Determinant sparse

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

Wetness of a metal surface The lime of wetness of the metal surface is an exceedingly complex, composite variable. It determines the duration of the electrochemical corrosion process. Firstly it involves a consideration of all the means by which an electrolyte solution can form in contact with the metal surface. Secondly, the conditions under which this solution is stable with respect to the ambient atmosphere must be considered, and finally the rate of evaporation of the solution when atmospheric conditions change to make its existence unstable. Attempts have been made to measure directly the time of wetness , but these have tended to use metals forming non-bulky corrosion products (see Section 20.1). The literature is very sparse on the r61e of insoluble corrosion products in extending the time of wetness, but considerable differences in moisture desorption rates are found for rusted steels of slightly differing alloy content, e.g. mild steel and Cor-Ten. 

Many in vitro studies have been undertaken to explore the relative effectiveness of isothiocyanates, and associated compounds, as inhibitors of cell growth and inducers of apoptosis in cell lines, and some of these are summarised in Table 4.2. Clonal survival studies are often used to determine whether isothiocyanates inhibit initial cell anchorage and subsequent growth of sparsely seeded cells. However, differences in cell number following challenge with the isothiocyanate could be due either to decreased cell division or increased cell loss, and the authors of some in vitro studies have failed to recognise this. Other studies have, however, considered the interacting... 

Often you need to carry forward data to a specific time point due to holes or sparseness of data. The previous example on determining baseline cholesterol level provides an excellent context for this problem. Assume that you have several cholesterol readings of HDL, LDL, and triglycerides for patients before they take an experimental pill designed to reduce cholesterol levels. For each cholesterol parameter, you want the last observation carried forward so long as the measures occur within a five-day window before the pill is taken. Here are some sample data that illustrate the problem ... 

The objectives of a formulator in performing a mixture design are to not only determine the component effects and blending relationships but also optimize the component levels to achieve a maximum or minimum response of a measured property. Unfortunately, the mixture design literature is sparse in references to mixture optimization. McLean and Anderson (9) in the classic flare example attempted to use Lagrange multipliers to maximize the equation describing the intensity of an ignited flare composition but obtained erroneous results. However, a secondary technique which was not discussed did produce the optimum. 

Aeolian sands are deposited by wind action, either in dunes or in extensive sheets ( cover sand areas ) Wind action is particularly effective in hot and dry regions such as deserts but sand dunes are also common in (sub)humid regions with sparse vegetation, notably in overgrazed areas and along beaches and fluvial braid plains . The (weathering) history of the parent materials in the source area determines whether the sands are rich in quartz and/or carbonates. 

Inlier detection methods, n - statistical tests which are conducted to determine if a spectmm resides within a region of the multivariate calibration space which is sparsely populated. 

One solution to the dipolar truncation of small coupling interactions by larger coupling interactions is to use sparsely labeled samples, e.g., using differently 13C-labeled glycerol in the expression as now used frequently for solid-state NMR structure determination, despite the need for more costly samples and extraction of structural parameters from several samples (which accordingly need to be identical). This approach has, along with recommendations on experimental protocols, recently been analyzed by Bayro et al. [144],... 

With these approximations the secular determinant for the energy eigenvalue e is reduced to sparse banded form for example, for a linear polyene... 

Modem LP solvers can solve very large LPs very quickly and reliably on a PC or workstation. LP size is measured by several parameters (1) the number of variables n, (2) the number of constraints m, and (3) the number of nonzero entries nz in the constraint matrix A. The best measure is the number of nonzero elements nz because it directly determines the required storage and has a greater effect on computation time than n or m. For almost all LPs encountered in practice, nz is much less than mn, because each constraint involves only a few of the variables jc. The problem density 100(nz/mn) is usually less than 1%, and it almost always decreases as m and n increase. Problems with small densities are called sparse, and real world LPs are always sparse. Roughly speaking, a problem with under 1000 nonzeros is small, between 1000 and 50,000 is medium-size, and over 50,000 is large. A small problem probably has m and n in the hundreds, a medium-size problem in the low to mid thousands, and a large problem above 10,000. 

As an example of the measurement of cross-correlated relaxation between CSA and dipolar couplings, we choose the J-resolved constant time experiment [30] (Fig. 7.26 a) that measures the cross-correlated relaxation of 1H,13C-dipolar coupling and 31P-chemical shift anisotropy to determine the phosphodiester backbone angles a and in RNA. Since 31P is not bound to NMR-active nuclei, NOE information for the backbone of RNA is sparse, and vicinal scalar coupling constants cannot be exploited. The cross-correlated relaxation rates can be obtained from the relative scaling (shown schematically in Fig. 7.19d) of the two submultiplet intensities derived from an H-coupled constant time spectrum of 13C,31P double- and zero-quantum coherence [DQC (double-quantum coherence) and ZQC (zero-quantum coherence), respectively]. These traces are shown in Fig. 7.26c. The desired cross-correlated relaxation rate can be extracted from the intensities of the cross peaks according to ... 

It is a somewhat surprising fact that information about vibrational frequencies and bond lengths in very common and important ions is very sparse. Clearly, many more such determinations are necessary. Only this year, for example, have data become available which enable metal-ligand displacements on electron removal for the aquo Fe(II)/Fe(III) and Co(II)/Co(III) systems to be obtained. In view of the importance of these partially unpublished data, I have reproduced them in Table VI. The values of A for the Fe and Co systems are 0.128-0.137, and 0.208 8 respectively. The variability of the Fe results points to the fact that caution must be exercised in using data obtained in crystal lattices for a solution environment. Spin-polarized neutron diffraction studies on the structure of solu-... 

In earlier work we pointed out that very rarely is Step 7 of Table I the rate-determining step the value calculated for L is almost always too large (J). This is an expected result, since on a sparsely covered surface two mole-... 

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