Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Regularization of ill-posed linear inverse problems via ? ₁ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ? ₁ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to ? ₁-constraints, using a gradient method, with projection on ? ₁-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.     

A globally convergent descent method for nonsmooth variational inequalities

We propose a descent method via gap functions for solving nonsmooth variational inequalities with a locally Lipschitz operator. Assuming monotone operator (not necessarily strongly monotone) and bounded domain, we show that the method with an Armijo-type line search is globally convergent. Finally, we report some numerical experiments. This work has been supported by the National Research Program PRIN/2005017083 “Innovative Problems and Methods in Nonlinear Optimization”.     

Linked-Cluster Expansion of the Ising Model

The linked-cluster expansion technique for the high-temperature expansion of spin modes is reviewed. A new algorithm for the computation of three-point and higher Green's functions is presented. Series are computed for all components of two-point Green's functions for a generalized 3D Ising model, to 25th order on the bcc lattice and to 23rd order on the sc lattice. Series for zero-momentum four-, six-, and eight-point functions are computed to 21st, 19th, and 17th order respectively on the bcc lattice.     

Continuous Frames, Function Spaces, and the Discretization Problem

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine, and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.     

Emulsions with structured continuous phases

Recent research on the role of the continuous phase in emulsions is reviewed. Special attention is given to the structured continuous phases such as lyotropic and thermotropic liquid crystals and gel networks. The implications in the formation, stability and properties of emulsions are discussed. Some recent applications, particularly in templating for the preparation of new materials, are also introduced.     

Stabilization of catalytic sol-gel entrapped perruthenate

A combination of more stable counter-cation (tetraphenylphosphonium in place of tetrapropylammonium), of local basic microenvironment, and of a non-solubilizing reaction medium (supercritical CO₂) improves the life-cycle and reusability of catalytic ORMOSILs doped with perruthenate in the oxidation of alcohols with O₂. A number of different bases were co-entrapped and their effect on catalysis assessed in the oxidative dehydrogenation of benzyl alcohol in dense phase carbon dioxide at 22 MPa and 75 °C. The optimized catalyst retains most of its activity after five consecutive runs when a normal ORMOSIL-entrapped TPAP has become inactive. Deactivation of TPAP could be ascribed by EPR analysis to the formation of catalytically inactive RuO₂.     

The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics.     

Non-centrosymmetric tetrameric assemblies of tetramethylammonium halides with uranyl salophen complexes in the solid state

Ditopic salophen-UO(2) receptors 1-4 and 7 co-crystallize with tetramethylammonium (TMA) chloride and fluoride salts producing good quality crystals amenable for X-ray diffraction characterization. The arrangement of the receptor and salt units in the crystal lattice is such that tetrameric ball-shaped assemblies are formed, where an inner cluster of four TMA cations are surrounded by an outer shell of four UO(2)-bound anions. These elaborate architectures, which occur in all cases, regardless of a certain degree of structural modification on the receptors, lead to lattices that belong to non-centrosymmetric (NCS) space groups. Interestingly, the tetragonal symmetry of the tetrameric ball-shaped assemblies is either retained (I4?) or lost (R3c and I4?3d) at the lattice level, without compromising the NCS nature of the crystal lattices. The principal X-ray investigation on TMAX (X = Cl/F) co-crystals, that is, 1-(TMA)Cl, 2-(TMA)Cl, 3-(TMA)Cl, 4-(TMA)Cl, 7-(TMA)Cl, and 7-(TMA)F, is accompanied by NMR and electrospray ionization (ESI) mass spectrometry studies to gather additional insight on the modality of formation of the solid state structures observed. The important role of cation-π interactions in the receptor-salt recognition process is renewed and strengthened by comparison with NMR titration data with a novel reference compound, the salophen-UO(2) complex 8. Given the importance of NCS and polar crystalline solids in the development of functional materials, this study shows that this property can be introduced into elaborate host-guest systems, as those which assemble in the architectures described here, thus expanding its field of applicability.     

Ytterbium triflate catalyzed synthesis of chlorinated lactones

The direct synthesis of chlorolactones from differently substituted alkenoic acids, using either sodium hypochlorite or chloramine T as the source of electrophilic chlorine and ytterbium triflate hydrate as the Lewis acid is described. In both cases the reactions proceeded in good yields affording selectively five- or six-membered chlorolactones.