Circularity of Finite Groups without Fixed Points

Let áA, B | A^m=1, Bⁿ=A^t, BAB^-1=A^r?\langle A, B\,\vert\, A^\mu=1,\, B^\nu=A^t,\, BAB^{-1}=A^\rho\rangle where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Hauptregelflächen einer Verallgemeinerten Regelfläche mit Zentralregelfläche

We consider generalized ruled surfaces in euclidean n-space ⁿ with k-dimensional generators and central ruled surface of dimension k–m+1 (O < m < k). Every orthogonal trajectoryy of the generators of defines a principal ruled surface _y with generators totally orthogonal to the generators of . In each generator of _y there exists an ellipsoid — called the indicatrix of the distribution parameters — which is defined by the distribution parameters of the tangent spaces to or _y. Formulars will be given for the distribution parameters of and _y .

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Herrn Prof. Dr. H.R. Müller zum 70. Geburtstag     

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 .     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.     

Striktionseigenschaften von Regelscharen imE n

Let ={e(u)|uI} be a one-parameter family of straight lines forming a ruledC ^r-2-surface E ⁿ (n2,r1) without singular generatorse(u) (uI). As a synopsis, a generalization and an improvement of various results already known about the strictional properties of ruled surfaces E ⁿ (especially in the casen=3) the author demonstrates a uniform geometrical way of defining and uniquely obtaining thestriction point S(u) and theparameter of distribution d(u) of a generatore(u) under the minimal assumptions thate(u)E ⁿ (n2) be noncylindrical andr1. Other methods of obtainingS(u) andd(u) are discussed in comparison, and special strictional properties ofskew ruled surfaces E ⁿ are proved.

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Herrn Prof. Dr. H. R. Müller zum 65. Geburtstag     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

Conditions for non-oscillation of singular linear differential equations of second order

Conditions are found in the fulfillment of which each non-trivial solution of the equation u+ (t)u+(t)u=0, where(t) L(a, b) and (t–a)(t–b)(t) L(a, b) has not more than one zero on the interval atb.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 633–639, November, 1969.     

On the existence of a cyclic vector for some families of operators

Under certain restrictions, it is proved that a family of self-adjoint commuting operatorsA=(A ) _where is a nuclear space, possesses a cyclic vector iff there exists a Hubert spaceH of full operator-valued measureE, where is the space dual to andE is the joint resolution of the identity of the familyA.Published in Ukrainskii Matematicheskii Zhurnal, Vol.45, No. 10, pp. 1362–1370, October, 1993.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Notes on Interpolation. X (Two Boundary Value Problems)

In this paper we use (0, 2) interpolational polynomials to give an approximate solution of the differential equation y(x) + A(x)y(x) = F(x), x I := [-1, 1] j in case when the boundary values are y(-1) = and y(1) = , , R.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Über die Anzahl der Gitterpunkte in verallgemeinerten Kreissektoren

In this paper we consider lattice points in domains bounded by algebraic curves of the formx ⁿ+yⁿ=Rⁿ fulfilling the additional condition where and are fixed positive real numbers. The number of these lattice points is estimated for largeR and it appears that for rational or badly approximable and the error term in the final result can be made smaller (at least forn3) than it is best possible when counting the lattice points without the additional condition indicated above.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Some fractal sets determined by stable processes

Summary LetY _i be independent stable subordinators in (, ,P) with indices 0< _i <1 andR _i are the ranges ofY _i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR ₁×R ₂, and whenever ₁ + ₂ 1/2, we deduce results for the vector sumR ₁R ₂={x+y:xR ₁,yR ₂}.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

A Selection Theorem for Strongly Regular Multivalued Mappings

We prove the following generalization of a theorem of Ferry concerning selections of strongly regular multivalued maps onto the class of paracompact spaces: Let : X (Z, ) be a map of a paracompact space X into a metric space (Z, ) whose values (x) are complete subspaces of Z and absolute extensors (AE), for every x X. Suppose that can be represented as = , where : X Y is a continuous singlevalued map of X onto some finite-dimensional paracompact space Y and : Y (Z, ) is a strongly regular map. Then for every closed subset A X and every selection r : A Z of the map |A : A Z, there exists an extension : X Z of r such that is a selection of the map . We also prove a local version of this theorem.     

The trace to subsets ofR n of Besov spaces in the general case

R ⁿ. , , , F R ⁿ, F , R ⁿ R ⁿ . ^p,q (Rⁿ), >0, 1, q, — ( ) Rⁿ. , ^p,q (Rⁿ) ^F Rⁿ. , q B ^p,q (F), = – (n–)/, >0, — « », ad — F, . , . : , F=R ^d,F— « » F— R ⁿ, « », F. .

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This work has been supported in part by the Swedish Natural Science Research Council.     

Die Weingarten-Regelflächen

Referring to articles of BELTRAMI (1865), DINI (1866) and CHARIAR (1978), but using a completely different approach, we determine allruled surfaces in Euclidean space ³, which are (at leastlocally) WEINGARTEN —-surfaces under theminimal assumption C². Theskew ruled WEINGARTEN —surfaces can be characterized by havingconstant invariants d O (parameter of distribution), k (skewness of distribution) and (striction angle); theirfunctional (WEINGARTEN-)relation between the mean curvature H and the Gaussian curvature K of is of the form H= (-K)^1/4 + (-K)^3/4 with arbitrary real constants ,. These facts allow various geometric interpretations.

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