26 research outputs found
Stationary processes whose filtrations are standard
We study the standard property of the natural filtration associated to a 0--1
valued stationary process. In our main result we show that if the process has
summable memory decay, then the associated filtration is standard. We prove it
by coupling techniques. For a process whose associated filtration is standard,
we construct a product type filtration extending it, based upon the usual
couplings and the Vershik's criterion for standardness.Comment: Published at http://dx.doi.org/10.1214/009117906000000151 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A strong pair correlation bound implies the CLT for Sinai Billiards
For Dynamical Systems, a strong bound on multiple correlations implies the
Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is
derived for dynamically Holder continuous observables of dispersing Billiards.
Here we weaken the regularity assumption and subsequently show that the bound
on multiple correlations follows directly from the bound on pair correlations.
Thus, a strong bound on pair correlations alone implies the CLT, for a wider
class of observables. The result is extended to Anosov diffeomorphisms in any
dimension.Comment: 13 page
Ruelle-Perron-Frobenius spectrum for Anosov maps
We extend a number of results from one dimensional dynamics based on spectral
properties of the Ruelle-Perron-Frobenius transfer operator to Anosov
diffeomorphisms on compact manifolds. This allows to develop a direct operator
approach to study ergodic properties of these maps. In particular, we show that
it is possible to define Banach spaces on which the transfer operator is
quasicompact. (Information on the existence of an SRB measure, its smoothness
properties and statistical properties readily follow from such a result.) In
dimension we show that the transfer operator associated to smooth random
perturbations of the map is close, in a proper sense, to the unperturbed
transfer operator. This allows to obtain easily very strong spectral stability
results, which in turn imply spectral stability results for smooth
deterministic perturbations as well. Finally, we are able to implement an Ulam
type finite rank approximation scheme thus reducing the study of the spectral
properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe
Duality Theorems in Ergodic Transport
We analyze several problems of Optimal Transport Theory in the setting of
Ergodic Theory. In a certain class of problems we consider questions in Ergodic
Transport which are generalizations of the ones in Ergodic Optimization.
Another class of problems is the following: suppose is the shift
acting on Bernoulli space , and, consider a fixed
continuous cost function . Denote by the set
of all Borel probabilities on , such that, both its and
marginal are -invariant probabilities. We are interested in the
optimal plan which minimizes among the probabilities on
.
We show, among other things, the analogous Kantorovich Duality Theorem. We
also analyze uniqueness of the optimal plan under generic assumptions on .
We investigate the existence of a dual pair of Lipschitz functions which
realizes the present dual Kantorovich problem under the assumption that the
cost is Lipschitz continuous. For continuous costs the corresponding
results in the Classical Transport Theory and in Ergodic Transport Theory can
be, eventually, different.
We also consider the problem of approximating the optimal plan by
convex combinations of plans such that the support projects in periodic orbits
A discrete time neural network model with spiking neurons II. Dynamics with noise
We provide rigorous and exact results characterizing the statistics of spike
trains in a network of leaky integrate and fire neurons, where time is discrete
and where neurons are submitted to noise, without restriction on the synaptic
weights. We show the existence and uniqueness of an invariant measure of Gibbs
type and discuss its properties. We also discuss Markovian approximations and
relate them to the approaches currently used in computational neuroscience to
analyse experimental spike trains statistics.Comment: 43 pages - revised version - to appear il Journal of Mathematical
Biolog
Factoring Products of Braids via Garside Normal Form
Braid groups are infinite non-abelian groups naturally arising from geometric braids. For two decades they have been proposed for cryptographic use. In braid group cryptography public braids often contain secret braids as factors and it is hoped that rewriting the product of braid words hides individual factors. We provide experimental evidence that this is in general not the case and argue that under certain conditions parts of the Garside normal form of factors can be found in the Garside normal form of their product. This observation can be exploited to decompose products of braids of the form ABC when only B is known. Our decomposition algorithm yields a universal forgery attack on WalnutDSA™, which is one of the 20 proposed signature schemes that are being considered by NIST for standardization of quantum-resistant public-key cryptography. Our attack on WalnutDSA™ can universally forge signatures within seconds for both the 128-bit and 256-bit security level, given one random message-signature pair. The attack worked on 99.8% and 100% of signatures for the 128-bit and 256-bit security levels in our experiments. Furthermore, we show that the decomposition algorithm can be used to solve instances of the conjugacy search problem and decomposition search problem in braid groups. These problems are at the heart of other cryptographic schemes based on braid groups.SCOPUS: cp.kinfo:eu-repo/semantics/published22nd IACR International Conference on Practice and Theory of Public-Key Cryptography, PKC 2019; Beijing; China; 14 April 2019 through 17 April 2019ISBN: 978-303017258-9Volume Editors: Sako K.Lin D.Publisher: Springer Verla