619 research outputs found
On The Isoperimetric Spectrum of Graphs and Its Approximations
In this paper we consider higher isoperimetric numbers of a (finite directed)
graph. In this regard we focus on the th mean isoperimetric constant of a
directed graph as the minimum of the mean outgoing normalized flows from a
given set of disjoint subsets of the vertex set of the graph. We show that
the second mean isoperimetric constant in this general setting, coincides with
(the mean version of) the classical Cheeger constant of the graph, while for
the rest of the spectrum we show that there is a fundamental difference between
the th isoperimetric constant and the number obtained by taking the minimum
over all -partitions. In this direction, we show that our definition is the
correct one in the sense that it satisfies a Federer-Fleming-type theorem, and
we also define and present examples for the concept of a supergeometric graph
as a graph whose mean isoperimetric constants are attained on partitions at all
levels. Moreover, considering the -completeness of the isoperimetric
problem on graphs, we address ourselves to the approximation problem where we
prove general spectral inequalities that give rise to a general Cheeger-type
inequality as well. On the other hand, we also consider some algorithmic
aspects of the problem where we show connections to orthogonal representations
of graphs and following J.~Malik and J.~Shi () we study the close
relationships to the well-known -means algorithm and normalized cuts method
Manifestly Finite Perturbation Theory for the Short-Distance Expansion of Correlation Functions in the Two Dimensional Ising Model
In the spirit of classic works of Wilson on the renormalization group and
operator product expansion, a new framework for the study of the theory space
of euclidean quantum field theories has been introduced. This formalism is
particularly useful for elucidating the structure of the short-distance
expansions of the -point functions of a renormalizable quantum field theory
near a non-trivial fixed point. We review and apply this formalism in the study
of the scaling limit of the two dimensional massive Ising model.
Renormalization group analysis and operator product expansions determine all
the non-analytic mass dependence of the short-distance expansion of the
correlation functions. An extension of the first order variational formula to
higher orders provides a manifestly finite scheme for the perturbative
calculation of the operator product coefficients to any order in parameters. A
perturbative expansion of the correlation functions follows. We implement this
scheme for a systematic study of correlation functions involving two spin
operators. We show how the necessary non-trivial integrals can be calculated.
As two concrete examples we explicitly calculate the short-distance expansion
of the spin-spin correlation function to third order and the spin-spin-energy
density correlation function to first order in the mass. We also discuss the
applicability of our results to perturbations near other non-trivial fixed
points corresponding to other unitary minimal models.Comment: 38 pages with 1 figure, UCLA/93/TEP/4
Ground State and Resonances in the Standard Model of Non-relativistic QED
We prove existence of a ground state and resonances in the standard model of
the non-relativistic quantum electro-dynamics (QED). To this end we introduce a
new canonical transformation of QED Hamiltonians and use the spectral
renormalization group technique with a new choice of Banach spaces.Comment: 50 pages change
Low-dimensional surgery and the Yamabe invariant
Assume that M is a compact n-dimensional manifold and that N is obtained by
surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants
\sigma(M) and \sigma(N) satisfy \sigma(N)\ge min (\sigma(M),\Lambda) for
\Lambda>0. We derive explicit lower bounds for \Lambda in dimensions where
previous methods failed, namely for (n,k)\in
{(4,1),(5,1),(5,2),(6,3),(9,1),(10,1)}. With methods from surgery theory and
bordism theory several gap phenomena for smooth Yamabe invariants can be
deduced.Comment: Version 2 contains new results: the case (n,k)=(6,3) is now solved,
Version 3: typos corrected, final version to appear in J. Math. Soc. Japa
Smoothed Particle Hydrodynamics in cosmology: a comparative study of implementations
We analyse the performance of twelve different implementations of Smoothed
Particle Hydrodynamics (SPH) using seven tests designed to isolate key
hydrodynamic elements of cosmological simulations which are known to cause the
SPH algorithm problems. In order, we consider a shock tube, spherical adiabatic
collapse, cooling flow model, drag, a cosmological simulation, rotating
cloud-collapse and disc stability. In the implementations special attention is
given to the way in which force symmetry is enforced in the equations of
motion. We study in detail how the hydrodynamics are affected by different
implementations of the artificial viscosity including those with a
shear-correction modification. We present an improved first-order
smoothing-length update algorithm that is designed to remove instabilities that
are present in the Hernquist and Katz (1989) algorithm.
For all tests we find that the artificial viscosity is the most important
factor distinguishing the results from the various implementations. The second
most important factor is the way force symmetry is achieved in the equation of
motion. Most results favour a kernel symmetrization approach. The exact method
by which SPH pressure forces are included has comparatively little effect on
the results. Combining the equation of motion presented in Thomas and Couchman
(1992) with a modification of the Monaghan and Gingold (1983) artificial
viscosity leads to an SPH scheme that is both fast and reliable.Comment: 30 pages, 26 figures and 9 tables included. Submitted to MNRAS.
Postscript version available at
ftp://phobos.astro.uwo.ca/pub/etittley/papers/sphtest.ps.g
On the optimal design of wall-to-wall heat transport
We consider the problem of optimizing heat transport through an
incompressible fluid layer. Modeling passive scalar transport by
advection-diffusion, we maximize the mean rate of total transport by a
divergence-free velocity field. Subject to various boundary conditions and
intensity constraints, we prove that the maximal rate of transport scales
linearly in the r.m.s. kinetic energy and, up to possible logarithmic
corrections, as the rd power of the mean enstrophy in the advective
regime. This makes rigorous a previous prediction on the near optimality of
convection rolls for energy-constrained transport. Optimal designs for
enstrophy-constrained transport are significantly more difficult to describe:
we introduce a "branching" flow design with an unbounded number of degrees of
freedom and prove it achieves nearly optimal transport. The main technical tool
behind these results is a variational principle for evaluating the transport of
candidate designs. The principle admits dual formulations for bounding
transport from above and below. While the upper bound is closely related to the
"background method", the lower bound reveals a connection between the optimal
design problems considered herein and other apparently related model problems
from mathematical materials science. These connections serve to motivate
designs.Comment: Minor revisions from review. To appear in Comm. Pure Appl. Mat
Analyzing Correlation Functions with Tesseral and Cartesian Spherical Harmonics
The dependence of inter-particle correlations on the orientation of particle
relative-momentum can yield unique information on the space-time features of
emission in reactions with multiparticle final states. In the present paper,
the benefits of a representation and analysis of the three-dimensional
correlation information in terms of surface spherical harmonics is presented.
The harmonics include the standard complex tesseral harmonics and the real
cartesian harmonics. Mathematical properties of the lesser-known cartesian
harmonics are illuminated. The physical content of different angular harmonic
components in a correlation is described. The resolving power of different
final-state effects with regarding to determining angular features of emission
regions is investigated. The considered final-state effects include identity
interference and strong and Coulomb interactions. The correlation analysis in
terms of spherical harmonics is illustrated with the cases of gaussian and
blast-wave sources for proton-charged meson and baryon-baryon pairs.Comment: 32 pages 10 figure
Energy flow polynomials: A complete linear basis for jet substructure
We introduce the energy flow polynomials: a complete set of jet substructure
observables which form a discrete linear basis for all infrared- and
collinear-safe observables. Energy flow polynomials are multiparticle energy
correlators with specific angular structures that are a direct consequence of
infrared and collinear safety. We establish a powerful graph-theoretic
representation of the energy flow polynomials which allows us to design
efficient algorithms for their computation. Many common jet observables are
exact linear combinations of energy flow polynomials, and we demonstrate the
linear spanning nature of the energy flow basis by performing regression for
several common jet observables. Using linear classification with energy flow
polynomials, we achieve excellent performance on three representative jet
tagging problems: quark/gluon discrimination, boosted W tagging, and boosted
top tagging. The energy flow basis provides a systematic framework for complete
investigations of jet substructure using linear methods.Comment: 41+15 pages, 13 figures, 5 tables; v2: updated to match JHEP versio
- …