A duality theorem for generalized local cohomology

We prove a duality theorem for graded algebras over a field that implies several known duality results: graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

     

Copper(II) complexes with pyridine-2,6-dicarboxylic acid from the oxidation of copper(I) iodide

Via an oxidation reaction of Cu(I) iodide with pyridine-2,6-dicarboxylic acid (H₂L) in DMF three copper(II) complexes, [(CH₃)₂NH₂]₂[CuL₂] (1), K₂[CuL₂]?H₂L?H₂O (2) and [Cu(L)(H₂O)]_n (3), were synthesized and characterized. The structures of 1–3 were determined by single crystal X-ray diffraction studies. In-situ DMF decomposition produces dimethylamine base under solvothermal conditions and a proton transfer reaction takes place for the complex formation of 1. 3-D networks are stabilized in 1 and 2 via hydrogen bonds. Complex 3 is a 1-D coordination polymer with Cu-O semi-coordination bonds. Thermal decomposition of the complexes results in the corresponding metal oxides. Also, the electrochemical behavior of 1 was determined to be a metal-centered and diffusion-controlled, one-electron reduction process.     

Super face d-antimagic labeling for disjoint union of toroidal fullerenes

The discovery of the fullerene molecules and related forms of carbon such as nanotubes has generated an explosion of activity in chemistry, physics, and materials science. Classical fullerene is an all-carbon molecule in which the atoms are arranged on a pseudospherical framework made up entirely of pentagons and hexagons. A toroidal fullerene (toroidal polyhex) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. In this paper we examine the existence of entire labeling, where face-weights of all 6-sided faces of disjoint union of toroidal fullerenes form an arithmetic progression with common difference \(\hbox {d}\in \{1,2,3\}\).     

Spinor Green’s functions via spherical means on products of space forms

We explicitly compute the Green’s function of the spinor Klein–Gordon equation on the Riemannian and Lorentzian manifolds of the form _M0×?×_MN$M_0\times ?\times M_N$, with each factor being a space of constant sectional curvature. Our approach is based on an extension of the method of spherical means to the case of spinor fields and on the use of Riesz distributions.     

Rotating waves within a hollow vortex core

The rotating waves within a hollow vortex core (polygonal patterns) are generated under shallow water conditions inside a cylindrical tank by a revolving disk at its bottom. We previously reported some basic features of these polygonal patterns during transition and at the equilibrium states. The present paper includes a more comprehensive investigation into the transition process of these polygonal patterns and expands the recent partial results that have been previously reported. We show that the parent wave (or the N-gon pattern) to disk frequencies ratio is around one-third (1/3) regardless of the flow conditions. A detailed insight into the transition process from the parent wave N-gon to daughter wave (N + 1)-gon is provided, which consists of two regimes, quasi-periodic and synchronized. Based on these observations, we explained the shrinking of the disk speed subintervals over which the N-gon patterns occur, when N increases.     

An integrated solution and analysis of bioluminescence tomography and diffuse optical tomography

While diffuse optical tomography (DOT) has been studied for years, bioluminescence tomography (BLT) is emerging as a promising optical molecular imaging tool. These two modalities have different goals. DOT is for reconstruction of optical parameters of a medium such as a breast from surface measurements induced by external sources. BLT is for reconstruction of a bioluminescent source distribution in a medium such as a mouse from surface measurements induced by internal bioluminescent sources. However, an important pre-requisite for BLT reconstruction is the knowledge on the distribution of optical parameters within the medium, which is the output of DOT. In this paper, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. The model is introduced through minimizing the difference between predicted quantities and boundary measurements, as well as incorporating regularization terms. Then, we show the solution existence, introduce numerical schemes and prove convergence of the numerical solution. We also present numerical results to illustrate the utility of our approach.     

Novel and Stereospecific Synthesis of (2S)‐3‐(2,4,5‐Trifluorophenyl)propane‐1,2‐diol from D‐Mannitol

A stereospecific synthesis of (2S)‐3‐(2,4,5‐trifluorophenyl)propane‐1,2‐diol from D ‐mannitol has been developed. The reaction of 2,3‐O‐isopropylidene‐D ‐glyceraldehyde with 2,4,5‐trifluorophenylmagnesium bromide gave [(4R)‐2,2‐dimethyl‐1,3‐dioxolan‐4‐yl](2,4,5‐trifluorophenyl)methanol in 65% yield as a mixture of diastereoisomers (1 : 1). The Ph₃P catalyzed reaction of the latter with C₂Cl₆ followed by reduction with Pd/C‐catalyzed hydrogenation gave (2S)‐3‐(2,4,5‐trifluorophenyl)propane‐1,2‐diol with >99% ee and 65% yield.