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This paper explores the numerical intergator of ODE based on combination of Appelroth's quadratization of dynamical systems with polynomial right-hand sides and Kahan's discretization method. Utilizing Appelroth's technique, we reduce any system of ordinary differential equations with a polynomial right-hand side to a quadratic form, enabling the application of Kahan's method. In this way, we get a difference scheme defining the one-to-one correspondence between the initial and final positions of the system (Cremona map). It provides important information about the Kahan method for differential equations with a quadratic right-hand side, because we obtain dynamical systems with a quadratic right-hand side that have movable branch points. We analyze algebraic properties of solutions obtained through this approach, showing that (1) the Kahan scheme describes the branch points as poles, significantly deviating from the behavior of the exact solution of the problem near these points, and (2) it disrupts algebraic invariant variety, in particular integral relations describing the relationship between old and Appelroth's variables. This study advances numerical methods, emphasizing the possibility of designing difference schemes whose algebraic properties differ significantly from those of the initial dynamical system. [ABSTRACT FROM AUTHOR]

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results. [ABSTRACT FROM AUTHOR]

The paper considers the possible relationship between the local integrability of an autonomous two-dimensional ODE system with polynomial right hand sides and its global integrability. A hypothesis is put forward that local integrability in a neighborhood of each point of some region of the phase space is necessary for the existence of the first integral in this region. By integrability in some domain of the phase space it is meant the existence there of a differentiable function which is constant along the orbit of the system. Using the example of a polynomial resonance case of the Liénard-type equation with parameters, we have written out the conditions for local integrability near stationary points and found restrictions on the parameters under which these conditions are satisfied. The resulting constraint is written as a system of algebraic equations for the ODE parameters. It is shown that for parameter values that are solutions of such an algebraic system, the ODE turns out to be integrable. In this way we have found several cases of integrability. We propose a heuristic method that allows one to a priori determine the cases of integrability of an autonomous ODE with a polynomial right-hand side. The paper has an experimental character. [ABSTRACT FROM AUTHOR]

Disaster management requires detailed data from past disasters for policy planning as well as for the generation of disaster exercises and simulations. Post-analysis of disasters is often distributed in official reports, which provide detailed analysis of the events that have happened. However, some information in such reports is often only given implicitly as part of natural language and thus not accessible to classical natural language processing-based text mining. To address this problem, in this paper, we propose to consider the information extraction tasks related to post-disaster report analysis as algebra stories, that can be treated with computer algebra systems together with natural language processing. We applied our enhanced information extraction approach in preliminary experiments to the report of a bushfire in 2009 in Victoria, Australia and used four different tools for solving fire-specific algebra story problems (ASP). Our evaluation shows that these tools have difficulty handling the occurring ASPs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering < on monomials. In classical mathematics, the well-quasiorder property of relation < is derived from Dickson's lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for < , which is based on a known method that we refer to here as the "pattern method." This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T. [ABSTRACT FROM AUTHOR]

Railway interlocking systems are crucial safety components in rail transportation, designed to prevent train collisions by regulating switch positions and signal indications. These systems delineate potential train movements within a railway station by connecting sections into routes, which are further divided into blocks. To ensure safety, the system prohibits the simultaneous allocation of the same block or intersecting routes to multiple trains. In this study, we characterize the 'interlocking problem' as a safety verification task for a single real-time station configuration, rather than a 'command and control' function. This is a matter of verification, not solution, typically managed by an interlocking system that receives movement authority requests. Over the years, we have developed various algebraic models to address this issue, suggesting the potential use of computer algebra systems in implementing interlocking systems. However, some of these models exhibit limitations. In this paper, we propose a novel algebraic model for decision-making in railway interlocking systems that overcomes the limitations of previous approaches, making it suitable for large railway stations. Our primary objective is to offer a mathematical solution to interlocking problems in linear time, which our approach accomplishes. [ABSTRACT FROM AUTHOR]

The Maxima computer algebra system, the open-source successor to MACSYMA, the first general-purpose computer algebra system that was initially developed at the Massachusetts Institute of Technology in the late 1960s and later distributed by the United States Department of Energy, has some remarkable capabilities, some of which are implemented in the form of add-on, “share” packages that are distributed along with the core Maxima system. One such share package is itensor, for indicial tensor manipulation. One of the more remarkable features of itensor is functional differentiation. Through this, it is possible to use itensor to develop a Lagrangian field theory and derive the corresponding field equations. In this paper, we demonstrate this capability by deriving Maxwell’s equations from the Maxwell Lagrangian, and exploring the properties of the system, including current conservation. [ABSTRACT FROM AUTHOR]

Decision-making in a railway station regarding the compatibility of the positions of the switches of the turnouts and the indications (proceed/stop) of the railway colour light signals is a safety-critical issue that is considered very labor-intensive. Different authors have proposed alternative solutions to automate its supervision, which is performed by the so-called railway interlocking systems. The classic railway interlocking systems are route-based and their compatibility is predetermined (usually by human experts): only some chosen routes are simultaneously allowed. Some modern railway interlocking systems are geographical and make decisions on the fly, but are unsuitable if the station is very large and the number of trains is high. In this paper, we present a completely new algebraic model for decision-making in railway interlocking systems, based on other computer algebra techniques, that bypasses the disadvantages of the approaches mentioned above (its performance does not depend on the number of trains in the railway station and can be used in large railway stations). The main goal of this work is to provide a mathematical solution to the interlocking problems. We prove that our approach solves it in linear time. Although our approach is interesting from a theoretical perspective, it has a significant limitation: it can hardly be adopted in an actual interlocking implementation, mainly due to the heavy certification requirements for this kind of safety-critical application. Nevertheless, the results may be useful for simulations that do not require certification credit. [ABSTRACT FROM AUTHOR]

We presented a new and complete algorithm for detecting isometries and symmetries of implicit algebraic surfaces. First, our method reduced the problem to the case of isometries fixing the origin. Second, using tools from elimination theory and polynomial factoring, we determined the desired isometries between the surfaces. We have implemented the algorithm in Maple to provide evidences of the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we describe an educational experience in the context of the Master's degree that is compulsory in Spain to become a secondary education mathematics teacher. Master's students from two universities in Madrid (Spain) attended lectures that addressed—emphasizing the concourse of a dynamic geometry software package—some historical, didactic and mathematical issues related to linkage mechanisms, such as those arising in the 18th and 19th centuries during the development of the steam engine. Afterwards, participants were asked to provide three different kinds of feedback: (i) working on an assigned group task, (ii) individually answering a questionnaire, and (iii) proposing some classroom activity, imagining it would be addressed to their prospective pupils. All three issues focused on the specific topic of the attended lectures. In the framework of Mason's reflective discourse analysis, the information supplied by the participants has been analyzed. The objective was to explore what they have learned from the experience and what their perception is of the potential interest in linkages as a methodological instrument for their future professional activity as teachers. This analysis is then the basis upon which to reflect on the opportunities (and problems) that this particular bar-joint linkages methodological approach could bring towards providing future mathematics teachers with attractive tools that would contribute to enhancing a STEAM-oriented education. Finally, the students' answers allow us to conclude that the experience was beneficial for these pre-service teachers, both in improving their knowledge on linkages history, mathematics, industrial, technological and artistic applications, and in enhancing the use in the classroom of this very suitable STEAM context. [ABSTRACT FROM AUTHOR]

Tensor formalism (and its special case—vector formalism) is a mathematical technique that is widely used in physical and engineering problems. Even though this formalism is fairy universal and suitable for describing many spaces, the application of other special mathematical techniques is sometimes required. For example, the problem of rotation in a 3D space is not very well described in tensor representation, and it is reasonable to use the formalism of Clifford algebra, in particular, quaternions and geometric algebra representations for its solution. In this paper, computer algebra is used to demonstrate the solution of the problem of rotation in a 3D space using both the quaternion and geometric algebra formalisms. It is shown that although these formalisms are fundamentally similar, the latter one seems to be clearer both for computations and interpretation of results. [ABSTRACT FROM AUTHOR]

Constructing algebraic polynomials using computer algebra techniques is believed to be state-of-the-art in analyzing gate-level arithmetic circuits. However, the existing approach applies algebraic rewriting directly to the gate-level netlist, which has potential memory explosion problem. This paper introduces an algebraic rewriting technique based on the and-inverter graph (AIG) representation of gate-level designs. Using AIG-based cut-enumeration and truth table computation, an efficient order of algebraic rewriting is identified, resulting in dramatic simplifications of the polynomial under construction. An automatic approach, which further reduces the complexity of algebraic rewriting by handling redundant polynomials, is also proposed. [ABSTRACT FROM AUTHOR]

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as { 2 , 5 } -inverses) of a matrix, which commute with it. This class of { 2 , 5 } -inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Inspired by a recent paper by Wares, we discuss an extension of another well-known geometric problem involving arcs which, since the solution is computer-assisted, could be a source of a mini-project for an introductory course on a computer algebra system. [ABSTRACT FROM AUTHOR]

Let n > 3 be a natural number. By a 1975 result of H. Strietz, the lattice Part(n) of all partitions of an n-element set has a four-element generating set. In 1983, L. Zádori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number v(n) of four-element generating sets of Part(n). We also present a computer-assisted statistical approach to v(n) for small values of n. In his 1983 paper, L. Zádori also proved that for n ≥ 7, the lattice Part(n) has a four-element generating set that is not an antichain. He left the problem whether such a generating set for n ∈ {5, 6} exists open. Here we solve this problem in negative for n = 5 and in affirmative for n = 6. Finally, the main theorem asserts that the direct product of some powers of partition lattices is four-generated. In particular, by the first part of this theorem, Part(n1) x Part(n2) is four-generated for any two distinct integers n1 and n2 that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product Part(n) x Part(n + 1) x ... x Part(3n - 14) is four-generated for each integer n ≥ 9. Also, for every positive integer u, the u-th the direct power of the direct product Part(n) x Part(n +1) x ... x Part(n + u - 1) is four-generated for all but finitely many n. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the 10127-th direct power of Part(1011) x Part(1012) x ... x Part(2020) is four-generated. [ABSTRACT FROM AUTHOR]

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approach to verifying higher-degree Galois-field (GF) arithmetic circuits. The proposed method describes GF arithmetic circuits using a mathematical graph-based representation and verifies them by a combination of algebraic transformations and a new verification method based on natural deduction for first-order predicate logic with equal sign. The natural deduction method can verify one type of higher-degree GF arithmetic circuit efficiently while the existing methods require an enormous amount of time, if they can verify them at all. In this paper, we first apply the proposed method to the design and verification of various Reed-Solomon (RS) code decoders. We confirm that the proposed method can verify RS decoders with higher-degree functions while the existing method needs a lot of time or fail. In particular, we show that the proposed method can be applied to practical decoders with 8-bit symbols, which are performed with up to 2,040-bit operands. We then demonstrate the design and verification of the Advanced Encryption Standard (AES) encryption and decryption processors. As a result, the proposed method successfully verifies the AES decryption datapath while an existing method fails. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the single traveling wave solutions of the fractional coupled nonlinear Schrödinger equation. By using the complete discriminant system method and computer algebra with symbolic computation, a series of new single traveling wave solutions are obtained, which include trigonometric function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, solitary wave solutions and rational function solutions. As you can see, we give all the classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation. The obtained results substantially improve or complement the corresponding conditions in the literature (Esen and Sulaiman in Optik 167:150-156, 2018), (Eslami in Appl. Math. Comput. 258:141-148, 2016), (Han et al. in Phys. Lett. 395:127217, 2021). Finally, in order to further explain the propagation of the fractional coupled nonlinear Schrödinger equation in nonlinear optics, two-dimensional and three-dimensional graphs are drawn. [ABSTRACT FROM AUTHOR]

Galois field (GF) arithmetic circuits find numerous applications in communications, signal processing, and security engineering. Formal verification techniques of GF circuits are scarce and limited to circuits with known bit positions of the primary inputs and outputs. They also require knowledge of the irreducible polynomial ${P(x)}$ , which affects final hardware implementation. This paper presents a computer algebra technique that performs verification and reverse engineering of GF(${2^{m}}$) multipliers directly from the gate-level implementation. The approach is based on extracting a unique irreducible polynomial in a parallel fashion and proceeds in three steps: 1) determine the bit position of the output bits; 2) determine the bit position of the input bits; and 3) extract the irreducible polynomial used in the design. We demonstrate that this method is able to reverse engineer GF(${2^{m}}$) multipliers in ${m}$ threads. Experiments performed on synthesized Mastrovito and Montgomery multipliers with different ${P(x)}$ , including NIST-recommended polynomials, demonstrate high efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a formal approach to verifying Error Correction Code (ECC) processors based on Galois-Field (GF) arithmetic. The proposed method describes GF arithmetic circuits by graph-based representation, called Galois-Field Arithmetic Circuits Graph (GF-ACG), and verifies them by a combination of an algebraic method with a new verification method based on natural deduction for the first-order predicate logic with equal sign. The natural deduction method is suitable for higher-degree GF arithmetic circuits such as in ECC decoders while the conventional methods requires enormous time to verify them or sometimes cannot verify them. In this paper, we apply the proposed method to the design and verifications of various Reed-Solomon (RS) code encoders and decoders. In particular, we confirm that the proposed method can verify RS code decoders with higher-degree functions while the conventional method fails. [ABSTRACT FROM AUTHOR]

This paper investigates the possibility of automatically linearizing nonlinear models. Constructing a linearised model for a nonlinear system is quite labor-intensive and practically unrealistic when the dimension is greater than 3. Therefore, it is important to automate the process of linearisation of the original nonlinear model. Based on the application of computer algebra, a constructive algorithm for the linearisation of a system of non-linear ordinary differential equations was developed. A software was developed on MatLab. The effectiveness of the proposed algorithm has been demonstrated on applied problems: an unmanned aerial vehicle dynamics model and a twolink robot model. The obtained linearized models were then used to test the stability of the original models. In order to account for possible inaccuracies in the measurements of the technical parameters of the model, an interval linearized model is adopted. For such a model, the procedure for constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. On the basis of the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the studied system was analyzed. [ABSTRACT FROM AUTHOR]

This paper proposes a formal approach to designing multiple-valued arithmetic circuits over Galois Fields (GFs). Our method introduces a high-level multiple-valued graph specified by variables and arithmetic formulae over specific GFs. The graph represents arithmetic circuits over GFs in a hierarchical manner. The proposed circuit description is formally verified by symbolic computations such as polynomial reduction using Gröbner Bases. In this paper, we propose the graph representation and show an example of its description and verification. The advantageous effects of the proposed approach are demonstrated through experimental designs of parallel multipliers over Galois field GF(2m) for different word-lengths and irreducible polynomials. An inversion circuit consisting of some multipliers is also designed and verified as a further application. The result shows that the proposed approach has a definite possibility of verifying practical arithmetic circuits where the conventional simulation and verification techniques failed. [ABSTRACT FROM AUTHOR]

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores. [ABSTRACT FROM AUTHOR]

This paper discusses some of the philosophical and historical underpinnings of the talk "The Mathieu Functions: Computational and Historical Perspectives" given at the Maple Conference 2022. I also comment on the role Problem Solving Environments (PSEs) play in curating the computational knowledge of the 19th century, which is so necessary for us to think of special functions as answers rather than questions. [ABSTRACT FROM AUTHOR]

This paper presents a formal method for designing cryptographic processor datapaths on the basis of arithmetic circuits over Galois fields (GFs). The proposed method describes GF arithmetic circuits in the form of hierarchical graph structures, where nodes represent sub-circuits whose functions are defined by arithmetic formulae over GFs, and edges represent data dependency between nodes. In this paper, we first introduce the application of graph representation to arithmetic circuits over extension fields of \mbi GF(\mbi p^\mbi m) (\mbi p \geq \bf 2) and composite fields, which are commonly used in the design of cryptographic processors. The newly proposed graph representation can be formally verified through symbolic computation techniques based on polynomial reduction and Gröbner basis. We then demonstrate the capabilities of the proposed approach through an experimental design of a 128-bit AES (Advanced Encryption Standard) datapath including multiplicative inversion circuits over the composite field \mbi GF(((2^2)^2)^2). The results show that the proposed method can describe such practical datapaths, as well as that complete verification of such a datapath can be carried out within a short period of time. [ABSTRACT FROM AUTHOR]

In algebraic geometry, a number of invariants for classifying algebraic varieties are obtained from the cohomology groups of coherent sheaves. Some typical algorithms to compute the dimensions of the cohomology groups have been proposed by Decker and Eisenbud, and their algorithms have been implemented over compute algebra systems such as Macaulay2 and Magma. On the other hand, Maruyama showed an alternative method to compute the dimensions in his textbook; we call the method Maruyama's method in this paper. However, Maruyama's method was not described in an algorithmic format, and it has not been implemented yet. In this paper, we give an explicit algorithm of his method to compute the dimensions and bases of the cohomology groups of coherent sheaves. We also analyze the complexity of our algorithm, and implemented it over Magma. By our implementation, we examine the computational practicality of our algorithm. Moreover, we give some possible applications of our algorithm in algebraic geometry over fields of positive characteristics. [ABSTRACT FROM AUTHOR]

The object of this research is the process of small oscillations of a three-dimensional elastic pendulum, tuned to a 1:1:2 resonance to vertical and horizontal oscillations. The purpose is to develop a symbolic algorithm for calculating small oscillations of the pendulum. The primary efforts are aimed at building a software package that generates formulas that approximate the movements of the pendulum with sufficient accuracy. The algorithms of this work are developed based on the resonant normal form method. The importance of the study is due to the wide range of applicability of the method of normal forms for constructing approximations of periodic and conditionally periodic local families of solutions of ODEs. For high-dimensional resonance systems, the technique is a generalization of the Poincare–Linstedt method, and for coarse systems, the Carleman linearization method. The reliability of the results is confirmed by comparisons with the results of numerical solutions. Results can be useful for specialists working at the interface of computational mathematics and continuum mechanics. Approaches and methodologies developed in this paper can be applied to solve various modeling problems. [ABSTRACT FROM AUTHOR]

Solution of the Global Navigation Satellite Systems (GNSS) phase ambiguity is considered as a global quadratic mixed integer programming task, which can be transformed into a pure integer problem with a given digit of accuracy. In this paper, three alternative algorithms are suggested. Two of them are based on local and global linearization via McCormic Envelopes, respectively. These algorithms can be effective in case of simple configuration and relatively modest number of satellites. The third method is a locally nonlinear, iterative algorithm handling the problem as {-1, 0, 1} programming and also lets compute the next best integer solution easily. However, it should keep in mind that the algorithm is a heuristic one, which does not guarantee to find the global integer optimum always exactly. The procedure is very powerful utilizing the ability of the numeric-symbolic abilities of a computer algebraic system, like Wolfram Mathematica and it is properly fast for minimum 4 satellites with normal configuration, which means the Geometric Dilution of Precision (GDOP) should be between 1 and 8. Wolfram Alpha and Wolfram Clouds Apps give possibility to run the suggested code even via cell phones. All of these algorithms are illustrated with numerical examples. The result of the third one was successfully compared with the LAMBDA method, in case of ten satellites sending signals on two carrier frequencies (L1 and L2) with weighting matrix used to weight the GNSS observation and computed as the inverse of the corresponding covariance matrix. [ABSTRACT FROM AUTHOR]

Fidelity is one of the most widely used quantities in quantum information that measures the distance of two quantum states through a noisy channel, a kind of quantum operations. In this paper, we consider the model of quantum Markov chain (QMC), in which transitions are weighted by super-operators to characterize quantum operations and the initial quantum state is left parametric. A quantum analogy of probabilistic computation tree logic, called QCTL, is introduced to take into account fidelity, instead of probability measure, over QMC. The key to the model checking problem lies in computing the fidelity of the super-operator valued measure specified by a path formula in QCTL. It is minimized over all initial quantum states, which is intended for analyzing the system performance in the worst case. We achieve it by a reduction to quantifier elimination in the existential theory of the reals. The method is absolutely exact, so that model checking QCTL formulas against QMCs is proved to be decidable in exponential time. [ABSTRACT FROM AUTHOR]

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group. [ABSTRACT FROM AUTHOR]

In this paper, we present new computer algebra based methods for testing the structural stability of n-D discrete linear systems (with n ≥ 2 ). More precisely, we show that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of C n ) is equivalent to the fact that a certain system of polynomials does not have real solutions. We then use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems. [ABSTRACT FROM AUTHOR]

One of the main problems of coding theory is to construct codes with best possible parameters. Cyclic codes and their various generalizations, such as quasi-twisted codes, have been a fruitful source in achieving this goal. Recently, a new generalization of cyclic codes that are known as skew cyclic codes have been introduced and some new codes obtained from this class. Unlike many other types of codes considered in algebraic coding theory, skew cyclic codes require one to work in a non-commutative ring called skew polynomial ring. In this paper, we present some new linear codes obtained from the class of skew cyclic codes and describe computational challenges in working with this class of codes. [ABSTRACT FROM AUTHOR]

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type 4 F 3 (1) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz's and Watson's identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method. [ABSTRACT FROM AUTHOR]

Railway interlocking systems are essential safety components in rail transportation, designed to prevent train collisions. They regulate the transitions between sections of a railway station using rail traffic control elements. An interlocking system can assess whether the configuration of these control elements poses a collision risk. Over the years, researchers have developed various algebraic models to tackle this issue, highlighting the potential use of computer algebra systems in implementing interlocking systems. In this work, we aim to enhance these systems' capabilities. Not only will they indicate whether a situation is dangerous, but if it is, they will also provide guidance on how to configure certain rail traffic control elements to ensure safety. In this paper, we introduce an algebraic model that represents the railway station through polynomials. This approach transforms the task of identifying dangerous situations into calculating the residue of a polynomial over a set of polynomials. The monomials contained in this residue polynomial encode all possible configurations that would render the situation safe.

In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation. [ABSTRACT FROM AUTHOR]

Considered are 80 sets of layer groups, each set consisting of four groups: ordinary single and double, and grey single and double layer groups. The structural properties of layer groups (factorization into cyclic subgroups and the existence of grading according to the sequence of halving subgroups) enable efficient symbolic computation (by the POLSym code) of the relevant properties, real and complex irreducible and allowed (half‐)integer (co‐)representations in particular. This task includes, as the first step, classification of the irreducible domains based on the group action in the Brillouin zone combined with torus topology. Also, the band (co‐)representations induced from the irreducible (co‐)representations of Wyckoff‐position stabilizers (site‐symmetry groups) are decomposed into the irreducible components. These, and other layer group symmetry related theoretical data relevant for physics, layered materials in particular, are tabulated and made available through the web site https://nanolab.group/layer/. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson's method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method. [ABSTRACT FROM AUTHOR]