Parallel definition

We shall show both from experimental evidence about gas-phase complexes and, to a lesser extent, from the results of electronic structure calculations that a parallel definition of the intermolecular halogen bond is appropriate The halogen bond is an attractive interaction between a halogen atom X and an atom or a group of atoms in different molecule(s), when there is evidence of bond formation. ... 

Notes 11 In this table, N is defined as moles/Z,2i, and c as moles/L3. Parallel definitions where Ni is in terms of M/L2t and c is M/L3t are easily developed. Definitions mixing moles and mass are infrequent. 

Hydration theory deals with aqueous solutions in terms of two parallel definitions of the set of components, the usual one in which the solutes are considered formally unhydrated and the amount of solvent is the nominal amount, and a second in which the solutes are formally hydrated and the amount of water is reduced from the nominal amount. The objective is to correct for hydration effects and obtain a model giving activity coefficients in terms of the usual set of components. Quantities pertaining to the second set of components will be denoted by an asterisk. The number of moles of water in the first set is given by... 

Conversely, there is a debate going on of the division between leading and lagging safety indicators. Hinze et al. (2013) summarize parallel definitions (Table 1). 

Equation 11.30 gives a sum over the quantum energy levels of a molecular system, the molecular partition function, when the molecular energies (Equation 11.1) are used for the e/s. (The parallel definition for the partition functions of systems of particles uses the... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

The equations for counterflow ate identical to equations for parallel flow except for the definitions of the terminal temperature differences. Counterflow heat exchangers ate much mote efficient, ie, these requite less area, than the parallel flow heat exchangers. Thus the counterflow heat exchangers ate always preferred ia practice. 

A low temperature of approach for the network reduces utihties but raises heat-transfer area requirements. Research has shown that for most of the pubhshed problems, utility costs are normally more important than annualized capital costs. For this reason, AI is chosen eady in the network design as part of the first tier of the solution. The temperature of approach, AI, for the network is not necessarily the same as the minimum temperature of approach, AT that should be used for individual exchangers. This difference is significant for industrial problems in which multiple shells may be necessary to exchange the heat requited for a given match (5). The economic choice for AT depends on whether the process environment is heater- or refrigeration-dependent and on the shape of the composite curves, ie, whether approximately parallel or severely pinched. In cmde-oil units, the range of AI is usually 10—20°C. By definition, AT A AT. The best relative value of these temperature differences depends on the particular problem under study. 

Thickness. The traditional definition of thermal conductivity as an intrinsic property of a material where conduction is the only mode of heat transmission is not appHcable to low density materials. Although radiation between parallel surfaces is independent of distance, the measurement of X where radiation is significant requires the introduction of an additional variable, thickness. The thickness effect is observed in materials of low density at ambient temperatures and in materials of higher density at elevated temperatures. It depends on the radiation permeance of the materials, which in turn is influenced by the absorption coefficient and the density. For a cellular plastic material having a density on the order of 10 kg/m, the difference between a 25 and 100 mm thick specimen ranges from 12—15%. This reduces to less than 4% for a density of 48 kg/m. References 23—27 discuss the issue of thickness in more detail. 

A Macromolecular Division of lUPAC was created in 1967, and it created a permanent Commission on Macromolecular Nomenclature, parallel to the other nomenclature commissions. The Commission over the years has issued recommendations on basic definitions, stereochemical definitions and notations, stmcture-based nomenclature for regular single-strand organic polymers and regular single-strand and quasisingle-strand inorganic and coordination polymers, source-based nomenclature for copolymers, and abbreviations for polymers. AH of these are coUected in a compendium referred to as the lUPAC Purple Book (99). 

The definition of fugacity of a species in solution is parallel to the definition of pure species fugacity. Equation 154 is analogous to the ideal gas expression ... 

One subtle, but major noise source is the output rectifier. The shape of the reverse recovery characteristic of the rectifiers has a direct affect on the noise generated within the supply. The abruptness or sharpness of the reverse recovery current waveform is often a major source of high-frequency noise. An abrupt recovery diode may need a snubber placed in parallel with it in order to lower its high-frequency spectral characteristics. A snubber will cost the designer in efficiency. Finding a soft recovery rectifier will definitely be an advantage in the design. 

To illustrate the relationship between the microscopic structure and experimentally accessible information, we compute pseudo-experimental solvation-force curves F h)/R [see Eq. (22)] as they would be determined in SEA experiments from computer-simulation data for T z [see Eqs. (93), (94), (97)]. Numerical values indicated by an asterisk are given in the customary dimensionless (i.e., reduced) units (see [33,75,78] for definitions in various model systems). Results are correlated with the microscopic structure of a thin film confined between plane parallel substrates separated by a distance = h. Here the focus is specifically on a simple fluid in which the interaction between a pair of film molecules is governed by the Lennard-Jones (12,6) potential [33,58,59,77,79-84]. A confined simple fluid serves as a suitable model for approximately spherical OMCTS molecules confined... 

As mentioned in the start of Chapter 4, the correlation between electrons of parallel spin is different from that between electrons of opposite spin. The exchange energy is by definition given as a sum of contributions from the a and /3 spin densities, as exchange energy only involves electrons of the same spin. The kinetic energy, the nuclear-electron attraction and Coulomb terms are trivially separable. 

In parallel with the new approaches to generating capacity additions, the utilities, with encouragement from regulators, introduced incentives during the 1980s for reducing load demand. Since the system peak hour load provided the inertia for capacity requirement definition, shaving of the peak became the focus of these incentives. 

This angle plate gravity separator removes suspensions of solids from a dilute liquid. The unit is more compact than a box-type settler due to the increased capacity achiev ed by the multiple parallel plates. The concept is fairly standard (U.S. Patent 1,458.805—year 1923) but there are variations in some details. For effective operation, the unit must receive the mixture with definite particles having a settling velocity. The units are not totally effective for flocculants or coagulated masses that may have a tendency to be buoyant. 

This must be a clear and definite intersection and not the situation represented in Figure 12-146 for a forward bladed fan system. Note that the limit curve parallels the static curve and has a very poor long tangential intersection. The... 

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