On a conjecture of R. E. Miles about the convex hull of random points

Denoty byp _d+i (B _d ,d+m) the probability that the convex hull ofd+m points chosen independently and uniformly from ad-dimensional ballB _d possessesd+i(i=1,...,m) vertices. We prove Mile's conjecture that, given any integerm, p _d+m (B _d ,d+m)»1 asd». This is obvious form=1 and was shown by Kingman form=2 and by Miles form=3. Further, a related result by Miles is generalized, and several consequences are deduced.Dedicated to Professor E. Halwaka on the occasion of his seventieth     

Random embeddings of Euclidean spaces in sequence spaces

It is proved that for some absolute constantd and forn≦dm mostn×m matrices with ± 1 entries are good embeddings ofl ₂ ⁿ intol ₁ ^m . Similar theorems are obtained wherel ₁ ^m is replaced by members of a wide class of sequence spaces. Supported in part by NSF Grant No. MCS-79-03042.     

On a division property of consecutive integers

Pillai and Brauer proved that form≧17 we can find blocksB _m ofm consecutive integers such that no element in the block is pairwise prime with each of the other elements. The following basic generalization is proved: For eachd>1 there is a numberG(d) such that for everym≧G(d) there exist infinitely many blocksB _m ofm consecutive integers, such that for eachr∈B _m there existss∈B _m , (r,s)≧d.     

Asymptotic consistency of fixed-width sequential confidence intervals for a multiple regression function

Summary Letm _n (x) be the recursive kernel estimator of the multiple regression functionm(x)=E[Y|X=x]. For given α (0<α<1) andd>0 we define a certain class of stopping timesN=N(α,d, x) and takeI _N,d (x)=[m _N (x)−d, m _N (x)+d] as a 2d-width confidence interval form(x) at a given pointx. In this paper it is shown that the probability P{m(x)∈I _N,d (x)} converges to α asd tends to zero.     

Star-shaped sets in normed spaces

We prove a generalization of the Krasnosel’ski theorem on star-shaped sets. Usingd-segments inn-dimensional Minkowski spaces instead of usual segments, the notions “d-visibility” and “d-star-shapedness” are introduced. Our main aim is to give necessary and sufficient conditions ford-star-shapedness in finite-dimensional normed spaces.     

Über die Symmetriegruppen von regulärseitigen Polytopen

Convex polytopes are called regular faced, if all their facets are regular. It is known, that all regular faced 3-polytopes have a nontrivial symmetry group, and also alld-polytopes with centrally symmetric facets. Here it is shown, that there ecist in fact regular facedd-polytopes with trivial symmetry group, but only ford=4. The corresponding class of polytopes is studied.     

Characterization of the Folded Johnson Graphs of Small Diameter by their Intersection Arrays

It is known that the folded Johnson graphs J(2m,m) withm ≥ 16 are uniquely determined as distance regular graphs by their intersection array. We show that the same holds form ≥ 6.     

The expected value of some functions of the convex hull of a random set of points sampled inR d

This paper presents formulas and asymptotic expansions for the expected number of vertices and the expected volume of the convex hull of a sample ofn points taken from the uniform distribution on ad-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional ton ^{(d−1)/(d+1)}, which generalizes Rényi and Sulanke’s asymptotic raten ^(1/3) ford=2 and agrees with Raynaud’s asymptotic raten ^{(d−1)/(d+1)} for the expected number of facets, as it should be, by Bárány’s result that the expected number ofs-dimensional faces has order of magnitude independent ofs. Our formulas agree with the ones Efron obtained ford=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown convex subset ofR ^d .     

Schwarz-Christoffel mapping of multiply connected domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.     

Thed-step conjecture for polyhedra of dimensiond<6

Two functions Δ and Δ _b , of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ(d, n) is the maximum diameter of convex polyhedra of dimensiond withn faces of dimensiond−1; similarly, Δ _b (d,n) is the maximum diameter of bounded polyhedra of dimensiond withn faces of dimensiond−1. The diameter of a polyhedronP is the smallest integerl such that any two vertices ofP can be joined by a path ofl or fewer edges ofP. It is shown that the boundedd-step conjecture, i.e. Δ _b (d,2d)=d, is true ford≤5. It is also shown that the generald-step conjecture, i.e. Δ(d, 2d)≤d, of significance in linear programming, is false ford≥4. A number of other specific values and bounds for Δ and Δ _b are presented. This revised version was published online in November 2006 with corrections to the Cover Date.     

On the regulation number of a multigraph

The regulation number of a multigraphG having maximum degreed is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG. It is shown that the regulation number of any multigraph is at most 3. The regulation number of a multidigraph is defined analogously and is shown never to exceed 2. A multigraphG has strengthm if every two distinct vertices ofG are joined by at mostm parallel edges. For a multigraphG of strengthm and maximum degreed, them-regulation number ofG is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG having strengthm. A sharp upper bound on the 2-regulation number of a multigraph is shown to be (d+5)/2, and a conjecture for generalm is presented. Research supported by a Western Michigan University faculty research fellowship. Research Professor of Electrical Engineering and Computer Science, Stevens Institute, Hoboken, NJ and Visiting Scholar, Courant Institute, New York University, Spring 1984. Research supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.     

The cubicald-polytopes with fewer than 2 d+1 vertices

A cubical polytope is a convex polytope of which every facet is a combinatorial cube. We give here a complete enumeration of all the cubicald-polytopes with fewer than 2 ^d+1 vertices, ford≥4.     

Convex polytopes whose projection bodies and difference sets are polars

In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of ad-simplex (d≥2) are polars. Obviously, ford=2 a convex domain has this property if and only if its difference set is bounded by a so-called Radon curve. A natural question emerges about further classes of convex bodies inR ^d (d≥3) inducing the mentioned polarity. The aim of this paper is to show that a convexd-polytope (d≥3) is a simplex if and only if its projection body and its difference set are polars.     

On partitions of {1,…,2m + 1}\{k} into differences d,…,d + m − 1: Extended Langford sequences of large defect

It is shown that for m = 2d ? 1, 2d, 2d + 1, and d ≥ 1, the set {1, 2,…, 2m + 2}, ? {2,k} can be partitioned into differences d,d + 1,…,d + m ? 1 whenever (m,k) ≡ (0,0), (1,d + 1), (2, 1), (3,d) (mod (4,2)) and (d,m,k) ≠ (1,1,3), (2,3,7) (where (x,y) ≡ (u,ν) mod (m,n) iff x ≡ u (mod m) and y ≡ ν (mod n)). It is also shown that if m ≥ 2d ? 1 and m ? [2d + 2, 8d ? 5], then the set {1, 2, …, 2m + 1} ? {k} can be partitioned into differences d,d + 1,…,d + m ? 1 whenever (m,k) ≡ (0, 1), (1,d), (2,0), (3,d + 1) mod (4,2). Finally, for d = 4 we obtain a complete result for when {1,…,2m + 1} ? {k} can be partitioned into differences 4,5,…,m + 3. © 2004 Wiley Periodicals, Inc.     

Genus of cartesian products of regular bipartite graphs

Let G(n, d) denote a connected regular bipartite graph on 2n vertices and of degree d. It is proved that any Cartesian product G(n, d) × G₁(n₁, d₁) × G₂(n₂, d₂) × ? × G_m(n_m, d_m), such that max {d₁, d₂,…, d_m} ≤ d ≤ d₁ + d₂ + ? + d_m, has a quadrilateral embedding, thereby establishing its genus, and thereby generalizing a result of White. It is also proved that if G is any connected bipartite graph of maximum degree D, if Q_m is the m-cube graph, and if m ≥ D then G × Q_m has a quadrilateral embedding.     

Topological complexity and real roots of polynomials

The topological complexity of an algorithm is the number of its branchings. In the paper we prove that the minimal topological complexity of an algorithm that approximately computes a root of a real polynomial of degreed equalsd/2 for evend, is greater than or equal to 1 for oddd>–3, and equals 1 ford=3 or 5.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 670–680, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00846a and by the INTAS under grant No. 4373.     

Finite group actions and asymptotic expansion ofe P(z)

We establish an asymptotic expansion for the number |Hom (G,S _n )| of actions of a finite groupG on ann-set in terms of the order |G|=m and the numbers _G (d) of subgroups of indexd inG ford|m. This expansion and related results on the enumeration of finite group actions follow from more general results concerning the asymptotic behaviour of the coefficients of entire functions of finite genus with finitely many zeros. As another application of these analytic considerations we establish an asymptotic property of the Hermite polynomials, leading to the explicit determination of the coefficientsC (;z) in Perron's asymptotic expansion for Laguerre polynomials in the cases =±1/2.Research supported by Deutsche Forschungsgemeinschaft through a Heisenberg-Fellowship.     

<Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript> weights and <Emphasis Type="Italic">d</Emphasis>-homogeneous measures on ahlfors <Emphasis Type="Italic">d</Emphasis>-regular space

Let X be an Ahlfors d-regular space and mad-regular measure on X . We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.     

A generalization of Brauer''s map of decomposition

We first reformulate Quillen's localization theorem forG-theory in complicial bi-Waldhausen category setting. Secondly, because of this reformulation, we are able to generalize Brauer's decomposition mapd ₀:G ₀(KG)G ₀(kG) to higherG-theory leveld _n :G _n (KG)G _n (kG),n=0, 1 ..., whereG is a finite group,R a Dedekind domain,m a maximal ideal ofR,K=quotient field ofR andk=R/m.     

The periodic Dirac operator is absolutely continuous

We consider the periodic Dirac operatorD inL ₂( ^d ). The magnetic potentialA and the electric potentialV are periodic. Ford=2 the absolute continuity ofD is established forA,VL _{r, loc} ,r>2; the proof is based on the estimates, obtained by the authors earlier [BSu2] for the periodic magnetic Schrödinger operatorM. Ford3 our considerations are based on the estimates forM, obtained in [So] forAC ^2d+3 . Under the same condition onA, forVC, the absolute continuity ofD, d3, is proved. ForA=0 the arguments of the paper give a new (and much simpler) proof of the main result of [D].The research was completed in the framework of the project INTAS-93-351.