Higher correlation inequalities

We prove a correlation inequality for n increasing functions on a distributive lattice, which for n = 2 reduces to a special case of the FKG inequality. The key new idea is to reformulate the inequalities for all n into a single positivity statement in the ring of formal power series. We also conjecture that our results hold in greater generality.     

Iodine‐Catalyzed Conversion of β‐Dicarbonyl Compounds into β‐Enaminones Within a Minute Under Solvent‐Free Conditions

Synthesis of β‐enaminones from β‐dicarbonyl compounds has been achieved in high yields within a minute using primary and aromatic amines and catalytic amounts of iodine under solvent‐free conditions at room temperature.     

Remarkable ESIPT induced NIR emission by a selective colorimetric dibenzimidazolo diimine sensor for acetate

A new dibenzimidazolo diimine sensor (DDS) has been designed and synthesized for selective detection of acetate ion. Significant naked eye recognized color change of DDS solution from light yellow to pink upon addition of only acetate ion is accompanied with near infra red (NIR) emission exploiting excited state intramolecular proton transfer (ESIPT).     

The FKG Inequality for Partially Ordered Algebras

The FKG inequality asserts that for a distributive lattice with log-supermodular probability measure, any two increasing functions are positively correlated. In this paper we extend this result to functions with values in partially ordered algebras, such as algebras of matrices and polynomials. This research was supported by an NSF grant.     

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.     

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.      

Single end-on azido bridged 1D chain copper(II) complex: Synthesis,X-ray crystal structure and catalytic activity in homogeneous epoxidation of cyclooctene

A new azido derivative of a NNO donor tridentate Schiff-base copper(II) complex, [CuL(μ_1,1-N₃)]_n (1) (HL = 1-(N-ortho-hydroxyacetophenimine)-2-(N-ethyl)aminoethane), containing a single end-on μ_1,1-azido bridged 1D infinite chain has been synthesized and characterized. X-ray single crystal structure analysis reveals that the complex [CuL(μ_1,1-N₃)]_n (1) is a 1D chain in which neighboring Cu(II) chromophores are related by glide planes and linked by single azido bridges in the EO mode. No measurable magnetic interaction was evidenced in the complex [CuL(μ_1,1-N₃)]_n (1) through variable temperature magnetic susceptibility measurements (10–300 K). The complex was used as an active catalyst for the epoxidation of cyclooctene using tert-butyl-hydroperoxide as an oxidant. The catalytic activity of the complex has been compared in a series of solvents. The results show that in acetonitrile medium, the epoxide was produced in high yield with high selectivity.     

ENO Reconstruction and ENO Interpolation Are Stable

We prove that the ENO reconstruction and ENO interpolation procedures are stable in the sense that the jump of the reconstructed ENO point values at each cell interface has the same sign as the jump of the underlying cell averages across that interface. Moreover, we prove that the size of these jumps after reconstruction relative to the jump of the underlying cell averages is bounded. Similar sign properties and the boundedness of the jumps hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on nonuniform meshes, indicate a remarkable rigidity of the piecewise polynomial ENO procedure.