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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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.

Packing four copies of a tree into a complete bipartite graph

Liqun Pu, Yuan Tang, Xiaoli Gao (2022)

Czechoslovak Mathematical Journal

Similarity:

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

Similarity:

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...

Translative packing of a convex body by sequences of its homothetic copies

Janusz Januszewski (2008)

Archivum Mathematicum

Similarity:

Every sequence of positive or negative homothetic copies of a planar convex body $C$ whose total area does not exceed $0.175$ times the area of $C$ can be translatively packed in $C$ .

Continuous rearrangements of the Haar system in $H_{p}$ for 0 < p < ∞

Krzysztof Smela (2008)

Studia Mathematica

Similarity:

We prove three theorems on linear operators $T_{τ, p} : H_{p} (ℬ) \to H_{p}$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{τ, p}$ to be continuous for 0 < p < ∞.

Construction of an Uncountable Difference between Φ(B) and $Φ_{f} (B)$

Josh Campbell, David Swanson (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We construct a set B and homeomorphism f where f and $f^{- 1}$ have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.

Characterization of local dimension functions of subsets of $ℝ^{d}$

L. Olsen (2005)

Colloquium Mathematicae

Similarity:

For a subset $E \subseteq ℝ^{d}$ and $x \in ℝ^{d}$ , the local Hausdorff dimension function of E at x is defined by $d i m_{H, l o c} (x, E) = l i m_{r ↘ 0} d i m_{H} (E \cap B (x, r))$ where $d i m_{H}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

On the asymptotics of counting functions for Ahlfors regular sets

Dušan Pokorný, Marc Rauch (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We deal with the so-called Ahlfors regular sets (also known as $s$ -regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit ${lim}_{ε \to 0 +} ε^{s} N (ε, K)$ exist, where $K$ is an $s$ -regular set and $N (ε, K)$ is for instance the $ε$ -packing number of $K$ ?

Mean value densities for temperatures

N. Suzuki, N. A. Watson (2003)

Colloquium Mathematicae

Similarity:

A positive measurable function K on a domain D in $ℝ^{n + 1}$ is called a mean value density for temperatures if $u (0, 0) = \int \int_{D} K (x, t) u (x, t) d x d t$ for all temperatures u on D̅. We construct such a density for some domains. The existence of a bounded density and a density which is bounded away from zero on D is also discussed.

Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

Similarity:

We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ${(A ₙ)}_{n \in ω}$ of X such that $f_{| A ₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

Similarity:

We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{δ}$ sections. ...

The effective Borel hierarchy

M. Vanden Boom (2007)

Fundamenta Mathematicae

Similarity:

Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is $Σ_{α}$ (respectively, $Π_{α}$ ) in the Borel hierarchy iff K is axiomatized by an infinitary $Σ_{α}$ (respectively, $Π_{α}$ ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective $Σ_{α}$ or effective $Π_{α}$ Borel set with a computable infinitary sentence of the same complexity....

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

Anders Nilsson, Peter Wingren (2007)

Studia Mathematica

Similarity:

A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(d i m_{H} (U), {\underset{̲}{d i m}}_{B} (U), {\bar{d i m}}_{B} (U)) = (r, s, t)$ . Moreover, $2^{- n} \leq H^{r} (K) \leq 2 n^{r / 2}$ .