A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ?M. Assume that the mean curvature H of the boundary ?M satisfies H≥(n?1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ?M is bounded from above by \(\frac{1}{k}\) and the upper bound is achieved if and only if M is isometric to an n-dimensional Euclidean ball of radius \(\frac{1}{k}\) .     

Means in metric spaces and the center of mass

In this article k-convex metric spaces are considered where a several variable mapping is provided as a limit point of an iteration scheme based on the midpoint map in the metric space itself. This mapping, considered as a mean of its variables, has some properties which relates it to the center of mass of these variables in the metric space. Sufficient conditions are given here for the two points to be identical, as well as upper bounds on their distances from one another. The asymptotic rate of convergence of the iterative process defining the mean is also determined here. The case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied here as examples of k-convex metric spaces.     

Propriétés de l'EDM pour un processus de Poisson d'intensité discontinue

The problem considered is a problem parameter estimation of a 2d-dimensional parameter of a Poisson process. The intensity function of the process is a smooth function with respect to first d variables and is discontinuous function of d other variables. We show the consistency and asymptotic normality of the minimum distance estimator. To cite this article: A.S. Dabye, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Covering Convex Bodies by Cylinders and Lattice Points by Flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1. K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

The slimmest arrangements of hyperplanes

We show how the results of Dowling and Wilson on Whitney numbers in ‘The slimmest geometric lattices’ imply minimum values for the numbers of k-dimensional flats and d-dimensional cells of a projective d-arrangement of hyperplanes and for the number of d-cells missed by an extra hyperplane. Their theorems also characterize the extremal arrangements. We extend their lattice results to doubly indexed Whitney numbers; thence we obtain minima for the number of k-dimensional cells and the number of pairs of flats with x \(\subseteq\) y and dim x=k, dim y=l. The lower bounds are in terms of the rank and number of points of the geometric lattice, or the dimension d and the number of hyperplanes of the arrangement. The minima for k-cells were conjectured by Grünbaum; R. W. Shannon proved the minima for k-dimensional flats and cells and characterized attainment for the latter by a more strictly geometric, non-latticial technique.     

A generalization of k-Cohen–Macaulay simplicial complexes

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CM _t , is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM _t (=1?CM _t ) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM _t property under the operation of join of two simplicial complexes. We show that a complex is k-CM _t if and only if the links of its non-empty faces are k-CM _t?1. We prove that for an integer s≤d, the (d?s?1)-skeleton of a (d?1)-dimensional k-CM _t complex is (k+s)-CM _t . This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.     

An Extension of a Theorem of Yao and Yao

In this paper we study N _d (k), the smallest positive integer such that any nice measure μ in $\mathbb{R}^{d}$ can be partitioned into N _d (k) convex parts of equal measure so that every hyperplane avoids at least k of them. A theorem of Yao and Yao states that N _d (1)≤2 ^d . Among other results, we obtain the bounds N _d (2)≤3?2 ^d?1 and N _d (1)≥C?2 ^d/2 for some constant C. We then apply these results to a problem on the separation of points and hyperplanes.     

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z(^k)(t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

Isoperimetric problems and roses of neighborhood for stationary flat processes

The paper yields necessary conditions for the directional distributions of stationary k–flat processes in ?^d that maximize their intersection density of order 2, that is, the mean (2k – d)–volume of their self–intersections in an observation window of unit d–volume. The conditions are given in terms of the rose of intersections (i.e., the intensity of intersections of the flat process with test flats). The notion of the rose of neighborhood is introduced which is an analogue of the rose of intersections for lower dimensional flat processes. Some properties of the rose of neighborhood are studied and an asymptotically unbiased estimator is given.     

A two-sided omega-theorem for an asymmetric divisor problem

In this article we study the arithmetic functiond _a,b (l,k;n) which is defined as the number of representationsn=v ^a w ^b withw lying in the residue classl modulok (a,b andl,k fixed positive integers). For the remainder term in the asymptotic formula for Σ_n≤xd_a,b(l,k,;n) we obtain an Ω_± (under a certain restriction onl andk) which is sharper than the known results for the corresponding “unrestricted” problem.     

Some notes on graphs whose index is close to 2

We consider two classes of graphs: (i) trees of order n and diameter d =n − 3 and (ii) unicyclic graphs of order n and girth g = n − 2. Assuming that each graph within these classes has two vertices of degree 3 at distance k, we order by the index (i.e. spectral radius) the graphs from (i) for any fixed k (1 ? k ? d − 2), and the graphs from (ii) independently of k.     

Local Particles Numbers in Critical Branching Random Walk

Critical catalytic branching random walk on an integer lattice ? ^d is investigated for all d∈?. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of the presence of particles at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman–Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on ? ^d .     

On the theta number of powers of cycle graphs

We give a closed formula for Lovász’s theta number of the powers of cycle graphs C _k ^d?1 and of their complements, the circular complete graphs K _k/d . As a consequence, we establish that the circular chromatic number of a circular perfect graph is computable in polynomial time. We also derive an asymptotic estimate for the theta number of C _k ^d .     

On a cone covering problem

Given a set of polyhedral cones C₁,…,Ck⊂Rd, and a convex set D, does the union of these cones cover the set D? In this paper we consider the computational complexity of this problem for various cases such as whether the cones are defined by extreme rays or facets, and whether D is the entire Rd or a given linear subspace Rt. As a consequence, we show that it is coNP-complete to decide if the union of a given set of convex polytopes is convex, thus answering a question of Bemporad, Fukuda and Torrisi.     

Balanced Metrics and Chow Stability of Projective Bundles over Kähler Manifolds II

In the previous article (Seyyedali, Duke Math. J. 153(3):573–605, 2010), we proved that slope stability of a holomorphic vector bundle E over a polarized manifold (X,L) implies Chow stability of $(\mathbb{P}E^{*},\mathcal{O}_{\mathbb{P}E^{*}}(1)\otimes\pi^{*} L^{k})$ for k?0 if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of 2πc ₁(L). In this article, using asymptotic expansions of the Bergman kernel on Sym ^d E, we generalize the main theorem of Seyyedali (Duke Math. J. 153(3):573–605, 2010) to polarizations $(\mathbb{P}E^{*},\mathcal {O}_{\mathbb{P}E^{*}}(d)\otimes\pi^{*} L^{k})$ for k?0, where d is a positive integer.     

Translational tilings by a polytope, with multiplicity

We study the problem of covering ? ^d by overlapping translates of a convex polytope, such that almost every point of ? ^d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile ? ^d . By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles ? ^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile ? ^d for some positive integer k.     

A class of graph-geodetic distances generalizing the shortest-path and the resistance distances

A new class of distances for graph vertices is proposed. This class contains parametric families of distances which reduce to the shortest-path, weighted shortest-path, and the resistance distances at the limiting values of the family parameters. The main property of the class is that all distances it comprises are graph-geodetic: d(i,j)+d(j,k)=d(i,k) if and only if every path from i to k passes through j. The construction of the class is based on the matrix forest theorem and the transition inequality.