Big Biases Amongst Products of Two Primes

We show that substantially more than a quarter of the odd integers of the form pqpq up to xx, with p,qp,q both prime, satisfy p≡q≡3 (mod4)p≡q≡3 (mod4)

Computation of Iwasawa Lambda invariants for imaginary quadratic fields

A method for computing the Iwasawa lambda invariants of an imaginary quadratic field is developed and used to construct a table of these invariants for discriminants up to 1,000 and primes up to 20,000

Quantum statistics on graphs

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics.Comment: 21 pages, 6 figure

Decompactifications and Massless D-Branes in Hybrid Models

A method of determining the mass spectrum of BPS D-branes in any phase limit of a gauged linear sigma model is introduced. A ring associated to monodromy is defined and one considers K-theory to be a module over this ring. A simple but interesting class of hybrid models with Landau-Ginzburg fibres over CPn are analyzed using special Kaehler geometry and D-brane probes. In some cases the hybrid limit is an infinite distance in moduli space and corresponds to a decompactification. In other cases the hybrid limit is at a finite distance and acquires massless D-branes. An example studied appears to correspond to a novel theory of supergravity with an SU(2) gauge symmetry where the gauge and gravitational couplings are necessarily tied to each other.Comment: PDF-LaTeX, 34 pages, 2 mps figure

Nonadiabatic dynamics in semiquantal physics

Every physical regime is some sort of approximation of reality. One lesser-known realm that is the semiquantal regime, which may be used to describe systems with both classical and quantum subcomponents. In the present review, we discuss nonadiabatic dynamics in the semiquantal regime. Our primary concern is electronic-nuclear coupling in polyatomic molecules, but we discuss several other situations as well. We begin our presentation by formulating the semiquantal approximation in quantum systems with degrees-of-freedom that evolve at different speeds. We discuss nonadiabatic phenomena, focusing on their relation to the Born-Oppenheimer approximation. We present several examples--including Jahn-Teller distortion in molecules and crystals and the dynamics of solvated electrons, buckyballs, nanotubes, atoms in a resonant cavity, SQUIDs, quantum particle-spin systems, and micromasers. We also highlight vibrating quantum billiards as a useful abstraction of semiquantal dynamics.Comment: 58 pages, 0 figures. To appear in Reports on Progress in Physic