Invariant operator

If Eq. (E.14) is satisfied for all elements of some point group G, A will be an invariant operator [13] (the Hermitian conjugate as well as the sum and/or product of two invariant operators are also invariant operators). Such an operator can be expanded in the form... 

Generators for use with the prime movers previously described will almost invariably operate in conjunction with an alternating current electrical system. Such a.c. generators will operate either in parallel with other sources of supply (i.e. a grid system or other generators) or in isolation providing the whole electrical supply to an installation. A.C. generators can be one of the two main types ... 

Species differences indicate that the AOS is not invariably operational prenatally, even though most peripheral and central neural units are in place and available for activation. Variability in the timing of maturation of the Organ-to-AOB linkage could well provide the necessary flexibility of response consistently associated with higher mammalian, and especially primate, neural systems. The onset of effective accessory... 

Soxtherm , FEXTRA and intermittent extractions. The newer methods (SFE, ASE , MAE) invariably operate at pressures above atmospheric. Micromethods of solvent extraction have also been described [57,58]. 

Geometric phase effect (GPE) (Continued) adiabatic states, conical intersections invariant operators, 735-737 Jahn-Teller theorem, 733-735 antilinear operator properties, 721-723 degenerate/near-degenerate vibration levels, 728-733... 

Invariant operators, permutational symmetry, conical intersection, adiabatic state, 735-737... 

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The last problem of general interest in algebraic theory is the evaluation of the eigenvalues of the invariant operators in the basis discussed in Section 2.4. As mentioned before, the invariant operators commute with all the Xs. As a result, they are diagonal in the basis [A,], A, ..., A.v],... 

Dynamic symmetries for chain (II) correspond to an expansion of the Hamiltonian in terms of invariant operators of 0(2). The linear invariant is... 

Consider first chain (I). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of the chain,... 

Invariant operators are important because they are related to conserved quantities, as the example of the angular momentum discussed in the text makes evident. 

Invariant operators were introduced by Casimir (1931) for SO(3). Racah (1950) generalized them to all orders. 

Mixed Bed NHjOH Cycle High pressure steam generators invariably operate on an All Volatile Treatment (AVT) or zero solids treatment which means that any conditioning of the water-steam circuit uses chemicals which do not increase the dissolved solids of the feedwater. One such chemical is ammonia which is dosed to give a feedwater pH of 8.8-9.6 depending upon the materials of construction, and being volatile is returned with the condensate. Polishing... 

It must be noted that the two invariants have to be compatible, i.e. the domain of is in the range of Ji moreover, the two invariant operators do not commute, i.e. 

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