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1.
Covering points by disjoint boxes with outliers   总被引:1,自引:0,他引:1  
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2.
We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body C , symmetric about the origin 0. This enables us to prove the following statement, which settles a problem of G. Halász. The maximum number of n-wise linearly independent lattice points in the n-dimensional ball r B n of radius r around 0 is O(rn/(n-1)). This bound cannot be improved. We also show that the order of magnitude of the number of diferent (n - 1)-dimensional subspaces induced by the lattice points in r&Bgr;n is rn/(n-1).  相似文献   

3.
In many point-line geometries, to cover all points except one, more lines are needed than to cover all points. Bounds can be given by looking at the dimension of the space of functions induced by polynomials of bounded degree.  相似文献   

4.
In this note we extend the main result of [8] concerning covering blocks of group algebras to the context of p-permutation algebras over arbitrary ground fields. We also establish several connections between the Green correspondence for points and the correspondences that arise using the Brauer homomorphism.  相似文献   

5.
6.
Consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines. We study the problem of covering all mobile clients using a set of k disks centered at k fixed centers. Each disk exists only at one instant and while it does, covers any client within its coverage radius. The task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk. In particular, we study the optimization problem of minimizing the maximum coverage radius. First we prove that, although the static version of the problem is polynomial, the kinetic version is NP-hard. Moreover, we show that the problem is not approximable by a constant factor (unless P = NP). We then present a generic framework to solve it for fixed values of k, which in turn allows us to solve more general optimization problems. Our algorithms are efficient for constant values of k.  相似文献   

7.
8.
Consider the Dvoretzky random covering on the circle T with a decreasing length sequence {?n}n?1 such that . We study, for a given β?0, the set Fβ of points which are asymptotically covered by a number βLn of the first n randomly placed intervals where . Three typical situations arise, delimited by two “phase transitions”, according to is zero, positive-finite or infinite, where . More precisely, if ?n tends to zero rapidly enough so that then, with probability one, dimHFβ=1 for all β?0; if ?n is moderate so that then, with probability one, we have for and Fβ=∅ for where and is the interval consisting of β's such that ; eventually, if ?n is so slow that then, with probability one, F1=T. This solves a problem raised by L. Carleson in a rather satisfactory fashion.Analogous results are obtained for the Poisson covering of the line, which is studied as a tool.  相似文献   

9.
The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results.  相似文献   

10.
For a closed connected triangulatedn-manifoldM, we study some numerical invariants (namedcategory andcovering numbers) ofM which are strictly related to the topological structure ofM. We complete the classical results of 3-manifold topology and then we prove some characterization theorems in higher dimensions. Finally some applications are given about the minimal number of critical points (resp. values) of Morse functions defined on a closed connected smoothn-manifold. Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. and financially supported by the M.P.I. of Italy within the project “Geometria delle Varietà Differenziabili”.  相似文献   

11.
Inverse function theorems for smooth nonlinear maps defined on convex cones in Banach spaces in a neighborhood of an irregular point are considered. The corresponding covering theorem is proved. The proofs are based on a Banach open mapping theorem for convex cones in Banach spaces, which is also proved in the paper. Sufficient conditions for tangency to the zero set of a nonlinear map without a priori regularity assumptions are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 483–497.Original Russian Text Copyright © 2005 by A. V. Arutyunov.This revised version was published online in April 2005 with a corrected issue number.  相似文献   

12.
The problems of minimum edge and minimum vertex covers by paths are discussed. The results relate to papers by Gallai-Milgram, Meyniel, Alspach-Pullman and others. One of the main results is concerned with partially ordered sets.  相似文献   

13.
Let to every elementx of a finite setM be associated some nonempty subsetM (x) ofM in such a way that the implicationyM(x)xM(y) is fulfilled. We prove two upper estimations for the least number of setsM(x) which are necessary to coverM. Several applications to number theory are presented.  相似文献   

14.
“If G is a 2-connected graph with n vertices and minimum degree d, then the vertices of G can be covered by less than n/d cycles. This settles a conjecture of Enomoto, Kaneko and Tuza for 2-connected graphs.”  相似文献   

15.
Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \(\mathcal{G }_q(n,r)\) by subspaces from the Grassmann graph \(\mathcal{G }_q(n,k)\) , \(k \ge r\) , are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, \(q\) -analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for \(q=2\) with \(r=2\) or \(r=3\) . We discuss the density for some of these coverings. Tables for the best known coverings, for \(q=2\) and \(5 \le n \le 10\) , are presented. We present some questions concerning possible constructions of new coverings of smaller size.  相似文献   

16.
For any space X, denote by dis(X) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis(X)?m, where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c. Here we show that this can be done if X is also hereditarily normal.Moreover, we prove the following mapping theorem that involves the cardinal function dis(X). If is a continuous surjection of a countably compact T2 space X onto a perfect T3 space Y then .  相似文献   

17.
An isometric path is merely any shortest path between two vertices. If the vertices of the hypercube Qn are represented by the set of 0–1 vectors of length n, an isometric path is obtained by changing the coordinates of a vector one at a time, never changing the same coordinate more than once. The minimum number of isometric paths required to cover the vertices of Qn is at least 2n/(n+1). We show that when n+1 is a power of 2, the lower bound is in fact the minimum. In doing so, we construct a family of disjoint isometric paths which can be used to find an upper bound for additional classes of hypercubes.  相似文献   

18.
Take a unit square and turn it into an annulus by cutting a parallel square hole in it. We prove that if the square hole has edge length $1 - 1/ \sqrt2 \approx 0.29$ and a finite number of strips cover the annulus, then after appropriate rearrangement the same strips can cover the unit square as well.  相似文献   

19.
In 2000 Bezdek asked which plane convex bodies have the property that whenever an annulus, consisting of the body less a sufficiently small scaled copy of itself, is covered by strips, the sum of the widths of the strips must still be at least the minimal width of the body. We characterise the polygons for which this is so.  相似文献   

20.
We study the number of solutions of the Diophantine equationn=x 1 x 2+x 2 x 3+x 3 x 4+...+x k x k+1 The combinatorial interpretation of this equation provides the name stacked lattices boxes. The study of these objects unites three separate threads in number theory: (1) the Liouville methods, (2) MacMahon's partitions withk different parts, (3) the asymptotics of divisor sums begun by Ingham.Partially supported by National Science Foundation Grant DMS-9206993, USA.  相似文献   

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