Search results for "Disjoint sets"
showing 10 items of 40 documents
Perturbations of surjective convolution operators
2002
Let μ 1 and μ 2 be (ultra)distributions with compact support which have disjoint singular supports. We assume that the convolution operator f → μ 1 *f is surjective when it acts on a space of functions or (ultra)distributions, and we investigate whether the perturbed convolution operator f→ (μ 1 + μ 2 ) * f is surjective. In particular we solve in the negative a question asked by Abramczuk in 1984.
Every triangle-free induced subgraph of the triangular lattice is(5m,2m)-choosable
2014
A graph G is (a,b)-choosable if for any color list of size a associated with each vertex, one can choose a subset of b colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer m>=1, every finite triangle-free induced subgraph of the triangular lattice is (5m,2m)-choosable.
A Polynomial Quantum Query Lower Bound for the Set Equality Problem
2004
The set equality problem is to tell whether two sets A and B are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any ω(1) query lower bound when sets A and B are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles \(\Omega(\sqrt[5]{\frac{n}{\ln n}})\) times, where n=|A|=|B|.
The Separable Complementation Property and Mrówka Compacta
2017
We study the separable complementation property for $C(K_{\cal A})$ spaces when $K_{\cal A}$ is the Mr\'owka compact associated to an almost disjoint family ${\cal A}$ of countable sets. In particular we prove that, if ${\cal A}$ is a generalized ladder system, then $C(K_{\cal A})$ has the separable complementation property ($SCP$ for short) if and only if it has the controlled version of this property. We also show that, when ${\cal A}$ is a maximal generalized ladder system, the space $C(K_{\cal A})$ does not enjoy the $SCP$.
Almost disjoint families of countable sets and separable complementation properties
2012
We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ with controlled and continuous SCP …
Improving table compression with combinatorial optimization
2002
We study the problem of compressing massive tables within the partition-training paradigm introduced by Buchsbaum et al. [SODA'00], in which a table is partitioned by an off-line training procedure into disjoint intervals of columns, each of which is compressed separately by a standard, on-line compressor like gzip. We provide a new theory that unifies previous experimental observations on partitioning and heuristic observations on column permutation, all of which are used to improve compression rates. Based on the theory, we devise the first on-line training algorithms for table compression, which can be applied to individual files, not just continuously operating sources; and also a new, …
Enumeration and Structure of Trapezoidal Words
2013
Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated …
A Classification of Trapezoidal Words
2011
Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.
Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes
2006
A multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infi…
An efficient method for clustered multi-metric learning
2019
Abstract Distance metric learning, which aims at finding a distance metric that separates examples of one class from examples of the other classes, is the key to the success of many machine learning tasks. Although there has been an increasing interest in this field, learning a global distance metric is insufficient to obtain satisfactory results when dealing with heterogeneously distributed data. A simple solution to tackle this kind of data is based on kernel embedding methods. However, it quickly becomes computationally intractable as the number of examples increases. In this paper, we propose an efficient method that learns multiple local distance metrics instead of a single global one.…