State space form implicit

We will first motivate this approach by considering the linearized form of the multibody equations (5.3.1) 

We will define a solution of the corresponding discretized linear system, cf. (5.3.2), 

Transformation to the state space form of (5.3.7), which in that case is globally defined. 

Discretization of the state space form by an implicit BDF method. The resulting system is - in contrast to the discretized form of (5.3.7) - a square linear system with a well defined solution. 

Back transformation to the full set of variables An. These values will 

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The implicit state space form is motivated by a transformation to state space form, discretization of the state space form equations and back transformation. Integrating constrained mechanical systems by first transforming them to state space form is used frequently, [HY90, Yen93]. This leads to approaches known as coordinate splitting methods. The method we discuss here does not carry out the transformation to state space form explicitly. It embeds the choice of state variables in the iterative solution of the nonlinear system (5.3.2), while the integration method always treats the full set of coordinates. 

We pointed out earlier that for equations of motion of constrained mechanical systems written in index-1 form the position and velocity constraints form integral invariants, see (5.1.16). Thus the coordinate projection and the implicit state space method introduced in the previous section can be viewed as numerical methods for ensuring that the numerical solution satisfies these invariants. 

In the preceding discussion we have expanded the density in terms of N < M Hilbert space such that their norms are less than or equal to one and the trace of the density is equal to N. All these expansions could in principle be exact there is no need for M = r =, as is clearly demonstrated in the KS procedure, where M = N Ai M <°o and i- = < , then new forms of auxiliary states, i.e. different from single determinantal ones, are implicitly introduced. 

As stated before, PGVL is too large to be fully enumerated practically. Therefore our strategy is to find a way to focus in a just-in-time manner on much smaller sub-regions ( 104) of PGVL for subsequent on-the-fly enumeration followed by standard similarity search against the same query molecule. It is intuitively evident that a virtual compound space built from parallel synthesis reaction protocols has inherent array structures in the form of implicit arrays of related just-in-time enumerated compounds, even if those compounds do not have their molecular structures yet enumerated at the time this inherent array structure is exploited. 

Half-Cells. The concept of a supported electrolyte has proven quite valuable in solution electrochemistry by allowing great theoretical simplification at (usually) only a small cost in accuracy. The several (often implicit) assumptions made in treating the electrolyte in a given cell as supported, however, deserve careful attention as they generally do not apply in the case of solid state electrochemical systems. It should also be noted that it is usually possible in solution electrochemistry to use a large, essentially kineticaUy reversible counterelectrode so that aU but a negligible fraction of the applied potential difference falls across the electrode-electrolyte interface of interest. In its simplest form, the supported approach assumes that all the potential difference in the system falls across the compact double layer—approximately one solvent molecule diameter in thickness— at this electrode, and the approach of the electroactive species to the boundary of the compact layer, the outer Helmholtz plane, occurs purely by diffusion. Corrections for the buildup of space... 

The diffusion problem solved in this section is a steady-state version of Equations 5.67 through 5.73, in which dcjdt=0, all variables are time independent, and the initial condition (Equation 5.68) is omitted. The normalized variables employed are defined earlier including the dimensionless homogeneous rate constant, K=k aVD. The main difference between this model and the ones described earlier is that concentrations of both forms of redox species (Cq and Cr) have to be computed by solving two partial differential equations. To save space, the diffusion from behind the shield has been neglected it can be taken into account by introducing additional boundary segments as discussed in Section 5.5.2. An implicit assumption of equal diffusion coefficients (Dq=D is not essential and can easily be eliminated. 

The spatial dependence of the electric field E(x, y, z) and the magnetic field H(x, y, z) of an optical waveguide is determined by Maxwell s equations. We assume an implicit time dependence exp( — iwt) in the field vectors, current density J and charge density a. The dielectric constant s(x, y, z) is related to the refractive index n(x, y, z) by e = n CQ, where Eq is the dielectric constant of free space. For the nonmagnetic materials which normally constitute an optical waveguide, the magnetic permeability p is very nearly equal to the free-space value Pq. Thus for convenience we assume p = Po throughout this book unless otherwise stated. Under these conditions. Maxwell s equations are expressible in the form[l]... 

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