http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2012, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ijct/2012/406250/We present two variations of the game 3-Euclid. The games involve a triplet of positive integers. Two players move alternately. In the first game, each move is to subtract a positive integer multiple of the smallest integer from one of the other integers as long as the result remains positive. In the second game, each move is to subtract a positive integer multiple of the smallest integer from the largest integer as long as the result remains positive. The player who makes the last move wins. We show that the two games have the same P-positions and positions of Sprague-Grundy value 1. We present three theorems on the periodicity of P-positions and positions of Sprague-Grundy value 1. We also obtain a theorem on the partition of Sprague-Grundy values for each game. In addition, we examine the misère versions of the two games and show that the Sprague-Grundy functions of each game and its misère version differ slightly.Nhan Bao HoCopyright © 2012 Nhan Bao Ho. All rights reserved.http://www.hindawi.com/journals/ijct/2011/403140/We tackle the combinatorics of coloured hard-dimer objects. This is achieved by identifying coloured hard-dimer configurations with a certain class of rooted trees that allow for an algebraic treatment in terms of noncommutative formal power series. A representation in terms of matrices then allows to find the asymptotic behaviour of these objects.Maria Simonetta Bernabei and Horst ThalerCopyright © 2011 Maria Simonetta Bernabei and Horst Thaler. All rights reserved.http://www.hindawi.com/journals/ijct/2012/154281/All extremal ternary self-dual codes of length 48 that have some automorphism of prime order p≥5 are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code.Gabriele NebeCopyright © 2012 Gabriele Nebe. All rights reserved.http://www.hindawi.com/journals/ijct/2011/824742/We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games.Grant Cairns and Nhan Bao HoCopyright © 2011 Grant Cairns and Nhan Bao Ho. All rights reserved.http://www.hindawi.com/journals/ijct/2011/389369/The k-tuple domination problem, for a fixed positive integer k, is to find a minimum size vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The case when k=2 is called 2-tuple domination problem or double domination problem. In this paper, the 2-tuple domination problem is studied on interval graphs from an algorithmic point of view, which takes O(n2) time, n is the total number of vertices of the interval graph.Tarasankar Pramanik, Sukumar Mondal, and Madhumangal PalCopyright © 2011 Tarasankar Pramanik et al. All rights reserved.http://www.hindawi.com/journals/ijct/2011/893061/Fuzzy sets, rough sets, and later on IF sets became useful mathematical tools for solving various decision making problems and data mining problems. Molodtsov introduced another concept soft set theory as a general frame work for reasoning about vague concepts. Since most of the data collected are either linguistic variable or consist of vague concepts so IF set and soft set help a lot in data mining problem. The aim of this paper is to introduce the concept of IF soft lower rough approximation and IF upper rough set approximation. Also, some properties of this set are studied, and also some problems of decision making are cited where this concept may help. Further research will be needed to apply this concept fully in the decision making and data mining problems.Sharmistha Bhattacharya (Halder) and Bijan DavvazCopyright © 2011 Sharmistha Bhattacharya (Halder) and Bijan Davvaz. All rights reserved.http://www.hindawi.com/journals/ijct/2011/101928/For any 2-distance set X in the n-dimensional binary Hamming space Hn, let ΓX be the graph with X as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in X. The binary spherical representation number of a graph Γ, or bsr(Γ), is the least n such that Γ is isomorphic to ΓX, where X is a 2-distance set lying on a sphere in Hn. It is shown that if Γ is a connected regular graph, then bsr(Γ)≥b−m, where b is the order of Γ and m is the multiplicity of the least eigenvalue of Γ, and the case of equality is characterized. In particular, if Γ is a connected strongly regular graph, then bsr(Γ)=b−m if and only if Γ is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph.Yury J. IoninCopyright © 2011 Yury J. Ionin. All rights reserved.http://www.hindawi.com/journals/ijct/2011/137356/We describe a construction of primitive 2-designs and strongly regular graphs from the simple groups L(3,5), U(5,2), and S(6,2). The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of L(3,5), U(5,2), and S(6,2). In addition, from the group S(6,2), we construct 2-designs with parameters (28,4,4) and (28,4,1) having the full automorphism group isomorphic to U(3,3):Z2.Dean Crnković and Vedrana Mikulić CrnkovićCopyright © 2011 Dean Crnković and Vedrana Mikulić Crnković. All rights reserved.http://www.hindawi.com/journals/ijct/2011/659567/Recently, we identified a hierarchy relation between trinucleotide comma-free codes and trinucleotide circular codes (see our previous works). Here, we extend our hierarchy with two new classes of codes, called DLD and LDL codes, which are stronger than the comma-free codes. We also prove that no circular code with 20 trinucleotides is a DLD code and that a circular code with 20 trinucleotides is comma-free if and only if it is a LDL code. Finally, we point out the possible role of the symmetric group ∑4 in the mathematical study of trinucleotide circular codes.Christian J. Michel and Giuseppe PirilloCopyright © 2011 Christian J. Michel and Giuseppe Pirillo. All rights reserved.http://www.hindawi.com/journals/ijct/2011/206930/Suppose that G is a finite group, such that |G|=27p, where p is prime. We show that if S is any generating set of G, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay (G;S).Ebrahim Ghaderpour and Dave Witte MorrisCopyright © 2011 Ebrahim Ghaderpour and Dave Witte Morris. All rights reserved.http://www.hindawi.com/journals/ijct/2011/787596/Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the vertex subsets. Implications between spectral and regularity properties are discussed.Marianna BollaCopyright © 2011 Marianna Bolla. All rights reserved.http://www.hindawi.com/journals/ijct/2011/649687/The graph Ramsey number R(F1,F2) is the smallest integer N with the property that any complete graph of at least N vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to F1 all of whose edges are red or a subgraph isomorphic to F2 all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine R(θ4,θk), R(θ5,θk) for k≥4. More specifically, we establish that R(θ4,θk)=R(θ5,θk)=2k-1 for k≥7. Furthermore, we determine R(θn,θn) for n≥5. In fact, we establish that R(θn,θn)=(3n/2)-1 if n is even, 2n-1 if n is odd.M. M. M. Jaradat, M. S. A. Bataineh, and S. M. E. RadaidehCopyright © 2011 M. M. M. Jaradat et al. All rights reserved.http://www.hindawi.com/journals/ijct/2011/208260/Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.Anthony SofoCopyright © 2011 Anthony Sofo. All rights reserved.http://www.hindawi.com/journals/ijct/2011/539030/Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahon's master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory.Philip Feinsilver and John McSorleyCopyright © 2011 Philip Feinsilver and John McSorley. All rights reserved.http://www.hindawi.com/journals/ijct/2011/872703/We study isolated vertices and connectivity in the random intersection graph G(n,m,p). A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When m=⌊nα⌋ and α>6, we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi random graphs.Yilun ShangCopyright © 2011 Yilun Shang. All rights reserved.http://www.hindawi.com/journals/ijct/2011/432738/We derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.Dae San KimCopyright © 2011 Dae San Kim. All rights reserved.http://www.hindawi.com/journals/ijct/2011/937941/Base sequences BS(m,n) are quadruples (A;B;C;D) of {±1}-sequences, with A and B of length m and C and D of length n, such that the sum of their nonperiodic autocorrelation functions is a δ-function. Normal sequences NS(n) are base sequences (A;B;C;D)∈BS(n,n) such that A=B. We introduce a definition of equivalence for normal sequences NS(n) and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of NS(n) for n≤40.Dragomir Ž. ÐokovićCopyright © 2011 Dragomir Ž. Ðoković. All rights reserved.http://www.hindawi.com/journals/ijct/2010/803210/A subset X in the k-dimensional Euclidean space ℝk that contains n points (elements) is called an n-point isosceles set if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that there exist exactly two 11-point isosceles sets in ℝ4 up to isomorphisms and that the maximum cardinality of isosceles sets in ℝ4 is 11.Hiroaki KidoCopyright © 2010 Hiroaki Kido. All rights reserved.http://www.hindawi.com/journals/ijct/2011/523806/Efficient and reliable power production is necessary to meet both the profitability of power systems operations and the electricity demand, taking also into account the environmental concerns about the emissions produced by fossil-fuelled power plants. The economic emission load dispatch problem has been defined and applied in order to deal with the optimization of these two conflicting objectives, that is, the minimization of both fuel cost and emission of generating units. This paper introduces and describes a solution to this famous problem using a new metaheuristic nature-inspired algorithm, called firefly algorithm, which was developed by Dr. Xin-She Yang at Cambridge University in 2007. A general formulation of this algorithm is presented together with an analytical mathematical modeling to solve this problem by a single equivalent objective function. The results are compared with those obtained by alternative techniques proposed by the literature in order to show that it is capable of yielding good optimal solutions with proper selection of control parameters.Theofanis Apostolopoulos and Aristidis VlachosCopyright © 2011 Theofanis Apostolopoulos and Aristidis Vlachos. All rights reserved.http://www.hindawi.com/journals/ijct/2010/842636/Dragomir Ž. ÐokovićCopyright © 2010 All rights reserved.http://www.hindawi.com/journals/ijct/2010/751861/We discuss some problems and permutation statistics involving two different types of random permutations. Under the usual model of random permutations, we prove that the shifted coverage of the elements of {1, 2, …, k} of a random permutation over {1, 2, …, n}; that is, the size of the union of the cycles containing these elements, excluding these elements themselves, follows a negative hypergeometric distribution. This fact gives a probabilistic model for the coverage via the canonical cycle representation. For a different random model, we determine some random permutation statistics regarding the problem of the lost boarding pass and its variations.Tamás LengyelCopyright © 2010 Tamás Lengyel. All rights reserved.http://www.hindawi.com/journals/ijct/2010/767361/The 157, 211 triangle-free symmetric 223 configurations are classified and some of their properties are examined. We conclude that each such configuration has a blocking set. Further properties like transitivity on lines, self-duality, and self-polarity are discussed.Abdullah Al-Azemi and Anton BettenCopyright © 2010 Abdullah Al-Azemi and Anton Betten. All rights reserved.http://www.hindawi.com/journals/ijct/2010/502989/It is known that every class-regular symmetric (μ,m)-net is tactical. Also it is known that all (μ,m)-nets with m=2 or μ=1 are tactical. In the work of Al-Kenani and Mavron (2003), it is proved that every symmetric net with m=3 is tactical if and only if it is class regular. In this paper, we construct (2,4)-net and show that it is class regular and therefore tactical. New necessary and sufficient conditions are given for a symmetric net to admit a nonidentity bitranslation.Ahmad N. Al-KenaniCopyright © 2010 Ahmad N. Al-Kenani. All rights reserved.http://www.hindawi.com/journals/ijct/2010/851857/Base sequences BS(n+1,n) are quadruples of {±1}-sequences (A;B;C;D), with A and B of length n+1 and C and D of length n, such that the sum of their nonperiodic autocor-relation functions is a δ-function. The base sequence conjecture, asserting that BS(n+1,n) exist for all n, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS(n+1,n) and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS(n+1,n) for n≤30. As the number of equivalence classes grows rapidly (but not monotonically) with n, the tables in the paper cover only the cases n≤13.Dragomir Ž. ÐokovićCopyright © 2010 Dragomir Ž. Ðoković. All rights reserved.http://www.hindawi.com/journals/ijct/2010/897196/A procedure for the construction of Costas arrays of infinite size representing configurations on non-attacking queens on the chessboard is presented.Konstantinos DrakakisCopyright © 2010 Konstantinos Drakakis. All rights reserved.http://www.hindawi.com/journals/ijct/2010/821078/The existence of a class regular symmetric transversal design STDλ[3λ;3] is equivalent to a generalized Hadamard matrix of order 3u over GF(3). Let nλ be the number of nonisomorphic STDλ[3λ;3]'s. It is known that n1=1, n2=1, n3=4, n4=1, n5=0, n6≥20, and n7≥5. In this paper, it is shown that n8≥24.Teppei Hatono and Chihiro SuetakeCopyright © 2010 Teppei Hatono and Chihiro Suetake. All rights reserved.http://www.hindawi.com/journals/ijct/2010/153621/It was shown by Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.Markus Kuba and Helmut ProdingerCopyright © 2010 Markus Kuba and Helmut Prodinger. All rights reserved.http://www.hindawi.com/journals/ijct/2009/624108/Recently, learning scheduling problems have received increasing attention. However, the majority of the research assume that the actual job processing time is a function of its position. This paper deals with the single-machine scheduling problem with a sum-of-processing-time-based learning effect. By the effect of sum-of-processing-time-based learning, we mean that the processing time of a job is defined by total normal processing time of jobs in front of it in the sequence. We show that the single-machine makespan problem remains polynomially solvable under the proposed model. We show that the total completion time minimization problem for a≥1 remains polynomially solvable under the proposed model. For the case of 0<a<1, we show that an optimal schedule of the total completion time minimization problem is V-shaped with respect to normal job processing times.Xingong Zhang and Guangle YanCopyright © 2009 Xingong Zhang and Guangle Yan. All rights reserved.http://www.hindawi.com/journals/ijct/2009/520923/Let H be an m×n real matrix and let Zi be the set of column indices of the zero entries of row i of H. Then the conditions |Zk∩(∪i=1k−1Zi)|≤1 for all k  (2≤k≤m) are called the (row) Zero Position Conditions (ZPCs). If H satisfies the ZPC, then H is said to be a (row) ZPC matrix. If HT satisfies the ZPC, then H is said to be a column ZPC matrix. The real matrix H is said to have a zero cycle if H has a sequence of at least four zero entries of the form hi1j1,hi1j2,hi2j2,hi2j3,…,hikjk,hikj1 in which the consecutive entries alternatively share the same row or column index (but not both), and the last entry has one common index with the first entry. Several connections between the ZPC and the nonexistence of zero cycles are established. In particular, it is proved that a matrix H has no zero cycle if and only if there are permutation matrices P and Q such that PHQ is a row ZPC matrix and a column ZPC matrix.Marina Arav, Frank Hall, Zhongshan Li, and Bhaskara RaoCopyright © 2009 Marina Arav et al. All rights reserved.