Intramolecular,Interrupted Homo-Nazarov Cascade Biscyclizations to Angular (Hetero)Aryl-Fused Polycycles

Herein, a SnCl₄-catalyzed intramolecular, interrupted homo-Nazarov cascade biscyclization to access angular (hetero)aryl-fused polycycles is reported. Subsequent decarboxylation of the readily enolizable products afforded the angular products in up to 71 % yield over two steps, with the trans-diastereomers as the major products. The cyclopropyl homo-Nazarov cyclization precursors were formed using a scalable and modular synthetic route that, ultimately, offers access to 6,6,6-, 6,6,5-, 6,5,6-, 6,6,5,6-, and 6,6,6,5-fused angular polycyclic products. To showcase the rigor and utility of the method, an 8-step total synthesis of (±)-1-oxoferruginol, an antibacterial aromatic abietane diterpenoid, was disclosed.     

The Regularity of Parametrized Integer Stationary Varifolds in Two Dimensions

We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of variations for the area functional. © 2020 Wiley Periodicals LLC     

A lower‐epiperimetric inequality for area‐minimizing surfaces

The epiperimetric inequality introduced by E. R. Reifenberg in [3] gives a rate of decay at a point for the decreasing k‐density of area of an area‐minimizing integral k‐cycle. While dilating the cycle at that point, this rate of decay holds once the configuration is close to a tangent cone configuration and above the limiting density corresponding to that configuration. This is why we propose to call the Reifenberg epiperimetric inequality an upper‐epiperimetric inequality. A direct consequence of this upper‐epiperimetric inequality is the statement that any point possesses a unique tangent cone. The upper‐epiperimetric inequality was proven by B. White in [5] for area‐minimizing 2‐cycles in ?ⁿ. In the present paper we introduce the notion of a lower‐epiperimetric inequality. This inequality gives this time a rate of decay for the decreasing k‐density of area of an area‐minimizing integral k‐cycle, while dilating the cycle at a point once the configuration is close to a tangent cone configuration and below the limiting density corresponding to that configuration. Our main result in the present paper is to prove the lower‐epiperimetric inequality for area‐minimizing 2‐cycles in ?ⁿ. As a consequence of this inequality we prove the “splitting before tilting” phenomenon for calibrated 2‐rectifiable cycles, which plays a crucial role in the proof of the regularity of 1‐1 integral currents in [4]. © 2004 Wiley Periodicals, Inc.     

Confinement Effects on the Benzene Orientational Structure

Liquids under confinement exhibit different properties compared with their corresponding bulk phases, for example, miscibility, phase transitions, and diffusion. The underlying cause is the local ordering of molecules, which is usually only studied using pure simulation methods. Herein, we derive experimentally the structure of benzene confined in MCM‐41 using total neutron scattering measurements. The study reveals a layering of molecules across a pore, and four concentric cylindrical shells can be distinguished for a pore with the radius of 18 Å. The nanoscale confinement of the liquid has a major effect on the spatial and orientational correlations observed between the molecules, when compared with the structure of the bulk liquid. These differences are most marked for molecules in parallel configurations, and this suggests differences in chemical reactivity between the confined and bulk liquids.     

On avoiding odd partial Latin squares and r-multi Latin squares

We show that for any positive integer k?4, if R is a (2k-1)×(2k-1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that if R contains at most nr/2 symbols and if there is an n×n Latin square L such that δn of the symbols in L cover the filled cells in R where 0<δ<1, then R is avoidable provided r is large enough.