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A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by n, the algorithm uses O(n1.43) field operations, breaking through the 3/2 barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require O(n1.63) field operations in general, and n3/2+o(1) field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in n5/3+o(1) operations, while previous ones run in O(n2) operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation. [ABSTRACT FROM AUTHOR]

We study the class of symmetric Dziobek configurations where there are p + q bodies with non-zero masses at the vertices of a regular (p + q - 1) -simplex and one more body with zero mass on the symmetric line going through the centers of a subsimplex with p vertices and its complementary subsimplex with q vertices. All bodies from the same subsimplex have equal masses and let μ ≠ 0 be the ratio of masses from different subsimplices. We consider such central configurations for homogeneous force laws with exponent a ≠ 0 . The Newtonian case of a = - 3 2 with μ > 0 is studied more intensively. Using elementary methods, we obtain results for general values of the parameters a, p, q, and μ . For a = - 3 2 , we count the real roots of a polynomial system in two variables with three parameters. We are able to reduce the number of parameters required in our symbolic computation to two or one and use some techniques to count roots rigorously for all parameter values. Our results extend many previously known cases. For example, the known three positions of the zero mass in the case of a = - 3 2 , p = q = 1 , μ > 0 are at the Lagrange points L 1 , L 2 , and L 3 discovered by Euler. [ABSTRACT FROM AUTHOR]

The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this approach suffers from "low-frequency breakdown," meaning that the accuracy of the calculated value of the integral deteriorates when the integrand is only slowly oscillating. Recently presented experimental evidence, however, suggests that if a Chebyshev spectral method is used to discretize the differential equation and the resulting linear system is solved via a truncated singular value decomposition, then no low-frequency breakdown occurs. Here, we provide a proof that this is the case, and our proof applies not only when the integrand is slowly oscillating, but even in the case of stationary points. Our result puts adaptive schemes based on the Levin method on a firm theoretical foundation and accounts for their behavior in the presence of stationary points. We go on to point out that by combining an adaptive Levin scheme with phase function methods for ordinary differential equations, a large class of oscillatory integrals involving special functions, including products of such functions and the compositions of such functions with slowly-varying functions, can be easily evaluated without the need for symbolic computations. Finally, we present the results of numerical experiments which illustrate the consequences of our analysis and demonstrate the properties of the adaptive Levin method. [ABSTRACT FROM AUTHOR]

In the paper, the existence of soliton and periodic solutions is studied for the perturbed sine–cosine–Gordon equation. The corresponding traveling wave system is converted to a regularly Hamilton system via using bifurcation theory of differential equations and the geometric singular perturbation theory. Then by using Melnikov's method and symbolic computation, periodic wave solutions, solitary solutions, and kink and antikink solutions are obtained. The wave speed selection principles are given, respectively. Numerical simulations are performed to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

The Supersymmetry Genomics project aims to classify supermultiplets by properties like adinkras and holoraumy. The project’s protocol is: (1) set up the SUSY transformation rules (and action) with unknown numerical coefficients, (2) solve those coefficients and (3) compute desired properties, e.g. non-closure geometry (the non-closure functions in the anticommutator of supercovariant derivatives) and holoraumy (the commutator of supercovariant derivatives). This paper provides the first broadly applicable computer algorithms for completing these computations, comprising a new Cadabra module we call “SusyPy”. We provide a significant extension of the available tensor arithmetic/canonicalization to include spinor-indexed expressions and NW–SE convention, and we provide a new Fierz expansion algorithm for spinor-indexed expressions. On top of this new tensor–spinor symbolic algebra, we have built algorithms for solving for the coefficients of a multiplet and its action, and for computing holoraumy, for both off-shell and on-shell 풩=1 multiplets of any dimension. We apply our tools to assimilate linearized 11D, 풩=1 supergravity into the Genomics project. We demonstrate solving the 11D, 풩=1 multiplet, as in Cremmer, Julia and Scherk, in a few lines of code, and we provide a comprehensive picture of the multiplet’s non-closure geometry. Further, we provide the first-ever computation of the holoraumy of 11D, 풩=1 supergravity. [ABSTRACT FROM AUTHOR]

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function's versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots. [ABSTRACT FROM AUTHOR]

In this paper, the symmetries and conservation laws of a variable-coefficient generalized Calogero–Bogoyavlenskii–Schiff (vcGCBS) equation are investigated by modeling the propagation of long waves in nonlinear optics, fluid dynamics, and plasma physics. A Painlevé analysis is applied using the Kruskal-simplified form of the Weiss–Tabor–Carnevale (WTC) method, which shows that the vcGCBS equation does not possess the Painlevé property. Under the compatibility condition ( a 1 (t) = a 2 (t) ), infinitesimal generators and a symmetry analysis are presented via the symbolic computation program designed. With the Lagrangian, the adjoint equation is analyzed, and the vcGCBS equation is shown to possess nonlinear self-adjointness. Based on its nonlinear self-adjointness, conservation laws for the vcGCBS equation are derived by means of Ibragimov's conservation theorem for each Lie symmetry. [ABSTRACT FROM AUTHOR]

In this article, the potential Kadomtsev-Petviashvili (pKP) type coupled system with variable coefficients is studied, which have many applications in wave phenomena and soliton interactions in a two-dimensional space with time. In this framework, Hirota bilinear form is applied to acquire diverse types of interaction lump solutions from the foresaid equation. Abundant lump, lump-periodic, lump-soliton and multi soliton solutions to the pKP system are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. Lump, lump-periodic, lump-soliton and multi soliton solutions are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. The movement role of the waves is investigated, and the theoretical analysis of the acquired solutions is discussed using the bilinear technique of all produced solutions with 2D and 3D plots with respective parameters. The computational difficulties and outcomes highlight the clarity, effectiveness, and simplicity of the approaches, suggesting that these schemes can be applied to a variety of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics as well as to other real-world situations and a wide range of academic fields. We used software Maple 2024 "(http://www.maplesoft.com/)". [ABSTRACT FROM AUTHOR]

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with p$p$‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with p=1,2,3,4$p=1,2,3,4$. No solutions for p>1$p>1$ were known previously in the literature. For p=1$p=1$, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) p$p$ symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations  as well as to nonlinear ODE systems of generalized Hénon–Heiles form. [ABSTRACT FROM AUTHOR]

Our paper reviews some key concepts in chemical reaction network theory and mathematical epidemiology, and examines their intersection, with three goals. The first is to make the case that mathematical epidemiology (ME), and also related sciences like population dynamics, virology, ecology, etc., could benefit by adopting the universal language of essentially non-negative kinetic systems as developed by chemical reaction network (CRN) researchers. In this direction, our investigation of the relations between CRN and ME lead us to propose for the first time a definition of ME models, stated in Open Problem 1. Our second goal is to inform researchers outside ME of the convenient next generation matrix (NGM) approach for studying the stability of boundary points, which do not seem sufficiently well known. Last but not least, we want to help students and researchers who know nothing about either ME or CRN to learn them quickly, by offering them a Mathematica package "bootcamp", including illustrating notebooks (and certain sections below will contain associated suggested notebooks; however, readers with experience may safely skip the bootcamp). We hope that the files indicated in the titles of various sections will be helpful, though of course improvement is always possible, and we ask the help of the readers for that. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non‐crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low‐symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank‐deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database. [ABSTRACT FROM AUTHOR]

Nowadays, people are interested in many nonlinear models in sciences and engineering, e.g., a (2+1)-dimensional generalized nonlinear evolution system in fluid dynamics, plasma physics, nonlinear optics and quantum mechanics. We purpose to study that system by means of constructing one set of the bilinear auto-Bäcklund transformations and one set of the similarity reductions. Our bilinear auto-Bäcklund transformations can link certain solutions for that system with other solutions for that system itself, while our similarity reductions go from that system to a known ordinary differential equation. Our results depend on all the coefficients in that system. The present work could assist the future investigations in fluid dynamics, quantum mechanics, nonlinear optics and plasma physics. [ABSTRACT FROM AUTHOR]

Hyperdimensional Computing (HDC), also known as Vector Symbolic Architectures (VSA), is a neuro-inspired computing framework that exploits high-dimensional random vector spaces. HDC uses extremely parallelizable arithmetic to provide computational solutions that balance accuracy, efficiency and robustness. The majority of current HDC research focuses on the learning capabilities of these high-dimensional spaces. However, a tangential research direction investigates the properties of these high-dimensional spaces more generally as a probabilistic model for computation. In this manuscript, we provide an approachable, yet thorough, survey of the components of HDC. To highlight the dual use of HDC, we provide an in-depth analysis of two vastly different applications. The first uses HDC in a learning setting to classify graphs. Graphs are among the most important forms of information representation, and graph learning in IoT and sensor networks introduces challenges because of the limited compute capabilities. Compared to the state-of-the-art Graph Neural Networks, our proposed method achieves comparable accuracy, while training and inference times are on average 14.6× and 2.0× faster, respectively. Secondly, we analyse a dynamic hash table that uses a novel hypervector type called circular-hypervectors to map requests to a dynamic set of resources. The proposed hyperdimensional hashing method has the efficiency to be deployed in large systems. Moreover, our approach remains unaffected by a realistic level of memory errors which causes significant mismatches for existing methods. [ABSTRACT FROM AUTHOR]

It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving the concept of resultants for bivariate skew polynomials. Then, we employ the derived resultant to incrementally eliminate indeterminates in skew polynomial systems, utilising both direct and modular approaches. Finally, we discuss some applications of the derived resultant, including cryptographic schemes (such as Diffie–Hellman) and combinatorial identities (such as Pascal's identity). We start by considering a bivariate skew polynomial system with two indeterminates; our intention is to isolate and eliminate one of the indeterminates to reduce the system to a simpler form (that is, relying only on one indeterminate in this case). The methodology is composed of two main techniques; in the first technique, we apply our definition of a (bivariate) resultant via a Sylvester-style matrix directly from the polynomials' coefficients, while the second is based on modular methods where we compute the resultant by using evaluation and interpolation approaches. The idea of this second technique is that instead of computing the resultant directly from the coefficients, we propose to evaluate the polynomials at a set of valid points to compute its corresponding set of partial resultants first; then, we can deduce the original resultant by combining all these partial resultants using an interpolation technique by utilising a theorem we have established. [ABSTRACT FROM AUTHOR]

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. [ABSTRACT FROM AUTHOR]

This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering. [ABSTRACT FROM AUTHOR]

This study investigates the dynamics of boomerang solitons and their interactions in three-mode fiber under the influence of inhomogeneous effects governed by three-coupled nonlinear Schrödinger (3-CNLS) equations in GVD and nonlinearity coefficients. Also, 3-CNLS equation can be used to investigate the optical soliton propagation in three-core fibers, which contain some potential applications in the optical fiber communication system. Using the Darboux transformation method with computer simulation, the two-bright boomerang soliton solutions are attained analytically for the dispersion and nonlinearity functions. The propagation and interaction properties of boomerang solitons are investigated, while the variable coefficients are adopted as linear functions. Various interaction scenarios are illustrated by plotted figures to reveal the importance of obtained soliton solutions. The physical significance of boomerang solitons can be understood from the displayed figures. The obtained results will be benefit soliton path control and soliton phase management in an inhomogeneous three-core optical fiber. [ABSTRACT FROM AUTHOR]

The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly. [ABSTRACT FROM AUTHOR]

The novel forms of the split octonionic Dirac equation and its corresponding Lagrangian are derived using symbolic computing techniques. [ABSTRACT FROM AUTHOR]

This study examines nonlinear duopoly games in the context of diseconomies of scale. The firms involved exhibit varied levels of rationality and can be classified as rational, boundedly rational, local monopoly approximation (LMA), or adaptive players. Using a systematic approach based on symbolic computation, we conduct a comprehensive analysis of the stability of these models. The results indicate that the equilibrium of these models is locally stable for all possible parameter values. We also compare the convergence factors of some models, which characterize the convergence speed of trajectories around the equilibrium. Insightfully, we find that for a heterogeneous duopoly market with one boundedly rational firm and one LMA firm, the convergence speed is determined by the efficiency-dominant (lower-cost) firm, regardless of whether the returns to scale are assumed to be decreasing or constant. [ABSTRACT FROM AUTHOR]

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota's method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper's aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems. [ABSTRACT FROM AUTHOR]

This study is focusing on the integrable (3+1)-dimensional equation that combines the potential Kadomtsev-Petviashvili (pKP) equation with B-type Kadomtsev-Petviashvili (BKP) equation, also known as the pKP-BKP equation. The idea of combining integrable equations has the potential to produce a variety of unexpected outcomes such as resonance of solitons. This article provides a wide range of alternative exact solutions for the pKP-BKP equation in three dimensional form, including dark solitons, singular solitons, singular periodic solutions, Jacobi elliptic function (JEF) solutions, rational solutions and exponential solution. The improved modified extended (IME) tanh function method is employed to investigate these solutions. All of the obtained solutions for the investigated model are presented using the Wolfram Mathematica program. To further help in understanding the solutions' physical characteristics and dynamic structure, the article provides visual representations of some derived solutions using 2D representation in addition to the 3D graphs via symbolic computation. This article aims to use a potent strategy using a powerful scheme to derive different solutions with various structures. Additionally, the results greatly improve and enhance the literature's solutions to a combined pKP-BKP equation and allow deep understanding of the nonlinear dynamic system through different exact solutions. [ABSTRACT FROM AUTHOR]

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from having them made separately for each problem via a machine learning model. This study reports lessons on such use of machine learning in symbolic computation, in particular on the importance of analysing datasets prior to machine learning and on the different machine learning paradigms that may be utilised. We present results for a particular case study, the selection of variable ordering for cylindrical algebraic decomposition, but expect that the lessons learned are applicable to other decisions in symbolic computation. We utilise an existing dataset of examples derived from applications which was found to be imbalanced with respect to the variable ordering decision. We introduce an augmentation technique for polynomial systems problems that allows us to balance and further augment the dataset, improving the machine learning results by 28% and 38% on average, respectively. We then demonstrate how the existing machine learning methodology used for the problem—classification—might be recast into the regression paradigm. While this does not have a radical change on the performance, it does widen the scope in which the methodology can be applied to make choices. [ABSTRACT FROM AUTHOR]

Abstract In this article, the potential Kadomtsev-Petviashvili (pKP) type coupled system with variable coefficients is studied, which have many applications in wave phenomena and soliton interactions in a two-dimensional space with time. In this framework, Hirota bilinear form is applied to acquire diverse types of interaction lump solutions from the foresaid equation. Abundant lump, lump-periodic, lump-soliton and multi soliton solutions to the pKP system are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. Lump, lump-periodic, lump-soliton and multi soliton solutions are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. The movement role of the waves is investigated, and the theoretical analysis of the acquired solutions is discussed using the bilinear technique of all produced solutions with 2D and 3D plots with respective parameters. The computational difficulties and outcomes highlight the clarity, effectiveness, and simplicity of the approaches, suggesting that these schemes can be applied to a variety of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics as well as to other real-world situations and a wide range of academic fields. We used software Maple 2024 “( http://www.maplesoft.com/ )”.

A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the Crystal code is presented, and the considerably improved computational efficiency over the previous implementation is illustrated. For this purpose, three different tasks involving the use of analytical forces are considered: (i) geometry optimization; (ii) harmonic frequency calculation; and (iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: (i) a metallic 2D model of the Cu(111) surface; (ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and (iii) a porous metal-organic crystal, namely the ZIF-8 zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched. [ABSTRACT FROM AUTHOR]

A symbolic C++ library—Tinned—has been developed for symbolic differentiation and manipulation in response theory. By recognizing different key building blocks in the density matrix‐based (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) and coupled‐cluster response theories, we have implemented their corresponding C++ symbolic classes, including but not limited to one‐ and two‐electron operators, exchange‐correlation energy and potential, and coupled‐cluster operator. Formulas of response theory can be well expressed in terms of the symbolic classes in the library Tinned. Their high‐order perturbation‐strength derivatives can be straightforwardly computed and extracted afterwards for numerical evaluation. The library Tinned will greatly facilitate the development work of response theory and may lead to a unified framework for response theory at different levels of electronic structure theory. [ABSTRACT FROM AUTHOR]

We investigate the rational solitons of the Kraenkel–Manna–Merle (KMM) system in ferrites via symbolic computation with the ansatz functions technique and logarithmic transformation. The KMM system describes the propagations of the ferromagnetic particles in ferrite materials. Multiwave solitons are reported by using the three-wave technique. Homocentric breathers (HBs) approach is also applied to build new forms of exact solutions. We construct M-shaped solitons and their dynamics are shown via the appropriate values of parameters. Two types of interactions for rational soliton with kink wave are investigated. Additionally, kink cross-rational solitons (KCRSs) and periodic cross-rational solitons (PCRSs) are examined via suitable transformation. [ABSTRACT FROM AUTHOR]

Given a monomial ideal I, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of I. In many cases we determine the asymptotic growth rate of these two functions. We also perform explicit computations by using the symbolic polyhedron. [ABSTRACT FROM AUTHOR]

Studies on the shallow water waves belong to the cutting-edge issues in sciences and engineering. In this paper, introducing symbolic computation, for a generalized nonlinear shallow water wave equation, with respect to the displacement and velocity of the water, we establish an auto-Bäcklund transformation with some solitonic solutions, as well as a set of the similarity reductions, the latter of which ought to be focused towards a known ordinary differential equation. Our results are seen to tie to the gravitational force and wave height. [ABSTRACT FROM AUTHOR]

We theoretically investigate the soliton dynamics of ferroelectric liquid crystal materials used in display devices. The approximations and extensions to existing device technology fields where mathematical analysis would improve optical device development are given special consideration. The ferroelectric liquid crystal maintained between two parallel electrical conductors is taken into consideration in order to illustrate bookshelf geometry. These liquid crystal molecules are influenced by an electric field due to their spontaneous polarization. The effect of electric fields and molecular excitations has been successfully established. We study Hirota's bilinearization method and the (G ′ / G) expansion method for ferroelectric liquid crystals, which can be expressed by the Sine–Gordon equation governing the soliton dynamics in crystal systems. We suggest that this kind of collisional solitonic humps in nematicon profiles indicates the wide range of optical switching strategies for network design and logic gates. This switching mechanism may have enormous applications in multiplexing liquid crystal displays with higher resolution. [ABSTRACT FROM AUTHOR]

We focused on solitonic phenomena in wave propagation which was extracted from a generalized breaking soliton system in (3 + 1)-dimensions. The model describes the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. Abundant double-periodic soliton, breather wave and the multiple rogue wave solutions to a generalized breaking soliton system by the Hirota bilinear form and a mixture of exponentials and trigonometric functions are presented. Periodic-soliton, breather wave and periodic are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. Through three-dimensional graph, density graph, and two-dimensional design using Maple, the physical features of double-periodic soliton and breather wave solutions are explained all right. The findings demonstrate the investigated model's broad variety of explicit solutions. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering. [ABSTRACT FROM AUTHOR]

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models. [ABSTRACT FROM AUTHOR]

Of current interest, there have recently appeared a series of the shallow-water papers in Mod. Phys. Lett. B. Illuminated by those papers and to make the story more complete, for the nonlinear long waves in the shallow water, this paper is scheduled to investigate a (2+1)-dimensional generalized Broer-Kaup system. As for the wave height and wave horizontal velocity, we employ symbolic computation, with the view of constructing out two groups of the similarity reductions. [ABSTRACT FROM AUTHOR]

This paper reports an application of the Lie algebra s e (3) of the Euclidean group S E (3) , which is isomorphic to the theory of screws in the velocity and acceleration analyses of serial manipulators. The symbolic computation of the infinitesimal kinematics allows one to obtain algebraic expressions related to the kinematic characteristics of the end effector of the serial manipulator, while in the case of complex manipulators, numerical computations are preferred owing to the emergence of long terms. The algorithm presented enables the symbolic computation of the velocity and acceleration characteristics of the end effector in serial manipulators in order to allow the compact and efficient modeling of velocity and acceleration analyses of both parallel and serial robotic manipulators. Unlike other algebras, these procedures can be extended without significant effort to higher-order analyses such as the jerk and jounce. [ABSTRACT FROM AUTHOR]

The rational solitons and other wave solutions demonstrate the existence of wave packets propagating through optical Kerr media, exhibiting non-diffractive and non-dispersive spatiotemporal localization. This paper focuses on constructing rational solitons, multi-wave solutions, and other solutions for a dimensionless time-dependent paraxial wave model (dt-pPWM). These solutions are derived through the application of symbolic computation and the ansatz function method, using both traveling wave and logarithmic transformations. Our approach encompasses three distinct methods: the positive quadratic function approach, the three-wave approach, and the double exponential approach. These wave results find practical applications and contribute to a deeper understanding of the physical phenomena described by this model. We have computed conservative quantities corresponding to the power, momentum, and energy of solitons. To assess the stability of the wave equation, we conducted modulational instability analysis, confirming the stability and accuracy of all soliton solutions. The structure of the solution provides a physical interpretation. By selecting appropriate parametric values, we generated 3D graphics illustrating various multi-wave solutions, including lump waves, rogue waves, and multipeak solitons. Additionally, we observed intriguing phenomena arising from the interaction of multi-waves, which have applications in telecommunications, quantum optics and other related fields. [ABSTRACT FROM AUTHOR]

The article studies the dynamics of Gaussian solitary waves in nano optical fibers which are described by perturbed and improved nonlinear Schrödinger equations which have logarithmic nonlinearities with and without attenuation terms. As a result, the Gaussian solitary waves have been extracted for the logarithmic perturbed nonlinear Schrödinger equation and the logarithmic perturbed improved nonlinear Schrödinger equation. Moreover, new logarithmic improved nonlinear Schrödinger equations are presented. With the Gaussian function and symbolic computation, we obtain the Gaussian solitary waves for such equations. [ABSTRACT FROM AUTHOR]

In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with M a p l e. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods. [ABSTRACT FROM AUTHOR]

The aim of this research is to design algorithms and develop software for the automated assessment of stability criteria in the dynamic behavior of unmanned aerial vehicles (UAVs), modeled through systems of ordinary differential equations (ODEs). By employing advanced symbolic computation techniques, an application has been created that facilitates the automatic linearization of nonlinear equation systems and the derivation of the characteristic polynomial for the resulting linearized ODE systems. The introduced methodologies and models provide substantial practical significance in the evaluation of stability across a broad range of both economic and technical systems. [ABSTRACT FROM AUTHOR]

We present a physics-informed machine learning (PIML) approach for the approximation of slow invariant manifolds of singularly perturbed systems, providing functionals in an explicit form that facilitates the construction and numerical integration of reduced-order models (ROMs). The proposed scheme solves the partial differential equation corresponding to the invariance equation (IE) within the geometric singular perturbation theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely, feedforward neural networks and random projection neural networks, with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely, the Michaelis--Menten, the target-mediated drug disposition reaction mechanism, and the 3D Sel'kov model. We show that the proposed PIML scheme provides approximations of equivalent or even higher accuracy than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance because there are many systems for which the gap between the fast and slow timescales is not that big, but still, ROMs can be constructed. A comparison of the computational costs between symbolic, automatic, and numerical approximation of the required derivatives in the learning process is also provided. [ABSTRACT FROM AUTHOR]

Considering validated Runge-Kutta schemes requires the computation of local truncation error. Computation of such error is expensive in term of complexity, thus few techniques have been proposed to compute or at least estimate it. For example, Runge-Kutta pairs provide an error estimation by computing the difference of two schemes of different orders while sharing the same stages. With the goal of proposing a validated scheme, an approximation is not sufficient, but could be helpful after some adjustments. In this paper, we compute a new pair and show its performance on a non-trivial problem. Two new techniques for the local truncation error computation are then presented: a symbolic based approach dedicated to a specific problem and a preliminary study of pair's approximation. [ABSTRACT FROM AUTHOR]

A theoretical inquiry delves into the presence of multiple solitons and breather configurations within a nonuniform, inhomogeneous, unmagnetized dusty plasma characterized by universally polarized force. This system encompasses ions, electrons, positively and negatively charged dust particles, distributed in accordance with a superthermal distribution. Employing the reductive perturbation technique (RPT), the Gardner equation (GE) is derived from the fundamental hydrodynamical model equations. The multi-solitons of the GE are synthesized utilizing the Hirota bilinear method (HBM), which further facilitates the exploration of breather solutions via the appropriate selection of complex wave numbers. The system exhibits both compressive and rarefactive multi-solitons. Moreover, the behavior of breather structures varies depending on associated plasma parameters; occasionally, they overlap, while at other times, they remain entirely disjoint. This research holds relevance for investigating the propagation of finite-amplitude waves in natural phenomena such as the atmosphere, oceans, optic fibers, and signal processing. Figures display the interaction between two-solitons (a), and breather structures (b), of the Gardner equation in dusty plasma with polarized force for the parameter values k i = 5 , k e = 2.5 , μ i = 0.7 , μ e = 0.3 , μ = 10 , ρ = 1.01 , σ i = 0.1 , R = 0.5 , Z = 0.01 , respectively. The manuscript presents some graphs to illustrate the propagation and interaction between two-soliton solutions, as well as the propagation of various breather structures in dusty plasma under polarized force. We explore the presented wave solutions using the symbolic computation software Mathematica. The diversity in two-solitons and breather structures depends on the plasma parameter and the choice of wave number, real or complex. This work analyzes, through graphical representation, how the plasma system affects the above nonlinear structures. [ABSTRACT FROM AUTHOR]

In this work, we use symbolic computation and ansatz function schemes, to investigate the soliton solutions for the nonlinear Schrödinger equation (NLSE) along with Kudryashov's quintuple self-phase modulation system (KQSPMS) including dual-form of nonlocal nonlinearity (DFNLN). We initially determine the ordinary differential (OD) form for this model through a variable transformation. Then we introduce numerous new dynamical soliton types: the M-shaped rational soliton, the M-shaped interaction between one and two stripe solitons, the periodic cross-M-shaped rational (PCMR) soliton, the periodic cross-kink (PCK) soliton, multi-waves, and the homoclinic breather soliton. Secondly, we determine the partial differential (PD) form for this model through a variable transformation. Moreover, a lump soliton, a periodic wave, a rogue wave, a lump interaction with a periodic and kink wave, and three different types of breather soliton will obtain. We'll demonstrate these solutions' unique structure and extremely interesting interaction behavior. We'll also use graphs (3-D and contour plots) to discuss the dynamics of the results after setting the parameters to the proper values. [ABSTRACT FROM AUTHOR]

The Zoomeron model covers particular kinds of solitons with distinctive properties that appear in several physical scenarios, such as, fluid dynamics, nonlinear optics and laser physics. First time utilising the mapping method, we determine the analytical solution to the described model, including several novel dynamical behaviours. Through symbolic computation, we are able to derive the breather waves, kink waves, dark soliton, singular soliton, periodic soliton and bright soliton of this model. Additionally, we encounter single kink waves and single breather waves. We find novel hyperbolic trigonometric, rational and elliptic functions. Modelling our observations with MATLAB tools and producing many 3D graphs. The results obtained will be crucial for further research on complicated nonlinear models. [ABSTRACT FROM AUTHOR]

In this work, we investigate the (2+1)-dimensional nonlinear electrical transmission line model (NETLM) and propose a modified generalized Riccati equation mapping (mGREM) approach for obtaining analytical soliton solutions. The NETLM characterizes wave distributions on network lines. We introduce the mGREM technique, which involves a traveling wave transformation to convert the equation into nonlinear ordinary differential equations. By applying this method, we extract various soliton solutions, including traveling waves, kink-type solitons, bell-shape solitons, and multi-solitons. These findings contribute to understanding the (2+1)-dimensional NETLM and provide valuable insights into wave distributions on network lines. To gain a comprehensive understanding of the solutions' dynamics, we conduct numerical simulations and present two-dimensional, three-dimensional, and contour graphs. These graphical illustrations offer valuable insights into voltage propagation patterns within the system. Our results contribute to the existing literature and open avenues for further investigations in the field of NETLM. The significance of this work lies in the successful application of the mGREM method to obtain diverse traveling wave solutions for the (2+1)-dimensional NETLM. By employing symbolic computation in Mathematica, we extract analytical soliton solutions, which go beyond previous efforts in the literature. The proposed approach provides a novel perspective on the NETLM equation and its soliton solutions. The obtained results enhance our understanding of the system's behavior and pave the way for future research in this domain. [ABSTRACT FROM AUTHOR]

show the power of Bruno Buchberger's seminal Gröbner Basis algorithm, interfaced, seamlessly, with what we call symbolic dynamical programming, to automatically generate algebraic equations satisfied by the generating functions enumerating so-called Generalized Dyck Walks, i.e. 2D walks that start and end on the x-axis, and never dip below it, for an arbitrary set of steps. More impressively, we combine it with calculus (that Maple knows very well!), to automatically compute generating functions for the sum-of-the-areas of these generalized Dyck paths, and even for the sum of any given power of the areas, enabling us to get statistical information about the area under a random generalized Dyck path. [ABSTRACT FROM AUTHOR]

Using symbolic computation with Maple, we can discover lots of (rigorously-proved!) facts about Standard Young Tableaux, in particular the distribution of the entries in any specific cell, and the sorting probabilities. [ABSTRACT FROM AUTHOR]