#### Sample records for algebraic cancer equation

1. A simple algebraic cancer equation: calculating how cancers may arise with normal mutation rates

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.

2. 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...

3. Differential Equations for Algebraic Functions

Bostan, Alin; Chyzak, Frédéric; Salvy, Bruno; Lecerf, Grégoire; Schost, Éric

2007-01-01

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove ...

4. Hopf algebras and Dyson-Schwinger equations

Weinzierl, Stefan

2016-06-01

In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.

5. Algebraic dynamics solution and algebraic dynamics algorithm of Burgers equations

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.

6. Solving Absolute Value Equations Algebraically and Geometrically

Shiyuan, Wei

2005-01-01

The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.

7. Asymptotic-group analysis of algebraic equations

Shamrovskii, A. D.; I. V. Andrianov; J. Awrejcewicz

2004-01-01

Both the method of asymptotic analysis and the theory of extension group are applied to study the Descates equation. The proposed algorithm allows to obtain various variants of simplification and can be easily generalized to their algebraic and differential equations.

8. Algebraic entropy for differential-delay equations

Viallet, Claude M.

2014-01-01

We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.

9. An Algebraic Approach to the Scattering Equations

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

2015-01-01

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

10. 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.

11. Algebraic differential equations associated to some polynomials

Barlet, Daniel

2013-01-01

We compute the Gauss-Manin differential equation for any period of a polynomial in \\ $\\C[x_{0},\\dots, x_{n}]$ \\ with \\ $(n+2)$ \\ monomials. We give two general factorizations theorem in the algebra \\ $\\C$ \\ for such a differential equations.

12. The kinematic algebras from the scattering equations

We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant

13. 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.

14. 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.

15. On solving equations of algebraic sum of equal powers

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.

16. Optical systolic solutions of linear algebraic equations

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

17. Algebras with Parastrophically Uncancellable Quasigroup Equations

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.

18. Characteristic Algebras of Fully Discrete Hyperbolic Type Equations

Habibullin, Ismagil T.

2005-01-01

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.

19. Wilsonian renormalization, differential equations and Hopf algebras

Thomas, Krajewski

2008-01-01

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Finally, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally...

20. Upper bounds on the solution of coupled algebraic riccati equation

2001-01-01

Upper bounds for eigenvalues of a solution to continuous time coupled algebraic Riccati equation (CCARE) and discrete time coupled algebraic Riccati equation (DCARE) are developed as special cases of bounds for the unified coupled algebraic Riccati equation (UCARE). They include bounds of the maximal eigenvalues, the sums of the eigenvalues and the trace.

1. Upper bounds on the solution of coupled algebraic riccati equation

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.

2. Laurent phenomenon algebras and the discrete BKP equation

Okubo, Naoto

2016-09-01

We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada–Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.

3. A Unified Algebraic Approach to Classical Yang-Baxter Equation

Bai, Chengming

2007-01-01

In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical ...

4. Algebraic dynamics solution to and algebraic dynamics algorithm for nonlinear advection equation

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.

5. Lie-algebraic symmetries of generalized Davey-Stewartson equations

We identify the full Lie-algebraic structure of the generalized Davey-Stewartson (GDS) system of equations with symmetries of a specific of continual Lie algebras. In particular, we show that they are related to two copies of the Poisson bracket continual Lie algebra.

6. Inhomogeneous linear equation in Rota-Baxter algebra

Pietrzkowski, Gabriel

2014-01-01

We consider a complete filtered Rota-Baxter algebra of weight $\\lambda$ over a commutative ring. Finding the unique solution of a non-homogeneous linear algebraic equation in this algebra, we generalize Spitzer's identity in both commutative and non-commutative cases. As an application, considering the Rota-Baxter algebra of power series in one variable with q-integral as the Rota-Baxter operator, we show certain Eulerian identities.

7. Quantum Einstein’s equations and constraints algebra

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.

8. Deformed algebras: examples and application to Lax equations

Magnot, Jean-Pierre

2014-01-01

We develop here a concept of deformed algebras through three examples and an application. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \\cite{Ma2013} is only an example among others, and how they often give rise to regular Fr\\"olicher Lie groups. Then, we show how such deformed algebras arise in a formal integration of Lax equations through time scaling. The infinite dime...

9. ON THE MINIMAL NONNEGATIVE SOLUTION OF NONSYMMETRIC ALGEBRAIC RICCATI EQUATION

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.

10. Algebraic Solutions of the Lam\\'e Equation, Revisited

Maier, Robert S.

2002-01-01

A minor error in the necessary conditions for the algebraic form of the Lam\\'e equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lam\\'e's differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was misse...

11. 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.

12. Clifford algebra and solution of Bargmann-Michel-Telegdi equation

The Clifford algebra structure of the Minkowski space is presented in the article. The method of solving motion equations within the frames of formalism of this algebra is described. The solution of the spin motion equation (the Bargmann-Michel-Telegdi equations) is obtained by means of the plotted algorithm for the case of the magnetic gyroscope motion in the constant and homogenous electromagnetic field

13. Counting equations in algebraic attacks on block ciphers

Knudsen, Lars Ramkilde; Miolane, Charlotte Vikkelsø

2010-01-01

This paper is about counting linearly independent equations for so-called algebraic attacks on block ciphers. The basic idea behind many of these approaches, e.g., XL, is to generate a large set of equations from an initial set of equations by multiplication of existing equations by the variables...

14. Affine Vertex Operator Algebras and Modular Linear Differential Equations

Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi

2016-05-01

In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.

15. Affine Vertex Operator Algebras and Modular Linear Differential Equations

Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi

2016-04-01

In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.

16. Algebraic methods for the solution of some linear matrix equations

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

1979-01-01

The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.

17. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

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.

18. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

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.

19. 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.

20. Algebraic properties of the Dirac equation in three dimensions

It is argued that, in three dimensions, spinors should have four components as a consequence of the algebraic structure realised from the Clifford algebra related to the Dirac equations. As an example, it is shown then that no induced mass appears in vacuum polarisation at 1-loop in 3 D quantum electrodynamics. (author)

1. Clifford algebras and the classical dynamical Yang-Baxter equation

Alekseev, Anton; Meinrenken, E.

2003-01-01

We describe a relationship of the classical dynamical Yang-Baxter equation with the following elementary problem for Clifford algebras: Given a vector space $V$ with quadratic form $Q_V$, how is the exponential of an element in $\\wedge^2(V)$ under exterior algebra multiplication related to its exponential under Clifford multiplication?

2. Hidden $sl_2$-algebra of finite-difference equations

Smirnov, Yuri; Turbiner, Alexander

1995-01-01

The connection between polynomial solutions of finite-difference equations and finite-dimensional representations of the $sl_2$-algebra is established (the talk given at the Wigner Symposium, Guadalajara, Mexico, August 1995, to be published in the Proceedings)

3. Linear boundary value problems for differential algebraic equations

Balla, Katalin; März, Roswitha

2003-01-01

By the use of the corresponding shift matrix, the paper gives a criterion for the unique solvability of linear boundary value problems posed for linear differential algebraic equations up to index 2 with well-matched leading coefficients. The solution is constructed by a proper Green function. Another characterization of the solutions is based upon the description of arbitrary affine linear subspaces of solutions to linear differential algebraic equations in terms of solutions to the adjoint ...

4. Difference algebraic relations among solutions of linear differential equations

Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael

2013-01-01

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrab...

5. Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra

In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs

6. Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations

The known (2+1)-dimensional breaking soliton equation, the coupled KP equation with three potentials and a new (3+1)-dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the (1+1)-dimensional AKNS equations. The Abel-Jacobi coordinates are introduced to straighten out the associated flows, from which algebraic-geometrical solutions of the (2+1)-dimensional breaking soliton equation, the coupled KP equation and the (3+1)-dimensional evolution equation are explicitly given in terms of the Riemann theta functions

7. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

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.

8. 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…

9. Exact solution of some linear matrix equations using algebraic methods

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

1979-01-01

Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

10. On algebraic relations between solutions of a generic Painleve equation

Nagloo, Ronnie

2011-01-01

We prove that if y" = f(y,y',t,\\alpha, \\beta,..) is a generic Painleve equation (i.e. an equation in one of the families PI-PVI but with the complex parameters \\alpha, \\beta,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school on "irreducibility" of the Painleve equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.

11. Linear spaces for index 2 differential-algebraic equations

Petry, Thomas

1995-01-01

In this paper we consider solution spaces of linear index-2-tractable differential algebraic equations. Relations between the solutions of the adjoint equations and the corresponding solution spaces are derived and, thus, a simple method for computing consistent initial values is provided.

12. Zeta functional equation on Jordan algebras of type II

Kayoya, J. B.

2005-02-01

Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation by using methods based on the algebraic properties of regular graded algebras which are in one-to-one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in this paper is a direct application of specific properties of Jordan algebras of type II.

13. 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.

14. Algebraic Approach to the Computation of the Defining Polynomial of the Algebraic Riccati Equation

Kitamoto, Takuya

The algebraic Riccati equation, which we denote by ’ARE’ in the rest of the paper, is one of the most important equations of the post modern control theory. It plays important role for solving H 2 and H ∞ optimal control problems.

15. Algebraic Reasoning in the Middle Grades: A View of Student Strategies in Pictorial and Algebraic System of Equations

Falcon, Raymond

2009-01-01

Teachers use action research in order to improve their teaching and student learning. This action research will analyze students' algebraic reasoning in finding values of variables in systems of equations pictorially and algebraically. This research will look at students solving linear systems of equations without knowing the algebraic algorithms.…

16. 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.

17. A Novel Partial Differential Algebraic Equation (PDAE) Solver

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

2004-01-01

accuracy and stability. The space-time CE/SE method is successfully implemented to solve PDAE systems through combining an iteration procedure for nonlinear algebraic equations. For illustration, chromatographic adsorption problems including convection, diffusion and reaction terms with a linear or......For solving partial differential algebraic equations (PDAEs), the space-time conservation element/solution element (CE/SE) method is addressed in this study. The method of lines (MOL) using an implicit time integrator is compared with the CE/SE method in terms of computational efficiency, solution...

18. On systems of algebraic equations with parametric exponents

Yokoyama, Kazuhiro

2004-01-01

We deal with systems of algebraic equations with parametric exponents. As the first step for solving such systems, we introduce a simple formulation and basic notions in ideal theory. Then we give a concrete method for most simple cases, univariate case and 0-dimensional case.

19. Matrix solutions of wave equations and Clifford algebras

We are extending the formation of matrix solutions un for linear and nonlinear wave equations by construction of unitary anti-Hermitian-anti-commuting matrices up to the eighth order. We use Clifford algebras C(0,n) with periodicity in modulo 8 to construct coupled matrix solutions. We also propose to use the matrix solutions for describing the intrinsic rotations of particles. (author)

20. 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.

1. 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.

2. Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras

Cherednik, Ivan

1998-01-01

This paper is the course of lectures delivered by the first author in Kyoto in 1996-97 and recorded by the others. We tried to follow closely the notes of the lectures not yielding to the temptation of giving more examples and names. The focus is on the relations of the Knizhnik-Zamolodchikov equations and Kac-Moody algebras to a new theory of spherical and hypergeometric functions based on affine and double affine Hecke algebras. Here mathematics and physics are closer than...

3. Dirac's equation in half of his algebra

Relativistic quantum physics has, since 1928, been introduced in textbooks by using 4x4 complex number matrices. Matrices are actually not needed at all. If used, then only the block diagonal 4x4 matrices are really needed for both the momentum operator and for the wavefunction. This new approach should 'simplify' the mathematics (no matrices), so that this wave-equation material can finally be pushed down into the Junior/Senior-level Modern Physics course. (author)

4. Algebra

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

5. Algorithm for solving polynomial algebraic Riccati equations and its application

Augusta, Petr; Augustová, Petra

2012-01-01

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

6. On Detectability of Switched Linear Differential-Algebraic Equations

Tanwani, Aneel; Trenn, Stephan

2015-01-01

This paper addresses the notion of detectability for continuous-time switched systems comprising linear differential-algebraic equations (DAEs). It relates to studying asymptotic stability of the set of state trajectories corresponding to zero input and zero output. Due to the nature of solutions of switched DAEs, the problem reduces to stability of the trajectories emanating from a non-vanishing unobservable subspace, for which we first derive a geometric expression. The stability of state t...

7. SPR0 SUBSTITUTIONS AND FAMILIES OF ALGEBRAIC RICCATI EQUATIONS

Fernández-Anaya, G.; Martínez-García, J.C.; Kučera, Vladimír; Aguilar-George, D.

Roč. 42, č. 5 ( 2006 ), s. 605-616. ISSN 0023-5954 R&D Projects: GA MŠk(CZ) 1M0567 Institutional research plan: CEZ:AV0Z10750506 Keywords : linear time invariant systems * positive real substitutions * properties preservation * algebraic Riccati equations * H-infinity-norm bounded systems Subject RIV: BC - Control Systems Theory Impact factor: 0.293, year: 2006

8. An Algorithm to Solve algebraic Riccati Equations with Polynomials

Augusta, Petr

Szczecin: West Pomeranian University of Technology, 2012, s. 409-414. ISBN 978-1-4673-2123-5. [The 17th International Conference on Methods and Models in Automation and Robotics . Międzyzdroje (PL), 27.08.2012-30.08.2012] R&D Projects: GA ČR GPP103/12/P494 Institutional support: RVO:67985556 Keywords : algebraic Riccati equations * spatially distributed systems * Fourier transforms Subject RIV: BC - Control Systems Theory

9. Prolongation Loop Algebras for a Solitonic System of Equations

Maria A. Agrotis

2006-11-01

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

10. Stability of the Exponential Functional Equation in Riesz Algebras

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.

11. How Structure Sense for Algebraic Expressions or Equations Is Related to Structure Sense for Abstract 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…

12. 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...

13. Validated Simulation of Differential Algebraic Equations with Runge-Kutta Methods

Alexandre Dit Sandretto, Julien; Chapoutot, Alexandre

2015-01-01

Differential Algebraic Equations (DAEs) are a general and implicit form of differential equations. This mathematical object is often used to represent physical systems such as dynamics of solid or chemical interactions. These equations are different from Ordinary Differential Equations (ODEs) in sense that some of the dependent variables occur without their derivatives. These variables are called algebraic variables'', which means free of derivatives and not with respect to abstract algebra...

14. First order linear ordinary differential equations in associative algebras

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.

15. Bilinear equations and q-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras

We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlevé equations. These cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation-period property. We will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation. (paper)

16. Solutions of Riccati-Abel equation in terms of characteristics of general complex algebra

The Riccati-Abel differential equation defined as an equation between the first order derivative and the cubic polynomial is explored. In the case of constant coefficients this equation is reduced into an algebraic equation. A method of derivation of a summation formula for solutions of the Riccati-Abel equation is elaborated. The solutions of the Riccati-Abel equation are expressed in terms of the characteristic functions of general complex algebra of the third order

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

Sari Saraswati; Ratu Ilma Indra Putri; Somakim

2016-01-01

This research aimed to describe how algebra tiles can support students’ understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students use the algebra tiles to find a method to solve linear equations with one variable. Design research was used as an approach in this study. It ...

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

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

19. Space-time algebra for the generalization of gravitational field equations

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.

20. 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.

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

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 ...

2. 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...

3. Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

Aghaei, S.; Chenaghlou, A.

2014-02-01

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

4. Algebraization of difference eigenvalue equations related to $U_q(sl_2)$

Wiegmann, P. B.; Zabrodin, A. V.

1995-01-01

A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polinomials (discrete polinomials). Such difference operators can be constructed by means of $U_q(sl_2)$, the quantum deformation of the $sl_2$ algebra. The roots of polinomials determine the spectrum and obey the Bethe Ansatz equations. A particular case of difference equations for $q$-hypergeometric and Askey-Wilso...

5. 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…

6. A Direct Algebraic Method in Finding Particular Solutions to Some Nonlinear Evolution Equations

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.

7. Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

BAO Yuan-Peng; RUAN Hang-Yu; XIE Wen-Fang; LI Zhi-Fang

2008-01-01

Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for (2+1)-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton: 1) the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2) the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.

8. Non-associative algebras, Yang-Baxter equations, and quantum computers

Non-associative algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras. The (quantum) Yang-Baxter equation and related structures are interesting topics, because they have applications in many areas of physics, mathematics, and computer science. Several new interpretations and results are presented in this paper

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

Nonlinear equations associated through a zero curvature type representation with Lie algebras S0Diff T2 and of infinitesimal diffeomorphisms of (S1)2, and also with a new infinite-dimensional Lie algebras. In particular, the general solution (in the sense of the Goursat problem) of the heavently equation which describes self-dual Einstein spaces with one rotational Killing symmetry is discussed, as well as the solutions to a generalized equation. The paper is supplied with Appendix containing the definition of the continuum graded Lie algebras and the general construction of the nonlinear equations associated with them. 11 refs

10. On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

Alvaro Alvarez-Parrilla

2012-01-01

Full Text Available A new technique for solving a certain class of systems of autonomous ordinary differential equations over n is introduced ( being the real or complex field. The technique is based on two observations: (1, if n has the structure of certain normed, associative, commutative, and with a unit, algebras over , then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2 a technique, previously introduced for solving differential equations over ℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.

11. An Application of Prolongation Algebras to Determine Bäcklund Transformations for Nonlinear Equations

Bracken, Paul

2015-01-01

Prolongation algebras which are determined by applying a version of the Wahlquist-Estabrook method to three different nonlinear partial differential equations can be employed to obtain not only Lax pairs but B\\"acklund transformations as well. By solving Maurer-Cartan equations for the related group specified by the prolongation algebra, a set of differential forms is obtained which can lead directly to these kinds of results. Although specific equations are studied, the approach should be ap...

12. 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.

13. Equations of motion for a time-dependent open system: An algebraic approach

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

14. Equations of motion for a time-dependent open system: An algebraic approach

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.

15. The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

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.

16. Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations

The ℓ-conformal Galilei algebra, denoted by gl(d), is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by gl(d) with a central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of gl(d) and vector field representations for d = 1, 2. (paper)

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

Sari Saraswati

2016-01-01

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

18. Computer algebra tools for higher symmetry analysis of nonlinear evolution equations

Paper presents a computer-aided approach and a software package for symbolic algebraic computation to solve the problem of verifying the existence of the canonical Lie-Backlund symmetries for multicomponent quasilinear evolution equations with polynomial-nonlinearity and computing a given order symmetry. In the presence of arbitrary numerical parameters the problem is reduced to investigation and solving of nonlinear algebraic equations in those parameters. 16 refs.; 2 tabs

19. Supersymmetric extension of the reflection equation algebra and integrable boundary conditions in doped spin-1 model

The reflection equation algebra of Sklyanin is extended to the supersymmetric case. A graded reflection equation algebra is proposed and the corresponding graded (supersymmetric) boundary quantum inverse scattering method (QISM) is formulated. As an application, integrable open-boundary conditions for the doped spin-1 chain of the supersymmetric t-J model are studied in the framework of the boundary QISM. Diagonal boundary K-matrices are found and four classes of integrable boundary terms are determined. (author)

20. Representations of ℓ-conformai Galilei algebra and hierarchy of invariant equation

The ℓ-conformai Galilei algebra, denoted by gℓ(d), is a particular non-semisimple Lie algebra specified by a positive integer d and a spin value ℓ. The algebra gℓ(d) admits central extensions. We study lowest weight representations, in particular Verma modules, of gℓ(d) with the central extensions for d = 1,2. We give a classification of irreducible modules over d = 1 algebras and a condition of the Verma modules over d = 2 algebras being reducible. As an application of the representation theory, hierarchies of differential equations are derived. The Lie group generated by gℓ(d) with the central extension is a kinematical symmetry of the differential equations.

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

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

2. 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

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

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.

4. Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra

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.

5. Hydra: Automatic algorithm exploration from linear algebra equations

Duchâteau, Alexandre; Padua, David; Barthou, Denis

2013-01-01

International audience Hydra accepts an equation written in terms of operations on matrices and automatically produces highly efficient code to solve these equations. Processing of the equation starts by tiling the matrices. This transforms the equation into either a single new equation containing terms involving tiles or into multiple equations some of which can be solved in parallel with each other. Hydra continues transforming the equations using tiling and seeking terms that Hydra know...

6. The Hamiltonian structure of soliton equations and deformed scr(W)-algebras

The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties and their relation to scr(W)-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel close-quote d endash Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand endash Dickey and the constrained KP hierarchies. Therefore, our results provide a unified description of the relation between the Hamiltonian structure of soliton equations and scr(W)-algebras, and it comprises almost all the results formerly obtained by other authors. The main result of this paper is an explicit general equation showing that the second Poisson bracket algebra is a deformation of the Dirac bracket algebra corresponding to the scr(W)-algebras obtained through Hamiltonian reduction. copyright 1997 Academic Press, Inc

7. The Algebra Solution to Mathematics Reform: Completing the Equation

Spielhagen, Frances R.

2011-01-01

How can we increase mathematics achievement among all students? This book provides a straightforward explanation of how changing mathematics tracking policies to provide algebra instruction to all students by at least eighth grade can bring about changes in both student achievement and teacher performance. Spielhagen chronicles the success of a…

8. New method for solving algebraic equations without the use of imaginary numbers

A new method for solving algebraic equations without the use of imaginary numbers has been proposed. Complex-conjugate roots can be expressed in terms of their absolute value and real part. This method simplifies the process of solving quadratic equations and provides simple approximate but sufficiently accurate formulas for the roots of cubic equations. This method can also be useful for solving equations of higher orders

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

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

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

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.

11. Stability of a Bi-Additive Functional Equation in Banach Modules Over a C⋆-Algebra

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.

12. Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1) Dimensions

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.

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

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.

14. On the algebraic Bethe ansatz for the XXX spin chain: creation operators 'beyond the equator'

Considering the XXX spin-1/2 chain in the framework of the algebraic Bethe ansatz, we make the following short comment: the product of the creation operators corresponding to the recently found solution of the Bethe equations 'on the wrong side of the equator' is just zero (not only its action on the pseudovacuum). (author). Letter-to-the-editor

15. Algebra

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.

16. Algebra

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.

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

Man, Yiu-Kwong, E-mail: ykman@ied.edu.h [Department of Mathematics and Information Technology, Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories (Hong Kong)

2010-10-15

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

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

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

19. Lie-algebraic approach for the Fokker-Planck equation with a nonlinear drift force

In this paper we have investigated the algebraic structure of the Fokker-Planck equation with a nonlinear drift force. Using the Lie-algebraic approach we have obtained an exact form of the time evolution operator which, in turn, enables us to derive the propagator readily. Since the propagator is for a very general Fokker-Planck equation with a nonlinear drift force, results for any special case can be easily deduced from it. (Copyright (c) 1999 Elsevier Science B.V., Amsterdam. All rights reserved.)

20. Gauss对解代数方程的贡献%Gauss' contribution to solving algebraic equation

王宵瑜

2011-01-01

目的 分析Gauss(1777-1855)解方程的主要思想和具体步骤,研究其对解代数方程的贡献.方法 从原始文献出发,寻求求解代数方程的内涵.结果 Gauss证明了分圆方程根式可解；通过对Lagrange(1736-1813)与Gauss关于解代数方程方法的比较,得出Gauss解方程的方法是对Lagrange解方程方法的应用；最后Gauss在解方程的过程中得出一个根式扩张塔.结论 Gauss不仅证明了分圆方程是根式可解的,且在解方程的过程中得到根式扩张塔,改变了方程可解的定义.%Aim To investigate Gauss'(1777-1855) main idea and procedures of solving the cyclotomic equations, and then establish Gauss' contribution to solving algebraic equation. Methods Search for the inner meaning of solving algebraic equations from the original literature. Results Gauss proved the algebraic solvability of cyclotomic equation; Lagrange's and Gauss' methods of solving equation being compared, it shows that Gauss' method is the application of Lagrange's method. Gauss obtained a tower of radical field extension in solving algebraic e-quation. Conclusion Gauss proved the algebraic solvability of cyclotomic equations and got a tower of radical field extension, which changed the definition of radical solvability of the equation.

1. Differential-algebraic equations and applications in circuit theory

Reich, Sebastian

2010-01-01

Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this p...

2. Quadratic algebras and position-dependent mass Schr\\"odinger equations

Quesne, C

2007-01-01

During recent years, exact solutions of position-dependent mass Schr\\"odinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of $d$-dimensional radial harmonic oscillators with $d\\ge2$ (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice. This provides us with a counterpart of the well-known su(1,1) Lie algebraic approach to the constant-mass oscillators.

3. Flipping an Algebra Classroom: Analyzing, Modeling, and Solving Systems of Linear Equations

Kirvan, Rebecca; Rakes, Christopher R.; Zamora, Regie

2015-01-01

The present study investigated whether flipping an algebra classroom led to a stronger focus on conceptual understanding and improved learning of systems of linear equations for 54 seventh- and eighth-grade students using teacher journal data and district-mandated unit exam items. Multivariate analysis of covariance was used to compare scores on…

4. Numerical integration of differential-algebraic equations with harmless critical points

Dokchan, Rakporn

2011-01-01

Algebro-Differentialgleichungen (engl. differential-algebraic equations - DAEs) sind implizite singuläre gewöhnliche Differentialgleichungen, die restringierte dynamische Prozesse beschreiben. Sie unterscheiden sich von expliziten gewöhnlichen Differentialgleichungen dahingehend, dass Anfangswerte nicht beliebig vorgegeben werden können. Weiterhin sind in einer DAE neben Integrations- auch Differentiationsaufgaben involviert. Der Differentiationsindex gibt an, wieviele Differentiationen zur L...

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

Aceto, Luca; Esik, Zoltan; Ingolfsdottir, Anna

2003-01-01

This paper shows that the collection of identities which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum is not finitely based. Moreover, it is proven that, for every n, the equations in at most n variables that hold in N do not form an...

6. 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…

7. On the application of certain interval iteration methods to solving a system of nonlinear algebraic equations

Vengerskii, P.S.; Kardash, A.I.; Seno, P.S. [Lvov State Univ. (Russian Federation)

1994-06-05

We consider the problem of finding real solutions of a system of nonlinear algebraic equations using interval analysis. Several versions of Newton and Runge interval iteration methods are presented. The computational aspects of their application are explained. 6 refs., 2 tabs.

8. 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.

9. A new algorithm for differential-algebraic equations based on HIDM

A new algorithm is proposed to solve differential-algebraic equations. The algorithm is an extension of the algorithm of general purpose HIDM (higher order implicit difference method). A computer program named HDMTDV and based on the new algorithm is constructed and its high performance is proved numerically through several numerical computations, including index-2 problem of differential-algebraic equations and connected rigid pendulum equations. The new algorithm is also secular error free when applied to dissipationless dynamical systems. This nature is demonstrated numerically by computation of the Kepler motion. The new code can solve the initial value problem O = L { φ(x), [dφ(x)]/dx, [d2φ(x)]/dx2, x }, where L and φ are vectors of length N. The values of first or second derivatives of φ(x) are not always necessary in the equations. (author)

10. On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations

Zhang, Yu-Feng; Tam, Honwah

2016-03-01

In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016

11. Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra

I.Yu. Krivsky; Simulik, V. M.

2010-01-01

We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance) the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration ...

12. Quadratic algebras and their application to the problem of non-commutative integration of the Klein-Gordon equation. The four-dimensional square-law algebras, containing the three-dimensional nilpotent Lie algebra

Investigation into the method of non-commutative integration of linear differential equations in particular derivatives is continued in the work. Solutions of the Klein-Gordon equation in the case when lots of non-commutating operators of the Klein-Gordon equation symmetry form square-law algebra comprising a single second order operator and the first order operators forming the Lie algebra, are obtained using the given method. 10 refs

13. Multiple Representations for Systems of Linear Equations Via the Computer Algebra System Maple

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.

14. Homomorphisms in Quasi-Banach Algebras Associated with a Pexiderized Cauchy-Jensen Functional Equation

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).

15. Statistical and algebraic analysis of a family of random Boolean equations

The statistical and algebraic properties of a family of random Boolean equations are studied in this paper. Based on studying the mechanism of influence propagation from some variable fixed to 1, the giant strongly connected component and magnetization of solutions are investigated, which exhibit the splitting phenomenon of solution space and a ferromagnetic transition. Furthermore, by analyzing the semi-group property of the solution space and the influence of propagation from some variable fixed to 0, the scale of generating elements is calculated, which undergoes linear, polynomial and exponential phases. Compared with the analysis by statistical mechanics, it is suggested that these different phases of algebraic complexity correspond to the structural complexity of the solution space as replica symmetry, one-step replica symmetry breaking and further-step replica symmetry breaking phases. It is supposed that the structural complexity of solution space can be interpreted from the viewpoint of algebra

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

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

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

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.

18. Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation

So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie-Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization. -- Highlights: → Generalized Hamiltonian fluid mechanics for the two-dimensional vorticity equation. → Enstrophy as Casimir of the Zeitlin (sine-bracket) truncation. → Algebraic construction of Nambu bracket by Bialynicki-Birula and Morrison method. → Infinite-mode limit yields continuum Nambu bracket. → Discussion of generalized Jacobi identity.

19. On a geometric interpretation of Heisenberg's commutation rule and the algebraic structure of the Pauli equation

It is shown that Heisenberg's commutation rule between the position co-ordinate and the corresponding canonically conjugate momentum may be interpreted by noncommuting geometrical structures. As in the absence of a magnetic field the Euclidean norm of the momentum space directly enters the kinetic energy, the momentum space can be mapped onto the quaternion field U2. Such a mapping preserves the norm of the momentum space. By that, the geometric and algebraic structure of the Pauli equation can be obtained and the relationship between the Pauli and the Dirac equation can be made apparent by noncommuting algebraic structures. In an appendix it will also be shown that the extension of the procedure to vector spaces equipped with Riemannian geometry makes no difficulties and a covariant quantization procedure can be formulated. (author)

20. On the vibrations of lumped parameter systems governed by differential-algebraic equations

Darbha, S.; Nakshatrala, K. B.; K.R. Rajagopal

2009-01-01

In this paper, we consider the vibratory motions of lumped parameter systems wherein the components of the system cannot be described by constitutive expressions for the force in terms of appropriate kinematical quantities. Such physical systems reduce to a system of differential-algebraic equations, which invariably need to be solved numerically. To illustrate the issues with clarity, we consider a simple system in which the dashpot is assumed to contain a "Bingham" fluid for which one canno...

1. Projected Runge-Kutta methods for differential algebraic equations of index 3

Schropp, Johannes

2003-01-01

In the present paper we introduce a new class of methods, Projected Runge-Kutta methods, for the solution of index 3 differential algebraic equations (DAEs) in Hessenberg form. The methods admit the integration of index 3 DAEs without any drift effects. This makes them particularly well suited for long term integration. Finally, implemented on the basis of the Radau5 code, the projected Runge-Kutta method admits larger step sizes for a prescribed tolerance than the corresponding classical sc...

2. Parabolic orbit determination. Comparison of the Olbers method and algebraic equations

Kuznetsov, V. B.

2016-05-01

In this paper, the Olbers method for the preliminary parabolic orbit determination (in the Lagrange-Subbotin modification) and the method based on systems of algebraic equations for two or three variables proposed by the author are compared. The maximum number of possible solutions is estimated. The problem of selection of the true solution from the set of solutions obtained both using additional equations and by the problem reduction to finding the objective function minimum is considered. The results of orbit determination of the comets 153P/Ikeya-Zhang and 2007 N3 Lulin are cited as examples.

3. Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras

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.

4. Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras

Grahovski, Georgi G., E-mail: G.Grahovski@leeds.ac.uk [Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT (United Kingdom); Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chausee, Sofia 1784 (Bulgaria); Mikhailov, Alexander V., E-mail: A.V.Mikhailov@leeds.ac.uk [Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT (United Kingdom)

2013-12-17

Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.

5. A new linear Dirac-like spin-3/2 wave equation using Clifford algebra

A new linear Dirac-like wave equation for spin-3/2 is derived, employing four of the seven irreducible eight-dimensional matrices obeying the Clifford algebra C7 with the wave function having the needed eight components only. Though this wave equation is not manifestly covariant and the wave function employed is not locally covariant, it is relativistically invariant and by its very derivation is connected to the Weaver, Hammer and Good (Phys. Rev.; 135: B241 (1964)) formalism for spin-3/2 by a chain of transformations which can be arbitrarily chosen to be either unitary or non-unitary. (author)

6. Hierarchies of finite-dimensional Lax equations with a spectral parameter on a Riemann surface and semisimple Lie algebras

Sheinman, O. K.

2015-12-01

Based on ℤ-gradings of semisimple Lie algebras and invariant polynomials on them, we construct hierarchies of Lax equations with a spectral parameter on a Riemann surface and prove the commutativity of the corresponding flows.

7. The exact solution of the Cauchy problem for two generalized linear vectorial Fokker-Planck equations: algebraic approach

The exact solutions of the Cauchy problem for two equations which are slight generalization of the so-called linear vectorial Fokker-Planck equation are found using the disentangling techniques of Feynman and algebraic (operational) methods. This approach may be considered as a generalization of the Suzuki's method for solving the one-dimensional linear Pokker-Planck equation

8. The Lie algebra of invariant group of the KdV, MKdV, or Burgers equation

In this paper it is proved that all generators of the invariant group of the KdV, MKdV or Burgers equation form a commutative Lie algebra, from which it follows that for any symmetry of these three equations, the evolution equation possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations). The authors conjecture that this result holds for a yet wider class of equations. (Auth.)

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

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.

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

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

11. Algebraic solution of the Lindblad equation for a collection of multilevel systems coupled to independent environments

Bolaños, Marduk; Barberis-Blostein, Pablo

2015-11-01

We consider the Lindblad equation for a collection of multilevel systems coupled to independent environments. The equation is symmetric under the exchange of the labels associated with each system and thus the open-system dynamics takes place in the permutation-symmetric subspace of the operator space. The dimension of this space grows polynomially with the number of systems. We construct a basis of this space and a set of superoperators whose action on this basis is easily specified. For a given number of levels, M, these superoperators are written in terms of a bosonic realization of the generators of the Lie algebra {sl}({M}2). In some cases, these results enable finding an analytic solution of the master equation using known Lie-algebraic methods. To demonstrate this, we obtain an analytic expression for the state operator of a collection of three-level atoms coupled to independent radiation baths. When analytic solutions are difficult to find, the basis and the superoperators can be used to considerably reduce the computational resources required for simulations.

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

Murat Osmanoglu

2013-01-01

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

13. A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra

Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei

2013-01-01

The analysis of the $\\mathbb{Z}_2^{3}$ Laplace-Dunkl equation on the $2$-sphere is cast in the framework of the Racah problem for the Hopf algebra $sl_{-1}(2)$. The related Dunkl-Laplace operator is shown to correspond to a quadratic expression in the total Casimir operator of the tensor product of three irreducible $sl_{-1}(2)$-modules. The operators commuting with the Dunkl Laplacian are seen to coincide with the intermediate Casimir operators and to realize a central extension of the Banna...

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

Papež, Jan; Liesen, J.; Strakoš, Z.

2014-01-01

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

15. Symmetries of the Schr\\"odinger Equation and Algebra/Superalgebra Duality

Toppan, Francesco

2014-01-01

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

16. Symmetries of the Schrödinger equation and algebra/superalgebra duality

Toppan, Francesco

2015-04-01

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

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

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

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

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.

19. Modeling boyciana-fish-human interaction with partial differential algebraic equations.

Jiang, Yushan; Zhang, Qingling; Wang, Haiyan

2016-07-01

Under the influence of human population distribution, the boyciana-fish ecological system is considered. First, the system can be described as a nonlinear partial differential algebraic equations system (PDAEs) with Neumann boundary conditions and ratio-dependent functional response. Second, we examine the system's persistence properties: the loacl stabilities of positive steady states, the absorbtion region and the global stability. And the proposed approach is illustrated by numerical simulation. Finally, by using the realistic data collected in the past fourteen years, the PDAEs parameter optimization model is built to predict the boyciana population. PMID:27155570

20. Graph theory, irreducibility, and structural analysis of differential-algebraic equation systems

Pryce, John D.; Nedialkov, Nedialko S.; Tan, Guangning

2014-01-01

The $\\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the DAE that can be exploited to speed up numerical solution. The paper compares this fine BTF with the usually coarser BTF derived from the sparsity pattern of the \\sigmx. It defines a Fine-Block Graph with weighted edges, which gives insight into the relatio...

1. Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model

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

2. 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.

3. Direct Method to Solve Differential-Algebraic Equations by Using the Operational Matrices of Chebyshev Cardinal Functions

M. Heydari

2013-05-01

Full Text Available A new and effective direct method to determine the numerical solution of linear and nonlinear differential-algebraic equations (DAEs is proposed. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a differentialalgebraic equation can be transformed to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique

4. Lie Algebraic Structures and Integrability of Long-Short Wave Equation in （2＋1）Dimensions

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.

5. On the extended real Clifford-Dirac algebra and new physically meaningful symmetries of the Dirac equations with nonzero mass

A 64-dimensional extended real Clifford-Dirac algebra is introduced. On its basis, new pure matrix symmetries of the Dirac equation in the Foldy-Wouthuysen representation was found. Finally, spin 1 Poincare symmetries both for the Foldy-Wouthuysen and standard Dirac equations with nonzero mass are found.

6. A Generalized Variable-Coefficient Algebraic Method Exactly Solving (3+1)-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is applied to construct several new families of exact solutions of physical interest for (3+1)-dimensional Kadomtsev-Petviashvilli (KP) equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.

7. Exact algebraization of the signal equation of spoiled gradient echo MRI

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.

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

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems. PMID:27330880

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

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

10. New solutions from algebraic equations for the Skyrme model coupled to a scalar meson

Braghin, Fabio L. [Universidade Federal do Rio Grande do Norte (IIF/UFRN), Natal, RN (Brazil). Inst. Internacional de Fisica

2010-07-01

Full text: In this work a modified Skyrme model is considered such as to incorporate the interaction of the hedgehog with a scalar field, based on a previous work. The Skyrme model is a model of the nucleon in which the baryon emerges as a topological soliton and its coupling to a scalar field can either correspond to the coupling to the lightest scalar isoscalar meson sigma and also to implement the spontaneous breakdown of chiral symmetry in a consistent way. Therefore it can be related to modifications of a dense interacting medium and it becomes suitable for investigating the role of the symmetry breaking and its restoration. A transcendental algebraic equation is found to be enough to extract a new class of profile solutions of the skyrmion in a constant background. The mass of the corresponding topological soliton was found to decrease considerably in the case small masses are associated to the scalar field. (author)

11. FAST SOLUTION FOR LARGE SCALE LINEAR ALGEBRAIC EQUATIONS IN FINITE ELEMENT ANALYSIS

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.

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

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.

13. Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions

Christiane Quesne

2007-05-01

Full Text Available An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrödinger equations.

14. The Stability of Two-Step Runge-Kutta Methods for Neutral Delay Integro Differential-Algebraic Equations with Many Delays

Haiyan Yuan; Jihong Shen

2014-01-01

This paper studies the asymptotic stability of the two-step Runge-Kutta methods for neutral delay integro differential-algebraic equations with many delays. It proves that A-stable two-step Runge-Kutta methods are asymptotically stable for neutral delay integro differential-algebraic equations with many delays.

15. An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order

In this paper, an extended algebraic method with symbolic computation is applied to construct a series of travelling wave solutions of the one-dimensional generalized BBM equation of any order with positive and negative exponents. As a result, the proposed method gives many explicit exact solutions such as solitary wave solutions, periodic solutions, solitary patterns solutions and compacton solutions

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

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.

17. New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy

A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-tilde2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.

18. Runge-Kutta方法对微分代数方程的正则性%Regularity properties of Runge-Kutta method for differential-algebraic equation

张春蕊; 郑宝东

2003-01-01

Runge-Kutta方法对微分代数方程是正则的,是指数值解的有限渐进值与方程本身的渐进值是相等的.给出了保证Runge-Kutta方法对微分代数方程是正则的条件,证明了Runge-Kutta方法是正则的充要条件是折叠方法是正则的.%The numerical method for differential-algebraic equation is regular if it has the same set offinite asymptotic values as the underlying differential-algebraic system. The conditions that guaranteeregular properties of Runge-Kutta method for differential-algebraic equation are considered. It isproved that the Runge-Kutta method for differential-algebraic equation is regular if and only if thefold method is regular for differential-algebraic equation.

19. Comparison of critical behaviors of elliptic and hyperbolic quadratic algebraic equations with variable coefficients

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.

20. Quadratic Algebra Approach to the Dirac Equation with Spin and Pseudospin Symmetry for the 4D Harmonic Oscillator and U (1) Monopole

In this paper, we study the Dirac equation with spin and pseudospin symmetry by the quadratic algebra approach for the 4-dimensional harmonic oscillator. By realization of the quadratic algebras in the deformed oscillator algebra, we obtain the relativistic energy spectrum. Also, by regarding the generalized Kustaanheimo–Stiefel transformation, we obtain the relativistic energy spectrum for the charge-dyon system with the U(1) monopole. (author)

1. Quadratic Algebra Approach to the Dirac Equation with Spin and Pseudospin Symmetry for the 4D Harmonic Oscillator and U(1) Monopole

Aghaei, S.; Chenaghlou, A.

2015-01-01

In this paper, we study the Dirac equation with spin and pseudospin symmetry by the quadratic algebra approach for the 4-dimensional harmonic oscillator. By realization of the quadratic algebras in the deformed oscillator algebra, we obtain the relativistic energy spectrum. Also, by regarding the generalized Kustaanheimo-Stiefel transformation, we obtain the relativistic energy spectrum for the charge-dyon system with the U(1) monopole.

2. Young Students Learning Formal Algebraic Notation and Solving Linear Equations: Are Commonly Experienced Difficulties Avoidable?

Hewitt, Dave

2012-01-01

This study looks at a mixed ability group of 21 Year 5 primary students (aged 9-10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software "Grid Algebra"…

3. 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…

4. Geometric interpretation of Heisenberg's commutation rule and the algebraic structure of the Pauli equation

Ulmer, W.; Hartmann, H. (Frankfurt Univ. (Germany, F.R.). Inst. fuer Physikalische und Theoretische Chemie)

1978-09-01

It is shown that Heisenberg's commutation rule between the position co-ordinate and the corresponding canonically conjugate momentum may be interpreted by noncommuting geometrical structures. As in the absence of a magnetic field the Euclidean norm of the momentum space directly enters the kinetic energy, the momentum space can be mapped onto the quaternion field U/sub 2/. Such a mapping preserves the norm of the momentum space. By that, the geometric and algebraic structure of the Pauli equation can be obtained and the relationship between the Pauli and the Dirac equation can be made apparent by noncommuting algebraic structures. In an appendix it will also be shown that the extension of the procedure to vector spaces equipped with Riemannian geometry makes no difficulties and a covariant quantization procedure can be formulated.

5. Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes by Algebraic Method

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.

6. A Dirac-Dunkl Equation on S 2 and the Bannai-Ito Algebra

De Bie, Hendrik; Genest, Vincent X.; Vinet, Luc

2016-05-01

The Dirac-Dunkl operator on the two-sphere associated to the Z23 reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

7. 代数Riccati方程可稳解的条件数%CONDITION NUMBERS OF THE STABILIZING SOLUTIONS OF THE ALGEBRAIC RICCATI EQUATIONS

刘新国

2001-01-01

The Condition numbers are defined for the stabilizing Solutionsof Continuoustime, discrete-time and the reverse discrete-time algebraic Riccati equations. The first-order perturbation expansions for the stabilizing Solutions are also obtained.

8. 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.

9. Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

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

10. Shape Optimization for Navier-Stokes Equations with Algebraic Turbulence Model: Existence Analysis

Bulíček, M.; Haslinger, J.; Málek, J.; Stebel, Jan

2009-01-01

Roč. 60, č. 2 (2009), s. 185-212. ISSN 0095-4616 R&D Projects: GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : optimal shape design * paper machine headbox * incompressible non-Newtonian fluid * algebraic turbulence model * outflow boundary condition Subject RIV: BA - General Mathematics Impact factor: 0.757, year: 2009

11. Chaos Control in Three Dimensional Cancer Model by State Space Exact Linearization Based on Lie Algebra

2016-01-01

This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM) by State Space Exact Linearization (SSEL) technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustne...

12. Symmetry Lie algebras and properties of linear ordinary differential equations with maximal dimension

Folly-Gbetoula, Mensah; Kara, A. H.

2015-04-01

Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the first-order source equation are obtained. Based on certain properties of iterative equations, finding the solutions is reduced to finding solutions of the second-order source equation. We have therefore found classes of solutions to the source equations by letting the parameters of the source equation be functions of a specific type such as monomials, functions of exponential and logarithmic type.

13. The exploration of Galois' algebraic equation theory%伽罗瓦的代数方程思想

赵晔; 王昌; 周畅

2011-01-01

目的 探讨在代数方程根式可解性理论的发展中,伽罗瓦(Evariste Galois,1811-1832)的代数方程理论思想发展过程.方法 采用历史考察与数理分析法.结果 伽罗瓦是通过引进"伽罗瓦群"、"正规子群"、"置换群"等概念开始建立他的理论,并且找出了根式扩张塔和可解群之间的对应关系,利用这种对应关系最终解决了代数方程根式可解性理论这一难题.结论 伽罗瓦继承了拉格朗日(J.L.Lagrange,1736-1813)问题转化的思想,并且把这一思想进行发展,使得人们对方程根式解问题的研究进入到对"结构"观念的研究,导致了抽象代数学科的诞生;伽罗瓦的研究思路是通过继承和发展前人的思想成果得出来的.%Aim To explore the theory of Galois' algebraic equation through the depth study on the radical solution theory. Methods Historical investigation and mathematical analysis. Results Galois established his theory by introducing some concepts such as "Galois group", "normal subgroup", "permutation group" etc, and identifying the corresponding relation between radical expansion tower and solvable group. He finally solved the problem of algebraic equation radical solution theory. Conclusion Galois inherited the thoughts of Lagrange's problem transforming, and he developed this thoughts so that people turned the research of algebraic equation radical solution theory into the study of structure concept. It caused the establishment of abstract algebra discipline. His research idea originated from inheriting and developing predecessors' achievements.

14. Desingularizing of Differential-Algebraic Equations%微分代数方程去奇异化分析

邹杰涛; 王瑞瑞; 杨志辉

2011-01-01

利用去奇异化方法讨论了拟线性微分代数方程在奇点邻域内光滑解的性质.通过尺度参数的微分同胚变换,将拟线性微分代数方程转化为相应的常微分方程,从而构造出在孤立奇点邻域内的初始微分代数方程的光滑解,给出解存在的充分条件,并进一步讨论了解的性质.%In this paper we use the desingularizing method to discuss smooth solutions of quasi-linear differential-algebraic equations (DAEs) in the neighborhood of singularities. Through a diffeomorphic transformation of the scale parameter, the original quasi-linear DAEs can be changed into a corresponding ODE from which we can construct smooth solutions of DAEs near the isolate singularities. Furthermore, a sufficient condition for the existence of smooth solutions of quasi-linear differential-algebraic equations is presented.

15. A wave equation including leptons and quarks for the standard model of quantum physics in Clifford Algebra

Daviau, Claude

2014-01-01

A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.

16. A Wave Equation including Leptons and Quarks for the Standard Model of Quantum Physics in Clifford Algebra

Daviau, Claude; Bertrand, Jacques

A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.

17. Shape optimization for Navier-Stokes equations with algebraic turbulence model : numerical analysis and computation

Haslinger, J.; Stebel, Jan

2011-01-01

Roč. 63, č. 2 (2011), s. 277-308. ISSN 0095-4616 R&D Projects: GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : optimal shape design * paper machine headbox * incompressible non-Newtonian fluid * algebraic turbulence model Subject RIV: BA - General Mathematics Impact factor: 0.952, year: 2011 http://link.springer.com/article/10.1007%2Fs00245-010-9121-x

18. On Lagrange＇s Algebraic Equation Theory and Its Influence%拉格朗日的代数方程求解理论及其影响

赵增逊

2012-01-01

Lagrange＇s theory of algebraic equations is an integral part of the resolution history of algebraic equations,and the theory has an important impact on later algebraist.To reveal the content of Lagrange＇s algebraic equations theory,express its far-reaching implications,the article,based on original literature, concisely describes the content of Lagrange＇s algebraic equations theory and its influence.Therefore, understanding Lagrange＇s theory of algebraic equations is not only conducive to the understanding of the auxiliary equation theory,the connotation of permutation theory,but also is more helpful to find out the entire resolution history of algebraic equation.%拉格朗日的代数方程求解理论是整个代数方程求解史中不可或缺的一部分,并且该理论对以后的代数学家产生了重要的影响。为展示拉格朗日代数方程求解理论的内容,说明该理论产生的深远影响,从原始文献出发,叙述了拉格朗日的代数方程求解理论的内容,重点阐述了该理论产生的重要影响。因此,清楚拉格朗日的代数方程求解理论不仅有利于了解辅助方程理论、置换思想的内涵,更有利于清楚整个代数方程的求解历史。

19. College algebra

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

20. The generalised continuous algebraic Riccati equation and impulse-free continuous-time LQ optimal control

Ferrante, Augusto; Ntogramatzidis, Lorenzo

2013-01-01

The purpose of this paper is to investigate the role that the continuous-time generalised Riccati equation plays within the context of singular linear-quadratic optimal control. This equation has been defined following the analogy with the discrete-time generalised Riccati equation, but, differently from the discrete case, to date the importance of this equation in the context of optimal control is yet to be understood. This note addresses this point. We show in particular that when the conti...

1. Generalized Flip-Flop Input Equations Based on a Four-Valued Boolean Algebra

Tucker, Jerry H.; Tapia, Moiez A.

1996-01-01

A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.

2. 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.…

3. Generating Generalized Bessel Equations by Virtue of Bose Operator Algebra and Entangled State Representations

FAN Hong-Yi; WANG Yong

2006-01-01

With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.

4. Electromagnetic energy-momentum equation without tensors: a geometric algebra approach

Sugon Jr., Quirino M.; McNamara, Daniel J.

2008-01-01

In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is ...

5. L-o cto-algebras

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.

6. A new general algebraic method with symbolic computation to construct new exact analytical solution for a (2 + 1)-dimensional cubic nonlinear Schroedinger equation

Based on a new general ansaetz, a new general algebraic method named improved Riccati equation rational expansion method is devised for constructing multiple nontravelling wave solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. With the aid of symbolic computation, we choose the (2 + 1)-dimensional cubic nonlinear Schroedinger equation to illustrate the method. As a result, six families of new exact analytical solutions for this equation are found, which include some new and more general exact rational form soliton-like solutions and triangular periodic-like solutions

7. Resolving G\\"odel's Incompleteness Myth: Polynomial Equations and Dynamical Systems for Algebraic Logic

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 ...

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

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.

9. Computer subroutine ISUDS accurately solves large system of simultaneous linear algebraic equations

Collier, G.

1967-01-01

Computer program, an Iterative Scheme Using a Direct Solution, obtains double precision accuracy using a single-precision coefficient matrix. ISUDS solves a system of equations written in matrix form as AX equals B, where A is a square non-singular coefficient matrix, X is a vector, and B is a vector.

10. Algebra Word Problem Solving Approaches in a Chemistry Context: Equation Worked Examples versus Text Editing

Ngu, Bing Hiong; Yeung, Alexander Seeshing

2013-01-01

Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…

11. Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy

Kazeykina, Anna

2012-01-01

We show that the Novikov--Veselov equation (an analog of KdV in dimension 2 + 1) at positive and negative energies does not have solitons with the space localization stronger than O(|x|^{-3}) as |x| \\to \\infty.

12. Realizations of Galilei algebras

All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations. (paper)

13. Picard-Fuchs equations, Integrable Systems, and higher Algebraic K-theory

del Angel, Pedro L.; Müller-Stach, Stefan

2002-01-01

This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman to our setup.

14. On the algebraic classification of perfect fluid solutions of Einstein's equations

This thesis concerns the Einstein-Petrov problem for perfect fluid space-times. This is the problem of finding exact solutions of Einstein's perfect fluid field equations for which the Weyl tensor is of a specific Petrov type. No examples of exact perfect fluid space-times having a Weyl tensor of Petrov type III have ever appeared in the literature. This is the only Petrov Class for which this is so. Consequently an example of such a solution is sought. Using the Newman-Penrose formalism a particular solution is obtained and its properties are discussed. (author)

15. Free Markov processes and stochastic differential equations in von Neumann algebras

Gao, Mingchu

2008-01-01

Free Markov processes are investigated in Voiculescu’s free probability theory. We show that Voiculescu’s free Markov property implies a property called “weak Markov property”, which is the classical Markov property in the commutative case; while, in the general case, the “weak Markov property” is the same as the Markov property defined by Bozejko, Kummer, and Speicher. We also show that a kind of stochastic differential equations driven by free Levy processes has solutions. The solutions are...

16. Linear algebra

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

17. Linear algebra

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

18. 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.

19. Linear algebra

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...

20. Phantom energy from graded algebras

Chaves, Max; Singleton, Douglas

2006-01-01

We construct a model of phantom energy using the graded Lie algebra SU(2/1). The negative kinetic energy of the phantom field emerges naturally from the graded Lie algebra, resulting in an equation of state with w

1. 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

2. Left Artinian Algebraic Algebras

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.

3. Colored Quantum Algebra and Its Bethe State

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

4. General solution and generalized Ulam-Hyers stability of a generalized n- type additive quadratic functional equation in Banach space and Banach algebra: direct and fixed point methods

Murthy, S; Arunkumar, M; V. Govindan

2015-01-01

In this paper, the authors introduce and investigate the general solution and generalized Ulam-Hyers stability of a generalized n-type additive-quadratic functional equation. g(x + 2y; u + 2v) + g(x 2y; u 2v) = 4[g(x + y; u + v) + g(x y; u v)] 6g(x; u) + g(2y; 2v) + g(2y;2v) 4g(y; v) 4g(y;v) Where  is a positive integer with , in Banach Space and Banach Algebras using direct and fixed point methods.

5. 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

6. Algebraic geometry

Lefschetz, Solomon

2012-01-01

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

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

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.

8. Exact Analytical Solution of the N-dimensional Radial Schrodinger Equation with Pseudoharmonic Potential via Laplace Transform Approach and Lie algebra of Ladder Operators

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.

9. Yang-Mills gauge fields conserving the symmetry algebra of the Dirac equation in a homogeneous space

We consider the Dirac equation with an external Yang-Mills gauge field in a homogeneous space with an invariant metric. The Yang-Mills fields for which the motion group of the space serves as the symmetry group for the Dirac equation are found by comparison of the Dirac equation with an invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The eigenfunctions and the corresponding eigenvalues for the Dirac equation are obtained in the space R2 × S2 by a noncommutative integration method

10. The Algebraic Way

Hiley, B. J.

In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.

11. Monomial algebras

Villarreal, Rafael

2015-01-01

The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.

12. A Deductive Approach towards Reasoning about Algebraic Transition Systems

Jun Fu; Jinzhao Wu; Hongyan Tan

2015-01-01

Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semanti...

13. Symplectic algebraic dynamics algorithm

2007-01-01

Based on the algebraic dynamics solution of ordinary differential equations andintegration of  ,the symplectic algebraic dynamics algorithm sn 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 sn 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.

14. Matrices and linear algebra

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

15. 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

16. Pre-Algebra Essentials For Dummies

Zegarelli, Mark

2010-01-01

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

17. Exceptional Vertex Operator Algebras and the Virasoro Algebra

Tuite, Michael P.

2008-01-01

We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Exam...

18. Proposition Algebra with Projective Limits

Bergstra, J A

2008-01-01

Sequential logic deviates from propositional logic by taking into account that atomic propositions yield different Boolean values at different times during the sequential evaluation of a single proposition. Reactive valuations capture this dynamics of a proposition's environment. This logic is phrased as an equationally specified algebra rather than in the form of proof rules. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The proposition algebra PRA is developed in a fashion similar to the process algebra ACP and the program algebra PGA via an algebraic specification which has a meaningful initial algebra for which a range of courser congruences are considered important as well. In addition infinite objects (that is propositions, processes and programs respectively) are preferably dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts ...

19. Quantum computation using geometric algebra

Matzke, Douglas James

This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.

20. Solitons as purely algebraic construction

A new purely algebraic method for finding soliton solutions for nonlinear equations Without using the inverse scattering method is elaborated. As the examples the soliton solutions are given explicitly for both the well known nonlinear equations and equations which have not been discussed earler. The symmetry basis of the method is connected with the infinite-dimensional internal symmetry Lie algebra of the system under consideration

1. Basic linear algebra

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:...

2. Supertropical algebra

Izhakian, Zur; Rowen, Louis

2008-01-01

We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...

3. Helmholtz algebraic solitons

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.

4. Helmholtz algebraic solitons

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.

5. Implementation of an algebraic bypass transition model into two-equation turbulence model for a finite volume method solver

Dobeš, J.; Fořt, J.; Příhoda, Jaromír

Praha: Institute of Thermomechanics AS CR, v. v. i., 2007 - (Jonáš, P.; Uruba, V.), s. 13-14 ISBN 978-80-87012-07-9. [Colloquium FLUID DYNAMICS 2007. Prague (CZ), 24.10.2007-26.10.2007] R&D Projects: GA ČR GA101/07/1508 Institutional research plan: CEZ:AV0Z20760514 Keywords : bypass transition modelling * two-equation turbulence model * finite volume method solver Subject RIV: BK - Fluid Dynamics

6. The Dirac equation in a non-Riemannian manifold III: An analysis using the algebra of quaternions and octonions

The geometrical properties of a flat tangent space-time local to the manifold of the Einstein--Schroedinger nonsymmetric theory to which an internal octonionic space is attached, is developed here. As an application of the theory, an octonionic Dirac equation for a spin-1/2 particle is also obtained, where is now used an octonionic-like gauge field. It is shown that the (quaternionic) nonsymmetric Yang--Mills theory can be easily recovered and from there, the usual gauge theory on a curved space

7. Semigroups and computer algebra in algebraic structures

Bijev, G.

2012-11-01

Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.

8. 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

9. Homogeneous conformal averaging operators on semisimple Lie algebras

Kolesnikov, Pavel

2014-01-01

In this note we show a close relation between the following objects: Classical Yang---Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular part of a solution of CYBE (in the operator form) on a Lie algebra $\\mathfrak g$ determines an averaging operator on the corresponding current conformal algebra $\\mathrm{Cur} \\mathfrak g$. For a finite-dimensional semisimple Lie algebra $\\mathfrak g$, we desc...

10. An infinite algebra of quantum Dirac brackets

A new algebraic approach to the theory with second-class constraints is proposed. The operator equations that generate automatically the infinite algebra of quantum Dirac brackets are formulated. First-class constraints are naturally involved into the new algebraic scheme. (orig.)