Determination of jumps in terms of derivative convolution operators

X. L. Shi and W. Wang [12] studied the question on determination of jumps for functions via conjugate convolution operators. In the present paper a different way to calculate jumps is discussed. We show that derivative convolution operators can be used to determine jumps for functions. Furthurmore, we point out that the assumption “? is even” in Shi–Wang’s theorem can be removed.     

An Affirmative Result of the Open Question on Determining Function Jumps by Spline Wavelets

We study the open question on determination of jumps for functions raised by Shi and Hu in 2009. An affirmative answer is given for the case that spline-wavelet series are used to approximate the functions.     

The perturbed compound Poisson risk model with two-sided jumps

In this paper, we consider a perturbed compound Poisson risk model with two-sided jumps. The downward jumps represent the claims following an arbitrary distribution, while the upward jumps are also allowed to represent the random gains. Assuming that the density function of the upward jumps has a rational Laplace transform, the Laplace transforms and defective renewal equations for the discounted penalty functions are derived, and the asymptotic estimate for the probability of ruin is also studied for heavy-tailed downward jumps. Finally, some explicit expressions for the discounted penalty functions, as well as numerical examples, are given.     

Determination of jumps for functions based on Malvar-Coifman-Meyer conjugate wavelets

In this paper we discuss determination of jumps for non-periodic function based on Malvar- Cofiman-Meyer (MCM) conjugate wavelets. We prove the equality of Lukacs type. Furthermore we establish several criteria on concentration factors for functions that satisfy weak-smoothness condition of Dini type. This work was supported by National Natural Science Foundation of China (Grant No. 10671062)     

Concentration factors for functions with harmonic bounded mean variation

We discuss determination of jumps for functions with generalized bounded variation. The questions are motivated by A. Gelb and E. Tadmor [1], F. Móricz [5] and [6] and Q. L. Shi and X. L. Shi [7]. Corollary 1 improves the results proved in B. I. Golubov [2] and G. Kvernadze [3]. Supported by NSFC 10671062.     

A Padé-based algorithm for overcoming the Gibbs phenomenon

Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Padé approximation. The new methods compare favorably in experiments with existing techniques.     

Transformations of diffusions with jumps

The paper deals with some transformations of diffusions with jumps. We consider the class of diffusions with jumps that is closed with respect to composition with invertible, twice continuously differentiable functions. A special random time change gives us again a diffusion with jumps. A result on transformation of a measure is valid for this class of diffusions with jumps. Bibliographty: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 79–100.     

Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients.

A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities.

In addition, we study the accuracy of the approximations and present some numerical examples.

     

Two-scale homogenization in transmission problems of elasticity with interface jumps

The elasticity problem in a periodic structure with prescribed interface jumps in displacements and tractions and oscillating Neumann condition on a part of the external boundary is considered. This work is just a generalization of inhomogeneous Dirichlet and Neumann conditions on the oscillating interface. Such interface jumps arise, e.g. in contact problems with known periodic contact interface. Two-scale approach was applied to the problem and the two-scale convergence was proven. This article also provides a detailed auxiliary analysis for Sobolev functions with interface jumps.     

Determination of jumps of distributions by differentiated means

Differentiated means are defined in order to find formulas for jumps of distributions. We analyze two types of jumps occurring in the notions of distributional jump behavior and symmetric jump behavior. We start by defining what we call Riesz differentiated means for numerical series, then the differentiated means are extended to distributional evaluations for the Schwartz class of tempered distributions. The jumps of tempered distributions are completely determined by the differentiated means of the Fourier transform. We also find formulas for the jumps in terms of the asymptotic behavior of partial derivatives of harmonic representations and harmonic conjugate functions. Applications to Fourier series are given. The second author gratefully acknowledges support by the Louisiana State Board of Regents grant LEQSF(2005-2007)-ENH-TR-21.     

Convergence of generalized conjugate partial Fourier integrals

Generalized conjugate partial Fourier integrals are used to find jumps of functions. The rate of convergence is studied and sharp results are obtained.     

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps, whose data (the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.     

Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations

In this paper we study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L. Wu, Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton–Jacobi–Bellman equations with integral-differential operators. For this proof, we adapt Peng’s BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure.     

The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points

In the paper, the asymptotic mean square stability of the zero solution for neutral stochastic delay differential equations with Poisson jumps is studied by fixed points theory without Lyapunov functions. The coefficient functions have not been asked for a fixed sign, and the sufficient condition for mean square stability has been obtained. Therefore, some well-known results are improved and generalized.     

Numerical Approximations of Impulsive Delay Differential Equations

The article deals with numerical approximations of impulsive delay differential equations with a non-fixed time of impulses. The right-hand side of the approximation is assumed to be Lipschitz with respect to the norm of the measurable functions, which allows us to estimate the distance between functions with different times of jumps. Illustrative examples are provided.     

符号函数的Legendre非线性逼近

1引言许多数学和物理工作者研究了逼近形式正交多项式级数的具有较好收敛性的非线性方法,如文献[2-5,9]．这些非线性逼近方法的一个共同点是使用了线性级数中正交多项式的母函数．众所周知,的符号函数具有很多的应用,如文献[7]利用符号函数的积分表示来分析相联存储器的回想过程．文献[1]及其中所引用的一些文献为了获得交迭格Dirac算子,讨论了符号函数的有理逼近和连分式展开．在本文中,我们研究符号函数的Lengendre     

Determination of jumps in terms of spectral data

Summary We discuss the concentration factor methods for determination of jumps in terms of spectral data. The results are relative to the classical Lukács Theorem and some recent results obtained by A. Gelb, E. Tadmor and F. Móricz.     

Statistical specification of jumps under semiparametric semimartingale models

We consider a semimartingale with jumps that are driven by a finite activity Lévy process. Suppose that the Lévy measure is completely unknown, and that the jump component has a Markovian structure depending on unknown parameters. This paper concentrates on estimating the parameters from continuous observations under the nonparametric setting on the Lévy measure. The estimating function is proposed by way of nonparametric approach for some regression functions. In the end, we can specify jumps of the underlying Lévy process and estimate some Lévy characteristics jointly.      

Problem of thermoelectromagnetoelasticity for a piezoelectric/piezomagnetic bimaterial with interface crack

We consider a plane strain problem for a piezoelectric/piezomagnetic bimaterial space with a crack in the region of the interface of the materials. At infinity, tensile and shear stresses and heat, electric, and magnetic flows are set. Using representations for all mechanical, thermal, and electromagnetic factors in terms of piecewise analytic functions, we formulate problems of linear conjugation that correspond to a model of an open crack and models taking into account the contact zone in the vicinity of a crack tip. Exact analytic solutions of the indicated problems are constructed. Expressions for stresses, the electric and magnetic inductions, jumps of derivatives of displacements, and electric and magnetic potentials on the interface are written. The coefficients of intensities of the indicated factors are presented. We derive a transcendental equation for the determination of the real length of the contact zone. The dependences of this length and the coefficients of intensity on the set external influences are investigated.     

Wave Propagation in an Elastic Medium Intersected by Systems of Parallel Fractures

An elastic anisotropic medium intersected by systems of parallel fractures is investigated. Every fracture is considered as a plane boundary with jumps of displacements and stresses, and these jumps are linear functions of displacements and stresses averaged on the boundary. For this medium, an effective model is constructed by the method of matrix averaging. The equations of this model describe wave propagation in the given medium and are more complicated than the equations of elasticity theory. In particular cases, the equations obtained are converted to the equations of elastic media. On the basis of the equations of the effective model, expressions for the densities of the kinetic and potential energies are derived, and conditions of absoption in the medium are established. Bibliography: 15 titles.