Rigorous results

Note [240] that the phase in Eq. (13) is gauge independent. Based on the above mentioned heuristic conjecture (but fully justified, to our mind, in the light of our rigorous results), Resta noted that Within a finite system two alternative descriptions [in temis of the squared modulus of the wave function, or in temis of its phase] are equivalent [247]. 

Gilliland s work should be an encouragement to us all in developing shortcut methods from rigorous results. The proliferation of computers makes our job relatively easy. 1 feel that when correlations of this type are developed, one shouldn t stop w ith the graph but continue until an equation is fitted to the data. Computers need equations. 

KleD81 Klein, D. J. Rigorous results for branched polymer models with excluded volume. J. Chem. Phys. 75 (1981) 5186-5189. 

A more rigorous result together with theoretical justification, has been given by Rothfield. ... 

Ruelle, D., Statistical Mechanics Rigorous Results, World Scientific Singapore, 1999... 

We have explored whether, on the basis of sound theoretical and physical reasoning, a semiempirical formula can be derived that would directly provide an accurate estimate of the dynamic correlation energy as a whole. Two known rigorous results are suggestive in this context (i) The dynamic correlation energy can be expressed as the expectation value of a perturbing correlation operator (65-67) and (ii) the correlation energy is known to be expressible (10) as sum of contributions of occupied orbital pairs, viz. 

Even with this enormous number of scientific papers on the subject, mathematically rigorous results on the subject are rare. Let us mention just ones aiming toward a rigorous justification of Taylor s dispersion model and its generalization to reactive flows. We could distinguish them by their approach... 

Finally ( 5.5), in order to illustrate an alternative asymptotic approach, available for systems with nowhere vanishing fixed charge density, we shall treat, following Please [4], a p — n junction (bipolar membrane). Presentation of this chapter is purely heuristic, based on numerical or formal asymptotic results. For some rigorous results concerning related matters, the reader is referred to [5]—[7]. 

It turns out that the expression derived in Eq. 12.49 is fortuitously close to the rigorous result for a fluid composed of hard spheres, which is... 

These two expressions differ only by the leading constant terms. The simple thermal conductivity expression derived here is roughly 40% the size of the rigorous result. It captures the functional dependence on temperature, molecular mass, heat capacity, and pressure (independent of pressure) of the exact result. Experimentally the thermal conductivity is generally found to be independent of pressure, except at very low pressures. The thermal conductivity is predicted to increase as the square root of temperature, which somewhat underestimates the actual temperature dependence. Consideration of interactions between molecules, as in the next section, brings the temperature dependence into better accord with observation. 

Actually an additional stability criterion is needed, see M.E. Fisher, Archives Rat. Mech. Anal. 17, 377 (1964) D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, New York 1969). A collection of point particles with mutual gravitational attraction is an example where this criterion is not satisfied, and for which therefore no statistical mechanics exists. 

Monoclinic. The essential symmetry for this class is one proper or improper twofold axis, that.is, either 2 or m (s2). This guarantees that one axis is perpendicular to the plane defined by the other two. It is important to appreciate that the mere occurrence of an axis that appears to be perpendicular (to within whatever accuracy, however great) does not make the crystal monoclinic. The perpendicularity must be the rigorous result of the symmetry,... 

There exist systems, namely systems which undergo processes involving hysteresis (plastic deformation or ferromagnetism, lor example) for which no equation of slate can he indicated. Although Ihe laws of thermodynamics may apply to such systems, the rigorous results of classical thermodynamics arc not applicable because the science of thermodynamics is developed on the assumption of the existence of the single-valued function. 

In general, this interpolative result is not identical with the rigorous result. Nevertheless, as we shall see in later sections, the information theory result often is in good agreement. Needless to say it would be better to use an optimization procedure which would simultaneously satisfy the moment theorems and give the correct A (0), but we have not been able to devise such a procedure. 

This is a rigorous result generalizing the random phase approximation, as will be shown in Sect. 5.4. 

Consequently, a parallel alignment of smectic layers is linearly stable against undulations even if the perpendicular alignment might be more preferable due to some thermodynamic considerations. As we have shown in Fig. 8, this rigorous result of standard smectic A hydrodynamics is weakened in our extended formulation of smectic A hydrodynamics. When the director can show independent dynamics, an appropriate anisotropy of the viscosity tensor can indeed reduce the threshold values of an undulation instability. 

Rigorous results for the performance of constrained MPC are lacking. However, there are a number of propositions on how the performance of MPC could be improved. Such propositions rely on (a) modifying the structure of MPC for robust performance, (b) tuning MPC for robust performance, and (c) developing efficient algorithms for the numerical solution of the MPC on-line optimization problem, thus enabling the formulation of more complex and realistic on-line optimization problems that would in turn improve performance. 

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