Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $\rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.     

Triacetic acid lactone as a bioprivileged molecule in organic synthesis

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Synthesis and antimycobacterial activity of purine conjugates with (S)-lysine and (S)-ornithine

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Stability of O3 + N2, O3 + O2 and O3 + CO2 double hydrates: DFT study

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Structural and magnetic properties of Ni/Pt multilayers

In this work, the variation of the magnetic moments of the Ni/Pt multilayers are studied using the linearized augmented plane waves (LAPW) method in the framework of the density functional theory (DFT) implemented in the version of WIEN2K program. The systems have been modeled by seven layers slab separated in z direction by a vacuum region of four substrate layers. We present the results of the dependence of the magnetic properties with respect to the thickness variation of the different multilayers. The modeling of these systems finds an important empirical support. Experiment and theory show the same trends for the magnetic moments: hybridization effects between Ni and Pt are mostly localized at the interface.     

Catalytic synthesis of single-crystalline gallium nitride nanobelts

Single-crystalline gallium nitride nanobelts have been synthesized through the reaction of gallium vapor with flowing ammonia using nickel as a catalyst. The as-synthesized products were characterized using X-ray powder diffraction (XRD), scanning electron microscopy, energy-dispersive X-ray spectroscopy, transmission electron microscopy, and selected-area electron diffraction (SAED). XRD and SAED results revealed that the products are pure, single-crystalline GaN with hexagonal structure. The widths and thickness of the nanobelts ranged from 80 to 200 nm, and 10 to 30 nm, respectively. The lengths were up to several tens of micrometers. The nanobelts had smooth surface with no amorphous sheath, and a sharp-tip end. The growth mechanism of nanobelts was discussed.     

Existence of the uniformly minimum risk equivariant estimators of parameters in a class of normal linear models

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)^α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.