Full counting statistics of weak measurement

A weak measurement consists in coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function for the moments of the output is provided. The outputs considered are not limited to the eigenstates of the pointer or of its conjugate variable, so that the results apply to any observable \Hat{o} of the probe. Furthermore, a family of well behaved complex quantities, the normal weak values, is introduced, in terms of which the statistics of the weak measurement can be described. It is shown that, within a good approximation, the whole statistics of weak measurement is described by a complex parameter, the weak value, and a real one.Comment: Expanded version: 9 pages, 3 Figs. Now the validity of the expansion for the moments is analysed. Introduced a one-parameter family of weak values, useful to express the correct characteristic function. More figures added. Thanks to Referee C of PRL for asking stimulating question

Conjugate two-dimensional electric potential maps

Two dimensional electric potential maps based on voltage detection in conducting paper are common practice in many physics courses in college. Most frequently, students work on `capacitor-like' geometries with current flowing between two opposite electrodes. A `topographical' investigation across the embedding medium (map of equipotential curves) allows to reassure a number of physical properties. This paper focuses on some less common configurations that bear pedagogical interest. We analyze `open-geometries' with electrodes in the form of long strips with slits. They provide a natural groundwork to bring the student to complex variable methods. Aided by this, we show that shaping the conducting paper board one may analyze finite size effects, as well as some meaningful discontinuities in the measured potential. The concept of conjugate electric potentials is exploited. Equipotentials and electric field lines acquire interchangeable roles and may be obtained in complementary `dual' experiments. A feasible theoretical analysis based on introductory complex variables and standardized numerics gives a remarkable quantification of the experimental results.Comment: 15 pages, 8 figure

Real-time growth rate for general stochastic SIR epidemics on unclustered networks

Networks have become an important tool for infectious disease epidemiology. Most previous theoretical studies of transmission network models have either considered simple Markovian dynamics at the individual level, or have focused on the invasion threshold and final outcome of the epidemic. Here, we provide a general theory for early real-time behaviour of epidemics on large configuration model networks (i.e. static and locally unclustered), in particular focusing on the computation of the Malthusian parameter that describes the early exponential epidemic growth. Analytical, numerical and Monte-Carlo methods under a wide variety of Markovian and non-Markovian assumptions about the infectivity profile are presented. Numerous examples provide explicit quantification of the impact of the network structure on the temporal dynamics of the spread of infection and provide a benchmark for validating results of large scale simulations.Comment: 45 pages, 8 figures, submitted to Mathematical Biosciences on 29/11/2014; Version 2: resubmitted on 15/04/2015; accepted on 17/04/2015. Changes: better explanations in introduction; restructured section 3.3 (3.3.3 added); section 6.3.1 added; more precise terminology; typos correcte

The geometry of modified Riemannian extensions

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3,3) whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four dimensional results in Osserman geometry