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SubjectsRecurrent sequences (4)Geometric patterns (2)Integer sequences (2)Mathematics (2)3-partition of a multiset (1)View MoreJournal

Electronic Notes in Discrete Mathematics (12)

AuthorsBagdasar, Ovidiu (8)Popa, Ioan-Lucian (4)Bagdasaryan, Armen G. (2)Korpelainen, Nicholas (2)Andrica, Dorin (1)View MoreYear (Issue Date)2018-06-14 (4)2016-12-01 (3)2018-12-06 (3)2016-12 (1)2018-06 (1)TypesArticle (12)
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Rostering air traffic controllers.

Conniss, Richard (Elsevier, 2018-06)

Many, if not most, real world scheduling problems fall into the class of NP. Classical, mathematically exact methods when applied to these problems often suffer from scaling issues, that prevent the computation of a solution in reasonable time. Similarly, real world problems can often be characterised by the requirement to dynamically change any prebuilt roster in reaction to unforeseen changes to the resources available, such as short notice staff absence or a change in the requirements of the task to be fulfilled. Metaheuristic methods have been successfully used to make approximations to optimal solutions which are good enough for practical use. In this paper an example of a novel and complex employee scheduling, or rostering, problem will be discussed and a simple metaheuristic method demonstrated to solve the problem.

On some new arithmetic functions involving prime divisors and perfect powers.

Bagdasar, Ovidiu; Tatt, Ralph-Joseph (Elsevier, 2018-12-06)

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 [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), 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 ≤ i, j ≤ 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) [5]. Finally, we discuss some other novel integer sequences.

On an arithmetic triangle of numbers arising from inverses of analytic functions.

Bagdasaryan, Armen G.; Bagdasar, Ovidiu (Elsevier, 2018-12-06)

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).

Some remarks on 3-partitions of multisets.

Andrica, Dorin; Bagdasar, Ovidiu (Elsevier, 2018-12-06)

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, ..., mn; α1, ..., αn), 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.

A boundary class for the k-path partition problem.

Korpelainen, Nicholas (Elsevier, 2018-06-14)

We establish the first known boundary class for the k-path partition problem and deduce that for a graph class defined by finitely many minimal forbidden induced subgraphs, the k-path partition problem remains NP-hard unless one of the forbidden induced subgraphs is a subcubic tree (a tree of maximum degree at most 3) with at most one vertex of degree 3.

Direct reading algorithm for hierarchical clustering

Murtagh, Fionn; Contreras, Pedro (Elsevier, 2016-12-01)

Reading the clusters from a data set such that the overall computational complexity is linear in both data dimensionality and in the number of data elements has been carried out through filtering the data in wavelet transform space. This objective is also carried out after an initial transforming of the data to a canonical order. Including high dimensional, high cardinality data, such a canonical order is provided by row and column permutations of the data matrix. In our recent work, we induce a hierarchical clustering from seriation through unidimensional representation of our observations. This linear time hierarchical classification is directly derived from the use of the Baire metric, which is simultaneously an ultrametric. In our previous work, the linear time construction of a hierarchical clustering is studied from the following viewpoint: representing the hierarchy initially in an m-adic, m =10, tree representation, followed by decreasing m to smaller valued representations that include p-adic representations, where p is prime and m is a non-prime positive integer. This has the advantage of facilitating a more direct visualization and hence interpretation of the hierarchy. In this work we present further case studies and examples of how this approach is very advantageous for such an ultrametric topological data mapping.

On the geometry of certain periodic non-homogeneous Horadam sequences.

Bagdasar, Ovidiu; Popa, Ioan-Lucian (Elsevier, 2016-12)

Horadam sequences are second-order recurrences depending on a family of four complex parameters: two initial conditions and two recurrence coefficients. The periodicity conditions, as well as the number and geometric structure of self-repeating Horadam sequences of fixed length have been recently investigated. In this paper various geometric properties of non-homogeneous Horadam sequences are explored, including periodicity conditions and the structure of certain periodic orbits.

A new graph construction of unbounded clique-width.

Korpelainen, Nicholas (Elsevier, 2016-12-01)

We define permutation-partition graphs by replacing one part of a 2K2-free bipartite graph (a bipartite chain graph) by an induced linear forest. We show that this hereditary graph class is of of unbounded clique-width (with a new graph construction of large clique-width). We show that this graph class contains no minimal graph class of unbounded clique-width, and give a conjecture for a contained boundary class for this property.

On the ratios and geometric boundaries of complex Horadam sequences.

Bagdasar, Ovidiu; Hedderwick, Eve; Popa, Ioan-Lucian (Elsevier, 2018-06-14)

Horadam sequences are second-order linear recurrences in the complex plane which depend on two initial conditions and two recurrence coefficients which are complex numbers. Recently, numerous papers have been devoted to the periodicity of these sequences, as well as to generalizations and applications. In this paper we investigate aspects related to the sequence of rations of consecutive terms and geometric bounds of Horadam sequences. We also propose some directions for further study.

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