The transfer of the Lyapunov integral criterion to two-dimensional periodic systems of second order

Two possibilities of the transfer of the Lyapunov integral criterion [1, p. 203] (see also [2; 3, p. 177]) for the boundedness of all the solutions of the scalar equation x+(t)x=0 with nonnegative-periodic function(t) to the two-dimensional systems x=A(t)x with piecewise continuous -periodic matrix coefficients are indicated.     

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

On the Existence of Solutions for Some Nondegenerate Nonlinear Wave Equations of Kirchhoff Type

Let be a bounded domain in #x211D;ⁿ with a smooth boundary . In this work we study the existence of solutions for the following boundary value problem:

Oscillation of nonlinear differential systems with retarded arguments

In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form

Das System Totalpolarer und eine Kennzeichnung des Winkels von Hyperebenen in Cayley/Klein-Räumen

In this paper at first we introduce thesystem of total polar subspaces of an arbitrary k-diniensional plane with respect to the absolute (configuration) of an arbitrary n- dimensional CAYLEY/KLEIN space as a generalization of the total polar set of a regular k-plane of . Using the system of total polar sets of the intersection (n–2)-plane of two hyperplanes and we give the followingnew characterization of the angle of . and : for any straight line g with and g=Ø the angle of and is equal to the distance of the two intersection points of g with and .

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Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

Boundedness in the mean of orthonormalized polynomials

For the polynomials {p_n(t)} ₀ , orthonormalized on [–1, 1] with weightp(t) = (1–t) (1+t) _v=1 ^m , we obtain necessary and sufficient conditions for boundedness of the sequences of norms: 1) 2) and 3) with the conditions that on [–1, 1] and (H,)^–1 L²(0, 2), where(H,) is the modulus of continuity in C(–1, 1) of function H.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 759–770, May, 1973.     

On a Binary Diophantine Inequality Involving Prime Numbers

Let 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all (N, 2N\ A with , the inequality has solutions in primes .     

On the essential approximate point spectrum of operators

If belongs to the essential approximate point spectrum of a Banach space operatorTB(X) and is a sequence of positive numbers with lim ^j a _j =0, then there existsxX such that for every polynomialp. This result is the best possible — if for some constantc>0 thenT has already a non-trivial invariant subspace, which is not true in general.     

New extremal properties for constructing conformal mappings

Summary It is well known that for a given simply connected regionR containing zero the uniform norm attains its minimum in the class of all holomorphic functions normalized byf(0)=0 andf(0)=1 only for the conformal mappingfRD(r)={z|z|}. It is shown that this theorem is still valid if one replaces the ordinary modulus | | on by any other norm on . For instance it is possible to obtain direct mappings ofR onto parallelograms, rectangles and ellipses. For the special norms |₁ and | this leads to a simple and fast computational technique involving linear programming methods. Several numerical examples are given.     

On relations among moduli of continuity in Lorentz spaces

Estimates are obtained among moduli of continuity of functions in several variables that belong to various Lorentz spaces. The functions considered are periodic with period 1 in each variable. More exactly the following theorem is proved: If 0<,<(t) and (t) are so-called -functions such that _;>₁>1, and

Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables

The series is studied, where S_n are the sums of independent equally distributed random variables, _n is a sequence of nonnegative numbers, >0, and >0 is an arbitrary positive number. For a broad class of sequences _n, the necessary and sufficient conditions are established for the convergence of this series for any >0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 770–784, June, 1993.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

A necessary condition for convergence of interpolating parabolic and cubic splines

Let the sequence of nets _n be such that , where h_i ⁽ⁿ⁾ are the lengths of the segments of a net. The bound is necessary in order that interpolating parabolic and cubic splines converge for any function in C ( = 0) or C(0 < < 1), where C is the class of functions satisfying a Lipschitz condition of order. It is also shown that this bound cannot essentially be weakened.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 165–178, February, 1976.The author thanks Yu. N. Subbotin for a useful discussion of the results obtained.     

Proper holomorphic mappings between generalized pseudoellipsoids

Summary We characterize the existence of proper holomorphic mappings between pseudoellipsoids of the form . If f:()() is any such mapping, the existence of a subgroup of Aut (()) such that is shown equivalent to a condition on the branch locus of f.     

Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Let (X _t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S _M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .     

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Prolongement Méromorphe des séries de Dirichlet à Support dans un sous-Ensemble semi-algébrique de Rn

We consider the Dirichlet series Z(P,A;s) = P^–s(m) (s C) where P R[X₁ ,..., X_n] and A is an open semi-algebraic subset of Rⁿ. We will say that Z(P,A;s) exists if this multiple series is absolutely convergent. In this paper we study the existence and several properties of meromorphic continuations of such series, under certain assumption on P and A. As an application, we show the existence of a finite asymptotic expansion of the counting function with support in A: N_p(A,t):= m A Zⁿ | P(m) t} when t +.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

Über Abstand 1 erhaltende Abbildungen in Minkowski-Ebenen

By using a computer the following theorem is proved: Consider K=GF(q), q {32,64,81,128}, :K² K² bijective such that P,Q K². Then is a semi-isometry. The assumption bijective can be dropped if q {32,128}.