Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction $F$ on ${\mathbb{R}}^{n}$, we construct a probability measure on the set of all solutions to the Cauchy problem $\dot x\in F(x)$ with $x(0)=0$. With probability one, the derivatives of these random solutions take values within the set $ext F(x)$ of extreme points for a.e.~time $t$. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.

Modelo de Kuramoto com campos aleatórios em redes complexas

Neste trabalho é estudado o modelo de Kuramoto num grafo completo, em redes scale-free com uma distribuição de ligações P(q) ~ q-Y e na presença de campos aleatórios com magnitude constante e gaussiana. Para tal, foi considerado o método Ott-Antonsen e uma aproximação "annealed network". Num grafo completo, na presença de campos aleatórios gaussianos, e em redes scale-free com 2 < y < 5 na presença de ambos os campos aleatórios referidos, foram encontradas transições de fase contínuas. Considerando a presença de campos aleatórios com magnitude constante num grafo completo e em redes scale-free com y > 5, encontraram-se transições de fase contínua (h < √2) e descontínua (h > √2). Para uma rede SF com y = 3, foi observada uma transição de fase de ordem infinita. Os resultados do modelo de Kuramoto num grafo completo e na presença de campos aleatórios com magnitude constante foram comparados aos de simulações, tendo-se verificado uma boa concordância. Verifica-se que, independentemente da topologia de rede, a constante de acoplamento crítico aumenta com a magnitude do campo considerado. Na topologia de rede scale-free, concluiu-se que o valor do acoplamento crítico diminui à medida que valor de y diminui e que o grau de sincronização aumenta com o aumento do número médio das ligações na rede. A presença de campos aleatórios com magnitude gaussiana num grafo completo e numa rede scale-free com y > 2 não destrói a transição de fase contínua e não altera o comportamento crítico do modelo de Kuramoto.

Distribuição de conteúdos em redes veiculares usando mecanismos de comunicação tolerantes ao atraso

The last couple of decades have been the stage for the introduction of new telecommunication networks. It is expected that in the future all types of vehicles, such as cars, buses and trucks have the ability to intercommunicate and form a vehicular network. Vehicular networks display particularities when compared to other networks due to their continuous node mobility and their wide geographical dispersion, leading to a permanent network fragmentation. Therefore, the main challenges that this type of network entails relate to the intermittent connectivity and the long and variable delay in information delivery. To address the problems related to the intermittent connectivity, a new concept was introduced – Delay Tolerant Network (DTN). This architecture is built on a Store-Carry-and-Forward (SCF) mechanism in order to assure the delivery of information when there is no end-to-end path defined. Vehicular networks support a multiplicity of services, including the transportation of non-urgent information. Therefore, it is possible to conclude that the use of a DTN for the dissemination of non-urgent information is able to surpass the aforementioned challenges. The work developed focused on the use of DTNs for the dissemination of non-urgent information. This information is originated in the network service provider and should be available on mobile network terminals during a limited period of time. In order to do so, four different strategies were deployed: Random, Least Number of Hops First (LNHF), Local Rarest Bundle First (LRBF) e Local Rarest Generation First (LRGF). All of these strategies have a common goal: to disseminate content into the network in the shortest period of time and minimizing network congestion. This work also contemplates the analysis and implementation of techniques that reduce network congestion. The design, implementation and validation of the proposed strategies was divided into three stages. The first stage focused on creating a Matlab emulator for the fast implementation and strategy validation. This stage resulted in the four strategies that were afterwards implemented in the DTNs software Helix – developed in a partnership between Instituto de Telecomunicac¸˜oes (IT) and Veniam R , which are responsible for the largest operating vehicular network worldwide that is located in Oporto city. The strategies were later evaluated on an emulator that was built for the largescale testing of DTN. Both emulators account for vehicular mobility based on information previously collected from the real platform. Finally, the strategy that presented the best overall performance was tested on a real platform – in a lab environment – for concept and operability demonstration. It is possible to conclude that two of the implemented strategies (LRBF and LRGF) can be deployed in the real network and guarantee a significant delivery rate. The LRBF strategy has the best performance in terms of delivery. However, it needs to add a significant overhead to the network in order to work. In the future, tests of scalability should be conducted in a real environment in order to confirm the emulator results. The real implementation of the strategies should be accompanied by the introduction of new types of services for content distribution.