Spectral partitioning of time-varying networks with unobserved edges

We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying network is different for every observation. In this fashion, the filtered graph signals can be interpreted as defined on a time-varying network. We model each of the underlying network realizations as generated by an independent draw from a latent stochastic blockmodel (SBM). To infer the partition of the latent SBM, we propose a simple spectral algorithm for which we provide a theoretical analysis and establish consistency guarantees for the recovery. We illustrate our results using numerical experiments on synthetic and real data, highlighting the efficacy of our approach.Comment: 5 pages, 2 figure

The Auslander-Reiten Translation in Submodule Categories

Let $\Lambda$ be an artin algebra and $S(\Lambda)$ the category of all embeddings $(A\subseteq B)$ where $B$ is a finitely generated $\Lambda$-module and $A$ is a submodule of $B$. Then $S(\Lambda)$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $S(\Lambda)$ can be computed within the category of $\Lambda$-modules by using our construction of minimal monomorphisms. If in addition $\Lambda$ is uniserial then any nonprojective indecomposable object in \Cal S(\Lambda) is invariant under the sixth power of the Auslander-Reiten translation.Comment: Dedicated to Idun Reite

On the non-integrability of the Popowicz peakon system

We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev\'e analysis provides strong evidence that the Popowicz system is non-integrable. Nevertheless, we are able to construct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N.Comment: 8 pages, AIMS class file. Proceedings of AIMS conference on Dynamical Systems, Differential Equations and Applications, Arlington, Texas, 200

Grothendieck ring of pretriangulated categories

We consider the abelian group $PT$ generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of "multiplication" $\bullet$ on the collection of DG categories which makes this abelian group into a commutative ring. A few applications are considered: representability of "standard" functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.Comment: 33 page

Residential Water Consumption: A Cross Country Analysis

Survey data from over 1,600 households in ten countries were used to analyse the determinants of residential water demand. Results show that in every country the price elasticity is negative and statistically significant. Households that do not have to pay for the water they use (volumetric water charges) consume about a third more water than similar households that do have to pay such charges. Consumers’ attitudes do not have a statistically significant effect on total water use, although they do increase the probability of households using some water saving behaviours. Volumetric water charges also have an impact on the adoption of water saving actions. Full-cost water pricing appears to be a highly effective instrument to manage residential water demand.water demand, water consumption, water pricing, Environmental Economics and Policy, Resource /Energy Economics and Policy, C21, Q25, Q50,