3,319 research outputs found
Non Thermal Equilibrium States of Closed Bipartite Systems
We investigate a two-level system in resonant contact with a larger
environment. The environment typically is in a canonical state with a given
temperature initially. Depending on the precise spectral structure of the
environment and the type of coupling between both systems, the smaller part may
relax to a canonical state with the same temperature as the environment (i.e.
thermal relaxation) or to some other quasi equilibrium state (non thermal
relaxation). The type of the (quasi) equilibrium state can be related to the
distribution of certain properties of the energy eigenvectors of the total
system. We examine these distributions for several abstract and concrete (spin
environment) Hamiltonian systems, the significant aspect of these distributions
can be related to the relative strength of local and interaction parts of the
Hamiltonian.Comment: RevTeX, 8 pages, 13 figure
Numerical Relativity Injection Infrastructure
This document describes the new Numerical Relativity (NR) injection
infrastructure in the LIGO Algorithms Library (LAL), which henceforth allows
for the usage of NR waveforms as a discrete waveform approximant in LAL. With
this new interface, NR waveforms provided in the described format can directly
be used as simulated GW signals ("injections") for data analyses, which include
parameter estimation, searches, hardware injections etc. As opposed to the
previous infrastructure, this new interface natively handles sub-dominant modes
and waveforms from numerical simulations of precessing binary black holes,
making them directly accessible to LIGO analyses. To correctly handle
precessing simulations, the new NR injection infrastructure internally
transforms the NR data into the coordinate frame convention used in LAL.Comment: 20 pages, 2 figures, technical repor
Polynomial dynamics and local analysis
We prove an analogue of the Manin-Mumford conjecture for polynomial dynamical
systems over number fields. In our setting the role of torsion points is taken
by the small orbit of a point . The small orbit of a point was
introduced by McMullen and Sullivan in their study of the dynamics of rational
maps where for a point and a polynomial it is given by
\begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta)
= f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}.
\end{align*} Our main theorem is a classification of the algebraic relations
that hold between infinitely pairs of points in when
everything is defined over the algebraic numbers and the degree of is
at least 2. Our proof relies on a careful study of localizations of the
dynamical system and follows an entirely different approach than previous
proofs in this area. At infinite places of we use known rigidity theorems
of Fatou and Levin to prove new such. These might be of independent interest in
complex dynamics. At finite places we introduce new non-archimedean methods to
study diophantine problems that might be applicable in other arithmetic
contexts. Our method at finite places allows us to classify all algebraic
relations that hold for infinitely pairs of points in the grand orbit
\begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta)
= f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\}
\end{align*} of if at a
finite place of good reduction co-prime to . This is an analogue of the
Mordell-Lang conjecture on finite rank groups for polynomial dynamics.Comment: All comments welcome
Unlikely intersections in semi-abelian surfaces
We consider a family, depending on a parameter, of multiplicative extensions
of an elliptic curve with complex multiplications. They form a 3-dimensional
variety which admits a dense set of special curves, known as Ribet curves,
which strictly contains the torsion curves. We show that an irreducible curve
in meets this set Zariski-densely only if lies in a fiber of the
family or is a translate of a Ribet curve by a multiplicative section. We
further deduce from this result a proof of the Zilber-Pink conjecture (over
number fields) for the mixed Shimura variety attached to the threefold ,
when the parameter space is the universal one.Comment: 20 pages. Appendix added, with a proof of the Zilber-Pink for the
Poincar\'e-biextension over a CM elliptic curv
A Manin-Mumford theorem for the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves
We study the intersection of an algebraic variety with the maximal compact
subgroup of a universal vectorial extension of a product of elliptic curves.
For this intersection we show a Manin-Mumford type statement. This answers some
questions posed by Corvaja-Masser-Zannier which arose in connection with their
investigation of the intersection of a curve with real analytic subgroups of
various algebraic groups. They prove finiteness in the situation of a single
elliptic curve. Using Khovanskii's zero-estimates combined with a
stratification result of Gabrielov-Vorobjov and recent work of the authors we
obtain effective bounds for this intersection that only depend on the degree of
the algebraic variety, and the dimension of the group. This seems new even if
restricted to the classical Manin-Mumford statement.Comment: 20 pages. Comments welcome! Fixed various mistakes and
inconsistencies in v
On the dynamical Bogomolov conjecture for families of split rational maps
We prove that Zhang's dynamical Bogomolov conjecture holds uniformly along
-parameter families of rational split maps and curves. This provides
dynamical analogues of recent results of Dimitrov-Gao-Habegger and K\"uhne. In
fact, we prove a stronger Bogomolov-type result valid for families of split
maps in the spirit of the relative Bogomolov conjecture. We thus provide first
instances of a generalization of a conjecture by Baker and DeMarco to higher
dimensions. Our proof contains both arithmetic and analytic ingredients. We
establish a characterization of curves that are preperiodic under the action of
a non-exceptional split rational endomorphism of
with respect to the measures of maximal entropy
of and , extending a previous result of Levin-Przytycki. We further
establish a height inequality for families of split maps and varieties
comparing the values of a fiber-wise Call-Silverman canonical height with a
height on the base and valid for most points of a non-preperiodic variety. This
provides a dynamical generalization of a result by Habegger and generalizes
results of Call-Silverman and Baker to higher dimensions. In particular, we
establish a geometric Bogomolov theorem for split rational maps and varieties
of arbitrary dimension.Comment: The proof of Theorems 4.1 and 4.3 relies on arXiv:2208.0159
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