Besov regularization,thresholding and wavelets for smoothing data

Smoothing of data is a problem very important for many applications ranging from 1-D signals (e.g., speech) to 2-D and 3-D signals (e.g., images). Many methods exist in the literature for facing the problem; in the present paper we point our attention on regularization. We shall treat regularization methods in a general framework which is well suited for wavelet analysis; in particular we shall investigate on the relation existing between thresholding methods and regularization. We shall also introduce a new regularization method (Besov regularization), which includes some known regularization and thresholding methods as particular cases. Numerical experiments based on some test problems will be performed in order to compare the performance of some methods of smoothing data.

AMS (MOS) Subject Classifications: 65R30, 41A60.     

Approximation of Multiple Stratonovich Fractional Integrals

Abstract

We study multiple Riemann-Stieltjes integral approximations to multiple Stratonovich fractional integrals. Two standard approximations (Wong-Zakai and Mollifier approximations) are considered and we show the convergence in the mean square sense and uniformly on compact time intervals of these approximations to the multiple Stratonovich fractional integral.     

On Double Stratonovich Fractional Integrals and Some Strong and Weak Approximations

Abstract

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived.

Riemann–Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.     

Existence and Besov regularity of the density for a class of SDEs with Volterra noise

By using a simple method based on the fractional integration by parts, we prove the existence and the Besov regularity of the density for solutions to stochastic differential equations driven by an additive Gaussian Volterra process. We assume weak regularity conditions on the drift. Several examples of Gaussian Volterra noises are discussed.     

Thermal analysis and calorimetry in the study of materials and processes for fundamental sciences and various applications in Central and Eastern Europe

We investigated the influence of B substitution for Al₂W₃O₁₂ on thermal changes of UV–Vis and Raman spectra, and colors. First, B-substituted Al₂W₃O₁₂ powder was synthesized by a solid-state reaction method. Single-phase Al_2?xB_xW₃O₁₂ powders with x = 0, 0.10 and 0.20 were successively prepared. B substitution promoted thermal changes of the UV–Vis spectra, resulting in a more pronounced color change of Al₂W₃O₁₂ in the range of 30–150 °C. Raman spectra of the Al_2?xB_xW₃O₁₂ powders with x = 0 and 0.20 indicated that the lattice vibrations of Al_2?xB_xW₃O₁₂ with x = 0.20 were larger than those of Al₂W₃O₁₂. The thermal change of the color phase (ΔE) in the range 30–150 °C of Al₂W₃O₁₂ was increased by B substitution. The color of the B-substituted Al₂W₃O₁₂ powders changed reversibly from pale white at 30 °C to light yellowish green at 150 °C.     

MTD-ADJ: A multiconformational minimal topologic difference for determining bioactive conformers using adjusted biological activities

The active conformation is part of a conformational mixture with experimental activity Yexp, and is used in QSAR studies to extract more information regarding the ligand-receptor interaction. To reflect the relative amount () of the active conformation, we adjust Yexp : Yadj = Yexp - log . We establish a quantitative structure-activity relationship (QSAR) between Yadj and 3D conformational characteristics for the acetylcholinesterase (AChE) hydrolysis rates of 25 acetic esters. The 3D-QSAR model was obtained using the adjusted multiconformational minimal steric/topologic difference (MTD-ADJ) method, optimizing the receptor map based on Yadj for each conformer. Yadj was updated during each step of the optimization process. and Yadj are based on the Boltzmann distribution calculated using AM1 (MOPAC 6.0) relative energies of the COSMIC 90 derived conformers. The MTD-ADJ results are: (i) the 3D-QSAR models obtained by this procedure have significant statistical parameters and are similar to the unadjusted (MTD-MC, using Yexp) models; (ii) the selected bioactive conformations are extended, occupy cavity vertices and, for the same structures, have the same MTD value; and (iii) the optimized conformational map of the neutral ligands obtained from the MTD-ADJ model fits well in the active site of the crystallographic structure of AChE (from Torpedo californica). We propose a neutral ligands binding site model for AchE. Our results show that MTD-ADJ, which can be implemented in any 3D-QSAR method, is capable of providing additional information regarding the active conformations, and can be used to gain further insight into the ligand-receptor models for which no structural data are available.     

Scaffold topologies. 2. Analysis of chemical databases

We have systematically enumerated graph representations of scaffold topologies for up to eight-ring molecules and four-valence atoms, thus providing coverage of the lower portion of the chemical space of small molecules (Pollock et al. J. Chem. Inf. Model., this issue). Here, we examine scaffold topology distributions for several databases: ChemNavigator and PubChem for commercially available chemicals, the Dictionary of Natural Products, a set of 2742 launched drugs, WOMBAT, a database of medicinal chemistry compounds, and two subsets of PubChem, "actives" and DSSTox comprising toxic substances. We also examined a virtual database of exhaustively enumerated small organic molecules, GDB (Fink et al. Angew. Chem., Int. Ed. 2005, 44, 1504-1508), and we contrast the scaffold topology distribution from these collections to the complete coverage of up to eight-ring molecules. For reasons related, perhaps, to synthetic accessibility and complexity, scaffolds exhibiting six rings or more are poorly represented. Among all collections examined, PubChem has the greatest scaffold topological diversity, whereas GDB is the most limited. More than 50% of all entries (13 000 000+ actual and 13 000 000+ virtual compounds) exhibit only eight distinct topologies, one of which is the nonscaffold topology that represents all treelike structures. However, most of the topologies are represented by a single or very small number of examples. Within topologies, we found that three-way scaffold connections (3-nodes) are much more frequent compared to four-way (4-node) connections. Fused rings have a slightly higher frequency in biologically oriented databases. Scaffold topologies can be the first step toward an efficient coarse-grained classification scheme of the molecules found in chemical databases.