Rational Functions Research Articles

A rational function quantum equivalence relation on the set of all prime numbers

In this paper, we prove new results which make it possible to vastly simplify solution of the problem of Nathanson concerning the enlargement of support bases and thus of the corresponding semigroup supports of rational function solutions to the quantum functional equations arising from multiplication of quantum integers. In particular, we establish a certain quantum equivalence relation on the set of all prime numbers which encodes the symmetry of these solutions and thus solve a problem which simplify in crucial ways the criteria for the problem posed by Nathanson concerning extensions of the support bases of these solutions.

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

On general fiber product rings, Poincaré series and their structure

The present paper deals with the investigation of the structure of general fiber product rings R × T S , where R, S and T are local rings with common residue field. We show that the Poincaré series of any R-module over the fiber product ring R × T S is bounded by a rational function. In addition, we give a description of depth ( R × T S ) , which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over R × T S obtained, we provide certain cases of the Cohen-Macaulayness of R × T S and, in particular, we show that R × T S is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over R × T S are also established.

On quadrirational pentagon maps

Abstract We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps R : ( x , y ) ↦ ( u ( x , y ) , v ( x , y ) ) , where u , v are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang–Baxter maps from entwining pentagon maps.

Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves.

Twisted Mahler Discrete Residues

Abstract Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^{p})-g(x)$ for some rational function $g(x)$ and an integer $p> 1$. Here we develop a notion of $\lambda $-twisted Mahler discrete residues for $\lambda \in \mathbb{Z}$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $p^{\lambda } g(x^{p})-g(x)$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations.

Chiral Amphiphilic Au23 Cluster and Its Specific Recognition to Remote Di-amines.

The atomic precision of metal nanoclusters and variability of surface ligands pave the way for their rational design and functionalization, whereas the property strengthening in multiple ways has been long challenging. Herein, improved amphiphilicity, chirality, thermostability, and strong CPL (circularly polarized luminescence) properties have been accomplished by facile ligand exchange of [Au23(CHT)16]- with HCapt (HCHT and HCapt denote cyclohexanethiol and captopril). In addition, the obtained chiral [Au23(SR)16]- (short for [Au23(CHT)16-x(Capt)x]-) clusters show specific binding affinity to remote-diamines (such as arginine and single/double strand DNA), originating from the hydrogen bonding and Van der Waals interaction among the surface Capt ligands and the di-amine groups.

Periodic, quasi-periodic, chaotic waves and solitonic structures of coupled Benjamin-Bona-Mahony-KdV system

Abstract In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg-de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system's solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.

Proper Maps of Ball Complements & Differences and Rational Sphere Maps

Proper Maps of Ball Complements & Differences and Rational Sphere Maps

Optical dromions for M-fractional Kuralay equation via complete discrimination system approach along with sensitivity analysis and quasi-periodic behavior

This paper investigates new analytical wave solutions for the fractional Kuralay-IIB equations (FK-IIBE), including a new definition of the fractional derivative. This model works in disciplines such as ferromagnetic materials, optical fibers, wave mixing, and nonlinear optics. The integrable motion of space curves can be determined by our governing model. This research holds great importance in the area of optical fibers, exact and analytical solitons solution. The polynomial method’s complete discrimination system (CDSPM) is used to identify a variety of exact and analytical soliton solutions such as periodic, hyperbolic, rational, Jacobian elliptic (JE), and trigonometric functions. Additionally, we also convert the JE function into a solitary wave (SW) solution. To provide a comprehensive understanding of the dynamic aspects of the system, we also address bifurcation points, quasi-periodic behavior, sensitivity analysis, Poincarè maps, time series profile, and critical solution conditions. The research offers straightforward, efficient algorithms that can consistently and effectively address FK-IIBE. Machine learning technologies are also utilized to satisfy the criteria defined by dynamic analysis. Python regression learner tools are used for reverse engineering to analyze the mathematical model’s equilibrium based on parameter intervals and thresholds determined by dynamical analysis.

Harmonic flow-field representations of quantum bits and gates

We describe a general procedure for mapping arbitrary n-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multiqubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and nonstandard computation. Published by the American Physical Society 2024

Pole dynamics and an integral of motion for multiple SLE(0)

We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation. These results are entirely self-contained and have no reliance on probabilistic objects, but make use of an integral of motion for the Loewner chain that is motivated by ideas from conformal field theory. We also show that the dynamics of the driving functions are a special case of the Calogero–Moser integrable system, restricted to a particular submanifold of phase space carved out by the Lax matrix. Our approach complements a recent result of Peltola and Wang, who showed that the real locus is the deterministic κ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa \\rightarrow 0$$\\end{document} limit of the multiple SLE(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\kappa )$$\\end{document} curves.

LinApart: Optimizing the univariate partial fraction decomposition

We present LinApart, a routine designed for efficiently performing the univariate partial fraction decomposition of large symbolic expressions. Our method is based on an explicit closed formula for the decomposition of rational functions with fully factorized denominators. We provide an implementation in the Wolfram Mathematica language, which can lead to very significant performance gains over the built-in Apart command. Furthermore, a C language library implementing the core functionality and suitable for interfacing with other software is also provided. Both codes are made available at https://github.com/fekeshazy/LinApart.

Scaled Global Operators and Fueter Variables on Non-zero Scaled Hypercomplex Numbers

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers Ht,t∈R∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_t,\\, t\\in \\mathbb {R}^*$$\\end{document}, of which the H-1=H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_{-1}=\\mathbb {H}$$\\end{document} is the space of quaternions and H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_{1}$$\\end{document} is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on Ht\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_t$$\\end{document}. Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function.

New periodic solutions and solitary wave solutions for the time-fractional differential equations

In this paper, we obtain many different types of exact solutions to the time-fractional Klein–Gordon equation and the time-fractional generalized Hirota-Satsuma coupled KdV system by using the modified rational function approach. Some new solutions such as the kink-periodic solution, the anti-kink-periodic solution and the concave-convex-periodic solution are constructed. Furthermore, the kink and the singular kink waves, the bell shaped soliton and the singular soliton solutions of the two equations also are found. Some numerical simulations are presented, these works can effectively reflect the propagation phenomena of time-fractional nonlinear systems, and also enable us to understand time-fractional nonlinear physical phenomena more clearly.

Submaximal force–velocity relationships during mountain ultramarathon: Data from the field

ABSTRACT This study presents a novel method for evaluating the submaximal velocity-force (V(F)) relationship in mountain ultramarathon races using crowdsourced data from Strava.com. The dataset includes positional data from 408 participants of the 171-km UTMB® 2023 race (9,850-m D+). The race was divided into 100-m segments. The mean net propulsive force and velocity were computed for each segment to describe the submaximal V(F) relationship as a rational function of three parameters. F1: propulsive force at 1 m · s−1; V0: theoretical maximum velocity on flat terrain; C: curvature parameter (the lower C, the more linear the V(F) relationship). The V(F) profile parameters were found to be F1 = 1.80 ± 0.33 N · kg−1, V0 = 2.36 ± 0.42 m · s−1, and C = 0.66 ± 1.81, with good independence between the parameters within a group of homogeneous performance. The best athletes had the highest F1, V0, and C values. V(F) parameters were affected by fatigue during the race, with decreases of 20.9%, 32.0%, and 59.8% between the first and second parts of the race respectively. These findings suggest that the V(F) relationship is an interesting original approach for studying performance and fatigability during mountain ultra-endurance races.

Novel complexiton, rational wave, multi-lumps and the kink solitary wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In this exploration, we aim to seek a number of new exact solutions to the new (3+1)-dimensional integrable fourth-order nonlinear equation, which is widely used to describe the shallow water waves. Employing the Cole-Hopf transformation, we develop its bilinear form. Then, taking advantage of the ansatz function method, a new functional form is utilized to probe the singular complexiton solutions. Based on which, the non-singular complexiton solutions are derived by imposing the constraint conditions. In addition, we find the rational wave solutions and multi-lumps solutions wielding the rational function method and new homoclinic method respectively. At the end, we investigate the kink solitary wave solutions using the variational approach that is based on the variational principle and Ritz method. Meanwhile, the Hamiltonian of the system is also elaborated. Correspondingly, the graphic descriptions of the extracted results are presented to unfold their dynamic behaviors through Maple. As we all know, the findings of this paper are firstly reported and can enlarge the exact solutions of the considered PDE.

Extracting non-equilibrium material properties in visco-hyperelastic polymers based on analytical stress solution

This study extracts non-equilibrium material properties in the visco-hyperelastic polymers using an analytical stress solution. At first, the analytical stress solution of visco-hyperelastic polymers is obtained from finite-strain creep test data based on the generalized Maxwell’s rheological model. The solution uses the low-computational-cost exponential generalized pseudospectral method and analytically brings the stress components in terms of time and unknown non-equilibrium material properties of Maxwell’s elements. The solution invokes the Ogden hyperelastic model for the elastic element of the generalized Maxwell’s model. The exponential generalized pseudospectral method, based on exponential–rational Lagrange functions suitable for the semi-infinite time interval [Formula: see text] and the rational Chebyshev roots as the collocation points, is applied to derive the stress components. The non-equilibrium properties are derived by minimizing a residual, which is obtained by making the derivatives of the residual in terms of the non-equilibrium properties as zero. Notably, the proposed analytical method requires a small number of exponent–rational Lagrange basis functions and collocation points. Besides, the proposed method does not require integration, discretization, or linearization. Two case studies are considered: the moderate-stretched ETFE polymer with the neo-Hookean model as the elastic element and the large-stretched rubber with the six-parameter Ogden hyperelastic model as the elastic branch. The results are validated using uniaxial creep modeling in ANSYS and available reports in the literature.

Exploring Soliton Solutions and Chaotic Dynamics in the (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation: A Generalized Rational Exponential Function Approach

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations.

The Extended Weierstrass Transformation Method for the Biswas–Arshed Equation with Beta Time Derivative

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions.