17,166 research outputs found
Effective Genericity and Differentiability
We prove that a real x is 1-generic if and only if every differentiable
computable function has continuous derivative at x. This provides a counterpart
to recent results connecting effective notions of randomness with
differentiability. We also consider multiply differentiable computable
functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction
A differentiable BLEU loss. Analysis and first results
In natural language generation tasks, like neural machine translation and image captioning, there is usually a mismatch between the optimized loss and the de facto evaluation criterion, namely token-level maximum likelihood and corpus-level BLEU score. This article tries to reduce this gap by defining differentiable computations of the BLEU and GLEU scores. We test this approach on simple tasks, obtaining valuable lessons on its potential applications but also its pitfalls, mainly that these loss functions push each token in the hypothesis sequence toward the average of the tokens in the reference, resulting in a poor training signal.Peer ReviewedPostprint (published version
Inverse velocity statistics in two dimensional turbulence
We present a numerical study of two-dimensional turbulent flows in the
enstrophy cascade regime, with different large-scale forcings and energy sinks.
In particular, we study the statistics of more-than-differentiable velocity
fluctuations by means of two recently introduced sets of statistical
estimators, namely {\it inverse statistics} and {\it second order differences}.
We show that the 2D turbulent velocity field, , cannot be simply
characterized by its spectrum behavior, . There
exists a whole set of exponents associated to the non-trivial smooth
fluctuations of the velocity field at all scales. We also present a numerical
investigation of the temporal properties of measured in different
spatial locations.Comment: 9 pages, 12 figure
The variation of invariant graphs in forced systems
In skew-product systems with contractive factors, all orbits asymptotically
approach the graph of the so-called sync function; hence, the corresponding
regularity properties primarily matter. In the literature, sync function
Lipschitz continuity and differentiability have been proved to hold depending
on the derivative of the base reciprocal, if not on its Lyapunov exponent.
However, forcing topological features can also impact the sync function
regularity. Here, we estimate the total variation of sync functions generated
by one-dimensional Markov maps. A sharp condition for bounded variation is
obtained depending on parameters, that involves the Markov map topological
entropy. The results are illustrated with examples
The dynamics of pseudographs in convex Hamiltonian systems
We study the evolution, under convex Hamiltonian flows on cotangent bundles
of compact manifolds, of certain distinguished subsets of the phase space.
These subsets are generalizations of Lagrangian graphs, we call them
pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By
this method, we find various orbits which connect prescribed regions of the
phase space. Our study is inspired by works of John Mather. As an application,
we obtain the existence of diffusion in a large class of a priori unstable
systems and provide a solution to the large gap problem. We hope that our
method will have applications to more examples
Fractional Calculus as a Macroscopic Manifestation of Randomness
We generalize the method of Van Hove so as to deal with the case of
non-ordinary statistical mechanics, that being phenomena with no time-scale
separation. We show that in the case of ordinary statistical mechanics, even if
the adoption of the Van Hove method imposes randomness upon Hamiltonian
dynamics, the resulting statistical process is described using normal calculus
techniques. On the other hand, in the case where there is no time-scale
separation, this generalized version of Van Hove's method not only imposes
randomness upon the microscopic dynamics, but it also transmits randomness to
the macroscopic level. As a result, the correct description of macroscopic
dynamics has to be expressed in terms of the fractional calculus.Comment: 20 pages, 1 figur
On Resonance in Periodically Forced Oscillators and Coupled Systems of Excitable Systems and Nonlinear Oscillators
We analyze some mathematical problems that arise in studies of phenomena observed in the cardiac action. We illustrate a method to characterize the response of a nonlinear oscillator to an external forcing, and introduce some numerical results. We also introduce some results of numerical computation in an example of a coupled system of an excitable system and a nonlinear oscillator
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