Alessandro Conflitti

Exponential sums over Mersenne numbers

We give estimates for exponential sums of the form P n6N �( n)exp(2 �iag n /m ), where m is a positive integer, a and g are integers relatively prime to m, andis the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form P p6N exp(2 �iaM p/m ), where Mp is the Mersenne number; Mp = 2 p !1 for any prime p. We also estimate some closely related sums, including P n6N µ(n)exp(2 �iag n /m ) and P n6N µ 2 (n)exp(2 �iag n /m ), where µ is the Mobius function.

ON THE LARGEST SIZE OF AN ANTICHAIN IN THE BRUHAT ORDER FOR A (2k,k)

We discuss a problem proposed by Brualdi and Deaett on the largest size of an antichain in the Bruhat order for the interesting combinatorial class of binary matrices of .

On the Largest Size of an Antichain in the Bruhat Order for Open image in new window

We discuss a problem proposed by Brualdi and Deaett on the largest size of an antichain in the Bruhat order for the interesting combinatorial class of binary matrices of .

NONCOMMUTATIVE HYPERGEOMETRIC AND BASIC HYPERGEOMETRIC EQUATIONS

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tiraos result, extended to the more general setting of hypergeometric functions over an abstract unital Banach algebra. We provide a similar (but more complicated-looking) result for a second type of noncommutative 2F1 Gauß hypergeometric function. We further give q-analogues for both types of noncommutative hypergeometric equations.

On Whitney numbers of the order ideals of generalized fences and crowns

We solve some recurrences given by E. Munarini and N. Zagaglia Salvi proving explicit formulas for Whitney numbers of the distributive lattices of order ideals of the fence poset and crown poset. Moreover, we get a method to obtain explicit formulas for Whitney numbers of lattices of order ideals of fences with higher asymmetric peaks.

On the multidimensional distribution of the subset sum generator of pseudorandom numbers

We show that for a random choice of the parameters, the subset sum pseudorandom number generator produces a sequence of uniformly and independently distributed pseudorandom numbers. The result can be useful for both cryptographic and quasi-Monte Carlo applications and relies on bounds of exponential sums.

On Elements of High Order in Finite Fields

We provide a more careful analysis of a construction of elements high order in finite fields which has recently been proposed by S. Gao. In particular, we improve and generalize one of his results.

Gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions

We present combinatorial Gray codes and explicit designs of efficient algorithms for lexicographical combinatorial generation of the sets of nonnesting and sparse nonnesting set partitions of length n.

On computation of the greatest common divisor of several polynomials over a finite field

We propose a probabilistic algorithm to reduce computing the greatest common divisor of m polynomials over a finite field (which requires computing m-1 pairwise greatest common divisors) to computing the greatest common divisor of two polynomials over the same field.

Gray codes for noncrossing and nonnesting partitions of classical types

In this paper, we present Gray codes for the sets of noncrossing partitions associated with the classical Weyl groups, and for the set of nonnesting partitions of type B. An algorithm for the generation of type D nonnesting partitions is developed in which a Gray code is given for those partitions having a zero-block, while the remaining are arranged in lexicographic order.