Conformal Theory of the Dimensions of Diffusion Limited Aggregates

We employ the recently introduced conformal iterative construction of Diffusion Limited Aggregates (DLA) to study the multifractal properties of the harmonic measure. The support of the harmonic measure is obtained from a dynamical process which is complementary to the iterative cluster growth. We use this method to establish the existence of a series of random scaling functions that yield, via the thermodynamic formalism of multifractals, the generalized dimensions D(q) of DLA for q >= 1. The scaling function is determined just by the last stages of the iterative growth process which are relevant to the complementary dynamics. Using the scaling relation D(3) = D(0)/2 we estimate the fractal dimension of DLA to be D(0) = 1.69 +- 0.03.Comment: 5 pages, 3 figures, submitted to Phys. Rev. Let

Isoscattering on surfaces

We give a number of examples of pairs of non-compact surfaces which are isoscattering, and which are exceptionally simple in one or more senses. We give examples which are of small genus with a small number of ends, and also examles which are congruence surfaces.Comment: 21 page

On Arithmetic Modular Categories

Modular categories are important algebraic structures in a variety of subjects in mathematics and physics. We provide an explicit, motivated and elementary definition of a modular category over a field of characteristic 0 as an equivalence class of solutions to a set of polynomial equations. We conclude that within each class of solutions, there is one which consists entirely of algebraic numbers. These algebraic solutions make it possible to discuss defining algebraic number fields of modular categories and their Galois twists. One motivation for such a definition is an arithmetic theory of modular categories which plays an important role in their classification. Another is to facilitate implementation of computer-based tools to resolve computational and classification problems intractible by other means. We observe some basic properties of Galois twists of modular categories and make conjectures about their relation to the the intrinsic data of modular categories.Comment: 57 page

Identification of the prebiotic translation apparatus within the contemporary ribosome

A structural element that could have existed independently in the prebiotic era was identified at the active site of the contemporary ribosome. It is suggested to have functioned as a proto-ribosome catalyzing peptide bond formation and non-coded elongation in the same manner that contemporary ribosomes exert positional catalysis, namely by accommodating the reactants in stereochemistry favourable for inline nucleophilic attack. This simple apparatus is a dimer of self-folding RNA units that could have assembled spontaneously into a symmetrical pocket-like structure, sufficiently efficient to be preserved throughout evolution as the active site of modern ribosomes, thus presenting a conceivable starting point for translation.Here we discuss the proto-ribosome emergence hypothesis and show that the tendency for dimerization, a prerequisite for obtaining the catalytic centre, is linked to the fold of its two components, indicating functional selection at the molecular level in the prebiotic era and supporting the existence of dimeric proto-ribosome