This paper deals with the Helmholtz-Hodge decomposition of a vector field in bounded domain. We present a practical algorithm to compute this decomposition in the context of divergence-free and curl-free wavelets satisfying suitable boundary conditions. The method requires the inversion of divergence-free and curl-free wavelet Gram matrices. We propose an optimal preconditioning which allows to...

A lot of machine learning algorithms are based on metric functions, which good functions lead to better results. Distance metric learning has been widely attracted by researchers in last decade. Kernel matrix is somehow a distance function which indicates the similarity between two instances in the feature space which contains high dimensions. Traditional distance metric learning approaches are...

Beside simplices, n-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n-cubes are not classified. In this work, we first show that there are no Coxeter n-cubes in H for n ≥ 10. Then, we show that the ideal ones exist only for n = 2 and 3, and provide a classification. The methods used are of combinatorial and algebraic nature, using prope...

In this paper we analyze the relationships between the eigenvalues of the m × m Gram matrix K for a kernel k(·, ·) corresponding to a sample x1, . . . ,xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem. We bound the differences between the two spectra and provide a performance bound on kernel PCA.

Fractional systems with Riemann-Liouville derivatives are considered. The initial memory value problem is posed and studied. We obtain explicit steering laws with respect to the values of the fractional integrals of the state variables. The Gramian is generalized and steering functions between memory values are characterized.

We consider the frame property of the Gabor system G(g, α, β) = {eg(t − αm) : m,n ∈ Z} for the case of rational oversampling, i.e. α, β ∈ Q. A ’rational’ analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, α, β) does not generate a frame if αβ = n−1 n . Special attention is paid to the first Hermite function h1(t) = te −πt .

Given a function ψ in L2(Rd), the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions {| det a|j/2ψ(ajx−γ) : j ∈ Z, γ ∈ Γ}. In this article we prove that the set of functions generating affine systems that are a Riesz basis of L2(Rd) is dense in L2(Rd). We also prove that a stronger result is true for affine systems that a...

In this paper, we construct Grammian-like quasideterminant solutions of a non-Abelian Hirota–Miwa equation. Through continuum limits of this non-Abelian Hirota–Miwa equation and its quasideterminant solutions, we construct a cascade of noncommutative differential-difference equations ending with the non-commutative KP equation. For each of these systems, the quasideterminant solutions are const...

The vorticity of a vector field on 3-dimensional Euclidean space is usually given by the curl field. In this paper, we extend concept to n-dimensional compact and oriented Riemannian manifold. We analyse many properties operation. prove that manifold admits unique Helmholtz decomposition establish every smooth an open neighbourhood Stokes’ type identity.