet structure generating algebra is obtained. We show that this algebra is a q-deformation of the Fourier series generating algebra, and reduces to this for certain value of the deformation parameter. This algebra is also homeomorphic with the q-deformed suq(2) algebra and some of its extensions. Through this algebraic approach new methods for obtaining the wavelets are introduced. (author). 20 refs
Central simple Poisson algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai; XU; Xiaoping
2004-01-01
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
El-Chaar, Caroline
2012-01-01
In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. U...
Introductory modern algebra a historical approach
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
Topics in quaternion linear algebra
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...
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...
Herranz, Francisco J.
1999-01-01
By starting from the non-standard quantum deformation of the sl(2,R) algebra, a new quantum deformation for the real Lie algebra so(2,2) is constructed by imposing the former to be a Hopf subalgebra of the latter. The quantum so(2,2) algebra so obtained is realized as a quantum conformal algebra of the (1+1) Minkowskian spacetime. This Hopf algebra is shown to be the symmetry algebra of a time discretization of the (1+1) wave equation and its contraction gives rise to a new $(2+1)$ quantum Po...
Chirvasitu, Alex; Smith, S. Paul
2015-01-01
This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "...
Kleene Algebra with Products and Iteration Theories
Kozen, Dexter; Mamouras, Konstantinos
2013-01-01
We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism.
Nonmonotonic logics and algebras
Institute of Scientific and Technical Information of China (English)
CHAKRABORTY Mihir Kr; GHOSH Sujata
2008-01-01
Several nonmonotonie logic systems together with their algebraic semantics are discussed. NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.
Kleyn, Aleks
2007-01-01
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
Kinds of Knowledge in Algebra.
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
Another Definition of Order—Sorted Algebra
Institute of Scientific and Technical Information of China (English)
何自强
1998-01-01
In this paper the definition of order-sorted algebra is generalized by introducing transformation functions between subtypes and supertypes.According to our definition,a type needn't be a subset of its supertype and a record model may form an order-sorted algebra.A new definition of equation is given.It has also been proved that equational theories and describing single inheritance have the initial model.
Exact WKB analysis and cluster algebras
International Nuclear Information System (INIS)
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)
Algebraic structure and Poisson method for a weakly nonholonomic system
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed.The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system.An example is given to illustrate the application of the result.
Figueredo, Grazziela P.; Peer-Olaf Siebers; Owen, Markus R.; Jenna Reps; Uwe Aickelin
2014-01-01
There is great potential to be explored regarding the use of agent-based modelling and simulation as an alternative paradigm to investigate early-stage cancer interactions with the immune system. It does not suffer from some limitations of ordinary differential equation models, such as the lack of stochasticity, representation of individual behaviours rather than aggregates and individual memory. In this paper we investigate the potential contribution of agent-based modelling and simulation w...
Mahé, Louis; Roy, Marie-Françoise
1992-01-01
Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...
Goze, Michel; Remm, Elisabeth
2006-01-01
A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product gxg...xg where g is a rigid Lie algebra.
Solvable quadratic Lie algebras
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.
Grabowski, Jan
2015-01-01
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating ...
Hochstadt, Harry
2012-01-01
Modern approach to differential equations presents subject in terms of ideas and concepts rather than special cases and tricks which traditional courses emphasized. No prerequisites needed other than a good calculus course. Certain concepts from linear algebra used throughout. Problem section at end of each chapter.
Cluster algebras in mathematical physics
International Nuclear Information System (INIS)
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
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are,in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A), and show an A∞-structure on E(A). Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos.
Li, Haisheng; Tan, Shaobin; Wang, Qing
2012-01-01
In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra with...
Grätzer, George
1979-01-01
Universal Algebra, heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .," has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well-selected additional bibliography of over 1250 papers and books which makes this a fine work for students, instructors, and researchers in the field. "This book will certainly be, in the years to come, the basic reference to the subject." --- The American Mathematical Monthly (First Edition) "In this reviewer's opinion [the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially sui...
Instrumentation of ICT-tools : the case of algebra in a computer algebra environment
Drijvers, P.; Herwaarden, van O.A.
2001-01-01
This paper describes a classroom experiment using hand held computer algebra for the learning of algebra. During a five week period students of the ninth grade (14-15 years old) used a symbolic calculator for solving systems of equations that contained parameters. In doing so, the aim was to develop
Yoneda algebras of almost Koszul algebras
Indian Academy of Sciences (India)
Zheng Lijing
2015-11-01
Let be an algebraically closed field, a finite dimensional connected (, )-Koszul self-injective algebra with , ≥ 2. In this paper, we prove that the Yoneda algebra of is isomorphic to a twisted polynomial algebra $A^!$ [ ; ] in one indeterminate of degree +1 in which $A^!$ is the quadratic dual of , is an automorphism of $A^!$, and = () for each $t \\in A^!$. As a corollary, we recover Theorem 5.3 of [2].
On the prolongation structures of Petrov type III vacuum spacetime equations
Ifidon, E. O.
2009-01-01
The universal covering symmetry algebra of the Robinson-Trautman equations of Petrov Type III is shown to include the infinite-dimensional affine Kac-Moody algebra A_1 as a prolongation algebra. This algebra has slower growth than the contragradient algebra K_2 obtained previously for this equation.
Algebra cohomology over a commutative algebra revisited
Pirashvili, Teimuraz
2003-01-01
The aim of this paper is to give a relatively easy bicomplex which computes the Shukla, or Quillen cohomology in the category of associative algebras over a commutative algebra $A$, in the case when $A$ is an algebra over a field.
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY
Institute of Scientific and Technical Information of China (English)
TaoChangli; LuShijie; ChenPeixin
2002-01-01
Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.
Institute of Scientific and Technical Information of China (English)
李邦河
2003-01-01
里特-吴特征集提供了用计算机解代数方程的有效方法,但迄今为止,还不能由这一方法给出孤立解的重数. 文章给出了孤立解的重数的两个定义,它们是等价的,并且在范德瓦尔登的定义有意义时与后者一致. 一个定义是在非标准分析的框架中,另一个则是标准分析的. 在证明与范德瓦尔登的定义一致时,非标准分析的定义是本质的. 通过再一次在计算机上应用里特-吴方法于由原方程得到的含无穷小参数的代数方程,可以得到原方程的孤立解的重数. 文中给出一个例子的计算机计算结果:首先得出有八个解,然后给出它们的重数:其中有两个的重数为六重,另六个为单根.%Ritt-Wu′s method of characteristic sets is an effective method to solve algebraic equations by computer, while one cannot get the multiplicity of an isolated solution by this method so far. We introduce two definitions for the multiplicity of an isolated solution, they are equivalent and coincide with that of Van der Waerden in the case the later works. One of our definitions is in the framework of nonstandard analysis, and another is in standard analysis. To prove the equivalence with that of Van der Waeden, the nonstandard one is essential. By using Ritt-Wu′s method on computer again to the algebraic equations with some infinitesimals as parameters obtained by modifying the original equations, we can then determine the multiplicity of an isolated solution of the original equations.An example by computer is given: first get 8 solutions, then compute their multiplicities-6 of them are simple roots, while each of the other two has multiplicity 6.
The Universal C*-Algebra of the Electromagnetic Field
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.
Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra
Dadbeh, Peruzan
2011-01-01
This paper presents equations for the inverse of a Clifford number in Clifford algebras of up to five dimensions. In presenting these, there are also presented formulas for the determinant and adjugate of a general Clifford number of up to five dimensions, matching the determinant and adjugate of the matrix representations of the algebra. These equations are independent of the metric used.
Enveloping algebras of some quantum Lie algebras
Pourkia, Arash
2014-01-01
We define a family of Hopf algebra objects, $H$, in the braided category of $\\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\\psi^2_{H\\otimes H}=id_{H\\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $\\psi^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analog...
The Yoneda algebra of a K2 algebra need not be another K2 algebra
Cassidy, T.; Phan, C.; Shelton, B.
2010-01-01
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
Workshop on Commutative Algebra
Simis, Aron
1990-01-01
The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results.
Idempotents of Clifford Algebras
Ablamowicz, R.; Fauser, B.; Podlaski, K.; Rembielinski, J.
2003-01-01
A classification of idempotents in Clifford algebras C(p,q) is presented. It is shown that using isomorphisms between Clifford algebras C(p,q) and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one sided ideals in Clifford algebras. Some low dimensional examples are discussed.
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
Generalized Quantum Current Algebras
Institute of Scientific and Technical Information of China (English)
ZHAO Liu
2001-01-01
Two general families of new quantum-deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enables one to define "tensor products" of these algebras. The standard quantum affine algebras turn out to be a very special case of the two algebra families, in which case the infinite Hopf family structure degenerates into a standard Hopf algebra. The relationship between the two algebraic families as well as thefr various special examples are discussed, and the free boson representation is also considered.
International Nuclear Information System (INIS)
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. They are a class of left-symmetric algebras with commutative right multiplication operators, which can be viewed as bosonic. Fermionic Novikov algebras are a class of left-symmetric algebras with anti-commutative right multiplication operators. They correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we commence a study on fermionic Novikov algebras from the algebraic point of view. We will show that any fermionic Novikov algebra in dimension ≤3 must be bosonic. Moreover, we give the classification of real fermionic Novikov algebras on four-dimensional nilpotent Lie algebras and some examples in higher dimensions. As a corollary, we obtain kinds of four-dimensional real fermionic Novikov algebras which are not bosonic. All of these examples will serve as a guide for further development including the application in physics
Institute of Scientific and Technical Information of China (English)
朱玲妹; 杨德贵; 王小灵
2003-01-01
In this paper, we investigate the growth of transcendental entire solutions of the following algebraic differential equation a(z)f ′2+(b 2(z)f2+b1(z)f+b0(z))f ′=d3(z)f3+d2(z)f2+d 1(z)f+d0(z), where a(z), bi(z) (0≤I≤2) and dj(z) (0≤j≤3) are all polynomials, and this equation relates closely to the following well -known algebraic differential equation C(z,w)w′2+B(z,w)w′+A(z,w)=0, where C(z,w)0, B(z,w) and A(z,w) are three polynomials in z and w. We give relationships between the growth of entire solutions and the degrees of the above three polynomials in detail.%研究了如下代数微分方程a(z)f ′2+(b2(z)f2+b1(z)f+b 0(z))f ′=d3(z)f3+d2(z)f2+d1(z)f+d0(z)(这里a(z),bi(z)(0≤i≤2)和dj(z) (0≤j≤3)是多项式)超越整函数解的增长性,这类方程与有名的代数微分方程C(z,w)w′2+B(z,w)w′+A(z,w)=0(C(z,w)0, B(z,w)和A(z ,w)是z和w的3个多项式)有紧密的关系.详细地给出了第1个方程的整函数解的增长性与它的3个多项式的次数之间的关系.
Energy Technology Data Exchange (ETDEWEB)
Kedem, Rinat [Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61821 (United States)], E-mail: rinat@uiuc.edu
2008-05-16
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)
Representing Feynman graphs on BV-algebras
van Suijlekom, Walter D
2008-01-01
We study the structure of renormalization Hopf algebras of gauge theories. We identify certain Hopf subalgebras in them, whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms. This can be understood physically as wave function renormalization and renormalization of the coupling constants, respectively. After taking into account the Slavnov-Taylor identities for the couplings as generators of a Hopf ideal, we find Hopf subalgebras in the corresponding quotient as well. In the second part of the paper, we explain the origin of these Hopf ideals by considering a coaction of the renormalization Hopf algebras on the Batalin-Vilkovisky (BV) algebras generated by the fields and couplings constants. The so-called classical master equation satisfied by the action in the BV-algebra implies the existence of the above Hopf ideals in the renormalization Hopf algebra. Finally, we exemplify our construction by applying it to Yang-Mills gauge theory.
Algebraically periodic translation surfaces
Calta, Kariane; Smillie, John
2007-01-01
Algebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically periodic surface. We show that for an algebraically periodic surface the slopes of the algebraically periodic directions are given by a number field which we call the periodic direction field. We show that translation surfaces with pseudo-Anosov automorphisms...
Clifford Algebra with Mathematica
Aragon-Camarasa, G.; Aragon-Gonzalez, G; Aragon, J. L.; Rodriguez-Andrade, M. A.
2008-01-01
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, a package for Clifford algebra calculations for the computer algebra program Mathematica is introduced through a presentation of the main ideas of Clifford algebras and illustrative examples. This package can be a useful computational tool since allows the manipulation of all these mathematical ob...
Institute of Scientific and Technical Information of China (English)
Jia-feng; Lü
2007-01-01
[1]Priddy S.Koszul resolutions.Trans Amer Math Soc,152:39-60 (1970)[2]Beilinson A,Ginszburg V,Soergel W.Koszul duality patterns in representation theory.J Amer Math Soc,9:473-525 (1996)[3]Aquino R M,Green E L.On modules with linear presentations over Koszul algebras.Comm Algebra,33:19-36 (2005)[4]Green E L,Martinez-Villa R.Koszul and Yoneda algebras.Representation theory of algebras (Cocoyoc,1994).In:CMS Conference Proceedings,Vol 18.Providence,RI:American Mathematical Society,1996,247-297[5]Berger R.Koszulity for nonquadratic algebras.J Algebra,239:705-734 (2001)[6]Green E L,Marcos E N,Martinez-Villa R,et al.D-Koszul algebras.J Pure Appl Algebra,193:141-162(2004)[7]He J W,Lu D M.Higher Koszul Algebras and A-infinity Algebras.J Algebra,293:335-362 (2005)[8]Green E L,Marcos E N.δ-Koszul algebras.Comm Algebra,33(6):1753-1764 (2005)[9]Keller B.Introduction to A-infinity algebras and modules.Homology Homotopy Appl,3:1-35 (2001)[10]Green E L,Martinez-Villa R,Reiten I,et al.On modules with linear presentations.J Algebra,205(2):578-604 (1998)[11]Keller B.A-infinity algebras in representation theory.Contribution to the Proceedings of ICRA Ⅸ.Beijing:Peking University Press,2000[12]Lu D M,Palmieri J H,Wu Q S,et al.A∞-algebras for ring theorists.Algebra Colloq,11:91-128 (2004)[13]Weibel C A.An Introduction to homological algebra.Cambridge Studies in Avanced Mathematics,Vol 38.Cambridge:Cambridge University Press,1995
Institute of Scientific and Technical Information of China (English)
2008-01-01
Analytical solutions of governing equations of various phenomena have their irre-placeable theoretical meanings. In addition, they can also be the benchmark solu-tions to verify the outcomes and codes of numerical solutions, and even to develop various numerical methods such as their differencing schemes and grid generation skills as well. A hybrid method of separating variables for simultaneous partial differential equation sets is presented. It is proposed that different methods of separating variables for different independent variables in the simultaneous equa-tion set may be used to improve the solution derivation procedure, for example, using the ordinary separating method for some variables and using extraordinary methods of separating variables, such as the separating variables with addition promoted by the first author, for some other variables. In order to prove the ability of the above-mentioned hybrid method, a lot of analytical exact solutions of two-buoyancy convection in porous media are successfully derived with such a method. The physical features of these solutions are given.
Institute of Scientific and Technical Information of China (English)
CAI RuiXian; LIU QiBin
2008-01-01
Analytical solutions of governing equations of various phenomena have their irre-placeable theoretical meanings. In addition, they can also be the benchmark solu-tions to verify the outcomes and codes of numerical solutions, and even to develop various numerical methods such as their differencing schemes and grid generation skills as well. A hybrid method of separating variables for simultaneous partial differential equation sets is presented. It is proposed that different methods of separating variables for different independent variables in the simultaneous equa-tion set may be used to improve the solution derivation procedure, for example, using the ordinary separating method for some variables and using extraordinary methods of separating variables, such as the separating variables with addition promoted by the first author, for some other variables. In order to prove the ability of the above-mentioned hybrid method, a lot of analytical exact solutions of two-buoyancy convection in porous media are successfully derived with such a method. The physical features of these solutions are given.
Sigma-models and Homotopy Algebras
Zeitlin, Anton M
2015-01-01
We review the relation between homotopy algebras of conformal field theory and geometric structures arising in sigma models. In particular we formulate conformal invariance conditions, which in the quasi-classical limit are Einstein equations with extra fields, as generalized Maurer-Cartan equations.
Titration Calculations with Computer Algebra Software
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,…
Classical Hamiltonian Dynamics and Lie Group Algebras
Aycock, B; Silverberg, J L; Widom, A
2008-01-01
The classical Hamilton equations of motion yield a structure sufficiently general to handle an almost arbitrary set of ordinary differential equations. Employing elementary algebraic methods, it is possible within the Hamiltonian structure to describe many physical systems exhibiting Lie group symmetries. Elementary examples include magnetic moment precession and the mechanical orbits of color charged particles in classical non-abelian chromodynamics.
Maps from the enveloping algebra of the positive Witt algebra to regular algebras
Sierra, Susan J.; Walton, Chelsea
2015-01-01
We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.
Induced Modules of Semisimple Hopf Algebras
Institute of Scientific and Technical Information of China (English)
Jun Hu; Yinhuo Zhang
2007-01-01
Let K be a field. Let H be a finite-dimensional K-Hopf algebra and D(H) be the Drinfel'd double of H. In this paper, we study Radford's induced module Hβ, whereβ is a group-like element in H*. Using the commuting pair established in [7], we obtain an analogue of the class equation for H*β when H is semisimple and cosemisimple. In case H is a finite group algebra or a factorizable semisimple cosemisimple Hopf algebra, we give an explicit decomposition of each Hβ into a direct sum of simple D(H)-modules.
Generalized Galilean algebras and Newtonian gravity
Directory of Open Access Journals (Sweden)
N. González
2016-04-01
Full Text Available 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.
Generalized Galilean Algebras and Newtonian Gravity
Albornoz, N L González; Salgado, P; Salgado, S
2016-01-01
The non-relativistic versions of the generalized Poincar\\'{e} algebras and generalized $AdS$-Lorentz algebras are obtained. This non-relativistic algebras are called, generalized Galilean algebras type I and type II and denoted by $\\mathcal{G}\\mathfrak{B}_{n}$ and $\\mathcal{G}\\mathfrak{L}_{_{n}}$ respectively. Using a generalized In\\"{o}n\\"{u}--Wigner contraction procedure we find that the generalized Galilean algebras type I can be obtained from the generalized Galilean algebras type II. The $S$-expansion procedure allows us to find the $\\mathcal{G}\\mathfrak{B}_{_{5}}$ algebra from the Newton--Hooke algebra with central extension. The procedure developed in Ref. \\cite{newton} allow us to show that the non-relativistic 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.
Filtering Algebraic Multigrid and Adaptive Strategies
Energy Technology Data Exchange (ETDEWEB)
Nagel, A; Falgout, R D; Wittum, G
2006-01-31
Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically.
Algebraic structure and Poisson's theory of mechanico-electrical systems
Institute of Scientific and Technical Information of China (English)
Liu Hong-Ji; Tang Yi-Fa; Fu Jing-Li
2006-01-01
The algebraic structure and Poisson's integral theory of mechanico-electrical systems are studied.The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained.The Lie algebraic structure and the Poisson's integral theory of Lagrange mechanico-electrical systems are derived.The Lie algebraic structure admitted and Poisson's integral theory of the Lagrange-Maxwell mechanico-electrical systems are presented.Two examples are presented to illustrate these results.
Brackets in representation algebras of Hopf algebras
Massuyeau, Gwenael; Turaev, Vladimir
2015-01-01
For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of so-called $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generaliza...
Samuel, Pierre
2008-01-01
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal
Boicescu, V; Georgescu, G; Rudeanu, S
1991-01-01
The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.
Osborn, J
1989-01-01
During the academic year 1987-1988 the University of Wisconsin in Madison hosted a Special Year of Lie Algebras. A Workshop on Lie Algebras, of which these are the proceedings, inaugurated the special year. The principal focus of the year and of the workshop was the long-standing problem of classifying the simple finite-dimensional Lie algebras over algebraically closed field of prime characteristic. However, other lectures at the workshop dealt with the related areas of algebraic groups, representation theory, and Kac-Moody Lie algebras. Fourteen papers were presented and nine of these (eight research articles and one expository article) make up this volume.
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.
Relation between dual S-algebras and BE-algebras
Directory of Open Access Journals (Sweden)
Arsham Borumand Saeid
2015-05-01
Full Text Available In this paper, we investigate the relationship between dual (Weak Subtraction algebras, Heyting algebras and BE-algebras. In fact, the purpose of this paper is to show that BE-algebra is a generalization of Heyting algebra and dual (Weak Subtraction algebras. Also, we show that a bounded commutative self distributive BE-algebra is equivalent to the Heyting algebra.
Homomorphisms between JC*-algebras and Lie C*-algebras
Institute of Scientific and Technical Information of China (English)
Chun Gil PARK; Jin Chuan HOU; Sei Qwon OH
2005-01-01
It is shown that every almost *-homomorphism h: A → B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r ＞ 1) for all x ∈ A, and that every almost linear mapping h: A → B is a *-homomorphism when h(2nu o y) = h(2nu) o h(y),h(3nu o y) = h(3nu) o h(y) or h(qnu o y) = h(qnu) o h(y) for all unitaries u ∈ A, all y ∈ A, and n = 0, 1, Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings.We prove that every almost *-homomorphism h: A → B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r ＞ 1) for all x ∈ A.
Directory of Open Access Journals (Sweden)
M. Sivasubramanian
2009-01-01
Full Text Available Problem statement: After formulating the special theory of relativity in 1905, Albert Einstein politely remarked: for velocities that are greater than light our deliberations become meaningless. In 1962, Sudarshan and his co-researchers proposed a hypothesis that particles/objects whose rest mass is imaginary can travel by birth faster than light. After the publication of Sudarshans research, many scholars began to probe into faster than light phenomena. In extended relativity, many properties of tachyons have been found. But still this micro second, the velocity of a free tachyon with respect to us is unknown. In this research the researchers found tachyon velocity. Approach: In this research, Einsteins variation of mass with velocity equation was transformed into quadratic equation. We introduced a new hypothesis to find the roots of the quadratic equation. Results: By introducing a new hypothesis in tachyon algebra, the researchers found that the velocity of superluminal objects with respect to us is v = c√3 where c is the velocity of the light. Conclusion/Recommendations: But the road to tachyon is too long. Hereafter it is up to experimental physicists to establish the existence/generation of tachyons.
Representations of twisted current algebras
Lau, Michael
2013-01-01
We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.
Hom-alternative algebras and Hom-Jordan algebras
Makhlouf, Abdenacer
2009-01-01
The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.
Cellularity of diagram algebras as twisted semigroup algebras
Wilcox, Stewart
2010-01-01
The Temperley-Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups. This theorem, which generalises a recent result of East about semigroup algebras of inverse semigroups, allows us to easily reproduce the cellularity of these algebras.
Institute of Scientific and Technical Information of China (English)
韩众; 张玉峰; 赵忠龙
2013-01-01
Based upon an improved unified algebra method and implement in the symbolic computation system Mathematica,the (2 + 1)-dimensional Zakharov-Kuznetsov modified equal width equation was considered.This method converted the work of constructing exact travelling wave solutions for an equation into solving a system of nonlinear algebra equations (NLAEs).After solving the system of nonlinear algebra equations,abundant general form solutions are obtained,including rational function solutions,trigonometric function solutions,hyperbolic function solutions,Jacobi elliptic function solutions,Weierstrass elliptic function solutions.The profiles of some obtained solutions are also given out.%利用一种改进的统一代数方法将构造(2+1)维ZK-MEW((2+1)-dimensional Zakharov-Kuznetsov modified equal width)方程精确行波解的问题转化为求解一组非线性的代数方程组.再借助于符号计算系统Mathematica求解所得到的非线性代数方程组,最终获得了方程的多种形式的精确行波解.其中包括有理解,三角函数解,双曲函数解,双周期Jacobi椭圆函数解,双周期Weierstrass椭圆形式解等.并给出了部分解的图形.
Cylindric-like algebras and algebraic logic
Ferenczi, Miklós; Németi, István
2013-01-01
Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.
Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra
Lagraa, M.; Touhami, N.
1997-01-01
We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.
Leinster, Tom
2000-01-01
We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the ...
Algebraic statistics computational commutative algebra in statistics
Pistone, Giovanni; Wynn, Henry P
2000-01-01
Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.
International Nuclear Information System (INIS)
We construct a special type of quantum soliton solutions for quantized affine Toda models. The elements of the principal Heisenberg subalgebra in the affinised quantum Lie algebra are found. Their eigenoperators inside the quantized universal enveloping algebra for an affine Lie algebra are constructed to generate quantum soliton solutions
Deficiently Extremal Gorenstein Algebras
Indian Academy of Sciences (India)
Pavinder Singh
2011-08-01
The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if / is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of / must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained.
Connecting Arithmetic to Algebra
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Khovanova, Tanya
2008-01-01
I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss other related sets of graphs. This construction can be used to build models of representations of simply-laced compact Lie groups.
Algebraic formulation of duality
International Nuclear Information System (INIS)
Two dimensional lattice spin (chiral) models over (possibly non-abelian) compact groups are formulated in terms of a generalized Pauli algebra. Such models over cyclic groups are written in terms of the generalized Clifford algebra. An automorphism of this algebra is shown to exist and to lead to the duality transformation
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers
Zheng, Quan
2016-05-01
We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.
Bases of Schur algebras associated to cellularly stratified diagram algebras
Bowman, C
2011-01-01
We examine homomorphisms between induced modules for a certain class of cellularly stratified diagram algebras, including the BMW algebra, Temperley-Lieb algebra, Brauer algebra, and (quantum) walled Brauer algebra. We define the `permutation' modules for these algebras, these are one-sided ideals which allow us to study the diagrammatic Schur algebras of Hartmann, Henke, Koenig and Paget. We construct bases of these Schur algebras in terms of modified tableaux. On the way we prove that the (quantum) walled Brauer algebra and the Temperley-Lieb algebra are both cellularly stratified and therefore have well-defined Specht filtrations.
Non RG logarithms via RG equations
Malyshev, Dmitry
2004-01-01
We compute complete leading logarithms in $\\Phi^4$ theory with the help of Connes and Kreimer RG equations. These equations are defined in the Lie algebra dual to the Hopf algebra of graphs. The results are compared with calculations in parquet approximation. An interpretation of the new RG equations is discussed.
Algebraic relaxation of a time correlation function
International Nuclear Information System (INIS)
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)
Jespers, Eric; Riley, David; Siciliano, Salvatore
2007-01-01
An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically generated GI-algebra locally finite?
Computer algebra and operators
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Indian Academy of Sciences (India)
Antonio J Calderón Martín; Manuel Forero Piulestán; José M Sánchez Delgado
2012-05-01
We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras is of the form $M=\\mathcal{U}+\\sum_jI_j$ with $\\mathcal{U}$ a subspace of the abelian Malcev subalgebra and any $I_j$ a well described ideal of satisfying $[I_j, I_k]=0$ if ≠ . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.
The Colombeau Quaternion Algebra
Cortes, W.; Ferrero, M. A.; Juriaans, S. O.
2008-01-01
We introduce the Colombeau Quaternio Algebra and study its algebraic structure. We also study the dense ideal, dense in the algebraic sense, of the algebra of Colombeau generalized numbers and use this show the existence of a maximal ting of quotions which is Von Neumann regular. Recall that it is already known that then algebra of COlombeau generalized numbers is not Von Neumann regular. We also use the study of the dense ideals to give a criteria for a generalized holomorphic function to sa...
Taghavi, Ali
2013-01-01
We study some properies of $Z^{*}$ algebras, thos C^* algebra which all positive elements are zero divisors. We show by means of an example that an extension of a Z* algebra by a Z* algebra is not necessarily Z* algebra. However we prove that an extension of a non Z* algebra by a non Z* algebra is again a Z^* algebra. As an application of our methods, we prove that evey compact subset of the positive cones of a C* algebra has an upper bound in the algebra.
2-Local derivations on matrix algebras over commutative regular algebras
Ayupov, Sh. A.; Kudaybergenov, K. K.; Alauadinov, A. K.
2012-01-01
The paper is devoted to 2-local derivations on matrix algebras over commutative regular algebras. We give necessary and sufficient conditions on a commutative regular algebra to admit 2-local derivations which are not derivations. We prove that every 2-local derivation on a matrix algebra over a commutative regular algebra is a derivation. We apply these results to 2-local derivations on algebras of measurable and locally measurable operators affiliated with type I von Neumann algebras.
Operator Algebras of Functions
Mittal, Meghna
2009-01-01
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.
Lectures on algebraic statistics
Drton, Mathias; Sullivant, Seth
2009-01-01
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
The graded Lie algebra of general relativity
Reiterer, Michael
2014-01-01
We construct a graded Lie algebra in which a solution to the vacuum Einstein equations is any element of degree 1 whose bracket with itself is zero. Each solution generates a cochain complex, whose first cohomology is linearized gravity about that solution. We gauge-fix to get a smaller cochain complex with the same cohomologies (deformation retraction). The new complex is much smaller, it consists of the solution spaces of linear homogeneous wave equations (symmetric hyperbolic equations). The algorithm that produces these gauges and wave equations is both for linearized gravity and the full Einstein equations. The gauge groupoid is the groupoid of rank 2 complex vector bundles.
Super Virasoro algebra and solvable supersymmetric quantum field theories
International Nuclear Information System (INIS)
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)
Perturbed linear rough differential equations
Coutin, Laure; Lejay, Antoine
2014-01-01
We study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebra more general than tensor algebra and to consider extensions of classical results such as the Mag...
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…
On Derivations Of Genetic Algebras
International Nuclear Information System (INIS)
A genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. In application of genetics this algebra often has a basis corresponding to genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. In this paper, we find the connection between the genetic algebras and evolution algebras. Moreover, we prove the existence of nontrivial derivations of genetic algebras in dimension two
On Generalized I-Algebras and 4-valued Modal Algebras
Figallo, Aldo V
2012-01-01
In this paper we establish a new characterization of 4-valued modal algebras considered by A. Monteiro. In order to obtain this characterization we introduce a new class of algebras named generalized I-algebras. This class contains strictly the class of C-algebras defined by Y. Komori as an algebraic counterpart of the infinite-valued implicative Lukasiewicz propositional calculus. On the other hand, the relationship between I-algebras and conmutative BCK-algebras, defined by S. Tanaka in 1975, allows us to say that in a certain sense G-algebras are also a generalization of these latter algebras
Omni-Lie Color Algebras and Lie Color 2-Algebras
Zhang, Tao
2013-01-01
Omni-Lie color algebras over an abelian group with a bicharacter are studied. The notions of 2-term color $L_{\\infty}$-algebras and Lie color 2-algebras are introduced. It is proved that there is a one-to-one correspondence between Lie color 2-algebras and 2-term color $L_{\\infty}$-algebras.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Stable endomorphism algebras of modules over special biserial algebras
Schröer, Jan; Zimmermann, Alexander
2002-01-01
We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle.
$L_{\\infty}$ algebra structures of Lie algebra deformations
Gao, Jining
2004-01-01
In this paper,we will show how to kill the obstructions to Lie algebra deformations via a method which essentially embeds a Lie algebra into Strong homotopy Lie algebra or $L_{\\infty}$ algebra. All such obstructions have been transfered to the revelvant $L_{\\infty}$ algebras which contain only three terms
Conformal and current algebras on a general Riemann surface
International Nuclear Information System (INIS)
Starting from a path integral formulation, Ward identifies are derived for conformal algebras on a general Riemann surface. An n + 1-point amplitude with energy-momentum tensor insertion is related by the Ward identity to an n-point amplitude and its derivative with respect to the modular parameters. This paper discusses how conformal algebra, defined in each coordinate path, is glued together to form a global algebraic structure on the Riemann surface. The authors discuss differential equations for correlation and partition functions of a theory corresponding to a degenerates representation of virasoro algebra
Lax operator algebras and Hamiltonian integrable hierarchies
Sheinman, Oleg K
2009-01-01
We consider the theory of Lax equations in complex simple and reductive classical Lie algebras with the spectral parameter on a Riemann surface of finite genus. Our approach is based on the new objects -- the Lax operator algebras, and develops the approach of I.Krichever treating the $\\gl(n)$ case. For every Lax operator considered as the mapping sending a point of the cotangent bundle on the space of extended Tyrin data to an element of the corresponding Lax operator algebra we construct the hierarchy of mutually commuting flows given by Lax equations and prove that those are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example we derive elliptic $A_n$, $C_n$, $D_n$ Calogero-Moser systems in frame of our approach.
International Nuclear Information System (INIS)
We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver's theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program. (general)
Evolution algebras and their applications
Tian, Jianjun Paul
2008-01-01
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
Poincare-Lie algebra and noncommutative differential calculus
International Nuclear Information System (INIS)
A realization of Poincare-Lie algebra in terms of noncommutative differential calculus is constructed. Corresponding relativistic quantum mechanics is considered. The important conclusion is that field equations appear in the integral form
Process Algebra Approach to Reasoning About Concurrent Actions
Institute of Scientific and Technical Information of China (English)
YuanFeng; Ming-ShengYing
2004-01-01
A reasonable transition rule is proposed for synchronized actions and some equational properties of bisimilarity and weak bisimilarity in the process algebra for reasoning about concurrent actions are presented.
Four Lie algebras associated with R6 and their applications
Zhang, Yufeng; Tam, Honwah
2010-09-01
The first part in the paper reads that a three-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which isospectral problems are established. By employing zero curvature equations, a modified Kaup-Newell (mKN) soliton hierarchy of evolution equations is obtained. The corresponding hereditary operator and Hamiltonian structure are worked out, respectively. Then two types of enlarging semisimple Lie algebras isomorphic to the linear space R6 are followed to construct, one of them is a complex Lie algebra. Their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mKN hierarchy. The hereditary operators, Hamiltonian structures of the hierarchies are produced again, respectively. The exact computing formulas of the constant γ appearing in the trace identity and the variational identity are derived under the semisimple algebras. The second part of this paper is devoted to constructing two kinds of Lie algebras by using product of complex vectors, which are also isomorphic to the linear space R6. Then we make use of the corresponding loop algebras to produce two integrable hierarchies along with bi-Hamiltonian structures. From various aspects, we give some ways for constructing Lie algebras which have extensive applications in generating integrable Hamiltonian systems.
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Relative Homological Algebra Volume 1
2011-01-01
This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists. The book is also suitable for an introductory course in commutative and ordinary homological algebra.
Finite-dimensional (*)-serial algebras
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let A be a finite-dimensional associative algebra with identity over a field k. In this paper we introduce the concept of (*)-serial algebras which is a generalization of serial algebras. We investigate the properties of (*)-serial algebras, and we obtain suficient and necessary conditions for an associative algebra to be (*)-serial.
Commutative combinatorial Hopf algebras
Hivert, F.; Novelli, J. -C.; Thibon, J. -Y.
2006-01-01
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary tre...
Algebraic nonlinear collective motion
Troupe, J.; Rosensteel, G.
1999-01-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real number $\\Lambda$. The $\\Lambda=0$ solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear g...
Brouder, Christian
2002-01-01
The Laplace Hopf algebra created by Rota and coll. is generalized to provide an algebraic tool for combinatorial problems of quantum field theory. This framework encompasses commutation relations, normal products, time-ordered products and renormalisation. It considers the operator product and the time-ordered product as deformations of the normal product. In particular, it gives an algebraic meaning to Wick's theorem and it extends the concept of Laplace pairing to prove that the renormalise...
Geometric Algebras and Extensors
Fernandez, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2007-01-01
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to met...
Symmetric Extended Ockham Algebras
Institute of Scientific and Technical Information of China (English)
T.S. Blyth; Jie Fang
2003-01-01
The variety eO of extended Ockham algebras consists of those algealgebra with an additional endomorphism k such that the unary operations f and k commute. Here, we consider the cO-algebras which have a property of symmetry. We show that there are thirty two non-isomorphic subdirectly irreducible symmetric extended MS-algebras and give a complete description of them.2000 Mathematics Subject Classification: 06D15, 06D30
How to be Brilliant at Algebra
Webber, Beryl
2010-01-01
How to be Brilliant at Algebra is contains 40 photocopiable worksheets designed to improve students' understanding of number relationships and patterns. They will learn about: odds and evens; patterns; inverse operations; variables; calendars; equations; pyramid numbers; digital root patterns; prime numbers; Fibonacci numbers; Pascal's triangle.
Infinite Charge Algebra of Gravitational Instantons
Hoppe, Jens; Park, Q-Han
1993-01-01
Using a formalism of minitwistors, we derive infinitely many conserved charges for the $sl(\\infty )$-Toda equation which accounts for gravitational instantons with a rotational Killing symmetry. These charges are shown to form an infinite dimensional algebra through the Poisson bracket which is isomorphic to two dimensional area preserving diffeomorphism with central extentions.
Infinite Charge Algebra of Gravitational Instantons
Hoppe, J; Hoppe, Jens
1994-01-01
Using a formalism of minitwistors, we derive infinitely many conserved charges for the $sl(\\infty )$-Toda equation which accounts for gravitational instantons with a rotational Killing symmetry. These charges are shown to form an infinite dimensional algebra through the Poisson bracket which is isomorphic to two dimensional area preserving diffeomorphism with central extentions.
Algebra 1Q, Mathematics: 5215.12.
Hirigoyen, Hector
This is the second of the six guidebooks on minimum course content for first-year algebra; it includes the ordered field properties of the real number system, solution of linear equations and inequalities, verbal problems, exponents and operations with polynomials. Overall goals for the course are stated; performance objectives for each unit, a…
Algebra 1p, Mathematics: 5215.11.
Strachan, Florence; Hirigoyen, Hector
This is the first of six guidebooks on minimum course content for first-year algebra; it introduces the language of sets, the fundamental operations and properties of the real number system, the use of variables, and the solution of simple linear equations and inequalities. Overall goals for the course are stated; then performance objectives, a…
Cayley-Dickson and Clifford Algebras as Twisted Group Algebras
Bales, John W.
2011-01-01
The effect of some properties of twisted groups on the associated algebras, particularly Cayley-Dickson and Clifford algebras. It is conjectured that the Hilbert space of square-summable sequences is a Cayley-Dickson algebra.
Kimura, Taro
2015-01-01
For a quiver with weighted arrows we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.
Kurosh, A G; Stark, M; Ulam, S
1965-01-01
Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. The book examines the parallelism between the theory of groups and the theory of rings; such examinations show the
Algebraic extensions of fields
McCarthy, Paul J
1991-01-01
""...clear, unsophisticated and direct..."" - MathThis textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamenta
Shafarevich, Igor Rostislavovich
2005-01-01
This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches
Solomon, Alan D
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean
Underwood, Robert G
2015-01-01
This text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras, and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields, and linearly recursive sequences. The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforw...
Transseries and real differential algebra
Hoeven, Joris
2006-01-01
Transseries are formal objects constructed from an infinitely large variable x and the reals using infinite summation, exponentiation and logarithm. They are suitable for modeling "strongly monotonic" or "tame" asymptotic solutions to differential equations and find their origin in at least three different areas of mathematics: analysis, model theory and computer algebra. They play a crucial role in Écalle's proof of Dulac's conjecture, which is closely related to Hilbert's 16th problem. The aim of the present book is to give a detailed and self-contained exposition of the theory of transseries, in the hope of making it more accessible to non-specialists.
Algebraic analysis of minimal representations
Kobayashi, Toshiyuki
2010-01-01
Small representations of a group bring us to large symmetries in a representation space. Analysis on minimal representations utilises large symmetries in their geometric models, and serves as a driving force in creating new interesting problems that interact with other branches of mathematics. This article discusses the following three topics that arise from minimal representations of the indefinite orthogonal group: 1. construction of conservative quantities for ultra-hyperbolic equations, 2. quantative discrete branching laws, 3. deformation of the Fourier transform with emphasis on the prominent roles of Sato's idea on algebraic analysis.
Relations Between BZMVdM-Algebra and Other Algebras
Institute of Scientific and Technical Information of China (English)
高淑萍; 邓方安; 刘三阳
2003-01-01
Some properties of BZMVdM-algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMVdM-algebra can produce a quasi-lattice implication algebra. The relations between BZMVdM-algebra and other algebras are discussed in detail. A pseudo-distance function is defined in linear BZMVdM-algebra, and its properties are derived.
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
Hijligenberg, N.W. van den; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
Zheng-xin CHEN; Ya-nan LIN
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)C1/I(A) of complex degenerate composition Lie algebras L(A)C1 by some ideals, where L(A)C1 is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)C1/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)C1 generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)C1 generated by simple A-modules.
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.
Universal Algebras of Hurwitz Numbers
A. Mironov; Morozov, A; Natanzon, S.
2009-01-01
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
Tilting theory and cluster algebras
Reiten, Idun
2010-01-01
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
Indian Academy of Sciences (India)
Cătălin Ciupală
2005-02-01
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: -algebras. We also define the Frölicher–Nijenhuis bracket in the non-commutative geometry on -algebras.
Symplectic $C_\\infty$-algebras
Hamilton, Alastair; Lazarev, Andrey
2007-01-01
In this paper we show that a strongly homotopy commutative (or $C_\\infty$-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\\infty$-algebra (an $\\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a $\\ci$-algebra and does not generalize to algebras over other operads.
On algebraic volume density property
Kaliman, Shulim; Kutzschebauch, Frank
2012-01-01
A smooth affine algebraic variety $X$ equipped with an algebraic volume form $\\omega$ has the algebraic volume density property (AVDP) if the Lie algebra generated by completely integrable algebraic vector fields of $\\omega$-divergence zero coincides with the space of all algebraic vector fields of $\\omega$-divergence zero. We develop an effective criterion of verifying whether a given $X$ has AVDP. As an application of this method we establish AVDP for any homogeneous space $X=G/R$ that admi...
(Quasi-)Poisson enveloping algebras
Yang, Yan-Hong; Yuan YAO; Ye, Yu
2010-01-01
We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
PARK Hong Goo; LEE Jeongsig; CHOI Seul Hee; CHEN XueQing; NAM Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m, m+n).
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
PARK; Hong; Goo; LEE; Jeongsig; CHOI; Seul; Hee; NAM; Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n).
Heinicke, C; Heinicke, Christian; Hehl, Friedrich W.
2001-01-01
We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some computer algebra systems and packages relevant to applications in gravitational physics. We conclude by presenting a couple of typical examples.
May, Robert D.
2016-01-01
Left and right "generalized Schur algebras", previously introduced by the author, are defined and analyzed. Filtrations of these algebras lead, in most cases, to parameterizations of the their irreducible representations over fields of characteristic 0 and fields of positive characteristic p.
Computing upper cluster algebras
Matherne, Jacob; Muller, Greg
2013-01-01
This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated. We include several examples of presentations produced by these methods.
Greuel, Gert-Martin
2000-01-01
Inhalte der Grundvorlesungen Lineare Algebra I und II im Winter- und Sommersemester 1999/2000: Gruppen, Ringe, Körper, Vektorräume, lineare Abbildungen, Determinanten, lineare Gleichungssysteme, Polynomring, Eigenwerte, Jordansche Normalform, endlich-dimensionale Hilberträume, Hauptachsentransformation, multilineare Algebra, Dualraum, Tensorprodukt, äußeres Produkt, Einführung in Singular.
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Algebraic Differential Characters
Esnault, H
1996-01-01
We give a construction of algebraic differential characters, receiving classes of algebraic bundles with connection, lifitng the Chern-Simons invariants defined with S. Bloch, the classes in the Chow group and the analytic secondary invariants if the variety is defined over the field of complex numbers.
Directory of Open Access Journals (Sweden)
Carlos C. Peña
2000-05-01
Full Text Available Topological algebras of sequences of complex numbers are introduced, endowed with a Hadamard product type. The complex homomorphisms on these algebras are characterized, and units, prime cyclic ideals, prime closed ideals, and prime minimal ideals, discussed. Existence of closed and maximal ideals are investigated, and it is shown that the Jacobson and nilradicals are both trivial.
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Symmetry Analysis of Telegraph Equation
Nadjafikhah, Mehdi; Hejazi, Seyed Reza
2010-01-01
Lie symmetry group method is applied to study the Telegraph equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
International Nuclear Information System (INIS)
Traces of powers of the zero mode in the W3 Algebra have recently been found to be of interest, for example in relation to Black Hole thermodynamics, and arise as the terms in an expansion of the full characters of the algebra. We calculate the first few such powers in two cases. Firstly, we find the traces in the 3-state Potts model by using null vectors to derive modular differential equations for the traces. Secondly, we calculate the exact results for Verma module representations. We compare our two methods with each other and the result of brute-force diagonalisation for low levels and find complete agreement
Integrable quantum systems and classical Lie algebras
International Nuclear Information System (INIS)
Six new infinite series of trigonometric solutions to triangule equations (quantum R-matrices) associated with the nonexceptional simple Lie algebras: sl(N), sp(N), o(N) have been obtained. R-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). These R-matrices provide an exact integrability of anisotropic generalizations of sl(N), sp(N), o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras
Unifying Clifford algebra formalism for relativistic fields
International Nuclear Information System (INIS)
It is shown that a Clifford algebra formalism provides a unifying description of spin-0, -1/2, and -1 fields. Since the operators and operands are both expressed in terms of the same Clifford algebra, the formalism obtains some results which are considerably different from those of the standard of formalisms for these fields. In particular, the conservation laws are obtained uniquely and unambiguously from the equations of motion in this formalism and do not suffer from the ambiguities and inconsistencies of the standard methods
Elements of mathematics algebra
Bourbaki, Nicolas
2003-01-01
This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...
Introduction to noncommutative algebra
Brešar, Matej
2014-01-01
Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
Kristen Abernathy; Jeremy Burke
2016-01-01
Despite improvements in cancer therapy and treatments, tumor recurrence is a common event in cancer patients. One explanation of recurrence is that cancer therapy focuses on treatment of tumor cells and does not eradicate cancer stem cells (CSCs). CSCs are postulated to behave similar to normal stem cells in that their role is to maintain homeostasis. That is, when the population of tumor cells is reduced or depleted by treatment, CSCs will repopulate the tumor, causing recurrence. In this pa...
Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras
Ashihara, Takahiro; Miyamoto, Masahiko
2008-01-01
If a vertex operator algebra $V=\\oplus_{n=0}^{\\infty}V_n$ satisfies $\\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In t...
The algebraic locus of Feynman integrals
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 is demonstrated by the two-loop vacuum diagram, where the algebraic locus can be interpreted as a threshold phenomenon and is given by the zeros of the Baikov polynomial. In addition, the symmetry group is interpreted geometrically as linear transformations of loop currents which preserve the subspace of squared currents within the space of quadratics in loop currents.
Lattice Landau Gauge and Algebraic Geometry
Mehta, Dhagash; von Smekal, Lorenz; Williams, Anthony G
2009-01-01
Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple 3x3 systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.
Successfully Transitioning to Linear Equations
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
Homotopy commutative algebra and 2-nilpotent Lie algebra
Dubois-Violette, Michel; Popov, Todor
2012-01-01
The homotopy transfer theorem due to Tornike Kadeishvili induces the structure of a homotopy commutative algebra, or $C_{\\infty}$-algebra, on the cohomology of the free 2-nilpotent Lie algebra. The latter $C_{\\infty}$-algebra is shown to be generated in degree one by the binary and the ternary operations.
The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra
Indian Academy of Sciences (India)
Vijay Kodiyalam; V S Sunder
2006-11-01
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.
Algebraic K-theory and algebraic topology
International Nuclear Information System (INIS)
This contribution treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers
The abc-conjecture for Algebraic Numbers
Institute of Scientific and Technical Information of China (English)
Jerzy BROWKIN
2006-01-01
The abc-conjecture for the ring of integers states that, for every ε＞ 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1+ε with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.In the present paper we propose an abc-conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number.From this abc-conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over Q.
Algebraic classification of Robinson-Trautman spacetimes
Podolsky, Jiri
2016-01-01
We consider a general class of four-dimensional geometries admitting a null vector field that has no twist and no shear but has an arbitrary expansion. We explicitly present the Petrov classification of such Robinson-Trautman (and Kundt) gravitational fields, based on the algebraic properties of the Weyl tensor. In particular, we determine all algebraically special subcases when the optically privileged null vector field is a multiple principal null direction (PND), as well as all the cases when it remains a single PND. No field equations are a priori applied, so that our classification scheme can be used in any metric theory of gravity in four dimensions. In the classic Einstein theory this reproduces previous results for vacuum spacetimes, possibly with a cosmological constant, pure radiation and electromagnetic field, but can be applied to an arbitrary matter content. As non-trivial explicit examples we investigate specific algebraic properties of the Robinson-Trautman spacetimes with a free scalar field, ...
A new algebra which transmutes to the braided algebra
Yildiz, A
1999-01-01
We find a new braided Hopf structure for the algebra satisfied by the entries of the braided matrix $BSL_q(2)$. A new nonbraided algebra whose coalgebra structure is the same as the braided one is found to be a two parameter deformed algebra. It is found that this algebra is not a comodule algebra under adjoint coaction. However, it is shown that for a certain value of one of the deformation parameters the braided algebra becomes a comodule algebra under the coaction of this nonbraided algebr...
Certain Clifford-like algebra and quantum vertex algebras
Li, Haisheng; Tan, Shaobin; Wang, Qing
2015-01-01
In this paper, we study in the context of quantum vertex algebras a certain Clifford-like algebra introduced by Jing and Nie. We establish bases of PBW type and classify its $\\mathbb N$-graded irreducible modules by using a notion of Verma module. On the other hand, we introduce a new algebra, a twin of the original algebra. Using this new algebra we construct a quantum vertex algebra and we associate $\\mathbb N$-graded modules for Jing-Nie's Clifford-like algebra with $\\phi$-coordinated modu...
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
The Stabilized Poincare-Heisenberg algebra: a Clifford algebra viewpoint
Gresnigt, N. G.; Renaud, P. F.; Butler, P. H.
2006-01-01
The stabilized Poincare-Heisenberg algebra (SPHA) is the Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after attempting to combine the Lie algebras of quantum mechanics and relativity which by themselves are stable, however not when combined. In this paper we show how the sixteen dimensional Clifford algebra CL(1,3) can be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional ...
Algebraic Signal Processing Theory
Pueschel, Markus; Moura, Jose M. F.
2006-01-01
This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a linear signal model defined as a triple (A, M, phi), where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively, and phi generalizes the concept of the z-transform to bijective linear mappings from a vector space of, e.g., signal samples, into the module M. A signal model provides the structure for a p...
Transgression and Clifford algebras
Rohr, Rudolf Philippe
2007-01-01
Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, >..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/)$ is isomorphic to a Clifford algebra $\\text{Cl}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by ...
Michael Roitman
2003-01-01
In this paper we prove that for any commutative (but in general non-associative) algebra $A$ with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra $V = V_0 \\oplus V_2 \\oplus V_3\\oplus ...$, such that $\\dim V_0 = 1$ and $V_2$ contains $A$. We can choose $V$ so that if $A$ has a unit $e$, then $2e$ is the Virasoro element of $V$, and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as well. In addition, the algebra $V$ can be chosen with...
Diassociative algebras and their derivations
International Nuclear Information System (INIS)
The paper concerns the derivations of diassociative algebras. We introduce one important class of diassociative algebras, give simple properties of the right and left multiplication operators in diassociative algebras. Then we describe the derivations of complex diassociative algebras in dimension two and three
Methods of algebraic geometry in control theory
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...
Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries
Sparano, G; Vinogradov, A M
2001-01-01
Vacuum gravitational fields invariant for a bidimensional non Abelian Lie algebra of Killing fields, are explicitly described. They are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order differential equation on the plane. Gravitational fields determined via the tortoise equation, are invariant for a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global gravitational fields out of local ones are also constructed.
Reinvention of early algebra : developmental research on the transition from arithmetic to algebra
Amerom, B.A. van
2002-01-01
In chapter 1 we give our reasons for carrying out this developmental research project on the transition from arithmetic to algebra, which includes the design of an experimental learning strand on solving equations. Chapter 2 describes the theoretical background of the book: current views on the teac
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.
The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
Directory of Open Access Journals (Sweden)
Karl Hallowell
2007-09-01
Full Text Available Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R. These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.
Riemann surfaces, Clifford algebras and infinite dimensional groups
International Nuclear Information System (INIS)
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.)
Irreducible Lie-Yamaguti algebras
Benito, Pilar; Elduque, Alberto; Martín-Herce, Fabián
2008-01-01
Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them...
Hom-power associative algebras
Yau, Donald
2010-01-01
A generalization of power associative algebra, called Hom-power associative algebra, is studied. The main result says that a multiplicative Hom-algebra is Hom-power associative if and only if it satisfies two identities of degrees three and four. It generalizes Albert's result that power associativity is equivalent to third and fourth power associativity. In particular, multiplicative right Hom-alternative algebras and non-commutative Hom-Jordan algebras are Hom-power associative.
Unitary spaces on Clifford algebras
Marchuk, N. G.; Shirokov, D. S.
2007-01-01
For the complex Clifford algebra Cl(p,q) of dimension n=p+q we define a Hermitian scalar product. This scalar product depends on the signature (p,q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra.