Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

Design of a reconfigurable transcendental function generator

In order to take into account the computing efficiency and flexibility of calculating transcendental functions, this paper proposes one kind of reconfigurable transcendental function generator. The generator is of a reconfigurable array structure composed of 30 processing elements （PEs）. The coordinate rotational digital computer （CORDIC） algorithm is implemented on this structure. Different functions, such as sine, cosine, inverse tangent, logarithmic, etc., can be calculated based on the structure by reconfiguring the functions of PEs. The functional simulation and field programmable gate array （FPGA） verification show that the proposed method obtains great flexibility with acceptable performance.

Arbitrary Waveforms Based on Front Panel of Agilent 33250A Function/Arbitrary Waveform Generator

Agilent 33200A family of function/arbitrary waveform generators are widely used in labs for creating arbitrary waveforms.Flexible applications of function/arbitrary waveform generator 33250A which is made by Agilent company are expatiated.There are three methods of transferring waveform data to arbitrary waveform generator 33250A,among which,the front panel method can produce a simple interface for arbitrary waveforms and is applicable to the composition of a small amount of linear waveform segment,and the progress of this method is explained in detail.This way is convenient and can be widely used,and it will offer some good guidance in library works.

Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

The Fourier Transform of a Sort of Generalized Function

The Fourier transforms of functions X ln e are given. where λ is a complex constant (λ≠-1,-2. …), a is a complex number. m=0. 1. 2,…,and

The Similarity Transformation of Regularizing Functionals and Simularity Property of Their Fourier Transformations

Let and denote respectively the functionswhere λ≥1, The author discusses the similarity transformation of the regularizing functionals of these functions and the similar property of their Fourier transformation.

The generalized Chapman-Richards function and applications to tree and stand growth

The generalized Chapman-Richards model was derived from the Chapman-Richards function in which parameters h, k and m were unconstrained. Based on the structure of solutions and biological interpretations, the model could be classified into eight cases (three categories) at all and among them only 4 kinds of cases are suitable in forestry that represent four typical growth patterns of trees and stands. For each of 4 equations, the model properties and biological interpretations for parameters were discussed in detail. The generalized Chapman-Richards model was capable of describing a wide range of growth curves that was asymptotic or nonasymptotic, with or without inflection point. In order to illustrate the versatility of the model, it was fitted to a group of data sets concerning the DBH growth of cryptomeria plantations with 4 initial densities and the DBH and height growth of natural Korean pine tree. Comparing the generalized Chapman-Richards function and the Schnute model, it was found that the parameters and expressions of the two models were interchangeable in theory, and the fitting results were explicitly identical in empirical applications.

CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.

THIRD-ORDER DIFFERENTIAL SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z（Bκ＋1^c f（z））′= κBκ^c f（z）-（κ- 1）Bκ＋1^c f（z）,where b, c, p ∈ C and κ = p ＋（b ＋ 1）/2 ∈ C ／ Z0^-（Z0^-= {0,-1,-2, … }）. The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.

Optimization of the Dressing Parameters in Cylindrical Grinding Based on a Generalized Utility Function

The existing studies, concerning the dressing process, focus on the major influence of the dressing conditions on the grinding response variables. However, the choice of the dressing conditions is often made, based on the experience of the qualified staff or using data from reference books. The optimal dressing parameters, which are only valid for the particular methods and dressing and grinding conditions, are also used. The paper presents a methodology for optimization of the dressing parameters in cylindrical grinding. The generalized utility function has been chosen as an optimization parameter. It is a complex indicator determining the economic, dynamic and manufacturing characteristics of the grinding process. The developed methodology is implemented for the dressing of aluminium oxide grinding wheels by using experimental diamond roller dressers with different grit sizes made of medium- and high-strength synthetic diamonds type AC32 and AC80. To solve the optimization problem, a model of the generalized utility function is created which reflects the complex impact of dressing parameters. The model is built based on the results from the conducted complex study and modeling of the grinding wheel lifetime, cutting ability, production rate and cutting forces during grinding. They are closely related to the dressing conditions （dressing speed ratio, radial in-feed of the diamond roller dresser and dress-out time）, the diamond roller dresser grit size/grinding wheel grit size ratio, the type of synthetic diamonds and the direction of dressing. Some dressing parameters are determined for which the generalized utility fimction has a maximum and which guarantee an optimum combination of the following： the lifetime and cutting ability of the abrasive wheels, the tangential cutting force magnitude and the production rate of the grinding process. The results obtained prove the possibility of control and optimization of grinding by selecting particular dressing parameters.

THE HILBERT BOUNDARY VALUE PROBLEM FOR GENERALIZED ANALYTIC FUNCTIONS IN CLIFFORD ANALYSIS

Let R0,n be the real Clifford algebra generated by e1, e2,... , en satisfying eiej＋ejei=-2δij,i,j=1,2…,ne0 is the unit element.Let Ω be an open set. A function f is called left generalized analytic in ft if f satisfies the equation Lf=0,where ……qi 〉0, i =-, 1, - ……, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in Rn＋1 will be investigated.

MONOTONICITY AND INEQUALITIES FOR THE GENERALIZED DISTORTION FUNCTION

The authors prove some monotonicity properties of functions involving the generalized Agard distortion function ηg（a,t）, and obtain some inequalities for ηk（a, t） and relative distortion functions.

ORTHOGONAL POLYNOMIALS AND DETERMINANT FORMULAS OF FUNCTION-VALUED PADE-TYPE APPROXIMATION USING FOR SOLUTION OF INTEGRAL EQUATIONS

To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.

Some inequalities for the generalized linear distortion function

In this paper, we establish several inequalities for the the generalized linear distortion function λ（a, K） by using the monotonicity and convexity of certain combinations λ（a, K）.

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

APPROXIMATE CONVEXITY AND CONCAVITY OFGENERALIZED GROTZSCH RING FUNCTION

In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa （r） by studying the monotonieity,convexity or concavity of certain composites of μa（r） are obtained.

Generating function of product of bivariate Hermite polynomials and their applications in studying quantum optical states

By virtue of the operator-Hermite-polynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are presented.

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m（x）,∞∑n=0sntm/n!m!H2n,2m（x,y）,and ∞∑n=0sntm/n!m!H2n＋l,2m＋k（x,y）.We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.

GENERALIZED BOCHNER'S THEOREM FOR RADIAL FUNCTION

A radial function can be expressed by its generator through The positive definite of the function plays an important rote in the radial basis interpolation. We can naturally use Bochner's Theorem to check if is positive definite. This requires however a n-dhnensiotial Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-di-mensional space. The completely monotone function:, which is discussed in [4] is positive definite for arbitrary space dimensions. With this technique tve can very easily characterize the positive definite, of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, whcih are very useful in application. Can we find some conditions (as the completely monotone function) only for the \-dimen simial Fourier transform of the generator epto characterize a radial function 9, which is positivedefinite in n-dimensional (fixed n) spacel In this paper we defined a kind of incompletelymonotone function of order a, for a= 0,,1/2 ,1,3/2,(we denote the function class by ICM) ,in this sence a normal positvie function is in ICM a positive monotone decreasing function is inICM and a positive monotone decreasing and convex function is in ICM2- Based on this definition we get a generalized Bochner's Theorem for radial function-. If dimensional Fouriertransform of the generator of a radial function can be written as , then corre-spending radial function (x) is positive definite as a n-variate function iff F is an incomplete-ly monotone function of order a= (n- 1 )/2 (or simply In this way we have characterized the positive definite of the radial function as a n-vari-ate function through its generator in the sense of the Bocher's Theorem.