Consider a feedforward network of single-server stations populated by multiple job types. Each job requires the completion of a number of tasks whose order of execution is determined by a set of deterministic precedence constraints. The precedence requirements allow some tasks to be done in parallel (in which case tasks would "fork") and require that others be processed sequentially (where tasks may "join"). Jobs of a. given type share the same precedence constraints, interarrival time distributions, and service time distributions, but these characteristics may vary across different job types. We show that the heavy traffic limit of certain processes associated with heterogeneous fork-join networks can be expressed as a semimartingale reflected Brownian motion with polyhedral state space. The polyhedral region typically has many more faces than its dimension, and the description of the state space becomes quite complicated in this setting. One can interpret the proliferation of additional faces in heterogeneous fork-join networks as (i) articulations of the fork and join constraints, and (ii) results of the disordering effects that occur when jobs fork and join in their sojourns through the network.

We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice $Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)$. In a first part, these processes are characterized as Gibbs states on path spaces of the form $C([0, T],R)Z^{d}$. In a second part, we study the Gibbsian character on $R^{Z}^{d}$ of $v^{t}$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law $v = v^{0}$ is Gibbsian.

We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion / when the initial distribution is Gibbsian.

The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. <br><br> AMS Classifications: 60G15 / 60G60 / 60H10 / 60J60

It is a research of the move trajectory of the ciliates while feeding the food, in order to estimate, differentiate from the movement behavior under different environments. First, discuss the differently distinguish with the single indicator. Second, discuss with integrate four kinds of indicator whether can distinguish differently. Finally, combine the indicator data and through different analysis technology look out the features of movement behavior, expect to be able to look out suitable information and knowledge from the indicator data. After deal with analytical technology, the result of decision tree is most suitable for predicted and have credibilities. If according to energy of biological, the analysis result is similar to optimal foraging theory. And learn from result under different condition, the movement behavior of the ciliates similar to the optimal foraging theory. In the matter of the result of analysis technology, data of the density of low food similar to data of the density of extremely high food. Besides, data of medium food and high food are analogous. The rule of decision tree can distinguish the density of different food, and can offer follow-up study to distinguish the environmental conditions. Those models are evaluated by predicting accuracies, and rules extracted from decision tree models are also of great help to prediction as well.

Lévy flights and Lévy walks are two mathematical models used to describe anomalous diffusion(i.e. those having mean square displacements nonlinearly related to time (as opposed to Brownian motion)). Lévy flights follow probability distributions p(|r|) yielding infinite mean square displacements since some rare steps are very long. Lévy walks, however, have coupled space-time probability distributions penalising very long steps. Both Lévy flights and Lévy walks are dominated by a few long steps, but most steps are much, much smaller. The semi-experimental part ofthis work dealt with how fluorescent probes moved in systems of cationic starch and latex/solutions of dodecyl trimethyl ammonium bromide, respectively. Visually, no Lévy walks couldbe detected. However, mathematical regression suggested enhanced diffusion and subdiffusion. Moreover, time-dependent diffusion coefficients were calculated. Also examined was how Microsoft Excel could be used to generate normal diffusion as well as anomalous diffusion. / Lévyflygningar och Lévypromenader är matematiska modeller som används för att beskriva anomal diffusion (i.e. dessa då medelvärdet av kvadratförflyttningarna är icke-linjärt relaterat tilltiden (till skillnad från Brownsk rörelse)). Lévyflygningar följer sannolikhetsfördelningar p(|r|)som ger oändliga medelkvadratförflyttningar eftersom vissa steg är väldigt långa. Lévypromenader,å andra sidan, har kopplade rum-tid-sannolikhetsfördelningar som kraftigt reducerar demycket långa stegen. Både Lévyflygningar och -promenader domineras av ett fåtal långa steg ävenom de flesta steg är mycket, mycket mindre. Den semiexperimentella delen av detta arbetestuderade hur fluorescerande prober rör sig i katjonisk stärkelse respektive latex/lösningar avdodecyltrimetylammoniumbromid. Inga Lévypromenader kunde ses. Emellertid taladematematisk regression för att superdiffusion och subdiffusion förelåg. Tidsberoende diffusionskoefficienter beräknades också. I detta arbete undersöktes även hur Microsoft Excel kan användas för att generera både normal och anomal diffusion.

Reciprocal processes, whose concept can be traced back to E. Schrödinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. Léonard.

Brownian dynamics (BD) simulation method has been widely used for studying problems in dispersed systems, such as polymer solutions, colloidal suspensions and more recently, complex (dusty) plasmas. The main problem addressed with this simulation technique is that of time scale separation, which occurs when one form of motion in the system is much faster than the other. This can be a serious problem in Molecular dynamics (MD) simulation where very short time steps are needed to handle the fast motions and thus, requiring very long time runs for the proper evolution of slower modes making the simulation very expensive. More importantly, the fast motions may not be of much interest within themselves, as will be the case in a dusty plasma. The motion of neutral atoms or molecules comprising the plasma occurs at a very fast time scale with respect to the motion of dust particles, and is usually of very little interest, though a large number of such neutrals are present. In such cases, an approximate method is usually adopted, whereby the neutral particles are omitted from the simulation and their effect upon the dynamics of dust particles modeled by a combination of random forces and frictional terms. This leads to a recasting of the Newton's Equation of motion solved in MD, to a Langevin equation, solved in BD. Adopting this approach, we simulate a system of charged dust particles interacting via Yukawa potential in a 2-Dimensional layer, and extract relevant equilibrium statistical features such as the radial distribution function, static structure factor and the low frequency dust wave modes. We then propose the use of a Generalized Hydrodynamical (GH) approach to provide a semi-analytical model for the dust collective modes, which not only provides us with good predictions of the wave dispersion but also provides reasonable estimates for wave-number dependent wave damping, both of which will be compared against the results obtained from BD simulation. Finally, through our simulations, we also observe the equilibrium configuration of dust particles in the presence of cold ions streaming perpendicularly into the 2-Dimensional layer of dust particles. This provides us with novel results in the regime of sub-sonic ion flow speeds.

Brownian dynamics (BD) simulation method has been widely used for studying problems in dispersed systems, such as polymer solutions, colloidal suspensions and more recently, complex (dusty) plasmas. The main problem addressed with this simulation technique is that of time scale separation, which occurs when one form of motion in the system is much faster than the other. This can be a serious problem in Molecular dynamics (MD) simulation where very short time steps are needed to handle the fast motions and thus, requiring very long time runs for the proper evolution of slower modes making the simulation very expensive. More importantly, the fast motions may not be of much interest within themselves, as will be the case in a dusty plasma. The motion of neutral atoms or molecules comprising the plasma occurs at a very fast time scale with respect to the motion of dust particles, and is usually of very little interest, though a large number of such neutrals are present. In such cases, an approximate method is usually adopted, whereby the neutral particles are omitted from the simulation and their effect upon the dynamics of dust particles modeled by a combination of random forces and frictional terms. This leads to a recasting of the Newton's Equation of motion solved in MD, to a Langevin equation, solved in BD. Adopting this approach, we simulate a system of charged dust particles interacting via Yukawa potential in a 2-Dimensional layer, and extract relevant equilibrium statistical features such as the radial distribution function, static structure factor and the low frequency dust wave modes. We then propose the use of a Generalized Hydrodynamical (GH) approach to provide a semi-analytical model for the dust collective modes, which not only provides us with good predictions of the wave dispersion but also provides reasonable estimates for wave-number dependent wave damping, both of which will be compared against the results obtained from BD simulation. Finally, through our simulations, we also observe the equilibrium configuration of dust particles in the presence of cold ions streaming perpendicularly into the 2-Dimensional layer of dust particles. This provides us with novel results in the regime of sub-sonic ion flow speeds.

In recent years, the field of Fractional Brownian motion, Fractional Gaussian noise and long-range dependent processes has gained growing interest. Fractional Brownian motion is of great interest for example in telecommunications, hydrology and the generation of artificial landscapes. In fact, Fractional Brownian motion is a basic continuous process through which we show that it is neither a semimartingale nor a Markov process. In this work, we will focus on the path properties of Fractional Brownian motion and will try to check the absence of the property of a semimartingale. The concept of volatility will be dealt with in this work as a phenomenon in finance. Moreover, some statistical method like R/S analysis will be presented. By using these statistical tools we examine the volatility of shares and we demonstrate empirically that there are in fact shares which exhibit a fractal structure different from that of Brownian motion.