Showing papers on "Elementary function published in 2014"

Fundamental Solutions in the Theory of Thermoelasticity for Solids with Double Porosity

Abstract: In this article, the linear theory of thermoelasticity for solids with double porosity is considered. The fundamental solutions for the systems of steady vibrations (including quasi-static case) and equilibrium equations are constructed by means of elementary functions; the basic properties of such solutions are also established.

Metalibm: A Mathematical Functions Code Generator

Abstract: There are several different libraries with code for mathematical functions such as exp, log, sin, cos, etc. They provide only one implementation for each function. As there is a link between accuracy and performance, that approach is not optimal. Sometimes there is a need to rewrite a function’s implementation with the respect to a particular specification.

Solitary wave solutions of the fourth order Boussinesq equation through the exp(–Ф(η))-expansion method

Abstract: The exp(–Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(–Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain solitary wave solutions in terms of the hyperbolic functions, the trigonometric functions and elementary functions. The results show that the exp(–Ф(η))-expansion method is straightforward and effective mathematical tool for the treatment of nonlinear evolution equations in mathematical physics and engineering.

Hybrid Model of Fixed and Floating Point Numbers in Secure Multiparty Computations

Abstract: This paper develops a new hybrid model of floating point numbers suitable for operations in secure multi-party computations. The basic idea is to consider the significand of the floating point number as a fixed point number and implement elementary function applications separately of the significand. This gives the greatest performance gain for the power functions (e.g. inverse and square root), with computation speeds improving up to 18 times in certain configurations. Also other functions (like exponent and Gaussian error function) allow for the corresponding optimisation.

A new approach to constructing Green’s functions and integral solutions in thermoelasticity

Abstract: This paper is devoted to a new approach for the derivation of main thermoelastic Green’s functions (MTGFs), based on their new integral representations via Green’s functions for Poisson’s equation. These integral representations have permitted us to derive in elementary functions new MTGFs and new Poisson-type integral formulas for a thermoelastic octant under mixed mechanical and thermal boundary conditions, which are formulated in a special theorem. Examples of validation of the obtained MTGFs are presented. The effectiveness of the obtained MTGFs and of the Poisson-type integral formula is shown on a solution in elementary functions of a particular BVP of thermoelasticity for octant. The graphical and numerical computer evaluation of the obtained MTGFs and of the thermoelastic displacements of the particular BVP for an octant is also presented. By using the proposed approach, it is possible to derive in elementary functions many new MTGFs and new Poisson-type integral formulas for many canonical Cartesian domains.

Solution in elementary functions to a bvp of thermoelasticity: green's functions and green's-type integral formula for thermal stresses within a half-strip

Abstract: This article presents new elementary Green's functions for displacements and stresses created by a unit heat source applied in an arbitrary interior point of a half-strip. We also obtain the corresponding new integration formulas of Green's and Poisson's types which directly determine the thermal stresses in the form of integrals of the products of internal distributed heat source, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of a theorem. Based on this theorem and on derived early by author general Green's type integral formula, we obtain a new solution to one particular boundary value problem of thermoelasticity for half-strip. The graphical presentation of thermal stresses created by a unit point heat source and of thermal stresses for one particular boundary value problem of thermoelasticity for half-strip is also included. The proposed method of constructing thermo...

G-Function Solutions for Schrödinger Equation in Cylindrical Coordinates System

Abstract: In this paper, the Schr?dinger equation is solved by Modified separation of variables (MSV) method suggested by Pishkoo and Darus. Using this method, Meijer’s G-function solutions are derived in cylindrical coordinate system for quantum particle in cylindrical can. All elementary functions and most of the special functions which are the solution of extensive problems in physics and engineering are special cases of Meijer’s G-functions.

Transverse vibrations of arbitrary non-uniform beams

Abstract: Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.

Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals

Abstract: This article presents in a closed form new influence functions of a unit point heat source on the displacements for three boundary value problems of thermoelasticity for a half-plane. We also obtain the corresponding new integral formulas of Green’s and Poisson’s types that directly determine the thermoelastic displacements and stresses in the form of integrals of the products of specified internal heat sources or prescribed boundary temperature and constructed already thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of three theorems. Based on these theorems and on derived early by author the general Green-type integral formula, we obtain in elementary functions new solutions to two particular boundary value problems of thermoelasticity for half-plane. The graphical presentation of the temperature and thermal stresses of one concrete boundary value problems of thermoelasticity for half-plane also is included. The proposed method of constructing thermoelastic Green’s functions and integral formulas is applicable not only for a half-plane, but also for many other two- and three-dimensional canonical domains of different orthogonal coordinate systems.

Enhanced Function Structure Applicability through Adaptive Function Templates

Abstract: The concept of function structures is well-established in early phases of engineering design for clarifying product functions. In addition, it forms the basis for the synthesis of solution principles with morphological boxes. Besides benefits, disadvantages remain with its application. First, function structures typically depend on the background their operator. If several persons with diverse backgrounds create function structures, results may be diverging. Second, function structures are non-reversible. It is easy to conceptualize function structures with given products in mind as well as to turn already existing structures into products. However, with unknown context and main functions of existing products there is no method to elaborate unambiguous function structures reversely from those products. Third, function structures are poorly applicable when mixed levels of product embodiment prevail. To overcome those shortages, an improved approach is proposed in this paper. It relies on function templates that are based on already existing function carriers. This is achieved by conceiving building blocks for functions not only consisting of the function description but also of possible function carriers and all subfunctions. For them the same principle of decomposition applies; down to the stage of elementary functions. The conception of pre-created templates helps generating unambiguous structures. By providing context information due to function carrier inclusion, solutions become traceable. Through the templates’ adaptive nature, mixed levels get manageable. The paper concludes with discussing the approach regarding the development of multi-technology machine tools. To resolve these challenges, function templates are chosen that are based on the technology chain elements of the manufacturing process.

A series solution of the general Heun equation in terms of incomplete Beta functions

Abstract: We show that in the particular case when a characteristic exponent of the singularity at infinity is zero the solution of the general Heun equation can be expanded in terms of the incomplete Beta functions. By means of termination of the series, closed-form solutions are derived for two infinite sets of the involved parameters. These finite-sum solutions are written in terms of elementary functions that in general are quasi-polynomials. The coefficients of the expansion obey a three-term recurrence relation, which in some particular cases may become two-term. We discuss the case when the recurrence relation involves two non-successive terms and show that the coefficients of the expansion are then calculated explicitly and the general solution of the Heun equation is constructed as a combination of several hypergeometric functions with quasi-polynomial pre-factors.

Application of dual series equations to wavy contact between piezoelectric materials and an elastic solid

Abstract: In this paper, the wavy contact between piezoelectric materials and an isotropic solid is considered. The Papkovich–Neuber potentials for the isotropic solid and three harmonic functions for piezoelectric materials are also presented. The stated problem is reduced to a pair of dual series equations and then recast as an integral equation of the Abel type. Employing the product relation for trigonometric functions and the Mehler integral yields an exact solution of the reduced Abel type integral equation. The relationship between contact length and the level of loading, and the distribution of the surface normal stress are given in terms of elementary functions. The derived results agree well with the previous ones for the purely elastic solid. It is found that a critical loading exists for the disturbance. For limiting cases, such as the low level of loading case and full contact case, corresponding contact behaviors are presented. Numerical analyses are done to reveal the influence of the level of loading on the contact behaviors.

Replacing Branches by Polynomials in Vectorizable Elementary Functions

Abstract: One of the goals for the mathematical function generator is to produce vectorizable codes. Therefore, in the generated code there should be no branching. As the most mathematical functions are implemented with domain splitting procedure and piecewise-polynomial approximation, there are several if-else statements in the final code to determine the corresponding polynomial coefficients. In this paper we propose a simple idea of replacing these if-else statements by the evaluation of a polynomial function. This is a novel approach that may not work for all the possible function implementation variants, and it needs to be improved with the use of some more sophisticated methods.

Optimized cubic chebyshev interpolator for elementary function hardware implementations

Abstract: —This paper presents a cubic interpolator for com-puting elementary functions using truncated-matrix arithmeticunits and an optimized number of coefficients bits. The pro-posed method optimizes the initial coefficient values found usinga Chebyshev series approximation, minimizing the maximumabsolute error of the interpolator output. The resulting designscan be utilized for approximating any function up to 53 -bitsof precision (IEEE double precision significand). Area, delay andpower estimates are given for 16 , 24 and 32 -bit cubic interpolatorsthat compute the reciprocal function, targeting a 65 nm CMOStechnology from IBM. Results indicate the proposed method usessmaller arithmetic units and has reduced lookup table sizes thanpreviously proposed methods. I. I NTRODUCTION Elementary function approximations are important for 3Dcomputer graphics, digital signal processing (DSP), scientificcomputing, artificial neural networks, and multimedia applica-tions. Unfortunately, elementary function approximations withhigh amounts of precision are required for many signal andimage processing algorithms. Due to the slowness of softwareroutines, which are used to approximation functions withhighly accurate results, hardware-based function evaluationsare desired over software because of their high-speed advan-tages.This paper utilizes an optimization method to localizeand find a closed-form solution for cubic-based interpola-tors allowing smaller realizable architectures. As opposed toprevious research [1], this paper focuses on using cubingunits compared to linear and quadratic-based interpolators.Our approach explores five different dimensions in optimizinghardware function evaluation: method, hardware optimization,power, area, and latency. Although the method presented in thispaper is directed at reciprocals, it can be used to obtain otherpopular elementary function approximations, such as sin(x)and e

Liouvillian integrability of gravitating static isothermal fluid spheres

Abstract: We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = −1 and n = −3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = −5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka–Volterra quadratic polynomial differential system in R2, and characterize the Liouvillian inte...

Generating Interior Sources for the Reissner-Nordström Metric

Abstract: We study the Einstein-Maxwell equations for isotropic pressure distributions. We postulate a relationship between the electric field intensity and one of the gravitational potentials. An algorithm is developed that allows us to systematically generate new classes of exact solutions for charged relativistic stars. The solutions are expressed in terms of simple elementary functions; it is possible to parametrize the solutions so that different values of a constant allows us to tabulate the models. For a particular class it is possible to generate models without any integration. We study the qualitative features of a particular solution, and show that it is physically reasonable in the region of a spherical shell surrounding the centre.

New set of universal functions for the two body-initial value problem

Abstract: In this paper, new set of universal functions, Y n (χ;α); based on Goodyear’s time transformation formula will be developed analytically and computationally for the two body-initial value problem, where n is non negative integer, χ, new independent variable, a kind of generalized anomaly, and α is the reciprocal of the orbit’s semi-major axis. For the analytical developments, the proofs of the linear independence of the Y’s functions, differential and recurrence formulae satisfied by them and their relations with the elementary function are given. Full, set of the identities for the Y’s functions are listed to serve as a ready reference when need. Exact analytical expressions of the general universal Kepler’s equation for the time t∈[t l ,t s ], together with some orbital parameters are developed as power series of χ; also some recurrence formulae are given to facilitate their computations. Finally, symbolical series solution of the general Kepler’s equation is established, and the literal analytical expressions of the coefficients of the series are listed in Horner form for efficient and stable evaluation. For the computational developments, algorithm for the evaluation of the Y’s functions is constructed. Computational initial value problem in terms of the Y’s functions is developed for which an efficient iterative method of arbitrary positive integer order of convergence ≥2 is established for the universal Kepler’s equation. The method is of dynamic natural in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. The applications of the method are illustrated by numerical examples of some test orbits of different eccentricities. The numerical results are accurate and efficient.

Multivalued Elementary Functions in Computer-Algebra Systems

Abstract: An implementation (in Maple) of the multivalued elementary inverse functions is described The new approach addresses the difference between the single-valued inverse function defined by computer systems and the multivalued function which represents the multiple solutions of the defining equation The implementation takes an idea from complex analysis, namely the branch of an inverse function, and defines an index for each branch The branch index then becomes an additional argument to the (new) function A benefit of the new approach is that it helps with the general problem of correctly simplifying expressions containing multivalued functions

Hybrid Model of Fixed and Floating Point Numbers in Secure Multiparty Computations.

Abstract: This paper develops a new hybrid model of floating point numbers suitable for operations in secure multi-party computations. The basic idea is to consider the significand of the floating point number as a fixed point number and implement elementary function applications separately of the significand. This gives the greatest performance gain for the power functions (e.g. inverse and square root), with computation speeds improving up to 18 times in certain configurations. Also other functions (like exponent and Gaussian error function) allow for the corresponding optimisation. We have proposed new polynomials for approximation, and implemented and benchmarked all our algorithms on the Sharemind secure multi-party computation framework.

On some Summation Formulae for the I-Function of Two Variables

Abstract: In this research paper, we aim to establish three interesting summation formulae for the Ifunction of two variables recently introduced in the literature. The results are derived with the help of classical summation theorems due to Watson, Dixon and Whipple. A few known results are also obtained as special cases of our main findings. Since the I-function of two variables is the most generalized function of two variables and it includes as special cases many of the known functions appearing in the literature, the results derived in this paper will therefore serve as the key formulas from which a large number of summation formulas including elementary functions can be obtained by specializing the parameters therein.

Physical Bounds for a Half-Infinite Plane Crack in an Infinite Thermo-Electro-Elastic Medium with Transverse Isotropy.

Abstract: This article presents the physical bounds for an infinite transversely isotropic thermo-electro-elastic medium, weakened by a half-infinite plane crack. The crack is assumed to be electrically permeable and is subjected to two identical point thermal loads. The original problem is transformed to a mixed boundary value problem of a half-space, which is solved by general solutions and the potential theory method. Exact and complete solutions are given in terms of elementary functions for the first time. The solutions along with the newest results in literature serve as upper and lower bounds for the corresponding crack problems in practice. Due to their desirable properties, the present solutions play an important role in clarifying the simplified analyses and checking the numerical codes.

A relativistic algorithm with isotropic coordinates

Abstract: We study spherically symmetric spacetimes for matter distributions with isotropic pressures. We generate new exact solutions to the Einstein field equations which also contains isotropic pressures. We develop an algorithm that produces a new solution if a particular solution is known. The algorithm leads to a nonlinear Bernoulli equation which can be integrated in terms of arbitrary functions. We use a conformally flat metric to show that the integrals may be expressed in terms of elementary functions. It is important to note that we utilise isotropic coordinates unlike other treatments.

Computational complexity of solving elementary differential equations over unbounded domains.

Abstract: In this paper we investigate the computational complexity of solving ordinary dierential equations (ODEs) y 0 = f(y) over unbounded domains. Contrarily to the bounded case, this problem has not been well-studied, apparently due to the \conventional wisdom" that it can always be reduced to the bounded case by using rescaling techniques. However, as we show in this paper, rescaling techniques do not seem to provide meaningful insights on the complexity of this problem, since the use of such techniques introduces a dependence on parameters which are hard to compute. Instead, we take another approach and study ODEs of the form y 0 = f(t;y), where each component of f is an elementary function in the sense of Analysis, i.e. each component can be built from the usual functions of Analysis: polynomials, trigonometric functions, the exponential, etc. through composition and basic operations (addition, product, etc.). We present algorithms which numerically solve these ODEs over unbounded domains. These algorithms have guaranteed precision, i.e. given some arbitrarily large time T and error bound " as input, they will output a value Y which satises ky(T ) Yk ". We analyze the complexity of solving these algorithms and show that they compute Y in time polynomial in several quantities like the time T or the precision of the output ". We consider both algebraic complexity and bit complexity.

Fourier Series Approximations to -Bounded Equatorial Orbits

Abstract: The current paper offers a comprehensive dynamical analysis and Fourier series approximations of -bounded equatorial orbits. The initial conditions of heterogeneous families of -perturbed equatorial orbits are determined first. Then the characteristics of two types of -bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts.