Über eine Verallgemeinerung des Plateauschen Problems

Let T be the domain in ^N defined by the inequalities O < ₁ < ... < _N < +. Put _N+k = /2(1+k) (k=1,2,3), _N+4=₁+2, and denote byF() the set of functions x=x(u,v)=(x¹(u,v),...,x^p(u,v)), (p2) of class , where B is the unit disk u²+v²<1, which maps the circular arcs _k={w=eⁱ:_k<<_K+1} (k=1,..., N+3) into the straight lines containing the edges a_k, a_k+1 (a_N+4=a₁) of a polygon IR^p. Then we show that the function ()= _inf ^xF() D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

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Hans Lewy und Charles B. Morrey gewidmet     

The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator

LetE be a Banach lattice which consists of some functionsu: over a fixed domain . This article is concerned with the local stability of a non-zero positive solutionu ^*E to the Hammerstein equationu(x)= x, wherek0, andf:×[0, )[0, ) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and forx,u>0. Under some mild additional hypotheses onE, k, andf it is proved that the spectral radius of the Fréchet derivative atu ^* of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.     

Приближение функции ограниченнойp-флукту адии полиномами по мульти пликативным система м

A new discrete modulus of continuity is introduced for functions of boundedp-fluctuation, and direct and converse theorems are proved on the approximation of these functions by polynomials with respect to multiplicative systems. Sufficient conditions for the convergence of Fourier series with respect to multiplicative systems are also obtained and these are the best possible in a certain sense.

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. 60-     

Quantum logics with the Radon-Nikodym property

Following the definition of S. Gudder and J. Zerbe, we say that a logicL has the Radon-Nikodym property (or, in short,L is an RN logic) if the following condition holds: Ifs, t are states onL ands is absolutely continuous with respect tot, then there is a central observablex such that for allaL. We first consider general RN logics. We establish their basic properties and show that they are closed under the formation of epimorphisms and products. Then we take up the RN property for concrete logics. We first show that in many cases the concrete RN logics have to be fully compatible (= Boolean -algebras). In contrast to that, we show that there are concrete RN logics with an arbitrary degree of noncompatibility. This extends the result of Navara and Pták and answers in full the question posed by Gudder and Zerbe.     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

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 . . -      

Convergence theorems for set-valued amarts and uniform amarts

( ) , — , - . .     

On ortbogonal polynomial bases in the spacesC andL

, >0 C L - ( ) {Q _n(x)} , Q _n (x)–v _n n ¹⁺ nn ₀ (). , =0.     

Stationary self-similar extremal processes

Summary Let ( _k ) _k =– be a stationary sequence of random variables, and, forA, let where _n is an affine transformation of (has the forma _n·+b _n,a _n>0,b _n). ThenM _n is a random sup measure, that is, for arbitrary collections of open setsG . We show that the possible limiting random sup measures for such sequences (M _n) are those which are stationary (M(·+b)= _d M forb) and self-similar (M(a·)= _d ^loga (M) fora>0, where is an affine transformation of ). By applying simple transformations, we need only study stationaryM such thatM(a·)= _d aM fora>0. We show that these processes retain some but not all of the properties of the classical case. In particular, we display a nontrivial example such thattM (0,t] is continuous wp1. The classical planar point process representation of extremal processes is a special case of the present approach, but is not adequate for describing all possible limits.     

On the summability of Fourier series with the method of lacunary arithmetic means

. f- ,S _n (f)— . {n _k }, n _k+1/n _k >1+ck ^– ,— , 0<1/2, f ₀, .     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

Poisson kernel is the unique approximate identity,asymptotically multiplicative on H∞

Let for anyf H(R), where (x): = ^–1(x^–1). Then (x) P (x + h) for some h R and > 0; P denotes the Poisson kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 82–89, 1989.     

On π-Solvable and Locally π-Solvable Groups with Factorization

We prove that, in a locally -solvable group G = AB with locally normal subgroups A and B, there exist pairwise-permutable Sylow - and p-subgroups A , A _p and B , B _p , p , of the subgroups A and B, respectively, such that A B is a Sylow -subgroup of the group G and, for an arbitrary nonempty set ,

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

ＱＵＡＤＲＡＴＩＣＥＳＴＩＭＡＴＯＲＳＯＦＱＵＡＤＲＡＴＩＣＦＵＮＣＴＩＯＮＳＷＩＴＨＰＡＲＡＭＥＴＥＲＳＩＮＮＯＲＭＡＬＬＩＮＥＡＲＭＯＤＥＬＳ￥ＷＵＱＩＧＵＡＮＧ（吴启光）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｅＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅ...     

Intersection Cohomology on Nonrational Polytopes⋆

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image R_* ( is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf , called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if is rational). Using we define the intersection cohomology space IH(). It is conjectured that a strictly convex piecewise linear function on acts as a Lefschetz operator on IH(). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.     

On the limit superior of analytic sets

. , A ₀,A ₁,— - lim supA _j - H, . , - - . , , ; , , . - . - .     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

On convergence of multiparameter strong submartingales in Banach lattices

- , ., B - — , B- {M _n ,nN ^d } n. .     

A high-indices theorem for strong Cesàro summability of double series

[4] , [C,(, )]_;- (, 0 1) s, [C,(+,+)]- s , 0 0< , (i) , ; > > 1 , (ii) ,>0; >=1 ,>1-1/, [C,(,)- [C,(+,+)]- .

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This work was done while the author was a visiting researcher at the Steklov Mathematical Institute, Moscow, U.S.S.R.