Generalization of a theorem on the quasi-analyticity of a function

It is a known fact that if a function, together with all of its derivatives, vanishes at a point, then the function will be zero in a neighborhood of the point if its successive derivatives satisfy certain estimates. We show that even if the function does not have a priori all of its derivatives but is such that its first derivative has a special sequence of majorizing functions, then in this case also the function will be equal to zero. We use our results to obtain theorems concerning the uniqueness of the solution of an abstract Cauchy problem.     

基于Beta函数及其偏导数的广义积分的高精度快速计算

考虑Beta函数偏导数的计算以及与此相关的广义积分的高精度快速计算问题.首先将Beta函数B(x,y)的定义扩展到整个复平面上,并建立了在整个复平面上Beta函数B(x,y)的偏导数的递推公式.对许多广义积分我们给出Beta函数偏导数的表示形式,因而利用Beta函数的偏导数计算这些广义积分.数值计算表明,算法无论从计算精度还是计算速度,远好于数值积分.另外,得到了B_(p,q)(x,y)存在闭形式的条件,并给出一些广义积分的闭形式.     

The Fractional Derivatives of a Fractal Function

The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.     

On the Hermite interpolation

Explicit representations for the Hermite interpolation and their derivatives of any order are provided.Furthermore,suppose that the interpolated function f has continuous derivatives of sufficiently high order on some sufficiently small neighborhood of a given point x and any group of nodes are also given on the neighborhood.If the derivatives of any order of the Hermite interpolation polynomial of f at the point x are applied to approximating the corresponding derivatives of the function f(x),the asymptotic representations for the remainder are presented.     

A globally convergent version of Robinson's algorithm for general nonlinear programming problems without using derivatives

Robinson's quadratically convergent algorithm for general nonlinear programming problems is modified in such a way that, instead of exact derivatives of the objective function and the constraints, approximations of these can be used which are computed by differences of function values. This locally convergent algorithm is then combined with a penalty function method to provide a globally and quadratically convergent algorithm that does not require the calculation of derivatives.     

近似函数高阶导数的高精度积分逼近方法

主要研究稳定计算近似函数的高阶导数的积分逼近方法,方法因由Lanczos提出故也称为Lanczos算法.利用Legendre多项式的正交性,提出了一类逼近近似函数高阶导数的高精度积分方法,即构造出一系列积分算子Dn,h(m)去逼近噪声函数的高阶导数,且这些积分算子具有O(δ(2n+2)/(2n+m+2))的收敛速度,其中δ为近似函数的噪声水平.数值模拟结果表明提出的方法是稳定而有效的.     

The Hamilton-Jacobi equation and its complementary form

The Hamilton-Jacobi equation is briefly reviewed and a complementary form of this equation is derived. Both equations involve the Hamiltonian function but, whereas in the Hamilton-Jacobi equation it is the kinetic energy of the Hamiltonian that is expressed in terms of the derivatives of the generating function, in the complementary Hamilton-Jacobi equation it is the potential energy of the Hamiltonian that is expressed in terms of the derivatives of the generating function. Three elementary problems are analyzed to illustrate the underlying theory.     

区域函数的广义导数及其应用

本文以[7]的基本概念为基础,并根据Clarke的广义导数[1],以及Lasotra和Strauss[6]的多值函数f(x)的广义微分D_f(x)的定义.从而建立了区域函数F(x)的广义导数D_F=∪∩{G(x)?B(R),?x∈B(R);G(x)=F_x=F(x)}讨论了区域函数F(x)的广义导数的存在性;建立了区域函数的广义Fréchet导数存在的必要充分条件.     

Two-loop Gell-Mann-Low function of the <Emphasis Type="Italic">N</Emphasis>=1 supersymmetric Yang-Mills theory regularized by higher covariant derivatives

We calculate the two-loop Gell-Mann-Low function for the N=1 supersymmetric Yang-Mills theory regularized by higher covariant derivatives. We show that the integrals determining this function reduce to total derivatives and can be easily calculated analytically. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 398–414, June, 2008.     

Bemerkungen zur glatten Interpolation

Summary The usual interpolation formulae for equidistant abscissae (Newton-Gregory, Bessel) define a piecewise polynomial approximation function whose first derivative is generally discontinuous at every meshpoint. It is shown how these formulae can be modified (without using higher derivatives of the given function) such that the piecewise polynomial approximation function hass continuous derivatives (wheres is a given integer).     

On the mean value property for polyharmonic functions in the ball

We obtain the mean value property for the normal derivatives of a polyharmonic function with respect to the unit sphere. We find the values of a polyharmonic function and its Laplacians at the center of the unit ball expressed via the integrals of the normal derivatives of this function over the unit sphere.     

A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force

This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang-Gao-Tenreiro Machado-Baleanu and Yang-Abdel-Aty-Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional-exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang-Abdel-Aty-Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang-Abdel-Aty-Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.     

Numerical differentiation with noisy signal

The aim of this paper is to present a new numerical method, which ables one to filter and compute numerical derivatives of a function whose values are known in some points from experimental measurements, inducing noisy data. We use a piecewise cubic spline interpolation to generate a function whose Fourier coefficients give an approximation of the numerical derivatives we are looking for. Error and stability analysis of this numerical algorithm are provided. Numerical results are presented for data smoothing and for the first and second derivatives computed from noisy data. They show that this method gives good numerical results. Comparison with other methods is done.     

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.     

Automatic differentiation of iterative processes

Automatic differentiation is used to compute the values of the derivative of a function. If the function is given by a computational graph or code list, then the derivative values can be obtained using the chain rule. An iterative process can be regarded as an infinite code list. It is well known from classical analysis that the limit of the derivatives of the code list is not necessarily equal to the derivative of the limit function. The limit of the derivatives is corect for an important class of iterative processes including generalized Newton methods.     

Differential and integral relations involving fractional derivatives of Airy functions and applications

Various differential and integral relations are deduced that involve fractional derivatives of the Airy function Ai(x) and the Scorer function Gi(x). Several new Wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. New fractional derivative conservation laws are derived for equations of the Korteweg-de Vries type.     

Uniqueness of entire function sharing a small function with its derivative

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J. Chang, Y. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2009) 491–496].     

Derivatives of the restrictions of harmonic functions on the Sierpinski gasket to segments

We give an explicit derivative computation for the restriction of a harmonic function on SG to segments at specific points of the segments: The derivative is zero at points dividing the segment in ratio 1:3. This shows that the restriction of a harmonic function to a segment of SG has the following curious property: The restriction has infinite derivatives on a dense subset of the segment (at junction points) and vanishing derivatives on another dense subset.     

Error bounds for quadrature methods involving lower order derivatives

Quadrature methods for approximating the definite integral of a function f(t) over an interval [a,b] are in common use. Examples of such methods are the Newton–Cotes formulas (midpoint, trapezoidal and Simpson methods etc.) and the Gauss–Legendre quadrature rules, to name two types of quadrature. Error bounds for these approximations involve higher order derivatives. For the Simpson method, in particular, the error bound involves a fourth-order derivative. Discounting the fact that calculating a fourth-order derivative requires a lot of differentiation, the main concern is that an error bound for the Simpson method, for example, is only relevant for a function that is four times differentiable, a rather stringent condition. This paper caters for functions for which derivatives exist only of order lower than normally required. A number of quadrature methods are considered and error bounds derived involving only lower order derivatives that can be used depending on the smoothness of the function.     

一类广义B-T方程边值问题解的存在性

研究一类含有两个分数阶导数的非线性分数阶微分方程在边值问题以及在边值非零情况下mild解的存在性,进而利用函数的初始值非零时Riemann-Liouville分数阶导数的奇异性,在加权连续函数空间上,运用Schauder不动点定理,得到关于这类问题的mild解存在的充分条件,完善和推广了某些已有结论.