Showing papers on "Elementary function published in 2008"

Moment Generating Functions of the Generalized η-μ and k-μ Distributions and Their Applications to Performance Evaluations of Communication Systems

Abstract: In this letter, we prove that in contrast with earlier reported results the moment generating functions (MGFs) of the generalized eta - mu and re k - mu distributions are expressed in terms of elementary functions. Thus by employing the MGF-based approach one can easily evaluate average bit error rates (ABERs) in fading radio channels characterized statistically by the eta - mu and re k - mu distributions either in terms of elementary functions or in terms of finite-range integrals of elementary functions.

Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld 3D Graphics Systems

Abstract: A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power (xy with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

Introduction to Analysis of the Infinite

Abstract: This chapter explains the origin of elementary functions and the impact of Descartes’s “Geometrie” on their calculation. The interpolation polynomial leads to Newton’s binomial theorem and to the infinite series for exponential, logarithmic, and trigonometric functions. The chapter ends with a discussion of complex numbers, infinite products, and continued fractions. The presentation follows the historical development of this subject, with the mathematical rigor of the period. The justification of dubious conclusions will be an additional motivation for the rigorous treatment of convergence in Chapter III.

Computing the far field scattered or radiated by objects inside layered fluid media using approximate Green's functions.

Abstract: A numerically efficient technique is presented for computing the field radiated or scattered from three-dimensional objects embedded within layered acoustic media. The distance between the receivers and the object of interest is supposed to be large compared to the acoustic wavelength. The method requires the pressure and normal particle displacement on the surface of the object or on an arbitrary circumscribing surface, as an input, together with a knowledge of the layered medium Green’s functions. The numerical integration of the full wave number spectral representation of the Green’s functions is avoided by employing approximate formulas which are available in terms of elementary functions. The pressure and normal particle displacement on the surface of the object of interest, on the other hand, may be known by analytical or numerical means or from experiments. No restrictions are placed on the location of the object, which may lie above, below, or across the interface between the fluid media. The proposed technique is verified through numerical examples, for which the near field pressure and the particle displacement are computed via a finite-element method. The results are compared to validated reference models, which are based on the full wave number spectral integral Green’s function.

Implementation of elementary functions for logarithmic number systems

Abstract: Computations in logarithmic number systems require realisations of four different elementary functions. In the current paper the authors use a recently proposed approximation method based on weighted sums of bit-products to realise these functions. It is shown that the considered method can be used to efficiently realise the different functions. Furthermore, a transformation is proposed to improve the results for functions with logarithmic characteristics. Implementation results shows that significant savings in area and power can be obtained using optimisation techniques.

Feynman Formulas for Particles with Position-Dependent Mass

Abstract: In this paper, we obtain Feynman formulas for solutions to equations describing the diffusion of particles with mass depending on the particle position and to Schrodinger-type equations describing the evolution of quantum particles with similar properties. Such particles (to be more precise, quasi-particles) arise in, e.g., models of semiconductors. Tens of papers studying such models have been published (see [1] and the references therein), but representations of solutions to the arising Schrodinger- and heat-type equations, which go back to Feynman, have not been considered so far. One of the possible reasons is that the traditional application of Feynman’s approach involves integrals with respect to diffusion processes whose transition probabilities have no explicit representation (in terms of elementary functions) in the situation under consideration. In this paper, instead of these transition probabilities, we use their approximations, which can be expressed in terms of elementary functions. Apparently, similar approximations were first applied in [4, 5] to study the diffusion and the quantum evolution of particles of constant mass on Riemannian manifolds. It turns out that the central idea of the approach developed in [4, 5] can also be applied (after appropriate modifications) to the situation considered in this paper. In what follows, we assume that solutions to the Cauchy problems for the equations under examination exist and are unique; thus, we can and shall consider not only solutions to equations but also the corresponding semigroups. Somewhat changing the terminology of [2, 6], we define a real (complex) Schrodinger semigroup as e – tH (respectively, e ith ), where H is a self-adjoint positive operator on a Hilbert space or the generator of a diffusion process.

Approximating real functions which possess nth derivatives of bounded variation and applications

Abstract: The main aim of this paper is to provide an approximation for the function f which possesses continuous derivatives up to the order n-1(n>=1) and has the nth derivative of bounded variation, in terms of the chord that connects its end points A=(a,f(a)) and B=(b,f(b)) and some more terms which depend on the values of the k derivatives of the function taken at the end points a and b, where k is between 1 and n. Natural applications for some elementary functions such as the exponential and the logarithmic functions are given as well.

An Efficient Approach for Extracting Poles of Green's Functions in General Multilayered Media

Abstract: A fast approach of closed-form Green's functions for mixed-potential integral equation analysis of both near- field and far-field in planar multilayered media is presented in this paper. Since all components of the Green's functions in spectral-domain are restructured concisely by four elementary functions, the surface wave pole extraction is very effective in terms of a function related to the generalized reflection coefficients. Meanwhile, the number of required two-level discrete complex image method (DCIM) with high-order Sommerfeld identities is also minimized under this scheme. The numerical results show that this approach is more efficient and accurate for the electromagnetic scattering by, and radiation in the presence of, electrically large three dimensional (3-D) objects in multilayered media. Although lossless examples are considered in the numerical examples, the method presented here works also for the lossy materials where the poles move away from the real axis in the complex plane.

Vortex images and q-elementary functions

Abstract: In the present paper, the problem of vortex images in the annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, and are located at sites of the q-lattice, where a dimensionless parameter q = r22/r21 is given by the square ratio of the cylinder radii. The resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. Our approach in this paper provides an efficient path to rediscover known solutions for the vortex–cylinder pair problem and yields new solutions as well. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we show that our solution coincides with the known one, obtained before by elliptic functions. The Schottky–Klein prime function for the annular domain is factorized explicitly in terms of q-exponents. The Hamiltonian, the Kirchhoff–Routh and the Green functions are constructed. As a new application of our approach, the uniformly rotating exact N-vortex polygon solutions with the rotation frequency expressed in terms of q-logarithms at Nth roots of unity are found. In particular, we show that a single vortex orbits the cylinders with constant angular velocity, given as the q-harmonic series. Vortex images in two particular geometries with only one cylinder as the q → ∞ limit are studied in detail.

The Riccati differential equation and a diffusion-type equation

Abstract: We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The heat kernel is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of a Riccati differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.

Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format

Abstract: We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10? 15ulp, and we give the worst ones. In particular, the worst case for |x| ? 3 ×10? 11is $\exp(9.407822313572878 \times 10^{-2}) = 1.098645682066338\,5\,0000000000000000\,278\ldots$. This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.

The Newtonian force experienced by a point mass near a finite cylindrical source

Abstract: The Newtonian gravitational force experienced by a point mass located at some external point from a thick-walled, hollow and uniform finite circular cylindrical body was recently solved by Lockerbie, Veryaskin and Xu (1993 Class. Quantum Grav. 10 2419). Their method of attack relied on the introduction of the circular cylindrical free-space Green function representation for the inverse distance which appears in the formulation of the Newtonian potential function. This ultimately leads Lockerbie et al to a final expression for the Newtonian potential function which is expressed as a double summation of even-ordered Legendre polynomials. However, the kernel of the cylindrical free-space Green function which is represented by an infinite integral of the product of two Bessel functions and a decaying exponential can be analytically evaluated in terms of a toroidal function. This leads to a simplification in the mathematical analysis developed by Lockerbie et al. Also, each term in the infinite series solution for the Newtonian potential function can be expressed in closed form in terms of elementary functions. The authors develop the Newtonian potential function by employing toroidal functions of zeroth order or Legendre functions of half-integral degree, (Bouwkamp and de Bruijn 1947 J. Appl. Phys.18 562, Cohl et al 2001 Phys. Rev.A 64 052509-1, Selvaggi et al 2004 IEEE Trans. Magn.40 3278). These functions are monotonically decreasing and converge rapidly (Moon and Spencer 1961 Field Theory for Engineers (New Jersey: Van Nostrand Company) pp 368?76, Cohl and Tohline 1999 Astrophys. J.527 86). The introduction of the toroidal harmonic expansion leads to an infinite series solution for which each term can be expressed as an elementary function. This enables one to easily compute the axial and radial forces experienced by an internal or an external point mass.

Arithmetic Circuit Optimization in Finite Word Length Approximation of Arbitrary Functions

Abstract: In digital circuit design using a hardware description language, some elementary functions and user defined functions can not be expressed directly. In such cases, the target function is approximated as a polynomial to be implemented by additions and multiplications. If an approximation error exceeds the permissible one, it can be corrected with a look-up table (LUT). However, considering discrete and nonlinear relation between the approximation error and the circuit area, optimization of reducing the total hardware cost would be more complex. In this research, for an arbitrary function, we propose the optimization technique of minimizing the hardware cost. The proposed method starts with multiple initial solutions based on two hardware models. Furthermore, for optimization algorithm, tabu search (TS) is used, and it is parallelized to make a global search.

An efficient method for evaluating polynomial and rational function approximations

Abstract: In this paper we extend the domain of applicability of the E-method [7, 8], as a hardware-oriented method for evaluating elementary functions using polynomial and rational function approximations. The polynomials and rational functions are computed by solving a system of linear equations using digit-serial iterations on simple and highly regular hardware. For convergence, these systems must be diagonally dominant. The E-method offers an efficient way for the fixed-point evaluation of polynomials and rational functions if their coefficients conform to the diagonal dominance condition. Until now, there was no systematic approach to obtain good approximations to f over an interval [a, b] by rational functions satisfying the constraints required by the E-method. In this paper, we present such an approach which is based on linear programming and lattice basis reduction. We also discuss a design and performance characteristics of a corresponding implementation.

Integrability of some classes of dynamic systems in terms of elementary functions

Abstract: The results of this paper are due to some previous studies of the application problem of motion of a rigid body in a resisting medium. During these studies, a transcendental integral expressed in terms of elementary functions was obtained for a particular case. This fact allowed one to completely analyze all phase trajectories and to indicate those properties that were “rough” and were retained for some more general systems. The integrability of such a system is related to latent symmetries. Therefore, the study of sufficiently wide classes of dynamic systems with similar symmetries is of interest.

Properties of sums of some elementary functions and modeling of transitional and other processes

Abstract: The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the presented findings have broader meaning and can be used for approximation of transitional and other processes in different areas of science and technology. We present discovered properties of sums of polynomial, power, and exponential functions of one variable. It is shown that for an equation composed of one type of function there is a corresponding equation composed of functions of the other type. The number of real solutions of such equations and the number of characteristic points of certain appropriate corresponding functions are closely related. In particular, we introduce a method similar to Descartes Rule of Signs that allows finding the maximum number of real solutions for the power equation and equation composed of sums of exponential functions. The discovered properties of these functions allow us to improve the adequacy of mathematical models of real phenomena.

Representations of Two-Variable Elementary Functions Using EVMDDs and their Applications to Function Generators

Abstract: This paper proposes a method to represent two-variable elementary functions using edge-valued multi-valued decision diagrams (EVMDDs), and presents a design method and an architecture for function generators using EVMDDs. To show the compactness of EVMDDs, this paper introduces a new class of integer-valued functions, l-restricted Mp-monotone increasing functions, and derives an upper bound on the number of nodes in an edge-valued binary decision diagram (EVBDD) for the l-restricted Mp-monotone increasing function. EVBDDs represent l-restricted Mp- monotone increasing functions more compactly than MTB- DDs and BMDs when p is small. Experimental results show that all the two-variable elementary functions considered in this paper can be converted into l-restricted Mp- monotone increasing functions with p = 1 or p = 3, and can be compactly represented by EVBDDs. Since EVMDDs have shorter paths and smaller memory size than EVBDDs, EVMDDs can produce fast and compact elementary function generators.

New Infinite Product Representations of Some Elementary Functions

Abstract: The derivation of Green’s functions is revisited in a trivial case of standard boundary value problems for the two-dimensional Laplace equation. Regions of a regular shape are considered, with Dirichlet and/or Neumann boundary conditions imposed. Classical closed analytic form of Green’s functions are reviewed and the method of images is used for obtaining their alternative representations in terms of infinite products. The latter are obviously less attractive compared to the closed form of Green’s functions. But the point, however, is that a surprising aspect was discovered when the two forms are compared. This brings some new ‘summation’ formulae for infinite products leading, in turn, to unlooked-for results in the approximation of elementary functions. Mathematics Subject Classification: 26A09, 35J25, 40A20

One-parameter families of conformally related asymptotically flat, static vacuum data

Abstract: Extending the results of Friedrich (2008 Class. Quantum Grav. 25 065012) we give a complete description of the asymptotically flat, conformally non-flat, static vacuum data which admit non-trivial, asymptotically smooth conformal mappings onto other such data. These data form a 3-parameter family which decomposes into 1-parameter families of data which are conformal to each other. The data and the associated static vacuum solutions are given explicitly in terms of elliptic and, in a special case, elementary functions.

Havriliak-Negami function for thermal system identification

Abstract: Fractional differentiation models have proven their usefulness in representing high dimensional systems with only few parameters. Generally, two elementary fractional functions are used in time-domain identification: Cole-Cole and Davidson-Cole functions. A third elementary function, called Havriliak-Negami, generalizes both previous ones and is particularly dedicated to dielectric systems. The use of this function is however not very popular in time-domain identification because it has no simple analytical impulse response. The only synthesis method of Havriliak-Negami elementary functions proposed in the literature is based on diffusive representation which sets restrictive conditions on fractional orders. A new synthesis method, with no such restrictions, is based on the splitting the Havriliak-Negami function into a Davidson-Cole function and a complementary one. Both functions are then synthesized in a limited frequency band using a recursive distribution of poles and zeros developed by [Ous95].

Feynman formula for a diffusion of particles with a variable mass in a domain

Abstract: In the present note we consider a class of second order parabolic equations with position dependent coefficients; such equations describe a diffusion of (quasi) particles with a variable mass. We represent a solution of Cauchy-Dirichlet problem for such class of equations in a bounded domain in the form of a limit of finite dimensional integrals of elementary functions. Such kind of a representation is usually called Feynman formula and can be used for calculations. Finite dimensional integrals in our Feynman formula give approximations for a functional integral over a probability measure on a set of trajectories in the domain where the solution of the considered problem is investigated; this measure is generated by a diffusion process with variable diffusion coefficient and absorption on the boundary, hence, to get Feynman formula also means to get a representation of the solution of the considered problem with the help of a functional integral (such kind of a representation is usually called Feynman-Kac formula).

Application of the principle of symmetry to the solution of the slichter problem

Abstract: This problem was first studied in [1] under the assumption that the water depth in channels is negligible However, the use of the Fourier solution method results in the following fact The convergence of the Fourier series obtained for components of the filtration rate becomes very slow as the ratio L/T increases, ie, in the most cases which are of interest for practice In [2] (P 181) one solves a similar problem with h = 0 by the method of conformal mappings In [3] one solves the problem by the same method, taking into account the presence of the water in channels However, the introduction of an auxiliary parametric variable for the conformal mapping of the flow domain and the plane of the velocity hodograph curve results in rather intricate formulas For this reason, no numeric calculations were performed and the influence of the parameters onto the character of the flow was not studied Let us apply the Riemann–Schwarz symmetry method ([4], P 147) and the known properties of elliptic functions ([5], pp 69–73) Below we show that this allows us to avoid difficulties connected with the use of the method of conformal mappings and to obtain an exact analytic solution to the problem by comparatively simple calculations (using the known special or elementary functions)

Constrained piecewise polinomial approximation for hardware implementation of elementary functions

Abstract: This paper presents a novel technique for designing piecewise polynomial interpolators for hardware implementation of elementary functions. In the proposed approach, we impose special constraints between polynomial coefficients of adjacent segments. This allows to significantly reduce look-up table size with respect to standard, unconstrained piecewise polynomial approximations, with negligible reduction in accuracy. The reduction of look-up table size improves performances in terms of area and speed. Implementations of linear and quadratic interpolators for the reciprocal function f(x)=1/x are presented and analyzed as an application example in the paper.

Nonholonomic Trajectory Optimization and the Existence of Twisted Matrix Logarithms

Abstract: Summary. The problem studied here is a shortest path problem of the type encountered in sub-Riemannian geometry. It is distinguished by special structures related to its Lie group setting and the Z 2 graded structure on the relevant Lie algebra. In spite of the fact that the first order necessary conditions lead to differential equations that are integrable in terms of elementary functions, in this case there remain questions related to the existence of appropriate values for the parameters which appear. In this paper we treat the problem in some generality but establish the existence of suitable parameter values only in the case of the general linear group of dimension two.

Overall parameter e.g. arithmetic and logic unit size, dimensioning method for e.g. controlling operation of power train of motor vehicle, involves determining parameter from elementary parameter and total number of executions

Abstract: The method involves decomposing an overall function into predefined elementary functions e.g. OR operation. Total number of executions necessary to the execution of the overall function is determined for each elementary function. An overall parameter of a logical controller is determined from an elementary parameter corresponding to each elementary function and the total number of executions. The decomposition and determining operations are repeated for each parameter.

Solving second order ordinary differential equations by extending the PS method

Abstract: An extension of the ideas of the Prelle-Singer procedure to second order differential equations is proposed. As in the original PS procedure, this version of our method deals with differential equations of the form y ′′ = M(x,y,y ′ )/N(x,y,y ′ ), where M and N are polynomials with coefficients in the field of complex numbers C. The key to our approach is to focus not on the final solution but on the first-order invariants of the equation. Our method is an attempt to address algorithmically the solution of SOODEs whose first integrals are elementary functions of x, y and y ′ .

Infinite block-structured matrixes in parametrical circuit theory

Abstract: In parametrical radio-circuits in case of simple input action a compound responses, that can not be expressed by finite number of elementary functions, occur. They should be expressed in form of infinite series, then the task is reduced to solution of infinite equations systems. It is considered their features and given solution advice on the example of parametrical circuit.