Reconstructing dimensions of intersections of convex sets

Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K _i:i T} and {K _i ^j :i T} are two finite families of convex sets inR ⁿ and if dim {K _i :i S} = dim {K _i ^j :i S}for eachS T such that|S| n + 1 then dim {K _i :i T} = dim {K _i : {i T}}.     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

Pencils of Hyperconics in Projective Planes of Characteristic Two

Let F be any field, finite or infinite, of characteristic 2. Put =PG(2,F). Let H ₁,H ₂ be hyperconics in . In this note we study the intersectionH ₁ H ₂. In particular we obtain canonical forms for H ₁,H ₂ in the cases where |H ₁ H ₂|=4,5,6. One interesting consequence is that the case |H ₁ H ₂|=6 can only occur if F contains a subfield of order 4. Related results concerning pencils of hyperconics are presented in Theorems 6 through 9. This work also leads to an extension to general fields of characteristic 2 of the well-known even intersection property for hyperovals in PG(2,4) which is pursued elsewhere ([2]).     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Weak convergence using higher-order cumulants

Denote byc _j (F) thejth cumulant (or semi-invariant) of the distribution functionF. We say thatF is specified by its higher-order cumulants if it is the unique distribution functionG having the following property: there exists a positive integerJ such thatc _j (G)=c _j (F) forj=1,2 andjJ. Let (F _n n1) be a sequence of distribution functions, and suppose that there existsJ such thatc _j (F _n )c _j (F) asn, forj=1,2 andjJ. It is proved thatF _n F so long asF is specified by its higher-order cumulants. It is an open problem to characterize the family of distributions which are specified by their higher-order cumulants.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

On the concept of code-isomorphy

LetF be a finite set of cardinality ¦F¦ =q 2,n 1 an integer and :F ⁿ×Fⁿ₀ theHamming metric. Acode isomorphism C D between two block codesC,D Fⁿ is defined as an isometry which can be extended to an isometry of the whole space Fⁿ. Any permutation S _n of the positions canonically induces a so-calledequivalence map Aut Fⁿ; any system (₁,₂,...,_n) ofn permutations of the character setF induces a so-calledconfiguration Aat Fⁿ. The group Aut Fⁿ of all isometries of Fⁿ turns out to be semidirect product of the configuration group with the symmetric group of degreen. The codeword estimating failure probability of a maximum likelihood codeword estimator for aq-nary symmetric channel does not depend on the transmitted codeword, if the automorphism group of the code acts transitively on the set of codewords. When using a systematic (n, k)-encoder, the symbol decoding failure probability does not depend on the transmitted symbol or on the time of transmission if the configuration group and the automorphism group act transitively on the set of codewords resp. on the set of thek information positions.In memoriam Giuseppe Tallini     

On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.     

Durchschnitte von Pjateckij-Shapiro-Folgen

The distribution of integers and prime numbers in sequences of the formF _c1F _c2 is investigated. HereF _c={[_n ^c]:n } withc>1.     

Continuity of the eigenvalues of self-adjoint operators with respect to the strong operator topology

Let T_n, T, and S be self-adjoint operators such that T_n converges to T in the sense of strong resolvent convergence. If S is bounded from below, _e(S) (–,O) = Ø, and T_n S for all n, then the negative eigenvalues of T_n converge to the negative eigenvalues of T. The corresponding eigenfunctions converge in norm. This generalizes a result due to T. Kato, where T_n T is assumed, and a recent result of the author for the case T_n T_n+1.     

On one question of B. Amberg

In the case where a group G is a productG=AB of Abelian subgroupsA andB one of which has a finite 0-rank, we prove that the Fitting subgroupF and the Hirsch-Plotkin radicalR admit the decompositionsF=(F A)(F B) andR=(RA)(R B), respectively. This gives the affirmative answer to one question of B. Amberg.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 457–461, April, 1994.This paper was supported by the Ukrainian State Foundation on Science and Technology.     

RC *-fields

It is stated that if a Boolean family W of valuation rings of a field F satisfies the block approximation property (BAP) and a global analog of the Hensel-Rychlick property (THR), in which case F, W is called an RC^*-field, then F is regularly closed with respect to the family W (The-orem 1). It is proved that every pair F, W, where W is a weakly Boolean family of valuation rings of a field F, is embedded in the RC^*-field F₀, W₀ in such a manner that R₀ R₀ F, R₀ W₀ is a continuous map, W₀ is homeomorphic over W to a given Boolean space, and R₀ is a superstructure of R₀ F for every R₀ W₀ (Theorem 2).Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 367–386, July-August, 1994.     

Inequalities for two set systems with prescribed intersections

Inequalities are presented for systems {(A _i ,B _i ):1 i m} of pairs of finite sets satisfyingA _i B _i = andA _i B _j orA _j B _i fori j.     

On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

In this paper, we prove the existence of an element of the group algebra A=F of a free groupF with two generatorsx andy over the field of complex numbersC such that, for any complexa andb for which ¦a¦=¦b¦=1, we haveA _a,b ()A=0, where _a,b ( is an automorphism ofA that mapsx,y intoax, by, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from Kourovka Notebook.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No.4, pp. 571–572, April, 1995.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

LAN of extreme order statistics

Consider an iid sampleZ ₁,...,Z _n with common distribution functionF on the real line, whose upper tail belongs to a parametric family {F : }. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Z _n–i+1n ) _i=1 ^k of the upperk=k(n) _n order statistics in the sample, if the family {F :} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZ _n–k+1n is the central sequence generating LAN. This implies thatZ _n–k+1n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter can be based on the single order statisticZ _n–k+1n . The rate at whichZ _n–k+1n becomes asymptotically sufficient is however quite poor.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.