Canonical example

In the canonical example, we could estimate the free energy difference between two runs by examining the overlap in their probability distributions. Similarly, in the grand canonical ensemble, we can estimate the pressure difference between the two runs. If the conditions for run I arc f//1. V. > ) and for run 2 (po, VjK), then... 

For non-linear chemical reactions that are fast compared with the local micromixing time, the species concentrations in fluid elements located in the same zone cannot be assumed to be identical (Toor 1962 Toor 1969 Toor and Singh 1973 Amerja etal. 1976). The canonical example is a non-premixed acid-base reaction for which the reaction rate constant is essentially infinite. As a result of the infinitely fast reaction, a fluid element can contain either acid or base, but not both. Due to the chemical reaction, the local fluid-element concentrations will therefore be different depending on their stoichiometric excess of acid or base. Micromixing will then determine the rate at which acid and base are transferred between fluid elements, and thus will determine the mean rate of the chemical reaction. 

Oxirene (oxacyclopropene) provides a canonical example of a molecule which even at the highest current levels of theory has declined to reveal its basic secret can it exist ( Oxirene to Be or Not to Be [53b]) . Very large basis sets and... 

Delocalization energy denotes the energy by which a molecule is stabilized or destabilized compared to a hypothetical reference compound in which electrons (usually ji electrons) are not as mobile. The canonical example is the energy of benzene compared to the hypothetical 1,3,5-cyclohexatriene in which there are three distinct double and three distinct triple bonds. With caveats, one measure of this energy is the heat of hydrogenation of benzene compared to three times the heat of hydrogenation of cyclohexene. As an electronic phenomenon, this lies outside the purview of MM. 

Continuing to use PVAc as the canonical example of a stable and easily reproducible polymer monolayer, we show how the two quantities, the static elasticity es from 77-A the isotherm, and the corresponding ej deduced from the SLS experiment, compare and contrast with each other for the time being, we defer to later the SLS results. This is shown in Fig. 12. Agreement between the two is remarkable up to respective maximum points. The observed deviation at higher 77 is not expected since the monolayer state is no longer maintained, hence the static elastic responses in macroscopic scales are not likely to be the same as the dynamic response to spontaneous capillary waves. 

The particular case of lithium acetylacetonate (acac), a canonic example of /I-diketone enolate, was also examined early. It was shown that its chelated (Z,Z) conformation was almost exclusive in methanol at — 60 °C and that dimers were probably formed in which one of the two lithium cations would be chelated by the two acac anions272. A somewhat similar dimer, obtained from the lithium enolate of ethyl acetoacetate complexed by a 2.1.1 cryptate, was characterized in one of the first 7Li NMR studies of enolates (Scheme 65)273. Note that the structure of the three ft -diketone mono- and dilithium enolates displayed in Scheme 64 has been studied, despite their poor solubility, in both THF-dg and DMSO-d6 by 13C NMR260. The data obtained for the monoenolates are consistent with rapidly equilibrating dimers, while the dimers of dienolates seem to form slowly on the NMR time scale. 

In real physical systems, such an explosive instability is usually opposed by the stabilizing influence of higher-order terms. Assuming that the system is still symmetric under x — x, the first stabilizing term must be x . Thus the canonical example of a system with a subcritical pitchfork bifurcation is... 

First-Principles Approach to Guinier-Preston Zones. We have already seen that the combination of first-principles calculations with Monte Carlo methods is a powerful synthesis which allows for the accurate analysis of structural questions. In chap. 6 we noted that with effective Hamiltonians deduced from a lower-level microscopic analysis it is possible to explore the systematics of phase diagrams with an accuracy that mimics that of the host microscopic model. An even more challenging set of related questions concern the emergence of microstructure in two-phase systems. An age-old question of this type hinted at in the previous chapter is the development of precipitates in alloys, with the canonical example being that of the Al-Cu system. 

We mention here also an excimer (originally short for excited dimer) which is a type of small radius self-trapped exciton state. In organic solids this excited state can be considered as a dimeric molecule formed from two molecules, when one of the molecules is in an electronic excited state. Excimers are usually formed between two molecules that would not bond if both were in the ground state and the molecule pyrene is one of the canonical examples of an excimer. The excimer states play an important role in applications and in photochemistry (more information can be found in (30)). 

Note that this is equivalent to saying that / factors via (1) an isomorphism of Y with a closed subprescheme of X, followed by (2) the canonical injection of the closed subprescheme into X. The canonical example of a closed subprescheme is this ... 

However, as it is applied to this wider range of systems, the validity of Redfield theory becomes more problematic than in NMR. A canonical example is the spin-boson Hamiltonian—two electronic levels linearly... 

An important phenomenon, particularly in the dense region of the spray, is the collision of drops. A canonical example is the head-on collision of binary drops. 

The canonical example is that of a Janus particle with one metallic and one insulating hemisphere [9], using the standard low-voltage model for electrokinetic motion of polarizable particles. In response to an applied electric field, the Janus particle rotates to align the interface between the two hemispheres with the field axis, due to both ICEP (electrohydrodynamics) and DEP (electrostatics). At the same time, for any orientation, the particle translates in the direction of its insulating end, propelled by ICEO flow on the metallic end, with a velocity... 

For homogeneous particles, the canonical example is that of an tmcharged, ideally polarizable particle of irregular shape in a weak, uniform DC field. In that case, the basic velocity scale for ICEP is... 

In this canonical example, broken symmetries generally lead to fluid flow past the object, if it is held fixed, or motion by induced-charge electrophoresis (ICEP), if it is freely suspended [3, 10],... 

Chandler s historical account of the evolution of capitalism in the U.S., UK, and Germany emphasizes the importance of scale and scope at the level of the firm. While the chemical industry is a canonical example of firm level scale economies, it also provides evidence of advantages of bigness at the level of the industry which arise from an increase in specialization and division of labor. 

Frustration results when contradictory constraints act upon the same site in a material. Our interest here is the case when the origin of the frustration results from the intersite geometry. The canonical examples are the equilateral triangle and the regular tetrahedron shown in Figure 2.1. 

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