Examples linearity

In order to improve the physical properties of HDPE and LDPE, copolymers of ethylene and small amounts of other monomers such as higher olefins, ethyl acrylate, maleic anhydride, vinyl acetate, or acryUc acid are added to the polyethylene. Eor example, linear low density polyethylene (LLDPE), although linear, has a significant number of branches introduced by using comonomers such as 1-butene or 1-octene. The linearity provides strength, whereas branching provides toughness. 

When water (a Newtonian liquid) is in an open-ended pipe, pressure can be applied to move it. Doubling the water pressure doubles the flow rate of the water. Water does not have a shear-thinning action. However, in a similar situation but using a plastic melt (a non-Newtonian liquid), if the pressure is doubled the melt flow may increase from 2 to 15 times, depending on the plastic used. As an example, linear low-density polyethylene (LLDPE), with a low shear-thinning action, experiences a low rate increase, which explains why it can cause more processing problems than other PEs. The higher-flow melts include polyvinyl chloride (PVC) and polystyrene (PS). 

The various independent variables can be the actual experimental variables or transformations of them. Dilferent transformations can be used for different variables. The independent variables need not be actually independent. For example, linear regression analysis can be used to fit a cubic equation by setting X, X and Z as the independent variables. 

The most representative characteristics are given. The Traditional Differential Equation (TDE) approach applies to the flow and solute module. Under "other" we may have for example linear analytic system solutions. 

Trost et al 1 have observed product distribution to be dependent in part on the steric and electronic properties of the substrate. For example, linear enyne 48 (Equation (30)) cyclized exclusively to the Alder-ene product 49, whereas branching at the allylic position led to the formation of 1,3-diene 50 (Equation (31)) under similar conditions. Allylic ethers also give 1,3-dienes this effect was determined not to be the result of chelation, as methyl ethers and tert-butyldimethylsilyl ethers both gave dialkylidene cyclopentanes despite the large difference in coordinating ability. 

In the second example, that of an industrial pyrolysis reactor, simplified material and energy balances were used to analyze the performance of the process. In this example, linear and nonlinear reconciliation techniques were used. A strategy for joint parameter estimation and data reconciliation was implemented for the evaluation of the overall heat transfer coefficient. The usefulness of sequential processing of the information for identifying inconsistencies in the operation of the furnace was further demonstrated. 

Re-expressions. These involved questions of what scale would serve to best simplify and improve the analysis of the data. Simple transformations, such as those presented earlier in this chapter, are used to simplify data behavior (for example, linearizing or normalizing) and clarify analysis. 

The retention properties of ultrafiltration membranes are expressed as Molecular Weight Cutoff (MWCO). This value refers to the approximate molecular weight (MW) of a dilute globular solute (i.e., a typical protein) which is 90% retained by the membrane. However, a molecule s shape can have a direct effect on its retention by a membrane. For example, linear molecules like DNA may find their way through pores that will retain a globular species of the same molecular weight. 

How is dimension reduction of chemical spaces achieved There are a number of different concepts and mathematical procedures to reduce the dimensionality of descriptor spaces with respect to a molecular dataset under investigation. These techniques include, for example, linear mapping, multidimensional scaling, factor analysis, or principal component analysis (PCA), as reviewed in ref. 8. Essentially, these techniques either try to identify those descriptors among the initially chosen ones that are most important to capture the chemical information encoded in a molecular dataset or, alternatively, attempt to construct new variables from original descriptor contributions. A representative example will be discussed below in more detail. 

Macromolecules having identical constitutional repeating units can nevertheless differ as a result of isomerism. For example, linear, branched, and crosslinked polymers of the same monomer are considered as structural isomers. Another type of structural isomerism occurs in the chain polymerization of vinyl or vinylidene monomers. Here, there are two possible orientations of the monomers when they add to the growing chain end. Therefore, two possible arrangements of the constitutional repeating units may occur ... 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

In some systems, the same polymer may be produced from different monomers, e.g., plastic sulfur can be formed by polymerizing SB as well as S8 molecules. In such systems, a polymer may be stable with respect to one monomer but labile with respect to another one. For example, linear siloxane (-Me2SiO-) , which could be formed from the cyclic tetramer (-Me2SiO-)4, decomposes into the cyclic trimer (-Me2SiO-)3 and the reaction 3 (-MegSiO-)4-> 4 (-Me2SiO-)3 proceeds, therefore, through a polymer intermediate. 

Fig. 10.2 Bilinear plot obtained from second-order sensor. In this example, linearized ISE response is plotted against absorbance from fiberoptic sensor. Point S represents an outlier ...

The molecular architecture of a polyphosphazene has a profound influence on properties. For example, linear and tri-star trifluoroethoxy-substituted polymers with the same molecular weight (1.2 x 104 or higher) have strikingly different properties.138 The linear polymers are white, fibrous materials that readily form films and fibers, whereas the tri-arm star polymers are viscous gums. One is crystalline and the other is amorphous. Cyclolinear polymers are usually soluble and flexible. Cyclomatrix polymers are insoluble and rigid. Linear polymers can be crystalline, but graft or comb polymers are usually amorphous. 

Now consider a fractional replica of type 215 11, which is the 1/2048-replica of a FUFE. It is pointless in this case to write down all aliased/confounded estimates as their number is enormous. As an example, linear effects are aliased/confounded with 105 even interactions. The design matrix of 215 11 is shown in Table 2.96. 

The different basis systems will be illustrated for two examples, linear polarization along the x-direction and right circular polarization. According to equ. (8.4) the electric field vector is represented as... 

For example, linear birefringence is principally contained in the matrix elements m42 and... 

One possibility is that there is differential susceptibility to both mutation and killing among the cells. If, for example, there are two populations of cells, the high-dose data reflect the mutation rate of the component more resistant to killing. If these cells are also resistant to mutation, the resistance of the total cell population is underestimated. Another possibility is that there is a mutation-repair mechanism that is turned on at high doses. Deviations from either of these causes can usually be detected by the shape and slope of the curve at high doses for example, linear extrapolation from high-dose... 

As an example, linear arrays of silicon photodiodes (photodiode arrays) are available as a package complete with the necessary circuitry to read out the array following exposure. Arrays are available with up to 2048 elements, and they are read out on an element by element basis. The scanning circuitry can access an element in 10 - 25 /rs, which suggests that the entire array can be read out in approximately 20 - 50 ms. Thus modulation frequencies can not exceed 50 Hz. However, as was discussed earlier, PEMs are driven at a resonant frequency which usually varies between 50 and 100 kHz, and this exceeds the readout rate of the photodiode array by three orders-of-magnitude. 

Interpretive methods will generally arrive at the global optimum after a limited number of experiments. However, (by definition) the recognition of the individual solutes is required in each experimental chromatogram. Also, the computational requirements are relatively high, especially if the simultaneous optimization of several parameters is considered. For example, (linear) ternary gradients (one parameter) will be much easier to optimize than quaternary gradients (two parameters). 

Effect Hierarchy Lower-order effects are more likely to be active than higher-order effects. For example, linear main effects are more likely to be active than quadratic main effects or interaction effects. 

Nevertheless the general conclusions discussed here, as well as the overall experimental design for their validation, still follow the same unifying trends. For example, linear extrathermodynamic expressions can be proposed between the free energy change of a polypeptide or protein molecule involved in such hydrophobic interactions and particular molecular property parameters %j. This relationship takes the form of... 

Variations in skeletal structure give rise to major differences in properties. For example, linear polyethylene has a melting point about 20 C higher than that of branched polyethylene. Unlike linear and branched polymers, network polymers do not melt... 

Copolymers are ubiquitous and important because they allow monomers to be combined in such a way so as to provide useful and sometimes unique properties. For example, linear polyethylene (PE) and isotactic polypropylene (i-PP) are both semi-crystalline plastics, but copolymers of ethylene and propylene (EPR) (usually with other comonomers) are rubbers at room temperature (depending on composition). The homopolymers are shown in the top two figures in Figure 6-2, and if you don t know which one s which by now you should collapse in deepest humiliation. A section of an EPR copolymer chain is shown at the bottom. 

Some structures were examined which were not included in the cubic section model used for testing. For example, linear and branched 3-alkyl cyclohexanols can change some forbidden cubes in col 3 to allowed. Also 4-alkyl cyclohexanols were found to extend the allowed area in layers 2, 3, and 4. These substrates were not used in the construction because they were studied under different reaction conditions or no values of enantiomeric excess were available. Addition of substrates containing hetereocyclic bicyclic rings (70-73) would change cubes OaO, laN, laP, laQ, and 3E to allowed (Figure 16). 

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