‎We present a Picone's identity for the‎ ‎$mathcal{A}_{p(x)}$-Laplacian‎, ‎which is an extension of the classic‎ ‎identity for the ordinary Laplace‎. ‎Also‎, ‎some applications of our‎ ‎results in Sobolev spaces with variable exponent are suggested.

This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux [11] and Carlen and Loss [10] for logarithmic Sobolev inequalities. Under local conditi...

where ' = ('1; : : : ; 'r) T is the unknown, M is an s£s dilation matrix with m = jdetM j, g = (g1; : : : ; gr) is a given compactly supported vector-valued functions on IR and a is a ̄nitely supported re ̄nement mask such that each a(®) is an r£ r (complex) matrix. In this paper, we characterize the optimal smoothness of a multiple re ̄nable function associated with homogeneous re ̄nement equation...

The problem of multi-target tracking of deforming objects in video sequences arises in many situations in image processing and computer vision. Many algorithms based on finite dimensional particle filters have been proposed. Recently, particle filters for infinite dimensional Shape Spaces have been proposed although predictions are restricted to a low dimensional subspace. We try to extend this...

Characterizations of the stability and orthonormality of a multivariate matrix refinable function Φ with arbitrary matrix dilation M are provided in terms of the eigenvalue and 1-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of Φ is equivalent to the order of the vanishing moment conditions of the matrix refinement ...

Quantum constraints of the typeQ|ψphys〉 = 0 can be straightforwardly implemented in cases where Q is a self-adjoint operator for which zero is an eigenvalue. In that case, the physical Hilbert space is obtained by projecting onto the kernel of Q, i.e. Hphys = ker Q = ker Q . It is, however, nontrivial to identify and project onto Hphys when zero is not in the point spectrum but instead is in th...

We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on RN . In particular, inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space of fractional order B(2 − 2 N , N) and, furthermore, that all continuity results in this s...

We define a Walsh space which contains all functions whose partial mixed derivatives up to order δ ≥ 1 exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain reproducing kernel Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital (t, α, s)-sequences achieve the ...

We consider fourth order singularly perturbed boundary value problems (BVPs) in one-dimension and the approximation of their solution by the hp version of the Finite Element Method (FEM). If the given problem’s boundary conditions are suitable for writing the BVP as a second order system, then we construct an hp FEM on the so-called Spectral Boundary Layer Mesh that gives a robust approximation...

where A is some set of multiindices α. A first crucial question is whether there is enough data to (uniquely) find f . Let u0 be a function in the Sobolev space H(Ω) with zero Cauchy data u0 = ∂νu0 = 0 on Γ0 and let f0 = −∆u0. Due to linearity, −∆(u+ u0) = f + f0. Obviously, u and u+ u0 have the same Cauchy data on Γ0, so f and f+f0 produce the same data (2), but they are different in Ω. It is ...