54,399 research outputs found
On isogeny classes of Edwards curves over finite fields
We count the number of isogeny classes of Edwards curves over finite fields,
answering a question recently posed by Rezaeian and Shparlinski. We also show
that each isogeny class contains a {\em complete} Edwards curve, and that an
Edwards curve is isogenous to an {\em original} Edwards curve over \F_q if
and only if its group order is divisible by 8 if , and 16
if . Furthermore, we give formulae for the proportion of
d \in \F_q \setminus \{0,1\} for which the Edwards curve is complete or
original, relative to the total number of in each isogeny class.Comment: 27 page
A study of the entanglement in systems with periodic boundary conditions
We define the local periodic linking number, LK, between two oriented closed
or open chains in a system with three-dimensional periodic boundary conditions.
The properties of LK indicate that it is an appropriate measure of entanglement
between a collection of chains in a periodic system. Using this measure of
linking to assess the extent of entanglement in a polymer melt we study the
effect of CReTA algorithm on the entanglement of polyethylene chains. Our
numerical results show that the statistics of the local periodic linking number
observed for polymer melts before and after the application of CReTA are the
same.Comment: 11 pages, 11 figure
Real-time non-equilibrium dynamics of quantum glassy systems
We develop a systematic analytic approach to aging effects in quantum
disordered systems in contact with an environment. Within the closed-time
path-integral formalism we include dissipation by coupling the system to a set
of independent harmonic oscillators that mimic a quantum thermal bath. After
integrating over the bath variables and averaging over disorder we obtain an
effective action that determines the real-time dynamics of the system. The
classical limit yields the Martin-Siggia-Rose generating functional associated
to a colored noise. We apply this general formalism to a prototype model
related to the spin-glass. We show that the model has a dynamic phase
transition separating the paramagnetic from the spin-glass phase and that
quantum fluctuations depress the transition temperature until a quantum
critical point is reached. We show that the dynamics in the paramagnetic phase
is stationary but presents an interesting crossover from a region controlled by
the classical critical point to another one controlled by the quantum critical
point. The most characteristic property of the dynamics in a glassy phase,
namely aging, survives the quantum fluctuations. In the sub-critical region the
quantum fluctuation-dissipation theorem is modified in a way that is consistent
with the notion of effective temperatures introduced for the classical case. We
discuss these results in connection with recent experiments in dipolar quantum
spin-glasses and the relevance of the effective temperatures with respect to
the understanding of the low temperature dynamics.Comment: 56 pages, Revtex, 17 figures include
Addition law structure of elliptic curves
The study of alternative models for elliptic curves has found recent interest
from cryptographic applications, once it was recognized that such models
provide more efficiently computable algorithms for the group law than the
standard Weierstrass model. Examples of such models arise via symmetries
induced by a rational torsion structure. We analyze the module structure of the
space of sections of the addition morphisms, determine explicit dimension
formulas for the spaces of sections and their eigenspaces under the action of
torsion groups, and apply this to specific models of elliptic curves with
parametrized torsion subgroups
Fluctuations of two-time quantities and time-reparametrization invariance in spin-glasses
This article is a contribution to the understanding of fluctuations in the
out of equilibrium dynamics of glassy systems. By extending theoretical ideas
based on the assumption that time-reparametrization invariance develops
asymptotically we deduce the scaling properties of diverse high-order
correlation functions. We examine these predictions with numerical tests in a
standard glassy model, the 3d Edwards-Anderson spin-glass, and in a system
where time-reparametrization invariance is not expected to hold, the 2d
ferromagnetic Ising model, both at low temperatures. Our results enlighten a
qualitative difference between the fluctuation properties of the two models and
show that scaling properties conform to the time-reparametrization invariance
scenario in the former but not in the latter.Comment: 17 pages, 5 figure
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