Puiseux Series Research Articles

Valuations and orderings on the real Weyl algebra

The first Weyl algebra A1(k) over a field k is the k-algebra with two generators x, y subject to [y,x] = 1 and was first introduced during the development of quantum mechanics. In this article, we classify all valuations on the real Weyl algebra A1(R) whose residue field is R. We then use a noncommutative version of the Baer-Krull theorem from real algebraic geometry to classify all orderings on A1(R) As a byproduct of our studies, we settle two open problems in noncommutative valuation theory. First, we show that not all valuations on A1(R) with residue field R extend to a valuation on a larger ring R[y;δ], where R is the ring of Puiseux series, introduced by Marshall and Zhang in [16], with the same residue field, and characterize the valuations that do have such an extension. Second, we show that for valuations on noncommutative division rings, Kaplansky’s theorem that extensions by limits of pseudo-Cauchy sequences are immediate fails in general.

Laurent Series and Puiseux Series in Maple

Let K be an algebraically closed field of characteristic zero. The field of fractions of the ring of formal multivariate power series over K, is called the field of formal multivariate Laurent series. In this document, we follow the ideas introduced by Monforte and Kauers in their paper Formal Laurent Series in Several Variables. Our objective is to report on a first implementation of formal multivariate Laurent series inside of Maple, and explain the challenges we had to overcome. In order to accomplish this goal, we make use of the already existing MultitivariatePowerSeries package, and its lazy evaluation scheme. In particular, we expose our ideas for adding and multiplying Laurent series with support inside different cones, where the support of a Laurent series is the set of all exponents of all non-zero monomials of our series. We also describe our biggest challenge, how to invert a Laurent series. Unfortunately, this problem cannot be completely solved in a lazy evaluation context. We describe some situations where we can solve the problem completely; our approach for the cases that fall outside of these situations; and how we let the user customize this approach, trading off between speed and the likelihood of an incorrect result.
 The algebraic closure of the field of formal multivariate Laurent series is call the field of formal multivariate Puiseux series. As an extension of our current work, we also present our ideas for an implementation of a multivariate Puiseux series object inside of Maple.

A Singularly Perturbed Dirichlet Problem for a Ring With Singular Boundaries

Uniform asymptotic expansions of small-parameter solutions of bisingular Dirichlet problems for a ring with any degree of accuracy are constructed by a generalized method of boundary functions and a small-parameter method. The investigated problems have two features: equations with a small parameter in the first derivatives and external solutions simultaneously have increasing features on the boundaries of the domain, i. e. limit equations have singularities at the same time on both boundaries of the ring. Formal asymptotic expansions are based on the maxima principle and the method of differential inequalities. The resulting asymptotic series are represented by the Puiseux series. The main term of the asymptotic expansions of the solutions has a negative fractional power with respect to the small parameter.

An effective algorithm for computing the asymptotes of an implicit curve

In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n-dimensional space.

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

A Novel Solution Approach for Time-Fractional Hyperbolic Telegraph Differential Equation with Caputo Time Differentiation

In the current analysis, a specific efficient and applicable novel solution approach, based on a fractional power series technique and Laplace transform operator, is considered to predict certain accurate approximate solutions (ASs) for a time-fractional hyperbolic telegraph equation by aid of time-fractional derivatives in a Caputo sense. The solutions are obtained in a fractional Maclurian series formula by solving the original problem in the Laplace space aided by a limit concept having fewer small iterations than the classical fractional power series technique. To confirm applicability and feasibility of the proposed approach, three appropriate initial value problems are considered. Consequently, some simulations of gained outcomes are numerically and graphically implemented to support the effect of the fractional-order parameter on the geometric behavior of the obtained solutions. In addition, graphical representations are also fulfilled to verify the convergence analysis of the fractional series solutions of the classical solution. The proposed technique is therefore proposed to be a straightforward, accurate and powerful approach for handling varied time-fractional models in various physical phenomena.

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψ Γ n α + 1 , where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the n th term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM.

On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of ⁎-palindromic matrix polynomials

We study the eigendecompositions of para-Hermitian matrices H(z), that is, matrix-valued functions that are analytic and Hermitian on the unit circle S1⊂C. In particular, we fill existing gaps in the literature and prove the existence of a decomposition H(z)=U(z)D(z)U(z)P where, for all z∈S1, U(z) is unitary, U(z)P=U(z)⁎ is its conjugate transpose, and D(z) is real diagonal; moreover, U(z) and D(z) are analytic functions of w=z1/N for some positive integer N, and U(z)P is the so-called para-Hermitian conjugate of U(z). This generalizes the celebrated theorem of Rellich for matrix-valued functions that are analytic and Hermitian on the real line. We also show that there exists a decomposition H(z)=V(z)C(z)V(z)P where C(z) is pseudo-circulant, V(z) is unitary and both are analytic in z. We argue that, in fact, a version of Rellich's theorem can be stated for matrix-valued function that are analytic and Hermitian on any line or any circle on the complex plane. Moreover, we extend these results to para-Hermitian matrices whose entries are Puiseux series (that is, on the unit circle they are analytic in w but possibly not in z). Finally, we discuss the implications of our results on the singular value decomposition of a matrix whose entries are S1-analytic functions of w, and on the sign characteristics associated with unimodular eigenvalues of ⁎-palindromic matrix polynomials.

Application of Fractional Power Series Method in Solving Fractional Differential Equations

Application of Fractional Power Series Method in Solving Fractional Differential Equations

Puiseux Series Expansion for Eigenvalue of the Generalized Friedrichs Model with the Perturbation of Rank One

Puiseux Series Expansion for Eigenvalue of the Generalized Friedrichs Model with the Perturbation of Rank One

Exact Solutions of Some Fractional Power Series

Exact Solutions of Some Fractional Power Series

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.

Fuzzy generalized fractional power series technique for simulating fuzzy fractional relaxation problem

Fuzzy generalized fractional power series technique for simulating fuzzy fractional relaxation problem

Analysis of a Fractional-Order COVID-19 Epidemic Model with Lockdown.

The outbreak of the coronavirus disease (COVID-19) has caused a lot of disruptions around the world. In an attempt to control the spread of the disease among the population, several measures such as lockdown, and mask mandates, amongst others, were implemented by many governments in their countries. To understand the effectiveness of these measures in controlling the disease, several mathematical models have been proposed in the literature. In this paper, we study a mathematical model of the coronavirus disease with lockdown by employing the Caputo fractional-order derivative. We establish the existence and uniqueness of the solution to the model. We also study the local and global stability of the disease-free equilibrium and endemic equilibrium solutions. By using the residual power series method, we obtain a fractional power series approximation of the analytic solution. Finally, to show the accuracy of the theoretical results, we provide some numerical and graphical results.

A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1 + 2)-physical models

Abstract In the present article, we geometrically and analytically examine the mutual impact of space-time Caputo derivatives embedded in (1 + 2)-physical models. This has been accomplished by integrating the residual power series method (RPSM) with a new trivariate fractional power series representation that encompasses spatial and temporal Caputo derivative parameters. Theoretically, some results regarding the convergence and the error for the proposed adaptation have been established by virtue of the Riemann–Liouville fractional integral. Practically, the embedding of Schrödinger, telegraph, and Burgers’ equations into higher fractional space has been considered, and their solutions furnished by means of a rapidly convergent series that has ultimately a closed-form fractional function. The graphical analysis of the obtained solutions has shown that the solutions possess a homotopy mapping characteristic, in a topological sense, to reach the integer case solution where the Caputo derivative parameters behave similarly to the homotopy parameters. Altogether, the proposed technique exhibits a high accuracy and high rate of convergence.

Fractional Power Series Method for Solving Fractional Differential Equations

Fractional Power Series Method for Solving Fractional Differential Equations

Semi-analytic solution of time-fractional Korteweg-de Vries equation using fractional residual power series method

In this paper, we have solved the non-linear Korteweg-de Vries equation by considering it in time-fraction Caputo sense and offered intrinsic properties of solitary waves. The fractional residual power series method is used to obtain the approximate solution of the aforesaid equation and compared the obtained results with Adomian Decomposition Method. Obtained results are efficient, reliable, and simple to execute on most of the non-linear fractional partial differential equations, which arise in various dynamical systems.

Differential equations for the closed geometric crystal chains

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is an extended Lotka–Volterra type system, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the continuous time Lax equations from a discrete time one are described in detail, where a novel method of taking a continuum limit by assuming asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series expansions is used.

Analytical Solution of Nonlinear Fractional Gradient-Based System Using Fractional Power Series Method

This paper adapted a reliable treatment technique, called the fractional residual power series, to the fractional gradient-based system in solving a class of nonlinear programming model in Caputo’s sense. The gradient-based system has been constructed to transform the nonlinear programming problem with equality constraints to unconstrained optimization problem, and then the fractional residual power series method is implemented to obtain the essential behavior of underlying problem. The proposed methods have been applied effectively to produce optimal solution in rapidly convergent fractional series representations without linearization, or any limitations. To confirm the performance of the proposed methods, some optimization problems are tested. Further, numerical comparisons with other existing methods are also given. The results exhibit that the FRPS method is easy, simple, effective, and fully compatible with the complexity of such models.

Degenerated Calabi–Yau varieties with infinite components, moduli compactifications, and limit toroidal structures

For any degenerating Calabi–Yau family, we introduce a new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi–Yau varieties, parametrized by the rational points of the Kontsevich–Soibelman’s essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications (Sect. 3), locally modelled on limit toric varieties (Sect. 2.4), the way to attach a tropicalized family to a given Calabi–Yau family (Sect. 4), which are weakly related to each other.