3,313 research outputs found
(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces
We develop a kind of pregeometry consisting of a web of overlapping fuzzy
lumps which interact with each other. The individual lumps are understood as
certain closely entangled subgraphs (cliques) in a dynamically evolving network
which, in a certain approximation, can be visualized as a time-dependent random
graph. This strand of ideas is merged with another one, deriving from ideas,
developed some time ago by Menger et al, that is, the concept of probabilistic-
or random metric spaces, representing a natural extension of the metrical
continuum into a more microscopic regime. It is our general goal to find a
better adapted geometric environment for the description of microphysics. In
this sense one may it also view as a dynamical randomisation of the causal-set
framework developed by e.g. Sorkin et al. In doing this we incorporate, as a
perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor
changes added relating to previous wor
Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type
We prove the existence of time-periodic, small amplitude solutions of
autonomous quasilinear or fully nonlinear completely resonant pseudo-PDEs of
Benjamin-Ono type in Sobolev class. The result holds for frequencies in a
Cantor set that has asymptotically full measure as the amplitude goes to zero.
At the first order of amplitude, the solutions are the superposition of an
arbitrarily large number of waves that travel with different velocities
(multimodal solutions). The equation can be considered as a Hamiltonian,
reversible system plus a non-Hamiltonian (but still reversible) perturbation
that contains derivatives of the highest order. The main difficulties of the
problem are: an infinite-dimensional bifurcation equation, and small divisors
in the linearized operator, where also the highest order derivatives have
nonconstant coefficients. The main technical step of the proof is the reduction
of the linearized operator to constant coefficients up to a regularizing rest,
by means of changes of variables and conjugation with simple linear
pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov
and Toland for standing water waves (ARMA 2005). Other ingredients are a
suitable Nash-Moser iteration in Sobolev spaces, and Lyapunov-Schmidt
decomposition.
(Version 2: small change in Section 2).Comment: 47 page
Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods
The resolution of many large-scale inverse problems using MCMC methods
requires a step of drawing samples from a high dimensional Gaussian
distribution. While direct Gaussian sampling techniques, such as those based on
Cholesky factorization, induce an excessive numerical complexity and memory
requirement, sequential coordinate sampling methods present a low rate of
convergence. Based on the reversible jump Markov chain framework, this paper
proposes an efficient Gaussian sampling algorithm having a reduced computation
cost and memory usage. The main feature of the algorithm is to perform an
approximate resolution of a linear system with a truncation level adjusted
using a self-tuning adaptive scheme allowing to achieve the minimal computation
cost. The connection between this algorithm and some existing strategies is
discussed and its efficiency is illustrated on a linear inverse problem of
image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio
Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion
We consider non-reversible perturbations of reversible diffusions that do not
alter the invariant distribution and we ask whether there exists an optimal
perturbation such that the rate of convergence to equilibrium is maximized. We
solve this problem for the case of linear drift by proving the existence of
such optimal perturbations and by providing an easily implementable algorithm
for constructing them. We discuss in particular the role of the prefactor in
the exponential convergence estimate. Our rigorous results are illustrated by
numerical experiments
A Numerical Analyst Looks at the "Cutoff Phenomenon" in Card Shuffling and Other Markov Chains
Diaconis and others have shown that certain Markov chains exhibit a "cutoff phenomenon" in which, after an initial period of seemingly little progress, convergence to the steady state occurs suddenly. Since Markov chains are just powers of matrices, how can such effects be explained in the language of applied linear algebra? We attempt to do this, focusing on two examples: random walk on a hypercube, which is essentially the same as the problem of Ehrenfest urns, and the celebrated case of riffle shuffling of a deck of cards. As is typical with transient phenomena in matrix processes, the reason for the cutoff is not readily apparent from an examination of eigenvalues or eigenvectors, but it is reflected strongly in pseudosprectra - provided they are measured in the 1-norm, not the 2-norm. We illustrate and explain the cutoff phenomenon with Matlab computations based in part on a new explicit formula for the entries of the "riffle shuffle matrix", and note that while the normwise cutoff may occur at one point, such as for the riffle shuffle, weak convergence may occur at an equally precise earlier point such as
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