A power transmission system can be represented by a network with nodes and links representing buses and electrical transmission lines, respectively. Each line can be given a weight, representing some electrical property of the line, such as line admittance or average power flow at a given time. We use a hierarchical spectral clustering methodology to reveal the internal connectivity structure of such a network. Spectral clustering uses the eigenvalues and eigenvectors of a matrix associated to the network, it is computationally very efficient, and it works for any choice of weights. When using line admittances, it reveals the static internal connectivity structure of the underlying network, while using power flows highlights islands with minimal power flow disruption, and thus it naturally relates to controlled islanding. Our methodology goes beyond the standard -means algorithm by instead representing the complete network substructure as a dendrogram. We provide a thorough theoretical justification of the use of spectral clustering in power systems, and we include the results of our methodology for several test systems of small, medium and large size, including a model of the Great Britain transmission network.

Abstract. We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C *

The main results of this paper show that various coarse ('large scale') geometric properties are closely related. In particular, we show that property A implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated.The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some 'bad' spaces. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting.

We prove that the asymptotic dimension of a finite-dimensional CAT.0/ cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT.0/ cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups. 20F65, 20F69, 54F45

Coarse median spaces simultaneously generalise the classes of hyperbolic spaces and median algebras, and arise naturally in the study of the mapping class groups and many other contexts. Their definition as originally conceived by Bowditch requires median approximations for all finite subsets of the space. Here we provide a simplification of the definition in the form of a 4-point condition analogous to Gromov's 4-point condition defining hyperbolicity. We give an intrinsic characterisation of rank in terms of the coarse median operator and use this to give a direct proof that rank 1 geodesic coarse median spaces are δhyperbolic, bypassing Bowditch's use of asymptotic cones. A key ingredient of the proof is a new definition of intervals in coarse median spaces and an analysis of their interaction with geodesics.