Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

On the abstract theory of additive and multiplicative Schwarz algorithms

Summary. In recent years, it has been shown that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space into a sum of subspaces and the splitting of the variational problem on into auxiliary problems on these subspaces. In this paper, we propose a modification of the abstract convergence theory of the additive and multiplicative Schwarz methods, that makes the relation to traditional iteration methods more explicit. The analysis of the additive and multiplicative Schwarz iterations can be carried out in almost the same spirit as in the traditional block-matrix situation, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical. In addition, we present a new bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the corresponding additive Schwarz operator. These results may be viewed as an appendix to the recent surveys [X], [Ys]. Received February 1, 1994 / Revised version received August 1, 1994     

On formal integration in composition rings of polynomials

This author wants to express his thanks to the School of Information Engineering at Teesside Polytechnic, Middlesbrough, England, for its support and hospitality during a visiting appointment of 3 months in 1989, when this paper was written.     

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H¹ norm best approximation error estimates for H² functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.     

On the equilibrium structures of the complexes H2C3H+ · Ar and c-C3H3(+) · Ar: results of explicitly correlated coupled cluster calculations

Explicitly correlated coupled cluster theory at the CCSD(T)-F12x (x = a, b) level [T. B. Adler et al., J. Chem. Phys. 127, 221106 (2007)] has been employed in a study of the potential energy surfaces for the complexes H(2)C(3)H(+) · Ar and c-C(3)H(3)(+) · Ar. For the former complex, a pronounced minimum with C(s) symmetry was found (D(e) ≈ 780 cm(-1)), well below the local "H-bound" minimum with C(2v) symmetry (D(e) ≈ 585 cm(-1)). The absorption at 3238 cm(-1) found in the recent infrared photodissociation spectra [A. M. Ricks et al., J. Chem. Phys. 132, 051101 (2010)] is, thus, interpreted as an essentially free acetylenic CH stretching vibration of the propargyl cation. A global minimum of C(s) symmetry was also obtained for c-C(3)H(3)(+) (D(e) ≈ 580 cm(-1)), but the energy difference with respect to the local C(2v) minimum is only 54 cm(-1).     

The Chiral Trimer and a Metastable Chiral Dimer of Achiral Hexafluoroisopropanol: A Multi‐Messenger Study

1,1,1,3,3,3‐hexafluoro‐propan‐2‐ol aggregates preferentially into an achiral dimer of achiral monomers, but the trimer is found to prefer three metastable chiral monomer units arranged into a strained OH???O hydrogen‐bonded ring, which is reinforced by secondary CH???FC interactions. This is shown by a combination of infrared, microwave, and Raman spectroscopy in supersonic jet expansions and supported by high‐level quantum chemical calculations. It involves an activation of the monomers by >15 kJ mol^?1, clearly driven by the much stronger hydrogen‐bond interaction available to the gauche and even more to the cis monomer units.     

Sweet like chocolate

A new crystalline form of αβ‐d ‐lactose prepared by oven drying a concentrated aqueous solution of d ‐lactose is a lesson in the power of observation and the rigorous analysis of powder samples.     

Fluorescence enhancement of several hallucinogenic drugs in cyclodextrin media

The fluorescence characteristics of selected hallucinogenic drugs dissolved in solutions of α-cyclodextrin and β-cyclodextrin are reported. Fluorescence intensity enhancements in cyclodextrin media relative to aqueous solution range from 1.2 to 4.0, probably because inclusion of the drugs into cyclodextrin increases the quantum yield. Calibration graphs are linear over 2–3 orders of magnitude; limits of detection are 6–13 μg l^?1 for ibogaine and N,N-dimethyltryptamine. The mescaline derivatives show limits of detection in the 0.8–1.4 ppm range. The role of cyclodextrin in enhancing the fluorescence intensities and some of the criteria for this fluorescence enhancement are discussed.     

Simple one-step process for immobilization of biomolecules on polymer substrates based on surface-attached polymer networks

For the miniaturization of biological assays, especially for the fabrication of microarrays, immobilization of biomolecules at the surfaces of the chips is the decisive factor. Accordingly, a variety of binding techniques have been developed over the years to immobilize DNA or proteins onto such substrates. Most of them require rather complex fabrication processes and sophisticated surface chemistry. Here, a comparatively simple immobilization technique is presented, which is based on the local generation of small spots of surface attached polymer networks. Immobilization is achieved in a one-step procedure: probe molecules are mixed with a photoactive copolymer in aqueous buffer, spotted onto a solid support, and cross-linked as well as bound to the substrate during brief flood exposure to UV light. The described procedure permits spatially confined surface functionalization and allows reliable binding of biological species to conventional substrates such as glass microscope slides as well as various types of plastic substrates with comparable performance. The latter also permits immobilization on structured, thermoformed substrates resulting in an all-plastic biochip platform, which is simple and cheap and seems to be promising for a variety of microdiagnostic applications.