Consistency conditions

The consistency condition for this set of equations to possess a (unique) solution is that the field intensity tensor defined in Eq. (99) is zero [72], which is also known as the curl condition and is written in an abbreviated form as... 

For the case of nonzero temperatures the vacuum averages in Eq.(7) should be replaced by thermal averages over phonon populations. Using (7) and (5) we obtain that the scattering of an exciton in the effective medium by the perturbation fi — v z)) is described by the following self-consistent condition... 

In order to explore the last case, note that, during inelastic loading, = 0 which leads to the consistency condition (5.12). Consequently, from (5.24), / = 0 and... 

The remainder of this section will be concerned with a particular case in which normality conditions hold. The constitutive equation for the internal state variables (5.11) involves the constitutive function a, and the normality conditions (5.56) and (5.57) involve an unknown scalar factor y. In some circumstances, a may be eliminated and y may be evaluated by using the consistency condition. These circumstances arise if b is nonsingular so that the normality condition in strain space (5.56j) may be solved for k... 

Substituting (5.11) into this equation, taking the inner product of each side with d jdk, and using the consistency condition (5.14)... 

The choice (5.77) for the evolution equation for the plastic strain sets the evolution equations for the internal state variables (5.78) into the form (5.11) required for continuity. The consistency condition in the stress space description may be obtained by differentiating (5.73), or directly by expanding (5.29)... 

In this case, may be eliminated and y may be evaluated by substituting (5.77) into the left-hand side of the normality condition, taking the inner product of each side with the term in chain brackets in (5.79), and using the consistency condition (5.79), which results in... 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

In order to find the self-consistent condition, we have to determine the order parameter... 

In spite of its simplicity this approach, supplemented with Blatt s correction [.3] for lattice distortion, was applied successfully for decades [4, 5] in studies of systematics in the residual resistivity. Its power was the exact treatment of the scattering and the use of the Friedel sum rule [1] as a self-consistency condition ensuring a correct valency difference between impurity and host atom. 

In order that consistent conditions can be obtained, the air speed over the thermometers should be not less than 1 m/s. This can be done with a mechanical aspiration fan (the Assmann psychrometer) or by rotating the thermometers manually on a radius arm (the sling psychrometer). If the thermometers cannot be in a moving airstream, they are shielded from draughts by a perforated screen and rely only on natural convection. In this case the wet bulb... 

It is an easy exercise to show that if Pn satisfies the Kolmogorov consistency conditions (equations 5.68) for all blocks Bj of size j < N, then T[N- N+LPN) satisfies the Kolmogorov consistency conditions for blocks Bj of size j < N + 1. Given a block probability function P, therefore, we can generate a set of block probability functions Pj for arbitrary j > N hy successive applications of the operator TTN-tN+i, this set is called the Bayesian extension of Pn-... 

Probabilities for blocks Bj with length j < N may be calculated by appealing to the Kolmogorov consistency conditions (equation 5.68). We now examine some low order LST approximations in more detail. 

Hhe other 2-block probabilities are obtained by appealing to the Kolmogorov consistency conditions, defined in equation 5.68 Pio = Pol = Pi — Pll-... 

A basis set of probabilities, B = p(i),P(2), >P(s) is selected for parameterizing arbitrary iV-block probabilities. It is a simple exercise to show that, because of the constraints imposed by the the Kolmogorov consistency conditions (equation 5.68, s -= 2 basis elements are necessary. 

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

The sum is taken over all 2 states, and P S S ) is a 2 x 2 matrix. Since p and P are both probabilities, we know that they must satisfy the following three consistency conditions... 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

Higher-order probability functions also obey a variety of consistency conditions which are best illustrated by means of examples such as the following ... 

The functions defined by Eq. (3-231) are obviously non-negative and have unit area. One set of consistency conditions that must be met is of the form... 

On the other hand, Eq. (3-233) states that A is the sum of two statistically independent, Poisson distributed random variables Ax and Aa with parameters n(t2 — tj) and n tx — t2) respectively. Consequently,49 A must be Poisson distributed with parameter n(t2 — tx) + n(t3 — t2) = n(t3 — tx) which checks our direct calculation. The fact that the most general consistency condition of the type just considered is also met follows in a similar manner from the properties of sums of independent, Poisson distributed random variables. 

The partition function (Eq. 26) may be regarded as a functional of external field H = H(r]),..., H(rl), which can be determined from the self-consistence condition [30]. Of utmost importance in finding the chemical correlators... 

Let us now turn to the case of a thermally thick solid. Of course thickness effects can be important, but only after the thermal penetration depth due to the flame heating reaches the back face, i.e. t = ff, 6j(t ) = d. As in the ignition case, if tf is relatively small, say 10 to even 100 s, the thermally thick approximation could even apply to solids of d < 1 cm. Again, we represent all of the processes by a thermal approximation involving the effective properties of Tlg, k, p and c. Materials are considered homogeneous and any measurements of their properties should be done under consistent conditions of their use. Other assumptions for this derivation are listed below ... 

The surface CP oc(E) in the chemisorbed system is determined by a self-consistency condition, which is found by substituting Gc(l,l) into the general CPA equation (6.39), to give... 

For variable-density flow, Muradoglu etal. (2001) identify a third independent consistency condition involving the mean energy equation. 

Muradoglu, M., S. B. Pope, and D. A. Caughey (2001). The hybrid method for the PDF equations of turbulent reactive flows Consistency conditions and correction algorithms. 

Combining these equation, the consistency condition of the stochastic model reads... 

The unambiguous identification of the extraction rate regime (diffusional, kinetic, or mixed) is difficult from both the experimental and theoretical viewpoints [12,13]. Experimental difficulties exist because a large set of different experimental information, obtained in self-consistent conditions and over a very broad range of several chemical and physical variables, is needed. Unless simplifying assumptions can be used, frequently the differential equations have no analytical solutions, and boundary conditions have to be detemtined by specific experiments. 

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