Tessellations generated by hyperplanes

Let be a locally finite system of hyperplanes in ^d with the property that the cells of the induced cell complex decomposition of ^d have uniformly bounded diameters. If is simple and the density of the vertices in exists, then the density of thek-cells in exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in exists and can be calculated. For certain nonsimple systems , corresponding inequalities are obtained.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Zur Beziehung zwischen Basen und Vektoren kürzester Länge in Gittern

LetR be an order of an algebraic number field of degreen over ,V ann-dimensional real vector space and the class of lattices inV which are free rank 1 modules overR. For certain ordersR and distance functionsd onV a method of computingd-minimal vectors of is described; further it is shown how to constructs anR-basis for by comparing thed-length of vectors of . An application to the computation of fundamental units and class numbers of real abelian number fields is mentioned.     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic

Given ringsR with prime power characteristicp ^k , quasivarieties (R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides ideald _R ofR. The main technical result is that (R) (S) makes all implications of the formd _s =S d_R=R true, for any such expressiond. The proof makes use of the known equivalence between (R) (S) and existence of an exact embedding functorR-Mod S -Mod. Fork 2, the ordered setW(p ^k ) of all lattice quasivarieties (R),R having characteristic p _K , is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(p ^k ) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerp ^k ,k 2, a ringR can be constructed so that (R) and (R ^op) for the opposite ringR ^op are distinct elements ofW(p ^k ).Presented by R. Freese.Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.     

A variational principle for eigenvalues of pencils of Hermitian matrices

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number is an eigenvalue ofH ifdet(A–B)=0 (we include = ifdetB=0). For nonsingularH (i.e., for which some is not an eigenvalue), we show precisely which eigenvalues can be characterized as _k ⁺ =sup{inf{*A:*B=1,S},SS _k},S _k being the set of subspaces of C ^N of codimensionk–1.Dedicated to the memory of our friend and colleague Branko NajmanResearch supported by NSERC of Canada and the I.W.Killam FoundationProfessor Najman died suddenly while this work was at its final stage. His research was supported by the Ministry of Science of CroatiaResearch supported by NSERC of Canada     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Paley-Wiener type theorems on harmonic extensions ofH-type groups

Letn=vz be anH-type group, and letna be the harmonic semidirect product ofn withaR. LetNA be the corresponding simply connected Lie group. If dimv=m and dimz=k, denoteQ=m/2+k. We prove that the spherical Fourier transform is a topological isomorphism between thep-Schwartz spacel ^p (N,A),(0<p2), (0<p2), and the space of holomorphic rapidly decreasing functions on the strip {sC:|Re(s)|<Q/2} with =2/p–1.     

A characterization of Thalesian orthogonality in Pappian planes of characteristic 2

In an affine plane over a field K, any Thalesian orthogonality relation is equivalent to _d for some dK{0}, where _d denotes the relation with constant of orthogonality d/it (i.e., after suitable coordinatization the slopes m, m*K{0} of orthogonal lines satisfy m·m*=d) (cf. [1], [5]). In the present paper we show that in a Pappian plane of characteristic two any orthogonality relation admitting the same group as ₁ is equivalent to ₁. This gives a characterization of Thalesian orthogonality over perfect fields of characteristic two.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

The threshold contact process: A continuum limit

Summary In the threshold contact process on thed-dimensional integer lattice with ranger, healthy sites become infected at rate if they have at least one infectedr-neighbour, and recover at rate 1. We show that the critical value _c (r) is asymptotic tor ^–d _c asr, where _c is the critical value of the birth rate for a continuum threshold contact process which may be described in terms of an oriented continuous percolation model driven by a Poisson process of rate ind+1 dimensions. We have bounds of 0.7320 < _c < 3 ford=1.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

Cutting disjoint disks by straight lines

Fork>0 letf(k) denote the minimum integerf such that, for any family ofk pairwise disjoint congruent disks in the plane, there is a direction such that any line having direction intersects at mostf of the disks. We determine the exact asymptotic behavior off(k) by proving that there are two positive constantsd ₁,d ₂ such thatd ₁k logkf(k)d ₂k logk. This result has been motivated by problems dealing with the separation of convex sets by straight lines.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).