Nondispersive solutions to the L2-critical half-wave equation

We consider the focusing $L^2$-critical half-wave equation in one space dimension $$i \partial_t u = D u - |u|^2 u,$$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u \|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2} \geq M_*$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page

The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear equations is constructed in the form of an expansion over particle clusters whose evolution is described by the corresponding order cumulant (semi-invariant) of evolution operators for the von Neumann equations. For the initial data from the space of sequences of trace class operators the existence of a strong and a weak solution of the Cauchy problem is proved. We discuss the relationships of this solution both with the $s$-particle statistical operators, which are solutions of the BBGKY hierarchy, and with the $s$-particle correlation operators of quantum systems.Comment: 26 page

Dynamical Boson Stars

The idea of stable, localized bundles of energy has strong appeal as a model for particles. In the 1950s John Wheeler envisioned such bundles as smooth configurations of electromagnetic energy that he called {\em geons}, but none were found. Instead, particle-like solutions were found in the late 1960s with the addition of a scalar field, and these were given the name {\em boson stars}. Since then, boson stars find use in a wide variety of models as sources of dark matter, as black hole mimickers, in simple models of binary systems, and as a tool in finding black holes in higher dimensions with only a single killing vector. We discuss important varieties of boson stars, their dynamic properties, and some of their uses, concentrating on recent efforts.Comment: 79 pages, 25 figures, invited review for Living Reviews in Relativity; major revision in 201

The violent youth of bright and massive cluster galaxies and their maturation over 7 billion years

In this study, we investigate the formation and evolution mechanisms of the brightest cluster galaxies (BCGs) over cosmic time. At high redshift (z ∼ 0.9), we selected BCGs and most massive cluster galaxies (MMCGs) from the Cl1604 supercluster and compared them to low-redshift (z ∼ 0.1) counterparts drawn from the MCXC meta-catalogue, supplemented by Sloan Digital Sky Survey imaging and spectroscopy. We observed striking differences in the morphological, colour, spectral, and stellar mass properties of the BCGs/MMCGs in the two samples. High-redshift BCGs/MMCGs were, in many cases, star-forming, late-type galaxies, with blue broad-band colours, properties largely absent amongst the low-redshift BCGs/MMCGs. The stellar mass of BCGs was found to increase by an average factor of 2.51 ± 0.71 from z ∼ 0.9 to z ∼ 0.1. Through this and other comparisons, we conclude that a combination of major merging (mainly wet or mixed) and in situ star formation are the main mechanisms which build stellar mass in BCGs/MMCGs. The stellar mass growth of the BCGs/MMCGs also appears to grow in lockstep with both the stellar baryonic and total mass of the cluster. Additionally, BCGs/MMCGs were found to grow in size, on average, a factor of ∼3, while their average Sérsic index increased by ∼0.45 from z ∼ 0.9 to z ∼ 0.1, also supporting a scenario involving major merging, though some adiabatic expansion is required. These observational results are compared to both models and simulations to further explore the implications on processes which shape and evolve BCGs/MMCGs over the past ∼7 Gyr

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page

The TiO2 Ru dcbpyH2 2 NCS 2 Au Schottky junction

Au TiO2 contacts modified with a monolayer of Ru dcbpyH2 2 NCS 2 molecules , Thin Solid Films 516 2008 7234 7236

Inline Co-Evolution between Users and Information Presentation for Data Exploration

International audienceThis paper presents an intelligent user interface model dedicated to the exploration of complex databases. This model is implemented on a 3D metaphor : a virtual museum. In this metaphor, the database elements are embodied as museum objects. The objects are grouped in rooms according to their semantic properties and relationships and the rooms organization forms the museum. Rooms organization is not predefi-ned but defined incrementally by taking into account not only the relationships between objects, but also the users centers of interest. The latter are evaluated in real-time through user interactions within the virtual museum. This interface allows for a personal reading and favors the discovery of unsuspec-ted links between data. In this paper, we present our model's formalization as well as its application to the context of cultural heritage