Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

We present criteria for the Cohen–Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen–Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen–Macaulayness of the second symbolic power or of all symbolic powers of a Stanley–Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen–Macaulay. In particular, all symbolic powers are Cohen–Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen–Macaulayness can pass from a symbolic power to another symbolic powers in different ways.     

Two-atom van der Waals interaction between polarizable/magnetizable atoms near magnetoelectric bodies

The van der Waals potential of two atoms in the presence of an arbitrary arrangement of dispersing and absorbing magnetoelectric bodies is studied. Starting from a polarizable atom placed within a given geometry, its interaction with a second polarizable/magnetizable atom is deduced from its Casimir-Polder interaction with a weakly polarizable/magnetizable test body. The general expressions for the van der Waals potential obtained thusly are illustrated by considering first the case of two atoms in free space, with special emphasis on the interaction between (i) two polarizable atoms and (ii) a polarizable and a magnetizable atom. Furthermore, the influence of magnetoelectric bodies on the van der Waals interaction is studied in detail for the example of two atoms placed near a perfectly reflecting plate or a magnetoelectric half space, respectively. The text was submitted by the authors in English.     

Cover Picture: Solid‐State Scrolls from Hierarchical Self‐Assembly of T‐Shaped Rod–Coil Molecules (Angew. Chem. Int. Ed. 9/2009)

Solid scrolls are reversibly formed by self‐assembly of rod‐shaped molecules with laterally attached coil units, in contrast to the layered structures formed from self‐assembly of planar molecules. As described by M. Lee and co‐workers in their Communication on page 1664 ff., the core structure of the scrolls, which are either filled cylinders or hollow tubes, can be controlled by variation of the length of the coil unit. The cover picture shows aligned tubular scrolls displaying well‐defined in‐plane ordering of the rod segments.

     

Suzuki Reaction of Aryl Bromides Using a Phosphine-Free Magnetic Nanoparticle-Supported Palladium Catalyst

A palladium catalyst immobilized on superparaganetic nanoparticles was prepared with a palladium loading of 0.30 mmol/g. The catalyst was characterized using X-ray diffraction, scanning electron microscopy, transmission electron microscopy, vibrating sample magnetometry, thermogravimetric analysis, Fourier transform infrared, atomic absorption spectrophotometry, and nitrogen adsorption. The immobilized palladium catalyst was an efficient catalyst without added phosphine ligands for the Suzuki cross-coupling reaction of several aryl bromides with phenylboronic acid. The recovery of catalyst was simply by magnetic decantation in the presence of a magnet. The immobilized palladium catalyst can be reused many times without significant degradation in catalytic activity. No leaching of active palladium species into the reaction solution was detected.     

Synthesis and Crystal Structure of Bis[<Emphasis Type="Italic">N</Emphasis>-{(<Emphasis Type="Italic">S</Emphasis>)-1-Phenylethyl}-<Emphasis Type="Italic">N</Emphasis>-{1-(2-Pyridinyl)Methyl}Amine] Zinc(II) Perchlorate

Abstract  

The reaction of N-[(S)-1-phenylethyl]-N-[1-(2-pyridinyl)methyl]amine (PEPMA) with Zn(ClO₄)₂ affords (PEPMA)₂Zn(ClO₄)₂ at room temperature. The complex is characterized by elemental analysis, ¹H-MNR and X-ray crystallography. The crystal system is orthorhombic with space group of P2₁2₁2₁ and unit cell parameters: a = 8.6911(4), b = 9.7755(7), c = 37.275(2) ?, V = 3166.9(3) ?³, Z = 4, D _x  = 1.445 Mg/m³. The crystal structure reveals that zinc metal is ligated by two PEPMA ligands in a distorted tetrahedral fashion. Interestingly, both nitrogen atoms of amine at PEPMA to the central zinc metal exhibit (R)-configuration.