Symbolic Computation Research Articles

Spacelike Ac-Slant Curves with Non-Null Principal Normal in Minkowski 3-Space

In this paper, we define a spacelike ac-slant curve whose scalar product of its acceleration vector and a unit non-null fixed direction is a constant in Minkowski 3-space. Furthermore, we give a characterization depending on the curvatures of the spacelike ac-slant curve. After that, we get the relationship between a spacelike ac-slant curve and several distinct types of curves, such as spacelike Lorentzian spherical curves, spacelike helices, spacelike slant helices, and spacelike Salkowski curves, enhancing our understanding of its geometric properties in Minkowski 3-space. Finally, we used Mathematica, a symbolic computation software, to support the notions of an ac-slant curve with attractive images.

Fast evaluation of generalized associated linear equations (GALEs) for nonlinear systems characterization and compensation

Nonlinear Output Frequency Response Functions (NOFRFs), which are a one-dimensional extension of the linear Frequency Response Function (FRF) to nonlinear cases, have been introduced for nonlinear system characterization and compensation. The evaluation of NOFRFs is accomplished by solving a series of linear difference or differential equations, known as Generalized Associated Linear Equations (GALEs). However, the derivation of GALEs often includes numerous recursive and symbolic calculations, posing significant challenges in both practical applications and computer programming. In the present study, a fast-computing algorithm, based on combinatorial calculations as opposed to recursive and symbolic calculations, is proposed for fast evaluation of GALEs. This allows for the evaluation of NOFRFs in a more streamlined and straightforward manner. The applications of the proposed fast-computing algorithm for nonlinear system characterization and compensation are demonstrated through several case studies including a High Power Amplifier (HPA) and a Duffing system. The results of the case studies show the effectiveness of applying the GALEs and NOFRFs for system design and safety control across a wide range of engineering applications.

An energy approach to asymptotic, higher-order, linear homogenization

A higher-order homogenization method for linear elastic structures is proposed. While most existing approaches to homogenization start from the equations of equilibrium, the proposed one works at the energy level. We start from an energy functional depending on microscopic degrees of freedom on the one hand and on macroscopic variables on the other hand; the homogenized energy functional is derived by relaxing the microscopic degrees of freedom and applying a formal two-scale expansion. This method delivers the the energy functional of the homogenized model directly, including boundary terms that have not been discussed in previous work. Our method is formulated in a generic setting which makes it applicable to a variety of geometries in dimension 1, 2 or 3, and without any particular assumption on material symmetry. An implementation using a symbolic calculation language is proposed and it is distributed as an open-source library. Simple illustrations to elastic trusses having pre-stress or graded elastic properties are presented. The approach is presented in the context of discrete elastic structures and the connection with previous work on the higher-order homogenization of period continua is discussed.

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In the present work, we aim to explore the new (3+1)-dimensional integrable fourth-order nonlinear equation(IFNE) for describing the shallow water waves. First, we study its N-soliton solutions via the bilinear form which is constructed by applying the Cole-Hopf transform. The resonance conditions of the soliton molecular are extracted and the soliton molecules are obtained. Second, the ansatz function method together with the symbolic computation, is implemented to develop the interaction wave solutions(IWSs). Finally, we take advantage of the Bernoulli sub-equation function method(BSFM) to look into the travelling wave solutions(TWSs). Different kinds of the TWSs like the singular-kink and kink solitary wave solutions are found. Correspondingly, the dynamic performances of the solutions are depicted graphically to present the physical interpretations. And for all we know, the solutions got in this work are all new and can be regarded as an extension of the solutions for the new (3+1) dimensional IFNE, which are expected to have practical significance for the application of these equations in physics.

Паралельний алгоритм розв’язування систем інтегральних рівнянь Вольтерри II роду

The problem of increasing the effectiveness of the study of integral models of dynamic systems is considered. A parallel algorithm for solving a system of Volterra linear integral equations of the second kind based on the quadrature method of numerical integration is proposed. The algo-rithm was implemented in the MATLAB computer mathematics system in the form of an m-function. The program uses the MATLAB Distributed Computing Toolbox infrastructure to manage workflows and distribute computations between them on multi-core processors. Com-putational experiments were conducted on a model example using the Symbolic Math Toolbox package for symbolic calculations and a comparison of the execution of parallel calculations with the execution time of the implementation of a sequential algorithm. The results showed a significant increase in the speed of research of integral models on multi-core processors when using the proposed algorithm and its computer implementation.

An exact method for free vibration of beams and frameworks using frequency-dependent mass, elastic and geometric stiffness matrices

The frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Bernoulli-Euler beam are developed through rigorous application of symbolic computation. These three matrices are related to the corresponding dynamic stiffness matrix so that free vibration analysis of axially loaded beams and frameworks can be carried out in an exact manner by applying the Wittrick-Williams algorithm as solution technique. Representative results from the proposed theory are presented for different boundary conditions of beams and frameworks, carrying tensile and compressive loads. Comparative results from finite element method are also presented. The duality between the free vibration and buckling problems is captured in that when the compressive load in a beam or frame approaches its critical buckling load, the fundamental natural frequency tends to zero and thus, buckling can be thoughtfully interpreted as free vibration at zero frequency. The investigation has opened the possibility of including damping in free vibration analysis of beams and frameworks when applying the dynamic stiffness method.

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

Algebraic School of V.M. Glushkov and Insertion Modeling

The Algebraic School of Viktor Mykhailovych Glushkov already has several generations of scientists, most of whom have already identified and are developing their own scientific directions. This school, started at the Institute of Cybernetics, was shaped by research that had the significance of inventions at the level of world science and the computer industry and in many ways predicted their further development. The work considers the formation of the paradigm of insertion modeling as a generalization of the main achievements of the Glushkov school, starting from the theory of automata and up to modern computer algebra systems. The main concepts from which insertion modeling was formed are considered, namely, symbolic computations, which were implemented even in the first personal machine of the MIR series. It is also the theory of agents and environments, parallel computing, and methods of artificial intelligence, developed at the Institute of Cybernetics, in the 60s and 70s. Numerous deployments of insertion modeling methods in cyber security, research in natural sciences, blockchain technology, in verification and testing of software and hardware, have shown the practicality and value of theoretical research conducted by followers of the Glushkov school. Keywords: symbolic computation, algebraic modeling, insertional modeling.

Collision dynamics of high-order localized waves in a novel [formula omitted]-dimensional integrable Boussinesq model

The pursuit of finding precise solutions for complex nonlinear systems in high dimensions has long been a significant goal in the fields of mathematics and physics. This paper focuses on investigating a novel equation, referred to as the (3+1)-dimensional Boussinesq equation, which accurately describes the behavior of gravity waves on the surface of water. Using the Hirota bilinear method, the study successfully derives multiple-soliton and multi-breather solutions for the Boussinesq equation. Moreover, it presents mixed solutions that combine solitons and breathers under specific parameter conditions. By considering the long wave limit for these solitons, rational and semi-rational solutions for the equation are obtained. These rational solutions encompass line rogue waves and lump waves, while the semi-rational solutions emerge from the interaction between line rogue waves and solitons. Additionally, employing symbolic computation, the paper introduces a class of multiple lumps for the (3+1)-d Boussinesq equation, showcasing intriguing triangular formations and circular patterns. Overall, the findings of this research enhance our understanding of unique nonlinear phenomena and offer valuable theoretical support for future experimental investigations.

Novel solitonary solutions of (2+1)-dimensional variable coefficient NLS equations

The objective of this paper is to obtain some solitary solutions of (2+1)-dimensional variable coefficient nonlinear Schrödinger (NLS) equations through computerized symbolic computation. By applying the [Formula: see text]-expansion and homogeneous balance method, two soliton solutions are constructed for the NLS equations, which provide a model of the interaction between two waves. Consequently, the solitary solutions are obtained in different forms of dynamical structures. Moreover, the propagation behavior of the resulting solitonary solutions is discussed. The results have rich physical structures that are helpful to explain the nonlinear soliton phenomena in nonlinear optics and plasma physics.

Oceanic shallow-water investigations on a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system

To date, with respect to water waves, researchers have studied certain systems, including a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system that describes, e.g., the dispersive long waves in the oceanic shallow water, which we study here. With respect to, e.g., the horizontal velocity of the water wave as well as the height of the deviation from the equilibrium position of the water, with symbolic computation, on the one hand, the system is found to pass the Painlevé test under some coefficient constraints, while on the other hand, two families of the bilinear forms and two families of the N-soliton solutions are constructed, with N as a positive integer. Related constraints are shown. Our bilinear forms and N-soliton solutions depend on the coefficients in the system.

Painlevé integrable property, Bäcklund transformations, Lax pair, and soliton solutions of a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid

In this paper, we focus our attention on a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid with symbolic computation. The Painlevé integrable property is derived. Via the Ablowitz–Kaup–Newell–Segur procedure, we obtain a Lax pair under the coefficient constraints. Based on the Hirota method, we obtain a bilinear form and a bilinear Bäcklund transformation under the coefficient constraints. We derive the auto-Bäcklund transformations based on the truncated Painlevé expansions. According to the bilinear form, we give the two-soliton solutions under the coefficient constraints. We also discuss the relation between the variable coefficients and soliton solutions, i.e., how the two solitons present different types with the different forms of the variable coefficients.

Symbolic Computation of an Arbitrary-Order Resonance Condition in a Hamiltonian System

Symbolic Computation of an Arbitrary-Order Resonance Condition in a Hamiltonian System

Soliton molecules, novel hybrid interaction solutions and periodic wave solutions to the (3 + 1)-dimensional nonlinear evolution equation for shallow-water waves

The central target of this research is looking into some novel solutions of the (3 + 1)-dimensional nonlinear evolution equation (NEE) for the shallow water waves. By manipulating the Hirota bilinear approach (HBA), the N-soliton solutions(N-SSs) are extracted. Then applying the resonance condition on the N-SSs, we successfully derive the soliton molecules (SMS) on the (x,y), (x,z) and (y,z)-planes. In addition, abundant novel hybrid interaction solutions are also studied by assigning the reasonable parameters, including the interaction between 1-soltion and the breather solution, interaction between 1-soltion and the soliton molecule(SM) with two solitons, interaction between two-breather solutions, interaction between breather solution and the SM of two solitons, interaction between two SMS of two solitons, interaction bwtween 1-soliton and the SM of three solitons. In the end, the periodic wave solutions (PWSs) are explored by taking advantage of the test function approach and symbolic calculation. The dynamic attributes of the obtained solutions are described graphically to unveil their physical behaviors. The discoveries in this work can enable us better understand the nonlinear dynamics of the considered equation.

Performance analysis of input shaped model reference adaptive control for a single-link flexible manipulator

Motion control for flexible link manipulators will have a great influence on their overall performance. Conventional control systems have the problem that they cannot compensate for the presence of uncertainty due to a variety of factors such as a change in the system itself-payload variations, degradation and modelling uncertainty. In addition, the underdamped structure of these systems entails motion-induced transient deflection and residual vibration. On the contrary, adaptive control, in particular Model Reference Adaptive Control (MRAC), has therefore been proposed to handle problems of this type dealing with other systems such as cranes, steering stability control for ground vehicles, airplanes and so on. This study presents a review and discussion on the MRAC and some individual issues of MRAC for these flexible link manipulators are addressed to specify: the state tracking error, the limitation of the adaptation values corresponding to the control gains and the control effort. Dealing with the last one, this work proves that shaping the command input to a Model Reference Adaptive Control (IS-MRAC) reduces the control effort necessary for the plant to follow the reference model and also mitigates the detrimental effects of flexibility. The IS-MRAC is implemented in a Single-Link Flexible Manipulator whose natural frequencies, experimentally and analytically estimated using the Transfer Matrix Method plus symbolic computation, vary significantly depending on payload mass and position. To this structured or parametric uncertainty, the proposed Lyapunov control law using the states associated with the low mode is designed for asymptotic stability and accommodates correctly. The state tracking, vibration suppression and control effort reduction performances of the proposed IS-MRAC combined controller are analysed via numerical simulations and experiments.

Influence of rationality levels on dynamics of heterogeneous Cournot duopolists with quadratic costs

This paper is intended to investigate the dynamics of heterogeneous Cournot duopoly games, where the first player adopts identical gradient adjustment mechanisms but the second player is endowed with distinct rationality levels. Based on tools of symbolic computations, we introduce a new approach and use it to establish rigorous conditions of the local stability for these models. We analytically investigate the bifurcations and prove that the period-doubling bifurcation is the only possible bifurcation that may occur for all the considered models. The most important finding of our study is regarding the influence of players' rational levels on the stability of heterogeneous duopolistic competition. It is derived that the stability region of the model where the second firm is rational is the smallest, while that of the one where the second firm is boundedly rational is the largest. This fact is counterintuitive and contrasts with relative conclusions in the existing literature. Furthermore, we also provide numerical simulations to demonstrate the emergence of dynamics such as periodic solutions with different orders and strange attractors.

Experimenting with discrete dynamical systems

We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of R k into itself.

On soliton solutions, periodic wave solutions, asymptotic analysis and interaction phenomena of the (3+1)-dimensional JM equation

In this paper, the M-lump solutions, the periodic type, and cross-kink wave solutions are acquired. Here, the Hirota bilinear operator is employed. By utilizing the symbolic computation and employing the utilized method, the (3+1)-dimensional Jimbo–Miwa (JM) equation is investigated. Based on the Hirota bilinear form, the soliton solution and periodic wave solution to the mentioned equation, respectively, are obtained. We gained plenty of multiple collisions of lumps. Next, the periodic wave and cross-kink wave have greatly enriched the existing literature on the JM equation. Through the three-dimensional designs, contour design, density design, and two-dimensional design by using Maple, the physical features of these soliton solutions are explained all right. The forms of the attained solutions are one-lump, two-lumps, and three-lumps wave solutions. Then, a class of rogue waves-type solutions to the (3+1)-dimensional JM equation within the frame of the bilinear equation is found. These results can help us better understand interesting physical phenomena and mechanisms.

Toward finiteness of central configurations for the planar six-body problem by symbolic computations. (I) Determine diagrams and orders

In a series of papers we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar n-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the 5-body problems. In their paper, bicolored graphs called zw-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving finiteness amounts to exclusions of central configurations associated to these diagrams. Following their method, the amount of computations becomes enormous when there are more than five bodies. In this paper we introduce matrix algebra for determination of possible diagrams and asymptotic orders, devise several criteria to reduce computational complexity, and determine possible zw-diagrams by automated deductions. For the planar six-body problem, we show that there are at most 86 zw-diagrams.

Lump Waves in a Spatial Symmetric Nonlinear Dispersive Wave Model in (2+1)-Dimensions

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion.