Derivatives linearity

Concerning isotopic equilibria between solute carbonates and gaseous CO2, Deines et al. (1974), on the basis of all the existing data, derived linear equations relating the fractionation factor to temperature ... 

For materials that are isotropic, that is, have the same properties in all directions, it can be shown that ay = 3ai. A material that has different properties in different directions is said to be anisotropic. Thus, a linear expansion coefficient, if no direction of measurement is explicitly stated, implies an isotropic material. Conversely, a volume thermal expansion coefficient implies an anisotropic material, and one should exercise caution when deriving linear thermal expansion coefficients from volume-based measurements. 

Nina A. Nedolya was born in Irkutsk (Russia) and educated in organic chemistry at the Irkutsk State University (Diploma 1972, PhD 1982, DSc 1998). From 1995 to 1999 she was associated with Prof. L. Brandsma at the Utrecht University (The Netherlands). In 1999 she obtained her second PhD from the Utrecht University. She is presently Head of the Research Group of Chemistry of Heterocyclic Compounds at A. E. Favorsky Irkutsk Institute of Chemistry. She is the author of over 210 review articles and research papers. She is also one of the inventors for 112 patents. She is interested in the chemistry of polyfunctional unsaturated heteroatomic systems (vinyl, allenyl, and alkynyl ethers and their derivatives, linear and cyclic heteropolyenes, hetero-cumulenes), including synthesis of important heterocycles, particularly pyrroles, thiophenes, thiazoles, imidazoles, dihydrofurans, dihydropyridines, pyridines, quinolines, dihydroazepines, and azepines, based on metallated allenes or alkynes and/or heterocumulenes. 

Hydroformylation of oct-l-ene (in the presence of 10 bar of synthesis gas at 60 °C) gave linear to branched product ratios higher than that obtained when only PPha ligands were used (PPha/Rh = 3). Effects of remote groups were noticeable for the variations R = H, Me, or Ph of the 2-ammebenzaldehyde derivative, linear to branched product ratios were 29, 57, and 8, respectively. The highest linear to branched ratios match those obtainable with xantphos as the ligand. 

The electron transfer reactivity of Ceo has been compared with that of p-benzoquinone which has a slightly more negative one-electron reduction potential ( °red relative to the SCE = -0.50 V) [44] than Ceo (E°red —0.43 V). The rate constants of electron transfer from Cgo and Ceo to electron acceptors such as allyl halides and manganese(III) dodecaphenylporphyrin [45] correlate well with those from semiquinone radical anions and their derivatives. Linear correlations are obtained between logarithms of rate constants and the oxidation potentials of... 

Alcohol sulfates are not stable in acid and are used in alkaline formulations. C12-C16 alcohol sulfates have excellent detergency, high foam, and good wetting properties. Alcohol sulfates are fully biodegradable under aerobic and anaerobic conditions and compete in performance with petrochemical-derived linear alkyl-benzene sulfonates (LABS). 

Another approach involves the application of derivative linear sweep voltammetry or DCV [Fig. 10(b)]. These techniques allow a precision of the order of 0.2 mV to be realized during peak potential measurements [29-31,36]. The feature giving the high precision is that the peak potential is measured at the point where the steeply descending first derivative crosses zero. 

The crosslinking process depends on the structure of the psoralen derivative. Linear furocoumarins form crosslinks efficiently with a yield up to 50% of the overall adducts formed. 

An interesting contribution to data processing Is the work by Bond [39], In which the performances of different variants of the voltammetrlc technique (seml-lntergral, seml-differentlal, direct current linear sweep, direct current derivative linear sweep and pulse and related techniques) were compared with the aid of computerized Instrumentation. 

The influence of exchangeable cations on zeolite framework vibrations was investigated by Maxwell and Baks [285], and, similar to the studies of Flanigen et al. [112], they derived linear relationships between the reciprocal of the sum of the cation radii and oxygen anion radius, l/frcation+fo -) and the wavenumbers of framework vibrations of zeoHtes A and X. 

As expected, the performance of all competing order-N methods depends on the system under investigation, the accuracy needed, the amount of experimental information available, the questions that need to be answered, and also computer-related parameters (processors, parallel architectures, etc.). All approaches have their pluses and minuses. It is also clear that the increased mathematical effort will "pay off" if at all) only beyond a critical number of atoms (around 100-1000 or so) below that, the normal route with cubic scaling is faster. Nonetheless, the same locality arguments may be used to derive linear-scaling methods for the extremely efficient calculation of electronic correlation (see Section 2.13). 

Crystal electric field parameters for YbPtBi have been derived from neutron scattering data (Robinson et al. 1993, 1995) and are interpreted in terms of strongly broadened crystal-field levels. The ground state is six-fold degenerate, consisting of a doublet (F7) and a quartet (Fg). Integration of the levels is in agreement with the experimentally derived linear specific-heat coefficient within an accuracy of 20%. Furthermore, ° Bi NMR... 

Using a finer grid with more points t, we may compute the solution x(t) at more points. The derived linear system would have greater size but the same tridiagonal structure. 

FIGURE 4. Plot of integral CL for different time intervals versus the number of PMN (tested). The data were derived from figure 3, The relationship of change in interval of integration to change in slope of the derived linear function is also presented. 

The experimentally determined Afla° values, measured with trace amounts (at zero surface coverage), can be correlated with their macroscopic sublimation enthalpies AH (Turler et al. 1999). From an empirically derived linear correlation between AH and AH, the sublimation enthalpy of Sg02Cl2 can be estimated as 127 kJ/moL This is an important quantity to estimate the sublimation enthalpy of Sg metal. Seaborgium is estimated to have an equal or even higher sublimation enthalpy than W, the least volatile element in the periodic table (Turler et al. 1999). 

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