Recurrence of the integral of an odd conditionally periodic function

We prove that the integral of a sufficiently smooth odd conditionally periodic function with zero mean and incommensurable frequencies recurs. Furthermore, we obtain the lower and upper bounds for smoothness guaranteeing the recurrence of the integral. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 570–577, April, 1997. Translated by N. K. Kulman     

Recurrence of the integral of a smooth three-frequency conditionally periodic function

We prove V. V. Kozlov's famous conjecture claiming that the integral of an analytic three-frequency conditionally periodic function with zero mean and incommensurable frequencies recurs. For a conditionally periodic function of classC ² onT ⁿ ,n=2, 3, we prove that the integral recurs uniformly with respect to the initial data. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 723–735, November, 1995.     

Differential equations with almost periodic or conditionally periodic coefficients: Recurrence and reducibility

Recurrence of an individual trajectory of a linear nonautonomous differential equation on a compact Lie group for the case of an almost periodic or conditionally periodic dependence of the right-hand side on time is proved. The relation between recurrence and reducibility is examined.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 229–237, August, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072.     

Nonrecursiveness of the integral of a conditionally periodic function

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 138–140, June, 1991.     

Problem of behavior of an integral of a conditionally periodic function

Translated from Matematicheskie Zametki, Vol. 50, No. 3, pp. 97–106, Spetember, 1991.     

Remarks on the universality of the periodic zeta function

We study the universality of a Dirichlet series with periodic coefficients. This property is proved in the case of multiplicative coefficients, and in the general case we establish universality in a certain set of analytic functions related to a probability distribution.     

Convergence in almost periodic cooperative systems with a first integral

This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of B. Tang, Y. Kuang and H. Smith in SIAM J. Math. Anal., 24 (1993), 1331-1339.

     

On integral representation of Bessel function of the first kind

A first kind Fredholm integral equation with nondegenerate kernel is given, which particular solution is the Bessel function of the first kind. This equation is solved by means of Mellin transform pair.     

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at −∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(⋅)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I−S)/2 is Fredholm on the variable Lebesgue space , then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces and with some exponents ql and qr lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.     

On the reducibility of a nonnegatively hamiltonian real periodic matrix to a block-diagonal form

In the paper, under certain conditions, it is shown that a Hamiltonian real periodic matrix can be block-diagonalized by a real matrix of the same period and, in the resulting block-diagonal form, the spectrum of one of the matrices on the diagonal belongs to the open left half-plane and that of the other to the open right half-plane. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 688–695, November, 1999.     

Exponential dichotomy of linear systems with almost periodic matrices

Using the Lyapunov direct method, we establish a condition of exponential dichotomy for the class of dynamical systems under weaker assumptions as compared to the case of an arbitrary continuous matrix on the derivative of the Lyapunov function along the trajectories. By way of application, we obtain a sufficient condition for the dichotomy of a second order almost periodic vector equation in terms of coefficients.     

On a conservative integral equation with two kernels

We study the solvability of the integral equation

The convergence of the Fourier integral of a unimodal distribution

We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine's criterion for unimodal distributions.

     

Abundant periodic and aperiodic solutions of a discontinuous three-term recurrence relation

In this note, we study a discontinuous three-term recurrence relation which arises from seeking the steady states of a cellular neural network with step control function. Several collections of periodic solutions are found. A necessary and sufficient condition for a solution to be periodic is stated and aperiodic solutions are found as consequences. We also show that any periodic solution can be derived from a primary periodic solution with least period not divisible by 5. Although the periodic or aperiodic solutions we found are not exhaustive, they are quite abundant and may reflect some of the rich physical phenomena in true biological systems. Our method in this note may also provide a general approach to analyze the periodicity of solutions of similar recurrence relations.     

On the maximum number of periodic solutions of piecewise smooth periodic equations by average method

In this paper, we prove smoothness of bifurcation function for a piecewise smooth periodic equation, and then use the bifurcation function together with its smoothness to study the maximum number of periodic solutions of the piecewise smooth periodic equation by the first order average.     

What is the complexity of periodic weakly singular integral equations?

A fast/quasifast solver for weakly singular periodic Fredholm integral equations is constructed in the situation where the information about the kernel and the free term of the equation is restricted to a finite number of sample values. Hence the complexity of weakly singular integral equations is the same as that for equations without singularities or close to that. AMS subject classification (2000)  65Y20, 65T05, 45B05     

Nuclearity of a nonnegative definite integral kernel on a separable metric space

The aim of this paper is to characterize the nuclearity of an integral operator, defined by a continuous non-negative definite square integrable kernel on a separable metric space, in terms of the integrability of the trace of the kernel function. Nuclearity here plays a role forU-statistics.