Rational Functions Research Articles

Exploring solitary wave solutions of extended Sakovich equations: A focus on nonlinear science

This aim of this work is to apply three proposed mathematical methods, namely, enhanced simple equation, [Formula: see text]-expansion and modified F-expansion to investigate solitary wave solutions for the two recently developed extended Sakovich equations in the context of [Formula: see text]- and [Formula: see text]-dimensional structures. The equations under this study belong to the category of Korteweg–de Vries (KdV) equations which are widely acknowledged as important components of fluid dynamics. Many scientific fields, including mathematics, physics, soliton theory, plasma physics, biology, chemistry, and nonlinear processes can benefit from the use of these equations. The derived solutions are in the form of Trigonometric, hyperbolic, exponential and rational functions. The Mathematica 13 computational software is used to represent some solutions graphically in two- and three-dimensional for the physical phenomena of concern models. Therefore, our work’s inventiveness is demonstrated by the application of many types of new solutions and newly employed creative ways. This facilitates additional research into nonlinear models that realistically capture important physical processes in daily life.

Dimensionless Machine Learning: Dimensional Analysis to Improve LSSVM and ANN models and predict bearing capacity of circular foundations

This research focuses on developing hybrid intelligence techniques to predict the bearing capacity of circular foundations using the most effective parameters. In this regard, a database involving 968 test circular foundations involving different rock properties, soil characteristics, and foundation radius has been prepared by laboratory tests for training and testing the new model presented in this study. For adequately considering various factors, the several parameters of depth (D) in m, density of soil (DS) in gr/cm3, internal angle of friction (IAF) in degree, cohesion of soil (CS) in kg/cm2, and foundation radius (FR) in m were considered as the input of the model and bearing capacity in kg/cm2 was considered as the target parameter. Three main strategies were addressed in this study. First, four well-known machine learning algorithms, Artificial Neural Network (ANN), Bagging Regressor (BR), Least Squares Support Vector Machine (LSSVM), and Gradient Boosting Regressor (GBR), were adopted to predict bearing capacity. Second, a novel and robust mathematical computation named dimensional analysis (DA) theorem was integrated with machine learning techniques to improve the accuracy and performance of the models by decreasing the number of inputs. Third, based on DA implementation, a rational mathematical formula using gene expression programming (GEP) was structured to anticipate bearing capacity. Furthermore, the sensitivity analysis process specified the impact of the five effective factors. The results of this study demonstrate the effectiveness of integrating DA with machine learning models to predict the BC of circular foundations. Among the developed models, the DA-LSSVM model showed superior performance, achieving an R² of 0.998 and 0.996 for training and testing, respectively, indicating high prediction accuracy. The results indicated that the IAF was the most sensitive factor with r = 0.709, while CS was the least sensitive with r = -0.087. A graphical user interface (GUI) app has been designed to apply the proposed models in this study conveniently. In the last step of this process, the GUI and the DA-based machine learning models were implemented by analyzing two examples. According to the findings, the GUI may be employed to make a reliable and speedy projection of the bearing capacity of circular foundations by considering a variety of input parameters.

Optimal harvest strategies with catch-dependent pricing for chub mackerel in South Korea

Chub mackerel (Scomber japonicus) is a key commercial species in South Korea. However, the catch volume of chub mackerel has experienced significant fluctuations over the past few decades, with current trends indicating a decline. Despite regulatory measures such as closed seasons, resource depletion remains a concern, thereby highlighting the requirement for effective management strategies. Numerous previous studies have proposed optimal harvest strategies by assuming constant prices. However, as large catches of mackerel tend to have lower prices, it is crucial to develop optimal harvest strategies that account for this decrease. Thus, we aim to develop a monthly optimal harvest strategy for chub mackerel that considers catch-dependent pricing. We define logarithmic, rational, and irrational catch-dependent price functions and their corresponding objective functions. In addition, we develop an optimal control system based on a discrete age-structured model. We use Pontryagin’s maximum principle to prove the necessary conditions for the optimal harvest strategy under the three catch-dependent pricing functions and perform simulations using the forward–backward sweep method. We compare the optimal harvest strategies under the three catch-dependent pricing scenarios with those under a constant price. The optimal harvest strategies with the rational and irrational price functions are similar to those with a constant price, where the fishing effort increases immediately after spawning and then gradually decreases. In contrast, the optimal harvest strategy with the logarithmic price function involves a gradual increase in fishing effort from July immediately after the spawning period, with the maximum effort in June before the next spawning season. In addition, we compare the effects of monthly closed seasons across the four pricing scenarios. A closed season in July immediately after spawning provides the highest resource recovery efficiency. In contrast, a closed season in June provides the highest catch and profit efficiencies. As the cost per unit of effort increases, the fishing effort, catch, and profit decrease, while the biomass increases, and the profit decrease is smallest under the logarithmic price function. Our method can improve monthly optimal harvest strategies for other species using catch-dependent pricing functions as well as significantly contribute to enhancing fishers’ profit.

In-depth analysis and exploration of rogue wave, lump wave, and different solitonic patterns to the Painlevé-integrable (3+1)D nonlinear evolution equations using a new extended methodology

This paper proposes a novel extended methodology, establishing the new extended Generalized Exponential Rational Function Method (GERFM), which is known as a “new extended GERFM” for solving the Painleve-integrable [Formula: see text]D generalized nonlinear evolution equation. We enhance GERFM by incorporating additional terms that combine functions and derivatives, making it a more generalized and efficient approach for solving nonlinear partial differential equations. Moreover, the unified method is applied to thoroughly delve deeper into the equation’s properties to understand its mathematical characteristics and potential solutions. This approach aims to systematically investigate and elucidate the equation’s inherent features, providing a more complete insight into its behavior and the nature of its solutions. Utilizing these methods, we have derived new exact solutions that involve exponential, trigonometric, hyperbolic, polynomial, and rational functions with additional parameters to the model. To emphasize the physical importance of these solutions, we present three-dimensional (3D) and contour plots, investigating different parameter choices. The graphs illustrate various wave patterns such as irregular periodic solitons, rogue waves, lumps, and the interaction of solitary and lump waves based on different parameter values. This visualization method enriches our comprehension of the solutions and facilitates a thorough exploration of how they could be applied to the generation of prolonged waves with small magnitudes in plasma physics, fluid dynamics, and other media with weak dispersion.

Generating Special Curves for Cubic Polynomials

An algorithmic method is proposed to generate all cubic polynomials with a critical orbit relation. We generate curves (polynomials of parameters) that correspond to those functions with critical orbit relations. The irreducibility of the polynomials obtained is left as an open problem. Our approach also works to generate critical orbit relations in all families of rational functions with active critical points.

Hidden Zeros Are Equivalent to Enhanced Ultraviolet Scaling, and Lead to Unique Amplitudes in Tr(ϕ3) Theory

We investigate the hidden amplitude zeros which describe a nontrivial vanishing of scattering amplitudes on special external kinematics. We first prove that every type of hidden zero is equivalent to what we call a “subset” enhanced scaling under Britto-Cachazo-Feng-Witten shifts for any rational function built from planar Lorentz invariants Xij=(pi+pi+1+⋯+pj−1)2. This directly applies to Tr(ϕ3), nonlinear sigma models, or Yang-Mills-scalar amplitudes, revealing a novel type of enhanced UV scaling in these theories. We also use this observation to prove the conjecture that Tr(ϕ3) amplitudes are uniquely fixed by the zeros, up to an overall normalization, when assuming an ordered and local propagator structure and trivial numerators. In this context, unitarity (residue factorization) may be viewed as a consequence of the zeros. For Yang-Mills theory, we conjecture the zeros, combined with the Bern-Carrasco-Johansson color-kinematic duality in the form of amplitude relations, uniquely fix the ⌊n/2⌋ distinct polarization structures of n-point gluon amplitudes. Our approach opens a new avenue for understanding previous similar uniqueness results, and also extending them beyond tree level for the first time. Published by the American Physical Society 2025

Collineation varieties of tensors

Abstract In this article, we introduce the k-th collineation variety of a third order tensor. This is the closure of the image of the rational map of size k minors of a matrix of linear forms associated to the tensor. We classify such varieties in the case of pencils of matrices, and nets of matrices of small size. We discuss the natural stratification of tensor spaces induced by the invariants and the geometric type of the collineation varieties.

Exploration of Soliton Solutions for the Kaup–Newell Model Using Two Integration Schemes in Mathematical Physics

ABSTRACTThis research deals with the Kaup–Newell model, a class of nonlinear Schrödinger equations with important applications in plasma physics and nonlinear optics. Soliton solutions are essential for analyzing nonlinear wave behaviors in different physical systems, and the Kaup–Newell model is also significant in this context. The model's ability to represent subpicosecond pulses makes it a significant tool for the research of nonlinear optics and plasma physics. Overall, the Kaup–Newell model is an important research domain in these areas, with ongoing efforts focused on understanding its various solutions and potential applications. A new version of the generalized exponential rational function method and ‐expansion function method are utilized to discover diverse soliton solutions. The generalized exponential rational function method facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to ‐expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability of the proposed model is examined, with numerical simulations complementing the analytical results to provide a better understanding of the solutions' dynamic behavior. These results offer a foundation for future research, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations; to our knowledge, for this equation, these methods of investigation have not been explored before. The accuracy of each solution has been verified using the Maple software program.

Discontinuous Structural Transitions in Fluids with Competing Interactions.

This paper explores how competing interactions in the intermolecular potential of fluids affect their structural transitions. This study employs a versatile potential model with a hard core followed by two constant steps, representing wells or shoulders, analyzed in both one-dimensional (1D) and three-dimensional (3D) systems. Comparing these dimensionalities highlights the effect of confinement on structural transitions. Exact results are derived for 1D systems, while the rational function approximation is used for unconfined 3D fluids. Both scenarios confirm that when the steps are repulsive, the wavelength of the oscillatory decay of the total correlation function evolves with temperature either continuously or discontinuously. In the latter case, a discontinuous oscillation crossover line emerges in the temperature-density plane. For an attractive first step and a repulsive second step, a Fisher-Widom line appears. Although the 1D and 3D results share common features, dimensionality introduces differences: these behaviors occur in distinct temperature ranges, require deeper wells, or become attenuated in 3D. Certain features observed in 1D may vanish in 3D. We conclude that fluids with competing interactions exhibit a rich and intricate pattern of structural transitions, demonstrating the significant influence of dimensionality and interaction features.

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.

Application of rational functions in primary and secondary thermometry

Application of rational functions in primary and secondary thermometry

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.

Précis of Why We Doubt

Abstract I selectively summarize some of the main ideas in my book Why We Doubt, focusing on the elements that are discussed in the replies from commentators. In the book, I investigated our skeptical intuitions which form the basis of the skeptic’s arguments. I argue that these intuitive judgments or inclinations to judge are produced by the deployment of a subconscious heuristic, pbs, which serves us well most of the time but misfires in esoteric cases like those associated with global skepticism. I do this by identifying the rational function of pbs and arguing that when we apply this rational function to global skepticism, it doesn’t yield a skeptical result.

On the Krull dimension of rings of integer-valued rational functions

On the Krull dimension of rings of integer-valued rational functions

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.

Bode gain-phase relation beyond the minimum phase condition

The Bode gain-phase relation links the phase shift introduced by a causal, linear and time-invariant system with its frequency-dependent magnitude gain. The relation, which is widely used in system theory and electronics, was first derived by H. W. Bode for a transfer function described, in the Fourier (Laplace) domain, by rational functions of frequency, where the numerator fulfills the so-called minimum-phase condition, according to which the polynomial numerator of the frequency-domain transfer function has no zeroes in the upper-half (right-half) of the complex ω-plane (s-plane). Here we discuss a general derivation that widens the range of applicability of the relation beyond the minimum phase condition, allowing the presence of zeroes and encompassing cases in which delays occur. Both these features are widely present in real physical systems, thus endowing the new proof with a broader range of validity.

Comment on: "Solving the conformable Huxley equation using the complex method" [Electron. Res. Arch., 31 (2023), 1303–1322

<p>Using the complex method, Guoqiang Dang and Qiyou Liu [Guoqiang Dang, Qiyou Liu, Electron. Res. Arch., 31 (2023), 1303–1322] have found some exact solutions of the conformable Huxley equation. In this comment, we first demonstrate that the elliptic function solutions and rational function solutions do not satisfy the complex conformable Huxley equation. Finally, all exact solutions of the conformable Huxley equation are given by us.</p>

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.