Dynamical singularities

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

Bornemann, F. A., Schiitte, Ch. On the Singular Limit of the Quantum-Classical Molecular Dynamics Model. Preprint SC 97-07 (1997) Konrad-Zuse-Zentrum Berlin. SIAM J. Appl. Math, (submitted)... 

F.A. Bornemann and Ch. Schiitte. On the singular limit of the quantum-classical molecular dynamics model. Preprint SC 97-07, ZIB Berlin, 1997. Submitted to SIAM J. Appl. Math. 

Singular points represent the positions of equilibrium of dynamical systems and merit further investigation. 

Dynamic programming, 305 Dynamical systems variational equations, 344 of singular points, 344 Djmamical variables characterizing a particle, 494 Dyson, F. J. 613 Dzyaloshinsky, /., 758... 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

If the inverse in Eq. (2.8) does not exist then the metric is singular, in which case the parameterization of the manifold of states is redundant. That is, the parameters are not independent, or splitting of the manifold occurs, as in potential curve crossing in quantum molecular dynamics. In both cases, the causes of the singularity must be studied and revisions made to the coordinate charts on the manifold (i.e. the way the operators are parameterized) in order to proceed with calculations. 

Cellular automata, then, are models, in the same sense that the Monte Carlo and molecular dynamics approaches are models, which can be employed for the purpose of simulating real systems. We shall use the term cellular automaton (singular) to refer to a model consisting of the following components ... 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

I want to emphasize, above all, that these theories of matter as stresses, strains, singularities, or vortices of ether, were mechanical (and even hydro-dynamic) theories. When scientists such as Crookes and Lodge, and Theoso-phists such as Besant and Leadbeater, melded physics with spiritual and psychic forces via theories of the ether (and the additional particles that Theosophy added to the equation), they were lending scientific credibility to spiritual ideas. Paradoxically, in their critique of scientific materialism, they asserted a mechanical theory of spirituality. Theosophy thus required a form of vitalism to counterbalance the mechanistic tendencies of its physics. 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

Eq. (5) shows that the classical dynamics depends on the scaled energy e = E Y-1/2. As it is clear from Eq. (5) the Hamiltonian has the singularity at f = 0. This singularity can be removed by performing the following transformations... 

The controllability analysis was conducted in two parts. The theoretical control properties of the three schemes were first predicted through the use of the singular value decomposition (SVD) technique, and then closed-loop dynamic simulations were conducted to analyze the control behavior of each system and to compare those results with the theoretical predictions provided by SVD. 

The minimum singular value is a measure of the invertibility of the system and therefore represents a measure of the potential problems of the system under feedback control. The condition number reflects the sensitivity of the system under uncertainties in process parameters and modelling errors. These parameters provide a qualitative assessment of the dynamic properties of a... 

The singular character of the diffusive modes allows their exponential relaxation at the rate given by the dispersion relation of diffusion. Their explicit construction can be used to perform an ab initio derivation of entropy production directly from the microscopic Hamiltonian dynamics [8, 9]. 

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