We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive re...

In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by their period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that we use is to solve a Jenkins-Serrin problem over a strip domain and to consider the conjugate minimal surface to the graph.

We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the half-plane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5]. In recent years, matroid theory has found connections with certain analytic properties of real multivariate polynomials. These properties are ...

Abstract Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure of the size of a set in the reference upper half-plane. This has useful harmonic and com...

For example, x1 + · · · + xd ∈ Ud (C). This follows from the fact that the upper half plane is a cone, so if σ1, . . . , σd are in the upper half plane then so is their sum. Another example is x1x2 − 1. If σ1 and σ2 are in the upper half plane then σ1σ2 ∈ C \ (0,∞), so σ1σ2 − 1 is not zero. U1 (C) is easily described. It is all polynomials in one variable whose roots are either real, or lie in ...

Let A be a bounded, relatively closed subset of the upper half plane H whose complement in H is simply connected. If Bt is a standard complex Brownian motion and τA = inf{t ≥ 0 : Bt 6∈ H \A}, the half-plane capacity hcap(A) is defined as hcap(A) := lim y→∞ y E [Im(BτA)] . This quantity arises in the study of Schramm-Loewner Evolutions (SLE). In this note, we show that hcap(A) is comparable to a...

The p-adic upper half plane X is a rigid analytic variety over a p-adic field K, on which the group GL2(K) acts, that Mumford introduced (as a formal scheme) as part of his efforts to generalize Tate’s p-adic uniformization of elliptic curves to curves of higher genus. The Cp–valued points of X are just P(Cp)−P(K), with GL2(K) acting by linear fractional transformations. Mumford showed that the...

Murota et al. have recently developed a theory of discrete convex analysis which concerns M -convex and L-convex functions on jump systems. We introduce here a family of M -concave functions arising naturally from polynomials (over the field of Puiseux series) with prescribed non-vanishing properties. This family contains several of the most studied M -concave functions in the literature. We al...

Elastic Optical Networks have emerged as a promising technology for the efficient use of optical network resources. Its adaptable characteristics and adjustable data rate enable operators to meet the diverse granularity of their clients needs. In order to automate an elastic optical network operation, a control plane is required. Wavelength Switched Optical Networks (WSON) may already rely on a...