Determining the time separation of events is a fundamental problem in the analysis, synthesis, and optimization of concurrent systems. Applications range from logic optimization of asynchronous digital circuits to evaluation of execution times of programs for real-time systems. We present an efficient algorithm to find exact (tight) bounds on the separation time of events in an arbitrary process graph without conditional behavior. This result is more general than the methods presented in several previously published papers as it handles cyclic graphs and yields the tightest possible bounds on event separations. The algorithm is based on a functional decomposition technique that permits the implicit evaluation of an infinitely unfolded process graph. Examples are presented that demonstrate the utility and efficiency of the solution. The algorithm will form a basis for exploration of timing-constrained synthesis techniques. Index terms: Abstract algebra, asynchronous systems, concurrent ...

Abstract. We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are F q 2-isomorphic to y q + y = x m for some m ∈ Z +. Goppa in [Go] showed how to construct linear codes from curves defined over finite fields. One of the main features of these codes is the fact that one can state a lower bound for the minimum distance of the codes. In fact, let CX(D, G) be a Goppa code defined over a curve X over the finite field Fq with q elements, where D = P1 +... + Pn,

We study the number of lattice points in integer dilates of the open rational polytope P = ( (x1 ; : : : ; xn ) 2 R n ?0 : n X k=1 x k a k ! 1 ) ; where a1 ; : : : ; an are positive integers. This polytope is closely related to the Frobenius problem: given relatively prime positive integers a1 ; : : : ; an , find the largest value of t (the Frobenius number) such that P n k=1 m k a k = 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 \Phi P n k=1 x k a k = t \Psi 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 computation of the 'constant' coefficient, a Dedekind-like finite Fourier sum appears. We find bounds for these generalized Dedekind sums and use them to give new bounds for the Frobenius number.

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 number-theoretic properties of the algorithm; lower bounds on Shellsort and Shellsort-based networks; average-case 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;

Abstract. We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. This paper continues the study, begun in [FT] and [FGT], of curves over finite fields with many rational points, based on Stöhr-Voloch’s approach [SV] to the Hasse-Weil bound by way of Weierstrass Point Theory and Frobenius orders. Some of the results were announced in [T]. A projective geometrically irreducible non-singular algebraic curve X |Fq of genus g is said to be optimal if #X(Fq) = max{#Y (Fq) : Y |Fq curve of genus g}. Optimal curves occupy a distinguished niche, for example, in coding theory after Goppa’s [Go]. We recall that #X(Fq) is bounded from above by the Hasse-Weil bound, namely q + 2g √ q + 1. The main goal of this paper is to sharpen and generalize some results in [FGT]. In that paper Garcia and us improved and generalized previous results obtained by Rück-

Abstract. Let �v0,..., �vk be vectors in Z k which generate Z k. We show that a body V ⊂ Z k with the vectors �v0,..., �vk as edge vectors is an almost minimal set with the property that every function f: V → R with periods �v0,..., �vk is constant. For k = 1 the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma. Suppose �0 is not a non-trivial linear combination of �v0,..., �vk with nonnegative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of �v0,..., �vk over Z≥0, but infinitely many points on each of its hyperfaces are not. For k =1theresult reduces to a formula of Sylvester corresponding to Frobenius ’ Coin-changing Problem in the case of coins of two denominations. 1.

Given a set of positive integers A = {a 1 ,...,a n }, we study the number p A (t)of nonnegative integer solutions (m 1 ,...,m n )to P n j=1 m j a j = t. We derive an explicit formula for the polynomial part of p A .

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 branch-and-bound 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.

Abstract. Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as ∑N i=1 aixi where x1,...,xN are non-negative integers. The condition that gcd(a1,...,aN) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1,..., aN is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often. 1.

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 Shaker-sort, for which an even stronger result is proved. 1.