We study eigenfunctions φj and eigenvalues Ej of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn with piecewise smooth boundary. We bound the distance between an arbitrary parameter E > 0 and the spectrum {Ej} in terms of the boundary L2-norm of a normalized trial solution u of the Helmholtz equation (∆ + E)u = 0. We also bound the L2-norm of the error of this trial solution from an eigenfun...

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation’s wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth ...

Aiming at properties of remote sensing image data such as high-dimension, nonlinearity and massive unlabeled samples, a kind of probability least squares support vector machine (PLSSVM) classification method based on hybrid entropy and L1 norm was proposed. Firstly, hybrid entropy was designed by combining quasi-entropy with entropy difference, which was used to select the most “valuable” sampl...

Aiming at properties of remote sensing image data such as high-dimension, nonlinearity and massive unlabeled samples, a kind of probability least squares support vector machine (PLSSVM) classification method based on hybrid entropy and L1 norm was proposed. Firstly, hybrid entropy was designed by combining quasi-entropy with entropy difference, which was used to select the most “valuable” sampl...

Alexandre Tsybakov (Paris VI, France) Estimation of high-dimensional low rank matrices Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m× T -matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the number mT of unknown entries can be much larger than the sample size N . Motivated by several application...

A straightforward inversion scheme (SIS) has been developed to interpret vertical electrical sounding data. This scheme does not require quasi-linearization of the inverse resistivity problem and thereby dispenses with the iterative process and the necessity of guessing the number of layers and their resistivities and thicknesses. The entire solution domain is divided into uniform thickness lay...

The inverse problem of determining the unknown coefficient of the non-linear differential equation of torsional creep is studied. The unknown coefficient g = g(ξ2) depends on the gradient ξ := |∇u| of the solution u(x), x ∈ Ω ⊂ Rn , of the direct problem. It is proved that this gradient is bounded in C-norm. This permits one to choose the natural class of admissible coefficients for the conside...

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation’s wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth ...

The minimization problem of an L2-sensitivity measure subject to L2-norm dynamic-range scaling constraints is formulated for a class of two-dimensional (2-D) state-space digital filters. First, the problem is converted into an unconstrained optimization problem by using linear-algebraic techniques. Next, the unconstrained optimization problem is solved by applying an efficient quasi-Newton algo...

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.