A catalyst-free one-step synthesis of N-pyrimidinyl amidines from endocyclic enamines and 4-azidopyrimidines

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Some Remarks on SLE Bubbles and Schramm’s Two-point Observable

Simmons and Cardy recently predicted a formula for the probability that the chordal SLE_8/3 path passes to the left of two points in the upper half-plane. In this paper we give a rigorous proof of their formula. Starting from this result, we derive explicit expressions for several natural connectivity functions for SLE_8/3 bubbles conditioned to be of macroscopic size. By passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting SLE_8/3 paths. The one-point version of this result has been predicted by Gamsa and Cardy. Finally, we derive an integral formula for the second moment of the area of an SLE_8/3 bubble conditioned to have radius 1. We evaluate the area integral numerically and relate its value to a hypothesis that the area follows the Airy distribution.     

Positive solutions of the porous medium equation with hysteresis

The problem under consideration is a degenerate parabolic equation with hysteretic terms. We establish existence and uniqueness of solutions with given initial data and investigate their main properties. An example of explicit self-similar solution is presented.     

Real zero polynomials and Pólya-Schur type theorems

Journal d'Analyse Mathématique -     

On Littlewood's Constants

In two papers, Littlewood studied seemingly unrelated constants:(i) the best such that for any polynomial f, of degree n, theareal integral of its spherical derivative is at most ·n,and (ii) the extremal growth rate rß of the lengthof Green's equipotentials for simply connected domains. Thesetwo constants are shown to coincide, thus greatly improvingknown estimates on . 2000 Mathematics Subject Classification30C50 (primary), 30C85, 30D35 (secondary).     

A Proof of Factorization Formula for Critical Percolation

We give mathematical proofs to a number of statements which appeared in the series of papers by Simmons et al. (Phys Rev E 76(4):041106, 2007; J Stat Mech Theory Exp 2009(2):P02067, 33, 2009) where they computed the probabilities of several percolation events.     

The critical pressure for flow in a porous medium

Bingham flow in a porous medium is considered. This can be modelled by a random structure whose dimensions are large compared with the local scale. The principal term of the asymptotic form of the critical pressure at which the liquid starts to move in this limit is computed explicitly.