Critical Exponents for the Heat-Conduction Equation with Nonlocal,Nonlinear Perturbations

We study the conditions for the existence and nonexistence of global in time (t > 0), nonnegative solutions of the problem

Asymptotic behaviour of renormalized solutions to elliptic equations with measure data and G-converging operators

We study the limit as n goes to +∞ of the renormalized solutions u ⁿ to the nonlinear elliptic problems

Small deviations of modified sums of independent random variables

Let S_n = X₁ + · · · + X _n , n ≥ 1, and S ₀ = 0, where X ₁, X ₂, . . . are independent, identically distributed random variables such that the distribution of S _n/B _n converges weakly to a nondeoenerate distribution F _α as n → ∞ for some positive B _n . We study asymptotic behavior of sums of the form

Global Behavior of Rational Sequences Involving Piecewise Power Function

In this paper, we are concerned with the global behavior ofthe solutions of the difference equation

On the Asymptotic Behavior of Solutions of Differential Systems

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality

On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

For a trigonometric series

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data

The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system

Results of Hurwitz Type for Five or More Squares

Let r _k(n) denote the number of representations of n as a sum of k squares. We give many cases (for k = 5, 6, 7) in which the generating function _n0 r _k(an + b)q ⁿ is a simple infinite product. For instance,

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form

Bounds for the Extreme Eigenvalues of Block 2 × 2 Hermitian Matrices

Let an n × n Hermitian matrix A be presented in block 2 × 2 form as , where A₁₂ ≠ 0, and assume that the diagonal blocks A₁₁ and A₂₂ are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Existence of Positive Solutions for a Class of Higher Order Neutral Functional Differential Equations

The higher order neutral functional differential equation

Асимптотические фор мулы для нулей функци и типа Миттаг-Леффлера

The paper is devoted to study the asymptotic behaviour of zerosz _n of an entire function of Mittag-Leffler's type

On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.