Gene regulatory network underlying neural crest formation

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First and second moments for self-similar couplings and Wasserstein distances

We study aspects of the Wasserstein distance in the context of self-similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of \emph{couplings}, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self-similar measures associated to equicontractive iterated function systems satisfying the open set condition and consisting of two maps on the unit interval. We are particularly interested in understanding the restricted family of \emph{self-similar} couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non-trivial upper and lower bounds for the 2nd Wasserstein distance.Comment: 14 pages, 3 figure

Remarks on the analyticity of subadditive pressure for products of triangular matrices

We study Falconer's subadditive pressure function with emphasis on analyticity. We begin by deriving a simple closed form expression for the pressure in the case of diagonal matrices and, by identifying phase transitions with zeros of Dirichlet polynomials, use this to deduce that the pressure is piecewise real analytic. We then specialise to the iterated function system setting and use a result of Falconer and Miao to extend our results to include the pressure for systems generated by matrices which are simultaneously triangularisable. Our closed form expression for the pressure simplifies a similar expression given by Falconer and Miao by reducing the number of equations needing to be solved by an exponential factor. Finally we present some examples where the pressure has a phase transition at a non-integer value and pose some open questions.Comment: 10 pages, 1 figure, to appear in Monatshefte f\"ur Mathemati

Inhomogeneous self-similar sets and box dimensions

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than the upper box dimension, Hausdorff dimension and packing dimension.Comment: To appear in Studia Mathematica, 20 pages, 4 figure

Assouad type dimensions and homogeneity of fractals

We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, like the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.Comment: 40 pages, 6 figure