20,865 research outputs found
Projective dynamics and first integrals
We present the theory of tensors with Young tableau symmetry as an efficient
computational tool in dealing with the polynomial first integrals of a natural
system in classical mechanics. We relate a special kind of such first
integrals, already studied by Lundmark, to Beltrami's theorem about
projectively flat Riemannian manifolds. We set the ground for a new and simple
theory of the integrable systems having only quadratic first integrals. This
theory begins with two centered quadrics related by central projection, each
quadric being a model of a space of constant curvature. Finally, we present an
extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure
A theoretical framework for supervised learning from regions
Supervised learning is investigated, when the data are represented not only by labeled points but also labeled regions of the input space. In the limit case, such
regions degenerate to single points and the proposed approach changes back to the classical learning context. The adopted framework entails the minimization
of a functional obtained by introducing a loss function that involves such regions. An additive regularization term is expressed via differential operators that model
the smoothness properties of the desired input/output relationship. Representer
theorems are given, proving that the optimization problem associated to learning
from labeled regions has a unique solution, which takes on the form of a linear
combination of kernel functions determined by the differential operators together
with the regions themselves. As a relevant situation, the case of regions given
by multi-dimensional intervals (i.e., âboxesâ) is investigated, which models prior
knowledge expressed by logical propositions
Classical Solutions for Two Dimensional QCD on the Sphere
We consider and gauge theory on the sphere. We express the
problem in terms of a matrix element of free fermions on a circle. This
allows us to find an alternative way to show Witten's result that the partition
function is a sum over classical saddle points. We then show how the phase
transition of Douglas and Kazakov occurs from this point of view. By
generalizing the work of Douglas and Kazakov, we find other `stringy' solutions
for the case in the large limit. Each solution is described by a net
charge. We derive a relation for the maximum charge for a given area and
we also describe the critical behavior for these new solutions. Finally, we
describe solutions for lattice which are in a sense dual to the
continuum solutions. (Parts of this paper were presented at the Strings
'93 Workshop, Berkeley, May 1993.)Comment: 26 pages, CERN-TH-7016, UVA-HET-93-0
From microscopic to macroscopic descriptions of cell\ud migration on growing domains
Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs
Nonlinear transverse cascade and sustenance of MRI-turbulence in Keplerian disks with an azimuthal magnetic field
We investigate magnetohydrodynamic turbulence driven by the magnetorotational
instability (MRI) in Keplerian disks with a nonzero net azimuthal magnetic
field using shearing box simulations. As distinct from most previous studies,
we analyze turbulence dynamics in Fourier (-) space to understand its
sustenance. The linear growth of MRI with azimuthal field has a transient
character and is anisotropic in Fourier space, leading to anisotropy of
nonlinear processes in Fourier space. As a result, the main nonlinear process
appears to be a new type of angular redistribution of modes in Fourier space --
the \emph{nonlinear transverse cascade} -- rather than usual direct/inverse
cascade. We demonstrate that the turbulence is sustained by interplay of the
linear transient growth of MRI (which is the only energy supply for the
turbulence) and the transverse cascade. These two processes operate at large
length scales, comparable to box size and the corresponding small wavenumber
area, called \emph{vital area} in Fourier space is crucial for the sustenance,
while outside the vital area direct cascade dominates. The interplay of the
linear and nonlinear processes in Fourier space is generally too intertwined
for a vivid schematization. Nevertheless, we reveal the \emph{basic subcycle}
of the sustenance that clearly shows synergy of these processes in the
self-organization of the magnetized flow system. This synergy is quite robust
and persists for the considered different aspect ratios of the simulation
boxes. The spectral characteristics of the dynamical processes in these boxes
are qualitatively similar, indicating the universality of the sustenance
mechanism of the MRI-turbulence.Comment: 32 pages, 17 figures, accepted for publication in Ap
All-Purpose Numerical Evaluation of One-Loop Multi-Leg Feynman Diagrams
A detailed investigation is presented of a set of algorithms which form the
basis for a fast and reliable numerical integration of one-loop multi-leg (up
to six) Feynman diagrams, with special attention to the behavior around
(possibly) singular points in phase space. No particular restriction is imposed
on kinematics, and complex masses (poles) are allowed.Comment: 56 page
Comparison theorems for conjugate points in sub-Riemannian geometry
We prove sectional and Ricci-type comparison theorems for the existence of
conjugate points along sub-Riemannian geodesics. In order to do that, we regard
sub-Riemannian structures as a special kind of variational problems. In this
setting, we identify a class of models, namely linear quadratic optimal control
systems, that play the role of the constant curvature spaces. As an
application, we prove a version of sub-Riemannian Bonnet-Myers theorem and we
obtain some new results on conjugate points for three dimensional
left-invariant sub-Riemannian structures.Comment: 33 pages, 5 figures, v2: minor revision, v3: minor revision, v4:
minor revisions after publicatio
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