Apollonian structure in the Abelian sandpile

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.     

Convexity Criteria and Uniqueness of Absolutely Minimizing Functions

We show that an absolutely minimizing function with respect to a convex Hamiltonian \({H : \mathbb{R}^{n} \rightarrow \mathbb{R}}\) is uniquely determined by its boundary values under minimal assumptions on H. Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in L^∞.     

UV promoted phenanthridine syntheses from oxime carbonate derived iminyl radicals

Oxime carbonates were found to be excellent precursors for the clean and direct generation of iminyl radicals under UV irradiation. Suitably functionalised iminyls underwent cyclisations yielding various phenanthridines and also substituted quinolines and isoquinolines. EPR and X-ray analyses of oxime carbonates provided insight into the mechanism.     

o-Nitrobenzylidene compounds. I—Competitive ortho interactions in the mass spectra of n-o-nitrobenzylidene-o-substituted anilines

The intense [M – 17]⁺ ion, which is a characteristic feature of the mass spectra of N-o-nitrobenzylideneaniline and its simple derivatives, may be substantially reduced in intensity when the aniline-derived ring is also ortho substituted: the intensity is lowest when this ortho substituent has a nucleophilic character and can itself interact with the CH?N group. The ortho substituents examined are Br, CH₃, OH, NH₂, SH, SC₆H₄CH₃(p), CO₂CH₃ and CO₂C₂H₅.