The NLS approximation for two dimensional deep gravity waves

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr\"odinger equation (NLS) on the natural cubic time scale.Comment: 23 page

Global bounds for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension

This article is concerned with the small data problem for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in $H^{0,1}$. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness.Comment: 15 pages. We fixed the proof of Lemma 2.

Bounded Coordinate-Descent for Biological Sequence Classification in High Dimensional Predictor Space

We present a framework for discriminative sequence classification where the learner works directly in the high dimensional predictor space of all subsequences in the training set. This is possible by employing a new coordinate-descent algorithm coupled with bounding the magnitude of the gradient for selecting discriminative subsequences fast. We characterize the loss functions for which our generic learning algorithm can be applied and present concrete implementations for logistic regression (binomial log-likelihood loss) and support vector machines (squared hinge loss). Application of our algorithm to protein remote homology detection and remote fold recognition results in performance comparable to that of state-of-the-art methods (e.g., kernel support vector machines). Unlike state-of-the-art classifiers, the resulting classification models are simply lists of weighted discriminative subsequences and can thus be interpreted and related to the biological problem

Two dimensional water waves in holomorphic coordinates II: global solutions

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small localized data leads to global solutions. This article is a continuation of authors' earlier paper arXiv:1401.1252.Comment: 21 pages. We have updated the authors' inf

The lifespan of small data solutions in two dimensional capillary water waves

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.Comment: Typos corrected, references updated, final versio

Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan

This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.Comment: 64 page

Enhanced Lifespan of Smooth Solutions of a Burgers-Hilbert Equation

We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.Comment: 13 pages, 1 figur

Some Reflections on the Liability of States for International Illicit Acts

The authors aim to foray in the institution of international responsibility, addressing constituent issues of international responsibility. Accountability is the general international law whichestablishes a legal relationship exclusively between two or more subjects of international law. International responsibility of a state can only be driven by another subject of international law whose international subjective right he violated (breach). If a Member suffers an injury directly and immediately, he may apply directly responsible for such State to obtain reparation. Rather, the injury suffered by a particular breach (violation) of international law does not provide, thereby, as a victim toobtain redress in international courts. A state - and can attract international responsibility only if the author of an international fact illegal. There is a wrongful act of the state, where: a) conduct consisting of an action or omission may be attributed (imputed), in accordance with international law, rule, and b)that conduct constitutes a breach of an international obligation of the State. The doctrine generally recognized international scope of these two elements gives rise to international responsibility of thestate

THE EUROPEAN BUSINESS CYCLE

The construction of European Monetary Union has raised several questions about the existence of a common business cycle, a European one. The lack of cyclical synchronization would complicate the monetary and fiscal policies in the Union, being a negativeEuropean business cycle, correlation, synchronization of business cycles