751 research outputs found
Variational electrodynamics of Atoms
We generalize Wheeler-Feynman electrodynamics by the minimization of a finite
action functional defined for variational trajectories that are required to
merge continuously into given past and future boundary segments. We prove that
the boundary-value problem is well-posed for two classes of boundary data and
show that the well-posed solution in general has velocity discontinuities,
henceforth broken extrema. Along regular segments, broken extrema satisfy the
Euler-Lagrange neutral differential delay equations with state-dependent
deviating arguments. At points where velocities are discontinuous, broken
extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta
are continuous. The electromagnetic fields of the variational trajectories are
derived quantities that can be extended only to a bounded region B of
space-time. For extrema with a finite number of velocity discontinuities,
extended fields are defined for all point in B with the exception of sets of
zero measure. The extended fields satisfy the integral laws of classical
electrodynamics for most surfaces and curves inside B. As an application, we
study globally bounded trajectories with vanishing far-fields for the
hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses
solutions of the neutral differential delay equations along regular segments
and a variational approximation for the collisional segments. Each hydrogenoid
model predicts a discrete set of finitely measured neighbourhoods of orbits
with vanishing far-fields at the correct atomic magnitude and in quantitative
and qualitative agreement with experiment and quantum mechanics, i.e., the
spacings between consecutive discrete angular momenta agree with Planck's
constant within thirty-percent, while orbital frequencies agree with a
corresponding spectroscopic line within a few percent.Comment: Full re-write using same equations and back to original title
(version 18 compiled with the wrong figure 5). A few commas introduced and
all paragraphs broken into smaller ones whenever possibl
Nonlinear Systems
The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers
Abstract book
Welcome at the International Conference on Differential and Difference Equations
& Applications 2015.
The main aim of this conference is to promote, encourage, cooperate, and bring
together researchers in the fields of differential and difference equations. All areas
of differential & difference equations will be represented with special emphasis on
applications. It will be mathematically enriching and socially exciting event.
List of registered participants consists of 169 persons from 45 countries.
The five-day scientific program runs from May 18 (Monday) till May 22, 2015
(Friday). It consists of invited lectures (plenary lectures and invited lectures in
sections) and contributed talks in the following areas:
Ordinary differential equations,
Partial differential equations,
Numerical methods and applications, other topics
Holographic quark gluon plasma with flavor
In this work I explore theoretical and phenomenological implications of
chemical potentials and charge densities inside a strongly coupled thermal
plasma, using the gauge/gravity correspondence. Strong coupling effects
discovered in this model theory are interpreted geometrically and may be taken
as qualitative predictions for heavy ion collisions at RHIC and LHC. In
particular I examine the thermodynamics, spectral functions, transport
coefficients and the phase diagram of the strongly coupled plasma. For example
stable mesons, which are the analogs of the QCD Rho-mesons, are found to
survive beyond the deconfinement transition. This paper is based on partly
unpublished work performed in the context of my PhD thesis. New results and
ideas extending significantly beyond those published until now are stressed.Comment: 45 figures, 166 page
Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential
The energy super-critical Gross--Pitaevskii equation with a harmonic
potential is revisited in the particular case of cubic focusing nonlinearity
and dimension d > 4. In order to prove the existence of a ground state (a
positive, radially symmetric solution in the energy space), we develop the
shooting method and deal with a one-parameter family of classical solutions to
an initial-value problem for the stationary equation. We prove that the
solution curve (the graph of the eigenvalue parameter versus the supremum) is
oscillatory for d = 13. Compared to the existing
literature, rigorous asymptotics are derived by constructing three families of
solutions to the stationary equation with functional-analytic rather than
geometric methods.Comment: 42 page
Ultracold atoms and the Functional Renormalization Group
We give a self-contained introduction to the physics of ultracold atoms using
functional integral techniques. Based on a consideration of the relevant length
scales, we derive the universal effective low energy Hamiltonian describing
ultracold alkali atoms. We then introduce the concept of the effective action,
which generalizes the classical action principle to full quantum status and
provides an intuitive and versatile tool for practical calculations. This
framework is applied to weakly interacting degenerate bosons and fermions in
the spatial continuum. In particular, we discuss the related BEC and BCS
quantum condensation mechanisms. We then turn to the BCS-BEC crossover, which
interpolates between both phenomena, and which is realized experimentally in
the vicinity of a Feshbach resonance. For its description, we introduce the
Functional Renormalization Group approach. After a general discussion of the
method in the cold atoms context, we present a detailed and pedagogical
application to the crossover problem. This not only provides the physical
mechanism underlying this phenomenon. More generally, it also reveals how the
renormalization group can be used as a tool to capture physics at all scales,
from few-body scattering on microscopic scales, through the finite temperature
phase diagram governed by many-body length scales, up to critical phenomena
dictating long distance physics at the phase transition. The presentation aims
to equip students at the beginning PhD level with knowledge on key physical
phenomena and flexible tools for their description, and should enable to embark
upon practical calculations in this field.Comment: 73 pages, 32 figures. Lecture notes for the 49th Schladming Winter
School `Physics at all scales: The Renormalization Group' (to appear in the
proceedings
Response theory and critical phenomena for noisy interacting systems
In this thesis we investigate critical phenomena for ensembles of identical interacting agents, namely weakly interacting diffusions. These interacting systems undergo two qualitatively different scenarios of criticality, critical transitions and phase transitions. The former situation conforms to the classical tipping point phenomenology that is observed in finite dimensional systems and originates from a setting where negative feedbacks that stabilise the system progressively loose their efficiency, resulting in amplified fluctuations and correlation properties of the system. On the other hand, \textit{phase transitions} stem from the complex interplay between the agents' own dynamics, the coupling among them and the noise, leading to macroscopic emergent behaviour of the system, and are only observed in the thermodynamic limit. Classically, \textit{phase transitions} are investigated with the use of suitable macroscopic variables, called order parameters, acting as effective reaction coordinates that capture the relevant features of the macroscopic dynamics. However, identifying an order parameter is not always possible. In this thesis we adopt a complementary point of view, based on Linear Response theory, to investigate such critical phenomena. We are able to identify the conditions leading either to a critical transition or a phase transition in terms of spectral properties of suitable response operators. We associate critical phenomena to settings where the response of the system breaks down. In particular, we are able to characterise these critical scenarios as settings where the complex valued susceptibility of the system develops a non analytical behaviour for real values of frequencies, resulting in a macroscopic resonance of the system. We provide multiple paradigmatic examples of equilibrium and nonequilibrium phase transitions where we are able to prove mathematically and numerically the clear signature of a singular behaviour of the susceptibility at the phase transition as the thermodynamic limit is reached. Being associated to spectral properties of suitable operators describing either correlation or response properties, these resonant phenomena do not depend on the specific details of the applied forcing nor on the observable under investigation, allowing one to bypass the problem of the identification of the order parameter for the system.Open Acces
Bounded Influence Approaches to Constrained Mixed Vector Autoregressive Models
The proliferation of many clinical studies obtaining multiple biophysical signals from several individuals repeatedly in time is increasingly recognized, a recognition generating growth in statistical models that analyze cross-sectional time series data. In general, these statistical models try to answer two questions: (i) intra-individual dynamics of the response and its relation to some covariates; and, (ii) how this dynamics can be aggregated consistently in a group. In response to the first question, we propose a covariate-adjusted constrained Vector Autoregressive model, a technique similar to the STARMAX model (Stoffer, JASA 81, 762-772), to describe serial dependence of observations. In this way, the number of parameters to be estimated is kept minimal while offering flexibility for the model to explore higher order dependence. In response to (ii), we use mixed effects analysis that accommodates modelling of heterogeneity among cross-sections arising from covariate effects that vary from one cross-section to another. Although estimation of the model can proceed using standard maximum likelihood techniques, we believed it is advantageous to use bounded influence procedures in the modelling (such as choosing constraints) and parameter estimation so that the effects of outliers can be controlled. In particular, we use M-estimation with a redescending bounding function because its influence function is always bounded. Furthermore, assuming consistency, this influence function is useful to obtain the limiting distribution of the estimates. However, this distribution may not necessarily yield accurate inference in the presence of contamination as the actual asymptotic distribution might have wider tails. This led us to investigate bootstrap approximation techniques. A sampling scheme based on IID innovations is modified to accommodate the cross-sectional structure of the data. Then the M-estimation is applied to each bootstrap sample naively to obtain the asymptotic distribution of the estimates.We apply these strategies to the extracted BOLD activation from several regions of the brain from a group of individuals to describe joint dynamic behavior between these locations. We used simulated data with both innovation and additive outliers to test whether the estimation procedure is accurate despite contamination
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