1,721,019 research outputs found
Arithmetische progressies in random kleuringen van de natuurlijke getallen
No full textDeze scriptie bespreekt voldoende voorwaarden opdat een random deelverzameling van de natuurlijke getallen willekeurig lange arithmetische progressies bevat. Deze random deelverzameling maken we door een random kleuring van de natuurlijke getallen te definiëren. Bij de resultaten onderscheiden we in een zwakke wet van de grote aantallen en een sterke versie. Dit doen we voor zowel onafhankelijke kleuringen als afhankelijke, in het bijzonder kleuringen volgens een Markovketen.Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Asymptotic behaviour of Markov processes with two time scales
No full textMarkov processes are interesting stochastic proceses with a wide range of applications. In this report we study Markov processes with two time scales where the difference between the time speed of the scales is asymptotically large. In such a situation the behaviour of the process can be estimated by substituting the "faster" process by its limiting process. We will introduce Markov processes and study in particular Markov processes with two time scales in the limit where one of the time scales is much faster than the other. Later we will prove statements about the behaviour for a general situation. Finally, two interesting applications of this scenario will be discussed.Electrical Engineering, Mathematics and Computer ScienceDelft Institute of Applied Mathematic
Dualiteit in een model van welvaartsverdeling
No full textDualiteit is een handige manier om Markovprocessen te bestuderen. Hierbij wordt een in zekere zin moeilijk proces op een bepaalde manier met een eenvoudiger proces verbonden. Deze techniek zal in dit verslag worden toegelicht en zal toegepast worden op het Immediate Exchange Model, een welvaartverdelingsmodel waarin handelaren telkens na een exponentiële tijd een random fractie van hun welvaart aan elkaar afstaan. Dit model zal in hoofdstuk 2 geïntroduceerd worden. Ook zal voor het geval dat de random fracties een Beta(s,t) verdeling hebben het zogenaamde duale proces gedefinieerd worden en zal de dualiteit worden aangetoond. Met deze dualiteit zullen we laten zien dat gammaverdelingen met vormparameter s+t optreden als invariante maten voor dit proces en zal er een ergodiciteitsresultaat volgen. Verder zullen we alle invariante maten karakteriseren. Ook zal het proces gedefinieerd worden voor meerdere handelaren en zullen we laten zien dat het product van de al gevonden invariante maten in bepaalde situaties ergodisch is voor het model met aftelbaar oneindig veel handelaren. Verder zullen we kort een variant op dit proces beschouwen en een vermoeden hierover formuleren. Vervolgens zullen we in hoofdstuk 3 laten zien dat het duale proces gezien kan worden als een discrete versie van het oorspronkelijke proces door de welvaart van de handelaren in het duale proces als fracties van een groot getal N te beschouwen en N naar oneindig te laten gaan. We zullen dit in eenvoudige gevallen direct uit de formules afleiden en voor de algemene situatie een meer kanstheoretische benadering gebruiken. Tenslotte zullen we in hoofdstuk 4 ook voor het duale proces invariante maten vinden en zullen we aantonen dat dit proces voor s=t=1 zelfduaal is. Voor dit alles is nog een algemene introductie om het kader waarin gewerkt wordt te schetsen en de belangrijkste definities en resultaten te geven en toe te lichten en wordt er een motivatie gegeven voor het onderzoek dat in dit verslag wordt gepresenteerd.Applied ProbabilityApplied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Large Deviations for Markov Jump Processes and Hamiltonian Trajectories
No full textThis master thesis is concerned with Large Deviation Theory in combination with Lagrangian and Hamiltonian dynamics. In particular, the Large Deviation behaviour of the empirical distribution of n independent two-state continuous-time Markov processes is studied. We start by looking at the general theory of Large Deviations in both the finite dimensional case as well as for infinite dimensional stochastic processes. After this, the connection is shown of Large Deviation Theory with Lagrangian and Hamiltonian dynamics. The Hamiltonian of the empirical distribution of n independent two-state Markov processes is derived and using this, via the Hamilton equations, the dynamics of this process are derived. That is, the most likely path that is taken when in time T we force the path to start in state a and end in state b. The main goal of this thesis is to find out more about the dynamics of this process (behaviour of the trajectories) and to derive an explicit equation for the so-called Action integral. We want to compute the Action for the general case. That is, the case in which the rate going from state one to state two can differ from the rate going from state two to state one. Once the Action integral is computed we look at the asymptotic behaviour of this Action integral. This is important as it says something about the probability of some trajectories occurring for T an extreme. We look at the asymptotics for both T in the limit to zero and T in the limit to infinity.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Hydrodynamic Limit for the Symmetric Inclusion Process
No full textThis thesis deals with the proof of the hydrodynamic limit for the symmetric inclusion process (SIP). After introduction of our process and some basic concepts on the theory of Markov processes we proved self-duality for the SIP. Once equipped with SIP self-duality, and together with an assumption on the initial configuration, we were able to prove that the evolution, on the diffusive scale, of the empirical density of the symmetric inclusion processes in is described by a weak solution of the heat equation.Electrical Engineering, Mathematics and Computer ScienceDIA
Dissipation in the Abelian Sandpile Model
No full textThe Abelian Sandpile model was originally introduced by Bak, Tang and Wiesenfeld in 1987 as a paradigm for self-organized criticality. In this thesis, we study a variant of this model, both from the point of view of mathematics, as well as from the point of view of physics. The effect of dissipation and creation of mass is investigated. By linking the avalanche dynamics of the infinite-volume sandpile model to random walks, we derive some criteria on the amount of dissipation and creation of mass in order for the model to be critical or non-critical. As an example we prove that a finite amount of conservative sites on a totally dissipative lattice is not critical, and more generally, if the distance to a dissipative site is uniformly bounded from above, then the model is not critical. We apply also applied a renormalisation method to the model in order to deduce its critical exponents and to determine whether a constant bulk dissipation destroys critical behaviour. Numerical simulations and a statistical analysis are performed to estimate critical exponents. Finally, we give a short discussion on self-organized criticality.Electrical Engineering, Mathematics and Computer ScienceDouble Bachelor Applied Mathematics/Applied Physic
Coupling, Concentration and Random Walks in Dynamic Random Environments
No full textIn this thesis we discuss concentration inequalities, relaxation to equilibrium of stochastic dynamics, and random walks in dynamic random environments. In stochastic systems one is interested in macroscopic and/or asymptotic properties as well as in fluctuations around typical behaviour. But the dependence structure induced by the interaction between the components of the system makes the analysis challenging. In order to overcome this in different settings a variety of methods are employed. Additive functionals of Markov processes play important roles in applications. In order to get exponential and moment estimates for their fluctuations a non-standard martingale approximation is used. The resulting general theorems do not require special properties like reversibility or a spectral gap. What is needed is some control on the expected evolution. That is, the difference of the evolution starting from two "adjacent'' configurations has to be controlled. Coupling methods are well suited to do perform this comparison. In concrete examples couplings are used to prove the conditions of the theorems. In statistical mechanics Gibbs measures and Markov random fields play important roles. The Poincaré inequality is an important property describing the regularity of the measure. We prove the Poincaré inequality via a martingale telescoping argument. To control the individual increments of the martingale we use a coupling method called disagreement percolation. If the clusters of this percolation are sufficiently small we obtain the Poincaré inequality. When interacting spin systems and their dynamics have a delicate connection to their ergodic measure(s) one has to take more care. We carefully study the graphical construction of the dynamics to understand how the influence of the measure can be preserved. An assumption is made that one can control how fast the system in equilibrium can compensate for a single spin flip. Under this assumption we obtain relaxation speed estimates for general functions. In attractive spin systems the condition can be reduced to the decay of auto-correlation of the spin at the origin. An application where this is of use is the low-temperature Ising model. Finally we look at random walks in dynamic random environments. Here a time-changing random environment drives the motion of a particle. The goal is to understand under which conditions the macroscopic behaviour of this random walk is like that of a Brownian motion. We use coupling to prove a law of large numbers as well as a functional central limit theorem for the position of the random walk. Only polynomial decay of correlations in time are needed for the environment, and the influence of the environment on the walk can be very general.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Prediction of Market Value of Used Commercial Aircraft
No full textAviation financers are interested in the current/future market value of used commercial aircraft, as this information is precious knowledge for them to support collateral position in the aircraft loan. In this paper, variables from general economy, airline industry and aviation fleet are explored to find out the factors predictive for used aircraft market. Two statistical methods- principal component regression and copula-are applied for building the prediction model of an aircraft.Risk and Environmental Modeling groupApplied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Symmetric interacting particle systems: Self-duality and hydrodynamics in dynamic random environment
No full textIn this thesis, we study scaling and detailed properties of a class of conservative interacting particle systems. In particular, in the first part we derive the hydrodynamic equation for the symmetric exclusion process in presence of dynamic random environment. The second part of the thesis focuses on a detailed property of conservative particle systems and, more generally, of Markov processes, called duality. There, we study and characterize duality and self-duality relations for so-called symmetric interacting particle systems
Random Polymers: Interaction with a Penetrable Surface
No full textWe introduce a polymer measure to describe a polymer that is attracted to the origin and repelled by the negative half-plane. We work out a direct expression for the partition function and look at properties of the free energy. We look at the model when repellance becomes attraction and when the repellance occurs in a random fashion. We prove several results concerning localization of the polymer, one of which is that the localized phase corresponds to positivity of the free energy. Finally, we construct a tailored Metropolis Algorithm to simulate the distribution of the polymer, and discuss preliminary results of the simulation.Probability TheoryApplied mathematicsElectrical Engineering, Mathematics and Computer Scienc
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