Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

In this paper, we investigate the long-time behavior of stochastic reaction–diffusion equations of the type \(\text {d}u = (Au + f(u))\text {d}t + \sigma (u) \text {d}W(t)\), where \(A\) is an elliptic operator, \(f\) and \(\sigma \) are nonlinear maps and \(W\) is an infinite-dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function \(f\) possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure, which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper, we expand the existing classes of nonlinear functions \(f\) and \(\sigma \) and elliptic operators \(A\) for which the invariant measure exists, in particular in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if \(A\) is the Shrödinger-type operator \(A = \frac{1}{\rho }(\text {div} \rho \nabla u)\) where \(\rho = \text {e}^{-|x|^2}\) is the Gaussian weight.     

Isomerization of free base porphyrin. Theoretical study

The photoinduced isomerization reaction of free base porphyrin molecule has been calculated using the DFT-B3LYP method combined with the 6-31G(d,p) basis set. To prove the accuracy of results, the 6-311G+(2p,2d) basis set was used. Two types of isomerization mechanisms were studied. It was found that the stepwise pathway of isomerization is the most appropriate. The geometric parameters of isomers and the transition structures of different multiplets are discussed. Zero-point energy and vibrational frequencies analysis are given.     

Projective functors for algebraic groups

We investigate the regular subquotient category introduced by Soergel in [W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (1–3) (2000) 311–335]. A detailed study of projective functors enables one to relate those categories for semi-simple algebraic group G and its subgroup schemata G_rT. As an application we derive some information about the characters of tilting modules for G.     

A challenging synthesis of new 1,3,4‐thiadiazole derivatives starting from 2‐acylamino‐3,3‐dichloroacrylonitriles

Available 2‐acylamino‐3,3‐dichloroacrylonitriles, when treated with hydrazine hydrate, provide 2‐alkyl‐ or 2‐aryl‐5‐hydrazino‐1,3‐oxazole‐4‐carbonitriles that readily add alkyl or aryl isothiocyanates and the adducts formed recyclize on heating. Finally, the synthesis results in 5‐alkyl(aryl)amino‐1,3,4‐thiadiazol‐2‐yl(acylamino)acetonitriles or the products of their further cyclization, 2‐(5‐amino‐1,3‐ oxazol‐2‐yl)‐1,3,4‐thiadiazole derivatives. The structures of the novel substituted 1,3,4‐thiadiazoles are corroborated spectroscopically as well as by X‐ray diffraction method. © 2004 Wiley Periodicals, Inc. Heteroatom Chem 15:454–458, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/hc.20041     

Insights into Modeling Approaches in Chemistry: Assessing Ligand-Protein Binding Thermodynamics Based on Rigid-Flexible Model Molecules

In the field of chemistry, model compounds find extensive use for investigating complex objects. One prime example of such object is the protein-ligand supramolecular interaction. Prediction the enthalpic and entropic contribution to the free energy associated with this process, as well as the structural and dynamic characteristics of protein-ligand complexes poses considerable challenges. This review exemplifies modeling approaches used to study protein-ligand binding (PLB) thermodynamics by employing pairs of conformationally constrained/flexible model molecules. Strategically designing the model molecules can reduce the number of variables that influence thermodynamic parameters. This enables scientists to gain deeper insights into the enthalpy and entropy of PLB, which is relevant for medicinal chemistry and drug design. The model studies reviewed here demonstrate that rigidifying ligands may induce compensating changes in the enthalpy and entropy of binding. Some “rules of thumb” have started to emerge on how to minimize entropy-enthalpy compensation and design efficient rigidified or flexible ligands.     

Chemical interaction in the B-BN system at high pressures and temperatures.: Synthesis of novel boron subnitrides

Chemical interaction and phase transformations in the B-BN system have been in situ studied by X-ray diffraction with synchrotron radiation at pressures up to 5.3 GPa and temperatures up to 2800 K using multianvil press. New rhombohedral boron subnitride B₁₃N₂ has been synthesized by crystallization from the B-BN melt at 5 GPa. The structure of B₁₃N₂ belongs to the R-3m space group (a=5.4455(2) Å, c=12.2649(9) Å) and represents a new structural type. The subnitride is an individual compound and not a solid solution, in contrast to boron carbide. Besides, the formation of two other boron-rich B-N phases denoted as “B₆N” and “B₅₀N₂” has been observed. Their structures seem to be much more sophisticated and have not been even resolved to present time.     

Synthesis,Anticancer, and QSAR Studies of 2‐Alkyl(aryl,hetaryl)quinazolin‐4(3H)‐thione's and [1,2,4]Triazolo[1,5‐c]quinazoline‐2‐thione's Thioderivatives

Considering the frightening high level of mortality from cancer, studies of anticancer agents are vital nowadays. The 24 thioderivatives of 2‐alkyl(aryl)‐quinazolin‐4(3H)‐thiones and 20 thioderivatives of [1,2,4]triazolo[1,5‐c]quinazoline‐2‐thiones were synthesized and evaluated for preliminary in vitro anticancer activity with subsequent in silico QSAR analysis. The substance 18 had the best results inhibiting growth of eight cancer cell lines: CCRF‐CEM of leukemia; SF‐539, SNB‐75, and U251 of CNS cancer; 786, RXF393, and UO‐31 of renal cancer; and MDA‐MB‐231/ATCC of breast cancer (?31.50 – 47.41% of cell growth) with low procancer effect. Calculated QSAR‐models for CCRF‐CEM of leukemia, T‐47D and HS 578T of breast cancer, and mean cell growth demonstrated good rate of anticancer activity prediction (r² = 0.7 – 0.8,  = 0.5 – 0.7).     

All-fiber Kerr cell

An all-fiber nanosecond Kerr light gate is described that was constructed using microstructured fibers. The switching voltage for a 20?cm long device is as low as Vπ～85 V at a 1.06?μm wavelength. The device is fully spliced. The active element is a three-hole fiber provided with internal electrodes in the side-holes and a liquid core of nitrobenzene, which is fully enclosed. This work allows the exploiting of electrically driven liquid-core fibers and demonstrated the removal of the major limitations of Kerr cells in the past, allowing for integration, safe use, and relatively low switching voltage.     

Quantification of local hydration at the surface of biomolecules using dual-fluorescence labels

By using four labels of the 3-hydroxyflavone family displaying selective sensitivity to hydrogen bond (HB) donors and poor response to other polar molecules, we developed an approach for measuring local water concentration [H(2)O](L) (or partial volume of water: W(A) = [H(2)O](L)/55.6) in the label surrounding both in solvent mixtures and in biomolecules by the intensity ratio of two emissive forms of the label, N*/T*. Using a series of binary water/solvent mixtures with limited preferential solvation effects, a linear dependence of log(N*/T*) on the local concentration of HB donor was obtained and then used as a calibration curve for estimating the W(A) values in the surroundings of the probes conjugated to biomolecules. By this approach, we estimated the hydration of the labels in different peptides and their complexes with DNAs. We found that W(A) values for the label at the peptide N-terminus are lower (0.63-0.91) than for free labels and depend strongly on the nature of the N-terminal amino acid. When complexed with different DNAs, the estimated hydration of the labels conjugated to the labeled peptides was much lower (W(A) = 0-0.47) and depended on the DNA nature and linker-label structure. Thus, the elaborated method allows a site-specific evaluation of hydration at the surface of a biomolecule through the determination of the partial volume of water. We believe the developed procedure can be successfully applied for monitoring hydration at the surface of any biomolecule or nanostructure.