Electronic Notes in Discrete Mathematics最新文献

{"title":"An identity relating Fibonacci and Lucas numbers of order k","authors":"Spiros D. Dafnis,&nbsp;Andreas N. Philippou,&nbsp;Ioannis E. Livieris","doi":"10.1016/j.endm.2018.11.006","DOIUrl":"10.1016/j.endm.2018.11.006","url":null,"abstract":"<div><p>The following relation between Fibonacci and Lucas numbers of order <em>k</em>,<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>[</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>+</mo><mo>(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mo>)</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>−</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>3</mn></mrow><mrow><mi>k</mi></mrow></munderover><mo>(</mo><mi>j</mi><mo>−</mo><mn>2</mn><mo>)</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>−</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>]</mo><mo>=</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>+</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>,</mo></math></span></span></span> is derived by means of colored tiling. This relation generalizes the well-known Fibonacci-Lucas identities, <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mn>3</mn></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mo>(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mo>)</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> of A.T. Benjamin and J.J. Quinn, D. Marques, and T. Edgar, respectively.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 37-42"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123659332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weighted t-way Sequences","authors":"Bernhard Garn,&nbsp;Dimitris E. Simos","doi":"10.1016/j.endm.2018.11.007","DOIUrl":"10.1016/j.endm.2018.11.007","url":null,"abstract":"<div><p>We define the notion of <em>weighted t-way sequences</em>, 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.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 43-48"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122613304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hypergraph Modeling and Visualisation of Complex Co-occurence Networks","authors":"X. Ouvrard ,&nbsp;J.M. Le Goff ,&nbsp;S. Marchand-Maillet","doi":"10.1016/j.endm.2018.11.011","DOIUrl":"10.1016/j.endm.2018.11.011","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 65-70"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122848731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On some new arithmetic functions involving prime divisors and perfect powers","authors":"Ovidiu Bagdasar,&nbsp;Ralph Tatt","doi":"10.1016/j.endm.2018.11.002","DOIUrl":"10.1016/j.endm.2018.11.002","url":null,"abstract":"<div><p>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 <span><math><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, counting the positive integers smaller than <em>n</em>, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msup></math></span> obtained for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>. 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, <span>http://oeis.org</span><svg><path></path></svg>, OEIS Foundation Inc. 2011.]. Finally, we discuss some other novel integer sequences.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 9-15"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115376671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On an arithmetic triangle of numbers arising from inverses of analytic functions","authors":"Armen G. Bagdasaryan,&nbsp;Ovidiu Bagdasar","doi":"10.1016/j.endm.2018.11.003","DOIUrl":"10.1016/j.endm.2018.11.003","url":null,"abstract":"<div><p>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).</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 17-24"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114092709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some remarks on 3-partitions of multisets","authors":"Dorin Andrica,&nbsp;Ovidiu Bagdasar","doi":"10.1016/j.endm.2018.11.001","DOIUrl":"10.1016/j.endm.2018.11.001","url":null,"abstract":"<div><p>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; <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>,…, <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>), 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.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 1-8"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115148144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic Techniques for Covering Arrays and related Structures","authors":"Bernhard Garn,&nbsp;Dimitris E. Simos","doi":"10.1016/j.endm.2018.11.008","DOIUrl":"10.1016/j.endm.2018.11.008","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 49-54"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127422339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Bounds for Quantum Error Correcting Codes over EJ-Integers","authors":"Eda Yildiz","doi":"10.1016/j.endm.2018.11.015","DOIUrl":"10.1016/j.endm.2018.11.015","url":null,"abstract":"<div><p>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., “<em>Codes over Eisenstein-Jacobi integers</em>”, Contemporary Mathematics <strong>168</strong> (1994), 165.]. There is an efficient algorithm for the encoding and decoding procedures of these codes [Huber, K., “<em>Codes over Eisenstein-Jacobi integers</em>”, Contemporary Mathematics <strong>168</strong> (1994), 165.]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers <em>p</em> = 7, 13, 19, 31, 37, 43, 61, … can be useful [Dong, X., C.B. Soh, E. Gunawan and L. Tang, “<em>Groups of Algebraic Integers used for Coding QAM Signals</em>”, Information Theory, IEEE <strong>44</strong> (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.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"70 ","pages":"Pages 89-94"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.11.015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125641415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A MILP formulation for a tire curing scheduling problem","authors":"Héctor Cancela ,&nbsp;Pedro Piñeyro,&nbsp;Joaquín Velázquez","doi":"10.1016/j.endm.2018.07.009","DOIUrl":"10.1016/j.endm.2018.07.009","url":null,"abstract":"<div><p>In this paper we consider the scheduling problem of the curing process for a tire factory. The objective is to determine the minimum makespan in order to meet the demand requirements of different tires, restricted by the number of parts, molds and heaters and allowed combinations of mold-mold and mold-heater. We provide a mixed-integer linear programming (MILP) for the problem and two different rules or estimators for determining an upper bound value of the planning horizon, needed for solving the model. In order to evaluate the suggested estimators, we carry out some numerical experiments over ten different instances based on real data. From the results of these numerical experiments we can conclude that the tightness of estimators have a significant impact on the resolution time of the model.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"69 ","pages":"Pages 61-68"},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.07.009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123874698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Models and cutting-plane strategies for the tree-star problem","authors":"Rafael Andrade,&nbsp;Jefferson Gurguri","doi":"10.1016/j.endm.2018.07.034","DOIUrl":"10.1016/j.endm.2018.07.034","url":null,"abstract":"<div><p>Let <em>G</em> = (<em>V</em>, <em>E</em>) be a connected graph of set of nodes <em>V</em> and set of edges <em>E</em>. Let <em>T</em> = (<em>V</em>_<em>T</em>, <em>E</em>_<em>T</em>), with <em>V</em>_<em>T</em> = <em>V</em> and <em>E</em>_<em>T</em> ⊆ <em>E</em>, be a spanning tree of <em>G</em>. With each edge <em>e</em> ∈ <em>E</em> there is associated a routing cost <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>R</mi></mrow></msubsup></math></span> if <em>e</em> connects two internal nodes of <em>T</em>; or an access cost <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>A</mi></mrow></msubsup></math></span>, otherwise. The problem is to determine a spanning tree (tree-star) considering access and routing edge costs of minimum cost. We present two new formulations and a cutting-plane algorithm. One is based on a classical spanning tree model. The novelty relies on the way we capture access and routing edges depending on the internal nodes of the tree. The second model is completely new and is based on the concept of dicycle to represent routing edges as quadratic variables that are linearized accordingly to obtain a tree-star equivalent structure. Computational experiments performed on benchmark instances for models <em>P</em>_{<em>Flow</em>} and <em>P</em>_<em>HR</em> from the literature and for the new ones (<em>P</em>_<em>ST</em> and <em>P</em>_<em>DC</em>) indicate that this problem is very difficult to deal with. Only a very small number of instances was solved to optimality in a given time limit. Models <em>P</em>_<em>DC</em> and <em>P</em>_<em>HR</em>, improved with cutting-plane strategies, although they do not solve optimally almost instances, performed better for this problem, with the dicycle-based model presenting the smallest gaps for instances for which some feasible solution was found.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"69 ","pages":"Pages 261-268"},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.07.034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133446938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}