2,901 research outputs found
A new representation for non--local operators and path integrals
We derive an alternative representation for the relativistic non--local
kinetic energy operator and we apply it to solve the relativistic Salpeter
equation using the variational sinc collocation method. Our representation is
analytical and does not depend on an expansion in terms of local operators. We
have used the relativistic harmonic oscillator problem to test our formula and
we have found that arbitrarily precise results are obtained, simply increasing
the number of grid points. More difficult problems have also been considered,
observing in all cases the convergence of the numerical results. Using these
results we have also derived a new representation for the quantum mechanical
Green's function and for the corresponding path integral. We have tested this
representation for a free particle in a box, recovering the exact result after
taking the proper limits, and we have also found that the application of the
Feynman--Kac formula to our Green's function yields the correct ground state
energy. Our path integral representation allows to treat hamiltonians
containing non--local operators and it could provide to the community a new
tool to deal with such class of problems.Comment: 9 pages ; 1 figure ; refs added ; title modifie
Toddler-Inspired Visual Object Learning
Real-world learning systems have practical limitations on the quality and quantity of the training datasets that they can collect and consider. How should a system go about choosing a subset of the possible training examples that still allows for learning accurate, generalizable models? To help address this question, we draw inspiration from a highly efficient practical learning system: the human child. Using head-mounted cameras, eye gaze trackers, and a model of foveated vision, we collected first-person (egocentric) images that represents a highly accurate approximation of the "training data" that toddlers' visual systems collect in everyday, naturalistic learning contexts. We used state-of-the-art computer vision learning models (convolutional neural networks) to help characterize the structure of these data, and found that child data produce significantly better object models than egocentric data experienced by adults in exactly the same environment. By using the CNNs as a modeling tool to investigate the properties of the child data that may enable this rapid learning, we found that child data exhibit a unique combination of quality and diversity, with not only many similar large, high-quality object views but also a greater number and diversity of rare views. This novel methodology of analyzing the visual "training data" used by children may not only reveal insights to improve machine learning, but also may suggest new experimental tools to better understand infant learning in developmental psychology
On the Path Integral in Imaginary Lobachevsky Space
The path integral on the single-sheeted hyperboloid, i.e.\ in -dimensional
imaginary Lobachevsky space, is evaluated. A potential problem which we call
``Kepler-problem'', and the case of a constant magnetic field are also
discussed.Comment: 16 pages, LATEX, DESY 93-14
Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
The main purpose of this paper is to approximate several non-local evolution
equations by zero-sum repeated games in the spirit of the previous works of
Kohn and the second author (2006 and 2009): general fully non-linear parabolic
integro-differential equations on the one hand, and the integral curvature flow
of an interface (Imbert, 2008) on the other hand. In order to do so, we start
by constructing such a game for eikonal equations whose speed has a
non-constant sign. This provides a (discrete) deterministic control
interpretation of these evolution equations. In all our games, two players
choose positions successively, and their final payoff is determined by their
positions and additional parameters of choice. Because of the non-locality of
the problems approximated, by contrast with local problems, their choices have
to "collect" information far from their current position. For integral
curvature flows, players choose hypersurfaces in the whole space and positions
on these hypersurfaces. For parabolic integro-differential equations, players
choose smooth functions on the whole space
Dynamics of a lattice Universe
We find a solution to Einstein field equations for a regular toroidal lattice
of size L with equal masses M at the centre of each cell; this solution is
exact at order M/L. Such a solution is convenient to study the dynamics of an
assembly of galaxy-like objects. We find that the solution is expanding (or
contracting) in exactly the same way as the solution of a
Friedman-Lema\^itre-Robertson-Walker Universe with dust having the same average
density as our model. This points towards the absence of backreaction in a
Universe filled with an infinite number of objects, and this validates the
fluid approximation, as far as dynamics is concerned, and at the level of
approximation considered in this work.Comment: 14 pages. No figure. Accepted version for Classical and Quantum
Gravit
The heart of a convex body
We investigate some basic properties of the {\it heart}
of a convex set It is a subset of
whose definition is based on mirror reflections of euclidean
space, and is a non-local object. The main motivation of our interest for
is that this gives an estimate of the location of the
hot spot in a convex heat conductor with boundary temperature grounded at zero.
Here, we investigate on the relation between and the
mirror symmetries of we show that
contains many (geometrically and phisically) relevant points of
we prove a simple geometrical lower estimate for the diameter of
we also prove an upper estimate for the area of
when is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic
and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6
Climate oscillations, glacial refugia, and dispersal ability: factors influencing the genetic structure of the least salmonfly, Pteronarcella badia (Plecoptera), in Western North America
Background: Phylogeographic studies of aquatic insects provide valuable insights into mechanisms that shape the genetic structure of communities, yet studies that include broad geographic areas are uncommon for this group. We conducted a broad scale phylogeographic analysis of the least salmonfly Pteronarcella badia (Plecoptera) across western North America. We tested hypotheses related to mode of dispersal and the influence of historic climate oscillations on population genetic structure. In order to generate a larger mitochondrial data set, we used 454 sequencing to reconstruct the complete mitochondrial genome in the early stages of the project. Results: Our analysis revealed high levels of population structure with several deeply divergent clades present across the sample area. Evidence from five mitochondrial genes and one nuclear locus identified a potentially cryptic lineage in the Pacific Northwest. Gene flow estimates and geographic clade distributions suggest that overland flight during the winged adult stage is an important dispersal mechanism for this taxon. We found evidence of multiple glacial refugia across the species distribution and signs of secondary contact within and among major clades. Conclusions: This study provides a basis for future studies of aquatic insect phylogeography at the inter-basin scale in western North America. Our findings add to an understanding of the role of historical climate isolations in shaping assemblages of aquatic insects in this region. We identified several geographic areas that may have historical importance for other aquatic organisms with similar distributions and dispersal strategies as P. badia. This work adds to the ever-growing list of studies that highlight the potential of next-generation DNA sequencing in a phylogenetic context to improve molecular data sets from understudied groups
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
Topology and Homoclinic Trajectories of Discrete Dynamical Systems
We show that nontrivial homoclinic trajectories of a family of discrete,
nonautonomous, asymptotically hyperbolic systems parametrized by a circle
bifurcate from a stationary solution if the asymptotic stable bundles
Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch
are twisted in different ways.Comment: 19 pages, canceled the appendix (Properties of the index bundle) in
order to avoid any text overlap with arXiv:1005.207
Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
We calculate very long low- and high-temperature series for the
susceptibility of the square lattice Ising model as well as very long
series for the five-particle contribution and six-particle
contribution . These calculations have been made possible by the
use of highly optimized polynomial time modular algorithms and a total of more
than 150000 CPU hours on computer clusters. For 10000 terms of the
series are calculated {\it modulo} a single prime, and have been used to find
the linear ODE satisfied by {\it modulo} a prime.
A diff-Pad\'e analysis of 2000 terms series for and
confirms to a very high degree of confidence previous conjectures about the
location and strength of the singularities of the -particle components of
the susceptibility, up to a small set of ``additional'' singularities. We find
the presence of singularities at for the linear ODE of ,
and for the ODE of , which are {\it not} singularities
of the ``physical'' and that is to say the
series-solutions of the ODE's which are analytic at .
Furthermore, analysis of the long series for (and )
combined with the corresponding long series for the full susceptibility
yields previously conjectured singularities in some , .
We also present a mechanism of resummation of the logarithmic singularities
of the leading to the known power-law critical behaviour occurring
in the full , and perform a power spectrum analysis giving strong
arguments in favor of the existence of a natural boundary for the full
susceptibility .Comment: 54 pages, 2 figure
- …