107 research outputs found
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
Exactness of the Original Grover Search Algorithm
It is well-known that when searching one out of four, the original Grover's
search algorithm is exact; that is, it succeeds with certainty. It is natural
to ask the inverse question: If we are not searching one out of four, is
Grover's algorithm definitely not exact? In this article we give a complete
answer to this question through some rationality results of trigonometric
functions.Comment: 8 pages, 2 figure
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
BKM Lie superalgebras from counting twisted CHL dyons
Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS
states that contribute to twisted helicity trace indices in four-dimensional
CHL models with N=4 supersymmetry. The generating functions of half-BPS states,
twisted as well as untwisted, are given in terms of multiplicative eta products
with the Mathieu group, M_{24}, playing an important role. These multiplicative
eta products enable us to construct Siegel modular forms that count twisted
quarter-BPS states. The square-roots of these Siegel modular forms turn out be
precisely a special class of Siegel modular forms, the dd-modular forms, that
have been classified by Clery and Gritsenko[arXiv:0812.3962]. We show that each
one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator
formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the
Weyl chamber are in one-to-one correspondence with the walls of marginal
stability in the corresponding CHL model for twisted dyons as well as untwisted
ones. This leads to a periodic table of BKM Lie superalgebras with properties
that are consistent with physical expectations.Comment: LaTeX, 32 pages; (v2) matches published versio
Three-variable Mahler measures and special values of modular and Dirichlet -series
In this paper we prove that the Mahler measures of the Laurent polynomials
, ,
and , for various values of , are of the form , where , is a CM newform of
weight 3, and is a quadratic character. Since it has been proved that
these Maher measures can also be expressed in terms of logarithms and
-hypergeometric series, we obtain several new hypergeometric evaluations
and transformations from these results
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
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