Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk

Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.

A sharp stability estimate for tensor tomography in non-positive curvature

We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $$L^2\mapsto H^{1/2}_{T}$$ L 2 ↦ H T 1 / 2 , where the $$H^{1/2}_{T}$$ H T 1 / 2 -space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.

The Skew Normal Distribution

This chapter considers the univariate skew-normal distribution, a generalization of the normal that includes the normal as a special case. The most natural parametrization is non-standard. This is because the Fisher information matrix is then singular at the true parameter value when the true model is the normal special case. The log-likelihood is then particularly flat in a certain coordinate direction. Standard theory cannot then be used to calculate the asymptotic distribution of all the parameter estimates. This problem can be handled using an alternative parametrization. There is another special case: the half/folded normal distribution. This occurs in the usual parametrization when the shape parameter is infinite. This is not a problem computationally and is easily handled. There are many generalizations to skew-t distributions and to tractable multivariate forms and regression versions. A very brief review is included of these.

Hausdorff measures, Hölder continuous maps and self-similar fractals

Let f: A → ℝn be Hölder continuous with exponent α, 0 < α ≼ 1, where A ⊂ ℝm has finite m-dimensional Lebesgue measure. Then, as is easy to see and well-known, the s-dimensional Hausdorif measure HS(fA) is finite for s = m/α. Many fractal-type sets fA also have positive Hs measure. This is so for example if m = 1 and f is a natural parametrization of the Koch snow flake curve in ℝ2. Then s = log 4/log 3 and α = log 3/log 4. In this paper we study the question of what s-dimensional sets in can intersect some image fA in a set of positive Hs measure where A ⊂ ℝm and f: A → ℝn is (m/s)-Hölder continuous. In Theorem 3·3 we give a general density result for such Holder surfacesfA which implies for example that Hs(E fA) = 0 for any totally disconnected self-similar set. E in this situation. In Theorem 32 we shall first show that such fA has positive s-dimensional lower density H8 almost everywhere.