We obtain formulae and estimates for character sums of the type S(χ, f, pm) = ∑pm x=1 χ(f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f = f1/f2 is a rational function over Z. In particular, if p is odd, d = deg(f1) + deg(f2) and d∗ = max(deg(f1), deg(f2)) then we obtain |S(χ, f, pm)| ≤ (d− 1)pm(1− 1 d∗ ) for any non constant f (mod p) and primitive ...

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see [8]. We prove that the universal character solves the Darboux chain. The N -periodic closing of the chain is equivalent to the Painlevé equation of type A N−1. Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters. Introduc...

Education and Development and at the Erasmus Institute during 2003-2004, my students, and the volume editors. Thanks to the Erasmus Institute for supporting a year of writing. Integrative Ethical Education Much of the debate over moral education in recent decades has centered around the advantages and disadvantages of two dominant educational approaches to the moral formation of children, refer...

We study recurrence relations and biorthogonality properties for polynomials and rational functions in the problem of the Padé interpolation in the usual scheme and in the scheme with prescribed poles and zeros. The main result is deriving explicit orthogonality and biorthogonality relations for polynomials and rational functions in both schemes. We show that the simplest linear restrictions in...

with y−s, y−s+1, . . . , y−1 ∈ (0,∞) and k, m ∈ {1, 2, 3, 4, . . .}, where s = k+m. We prove that if gcd(2k,m) = 1, then solutions to this equation are asymptotically 2k-periodic. This generalizes the corresponding result for m = 3 and k = 1, proved in K. S. Berenhaut et al., Periodic Solutions of the Rational Difference Equation yn = yn−3+yn−4 yn−1 ,Journal of Difference Equations and Applicat...

AUTOMATE is a package for symbolic computation on nite automata, extended rational expressions and nite semigroups. On the one hand, it enables one to compute the deterministic minimal automaton of the language represented by a rational expression or given by its table. On the other hand, given the transition table of a deterministic automaton, AUTOMATE computes the associated transition monoid...