Partial measures

We study sigma-additive set functions defined on a hereditary subclass of a sigma-algebra and taken values in the extended real line. Analogs of the Jordan decomposition theorem and the Radon-Nikodym theorem are obtained.Comment: 4 pages. Submitted to Lobachevskii Journal of Mathematics ( http://ljm.ksu.ru

Radon-Nikodym derivatives of quantum operations

Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all CP maps $S$ such that $T-S$ is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon-Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic treatments. We also comment on how the Radon-Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.Comment: 22 pages; final versio

El teorema de Radon-Nikodym y sus consecuencias

El propósito de este trabajo es dar una prueba del Teorema de Radon-Nikodym, y también presentar algunas de sus consecuencias más importantes. En primer lugar, presentamos algunas propiedades preliminares sobre medidas. Para entender el Teorema de Radon-Nikodym definimos la continuidad absoluta de una medida. Una vez tenemos las herramientas necesarias, probamos el Teorema de Radon-Nikodym para una medida positiva y finita de dos formas diferentes. Después de esto, extendemos el teorema a otros tipos de medidas y damos algunas consecuencias importantes, como por ejemplo el Teorema de Descomposición de Jordan. Con el Teorema de Radon-Nikodym somos capaces de probar un resultado muy importante: la identificación isométrica del dual de los espacios L p para 1 ≤ p < ∞. Después de presentar la teoría para medidas positivas, reales y complejas, proporcionamos algunas definiciones importantes y resultados sobre medidas vectoriales, y probamos que en este caso el Teorema de Radon-Nikodym no se cumple en general. Finalmente, presentamos alguna condiciones bajo las cuales se cumple el Teorema de Radon Nikodym para medidas vectoriales.The aim of this project is to give a proof of the Radon-Nikodym Theorem; we also present some of its most important consequences. First of all, we present some preliminary properties about measures. In order to understand the Radon-Nikodym Theorem, we define the absolute continuity of a measure. Once we have all the necessary tools, we prove the Radon-Nikodym Theorem for a positive and finite measure in two different ways. After that, we extend the theorem to differents types of measures and we give some important consequences, like, for example, the Jordan Descomposition Theorem. With the Radon-Nikodym Theorem we are able to prove a very important result: the isometric identification of the dual of the L p spaces for 1 ≤ p < ∞. After presenting the theory for positive, real and complex measures, we provide some important definitions and results for vector measures and we prove that in this case the Radon-Nikodym Theorem does not hold in general. Finally, present some conditions which ensure the validity of the Radon Nikodym Theorem for vector measuresUniversidad de Sevilla. Grado en Matemática

Modeling micro-macro pedestrian counterflow in heterogeneous domains

We present a micro-macro strategy able to describe the dynamics of crowds in heterogeneous media. Herein we focus on the example of pedestrian counterflow. The main working tools include the use of mass and porosity measures together with their transport as well as suitable application of a version of Radon-Nikodym Theorem formulated for finite measures. Finally, we illustrate numerically our microscopic model and emphasize the effects produced by an implicitly defined social velocity. Keywords: Crowd dynamics; mass measures; porosity measure; social network