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.
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
Algebraic solution of master equations
R. Rangel; L. Carvalho
2003-01-01
We present a simple analytical method to solve master equations for finite temperatures and any initial conditions, which consists in the expansion of the density operator into normal modes. These modes and the expansion coefficients are obtained algebraically by using ladder superoperators. This algebraic technique is successful in cases in which the Liouville superoperator is quadratic in the creation and annihilation operators.
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...
Reflection algebra and functional equations
Galleas, W.; Lamers, J.
2014-01-01
In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partitio
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.
General Theory of Algebraic Equations
Bezout, Etienne
2008-01-01
This book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations. It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bézout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bézout's theorem," is stat
Reflection algebra and functional equations
Energy Technology Data Exchange (ETDEWEB)
Galleas, W., E-mail: w.galleas@uu.nl; Lamers, J., E-mail: j.lamers@uu.nl
2014-09-15
In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations satisfied by the partition function.
RICCATI EQUATIONS ON NONCOMMUTATIVE BANACH ALGEBRAS
Curtain, Ruth
2011-01-01
Conditions for the existence of a solution of a Riccati equation to be in some prescribed noncommutative involutive Banach algebras are given. The Banach algebras are inverse-closed subalgebras of the space of bounded linear operators on some Hilbert space, and the Riccati equation has an exponentia
An Algebraic Approach to the Scattering Equations
Huang, Rijun; 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.
Algebraic Approaches to Partial Differential Equations
Xu, Xiaoping
2012-01-01
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations...
On a Equation in Finite Algebraically Structures
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
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.
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.
On Riccati equations in Banach algebras
Curtain, Ruth F
2010-01-01
Let $R$ be a commutative complex Banach algebra with the involution $\\cdot ^\\star$ and suppose that $A\\in R^{n\\times n}$, $B\\in R^{n\\times m}$, $C\\in R^{p\\times n}$. The question of when the Riccati equation
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.
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.
Elimination and nonlinear equations of Rees algebra
Busé, Laurent; Simis, Aron
2009-01-01
A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, ...
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
Kersten, Paul H.M.; Gragert, Peter K.H.
1983-01-01
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.
Surveys in differential-algebraic equations IV
Reis, Timo
2017-01-01
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs) which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - History of DAEs - DAE aspects of mechanical multibody systems - Model reduction of DAEs - Observability for DAEs - Numerical Analysis for DAEs The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
Noncommutative Algebraic Equations and Noncommutative Eigenvalue Problem
Schwarz, A
2000-01-01
We analyze the perturbation series for noncommutative eigenvalue problem $AX=X\\lambda$ where $\\lambda$ is an element of a noncommutative ring, $ A$ is a matrix and $X$ is a column vector with entries from this ring. As a corollary we obtain a theorem about the structure of perturbation series for Tr $x^r$ where $x$ is a solution of noncommutative algebraic equation (for $r=1$ this theorem was proved by Aschieri, Brace, Morariu, and Zumino, hep-th/0003228, and used to study Born-Infeld lagrangian for the gauge group $U(1)^k$).
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.
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.
Differential-algebraic solutions of the heat equation
Buchstaber, Victor M.; Netay, Elena Yu.
2014-01-01
In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.
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.
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.
Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations
2014-07-01
non- linear hybrid systems by linear algebraic methods. In Radhia Cousot and Matthieu Martel, editors, SAS, volume 6337 of LNCS, pages 373–389. Springer...Tarski. A decision method for elementary algebra and geometry. Bulletin of the American Mathematical Society, 59, 1951. [36] Wolfgang Walter. Ordinary...Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 July 2014
Dual number coefficient octonion algebra, field equations and conservation laws
Chanyal, B. C.; Chanyal, S. K.
2016-08-01
Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.
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.
Quasi-lisse vertex algebras and modular linear differential equations
Arakawa, Tomoyuki
2016-01-01
We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level $-h^{\\vee}/6-1$, which expresses the homogeneous Schur limit of the superconformal index of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Hidden sl$_{2}$-algebra of finite-difference equations
Smirnov, Yu F; 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. (Talk presented at the Wigner Symposium, Guadalajara, Mexico, August 1995; to be published in Proceedings)
Comparative study of homotopy continuation methods for nonlinear algebraic equations
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
Newton—Like Iteration Method for Solving Algebraic Equations
Institute of Scientific and Technical Information of China (English)
JihuanHE
1998-01-01
In this paper,a Newton-like iteration method is proposed to solve an approximate solution of an algebraic equation.The iteration formula obtained by homotopy perturbation method contains the well-known Newton iteration formulain logic.
Algebraic and geometric structures of analytic partial differential equations
Kaptsov, O. V.
2016-11-01
We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.
Positive Stabilization of Linear Differential Algebraic Equation System
Directory of Open Access Journals (Sweden)
Muhafzan
2016-01-01
Full Text Available We study in this paper the existence of a feedback for linear differential algebraic equation system such that the closed-loop system is positive and stable. A necessary and sufficient condition for such existence has been established. This result can be used to detect the existence of a state feedback law that makes the linear differential algebraic equation system in closed loop positive and stable.
2008-01-01
Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the s.c. "gravitational theories with covariant and contravariant connection and metrics", it is shown that a wide variety of third, fourth, fifth, seventh, tenth- degree algebraic equations exists in gravity theory. This is important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Since the obtained...
A Geometric Treatment of Implicit Differential-Algebraic Equations
Rabier, P. J.; Rheinboldt, W. C.
A differential-geometric approach for proving the existence and uniqueness of implicit differential-algebraic equations is presented. It provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems. The differential-algebraic equation is transformed into an explicit ordinary differential equation by a reduction process that can be abstractly defined for specific submanifolds of tangent bundles here called reducible π-submanifolds. Local existence and uniqueness results for differential-algebraic equations then follow directly from the final stage of this reduction by means of an application of the standard theory of ordinary differential equations.
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…
Reduced minimax filtering by means of differential-algebraic equations
Mallet, V.; Zhuk, S.
2011-01-01
A reduced minimax state estimation approach is proposed for high-dimensional models. It is based on the reduction of the ordinary differential equation with high state space dimension to the low-dimensional Differential-Algebraic Equation (DAE) and on the subsequent application of the minimax state
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.
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.
The Algebraic Riccati Matrix Equation for Eigendecomposition of Canonical Forms
Directory of Open Access Journals (Sweden)
M. Nouri
2013-01-01
Full Text Available The algebraic Riccati matrix equation is used for eigendecomposition of special structured matrices. This is achieved by similarity transformation and then using the algebraic Riccati matrix equation to the triangulation of matrices. The process is the decomposition of matrices into small and specially structured submatrices with low dimensions for easy finding of eigenpairs. Here, we show that previous canonical forms I, II, III, and so on are special cases of the presented method. Numerical and structural examples are included to show the efficiency of the present method.
Involutive characteristic sets of algebraic partial differential equation systems
Institute of Scientific and Technical Information of China (English)
陈玉福; 高小山
2003-01-01
This paper presents an algorithm to reduce a nonlinear algebraic partial differential equation system into the involutive characteristic set with respect to an abstract involutive prolongation direction, which covers the existing algorithms based on Riquier method, Thomas method, and Pommaret method. It also provides new algorithms for computing involutive characteristic sets due to the existence of new involutive directions. Experiments show that these new algorithms may be used to significantly reduce the computational steps in Wu-Ritt's characteristic set method for algebraic partial differential equations.
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.
Null controllability and the algebraic Riccati equation in Banach Spaces
Van Neerven, J.M.A.M.
2005-01-01
By a recent result of Priola and Zabczyk, a null controllable linear system [y'(t) = Ay(t) + Bu(t)] in a Hilbert space E is null controllable with vanishing energy if and only if it is null controllable and the only positive self-adjoint solution of the associated algebraic Riccati equation [XA + A*
Bounds for the eigenvalues of the continuous algebraic Riccati equation
Liu, Jianzhou; Zhang, Juan
2011-10-01
By using singular value decomposition and majorisation inequalities, we propose new upper and lower bounds for summations of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation. These bounds improve and extend some of the previous results. Finally, we give corresponding numerical examples to illustrate the effectiveness of our results.
Application of Computer Algebra in Solving Chaffee Infante Equation
Xie, Fu-Ding; Liu, Xiao-Dan; Sun, Xiao-Peng; Tang, Di
2008-04-01
In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find multiple soliton solutions of nonlinear partial differential equations, this approach is constructive and pure algebraic. The results found here are tested on computer and therefore their validity is ensured.
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.
All Meromorphic Solutions of Some Algebraic Differential Equations
Institute of Scientific and Technical Information of China (English)
Wenjun Yuan; Zifeng Huang; Jinchun Lai; Jianming Qi
2014-01-01
In this article, we introduce some results with respect to the integrality and exact solutions of some 2nd order algebraic DEs. We obtain the sufficient and neces-sary conditions of integrable and the general meromorphic solutions of these equa-tions by the complex method, which improves the corresponding results obtained by many authors. Our results show that the complex method provides a powerful math-ematical tool for solving a large number of nonlinear partial differential equations in mathematical physics.
Solvability of the $H^\\infty$ algebraic Riccati equation in Banach algebras
Sasane, Amol
2011-01-01
Let $R$ be a commutative complex unital semisimple Banach algebra with the involution $\\cdot ^\\star$. Sufficient conditions are given for the existence of a stabilizing solution to the $H^\\infty$ Riccati equation when the matricial data has entries from $R$. Applications to spatially distributed systems are discussed.
Algebraic matrix equations in two unknowns
Bourgeois, Gerald
2011-01-01
Let r1,r2,s1,s2 be integers such that gcd(r1,r2)=1 and gcd(s1,s2)=1. We solve the matrix equation A^{r1}B^{s1}A^{r2}B^{s2}=+-Identity where A,B are 2,2 complex matrices that have no common eigenvectors. Let p,q be coprime integers such that |p|+|q|>2. We study the matrix equation B^{-1}A^pB=A^q where A,B are n,n complex invertible matrices. We show that such matrices satisfy B^{-1}AB and A commute. We provide a necessary and sufficient condition for similarity of A^p and A^q. We explicitly solve this problem when A has n distinct eigenvalues and in other particular cases.
Yang-Baxter equation and quantum enveloping algebras
Ma Zhong Qi
1993-01-01
The exact solution of C N Yang's one-dimensional many-body problem with repulsive delta-function interactions and R J Baxter's eight-vertex statistical model are brilliant achievements in many-body statistical physics. A nonlinear equation, now known as the Yang-Baxter equation, is the key to the solution of both problems. The Yang-Baxter equation has also come to play an important role in such diverse topics as completely integrable statistical models, conformal and topological field theories, knots and links, braid groups and quantum enveloping algebras.This pioneering textbook attempts to m
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
A Novel Partial Differential Algebraic Equation (PDAE) Solver
DEFF Research Database (Denmark)
Lim, Young-il; Chang, Sin-Chung; Jørgensen, Sten Bay
2004-01-01
For solving partial differential algebraic equations (PDAEs), the space-time conservation element/solution element (CE/SE) method is addressed in this study. The method of lines (MOL) using an implicit time integrator is compared with the CE/SE method in terms of computational efficiency, solution...... 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 nonlinear adsorption isotherm are solved by the two methods. The CE/SE method enforces both local and global flux conservation in space and time, and uses a simple stencil structure (two points at the previous time level and one point at the present time level). Thus, accurate and computationally...
Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Petzold, L; Cao, Y; Li, S; Serban, R
2005-08-09
Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.
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.
Universal Jensen's Equations in Banach Modules over a C*-Algebra and Its Unitary Group
Institute of Scientific and Technical Information of China (English)
Chun Gil PARK
2004-01-01
In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen's equations in Banach modules over a unital C*-algebra. It is applied to show the stability of universal Jensen's equations in a Hilbert module over a unital C*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital C*-algebra.
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…
Elliptic grid generation based on Laplace equations and algebraic transformations
Energy Technology Data Exchange (ETDEWEB)
Spekreuse, S.P. [National Aerospace Lab., Amsterdam (Netherlands)
1995-04-01
An elliptic grid generation method is presented to generate boundary conforming grids in domains in 2D and 3D physical space and on minimal surfaces and parametrized surfaces in 3D physical space. The elliptic grid generation method is based on the use of a composite mapping. This composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation. The elliptic transformation is based on the Laplace equations for domains, or on the Laplace-Beltrami equations for surfaces. The algebraic transformation maps the computational space one to-one onto a parameter space. The elliptic transformation maps the parameter space one-to-one onto the domains or surfaces. The composition of these two mapping is a differentiable one-to-one mapping from computational space onto the domains or surfaces and has a nonvanishing Jacobian. This composite mapping defines the grid point distribution in the interior of the domains or surfaces. For domains and minimal surfaces, the composite mapping obeys a nonlinear elliptic Poisson system with control functions completely defined by the algebraic transformation. The solution of the Poisson systems is obtained by Picard iteration and black-box multigrid solvers. For parametrized curved surfaces, it is not necessary to define and solve a nonlinear elliptic Poisson system. Instead a linear elliptic system and an inversion problem is solved to generate the grid in the interior of the surface.
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...
The Generalised Discrete Algebraic Riccati Equation in LQ optimal control
Ferrante, Augusto
2012-01-01
This paper investigates the properties of the solutions of the generalised discrete algebraic Riccati equation arising from the solution of the classic infinite-horizon linear quadratic control problem. In particular, a geometric analysis is used to study the relationship existing between the solutions of the generalised Riccati equation and the output-nulling subspaces of the underlying system and the corresponding reachability subspaces. This analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem, which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil. In establishing themain results of this paper, several ancillay problems on the discrete Lyapunov equation and spectral factorisation are also addressed and solved.
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.
Symmetries of the Schrodinger Equation and Algebra/Superalgebra Duality
Energy Technology Data Exchange (ETDEWEB)
Toppan, Francesco
2014-12-15
Some key features of the symmetries of the Schroedinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving rst and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation dependent notion of on-shell symmetry is introduced. The difference in associating the time derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric. (author)
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.
The Shifting Technique for Solving a Nonsymmetric Algebraic Riccati Equation
Chiang, Chun-Yueh
2011-01-01
This paper analyzes a special instance of nonsymmetric algebraic matrix Riccati equations arising from transport theory. Traditional approaches for finding the minimal nonnegative solution of the matrix Riccati equations are based on the fixed point iteration and the speed of the convergence is linear. Relying on simultaneously matrix computation, a structure-preserving doubling algorithm (SDA) with quadratic convergence is designed for improving the speed of convergence. The difficulty is that the double algorithm with quadratic convergence cannot guarantee to work all the time. Our main trust in this work is to show that applied with a suitable shifted technique, the SDA is guaranteed to converge quadratically with no breakdown. Also, we modify the conventional simple iteration algorithm in the critical case to dramatically improve the speed of convergence. Numerical experiments strongly suggest that the total number of computational steps can be significantly reduced via the shifting procedure.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Wackerbauer, R. (Max-Planck-Institut fuer Extraterrestrische Physik, Garching (Germany)); Huebler, A. (Illinois Univ., Urbana, IL (United States). Center for Complex Systems Research); Mayer-Kress, G. (Los Alamos National Lab., NM (United States) California Univ., Santa Cruz, CA (United States). Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
Energy Technology Data Exchange (ETDEWEB)
Zhang Huiqun [College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071 (China)], E-mail: hellozhq@yahoo.com.cn
2009-02-15
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
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 ...
A Higher Dimensional Loop Algebra and Integrable Couplings System of Evolution Equations Hierarchy
Institute of Scientific and Technical Information of China (English)
夏铁成; 于发军; 陈登远
2005-01-01
An extension of the Lie algebra An-1 has been proposed [ Phys. Lett. A, 2003, 310 : 19-24 ]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra G～. Based on the loop algebra G～, the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.
On the Lie Symmetry Algebras of the Stationary Schrödinger and Pauli Equations
Boldyreva, M. N.; Magazev, A. A.
2017-02-01
A general method for constructing first-order symmetry operators for the stationary Schrödinger and Pauli equations is proposed. It is proven that the Lie algebra of these symmetry operators is a one-dimensional extension of some subalgebra of an e(3) algebra. We also assemble a classification of stationary electromagnetic fields for which the Schrödinger (or Pauli) equation admits a Lie algebra of first-order symmetry operators.
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...
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.
Applications of algebraic method to exactly solve some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)]. E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)]. E-mail: aramady@yahoo.com
2007-08-15
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear evolution equations is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDE's) are obtained. Graphs of the solutions are displayed.
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…
The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
Energy Technology Data Exchange (ETDEWEB)
Gurau, Razvan, E-mail: rgurau@perimeterinstitute.ca [Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo (Canada)
2012-12-01
Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson equations, generalizing the loop equations of matrix models, translate into constraints satisfied by the partition function. The constraints have been shown, in the large N limit, to close a Lie algebra indexed by colored rooted D-ary trees yielding a first generalization of the Virasoro algebra in arbitrary dimensions. In this paper we complete the Schwinger Dyson equations and the associated algebra at all orders in 1/N. The full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.
A multilayer recurrent neural network for solving continuous-time algebraic Riccati equations.
Wang, Jun; Wu, Guang
1998-07-01
A multilayer recurrent neural network is proposed for solving continuous-time algebraic matrix Riccati equations in real time. The proposed recurrent neural network consists of four bidirectionally connected layers. Each layer consists of an array of neurons. The proposed recurrent neural network is shown to be capable of solving algebraic Riccati equations and synthesizing linear-quadratic control systems in real time. Analytical results on stability of the recurrent neural network and solvability of algebraic Riccati equations by use of the recurrent neural network are discussed. The operating characteristics of the recurrent neural network are also demonstrated through three illustrative examples.
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
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.
Hartwig, J.T.; Stokman, J.V.
2013-01-01
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use co
A Note on Four-Dimensional Symmetry Algebras and Fourth-Order Ordinary Differential Equations
Directory of Open Access Journals (Sweden)
A. Fatima
2013-01-01
Full Text Available We provide a supplementation of the results on the canonical forms for scalar fourth-order ordinary differential equations (ODEs which admit four-dimensional Lie algebras obtained recently. Together with these new canonical forms, a complete list of scalar fourth-order ODEs that admit four-dimensional Lie algebras is available.
Sesiano, Jacques
2009-01-01
This text should not be viewed as a comprehensive history of algebra before 1600, but as a basic introduction to the types of problems that illustrate the earliest forms of algebra. It would be particularly useful for an instructor who is looking for examples to help enliven a course on elementary algebra with problems drawn from actual historical texts. -Warren Van Egmond about the French edition for MathSciNet This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever
Quantum Yang-Baxter equation and constant R-matrix over Grassmann algebra
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
Constant solutions to Yang-Baxter equation are investigated over Grassmann algebra for the case of 6-vertex R-matrix.The general classification of all possible solutions over Grassmann algebra and particular cases with 2,3,4 generators are studied.As distinct from the standard case, when R-matrix over number field can have a maximum 5 nonvanishing elements, we obtain over Grassmann algebra a set of new full 6-vertex solutions. The solutions leading to regular R-matrices which appear in weak Hopf algebras are considered.
Directory of Open Access Journals (Sweden)
Bapurao C. Dhage
2006-03-01
Full Text Available In this paper, we prove an existence theorem for hyperbolic differential equations in Banach algebras under Lipschitz and Caratheodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions.
Existence and Uniqueness of Solution of Schrodinger equation in extended Colombeau algebra
Directory of Open Access Journals (Sweden)
Fariba Fattahi
2014-09-01
Full Text Available In this paper, we establish the existence and uniquenessresult of the linear Schr¨odinger equation with Marchaudfractional derivative in Colombeau generalized algebra.The purpose of introducing Marchaud fractional derivativeis regularizing it in Colombeau sense.
Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra
Directory of Open Access Journals (Sweden)
Zhaolin Jiang
2014-01-01
Full Text Available The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of n×n complex skew-circulant matrices are displayed in this paper.
Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups
Breev, A. I.; Mosman, E. A.
2016-12-01
The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.
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
Kondo, Kenichi
2013-11-01
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.
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.
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.
Runge-Kutta collocation methods for differential-algebraic equations of indices 2 and 3
Skvortsov, L. M.
2012-10-01
Stiffly accurate Runge-Kutta collocation methods with explicit first stage are examined. The parameters of these methods are chosen so as to minimize the errors in the solutions to differential-algebraic equations of indices 2 and 3. This construction results in methods for solving such equations that are superior to the available Runge-Kutta methods.
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.
Some Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation
Directory of Open Access Journals (Sweden)
Zübeyde Ulukök
2013-01-01
Full Text Available We propose diverse upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE by building the equivalent form of the CARE and using some matrix inequalities and linear algebraic techniques. Finally, numerical example is given to demonstrate the effectiveness of the obtained results in this work as compared with some existing results in the literature. These new bounds are less restrictive and provide more efficient results in some cases.
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.
Grid generation: Algebraic and partial differential equations techniques revisited
Soni, Bharat K.
A systematic procedure for grid generation which can provide compuational grids for a wide range of geometries related to internal/external flow configuration is developed by combining the best features of algebraic and elliptic grid generation systems. The algebraic and elliptic grid generation system are well developed in the literature. A revisit to these techniques is given in this paper in view of economy and efficiency of the grid generation process. A technique to automatically calculate slopes and twist vectors required in hermite transfinite interpolation is developed. The weighted transfinite interpolation is combined with automatically created Bezier, B-spline curves, and Non-Uniform Rational B-spline (NURB) curves to generate well-distributed, smooth and near orthogonal grid patches (sub-blocks). A novel approach to evaluate control functions for elliptic generation systems is developed. This approach allows a quick refinement utilizing elliptic system. Computational examples are presented to demonstrate the success of these methodologies.
A New General Algebraic Method and Its Application to Shallow Long Wave Approximate Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
A new general algebraic method is presented to uniformly construct a series of exact solutions for nonlinear evolution equations (NLEEs). For illustration, we apply the new method to shallow long wave approximate equations and successfully obtain abundant new exact solutions, which include rational solitary wave solutions and rational triangular periodic wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.
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.
On the economical solution method for a system of linear algebraic equations
Directory of Open Access Journals (Sweden)
Jan Awrejcewicz
2004-01-01
Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22. The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.
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Shulin Wu
2009-01-01
Full Text Available We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed-point equations with vector form. Based on the fixed-point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form.
Extended Fan's Algebraic Method and Its Application to KdV and Variant Boussinesq Equations
Institute of Scientific and Technical Information of China (English)
YANG Xian-Lin; TANG Jia-Shi
2007-01-01
An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinear partial differential equations. The key idea of this method is to introduce an auxiliary ordinary differential equation which is regarded as an extended elliptic equation and whose degree r is expanded to the case of r ＞ 4. The efficiency of the method is demonstrated by the KdV equation and the variant Boussinesq equations. The results indicate that the method not only offers all solutions obtained by using Fu's and Fan's methods, but also some new solutions.
Solving differential–algebraic equation systems by means of index reduction methodology
DEFF Research Database (Denmark)
Sørensen, Kim; Houbak, Niels; Condra, Thomas
2006-01-01
of a number of differential equations and algebraic equations — a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of ordinary differential equations — ODEs....
Solving differential-algebraic equation systems by means of index reduction methodology
DEFF Research Database (Denmark)
Sørensen, Kim; Houbak, Niels; Condra, Thomas Joseph
2006-01-01
of a number of differential equations and algebraic equations - a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of Ordinary- Differential-Equations - ODE’s....
Convergence of Galerkin Solutions for Linear Differential Algebraic Equations in Hilbert Spaces
Matthes, Michael; Tischendorf, Caren
2010-09-01
The simulation of complex systems describing different physical effects becomes more and more of interest in various applications. Examples are couplings describing interactions between circuits and semiconductor devices, circuits and electromagnetic fields, fluids and structures. The modeling of such complex processes [1, 2, 3, 4, 7, 8] often leads to coupled systems that are composed of ordinary differential equations (ODEs), differential-algebraic equations (DAEs) and partial differential equations (PDEs). Such coupled systems can be regarded in the general framework of abstract differential-algebraic equations. Here, we discuss a Galerkin approach for handling linear abstract differential-algebraic equations with monotone operators. It is shown to provide solutions that converge to the unique solution of the abstract differential-algebraic system. Furthermore, the solution is proved to depend continuously on the data. The most interesting point of the Galerkin approach is the choice of basis functions. They have to be chosen in proper subspaces in order to guarantee that the solution satisfies the non-dynamic constraints. In contrast to other approaches as e.g. [5, 6], this approach allows time dependent operators but needs monotonicity.
Counting equations in algebraic attacks on block ciphers
DEFF Research Database (Denmark)
Knudsen, Lars Ramkilde; Miolane, Charlotte Vikkelsø
2010-01-01
in the system. One of the most difficult tasks is to determine the exact number of linearly independent equations one obtain in the attacks. In this paper, it is shown that by splitting the equations defined over a block cipher (an SP-network) into two sets, one can determine the exact number of linearly...
Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method
Directory of Open Access Journals (Sweden)
Chein-Shan Liu
2013-01-01
Full Text Available We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.
Institute of Scientific and Technical Information of China (English)
Yao-lin Jiang
2003-01-01
In this paper we presented a convergence condition of parallel dynamic iteration methods for a nonlinear system of differential-algebraic equations with a periodic constraint.The convergence criterion is decided by the spectral expression of a linear operator derivedfrom system partitions. Numerical experiments given here confirm the theoretical work ofthe paper.
Matrix Bounds for the Solution of the Continuous Algebraic Riccati Equation
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Juan Zhang
2010-01-01
Full Text Available We propose new upper and lower matrix bounds for the solution of the continuous algebraic Riccati equation (CARE. In certain cases, these lower bounds improve and extend the previous results. Finally, we give a corresponding numerical example to illustrate the effectiveness of our results.
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.
STABILITY RADIUS OF NON-SMOOTH PRITCHARD-SALAMON SYSTEMS AND THE ALGEBRAIC RICCATI EQUATION
Institute of Scientific and Technical Information of China (English)
Weisheng JIANG; Falun HUANG; Tingyu ZHU
2009-01-01
The authors discuss the stability radius of the non-smooth Pritchard-Salamon systems under structured perturbations. A formula for the stability radius in terms of the norm of a certain input-output operator is obtained. Furthermore, the relationship between stability radius and the solvability of some type of algebraic Riccati equations is given.
Order of growth of solutions to algebraic differential equations in the unit disk
Directory of Open Access Journals (Sweden)
D. Benbourenane
2004-01-01
Full Text Available S. B. Bank has shown that there is no uniform growth estimate for meromorphic solutions of algebraic differential equations with meromorphic coefficients in the unit disk. We give conditions under which such solutions must have a finite order of growth.
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…
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...
On a modification of minimal iteration methods for solving systems of linear algebraic equations
Yukhno, L. F.
2010-04-01
Modifications of certain minimal iteration methods for solving systems of linear algebraic equations are proposed and examined. The modified methods are shown to be superior to the original versions with respect to the round-off error accumulation, which makes them applicable to solving ill-conditioned problems. Numerical results demonstrating the efficiency of the proposed modifications are given.
Foley, Greg
2011-01-01
Continuous feed and bleed ultrafiltration, modeled with the gel polarization model for the limiting flux, is shown to provide a rich source of non-linear algebraic equations that can be readily solved using numerical and graphical techniques familiar to undergraduate students. We present a variety of numerical problems in the design, analysis, and…
Gasyna, Zbigniew L.
2008-01-01
Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)
Iftime, OV; Zwart, HJ; Curtain, RF
2005-01-01
We obtain a representation of all self-adjoint solutions of the control algebraic Riccati equation associated to the infinite-dimensional state linear system Sigma(A, B, C) under the following assumptions: A generates a C-0-group, the system is output stabilizable, strongly detectable and the dual R
Tonisson, Eno
2015-01-01
Sometimes Computer Algebra Systems (CAS) offer an answer that is somewhat different from the answer that is probably expected by the student or teacher. These (somewhat unexpected) answers could serve as a catalyst for rich mathematical discussion. In this study, over 120 equations from school mathematics were solved using 8 different CAS. Many…
Lie algebras of conservation laws of variational ordinary differential equations
Fiorani, Emanuele; Spiro, Andrea
2015-02-01
We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler-Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely they simultaneously preserve the holonomy distribution of the jets space and the action from which the Euler-Lagrange equations are derived.
The Wheeler-DeWitt Equation in Filćhenkov Model: The Lie Algebraic Approach
Panahi, H.; Zarrinkamar, S.; Baradaran, M.
2016-11-01
The Wheeler-DeWitt equation in Filćhenkov model with terms related to strings, dust, relativistic matter, bosons and fermions, and ultra stiff matter is solved in a quasi-exact analytical manner via the Lie algebraic approach. In the calculations, using the representation theory of sl(2), the general (N+1)-dimensional matrix equation is constructed whose determinant yields the solutions of the problem.
Theory and Application of a Class of Abstract Differential-Algebraic Equations
Pierson, Mark A
2005-01-01
We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy ...
Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra
Directory of Open Access Journals (Sweden)
J. C. Ndogmo
2016-01-01
Full Text Available An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution.
Energy Technology Data Exchange (ETDEWEB)
Bessis, N.; Bessis, G. [Laboratoire de Physique des Lasers, U.R.A. 282 du C.N.R.S., Universite Paris-Nord, avenue J. B. Clement, 93430 Villetaneuse (France)
1997-11-01
An algebraic procedure is proposed for the analytical solution of Schr{umlt o}dinger equations that can be viewed as a factorizable equation with an additional potential V(x). Once V(x) has been expanded in a series of suitable x-basis functions u=u(x), which are specific to each factorization type, the solution of the Riccati equation associated with the given equation is performed by means of an open perturbation technique, i.e., at each order of the perturbation, an additional balance u-dependent term is introduced so that the resulting equation becomes solvable. Since the unperturbed potential involves the whole given potential and since the balance term is expected to be small, improved results are expected at low orders of the perturbation, even at the zeroth order. The procedure, well adapted to the use of computer algebra, is applied to the solution of the gx{sup 4}-anharmonic oscillator equation: by means of very simple algebraic manipulations, the trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g=0.002 to g=20000). {copyright} {ital 1997 American Institute of Physics.}
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).
Algebraic Stress Model with RNG ε-Equation for Simulating Confined Strongly Swirling Turbulent Flows
Institute of Scientific and Technical Information of China (English)
Xu Jiangrong; Yao Qiang; Cao Xingyu; Cen Kefa
2001-01-01
Strongly swirl flow simulation are still under developing. In this paper, ε equation based on the Renormalization Group theory is used into algebraic stress model. Standard k-ε model, algebraic stress model by Jiang Zhang[5]and present model (RNG-ASM) are applied simultaneously to simulating the confined strongly swirling flow.The Simulating results by RNG-ASM model are compared to the results by other two model, it is shown that the predictions by this model display reasonable agreement with experimental data, and lead to greater improvement than Zhang's ASM turbulence model[5].
Multiple Representations for Systems of Linear Equations Via the Computer Algebra System Maple
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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.
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.
Directory of Open Access Journals (Sweden)
Qiang Zang
2013-01-01
Full Text Available For nonlinear differential-algebraic-equation subsystems, whose index is one and interconnection input is locally measurable, the problem of invertibility is discussed and the results are applied to the power systems component decentralized control. The inverse systems’ definitions for such a class of differential-algebraic-equation subsystems are put forward. A recursive algorithm is proposed to judge whether the controlled systems are invertible. Then physically feasible α-order integral right inverse systems are constructed, with which the composite systems are linearizaed and decoupled. Finally, decentralized excitation and valve coordinative control for one synchronous generator within multimachine power systems are studied and the simulation results based on MATLAB demonstrate the effectiveness of the control scheme proposed in this paper.
Tracking children's mental states while solving algebra equations.
Anderson, John R; Betts, Shawn; Ferris, Jennifer L; Fincham, Jon M
2012-11-01
Behavioral and function magnetic resonance imagery (fMRI) data were combined to infer the mental states of students as they interacted with an intelligent tutoring system. Sixteen children interacted with a computer tutor for solving linear equations over a six-day period (days 0-5), with days 1 and 5 occurring in an fMRI scanner. Hidden Markov model algorithms combined a model of student behavior with multi-voxel imaging pattern data to predict the mental states of students. We separately assessed the algorithms' ability to predict which step in a problem-solving sequence was performed and whether the step was performed correctly. For day 1, the data patterns of other students were used to predict the mental states of a target student. These predictions were improved on day 5 by adding information about the target student's behavioral and imaging data from day 1. Successful tracking of mental states depended on using the combination of a behavioral model and multi-voxel pattern analysis, illustrating the effectiveness of an integrated approach to tracking the cognition of individuals in real time as they perform complex tasks.
A Modern Introduction to Cardano and Ferrari Formulas in the Algebraic Equations
Fujii, K
2003-01-01
We give a modern approach to the famous Cardano and Ferrari formulas in the algebraic equations with three and four degrees. Namely, we reconstruct these formulas from the point of view of superposition principle in quantum computation based on three and four level systems which are being developed by the author. We also present a problem on some relation between Galois theory and Qudit theory.
Directory of Open Access Journals (Sweden)
Yu Liu
2009-01-01
Full Text Available By using diagonalizable matrix decomposition and majorization inequalities, we propose new trace bounds for the product of two real square matrices in which one is diagonalizable. These bounds improve and extend the previous results. Furthermore, we give some trace bounds for the solution of the algebraic Riccati equations, which improve some of the previous results under certain conditions. Finally, numerical examples have illustrated that our results are effective and superior.
Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations
Koutschan, Christoph; Schoeberl, Joachim
2011-01-01
We consider the numerical discretization of the time-domain Maxwell's equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approximations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of recurrence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with computer algebra methods reminiscent of the holonomic systems approach.
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
Grahovski, Georgi G.; Mikhailov, Alexander V.
2013-12-01
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 Schrodinger Equations on Grassmann Algebras
Grahovski, Georgi G
2013-01-01
Integrable discretisations for a class of coupled nonlinear Schrodinger (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 Grassmamm generalisations of the difference Toda and NLS equations. The resulting discrete systems will have Lax pairs provided by the set of two consistent Darboux transformations.
Energy Technology Data Exchange (ETDEWEB)
Davies, Richard [School of Technology, University of Glamorgan, Pontypridd, CF37 1DL (United Kingdom)]. E-mail: rkdavies@glam.ac.uk; Peng Shi [School of Technology, University of Glamorgan, Pontypridd, CF37 1DL (United Kingdom)]. E-mail: pshi@glam.ac.uk; Wiltshire, Ron [School of Technology, University of Glamorgan, Pontypridd, CF37 1DL (United Kingdom)]. E-mail: rjwiltsh@glam.ac.uk
2007-04-15
In dynamical systems studies, the so-called Riccati and Lyapunov equations play an important role in stability analysis, optimal control and filtering design. In this paper, upper matrix bounds for the perturbation of the stabilizing solution of the continuous algebraic Riccati equation (CARE) are derived for the case when one, or all the coefficient matrices are subject to small perturbations. Comparing with existing works on this topic, the proposed bounds are less restrictive. In addition to these bounds, iterative algorithms are also derived to obtain more precise estimates.
Coadjoint orbits of the Virasoro algebra and the global Liouville equation
Balog, J; Palla, L
1998-01-01
The classification of the coadjoint orbits of the Virasoro algebra is reviewed and is then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well-known that the Liouville equation for a smooth, real field $\\varphi$ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field $g \\in SL(2,R)$ is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field $Q=\\kappa g_{22}$ where $\\kappa\
Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software
Directory of Open Access Journals (Sweden)
L. L. Williams
2015-01-01
Full Text Available This paper calculates the Kaluza field equations with the aid of a computer package for tensor algebra, xAct. The xAct file is provided with this paper. We find that Thiry’s field equations are correct, but only under limited circumstances. The full five-dimensional field equations under the cylinder condition are provided here, and we see that most of the other references miss at least some terms from them. We go on to establish the remarkable Kaluza Lagrangian, and verify that the field equations calculated from it match those calculated with xAct, thereby demonstrating self-consistency of these results. Many of these results can be found scattered throughout the literature, and we provide some pointers for historical purposes. But our intent is to provide a definitive exposition of the field equations of the classical, five-dimensional metric ansatz of Kaluza, along with the computer algebra data file to verify them, and then to recover the unique Lagrangian for the theory. In common terms, the Kaluza theory is an “ω=0” scalar field theory, but with unique electrodynamic couplings.
N=2 Super - $W_{3}$ Algebra and N=2 Super Boussinesq Equations
Ivanov, E; Malik, R P
1995-01-01
We study classical $N=2$ super-$W_3$ algebra and its interplay with $N=2$ supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs - covariant reduction approach. These techniques have been previously applied by us in the bosonic $W_3$ case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general $N=2$ super Boussinesq equation and two kinds of the modified $N=2$ super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear $N=2$ super-$W_3^{\\infty}$ symmetry associated with $N=2$ super-$W_3$. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second hamiltonian structure.
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.
Non-Linear Integral Equations for complex Affine Toda associated to simply laced Lie algebras
Zinn-Justin, P
1998-01-01
A set of coupled non-linear integral equations is derived for a class of models connected with the quantum group $U_q(\\hat g)$ ($q=e^{i\\gamma}$ and $g$ simply laced Lie algebra), which are solvable using the Bethe Ansatz; these equations describe arbitrary excited states of a system with finite spatial length $L$. They generalize the Destri-De Vega equation for the Sine-Gordon/massive Thirring model to affine Toda field theory with imaginary coupling constant. As an application, the central charge and all the conformal weights of the UV conformal field theory are extracted in a straightforward manner. The quantum group truncation for rational values of $\\gamma/\\pi$ is discussed in detail; in the UV limit we recover through this procedure the RCFTs with extended $W(g)$ conformal symmetry.
Dependence of solutions of nonsmooth differential-algebraic equations on parameters
Stechlinski, Peter G.; Barton, Paul I.
2017-02-01
The well-posedness of nonsmooth differential-algebraic equations (DAEs) is investigated. More specifically, semi-explicit DAEs with Carathéodory-style assumptions on the differential right-hand side functions and local Lipschitz continuity assumptions on the algebraic equations. The DAEs are classified as having differential index one in a generalized sense; solution regularity is formulated in terms of projections of generalized (Clarke) Jacobians. Existence of solutions is derived under consistency and regularity of the initial data. Uniqueness of a solution is guaranteed under analogous Carathéodory ordinary-differential equation uniqueness assumptions. The continuation of solutions is established and sufficient conditions for continuous and Lipschitzian parametric dependence of solutions are also provided. To accomplish these results, a theoretical tool for analyzing nonsmooth DAEs is provided in the form of an extended nonsmooth implicit function theorem. The findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth DAEs.
Upper solution bounds of the continuous coupled algebraic Riccati matrix equation
Liu, Jianzhou; Zhang, Juan
2011-04-01
In this article, by using some matrix identities, we construct the equivalent form of the continuous coupled algebraic Riccati equation (CCARE). Further, with the aid of the eigenvalue inequalities of matrix's product, by solving the linear inequalities utilising the properties of M-matrix and its inverse matrix, new upper matrix bounds for the solutions of the CCARE are established, which improve and extend some of the recent results. Finally, a corresponding numerical example is proposed to illustrate the effectiveness of the derived results.
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.
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.
Meshkat, Nicolette; DiStefano, Joseph J
2012-01-01
Differential algebra approaches to structural identifiability analysis of a dynamic system model in many instances heavily depend upon Ritt's pseudodivision at an early step in analysis. The pseudodivision algorithm is used to find the characteristic set, of which a subset, the input-output equations, is used for identifiability analysis. A simpler algorithm is proposed for this step, using Gr\\"obner Bases, along with a proof of the method that includes a reduced upper bound on derivative requirements. Efficacy of the new algorithm is illustrated with two biosystem model examples.
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
D'Alfonso, Lisi; Ollivier, François; Sedoglavic, Alexandre; Solernó, Pablo
2010-01-01
This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely algebraic (polynomial) equation plus an under-determined ODE (that is, a semi-explicit DAE system of differentiation index 1) in as many variables as the order of the input system. This can be done by means of a Kronecker-type algorithm with bounded complexity.
Borzykh, A. N.
2017-01-01
The Seidel method for solving a system of linear algebraic equations and an estimate of its convergence rate are considered. It is proposed to change the order of equations. It is shown that the method described in Faddeevs' book Computational Methods of Linear Algebra can deteriorate the convergence rate estimate rather than improve it. An algorithm for establishing the optimal order of equations is proposed, and its validity is proved. It is shown that the computational complexity of the reordering is 2 n 2 additions and (12) n 2 divisions. Numerical results for random matrices of order 100 are presented that confirm the proposed improvement.
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.
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.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
A generalized variable-coefficient algebraic method is applied to construct several new families of exact solutions of physical interestfor (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.
Fixed Points and Stability of an Additive Functional Equation of n-Apollonius Type in C∗-Algebras
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Fridoun Moradlou
2008-01-01
Full Text Available Using the fixed point method, we prove the generalized Hyers-Ulam stability of C∗-algebra homomorphisms and of generalized derivations on C∗-algebras for the following functional equation of Apollonius type ∑i=1nf(z−xi=−(1/n∑1≤i
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.
Exact algebraization of the signal equation of spoiled gradient echo MRI
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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.
A subspace shift technique for solving close-to-critical nonsymmetric algebraic Riccati equations
Iannazzo, Bruno
2010-01-01
The worst situation in computing the minimal nonnegative solution $X_*$ of a nonsymmetric algebraic Riccati equation $\\mathcal R(X)=0$ associated with an M-matrix occurs when the derivative of $\\mathcal R$ at $X_*$ is near to a singular matrix. When the derivative of $\\mathcal R$ at $X_*$ is singular, the problem is ill-conditioned and the convergence of the algorithms based on matrix iterations is slow; however, there exist some techniques to remove the singularity and restore well-conditioning and fast convergence. This phenomenon is partially shown also in the close-to-critical case, but the techniques used for the null recurrent case cannot be applied to this setting. We present a new method to accelerate the convergence and amend the conditioning in close-to-critical cases. The numerical experiments confirm the efficiency of the new method.
Novel insights on the stabilising solution to the continuous-time algebraic Riccati equation
Rojas, A. J.
2014-11-01
In the present paper we present a closed-form solution, as a function of the closed-loop poles, for the continuous-time algebraic Riccati equations (CAREs) related to single-input single-output systems with non-repeated poles. The proposed solution trades the standard numerical algorithm approach for one based on a spectral factorisation argument, offering potential insight into any control technique based on a CARE and its solution. As an example, we present the equivalence of two fairly recent control over network results. Furthermore we apply the proposed result to the formula for the optimal regulator gain matrix k (or equivalently the Luenberger's observer gain l) and present an example. Finally, we conclude by discussing the possible extension of the proposed closed-form solution to the repeated eigenvalues case and to the case when the CARE is related to multiple-input multiple-output systems.
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
It was proved numerically that the Domain Decomposition Method (DDM) with one layer overlapping grids is identical to the block iterative method of linear algebra equations. The results obtained using DDM could be in reasonable aggeement with the results of full-domain simulation. With the three dimensional solver developed by the authors, the flow field in a pipe was simulated using the full-domain DDM with one layer overlapping grids and with patched grids respectively. Both of the two cases led to the convergent solution. Further research shows the superiority of the DDM with one layer overlapping grids to the DDM with patched grids. A comparison between the numerical results obtained by the authors and the experimental results given by Enayet[3] shows that the numerical results are reasonable.
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)
Liu Jianzhou
2009-01-01
Full Text Available By using singular value decomposition and majorization inequalities, we propose new inequalities for the trace of the product of two arbitrary real square matrices. These bounds improve and extend the recent results. Further, we give their application in the algebraic Riccati equation. Finally, numerical examples have illustrated that our results are effective and superior.
Bulatov, M. V.; Ming-Gong, Lee; Solovarova, L. S.
2010-11-01
Difference schemes of the Euler and trapezoidal types for the numerical solution of the initial-value problem for linear differential-algebraic equations are examined. These schemes are analyzed for model examples, and their superiority over the familiar first- and second-order implicit methods is shown. Conditions for the convergence of the proposed algorithms are formulated.
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.
Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions
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Viorica Mariela Ungureanu
2015-01-01
Full Text Available We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs affected simultaneously by multiplicative white noise (MN and Markovian jumps (MJs. Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.
New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-2M. 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)
柳银萍; 李志斌
2003-01-01
Based on a 0 of elliptic equation, a new algebraic method to construct a series of exact solutions for nonlinear evolution equations is proposed, meanwhile, its complete implementation TRWS in Maple is presented. The TRWS can output a series of travelling wave solutions entirely automatically, which include polynomial solutions, exponential function solutions, triangular function solutions, hyperbolic function solutions, rational function solutions, Jacobi elliptic function solutions, and Weierstrass elliptic function solutions. The effectiveness of the package is illustrated by applying it to a variety of equations. Not only are previously known solutions recovered but also new solutions and more general form of solutions are obtained.
Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras
Energy Technology Data Exchange (ETDEWEB)
Schlichenmaier, M [Mathematics Laboratory, University of Luxembourg (Luxembourg); Sheinman, O K [Independent University of Moscow, Moscow (Russian Federation)
2004-08-31
In this paper a global operator approach to the Wess-Zumino-Witten-Novikov theory for compact Riemann surfaces of arbitrary genus with marked points is developed. The term 'global' here means that Krichever-Novikov algebras of gauge and conformal symmetries (that is, algebras of global symmetries) are used instead of loop algebras and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous paper of the authors (Russ. Math. Surveys 54:1 (1999)). The present paper gives a construction of the conformal blocks and of a projectively flat connection on the bundle formed by them.
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.
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…
Mastering algebra retrains the visual system to perceive hierarchical structure in equations.
Marghetis, Tyler; Landy, David; Goldstone, Robert L
2016-01-01
Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.
Yukhno, L. F.
2007-12-01
The use of modifications of certain well-known methods of the conjugate direction type for solving systems of linear algebraic equations with rectangular matrices is examined. The modified methods are shown to be superior to the original versions with respect to the round-off accumulation; the advantage is especially large for ill-conditioned matrices. Examples are given of the efficient use of the modified methods for solving certain fairly large ill-conditioned problems.
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.
代数微分方程组的分量允许解%Components Admissible Solution of Algebraic Differential Equations Systems
Institute of Scientific and Technical Information of China (English)
李国望; 高凌云
2011-01-01
Using the Nevanlinna value distribution theory of meromorphic function, we investigate the existence problem of admissible solutions of higher-order algebraic differential equations systems, and obtain a result concerning admissible components of solution.
Ivanov, I. G.; Netov, N. C.; Bogdanova, B. C.
2015-10-01
This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived.
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.
McKeague, Charles P
1981-01-01
Elementary Algebra 2e, Second Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first tackles the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the substitution method, solving linear systems by graphing, solutions to linear equations in two variables, multiplication property of equality, word problems, addition property of equality, and subtraction, addition, multiplication, and division of real numbers. The manuscript then examines exponents and polynomials, factoring, and rational e
McKeague, Charles P
1986-01-01
Elementary Algebra, Third Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first ponders on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the elimination method, solving linear systems by graphing, word problems, addition property of equality, solving linear equations, linear inequalities, addition and subtraction of real numbers, and properties of real numbers. The text then takes a look at exponents and polynomials, factoring, and rational expressions. Topics include reducing ra
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
Liu, Jianzhou; Zhang, Juan
2011-08-01
In this article, applying the properties of M-matrix and non-negative matrix, utilising eigenvalue inequalities of matrix's sum and product, we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic Riccati equation (DCARE). Secondly, we discuss the solution existence uniqueness condition of the DCARE using the developed upper and lower matrix bounds and a fixed point theorem. Thirdly, a new fixed iterative algorithm of the solution for the DCARE is shown. Finally, the corresponding numerical examples are given to illustrate the effectiveness of the developed results.
An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).
Yu, Zhang; Zhang, Yufeng
2009-01-30
Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified KdV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of solition equations.
Existence Condition on Solutions to the Algebraic Riccati Equation%代数Riccati方程解的存在性条件
Institute of Scientific and Technical Information of China (English)
倪茂林
2008-01-01
First, the existence conditions on the solutions to the algebraic Riccati equation are reviewed. Then, a strict proof is presented for a necessary and sufficient condition on the existence of a unique optimal positive definite solution to this equation. By using this condition, some untrue results on the design of robust decentralized controllers are corrected.
Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.
2016-08-01
We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.
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
Nordtvedt, Kenneth
2015-01-01
A method for constructing metric gravity's N-body Lagrangian is developed which uses iterative, liner algebraic euqations which enforce invariance properties of gravity --- exterior effacement, interior effacement, and the time dilation and Lorentz contraction of matter under boosts. The method is demonstrated by obtaining the full 1/c^4 order Lagrangian, and a combination of exterior and interior effacement enforcement permits construction of the full Schwarzschild temporal and spatial metric potentials.
A new variable coefficient algebraic method and non-traveling wave solutions of nonlinear equations
Institute of Scientific and Technical Information of China (English)
Lu Bin; Zhang Hong-Qing
2008-01-01
In this paper,a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics,which is direct and more powerful than projective Riccati equation method.In order to illustrate the validity and the advantages of the method,(2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation is employed and many new double periodic non-travelling wave solutions are obtained.This algorithm can also be applied to other nonlinear differential equations.
The ⋆-value equation and Wigner distributions in noncommutative Heisenberg algebras
Rosenbaum, Marcos; Vergara, J. David
2006-04-01
We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.
The $\\star$-value Equation and Wigner Distributions in Noncommutative Heisenberg algebras
Rosenbaum, M; Rosenbaum, Marcos
2005-01-01
We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.
Institute of Scientific and Technical Information of China (English)
夏铁成; 于发军; 陈登远
2005-01-01
An extension of the Lie algebra An-1 has been proposed [Phys. Lett. A, 2003, 310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra G. Based on the loop algebra G, the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.
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.
Yang-Mills gauge fields conserving symmetry algebra of Dirac equation in homogeneous space
Breev, A I
2014-01-01
We consider the Dirac equation with external Yang-Mills gauge field in a homogeneous space with invariant metric. The Yang-Mills fields for which the motion group of the space serves as symmetry group of the Dirac equation are found by comparison of the Dirac equation with a invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The basis of eigenfunctions and corresponding spectrum are obtained for the Dirac equation in the space $\\mathbb{R}^2 \\times \\mathbb{S}^2$ in the framework of the noncommutative integration method.
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.…
RICCATI EQUATIONS ON NONCOMMUTATIVE BANACH ALGEBRAS (vol 49, pg 2542, 2011)
Curtain, Ruth
2013-01-01
This paper contains corrections to [R. Curtain, SIAM J. Control Optim., 49 (2011), pp. 2542-2557]. While all claims remain valid, and the proof for the linear quadratic Riccati equations is correct, this proof does not cover the positive-real and bounded-real Riccati equations. Here a correct proof
Directory of Open Access Journals (Sweden)
Mohammad Shahzad
2016-05-01
Full Text Available 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 robustness of the technique implemented on the chosen chaotic system.
Samuel, Koji; Mulenga, H. M.; Angel, Mukuka
2016-01-01
This paper investigates the challenges faced by secondary school teachers and pupils in the teaching and learning of algebraic linear equations. The study involved 80 grade 11 pupils and 15 teachers of mathematics, drawn from 4 selected secondary schools in Mufulira district, Zambia in Central Africa. A descriptive survey method was employed to…
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.
Stability of the Cauchy-Jensen Functional Equation in CÃ¢ÂˆÂ—-Algebras: A Fixed Point Approach
Directory of Open Access Journals (Sweden)
Jong Su An
2008-06-01
Full Text Available we prove the Hyers-Ulam-Rassias stability of CÃ¢ÂˆÂ—-algebra homomorphisms and of generalized derivations on CÃ¢ÂˆÂ—-algebras for the following Cauchy-Jensen functional equation 2f((x+y/2+z=f(x+f(y+2f(z, which was introduced and investigated by Baak (2006. The concept of Hyers-Ulam-Rassias stability originated from the stability theorem of Th. M. Rassias that appeared in (1978.
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
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...
Jamal, Sameerah; Shabbir, Ghulam
2017-02-01
We study the geometric properties of generators for the Klein-Gordon equation in Kantowski-Sachs and certain Bianchi-type spaces. Several versions of the Klein-Gordon equation are derived from its dependence on a potential function. The criteria for different versions of the (1+3) Klein-Gordon equation originates from analyzing three sources, viz. through generators that are identically the Killing algebra, or with the Killing vector fields that are recast into linear combinations and thirdly, real sub-algebras within the conformal algebra. In turn, these equations admit a catalogue of infinitesimal symmetries that are equivalent to the corresponding Killing vector fields in Kantowski-Sachs, Bianchi type III, IX, VIII, VI0 and VII0 space-times, with the exception of a linear vector W=upartialu in every case. The sheer number of results are displayed in appropriate tables. Subsequently, in application, we derive some Noetherian conservation laws and identify some exact solutions by quadratures.
Energy Technology Data Exchange (ETDEWEB)
Morgansen, K.A.; Pin, F.G.
1995-03-01
This paper describes an enhanced version of the code for the Full Space Parameterization (FSP) method that has recently been presented for determining optimized (and possibly constrained) solutions, x, to underspecified system`s of algebraic equations b = Ax. The enhanced code uses the conditions necessary for linear independence of the m {minus} n + 1 vectors forming the solution as a basis for an efficient search pattern to quickly find the full set of solution vectors. A discussion is made of the complications which may be present due to the particular combination of the matrix A and the vector b. The first part of the code implements the various methods needed to handle these particular cases before the solution vectors are calculated so that computation time may be decreased. The second portion of the code implements methods which can be used to calculate the necessary solution vectors. The respective expressions of the full solution space, S, for the cases of the matrix A being full rank and rank deficient are given. Finally, examples of the resolution of particular cases are provided, and a sample application to the joint motion of a mobile manipulator for a given end-effector trajectory is presented.
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.
Li, Jing; Hong, Wenxue
2014-12-01
The feature extraction and feature selection are the important issues in pattern recognition. Based on the geometric algebra representation of vector, a new feature extraction method using blade coefficient of geometric algebra was proposed in this study. At the same time, an improved differential evolution (DE) feature selection method was proposed to solve the elevated high dimension issue. The simple linear discriminant analysis was used as the classifier. The result of the 10-fold cross-validation (10 CV) classification of public breast cancer biomedical dataset was more than 96% and proved superior to that of the original features and traditional feature extraction method.
Institute of Scientific and Technical Information of China (English)
张晓梅; 孙道椿
2011-01-01
利用待定系数的方法研究了一类二阶线性齐次亚纯系数复微分方程的亚纯解及代数元素解的存在性.%The existence of meromorphic and algebraic element solutions for a kind of secondorder complex homogeneous linear differential equations is investigated by using the method of undetermined coefficients.
Algebra for Gifted Third Graders.
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Institute of Scientific and Technical Information of China (English)
CHENYong; WANGQi; LIBiao
2004-01-01
Making use of a new and more general ansatz, we present the generalized algebraic method to uniformly construct a series of new and general travelling wave solution for nonlinear partial differential equations. As an application of the method, we choose a (1+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by the method proposed by Fan [E. Fan, Comput. Phys. Commun. 153 (2003) 17] and find other new and more general solutions at the same time, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic and soliton solutions, Jacobi and Weierstrass doubly periodic wave solutions.
Shafarevich, I
1994-01-01
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
Schlichenmaier, Martin; Sheinman, Oleg K.
2003-01-01
This paper continues the same-named article, Part I (math.QA/9812083). We give a global operator approach to the WZWN theory for compact Riemann surfaces of an arbitrary genus g with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The elements of this global approach are described in Part I. In the present paper we give the ...
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
Prediction of Algebraic Instabilities
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
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.
Algebraic partial Boolean algebras
Energy Technology Data Exchange (ETDEWEB)
Smith, Derek [Math Department, Lafayette College, Easton, PA 18042 (United States)
2003-04-04
Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A{sub 5} sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E{sub 8}.
Norén, Patrik
2013-01-01
Algebraic statistics brings together ideas from algebraic geometry, commutative algebra, and combinatorics to address problems in statistics and its applications. Computer algebra provides powerful tools for the study of algorithms and software. However, these tools are rarely prepared to address statistical challenges and therefore new algebraic results need often be developed. This way of interplay between algebra and statistics fertilizes both disciplines. Algebraic statistics is a relativ...
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
Developable algebraic surfaces
Institute of Scientific and Technical Information of China (English)
CHEN Dongren; WANG Guojin
2004-01-01
An algebraic surface can be defined by an implicit polynomial equation F(x,y,z)=0. In this paper, general characterizations of developable algebraic surfaces of arbitrary degree are presented. Using the shift operators of the subscripts of Bézier ordinates, the uniform apparent discriminants of developable algebraic surfaces to their Bézier ordinates are given directly. To degree 2 algebraic surfaces, which are widely used in computer aided geometric design and graphics, all possible developable surface types are obtained. For more conveniently applying algebraic surfaces of high degree to computer aided geometric design, the notion of ε-quasi-developable surfaces is introduced, and an example of using a quasi-developable algebraic surface of degree 3 to interpolate three curves of degree 2 is given.
A partial differential equation model of metastasized prostatic cancer.
Friedman, Avner; Jain, Harsh Vardhan
2013-06-01
Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castration-resistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2-dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.
Lefschetz, Solomon
2005-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
Network algebra for synchronous and asynchronous dataflow
Bergstra, J.A.; Stefanescu, G.
2008-01-01
Network algebra (NA) is proposed as a uniform algebraic framework for the description (and analysis) of dataflow networks. The core of this algebraic setting is provided by an equational theory called Basic Network Algebra (BNA). It constitutes a selection of primitives and identities from the algeb
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.
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.
Schlichenmaier, Martin; Sheinman, Oleg K.
1998-01-01
Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with $N$ marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector fiel...
Institute of Scientific and Technical Information of China (English)
李向正
2012-01-01
The bounded bell shape algebraic solitary wave solutions of nonlinear evolution equations are researched in this paper. The Kolmogorov-Petrovskii-Piskunov (KPP for short) equation,compound KdV-mKdV equation and mKdV equation are chose to as examples. The theory of planar dynamical systems is applied to study the existence conditions of algebraic solitary wave solutions. The algebraic solitary wave solutions of these three equations are obtained respectively. And a method for solving this type solutions is proposed, which is called algebraic solitary wave solution method(ASW method for short).%本文以非线性发展方程的有界钟状代数孤波解为研究对象,以Kolmogorov-Petrovskii-Piskunov(简称KPP)方程、组合KdV-mKdV方程和mKdV方程为例,利用平面动力系统知识,分析有界钟状代数孤立波解出现的条件,提出求解的方法,称之为代数孤波解解法(简称ASW解法),分别获得这三个方程的代数孤立波解.
Irreducible representations of Birman-Wenzl algebras
Institute of Scientific and Technical Information of China (English)
潘峰
1995-01-01
Irreducible representations of Birman-Wenzl algebras are constructed by using the induced representation and the linear equation method. Self-adjoint representations of Birman-Wenzl algebras Cf (r, q) with f≤4 are presented.
微分代数系统的数值仿真算法%Numerical Algorithm for differential-algebraic equations
Institute of Scientific and Technical Information of China (English)
宋晓秋
2000-01-01
介绍了微分代数系统DAE的基本概念及仿真算法，特别指出了用BDF方法 求解高指标常系数线性DAE系统时的数值稳定性缺陷。最后，针对飞行器轨道约束 实时控制问题，给出了3阶收敛的代数约束算法。%In this paper, the basic concept of differential-algebraic equations is introduced and its numerical algorithm is discussed. The defect for numerical stability of BDF method is shown when it is applied to linear constant coefficient DAE's with high index. Finally, a new algorithm with three orders for aerospace real-time control problem is given out.
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
Beginning algebra a textworkbook
McKeague, Charles P
1985-01-01
Beginning Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in algebra. The publication first elaborates on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on solving linear systems by graphing, elimination method, graphing ordered pairs and straight lines, linear and compound inequalities, addition and subtraction of real numbers, and properties of real numbers. The text then examines exponents and polynomials, factoring, and rational expressions. Topics include multiplication and division
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
Generalized braided Hopf algebras
Institute of Scientific and Technical Information of China (English)
LU Zhong-jian; FANG Xiao-li
2009-01-01
The concept of (f, σ)-pair (B, H)is introduced, where B and H are Hopf algebras. A braided tensor category which is a tensor subcategory of the category HM of left H-comodules through an (f, σ)-pair is constructed. In particularly, a Yang-Baxter equation is got. A Hopf algebra is constructed as well in the Yetter-Drinfel'd category HHYD by twisting the multiplication of B.
Directory of Open Access Journals (Sweden)
M. Legua
2008-01-01
Full Text Available In signal processing, a pulse means a rapid change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. A square wave function may be viewed as a pulse that repeats its occurrence periodically but the return to the baseline value takes some time to happen. When these periodic functions act as inputs in dynamic systems, the standard tool commonly used to solve the associated initial value problem (IVP is Laplace transform and its inverse. We show how a computer algebra system may also provide the solution of these IVP straight forwardly by adequately introducing the periodic input.
Energy Technology Data Exchange (ETDEWEB)
Magnin, H.; Coulomb, J.L. (Laboratoire d' electrotechnique de Grenoble (UA CNRS 355) E.N.S.I.E.G. BP 46 38402 St. Martin d' Heres (FR))
1989-07-01
Electromagnetic field analysis by finite elements methods needs solving of large sparse systems of linear equations. Though no discernible structure for the distribution of non-zero elements can be found (e.g. multidiagonal structures,...), subsets of independent equations can be determined. Equations that are in a same subset are then solved in parallel. A good choice for the storage scheme of sparse matrices is also very important to speedup the resolution by vectorization. The modifications the authors made to data structures are presented, and the possibility to use some other schemes is discussed.
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:...
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
2013-01-01
The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.
Linear algebraic methods applied to intensity modulated radiation therapy.
Crooks, S M; Xing, L
2001-10-01
Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.
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.
Templates for Linear Algebra Problems
Bai, Z.; Day, D.; Demmel, J.; Dongarra, J.; Gu, M.; Ruhe, A.; Vorst, H.A. van der
2001-01-01
The increasing availability of advanced-architecture computers is having a very signicant eect on all spheres of scientic computation, including algorithm research and software development in numerical linear algebra. Linear algebra {in particular, the solution of linear systems of equations and eig
Lloris Ruiz, Antonio; Parrilla Roure, Luis; García Ríos, Antonio
2014-01-01
This book presents a complete and accurate study of algebraic circuits, digital circuits whose performance can be associated with any algebraic structure. The authors distinguish between basic algebraic circuits, such as Linear Feedback Shift Registers (LFSRs) and cellular automata, and algebraic circuits, such as finite fields or Galois fields. The book includes a comprehensive review of representation systems, of arithmetic circuits implementing basic and more complex operations, and of the residue number systems (RNS). It presents a study of basic algebraic circuits such as LFSRs and cellular automata as well as a study of circuits related to Galois fields, including two real cryptographic applications of Galois fields.
Ebert, U.; Saarloos, W. van; Peletier, L.A.
2000-01-01
We analyze the front structures evolving under the difference-diffe-ren-tial equation $partial_tC_j=-C_j+C_{j-1^2$ from initial conditions $0le C_j(0)le1$ such that $C_j(0)to1$ as $jtoinfty$ sufficiently fast. We show that the velocity $v(t)$ of the front converges to a constant value $v^*$ accordin
New Matrix Loop Algebra and Its Application
Institute of Scientific and Technical Information of China (English)
DONG Huan-He; XU Yue-Cai
2008-01-01
A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly, as its appfication, the multi-component TC equation hierarchy is obtained, then by use of trace identity the Hamiltonian structure of the above system is presented. Finally, the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.
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
Adaptive Algebraic Multigrid Methods
Energy Technology Data Exchange (ETDEWEB)
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
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
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.
Quantitative Algebraic Reasoning
DEFF Research Database (Denmark)
Mardare, Radu Iulian; Panangaden, Prakash; Plotkin, Gordon
2016-01-01
We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We deﬁne an equality relation indexed by rationals: a =ε b which we think of as saying that “a is approximately equal to b up to an error of ε”. We have 4 interesting examples where we have a quantitative...
Issa, A Nourou
2010-01-01
Non-Hom-associative algebras and Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms.
Directory of Open Access Journals (Sweden)
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
Energy Technology Data Exchange (ETDEWEB)
Odesskii, A V [L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow (Russian Federation)
2002-12-31
This survey is devoted to associative Z{sub {>=}}{sub 0}-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). Examples are considered of such algebras, depending on two continuous parameters (namely, on an elliptic curve and a point on it), that are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces, and other directions of modern investigations.
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.
On deformation theory of quantum vertex algebras
Grosse, H; Grosse, Harald; Schlesinger, Karl-Georg
2005-01-01
We study an algebraic deformation problem which captures the data of the general deformation problem for a quantum vertex algebra. We derive a system of coupled equations which is the counterpart of the Maurer-Cartan equation on the usual Hochschild complex of an assocative algebra. We show that this system of equations results from an action principle. This might be the starting point for a perturbative treatment of the deformation problem of quantum vertex algebras. Our action generalizes the action of the Kodaira-Spencer theory of gravity and might therefore also be of relevance for applications in string theory.
Independence-friendly cylindric set algebras
Mann, Allen L
2007-01-01
Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic. We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every one-dimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlyin...
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
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
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...
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Algebra-Geometry of Piecewise Algebraic Varieties
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.
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,...
Generalized NLS Hierarchies from Rational $W$ Algebras
Toppan, F
1994-01-01
Finite rational $\\cw$ algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of equations, by showing how to associate to a rational $\\cw$ algebra its corresponding hierarchy. We work out two examples: the $sl(2)/U(1)$ coset, leading to the Non-Linear Schr\\"{o}dinger hierarchy, and the $U(1)$ coset of the Polyakov-Bershadsky $\\cw$ algebra, leading to a $3$-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies.
Chisolm, Eric
2012-01-01
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric al...
Garrett, Paul B
2007-01-01
Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal
Elementary Algebra Connections to Precalculus
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
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.
Rota-Baxter operators on Witt and Virasoro algebras
Gao, Xu; Liu, Ming; Bai, Chengming; Jing, Naihuan
2016-10-01
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
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
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
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
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.
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
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.
Marchuk, Nikolay
2011-01-01
Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann alg...
Edwards, Harold M
1995-01-01
In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject
DEFF Research Database (Denmark)
2007-01-01
of algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a "general audience" of algebraic group......-theorists, and to stimulate contacts between participants. Each of the first four days was dedicated to one area of research that has recently seen decisive progress: \\begin{itemize} \\item structure and classification of wonderful varieties, \\item finite reductive groups and character sheaves, \\item quantum cohomology...... of homogeneous varieties, \\item representation categories and their connections to orbits and flag varieties. \\end{itemize} The first three days started with survey talks that will help to make the subject accessible to the next generation. The talks on the last day introduced to several recent advances...
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.)
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.
Directory of Open Access Journals (Sweden)
G.C. Rao
2012-11-01
Full Text Available A C- algebra is the algebraic form of the 3-valued conditional logic, which was introduced by F. Guzman and C. C. Squier in 1990. In this paper, some equivalent conditions for a C- algebra to become a boolean algebra in terms of congruences are given. It is proved that the set of all central elements B(A is isomorphic to the Boolean algebra of all C-algebras Sa, where a B(A. It is also proved that B(A is isomorphic to the Boolean algebra of all C-algebras Aa, where a B(A.
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 ...
Jacobson, Nathan
2009-01-01
A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as L
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
Luo, Zhaohua
2008-01-01
The primary goal of this paper is to present a unified way to transform the syntax of a logic system into certain initial algebraic structure so that it can be studied algebraically. The algebraic structures which one may choose for this purpose are various clones over a full subcategory of a category. We show that the syntax of equational logic, lambda calculus and first order logic can be represented as clones or right algebras of clones over the set of positive integers. The semantics is then represented by structures derived from left algebras of these clones.
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.
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
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.
Energy Technology Data Exchange (ETDEWEB)
Gonzalez Herrera, Juan Anibal
1996-10-01
This work presents the construction, analysis and solution of an equipment`s network in steady and dynamic state from: a) The mathematical models of individual equipment and of their geometry. b) The topology let interconnections between equipment. c) The selection of a numerical method to solve simultaneously the mathematical models. The selected mathematical models represent the cycle boiler-superheater. These models were taken from the MICROTERM-300 modular simulator, which contains the simplified models of the process (feedwater, boiler, turbines, etc.) of the thermoelectric plant Francisco Perez Rios from Tula Hidalgo, Mexico. This work was developed in the following stages: 1.- The selection of an appropiate numerical integration method to solve simultaneously the algebraic and differential equations of the equipment conforming the cycle boiler-superheater. 2.- The adaptation of individual mathematical models to allow changes in their geometry, operating conditions and different forms of connection. Also, this models were modified to have a representation of the equations to allow their analysis and an efficient organization to get their solution. 3.- The application of two computer-aided tools to trace possible coding errors in the mathematical models: a) A syntax analyzer which detect assignation and reference errors of variables. b) A structural analyzer to obtain the structural matrix, which relate the variables and the equations in a model. During this stage some improvements to these computer-aided tools were suggested. 4.- The individual testing of each mathematical model in steady and dynamic state in order to: a) Validate the mathematical models. b) Analyze the behavior of the variables of the mathematical models with different parameters, different operating conditions and different initial conditions. 5.- Lastly, the coupling between equipment analyzed to form an equipment`s network what represent the cycle boiler-superheater and the testing in
An algebraic Birkhoff decomposition for the continuous renormalization group
Girelli, F; Martinetti, P
2004-01-01
This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited.
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 ...
INPUT-OUTPUT STRUCTURE OF LINEAR-DIFFERENTIAL ALGEBRAIC SYSTEMS
KUIJPER, M; SCHUMACHER, JM
1993-01-01
Systems of linear differential and algebraic equations occur in various ways, for instance, as a result of automated modeling procedures and in problems involving algebraic constraints, such as zero dynamics and exact model matching. Differential/algebraic systems may represent an input-output relat
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.
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
Function theory for a beltrami algebra
Directory of Open Access Journals (Sweden)
B. A. Case
1985-01-01
Full Text Available Complex functions are investigated which are solutions of an elliptic system of partial differential equations associated with a real parameter function. The functions f associated with a particualr parameter function g on a domain D form a Beltrami algebra denoted by the pair (D,g and a function theory is developed in this algebra. A strong conformality property holds for all functions in a (D,g algebra. For g≡|z|=r the algebra (D,r is that of the analytic functions.
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...
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.
Math for All Learners: Algebra.
Meader, Pam; Storer, Judy
This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…
Process algebra for synchronous communication
Bergstra, J.A.; Klop, J.W.
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
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.
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.
Iachello, F
1995-01-01
1. The Wave Mechanics of Diatomic Molecules. 2. Summary of Elements of Algebraic Theory. 3. Mechanics of Molecules. 4. Three-Body Algebraic Theory. 5. Four-Body Algebraic Theory. 6. Classical Limit and Coordinate Representation. 8. Prologue to the Future. Appendices. Properties of Lie Algebras; Coupling of Algebras; Hamiltonian Parameters
Algebraic Systems and Pushdown Automata
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
Energy Technology Data Exchange (ETDEWEB)
Casasent, D.; Ghosh, A.
1983-01-01
Many of the linear algebra operations and algorithms possible on optical matrix-vector processors are reviewed. Emphasis is given to the use of direct solutions and their realization on systolic optical processors. As an example, implicit and explicit solutions to partial differential equations are considered. The matrix-decomposition required is found to be the major operation recommended for optical realization. The pipelining and flow of data and operations are noted to be key issues in the realization of any algorithm on an optical systolic array processor. A realization of the direct solution by householder qr decomposition is provided as a specific case study. 19 references.
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...
Solvable quadratic Lie algebras
Institute of Scientific and Technical Information of China (English)
ZHU; Linsheng
2006-01-01
A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.
DÍaz, R.; Rivas, M.
2010-01-01
In order to study Boolean algebras in the category of vector spaces we introduce a prop whose algebras in set are Boolean algebras. A probabilistic logical interpretation for linear Boolean algebras is provided. An advantage of defining Boolean algebras in the linear category is that we are able to study its symmetric powers. We give explicit formulae for products in symmetric and cyclic Boolean algebras of various dimensions and formulate symmetric forms of the inclusion-exclusion principle.
Algebras and algebraic curves associated with PDEs and Bäcklund transformations
Igonin, S.A.
2010-01-01
Using the language of jet spaces, for any analytic PDE E we define, in a coordinatefree way, a family of associative algebras A(E). In the considered examples, which include the KdV, Krichever-Novikov, nonlinear Schr¨odinger, Landau-Lifshitz equations, the algebras A(E) are commutative and are isomo
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
The algebraic combinatorics of snakes
Josuat-Vergès, Matthieu; Thibon, Jean-Yves
2011-01-01
Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.
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.
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...
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].
Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra
Institute of Scientific and Technical Information of China (English)
ZHANG Yu-Feng; LIU Jing
2008-01-01
By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 (2006) 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.
Energy Technology Data Exchange (ETDEWEB)
Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado
1997-10-01
The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory.
Marchuk, Nikolay
2011-01-01
Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann algebra and Clifford algebra. Specialists consider models of gravity that based on a mathematical formalism with two metric tensors. We hope that the considered in this paper 2-metric exterior algebra can be useful for development of this model in gravitation theory. Especially in description of fermions in presence of a gravity field.
Discrete integrable systems and deformations of associative algebras
Energy Technology Data Exchange (ETDEWEB)
Konopelchenko, B G [Dipartimento di Fisica, Universita del Salento and INFN, Sezione di Lecce, 73100 Lecce (Italy)], E-mail: konopel@le.infn.it
2009-10-30
Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. Theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations. It is demonstrated that many discrete equations such as discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BKP equations, discrete Hirota-Miwa equations for KP and BKP hierarchies are particular realizations of the central system. An interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful. An interpretation of the Menelaus relation (discrete Schwarzian KP equation), discrete Hirota-Miwa equation for KP hierarchy, consistency around the cube as the associativity conditions and the concept of gauge equivalence, for instance, between the Menelaus and KP configurations are particular examples.
Closed nominal rewriting and efficiently computable nominal algebra equality
Fernández, Maribel; 10.4204/EPTCS.34.5
2010-01-01
We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of the lambda-calculus and first-order logic.
Closed nominal rewriting and efficiently computable nominal algebra equality
Directory of Open Access Journals (Sweden)
Maribel Fernández
2010-09-01
Full Text Available We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of the lambda-calculus and first-order logic.
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY
Institute of Scientific and Technical Information of China (English)
TaoChangli; LuShijie; ChenPeixin
2002-01-01
Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.
Rigidification of algebras over essentially algebraic theories
Rosicky, J
2012-01-01
Badzioch and Bergner proved a rigidification theorem saying that each homotopy simplicial algebra is weakly equivalent to a simplicial algebra. The question is whether this result can be extended from algebraic theories to finite limit theories and from simplicial sets to more general monoidal model categories. We will present some answers to this question.
Nishiyama, Seiya
2014-01-01
In this paper we present the induced representation of SO(2N) canonical transformation group and introduce SO(2N)/U(N) coset variables. We give a derivation of the time dependent Hartree-Bogoliubov (TDHB) equation on the Kaehler coset space G/H=SO(2N)/U(N) from the Euler-Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of TDRHB equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Moebius transformation together with the geodesic flow on the manifold.
Nishiyama, Seiya; da Providência, João
2015-02-01
In this paper we present the induced representation of SO(2N) canonical transformation group and introduce (SO(2N))/(U(N)) coset variables. We give a derivation of the time-dependent Hartree-Bogoliubov (TDHB) equation on the Kähler coset space (G)/(H) = (SO(2N))/(U(N)) from the Euler-Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati-Hartree-Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.
The Yoneda algebra of a K_2 algebra need not be another K_2 algebra
Cassidy, T.; Phan, Van C.; Shelton, B.
2008-01-01
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
Algebraic cobordism theory attached to algebraic equivalence
Krishna, Amalendu
2012-01-01
After the algebraic cobordism theory of Levine-Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the zero-th semi-topological K-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional or special types of varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.
Workshop on Commutative Algebra
Simis, Aron
1990-01-01
The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results.
Probabilistic Concurrent Kleene Algebra
Directory of Open Access Journals (Sweden)
Annabelle McIver
2013-06-01
Full Text Available We provide an extension of concurrent Kleene algebras to account for probabilistic properties. The algebra yields a unified framework containing nondeterminism, concurrency and probability and is sound with respect to the set of probabilistic automata modulo probabilistic simulation. We use the resulting algebra to generalise the algebraic formulation of a variant of Jones' rely/guarantee calculus.
Finite and Infinite W Algebras and their Applications
Tjin, T
1993-01-01
In this paper we present a systematic study of $W$ algebras from the Hamiltonian reduction point of view. The Drinfeld-Sokolov (DS) reduction scheme is generalized to arbitrary $sl_2$ embeddings thus showing that a large class of W algebras can be viewed as reductions of affine Lie algebras. The hierarchies of integrable evolution equations associated to these classical W algebras are constructed as well as the generalized Toda field theories which have them as Noether symmetry algebras. The problem of quantising the DS reductions is solved for arbitrary $sl_2$ embeddings and it is shown that any W algebra can be embedded into an affine Lie algebra. This also provides us with an algorithmic method to write down free field realizations for arbitrary W algebras. Just like affine Lie algebras W algebras have finite underlying structures called `finite W algebras'. We study the classical and quantum theory of these algebras, which play an important role in the theory of ordinary W algebras, in detail as well as s...
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.
Generalized Quantum Current Algebras
Institute of Scientific and Technical Information of China (English)
ZHAO Liu
2001-01-01
Two general families of new quantum-deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enables one to define "tensor products" of these algebras. The standard quantum affine algebras turn out to be a very special case of the two algebra families, in which case the infinite Hopf family structure degenerates into a standard Hopf algebra. The relationship between the two algebraic families as well as thefr various special examples are discussed, and the free boson representation is also considered.
El-Chaar, Caroline
2012-01-01
In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. Using this fourth realization, we explicitly describe all its ideals.
Perturbations of planar algebras
Das, Paramita; Gupta, Ved Prakash
2010-01-01
We introduce the concept of {\\em weight} of a planar algebra $P$ and construct a new planar algebra referred as the {\\em perturbation of $P$} by the weight. We establish a one-to-one correspondence between pivotal structures on 2-categories and perturbations of planar algebras by weights. To each bifinite bimodule over $II_1$-factors, we associate a {\\em bimodule planar algebra} bimodule corresponds naturally with sphericality of the bimodule planar algebra. As a consequence of this, we reproduce an extension of Jones' theorem (of associating 'subfactor planar algebras' to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. We show that the perturbation class of a bimodule planar algebra contains a unique spherical unimodular bimodule planar algeb...
A Note on Indefinite Stochastic Riccati Equations
Qian, Zhongmin
2012-01-01
An indefinite stochastic Riccati Equation is a matrix-valued, highly nonlinear backward stochastic differential equation together with an algebraic, matrix positive definiteness constraint. We introduce a new approach to solve a class of such equations (including the existence of solutions) driven by one-dimensional Brownian motion. The idea is to replace the original equation by a system of BSDEs (without involving any algebraic constraint) whose existence of solutions automatically enforces the original algebraic constraint to be satisfied.
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.
Yangians and transvector algebras
Molev, A. I.
1998-01-01
Olshanski's centralizer construction provides a realization of the Yangian for the Lie algebra gl(n) as a subalgebra in the projective limit of a chain of centralizers in the universal enveloping algebras. We give a modified version of this construction based on a quantum analog of Sylvester's theorem. We then use it to get an algebra homomorphism from the Yangian to the transvector algebra associated with the general linear Lie algebras. The results are applied to identify the elementary rep...
Quiney, H. M.; Glushkov, V. N.; Wilson, S.; Sabin,; Brandas, E
2001-01-01
A comparison is made of the accuracy achieved in finite difference and finite basis set approximations to the Dirac equation for the ground state of the hydrogen molecular ion. The finite basis set calculations are carried out using a distributed basis set of Gaussian functions the exponents and pos
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.
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.
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 th...
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.
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.
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
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.
Goldmann, H
1990-01-01
The first part of this monograph is an elementary introduction to the theory of Fréchet algebras. Important examples of Fréchet algebras, which are among those considered, are the algebra of all holomorphic functions on a (hemicompact) reduced complex space, and the algebra of all continuous functions on a suitable topological space.The problem of finding analytic structure in the spectrum of a Fréchet algebra is the subject of the second part of the book. In particular, the author pays attention to function algebraic characterizations of certain Stein algebras (= algebras of holomorphic functions on Stein spaces) within the class of Fréchet algebras.
Single axioms for Boolean algebra.
Energy Technology Data Exchange (ETDEWEB)
McCune, W.
2000-06-30
Explicit single axioms are presented for Boolean algebra in terms of (1) the Sheffer stroke; (2) disjunction and negation; (3) disjunction, conjunction, and negation; and (4) disjunction, conjunction, negation, 0, and 1. It was previously known that single axioms exist for these systems, but the procedures to generate them are exponential, producing huge equations. Automated deduction techniques were applied to find axioms of lengths 105, 131, 111, and 127, respectively, each with six variables.
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.
Shilov, Georgi E
1977-01-01
Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.
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
Abian, Alexander
1973-01-01
Linear Associative Algebras focuses on finite dimensional linear associative algebras and the Wedderburn structure theorems.The publication first elaborates on semigroups and groups, rings and fields, direct sum and tensor product of rings, and polynomial and matrix rings. The text then ponders on vector spaces, including finite dimensional vector spaces and matrix representation of vectors. The book takes a look at linear associative algebras, as well as the idempotent and nilpotent elements of an algebra, ideals of an algebra, total matrix algebras and the canonical forms of matrices, matrix
Clifford Algebra with Mathematica
Aragon-Camarasa, G; Aragon, J L; Rodriguez-Andrade, M A
2008-01-01
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, a package for Clifford algebra calculations for the computer algebra program Mathematica is introduced through a presentation of the main ideas of Clifford algebras and illustrative examples. This package can be a useful computational tool since allows the manipulation of all these mathematical objects. It also includes the possibility of visualize elements of a Clifford algebra in the 3-dimensional space.
Institute of Scientific and Technical Information of China (English)
PENG Jia-yin
2011-01-01
The notions of norm and distance in BCI-algebras are introduced,and some basic properties in normed BCI-algebras are given.It is obtained that the isomorphic(homomorphic)image and inverse image of a normed BCI-algebra are still normed BCI-algebras.The relations of normaled properties between BCI-algebra and Cartesian product of BCIalgebras are investigated.The limit notion of sequence of points in normed BCI-algebras is introduced,and its related properties are investigated.
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.
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.
Superconformal Algebras and Supersymmetric Integrable Flows
Sachse, Christoph; Devchand, Chandrasekhar
2009-01-01
After a comprehensive review of superconformal algebras, super-diffeomorphisms and supervector fields on supercircles S^{1|n} we study various supersymmetric extensions of the KdV and Camassa-Holm equations. We describe their (super) Hamiltonian structures and their connection to bihamiltonian geometry. These are interpreted as geodesic flows on various superconformal groups. We also give an example of superintegrable systems of Ramond type. The one-parameter family of equations shown by Degasperis, Holm and Hone (DHH) to possess multi-peakon solutions is identified as a geodesic flow equation on a one-parameter deformation of the group of diffeomorphisms of the circle, with respect to a right-invariant Sobolev H^1--metric. A supersymmetrisation of the algebra of deformed vector fields on S^1 yields supersymmetric DHH equations (also known as b-field equations), which include the supersymmetric Camassa--Holm equation as a special case.
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.
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.
Automatic Construction of Finite Algebras
Institute of Scientific and Technical Information of China (English)
张健
1995-01-01
This paper deals with model generation for equational theories,i.e.,automatically generating (finite)models of a given set of (logical) equations.Our method of finite model generation and a tool for automatic construction of finite algebras is described.Some examples are given to show the applications of our program.We argue that,the combination of model generators and theorem provers enables us to get a better understanding of logical theories.A brief comparison betwween our tool and other similar tools is also presented.
Mathematical methods linear algebra normed spaces distributions integration
Korevaar, Jacob
1968-01-01
Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector
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
1997-01-01
We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.
Categories and Commutative Algebra
Salmon, P
2011-01-01
L. Badescu: Sur certaines singularites des varietes algebriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algebriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de series formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all'algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.- C.B. Winters: An elementary lecture on algebraic spaces.
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.
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…
Bergstra, J.A.; Fokkink, W.J.; Middelburg, C.A.
2008-01-01
Timed frames are introduced as objects that can form a basis of a model theory for discrete time process algebra. An algebraic setting for timed frames is proposed and results concerning its connection with discrete time process algebra are given. The presented theory of timed frames captures the ba
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.
REAL PIECEWISE ALGEBRAIC VARIETY
Institute of Scientific and Technical Information of China (English)
Ren-hong Wang; Yi-sheng Lai
2003-01-01
We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set, etc., we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in Cμ spline ring.
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.
Division algebras and extended super KdVs
Energy Technology Data Exchange (ETDEWEB)
Toppan, F. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil). Coordenacao de Teoria de Campos e Particulas]. E-mail: toppan@cbpf.br
2001-05-01
The division algebras R, C, H, O are used to construct and analyze the N = 1, 2, 4, 8 supersymmetric extensions of the KdV hamiltonian equation. In particular a global N = 8 super-KdV system is introduced and shown to admit a Poisson bracket structure given by the 'Non-Associate N = 8 Superconformal Algebra'. (author)
On W algebras commuting with a set of screenings
Litvinov, Alexey
2016-01-01
We consider the problem of classification of all W algebras which commute with a set of exponential screening operators. Assuming that the W algebra has a nontrivial current of spin 3, we find equations satisfied by the screening operators and classify their solutions.
On W algebras commuting with a set of screenings
Litvinov, Alexey; Spodyneiko, Lev
2016-11-01
We consider the problem of classification of all W algebras which commute with a set of exponential screening operators. Assuming that the W algebra has a nontrivial current of spin 3, we find equations satisfied by the screening operators and classify their solutions.
Lie symmetry algebra of one-dimensional nonconservative dynamical systems
Institute of Scientific and Technical Information of China (English)
Liu Cui-Mei; Wu Run-Heng; Fu Jing-Li
2007-01-01
Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1-1 mapping,the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion.
Modular operads and the quantum open-closed homotopy algebra
Doubek, Martin; Jurčo, Branislav; Münster, Korbinian
2015-12-01
We verify that certain algebras appearing in string field theory are algebras over Feynman transform of modular operads which we describe explicitly. Equivalent description in terms of solutions of generalized BV master equations are explained from the operadic point of view.
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.
Algorithms in Algebraic Geometry
Dickenstein, Alicia; Sommese, Andrew J
2008-01-01
In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its
Noncommutative physics on Lie algebras, Z_2^n lattices and Clifford algebras
Majid, S
2004-01-01
We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, $\\theta$-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice $(Z_2)^n$ and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on $Z_2\\times Z_2$ in a path integral approach.
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…
The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
Karl Hallowell; Andrew Waldron
2007-01-01
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 resu...
Lectures on algebraic statistics
Drton, Mathias; Sullivant, Seth
2009-01-01
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
The Boolean algebra and central Galois algebras
Directory of Open Access Journals (Sweden)
George Szeto
2001-01-01
Full Text Available Let B be a Galois algebra with Galois group G, Jg={b∈B∣bx=g(xb for all x∈B} for g∈G, and BJg=Beg for a central idempotent eg. Then a relation is given between the set of elements in the Boolean algebra (Ba,≤ generated by {0,eg∣g∈G} and a set of subgroups of G, and a central Galois algebra Be with a Galois subgroup of G is characterized for an e∈Ba.
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.
Lax operator algebras and Hamiltonian integrable hierarchies
Energy Technology Data Exchange (ETDEWEB)
Sheinman, Oleg K [Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)
2011-02-28
This paper considers the theory of Lax equations with a spectral parameter on a Riemann surface, proposed by Krichever in 2001. The approach here is based on new objects, the Lax operator algebras, taking into consideration an arbitrary complex simple or reductive classical Lie algebra. For every Lax operator, regarded as a map sending a point of the cotangent bundle on the space of extended Tyurin data to an element of the corresponding Lax operator algebra, a hierarchy of mutually commuting flows given by the Lax equations is constructed, and it is proved that they are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example, elliptic A{sub n}, C{sub n}, and D{sub n} Calogero-Moser systems are derived in the framework of our approach. Bibliography: 13 titles.
Infinite Dimensional Free Algebra and the Forms of the Master Field
Halpern, M B
1999-01-01
We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and the dual basis is closely related to the planar connected parts. This leads to a number of free-algebraic forms of the master field including an algebraic derivation of the Gopakumar-Gross form. For action theories, these forms of the master field immediately give a number of new free-algebraic packagings of the planar Schwinger-Dyson equations.
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.
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.
The Hall Algebra of Cyclic Serial Algebra
Institute of Scientific and Technical Information of China (English)
郭晋云
1994-01-01
In this paper, orders <1 and <2 on ((Z)+)nm are introduced and also regarded as orders on the isomorphism classes of finite modules of finite .cyclic algebra R with n simple modules and all the indecomposable projective modules have length m through a one-to-one correspondence between ((Z)+)nm and the isomorphism classes of finite R modules. Using this we prove that the Hall algebra of a cyclic serial algebra is identified with its Loewy subalgebra, and its rational extension has a basis of BPW type, and is a (((Z)+)nm, <2) filtered ring with the associated graded ring as an iterated skew polynomial ring. These results are also generalized to the Hall algebra of a tube over a finite field.
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.
Dynamical Algebraic Approach to the Modified Jaynes－Cummings Model
Institute of Scientific and Technical Information of China (English)
许晶波; 邹旭波
2001-01-01
The modified Jaynes-Cummings model of a single two-level atom placed in the common domain of two cavities or interacting with two quantized modes is studied by a dynamical algebraic method. With the help of an SU(2) algebraic structure, we then obtain the eigenvalues, eigenstates, time evolution operator and atomic inversion operator for the system. We proceed to investigate the modified Jaynes-Cummings model governed by the Milburn equation and present the exact solution of the Milburn equation.
Nonlinear realizations of the W$_{3}^{(2}$) algebra
Bellucci, S; Krivonos, S; Pashnev, A I
1994-01-01
In this letter we consider the nonlinear realizations of the classical Polyakov's algebra $W_3^{(2)}$. The coset space method and the covariant reduction procedure allow us to deduce the Boussinesq equation with interchanged space and evolution coordinates. By adding one more space coordinate and introducing two copies of the $W_3^{(2)}$ algebra, the same method yields the $sl(3,R)$ Toda lattice equations.
Invariant algebraic surfaces for a virus dynamics
Valls, Claudia
2015-08-01
In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.
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, 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.
Short single axioms for boolean algebra.
Energy Technology Data Exchange (ETDEWEB)
McCune, W.; Veroff, R.; Fitelson, B.; Harris, K.; Feist, A.; Wos, L.; Mathematics and Computer Science; Univ. of New Mexico; Univ. of Wisconsin at Madison; Duke Univ.
2002-01-01
We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.
Learning Activity Package, Algebra-Trigonometry.
Holland, Bill
A series of ten teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, the units cover logic; absolute value, inequalities, exponents, and complex numbers; functions; higher degree equations and the derivative; the trigonometric function; graphs and applications of the trigonometric functions; sequences and…
The Rees Algebra for Certain Monomial Curves
Mukhopadhyay, Debasish
2010-01-01
In this article, we find the equations defining the Rees algebra for certain Monomial Curves explicitly and use them to prove that the blowup scheme is not smooth. This proves a conjecture of Francia in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.
Finite-dimensional (*)-serial algebras
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let A be a finite-dimensional associative algebra with identity over a field k. In this paper we introduce the concept of (*)-serial algebras which is a generalization of serial algebras. We investigate the properties of (*)-serial algebras, and we obtain suficient and necessary conditions for an associative algebra to be (*)-serial.
Directory of Open Access Journals (Sweden)
R. A. Borzooei
2006-01-01
Full Text Available We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
Borzooei, R. A.; Dudek, W. A.; Koohestani, N.
2006-01-01
We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
On the Toroidal Leibniz Algebras
Institute of Scientific and Technical Information of China (English)
Dong LIU; Lei LIN
2008-01-01
Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras gOC[t±11 ,...,t±v1] in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Algebra Automorphisms of Quantized Enveloping Algebras Uq(■)
Institute of Scientific and Technical Information of China (English)
查建国
1994-01-01
The algebra automorphisms of the quantized enveloping algebra Uq(g) are discussed, where q is generic. To some extent, all quantum deformations of automorphisms of the simple Lie algebra g have been determined.
Cayley-Dickson and Clifford Algebras as Twisted Group Algebras
Bales, John W
2011-01-01
The effect of some properties of twisted groups on the associated algebras, particularly Cayley-Dickson and Clifford algebras. It is conjectured that the Hilbert space of square-summable sequences is a Cayley-Dickson algebra.
Transformation of time dependence to linear algebra
Menšík, Miroslav
2005-10-01
Reduced density matrix and memory function in the Nakajima-Zwanzig equation are expanded in properly chosen basis of special functions. This trick completely transforms time dependence to linear algebra. Then, the master equation for memory function is constructed and expanded in the same basis functions. For the model of a simple harmonic oscillator it is shown that this trick introduces infinite partial summation of the memory function in the system-bath interaction.
Algebraic Bethe ansatz for a singular solution
Nepomechie, Rafael I
2013-01-01
The Bethe equations for the spin-1/2 Heisenberg chain with N sites have a "two-string" solution i/2, -i/2 that is singular: both the corresponding energy and algebraic Bethe ansatz vector are divergent. We show that this solution must be carefully regularized in order to obtain the correct eigenvector. This regularization involves a parameter that can be determined using a generalization of the Bethe equations. It follows that this solution must be excluded for odd N.
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
Krichever, Igor M.; Sheinman, Oleg K.
2007-01-01
In this paper we develop a general concept of Lax operators on algebraic curves introduced in [1]. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct the orthogonal and symplectic analogs of Lax operators, prove that they constitute almost graded Lie algebras and construct local central extensions of those Lie algebras.
Noncommutative Pfaffians associated with the orthogonal algebra
Artamonov, Dmitrii V.; Golubeva, Valentina A.
2012-12-01
Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of \\mathfrak{o}_N. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of \\mathfrak{o}_N is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in U(\\mathfrak{o}_N), which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations. Bibliography: 18 titles.
Noncommutative Pfaffians associated with the orthogonal algebra
Energy Technology Data Exchange (ETDEWEB)
Artamonov, Dmitrii V; Golubeva, Valentina A
2012-12-31
Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of o{sub N}. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of o{sub N} is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in U(o{sub N}), which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations. Bibliography: 18 titles.
Directory of Open Access Journals (Sweden)
Frank Roumen
2017-01-01
Full Text Available We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.
Algebraic extensions of fields
McCarthy, Paul J
1991-01-01
""...clear, unsophisticated and direct..."" - MathThis textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamenta
Balan, Adriana
2010-01-01
We extend Barr's well-known characterization of the final coalgebra of a $Set$-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a $Set$-monad $\\mathbf{M}$ for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad $\\mathbf{M}$ and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.
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
Underwood, Robert G
2015-01-01
This text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras, and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields, and linearly recursive sequences. The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforw...
Solomon, Alan D
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean
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
Generalized supersymmetry and L\\'evy-Leblond equation
Aizawa, N; Tanaka, H; Toppan, F
2016-01-01
The symmetries of the L\\'evy-Leblond equation are investigated beyond the standard Lie framework. It is shown that the equation has two remarkable symmetries. One is given by the super Schr\\"odinger algebra and the other one by a $\\ZZ$ graded Lie algebra. The $\\ZZ$ graded Lie algebra is achieved by transforming bosonic into fermionic operators in the super Schr\\"odinger algebra and introducing second order differential operators as generators of symmetry.
Directory of Open Access Journals (Sweden)
Grazziela P Figueredo
Full Text Available 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 when contrasted with stochastic versions of ODE models using early-stage cancer examples. We seek answers to the following questions: (1 Does this new stochastic formulation produce similar results to the agent-based version? (2 Can these methods be used interchangeably? (3 Do agent-based models outcomes reveal any benefit when compared to the Gillespie results? To answer these research questions we investigate three well-established mathematical models describing interactions between tumour cells and immune elements. These case studies were re-conceptualised under an agent-based perspective and also converted to the Gillespie algorithm formulation. Our interest in this work, therefore, is to establish a methodological discussion regarding the usability of different simulation approaches, rather than provide further biological insights into the investigated case studies. Our results show that it is possible to obtain equivalent models that implement the same mechanisms; however, the incapacity of the Gillespie algorithm to retain individual memory of past events affects the similarity of some results. Furthermore, the emergent behaviour of ABMS produces extra patters of behaviour in the system, which was not obtained by the Gillespie algorithm.
Figueredo, Grazziela P; Siebers, Peer-Olaf; Owen, Markus R; Reps, Jenna; Aickelin, Uwe
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 when contrasted with stochastic versions of ODE models using early-stage cancer examples. We seek answers to the following questions: (1) Does this new stochastic formulation produce similar results to the agent-based version? (2) Can these methods be used interchangeably? (3) Do agent-based models outcomes reveal any benefit when compared to the Gillespie results? To answer these research questions we investigate three well-established mathematical models describing interactions between tumour cells and immune elements. These case studies were re-conceptualised under an agent-based perspective and also converted to the Gillespie algorithm formulation. Our interest in this work, therefore, is to establish a methodological discussion regarding the usability of different simulation approaches, rather than provide further biological insights into the investigated case studies. Our results show that it is possible to obtain equivalent models that implement the same mechanisms; however, the incapacity of the Gillespie algorithm to retain individual memory of past events affects the similarity of some results. Furthermore, the emergent behaviour of ABMS produces extra patters of behaviour in the system, which was not obtained by the Gillespie algorithm.
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.
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.
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.
Perturbed Einstein field equations using Maple
De Campos, M
2003-01-01
We obtain the perturbed components of affine connection and Ricci tensor using algebraic computation. Naturally, the perturbed Einstein field equations for the vacuum can written. The method can be used to obtain perturbed equations of the superior order.
Auxiliary equation method for solving nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Sirendaoreji,; Jiong, Sun
2003-03-31
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
PARK; Hong; Goo; LEE; Jeongsig; CHOI; Seul; Hee; NAM; Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n).
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
PARK Hong Goo; LEE Jeongsig; CHOI Seul Hee; CHEN XueQing; NAM Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m, m+n).
Partial Differential Equations An Introduction
Choudary, A. D. R.; Parveen, Saima; Varsan, Constantin
2010-01-01
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie algebras of vector fields and their algebraic-geometric representations are involved in solving overdetermined of PDE and getting integral representation of stochastic differential equations (SDE). It is addressing to all scientists using PDE in treating mathe...
Partial Differential Equations An Introduction
Choudary, A D R; Varsan, Constantin
2010-01-01
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie algebras of vector fields and their algebraic-geometric representations are involved in solving overdetermined of PDE and getting integral representation of stochastic differential equations (SDE). It is addressing to all scientists using PDE in treating mathematical methods.
Derived equivalence of algebras
Institute of Scientific and Technical Information of China (English)
杜先能
1997-01-01
The derived equivalence and stable equivalence of algebras RmA and RmB are studied It is proved, using the tilting complex, that RmA and RmB are derived-equivalent whenever algebras A and B are derived-equivalent
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Herriott, Scott R.; Dunbar, Steven R.
2009-01-01
The common understanding within the mathematics community is that the role of the college algebra course is to prepare students for calculus. Though exceptions are emerging, the curriculum of most college algebra courses and the content of most textbooks on the market both reflect that assumption. This article calls that assumption into question…
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…
Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach
Panahi, H.; Baradaran, M.; Savadi, A.
2015-10-01
Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.
Powell, Sarah R.; Fuchs, Lynn S.
2014-01-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2nd- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Ba...
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…
Institute of Scientific and Technical Information of China (English)
臧强; 张凯锋; 戴先中; 周颖
2012-01-01
For a class of nonlinear differential - algebraic equations ( DAE) subsystems: whose index is one and interconnection is local measurable, the inverse system control method is studied in this paper. The result is applied to the components control of power systems. At first the background and the particularities of such systems are expatiated. Then the definition of - order right inverse system is put forward. A recursive algorithm is given, with which to identify whether the nonlinear DAE subsystems are invertible. An - order right inverse system is realized by both state - feedback and dynamic compensation, with which the nonlinear DAE subsystems are decoupled and linearized. Finally, an excitation controller is designed for one of the synchronous generators in the multi - machine power systems based on the proposed method in this paper. The simulation is conducted and the results demonstrate the effectiveness of the proposed control scheme.%对于指数1且关联可测的非线性微分-代数子系统,研究其逆系统控制方法,并将结果应用于电力系统元件分散控制.首先描述了此类非线性微分-代数子系统的物理背景和系统特性,并给出了非线性微分-代数子系统的α阶积分右逆系统和可逆的定义;然后给出了一种递归算法,以此来判别被控系统的可逆性,并构造出由状态反馈和动态补偿实现的α阶积分右逆系统,实现了复合系统的线性化解耦；最后针对多机电力系统中的一台同步发电机,应用所提出的方法研究其励磁控制电压问题.仿真结果验证了所提出方法的有效性.
Introduction to noncommutative algebra
Brešar, Matej
2014-01-01
Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
Elements of mathematics algebra
Bourbaki, Nicolas
2003-01-01
This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...
Fioravanti, D; Fioravanti, Davide; Rossi, Marco
2001-01-01
A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, {\\it in the cylinder continuum limit}, to a Perturbed Minimal Conformal Field Theory description is analysed and supported.
Stability of algebraically unstable dispersive flows
King, Kristina R.; Weinstein, Steven J.; Zaretzky, Paula M.; Cromer, Michael; Barlow, Nathaniel S.
2016-11-01
A largely unexplored type of hydrodynamic instability is examined: long-time algebraic growth. Such growth is possible when the dispersion relation extracted from classical stability analysis indicates neutral stability. A physically motivated class of partial differential equations that describes the response of a system to disturbances is examined. Specifically, the propagation characteristics of the response are examined in the context of spatiotemporal stability theory. Morphological differences are identified between system responses that exhibit algebraic growth and the more typical case of exponential growth. One key attribute of predicted algebraically growing solutions is the prevalence of transient growth in almost all of the response, with the long-time growth occurring asymptotically at precisely one wave speed.
Computational algebraic geometry of epidemic models
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
Exact linear modeling using Ore algebras
Schindelar, Kristina; Zerz, Eva
2010-01-01
Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.
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, ...
The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra
Indian Academy of Sciences (India)
Vijay Kodiyalam; V S Sunder
2006-11-01
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.
Graded Lie Algebra Generating of Parastatistical Algebraic Relations
Institute of Scientific and Technical Information of China (English)
JING Si-Cong; YANG Wei-Min; LI Ping
2001-01-01
A new kind of graded Lie algebra (We call it Z2,2 graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the Z2,2 graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.
Leibniz algebras associated with representations of filiform Lie algebras
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
Directory of Open Access Journals (Sweden)
Sinan AYDIN
2009-04-01
Full Text Available Linear algebra is a basic course followed in mathematics, science, and engineering university departments.Generally, this course is taken in either the first or second year but there have been difficulties in teachingand learning. This type of active algebra has resulted in an increase in research by mathematics educationresearchers. But there is insufficient information on this subject in Turkish and therefore it has not beengiven any educational status. This paper aims to give a general overview of this subject in teaching andlearning. These education studies can be considered quadruple: a the history of linear algebra, b formalismobstacles of linear algebra and cognitive flexibility to improve teaching and learning, c the relation betweenlinear algebra and geometry, d using technology in the teaching and learning linear algebra.Mathematicseducation researchers cannot provide an absolute solution to overcome the teaching and learning difficultiesof linear algebra. Epistemological analyses and experimental teaching have shown the learning difficulties.Given these results, further advice and assistance can be offered locally.
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
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...
Directory of Open Access Journals (Sweden)
Ali Ghaffari
2014-01-01
Full Text Available The main objective of this paper is to propose an optimal finite duration treatment method for cancer. A mathematical model is proposed to show the interactions between healthy and cancerous cells in the human body. To extend the existing models, the effect of vaccine therapy and chemotherapy are also added to the model. The equilibrium points and the related local stability are derived and discussed. It is shown that the dynamics of the cancer model must be changed and modified for finite treatment duration. Therefore, the vaccine therapy is used to change the parameters of the system and the chemotherapy is applied for pushing the system to the domain of attraction of the healthy state. For optimal chemotherapy, an optimal control is used based on state dependent Riccati equation (SDRE. It is shown that, in spite of eliminating the treatment, the system approaches the healthy state conditions. The results show that the development of optimal vaccine-chemotherapy protocols for removing tumor cells would be an appropriate strategy in cancer treatment. Also, the present study states that a proper treatment method not only reduces the population of the cancer cells but also changes the dynamics of the cancer.
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
On dibaric and evolution algebras
Ladra, M; Rozikov, U A
2011-01-01
We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps $f, g$ of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. Using the pair $(f,g)$ we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a ...
Classification of Noncommutative Domain Algebras
Arias, Alvaro
2012-01-01
Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of $\\C^n$ defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete classification of these algebras based upon techniques inspired by multivariate complex analysis, and more specifically the classification of domains in hermitian spaces up to biholomorphic equivalence.
Process algebra for Hybrid systems
Bergstra, J.A.; Middelburg, C.A.
2008-01-01
We propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from Baeten and Middelburg [Process Algebra with Timing, Springer, Chap. 4, 2002] and the process algebra with propositional signals from Baeten and Bergstra [Theoretical Computer
Process algebra for hybrid systems
Bergstra, J.A.; Middelburg, C.A.
2005-01-01
We propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from Baeten and Middelburg (Process Algebra with Timing, Springer,Berlin, 2002, Chapter 4), and the process algebra with propositional signals from Baeten and Bergstra(Theoret. Com
Cohen, A.M.; Liu, S.
2015-01-01
For each n ≥ 1, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular struc
Directory of Open Access Journals (Sweden)
J. W. Kitchen
1994-01-01
Full Text Available We study bundles of Banach algebras π:A→X, where each fiber Ax=π−1({x} is a Banach algebra and X is a compact Hausdorff space. In the case where all fibers are commutative, we investigate how the Gelfand representation of the section space algebra Γ(π relates to the Gelfand representation of the fibers. In the general case, we investigate how adjoining an identity to the bundle π:A→X relates to the standard adjunction of identities to the fibers.
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.
The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
Hallowell, Karl; Waldron, Andrew
2007-09-01
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.
Operator product expansion algebra
Energy Technology Data Exchange (ETDEWEB)
Holland, Jan [School of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG (United Kingdom); Hollands, Stefan [School of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG (United Kingdom); Institut für Theoretische Physik, Universität Leipzig, Brüderstr. 16, Leipzig, D-04103 (Germany)
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
Maxwell Equations and the Redundant Gauge Degree of Freedom
Wong, Chun Wa
2009-01-01
On transformation to the Fourier space (k,[omega]), the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the…
On isomorphisms of integral table algebras
Institute of Scientific and Technical Information of China (English)
FAN; Yun(樊恽); SUN; Daying(孙大英)
2002-01-01
For integral table algebras with integral table basis T, we can consider integral R-algebra RT over a subring R of the ring of the algebraic integers. It is proved that an R-algebra isomorphism between two integral table algebras must be an integral table algebra isomorphism if it is compatible with the so-called normalizings of the integral table algebras.
Cameron, Peter J
2007-01-01
This Second Edition of a classic algebra text includes updated and comprehensive introductory chapters,. new material on axiom of Choice, p-groups and local rings, discussion of theory and applications, and over 300 exercises. It is an ideal introductory text for all Year 1 and 2 undergraduate students in mathematics. - ;Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with. applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the th...
Indian Academy of Sciences (India)
Vijay Kodiyalam; R Srinivasan; V S Sunder
2000-08-01
In this paper, we study a tower $\\{A^G_n(d):n≥ 1\\}$ of finite-dimensional algebras; here, represents an arbitrary finite group, denotes a complex parameter, and the algebra $A^G_n(d)$ has a basis indexed by `-stable equivalence relations' on a set where acts freely and has 2 orbits. We show that the algebra $A^G_n(d)$ is semi-simple for all but a finite set of values of , and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the `generic case'. Finally we determine the Bratteli diagram of the tower $\\{A^G_n(d): n≥ 1\\}$ (in the generic case).
Markarian, Nikita
2017-03-01
We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.
Axler, Sheldon
2015-01-01
This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the ...
Parametrizing Algebraic Curves
Lemmermeyer, Franz
2011-01-01
We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.
Overview of parallel algorithms in numerical linear algebra
Energy Technology Data Exchange (ETDEWEB)
Sameh, A.
1983-01-01
The author gives a brief survey of the development of multiprocessor algorithms for: (i) the direct solution of linear systems, (ii) the algebraic eigenvalue problem, and (iii) the direct and iterative methods for solving the finite-difference or finite-element linear systems of equations arising from the discretization of linear partial differential equations. 66 references.
Continual Lie algebras and noncommutative counterparts of exactly solvable models
Zuevsky, A.
2004-01-01
Noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of Saveliev-Vershik continual Lie algebras. Examples of noncommutative Liouville and sin/h-Gordon equations are given. The simplest soliton solution to the noncommutative sine-Gordon equation is found.
Super-Poincare' algebras, space-times and supergravities (II)
Santi, Andrea
2011-01-01
The presentation of supergravity theories of our previous paper "Super-Poincare' algebras, space-times and supergravities (I)" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia and Scherk's theory of 11D-supergravity are equivalent to manifestly covariant equations on a supermanifold.
Institute of Scientific and Technical Information of China (English)
Antonio AIZPURU; Antonio GUTI(E)RREZ-D(A)VILA
2004-01-01
In this paper we will study some families and subalgebras ( ) of ( )(N) that let us characterize the unconditional convergence of series through the weak convergence of subseries ∑i∈A xi, A ∈ ( ).As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.
Introduction to abstract algebra
Nicholson, W Keith
2012-01-01
Praise for the Third Edition ". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."-Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately be
Noncommutative algebra and geometry
De Concini, Corrado; Vavilov, Nikolai 0
2005-01-01
Finite Galois Stable Subgroups of Gln. Derived Categories for Nodal Rings and Projective Configurations. Crowns in Profinite Groups and Applications. The Galois Structure of Ambiguous Ideals in Cyclic Extensions of Degree 8. An Introduction to Noncommutative Deformations of Modules. Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II. Quotient Grothendieck Representations. On the Strong Rigidity of Solvable Lie Algebras. The Role of Bergman in Invesigating Identities in Matrix Algebras with Symplectic Involution. The Triangular Structure of Ladder Functors.
Andrilli, Stephen
2010-01-01
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, expl
Preliminary group classification of quasilinear third-order evolution equations
Institute of Scientific and Technical Information of China (English)
Ding-jiang HUANG; Hong-qing ZHANG
2009-01-01
Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transfor-mations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non-equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.
Ten-Year-Old Students Solving Linear Equations
Brizuela, Barbara; Schliemann, Analucia
2004-01-01
In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…
Directory of Open Access Journals (Sweden)
Ivone Patrão
Full Text Available Objective: Women diagnosed with breast cancer are confronted with different stressors throughout the illness trajectory, e.g. awaiting diagnosis, having surgery, anticipating the possibility that the cancer has spread and coping with side effects. The aim of this study was to assess the impact of the psychosocial pathway of breast cancer. Methods: A total of 360 women diagnosed with breast cancer were evaluated, on 3 separate occasions, regarding: distress, emotional control, neuroticism, social support, coping, quality of life (QoL and demographic characteristics. We used structural equation modeling (SEM to examine the relationships among all the variables. Results: The emotional, cognitive response, and the QoL suffered significant changes concerning diagnosis (time 1, surgery (time 2 and treatments (time 3. Furthermore, results indicate that an adapted emotional response is associated to efficient coping strategy, and satisfaction with the perceived social support and good QoL. This is particularly the case when women are undergoing a psychological intervention. Conclusion: To help breast cancer patients adjust to their situation the clinical psychologist should encourage the patient to adopt more efficient coping strategies. By doing so, patients may indeed experience less psychological distress and a higher quality of life, thereby increasing their overall sense of well-being.
Basic math and pre-algebra for dummies
Zegarelli, Mark
2014-01-01
Tips for simplifying tricky basic math and pre-algebra operations Whether you're a student preparing to take algebra or a parent who wants or needs to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary math skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations. Explanations and practical examples that mirror today's teaching methodsRelevant cultural vernacular and referencesStandard For Dummies materials that
Symmetric linear systems - An application of algebraic systems theory
Hazewinkel, M.; Martin, C.
1983-01-01
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
A generalization of the Virasoro algebra to arbitrary dimensions
Energy Technology Data Exchange (ETDEWEB)
Gurau, Razvan, E-mail: rgurau@perimeterinstitute.ca [Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo (Canada)
2011-11-21
Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.
Differential Hopf algebra structures on the universal enveloping algebra of a lie algebra
Hijligenberg, van den, N.W.; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g)$. The construction of such differential structures is interpreted in terms of colour Lie superalgebras.
Topological ∗-algebras with *-enveloping Algebras II
Indian Academy of Sciences (India)
S J Bhatt
2001-02-01
Universal *-algebras *() exist for certain topological ∗-algebras called algebras with a *-enveloping algebra. A Frechet ∗-algebra has a *-enveloping algebra if and only if every operator representation of maps into bounded operators. This is proved by showing that every unbounded operator representation , continuous in the uniform topology, of a topological ∗-algebra , which is an inverse limit of Banach ∗-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-* algebra () of . Given a *-dynamical system (, , ), any topological ∗-algebra containing (, ) as a dense ∗-subalgebra and contained in the crossed product *-algebra *(, , ) satisfies ()=*(, , ). If $G = \\mathbb{R}$, if is an -invariant dense Frechet ∗-subalgebra of such that () = , and if the action on is -tempered, smooth and by continuous ∗-automorphisms: then the smooth Schwartz crossed product $S(\\mathbb{R}, B, )$ satisfies $E(S(\\mathbb{R}, B, )) = C^*(\\mathbb{R}, A, )$. When is a Lie group, the ∞-elements ∞(), the analytic elements () as well as the entire analytic elements () carry natural topologies making them algebras with a *-enveloping algebra. Given a non-unital *-algebra , an inductive system of ideals is constructed satisfying $A = C^*-\\mathrm{ind} \\lim I_$; and the locally convex inductive limit $\\mathrm{ind}\\lim I_$ is an -convex algebra with the *-enveloping algebra and containing the Pedersen ideal of . Given generators with weakly Banach admissible relations , we construct universal topological ∗-algebra (, ) and show that it has a *-enveloping algebra if and only if (, ) is *-admissible.
Schaum's outline of theory and problems of linear algebra
Lipschutz, Seymour
2001-01-01
This third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.
Universal Algebra Applied to Hom-Associative Algebras, and More
Hellström, Lars; Makhlouf, Abdenacer; Silvestrov, Sergei D.
2014-01-01
The purpose of this paper is to discuss the universal algebra theory of hom-algebras. This kind of algebra involves a linear map which twists the usual identities. We focus on hom-associative algebras and hom-Lie algebras for which we review the main results. We discuss the envelopment problem, operads, and the Diamond Lemma; the usual tools have to be adapted to this new situation. Moreover we study Hilbert series for the hom-associative operad and free algebra, and describe them up to total...
Axis Problem of Rough 3-Valued Algebras
Institute of Scientific and Technical Information of China (English)
Jianhua Dai; Weidong Chen; Yunhe Pan
2006-01-01
The collection of all the rough sets of an approximation space has been given several algebraic interpretations, including Stone algebras, regular double Stone algebras, semi-simple Nelson algebras, pre-rough algebras and 3-valued Lukasiewicz algebras. A 3-valued Lukasiewicz algebra is a Stone algebra, a regular double Stone algebra, a semi-simple Nelson algebra, a pre-rough algebra. Thus, we call the algebra constructed by the collection of rough sets of an approximation space a rough 3-valued Lukasiewicz algebra. In this paper,the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras, are studied. Whether the rough 3-valued Lukasiewicz algebra is a axled 3-valued Lukasiewicz algebra is examined.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-07
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
De Sole, Alberto; Kac, Victor G.; Valeri, Daniele
2013-10-01
We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical -algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.
Numerical linear algebra for reconstruction inverse problems
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
Identifying Causal Effects with Computer Algebra
García-Puente, Luis David; Sullivant, Seth
2010-01-01
The long-standing identification problem for causal effects in graphical models has many partial results but lacks a systematic study. We show how computer algebra can be used to either prove that a causal effect can be identified, generically identified, or show that the effect is not generically identifiable. We report on the results of our computations for linear structural equation models, where we determine precisely which causal effects are generically identifiable for all graphs on three and four vertices.