Symbolic Computation Research Articles

Newly formed center-controlled rouge wave and lump solutions of a generalized (3+1)-dimensional KdV-BBM equation via symbolic computation approach

In this article, we investigate the generalized (3+1)-dimensional KdV-Benjamin-Bona-Mahony equation governed with constant coefficients. It applies the Painlevé analysis to test the complete integrability of the concerned KdV-BBM equation. The symbolic computational approach provides first-order, second-order rogue wave and lump solutions with center-controlled parameters. The rogue waves localized in space and time have a significant amplitude, and lumps are of rational form solution, localized decaying solutions in all space directions rationally. Utilizing a symbolic computation approach, we get the bilinear equation of the KdV-Benjamin-Bona-Mahony equation and show the center-controlled rogue waves and lumps. We employ the symbolic system software Mathematica to do the symbolic computations, form the first and second-order rogue waves, and lump solutions with appropriate values of constant coefficients. The KdV-Benjamin-Bona-Mahony equation analyses the evolution of long waves with modest amplitudes propagating in plasma physics and the motion of waves in fluids and other weakly dispersive mediums. Moreover, rogue waves and lumps occur in several scientific areas, such as fluid dynamics, optical fibers, dusty plasma, oceanography, water engineering, and other nonlinear sciences.

Novel dynamics of the Zoomeron model via different analytical methods

We apply the novel Kudryashov scheme, the 1G′ approach, and the G′G′+G+A technique to handle the Zoomeron model for the first time, which yields several kinds of soliton solutions with some novel dynamic properties. The proposed model handles specific incidents of soliton structures with distinctive characteristics that arise in laser sciences, fluid mechanics, and nonlinear optics. We observe dark soliton, bright soliton, periodic waves, kink waves, anti-kink waves, and breather waves for the mentioned equation by symbolic calculation. The suggested model also produces lump-type breather waves. Density, 3D, and 2D images are used to exhibit the dynamics of the adopted outcomes. The results will be crucial for future research on higher-order nonlinear models.

Checked corecursive streams: Expressivity and completeness

Checked corecursive streams are a novel approach to stream definitions relying on a semantics of function application detecting cyclic calls, and a well-definedness check ensuring that access to an arbitrary index will always return a result. We extend the technique beyond the simple stream operators considered in previous work, notably by adding an interleaving combinator which has a non-trivial recursion scheme. We show that this actually enhances expressive power, since the interleaving operator cannot be expressed by the others, and that it is still possible to perform a sound and complete well-definedness check, through a symbolic computation which mimics access to an arbitrary index.

Bifurcation studies, chaotic pattern, phase diagrams and multiple optical solitons for the ([formula omitted])-dimensional stochastic coupled nonlinear Schrödinger system with multiplicative white noise via Itô calculus

The stochastic coupled nonlinear Schrödinger systems are very important equations which can be wildly used in the fields of the optical-fiber communications, nonlinear optics, plasma physics, ecological system, statistical mechanics and so on. This work mainly focuses on dynamical behavior, phase portraits, chaotic behavior and multiple optical solitons for the (2+1)-dimensional stochastic coupled nonlinear Schrödinger system with multiplicative white noise. Here, we analytically deduced bright solitons, dark solitons and periodic solutions through the bifurcation theory. Additionally, some other bounded traveling wave solutions which include Jacobi elliptic function solutions, trigonometric function solutions, rational function solutions, hyperbolic function solutions and solitary wave solutions are also obtained by using the symbolic computation as well as the complete discriminant system method. It is worth noting that we give the classification of all single traveling wave solutions at the same time. Finally, in order to further explore the propagation of the (2+1)-dimensional stochastic coupled nonlinear Schrödinger system in nonlinear optics, three-dimensional, two-dimensional, density graphs and contour graphs are also given.

Resonance Y-type soliton and hybrid solutions for a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma

In this paper, a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma is investigated. Via the Hirota method and symbolic computation, we work out some two-resonance Y-type soliton solutions as well as some hybrid solutions composed of the two-resonance Y-type solitons and solitons/breathers. Graphically, we display some two-resonance Y-type solitons. We present the interactions between the two-resonance Y-type soliton and one soliton, among the two-resonance Y-type soliton and two solitons, between the two-resonance Y-type soliton and first-order breather as well as among the two-resonance Y-type soliton and second-order breathers.

LSMMD-MA: scaling multimodal data integration for single-cell genomics data analysis.

Modality matching in single-cell omics data analysis-i.e. matching cells across datasets collected using different types of genomic assays-has become an important problem, because unifying perspectives across different technologies holds the promise of yielding biological and clinical discoveries. However, single-cell dataset sizes can now reach hundreds of thousands to millions of cells, which remain out of reach for most multimodal computational methods. We propose LSMMD-MA, a large-scale Python implementation of the MMD-MA method for multimodal data integration. In LSMMD-MA, we reformulate the MMD-MA optimization problem using linear algebra and solve it with KeOps, a CUDA framework for symbolic matrix computation in Python. We show that LSMMD-MA scales to a million cells in each modality, two orders of magnitude greater than existing implementations. LSMMD-MA is freely available at https://github.com/google-research/large_scale_mmdma and archived at https://doi.org/10.5281/zenodo.8076311.

Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods

The Schamel-Korteweg-de Vries (S-KdV) equation including a square root nonlinearity is very important pattern for the research of ion-acoustic waves in plasma and dusty plasma. As known, it is significant to discover the traveling wave solutions of such equations. Therefore, in this paper, some new traveling wave solutions of the S-KdV equation, which arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space, are constructed. For this purpose, the Bernoulli Sub-ODE and modified auxiliary equation methods are used. It has been shown that the suggested methods are effective and give different types of function solutions as: hyperbolic, trigonometric, power, exponential, and rational functions. The applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations. The results found in the paper are of great interest and may also be used to discover the wave sorts and specialities in several plasma systems.

Implementation of the Adams method for solving ordinary differential equations in the Sage computer algebra system

This work is devoted to the implementation and testing of the Adams method for solving ordinary differential equations in the Sage computer algebra system. The Sage computer algebra system has, to some extent, trivial means for numerical integration of ordinary differential equations, but at the same time, it is worth noting that this environment is convenient and practical for conducting computer experiments related to symbolic numerical calculations in it. The article presents the FDM package developed on the basis of the RUDN, which contains the developments of recent years, performed by M. D. Malykh and his students, for numerical integration of differential equations. In this package, attention is paid to the visualization of the calculation results, including the construction of various kinds of auxiliary diagrams, such as Richardson diagrams, as well as graphs of dependence, for example, the value of a function or step from a moment in time. The implementation of the Adams method will be considered from this package. In this article, this implementation of the Adams method will be tested on various examples of input data, and the method will also be compared with the Jacobi system. Exact and approximate values will be found and compared, and an estimate for the error will be obtained.

Resolviendo ecuaciones diferenciales ordinarias con Symbolic Math Toolbox™ (Matlab) y SymPy (Python)

This paper shows solutions of ordinary differential equations (EDOS) obtained by using two symbolic packages: Symbolic Math Toolbox™ (Matlab) and SymPy (Python). The basic instructions to obtain solutions of both packages are explained step by step, through a group of examples from a traditional ordinary differential equations course. Differential equations that are solved with methods such as: separable variables, linear equations, indeterminate coefficients, variation of parameters, power series, Laplace transform, and numerical solutions are included. By means of the symbolic computation carried out with these packages it is possible to obtain the solution of linear systems, as well as the visualization of the direction field of a differential equation or of a non-linear system of differential equations. The main contribution of this work consists in providing the reader with a practical guide that allows him to start the study of differential equations assisted by Symbolic Math Toolbox™ or SymPy. Among the benefits of using these computational tools in teaching and/or learning practices, it is shown how the use of symbolic or numerical computation saves us effort in the computation of tedious calculations; focusing attention on important ideas and concepts such as: the relationship between the mathematical model and its physical counterpart, asymptotic behavior and qualitative analysis of the solutions.

On a variable-coefficient AB system in a baroclinic flow: Generalized Darboux transformation and non-autonomous localized waves

In this paper, we focus our attention on a variable-coefficient AB system, which models the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we construct an N-fold generalized Darboux transformation (GDT) via symbolic computation, where N is a positive integer. Via the obtained GDT, several kinds of the non-autonomous localized waves, e.g., the multi-pole solitons, multi-pole breathers, rogue waves and their interactions, are investigated. We have selected four types of the variable coefficients, i.e., constant, linear about t, quadratic about t and trigonometric about t, where t is the normalized retarded time coordinate. We explore the influence of those selected variable coefficients on the non-autonomous localized waves. Moreover, we discuss the bound states among the non-autonomous multiple-pole solitons and one soliton, which exhibit the periodic attractions and repulsions between the adjacent solitons.

Multiple soliton solutions, lump, rogue wave and breather solutions of high dimensional equation for describing Rossby waves

In the past, we haven't paid much attention to higher-dimensional models, which are actually more consistent with the real atmosphere. In this manuscript, we derive a high-dimensional Kadomtsev-Petviashvili equation from a fluid system based on the Gardner-Morikawa transformation and the small parameter perturbation method, which is the first time to characterize the propagation process of Rossby waves in Large-scale atmospheric. The exact solutions such as soliton, lump, rogue wave and breather of this equation are achieved according to Hirota bilinear method and symbolic computation approach. Further we utilize the composite diagram of these solutions to observe the propagation properties of Rossby waves with assistance of Mathematica. The results enrich the analysis of Rossby waves in marine engineering and fluid dynamics.

Symbolic computation and Novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach

Symbolic computation and Novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach

Multi-type solitary wave solutions of Korteweg–de Vries (KdV) equation

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 [Formula: see text]. 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.

Normal Cayley digraphs of dihedral groups with CI-property

A Cayley (di)graph Cay(G,S) of a group G with respect to a set S ⊆ G is said to be normal if the image of G under its right regular representation is normal in the automorphism group of Cay(G,S), and is called a CI-(di)graph if for every T ⊆ G with Cay(G,S)≅ Cay(G,T), there is α∈ Aut(G) such that Sα = T. A finite group G is called a DCI-group or an NDCI-group if all Cayley digraphs or all normal Cayley digraphs of G are CI-digraphs, respectively, and is called a CI-group or an NCI-group if all Cayley graphs or all normal Cayley graphs of G are CI-graphs, respectively. Motivated by a conjecture proposed by Ádám in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It took about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order 2n is an NCI-group or an NDCI-group if and only if n = 2, 4 or n is odd. As a direct consequence, we have that if a dihedral group D2n of order 2n is a DCI-group then n = 2 or n is odd-square-free, and that if D2n is a CI-group then n = 2, 9 or n is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups. As a byproduct, we construct a non-CI Cayley graph of the dihedral group D8, but Holt and Royle [A census of small transitive groups and vertex-transitive graphs, J. Symbolic Comput. 101 (2020) 51-60] claimed that D8 is a CI-group by an algorithm there.

Multisolitons in the surface gravity waves and internal waves

In this paper, a five-order Korteweg–de Vries (KdV) equation is studied, which is used to describe the nonlinear phenomena in the fluids, especially those of the surface gravity waves and internal waves in the stratified fluids. (a) Via the symbolic calculation, this KdV equation cannot pass the Painlevé test without any constraint conditions. By virtue of the ansatz method, bell-shape and kink soliton solutions of this KdV equation are attained. (b) Via the bilinear method, multisoliton solutions of this KdV equation are obtained under some constraint conditions. Propagation and interaction of the multisoliton are discussed. Soliton interaction is elastic, that is to say, they have no effect on each other’s amplitude and speed except for phase shift. We hope that our results will be useful for experimental studies of surface gravity waves and internal waves since the coefficients of this KdV equation are all expressed in terms of physical constants, depths, and densities of the fluid.

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics.

Is Nonsymbolic Arithmetic Truly "Arithmetic"? Examining the Computational Capacity of the Approximate Number System in Young Children.

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.

Faithful Real-Time Animation of Parametrized (Semi-) Algebraic Expressions via Cylindrical Algebraic Decomposition

We present some current improvements, implemented in the software package GeoGebra Discovery, that combine symbolic computation and graphics algorithms to faithfully visualize (semi-)algebraic expressions. Our implementation allows fluid animation of set of (semi-)algebraic sets of dimension 1 in a desktop application or a web browser. We use the Tarski library to create a cylindrical algebraic decomposition of the input, and its plot2d command, which is processed further in GeoGebra Discovery, to provide the user with a familiar look and feel.

Invariants for neural automata

AbstractComputational modeling of neurodynamical systems often deploys neural networks and symbolic dynamics. One particular way for combining these approaches within a framework called vector symbolic architectures leads to neural automata. Specifically, neural automata result from the assignment of symbols and symbol strings to numbers, known as Gödel encoding. Under this assignment, symbolic computation becomes represented by trajectories of state vectors in a real phase space, that allows for statistical correlation analyses with real-world measurements and experimental data. However, these assignments are usually completely arbitrary. Hence, it makes sense to address the problem which aspects of the dynamics observed under a Gödel representation is intrinsic to the dynamics and which are not. In this study, we develop a formally rigorous mathematical framework for the investigation of symmetries and invariants of neural automata under different encodings. As a central concept we define patterns of equality for such systems. We consider different macroscopic observables, such as the mean activation level of the neural network, and ask for their invariance properties. Our main result shows that only step functions that are defined over those patterns of equality are invariant under symbolic recodings, while the mean activation, e.g., is not. Our work could be of substantial importance for related regression studies of real-world measurements with neurosymbolic processors for avoiding confounding results that are dependant on a particular encoding and not intrinsic to the dynamics.

Summation identities for the Kummer confluent hypergeometric function 1F1(a; b;z)

The role which hypergeometric functions have in the numerical and symbolic calculation, especially in the fields of applied mathematics and mathematical physics motivated research in this paper. In this note, a general formula for the sum, with the Kummer confluent hypergeometric function 1F1(a; b; z) is derived and given in terms of the function 2F2(a; b; z). The idea for this investigation comes from the theory of generalized Gauss-Rys quadrature formulas developed recently by (Milovanović, 2018) and (Milovanović and Vasović, 2022). Several results are obtained as special cases of the main result.