The Gray-code filter kernels

Abstract In this paper we introduce a family of filter kernels -the Gray-Code Kernels (GCK) and demonstrate their use in image analysis. Filtering an image with a sequence of Gray-Code Kernels is highly efficient and requires only 2 operations per pixel for each filter kernel, independent of the size or dimension of the kernel. We show that the family of kernels is large and includes the Walsh-Hadamard kernels amongst others. The GCK can be used to approximate any desired kernel and as such forms a complete representation. The efficiency of computation using a sequence of GCK filters can be exploited for various real-time applications, such as, pattern detection, feature extraction, texture analysis, texture synthesis, and more

Global well-posedness of the KP-I initial-value problem in the energy space

We prove that the KP-I initial value problem is globally well-posed in the natural energy space of the equation

Asymptotic Lower Bounds for a class of Schroedinger Equations

We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - \Delta u + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.Comment: 24 pages. to appear on Comm. Math. Phy

Structural resolvent estimates and derivative nonlinear Schrodinger equations

A refinement of uniform resolvent estimate is given and several smoothing estimates for Schrodinger equations in the critical case are induced from it. The relation between this resolvent estimate and radiation condition is discussed. As an application of critical smoothing estimates, we show a global existence results for derivative nonlinear Schrodinger equations.Comment: 21 page

A class of Schrodinger operators with decaying oscillatory potentials

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous spectrum and give bounds on the Hausdorff dimension of the singular part of the spectral measure. We also discuss the analogs for orthogonal polynomials on the real line and unit circle.Comment: 18 page

Quantum Tomography under Prior Information

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d-4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log_2(d) smaller than this upper bound.Comment: v3: There was a mistake in the derived finer upper bound in Theorem 3. The corrected upper bound is +1 to the earlier versio

Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption

International audienceThe behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption ∂_t u − ∆_p u + |∇u|^{p−1} = 0 in (0, ∞) × R^N , and fast diffusion 2N/(N + 1) < p < 2. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as |x| → ∞, it is shown that the corresponding solution u to the above equation approaches a uniquely determined separate variable solution of the form U (t, x) = (T_e − t)^{1/(2−p)} f_* (|x|), (t, x) ∈ (0, T_e) × R^N , as t → T_e , where T_e denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile f_* is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional

Encoding order and developmental dyslexia:a family of skills predicting different orthographic components

We investigated order encoding in developmental dyslexia using a task that presented nonalphanumeric visual characters either simultaneously or sequentially—to tap spatial and temporal order encoding, respectively—and asked participants to reproduce their order. Dyslexic participants performed poorly in the sequential condition, but normally in the simultaneous condition, except for positions most susceptible to interference. These results are novel in demonstrating a selective difficulty with temporal order encoding in a dyslexic group. We also tested the associations between our order reconstruction tasks and: (a) lexical learning and phonological tasks; and (b) different reading and spelling tasks. Correlations were extensive when the whole group of participants was considered together. When dyslexics and controls were considered separately, different patterns of association emerged between orthographic tasks on the one side and tasks tapping order encoding, phonological processing, and written learning on the other. These results indicate that different skills support different aspects of orthographic processing and are impaired to different degrees in individuals with dyslexia. Therefore, developmental dyslexia is not caused by a single impairment, but by a family of deficits loosely related to difficulties with order. Understanding the contribution of these different deficits will be crucial to deepen our understanding of this disorder