Fast variable

Coupling Fast Variable Selection Methods to Neural Network-Based Classifiers Application to Multi-Sensor Systems. 

The r dependence of the (r) and V/ (r) arises from the r dependence of the potential energy, V. The fast variable quantum state wavefunctions, (r), also depend on the fast variable coordinates, but this is not explicitly shown. 

Transitions can occur between different adiabatic states for the fast variable subsystem, since the adiabatic approximation ignores the action of T on the Yjf. The ignored terms act as the coupling between the different fast variable states. T" involves derivatives with respect to slow variable coordinates. In the discussion below, the coupling between the fast variable states is given by the nonadiabatic coupling vector... 

Sometimes it is useful to employ a diabatic representations for the fast variable quantum states, rather than the adiabatic representation. In this work we define a diabatic representation as one for which < /j V /i > = 0, where the superscript d indicates the fast variable states in the diabatic representation, There are off-diagonal matrix elements of the fast variable Hamiltonian, Vy(r) = < rf /j >, in this representation. In contrast, the off-diagonal elements of If are all zero in the adiabatic representation, since the /j are eigenfimction of in this case. 

If the initial state of the system is described by t /ift /os(io), where /oXi o) is the initial wavefunction for the slow variable subsystem at t = 0, then the k-hop contribution to the component of the system in fast variable state //at time t is given by... 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

In Sections V and VI, a brief history of the developments of the MCT from the hydrodynamic approach (Critical Phenomena) and the renormalized kinetic theory approach has been presented. The basic concept of MCT is to use the product of the slow (hydrodynamic) variables to span the orthogonal subspace of the fast variables. 

As seen, a fast variable here must consider only 8Z. At any 0BZ and cA 0, the equation for 0Z has a unique and asymptotically globally stable steady-state solution... 

A fast variable here is cA. At fixed 6Z and 0BZ, it is necessary to examine a system of fast motions. This has the steady states... 

As mentioned before, obtaining an explicit variable separation for the system in Equation (2.36) requires a nonlinear coordinate transformation. The fact that k(x) = 0 in the slow time scale t and k(x) 0 in the fast time scale r indicates that the functions fcj(x) should be used in such a coordinate transformation as fast variables. Then, it can be shown (see, e.g., Kumar and Daoutidis 1999a) that a coordinate change of the form... 

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