The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

We show that the critical Kac–Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac–Ward operator on any planar graph is given by what we call a fermionic generating function. We also present a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac–Ward operators on isoradial graphs.     

Special homogeneous linear systems on Hirzebruch surfaces

The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces mathbbF_n{mathbb{F}_n}. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8.     

Covalent Template-Directed Synthesis of a Spoked 18-Porphyrin Nanoring**

Rings of porphyrins mimic natural light-harvesting chlorophyll arrays and offer insights into electronic delocalization, providing a motivation for creating larger nanorings with closely spaced porphyrin units. Here, we demonstrate the first synthesis of a macrocycle consisting entirely of 5,15-linked porphyrins. This porphyrin octadecamer was constructed using a covalent six-armed template, made by cobalt-catalyzed cyclotrimerization of an H-shaped tolan with porphyrin trimer ends. The porphyrins around the circumference of the nanoring were linked together by intramolecular oxidative meso-meso coupling and partial β-β fusion, to give a nanoring consisting of six edge-fused zinc(II) porphyrin dimer units and six un-fused nickel(II) porphyrins. STM imaging on a gold surface confirms the size and shape of the spoked 18-porphyrin nanoring (calculated diameter: 4.7 nm).     

Mechanochemical Synthesis of Fluorinated Imines

A number of imines, including 12 new compounds, previously not reported in the literature, derived from variously fluorinated benzaldehydes and different anilines or chiral benzylamines were synthesized by a solvent-free mechanochemical method, which was based on the manual grinding of equimolar amounts of the substrates at the room temperature. In a very short reaction time of only 15 min, the method produced the expected products with good-to-excellent yields. The yields were comparable or significantly higher than those reported in the literature for the imines synthesized by other methods. Importantly, the conditions used for the reactions with aniline derivatives also resulted in the high yields of imines obtained from chiral benzylamines, and can be extended to the synthesis with other similar amines. Structures of all imines were confirmed by NMR spectroscopy: ¹H, ¹³C and ¹⁹F. For four compounds, X-ray structures were also obtained. The synthetic approach presented in this paper contributes to the prevention of environmental pollution and can be easily extended for larger-scale syntheses. The mechanochemical solvent-free method provides a convenient strategy particularly useful for the preparation of fluorinated imines being versatile intermediates or starting material in the synthesis of drugs and other fine chemicals.     

Microwave-assisted palladium-catalyzed cross-coupling of aryl and vinyl halides with H-phosphonate diesters

A general and efficient method for the microwave-assisted formation of the C-P bond was developed. Using a prevalent palladium catalyst, Pd(PPh3)4, a quantitative cross-coupling of various H-phosphonate diesters with aryl and vinyl halides was achieved in less than 10 min. The reactions occurred with retention of configuration at the phosphorus center and in the vinyl moiety. Using this protocol, several C-phosphonates, including those bearing nucleoside and cholesteryl moieties, were prepared in high yields.