Mean field spin glasses treated with PDE techniques

Following an original idea of Guerra, in these notes we analyze the Sherrington-Kirkpatrick model from different perspectives, all sharing the underlying approach which consists in linking the resolution of the statistical mechanics of the model (e.g. solving for the free energy) to well-known partial differential equation (PDE) problems (in suitable spaces). The plan is then to solve the related PDE using techniques involved in their native field and lastly bringing back the solution in the proper statistical mechanics framework. Within this strand, after a streamlined test-case on the Curie-Weiss model to highlight the methods more than the physics behind, we solve the SK both at the replica symmetric and at the 1–RSB level, obtaining the correct expression for the free energy via an analogy to a Fourier equation and for the self-consistencies with an analogy to a Burger equation, whose shock wave develops exactly at critical noise level (triggering the phase transition). Our approach, beyond acting as a new alternative method (with respect to the standard routes) for tackling the complexity of spin glasses, links symmetries in PDE theory with constraints in statistical mechanics and, as a novel result from the theoretical physics perspective, we obtain a new class of polynomial identities (namely of Aizenman-Contucci type, but merged within the Guerra’s broken replica measures), whose interest lies in understanding, via the recent Panchenko breakthroughs, how to force the overlap organization to the ultrametric tree predicted by Parisi.     

Ring expansion of cyclic β-amino alcohols induced by diethylaminosulfur trifluoride: synthesis of cyclic amines with a tertiary fluorine at C3

As the replacement of a hydrogen atom by a fluorine atom in a compound can have an important impact on its biological properties, the development of methods allowing the introduction of a fluorine atom is of great importance. The scope and limitations of the ring expansion of cyclic 2-hydroxymethyl amines induced by diethylaminosulfur trifluoride (DAST) to produce cyclic β-fluoro amines was studied as well as the enantioselectivity of the process.     

Titanium(IV) imido complexes of imine imidazol-2-imine ligands

Free imine imidazol-2-imine ligands with two different substitution patterns have been isolated for the first time and they were found to exist as an equilibrium mixture of geometric and mesomeric isomers. The relative ratios of these isomers are dependent on both the nature of the substituents and of the solvent. The synthesis of the titanium(IV) alkyl and arylimido complexes of these ligands was unexpectedly found to be very selective and was successfully achieved only with the lesser sterically-demanding 2,4,6-trimethylphenyl derivative IMesN^Imine 2a. The solid-state structure of the alkylimido complex further confirms the zwitterionic character of the ligand. The isolated titanium imido complexes were found to be active catalysts for the polymerisation of ethylene.     

Ring-opening radical clock reactions: many density functionals have difficulty keeping time

The ability of several density-functional theory methods to describe the kinetics and energetics of a series of ring-opening reactions of cyclopropyl and cyclobutyl-type radicals was explored. PBE, B971 and B3LYP perform quite well in their ability to replicate experiment, based upon the ring opening of cyclopropylcarbinyl, two α-trialkylsilyloxycyclopropylmethyl radicals, pentamethylcyclopropylcarbinyl, cyclobutylcarbinyl and 1-cyclobutylethylcarbinyl. The other functionals tested, which includes BLYP, CAM-B3LYP, BHandHLYP, B2PLYP and B2PLYP-D, as well as functionals designed for kinetics applications, namely MPW1K, BMK and M06-2X, all perform poorly. The latter of these functionals display some integration grid dependencies.     

Regioselective cobalt-catalyzed formation of bicyclic 3- and 4-aminopyridines

Bimolecular cobalt-catalyzed [2 + 2 + 2] cycloadditions between yne-ynamides and nitriles afford bicyclic 3- or 4-aminopyridines in up to 100% yield. The high regioselectivity observed depends on the substitution pattern at the starting ynamide. Aminopyridines bearing TMS and Ts groups are efficiently deprotected in an orthogonal fashion.     

Click-chemistry as an efficient synthetic tool for the preparation of novel conjugated polymers

The Cu(I)-catalysed 1,3-dipolar "click" cycloaddition is utilised as an efficient reaction for the preparation of novel fluorene-based conjugated polymers.     

Line soliton interactions of the Kadomtsev-Petviashvili equation

We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4u(t)+6uu(x)+3u(xxx))(x)+u(yy)=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y-->infinity coincide with those of the N asymptotic line solitons as y-->-infinity. We also show that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and we calculate the soliton position shifts arising from the interactions.