A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces $H^s(\mathbb{R}^2)$, where $s$ is greater than the critical scaling index $s_k=1-2/k$. For $k\geq 3$ we also establish a sharp criteria to obtain global $H^1(\R^2)$ solutions. A nonlinear scattering result in $H^1(\R^2)$ is also established assuming the initial data is small and belongs to a suitable Lebesgue space

Large data scattering for the defocusing supercritical generalized KdV equation

We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u-\partial_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then the corresponding solution is global and scatters in $H^1$. Our method of proof is inspired on the compactness method introduced by C. Kenig and F. Merle.Comment: 36 page

A programmable VLSI filter architecture for application in real-time vision processing systems

An architecture is proposed for the realization of real-time edge-extraction filtering operation in an Address-Event-Representation (AER) vision system. Furthermore, the approach is valid for any 2D filtering operation as long as the convolutional kernel F(p,q) is decomposable into an x-axis and a y-axis component, i.e. F(p,q)=H(p)V(q), for some rotated coordinate system [p,q]. If it is possible to find a coordinate system [p,q], rotated with respect to the absolute coordinate system a certain angle, for which the above decomposition is possible, then the proposed architecture is able to perform the filtering operation for any angle we would like the kernel to be rotated. This is achieved by taking advantage of the AER and manipulating the addresses in real time. The proposed architecture, however, requires one approximation: the product operation between the horizontal component H(p) and vertical component V(q) should be able to be approximated by a signed minimum operation without significant performance degradation. It is shown that for edge-extraction applications this filter does not produce performance degradation. The proposed architecture is intended to be used in a complete vision system known as the Boundary-Contour-System and Feature-Contour-System Vision Model, proposed by Grossberg and collaborators. The present paper proposes the architecture, provides a circuit implementation using MOS transistors operated in weak inversion, and shows behavioral simulation results at the system level operation and electrical simulation and experimental results at the circuit level operation of some critical subcircuits

Hierarchy of inequalities for quantitative duality

We derive different relations quantifying duality in a generic two-way interferometer. These relations set different upper bounds to the visibility V of the fringes measured at the output port of the interferometer. A hierarchy of inequalities is presented which exhibits the influence of the availability to the experimenter of different sources of which-way information contributing to the total distinguishability D of the ways. For mixed states and unbalanced interferometers an inequality is derived, V^2+ Xi^2 \leq 1, which can be more stringent than the one associated with the distinguishability (V^2+ D^2 \leq 1).Comment: 7 pages, 4 figure