Stationary binary subdivision schemes using radial basis function interpolation

A new family of interpolatory stationary subdivision schemes is introduced by using radial basis function interpolation. This work extends earlier studies on interpolatory stationary subdivision schemes in two aspects. First, it provides a wider class of interpolatory schemes; each 2L-point interpolatory scheme has the freedom of choosing a degree (say, m) of polynomial reproducing. Depending on the combination (2L,m), the proposed scheme suggests different subdivision rules. Second, the scheme turns out to be a 2L-point interpolatory scheme with a tension parameter. The conditions for convergence and smoothness are also studied. Dedicated to Prof. Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 41A05, 41A25, 41A30, 65D10, 65D17. Byung-Gook Lee: This work was done as a part of Information & Communication fundamental Technology Research Program supported by Ministry of the Information & Communication in Republic of Korea. Jungho Yoon: Corresponding author. Supported by the Korea Science and Engineering Foundation grant (KOSEF R06-2002-012-01001).     

A family of non-oscillatory 6-point interpolatory subdivision schemes

In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Power _p schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Power _p schemes and it is based on a weighted analog of the Power _p mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Power _p schemes.     

Convergence of subdivision schemes and smoothness of limit functions

Starting with vector λ=(λ(k))_k∈Z∈?p(Z), the subdivision scheme generates a sequence of vectors by the subdivision operator

     

Conditionally positive definite kernels on Euclidean domains

This paper characterizes several classes of conditionally positive definite kernels on a domain Ω of either or . Among the classes is that composed of strictly conditionally positive definite kernels. These kernels are known to be useful in the solution of variational interpolation problems on Ω. Our study covers the case in which Ω is the sphere S^l−1 of or a similar manifold. Among other things, our results imply that the characterization of (strict) conditional positive definiteness on Ω can be obtained from a characterization of (strict) positive definiteness on Ω. The bi-zonal strictly conditionally positive definite kernels on S^l−1, l?3, are described.     

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

Estimates of the coefficients of the Jacobi expansion by measures of smoothness

For expansion by Jacobi polynomials we relate smoothness given by appropriate K-functionals in Lp, 1?p?2, to estimates on the coefficients in the ?q form. As a corollary for 1<p?2, and an the coefficients of the Legendre expansion of f∈Lp[−1,1], we obtain the estimate

     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes.To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice L we describe a formal deformation of the barycentric subdivision of the atom crosscut complex Γ(L) to its order complex . We proceed by proving that the complex of sets bounded from below J(L) can also be collapsed to .Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph G, the neighborhood complex N(G) and the polyhedral complex Hom(K₂,G) have the same simple homotopy type in the sense of Whitehead.     

Several remarks on dimensions modulo ANR-compacta

We investigate a dimension function L-dim (L is a class of ANR-compacta). Main results are as follows.Let L be an ANR-compactum.(1) If L*L is not contractible, then for every n?0 there is a cube Im with .(2) If L is simply connected and f:X→Y is an acyclic mapping from a finite-dimensional compact Hausdorff space X onto a finite-dimensional space Y, then .(3) If L is simply connected and L*L is not contractible, then for every n?2 there exists a compact Hausdorff space such that , and for an arbitrary closed set either or .     

On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary

We study the differentiability of very weak solutions v∈L¹(Ω) of ₀(v,L^?φ)=₀(f,φ) for all vanishing at the boundary whenever f is in L¹(Ω,δ), with δ=dist(x,∂Ω), and L^* is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function of f.     

Basic deformation theory of smooth formal schemes

We provide the main results of a deformation theory of smooth formal schemes as defined in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Algebra 35 (2007) 1341-1367]. Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext¹ group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f₀:X₀→Y₀ of noetherian formal schemes and a closed immersion Y₀?Y given by a square zero ideal I, we prove that the set of isomorphism classes of smooth formal schemes lifting X₀ over Y is classified by and that there exists an element in which vanishes if and only if there exists a smooth formal scheme lifting X₀ over Y.     

On the intrinsic and the spatial numerical range

For a bounded function f from the unit sphere of a closed subspace X of a Banach space Y, we study when the closed convex hull of its spatial numerical range W(f) is equal to its intrinsic numerical range V(f). We show that for every infinite-dimensional Banach space X there is a superspace Y and a bounded linear operator such that . We also show that, up to renormig, for every non-reflexive Banach space Y, one can find a closed subspace X and a bounded linear operator T∈L(X,Y) such that .Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobás property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property.     

Riesz transforms associated to Schrödinger operators on weighted Hardy spaces

Let w be some Ap weight and enjoy reverse Hölder inequality, and let L=−Δ+V be a Schrödinger operator on Rn, where is a non-negative function on Rn. In this article we introduce weighted Hardy spaces associated to L in terms of the area function characterization, and prove their atomic characters. We show that the Riesz transform ∇L−1/2 associated to L is bounded on for 1<p<2, and bounded from to the classical weighted Hardy space .     

A new cross subdivision scheme for surface design

In this paper, a new primal approximation -subdivision scheme for surface design, being named the cross scheme or C-scheme for short, is presented. The C²-continuity of the C-scheme is mathematically analyzed. The new scheme can be effectively applied to any 3D mesh of quadrilaterals. Extraordinary vertices and miscellaneous boundary scenarios are handled as well. Extensive implementation shows that it performs better than the classical Catmull-Clark subdivision scheme both at nearby vertices and along the non-vertical and non-horizontal directions.     

Families of low dimensional determinantal schemes

A scheme X⊂Pⁿ of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (f_ij). Given integers a₀≤a₁≤?≤a_t+c−2 and b₁≤?≤b_t, we denote by the stratum of standard determinantal schemes where f_ij are homogeneous polynomials of degrees a_j−b_i and is the Hilbert scheme (if n−c>0, resp. the postulation Hilbert scheme if n−c=0).Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of in and we show that is generically smooth along under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of appearing in Kleppe and Miró-Roig (2005) [25].     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.