31,459 research outputs found
Unifying discrete and continuous Weyl-Titchmarsh theory via a class of linear Hamiltonian systems on Sturmian time scales
In this study, we are concerned with introducing Weyl-Titchmarsh theory for a
class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian
time scales. After developing fundamental properties of solutions and regular
spectral problems, we introduce the corresponding maximal and minimal operators
for the system. Matrix disks are constructed and proved to be nested and
converge to a limiting set. Some precise relationships among the rank of the
matrix radius of the limiting set, the number of linearly independent square
summable solutions, and the defect indices of the minimal operator are
established. Using the above results, a classification of singular dynamic
linear Hamiltonian nabla systems is given in terms of the defect indices of the
minimal operator, and several equivalent conditions on the cases of limit point
and limit circle are obtained, respectively. These results unify and extend
certain classic and recent results on the subject in the continuous and
discrete cases, respectively, to Sturmian time scales.Comment: 34 page
Dynamic universality class of the QCD critical point
We show that the dynamic universality class of the QCD critical point is that
of model H and discuss the dynamic critical exponents. We show that the baryon
diffusion rate vanishes at the critical point. The dynamic critical index
is close to 3.Comment: 12 pages. To be published in PRD. Appendix about isospin density
added, introduction expande
Flowing in Group Field Theory Space: a Review
We provide a non-technical overview of recent extensions of renormalization
methods and techniques to Group Field Theories (GFTs), a class of
combinatorially non-local quantum field theories which generalize matrix models
to dimension . More precisely, we focus on GFTs with so-called
closure constraint, which are closely related to lattice gauge theories and
quantum gravity spin foam models. With the help of recent tensor model tools, a
rich landscape of renormalizable theories has been unravelled. We review our
current understanding of their renormalization group flows, at both
perturbative and non-perturbative levels
Exact solutions to the nonlinear dynamics of learning in deep linear neural networks
Despite the widespread practical success of deep learning methods, our
theoretical understanding of the dynamics of learning in deep neural networks
remains quite sparse. We attempt to bridge the gap between the theory and
practice of deep learning by systematically analyzing learning dynamics for the
restricted case of deep linear neural networks. Despite the linearity of their
input-output map, such networks have nonlinear gradient descent dynamics on
weights that change with the addition of each new hidden layer. We show that
deep linear networks exhibit nonlinear learning phenomena similar to those seen
in simulations of nonlinear networks, including long plateaus followed by rapid
transitions to lower error solutions, and faster convergence from greedy
unsupervised pretraining initial conditions than from random initial
conditions. We provide an analytical description of these phenomena by finding
new exact solutions to the nonlinear dynamics of deep learning. Our theoretical
analysis also reveals the surprising finding that as the depth of a network
approaches infinity, learning speed can nevertheless remain finite: for a
special class of initial conditions on the weights, very deep networks incur
only a finite, depth independent, delay in learning speed relative to shallow
networks. We show that, under certain conditions on the training data,
unsupervised pretraining can find this special class of initial conditions,
while scaled random Gaussian initializations cannot. We further exhibit a new
class of random orthogonal initial conditions on weights that, like
unsupervised pre-training, enjoys depth independent learning times. We further
show that these initial conditions also lead to faithful propagation of
gradients even in deep nonlinear networks, as long as they operate in a special
regime known as the edge of chaos.Comment: Submission to ICLR2014. Revised based on reviewer feedbac
Nonintegrability, Chaos, and Complexity
Two-dimensional driven dissipative flows are generally integrable via a
conservation law that is singular at equilibria. Nonintegrable dynamical
systems are confined to n*3 dimensions. Even driven-dissipative deterministic
dynamical systems that are critical, chaotic or complex have n-1 local
time-independent conservation laws that can be used to simplify the geometric
picture of the flow over as many consecutive time intervals as one likes. Those
conserevation laws generally have either branch cuts, phase singularities, or
both. The consequence of the existence of singular conservation laws for
experimental data analysis, and also for the search for scale-invariant
critical states via uncontrolled approximations in deterministic dynamical
systems, is discussed. Finally, the expectation of ubiquity of scaling laws and
universality classes in dynamics is contrasted with the possibility that the
most interesting dynamics in nature may be nonscaling, nonuniversal, and to
some degree computationally complex
General Relativity solutions in modified gravity
Recent gravitational wave observations of binary black hole mergers and a
binary neutron star merger by LIGO and Virgo Collaborations associated with its
optical counterpart constrain deviation from General Relativity (GR) both on
strong-field regime and cosmological scales with high accuracy, and further
strong constraints are expected by near-future observations. Thus, it is
important to identify theories of modified gravity that intrinsically possess
the same solutions as in GR among a huge number of theories. We clarify the
three conditions for theories of modified gravity to allow GR solutions, i.e.,
solutions with the metric satisfying the Einstein equations in GR and the
constant profile of the scalar fields. Our analysis is quite general, as it
applies a wide class of single-/multi-field scalar-tensor theories of modified
gravity in the presence of matter component, and any spacetime geometry
including cosmological background as well as spacetime around black hole and
neutron star, for the latter of which these conditions provide a necessary
condition for no-hair theorem. The three conditions will be useful for further
constraints on modified gravity theories as they classify general theories of
modified gravity into three classes, each of which possesses i) unique GR
solutions (i.e., no-hair cases), ii) only hairy solutions (except the cases
that GR solutions are realized by cancellation between singular coupling
functions in the Euler-Lagrange equations), and iii) both GR and hairy
solutions, for the last of which one of the two solutions may be selected
dynamically.Comment: 9 pages; version to appear in Phys.Lett.
Singularities and Quantum Gravity
Although there is general agreement that a removal of classical gravitational
singularities is not only a crucial conceptual test of any approach to quantum
gravity but also a prerequisite for any fundamental theory, the precise
criteria for non-singular behavior are often unclear or controversial. Often,
only special types of singularities such as the curvature singularities found
in isotropic cosmological models are discussed and it is far from clear what
this implies for the very general singularities that arise according to the
singularity theorems of general relativity. In these lectures we present an
overview of the current status of singularities in classical and quantum
gravity, starting with a review and interpretation of the classical singularity
theorems. This suggests possible routes for quantum gravity to evade the
devastating conclusion of the theorems by different means, including modified
dynamics or modified geometrical structures underlying quantum gravity. The
latter is most clearly present in canonical quantizations which are discussed
in more detail. Finally, the results are used to propose a general scheme of
singularity removal, quantum hyperbolicity, to show cases where it is realized
and to derive intuitive semiclassical pictures of cosmological bounces.Comment: 41 pages, lecture course at the XIIth Brazilian School on Cosmology
and Gravitation, September 200
Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows
In self-gravitating stars, two dimensional or geophysical flows and in
plasmas, long range interactions imply a lack of additivity for the energy; as
a consequence, the usual thermodynamic limit is not appropriate. However, by
contrast with many claims, the equilibrium statistical mechanics of such
systems is a well understood subject. In this proceeding, we explain briefly
the classical approach to equilibrium and non equilibrium statistical mechanics
for these systems, starting from first principles. We emphasize recent and new
results, mainly a classification of equilibrium phase transitions, new
unobserved equilibrium phase transition, and out of equilibrium phase
transitions. We briefly discuss what we consider as challenges in this field
Primordial brusque bounce in Born-Infeld determinantal gravity
We study a particular exact solution to the Born-Infeld determinantal gravity
consisting of a cosmological model which undergoes a brusque bounce. The latter
consists of an event characterized by a non-null (but finite) value of the
squared Hubble rate occurring at a minimum (non-null) scale factor. The energy
density and pressure of the fluid covering the whole manifold are perfectly
well behaved in such an event, but the curvature invariants turn out to be
undefined there because of the undefined character of the time derivative of H.
It is shown that the spacetime results geodesically complete and singularity
free, and that it corresponds to a picture of an eternal Universe in which a
(somewhat unconventional) bounce replaces the standard Big Bang singularity.
This example tends to emphasize that, beyond Einstein's theory of General
Relativity, and in the context of extended theories of gravity formulated by
purely torsional means, the criterion of a singularity based on pathologies of
scalars constructed upon the Riemann curvature tensor, becomes objectionable.Comment: 8 pages, one figure. Typos corrected, some references added and
updated. Final version to appear in Phys. Rev.
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