Order of a class

Henoe the inverses of the elements of a class belong to a class (it is sometimes the same one) and the order of a class of elements and the order of the class of their inverses is the same. 

The order of a class is defined as the number of elements in the class. For example, the order of the class of the reflection operations in C3v is 3, and the order of the class of the rotation operations is 2. The order of a class, or a subgroup, is an integral divisor of the order of the group. 

Discovery of nerve agents in Germany led to the availability of a class of compounds at least one order of magnitude more lethal than previously known where death might occur in a matter of minutes instead of hours. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

Both in work reviewed herein and in theoretical research by other workers[10-30], a consensus has been reached that anomalous properties (compared to graphite and normal graphitic nanotubes) can be obtained with graphitic nanotubes with diameters of the order of a nanometer. In terms of electronic properties, the nanotubes are expected to fall into two major classes on one hand the moderate band gap nanotubes that do not satisfy the Wj — = 3 /w condition in the... 

Although these potential barriers are only of the order of a few thousand calories in most circumstances, there are a number of properties which are markedly influenced by them. Thus the heat capacity, entropy, and equilibrium constants contain an appreciable contribution from the hindered rotation. Since statistical mechanics combined with molecular structural data has provided such a highly successful method of calculating heat capacities and entropies for simpler molecules, it is natural to try to extend the method to molecules containing the possibility of hindered rotation. Much effort has been expended in this direction, with the result that a wide class of molecules can be dealt with, provided that the height of the potential barrier is known from empirical sources. A great many molecules of considerable industrial importance are included in this category, notably the simpler hydrocarbons. 

Notice that the common feature of many evolutionary transitions identified by Maynard Smith and Szathmary involves a change in the status of replicators. Once the replication of a class of entities becomes dependent on a larger whole, they do not become independent replicators again. So new replicators at the emergent level must exist and replicate independently in order for there to be a still higher level transition. Given the nature of replicators as analyzed by both Dawkins and Hull, there is a problem with this characterization of the common feature only one or possibly two evolutionary transitions are at all likely to have occurred in the history of life. 

Nitrogen-14, with its natural abundance of 99.6%, is one of the most ubiquitous and, until recently, least studied NMR-active nuclei. Due to the integer spin number (/ = 1), its single-quantum transitions are affected by first-order quadrupolar broadening, which in most materials is on the order of a few megahertz. A new class of 2D HETCOR protocols has been recently developed, which makes it possible to indirectly observe well-resolved 14N sites via their spin-1/2 neighbors and obtain the related parameters of the quadrupolar tensors. 

A group will be denoted by a capital German letter ( ), its order by the corresponding lower-case Latin letter (g), and its elements by lower-case Latin letters s, t, etc. The reader is assumed familiar with the concepts of subgroup and class, and with their elementary properties, e.g., that the order of a subgroup, or of a class, must be a divisor of g. A class will be denoted by a script c, its order by c, sometimes with a subscript on both. 

J. Alvarez-Raimrez, R. Femat, and J. Gonzalez-Trejo. Robust control of a class of uncertain first-order systems with least prior knowledge. Chem. Eng. Sci., 53(15) 2701-2710, 1998. 

In order to determine the age of the solar system, we must first decide what is really meant by its age. Astronomers and theoreticians tell us that the process of forming a star from a dense core of gas and dust in a molecular cloud takes on the order of a few million years. The process can be divided into four stages that are defined by the characteristics of four classes... 

The allyl cation (9) is the simplest member of the class of resonance-stabilized cations that includes the alkyl-substituted cyclopentenyl cations. But one could also say that the carbenium ion (CH3) is the simplest member of a class of cations that includes the trityl cation. In each case, 10 or so orders of magnitude of acidity separate the primitive member from its more elaborate derivatives. 

The majority of FPTRMS investigations have been of the reaction of a free radical with an excess of a stable molecule, making the radical decay pseudo-first order. Of this class of reaction, the most frequently studied has been association with 02 to form a peroxy radical. These reactions are of importance in combustion and in the atmosphere. Bayes and coworkers have reported a series of FPTRMS investigations of radical/02 association reactions, following the decay of ions formed by photoionization of the radicals. 

Using DQF-COSY and TOCSY we can link all of the protons within a single spin system, which corresponds to a single amino acid residue. We can classify each spin system as a pattern of chemical shifts unique to one amino acid or as a member of a class AMX or five spin. In order to get sequence-specific assignments, however, we have to have some way to correlate protons in one residue to protons in the next residue in the sequence. For unlabeled proteins this is done by NOE interactions certain protons in one residue are constrained by the peptide bond to be close in space to certain protons in the next residue. These NOE correlations are called sequential or z, i + 1 because they correlate a proton in residue z with a proton in the next residue in the sequence, residue z + 1. Specifically, we expect to see NOE correlations between Ha of residue z and Hn of residue z + 1 (Fig. 12.15) and sometimes between the protons of residue z and the Hn of the next residue. Because the DQF-COSY and TOCSY spectra correlate protons within a residue, we can move from... 

More than 140 different alkenes have been identified in the atmosphere [27]. The sources of alkenes are similar to those for the alkanes with combustion of fossil fuel being a major source. The presence of unsaturated bonds makes these compounds much more reactive than the alkanes. The most persistent member of this class of compounds (ethene) has an atmospheric lifetime of the order of a day, while more typically the lifetimes for alkenes are measured in hours. As a result of their short lifetimes the atmospheric concentrations of alkenes are highly variable and decrease dramatically away from their source locations. The mechanisms of atmospheric oxidation of alkenes have recently been reviewed [55]. As with the alkanes the reaction of OH radicals is an important loss mechanism. This reaction proceeds mainly via addition to the unsaturated bond as illustrated for ethene in Fig. 4. In one atmosphere of air at 298 K the dominant atmospheric fate of the alkoxy radical HOCH2CH2O is decomposition via C - C bond scission, while reaction with O2 makes a 20% contribution [56]. The fate of alkoxy radicals resulting from addition of OH to alkenes is generally decomposition via C - C bond scission [8]. Thus, the OH radical initiated oxidation of propene gives acetaldehyde and HCHO, oxida-... 

The solution and gas-phase acidities of C—H acids are of particular interest because of the wide structural variations that are possible in this class of compounds [113, 123]. A qualitative ordering of a selection of C—H acids (and some O—H acids for comparison) gives the following sequence of increasing acidity in the gas phase [120] ... 

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