21,911 research outputs found
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
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