Direct methods discrete-valued functions

When there are only a few discrete-valued decision variables in a model, the most effective method of analysis is usually the most direct one total enumeration of all the possibilities. For example, a model with only eight 0-1 variables could be enumerated by trying all 2 = 256 combinations of values for the different variables. If the model is pure discrete, it is only necessary to check whether each possible assignment of values to discrete variables is feasible and to keep track of the feasible solution with best objective function value. For mixed models the process is more complicated because each choice of discrete values yields a residual optimization problem over the continuous variables. Each such continuous problem must be solved or shown infeasible to establish an optimtil solution for the full mixed problem. 

The Lattice Boltzmann Method (LBM), its simple form consist of discreet net (lattice), each place (node) is represented by unique distribution equation, which is defined by particle s velocity and is limited a discrete group of allowed velocities. During each discrete time step of the simulation, particles move, or hop, to the nearest lattice site along their direction of motion, where they "collide" with other particles that arrive at the same site. The outcome of the collision is determined by solving the kinetic (Boltzmann) equation for the new particle-distribution function at that site and the particle distribution function is updated (Chen Doolen, 1998 Wilke, 2003). Specifically, particle distribution function in each site f[(x,t), it is defined like a probability of find a particle with direction velocity. Each value of the index i specifies one of the allowed directions of motion (Chen et al., 1994 ThAurey, 2003). 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

To solve this equation, we use the discrete mapping method and select the simplest scale value of K = 2, corresponding to a block of two interacting spins. The excess partition functions for blocks of K = 2 and = 2 = 4 spins easily can be calculated exactly by direct summation ... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectrum. ° In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator,which circumvent the cumbersome root-search problem in boundary-value based semiclassical methods, have been successfully applied to a variety of systems (see, for example, the reviews Refs. 85, 86 and references therein). These methods cannot directly be applied to nonadiabatic dynamics, though, because the Hamilton operator for the vibronic coupling problem [Eq. (1)] involves discrete degrees of freedom (discrete electronic states) which do not possess an obvious classical counterpart. 

In particle characterization experiments, q(x) is obtained at the x values that result either from a calibrated direct measurement or from a conversion process involving predetermined x values. In the other words, the x values may be preset, extracted, or measured directly. The choices of the x values where q(x) are to be obtained depend on the characterization technology and the property of the sample. From a statistical point of view, the x values should be arranged in a way that the distances (or steps) between consecutive x values are all the same or similar (i.e.. Ax is spaced more or less equally) thus giving a good representation of the true distribution. Although, because of the nearly infinite number of particles in a sample, the distribution function q(x) can be continuous or discrete, but the data points (the locations of x), sampled from q(x) in an experiment are often finite, except in chromatography and a few other methods where a continuous measurement of q(x) is possible. Thus, except when an... 

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