On the Frobenius number of Fibonacci numerical semigroups.
Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are J-Bessel and K-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula are given. The inverse Laplace transform of Weber's first exponential integral is the basic tool to derive the integral representations. As an application, we give a new proof of the Fourier series expansion...
Let be an expanding matrix, a set with elements and define via the set equation . If the two-dimensional Lebesgue measure of is positive we call a self-affine plane tile. In the present paper we are concerned with topological properties of . We show that the fundamental group of is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of . Furthermore, we give a short proof of the fact that the closure of each component of is a locally...
Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field ℚ(ϵ) is Galois. Let ϵ, ϵ' and ϵ'' be the three real conjugates of ϵ. We tackle the problem of whether {ϵ,ϵ'} is a system of fundamental units of the cubic order ℤ[ϵ,ϵ',ϵ'']. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic polynomials...
Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed , Filaseta and Lam have shown that the th degree Generalized Laguerre Polynomial is irreducible for all large enough . We use our criterion to show that, under these conditions, the Galois group of is either the alternating or symmetric group on letters, generalizing results of Schur for .
Let be a finite abelian extension of , with the ring of algebraic integers of . We investigate the Galois structure of the unique fractional -ideal which (if it exists) is unimodular with respect to the trace form of .
In this note, we estimate the distance between two -nomial coefficients , where and is an integer.