Cyclic Difference Packing and Covering Arrays

Let n and k be positive integers. Let C_q be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (a_ij) with entries a_ij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from C_q such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of C_q occurs in the difference list at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.     

Simultaneous Packing and Covering in Sequence Spaces

We adapt a construction of Klee (1981) to find a packing of unit balls in ℓ _p (1≤p<∞) which is efficient in the sense that enlarging the radius of each ball to any R>2^1−1/p covers the whole space. We show that the value 2^1−1/p is optimal. Dedicated to the memory of Victor Klee.     

Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles

According to the classical Erd?s–Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most vertices that hits all its cycles. Robertson and Seymour (J Comb Theory Ser B 41 (1986), 92–114) generalized this result in the best possible way. More specifically, they showed that if is the class of all graphs that can be contracted to a fixed planar graph H, then every graph G either contains a set of k vertex‐disjoint subgraphs of G, such that each of these subgraphs is isomorphic to some graph in or there exists a set S of at most vertices such that contains no subgraph isomorphic to any graph in . However, the function f is exponential. In this note, we prove that this function becomes quadratic when consists all graphs that can be contracted to a fixed planar graph . For a fixed c, is the graph with two vertices and parallel edges. Observe that for this corresponds to the classical Erd?s–Pósa theorem.     

The Set of Packing and Covering Densities of Convex Disks

For every convex disk $K$ (a convex compact subset of the plane, with non-void interior), the packing density $\delta (K)$ and covering density ${\vartheta (K)}$ form an ordered pair of real numbers, i.e., a point in $\mathbb{R }^2$ . The set $\varOmega $ consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on $\delta (K)$ and ${\vartheta (K)}$ jointly outline a relatively small convex polygon $P$ that contains $\varOmega $ , while the exact shape of $\varOmega $ remains a mystery. Here we describe explicitly a leaf-shaped convex region $\Lambda $ contained in $\varOmega $ and occupying a good portion of $P$ . The sets $\varOmega _T$ and $\varOmega _L$ of translational packing and covering densities and lattice packing and covering densities are defined similarly, restricting the allowed arrangements of $K$ to translated copies or lattice arrangements, respectively. Due to affine invariance of the translative and lattice density functions, the sets $\varOmega _T$ and $\varOmega _L$ are compact. Furthermore, the sets $\varOmega , \,\varOmega _T$ and $\varOmega _L$ contain the subsets $\varOmega ^\star , \,\varOmega _T^\star $ and $\varOmega _L^\star $ respectively, corresponding to the centrally symmetric convex disks $K$ , and our leaf $\Lambda $ is contained in each of $\varOmega ^\star , \,\varOmega _T^\star $ and $\varOmega _L^\star $ .     

On the Packing Densities and the Covering Densities of the Cartesian Products of Convex Bodies

In this article we study the following problem: Is the covering (packing) density of a Cartesian product of two convex bodies always equal to the product of their corresponding covering (packing) densities? For the covering case we get a negative answer. For the packing case we get a combinatorial version which seems to be important for its own interest.     

On the Packing Densities and the Covering Densities of the Cartesian Products of Convex Bodies

In this article we study the following problem: Is the covering (packing) density of a Cartesian product of two convex bodies always equal to the product of their corresponding covering (packing) densities? For the covering case we get a negative answer. For the packing case we get a combinatorial version which seems to be important for its own interest.     

Packing and covering with integral feasible flows in integral supply-demand networks

Polynomial-time algorithms are presented for solving combinatorial packing and covering problems defined from the integral feasible flows in an integral supply-demand network. These algorithms are also shown to apply to packing and covering problems defined by the minimal integral solutions to general totally unimodular systems. Analogous problems arising from matroid bases are also discussed and it is demonstrated that a means for solving such problems is provided by recent work of Cunningham.     

An Inequality on the Coding Gain of Densest Lattice Packings in Successive Dimensions

A lower bound of the form is derived on the coding gain of the densest n-dimensional (n-D) lattice(s). The bound is obtained based on constructing an n-D lattice which consists of parallel layers. Each layer isselected as a translated version of a densest ( n-1)-D lattice.0The relative positioning of the layers is adjusted to make the coding gainas large as possible. For large values of n, the bound isimproved through tightening Ryskov's inequality on covering radius andminimum distance of a lattice.     

两类拟共形映射的掩盖定理

In this paper we define two classes of quasiconformal mappings,and study their covering properties by methods of module.We obtain some new results.In the meantime,we give new methods to prove Koebe 1/2 covering theorem on convex conformal mappings.     

The Densest Packing of 19 Congruent Circles in a Circle

Dense packings of n congruent circles in a circle were given by Kravitz in 1967 for n = 2,..., 16. In 1969 Pirl found the optimal packings for n 10, he also conjectured the dense configurations for 11 n 19. In 1994, Melissen provided a proof for n = 11. In this paper we exhibit the densest packing of 19 congruent circles in a circle with the help of a technique developed by Bateman and Erdös.     

Packing Densities and Simulated Tempering for Hard Core Gibbs Point Processes

Monotonicity and convergence properties of the intensity of hard core Gibbs point processes are investigated and compared to the closest packing density. For such processes simulated tempering is shown to be an efficient alternative to commonly used Markov chain Monte Carlo algorithms. Various spatial characteristics of the pure hard core process are studied based on samples obtained with the simulated tempering algorithm.     

Packing superballs from codes and algebraic curves

In the present paper, we make use of codes with good parameters and algebraic curves over finite fields with many rational points to construct dense packings of superballs. It turns out that our packing density is quite reasonable. In particular, we improve some values for the best-known lower bounds on packing density.     

A Bound on the Ratio between the Packing and Covering Densities of a Convex Body

It is shown that if K is a compact convex set which is centrally symmetric and has a nonempty interior, then the density of the tightest lattice packing with copies of K in Euclidean 3-space divided by the density of the thinnest lattice covering of Euclidean 3-space with copies of K is greater than or equal to 1/4. It is likely this bound can be improved, though not beyond approximately 1/2. Received October 8, 1998, and in revised form December 30, 1998.     

The Pointwise Densities of the Cantor Measure

Let be the classical middle-third Cantor set and let μ be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x the upper and lower s-densities Θ*^s(μ , x), Θ*^s(μ , x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F such that 9(Θ*^s(μ , x))^{− 1/s} + (Θ*^s(μ , x))^{− 1/s} = 16 holds for x \F. Furthermore, for μ_C almost all x, Θ*^s(μ , X) − 2 · 4^{− s} and Θ*^s(μ , x) = 4^{− s}. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set is 4^s.     

The Covering Theorems for Normalized Holomorphic Mappings on Bounded Starlike Circular Domains in C~n

In this paper, we will investigate convex domains and starlike domains which contain the image set of normalized holomorphic mappings on bounded starlike circular domains in Cn.     

GA-凸函数的Hadamard型不等式的改进和应用

通过研究函数的凸性、单调性及相关理论,建立了关于GA-凸函数的一些新的Hadamard型不等式,这些不等式推广了最近文献中的有关结果.     

Packing triangles in a graph and its complement

How few edge‐disjoint triangles can there be in a graph G on n vertices and in its complement ? This question was posed by P. Erd?s, who noticed that if G is a disjoint union of two complete graphs of order n/2 then this number is n²/12 + o(n²). Erd?s conjectured that any other graph with n vertices together with its complement should also contain at least that many edge‐disjoint triangles. In this paper, we show how to use a fractional relaxation of the above problem to prove that for every graph G of order n, the total number of edge‐disjoint triangles contained in G and is at least n²/13 for all sufficiently large n. This bound improves some earlier results. We discuss a few related questions as well. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 203–216, 2004     

The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space

Summary We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in<span lang=EN-US style='font-size:10.0pt; mso-ansi-language:EN-US'>[12]and [13]. The optimal hyperball packings of these tilings are generated by the ``inscribed hyperspheres' whose metric data can be calculated by our method -- based on the projective interpretation of the hyperbolic geometry -- by the volume formulas of J. Bolyai and R. Kellerhals, respectively. We summarize in some tables the data and the densities of the optimal hyperball packings to each prism tiling in the hyperbolic space H³.     

一个不等式的进一步推广和加强

利用幂级数展式和凸函数的性质把关于一个不等式的推广和强化的两个最新结果推广到更加一般的情形p(p -1 ) d ap- 1pn+1-ap- 1pm <∑nk=m1a1pk相似文献   

An equivalent condition for the self similar sets on the real line to have best coverings

In this paper, an equivalent condition for the self similar sets on the real line to have best coverings is given. As a result, it partly gives answer to the conjecture which was posed by Zhou and Feng [Zhou, Z. L., Feng, L.： Twelve open problems on the exact value of the Hausdorff measure and on topological entropy： A brief survey of recent results. Nonlinearity, 17（2）, 493-502 （2004）].