Computer Algebra Research Articles

The Building of Mathematical Concepts When Solving Mathematical Tasks in Traditional Way and Using Graphing Calculators

This article aims to understand how students learn mathematical concepts related to the study of functions and differential calculus using graphing calculators with the computer algebra system (GCC) and traditional methods. With a qualitative methodology, we show that students can make the transition to mathematical meanings of concepts using both processes.

ТЕХНОЛОГІЇ ЗАСТОСУВАННЯ СЕРЕДОВИЩ ДИНАМІЧНОЇ ГЕОМЕТРІЇ

The paper considers some problems and technologies of teaching geometry using dynamic geometry environments to students of advanced classes and specialized study of mathematics. Integrated application of knowledge of geometry, algebra and mathematical analysis and computer science provides implementation of intersubject relations and introduces to students the elements of the research approach. The author emphasizes the urgency of creating computer-oriented technologies in teaching mathematics, making appropriate amendments to the curriculum and textbooks, conducting systematic research of the effectiveness of their application considering educational risks.

Shannon Entropy Computations in Navier–Stokes Flow Problems Using the Stochastic Finite Volume Method

The main aim of this study is to achieve the numerical solution for the Navier–Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation technique and the Monte Carlo scheme, are engaged for this purpose. It is implemented with the aid of the polynomial bases for the pressure–velocity–temperature (PVT) solutions, for which the weighted least squares method (WLSM) algorithm is applicable. The deterministic problem is solved using the freeware OpenFVM, the computer algebra software MAPLE 2019 is employed for the LSM local fittings, and the resulting probabilistic quantities are computed. The first two probabilistic moments, as well as the Shannon entropy spatial distributions, are determined with this apparatus and visualized in the FEPlot software. This approach is validated using the 2D heat conduction benchmark test and then applied for the probabilistic version of the 3D coupled lid-driven cavity flow analysis. Such an implementation of the SFVM is applied to model the 2D lid-driven cavity flow problem for statistically homogeneous fluid with limited uncertainty in its viscosity and heat conductivity. Further numerical extension of this technique is seen in an application of the artificial neural networks, where polynomial approximation may be replaced automatically by some optimal, and not necessarily polynomial, bases.

Online energy consumption forecast for battery electric buses using a learning-free algebraic method

Accurately predicting the energy consumption plays a vital role in battery electric buses (BEBs) route planning and deployment. Based on the algebraic derivative estimation, we present a novel method to forecast the energy consumption in real time. In contrast to the mainstream machine-learning-based methods, the proposed method does not require access to the historical energy consumption data. It eliminates the time-consuming and computationally expensive offline training. Consequently, its prediction performance is not constrained by the quantity and quality of the training data. Moreover, the method can swiftly adapt to new situations not included in the previous driving cycles, which makes it especially suitable for emerging transport modes, e.g., on-demand transit services. In addition, its online execution only involves algebraic calculations, yielding superior calculation efficiency. Using real-world data, we comprehensively compare the performance of the proposed learning-free algebraic method with multiple representative machine-learning-based methods. Finally, the advantages and limitations of the proposed method are discussed in detail.

Accelerating machine learning queries with linear algebra query processing

The rapid growth of large-scale machine learning (ML) models has led numerous commercial companies to utilize ML models for generating predictive results to help business decision-making. As two primary components in traditional predictive pipelines, data processing, and model predictions often operate in separate execution environments, leading to redundant engineering and computations. Additionally, the diverging mathematical foundations of data processing and machine learning hinder cross-optimizations by combining these two components, thereby overlooking potential opportunities to expedite predictive pipelines. In this paper, we propose an operator fusion method based on GPU-accelerated linear algebraic evaluation of relational queries. Our method leverages linear algebra computation properties to merge operators in machine learning predictions and data processing, significantly accelerating predictive pipelines by up to 317x. We perform a complexity analysis to deliver quantitative insights into the advantages of operator fusion, considering various data and model dimensions. Furthermore, we extensively evaluate linear algebra query processing and operator fusion utilizing the widely-used Star Schema and TPC-DI benchmarks. Through comprehensive evaluations, we demonstrate the effectiveness and potential of our approach in improving the efficiency of data processing and machine learning workloads on modern hardware.

SMARANDACHE ANTI ZERO DIVISORS

In this paper, we study and discuss the concept of Smarandache anti zero divisor (SAZD) element of the ring and the group ring , where is a cyclic group of order generated by . Moreover, we introduce and discuss the concept of SAZD ideal of the ring . Some results related to the given concepts are proved in detail. Accordingly, a Computer Algebra System (GAP) is used to verify the results of this study

Hybrid Laplace-time domain model for coupled dynamic analysis of multibody mating system in topside float-over installations

The mating operation is the most critical phase during the topside float-over installation for jacket platforms, involving complex load transfer process and nonlinear couplings among the topside, float-over barge, jacket, and other buffering devices. It is very important and yet challenging to accurately and efficiently capture the coupled dynamics of the multibody mating system, as the time domain (TD) model requires a sufficiently small time interval to clearly capture the strong impacts on the buffering devices, resulting in a worse computational efficiency. This paper innovatively proposes a hybrid Laplace-time domain (HLTD) model to improve the computational efficiency in the coupled dynamic analysis while the computational accuracy is remained. The HLTD model computes the coupled dynamics by the time-domain iterative operations, mixing the Laplace-domain pole-residue operations for computing the motion responses in each iteration. Since the tedious differential and integral solutions in the TD model are replaced by the simple algebraic pole-residue calculations, the HLTD model has much higher computational efficiency with the same computational accuracy as the TD model. By comparing with the commercial software OrcaFlex, the proposed HLTD model shows satisfactory accuracy and superior efficiency in the heave-only mating analysis during the topside float-over installation of a jacket platform.

Cadabra and python algorithms in general relativity and cosmology I: Generalities

The aim of this work is to present a series of concrete examples which illustrate how the computer algebra system Cadabra can be used to manipulate expressions appearing in General Relativity and other gravitational theories. We highlight the way in which Cadabra's philosophy differs from other systems with related functionality. The use of various new built-in packages is discussed, and we show how such packages can also be created by end-users directly using the notebook interface. The current paper focuses on fairly generic applications in gravitational theories, including the use of differential forms, the derivation of field equations and the construction of their solutions.

Developing of neural network computing methods for solving inverse elasticity problems

This paper examines the use of neural network methods to solve inverse problems in the mechanics of elastic materials. The aim is to design physics-informed neural networks that can predict the parameters of structural components, and the physical properties of materials based on a specified displacement distribution. A key feature of the specified neural networks is the integration of differential equations and boundary conditions into the loss function calculation. This approach ensures that the error in approximating unknown functions has a direct impact on optimizing the network's weights. As a result, the resulting neural network approximations of unknown functions comply with the differential equations and boundary conditions. To test the capabilities of the designed neural networks, inverse problems involving the bending of plates and beams have been solved, focusing on determining one or two unknown parameters. Comparison of predicted and exact values demonstrates high accuracy of the constructed neural network models, with a relative prediction error of less than 3 % across all cases. Unlike analytical methods for solving inverse problems, the primary advantage of physics-informed neural networks is their flexibility when addressing both linear and nonlinear problems. For instance, the same network architecture can be employed to solve various boundary-value problems without modification. Compared to classical numerical methods, the parallelization capability of neural networks is inherently supported by modern software libraries. Therefore, the application of physics-informed neural networks for solving inverse elasticity problems of plates and beams is effective, as evidenced by the achieved relative errors and the computational robustness of the method. In practice, the proposed solution can be used for relevant calculations during the design of structural elements. The developed software code can also be integrated into automated design systems or computer algebra systems

Algorithm xxx: ellipFor , a Fortran software library for Legendre elliptic integrals and Jacobi elliptic functions with generalized input arguments

Legendre elliptic integrals and Jacobi elliptic functions arise in multiple applications within the physical sciences, including oscillations, celestial mechanics, and geodynamics. In this study, we describe the Fortran library ellipFor capable of evaluating the following for generalized input values: 1) the complete Legendre elliptic integrals of the first and second kinds, 2) the incomplete Legendre elliptic integrals of the first and second kinds, and 3) the principal Jacobi elliptic functions. Our software builds upon previously developed Fortran routines, which were designed with restrictions on input parameters that may be limiting in applications. Our routines apply multiple transformations to allow for more general input values, such as elliptic moduli greater than unity for points 1–3, arbitrary real Jacobi amplitudes for points 1–2, and complex first arguments for point 3. In addition, our routines are thread-safe, allowing for parallel computations. Our routines were compared with values from the computer algebra system SageMath over a wide range of input parameters. Values from ellipFor and SageMath agreed to within tolerances commensurate with the limitations of floating-point arithmetic used for the elliptic integrals and Jacobi elliptic functions listed in points 1, 2, and 3 above for generalized input arguments.

An Algebraic Computation of the Height of the First Chern Class of the Canonical K-Plane Bundle over the Complex Grassmannian CG_(n,k)

We prove algebraically that the height of the first Chern class of the canonical k-plane bundle over the complex Grassmannian CG_(n,k) is k(n-k) in H^* ( CG_(n,k);Q) for k=2,3,4, and any positive integer n≥2k. We use the technique of R. E. Stong, of embedding the rational cohomology ring of the complex Grassmannian CG_(n,k) into the rational cohomology ring of the complete complex flag manifold F_C (⏟(1,…,1)┬n) .

A Class of Binary Codes Using a Specific Automorphism Group

In this article, we showcase PSL(3,4) as the automorphism group for a specific class of three linear binary codes, C1, C2 and C3, with dimension 9. The demonstration involves leveraging the action of the group PSL(3,4), represented by invertible matrices of size 9 by 9 up to isomorphism, on the vector space F29. Additionally, we establish that these codes exhibit a three-weight self-orthogonal property. All computations presented in this paper were performed using the guava package of GAP (Groups, Algorithms, Programming) a system designed for computational discrete algebra.

Natural frequencies and mode shapes for axisymmetric vibration of the shells of revolution using Carrera Unified Formulation

Here, natural frequencies and mode shapes for axisymmetric vibration of the composite laminate shells of revolution have been considered using the Unified Carrera Formulation (CUF) approach. For the first time, results of natural frequencies and mode shapes of complex geometry shells such as spherical, parabolic, elliptical, hyperbolic, catenoidal, toroidal and pseudospherical have been presented. Calculations have been done for the first, second, third, fourth and fifth order models and a comparison with the classical Timoshenko model. We consider thick, moderately thick, moderately thin and thin composite laminate shells. The higher-order layer-wise models of elastic composite multilayer shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic elastodynamics and generalized series in the thickness coordinates. The higher-order composite axisymmetric spherical, paraboloidal, elliptical, hyperbolic, catenoidal, toroidal and pseudo-spherical shell fixed at the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations have been used for theoretical analysis and calculation of the eigenvalues and eigenmodes of the higher-order shells of revolution used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher-order composite laminate shells.

Perturbation Methods in Solving the Problem of Two Bodies of Variable Masses with Application of Computer Algebra

The classical many-body problem is not integrable, so perturbation theory based on an exact solution to the two-body problem is usually applied to investigate the dynamics of planetary systems. However, in the case of variable masses, the two-body problem is not integrable, in general, and application of perturbation theory is required to investigate it, as well. In the present paper, we use the perturbation theory to derive the differential equations determining the orbital elements of the relative motion of one body around the other. Two models of the perturbed aperiodic motion on conic and quasi-conic sections are considered and compared. Special attention is paid to the practically important case of small eccentricities, when the perturbing forces may be replaced by the corresponding power series expansions. The differential equations of the perturbed motion are averaged over the mean anomaly, and the evolutionary equations describing the behavior of the orbital elements over long periods of time are obtained for two models. Comparing the corresponding solutions to the evolutionary equations, we have shown that both models demonstrate similar behavior with regard to the secular perturbations of the orbital elements. However, the second model, based on the aperiodic motion on a quasi-conic section, is more appropriate for generalization to the many-body problem with variable masses. All the relevant symbolic and numerical calculations are performed with the computer algebra system Wolfram Mathematica.

CAN SYMBOLIC COMPUTATION AND FORMALIST SYSTEMS ENHANCE MATH EDUCATION WITH ARTIFICIAL INTELLIGENCE?

In recent years, a solution developed using deep learning methods has been used to solve difficult problems in a field. The capability of deep learning models is that they require large and heavily sampled data sets. Computer Algebra Systems developed over time have made significant progress, especially in the field of symbolic mathematics solutions solved by machine learning. It is a persistent problem how appropriate it is to use such formal systems in some aspects of algorithmic decision-making. In this paper, we discussed the suitability of artificial intelligence applications to formal propositions by evaluating a deep learning study conducted especially in the field of symbolic mathematics and Math education. Symbolic computation systems have a strong potential for enhancing math education. Furthermore, within the framework of the Incompleteness Theorem, to show why the construction of a mathematical grammar is not a complete solution for Mathematics education systems.

Practical Canonical Labeling of Multi-Digraphs via Computer Algebra

Practical algorithms for computing canonical forms of multi-digraphs do not exist in the literature. This paper proposes two practical approaches for finding canonical forms, from the perspective of nD symbolic computation. Initially, the approaches turn the problem of finding canonical forms of multi-digraphs into computing canonical forms of indexed monomials in computer algebra. Then, the first approach utilizes the double coset representative method in computational group theory for canonicalization of indexed monomials and shows that finding the canonical forms of a class of multi-digraphs in practice has polynomial complexity of approximately O((k+p)2) or O(k2.1) by the computer algebra system (CAS) tool Tensor-canonicalizer. The second approach verifies the equivalence of canonicalization of indexed monomials and finding canonical forms of (simple) colored tripartite graphs. It is found that the proposed algorithm takes approximately O((k+2p)4.803) time for a class of multi-digraphs in practical implementation, combined with one of the best known graph isomorphism tools Traces, where k and p are the vertex number and edge number of a multi-digraph, respectively.

GATlab: Modeling and Programming with Generalized Algebraic Theories

Categories and categorical structures are increasingly recognized as useful abstractions for modeling in science and engineering. To uniformly implement category-theoretic mathematical models in software, we introduce GATlab, a domain-specific language for algebraic specification embedded in a technical programming language. GATlab is based on generalized algebraic theories (GATs), a logical system extending algebraic theories with dependent types so as to encompass category theory. Using GATlab, the programmer can specify generalized algebraic theories and their models, including both free models, based on symbolic expressions, and computational models, defined by arbitrary code in the host language. Moreover, the programmer can define maps between theories and use them to declaratively migrate models of one theory to models of another. In short, GATlab aims to provide a unified environment for both computer algebra and software interface design with generalized algebraic theories. In this paper, we describe the design, implementation, and applications of GATlab.Comment: 14 pages plus references and appendix. To appear at MFPS 2024

Identifying Markov Chain Models from Time-to-Event Data: An Algebraic Approach.

Many biological and medical questions can be modeled using time-to-event data in finite-state Markov chains, with the phase-type distribution describing intervals between events. We solve the inverse problem: given a phase-type distribution, can we identify the transition rate parameters of the underlying Markov chain? For a specific class of solvable Markov models, we show this problem has a unique solution up to finite symmetry transformations, and we outline a recursive method for computing symbolic solutions for these models across any number of states. Using the Thomas decomposition technique from computer algebra, we further provide symbolic solutions for any model. Interestingly, different models with the same state count but distinct transition graphs can yield identical phase-type distributions. To distinguish among these, we propose additional properties beyond just the time to the next event. We demonstrate the method's applicability by inferring transcriptional regulation models from single-cell transcription imaging data.

Topological Properties of the Intersection Curves Between a Torus and Families of Parabolic or Elliptical Cylinders

This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The parabolic cylinders under analysis have generatrices parallel to the axis of the torus, whereas the elliptical cylinders, centered at the same point as the torus, have axes either aligned with or orthogonal to the torus’s axis. For the topological classification of these intersection curves, we consider their number of connected components and self-intersection points. GeoGebra, which was used to create the 3D visual geometric representations of the intersection curves, and Maple, which was used to perform the essential symbolic algebraic calculations, were critical computational tools in the development of this work. Theoretical and computational approaches are interwoven throughout this study, with the computational work serving as the foundation for exploration and providing insights that contributed to the theoretical validation of the results revealed through GeoGebra simulations.

Solving the Master Equation on river networks: A computer algebra approach

Solving the Master Equation on river networks: A computer algebra approach