Equivalent link

Lignans are a class of secondary metabolites widely encountered in the plant kingdom. The term lignan was originally introduced by Harworth [1] to describe a group of plant phenols whose structure was determined by union of two cinnamic acid residues or their biogenetic equivalents, linked B,B , Fig. (1). 

The measurement of the volume of gaseous fuels using turbine meters, or modem equivalents, linked to devices which correct for the temperature and pressure of the gas has proved to be fully satisfactory. Surprisingly, the CV of natural gas can be variable, owing to supply from two or more gas fields. A number of producers have found it necessary to install CV meters on site. 

If the persistence length is 50 nm and the contour length is 50pm, then the contour length is on the order of 100 times the persistence length. Such a chain is coiled in solution. Another way of expressing this model is to say that xanthan (M=107) behaves hydrodynamically like a flexible chain with about 50 freely-jointed Kuhn-equivalent links each about 100 nm long. 

Since the equivalent chain is not a real physical polymer chain, and since the equivalent links do not represent real chemical bonds, we are free to choose t(R,) for the equivalent link in any manner we choose. One possibility is the freely hinged chain of (2.15), but that leads to fairly complicated mathematical expressions ) for the distribution... 

Thus the use of an equivalent gaussian link chain, which appeared artificial in the case of the freely hinged chain, is quite acceptable for the freely rotating chain, the chain with hindered internal rotation, or the rotational isomeric model. In these cases there is no pretense of reality of the equivalent links, so they need not be given fixed lengths /eit- This equivalent chain is therefore a summary of the chainlike character of flexible polymer chains, and its simplicity enables the investigation of those properties of polymer systems which are solely dependent on the flexible chainlike nature of the polymer molecules. It is then only natural to attempt to formulate simple models which are characteristic of stiff... 

As in (2.2) and (2.8), this potential energy gives rise to the Boltzmann factor exp (—/(Ifa). If gaussian equivalent links are used for the discrete chain, (2.8) and (5.7) give... 

Consider a polymer chain with n equivalent links, each of rms length /, for which there is some short-range repulsive interaction... 

Again we consider the simplest case of gaussian equivalent links. The interaction (6.1), or (2.1), implies that the thermal Green s function for the entire chain is... 

We shall now prove that the S-forms associated to stably equivalent linking formations are stably isomorphic. 

In particular stably equivalent linking formations represent the same element in the Witt group. There are defined forgetfu maps... 

Since the 1960 s, two ideas have gained our attention the struggle against pollution before the first oil crisis of 1973 and the diminution of consumption since. One can consider, in fact, that the two objectives are linked. Indeed, any maladjustment of a fuel admission system will modify the equivalence ratio of the mix. The consequences are modifications, on one hand, of the consumption and on the other, of the nature and the quantity of pollutants emitted CO, NO, and unburned hydrocarbons. 

Truncating this series after the first derivative and integrating provides the basis for the hermodynamic integration approach. Moreover, if the Taylor series expansion is continued intil it converges then Equation (11.45) is equivalent to the thermodynamic perturbation brmula, so providing a link between the two approaches. In practice, it is always necessary... 

No polymer is ever 100% crystalline at best, patches of crystallinity are present in an otherwise amorphous matrix. In some ways, the presence of these domains of crystallinity is equivalent to cross-links, since different chains loop in and out of the same crystal. Although there are similarities in the mechanical behavior of chemically cross-linked and partially crystalline polymers, a significant difference is that the former are irreversibly bonded while the latter are reversible through changes of temperature. Materials in which chemical cross-linking is responsible for the mechanical properties are called thermosetting those in which this kind of physical cross-linking operates, thermoplastic. 

Polycarbonates. Currently, all audio CDs (CD-AD), all CD-ROM, and the biggest fraction of substrate disks for WORM and EOD worldwide are manufactured from a modified bisphenol A—polycarbonate (BPA-PC) (3). In 1991, some 1.3 x 10 compact disks were produced, equivalent to an annual amount of about 35,000 t BPA-PC. WORM and EOD disks are manufactured mainly from BPA-PC for sizes of 5.25 in. and below, and glass for larger form factors (eg, 12 in.), partially also from BPA-PC, and in some cases from aluminum or from cross-linked polymers (epoxy resins) (190). 

Two or more linked enzyme reactions can lead to a change in the concentration of NADH or NADPH that is equivalent to the concentration of the original analyte. The reference glucose measurement using hexokinase [9001-51-8] and glucose-6-phosphate dehydrogenase [9001-40-5] is an example ... 

The second largest use at 21% is for unsaturated polyester resins, which are the products of polycondensation reactions between molar equivalents of certain dicarboxyhc acids or thek anhydrides and glycols. One component, usually the diacid or anhydride, must be unsaturated. A vinyl monomer, usually styrene, is a diluent which later serves to fully cross-link the unsaturated portion of the polycondensate when a catalyst, usually a peroxide, is added. The diacids or anhydrides are usually phthahc anhydride, isophthahc acid, and maleic anhydride. Maleic anhydride provides the unsaturated bonds. The exact composition is adjusted to obtain the requked performance. Resins based on phthahc anhydride are used in boat hulls, tubs and spas, constmction, and synthetic marble surfaces. In most cases, the resins contain mineral or glass fibers that provide the requked stmctural strength. The market for the resins tends to be cychcal because products made from them sell far better in good economic times (see Polyesters,unsaturated). 

Both chloramphenicol and thiamphenicol cause reversible bone marrow suppression (9). The irreversible, often fatal, aplastic anemia, however, is only seen for chloramphenicol (9). This rare (1 in 10,000—45,000) chloramphenicol toxicity has been linked to the nitroaromatic function (1,9). Thiamphenicol, which is less toxic than chloramphenicol in regard to aplastic anemia, lacks potency as can be seen in Table 1, and thiamphenicol has never found much usage in the United States. An analogue of thiamphenicol having antimicrobial potencies equivalent to chloramphenicol was sought. Florfenicol (2) was selected for further development from a number of closely related stmctures. 

The main producers of organic accelerators for mbber vulcanization are shown in Table 3. This table is not meant to be completely comprehensive, but rather to indicate the principal historical suppHers to the mbber industry. Most producers offer chemical equivalents in the largest-volume products. Within the range of smaHer-volume specialty accelerators, chemical equivalents become less common. Each producer may offer different products to achieve the same purpose of rapid cross-linking, resistance to thermal degradation, or other performance characteristics. Many offer proprietary blends of accelerators. 

Multifunctional Hydroxy, Mercapto, and Amino Compounds. These are used to cross-link halogenated polymers. Depending on the labihty of the halogen, the cross-linking agents can be capped to reduce reactivity or used in combination with accelerators to increase the rate of reaction. Benzoyl capping is common with hydroxy and mercapto compounds forming the carbamate by reaction with one equivalent of carbon dioxide is used with diamines. 

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