Space-modulated Stability and Averaged Dynamics

In this brief note we give a brief overview of the comprehensive theory, recently obtained by the author jointly with Johnson, Noble and Zumbrun, that describes the nonlinear dynamics about spectrally stable periodic waves of parabolic systems and announce parallel results for the linearized dynamics near cnoidal waves of the Korteweg-de Vries equation. The latter are expected to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees partielles", Roscoff 201

Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation

We provide a detailed study of the dynamics obtained by linearizing the Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability, we prove global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. Furthermore, we provide both a leading-order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters. This requires a global-in-time study of the dynamics generated by a non normal operator with non constant coefficients. On the road we also prove estimates on oscillatory integrals particularly suitable to derive large-time asymptotic systems that could be of some general interest

PSI-20 and global indexes stock market efficiency

This paper is an abstract from my Master degree in Finance. The dissertation discusses the hypothesis that world financial markets indexes are efficient in their weak form.Random Walk I, II, III, Martingale, Efficiency, variance ratios, Arch and Garch.

Regression with R

This document aims to explain how to use R matrix capacity in the context of regression analysis.R, Matrices, Regression

Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow

A technical obstruction preventing the conclusion of nonlinear stability of large-Froude number roll waves of the St. Venant equations for inclined thin film flow is the "slope condition" of Johnson-Noble-Zumbrun, used to obtain pointwise symmetrizability of the linearized equations and thereby high-frequency resolvent bounds and a crucial H s nonlinear damping estimate. Numerically, this condition is seen to hold for Froude numbers 2 \textless{} F 3.5, but to fail for 3.5 F. As hydraulic engineering applications typically involve Froude number 3 F 5, this issue is indeed relevant to practical considerations. Here, we show that the pointwise slope condition can be replaced by an averaged version which holds always, thereby completing the nonlinear theory in the large-F case. The analysis has potentially larger interest as an extension to the periodic case of a type of weighted "Kawashima-type" damping estimate introduced in the asymptotically-constant coefficient case for the study of stability of large-amplitude viscous shock waves

Whitham's equations for modulated roll-waves in shallow flows

This paper is concerned with the detailed behaviour of roll-waves undergoing a low-frequency perturbation. We rst derive the so-called Whitham's averaged modulation equations and relate the well-posedness of this set of equations to the spectral stability problem in the small Floquet-number limit. We then fully validate such a system and in particular, we are able to construct solutions to the shallow water equations in the neighbourhood of modulated roll-waves proles that exist for asymptotically large time

Multi-modal Image Processing based on Coupled Dictionary Learning

In real-world scenarios, many data processing problems often involve heterogeneous images associated with different imaging modalities. Since these multimodal images originate from the same phenomenon, it is realistic to assume that they share common attributes or characteristics. In this paper, we propose a multi-modal image processing framework based on coupled dictionary learning to capture similarities and disparities between different image modalities. In particular, our framework can capture favorable structure similarities across different image modalities such as edges, corners, and other elementary primitives in a learned sparse transform domain, instead of the original pixel domain, that can be used to improve a number of image processing tasks such as denoising, inpainting, or super-resolution. Practical experiments demonstrate that incorporating multimodal information using our framework brings notable benefits.Comment: SPAWC 2018, 19th IEEE International Workshop On Signal Processing Advances In Wireless Communication

Generalized Bregman Divergence and Gradient of Mutual Information for Vector Poisson Channels

We investigate connections between information-theoretic and estimation-theoretic quantities in vector Poisson channel models. In particular, we generalize the gradient of mutual information with respect to key system parameters from the scalar to the vector Poisson channel model. We also propose, as another contribution, a generalization of the classical Bregman divergence that offers a means to encapsulate under a unifying framework the gradient of mutual information results for scalar and vector Poisson and Gaussian channel models. The so-called generalized Bregman divergence is also shown to exhibit various properties akin to the properties of the classical version. The vector Poisson channel model is drawing considerable attention in view of its application in various domains: as an example, the availability of the gradient of mutual information can be used in conjunction with gradient descent methods to effect compressive-sensing projection designs in emerging X-ray and document classification applications