Two-point correlation function for Dirichlet L-functions

The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy-Littlewood conjecture for pairs of primes in arithmetic progression. The result matches the conjectured Random-Matrix form in the limit as $E\rightarrow\infty$ and, importantly, includes finite-E corrections. These finite-E corrections differ from those in the case of the Riemann zeta-function, obtained in (1996 Phys. Rev. Lett. 77 1472), by certain finite products of primes which divide the modulus of the primitive character used to construct the L-function in question.Comment: 10 page

Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions \lambda of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3 recurrence relation where l is the length of the partition \lambda. Explicit expressions for such recurrence relations are given.Comment: 25 pages, typed in AMSTe

Constructing a partially transparent computational boundary for UPPE using leaky modes

In this paper we introduce a method for creating a transparent computational boundary for the simulation of unidirectional propagation of optical beams and pulses using leaky modes. The key element of the method is the introduction of an artificial-index material outside a chosen computational domain and utilization of the quasi-normal modes associated with such artificial structure. The method is tested on the free space propagation of TE electromagnetic waves. By choosing the material to have appropriate optical properties one can greatly reduce the reflection at the computational boundary. In contrast to the well-known approach based on a perfectly matched layer, our method is especially well suited for spectral propagators.Comment: 32 pages, 19 figure

Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201