Crump-Mode-Jagers Branching Process: Modelling and Application for Human Population

2010 Mathematics Subject Classification: 60J85, 92D25.

Mutually catalytic branching at infinite rate

The purpose of this doctoral thesis is to prove existence for a mutually catalytic random walk with infinite branching rate on countably many sites. The process is defined as a weak limit of an approximating family of processes. An approximating process is constructed by adding jumps to a deterministic migration on an equidistant time grid. As law of jumps we need to choose the invariant probability measure of the mutually catalytic random walk with a finite branching rate in the recurrent regime. This model was introduced by Dawson and Perkins (1998) and this thesis relies heavily on their work. Due to the properties of this invariant distribution, which is in fact the exit distribution of planar Brownian motion from the first quadrant, it is possible to establish a martingale problem for the weak limit of any convergent sequence of approximating processes. We can prove a duality relation for the solution to the mentioned martingale problem, which goes back to Mytnik (1996) in the case of finite rate branching, and this duality gives rise to weak uniqueness for the solution to the martingale problem. Using standard arguments we can show that this solution is in fact a Feller process and it has the strong Markov property. For the case of only one site we prove that the model we have constructed is the limit of finite rate mutually catalytic branching processes as the branching rate approaches infinity. Therefore, it seems naturalto refer to the above model as an infinite rate branching process. However, a result for convergence on infinitely many sites remains open.

Branching Processes in Autoregressive Random Environment

2000 Mathematics Subject Classification: 60J80, 60K05.

A New Class of Processes for Formalizing and Generalizing Individual-Based Models: The Semi-Semi-Markov Processes

2000 Mathematics Subject Classification: 60K15, 60K20, 60G20,60J75, 60J80, 60J85, 60-08, 90B15.

A Note on the Extinction Problem for Controlled Multitype Branching Processes

2000 Mathematics Subject Classification: 60J80, 60J10.

Branching processes and computational collapse of discretized unimodal mappings

In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.

Similarities and dissimilarities of branching and septation during lung development.

The lungs of small premature babies are at a developmental stage of finalizing their airway tree by a process called branching morphogenesis, and of creating terminal gas exchange units by a mechanism called septation. If the branching process is disturbed, the lung has a propensity to be hypoplastic. If septation is impaired, the terminal gas exchange units, the alveoli, tend to be enlarged and reduced in number, an entity known as bronchopulmonary dysplasia. Here, we review current knowledge of key molecules influencing branching and septation. In particular, we discuss the molecular similarities and dissimilarities between the two processes of airspace enlargement. Understanding of the molecular mechanisms regulating branching and septation may provide perinatologists with targets for improving lung growth and maturation.

Using Normative Markov Decision Processes for Evaluating Electronic Contracts: A Case Study in a Simulated Aerospace Aftermarket

Before signing electronic contracts, a rational agent should estimate the expected utilities of these contracts and calculate the violation risks related to them. In order to perform such pre-signing procedures, this agent has to be capable of computing a policy taking into account the norms and sanctions in the contracts. In relation to this, the contribution of this work is threefold. First, we present the Normative Markov Decision Process, an extension of the Markov Decision Process for explicitly representing norms. In order to illustrate the usage of our framework, we model an example in a simulated aerospace aftermarket. Second, we specify an algorithm for identifying the states of the process which characterize the violation of norms. Finally, we show how to compute policies with our framework and how to calculate the risk of violating the norms in the contracts by adopting a particular policy.

Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent t~3=2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

Green-Markov models - new optimization strategies: a case study for user allocation in co-channel macro/femto networks

ABSTRACT: The femtocell concept aims to combine fixed-line broadband access with mobile telephony using the deployment of low-cost, low-power third and fourth generation base stations in the subscribers' homes. While the self-configuration of femtocells is a plus, it can limit the quality of service (QoS) for the users and reduce the efficiency of the network, based on outdated allocation parameters such as signal power level. To this end, this paper presents a proposal for optimized allocation of users on a co-channel macro-femto network, that enable self-configuration and public access, aiming to maximize the quality of service of applications and using more efficiently the available energy, seeking the concept of Green networking. Thus, when the user needs to connect to make a voice or a data call, the mobile phone has to decide which network to connect, using the information of number of connections, the QoS parameters (packet loss and throughput) and the signal power level of each network. For this purpose, the system is modeled as a Markov Decision Process, which is formulated to obtain an optimal policy that can be applied on the mobile phone. The policy created is flexible, allowing different analyzes, and adaptive to the specific characteristics defined by the telephone company. The results show that compared to traditional QoS approaches, the policy proposed here can improve energy efficiency by up to 10%.

Green markov - uma abordagem híbrida de política markoviana e simulação discreta para planejamento de alocação de usuários em redes macro-femto

O uso da comunicação de voz e dados através de dispositivos móveis vem aumentando significativamente nos últimos anos. Tal expansão traz algumas dificuldades inerentes, tais como: ampliação constante de capacidade das redes e eficiência energética. Neste contexto, vem se consolidando o conceito de Green networks, que se concentra no esforço para economia de energia e redução de CO2. Neste sentido, este trabalho propõe validar um modelo de uma política baseado em processo markoviano de decisão, visando a otimizar o consumo de energia, QoS e QoE, na alocação de usuários em redes macrocell e femtocell. Para isso o modelo foi inserido no simulador NS-2, aliando a solução analítica markoviana à flexibilidade característica da simulação discreta. A partir dos resultados apresentados na simulação, a política obteve uma economia significativa no consumo energético, melhorando a eficiência energética em até 4%, além de melhorar a qualidade de serviço em relação às redes macrocell e femtocell, demonstrando-se eficaz, de modo a alterar diretamente as métricas de QoS e de QoE.

Budding Process During the Organogenesis of the Ventral Prostatic Lobe in Mongolian Gerbil

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Charakterisierung und Konvergenz mehrdimensionaler kontinuierlicher Verzweigungsprozesse

In dieser Arbeit wird eine Klasse von stochastischen Prozessen untersucht, die eine abstrakte Verzweigungseigenschaft besitzen. Die betrachteten Prozesse sind homogene Markov-Prozesse in stetiger Zeit mit Zuständen im mehrdimensionalen reellen Raum und dessen Ein-Punkt-Kompaktifizierung. Ausgehend von Minimalforderungen an die zugehörige Übergangsfunktion wird eine vollständige Charakterisierung der endlichdimensionalen Verteilungen mehrdimensionaler kontinuierlicher Verzweigungsprozesse vorgenommen. Mit Hilfe eines erweiterten Laplace-Kalküls wird gezeigt, dass jeder solche Prozess durch eine bestimmte spektral positive unendlich teilbare Verteilung eindeutig bestimmt ist. Umgekehrt wird nachgewiesen, dass zu jeder solchen unendlich teilbaren Verteilung ein zugehöriger Verzweigungsprozess konstruiert werden kann. Mit Hilfe der allgemeinen Theorie Markovscher Operatorhalbgruppen wird sichergestellt, dass jeder mehrdimensionale kontinuierliche Verzweigungsprozess eine Version mit Pfaden im Raum der cadlag-Funktionen besitzt. Ferner kann die (funktionale) schwache Konvergenz der Prozesse auf die vage Konvergenz der zugehörigen Charakterisierungen zurückgeführt werden. Hieraus folgen allgemeine Approximations- und Konvergenzsätze für die betrachtete Klasse von Prozessen. Diese allgemeinen Resultate werden auf die Unterklasse der sich verzweigenden Diffusionen angewendet. Es wird gezeigt, dass für diese Prozesse stets eine Version mit stetigen Pfaden existiert. Schließlich wird die allgemeinste Form der Fellerschen Diffusionsapproximation für mehrtypige Galton-Watson-Prozesse bewiesen.

Modeling the survival of T-cell lymphocytes using compound branching processes

Radiotherapy has been a method of choice in cancer treatment for a number of years. Mathematical modeling is an important tool in studying the survival behavior of any cell as well as its radiosensitivity. One particular cell under investigation is the normal T-cell, the radiosensitivity of which may be indicative to the patient's tolerance to radiation doses.^ The model derived is a compound branching process with a random initial population of T-cells that is assumed to have compound distribution. T-cells in any generation are assumed to double or die at random lengths of time. This population is assumed to undergo a random number of generations within a period of time. The model is then used to obtain an estimate for the survival probability of T-cells for the data under investigation. This estimate is derived iteratively by applying the likelihood principle. Further assessment of the validity of the model is performed by simulating a number of subjects under this model.^ This study shows that there is a great deal of variation in T-cells survival from one individual to another. These variations can be observed under normal conditions as well as under radiotherapy. The findings are in agreement with a recent study and show that genetic diversity plays a role in determining the survival of T-cells. ^

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.