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18
The Frobenius Problem, Rational Polytopes, and FourierDedekind Sums
, 2003
"... We study the number of lattice points in integer dilates of the rational polytope P = (x1,..., xn) ∈ R n n∑ ≥0: xkak ≤ 1, where a1,..., an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,..., an, fin ..."
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Cited by 28 (13 self)
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We study the number of lattice points in integer dilates of the rational polytope P = (x1,..., xn) ∈ R n n∑ ≥0: xkak ≤ 1, where a1,..., an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,..., an, find the largest value of t (the Frobenius number) such that m1a1 + · · · + mnan = t has no solution in positive integers m1,..., mn. This is equivalent to the problem of finding the largest dilate tP such that the facet n k=1 xkak = t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(P, t): = #(tP ∩ Z n) and L(P ◦ , t):= #(tP ◦ ∩ Z n). Within the computations a Dedekindlike finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of k=1
Analysis of Shellsort and related algorithms
 ESA ’96: Fourth Annual European Symposium on Algorithms
, 1996
"... This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellso ..."
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Cited by 26 (0 self)
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This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellsort and Shellsortbased networks; averagecase results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;
Frobenius Problem for Semigroups S (d1, d2, d3)
, 2008
"... The matrix representation of the set ∆(d 3), d 3 = (d1, d2, d3), of the integers which are unrepresentable by d1, d2, d3 is found. The diagrammatic procedure of calculation of the generating function Φ ( d 3; z) for the set ∆(d 3) is developed. The Frobenius number F ( d 3) , genus G ( d 3) and Hilb ..."
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Cited by 2 (2 self)
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The matrix representation of the set ∆(d 3), d 3 = (d1, d2, d3), of the integers which are unrepresentable by d1, d2, d3 is found. The diagrammatic procedure of calculation of the generating function Φ ( d 3; z) for the set ∆(d 3) is developed. The Frobenius number F ( d 3) , genus G ( d 3) and Hilbert series H(d 3; z) of a graded subring for non–symmetric and symmetric semigroups S ( d 3) are found. The upper bound for the number of non–zero coefficients in the polynomial numerators of Hilbert series H(d m; z) of graded subrings for non–symmetric semigroups S (d m) of dimension, m ≥ 4, is established.
On the Number of Representations of an Integer by a Linear Form
"... Let a1,..., ak be positive integers generating the unit ideal, and j be a residue class modulo L = lcm(a1,..., ak). It is known that the function r(N) that counts solutions to the equation x1a1 +... + xkak = N in nonnegative integers xi is a polynomial when restricted to nonnegative integers N ≡ j ..."
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Cited by 2 (2 self)
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Let a1,..., ak be positive integers generating the unit ideal, and j be a residue class modulo L = lcm(a1,..., ak). It is known that the function r(N) that counts solutions to the equation x1a1 +... + xkak = N in nonnegative integers xi is a polynomial when restricted to nonnegative integers N ≡ j (mod L). Here we give, in the case of k = 3, exact formulas for these polynomials up to the constant terms, and exact formulas including the constants for q = gcd(a1, a2) · gcd(a1, a3) · gcd(a2, a3) of the L residue classes. The case q = L plays a special role, and it is studied in more detail. 1
Some experimental results on the Frobenius problem
 the electronic journal of combinatorics 12 (2005), #R27 36 2.5 1.5 n � 8 1.35
"... Abstract. We study the Frobenius problem: given relatively prime positive integers a1,..., ad, find the largest value of t (the Frobenius number) such that ∑ d k=1 mkak = t has no solution in nonnegative integers m1,..., md. Based on empirical data, we conjecture that except for some special cases t ..."
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Abstract. We study the Frobenius problem: given relatively prime positive integers a1,..., ad, find the largest value of t (the Frobenius number) such that ∑ d k=1 mkak = t has no solution in nonnegative integers m1,..., md. Based on empirical data, we conjecture that except for some special cases the Frobenius number can be bounded from above by √ a1a2a3 5/4 − a1 − a2 − a3. 1
Analytic Representations in the 3dim Frobenius Problem
, 2008
"... We consider the Diophantine problem of Frobenius for semigroup S ( d3) where d3 denotes the tuple (d1, d2, d3), gcd(d1, d2, d3) = 1. Based on the Hadamard product of analytic ( functions [17] 3 we have found the analytic representation for the diagonal elements akk d) of the Johnson’s matrix of min ..."
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Cited by 1 (1 self)
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We consider the Diophantine problem of Frobenius for semigroup S ( d3) where d3 denotes the tuple (d1, d2, d3), gcd(d1, d2, d3) = 1. Based on the Hadamard product of analytic ( functions [17] 3 we have found the analytic representation for the diagonal elements akk d) of the Johnson’s matrix of minimal relations [12] in terms of d1, d2, d3. Bearing in mind the results of the recent
Gaps in Nonsymmetric Numerical Semigroups
, 2008
"... There exist two different sorts of gaps in the nonsymmetric numerical additive semigroups finitely generated by a minimal set of positive integers {d1,..., dm}. The h–gaps are specific only for the nonsymmetric semigroups while the g–gaps are possessed by both, symmetric and nonsymmetric semigroups. ..."
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There exist two different sorts of gaps in the nonsymmetric numerical additive semigroups finitely generated by a minimal set of positive integers {d1,..., dm}. The h–gaps are specific only for the nonsymmetric semigroups while the g–gaps are possessed by both, symmetric and nonsymmetric semigroups. We derive the generating functions for the corresponding sets of gaps, ∆H (d m) and ∆G (d m), and prove several statements on the minimal and maximal values of the h–gaps. Detailed description of both sorts of gaps is given for three special kinds of nonsymmetric semigroups: semigroups with maximal embedding dimension, semigroups of maximal and almost maximal length, and pseudo–symmetric semigroups.
Power Sums Related to Semigroups S (d1, d2, d3)
, 2005
"... The explicit formulas for the sums of positive powers of the integers si unrepresentable by the triple of integers d1, d2, d3 ∈ N, gcd(d1, d2, d3) = 1, are derived. Let S (d1,..., dm) be the semigroup generated by a set of integers {d1,..., dm} such that 1 < d1 <... < dm, gcd(d1,..., dm) = 1. (1) ..."
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The explicit formulas for the sums of positive powers of the integers si unrepresentable by the triple of integers d1, d2, d3 ∈ N, gcd(d1, d2, d3) = 1, are derived. Let S (d1,..., dm) be the semigroup generated by a set of integers {d1,..., dm} such that 1 < d1 <... < dm, gcd(d1,..., dm) = 1. (1) For short we denote the tuple (d1,..., dm) by d m and consider the generating function Φ (d m; z) Φ (d m; z) = � si ∈ ∆(d m) z si = z + z s2 +... + z s G(d m) , (2) for the set ∆ (d m) of the integers s which are unrepresentable as s = � m i=1 xidi, xi ∈ N ∪ {0} ∆ (d m) = {s1, s2,..., sG(d m)} , s1 = 1. (3) The integer G (dm) is known as the genus for semigroup S (dm). Recall the relation of Φ (dm; z) with the Hilbert series H(dm; z) of a graded monomial subring k � zd1 � dm,..., z [1] H(d m; z) + Φ(d m; z) = 1 1 − z, where H(dm; z) = � 1 s ∈ S(d m) z s = Q(dm; z), (4) (1 − zdj) � m j=1 and Q(d m; z) is a polynomial in z. The calculation of the power sums was performed in [2] for m = 2
and
, 2008
"... The explicit formulas for the sums of positive powers of the integers si unrepresentable by the triple of integers d1,d2,d3 ∈ N, gcd(d1,d2,d3) = 1, are derived. Key words: Non–symmetric and symmetric semigroups, Power sums. 2000 Math. Subject Classification: Primary 11P81; Secondary 20M99 1 Let S ..."
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The explicit formulas for the sums of positive powers of the integers si unrepresentable by the triple of integers d1,d2,d3 ∈ N, gcd(d1,d2,d3) = 1, are derived. Key words: Non–symmetric and symmetric semigroups, Power sums. 2000 Math. Subject Classification: Primary 11P81; Secondary 20M99 1 Let S (d1,...,dm) be the semigroup generated by a set of integers {d1,..., dm} such that 1 < d1 <... < dm, gcd(d1,...,dm) = 1. (1) For short we denote the tuple (d1,..., dm) by d m and consider the generating function Φ (d m; z) Φ (d m; z) = si ∈ ∆(d m) z si = z + z s2 +... + z s G(d m), (2) for the set ∆ (d m) of the integers s which are unrepresentable as s = ∑ m i=1 xidi, xi ∈ N ∪ {0} ∆ (d m) = {s1, s2,..., sG(d m)} , s1 = 1. (3) The integer G (dm) is known as the genus for semigroup S (dm). Recall the relation of Φ (dm; z) with the Hilbert series H(dm; z) of a graded monomial subring k [ zd1] dm,...,z [1] H(d m; z) + Φ(d m; z) = 1 1 − z, where H(dm; z) = ∑ s ∈ S(d m) and Q(d m; z) is a polynomial in z. The calculation of the power sums was performed in [2] for m = 2
Perfect Pairs of Ideals and Duals in Numerical
, 2005
"... set 1 Abstract. This paper considers numerical semigroups S that have a nonprincipal relative ideal I such that µS(I)µS(S − I) = µS(I + (S − I)). We show the existence of an infinite family of such pairs (S,I) in which I + (S − I) = S\{0}. We also show examples of such pairs that are not members ..."
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set 1 Abstract. This paper considers numerical semigroups S that have a nonprincipal relative ideal I such that µS(I)µS(S − I) = µS(I + (S − I)). We show the existence of an infinite family of such pairs (S,I) in which I + (S − I) = S\{0}. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them. 1. Definitions, Notation and Background (1.1) Definitions/Notations: (a) A numerical semigroup S is a subset of the nonnegative integers N which contains 0, is closed under addition, and such that N\S is finite. If G is the smallest subset of S such that every element of S is a sum of elements from G, then we say G is the minimal generating set of S and we write