Computer Algebra Software Research Articles

"HYPERELLIPTIC DIOPHANTINE EQUATIONS FROM THE STUDY OF PARTITIONS"

"We present two Diophantine equations that arise from some new results in the theory of partitions with equal sums. We link these to the problem of finding rational points on some hyperelliptic curves and we solve the latter, assisted by computer algebra packages, using a p-adic method pioneered by Chabauty and Coleman."

Composite refinement techniques for solving linear systems

A composite refinement approach for stationary iterative methods is introduced. Two new formulas (RJGS and RGSJ) are compared with the classical forms. Rates of convergence of the introduced composite formulas (RJGS and RGSJ) are well established. The efficient performance of the new forms is established theoretically and confirmed through numerical examples. The decrease in the required number of iterations for convergence is established through the calculation of the spectral radius of the iteration matrices. The algorithmic structure of the new formulas is announced. Three numerical examples with different convergent properties are considered. The calculations are performed with the help of computer algebra software Mathematica.

A Note on the Accurate Computation of Structural Properties for Dynamic Control Systems

In analysing the observability, identifiability, and controllability of smooth (possibly non-linear) dynamic control systems, the authors identified numerous ways to improve both the accuracy and efficiency of their computations. We summarize a few techniques that help to pinpoint exactly what kind of linear dependencies exist between parametric output sensitivities. These relations, in turn, can be traced back with the help of computer algebra software to an exact re-parametrizations of the original model. The suggested techniques to increase the accuracy of results will be presented on the basis of a few examples.

Vibrations in a Growing Nonlinear Chain

A one-dimensional chain describing the linear statistical increment of growing homogeneous atoms was arbitrarily built and investigated using an energy potential function. The analytic form of the considered potential has two exponential terms which describes chaotic behavior when the chain was excited. In order to investigate the dynamics of statistical attachment of individual atoms in the slender gold chain, the total energy of the entire system was changed by increasing the kinetic energy upon increment of homogeneous atoms in the chain. This resulted in a corresponding increase of the total energy in the system. On the other hand, the potential energy of the system on increment of homogeneous atoms equals zero, because the distance between corresponding atoms equals to the molecular distance (minimum potential distance). We considered the dynamical system with linear damping and without linear damping. \\ Different initial points were investigated to obtain trends of vibration that includes chaotic and regular oscillations. At some initial point(s), the attached atom experiences an infinite jump which means it falls off the nonlinear slender chain and the chain was broken. The interpretation of this phenomenon means the gold chain will result into an unstable nanostructure. We compared the numerical simulation of the system with different built-in ordinary differential equation solvers of various computer algebra software. Numerical simulation were carried out by plotting the system of growing atoms' displacement against time. The system of linearly attached atoms were numerically simulated and inferences were stated from the study. In all cases considered, we inferred that amplitude of oscillation significantly increased at the end of the chain (terminal point) as compared to the initial point the oscillation started.

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.

Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

Calculation of elastic-creeping characteristics of a beam made of a layered composite material

Multilayer composite materials are often used in building structures. The direct calculation of layered structures requires large expenditures of computer time. Therefore, the homogenization method is used. This method reduces the problem of a layered material with isotropic layers to the problem of a homogeneous transversely isotropic medium. The material considered in the article is also elastic-creeping. In the equations of state of such a material, terms of the convolution type with difference creep (relaxation) kernels are added to the terms of the usual theory of elasticity. The creep (relaxation) kernels are represented by decreasing exponential functions depending on two parameters. This problem becomes a problem of the theory of elasticity with a parameter after applying the Laplace transform in time to it. The inverse Laplace transform can be done in a computer algebra package, for example, Wolfram Mathematica, Wolfram-alpha. The obtained characteristics of the material are used to solve the problem of a layered elastic-creeping beam with hinge support. Formulas are given for determining displacements in the case of layers parallel to the beam axis.

Comment on “Dr. Bertlmann’s Socks in a Quaternionic World of Ambidextral Reality”

I point out critical errors in the paper "Dr. Bertlmann's Socks in a Quaternionic World of Ambidextral Reality" by Joy~Christian, published in IEEE Access. Christian's model does not generate the singlet correlations but in fact simply reproduces the Bertlmann effect. John Bell's friend and colleague Reinhold Bertlmann of CERN, in his younger days, always wore one pink and one blue sock, at random. The moment you saw his left foot, you knew what colour sock would be on his right foot. Action at a distance? As John Bell liked to explain, quantum entanglement cannot be explained away in such an easy way. Yet Christian's model assigns the two particles of the EPR-B experiment an equal and opposite spin at the source, the choice being determined by a fair coin toss. However they are measured, these spins are recovered. Christian's computer simulation works by not actually simulating his model at all but by almost directly tracing the negative cosine built into his computer algebra package. Bell's theorem has not been disproved. Debate as to what it means for the foundations of physics as well as for quantum information engineering (quantum communication, computation) is more lively today than ever before. Christian's paper, alas, does not contribute to the debate, but distracts from it. A possible role for Geometric Algebra is still wide open and deserves further investigation, informed by a proper understanding of the mathematical content of Bell's theorem.

一类反应扩散方程的孤立周期波和局部临界周期分支

The small-amplitude solitary periodic wave solutions and the local critical periodic bifurcations of the traveling wave equations for a class of reaction-diffusion equations with quintic nonlinear reaction terms and constant diffusion terms were studied. First, the reaction-diffusion equation was transformed into the corresponding traveling wave system through traveling wave transformation. The first 8 singular point quantities of the system were calculated with the singular point value method and the computer algebra software MATHEMATICA. Then, 2 center conditions for the singular point of the system were obtained, 8 limit cycles were proved to bifurcate at the origin of the traveling wave system, and 8 small-amplitude solitary periodic wave solutions were found to exist in the corresponding nonlinear reaction-diffusion equation. Furthermore, through computation of the period constants, the weak center order for the origin of the traveling wave system was derived. Then, the system was proved to have at most 3 local critical periodic bifurcations and be able to reach the 3 bifurcations. Moreover, the analysis of the critical periodic bifurcations of the traveling wave system reveals that the reaction-diffusion equation has 3 critical periodic wavelengths.

On some configurations of oppositely charged trapped vortices in the plane

Our aim in the present work is to identify all the possible standing wave configurations involving few vortices of different charges in an atomic Bose-Einstein condensate (BEC). In this effort, we deploy the use of a computational algebra approach in order to identify stationary multi-vortex states with up to 6 vortices. The use of invariants and symmetries enables deducing a set of equations in elementary symmetric polynomials, which can then be fully solved via computational algebra packages. We retrieve a number of previously identified configurations, including collinear ones and polygonal (e.g. quadrupolar and hexagonal) ones. However, importantly, we also retrieve a configuration with 4 positive charges and 2 negative ones, which is unprecedented, to the best of our knowledge, in BEC studies. We corroborate these predictions via numerical computations in the fully two-dimensional PDE system of the Gross-Pitaevskii type which characterizes the BEC at the mean-field level.

Teaching multivariable calculus and tensor calculus with computer algebra software

To go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.

A DOMIAN DECOMPOSITION METHOD FOR STEADY FREE CONVECTIVE COUETTE FLOW IN A VERTICAL CHANNEL WITH NON-LINEAR THERMAL RADIATION, DYNAMIC VISCOSITY AND DYNAMIC THERMAL CONDUCTIVITY EFFECTS

In this paper, we investigate steady free convective Couette flow in a vertical channel with nonlinear thermal radiation, dynamic viscosity and dynamic thermal conductivity effects. The investigation is motivated by the studies of some researchers which assumed linear thermal radiation and constant fluid properties. However, this is uncalled for; as these assumptions do not reflect true behavior of the flow. For instance; increase in temperature affects fluid viscosity, thermal conductivity thereby changing the transport phenomenon. Here; the investigation considers both the fluid viscosity and thermal conductivity to be dependent on temperature with the thermal radiation adopting nonlinear form. Due to this reasons, the associated flow equations are highly nonlinear and exhibit no analytical solution and therefore require the use of Adomian decomposition method (ADM) of solution. The attained ADM solution is then coded into computer algebra package of mathematica where results under the parameters of interest are presented and discussed. Results of the investigation show that raising the thermal radiation leads to corresponding rise in both the velocity and temperature of the fluid in the channel. Furthermore; lessening the viscosity and thermal conduction of the fluid were identified to escalate both velocity and temperature of the fluid.

The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra

In Ekhad and Zeilberger (Personal J Shalosh B Ekhad and Doron Zeilberger, 2020) an exciting case study has been initiated in which experimental mathematics and symbolic computation are utilized to discover new properties concerning the so-called Absent-Minded Passengers Problem. Based on these results, Doron Zeilberger raised some challenging tasks to gain further probabilistic insight. In this note we report on this enterprise. In particular, we demonstrate how the computer algebra packages of RISC can be used to carry out the underlying heavy calculations.

Computation of higher order Lie derivatives on the Infinity Computer

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the Infinity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique.

Singular elastic solutions in corners with spring boundary conditions under anti-plane shear

A new analytical procedure is developed for the deduction of the asymptotic series of the singular solutions in displacements and stresses near the vertex of the linear elastic isotropic corners with the Dirichlet–Robin (fixed-spring) and Neumann–Robin (free-spring) boundary conditions. Under the assumption of antiplane shear loading, the corresponding elastic problem reduces to the Laplace equation for the out-of-plane displacement. In the deduction of such singular solution, the complex variable is used to propose a harmonic function in the form of an asymptotic series including both power and logarithmic terms. This original procedure is suitable for its implementation in a computer algebra software which makes all the necessary symbolic computing, simplifications and rearrangements. This is a key issue due to the fact that the complexity of terms in these series may increase with increasing order of terms. These series are composed by the main terms (also called main singularities), solutions of the corresponding Dirichlet–Neumann or Neumann–Neumann problems, and the associated finite or infinite series of the so-called shadow terms (also called shadow singularities). These terms are determined by solving systems of recursive inhomogeneous Dirichlet–Neumann or Neumann–Neumann problems, respectively. A general classification of the behaviours of the asymptotic series covering all the considered corner problems is introduced. A few examples of the asymptotic series for corners with Dirichlet–Robin and Neumann–Robin boundary conditions are presented to illustrate the capabilities of this procedure.

PyScrew4Mobility: Modul Python untuk Penentuan Mobilitas Manipulator Paralel Berbasis Teori Screw

This paper addresses the mobility determination of parallel manipulators (PMs) using screw theory with algebra methods on an open-source computer algebra package: Sympy (symbolic python). The screw theory can specify the number and the type of motions owned by PMs with over-constrained or non-over constrained kinematic structures. The algebra methods are applied to obtain reciprocal of a screw or a screw system and basis of a screw system using null-spaces and the row/column-spaces technique, respectively. Hence, an object-oriented python module, called a pyScrew4Mobility module, is designed to realize such implementation. The module consists of three classes namely a Screw, a ScrewSystem and a ManipulatorMobility. A screw is constructed by the class of Screw, including its algebraic calculation such as negation, addition, multiplication, and product of two screws. Then, the ScrewSystem is used to construct a system of screws. It can be used to find reciprocal screws, unique screws within the system of screws, and calculate the products of two systems of screws. The last class or ManipulatorMobility has a direct implementation to determine the mobility of PMs. It uses information from the list of unit screw direction, position, and pitch. Finally, the designed screw module is tested to demonstrate its capability to determine the mobility of four well-known PMs, i.e. 3-PRRR; 3-PR(Pa)R; 4-PRRU; and 6-UPS, including the respective time spent for calculation.

Calculation and Properties of Zonal Polynomials

We investigate the zonal polynomials, a family of symmetric polynomials that appear in many mathematical contexts, such as multivariate statistics, differential geometry, representation theory, and combinatorics. We present two computer algebra packages, in SageMath and in Mathematica, for their computation. With the help of these software packages, we carry out an experimental mathematics study of some properties of zonal polynomials. Moreover, we derive and prove closed forms for several infinite families of zonal polynomial coefficients.

Computing real solutions of fuzzy polynomial systems

This paper presents an efficient algorithm called SolveFuzzySystem, or SFS, for finding real solutions of polynomial systems whose coefficients are symmetrical L-R fuzzy numbers with bounded support for which the spread functions L and R are bijective. The real solutions of such a system are deduced from solutions of some polynomial systems with real coefficients. This algorithm is based on new results that are universal because they are independent from the spread functions. These theoretical results include the management of the fuzzy system's solutions signs. An implementation in the Fuzzy package of the free computer algebra software SageMath and a parallel version of the algorithm are described.

Fault Detection and Identification for Nonlinear MIMO Systems Using Derivative Estimation

In this paper a method for fault detection and identification of affine input nonlinear systems is presented, which is based on derivative estimation with orthonormal Jacobi polynomials. A systematic approach is presented to derive a residual and a differential algebraic expression of the fault from the system description, which solely depends on measurable input and output signals as well as on their time derivatives. For this, a systematic algorithm is provided, which can be directly implemented in computer algebra packages. Furthermore, arbitrary disturbances are taken into account, by making use of a disturbance decoupling. Fault detection and identification is then achieved by polynomial approximation of the determined fault or residual expression. The results are illustrated for a faulty point-mass satellite model.

Combinatorial identities related to 2 × 2 submatrices of recursive matrices

Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of 2 × 2 minors of certain recursive matrices, the alternating sums of their 2 × 2 minors, and the sums of their 2 × 2 permanents. We obtain some combinatorial identities related to these sums, which generalized the work of Sun and Ma (2014) [23,24]. With the help of the computer algebra package HolonomicFunctions, we further get some new identities involving Narayana polynomials.