Computer Algebra System Research Articles

New Methods of Series Expansions between Three Anomalies

The calculation of satellite orbit involves some very complex formula derivations and expansions, which are very difficult to manually derive and prone to errors. And the efficiency of manual derivation is not high. We can use computer algebra systems to derive complex formulas related to satellite orbits. This can avoid some of the drawbacks of manual derivation and significantly improve computational efficiency and accuracy. In the past, the relationship among three anomalies was generally represented in the form of a trigonometric series with the first eccentricity e as the parameter. In this paper, the trigonometric series with the parameter m=1−1−e2e is used, as determined by the Lagrange conjugate series. We can use the formula of the Lagrange conjugate series to derive the relationship between the true anomaly and elliptic anomaly. And the relationship between the elliptic anomaly and the mean anomaly is derived by using the symbolic iteration method. In this research paper, we calculated the accuracy of the trigonometric series expansion among three types of anomalies at the first eccentricity e equal to values of 0.01, 0.1, and 0.2. The calculation results indicate that the accuracy of the trigonometric series expansion with m as the parameter is better than 10−5. Moreover, in some cases, the trigonometric series expansion among the three anomalies with m as a parameter is simpler in form than the expansion expressed with parameter e. This paper also derived and calculated the symbolic expressions and extreme values of the difference among three anomalies and expressed the extreme values of the difference in the form of a power series of e. It can be seen that the extreme value increases with the increase in eccentricity e. And the absolute values of the extreme value of the difference between the elliptic anomaly and the mean anomaly, the true anomaly and the elliptic anomaly, and the true anomaly and the mean anomaly increase in this order. When the eccentricity is small, the absolute value of the extreme value of the difference between the true anomaly and the mean anomaly is about twice as large as the elliptic anomaly and the mean anomaly and the true anomaly and the mean anomaly.

Exact Bayesian Inference for Loopy Probabilistic Programs using Generating Functions

We present an exact Bayesian inference method for inferring posterior distributions encoded by probabilistic programs featuring possibly unbounded loops . Our method is built on a denotational semantics represented by probability generating functions , which resolves semantic intricacies induced by intertwining discrete probabilistic loops with conditioning (for encoding posterior observations). We implement our method in a tool called Prodigy; it augments existing computer algebra systems with the theory of generating functions for the (semi-)automatic inference and quantitative verification of conditioned probabilistic programs. Experimental results show that Prodigy can handle various infinite-state loopy programs and exhibits comparable performance to state-of-the-art exact inference tools over loop-free benchmarks.

ПАКЕТ GAP ДЛЯ РОЗЩЕПЛЕННЯ ЛІНІЙНИХ СИСТЕМ

There are a number of problems, the solution of which requires the breakdown of the initial system of equations using algebraic decoupling methods. This means reducing the matrix of coefficients to block-diagonal (or block-triangular) form by means of substitution of variables. The main computational tasks when using such methods are finding the centralizer of several matrices or compiling the algebra generated by these matrices. For calculations, it is convenient to use the GAP computer algebra system because the system itself is designed for discrete algebra calculations. The problem is that the GAP program does not support calculations with real numbers. For practical problems you can try to replace them (with some accuracy) by rational numbers. At the same time, the decision may turn out to be excessively cumbersome. On the other hand, the advantage of GAP is the complete absence of rounding errors.

Programming the Minimal Model Program: a proposal

AbstractThe aim of this paper is to propose a strategy to implement the Minimal Model Program in modern computer algebra systems.

IMPLEMENTATION IN SСAS KIDYM OF SOFTWARE DETERMINATION OF DIMENSIONS OF UNKNOWNS IN PROBLEMS OF MECHANICS BASED ON ENERGY RELATIONS

The paper presents the results of the study of an algorithm for processing measurement units of physical and geometric quantities in the problems of mechanics implemented in the specialized computer algebra system (SCAS) KiDyM. The KiDyM software complex is used for solving problems of mechanics of engineering and scientific orientation of any complexity. For the completeness of the initial data preparation process for such tasks, the use of measurement units in the initial and resulting data is essential. The output data of KiDyM tasks have a rather slow appearance due to their ana- lytical form. To provide assistance to the user, KiDyM programs have a special diagnostic block of output data. Therefore, the implementation of the dimensions used in KiDyM data pursues the additional goal of increasing the diagnostic capabilities of the system. Dimensional formulas are imple- mented here as ordinary analytical expressions, which enables the built-in computer algebra system to form expressions of data measurement units during their computer analytical transformations in the process of solving problems of dynamics, statics and kinematics. Thus, working out the meas- urement units in the SCAS KiDyM includes reading the values of the variables with the dimensions specified in formula form the initial data, diagnos- ing the correctness of dimensionising and compliance of the dimensions with the data, storing them in instances of the "variable" and "element" classes, calculating the dimensions according to data conversion formulas to obtain calculation results. The paper demonstrates how to efficiently es- tablish dimensions of geometrical and physical quantities using energy relations between coordinates and characteristics of elements of the mechanical model. For the demonstration of the logic of the implemented algorithm, simple compact examples from educational problems of kinematics, statics, and dynamics of flat and spatial systems are provided. The article shows how KiDyM constructs the necessary equations for solving tasks, and how units of measurement for equation components and their solutions can be obtained. Since dimensions are represented as formulas here, their simplifica- tion – reduction and substitution with derivative units – can be easily implemented. Moreover, since the variables involved in dimension formulas and the variables of the task itself, within KiDyM, are situated in different data spaces, they can have the same names without any confusion.

A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem (Student Abstract)

The problem of finding the minimum three-dimensional Kochen–Specker (KS) vector system, an important problem in quantum foundations, has remained open for over 55 years. We present a new method to address this problem based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS). Our approach improved the lower bound on the size of a KS system from 22 to 24. More importantly, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a proof size of 41.6 TiB for order 23. The efficiency is due to the powerful combination of SAT solvers and CAS-based orderly generation.

A SAT + Computer Algebra System Verification of the Ramsey Problem R(3, 8) (Student Abstract)

The Ramsey problem R(3,8) asks for the smallest n such that every red/blue coloring of the complete graph on n vertices must contain either a blue triangle or a red 8-clique. We provide the first certifiable proof that R(3,8) = 28, automatically generated by a combination of Boolean satisfiability (SAT) solver and a computer algebra system (CAS). This SAT+CAS combination is significantly faster than a SAT-only approach. While the R(3,8) problem was first computationally solved by McKay and Min in 1992, it was not a verifiable proof. The SAT+CAS method that we use for our proof is very general and can be applied to a wide variety of combinatorial problems.

Sorting in construction of resolutions of combinatorial designs

There are different kinds of sorting algorithms. Each algorithm has its own advantages and disadvantages depending on the data they process. We are interested in the performance of sorting algorithms in the case of construction of resolutions of combinatorial designs. Studying their performance for this special class of problems will give us the opportunity to improve, if possible, the speed of our software for solving similar problems. We use C++ and, for part of our investigations, the computer algebra system GAP.

Analytical investigations of propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide by three computational ideas

This paper mainly concentrates on obtaining solutions and other exact traveling wave solutions using the generalized G-expansion method. Some new exact solutions of the coupled nonlinear Schrödinger system using the mentioned method are extracted. This method is based on the general properties of the nonlinear model of expansion method with the support of the complete discrimination system for polynomial method and computer algebraic system (AS) such as Maple or Mathematica. The nonparaxial solitons with the propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide is studied. To attain this, an illustrative case of the coupled nonlinear Helmholtz (CNLH) system is given to illustrate the possibility and unwavering quality of the strategy utilized in this research. These solutions can be significant in the use of understanding the behavior of wave guides when studying Kerr medium, optical computing and optical beams in Kerr like nonlinear media. Physical meanings of solutions are simulated by various Figures in 2D and 3D along with density graphs. The constraint conditions of the existence of solutions are also reported in detail. Finally, the modulation instability analysis of the CNLH equation is presented in detail.

On the labelling of characters of Weyl groups of type $$F_4$$

AbstractIn the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type $$F_4$$ F 4 . We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system , we spotted an error in Spaltenstein’s tables for the generalised Springer correspondence in type $$E_7$$ E 7 .

Modeling of Some Classes of Extended Oscillators: Simulations, Algorithms, Generating Chaos, and Open Problems

In this article, we propose some extended oscillator models. Various experiments are performed. The models are studied using the Melnikov approach. We show some integral units for researching the behavior of these hypothetical oscillators. These will be implemented as add-on sections of a thoughtful main web-based application for researching computations. One of the main goals of the study is to share the difficulties that researchers (who are not necessarily professional mathematicians) encounter in using contemporary computer algebraic systems (CASs) for scientific research to examine in detail the dynamics of modifications of classical and newer models that are emerging in the literature (for the large values of the parameters of the models). The present article is a natural continuation of the research in the direction that has been indicated and discussed in our previous investigations. One possible application that the Melnikov function may find in the modeling of a radiating antenna diagram is also discussed. Some probability-based constructions are also presented. We hope that some of these notes will be reflected in upcoming registered rectifications of the CAS. The aim of studying the design realization (scheme, manufacture, output, etc.) of the explored differential models can be viewed as not yet being met.

Simulation of spring-friction set of freight car truck, taking into account track profile

The purpose of this article is the mathematical modeling of a spring-friction set of a freight car trolley depending on the profile of the railway track. The article uses the methodology of the results of the generalized model of power interaction of friction pairs "pressure beam-friction wedge" and "friction wedge-friction bar" of the spring-friction set of the cart 18-578 in the mechanical system "track-car-load," in a particular case cart from which such models of the e of the freight car 18-100 can be obtained. The manuscript contains a software study using a computer simulation of a freight wagon trolley, which complements the experimental studies. The mathematical (symbolic) method in the computer algebra system MathCAD (a type of computer-aided design system) is used to solve the equations of equilibrium of the supercharger beam as a real object. In the course of the experimental study, a computational mathematical model of the force action of friction pairs of spring-absorbing trolley apparatuses was developed, and the results of calculations of the reaction of the friction bar to the friction wedge were mathematically obtained. The optimal parameters were obtained analytically, and the indicators of the angles of inclination of the contact surfaces of the trolley of the freight car were justified.

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

Analytic solution to functional differential equations via Bell’s polynomials

Abstract It is shown how to approximate the solution of functional differential equations in terms of Bell’s polynomials. Some numerical checks are shown, by using the computer algebra system Mathematica © {{}^{\copyright}} .

Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems

In this paper, we propose a new class of micro-electromechanical oscillators. Some investigations based on Melnikov’s approach are applied for identifying some chaotic possibilities. We demonstrate also some specialized modules for investigating the dynamics of these oscillators. This will be included as an integral part of a planned much more general Web-based application for scientific computing. It turns out that the theoretical apparatus for studying the circuit implementation (design, fabricating, etc.) of the considered differential model for large parameter values is extremely complex and requires a serious investigation. This is the reason to offer this model to the attention of specialists working in this scientific direction. Some open problems related to the use of existing computer algebraic systems for the study of this class of oscillators for large values of n,m and N are also posed. In general, the entire article is subordinated to this frank conversation with the readers with the sole purpose being the professional upgrading of the specialized modules provided for this purpose in subsequent licensed versions of CAS.

Solutions of Time-Space Fractional Partial Differential Equations Using Picard's Iterative Method

Abstract In this paper, we present Picard's iterative method (PIM) for solving time–space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The proposed method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.

Biomechanical properties of nasal tissues

Introduction. Plastic rhinosurgery and augmentation rhinoplasty are very relevant today. Especially in relation to patients with congenital saddle deformity of the nasal dorsum, as well as patients with iatrogenic disorders of the nose shape, which dramatically reduce the quality of human life. There are violations of the function of the nose. The purpose of the work. Purpose. Within the framework of the necessity of performing silicone implantation rhinoplasty, the strength, elastic and hyperelastic properties of the soft tissues of the nasal fascia and periosteum were studied, the dynamic viscosity of the tissues of the nasal fascia and periosteum was evaluated, determining their relaxation properties using elastic, hyperelastic and rheological models. Methods. Linear, bilinear (with two Young modules), exponential, hyperelastic (neohookean, Mooney-Rivlin, Ogden, Yeoh, polynomial and Veronda-Westmann) and elastic-viscous (Maxwell) models of biological tissues were used. We used the Mathcad 15.0 computer algebra system and the universal package of interdisciplinary programs ANSYS Multiphysics Software (version 2022 R2). The accuracy of replacing the properties of real tissues with the results of calculations was calculated on the basis of indicators of descriptive statistics (standard deviation, maximum absolute error, maximum relative error and correlation coefficient). Results. It is proved that the bilinear model for the exact reproduction of the stress-strain curve assumes at the molecular level of tissues an initial linear reaction of elastin fibers, passing at ε=εcr into the final linear reaction of the collagen matrix. It was found that the properties of the periosteum (correlation coefficient R=0.9999) and nasal fascia (correlation coefficient R=0.9999) are described most adequately by the 5-parametric polynomial model of the 2nd order and the Yeoh model of the 3rd order (R=0.999 and R=1, respectively), the least accurately by the simple 1-parameter neohookean model (R=0.898 for the periosteum and R=0.905 for the fascia). Among elastic models, the exponential dependence characterizes the behavior of nasal materials quite well. Conclusion. Biomechanically verified that the periosteum is thicker, stronger and more rigid than the fascia. The established very high viscosity values indicate the slowness of relaxation processes in tissues. The periosteum relaxes tension more slowly, which improves the result of implant fixation. On the contrary, the relaxation time of the periosteum is higher than that of the fascia, from which it is concluded that it is preferable not to cut the periosteum during subcostal implantation.

Minimization of hypersurfaces

Let F ∈ Z [ x 0 , … , x n ] F \in \mathbb {Z}[x_0, \ldots , x_n] be homogeneous of degree  d d and assume that F F is not a ‘nullform’, i.e., there is an invariant  I I of forms of degree  d d in n + 1 n+1  variables such that I ( F ) ≠ 0 I(F) \neq 0 . Equivalently, F F is semistable in the sense of Geometric Invariant Theory. Minimizing  F F at a prime  p p means to produce T ∈ M a t ( n + 1 , Z ) ∩ G L ( n + 1 , Q ) T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e ∈ Z ≥ 0 e \in \mathbb {Z}_{\ge 0} such that F 1 = p − e F ( [ x 0 , … , x n ] ⋅ T ) F_1 = p^{-e} F([x_0, \ldots , x_n] \cdot T) has integral coefficients and v p ( I ( F 1 ) ) v_p(I(F_1)) is minimal among all such  F 1 F_1 . Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w ∈ Z ≥ 0 n + 1 w \in \mathbb {Z}_{\ge 0}^{n+1} to  F F . We show that for any dimension  n n and degree  d d , there is a complete set of weight vectors consisting of [ 0 , w 1 , w 2 , … , w n ] [0,w_1,w_2,\dots ,w_n] with 0 ≤ w 1 ≤ w 2 ≤ ⋯ ≤ w n ≤ 2 n d n − 1 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1} . When n = 2 n = 2 , we improve the bound to  d d . This answers a question raised by Kollár. These results are valid in a more general context, replacing Z \mathbb {Z} and  p p by a PID  R R and a prime element of  R R . Based on this result and a further study of the minimization process in the planar case n = 2 n = 2 , we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree  d d . We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma.

Information modeling of the normal forms of logical functions in Maple system

The article discusses possible ways to use the Maple computer algebra system to obtain perfect conjunctive and disjunctive normal forms. Computer algebra systems make it possible to expand the possibilities for creating, applying and using mathematical models in the daily activities of engineers, researchers, and also contribute to improving the education of students of various training profiles. In addition, the use of computer technology can help to apply various methods and expand approaches to the representation of the studied objects (Boolean functions) of discrete form, including in education. Computer algebra systems such as Maple, Mathematica, etc. are the most adapted for studying various sections of discrete mathematics and mathematics in general. The possibilities of the Maple program are shown for use at the initial stages of mastering the discrete mathematics section of the university course - Boolean functions, in particular, for work on the construction and study of perfect conjunctive and disjunctive normal forms according to the truth table.

A novel way to build expert systems with infinite-valued attributes

<abstract><p>An expert system is a computer program that uses the knowledge of an expert to solve problems in a specific domain. Expert systems are used in a wide variety of fields, such as medicine, financial diagnosis and engineering. The attributes of an expert system are the characteristics of the problems that the system can solve. In traditional expert systems, attributes typically have a finite number of possible values. However, in scenarios where an attribute can assume a value from an infinite (or significantly large finite) set, the expert system cannot be represented using propositional logic. Until now, no method had been identified to implement such a system on a Computer Algebra System. Here, we break new ground by presenting a model that not only addresses this gap but also provides a fresh perspective on previous results. In fact, these prior results can be viewed as specific instances within the broader framework of our proposed solution. In this paper, we put forth an algebraic approach for the development of expert systems capable of handling attributes with infinite values, thereby expanding the problem-solving capacity of these systems.</p></abstract>