Computer Algebra System Research Articles

Poisson’s ratio of fluorapatite crystals

Introduction. It is known that in the pathogenesis of caries and non-carious lesions of teeth, an important role is played by enamel resistance, which is determined by numerous factors, among which the structure and physical and mechanical properties of tooth tissues are considered to be among the most important. This report summarizes the literature data on the Poisson’s ratio of fluorapatite (FAp), a biomaterial and one of the mineral components of dental hard tissues. The literature search was carried out in electronic databases Scopus, Web of Science, PubMed, Medline, Elibrary, MatWeb for the period from January 1970 to March 2023 inclusive. Some of the data are supplemented by our own calculations based on formulas for crystalline hexagonal systems (systems). As is known, Poisson’s ratio determines the volumetric response of materials and biological tissues to mechanical load, but its anisotropic properties have not been studied in detail. Material and methods. Calculations of the Poisson’s ratio of fluorapatite were carried out in the computer algebra system Mathcad 15.0, using the online analysis of elasticity tensors ELATE, and an integrated molecular modeling system was used - the graphic package VESTA (Visualization for Electronic and STructural Analisis). Results and discussion. The Poisson’s ratio values of fluorapatite crystals varied: minimum values 0.057–0.283, maximum values 0.302–0.494, average values obtained by integration in all directions 0.24–0.308. The coefficient of elastic anisotropy of the fluorapatite crystal lattice, calculated based on the values of Poisson’s ratio, is 1.21–5.29. The results obtained were compared with those of hydroxyapatite (HAp) crystals, dentin, enamel and filling materials. The highest values of the parameters μ and μmax are found in dentin (0.312 and 0.54), the lowest μmin are also found in dentin (0.13). Conclusions. Fluorapatite crystals are elastically anisotropic, and Poisson’s ratio μmax/μmin varies over a wide range. The Poisson’s ratio of fluorapatite showed similarity with the transverse strain coefficient of isomorphic hydroxyapatite crystals (0.207–0.374), dentin (0.13–0.54), enamel (0.16–0.47) and dental composite filling materials (0.24–0 ,35). The latter should help improve the quality of restorations using filling materials for replacing hard dental tissues. It has been established that the organic component of dentin increases its Poisson’s ratio as a biocomposite and the anisotropy of elastic properties.

Compiling vocabularies of nonoverlapping codons with graph theory and SageMath

We consider the compilation of complete (nonextensible) dictionaries of pairwise nonoverlapping codons containing the maximum possible number of such codons. This includes three subcases: (a) codons that do not allow overlapping by two letters, but allowing overlapping by one letter (e.g., ATC and CTG); (b) on the contrary, codons that do not allow only overlapping by one letter, but allow overlapping by two letters (e.g., ATC and TCG); (c) when there can be no overlap (as ATC and GTC). The last subcase is also an example of the nonoverlapping code that is used in coding theory and automata theory. Herein, we use graph theory and the SageMath computer algebra system as working tools.

Solving the Applied Problem of Random-Dot Images Analysis Using a Multidimensional Extension of Classic Catalan Numbers

The paper introduces the concept of generalized Catalan numbers as a useful tool for solving many theoretical and applied combinatorial-probabilistic problems. Combined with analytical transformation software algorithms, the generalized Catalan numbers simplify the solution of many problems in computer science and applied mathematics. In particular, they turn out to be an effective tool for solving problems related to the registration of random-dot images, in signal transformations of various degrees of smoothness, and when developing speed-optimal algorithms for searching for pulsed-point objects with a random generation time of super short pulses. The proposal of the authors to formulate the problems of enumerative combinatorics in a word-symbolic form naturally leads to multidimensional extensions of the classical Catalan numbers and has several advantages. The combined use of multidimensional Catalan numbers and high-performance computer algebra systems enables solving a number of complex applied problems related to the reliability of the registration of random-dot images.

Computer assisted investigation of alienness of linear functional equations

AbstractIn this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations.

Trigonometric Functions and the Sensations of Tone

Music provides an excellent setting for students to explore trigonometric functions in a setting which is independent of their utility in resolving triangles. In this paper, we present several investigations using trigonometric functions to model musical phenomena. From a mathematical technology point of view, most examples require only function graphing and can be explored using a modern interactive graphing application. One example, however, is an excellent application of trigonometric simplification, for which a Computer Algebra System is a useful tool. A final application is suitable for calculus students and provides an exercise in integration by parts. This can be done by hand, or with the aid of an algebra system.

On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

This paper presents an original package for investigating numerical solutions of ordinary differential equations, which is built in the Sage computer algebra system. This project is focused on a closer integration of numerical and symbolic methods while primarily aiming to create a convenient tool for working with numerical solutions in Sage. The package defines two new classes: initial problems and approximate solutions. The first class defines tools for symbolic computations related to initial problems, while the second class defines tools for interpolating values of symbolic expressions on an approximate solution and estimating the error with the use of the Richardson method. An implementation of the Runge–Kutta method is briefly described, with its main feature being the possibility of working with arbitrary Butcher tableaux and arbitrary numeric fields.

Resonances and Periodic Motions of Atwood’s Machine with Two Oscillating Weights

The problem of constructing periodic solutions to the equations of motion of Atwood’s machin in which both weights have the same mass and can oscillate in the vertical plane is discussed. Differential equations governing the motion of this system are derived, and an algorithm for calculating their solutions that determine periodic oscillations of the weights under the condition that the oscillation frequencies are in resonance nω1 = mω2, where n and m are natural numbers, is proposed. These solutions are obtained in the form of series in a small parameter. The comparison of the results with numerical solutions of the equations of motion confirm the validity of the obtained solutions. All computations are performed using the computer algebra system Wolfram Mathematica.

Revealing patterns of thermophysical parameters in the designed energy-saving structures for external fencing with air channels

This study reports the design of new models of energy-saving enclosing structures with air channels. To calculate the thermophysical parameters of external fences, the Maple computer algebra system was used; the value of thermal resistance of structures was determined on the basis of a finite-element method in ANSYS. The result of the structure analysis showed that the value of thermal inertia of the traditional design and the average value of the thermal inertia of the developed structures were equal. However, the vibration amplitude of the designed enclosing structures was up to 20.72 % more efficient than the traditional one. At the same time, it was revealed that the air gaps did not affect the thermal inertia of the strucure, and its parameters depended only on the total thickness of the material. The analysis showed that the vapor permeability of the inner wall of the designed structures was equal to the traditional one. However, the value of resistance to vapor permeation of the fence of the developed structures was 3.21 % more effective. At the same time, the use of a closed air layer with a heat-reflecting screen makes it possible to shift the possible condensation zone towards the outer surface of the fence. An analysis of the check for the non-condensation of condensate in the ventilated air gap showed that condensate did not fall out in the ventilated air gap in all the considered schemes, and the results of the analysis by the air permeability value showed that all fencing schemes met the requirements for air permeability. Solving the problems of energy saving in construction through the development of new energy-efficient designs of enclosing structures help reduce the cost of thermal energy of buildings, which is an urgent task all over the world today

Bibliographic Study on the Computer Algebra System Maple

The authors were surprised by the number of bibliographic citations to the computer algebra system DERIVE long after it was discontinued and developed a small bibliographic study about its evolution. They later carried out a similar one about the successful dynamic geometry system GeoGebra. Finally, they presented at Maple Conference 2022 a comparative small bibliographic study about Maple and other computer algebra systems, which can be found here.

Exploiting the Abstract Calculus Pattern for the Integration of Ordinary Differential Equations for Dynamics Systems: An Object-Oriented Programming Approach in Modern Fortran

This manuscript relates to the exploiting of the abstract calculus pattern (ACP) for the (numerical) solution of ordinary differential equation (ODEs) systems, which are ubiquitous mathematical formulations of many physical (dynamical) phenomena. We present FOODIE, a software suite aimed to numerically solve ODE problems by means of a clear, concise, and efficient abstract interface. The results presented prove manifold findings, in particular that our ACP approach enables ease of code development, clearness and robustness, maximization of code re-usability, and conciseness comparable with computer algebra system (CAS) programming (interpreted) but with the computational performance of compiled programming. The proposed programming model is also proven to be agnostic with respect to the parallel paradigm of the computational architecture: the results show that FOODIE applications have good speedup with both shared (OpenMP) and distributed (MPI, CAF) memory architectures. The present paper is the first announcement of the FOODIE project: the current implementation is extensively discussed, and its capabilities are proved by means of tests and examples.

The face-on projection of the Miyamoto & Nagai discs

ABSTRACT The face-on projected density profile of the Miyamoto & Nagai discs of arbitrary flattening is obtained analytically in terms of incomplete elliptic integrals of first and second type, by using two complementary approaches, and then checked against the results of numerical integration. As computer algebra systems do not seem able to obtain the resulting formula in any straightforward way, the relevant mathematical steps are provided. During this study, three wrong identities in the Byrd & Friedman tables of elliptic integrals have been identified, and their correct expression is given.

Computing error bounds for asymptotic expansions of regular P-recursive sequences

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

Generic Gröbner basis of a parametric ideal and its application to a comprehensive Gröbner system

A relationship between a parametric ideal and its generic Gröbner basis is discussed, and a new stability condition of a Gröbner basis is given. It is shown that the ideal quotient of the parametric ideal by the ideal generated using the generic Gröbner basis provides the stability condition. Further, the stability condition is utilized to compute a comprehensive Gröbner system, and hence as an application, we obtain a new algorithm for computing comprehensive Gröbner systems. The proposed algorithm has been implemented in the computer algebra system Risa/Asir and experimented with a number of examples. Its performance (execution timings) has been compared with the Kapur-Sun-Wang’s algorithm for computing comprehensive Gröbner systems.

Prediction of Refractive Index of Petroleum Fluids by Empirical Correlations and ANN

The refractive index is an important physical property that is used to estimate the structural characteristics, thermodynamic, and transport properties of petroleum fluids, and to determine the onset of asphaltene flocculation. Unfortunately, the refractive index of opaque petroleum fluids cannot be measured unless special experimental techniques or dilution is used. For that reason, empirical correlations, and metaheuristic models were developed to predict the refractive index of petroleum fluids based on density, boiling point, and SARA fraction composition. The capability of these methods to accurately predict refractive index is discussed in this research with the aim of contrasting the empirical correlations with the artificial neural network modelling approach. Three data sets consisting of specific gravity and boiling point of 254 petroleum fractions, individual hydrocarbons, and hetero-compounds (Set 1); specific gravity and molecular weight of 136 crude oils (Set 2); and specific gravity, molecular weight, and SARA composition data of 102 crude oils (Set 3) were used to test eight empirical correlations available in the literature to predict the refractive index. Additionally, three new empirical correlations and three artificial neural network (ANN) models were developed for the three data sets using computer algebra system Maple, NLPSolve with Modified Newton Iterative Method, and Matlab. For Set 1, the most accurate refractive index prediction was achieved by the ANN model, with %AAD of 0.26% followed by the new developed correlation for Set 1 with %AAD of 0.37%. The best literature empirical correlation found for Set 1 was that of Riazi and Daubert (1987), which had %AAD of 0.40%. For Set 2, the best performers were the models of ANN, and the new developed correlation of Set 2 with %AAD of refractive index prediction was 0.21%, and 0.22%, respectively. For Set 3, the ANN model exhibited %AAD of refractive index prediction of 0.156% followed by the newly developed correlation for Set 3 with %AAD of 0.163%, while the empirical correlations of Fan et al. (2002) and Chamkalani (2012) displayed %AAD of 0.584 and 0.552%, respectively.

Equation generator for equation-of-motion coupled cluster assisted by computer algebra system

We present an equation generator algorithm that utilizes second-quantized operators in normal order with respect to a correlated or non-correlated reference and the corresponding Wick theorem. The algorithm proposed here, written with Mathematica, enables the generation of non-redundant strings of second-quantized operators that, after classification, are directly assigned to many-body term quantities used to construct the many-body Hamiltonian. We demonstrate the capabilities of the algorithm by computing the coupled-cluster amplitude equations and various blocks of the equation-of-motion many-body Hamiltonian. A comprehensive description of this four-step algorithm is provided alongside concrete examples.

Solving plane stress problems using EsfPlano2D educational software

An educational computational application for analyzing plane stress problems oriented to engineering students called EsfPlano2D is presented to motivate students to study the subject and strengthen didactic strategies in the classroom. The application assists the step-by-step calculation of plane stress problems, which is a fundamental topic in teaching the subject of Mechanics of Materials in engineering careers. The application is a computer program developed in Fortran, linked to the free Computer Algebra System MAXIMA to plot Mohr’s circle of stresses. The developed tool is user-friendly and completely solves the plane stress problem, the associated deformations, and the strain energy by drawing Mohr’s circle of plane stresses. A validation case is presented with a typical example of the mechanics of materials. The results show the feasibility of using custom-designed tools to strengthen classroom learning and autonomous work.

Exact Solutions of Systems of Nonlinear Time-Space Fractional Partial Differential Equations Using an Iterative Method

Abstract Fractional partial differential equations are useful tools to describe transportation, anomalous, and non-Brownian diffusion. In the present paper, we propose the Daftardar-Gejji and Jafari method along with its error analysis for solving systems of nonlinear time–space fractional partial differential equations (PDEs). Moreover, we solve a variety of nontrivial time–space fractional systems of PDEs. The obtained solutions either occur in exact form or in the form of a series, which converges to a closed form. The proposed method is free from linearization and discretization and does not include any tedious calculations. Moreover, it is easily employable using the computer algebra system such as Mathematica, Maple, etc.

Simple algorithm for judging equivalence of differential-algebraic equation systems

Mathematical formulas play a prominent role in science, technology, engineering, and mathematics (STEM) documents; understanding STEM documents usually requires knowing the difference between equation groups containing multiple equations. When two equation groups can be transformed into the same form, we call the equation groups equivalent. Existing tools cannot judge the equivalence of two equation groups; thus, we develop an algorithm to judge such an equivalence using a computer algebra system. The proposed algorithm first eliminates variables appearing only in either equation group. It then checks the equivalence of the equations one by one: the equations with identical algebraic solutions for the same variable are judged equivalent. If each equation in one equation group is equivalent to an equation in the other, the equation groups are judged equivalent; otherwise, non-equivalent. We generated 50 pairs of equation groups for evaluation. The proposed method accurately judged the equivalence of all pairs. This method is expected to facilitate comprehension of a large amount of mathematical information in STEM documents. Furthermore, this is a necessary step for machines to understand equations, including process models.

Closed-form expressions for monoprotic weak acid aqueous solutions

A general closed-form expression for {[textrm{H}_{3}textrm{O}^+]} of solutions of a weak acid and a weak acid buffer, as well as their titration with a strong base, has been obtained and mathematically analyzed with the aid of computer algebra systems. This expression is used to evaluate, without the use of numerical approximations, the precision and accuracy of different approximations commonly employed in general chemistry and chemical analysis courses. The closed-form expression for {{[textrm{H}_{3}textrm{O}^+]}} of a buffer solution is used to obtain an analytical expression for the pH stability when a strong base is added. Finally, it is shown that the {[textrm{H}_{3}textrm{O}^+]} expressions for all the systems under study can be obtained from a general closed-form expression in terms of an effective weak acid constant and an effective acid concentration. The formulas found in this work do not display numerical rounding errors and do not require the use of numerical or graphical methods for their evaluation.Graphical abstractGraphical TOC: A closed-form expression for the weak acid and weak acid buffersolutions and their titrations with a strong base

Computer Experiment in Teaching Mathematics

A mathematical experiment has always been a key source for a mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing mathematical education closer to mathematical research for the majority of students. This tendency is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the authors’ work over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also discussed. A project concerning schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.