Well-posedness for a coagulation multiple-fragmentation equation

We consider a coagulation multiple-fragmentation equation, which describes the concentration $c\_t(x)$ of particles of mass $x \in (0,\infty)$ at the instant $t \geq 0$ in a model where fragmentation and coalescence phenomena occur. We study the existence and uniqueness of measured-valued solutions to this equation for homogeneous-like kernels of homogeneity parameter $\lambda \in (0,1]$ and bounded fragmentation kernels, although a possibly infinite total fragmentation rate, in particular an infinite number of fragments, is considered. This work relies on the use of a Wasserstein-type distance, which has shown to be particularly well-adapted to coalescence phenomena. It was introduced in previous works on coagulation and coalescence

Stochastic Coalescence Multi-Fragmentation Processes

We study infinite systems of particles which undergo coalescence and fragmentation, in a manner determined solely by their masses. A pair of particles having masses $x$ and $y$ coalesces at a given rate $K(x,y)$. A particle of mass $x$ fragments into a collection of particles of masses $\theta\_1 x, \theta\_2 x, \ldots$ at rate $F(x) \beta(d\theta)$. We assume that the kernels $K$ and $F$ satisfy H\"older regularity conditions with indices $\lambda \in (0,1]$ and $\alpha \in [0, \infty)$ respectively. We show existence of such infinite particle systems as strong Markov processes taking values in $\ell\_{\lambda}$, the set of ordered sequences $(m\_i)\_{i \ge 1}$ such that \sum\_{i \ge 1} m\_i^{\lambda} \textless{} \infty. We show that these processes possess the Feller property. This work relies on the use of a Wasserstein-type distance, which has proved to be particularly well-adapted to coalescence phenomena.Comment: arXiv admin note: substantial text overlap with arXiv:1301.193

The Most Exigent Eigenvalue: Guaranteeing Consensus under an Unknown Communication Topology and Time Delays

This document aims to answer the question of what is the minimum delay value that guarantees convergence to consensus for a group of second order agents operating under different protocols, provided that the communication topology is connected but unknown. That is, for all the possible communication topologies, which value of the delay guarantees stability? To answer this question we revisit the concept of most exigent eigenvalue, applying it to two different consensus protocols for agents driven by second order dynamics. We show how the delay margin depends on the structure of the consensus protocol and the communication topology, and arrive to a boundary that guarantees consensus for any connected communication topology. The switching topologies case is also studied. It is shown that for one protocol the stability of the individual topologies is sufficient to guarantee consensus in the switching case, whereas for the other one it is not

Smoluchowski's equation: rate of convergence of the Marcus-Lushnikov process

We derive a satisfying rate of convergence of the Marcus-Lushnikov process toward the solution to Smoluchowski's coagulation equation. Our result applies to a class of homogeneous-like coagulation kernels with homogeneity degree ranging in $(-\infty,1]$. It relies on the use of a Wasserstein-type distance, which has shown to be particularly well-adapted to coalescence phenomena.Comment: 34 Page

Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function

This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments

Report of Investigations No. 135 Oligocene Volcanism and Multiple Caldera Formation in the Chinati Mountains, Presidio County, Texas

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