Convergence of Eigenfunction Expansions of a Differential Operator with Integral Boundary Conditions

For a second-order ordinary differential operator on an interval of the real line with integral boundary conditions, conditions for the unconditional basis property and uniform convergence of the expansion of a function in terms of the eigen- and associated functions of this operator are established. The convergence and equiconvergence rates of this expansion and the equiconvergence rate of the trigonometric Fourier expansion of this function are estimated. The uniform convergence of its expansion in the adjoint system is studied.

     

Malliavin calculus, geometric mixing, and expansion of diffusion functionals

Under geometric mixing condition, we presented asymptotic expansion of the distribution of an additive functional of a Markov or an ε-Markov process with finite autoregression including Markov type semimartingales and time series models with discrete time parameter. The emphasis is put on the use of the Malliavin calculus in place of the conditional type Cramér condition, whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach. In the second part, by means of the perturbation method and the operational calculus, we proved the geometric mixing property for non-symmetric diffusion processes, and presented a sufficient condition which is easily checked in practice. Accordingly, we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions, e.g., without the strong contractivity condition. Received: 7 September 1997 / Revised version: 17 March 1999     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system

In this paper, we first study the analytical property of the first Melnikov function for general Hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4, 5 or 6 limit cycles in a neighborhood of loop. As an application of our main results, we consider some polynomial Lienard systems and find 4, 5 or 6 limit cycles.     

An adaptive boundary collocation method for linear PDEs

The article proposes an adaptive algorithm based on a boundary collocation method for linear PDEs satisfying the maximal principle with possibly nonlinear boundary conditions. Given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficients by collocation of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error tolerance, additional expansion terms and boundary collocation points are added and the process repeated until the tolerance is satisfied. The performance of the algorithm is illustrated by an example of the potential flow past a cylinder placed between parallel walls. © 1995 John Wiley & Sons, Inc.     

Loose almost blowing-up and related properties

It is shown that finitely-fixed processes have -metric versions of the almost blowing-up property, the approximate-match waiting-time property, and the strong form of extremality. The proofs make use of a new result that connects, in a local almost sure way, fixed-length blocks in a process with fixed-length blocks in an expansion of the process. Supported in part by joint NSF-Hungarian Academy grant INT-9515485.     

Markov process functionals in finance and insurance

The Markov property of Markov process functionals which are frequently used in economy, finance, engineering and statistic analysis is studied. The conditions to judge Markov property of some important Markov process functionals are presented, the following conclusions are obtained： the multidimensional process with independent increments is a multidimensional Markov process; the functional in the form of path integral of process with independent increments is a Markov process; the surplus process with the doubly stochastic Poisson process is a vector Markov process. The conditions for linear transformation of vector Markov process being still a Markov process are given.     

Series expansions of analytic functions

Many of the classical polynomial expansions of analytic functions share a common property: the space of “expandable” functions is a Banach space isometrically isomorphic to the space of complex sequences with limit 0. Under the isometries, these polynomial expansions all correspond to essentially the same biorthogonal expansion in this sequence space. Sufficient conditions for such an isometry to exist are obtained, and convergence properties of the expansions are studied. The results obtained also apply to expansions other than polynomial expansions.     

Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence

In this paper, we propose a modified extragradient method for solving variational inequalities (VI) which has the following nice features: (i) The generated sequence possesses an expansion property with respect to the starting point; (ii) the existence of the solution to a VI problem can be verified through the behavior of the generated sequence from the fact that the iterative sequence diverges to infinity if and only if the solution set is empty. Global convergence of the method is guaranteed under mild conditions. Our preliminary computational experience is also reported.     

The method of homogeneous solutions in an axisymmetric problem of statics for a finite cylinder with homogeneous conditions on the ends

Using the property of Papkovich generalized orthogonality of eigenfunctions, we develop a method of satisfying the boundary conditions on the lateral surface of a cylinder. The stresses and displacements in a finite cylinder with homogeneous conditions on the ends are represented in terms of the axial displacement. The solution is constructed as an expansion in a series of eigenfunctions of the corresponding homogeneous boundary-value problem. We find a class of boundary conditions that admits a solution of the problem without reduction to an infinite system of algebraic equations. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 135–139.     

New sufficient conditions of existence,moment estimations and non confluence for SDEs with non-Lipschitzian coefficients

The objective of the present paper is to find new sufficient conditions for the existence of unique strong solutions to a class of (time-inhomogeneous) stochastic differential equations with random, non-Lipschitzian coefficients. We give an example to show that our conditions are indeed weaker than those relevant conditions existing in the literature. We also derive moment estimations for the maximum process of the solution. Finally, we present a sufficient condition to ensure the non confluence property of the solution of time-homogeneous SDE which, in one dimension, is nothing but stochastic monotone property of the solution.     

Multiple stochastic integrals constructed by special expansions of products of the integrating stochastic processes

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some necessary and sufficient conditions are obtained for the existence of multiple stochastic integrals defined by an expansion of the product of Wiener processes. It was obtained a recurrent representation for the Wiener stochastic integral as an analog of the Hu–Meyer formula.     

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

This paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.     

Korovkin type theorems and approximate Hermite–Hadamard inequalities

The main results of this paper offer sufficient conditions in order that an approximate lower Hermite–Hadamard type inequality implies an approximate convexity property. The failure of such an implication with constant error term shows that functional error terms should be considered for the inequalities and convexity properties in question. The key for the proof of the main result is a Korovkin type theorem which enables us to deduce the approximate convexity property from the approximate lower Hermite–Hadamard type inequality via an iteration process.     

Non-conservative oscillations of a tool for deep hole honing

The dynamics of a rotating tool, commonly employed in deep hole honing, is considered. A mathematical model of the process including a dynamic representation of tool, workpiece surface and honing stones interaction is suggested and analyzed. It is shown that interaction forces are non-conservative. The honing tool is modeled as a rotating continuous slender beam with a mandrel attached at an intermediate cross-section. The transverse oscillations of tool shaft cause variation in expansion pressure and change the interaction forces on the surface of the workpiece. The interaction forces are nonlinear and non-conservative in the general case including delayed functions. The expansion pressure of stones turns out to be a critical parameter in a honing process. Since the removal of stock increases linearly with pressure, the productivity can be improved by increasing expansion pressure, but in certain conditions may cause dynamic instability of the tool shaft. On the other hand, the lower expansion pressure and feed rates increase accuracy. Another source of shaft instability is due to the asymmetry of the tool that leads to discrepancy of the system stiffness in the transverse directions. As a result, under certain conditions unstable parametric vibrations may occur. Needless to say, the dynamic behavior of the tool can considerably influence the machined surface formation. In this study a model of honing surface formation is introduced and integrated into the model of process simulation. The system response is studied for different parameters in a non-dimensional form. Thus it is possible to analyze a set of real processes by applying the similarity conditions. The variation in the machined surface having various initial discrepancies from ideal cylinder is also analyzed.     

Functions harmonic for a Markov process

The compactness property of a family of functions harmonic for a Markov process is studied and, in particular, an inequality of Harnack type is derived. It is shown that under broad conditions the property that a function be locally harmonic implies that it is harmonic.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 587–596, April, 1973.     

Implementing the Nelder-Mead simplex algorithm with?adaptive parameters

In this paper, we first prove that the expansion and contraction steps of the Nelder-Mead simplex algorithm possess a descent property when the objective function is uniformly convex. This property provides some new insights on why the standard Nelder-Mead algorithm becomes inefficient in high dimensions. We then propose an implementation of the Nelder-Mead method in which the expansion, contraction, and shrink parameters depend on the dimension of the optimization problem. Our numerical experiments show that the new implementation outperforms the standard Nelder-Mead method for high dimensional problems.     

Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model

With the help of a general methodology of asymptotic expansions for mixing processes, we obtain the Edgeworth expansion for log-returns of a stock price process in Barndorff-Nielsen and Shephard's stochastic volatility model, in which the latent volatility process is described by a stationary non-Gaussian Ornstein–Uhlenbeck process (OU process) with invariant selfdecomposable distribution on ${\mathbb{R}}_{+}$. The present result enables us to simultaneously explain non-Gaussianity for short time-lags as well as approximate Gaussianity for long time-lags. The Malliavin calculus formulated by Bichteler, Gravereaux and Jacod for processes with jumps and the exponential mixing property of the OU process play substantial roles in order to ensure a conditional type Cramér condition under a certain truncation. Owing to several inherent properties of OU processes, the regularity conditions for the expansions can be verified without any difficulty, and the coefficients of the expansions up to any order can be explicitly computed.     

Mathematical modeling and process simulation of perlite grain expansion in a vertical electrical furnace

Expanded perlite is a lightweight material with remarkable thermal and acoustic insulation properties, rendering it widely useful in the construction and manufacturing industries. Currently applied perlite expansion technology suffers numerous technical disadvantages, which adversely affect product quality and limit the range of its applications. To overcome these established drawbacks, a new perlite expansion process has been designed on the basis of a vertical electrically heated expansion furnace. The novel furnace enables precise control of experimental conditions, in order to allow for efficient adjustment of particle residence time and internal temperature. The quality of expanded perlite strongly depends on raw material thermophysical properties as well as furnace operating conditions, and the experimental investigation of the isolated effect of each parameter on expanded product quality is technically cumbersome and extremely time-consuming and expensive.A mathematical model for perlite grain expansion has been developed in order to perform a detailed numerical investigation of process efficiency, toward the optimization of the expansion process in the novel pilot-scale furnace. The dynamic model consists of ordinary differential equations for both air and particle heat and momentum balances, as well as nonlinear algebraic equations for both air and perlite melt thermophysical and transport properties, probing the air temperature distribution within the vertical electrical furnace as well as the particle velocity, temperature and size along its trajectory inside the heating chamber. The effect of raw material physical properties (raw feed origin, initial particle size, effective water content) as well as operating parameters (air inlet temperature and flowrate, furnace wall temperature) on evolution of the particle state variables is presented and discussed. Model results indicate perlite expansion is strongly affected by raw ore feed origin, size and water content. Moreover, operating conditions affect expansion considerably, and furnace wall temperature has the strongest effect on the final particle expansion ratio attained. The new dynamic model is instrumental towards achieving a detailed comprehension of perlite expansion in the vertical electrical furnace towards multi-parametric sensitivity analysis, process optimization and efficient control.     

遗传中紧空间与散射分解

本文证明了可数仿紧(中紧、亚紧)空间有类似Junnila的刻画，遗传中紧空间不具有类似Junnila的刻画，并给出了每个散射分解有紧有限的开膨胀的充要条件．