216 research outputs found
Enumerating Permutation Polynomials over finite fields by degree
We prove an asymptotic formula for the number of permutation for which the
associated permutation polynomial has degree smaller than .Comment: LaTeX2e amsart 5 page
Pseudo-elliptic integrals, units, and torsion
We remark on pseudo-elliptic integrals and on exceptional function fields,
namely function fields defined over an infinite base field but nonetheless
containing non-trivial units. Our emphasis is on some elementary criteria that
must be satisfied by a squarefree polynomial whose square root generates a
quadratic function field with non-trivial unit. We detail the genus 1 case.Comment: Submitted preprin
Divisibility of reduction in groups of rational numbers
Given a multiplicative group of non zero rational numbers and a
positive integer
m
, we consider the problem of determining the density
of the set of primes
p
for which the order of the reduction modulo
p
of the group is divisible by
m
. In the case when the group is nitely
generated the density is explicitly computed. Some example of groups
with innite rank are considered
Entanglement dynamics and chaos in long-range quantum systems
Over the past twenty years, experimental and technological progresses have motivated a renewed attention to the study of non-equilibrium isolated many-body systems, leading to a relatively well-established paradigm in the case of local Hamiltonians.
In the present thesis, I have used quantum information theoretical tools to study out-of-equilibrium dynamics, with particular attention on long-range interacting many-body systems. I have explored the dynamics of bipartite and multipartite entanglement in connection to chaos and scrambling in various long-range (clean and disordered) models. The results contained in this thesis contribute to establishing semi-classical tools as powerful techniques for the description of the quantum information spreading in long-range systems.
I have further considered a different, yet connected question, concerning the multipartite entanglement structure of chaotic eigenstates and its generic evolution
Quantum bounds on the generalized Lyapunov exponents
We discuss the generalized quantum Lyapunov exponents , defined from the
growth rate of the powers of the square commutator. They may be related to an
appropriately defined thermodynamic limit of the spectrum of the commutator,
which plays the role of a large deviation function, obtained from the exponents
via a Legendre transform. We show that such exponents obey a generalized
bound to chaos due to the fluctuation-dissipation theorem, as already discussed
in the literature. The bounds for larger are actually stronger, placing a
limit on the large deviations of chaotic properties. Our findings at infinite
temperature are exemplified by a numerical study of the kicked top, a
paradigmatic model of quantum chaos.Comment: The paper is dedicated to Professor Giulio Casati on the Occasion of
his 80th Birthday, submitted to the Special Issue of Entropy: "Quantum chaos
- dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday
On ther-rank Artin Conjecture, II
AbstractFor any finitely generated subgroupΓofQ* we compute a formula for the density of the primes for which the reduction modulopofΓcontains a primitive root modulop. We use this to conjecture a characterization of “optimal” subgroups (i.e., subgroups that have maximal density). We also improve the error term in the asymptotic formula of Pappalardi's Theorem 1.1 (Math. Comp.66(1997), 853–868)
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