Set Of Elementary Functions Research Articles

Parameter N Analysis on the Rational-Talbot Algorithm for Numerical Inversion of Laplace Transform

This article investigates the numerical inversion of the Laplace Transform by the Rational-Talbot method and analyzes the influence on the variation of the free parameter N established by the technique when applied to certain functions. The set of elementary functions, for which the method is tested, has exponential and oscillatory characteristics. Based on the results obtained, it was concluded that the Rational-Talbot method is e cient for the inversion of decreasing exponential functions. At the same time, to perform the inversion process effectively for trigonometric forms, the algorithm requires a greater amount of terms in the sum. For higher values of N, the technique works well. In fact, this is observed in inverting the functions transform, that combine trigonometric and polynomial factors. The method numerical results have a good precision for the treatment of decreasing exponential functions when multiplied by trigonometric functions.

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

The paper aims to employ a new effective methodology to build exact fractional solutions to the generalized nonlinear Schrödinger equation with a local fractional operator defined on Cantor sets. The equation contains group velocity dispersion and second‐order spatiotemporal dispersion coefficients. We obtain exact solutions of the equation via the generalized version of the exponential rational function method. This new version of the method uses a set of elementary functions that are adopted on the contour set. To study the dynamic behavior of the obtained results, extensive numerical simulations are provided. We observe that the employed method is simple but quite efficient for determining the exact solutions of the problem in the local sense. Moreover, they are practically compatible with solving various classes of nonlinear problems arising in mathematical physics. All computations are carried out using the Maple package.

On the nondifferentiable exact solutions to Schamel's equation with local fractional derivative on Cantor sets

AbstractThe various aspects of differential calculus are always on the path to progress and excellence, and these trends have been more highlighted in recent decades. More specifically, tremendous advances have been made in the field of fractional calculus. One of the main branches in this field is the local fractional derivative, which has been used successfully to describe many real‐world phenomena in science, and engineering. The present contribution aims to adopt a newly proposed analytical technique to construct exact solutions to local fractional Schamel's equation defined on Cantor sets. To this end, a set of elementary functions are constructed on Cantor sets is defined. Furthermore, the combination of these modified functions is utilized to constitute a formal representation of the searched exact solution for the equation. Numerical simulations related to some of the obtained solutions are also included. The acquired results confirm that the method used is not only very straightforward but also efficient in terms of application. In order to perform complicated and tedious calculations while solving this problem, the use of a symbolic computational packages in Maple or Mathematica is inevitable. The work emphasizes the power of the employed method in providing various exact solutions to different physical problems involving local fractional derivatives.

On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

One of the most interesting branches of fractional calculus is the local fractional calculus, which has been used successfully to describe many fractal problems in science and engineering. The main purpose of this contribution is to construct a novel efficient technique to retrieve exact fractional solutions to local fractional Gardner's equation defined on Cantor sets by an effective numerical methodology. In the framework of this technique, first a set of elementary functions are defined on the contour set. Taking these functions into account, the general structure of the exact solution for the equation is suggested as a specific combination of the defined basis functions. By determining the unknown coefficients in this expansion and by placing them in the original equation, specific forms of solutions to the fractional equation are determined. Several interesting numerical simulations of the achieved results are also presented in the article to give a better understanding of the dynamic behavior of these results. The results obtained in this research confirm that the method used is very simple and efficient in terms of application. Moreover, it is accurate and free of any errors in terms of calculation. Therefore, it can be employed to handle other partial differential equations including local fractional derivatives.

Abstract Convexity of Functions with Respectto the Set of Lipschitz (Concave) Functions

The paper is devoted to the abstract $${\mathcal{H}}$$ -convexity of functions (where $${\mathcal{H}}$$ is a given set of elementary functions) and its realization in the cases when $${\mathcal{H}}$$ is the space of Lipschitz functions or the set of Lipschitz concave functions. The notion of regular $${\mathcal{H}}$$ -convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) $${\mathcal{H}}$$ -minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support $${\mathcal{H}}$$ -minorants at a point and the set of lower $${\mathcal{H}}$$ -support points. Using these tools, we formulate both a necessary condition and a sufficient one for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of $${\mathcal{H}}$$ -convexity are realized in the specific cases when functions are defined on a metric or normed space $$X$$ and the set of elementary functions is the space $${\mathcal{L}}(X,{\mathbb{R}})$$ of Lipschitz functions or the set $${\mathcal{L}}\widehat{C}(X,{\mathbb{R}})$$ of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function lower bounded by a Lipschitz function, the set of lower $${\mathcal{L}}$$ -support points and the set of lower $${\mathcal{L}}\widehat{C}$$ -support points coincide and are dense in the effective domain of the function. These results extend the known Brondsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.

Exponential Sums and Correctly-Rounded Functions

The 2008 revision of the IEEE-754 standard, which governs floating-point arithmetic, recommends that a certain set of elementary functions should be correctly rounded. Successful attempts for solving the Table Maker's Dilemma in binary64 made it possible to design CRlibm , a library which offers correctly rounded evaluation in binary64 of some functions of the usual libm . It evaluates functions using a two step strategy, which relies on a folklore heuristic that is well spread in the community of mathematical functions designers. Under this heuristic, one can compute the distribution of the lengths of runs of zeros/ones after the rounding bit of the value of the function at a given floating-point number. The goal of this paper is to change, whenever possible, this heuristic into a rigorous statement. The underlying mathematical problem amounts to counting integer points in the neighborhood of a curve, which we tackle using so-called exponential sums techniques, a tool from analytic number theory.

Exact Hausdorff and packing measures of Cantor sets with overlaps

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that $$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$, and $$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$ centered in $K$, where ${\mathcal{H}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional Hausdorff measure and ${\mathcal{P}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these density theorems, we describe a scheme for computing ${\mathcal{H}}^{{\it\alpha}}(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing ${\mathcal{P}}^{{\it\alpha}}(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in $\mathbb{R}$ with overlaps.

Redundant Floating-Point Decimal CORDIC Algorithm

In this work, we propose a new decimal redundant CORDIC algorithm to manage transcendental functions, using floating-point representation. The algorithms determine the direction of the elementary rotation using sign estimations. Unlike binary redundant CORDIC, repetition of iterations are not required to ensure convergence since novel decimal codes have been carefully selected with sufficient redundancy to prevent any repetition. The algorithms are mapped to a low-cost unit based on a decimal 3-2 carry-save adder which can also be used as a floating-point decimal division unit. Compared to current decimal floating-point units, the implementation of our algorithm involves minor modifications of the native hardware, while providing a huge set of elementary functions.

Modeling electron flow produced by a three-dimensional spatially periodic field emitter

We study the space charge limited field emission from an emitter whose surface has a simple periodic structure with bumps. The shape of each bump is represented by a smooth function and the emission is governed by the Fowler–Nordheim–Schottky law. A mathematical scheme for modeling the potential and current structure by a set of elementary functions is developed and implemented numerically with the help of a special least square procedure. Our results show that such emitters are more efficient than emitters with long ridges only in weak electric fields. In stronger fields the latter give larger currents and they should be more durable. The emission by an individual bump in our periodic structure is compared also with that of a single emitter bump of the same shape, they appear to be quite close.

The Matching Pursuit Method for Extracting Feature Based on DT-CWT and its Application

A matching pursuit method based on Dual-Tree Complex Wavelet Transform (DT-CWT) is proposed for extracting feature. Many new orthogonal wavelet bases formed Hilbert transform pairs is constructed by the method which is based on the sufficient and necessary condition on constructing wavelet, via the flat delay filter, and translated the problem into resolving algebraic equations. And taking these wavelets as choice object, a matching pursuit method based on DT-CWT is used for extracting feature. The matching pursuit method is based on series expansion of the signal by a set of elementary functions of orthogonal wavelets formed Hilbert transform pairs to match feature more effectively. Simulation testing and field experiments confirm that the proposed method is effective especially in extracting impulsive feature on high intensity noise, which matching pursuit method based on Discrete Wavelet Transform and other wavelet de-noising methods based on threshold and frequency-band, etc cannot do it completely.

Sparse Representation of Musical Signals Using Source-Specific Dictionaries

The sparse representation of music sounds that consist of a single note at a time was examined in. Here, we extend the results to a more generic setting where music sounds may contain multiple notes (or chords) at the same time. The basic idea is to determine a set of elementary functions, called source-specific atoms, that efficiently capture music signal characteristics. We first decompose basic components of musical signals (i.e,, musical notes) into a set of Gabor atoms. Then, these Gabor atoms are prioritized according to their approximation capability to music signals of interest, and the prioritized Gabor atoms are used to synthesize source-specific atoms. To find a sparse representation for musical chords, we generate new atoms by regrouping source-specific atoms. This technique is applied to the approximation of real piano recordings, and its effectiveness in terms of good approximation capability and low computational complexity is demonstrated by experiments.

Further investigation of abstract convexity with respect to the class of general min-type functions

In this article we continue the examination of the basic concepts of abstract convexity, taking for the set of elementary functions the totality of the so-called min-type functions which are defined on the n-dimensional Euclidean space Rn as the pointwise minimum of finite collections of linear functions. We examine compact -convex sets, -convex hull and -extreme points of a subset of . We show that among global maximizers of a convex-along-rays function over an -convex set there are -extreme points of the set. We introduce the positive and the negative polar sets and examine their properties. We present the description of the bipolar sets and show that a subset of is -convex if it is the intersection of its positive and negative bipolar sets. We describe the support set of the upper envelope of a subset of .

The polynomial approximation in the finite element method

In the analysis of a finite element method (FEM) we can describe the shape of a given element by a set of elementary functions known as shape functions. The approaches describing these functions are quite different ones. In the plane ( x 1, x 2), these functions are a product of Lagrangian polynomials when the coordinate system can be chosen with the axes parallel to the sides of the element, otherwise a system of barycentric coordinates (sometimes called area coordinates) could be introduced. The aim of this paper is the description and the representation of shape functions when the element has triangular shape (the simplest). The representation has been done by using two algorithmic schemes: Neville-Aitken and De Casteljau. For these schemes we have deduced very important properties.

Algebraic Properties of the Elementary Functions of Analysis

The elementary functions of a complex variable z are those functions built up from the rational functions of z by exponentiation, taking logarithms, and algebraic operations. The purpose of this paper is first, to prove a structure theorem which shows that if an algebraic relation holds among a set of elementary functions, then they must satisfy an algebraic relation of a special kind. Then we make four applications of this theorem, obtaining both new and old results which are described here briefly (and imprecisely). (1) An algorithm is given for telling when two elementary expressions define the same function. (2) A characterization is derived of those ordinary differential equations having elementary solutions. (3) The four basic functions of elementary calculus-exp, log, tan, tan-' -are shown to be irredundant. (4) A characterization is given of elementary functions possessing elemen- tary inverses.