Sparse large

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

This factorization might be preferable to the LQ factorization for sparse large systems. 

In sparse large-scale problems, it may be computationally onerous to project a ( ) vector using either (13.41) or LQ factorization. 

Most of the discretization methods for time-dependent PDFs are extensions of the well-known method of lines based on first space and then time discretization. Discretizing first in space, the initial problem (1.1) is transformed into an ODF- or DAF-system which is nonlinear, stiff, sparse, large and block-structured in industrial applications (see Figure 1.1). 

For a very large number of variables, the question of storing the approximate Hessian or inverse Hessian F becomes important. Wavefunction optimization problems can have a very large number of variables, a million or more. Geometry optimization at the force field level can also have thousands of degrees of freedom. In these cases, the initial inverse Hessian is always taken to be diagonal or sparse, and it is best to store the... 

This left 20 X 20 X 8 = 3200 classes, with some classes being very sparsely populated. For such classes, the error term is unacceptably large,... 

The LIN method (described below) was constructed on the premise of filtering out the high-frequency motion by NM analysis and using a large-timestep implicit method to resolve the remaining motion components. This technique turned out to work when properly implemented for up to moderate timesteps (e.g., 15 Is) [73] (each timestep interval is associated with a new linearization model). However, the CPU gain for biomolecules is modest even when substantial work is expanded on sparse matrix techniques, adaptive timestep selection, and fast minimization [73]. Still, LIN can be considered a true long-timestep method. 

X-ray diffraction work (11,15) shows that there is an ionomer peak at 4°C which is absent in the acid precursor. This low, broad peak is not affected by annealing or ion type and persists up to 300°C. Since the 4°C peak corresponds to a spacing of about 2.5 nm, it is reasonable to propose a stmctural feature of this dimension in the ionomer. The concept of ionic clusters was initially suggested to explain the large effects on properties of relatively sparse ionic species (1). The exact size of the clusters has been the subject of much debate and has been discussed in a substantial body of Hterature (3,4,18—20). A theoretical treatment has shown that various models can give rise to supramoleculat stmctures containing ionic multiplets which ate about 10 nm in diameter (19). 

To avoid such small time steps, which become smaller as Ax decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an /i X n grid to a series of 2n one-dimensional problems on an n grid. 

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... 

Different processes like eddy turbulence, bottom current, stagnation of flows, and storm-water events can be simulated, using either laminar or turbulent flow model for simulation. All processes are displayed in real-time graphical mode (history, contour graph, surface, etc.) you can also record them to data files. Thanks to innovative sparse matrix technology, calculation process is fast and stable a large number of layers in vertical and horizontal directions can be used, as well as a small time step. You can hunt for these on the Web. 

The highest level of integration would be to establish one large set of equations and to apply one solution process to both thermal and airflow-related variables. Nevertheless, a very sparse matrix must be solved, and one cannot use the reliable and well-proven solvers of the present codes anymore. Therefore, a separate solution process for thermal and airflow parameters respectively remains the most promising approach. This seems to be appropriate also for the coupling of computational fluid dynamics (CFD) with a thermal model. ... 

Specific details regarding the accident and its causes are sparse because, as is usually the case in such a miumiioth disaster, none of the phmt operators present at tlie time sur ived. The large buildings containing tlie anunonium sulfonitrate disappeared entirely, mid nothing was left in their place... 

More often, however, s.a. methods are applied at the outset, with more or less arbitrary initial approximations, and without previous application of a direct method. Usually this is done if the matrix is extremely large and extremely sparse, which is most often true of the matrices that arise in the algebraic representation of a functional equation, especially a partial differential equation. The advantage is that storage requirements are much more modest for s.a. methods. 

The obvious question then arises as to whether the effective double layer exists before current or potential application. Both XPS and STM have shown that this is indeed the case due to thermal diffusion during electrode deposition at elevated temperatures. It is important to remember that most solid electrolytes, including YSZ and (3"-Al2C)3, are non-stoichiometric compounds. The non-stoichiometry, 8, is usually small (< 10 4)85 and temperature dependent, but nevertheless sufficiently large to provide enough ions to form an effective double-layer on both electrodes without any significant change in the solid electrolyte non-stoichiometry. This open-circuit effective double layer must, however, be relatively sparse in most circumstances. The effective double layer on the catalyst-electrode becomes dense only upon anodic potential application in the case of anionic conductors and cathodic potential application in the case of cationic conductors. 

The major types of cytoskeletal filaments are 7-nm-thick microfilaments. 25-nm-thick microtubules, and 10-nm-thick intermediate filaments (IPs). These are respectively composed of actin, tubulin, and a variety of interrelated sparsely soluble fibrous proteins termed intermediate filament proteins. In addition, thick myosin filaments are present in large numbers in skeletal and heart muscle cells and in small numbers in many other types of eukaryotic cells. 

REX-CPHMD simulations have also been applied to understand the mechanism of the formation of protein intermediate states. Recent solution NMR data revealed a sparsely populated intermediate in the villin headpiece domain, in which the N-terminal subdomain is largely random but the C-terminal subdomain adopts a nativelike fold [34], Interestingly, H41 in this intermediate state titrates at a pH value of... 

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