2,951 research outputs found
Threshold functions and Poisson convergence for systems of equations in random sets
We present a unified framework to study threshold functions for the existence
of solutions to linear systems of equations in random sets which includes
arithmetic progressions, sum-free sets, -sets and Hilbert cubes. In
particular, we show that there exists a threshold function for the property
" contains a non-trivial solution of
", where is a random set and each of
its elements is chosen independently with the same probability from the
interval of integers . Our study contains a formal definition of
trivial solutions for any combinatorial structure, extending a previous
definition by Ruzsa when dealing with a single equation.
Furthermore, we study the behaviour of the distribution of the number of
non-trivial solutions at the threshold scale. We show that it converges to a
Poisson distribution whose parameter depends on the volumes of certain convex
polytopes arising from the linear system under study as well as the symmetry
inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page
Sequential Compressed Sensing
Compressed sensing allows perfect recovery of sparse signals (or signals
sparse in some basis) using only a small number of random measurements.
Existing results in compressed sensing literature have focused on
characterizing the achievable performance by bounding the number of samples
required for a given level of signal sparsity. However, using these bounds to
minimize the number of samples requires a-priori knowledge of the sparsity of
the unknown signal, or the decay structure for near-sparse signals.
Furthermore, there are some popular recovery methods for which no such bounds
are known.
In this paper, we investigate an alternative scenario where observations are
available in sequence. For any recovery method, this means that there is now a
sequence of candidate reconstructions. We propose a method to estimate the
reconstruction error directly from the samples themselves, for every candidate
in this sequence. This estimate is universal in the sense that it is based only
on the measurement ensemble, and not on the recovery method or any assumed
level of sparsity of the unknown signal. With these estimates, one can now stop
observations as soon as there is reasonable certainty of either exact or
sufficiently accurate reconstruction. They also provide a way to obtain
"run-time" guarantees for recovery methods that otherwise lack a-priori
performance bounds.
We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli)
random measurement ensembles, both for exactly sparse and general near-sparse
signals, and with both noisy and noiseless measurements.Comment: to appear in IEEE transactions on Special Topics in Signal Processin
A nonlinear vehicle-structure interaction methodology with wheel-rail detachment and reattachment
. A vehicle-structure interaction methodology with a nonlinear contact formulation
based on contact and target elements has been developed. To solve the dynamic equations of
motion, an incremental formulation has been used due to the nonlinear nature of the contact
mechanics, while a procedure based on the Lagrange multiplier method imposes the contact
constraint equations when contact occurs. The system of nonlinear equations is solved by an
efficient block factorization solver that reorders the system matrix and isolates the nonlinear
terms that belong to the contact elements or to other nonlinear elements that may be incorporated
in the model. Such procedure avoids multiple unnecessary factorizations of the linear
terms during each Newton iteration, making the formulation efficient and computationally
attractive. A numerical example has been carried out to validate the accuracy and efficiency
of the present methodology. The obtained results have shown a good agreement with the results
obtained with the commercial finite element software ANSY
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