Search results for "rete"

Universal Lyndon Words

2014

A word w over an alphabet Σ is a Lyndon word if there exists an order defined on Σ for which w is lexicographically smaller than all of its conjugates (other than itself). We introduce and study universal Lyndon words, which are words over an n-letter alphabet that have length n! and such that all the conjugates are Lyndon words. We show that universal Lyndon words exist for every n and exhibit combinatorial and structural properties of these words. We then define particular prefix codes, which we call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in bijection with the set of the shortest unrepeated prefixes of the conjugates of a universal Lyndon word. This allows us t…

Sobolev embeddings, extensions and measure density condition

2008

AbstractThere are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density condition. The second main result, Theorem 5, provides several characterizations of the Wm,p-extension domains for 1<p<∞. As a corollary we prove that the property of being a W1,p-extension domain, 1<p⩽∞, is invariant under bi-Lipschitz mappings, Theorem 8.

Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals

2010

The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra \({\mathfrak{A}_0}\) of a topological *-algebra \({\mathfrak{A}}\), satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given.

Lipschitz conditions,b-arcwise connectedness and conformal mappings

1982

On the computational power of affine automata

2017

We investigate the computational power of affine automata (AfAs) introduced in [4]. In particular, we present a simpler proof for how to change the cutpoint for any affine language and a method how to reduce error in bounded error case. Moreover, we address to the question of [4] by showing that any affine language can be recognized by an AfA with certain limitation on the entries of affine states and transition matrices. Lastly, we present the first languages shown to be not recognized by AfAs with bounded-error.

On Packing Colorings of Distance Graphs

2014

International audience; The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=…

Understanding Quantum Algorithms via Query Complexity

2017

Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we su…

On Physical Problems that are Slightly More Difficult than QMA

2013

We study the complexity of computational problems from quantum physics. Typically, they are studied using the complexity class QMA (quantum counterpart of NP) but some natural computational problems appear to be slightly harder than QMA. We introduce new complexity classes consisting of problems that are solvable with a small number of queries to a QMA oracle and use these complexity classes to quantify the complexity of several natural computational problems (for example, the complexity of estimating the spectral gap of a Hamiltonian).

The Alternating BWT: an algorithmic perspective

2020

Abstract The Burrows-Wheeler Transform (BWT) is a word transformation introduced in 1994 for Data Compression. It has become a fundamental tool for designing self-indexing data structures, with important applications in several areas in science and engineering. The Alternating Burrows-Wheeler Transform (ABWT) is another transformation recently introduced in Gessel et al. (2012) [21] and studied in the field of Combinatorics on Words. It is analogous to the BWT, except that it uses an alternating lexicographical order instead of the usual one. Building on results in Giancarlo et al. (2018) [23] , where we have shown that BWT and ABWT are part of a larger class of reversible transformations, …

Polyhedral results for a vehicle routing problem

1991

Abstract The Vehicle Routing Problem is a well known, and hard, combinatorial problem, whose polyhedral structure has deserved little attention. In this paper we consider the particular case in which all the demands are equal (since in the general case the associated polytope may be empty). From a known formulation of the problem we obtain the dimension of the corresponding polytope and we study the facetial properties of every inequality in it.