Error analysis of large-time approximations for decay chains

Consider a radioactive decay chain X₁ → ? → X_n→ and let N_n(t) be the amount of X_n at time t. This paper establishes error bounds for large-time approximations to N_n(t) that include and generalize the transient equilibrium approximations and other known approximations. The error bounds allow one to find the range of t for which these approximations can be used with a given degree of precision.     

卷积算子的Gibbs现象

本文建立了一般卷积算子在函数的第一类间断点处有Gibbs现象的刻画定理,并且给出了某些例子.     

关于Erlang分布的一个注记

利用递推的方法得到了尺度参数不同的两个相互独立的Erlang分布的卷积的表达形式.作为应用得到了Beta分布的Laplace变换的表达式.     

Singular Integral Equations of Convolution Type with Reflection and Translation Shifts

This article deals with some kinds of singular integral equations of convolution type with reflection in class {0}. Such equations are transformed into the Riemann boundary value problems with both discontinuous coefficients and reflection by Fourier transform. For such problems, we propose one method different from classical one, by which the explicit solutions and the conditions of solvability are obtained. Finally, we propose and discuss singular integral equations with reflection and translation shifts.     

Convolution theorems associated with some integral operators and convolutions

In this article, various convolution theorems involving certain weight functions and convolution products are derived. The convolution theorems we obtain are more general, convenient, and efficient than the complicated convolution theorem of the Hartley transform. Further results involving new variants of generalizations of Fourier and Hartley transforms are also discussed.     

广义次指数密度的两个新的等价条件

得到了广义次指数密度的两个新的等价条件,刻划了它们的卷积封闭性和卷积根封闭性.     

On the Convolution of the Generalized Finite Legendre Transform

The translation operator and the convolution for the finite Legendre transformation are investigated in the space ??(?1,1) of testing-functions and its dual through an approach that emphasizes the close similarity existing between this transform and the infinite Mehler - Fock transformation. The theory developed is used in solving some distributional boundary-value problems.     

Asymptotic behavior of the resolvent of an unstable volterra equation with kernel depending on the difference of the arguments

We present the structure of the resolvent of a difference kernel, which allows us to study the asymptotic behavior of the solution of the renewal equation for a given asymptotic behavior of the constant term. An asymptotic representation for the resolvent is obtained under minimal requirements on the moments of the kernel. Similar results are given for integro-differential equations. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 88–94, July, 1997. Translated by M. A. Shishkova     

Some New Results for McKean’s Graphs with Applications to Kac’s Equation

The main goal of the present paper is to sharpen some results about the error made when the Wild sums, used to represent the solution of the Kac analog of Boltzmann’s equation, are truncated at the n-th stage. More precisely, in Carlen, Carvalho and Gabetta (J. Funct. Anal. 220: 362–387 (2005)), one finds a bound for the above-mentioned error which depends on (an ^Λ+ε). On the one hand, it is shown that Λ, the least negative eigenvalue of the linearized collision operator, is the best possible exponent. On the other hand, ε is an extra strictly positive number and a a positive coefficient which depends on ε too. Thus, it is interesting to check whether ε can be removed from the above bound. According to the aforesaid reference, this problem is studied here by means of the probability distribution of the depth of a leaf in a McKean random tree. In fact, an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with (a′ n ^Λ).     

Spectral Line Profile Correction of the Detector Effect: for a SOHO/UVCS Spectrometer

It has been noticed that observed spectral line profiles by the SOHO ultraviolet coronagraph spectrometer (SOHO/UVCS) are contaminated by a so-called detector effect, and an iterative correction of observed profiles by convolution with a zero-integral function has been employed to remove this unwanted effect on UVCS detectors. It is shown here that this iteration procedure using a zero-integral function can be replaced by an only-once application of convolution. For zero-integral functions of exp[- |x|/L]-type, the convolution function in this method is proved to have exactly the same, but narrower (steeper), functional form as the original one. These results are also valid, with some modifications, for the discrete case used in numerical calculation.