pinecone/core-2020-05-10-deduplication huggingface dataset:
Asymptotic expansion of solutions to the drift-diffusion equation with
  fractional dissipation

The initial-value problem for the drift-diffusion equation arising from the
model of semiconductor device simulations is studied. The dissipation on this
equation is given by the fractional Laplacian. When the exponent of the
fractional Laplacian is large, large-time behavior of solutions is known.
However, when the exponent is small, the perturbation methods used in the
preceding works would not work. Large-time behavior of solutions to the
drift-diffusion equation with small exponent is discussed. Particularly, the
asymptotic expansion of solutions with high-order is derived
The divisible sandpile at critical density

The divisible sandpile starts with i.i.d. random variables ("masses") at the
vertices of an infinite, vertex-transitive graph, and redistributes mass by a
local toppling rule in an attempt to make all masses at most 1. The process
stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1,
where $m$ is the mean mass per vertex. The main result of this paper is that in
the critical case m=1, if the initial masses have finite variance, then the
process almost surely does not stabilize. To give quantitative estimates on a
finite graph, we relate the number of topplings to a discrete biLaplacian
Gaussian field.Comment: 34 pages, to appear in Annales Henri Poincar
On the center of mass of asymptotically hyperbolic initial data sets

We define the (total) center of mass for suitably asymptotically hyperbolic
time-slices of asymptotically anti-de Sitter spacetimes in general relativity.
We do so in analogy to the picture that has been consolidated for the (total)
center of mass of suitably asymptotically Euclidean time-slices of
asymptotically Minkowskian spacetimes (isolated systems). In particular, we
unite -- an altered version of -- the approach based on Hamiltonian charges
with an approach based on CMC-foliations near infinity. The newly defined
center of mass transforms appropriately under changes of the asymptotic
coordinates and evolves in the direction of an appropriately defined linear
momentum under the Einstein evolution equations
Nonequilibrium statistical mechanics of weakly stochastically perturbed
  system of oscillators

We consider a finite region of a $d$-dimensional lattice, $d\in\mathbb{N}$,
of weakly coupled harmonic oscillators. The coupling is provided by a
nearest-neighbour potential (harmonic or not) of size $\varepsilon$. Each
oscillator weakly interacts by force of order $\varepsilon$ with its own
stochastic Langevin thermostat of arbitrary positive temperature. We
investigate limiting as $\varepsilon\rightarrow 0$ behaviour of solutions of
the system and of the local energy of oscillators on long-time intervals of
order $\varepsilon^{-1}$ and in a stationary regime. We show that it is
governed by an effective equation which is a dissipative SDE with nondegenerate
diffusion. Next we assume that the interaction potential is of size
$\varepsilon\lambda$, where $\lambda$ is another small parameter, independent
from $\varepsilon$. Solutions corresponding to this scaling describe small low
temperature oscillations. We prove that in a stationary regime, under the limit
$\varepsilon\rightarrow 0$, the main order in $\lambda$ of the averaged
Hamiltonian energy flow is proportional to the gradient of temperature. We show
that the coefficient of proportionality, which we call the conductivity, admits
a representation through stationary space-time correlations of the energy flow.
Most of the results and convergences we obtain are uniform with respect to the
number of oscillators in the system.Comment: 52 page
Convergence of Clock Processes and Aging in Metropolis Dynamics of a Truncated REM

International audienceWe study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud's REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence result of a new type where the number of summands is itself a clock process. This reflects the fact that the exploration process of Metropolis dynamics is itself an aging process, governed by its own clock. Both clock processes are shown to converge to stable subor-dinators below certain critical lines in their timescale and temperature domains, almost surely in the random environment
Applications of Fixed Point Theorems to the Vacuum Einstein Constraint
  Equations with Non-Constant Mean Curvature

In this paper, we introduce new methods for solving the vacuum Einstein
constraints equations: the first one is based on Schaefer's fixed point theorem
(known methods use Schauder's fixed point theorem) while the second one uses
the concept of half-continuity coupled with the introduction of local
supersolutions. These methods allow to: unify some recent existence results,
simplify many proofs (for instance, the main theorem in arXiv:1012.2188) and
weaken the assumptions of many recent results.Comment: In this version, I change from 3-dimensional case to n-dimensional
  cas
Local inverse scattering at a fixed energy for radial Schr{\"o}dinger
  operators and localization of the Regge poles

We study inverse scattering problems at a fixed energy for radial
Schr\"{o}dinger operators on $\R^n$, $n \geq 2$. First, we consider the class
$\mathcal{A}$ of potentials $q(r)$ which can be extended analytically in $\Re z
\geq 0$ such that $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho
\textgreater{} \frac{3}{2}$. If $q$ and $\tilde{q}$ are two such potentials and
if the corresponding phase shifts $\delta\_l$ and $\tilde{\delta}\_l$ are
super-exponentially close, then $q=\tilde{q}$. Secondly, we study the class of
potentials $q(r)$ which can be split into $q(r)=q\_1(r) + q\_2(r)$ such that
$q\_1(r)$ has compact support and $q\_2 (r) \in \mathcal{A}$. If $q$ and
$\tilde{q}$ are two such potentials, we show that for any fixed
$a\textgreater{}0$, ${\ds{\delta\_l - \tilde{\delta}\_l \ = \ o \left(
\frac{1}{l^{n-3}} \ \left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ when $l
\rightarrow +\infty$ if and only if $q(r)=\tilde{q}(r)$ for almost all $r \geq
a$. The proofs are close in spirit with the celebrated Borg-Marchenko
uniqueness theorem, and rely heavily on the localization of the Regge poles
that could be defined as the resonances in the complexified angular momentum
plane. We show that for a non-zero super-exponentially decreasing potential,
the number of Regge poles is always infinite and moreover, the Regge poles are
not contained in any vertical strip in the right-half plane. For potentials
with compact support, we are able to give explicitly their asymptotics. At
last, for potentials which can be extended analytically in $\Re z \geq 0$ with
$\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}$, $\rho \textgreater{}1$ , we
show that the Regge poles are confined in a vertical strip in the complex
plane.Comment: 52 page
Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes
  from Group-Valued Moment Maps

We present a generalization of Minkowski's classic theorem on the
reconstruction of tetrahedra from algebraic data to homogeneously curved
spaces. Euclidean notions such as the normal vector to a face are replaced by
Levi-Civita holonomies around each of the tetrahedron's faces. This allows the
reconstruction of both spherical and hyperbolic tetrahedra within a unified
framework. A new type of hyperbolic simplex is introduced in order for all the
sectors encoded in the algebraic data to be covered. Generalizing the phase
space of shapes associated to flat tetrahedra leads to group valued moment maps
and quasi-Poisson spaces. These discrete geometries provide a natural arena for
considering the quantization of gravity including a cosmological constant. A
concrete realization of this is provided by the relation with the spin-network
states of loop quantum gravity. This work therefore provides a bottom-up
justification for the emergence of deformed gauge symmetries and quantum groups
in 3+1 dimensional covariant loop quantum gravity in the presence of a
cosmological constant.Comment: 38 pages and 9 figure
Anderson transition at 2 dimensional growth rate on antitrees and
  spectral theory for operators with one propagating channel

We show that the Anderson model has a transition from localization to
delocalization at exactly 2 dimensional growth rate on antitrees with
normalized edge weights which are certain discrete graphs. The kinetic part has
a one-dimensional structure allowing a description through transfer matrices
which involve some Schur complement. For such operators we introduce the notion
of having one propagating channel and extend theorems from the theory of
one-dimensional Jacobi operators that relate the behavior of transfer matrices
with the spectrum. These theorems are then applied to the considered model. In
essence, in a certain energy region the kinetic part averages the random
potentials along shells and the transfer matrices behave similar as for a
one-dimensional operator with random potential of decaying variance. At $d$
dimensional growth for $d>2$ this effective decay is strong enough to obtain
absolutely continuous spectrum, whereas for some uniform $d$ dimensional growth
with $d<2$ one has pure point spectrum in this energy region. At exactly
uniform $2$ dimensional growth also some singular continuous spectrum appears,
at least at small disorder. As a corollary we also obtain a change from
singular spectrum ($d\leq 2$) to absolutely continuous spectrum ($d\geq 3)$ for
random operators of the type $\mathcal{P}_r \Delta_d \mathcal{P}_r+\lambda
\mathcal{V}$ on $\mathbb{Z}^d$, where $\mathcal{P}_r$ is an orthogonal radial
projection, $\Delta_d$ the discrete adjacency operator (Laplacian) on
$\mathbb{Z}^d$ and $\lambda \mathcal{V}$ a random potential.Comment: 38 pages, 1 figure; Introduction reorganized, Corollary 1.3 added and
  almost sure essential spectrum now characterized (Proposition 1.4
Born-Oppenheimer approximation for an atom in constant magnetic fields

We obtain a reduction scheme for the study of the quantum evolution of an
atom in constant magnetic fields using the method developed by Martinez, Nenciu
and Sordoni based on the construction of almost invariant subspace. In
Martinez-Sordoni \cite{MaSo2} such a case is also studied but their reduced
Hamiltonian includes the vector potential terms. In this paper, using the
center of mass coordinates and constructing the almost invariant subspace
different from theirs, we obtain the reduced Hamiltonian which does not include
the vector potential terms. Using the reduced evolution we also obtain the
asymptotic expantion of the evolution for a specific localized initial data,
which verifies the straight motion of an atom in constatnt magnetic fields
On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
The spectral properties of the strongly coupled Sturm Hamiltonian of
  eventually constant type

We study the spectral properties of the Sturm Hamiltolian of eventually
constant type, which includes the Fibonacci Hamiltonian. Let $s$ be the
Hausdorff dimension of the spectrum. For $V>20$, we show that the restriction
of the $s$-dimensional Hausdorff measure to the spectrum is a Gibbs type
measure; the density of states measure is a Markov measure. Based on the fine
structures of these measures, we show that both measures are exact dimensional;
we obtain exact asymptotic behaviors for the optimal H\"older exponent and the
Hausdorff dimension of the density of states measure and for the Hausdorff
dimension of the spectrum. As a consequence, if the frequency is not silver
number type, then for $V$ big enough, we establish strict inequalities between
these three spectral characteristics.
  We achieve them by introducing an auxiliary symbolic dynamical system and
applying the thermodynamical and multifractal formalisms of almost additive
potentials.Comment: This is a revised version. We generalize the results for all the
  frequencies of eventurally constant type. Accepted by Annales Henri Poincar
Indefinite Kasparov modules and pseudo-Riemannian manifolds

We present a definition of indefinite Kasparov modules, a generalisation of
unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g.
hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov
module we can associate a pair of (genuine) Kasparov modules, and that this
process is reversible. We present three examples of our framework: the Dirac
operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an
indefinite metric), the harmonic oscillator, and the construction via the
Kasparov product of an indefinite spectral triple from a family of spectral
triples. This last construction corresponds to a foliation of a globally
hyperbolic spacetime by spacelike hypersurfaces.Comment: 24 pages, Annales Henri Poincar\'e, online version 201
Quantum Hamiltonians with weak random abstract perturbation. I. Initial
  length scale estimate

We study random Hamiltonians on finite-size cubes and waveguide segments of
increasing diameter. The number of random parameters determining the operator
is proportional to the volume of the cube. In the asymptotic regime where the
cube size, and consequently the number of parameters as well, tends to
infinity, we derive deterministic and probabilistic variational bounds on the
lowest eigenvalue, i.e. the spectral minimum, as well as exponential
off-diagonal decay of the Green function at energies above, but close to the
overall spectral bottom
Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the
  Klein-Gordon Equation

We analyze spin-0 relativistic scattering of charged particles propagating in
the exterior, $\Lambda \subset \mathbb{R}^3$, of a compact obstacle $K \subset
\mathbb{R}^3$. The connected components of the obstacle are handlebodies. The
particles interact with an electro-magnetic field in $\Lambda$ and an
inaccessible magnetic field localized in the interior of the obstacle (through
the Aharonov-Bohm effect). We obtain high-momenta estimates, with error bounds,
for the scattering operator that we use to recover physical information: We
give a reconstruction method for the electric potential and the exterior
magnetic field and prove that, if the electric potential vanishes, circulations
of the magnetic potential around handles (or equivalently, by Stokes' theorem,
magnetic fluxes over transverse sections of handles) of the obstacle can be
recovered, modulo $2 \pi$. We additionally give a simple formula for the
high-momenta limit of the scattering operator in terms of certain magnetic
fluxes, in the absence of electric potential. If the electric potential does
not vanish, the magnetic fluxes on the handles above referred can be only
recovered modulo $\pi$ and the simple expression of the high-momenta limit of
the scattering operator does not hold true
Energy-time uncertainty principle and lower bounds on sojourn time

One manifestation of quantum resonances is a large sojourn time, or
autocorrelation, for states which are initially localized. We elaborate on
Lavine's time-energy uncertainty principle and give an estimate on the sojourn
time. For the case of perturbed embedded eigenstates the bound is explicit and
involves Fermi's Golden Rule. It is valid for a very general class of systems.
We illustrate the theory by applications to resonances for time dependent
systems including the AC Stark effect as well as multistate systems.Comment: Version to appear in Annales Henri Poincar\'
Elliptic Genera and 3d Gravity

We describe general constraints on the elliptic genus of a 2d supersymmetric
conformal field theory which has a gravity dual with large radius in Planck
units. We give examples of theories which do and do not satisfy the bounds we
derive, by describing the elliptic genera of symmetric product orbifolds of
$K3$, product manifolds, certain simple families of Calabi-Yau hypersurfaces,
and symmetric products of the "Monster CFT." We discuss the distinction between
theories with supergravity duals and those whose duals have strings at the
scale set by the AdS curvature. Under natural assumptions we attempt to
quantify the fraction of (2,2) supersymmetric conformal theories which admit a
weakly curved gravity description, at large central charge.Comment: 50 pages, 9 figures, v2: minor corrections to section
On the stationary Navier-Stokes equations in the half-plane

We consider the stationary incompressible Navier-Stokes equation in the
half-plane with inhomogeneous boundary condition. We prove existence of strong
solutions for boundary data close to any Jeffery-Hamel solution with small flux
evaluated on the boundary. The perturbation of the Jeffery-Hamel solution on
the boundary has to satisfy a nonlinear compatibility condition which
corresponds to the integral of the velocity field on the boundary. The first
component of this integral is the flux which is an invariant quantity, but the
second, called the asymmetry, is not invariant, which leads to one
compatibility condition. Finally, we prove existence of weak solutions, as well
as weak-strong uniqueness for small data.Comment: 28 pages, 4 figure
Matrix models from operators and topological strings, 2

The quantization of mirror curves to toric Calabi--Yau threefolds leads to
trace class operators, and it has been conjectured that the spectral properties
of these operators provide a non-perturbative realization of topological string
theory on these backgrounds. In this paper, we find an explicit form for the
integral kernel of the trace class operator in the case of local P1xP1, in
terms of Faddeev's quantum dilogarithm. The matrix model associated to this
integral kernel is an O(2) model, which generalizes the ABJ(M) matrix model. We
find its exact planar limit, and we provide detailed evidence that its 1/N
expansion captures the all genus topological string free energy on local P1xP1.Comment: 37 pages, 4 figures; v2: misprints corrected, comments and Appendix
  adde
Semi-classical Dirac vacuum polarisation in a scalar field

We study vacuum polarisation effects of a Dirac field coupled to an external
scalar field and derive a semi-classical expansion of the regu-larised vacuum
energy. The leading order of this expansion is given by a classical formula due
to Chin, Lee-Wick and Walecka, for which our result provides the first rigorous
proof. We then discuss applications to the non-relativistic large-coupling
limit of an interacting system, and to the stability of homogeneous systems.Comment: Revised version to appear in AHP (DOI: 10.1007/s00023-016-0472-y
Multi-component Ginzburg-Landau theory: microscopic derivation and
  examples

This paper consists of three parts. In part I, we microscopically derive
Ginzburg--Landau (GL) theory from BCS theory for translation-invariant systems
in which multiple types of superconductivity may coexist. Our motivation are
unconventional superconductors. We allow the ground state of the effective gap
operator $K_{T_c}+V$ to be $n$-fold degenerate and the resulting GL theory then
couples $n$ order parameters. In part II, we study examples of multi-component
GL theories which arise from an isotropic BCS theory. We study the cases of (a)
pure $d$-wave order parameters and (b) mixed $(s+d)$-wave order parameters, in
two and three dimensions. In part III, we present explicit choices of
spherically symmetric interactions $V$ which produce the examples in part II.
In fact, we find interactions $V$ which produce ground state sectors of
$K_{T_c}+V$ of arbitrary angular momentum, for open sets of of parameter
values. This is in stark contrast with Schr\"odinger operators $-\nabla^2+V$,
for which the ground state is always non-degenerate. Along the way, we prove
the following fact about Bessel functions: At its first maximum, a half-integer
Bessel function is strictly larger than all other half-integer Bessel
functions.Comment: 57 pages, 2 tables and 1 figure. Final version to appear in Ann. H.
  Poincar\'
Asymptotics for Erdos-Solovej Zero Modes in Strong Fields

We consider the strong field asymptotics for the occurrence of zero modes of
certain Weyl-Dirac operators on $\mathbb{R}^3$. In particular we are interested
in those operators $\mathcal{D}_{B}$ for which the associated magnetic field
$B$ is given by pulling back a $2$-form $\beta$ from the sphere $\mathbb{S}^2$
to $\mathbb{R}^3$ using a combination of the Hopf fibration and inverse
stereographic projection. If $\int_{\mathbb{S}^2}\beta\neq0$ we show that \[
\sum_{0\le t\le T}\mathrm{dim}\,\mathrm{Ker}\,\mathcal{D}_{tB}
=\frac{T^2}{8\pi^2}\,\biggl\lvert\int_{\mathbb{S}^2}\beta\biggr\rvert\,\int_{\mathbb{S}^2}\lvert{\beta}\rvert+o(T^2)
\] as $T\to+\infty$. The result relies on Erd\H{o}s and Solovej's
characterisation of the spectrum of $\mathcal{D}_{tB}$ in terms of a family of
Dirac operators on $\mathbb{S}^2$, together with information about the strong
field localisation of the Aharonov-Casher zero modes of the latter.Comment: 24 pages, typos corrected, some minor rewordin
Cohomology with causally restricted supports

De Rham cohomology with spacelike compact and timelike compact supports has
recently been noticed to be of importance for understanding the structure of
classical and quantum Maxwell theory on curved spacetimes. Similarly causally
restricted cohomologies of different differential complexes play a similar role
in other gauge theories. We introduce a method for computing these causally
restricted cohomologies in terms of cohomologies with either compact or
unrestricted supports. The calculation exploits the fact that the de
Rham-d'Alembert wave operator can be extended to a chain map that is homotopic
to zero and that its causal Green function fits into a convenient exact
sequence. As a first application, we use the method on the de Rham complex,
then also on the Calabi (or Killing-Riemann-Bianchi) complex, which appears in
linearized gravity on constant curvature backgrounds. We also discuss
applications to other complexes, as well as generalized causal structures and
functoriality.Comment: 26 pages, no figures, BibTeX; v2: some sections rearranged, slightly
  modified notation; close to published versio