On the rate of change of the sharp constant in the Sobolev–Poincaré inequality

The rate of change of the sharp constant in the Sobolev–Poincaré or Friedrichs inequality is estimated for a Euclidean domain that moves outward. The key ingredients are a Hadamard variation formula and an inequality that reverses the usual Hölder inequality.     

Cerium (IV) ammonium nitrate (CAN)-mediated regioselective synthesis and anticancer activity of 6-substituted 5,8-dimethoxy-1,4-naphthoquinone

6-Substituted 5,8-O-dimethyl-1,4-naphthoquinones (6-DMNQ), the promising anticancer scaffolds, were selectively generated by oxidative demethylation of 2-substituted 1,4,5,8-tetramethoxynaphthalenes with CAN in EtOAc/H₂O in comparatively high yields. An interesting finding was that apart from the reported electron-withdrawing effects of substituents on position 2 of naphthalene ring, regioselective synthesis of 6-DMNQ was largely dependent on the steric effects in CAN-mediated oxidation. The selective cytotoxicities of 6-DMNQ from the in vitro cell-based assays were exhibited between the cancer cells and normal cells. Moreover, most of sulfur-containing 6-DMNQ derivatives displayed better anticancer activities than the corresponding oxygen-containing ones, which could provide an available strategy for the design of 6-DMNQ derivatives as potential anticancer agents.     

Functions averages over the roots of unity,cauchy problems and addition formulas

An averaging operator over the roots of unity is defined on a class of analytic functions and its algebraic and analytic properties are investigated. A Cauchy like integral formula for this is obtained. This operator and its properties are then employed to solve higher order Cauchy problems, to derive addition formulas for hypergeometric functions and to obtain integral representations for special classes of hypergeometric functions.     

Cousin I condition and Stein spaces

Here we give a few n-dimensional extensions of a recent result of M. Abe concerning a Cousin I characterization of two-dimensional Stein manifolds.     

From Two‐ to Three‐Dimensional Structures of a Supertetrahedral Boran Using Density Functional Calculations

With help of the DFT calculations and imposing of periodic boundary conditions the geometrical and electronic structures were investigated of two‐ and three‐dimensional boron systems designed on the basis of graphane and diamond lattices in which carbons were replaced with boron tetrahedrons. The consequent studies of two‐ and three‐layer systems resulted in the construction of a three‐dimensional supertetrahedral borane crystal structure. The two‐dimensional supertetrahedral borane structures with less than seven layers are dynamically unstable. At the same time the three‐dimensional superborane systems were found to be dynamically stable. Lack of the forbidden electronic zone for the studied boron systems testifies that these structures can behave as good conductors. The low density of the supertetrahedral borane crystal structures (0.9 g cm⁻³) is close to that of water, which offers the perspective for their application as aerospace and cosmic materials.     

On solitary wave solutions to the nonlinear Schrödinger equations with two parameters

In this paper, we discuss the existence and multiplicity of positive solitary wave solutions for nonlinear Schrödinger equations with two parameters. The proof is based on the method of upper and lower solutions and the fixed point index.