Rational Functions Research Articles

Tailoring Spectral Response of Grating-Assisted Co-Directional Couplers with Weighting Techniques and Rational Transfer Functions: Theory and Experiment

This work addresses the tailoring spectral response of grating-assisted co-directional couplers (GADCs) in the context of wavelength filtering for fiber-to-the-home (FTTH) applications. Design methods for spectral response engineering by means of coupling profile apodization-type weighting techniques and also more advanced rational transfer functions fitting a predefined spectral window template are presented. Modeling results based on coupled mode theory are then applied for the design and experimental fabrication of InGaAsP/InP GADCs targeting 1.3+/1.3− µm diplexer application in FTTH access networks. The experimental results are found to be in good agreement with the modeling predictions. The design tools presented are quite general and can be easily adapted to other technology platforms, such as silicon photonics for the use of GADCs as add-drop wavelength division multiplexers. The field of parity–time symmetry is another avenue where these types of gain–loss-assisted GADCs as active components are of interest for switching applications, and the design methods presented here may find utility.

Entire maps with rational preperiodic points and multipliers

Abstract Given a number field $\mathbb {K} \subset \mathbb {C}$ that is not contained in $\mathbb {R}$ , we prove the existence of a dense set (with respect to the topology of local uniform convergence) of entire maps $f \colon \mathbb {C} \rightarrow \mathbb {C}$ whose preperiodic points and multipliers all lie in $\mathbb {K}$ . This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb {Q}$ .

Nonlinear wave dynamics of (1+1)- dimensional conformable coupled nonlinear Higgs equation using modified ( G′G2)-expansion method

Abstract The conformable coupled nonlinear Higgs equation has been analyzed here using the modified \((\frac{G'}{G^2})\)-expansion method. This equation describes the dynamics of conserved scalar nucleons interacting with neutral scalar mesons. Accordingly, new exact solutions have been obtained. Functions derived from trigonometry, hyperbolic operations, and rational functions are used to define the traveling waveform responses. These solutions are illustrated with various interesting figures for different parameters. Additionally, physical interpretations of these results are provided. The solutions uncovered in this study are more thorough, wide-ranging, and, in some cases, novel, as they have not been discovered in previous studies. {\color{red}These solutions may be useful in understanding nonstandard interactions or forces and potentially addressing phenomena beyond the standard model.} The implemented strategies are efficient, align with computer algebraic expressions, and provide a sufficient number of diverse wave solutions.

Relative Openness of the Gabor Frame Set on the Hyperbolas

We prove that for the functions of the form g(x)=h(x)+Cx+i, where h belongs to the continuous Wiener algebra W0, the intersection of the Gabor frame set Fg with every hyperbola {α,β>0∣αβ=c} 0 \\mid \\alpha \\beta = c\\}$$\\end{document}]]> is open in the relative topology. In particular, this applies to all rational functions g. We also show that the same phenomenon occurs for the functions from the Wiener algebra W which are discontinuous at one point.

Shape preserving monotonic and convex data interpolation using rational cubic ball functions

In this study, a rational cubic Ball function has been used to preserve the shape of monotonic and convex data. Conditions for shape preservation were drawn from the data and imposed on the free parameters of the interpolant function in such a way as to preserve the shape of the data. The interpolant is C1, which is continuous and visually pleasant function. The outputs of a number of numerical examples are presented.

A time-domain finite element formulation of the equivalent fluid model for the acoustic wave equation

Sound-absorptive materials such as foam can be described by the equivalent fluid (EF) model. The homogenized fluid’s acoustic behavior is thereby described by complex-valued, frequency-dependent acoustic material parameters. When transforming the acoustic wave equation for the EF model from the frequency domain to the time domain, convolution integrals arise. The auxiliary differential equation (ADE) method is used to circumvent the direct calculation of these convolution integrals. The wave equation and the coupled set of ordinary ADEs are solved in the time domain using the finite element (FE) method. The approach relies on approximating the complex-valued frequency response functions of the inverse equivalent bulk modulus and density by a sum of rational functions consisting of real and complex poles. The order of the rational function approximation defines the number of additionally introduced auxiliary variables per nodal degree of freedom. The presented FE formulation includes a narrow-band non-reflecting boundary condition (NRBC) for normal incidence. The implementation in openCFS shows optimal temporal and spatial convergence for a semi-infinite duct based on the analytic plane wave solution for harmonic excitation. The simulation of a pressure pulse propagating in an infinite EF domain with a scatterer demonstrates the capability for multidimensional, actual transient problems.

On existence of primitive pairs (ξ + ξ −1, f(ξ)) with arbitrary traces

Let F q n be a finite extension of the finite field F q of degree n, where q is a prime power and n is a positive integer. Suppose a , b ∈ F q . In this article we search for pairs (q, n) such that there exists a primitive pair ( ξ + ξ − 1 , f ( ξ ) ) with Tr F q n / F q ( ξ ) = a and Tr F q n / F q ( ξ − 1 ) = b , where ξ is a non-zero element of F q n and f is any non-exceptional rational function in F q n ( x ) .

Novel insights into high-order dispersion and soliton dynamics in optical fibers via the perturbed Schrödinger–Hirota equation

This study offers a comprehensive analysis of the Perturbed Schrödinger -Hirota Equation (PSHE), crucial for understanding soliton dynamics in modern optical communication systems. We extended the traditional Nonlinear Schrödinger Equation (NLSE) to include higher-order nonlinearities and spatiotemporal dispersion, capturing the complexities of light pulse propagation. Employing the modified auxiliary equation method and Adomian Decomposition Method (ADM), we derived a spectrum of exact traveling wave solutions, encompassing exponential, rational, trigonometric, and hyperbolic functions. These solutions provide insights into soliton behaviors across diverse parameters, essential for optimizing fiber optic systems. The precision of our analytical solutions was validated through numerical solutions, and we explored modulation instability, revealing conditions for soliton formation and evolution. The findings have significant implications for the design and optimization of next-generation optical communication technologies.

A quantitative version of Northcott's theorem on points of bounded height: The function field case

AbstractLet be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let denote an algebraic closure of . For given integers and we count points in projective space with absolute logarithmic height and generating an extension of degree over . Specifically, we derive an asymptotic estimate for the number of such points as when and orders of growth for the number of such points when .

High-Accuracy Solutions to the Time-Fractional KdV–Burgers Equation Using Rational Non-Polynomial Splines

A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in 2D and 3D graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.

Dynamical insights into bifurcation, chaos and solitons in the space-time fractional symmetric regularized long wave equation

Abstract The nonlinear conformable time-fractional Symmetric Regularized Long Wave (SRLW) equation is a significant model in physics, particularly for describing ion acoustic and space charge waves with weak nonlinearity. In this study, we apply the modified Sardar sub equation method and the modified extended auxiliary equation method to solve the SRLW equation. We effectively manage the fractional derivatives in the equation by employing Jumarie’s modified Riemann-Liouville derivative. Various types of soliton solutions, including kink, periodic, dark, bright, and singular solitary waves, are obtained in forms such as rational, hyperbolic, trigonometric, and exponential functions. A comparative analysis of the methods and the results is conducted, along with an exploration of fractional derivative effects by varying their values. The study also includes 2D and 3D plots that show how the solutions behave over time. It demonstrates that the methods used can be applied to other nonlinear models in mathematical physics. The research looks closely at the model’s behavior, including bifurcation, chaos, and stability. Phase portrait analysis at key points shows changes in the system’s behavior, and adding an external periodic force leads to chaotic patterns. The stability analysis proves that these methods
are reliable for studying phase portraits and soliton solutions in different nonlinear systems.

Galois groups of random polynomials over the rational function field

AbstractFor a fixed prime power and natural number , we consider a random polynomial with drawn uniformly and independently at random from the set of all polynomials in of degree . We show that with probability tending to 1 as the Galois group of over is isomorphic to , where is cyclic, and are small quantities with a simple explicit dependence on . As a corollary we deduce that as . Thus, we are able to overcome the versus ambiguity in the most natural restricted coefficients random polynomial model over , which has not been achieved over so far.

The propagation of hyper-geometric optical soliton waves in plasma dynamics for the integrable Zhanbota-IIA equation

This research utilizes the new auxiliary equation method to investigate solitary wave patterns in the integrable Zhanbota-IIA equation, a model relevant to phenomena like tidal waves and tsunamis. This method simplifies complex nonlinear-coupled partial differential equations into ordinary differential equation through a traveling wave transformation. Previous studies have not achieved such solutions. The main aim is to enhance comprehension of the integrable Zhanbota-IIA equation behavior under various conditions. Prior to this study, there does not exist any study in which such kinds of solutions are discussed by using the new auxiliary equation method. The new auxiliary equation method is utilized to derive analytical solutions for the underlying model equation. These solutions are expressed in the form of hyperbolic, trigonometric, exponential, and rational functions such as dark, bright, periodic, dark–bright, smooth topological with high peaks, dark-singular, bright singular and periodic singular. The key advantage of this method compared to others lies in its capacity to yield more comprehensive solutions with certain flexible parameters. To illustrate the dynamic structures of these solutions, 3D and 2D graphs are employed with suitable parameter values. This study explores soliton behavior and its telecommunication applications, integrating mathematics, computer science, and physics for technological advancements.

The Use of Educational Technology (Ed Tech) to Improve Student’s Learning Rational Functions: Conceptual and Procedural Understanding

Abstract The purpose of this study was to ascertain how teaching rational function to students using the GeoGebra and cycle model affected their conceptual and procedural comprehension. To achieve the purpose, the study used a quasi-experimental pre-posttest design with a quantitative methodology. Thirty four learners in grade eleven in Ethiopia school comprised the study’s sample, of which sixteen were in the experimental (eg) and eighteen were in the control groups (cg). The data obtained from pretest and post-test were analyzed using spss v.20. The study showed that difference between the eg’s and cg’s achievement was statistically significant, with the eg’s effect size being 0.38 and the cg’s being 0.544 following procedural and conceptual interventions, respectively. The study advise Ethiopian schools should embrace GeoGebra software in conjunction with the three theories through a continuous special capacity building approach, keeping in mind the software’s limitations and constraints as these results are in line with international research.

A Characterization of Rational Functions

We give a characterization of rational functions among meromorphic functions in the complex plane, which extends a characteristic of polynomials and also the Fundamental Theorem of Algebra to rational functions.

Propagation of Non-stationary Skew-Symmetric Waves from a Spherical Cavity in a Porous-elastic Half-space

Problems of propagation and diffraction of non-stationary waves in porous-elastic mediums are of great theoretical and practical importance in such fields of science and technology as geophysics, seismic exploration of minerals, seismic resistance of structures, and many others. The work considers the problem of propagation of non-stationary skew-symmetric waves from a spherical cavity in a porous-elastic half-space saturated with liquid. To solve the problem, the integral Laplace transform in dimensionless time and the method of incomplete separation of variables were used. In the space of Laplace images in time, known and unknown functions are expanded into Gegenbauer polynomials. The problem is reduced to solving an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Recurrence relations for the coefficients of the series and initial conditions for them are obtained, which makes it possible to obtain a solution to the infinite system without using the reduction method. Recurrence relations make it possible to determine the coefficients of a series in the form of rational functions, which makes it possible to calculate their originals using the theory of residues. In image space, formulas are obtained for the coefficients of the series of components of the displacement vector and stress tensor. Numerical experiments were carried out, the results of which are presented in the form of graphs. The results obtained can be used in geophysics, seismology, and design organizations during the construction of structures, as well as in the design of underground reservoirs.

Rational function solutions of higher‐order dispersive cubic‐quintic nonlinear Schrödinger dynamical model and its applications in fiber optics

The study explores a series of cubic‐quintic nonlinear Schrödinger equation with higher‐order dispersive characteristics. This equation is also a fundamental equation in nonlinear physics that is used to depict the dynamics of femtosecond light pulses propagating through a medium with a nonlinearity profile characterized by a parabolic function. Symbolic computation is utilized, and the double ‐expansion technique is applied to investigate the mathematical characteristics of this equation. Novel solitons and rational function solutions in various forms of the high‐order dispersive cubic‐quintic nonlinear Schrödinger equation are derived. These solutions have applications in engineering, nonlinear physics and fiber optics, providing insights into the physical nature of wave propagation in dispersive optics media. The results obtained form a basis for understanding complex physical phenomena in the described dynamical model. The computational approach employed is demonstrated to be straightforward, versatile, potent, and effective. Additionally, the presented solutions showcase various intriguing patterns, including kink‐type periodic waves, combined bright‐dark periodic waves, multipeak solitons, and breather‐type waves. This diverse set of solutions contributes to the interpretation of the dynamical model, illustrating its complexity. Moreover, the simplicity and effectiveness of our computational technique make it applicable to solving similar models in physics and other fields of applied science.

Introducing and analyzing a new combined version of the unstable Schrödinger equations with strong and weak stability effects

In the literature, two types of unstable nonlinear Schrodinger equations have been independently developed and studied. Each was derived by incorporating either a self-effect term or a time-space dispersion term into the standard nonlinear Schrodinger equation. Both models describe the time evolution of disturbances in unstable media. The primary contribution of this work is the combination of these two types into a single, new unstable version of the nonlinear Schrodinger equation. This new model is analyzed using two effective methods: the rational sine-cosine and the rational sinh-cosh functions. Additionally, a comparison test of the embedded unstable terms is conducted to assess their respective impacts on the stability of the Schrodinger model. Finally, graphical analyses, including 2D and 3D plots, are performed to validate the study’s findings.

Investment Market Trend Forecasting and Strategy Optimization based on Big Data Analytics

The financial market is an important part of the global economy, and there exists a certain special internal regularity in its changes. With the rapid development of big data technology, the huge amount of data in the financial market provides new opportunities for the prediction of investment trends and strategy optimization. Based on big data analytics technology, combined with machine learning models, the study explores how to construct investment market trend prediction models through effective data preprocessing and feature engineering, and evaluates and compares the performance of different prediction models. How to accurately predict the direction of the financial market will be an important reference value for relevant practitioners as well as researchers. Traditional financial time series research mainly focuses on linear trend fitting based on statistical knowledge, however, the trend of the financial market is not limited to pure mathematical and rational function curves, the financial market is affected by a combination of various factors from all walks of life. Big data analytics has significant application potential in the investment market, providing investors with more scientific and accurate decision support.

Formulas and Finite Sums Covering Beta-type Rational Functions and Euler-Frobenius-type Polynomials

The aim of this article is to derive some novel formulas and finite sums covering Stirling type numbers, the Frobenius Euler numbers and polynomials, the beta-type rational functions, and combinatorial numbers with the help of both generating functions and their functional equations, and also some special identities associated with special numbers and polynomials.