Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

Campo-Montalvo, Elena; Sevilla, Marian Fernández de; Magdalena‐Benedito, Rafael; Pérez–Díaz, Sonia

doi:10.3390/sym15010069

Cited by 3 publications

(2 citation statements)

References 36 publications

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“…The applications of algebraic curves encompass several domains in the mathematical sciences and in the computational sciences [12,13]. There are interplays between algebraic curves and the concepts of topology that determine various interesting algebraic as well as topological properties [12]. The various concepts of topology can be applied to the Bézout theorem to gain deeper understandings.…”

Section: Motivationsmentioning

confidence: 99%

A Topological Approach to the Bézout’ Theorem and Its Forms

Bagchi

2023

Symmetry

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The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties.

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Section: Motivationsmentioning

confidence: 99%

A Topological Approach to the Bézout’ Theorem and Its Forms

Bagchi

2023

Symmetry

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“…This allows flexibly control of the accuracy of computations and identification of zero coefficients even with numerical integration. CAS are intensively used in research in various fields of mathematics [32][33][34][35][36][37], physics [31,[38][39][40][41][42], engineering [43], education [44], and other areas.…”

Section: Introductionmentioning

confidence: 99%

Computational Technology for the Basis and Coefficients of Geodynamo Spectral Models in the Maple System

Vodinchar

Feshchenko

2023

Mathematics

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Spectral models are often used in the study of geodynamo problems. Physical fields in these models are presented as stationary basic modes combinations with time-dependent amplitudes. To construct a model it is necessary to calculate the modes parameters, and to calculate the model coefficients (the Galerkin coefficients). These coefficients are integrals of complex multiplicative combinations of modes and differential operators. The paper proposes computing technology for the calculation of parameters, the derivation of integrands and the calculation of the integrals themselves. The technology is based on computer algebra methods. The main elements for implementation of technology in the Maple system are described. The proposed computational technology makes it possible to quickly and accurately construct fairly wide classes of new geodynamo spectral models.

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Computational Technology for Shell Models of Magnetohydrodynamic Turbulence Constructing

Vodinchar,

Feshchenko

2024

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The paper discusses the computational technology for constructing one type of small-scale magnetohydrodynamic turbulence models – shell models. Any such model is a system of ordinary quadratic nonlinear differential equations with constant coefficients. Each phase variable is interpreted in absolute value as a measure of the intensity of one of the fields of the turbulent system in a certain range of spatial scales (scale shell). The equations of any shell model must have several quadratic invariants, which are analogues of conservation laws in ideal magnetohydrodynamics. The derivation of the model equations consists in obtaining such expressions for constant coefficients for which the predetermined quadratic expressions will indeed be invariants. Derivation of these expressions «manually» is quite cumbersome and the likelihood of errors in formula transformations is high. This is especially true for non-local models in which large-scale shells that are distant in size can interact. The novelty and originality of the work lie in the fact that the authors proposed a computational technology that allows one to automate the process of deriving equations for shell models. The technology was implemented using computer algebra methods, which made it possible to obtain parametric classes of models in which the invariance of given quadratic forms is carried out absolutely accurately – in formula form. The determination of the parameter values in the resulting parametric class of models is further carried out by agreement with the measures of the interaction of shells in the model with the probabilities of their interaction in a real physical system. The idea of the described technology and its implementation belong to the authors. Some of its elements were published by the authors earlier, but in this work, for the first time, its systematic description is given for models with complex phase variables and agreement of measures of interaction of shells with probabilities. There have been no similar works by other authors previously. The technology allows you to quickly and accurately generate equations for new non-local turbulence shell models and can be useful to researchers involved in modeling turbulent systems.

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Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

Cited by 3 publications

References 36 publications

A Topological Approach to the Bézout’ Theorem and Its Forms

A Topological Approach to the Bézout’ Theorem and Its Forms

Computational Technology for the Basis and Coefficients of Geodynamo Spectral Models in the Maple System

Computational Technology for Shell Models of Magnetohydrodynamic Turbulence Constructing

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