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36
IRREDUCIBLE NUMERICAL SEMIGROUPS
 PACIFIC JOURNAL OF MATHEMATICS VOL. 209, NO. 1, 2003
, 2003
"... We give a characterization for irreducible numerical semigroups. From this characterization we obtain that every irreducible numerical semigroup is either a symmetric or pseudosymmetric numerical semigroup. We study the minimal presentations of an irreducible numerical semigroup. Separately, we deal ..."
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Cited by 18 (6 self)
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We give a characterization for irreducible numerical semigroups. From this characterization we obtain that every irreducible numerical semigroup is either a symmetric or pseudosymmetric numerical semigroup. We study the minimal presentations of an irreducible numerical semigroup. Separately, we deal with the cases of maximal embedding dimension and multiplicity 3 and 4.
Representations of integers by linear forms in nonnegative integers
 J. Number Theory
, 1972
"... Let Sz be the set of positive integers that are omitted values of the form f = z” = *1 a.x. $1) where the a, are fixed and relatively prime natural numbers and the xi are variable nonnegative integers. Set w = #Q and K = max 0 + 1 (the conductor). Properties of w and K are studied, such as an estima ..."
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Cited by 16 (1 self)
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Let Sz be the set of positive integers that are omitted values of the form f = z” = *1 a.x. $1) where the a, are fixed and relatively prime natural numbers and the xi are variable nonnegative integers. Set w = #Q and K = max 0 + 1 (the conductor). Properties of w and K are studied, such as an estimate for w (similar to one found by Brauer) and the inequality 2w> K. The socalled Gorenstein condition is shown to be equivalent to 2w = K. 1.
Finite Group Actions and Asymptotic Expansion of . . .
, 1995
"... We establish an asymptotic expansion for the number jHom(G; S n )j of actions of a finite group G on an nset in terms of the order jGj = m and the number s G (d) of subgroups of index d in G for djm: This expansion and related results on the enumeration of finite group actions follow from more ge ..."
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Cited by 10 (6 self)
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We establish an asymptotic expansion for the number jHom(G; S n )j of actions of a finite group G on an nset in terms of the order jGj = m and the number s G (d) of subgroups of index d in G for djm: This expansion and related results on the enumeration of finite group actions follow from more general results concerning the asymptotic behaviour of the coefficients of entire functions of finite genus with finitely many zeros. As another application of these analytic considerations we establish an asymptotic property of the Hermite polynomials, leading to the explicit determination of the coefficients C (ff; z) in Perron's asymptotic expansion for Laguerre polynomials in the cases ff = \Sigma1=2:
Hard Equality Constrained Integer Knapsacks
, 2005
"... We consider the following integer feasibility problem: “Given positive integer numbers a0, a1,..., an, with gcd(a1,..., an) = 1 and a = (a1,..., an), does there exist a vector x ∈ Z n ≥0 satisfying ax = a0? ” We prove that if the coefficients a1,..., an have a certain decomposable structure, then t ..."
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Cited by 10 (0 self)
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We consider the following integer feasibility problem: “Given positive integer numbers a0, a1,..., an, with gcd(a1,..., an) = 1 and a = (a1,..., an), does there exist a vector x ∈ Z n ≥0 satisfying ax = a0? ” We prove that if the coefficients a1,..., an have a certain decomposable structure, then the Frobenius number associated with a1,..., an, i.e., the largest value of a0 for which ax = a0 does not have a nonnegative integer solution, is close to a known upper bound. In the instances we consider, we take a0 to be the Frobenius number. Furthermore, we show that the decomposable structure of a1,..., an makes the solution of a lattice reformulation of our problem almost trivial, since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small. For branchandbound such instances are difficult to solve, since they are infeasible and have large values of a0/ai, 1 ≤ i ≤ n. We illustrate our results by some computational examples.
Frobenius numbers by lattice point enumeration
 Integers
"... The Frobenius number g(A) of a set A = (a1, a2,..., an) of positive integers is the largest integer not representable as a nonnegative linear combination of the ai. We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for ..."
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Cited by 7 (0 self)
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The Frobenius number g(A) of a set A = (a1, a2,..., an) of positive integers is the largest integer not representable as a nonnegative linear combination of the ai. We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for computing it. The algorithm appears to run in average time that is softly quadratic and we prove that this is the case for almost all of the steps. In practice, the algorithm is very fast: examples with n = 4 and the numbers in A having 100 digits take under one second. The running time increases with dimension and we can succeed up to n = 11. We use the geometric structure of a fundamental domain D, having a1 points, related to a lattice constructed from A. The domain encodes information needed to find the Frobenius number. One cannot generally store all of D, but it is possible to encode its shape by a small set of vectors and that is sufficient to get g(A). The ideas of our algorithm connect the Frobenius problem to methods in integer linear programming and computational algebra. A variation of these ideas works when n = 3, where D has much more structure. An integer programming method of Eisenbrand and Rote can be used to design an algorithm
THE FROBENIUS PROBLEM IN A FREE MONOID
, 2008
"... The classical Frobenius problem over N is to compute the largest integer g not representable as a nonnegative integer linear combination of nonnegative integers x1, x2,..., xk, where gcd(x1, x2,..., xk) = 1. In this paper we consider novel generalizations of the Frobenius problem to the noncommu ..."
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Cited by 7 (3 self)
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The classical Frobenius problem over N is to compute the largest integer g not representable as a nonnegative integer linear combination of nonnegative integers x1, x2,..., xk, where gcd(x1, x2,..., xk) = 1. In this paper we consider novel generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for several analogues of g, with the precise bound depending on the particular measure chosen.
On comparing two chains of numerical semigroups and detecting Arf semigroups, Semigroup Forum 63
, 2001
"... If T is a numerical semigroup with maximal ideal N, define associated semigroups B(T):= (N − N) and L(T) = ∪{(hN − hN) : h ≥ 1}. If S is a numerical semigroup, define strictly increasing finite sequences {Bi(S) : 0 ≤ i ≤ β(S)} and {Li(S) : 0 ≤ i ≤ λ(S)} of semigroups by B0(S): = S =: L0(S), B β(S)( ..."
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Cited by 6 (5 self)
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If T is a numerical semigroup with maximal ideal N, define associated semigroups B(T):= (N − N) and L(T) = ∪{(hN − hN) : h ≥ 1}. If S is a numerical semigroup, define strictly increasing finite sequences {Bi(S) : 0 ≤ i ≤ β(S)} and {Li(S) : 0 ≤ i ≤ λ(S)} of semigroups by B0(S): = S =: L0(S), B β(S)(S): = N =: L λ(S)(S), Bi+1(S): = B(Bi(S)) for 0 < i < β(S), Li+1(S): = L(Li(S)) for 0 < i < λ(S). It is shown, contrary to recent claims and conjectures, that B2(S) need not be a subset of L2(S) and that β(S)−λ(S) can be any preassigned integer. On the other hand, B2(S) ⊆ L2(S) in each of the following cases: S is symmetric; S has maximal embedding dimension; S has embedding dimension e(S) ≤ 3. Moreover, if either e(S) = 2 or S is pseudosymmetric of maximal embedding dimension, then Bi(S) ⊆ Li(S) for each i, 0 ≤ i ≤ λ(S). For each integer n ≥ 2, an example is given of a (necessarily nonArf) semigroup S such that β(S) = λ(S) = n, Bi(S) = Li(S) for all 0 ≤ i ≤ n − 2, and Bn−1(S) � Ln−1(S). 1
The computational complexity of the local postage stamp problem
 ACM SIGACT News
, 2002
"... The wellstudied local postage stamp problem (LPSP) is the following: given a positive integer k, a set of positive integers 1 = a1 < a2 < · · · < ak and an integer h ≥ 1, what is the smallest positive integer which cannot be represented as a linear combination ∑ 1≤i≤k xiai where ∑ 1≤i≤k xi ≤ h an ..."
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Cited by 5 (1 self)
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The wellstudied local postage stamp problem (LPSP) is the following: given a positive integer k, a set of positive integers 1 = a1 < a2 < · · · < ak and an integer h ≥ 1, what is the smallest positive integer which cannot be represented as a linear combination ∑ 1≤i≤k xiai where ∑ 1≤i≤k xi ≤ h and each xi is a nonnegative integer? In this note we prove that LPSP is NPhard under Turing reductions, but can be solved in polynomial time if k is fixed. 1
The worst case in Shellsort and related algorithms
 Journal of Algorithms
, 1993
"... Abstract. We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N 1+c/ √ m comparisons in the worst case, for some constant c> 0. For m ≤ (log N / log log N) 2 we obtain an upper bound of the same form. We also prov ..."
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Cited by 5 (0 self)
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Abstract. We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N 1+c/ √ m comparisons in the worst case, for some constant c> 0. For m ≤ (log N / log log N) 2 we obtain an upper bound of the same form. We also prove that Ω(N(log N / log log N) 2) comparisons are needed regardless of the number of increments. Our approach is general enough to apply to other sorting algorithms, including Shakersort, for which an even stronger result is proved. 1.