Linear Fraction

The linear fraction is the fraction of a line in a specified direction that is covered by atoms whose centers lie on the line. Since the width of an atom is 2R, the linear atomic fraction of both the (101) family in fee and the (111) fraction in bcc will be 1. 

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A much better agreement between theory and experiment is found in the closely-related field of macrocyclisation equilibria. Investigations of the cyclic populations in ring-chain equilibrates set up in typical polymeric systems such as polyesters, polyethers, polysiloxanes, and polyamides take a major advantage from the relative ease with which the cyclic fraction can be separated from the linear fraction and analysed for the relative abundance of the individual oligomeric rings. This is conveniently done by means of modern analytical techniques such as gas-liquid and gel-permeation chromatography (Semiyen, 1976). 

Titanium-calcium. The first evidence for isotopic anomalies in the iron-group was found in Ti showing up to 10% excesses of Ti in hibonites from the Murray CM2 meteorite (Hutcheon et al. 1983 Fahey et al. 1985 Ireland et al. 1985 Hinton et al. 1987). Further studies in Murchison showed that Ti extended from -7% to +27% associated with Ca variation from -6% to +10% (Ireland 1988 Ireland 1990). Except for the magnitude of the variations, this is similar to the results from Allende inclusions. Only a few samples display mass-dependent fractionation for which it ranges up to 1.3 %/amu. In the majority of the samples, it is absent or very low (less than 1 %o/amu) for Ca-Ti. There is no correlation between the presence of linear fractionation and the magnitude of Ti effects. Ti variations are also present, but about an order of magnitude smaller than Ti. Variations affecting these two isotopes are related but not strictly correlated (Ireland 1988). 

Ravi, V. and Reddy, P.J. (1998) Fuzzy linear fractional goal programming applied to refinery operations planning. Fuzzy Sets and Systems, 96, 173. 

Most potato starches are composed of a mixture of two polysaccharides, a linear fraction, amylose, and a highly branched fraction, amylopectin. The content of amylose is between 15 and 25% for most starches. The ratio of amylose to amylopectin varies from one starch to another. The two polysaccharides are homoglucans with only two types of chain linkage, a-(l 4) in the main chain and a-(l 6)-linked branch chains. Physicochemical properties of potato and its starch are believed to be influenced by amylose and amylopectin content, molecular weight, and molecular weight distribution, chain length and its distribution, and phosphorus content (Jane and Chen, 1992). 

To determine the amylose content of starch, the iodine reaction has been most commonly used because amylose and amylopectin have different abilities to bind iodine. The methods such as blue value (absorbance at 680 nm for starch-iodine complex using amylose and amylopectin standards), and potentiometric and amperometric titration have been used for more than 50 years. These procedures are based on the capacity of amylose to form helical inclusion complexes with iodine, which display a blue color characterized by a maximum absorption wavelength (kmax) above 620 nm. During the titration of starch with iodine solution, the amount (mg) of iodine bound to 100 mg of starch is determined. The value is defined as iodine-binding capacity or iodine affinity (lA). The amylose content is based on the iodine affinity of starch vs. purified linear fraction from the standard 100 mg pure linear amylose fraction has an iodine affinity of 19.5-21.0mg depending on amylose source. Amylopectin binds 0-1.2mg iodine per 100mg (Banks and Greenwood, 1975). The amylose content determined by potentiometric titration is considered an absolute amylose content if the sample is defatted before analysis. 

Fraction DII is a polystyrene containing 2.2 branches per molecule. DII has been prepared together with the absolutely linear fraction D IV by Henrici-Oliv and Oliv6 (120). Apparently, the effect of the poly-dispersity of sample D II prevails over the opposite effect of branching. In this respect the investigation of anionic star-molecules would be extremely interesting. 

Fig. 25. Composition analysis of technical PDADMAC ( experimental values - - - Gaussian fit linear fraction -branched fraction density) (Data taken from [35])...

Figure 7.6 Isotopic compositions of meteorite (Murray), solar, and terrestrial Xe, displayed as per mil variation of observed isotope ratios (normalized to 130Xe) in air and the carbonaceous chondrite Murray from the corresponding ratios SUCOR, a solar Xe composition calculated to be surface-correlated Xe in a lunar mare soil. The dashed line, illustrating linear fractionation, is primarily for reference. Reproduced from Podosek (1978).

Source-receptor analysis was performed to allocate the air chemistry parameters to the individual emission sources. To estimate the impact of a source group on a certain pollutant, several simulations have to be accomplished. To minimize the associated uncertainties (non-linearity of chemical processes), the source group was suppressed. Due to the non-linear chemical processes, background concentrations and advection a non-linear fraction has to be introduced (DG-ENV 2001). The source-receptor analysis is an important tool for abatement and emission reduction strategies. 

If the linear fraction of siloxane oHgomers is used directiy in the manufacture of siHcone polymers, extremely pure (greater than 99.99%) dimethyldichlorosilane is required. A higher content of methyltrichlorosilane can produce significant amounts of ttifunctional units and considerably affect the physical properties of the final products. If such high purity dimethyldichlorosilane is not achieved, an additional step, called cracking, must be included in the production scheme (66). 

Linear fraction (amylose) -1,4-linked D-glucose residues Branched fraction (amylopectin) ... 

For linear fractions of dextrans in water, Senti, et al. 12) reported values of 97.8 X 10 and 0.50 for the constants, and for branched dextrans the value of the exponent was 0.20. Cerney, et al. 16) reported values of 10.3 X 10 and 0.25 for branched dextrans in methanol-water. The suppliers data in 0.05 M Na2S04 yields values of 9 X 10 and 0.50 at 30 °C for a broad range of molecular weights 17). The results of this work are in line with those reported for dextrans, and the exponent corresponds to that for a highly branched coil about 0.0-0.5 18). 

The solution viscosity of the B-fraction is substantially higher than that of the A-fraction. Coupled with sedimentation data and alkali lability, it appears fairly certain that the branched component possesses a higher molecular weight than the linear fraction. 

Extensive reviews on global optimization can be found in Horst (1990) and Horst and Tuy (1990). In this section we present a summary of a global optimization method that has been developed by Quesada and Grossmann for solving nonconvex NLP problems which have the special structure that they involve linear fractional and bilinear terms. It should be noted that global optinuzation has clearly become one of the new trends in optimization and synthesis, and active workers involved in this area include Floudas and Visweswaran (1990), Swaney (1990), Manousiouthakis and Sourlas (1992), and Sahinidis (1993). 

As shown above, the objective function and the constraints generally involve linear fractional and bilinear terms corresponding to the two summation terms, while the last term h x, y, z) is assumed to correspond to a convex function. This type of problem arises, for instance, in the optimization of heat-exchanger... 

In the method proposed by Quesada and Grossmann (1995b), the main idea is to replace the bilinearities and linear fractional terms by valid under- and over-estimators which will yield a convex NLP (or LP) whose solution provides a lower bound to the global optimum. Consider, for instance, fractional terms with positive coefficients. By introducing the variables we can express the fractional term as the constraint... 

The kinetics and thermodynamics of this process show close similarities to the polymerization of D4 initiated by CF3SO3H [3,4]. Both processes involve simultaneous formation of cyclic oligomers and polymer and lead to equilibrium, with similar proportions of cyclic-to-linear fractions [3-5], They also show similar thermodynamic parameters and a similar effect of water addition on the initial rate of polymerization. The specific feature of the polymerization of >2 is that cyclic oligomers 03 and D4 are formed simultaneously with the polymer fraction, but they equilibrate with monomer much faster than the polymer fraction. This behavior is best understood assuming formation of the tertiary oxonium ion intermediate, which isomerizes by ring expansion-ring contraction [3]. These kinetic features of the polymerization of make this monomer an interesting model for deeper studies on the cyclic trisilyloxonium ion question. 

We then use the hypercube representation to carry out a nonlinear dynamical analysis of these networks. The key insight is that quantitative aspects of flows in phase space can be computed from linear fractional maps that represent the flows between boundaries on the hypercube. Analysis is possible because the composition of two linear fractional maps is a hnear fractional map. This analysis is useful for analyzing steady states, limit cycles, and chaotic dynamics in these networks. 

In many cases the dynamics in Eq. (5) are amenable to theoretical analysis. The main theoretical insight is that if all the decay constants y, are equal, the maps that take the flows from one orthant boundary to the next have a simple form called a linear fractional map ... 

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