Showing papers on "Elementary function published in 2019"

Cascaded Fluctuating Two-Ray Fading Channels

Abstract: In this letter, we derive the exact probability density function (PDF), cumulative distribution function (CDF), and generalized moment generating function of a product of $N$ independent fluctuating two-ray (FTR) random variables. Capitalizing on the derived expressions, new expressions for the PDF and CDF of the FTR fading model, i.e., $N=1$ , are obtained in terms of elementary functions, where the existing ones are expressed in terms of special functions. The derived distributions are then used to derive exact analytical expressions for the outage probability and average bit error rate. Monte Carlo simulations have been carried out to validate the correctness of our results.

On Path-Complete Lyapunov Functions: Geometry and Comparison

Abstract: We study optimization-based criteria for the stability of switching systems, known as path-complete Lyapunov functions, and ask the question “can we decide algorithmically when a criterion is less conservative than another?”. Our contribution is twofold. First, we show that a path-complete Lyapunov function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each path-complete criterion an implied invariant set. Second, we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given path-complete Lyapunov functions.

The frequency response function of the creep compliance

Abstract: Motivated from the need to convert time-dependent rheometry data into complex frequency response functions, this paper studies the frequency response function of the creep compliance that is coined the complex creep function. While for any physically realizable viscoelastic model the Fourier transform of the creep compliance diverges in the classical sense, the paper shows that the complex creep function, in spite of exhibiting strong singularities, it can be constructed with the calculus of generalized functions. The mathematical expressions of the real and imaginary parts of the Fourier transform of the creep compliance of simple rheological networks derived in this paper are shown to be Hilbert pairs; therefore, returning back in the time domain a causal creep compliance. The paper proceeds by showing how a measured creep compliance of any solid-like or fluid-like viscoelastic material can be decomposed into elementary functions with parameters that can be identified from best fit of experimental data. The proposed technique allows for a direct determination of the sufficient parameters needed to approximate an experimentally measured creep compliance and the presented mathematical formulae offers dependable expressions of the corresponding complex-frequency response functions.

A New Approach to the Statistical Analysis of Non-Central Complex Gaussian Quadratic Forms With Applications

Abstract: This paper proposes a novel approach to the statistical characterization of non-central complex Gaussian quadratic forms (CGQFs). Its key strategy is the generation of an auxiliary random variable that replaces the original CGQF and converges in distribution to it. This technique is valid for both definite and indefinite CGQFs and yields simple expressions of the probability density function (PDF) and the cumulative distribution function (CDF) that only involve elementary functions. This overcomes a major limitation of previous approaches, where the complexity of the resulting PDF and CDF does not allow for further analytical derivations. Additionally, the mean square error between the original CGQF and the auxiliary one is provided in a simple closed-form formulation. These new results are then leveraged to analyze the outage probability and the average bit error rate of maximal ratio combining systems over correlated Rician channels.

Exact Solutions of the Nonlinear Diffusion Equation

Abstract: We construct new radially symmetric exact solutions of the multidimensional nonlinear diffusion equation, which can be expressed in terms of elementary functions, Bessel functions, Jacobi elliptic functions, Lambert W-function, and the exponential integral. We find new self-similar solutions of a spatially one-dimensional parabolic equation similar to the nonlinear heat equation. Our exact solutions can help verify difference schemes and numerical calculations used in the mathematical modeling of processes and phenomena described by these equations.

Sound Approximation of Programs with Elementary Functions

Abstract: Elementary function calls are a common feature in numerical programs. While their implementations in mathematical libraries are highly optimized, function evaluation is nonetheless very expensive compared to plain arithmetic. Full accuracy is, however, not always needed. Unlike arithmetic, where the performance difference between for example single and double precision floating-point arithmetic is relatively small, elementary function calls provide a much richer tradeoff space between accuracy and efficiency. Navigating this space is challenging, as guaranteeing the accuracy and choosing correct parameters for good performance of approximations is highly nontrivial. We present a fully automated approach and a tool which approximates elementary function calls inside small programs while guaranteeing overall user given error bounds. Our tool leverages existing techniques for roundoff error computation and approximation of individual elementary function calls and provides an automated methodology for the exploration of parameter space. Our experiments show that significant efficiency improvements are possible in exchange for reduced, but guaranteed, accuracy.

Fundamental solution of the system of equations of pseudo oscillations in the theory of thermoelastic diffusion materials with double porosity

Abstract: Purpose The purpose of this paper to construct the fundamental solution of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity. Design/methodology/approach The paper deals with the study of pseudo oscillations in the generalized theory of thermoelastic diffusion materials with double porosity. Findings The paper finds the fundamental solution of partial differential equations in terms of elementary functions. Originality/value Assuming the displacement vector, volume fraction fields, temperature change and chemical potential functions in terms of oscillation frequency in the governing equations, pseudo oscillations have been studied and finally the fundamental solution of partial differential equations in case of pseudo oscillations in terms of elementary functions has been constructed.

Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions

Abstract: In Cauchy problems with blow-up solutions there exists a singular point whose position is unknown a priori (for this reason, the application of standard fixed-step numerical methods for solving such problems can lead to significant errors). In this paper, we describe a method for numerical integration of blow-up problems for non-linear systems of coupled ordinary differential equations of the first order ( x m ) t ′ = f m ( t , x 1 , … , x n ) , m = 1 , … , n , based on the introduction a new non-local independent variable ξ , which is related to the original variables t and x 1 , … , x n by the equation ξ t ′ = g ( t , x 1 , … , x n , ξ ) . With a suitable choice of the regularizing function g , the proposed method leads to equivalent problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore, the transformed problems admit the use of standard numerical methods with a fixed stepsize in ξ . Several test problems are formulated for systems of ordinary differential equations that have monotonic and non-monotonic blow-up solutions, which are expressed in elementary functions. Comparison of exact and numerical solutions of test problems showed the high efficiency of numerical methods based on non-local transformations of a special kind. The qualitative features of numerical integration of blow-up problems for single ODEs of higher orders with the use of non-local transformations are described. The efficiency of various regularizing functions is compared. It is shown that non-local transformations in combination with the method of lines can be successfully used to integrate initial–boundary value problems, described by non-linear parabolic and hyperbolic PDEs, that have blow-up solutions. We consider test problems (admitting exact solutions) for nonlinear partial differential equations such as equations of the heat-conduction type and Klein–Gordon type equations, in which the blowing-up occurs both in an isolated point of space x = x ∗ , and on the entire range of variation of the space variable 0 ≤ x ≤ 1 . The results of numerical integration of test problems, obtained when approximating PDEs by systems with a different number of coupled ODEs, are compared with exact solutions.

Synthesizing Efficient Low-Precision Kernels

Abstract: In this paper, we present a fully automated approach for synthesizing fast numerical kernels with guaranteed error bounds. The kernels we target contain elementary functions such as sine and logarithm, which are widely used in scientific computing, embedded as well as machine-learning programs. However, standard library implementations of these functions are often overly accurate and therefore unnecessarily expensive. Our approach trades superfluous accuracy against performance by approximating elementary function calls by polynomials and by implementing arithmetic operations in low-precision fixed-point arithmetic. Our algorithm soundly distributes and guarantees an overall error budget specified by the user. The evaluation on benchmarks from different domains shows significant performance improvements of 2.23\(\times \) on average compared to state-of-the-art implementations of such kernel functions.

Analytical and numerical methods for efficient calculation of edge diffraction by an arbitrary incident signal.

Abstract: In the present study, diffraction of plane and spherically spreading signals by half-planes is considered. An existing analytical impulse response is investigated, which is exact for plane and approximate for spherical incident signals. It is shown that all its primitive functions with respect to time exist and have an explicit form involving elementary functions. The primitive functions are employed to (i) prove that the convolution of the impulse response with any bounded signal is also bounded for all times, (ii) obtain analytically the diffraction response as a combination of elementary functions for any incident signal approximated piecewise by fitting polynomials, (iii) improve the performance of the numerical convolution by orders of magnitude, and (iv) handle the convolution of very coarsely sampled incident signals. An impulse response is presented for finite-length edges, which, unlike the traditional integration formulas along the edge, is an explicit form of time. Because it is based on the impulse response for infinite edges, it inherits all aforementioned benefits associated with its primitive functions. Furthermore, it offers a substantial computational benefit compared to traditional integration formulas along the edge. Finally, the requirements for the direct application of the presented results to other impulse responses are discussed.

On degenerate sigma-functions in genus 2

Abstract: We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus 1 sigma-function and elementary functions as solutions of a system of linear partial differential equations satisfied by the sigma-function. By way of application, we derive a solution for a class of generalized Jacobi inversion problems on elliptic curves, a family of Schrodinger-type operators on a line with common spectrum consisting of a point and two segments, explicit construction of a field of three-periodic meromorphic functions. Generators of rank 3 lattice in ℂ2 are given explicitly.

Simplified error bounds for turning point expansions

Abstract: Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of F. W. J. Olver.

Integrability in Finite Terms and Actions of Lie Groups

Abstract: According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.

До розрахунку вільних коливань дисипативного осцилятора з нелінійною пружністю

Abstract: An approximate method for calculating the amplitudes of free damping oscillations of an oscillator with a nonlinear power term in terms of the elastic characteristic under the force of linear viscous resistance is presented. The method does not require the construction of a solution of a nonlinear differential equation of motion and is based on the well-known assumption that the envelope graph of the free damping oscillations of a dissipative Duffing oscillator is approximately described by an exponential function, as in a linear oscillator. Based on this position, the calculation of the damping oscillations of the oscillator with power nonlinearity in the expression of elasticity, where there is also a linear term, is reduced to recurrent relations. In the case of a rigid power characteristic, the two-digit Lambert function of the negative argument, where its first branch is used, is included in the ratio. For a soft power characteristic, the recurrence relation has a Lambert function of a positive argument, which is unambiguous. In order to simplify the numerical implementation of the analytical solutions obtained, it is recommended to use known tables of this special function, and in the case of small modular values of the argument, the proposed approximation by its elementary functions. In order to provide information on the actual errors of the proposed calculation method, examples are given where the results to which it results are compared with the results of numerical computer integration of the differential equation of motion for quadratic and cubic nonlinearities. The satisfactory consistency of the comparison results confirmed the suitability of using the proposed method in engineering calculations. Calculations showed that the influence of the nonlinear component of elasticity weakens during oscillation damping and, at small amplitudes, they are close to the terms of the geometric progression, as in a linear oscillator.

A handy, accurate, invertible and integrable expression for Dawson’s function

Abstract: This article proposes a handy, accurate, invertible and integrable expression for Dawson’s function. It can be observed that the biggest relative error committed, employing the proposed approximation here, is about 2.5%. Therefore, it is noted that this integral approximation to Dawson’s function, expressed only in terms of elementary functions, has a maximum absolute error of just 7 × 10-3. As a case study, the integral approximation proposed here will be applied to a nonclassical heat conduction problem, contributing to obtain a handy, accurate, analytical approximate solution for that problem

Elementary function representations for the moments of the Meyer-König and Zeller operators

Abstract: We calculate the moments of the Meyer-Konig and Zeller operators in terms of elementary functions and polylogarithms.

Higher-Order Bessel Equations Integrable in Elementary Functions

Abstract: The eigenfunction problem for a scalar Euler operator leads to an ordinary differential equation, which is an analog of higher-order Bessel equations. Its solutions are expressed through elementary functions in the case where the corresponding Euler operator can be factorized in a certain appropriate way. We obtain a formula describing such solutions. We consider the problem on common eigenfunctions of two Euler operators and present commuting Euler operators of orders 4, 6, and 10 and a formula for their common eigenfunction and also commuting operators of orders 6 and 9.

Comments on J. F. Ritt's book "Integration in Finite Terms"

Abstract: The First and Second Liouville's Theorems provide correspondingly criterium for integrability of elementary functions "in finite terms" and criterium for solvability of second order linear differential equations by quadratures. The brilliant book of J.F.~Ritt contains proofs of these theorems and many other interesting results. This paper was written as comments on the book but one can read it independently. The first part of the paper contains modern proofs of The First Theorem and of a generalization of the Second Theorem for linear differential equations of any order. In the second part of the paper we present an outline of topological Galois theory which provides an alternative approach to the problem of solvability of equations in finite terms. The first section of this part deals with a topological approach to representability of algebraic functions by radicals and to the 13-th Hilbert problem. This section is written with all proofs. Next sections contain only statements of results and comments on them (basically no proofs are presented there).

Erdősian functions and an identity of Gauss

Abstract: A famous identity of Gauss gives a closed form expression for the values of the digamma function $\psi(x)$ at rational arguments $x$ in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for $n$th Catalan number in terms of these functions.

Solutions to Equations of a Three-Dimensional Electron Beam in Elementary Functions

Abstract: —The exact three-dimensional solutions describing nonstationary nonrelativistic electron flows in a uniform magnetic field are given, with preference given to the study of oscillatory regimes.

Jacob's ladders and the set of elementary meta-functional equations giving new kind of interactions of $\zeta(s)$ with itself

Abstract: In this paper we obtain the set of grafts from some set of 12 elementary functions on 12 critical strips. We make use this directly for grafting of elements of the corresponding set of $\zeta$-factorization formulas. This procedure gives the set of elementary meta-functional equations that represents the set of elementary interactions of the Riemann's zeta-function with itself.

Power Series Solutions

Abstract: Generally, second-order differential equations with variable coefficients cannot be solved in terms of the known functions. However, there is a fairly large class of differential equations whose solutions can be expressed either in terms of power series, or as simple combination of power series and elementary functions [1, 2, 3]. It is this class of differential equations that we shall study in this chapter.

Series Solutions of Second-Order Ordinary Differential Equations

Abstract: When a second-order ODE has constant coefficients, a systematic procedure for determining fundamental solutions has been given in the previous chapter However, a larger class of equations have variable coefficients Therefore, we have to find solutions of ODEs using methods other than the familiar elementary functions

Pendulum Analysis by Leaf Functions and Hyperbolic Leaf Functions

Abstract: The mathematical model representing the equation of motion of a pendulum is nonlinear. Solutions that satisfy the equation cannot be represented by elementary functions, such as trigonometric functions. To solve such problems, it is common to linearize the nonlinear equations and derive approximate numerical solutions and exact solutions. Applying such linearization is limited to cases in which the angle of the pendulum is relatively small. In cases where the angle of the pendulum is large, various methods have been presented that rely on numerical solutions and the exact solutions based on Jacobian elliptic functions, to name one example. On the other hand, the author has been studying a certain type differential equation. The second derivative of a function is equal to the term multiplied by $-n$(or $n$) for terms whose function is raised to $2n-1$. Curves based on solving this differential equation are constructed as regular waves with periods. The author has termed the function satisfying this differential equation the leaf function (or the hyperbolic leaf function). In this paper, we attempt to apply this leaf function (or hyperbolic leaf function) to undamped pendulums and overdamped pendulums for large angles.

Taylor Series Revisited

Abstract: We propose a renovated approach around the use of Taylor expansions to provide polynomial approximations. We introduce a coinductive type scheme and finely-tuned operations that altogether constitute an algebra, where our multivariate Taylor expansions are first-class objects. As for applications, beyond providing classical expansions of integro-differential and algebraic expressions mixed with elementary functions, we demonstrate that solving ODE and PDE in a direct way, without external solvers, is also possible. We also discuss the possibility of computing certified errors within our scheme.

Elementary hypergeometric functions, Heun functions, and moments of MKZ operators

Abstract: We consider some hypergeometric functions and prove that they are elementary functions. Consequently, the second order moments of Meyer-Konig and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary functions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions.