Inequality: consequences

In this section we analyse some approximation methods for variational inequalities considered in Section 1.2. We discuss the penalty and the projection methods and their consequences. As for numerical methods, we refer the reader to (Glowinski et al., 1976). 

Consequently, the substitution of these values w,M in (2.63) provides the fulfilment of the inequality... 

This variational inequality provides the property x = xU Q) Consequently,... 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

By (1 -I- 2cc) < 1, the right-hand side of this inequality converges to zero as n —> 00. Consequently,... 

The most important consequence of bound smoothing is the transfer of infonnation from those atoms for which NMR data are available to those that cannot be observed directly in NMR experiments. Within the original experimental bounds, the minimal distance intervals are identified for which all triangle inequalities can be satisfied. A distance chosen outside these intervals would violate at least one triangle inequality. Eor example, an NOE between protons pi and pj and the covalent bond between pj and carbon Cj imposes upper and lower bounds on the distance between pi and Cy, although this distance is not observable experimentally nor is it part of E hem-... 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

All of Bell s papers on the conceptual and philosophical problems of quantum mechanics, including his landmark PIPR paper in which he derives his famous inequality, are collected in [bell87. An excellent collection of papers exploring the philosophical consequences of Bell s theorem appears in a volume edited by Crushing and McMullin [cush89. ... 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

It is clear that J-diffusion is a good approximation for rotational relaxation as a whole, if the centre of equilibrium distribution over J is within the limits of non-adiabatic theory. In the opposite case m-diffusion is preferable. Consequently, the J-diffusion model is applicable, if the following inequality holds ... 

It should be noted, however, that increases quadratically with /. Starting with / = L such that rLTj = 1, inequality (2.30) is reversed, and Eq. (2.24) is no longer valid. Consequently, in series (2.36) only the... 

The situation is more complicated in the adiabatic limit when this inequality is reversed. According to Eq. (4.36) and Eq. (4.4) the off-diagonal parts of T and y are different. To elucidate this difference and explore its consequences we shall examine the spectra of the four-level system passing from non-adiabatic to adiabatic broadening. 

To prove this fact, we make use of estimate (41) and the inequality which is a consequence of... 

Thus the observed rate coefficient is equivalent to feiA 2/fc i. Because of the limitations of the kinetic procedure, there is some small doubt over the inclusion of a term in Ce(IV) concentration in the denominator of (12.4). However, the relative insignificance of this term means that A i[Ce(III)] > A 2[Ce(IV)]. Also, the inequality A 3[Ce(rV)] 2[Ce(Iir)] is a consequence of the observed kinetics. As in the Cr(VI)-l-Fe(II) system , the slow stage involves the inter-... 

In addressing the question of how income inequality is linked to population health, Muntaner and Lynch (1999) identified two strands of causation, thus combining the approaches of Kaplan and Wilkinson. First, they claim that income inequality is associated with a set of social processes and economic policies that systematically under-invest in physical and social infrastructure (e.g. education). Secondly, large disparities in income distribution may have direct consequences on people s perceptions of their relative place in the social environment, which leads to behavioral and cognitive states that influence health. 

One consequence of the positivity of a is that A A < (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A > (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that... 

The given inequalities are no longer satisfied and additional measures are required. To trigger an event (excess over design limits), at least two failures are necessary. Consequently the causal chain detailed under II a) and II b) must be interrupted by one measure in each case. These measures must be effective independently of one another. The same applies in respect of the measures concerning the failure mode which leads to a reduction in the solvent quantity D. 

The properties of the minors of the secular determinant of an alternant hydrocarbon may again be used to show that the integrals for which the index is even in (44) and odd in (45) and (46) are zero. It follows that the finite change Aq is an odd function, of Sa, while AFg and Apgt are even. Any inequalities between values of any index for two different positions u), as defined in equations (31) to (34) which arise as first terms of the corresponding infinite series in (44) to (46), persist term-by-term in the expression for the exact finite changes (Baba, 1957). In consequence, the broad agreement with experiment found earlier in the description of ionic and radical reactions by the approximate method carries over to the exact form. 

More accurate estimate can be produced from inequalities (93), if it is necessary. For our goals it is sufficient to use the following consequence of Equation (94) ... 

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