Stochastic counterpart

The main diflierence between looking at stochastic nets rather than their deterministic counterparts is stochastic nets force us to shift our focus of attention since under a stochastic rule the same initial states generally evolve into diflierent final states, we are in the stochastic case not so much interested in the final state of the... 

In this paper we consider the QCD counterpart of this problem. Namely, we address the problem of regular and chaotic motion in periodically driven quarkonium. Using resonance analysis based on the Chirikov criterion of stochasticity we estimate critical values of the external field strength at which quarkonium motion enters into chaotic regime. 

U (/) = U(x1 (t), t) is illustrated in Fig. 2.3.1 Note that the Lagrangian velocity varies more slowly with time than the Eulerian counterpart shown in Fig. 2.2 (Yeung 2002). This fact has important ramifications on stochastic models for the Lagrangian velocity discussed in Chapter 6. 

To ensure that the original information structure associated with the decision process sequence is honored, for each of the products whose demand is uncertain, the number of new constraints to be added to the stochastic model counterpart, replacing the original deterministic constraint, corresponds to the number of scenarios. Herein lies a demonstration of the fact that the size of a recourse model increases exponentially since the total number of scenarios grows exponentially with the number of random parameters. In general, the new constraints take the form ... 

Due to the complexity of numerical integration and the exponential increase in sample size with the increase of the random variables, we employ an approximation scheme know as the sample average approximation (SAA) method, also known as stochastic counterpart. The SAA problem can be written as ... 

The main advantage of the stochastic matrix approach is the simplicity for its computer implementation. Equation 17 directly provides the desired result, and Equation 28 is the basis of a validation test which my or may not be performed according to previous experience. In other words, the proposed method is conceptually and practically easier to implement than the Kalman counterpart. The... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

The macroscopic moduli ah(x), etc., can be found by solving cell local problems, being stochastic counterpart of the local problems formulated in [14] for the periodic case. For instance, we have... 

All diffraction-unlimited nanoscopy methods can provide improved axial resolution even when implemented with a single lens [77,83,120,131,132[. However, because it starts out from less favorable values, the -resolution usually remains worse than its focal plane counterpart. The coherent use of opposing lenses pioneered in 4Pi microscopy and FM, however, facilitates an independent resolution improvement factor by 3-7-fold along the optic axis as has already been demonstrated with STED [83,84,87]. In the stochastic single molecule switching modalities, a similar gain in resolution will take place by the coherent use of opposing lenses [133]. Thus, while 4Pi microscopy and FM did not break the diffraction barrier, they remain cornerstones of far-field fluorescence nanoscopy in the future. 

Modeling intracellular phenomena therefore demands an a priori choice of methods. As long as fluctuations are negligible, deterministic equations correctly capture the dynamics [10, 17]. However, these approaches break down in the presence of noise. Comparisons between stochastic models and their deterministic counterparts have revealed that noise can induce a dynamical behavior that is not present in the absence of fluctuations. For instance, the MinCDE system only oscillates if the experimentally observed small number of interacting molecules is respected [16]. The deterministic equations decay to a fixed point. 

This approach to the numerical solution of stochastic optimization problems is a natural outgrowth of the Monte Carlo method of estimation of the expected value of a random function. The method is known by various names, and it is difircirlt to point out who was the first to suggest this approach. In the recent literature a variant of this method, based on the likelihood ratio estimator g (x), was suggested in Rubinstein and Shapiro (1990) imder the name stochastic counterpart method (tilso see Rubinstein and Shapiro 1993 for a thorough discussion of such a Ukelihood ratio-stunple approximation approach). In Robinson (1996) such an approach is called the sample path method. This idea can also be applied to cases in which the set x is finite, that is, to stochastic discrete optimization problems (Kleywegt and Shapiro 1999). 

Stimulation, environmental vs. task, 1357, 1358 STL (stereo lithography format), 208 Stochastic approximation, 2634-2635 Stochastic counterpart method, 2635 Stochastic decision trees, 2384, 2385 Stochastic models, 2146-2170 benefits of mathematical analysis of, 2146 definition of, 2146, 2150 Markov chains, 2150-2156 in continuous time, 2154-2156 and Markov property, 2150-2151 queueing model based on, 2153-2154... 

The remainder of this section introduces the relevant notation with an additional focus on the extension to heteroscedastic models (so-called (G)ARCH and ARMA-GARCH models) as these can be seen as the discrete-time counterpart of continuous stochastic processes formulated in terms of SDEs. 

This subsection briefly introduces GARCH models as discrete counterpart of continnous stochastic processes. In contrast to ARMA models, the basic idea is that the variance/ volatility in time is no longer deterministic and constant bnt depends on previous errors and volatility, i.e. 

In a CTNHSMP, transitions between two states may depend not only on such states and on the sojourn times (x) (as it occurs with the homogeneous counterpart), but also on both times of the last (v) and next (t) transitions, with x = t — t. The time variable t is also known as the most recent arrival or last entry time, and t is the calendar or process time. Thus, CTNHSMP extend other models such as homogeneous semi-Markov, (non-) homogeneous ordinary Markov and other point stochastic processes. 

Since chemical reaction is considered as a stochastic process, and furthermore as a thermodynamic process, it is a natural question to ask what are the counterparts of the statements of the fluctuation theory of nonequilibrium thermodynamics. In the theory of thermodynamics the fluctuation-dissipation theorem is associated with the observation that the dissipative process leading to equilibrium is connected with fluctuations around that equilibrium. This fact was pointed out in a particular case (related to Brownian movement) by Einstein. Different representations of the theorem exist for linear thermodynamic processes (Callen Welton, 1951 Greene Callen, 1951 Kubo 1957 Lax, 1960 van Vliet Fasset, 1965 van Kampen, 1965.)... 

According to the traditional point of view fluctuations are averaged out. It was clearly demonstrated that noise can support the transition of a system from a stable state to another stable regime. Since stochastic models might exhibit qualitatively different behaviour than their deterministic counterparts, external noise can support transitions to states which are not available (or even do not exist) in a deterministic framework. (The theory of noise-induced transition, as well as its applications are discussed in the book of Horsthemke Lefever (1984b). 

The influence of external fluctuations on two-dimensional oscillatory systems has been studied by Ebeling Engel-Herbert (1980). The stochastic counterpart of the appearence of a limit cycle is strongly connected with the formation of a probability crater on the stationary probability surface. They studied particular cases when the system has the form... 

To define the stochastic counterpart of a deterministic model of a reaction, the elementary reactions and the reaction rates of the elementary reactions have to be known. As this has been given above, the stochastic model may be considered to have been defined. Several remarks still may be appropriate. 

To assess the importance of the stochastic formulation, the deterministic model using the nominal demand values has been first solved and compared with its stochastic counterpart. The results obtained are detailed in Table 1. The schedules are not shown for space reasons, but they are significantly different. In addition, it is noted that the makespan of the deterministic model is shorter, because the model does not generate inventory to hedge from adverse scenarios, as the stochastic one does. 

---