About some new integral inequalities of Wendroff type for discontinuous functions

In this article, we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables (Wendroff type) by including also inequalities with delay. We deduce new generalizations of earlier results given by R.P. Agarwal, R. Bellman, I. Bihari, B.K. Bondge, V. Lakshmikantham, S. Leela, B.G. Pachpatte for continuous and discrete functions. Furthermore, generalizations of some results for integro-sum inequalities are obtained as well.     

一个推广的二变量时滞积分不等式及其应用

建立了一类二变量的时滞积分不等式,不等式包含一个一重积分和两个二重积分,二重积分内包含两个不同的没有假设单调性的未知函数的复合函数.使用单调化技术,给出积分不等式中未知函数的估计.结果能对相关文献中考虑的积分不等式中未知函数进行估计.进一步,结果给出了一类积分-微分方程解的估计.     

一类二变量积分不等式及其在积分-微分方程中的应用

建立了一类二变量的积分不等式,该不等式包含了一个一重积分和两个二重积分.利用分析技巧,给出了积分不等式中未知函数的估计.这一结果可以作为研究积分-微分方程解的定性性质的工具.     

Probabilistic inequalities and remarks

The authors use their recently proved integral inequality to obtain bounds for the covariance of two random variables

• 1.in a general setup and
• 2.for a class of special joint distributions.

The same inequality is also used to estimate the difference of the expectations of two random variables. Finally, the authors study the attainability of a related inequality.     

Nouvelles inégalités intégrales et applications à la stabilisation des systèmes distribués non dissipatifs

First we prove some new integral inequalities to obtain an estimate on behavior at infinity of a positive and non necessarly decreasing function. This extends some integral inequalities due to A. Haraux, V. Komornik and P. Martinez concerning decreasing functions. Then we give applications to (internal or boundary) stabilization of certain nondissipative distributed systems. To cite this article: A. Guesmia, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On projective methods of approximate solution of singular integral equations

The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov-Galerkin and collocation methods are investigated.This author is also known as A. V. Dzhishkariani.     

Legendre singularities in systems of implicit ODEs and slow-fast dynamical systems

Generic properties of regular first integrals of systems of implicit differential equations are considered. In particular, for systems of two equations with two phase variables, a classification of generic bifurcations of integral level surfaces is described. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 261, pp. 140–153.     

High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures

We estimate d-dimensional least squares approximations of an arbitrary d-regular measure μ via discrete curvatures of d+2 variables. The main result bounds the least squares error of approximating μ (or its restrictions to balls) with a d-plane by an average of the discrete Menger-type curvature over a restricted set of simplices. Its proof is constructive and even suggests an algorithm for an approximate least squares d-plane. A consequent result bounds a multiscale error term (used for quantifying the approximation of μ with a sufficiently regular surface) by an integral of the discrete Menger-type curvature over all simplices. The preceding paper (part I) provided the opposite inequalities of these two results. This paper also demonstrates the use of a few other discrete curvatures which are different from the Menger-type curvature. Furthermore, it shows that a curvature suggested by Léger (Ann. Math. 149(3), pp. 831–869, 1999) does not fit within our framework.     

Branching of solutions for the problem of mean-square approximation of a real finite function of two variables by the modulus of double Fourier transformation

We investigate the branching of solutions of a Hammerstein-type nonlinear two-dimensional integral equation that arises in the problems of mean-square approximation of a real finite nonnegative function of two variables by the modulus of double Fourier integral, depending on two parameters [Mat. Metody Fiz.-Mekh. Polya, 51, No. 1, 53–64; No. 4, 80–85 (2008)]. We have derived analytical expressions for eigenfunctions of the corresponding linear homogeneous integral equation, necessary for the construction of branched-off solutions, and obtained systems of transcendental equations for finding the points of their branching. We also present, in the first approximation, the analytical representations of complex solutions branched-off from the real solution for the two-dimensional case of branching.     

Collocation discretization for an integral equation in ocean acoustics with depth‐dependent speed of sound

We analyze two collocation schemes for the Helmholtz equation with depth‐dependent sonic wave velocity, modeling time‐harmonic acoustic wave propagation in a three‐dimensional inhomogeneous ocean of finite height. Both discretization schemes are derived from a periodized version of the Lippmann‐Schwinger integral equation that equivalently describes the sound wave. The eigenfunctions of the corresponding periodized integral operator consist of trigonometric polynomials in the horizontal variables and eigenfunctions to some Sturm‐Liouville operator linked to the background profile of the sonic wave velocity in the vertical variable. Applying an interpolation projection onto a space spanned by finitely many of these eigenfunctions to either the unknown periodized wave field or the integral operator yields two different collocation schemes. A convergence estimate of Sloan [J. Approx. Theory, 39:97–117, 1983] on non‐polynomial interpolation allows to show converge of both schemes, together with algebraic convergence rates depending on the smoothness of the inhomogeneity and the source. Copyright © 2016 John Wiley & Sons, Ltd.     

Algorithm for calculating the roots of polynomials with coefficients in the ring of polynomials over an arbitrary integral domain

A deterministic algorithm for calculating the roots of polynomials in one variable with coefficients in the ring of polynomials in several variables over an arbitrary integral domain is constructed. An estimate for the arithmetic complexity of the algorithm in the worst case is obtained.     

Asymptotic expansion of perturbative Chern-Simons theory via Wiener space

The Chern-Simons integral is divided into a sum of finitely many resp. infinitely many contributions. A mathematical meaning is given to the “finite part” and an asymptotic estimate of the other part is given, using the abstract Wiener space setting. The latter takes the form of an asymptotic expansion in powers of a charge, using the infinite-dimensional Malliavin-Taniguchi formula for a change of variables.     

Dyadic approximation of double integrals with respect to symmetric stable processes

Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index α in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that it can also be defined as the limit of integrals of step functions and that the two approaches are equivalent. For many purposes this result reduces the study of double integrals to that of quadratic forms in independent stable random variables.     

Estimation of the distribution density by spline functions

One finds the mean square integral error of the histogram estimate of the density of an n-dimensional random variable and an estimate of the density of a one-dimensional random variable by splines of the first and second degree.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 27–36, 1986.     

On the best mean-square approximation of a real nonnegative finite continuous function of two variables by the modulus of a double Fourier integral. I

We study the nonlinear problem of mean-square approximation of a real finite nonnegative continuous function of two variables by the modulus of a double Fourier integral depending on two parameters. The solution of this problem is reduced to the solution of a nonlinear two-dimensional integral equation of the Hammerstein type. Numerical algorithms for determination of branching lines and branched solutions of equation are constructed and substantiated. Some numerical examples are given. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 53–64, January–March, 2008.     

Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'ki?  type. In order to illustrate the results obtained here, an example is given.     

The application of the Shimura lifting to the representation of large numbers of ternary quadratic forms

One obtains an asymptotic formula for the number of the representations of numbers, divisible by a large square, by a positive ternary integral quadratic form. One gives an estimate of the remainder, unimprovable with respect to the quadratic part and uniform with respect to the square-free part of the represented number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 21–26, 1985.     

Representations of an even zeta function on theta series

Representation matrices are computed for the Hecke operators corresponding to the coefficients of an even zeta function of the symplectic group on the theta series of positive integral quadratic forms with an even number of variables belonging to a fixed genus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 5–14, 1984.     

Integral representations and asymptotic expansions of the incomplete probability integral

The incomplete probability integral is defined and its connection with Wen's integral is established. For the integral are obtained various integral representations and transformations with respect to both variables, on the basis of which asymptotic expansions are deduced for various values of the parameters.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 213–220, August, 1972.The authors wish to express their deep gratitude to L. N. Bol'shev for discussion of the results.