Use Of Computer Algebra Research Articles

DEVELOPMENT OF ALGORITHMS AND SOFTWARE FOR MODELING CONTROLLED DYMAMIC SYSTEMS USING SYMBOLIC COMPUTATIONS AND STOCHASTIC METHODS

The development of software for synthesizing and analyzing models of controlled systems taking into account their deterministic and stochastic description is an important direction of research. Results of the development of software for modeling dynamic systems the behavior of which can be described by onestep processes are presented. Models of population dynamics are considered as an example. The software uses a deterministic description of the model at its input to obtain a corresponding stochastic model in symbolic form and also analyze the model in detail (calculate trajectories in the deterministic and stochastic cases, find control functions, and visualize the results). An important aspect of the development is the use of computer algebra for analyzing the model and synthesizing controls. Methods and algorithms based on deterministic and stochastic Runge–Kutta methods, stability and control theory, methods for designing self-consistent stochastic models, numerical optimization algorithms, and artificial intelligence are implemented. The software was developed using high-level programming languages Python and Julia. As the basic tools, high-performance libraries for vector–matrix computations, symbolic computation libraries, libraries for the numerical solution of ordinary differential equations, and libraries of global optimization algorithms are used.

Enumeration of labelled 4-regular planar graphs II: Asymptotics

This work is a follow-up of the article (Noy et al., 2019), where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number gn of labelled 4-regular planar graphs on n vertices. Our estimate is of the form gn∼g⋅n−7/2ρ−nn!, where g>0 is a constant and ρ≈0.24377 is the radius of convergence of the generating function ∑n≥0gnxn/n!, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.

RETRACTED: New interaction solutions to the (2 + 1)-dimensional Hirota–Satsuma–Ito equation

This study aims to determine novel analytical lump solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equation through conducting symbolic computations using the Hirota direct method. This nonlinear model describes the propagation of unidirectional shallow-water waves and interactions of two long waves with different dispersion forms. Some informative descriptions of physical behavior related to the solutions obtained in this article have also been included through several 3D figures and 2D contour plots. The acquired results in this research may be beneficial for better understanding the interaction phenomena of localized nonlinear waves in different research fields of nonlinear science. It is notable that the use of computer algebra in the calculations required by the article is inevitable. Accordingly, in this work, we have used the symbolic package Mathematica. Our results will be meaningful for the investigation of the future development of lump solitons in many physical systems.

Dynamic Geometry Software Supplemented with a Computer Algebra System as a Proving Tool

The topic of this contribution is aimed at lower and upper secondary school mathematics teaching as well as at university training of teachers of mathematics. Joint use of computer algebra (CAS) and dynamic geometry software (DGS) or even an incorporation of CAS into DGS brings new possibilities to the teaching of mathematics, such as experimentation, the modelling of real-world situations or the deriving and proving of conjectures. We will deal with all these issues and how they can be realized with the free dynamic mathematics software GeoGebra, mainly in terms of the use of its tools of automated theorem proving. Particular examples of possible ways of utilizing the methods and tools of automated theorem proving will be presented with the aim of illustrating that dynamic geometry enriched with algebraic features can contribute to the effective fulfilment of the requirements for educational proof, from a decision on the correctness of a geometric property to the modelling of real-world phenomena.

TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++

TOPAS and its academic variant TOPAS-Academic are nonlinear least-squares optimization programs written in the C++ programming language. This paper describes their functionality and architecture. The latter is of benefit to developers seeking to reduce development time. TOPAS allows linear and nonlinear constraints through the use of computer algebra, with parameter dependencies, required for parameter derivatives, automatically determined. In addition, the objective function can include restraints and penalties, which again are defined using computer algebra. Of importance is a conjugate gradient solution routine with bounding constraints which guide refinements to convergence. Much of the functionality of TOPAS is achieved through the use of generic functionality; for example, flexible peak-shape generation allows neutron time-of-flight (TOF) peak shapes to be described using generic functions. The kernel of TOPAS can be run from the command line for batch mode operation or from a closely integrated graphical user interface. The functionality of TOPAS includes peak fitting, Pawley and Le Bail refinement, Rietveld refinement, single-crystal refinement, pair distribution function refinement, magnetic structures, constant wavelength neutron refinement, TOF refinement, stacking-fault analysis, Laue refinement, indexing, charge flipping, and structure solution through simulated annealing.

Ising and Gross-Neveu model in next-to-leading order

We study scalar and chiral fermionic models in next-to-leading order with the help of the functional renormalisation group. Their critical behaviour is of special interest in condensed matter systems, in particular graphene. To derive the beta functions, we make extensive use of computer algebra. The resulting flow equations were solved with pseudo-spectral methods to guarantee high accuracy. New estimates on critical quantities for both the Ising and the Gross-Neveu model are provided. For the Ising model, the estimates agree with earlier renormalisation group studies of the same level of approximation. By contrast, the approximation for the Gross-Neveu model retains many more operators than all earlier studies. For two Dirac fermions, the results agree with both lattice and large-$N_f$ calculations, but for a single flavour, different methods disagree quantitatively, and further studies are necessary.

Equation-based languages – A new paradigm for building energy modeling, simulation and optimization

Most of the state-of-the-art building simulation programs implement models in imperative programming languages. This complicates modeling and excludes the use of certain efficient methods for simulation and optimization. In contrast, equation-based modeling languages declare relations among variables, thereby allowing the use of computer algebra to enable much simpler schematic modeling and to generate efficient code for simulation and optimization.We contrast the two approaches in this paper. We explain how such manipulations support new use cases. In the first of two examples, we couple models of the electrical grid, multiple buildings, HVAC systems and controllers to test a controller that adjusts building room temperatures and PV inverter reactive power to maintain power quality. In the second example, we contrast the computing time for solving an optimal control problem for a room-level model predictive controller with and without symbolic manipulations. Exploiting the equation-based language led to 2200 times faster solution.

Methods of calculating the aircraft wing characteristics with influence aeroelasticity

Aeroelasticity is important in many areas of engineering, especially in aviation, where the resulting large size and velocity of the aircraft aeroelasticity is the most significant (negative) factor taken into account when creating the new aircraft. When designing the wing one of the objectives is to eliminate and reduce the harmful effects of aeroelasticity his design during the flight. This makes it necessary to study many variations of designs, hence the relevance of using new computer methods of calculation. In this article introduce new technique that ensures the creation of a mathematical model of elastic wings under aerodynamic and mass loads, which will describe the interaction of the wings of the air flow in an analytical way. The technique is to divide the wings into sections and replacing them with the corresponding finite elements, characterized by its rigidity, strength and weight lifting. These characteristics are described by the matrices aerodynamic impact, stiffness and mass, in analytical form, which are oriented in their element coordinate systems. Each individual section is characterized by the provisions of the centers of mass and stiffness of pressure that used to bring all the elements into one global coordinate system. Based on this design model is constructed wing that looks consistent and characteristic elements connected in parallel, with subsequent transformation and analysis. There have been several studies on its use for finding the critical velocity divergence wing aircraft and computing console wing movements under load. The method has proven quite high efficiency and accuracy of the results. This calculation method can be used during the preliminary design of the aircraft, it will help to simplify the automation of this process through the widespread use of computer algebra, increase economic efficiency by reducing design time costs, human and machine resources. Will open new opportunities to simplify the process of optimizing mass characteristics of the aircraft projected. Also, the technique will allow a simple interpretation of interim results and further analytical studies.

Relative parametrization of linear multidimensional systems

In the last chapter of his book “The Algebraic Theory of Modular Systems” published in 1916, F. S. Macaulay developped specific techniques for dealing with “unmixed polynomial ideals” by introducing what he called “inverse systems”. The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential or partial differential equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients. The first and main idea is to replace unmixed polynomial ideals by pure differential modules. The second idea is to notice that a module is \(0\)-pure if and only if it is torsion-free and thus if and only if it admits an “absolute parametrization” by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an “absolute localization”. The third idea is to refer to a difficult theorem of algebraic analysis saying that an \(r\)-pure module can be embedded into a module of projective dimension \(r\), that is a module admitting a projective resolution with exactly \(r\) operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of “involution” and a ‘relative localization” leading to a “relative parametrization”. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra.

Three-loop calculation of the higgs boson mass in supersymmetry

A Key feature of the minimal supersymmetric extension of the Standard Model (mssm) is the existence of a light Higgs boson, the mass of which is not a free parameter but an observable that can be predicted from the theory. Given that the lhc is able to measure the mass of a light Higgs with very good accuracy, a lot of effort has been put into a precise theoretical prediction. We present a calculation of the susy-qcd corrections to this observable to three-loop order. We perform multiple asymptotic expansions in order to deal with the multi-scale three-loop diagrams, making heavy use of computer algebra and keeping a keen eye on the numerical error introduced. We provide a computer code in the form of a Mathematica package that combines our three-loop susy-qcd calculation with the literature of one- and two-loop corrections to the Higgs mass, providing a state-of-the-art prediction for this important observable.

Invariants and symbolic calculations in the theory of quadratic differential systems

While quadratic differential systems intervene in many areas of applied mathematics and they also have theoretical importance, the topological classification of this class remains an extremely hard problem. However, in recent years much progress has been achieved due to the use of computer algebra and numerical calculations for obtaining complete classifications of some families of quadratic systems by effectively computing polynomial invariants and by an interplay between computer algebra and numerical computations. We illustrate on a specific family how these techniques yield the complete classification of the family within the 12-dimensional space of the coefficients of the systems.

Symbol-crunching the Harborth graph

The Harborth graph is widely regarded as the smallest known example of a 4-regular planar unit-distance graph. In this paper we sketch an exact proof of this property, and present some of the minimal polynomials of the coordinates of the vertices of its most prominent embedding, the calculations of which give an example for the heavy use of computer algebra in the area of geometric graph theory. Finally we use the particular algebraic structure of these polynomials to deduce some geometric properties of the Harborth graph.

Using Maple Worksheets to Enable Student Explorations of Cryptography

This paper explores the use of Maple worksheets in a course on modern cryptography for undergraduate sophomores and juniors. The worksheets allow students to explore deep topics without requiring many prerequisites. A sample worksheet, where students examine the effectiveness of the Fermat test is considered in some detail, and a sample list of other explorations done by the students is given. The paper concludes with the author's evaluation of the use of computer algebra in teaching cryptography.

A Maple package for verifying ultradiscrete soliton solutions

We present a computer algebra program for verifying soliton solutions of ultradiscrete equations in which both dependent and independent variables take discrete values. The package is applicable to equations and solutions that include the max function. The program is implemented using Maple software. Program summary Program title: Ultde Catalogue identifier: AEDB_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEDB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3171 No. of bytes in distributed program, including test data, etc.: 13 633 Distribution format: tar.gz Programming language: Maple 10 Computer: PC/AT compatible machine Operating system: Windows 2000, Windows XP RAM: Depends on the problem; minimum about 1 GB Word size: 32 bits Classification: 5 Nature of problem: The existence of multi-soliton solutions strongly suggest the integrability of nonlinear evolution equations. However enormous calculation is required to verify multi-soliton solutions of ultradiscrete equations. The use of computer algebra can be helpful in such calculations. Solution method: Simplification by using the properties of max-plus algebra. Restrictions: The program can only handle single ultradiscrete equations. Running time: Depends on the complexity of the equation and solution. For the examples included in the distribution the run times are as follows. (Core 2 Duo 3 GHz, Windows XP) • Example 1: 2725 sec. • Example 2: 33 sec. • Example 3: 1 sec.

Use of computer algebra in Hamiltonian calculations

The use of Hamilton’s Principle in determining the governing partial differential equations of motion for such problems as the free vibration of complex elastodynamic systems has been known for many years, but the implementation of the integration procedure using symbolic computation does not appear to have been solved. In this paper we present the application of a computer algebra package called REDUCE that takes away the human cost. We show the use of this package in a simple example, and for a twisted beam such as an idealised helicopter blade.

A NEW ALGEBRAIC APPROACH TO PERTURBATION THEORY

An algebraic nonperturbative approach is proposed for the analytical treatment of Schrödinger equations with a potential that can be expressed in terms of an exactly solvable piece with an additional potential. Avoiding disadvantages of standard approaches, new handy recursion formulas with the same simple form both for ground and excited states have been obtained. As an illustration the procedure, well adapted to the use of computer algebra, is successfully applied to quartic anharmonic oscillators by means of very simple algebraic manipulations. The trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g = 0.001–10000).

Applications of Computer Algebra in Solving Nonlinear Evolution Equations**The project supported by the National Key Basic Research Development Project Program under Grant No. G1998030600, and the Foundation of Liaoning Normal University

With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV–Burgers equation and KP–B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.

Assessing mathematics automatically using computer algebra and the internet

This paper reports some recent developments in mathematical computer aided assessment which employs computer algebra to evaluate students’ work using the internet. Technical and educational issues raised by this use of computer algebra are addressed. Working examples from core calculus and algebra which have been used with first year university students are described.

Application of Gröbner basis techniques to enzyme kinetics

It is possible to write down the equations governing a one-stage enzyme-catalysed reaction (according to Michaelis–Menten kinetics) quite easily, and deduce information about the steady-state flux in such a system. The situation is somewhat more complicated if several such reactions form a linear chain. We have applied Gröbner basis techniques to solve such systems and hence demonstrate the use of computer algebra as a powerful technique in enzyme kinetic analysis.

SCALE INVARIANT COSMOLOGY II: EXTRA DIMENSIONS AND THE REDSHIFT OF DISTANT SUPERNOVAE

The cosmological problem is studied here starting from a metric with a variable scale factor for time and two different ones for space, noted R for ordinary space and S for the extra dimensions. Once R has been obtained from physical principles, one is left for S with a differential equation which can be solved through Lie group analysis and dynamical systems theory, and a frequent use of computer algebra has been made in the ensuing calculations. This leads to a good agreement with the recent observational results relating to distant supernovae. However, the solution obtained for S is particularly simple if spacetime has a dimension at least equal to ten, a result to be compared with those given by superstring or M theories.