State invariant

The model is usually improved by defining states some specifications become simplified by directly referring to states (including parameterized states), and letting the state invariants implicitly deal with the details, e.g. 

As we have written it in Equation (8), the DMRG wave function contains redundant variational parameters. This means that the set of variational tensors f/ni... ij/Hk in the DMRG wave function is not unique, because we can find another set of tensors whose matrix product yields an identical state. This redundancy is analogous to the redundancy of the orbital parametrization of the Hartree-Fock determinant. In the case of the DMRG wave function, we can insert a matrix T and its inverse between any two variational tensors and leave the state invariant... 

Analysis of Spectra. An understanding of the mechanism of polymer degradation must involve identification of the radical intermediates. However, anisotropy due to spin lattice interactions in the solid state invariably results in broad, poorly resolved ESR spectra and together with the low concentration of radicals which is usually present, can result in major problems with analysis. We have developed two approaches to this problem 1) increasing resolution and 2) sophisticated analysis routines. 

One factor must, however, be kept in mind. Triplet states invariably lie at lower energies than do the corresponding singlet states131. While it may be true that the multiplicity per se will not affect reactivity for many processes, the difference in energy may be decisive and some reactions can occur with singlet states which would not be possible with triplet states. 

Phosphorescence is also characterized by an excitation and an emission spectrum. Because the lowest triplet state invariably lies lower in energy than the lowest excited singlet state of the same molecule, phosphorescence will occur at wavelengths longer than those of fluorescence and, therefore, at wavelengths much longer than those of the excitation spectrum. As in the case of fluorescence, the phosphorescence spectrum and the phosphorescence excitation spectrum are distorted by the instrumental components and therefore do not represent true spectra. 

Numerous rigorous results have been obtained for gradient systems. One natural method of investigation of gradient systems is elementary catastrophe theory the field of catastrophe theory dealing with an examination of gradient systems. In the case of the gradient system of equations (1.8), properties of a stationary state, that is the state invariant with time, may be readily studied... 

Increasing Energy Dissipation Characterizes the Cell Attachment Process. The process of cell attachment as monitored by the QCM is rather lengthy, involving many hours of change before steady state invariant behavior is observed. We have followed this process for normal ECs as well as for other transformed cell types, but our most detailed studies have been performed... 

In standard QM, the reversibility in time is a manifestation of a Hermifian (self-adjoint) system with stationary states and is reflected in the unitarity of the S-matrix. Unitarity entails the inclusion of the contribution of fime-reversed states. In other words, for a stationary state, invariance under time-reversal implies that if is a stationary wavefunction, then so is A major tool for deriving results in the framework of a Hermitian formalism, explicitly or implicitly, is the resolution of the identity operator, I, on the real axis, which is a Hermitian projection operator. 

Solid state invariant equilibria were determined by [1938Egg]. However, the eutectoid at 920°C involved the Fe3Nb2 phase, which is not present in the accepted binary Fe-Nb phase diagram. Owing to uneertainties in the binary systems available at that time these ternary invariants are ignored in this assessment. 

The A4 amplitude corresponds to Ti state, invariant since its lifetime is much longer than the timescale femtosecond experiment. Adequate mathematical treatment leads to the full assignment of the transient spectra and of the rate of conversion, so that the complete state diagram is reconstmcted (see Fig. 6.9). 

We also concentrated on verifying simple state invariant properties only. The traditional BDD based model checking techniques were too time consuming on our case study model. Most bounded model checking techniques do not give proofs when specifications are true. We used the k-induction algorithm implemented in NuSMV that can also prove properties in some cases. 

A timed automata may contain an arbitrary number of clocks, which run at the same rate. (There are also extensions of timed automata where clocks can have different rates (Daws Yovine, 1995).) The clocks may be reset to zero, independently of each other, and used in conditions on state transitions and state invariants. A simple yet illustrative example is presented in Figure 2, from the UppAal tool. The automaton changes state from A to B if event a occurs twice within 2 time units. There is a clock, f, which is reset after an initial occurrence of event a. If the clock reaches 2 time units before any additional event a arrives, the invariant on the middle state forces a state transition back to the initial state A. 

The header of the state schema includes the class name (C) and the formal parameters a,b for the class. The declarations (e.g Xi Si) introduce the state conqxNients (Xi) and their types (Si). The state invariants represent any constraints on the values of the state components md the initialisation conditions represent any additional constraints to be satisfied on initialisation. 

Event ej(fmivener.r. For every operation, it must be shown that, for every consistent state and parameter setting in which the operation s precondition holds, its postcondition must satisfy the state invariant. (This must also be proved for the "initialisation" of each schema, which may be thought as an anonymous operation, with no ivecondition, which occurs when a system satisfying the schema comes into existence.)... 

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