Finite dimensional state vector

The finite dimensional state vector can accommodate the description of particles with considerable internal structure. For example, consider a cell with m compartments. Each compartment may be considered as well mixed containing a total of n quantities. Suppose now the cell changes its state by interaction between its compartments and with the environment. The particle state can be described by a partitioned vector [x, X2,..., x ] where x represents the vector of n components in the ith compartment. 

The NDF is very similar to the PDFs introduced in the previous section to describe turbulent reacting flows. However, the reader should not confuse them and must keep in mind that they are introduced for very different reasons. The NDF is in fact an extension of the finite-dimensional composition vector laminar flow where the PDFs are not needed, the NDF still introduces an extra dimension (1) to the problem description. The choice of the state variables in the CFD model used to solve the PBE will depend on how the internal coordinate is discretized. Roughly speaking (see Ramkrishna (2000) for a more complete discussion), there are two approaches that can be employed ... 

In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes. The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., liquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical applications to assume that the particle state can be described by a finite dimensional vector. ... 

Finally we have the tools to state the answer to our original question what is the final state of a measured particle Consider a measurement A on a finite-dimensional vector space V, possibly with multiplicities. Suppose that a particle enters the measuring device in a state [u] and the measurement yields the result A. Let denote the subspace of states whose measurement is sine to yield A. Note that [u] because there is a nonzero chance... 

In Section 5, we have introduced the finite-dimensional Lanczos subspace CM cH spanned by the Lanczos state vectors via ... 

Thus, we obtain only finite dimensional unirreps for so(4) and only infinite dimensional ones for so(3,1). For our applications to perturbation theory we shall only need the so called hydrogenic case (cf. Section VII) where V is the Laplace-Runge-Lenz vector. For the realization of the generators in this manner we shall show that j0 — 0 and q is the principal quantum number. The unirreps of so(3,1) may be of interest in scattering problems which deal with the continuum states of the hydrogen atom. 

In this section we briefly describe a linear state space model that serves as a building block for the main models of collaborative forecasting processes we present in this chapter. We then present a well-known forecasting technique associated with this model namely, the Kalman filter. Let Xt be a finite, n-dimensional vector process called the state of the system. In the context of inventory management, this vector may consist of early indicators of future demand in the channel, actual demand realizations at various points of the channel, and so forth. Suppose that the state vector evolves according to ... 

As was mentioned earlier, sometimes the state is described by a function (e.g. in polymers and oils) and not by a vector, and so the concept of a continuously changing component can be introduced. For the sake of the general mathematical treatment we mention that a finite-dimensional vector... 

According to the conventional treatment of pure homogeneous reaction kinetics the state is a finite-dimensional vector and the only constitutive quantities are the reaction rates. 

The state of the system we investigate in the present book is almost exclusively characterised by a finite dimensional vector. Sometimes it may prove insuflScient. The usual stochastic model can be conceived of as one where the state is a random variable, i.e. an element of an infinite dimensional linear space. If one wants to describe spatial effects (in a deterministic model) then a possible way to do so is to characterise the state by a function (by the mass density) and to write down a differential equation for the time versus mass density function. This will be a differential equation for a function with values in an infinite dimensional state space. (Memory effects can also be taken into consideration using an infinite dimensional state space. Cf. Atlan Weisbuch, 1973.)... 

We shall consider here a population of particles distinguished from one another by a finite dimensional vector x of internal coordinates and distributed uniformly in space. Further, we shall be concerned with the open system of Section 2.8 whose behavior is dictated by the population balance equation (2.8.3). Thus the number density in the feed,/i jn(x), may be assumed to be Nff x) where Nf is the total number density in the feed stream and /(x) is probability density of particle states in it. It will also be assumed that the continuous phase plays no role in the behavior of the system. Relaxing this assumption does not add to any conceptual difficulty, although it may increase the computational burden of the resulting simulation procedure. 

In a lattice-gas the individual cells are the structural units of a D-dimensional regular lattice. Each cell is defined by its position vector r on the discrete space, a finite number of states s(r ) and a set of transition rules E that map the state of the cell at time t into the state at time t + 1. A finite number of particles reside in each cell. A discrete velocity c, with / = 1,..., k is associated to each particle. This velocity is chosen such that the particle can propagate to a neighboring cell in unit time. Each velocity direction is subject to an exclusion principle of utmost single occupancy. The combinations of occupancies define the set of possible states associated with each cell. The configuration of each cell is defined by the Boolean field... 

Vector opmtions dominate in irrany quantum mechanical calculatitHis and the advent of computers with a design that greatly enhances the rate at which they can be performed has made it feasible to consider also some very demanding problems, such as die determinatitm of state-to-state cross sections in reactive scattering theo. My report here concerns some aspects of a three-dimensional treatment of atom-diatom collisions based on the Finite Element Method employed in a hyperspheiical coordinate formulation. 

The N-SmA-SmC problem bears strong similarities with the transition from paramagnetic to helimagnetic states [47]. Fluctuations are maximum at a finite m-dimensional vector d-dimensional space. Mean field N-SmA-SmC theory predicts a second oder N-SmC transition as shown. Fluctuations, however, are believed to lead to a first-order transition when m = d or d-l. The A(-SmC transition with d = 3 and m = 2 is therefore expected to be first order. 

At finite temperature, stochastic fluctuations of the membrane due to thermal motion affect the dynamics of vesicles. Since the calculation of thermal fluctuations under flow conditions requires long times and large membrane sizes (in order to have a sufficient range of undulation wave vectors), simulations have been performed for a two-dimensional system in the stationary tank-treading state [213]. For comparison, in the limit of small deviations from a circle, Langevin-type equations of motion have been derived, which are highly nonlinear due to the constraint of constant perimeter length [213]. 

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