A Strong Law for Mixing Random Variables

A sharp almost sure bound is derived for limit points of average sum of weakly dependent random variables, which ensures strong laws of large numbers for and -mixing random variables, without assumptions on rate of tending to zero of and -mixing parameters _n and _n.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Group Classification of Nonlinear Schrödinger Equations

We propose an approach to problems of group classification. By using this approach, we perform a complete group classification of nonlinear Schrödinger equations of the form i _t + + F(, *) = 0.     

Jordan-Basen projektiver Räume bezüglich linearer Selbstabbildungen

In vector spaces the notion of Jordan base is useful in considerations of the Jordan normal form of matrices. The notion of Jordan base with respect to a given linear self-mapping of a projective space is defined, and the existence of such a Jordan base is demonstrated under weak assumptions about .

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Herrn Prof. Dr. Walter Wunderlich zum 80. Geburtstag gewidmet     

Free Algebras over Biregularizations of Varieties

Let : F N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of nonnegative integers. An identity of type is called biregular if the sets of variables in and are identical and the sets of fundamental operation symbols in and are identical. If K is a variety of type , we denote by K_b the variety of type defined by all biregular identities from Id(K). K_b will be called the biregularization of K. In this paper we give a representation of free algebras over K_b by means of free algebras over K.     

Diameters of a class of smooth functions in the space L2

The class V, consisting of the smooth functions f(t), ot1, satisfying the condition ₀ ¹ [f ^(r) (t)]dt1, where the function (t) is nonnegative and r is a natural number, is studied. Under certain restrictions on the function (t) ensuring the compactness of the class V, the order of decrease of the Kolmogorov diameters d_n(V) is computed. The analogous problem for the case r=1 is solved also for functions of several variables.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 671–678, November, 1977.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

When do Toeplitz and Hankel operators commute?

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolz, where is the symbol of the Toeplitz operator, and is an affine function of the characteristic function of certain anti-symmetric sets of the unit circle.     

An algorithm for constructing an optimal structural schedule

The article is devoted to the problem of finding an optimal schedule for a class of functionals ƒ which allows for the existence of a structural set of activities. The functionalƒ(R), where, is defined in the following way: where {_i(t)} is a structural set of functions, and the function F is defined on any finite set of arguments and satisfies the following conditions: 1)F(x)=(x); 2) F(x₁,x₂)=(x₁,x₂), F(x₁,x₂,...x₃)= (x₁, F(x₂,...,x_s)), S2; 3) and do not decrease in each of their arguments, and moreover, 3a) strictly increases with the increase of both arguments, 3b) if (x₁,x₂)>(x₁, x₂ (x₂, x₃)> (x₂,x₃), then F(x₁,x₂,x₃)>F(x₁,x₂,x₃).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 5–20, 1983.     

Büschel linearer Abbildungen

If , , are linear mappings out of a projective space (P,G) into a projective space (P', G') and , then is said to belong to the pencil <,<> of linear mappings spanned by and if in the main (x), (x), (x) are collinear for all x P. We give some sufficient conditions for x P and , , such that (x) is uniquely determined by giving, and (z), z P.

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Herrn Prof. Dr.Helmut Karzel zum 60. Geburtstag gewidmet     

Asymptotic formulas for Toeplitz and Wiener-Hopf operators

For certain analytic functions f, the expression trace {f(T_n[]) – f(T_n[]T_n[])} is computed asymptotically. Here T_n[] is the finite Toeplitz matrix generated by the function . The analogous expression for Wiener-Hopf operators is also computed asymptotically. These results in turn yield information concerning the asymptotic behavior of determinants of finite Toeplitz and Wiener-Hopf operators with discontinuous generating function.     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

Inclusion statements for operator equations by a continuity principle

The paper is concerned with Range-Domain Implications MvCvK, where M is a given operator and C,K denote given sets. Sufficient conditions are derived by a very general continuity principle. Various special cases are considered such as inverse-positivity, MvMwvw, and a generalization H(,[,])MvH(,[,]) v, where Mu=H(u,u) and [,] denotes an order interval. These results are applied to differential operators related to boundary or initial value problems. The goal is to furnish a simple uniform approach, to explain its application, and to provide a kind of survey on what problems have been treated in this way.     

Topologically invariant linear forms on spaces of abstract distributions on a locally compact abelian group

A special case of the main result proved in this paper is the following. IfG is a locally compact, -compact, non-compact connected abelian group, thenL ² (G)={f–*f:fL ² (G), L ¹ (G), 0 and _G =1}. In this case, any topologically invariant linear form onL ² (G) is 0.     

The continuous solution of a functional equation of Abel

Summary The functional equation(x) + (y) = (xf(y) + yf(x)) (1) for the unknown functionsf, and mapping reals into reals appears in the title of N. H. Abel's paper [1] from 1827 and its differentiable solutions are given there. In 1900 D. Hilbert pointed to (1), and to other functional equations considered by Abel, in the second part of his fifth problem. He asked if these equations could be solved without, for instance, assumption of differentiability of given and unknown functions. Hilbert's question was recalled by J. Aczél in 1987, during the 25th International Symposium on Functional Equations in Hamburg-Rissen. In particular Aczél asked for all continuous solutions of (1). An answer to his question is contained in our paper. We determine all continuous functionsf: I ,: A _f (I × I) and: I that satisfy (1). HereI denotes a real interval containing 0 andA _f (x,y) := xf(y) + yf(x), x, y I. The list contains not only the differentiable solutions, implicitly described by Abel, but also some nondifferentiable ones.Applying some results of C. T. Ng and A. Járai we are able to obtain even a more general result. For instance, the assertion (i.e. the list of solutions) remains unchanged if we replace continuity of and by local boundedness of orf(0)I from above or below. Strengthening a bit the assumptions onf we can preserve a large part of the assertion requiring only the measurability of either orf(0)I.     

Asymptotic normality of p-norm estimators in multiple regression

Summary The consistency and asymptotic normality of p-norm estimators (1<p<2) is established by applying some of the ideas of Huber (1973), where asymptotic normality of the so-called M-estimators for regression parameters is shown. A central role is played by a weight function . Huber assumed that , and are bounded. This is, however, not the case for p-norm estimators with 1<p<2, but some of his ideas can still be applied.     

Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(n×nK) of general type S is of the form B(,) = (x)(x)d(x), , sMpMp (n), where is a positive {M}-tempered measure, i.e., for every > 0 exp[-M(|x|)] d(x) < . To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.     

Bari bases of subspaces

We shall establish certain characteristic properties of Bari^* bases of subspaces. We shall show that a complete sequence of finite-dimensional subspaces {N _j}₁ is a Bari basis if and only if each sequence {_j{₁ (_j€N _j, _j=1) is a Bari basis of its own closed linear hull.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 461–469, April, 1969.