1,403 research outputs found
Extended matter coupled to BF theory
Recently, a topological field theory of membrane-matter coupled to BF theory
in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss
various aspects of the four-dimensional theory. Firstly, we study classical
solutions leading to an interpretation of the theory in terms of strings
propagating on a flat spacetime. We also show that the general classical
solutions of the theory are in one-to-one correspondence with solutions of
Einstein's equations in the presence of distributional matter (cosmic strings).
Secondly, we quantize the theory and present, in particular, a prescription to
regularize the physical inner product of the canonical theory. We show how the
resulting transition amplitudes are dual to evaluations of Feynman diagrams
coupled to three-dimensional quantum gravity. Finally, we remove the regulator
by proving the topological invariance of the transition amplitudes.Comment: 27 pages, 7 figure
Navigability is a Robust Property
The Small World phenomenon has inspired researchers across a number of
fields. A breakthrough in its understanding was made by Kleinberg who
introduced Rank Based Augmentation (RBA): add to each vertex independently an
arc to a random destination selected from a carefully crafted probability
distribution. Kleinberg proved that RBA makes many networks navigable, i.e., it
allows greedy routing to successfully deliver messages between any two vertices
in a polylogarithmic number of steps. We prove that navigability is an inherent
property of many random networks, arising without coordination, or even
independence assumptions
Reducible quasi-periodic solutions for the Non Linear Schr\"odinger equation
The present paper is devoted to the construction of small reducible
quasi--periodic solutions for the completely resonant NLS equations on a
--dimensional torus \T^d. The main point is to prove that prove that the
normal form is reducible, block diagonal and satisfies the second Melnikov
condition block wise. From this we deduce the result by a KAM algorithm.Comment: 48 page
An Algorithmic Theory of Integer Programming
We study the general integer programming problem where the number of
variables is a variable part of the input. We consider two natural
parameters of the constraint matrix : its numeric measure and its
sparsity measure . We show that integer programming can be solved in time
, where is some computable function of the
parameters and , and is the binary encoding length of the input. In
particular, integer programming is fixed-parameter tractable parameterized by
and , and is solvable in polynomial time for every fixed and .
Our results also extend to nonlinear separable convex objective functions.
Moreover, for linear objectives, we derive a strongly-polynomial algorithm,
that is, with running time , independent of the rest of
the input data.
We obtain these results by developing an algorithmic framework based on the
idea of iterative augmentation: starting from an initial feasible solution, we
show how to quickly find augmenting steps which rapidly converge to an optimum.
A central notion in this framework is the Graver basis of the matrix , which
constitutes a set of fundamental augmenting steps. The iterative augmentation
idea is then enhanced via the use of other techniques such as new and improved
bounds on the Graver basis, rapid solution of integer programs with bounded
variables, proximity theorems and a new proximity-scaling algorithm, the notion
of a reduced objective function, and others.
As a consequence of our work, we advance the state of the art of solving
block-structured integer programs. In particular, we develop near-linear time
algorithms for -fold, tree-fold, and -stage stochastic integer programs.
We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified
proximity-scaling algorith
Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
We introduce "puzzles of quasi-finite type" which are the counterparts of our
subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of
combinatorial puzzles as defined in complex dynamics. We are able to analyze
these dynamics defined by entropy conditions rather completely, obtaining a
complete classification with respect to large entropy measures and a
description of their measures with maximum entropy and periodic orbits. These
results can in particular be applied to entropy-expanding maps like
(x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the
meromorphy of the Artin-Mazur zeta function on a large disk. This follows from
a similar new result about strongly positively recurrent Markov shifts where
the radius of meromorphy is lower bounded by an "entropy at infinity" of the
graph.Comment: accepted by Annales de l'Institut Fourier, final revised versio
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