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1.
Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS 3, connected sums of handlesS 1 × S 2 , euclidean and spherical space forms, and other examples with non-trivial fundamental group.  相似文献   

2.
We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope Pn is the convex hull of the set of incidence vectors of all partitions n=x1+2x2++nxn. We show that the sequence P1,P2,…,Pn,… can be treated as an embedded chain. The dynamics of behavior of the vertices of Pn, as n increases, is established. Some sufficient and some necessary conditions for a point of Pn to be its vertex are proved. Representation of the partition polytope as a polytope on a partial algebra—which is a generalization of the group polyhedron in the group theoretic approach to the integer linear programming—allows us to prove subadditive characterization of the nontrivial facets of Pn. These facets correspond to extreme rays of the cone of subadditive functions with additional requirements p0=pn and pi+pni=pn,1≤i<n. The trivial facets are explicitly indicated. We also show how all vertices and facets of the polytopes of constrained partitions—in which some numbers are forbidden to participate—can be obtained from those of the polytope Pn. All vertices and facets of Pn for n≤8 and n≤6, respectively, are presented.  相似文献   

3.
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.  相似文献   

4.
A convex polytope in real Euclidean space islattice-free if it intersects some lattice in space exactly in its vertex set. Lattice-free polytopes form a large and computationally hard class, and arise in many combinatorial and algorithmic contexts. In this article, affine and combinatorial properties of such polytopes are studied. First, bounds on some invariants, such as the diameter and layer-number, are given. It is shown that the diameter of ad-dimensional lattice-free polytope isO(d 3). A bound ofO(nd+d 3) on the diameter of ad-polytope withn facets is deduced for a large class of integer polytopes. Second, Delaunay polytopes and [0, 1]-polytopes, which form major subclasses of lattice-free polytopes, are considered. It is shown that, up to affine equivalence, for anyd≥3 there are infinitely manyd-dimensional lattice-free polytopes but only finitely many Delaunay and [0, 1]-polytopes. Combinatorial-types of lattice-free polytopes are discussed, and the inclusion relations among the subclasses above are examined. It is shown that the classes of combinatorial-types of Delaunay polytopes and [0,1]-polytopes are mutually incomparable starting in dimension six, and that both are strictly contained in the class of combinatorial-types of all lattice-free polytopes. This research was supported by DIMACS—the Center for Discrete Mathematics and Theoretical Computer Science at Rutgers University.  相似文献   

5.
Suppose we toss an independent coin with probability of success p for each subset of [n]={1,…,n}, and form the random hypergraph (n,p) by taking as hyperedges the subsets with successful coin tosses. We investigate the cardinality of the largest Sperner family contained in (n,p). We obtain a sharp result for the range of p=p(n) in which this Sperner family has cardinality comparable to the cardinality of (n,p).  相似文献   

6.
In 1974, Sen proved weak convergence of the empirical processes (in the J1-topology on Dp[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φn}, i.e., φn = O(n−1−δ) for some δ > 0.  相似文献   

7.
Let = {Ut: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {Utn1 * μ2*…*μn)*δxn}, where {μn}Q, {xn}X and measures Utnμj (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes Lm( ) as follows: L0( )= ( (X); ), Lm( )= (Lm−1( ); ) for m=1, 2,… and L( )=m=0Lm( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from Lm( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.  相似文献   

8.
For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δa,r,C,D(X) = (I − (ar(|X|2)) D−1/2CD1/2 |X|−2)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X1, X2, …, Xn are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X1, X2, …, Xn), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.  相似文献   

9.
Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1,…,n}, then with probability tending to 1 as n→∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components.  相似文献   

10.
Martin Bokler   《Discrete Mathematics》2003,270(1-3):13-31
In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r2(q) be the number such that q+r2(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most qm+qm−1+…+q+1+r2(q)·(∑i=2mnm−1qi) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals qm+qm−1+…+q+1+r2(q)(∑i=2mnm−1qi). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<qm+qm−1+…+q+1+r2(q)(qm−1+qm−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. For q=p3h, p7 and q not a square we show this assertion for |B|qm+qm−1+…+q+1+q2/3·(qm−1+…+1).  相似文献   

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