Using divergence-free and curl-free wavelets for the simulation of turbulent flows

We present a numerical method based on divergence-free and curl-free wavelets to solve the incompressible Navier-Stokes equations. We introduce a new scheme which uses anisotropic divergence-free wavelets for the decomposition of the velocity, and which only needs Fast Wavelet Transform algorithms. Numerical results finally show the feasibility of the method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

Recently, error estimates have been made available for divergence-free radial basis function (RBF) interpolants. However, these results are only valid for functions within the associated reproducing kernel Hilbert space (RKHS) of the matrix-valued RBF. Functions within the associated RKHS, also known as the ``native space' of the RBF, can be characterized as vector fields having a specific smoothness, making the native space quite small. In this paper we develop Sobolev-type error estimates when the target function is less smooth than functions in the native space.

     

Craya decomposition using compactly supported biorthogonal wavelets

We present a new local Craya–Herring decomposition of three-dimensional vector fields using compactly supported biorthogonal wavelets. Therewith vector-valued function spaces are split into two orthogonal components, i.e., curl-free and divergence-free spaces. The latter is further decomposed into toroidal and poloidal parts to decorrelate horizontal from vertical contributions which are of particular interest in geophysical turbulence. Applications are shown for isotropic, rotating and stratified turbulent flows. A comparison between isotropic and anisotropic orthogonal Craya–Herring wavelets, built in Fourier space and thus not compactly supported, is also given.     

The Neumann problem and Helmholtz decomposition in convex domains

We show that the Neumann problem for Laplace's equation in a convex domain Ω with boundary data in Lp(∂Ω) is uniquely solvable for 1<p<∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in Lp(Ω,Rd).     

Bodies of constant width in arbitrary dimension

We give a number of characterizations of bodies of constant width in arbitrary dimension. As an application, we describe a way to construct a body of constant width in dimension n, one of its (n – 1)‐dimensional projection being given. We give a number of examples, like a four‐dimensional body of constant width whose 3D‐projection is the classical Meissner's body. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Percolation of arbitrary words in one dimension

We consider a type of long‐range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and we provide partial results in others. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

On the Helmholtz decomposition in general unbounded domains

It is well known that the Helmholtz decomposition of L^q-spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C²-type in if we replace the space L^q, 1 < q < ∞, by L² ∩ L^q for q > 2 and by L^q + L² for 1 < q < 2. In this paper, we generalize this new approach from the three-dimensional case to the n-dimensional case, n ≥ 2. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform C²-type. Received: 15 February 2006     

A discrete weighted Helmholtz decomposition and its application

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted $L^2$ inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxwell system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner.     

Approximation and regularisation of arbitrary sets in finite dimension

We define the regularisation of a set by using two kernels. This procedure generalizes that defined by Laory-Lions for the functions. We privilege the geometrical point of view and we obtain in particular some new results about the asymptotic behaviour of Clarke's normal cone at a point of a set belonging to the regularized family     

Constant Q-curvature metrics in arbitrary dimension

Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order non-linear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.     

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

We exhibit the necessary range for which functions in the Sobolev spaces $L_p^s$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.     

Construction and decomposition of biorthogonal vector-valued wavelets with compact support

In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale. The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed. A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory. The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory. Three biorthogonality formulas regarding them are presented.     

Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension

We construct symmetric numerical semigroups for every minimal number of generators and multiplicity , . Furthermore we show that the set of their defining congruence is minimally generated by elements.

     

Remark on the Helmholtz decomposition in domains with noncompact boundary

Let \(\varOmega \) be a domain in \(\mathbb {R}^{d+1}\) whose boundary is given as a uniform Lipschitz graph \(x_{d+1}=\eta (x)\) for \(x \in \mathbb {R}^d\) . For such a domain, it is known that the Helmholtz decomposition is not always valid in \(L^p(\varOmega )\) except for the energy space \(L^2 (\varOmega )\) . In this paper we show that the Helmholtz decomposition still holds in certain anisotropic spaces which include vector fields decaying slowly in the \(x_{d+1}\) variable. In particular, these classes include some infinite energy vector fields. For the purpose, we develop a new approach based on a factorization of divergence form elliptic operators whose coefficients are independent of one variable.     

Two results on the equivariant Ginzburg-Landau vortex in arbitrary dimension

We characterize the O(N)-equivariant vortex solution for Ginzburg-Landau type equations in the N-dimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional.     

Statistics of energy partitions for many-particle systems in arbitrary dimension

In some previous articles, we defined several partitions of the total kinetic energy T of a system of N classical particles in ? _d into components corresponding to various modes of motion. In the present paper, we propose formulas for the mean values of these components in the normalization T = 1 (for any d and N) under the assumption that the masses of all the particles are equal. These formulas are proven at the “physical level” of rigor and numerically confirmed for planar systems (d = 2) at 3 ? N ? 100. The case where the masses of the particles are chosen at random is also considered. The paper complements our article of 2008 [Russian J. Phys. Chem. B, 2(6):947–963] where similar numerical experiments were carried out for spatial systems (d = 3) at 3 ? N ? 100.     

A gradient-projective basis of compactly supported wavelets in dimension n > 1

A given set W = {W _X } of n-variable class C ¹ functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm $\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} $ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W _x } of compactly supported class C ^2?? functions on ? ⁿ such that where X has the structure $\chi = (\tilde \chi ,\nu )$ , ν ∈ ?. W is a wavelet set in the sense that the functions indexed by $\tilde \chi $ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by $\tilde \chi $ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy — just anti-differentiate the Haar functions.)