Theoretical elastic moduli for disordered packings of interconnected spheres

A theoretical model has been developed which provides analytical expressions for the elastic moduli of disordered isotropic ensembles of spheres interconnected by physical bonds. Young's and shear moduli have been derived assuming an ideal random isotropic network and the radial distribution function for disordered packings of spheres. The interparticle interactions are accounted for in terms of surface forces for the two distinct cases of perfectly rigid spheres and spheres deformable at contact. A theoretical expression is also derived in a similar way for the bulk or compressibility modulus. In this case, an atomistic approach has been followed based on the analogy with noble gas solids and colloidal crystals. Also in this case, disordered spatial distribution of the spheres is described statistically. For the case of colloidal aggregates, a total two-body mean-field interaction potential is used which includes the Born repulsion energy. This latter contribution plays an essential role in determining the compression behavior of systems of particles aggregated in the primary minimum of the potential well and, therefore, must not be neglected. Both the expression of the Young's modulus and that of the compressibility modulus derived in this work are found to be consistent with two distinct sets of experimental data which recently appeared in the literature.     

Magnetic couplings vs. stress and strain in epitaxial (La, Sr)MnO3

Based on ab initio calculations, we quantify the magnetic couplings and the stress tensor in ferromagnetic Sr-doped LaMnO₃ upon combined application of built-in epitaxial and external uniaxial strains. We suggest YAlO₃ as a substrate suited to change magnetic order in manganite film with practicable external strains. The effect could lead to strain-activated switches based on piezoelectric-piezomagnetic heterojunctions.     

A Simple and Versatile Re‐Catalyzed Meyer–Schuster Rearrangement of Propargylic Alcohols to α,β‐Unsaturated Carbonyl Compounds

Rhenium does the job! A readily available rhenium complex efficiently catalyzed the direct Meyer–Schuster‐like rearrangement of different alkyl‐ and aryl‐substituted propargylic secondary and tertiary alcohols to the corresponding α,β‐unsaturated compounds, which were produced with virtually complete E stereoselectivity. The reaction proceeded under neutral conditions and no racemization of potentially enolizable stereocenters was observed.

     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

A note on the regularity of the free boundaries in the optimal partial transport problem

In this work we study the global regularity of the free boundaries arising in the optimal partial transport problem. Assuming the supports of both the source and the target measure to be convex, we show that the free boundaries of the active regions are globally C ^0,1/2.      

Three equivalent conjectures on the birational geometry of Fano 3-folds

I propose three equivalent conjectures on the birational geometry of Fano 3-folds. Roughly speaking, they suggest that ergodic, or chaotic, behaviour does not occur for Fano 3-folds.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

We prove C ¹ regularity of c-convex weak Alexandrov solutions of a Monge–Ampère type equation in dimension two, assuming only a bound from above on the Monge–Ampère measure. The Monge–Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C ¹ regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampère equation det D ² u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].