Hausdorff Calculus and Some of Its Applications

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论平面上的可求长曲线

本文利用Hausdorff分数型导数与积分理论给出平面上可求长曲线的积分表 达式.     

Stochastic Functional Inclusion Driven by Semimartingale

We define a set-valued stochastic process and an integral of a such process with respect to a semimartingale. Next we consider a stochastic integral inclusion. Using fixed point methods we prove some results on the existence of its solutions     

On a problem of Dynkin

Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

     

A comparison of two equivalent solutions of the diffusion equation

Plane radial flow of water in a closed aquifer towards a circular well or heat flow through a homogeneous conducting solid from a circular hole to infinity are well known problems that were solved long ago. The solution is expressed in terms of an integral of ordinary Bessel functions. A new solution, which is expressed in terms of an integral of modified Bessel functions, is derived by means of Laplace transformation instead of integration of Green's function. One particular choice of the Bromwich contour in the invers transformation, which appears unnecessarily complicated, gives a solution the integral of which converges much more rapidly than the corresponding integral that is given in the literature. In contrast with the classical solution, the new one gives an explicit expression for the transient part. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 657–671, 1999     

Quasi-Homeomorphisms and Measures of Finite Energy Integrals of Generalized Dirichlet Forms

We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover, we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is ɛ-exceptional if and only if μ (B) = 0 for any measure μ of finite energy integral. Received May 28, 1999, Revised September 8, 1999, Accepted December 10, 1999     

Mixed Functionals of Convex Bodies

We define a class of real functions on tuples of convex bodies. They are a common generalization of mixed volumes and of certain functionals which have been studied in translative integral geometry. For polytopes, these functionals have various explicit representations in terms of volumes of lower-dimensional faces. For the mentioned functionals from integral geometry, these representations generalize a result of Weil and answer a question posed by Janson. Received January 5, 1999, and in revised form March 12, 1999. Online publication May 16, 2000.     

Integrals of certain random functions with respect to general random measures

For random functions that are sums of random functional series, we determine an integral over a general random measure and prove limit theorems for this integral. We consider the solution of an integral equation with respect to an unknown random measure. National University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1087–1095, August, 1999.     

A new approach to the theory of functional integral equations of fractional order

The aim of the paper is to present a new approach to the theory of functional integral equations of fractional order. That approach depends on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type. It turns out that the study of functional integral equations of Volterra-Stieltjes type is more convenient and effective than the study of functional integral equations of fractional order. An example illustrating our approach is also discussed.     

S~O-类函数的积分性质及其在分形中的应用

利用非标准分析的方法,给出了S~0-类实函数的一个积分不等式和一个积分等式:(1)设A是标准的完备度量空间(X,d)中的一个准标准内的子集,μ是X上的一个Borel正则有限的标准外测度.若f是X上的一个非负的S~0-类实函数,则有°(∫_Aμ)≤∫_(°A)°fdμ.特殊情况当f≡1时,有°(μ(A))≤μ(°A);(2)设E是标准的s-紧集,若f是E上的一个S~0-类实函数,则°(∫_E fdH~s)=∫_E°fdH~s,这里°(*)指的是*的影子.并给出了这些结果在分形几何中用以判断一个分形集其内部是否非空的方法及一个分形函数在Hausdorff测度空间上的积分的计算方法,并给出了相应的实例加以验证.     

Representations of the additive group of a nuclear space in terms of self-adjoint operators

We prove theorems on integral representations of the additive group of a real nuclear space in terms of self-adjoint operators. We assume that algebraic relations are realized in a dense invariant set of integral vectors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol.51, No. 2, pp. 271–274, February, 1999.     

Monotonic solutions of a quadratic integral equation of fractional order

In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.     

On primitively universal quadratic forms

In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of quadratic forms and investigate a primitive counterpart to the Fifteen Theorem. In particular, we give an efficient method for deciding whether a positive definite integral quadratic form in four or more variables with odd square-free determinant is almost primitively universal.     

Decomposition of Polytopes and Polynomials

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes. Received December 2, 1999, and in revised form November 6, 2000. Online publication May 4, 2001.     

Normality and paracompactness of the Fell topology

Let be a Hausdorff topological space and the hyperspace of all closed nonempty subsets of . We show that the Fell topology on is normal if and only if the space is Lindelöf and locally compact. For the Fell topology normality, paracompactness and Lindelöfness are equivalent.

     

The well‐posedness of the electric field integral equation for transient scattering from a perfectly conducting body

We consider the scattering of a transient electromagnetic field incident on a body with a smooth, perfectly conducting surface. A standard numerical method for calculating the scattered field is to use a time dependent, surface integral equation (called the electric field integral equation) to calculate the surface currents and charges induced by the incident field—these currents and charges then yield the scattered fields by means of standard integral representations (vector and scalar potentials). In this paper we show that the time‐dependent electric field integral equation is well‐posed in a suitable function space setting. We also investigate the behaviour of the solutions at large time. Copyright © 1999 John Wiley & Sons, Ltd.     

Sums and Differences of Two Cubic Polynomials

 When f(x) is a cubic polynomial with integral coefficients, we show that almost all integers represented as the sum or difference of two values of f(x), with , are thus represented essentially uniquely. (Received 18 January 1999; in revised form 17 May 1999)     

$C^1$ error estimation on the boundary for an exterior Neumann problem in $\mathbb R^3$

Summary.   In this paper we establish a error estimation on the boundary for the solution of an exterior Neumann problem in . To solve this problem we consider an integral representation which depends from the solution of a boundary integral equation. We use a full piecewise linear discretisation which on one hand leads to a simple numerical algorithm but on the other hand the error analysis becomes more difficult due to the singularity of the integral kernel. We construct a particular approximation for the solution of the boundary integral equation, for the solution of the Neumann problem and its gradient on the boundary and estimate their error. Received May 11, 1998 / Revised version received July 7, 1999 / Published online August 24, 2000     

Asymptotic behaviour and stability assessment of marine risers

An integral inequality is used to derive sufficient conditions for the stability of marine risers. Moreover, an estimate of the rate of decay of the solutions is obtained. © 1999 John Wiley & Sons, Ltd.