Constructing universally small subsets of a given packing index in Polish groups

Taras Banakh; Nadya Lyaskovska

Displaying similar documents to “Constructing universally small subsets of a given packing index in Polish groups”

Dimensions of non-differentiability points of Cantor functions

Yuanyuan Yao, Yunxiu Zhang, Wenxia Li (2009)

Studia Mathematica

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For a probability vector (p₀,p₁) there exists a corresponding self-similar Borel probability measure μ supported on the Cantor set C (with the strong separation property) in ℝ generated by a contractive similitude $h_{i} (x) = a_{i} x + b_{i}$ , i = 0,1. Let S denote the set of points of C at which the probability distribution function F(x) of μ has no derivative, finite or infinite. The Hausdorff and packing dimensions of S have been found by several authors for the case that $p_{i} > a_{i}$ , i = 0,1. However, when p₀ < a₀...

Packing of nonuniform hypergraphs - product and sum of sizes conditions

Paweł Naroski (2009)

Discussiones Mathematicae Graph Theory

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Hypergraphs $H ₁, . . ., H_{N}$ of order n are mutually packable if one can find their edge disjoint copies in the complete hypergraph of order n. We prove that two hypergraphs are mutually packable if the product of their sizes satisfies some upper bound. Moreover we show that an arbitrary set of the hypergraphs is mutually packable if the sum of their sizes is sufficiently small.

Packing constant for Cesàro-Orlicz sequence spaces

Zhen-Hua Ma, Li-Ning Jiang, Qiao-Ling Xin (2016)

Czechoslovak Mathematical Journal

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The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesàro-Orlicz sequence spaces ( ${ces}_{φ}$ ) defined by an Orlicz function $φ$ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant...

Some properties of packing measure with doubling gauge

Sheng-You Wen, Zhi-Ying Wen (2004)

Studia Mathematica

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Let g be a doubling gauge. We consider the packing measure $^{g}$ and the packing premeasure $₀^{g}$ in a metric space X. We first show that if $₀^{g} (X)$ is finite, then as a function of X, $₀^{g}$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $λ s u p ₀^{g} (F) \leq^{g} (B) \leq s u p ₀^{g} (F)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite $₀^{g}$ -premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling...

Packings of pairs with a minimum known number of quadruples

Jiří Novák (1995)

Mathematica Bohemica

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Let $E$ be an $n$ -set. The problem of packing of pairs on $E$ with a minimum number of quadruples on $E$ is settled for $n < 15$ and also for $n = 36 t + i$ , $i = 3$ , $6$ , $9$ , $12$ , where $t$ is any positive integer. In the other cases of $n$ methods have been presented for constructing the packings having a minimum known number of quadruples.

A note on dual approximation algorithms for class constrained bin packing problems

Eduardo C. Xavier, Flàvio Keidi Miyazawa (2009)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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In this paper we present a dual approximation scheme for the class constrained shelf bin packing problem. In this problem, we are given bins of capacity $1$ , and $n$ items of $Q$ different classes, each item $e$ with class $c_{e}$ and size $s_{e}$ . The problem is to pack the items into bins, such that two items of different classes packed in a same bin must be in different shelves. Items in a same shelf are packed consecutively. Moreover, items in consecutive shelves must be separated by shelf divisors...

The s-packing chromatic number of a graph

Wayne Goddard, Honghai Xu (2012)

Discussiones Mathematicae Graph Theory

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Let S = (a₁, a₂, ...) be an infinite nondecreasing sequence of positive integers. An S-packing k-coloring of a graph G is a mapping from V(G) to 1,2,...,k such that vertices with color i have pairwise distance greater than $a_{i}$ , and the S-packing chromatic number $χ_{S} (G)$ of G is the smallest integer k such that G has an S-packing k-coloring. This concept generalizes the concept of proper coloring (when S = (1,1,1,...)) and broadcast coloring (when S = (1,2,3,4,...)). In this paper, we consider...

Some results on packing in Orlicz sequence spaces

Y. Q. Yan (2001)

Studia Mathematica

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We present monotonicity theorems for index functions of N-fuctions, and obtain formulas for exact values of packing constants. In particular, we show that the Orlicz sequence space $l^{(N)}$ generated by the N-function N(v) = (1+|v|)ln(1+|v|) - |v| with Luxemburg norm has the Kottman constant $K (l^{(N)}) = N^{- 1} (1) / N^{- 1} (1 / 2)$ , which answers M. M. Rao and Z. D. Ren’s [8] problem.

A note on the open packing number in graphs

Mehdi Mohammadi, Mohammad Maghasedi (2019)

Mathematica Bohemica

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A subset $S$ of vertices in a graph $G$ is an open packing set if no pair of vertices of $S$ has a common neighbor in $G$ . An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The maximum cardinality of an open packing set is called the open packing number and is denoted by $ρ^{o} (G)$ . A subset $S$ in a graph $G$ with no isolated vertex is called a total dominating set if any vertex of $G$ is adjacent to some vertex of $S$ . The total domination number...

Packing four copies of a tree into a complete bipartite graph

Liqun Pu, Yuan Tang, Xiaoli Gao (2022)

Czechoslovak Mathematical Journal

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In considering packing three copies of a tree into a complete bipartite graph, H. Wang (2009) gives a conjecture: For each tree $T$ of order $n$ and each integer $k \geq 2$ , there is a $k$ -packing of $T$ in a complete bipartite graph $B_{n + k - 1}$ whose order is $n + k - 1$ . We prove the conjecture is true for $k = 4$ .

Perturbing the hexagonal circle packing: a percolation perspective

Itai Benjamini, Alexandre Stauffer (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the hexagonal circle packing with radius $1 / 2$ and perturb it by letting the circles move as independent Brownian motions for time $t$ . It is shown that, for large enough $t$ , if $𝛱_{t}$ is the point process given by the center of the circles at time $t$ , then, as $t \to \infty$ , the critical radius for circles centered at $𝛱_{t}$ to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly...