Weierstrass Function Research Articles

A Malmquist–Steinmetz Theorem for Difference Equations

It is shown that if the equation f(z+1)n=R(z,f),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f(z+1)^n=R(z,f), \\end{aligned}$$\\end{document}where R(z, f) is rational in both arguments and degf(R(z,f))≠n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg _f(R(z,f))\ ot =n$$\\end{document}, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case degf(R(z,f))=n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg _f(R(z,f))=n$$\\end{document} of the equation above and thus provide a complete difference analogue of Steinmetz’ generalization of Malmquist’s theorem.

Optical Solitons and traveling wave solutions to Kudryashov’s equation

This paper employs a generalized transformation to investigate Kudryashov’s equation with four nonlinear terms in order to construct traveling wave solutions and optical solitons. The proposed unified ansätze framework is manipulated to derive analytical solutions, which are expressed as a combination of general solutions to simpler equations or systems of equations that are integrable or have unique known solutions. Using this technique, one can produce families of new exact solutions as well as solitary solutions. In addition to other model solutions, the scheme reveals the elliptic functions of Jacobi, Weierstrass, and Riccati. These are categorized as singular, bright, and dark optical solitons.

Bright, dark and kink exact soliton solutions for perturbed Gerdjikov–Ivanov equation with full nonlinearity

In this paper, the perturbed Gerdjikov-Ivanov equation (PGI) which describes the dynamics of the soliton in optical fiber is investigated. Using traveling wave transformation, the nonlinear PGI equation is transformed into two nonlinear ordinary differential equations. Consequently, these equations are reduced to a first -order ordinary differential equation that has a well-known exact solution. The exact solutions for the PGI equation will be introduced including Weierstrass, Jacobi elliptic, and hyperbolic functions. Various types of optical soliton solutions were raised such as bright, dark, and kink soliton solution. The obtained optical soliton solutions are presented graphically to clarify some of its physical parameters.

Traveling wave solutions for the Radhakrishnan–Kundu–Lakshmanan equation with a quadrupled nonlinearity law

This study investigates the optical soliton propagation for the new version of the Radhakrishnan–Kundu–Lakshmanan equation with a quadrupled-power law structure of self-phase modulation. An improved modified extended tanh-function procedure is implemented to procure novel optical solitons and several solution types for the governing model. The obtained solitons are bright, dark, and singular, along with the rational, exponential, and Weierstrass function solutions. The retrieved solitons emerged with their constraint conditions.

Chaotic oscillators with two types of semi-fractal equilibrium points: Bifurcations, multistability, and fractal basins of attraction

Two new three-dimensional chaotic oscillators are introduced in this paper. Each oscillator has a different type of semi-fractal equilibrium curve: one with an R domain semi-fractal curve and one with a circular parametric semi-fractal curve. Both oscillators have the Weierstrass function as a basis in their equations. Different properties of these oscillators, such as bifurcation, multistability, and fractal basins of attraction, are investigated. The proposed system, like the chaotic systems of the references (such as systems with no equilibria and systems with a stable equilibrium) is typical. We believe such a chaotic system with fractal equilibria was not proposed before.

Combining the global trends of DFA or CDFA of different orders

When dealing with the detrended fluctuation analysis (DFA) and its variants such as the higher-order DFA and the continuous DFA (CDFA), removing the mean, integrating the signal and detrending it amount to applying an equivalent linear filter characterized by its frequency response when processing a wide-sense stationary random process. By studying the frequency response of each variant, one can better understand its behavior. Thus, in this paper, we first compare the frequency responses of two DFAs whose orders are different. Then, by deriving the higher-order CDFA using Lagrange multipliers, we can show how the frequency responses of the equivalent filters of the CDFA and the DFA are related to each other. Finally, we propose to combine the global trends obtained with the DFA or the CDFA of different orders to derive a new variant. Once again, the behavior is studied using the frequency response of the equivalent filter. Illustrations and comments on ARFIMA processes and Weierstrass functions are also given to evaluate their long range dependence.

Mathcal{N} = 2* Schur indices

We find closed-form expressions for the Schur indices of 4d mathcal{N} = 2* super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.

Neural Network Approximation of Refinable Functions

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions and, on the other hand, very rough functions such as the Weierstrass function, which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU neural networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

Approximation by polynomials composed of Weierstrass doubly periodic functions

The problem of describing classes of functions in terms of the rate of approximation of these functions by polynomials, rational functions, splines entered in the theory of approximation more than 100 years ago and still retains its relevance. Among a large number of problems related to approximation, we considered the problem of polynomial approximation in two variables of a function defined on the continuum of an elliptic curve in C2 and holomorphic in its interior. The formulation of such a question led to the need to study the approximation of a function that is continuous on the continuum of the complex plane and analytic in its interior, using polynomials in doubly periodic Weierstrass functions and their derivatives. This work is devoted to the development of this topic.

On the scaling properties of oscillatory modes with balanced energy.

Animal bodies maintain themselves with the help of networks of physiological processes operating over a wide range of timescales. Many physiological signals are characterized by 1/f scaling where the amplitude is inversely proportional to frequency, presumably reflecting the multi-scale nature of the underlying network. Although there are many general theories of such scaling, it is less clear how they are grounded on the specific constraints faced by biological systems. To help understand the nature of this phenomenon, we propose to pay attention not only to the geometry of scaling processes but also to their energy. The first key assumption is that physiological action modes constitute thermodynamic work cycles. This is formalized in terms of a theoretically defined oscillator with dissipation and energy-pumping terms. The second assumption is that the energy levels of the physiological action modes are balanced on average to enable flexible switching among them. These ideas were addressed with a modelling study. An ensemble of dissipative oscillators exhibited inverse scaling of amplitude and frequency when the individual oscillators' energies are held equal. Furthermore, such ensembles behaved like the Weierstrass function and reproduced the scaling phenomenon. Finally, the question is raised whether this kind of constraint applies both to broadband aperiodic signals and periodic, narrow-band oscillations such as those found in electrical cortical activity.

Variational and non-variational approaches with Lie algebra of a generalized (3 + 1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation in Engineering and Physics

Nonlinear partial differential equations emerge in an extensive variants of physical problems inclusive of fluid dynamics, solid mechanics, plasma physics, quantum field theory as well as mathematics and engineering. It has also been noticed that systems of nonlinear partial differential equations arise in biological and chemical applications. This article presents the analytical investigation of a completely generalized (3 + 1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation which has applications in the fields of engineering and physics. The generalized version of the potential Yu-Toda-Sasa-Fukuyama equation is more comprehensively studied in this paper compared to other research work previously done on the equation, with various new solutions of interests achieved. The theory of Lie group is applied to the nonlinear partial differential equation to basically reduce the equation to an integrable form which consequently allows for direct integration of the result. The rigorous process involved in performing a comprehensive reduction of the model under consideration using its Lie algebra makes it possible to achieve various nontrivial solutions. Besides, more general solutions are found via a well-known standard technique. In consequence, we secured diverse solitons and solutions of great interest including topological kink solitons, singular solitons, algebraic functions, exponential function, rational function, Weierstrass function, Jacobi elliptic function as well as series solutions of the underlying equation. Moreover, the completeness of the result was ascertained by presenting the solutions graphically. In addition, discussions of the pictorial representations of the results are done. Conclusively, we constructed conserved quantities of the underlying equation via both the variational and non-variational approaches using the classical Noether’s theorem as well as the standard multiplier technique respectively. In addition, some pertinent observations made from the secured results via both techniques are analyzed.

A unified presentation of traveling wave solutions for the Mukherjee–Kundu equation

This paper deals with a unified presentation traveling wave solutions for the Mukherjee–Kundu equation which is a variant of the well-known nonlinear Schrödinger equation. A Weierstrass function approach has been employed to derive a Weierstrass elliptic function solutions, which can be reduced to the Jacobian elliptic function solutions through the conversion transformations. Due to the reductions of the Jacobian elliptic function solutions, the solitary wave solution and the trigometric period wave solution can be found by [Formula: see text] and [Formula: see text], respectively. In addition, the features of the obtained traveling wave solutions are represented graphically with the appropriate parameters.

APPROXIMATION WITH FRACTAL FUNCTIONS BY FRACTAL DIMENSION

On the basis of previous studies, we explore the approximation of continuous functions with fractal structure. We first give the calculation of fractal dimension of the linear combination of continuous functions with different Hausdorff dimension. Fractal dimension estimation of the linear combination of continuous functions with the same Hausdorff dimension has also been discussed elementary. Then, based on Weierstrass Theorem and the related results of Weierstrass function, we give the conclusion that the linear combination of polynomials with the same Hausdorff dimension approximates the objective function. The corresponding results with noninteger and integer Hausdorff dimensions have been investigated. We also give the preliminary applications of the theory in the last section.

Continuous Maps Admitting No Tangent Lines: A Centennial of Besicovitch Functions

One of the most influential examples in analysis is a Weierstrass function from to that is continuous but differentiable at no point. However this map, as well most of the others among the myriad similar examples, still admits vertical tangent lines. The examples of continuous maps that admit no tangent line in any direction are also known; however, all currently existing presentations of such maps are not easily accessible due to their very complicated descriptions and hard-to-follow proofs of their desired properties. The goal of this article is to present in an accessible way two such examples. The first—a coordinate of the classical Peano space-filling curve—is simpler, but admits one-sided vertical tangent lines at some points. The second is a variation of a function from a 1924 paper of Besicovitch, which is continuous but admits no one-sided tangent line in any direction. The proofs of nondifferentiability of these two examples will be facilitated by a simple yet general lemma that also implies nondifferentiability of other similar maps, including those of Takagi and van der Waerden.

On the Solution of a Conformal Mapping Problem by Means of Weierstrass Functions

The conformal mapping problem for the section of a channel filled with porous material under a rectangular dam onto the upper half-plane is considered. Similar problems arise in computing of fluid flow in hydraulic structures. As a solution method, the representation of Christoffel–Schwartz elliptic integral in terms of Weierstrass functions is used. The calculation is based on Taylor series for the sigma function, the coefficients of which are determined recursively. A simple formula for a conformal mapping is obtained, which depends on four parameters and uses the sigma function. A numerical experiment was carried out for a specific area. The degeneration of the region, which consists in the dam width tending to zero, is considered, and it is shown that the resulting formula has a limit that implements the solution of the limiting problem. A refined proof of Weierstrass recursive formula for the coefficients of Taylor series of the sigma function is presented.

The cosmological constant vs adiabatic invariance

The property of adiabatic invariance is studied for the generalized potential satisfying the condition of identity of sphere's and point mass's gravity. That function contains a second term corresponding to the cosmological constant as weak-field General Relativity and enables to describe the dynamics of groups and clusters of galaxies and the Hubble tension as a result of two flows, local and global ones. Using the adiabatic invariance approach we derive the orbital parameters via Weierstrass functions, including the formula for the eccentricity which explicitly reveals the differences from the Kepler problem.

APPROXIMATION OF THE SAME BOX DIMENSION IN CONTINUOUS FUNCTIONS SPACE

In this paper, we make research on the approximation of functions with fractal dimension in continuous functions space. We first investigate fractal dimension of the linear combination of continuous functions with different fractal dimensions. Then, fractal winding of continuous functions has been given. Furthermore, based on Weierstrass theorem and Weierstrass function, we establish a theorem that continuous functions with the non-integer fractal dimension can be approximated by the linear combination of trigonometric polynomials with the same fractal dimension. Condition about continuous functions with fractal dimension one and two has also been discussed elementary.

Optical solutions and conservation laws of the Chen–Lee–Liu equation with Kudryashov's refractive index via two integrable techniques

The fundamental objective of this study is to find optical solutions to the perturbed Chen–Lee–Liu equation with Kudryashov's arbitrary refractive index model. Therefore, we applied the enhanced Kudryashov's and improved extended tanh-function techniques to the model. We obtained bright soliton, singular soliton, singular periodic wave, Jacobi elliptic function, exponential function, Weierstrass function, and dark soliton solutions. The obtained results are different from the other solutions available in the literature. As a result of the study, we can say that the proposed methods are effective and powerful for securing exact solutions of the nonlinear partial differential equations. Finally, using a new conservation theorem, we were able to establish the conservation laws for the current model.

Weierstrass function methodology for uncertainty analysis of random media criticality with spectrum range control

Randomized Weierstrass function (RWF) has been under development for evaluating the uncertainty of random media criticality due to the material mixture in disorder. In this work, the modelling capability of RWF is refined so that the spectrum range can be controlled by specifying its lower and upper ends of the frequency domain variable. As a result, it becomes possible to make fair criticality comparison among replicas of random media under inverse power law power spectra. Technically, the infinite sum of trigonometric terms in RWF is formally extended to cover the arbitrarily low frequency domain and then truncated to finite terms for the sole purpose of spectrum range control. This means that the refined RWF is free of the issue of convergence towards a reference fractal characteristic of Weierstrass function and thus termed Incomplete Randomized Weierstrass function (IRWF). As a demonstration, a three-dimensional version of IRWF is applied to the mixture of three fuels with different burnups in a sufficiently water-moderated environment. Monte Carlo criticality calculations are carried out to evaluate the uncertainty of neutron effective multiplication factor (keff) due to the indeterminacy of mixture formation.

Pressure wave characteristics in a bubble-liquid mixture via Kudryashov–Sinelshchikov equation

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system.