Electronic Notes in Discrete Mathematics最新文献

The asymptotics of the number of possible endpoints of a random walk on a metric graph with incommensurable edge lengths is found.

We define the notion of weighted t-way sequences, which is built upon sequence covering arrays. The integration of a weight-based modelling formalism together with partitions of positive integers increases the expressiveness of the generated sequences considerably, and makes them applicable as abstract test sequences for real-world sequence testing problems. Applicability of this concept to real-world testing scenarios is investigated.

Finding inherent or processed links within a dataset allows to discover potential knowledge. The main contribution of this article is to define a global framework that enables optimal knowledge discovery by visually rendering co-occurences (i.e. groups of linked data instances attached to a metadata reference) – either inherently present or processed – from a dataset as facets. Hypergraphs are well suited for modeling co-occurences since they support multi-adicity whereas graphs only support pairwise relationships. This article introduces an efficient navigation between different facets of an information space based on hypergraph modelisation and visualisation.

Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora's theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [Wiles, A.J., Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics, 141 (1995), 443–551.]) or Catalan (solved in 2002 by P. Mihăilescu [Mihăilescu, P., Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math., 572 (2004), 167–195.]). The purpose of this paper is two-fold. First, we present some new integer sequences $a\left(n\right)$, counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers $i^j$ obtained for $1\le i,j\le n$. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [The On-Line Encyclopedia of Integer Sequences, http://oeis.org, OEIS Foundation Inc. 2011.]. Finally, we discuss some other novel integer sequences.

The Lagrange inversion formula is a fundamental tool in combinatorics. In this work, we investigate an inversion formula for analytic functions, which does not require taking limits. By applying this formula to certain functions we have found an interesting arithmetic triangle for which we give a recurrence formula. We then explore the links between these numbers, Pascal's triangle, and Bernoulli's numbers, for which we obtain a new explicit formula. Furthermore, we present power series and asymptotic expansions of some elementary and special functions, and some links to the Online Encyclopedia of Integer Sequences (OEIS).

Partitions play an important role in numerous combinatorial optimization problems. Here we introduce the number of ordered 3-partitions of a multiset M having equal sums denoted by S(m₁,…, m_n; ${\alpha }_1$,…, ${\alpha }_n$), for which we find the generating function and give a useful integral formula. Some recurrence formulae are then established and new integer sequences are added to OEIS, which are related to the number of solutions for the 3-signum equation.

In this paper, we extend an existing algebraic modelling technique for covering arrays by considering additional properties which are required when these structures are applied in practice in a branch of software testing called combinatorial testing. Corresponding to these properties, we give semantically equivalent systems of multivariate polynomial equations.

There are some differences between quantum and classical error corrections [Nielsen M.A., and I.L. Chuang, “Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge, 2002.]. Hence, these differences should be considered when a new procedure is performed. In our recent study, we construct new quantum error correcting codes over different mathematical structures. The classical codes over Eisenstein-Jacobi(EJ) integers are mentioned in [Huber, K., “Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165.]. There is an efficient algorithm for the encoding and decoding procedures of these codes [Huber, K., “Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165.]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers p = 7, 13, 19, 31, 37, 43, 61, … can be useful [Dong, X., C.B. Soh, E. Gunawan and L. Tang, “Groups of Algebraic Integers used for Coding QAM Signals”, Information Theory, IEEE 44 (1998), 1848–1860.]. Thus, in this study, we introduce quantum error correcting codes over EJ-integers. This type of quantum codes may lead to codes with some new and good parameters.

This work investigates the tour scheduling problem with a multi-activity context, a challenging problem that often arises in personnel scheduling. We propose a hybrid heuristic, which combines tabu search and large neighborhood search techniques. We present computational experiments on weekly time horizon problems dealing with up to five work activities. The results show that the proposed approach is able to find good quality solutions.

Let D = (V, A) be a directed graph with set of vertices V and set of arcs A, and let each arc (i, j) ∈ A, with i, j ∈ V, be associated with a non-negative cost. The constrained shortest path tour problem (CSPTP) is NP-Hard and consists in finding a shortest path between two distinct vertices s ∈ V and t ∈ V such that the path does not include repeated arcs and must visit a sequence of vertex disjoint subsets T₁, …, T_N in this order. In this work, we formulate the CSPTP as an integer linear programming (ILP) model and present valid inequalities for the problem. Computational experiments performed on benchmark data sets from the literature show that our ILP model consistently outperforms existing exact algorithms for the CSPTP and finds optimal solutions for most instances.