A construction of integer-valued polynomials with prescribed sets of lengths of factorizations

For an arbitrary finite set S of natural numbers greater 1, we construct an integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of lengths of f is the set of all natural numbers n, such that f has a factorization as a product of n irreducibles in Int(Z)={g in Q[x] | g(Z) contained in Z}.Comment: To appear in Monatshefte f\"ur Mathematik; 11 page

Impulsive waves in electrovac direct product spacetimes with Lambda

A complete family of non-expanding impulsive waves in spacetimes which are the direct product of two 2-spaces of constant curvature is presented. In addition to previously investigated impulses in Minkowski, (anti-)Nariai and Bertotti-Robinson universes, a new explicit class of impulsive waves which propagate in the exceptional electrovac Plebanski-Hacyan spacetimes with a cosmological constant Lambda is constructed. In particular, pure gravitational waves generated by null particles with an arbitrary multipole structure are described. The metrics are impulsive members of a more general family of the Kundt spacetimes of type II. The well-known pp-waves are recovered for Lambda=0.Comment: 6 pages, 1 figure, LaTeX 2e. To appear in Class. Quantum Gra

Integral closure of rings of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $\mu_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $\mu_a(X)$, $a\in A$, into elements that are integral over $D$. The result is obtained by identifying $A$ with a $D$-subalgebra of the matrix algebra $M_n(K)$ for some $n$ and then considering polynomials which map a matrix to a matrix integral over $D$. We also obtain information about polynomially dense subsets of these rings of polynomials.Comment: Keywords: Integer-valued polynomial, matrix, triangular matrix, integral closure, pullback, polynomially dense set. accepted for publication in the volume "Commutative rings, integer-valued polynomials and polynomial functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 201

The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields

The spectrum and degeneracies of the Dirac operator are analysed on compact coset spaces when there is a non-zero homogeneous background gauge field which is compatible with the symmetries of the space, in particular when the gauge field is derived from the spin-connection. It is shown how the degeneracy of the lowest Landau level in the recently proposed higher dimensional quantum Hall effect is related to the Atiyah-Singer index theorem for the Dirac operator on a compact coset space.Comment: 25 pages, typeset in LaTeX, uses youngtab.st

Deformation Quantization of Geometric Quantum Mechanics

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \over \hbar}$.Comment: 27+1 pages, harvmac file, no figure