On spectral resolutions connected with self-adjoint differential vector-operators in a Hilbert space

We show that spectral resolutions of differential vector-operators may be represented as a specific direct sum integral operator with a kernel written in terms of generalized vector-operator eigenfunctions. Then we prove that a generalized eigenfunction measurable with respect to the spectral parameter may be decomposed using a set of analytical defining systems of coordinate operators.     

Certain algebraic surfaces with Eisenbud‐Harris general fibration of genus 4

We classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.     

Cramér's asymptotics in systems with fast and slow motions

This paper is devoted to the averaging principle for stochastic systems with slow and intermixing fast motions. Here we (i) prove the existence of the Cramér type asymptotics for the probabilities of large deviations from an averaged motion, which implies the central limit theorem, and (ii) develop an analytic procedure for computation of this asymptotics. We use general apparatus of superregular perturbations of fiber ergodic semigroups to investigate two systems in the same way. The first of them is a cascade in which slow motions are determined by a vector field depending both on slow and fast variables, and fast motions compose a Markov chain depending on the slow variable. The second is a process defined by a system of two stochastic differential equations.     

Comparison principles applied to obstacle problems with penalties

Penalty methods form a well known technique to embed elliptic variational inequality problems into a family of variational equations (cf. [6], [13], [17]). Using the specific inverse monotonicity properties of these problems L _∞-bounds for the convergence can be derived by means of comparison solutions. Lagrange duality is applied to estimate parameters involved.

For piecewise linear finite elements applied on weakly acute triangulations in combination with mass lumping the inverse monotonicity of the obstacle problems can be transferred to its discretization. This forms the base of similar error estimations in the maximum norm for the penalty method applied to the discrete problem.

The technique of comparison solutions combined with the uniform boundedness of the Lagrange multipliers leads to decoupled convergence estimations with respect to the discretization and penalization parameters.     

Synthesis of 29H,31H-phthalocyanine and chloro (29H,31H-phthalocyaninato) aluminum derivatives showed anti-cancer and anti-bacterial actions

Continuous administration of most chemotherapeutic drugs can induce different types of side effects. There has been growing interest in exploring an alternative approach to synthesizing compounds that are most effective and have fewer side effects. We synthesized 29H,31H-Phthalocyanine, and Chloro (29H,31H- phthalocyaninato) aluminum at low temperatures using lithium in the present study with diisopropylamide as the nucleophile. The physical characteristics of 29H,31H-Phthalocyanine, and Chloro (29H,31H- phthalocyaninato) aluminum were confirmed by FT-IR method, XRD, SEM, and the impact of these compounds on human colorectal carcinoma (HCT-116) and human cervical cells (HeLa) was examined. Treatment with 29H,31H-Phthalocyanine significantly decreased cancer cell growth and proliferation, as determined by MTT and DAPI staining analysis. In contrast, Chloro (29H,31H- phthalocyaninato) aluminum treatment did not show any inhibitory action on colon or cervical cancer cells. We also calculated the inhibitory concentration (IC50) of 29H,31H-Phthalocyanine, which was 30 µg/ml (HCT-116) and 33 µg/ml (HeLa cells). The antibacterial effectiveness of 29H,31H-Phthalocyanine, and chloro (29H,31H- phthalocyaninato) aluminum was studied using Enterococcus faecalis (E. faecalis). The CFU (colony frequency unit) assay confirmed significant activity against the test bacterium after treatment with 29H,31H-Phthalocyanine. However, no activity was seen upon treatment with chloro (29H,31H- phthalocyaninato) aluminum against E. faecalis.     

Approximation theorems for localized szsz–Mirakjan operators

In the present paper, we investigate the convergence and the approximation order of the localized Szsz–Mirakjan operators, and obtain some new results to improve the results due to Omey [Note on operators of Szsz–Mirakjan type, J. Approx. Theory 47 (1986) 246–254].     

Updating the principal angle decomposition

A class of fast Householder-based sequential algorithms for updating the Principal Angle Decomposition is introduced. The updated Principal Angle Decomposition is of key importance in the adaptive implementation of several fundamental operations on correlated processes, such as adaptive Wiener filtering, rank-adaptive system identification, and rank and data compression concepts using canonical coordinates. An instructive example of rank-adaptive system identification is examined experimentally.     

Spectroscopic studies of triethoxysilane sol-gel and coating process

Silica sol-gels have been prepared under different conditions using triethoxysilane (TES) as precursor. The prepared sol-gels have been used to coat aluminum for corrosion protection. Vibrational assignments have been made for most vibration bands of TES, TES sol-gel, TES sol-gel-coated aluminum and xerogel. It has been noticed that air moisture may have helped the hydrolysis of the thin coating films. Xerogels have been obtained from the sol-gel under different temperature conditions and the resulting samples have been characterized by using infrared and Raman spectroscopic methods. IR data indicate that the sol-gel process is incomplete under the ambient conditions although an aqueous condition can have slightly improved the process. Two nonequivalent silicon atoms have been identified from the collected (29)Si NMR spectra for the sol-gel, supporting the result derived from the IR data. The frequency of SiH bending vibration has been found to be more sensitive to the skeletal structure than that of the SiH stretching vibration. A higher temperature condition could favor the progression of hydrolysis and condensation. A temperature higher than 300 degrees C would cause sample decomposition without seriously damaging the silica network. From infrared intensity measurements and thermo-gravimetric analyses, the fractions of incomplete hydrolysis and condensation species have been estimated to be 4% and 3%, respectively. Electrochemical data have shown that the sol-gel coating significantly improves the corrosion protection properties of aluminum.     

High stable sandwich-type polyoxometallates based on . Synthesis, chemical properties and characterization of [(A-β-SiW9O34)2(MOH2)3CO3] (M = Y and Yb)

Highly hydrolytic and thermally stable sandwich-type polyoxometallates of [(A-β-SiW₉O₃₄)₂(MOH₂)₃CO₃]¹³⁻ (M = Y³⁺ and Yb³⁺) have been synthesized at room temperature by stoichiometric reactions of the trilacunary ligand with M³⁺ in 0.1 M carbonate solution. The new complexes were isolated as sodium and mixed sodium/potassium salts and were characterized by elemental analysis, IR, ¹³C and ²⁹Si NMR, UV–Vis spectroscopy, TGA, DSC and single crystal structure analysis. The crystal structure of the complexes consist of two lacunary Keggin moieties which are linked by a (H₂OMO)₃C belt into an assembly of virtual C₂ symmetry. Each M³⁺ ion adopts a mono-capped trigonal-prismatic coordination. The C₂ axis of the complexes and the local 3-fold axis of the MO₆ group lies in the (H₂OMO)₃C belt plane. The trigonal prismatic geometry is achieved by the two terminal oxygen atoms of an edge shared pair of WO₆ octahedra from each moiety and two oxygen from the belt, and the cap by one external water ligand. The hydrolytic and thermal stabilities of the complexes and the reasons that prove the retention of the isomeric form of the trilacunary ligand upon complexation are discussed.