Spherical decomposition

Invariants and Spherical Decomposition of a Second-Order Real-Valued Symmetric Tensor... 

S.3.2 Geometrical Interpretation of Spherical Decomposition in the Principal Space... 

This implies that, by spherical decomposition, cylindrical polar coordinates are introduced in terms of the hydrostatic axis. 

S.3.3 Spherical Decompositions of Stress and Strain and the Response of an Isotropic Elastic Solid... 

The stress o is a second-order real-valued symmetric tensor, and the spherical decomposition is given as follows ... 

Wlien the atom-atom or atom-molecule interaction is spherically symmetric in the chaimel vector R, i.e. V(r, R) = V(/-,R), then the orbital / and rotational j angular momenta are each conserved tln-oughout the collision so that an i-partial wave decomposition of the translational wavefiinctions for each value of j is possible. The translational wave is decomposed according to... 

When solving difference boundary-value problems for Poisson s equation in rectangular, angular, cylindrical and spherical systems of coordinates direct economical methods are widely used that are known to us as the decomposition method and the method of separation of variables. The calculations in both methods for two-dimensional problems require Q arithmetic operations, Q — 0 N og. N), where N is the number of the grid nodes along one of the directions. 

In electrocatalysis, the activated carbons, glassy carbon, and carbon black are the transitional forms used. Carbon black is the product of incomplete combustion or decomposition of organic compounds. The shape of its particles is close to spherical. They contain several carbon atom lattice fragments arranged without order. Various types of carbon black serve as substrates for metal catalysts, the properties of the carbon blacks themselves having a strong elfect on the catalytic activity of the combined catalysts thus obtained. 

One successful technique involves the addition of urea to metal salts in aqueous solution under conditions of pH which facilitate the decomposition of the urea and precipitation of metal hydroxycarbonates.89-100 This has led, for example, to the preparation of spherical particles of Y203 Eu particles of 100, 200, 300, 400, and 500 nm dimensions having a size distribution that can be controlled to within 5%.92-97 Gd203 Eu has been prepared similarly, giving particles of sizes between 70 nm and 250 nm108... 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

Kimura et al. [5.9] produced spherical YBa2Cu307-delta material with a diameter between 20-30 pm from freeze dried nitrate solutions and its thermal decomposition. 

ZnO particle morphologies are very complex and diversiform in comparison with Ti02. Thus, monodispersed ZnO particles with well-defined morphological characteristics, such as spherical, ellipsoidal, needle, prismatic, and rod-like shapes, have been obtained. Aggregates composed of these basic shape particles have also been achieved. The methods used for synthesis of these ZnO powders include alkali precipitation [214-216], thermal decomposition [217], hydrothermal synthesis [218], organo-zinc hydrolysis [219], spray pyrolysis [220], and other routes. 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

Before we examine some specific solvation effects on cooperativity we must first consider various aspects of the solvation Gibbs energy of a macromolecule a. We present here one possible decomposition of AG which will be useful for our purposes. Consider a globular protein a which, for simplicity, is assumed to be compactly packed so that there are no solvent molecules within some spherical region to which we refer as the hard core of the protein. The interaction energy between a and the fth solvent molecule (the solvent is presmned to be water, w) is written as... 

Function approximation comes naturally with the Fourier transition. Since tiny details of a function in real space relate to high-frequency components in Fourier space, restricting to low-order components when transforming back to real space (low-pass filtering) effectively smoothes the function to any desirable degree. There are special function decomposition schemes, like spherical harmonics, which especially build on this ability [128]. 

While the evidence for dual phase continuity provided by Figure 5 does not indicate directly any mechanism for phase separation, or the shape of the phases, dispersed, spherical polystyrene domains probably would not yield results of this type. By hind sight, the data are consistent with the notion of spinodal decomposition and cylindrical domains. 

The collage of electron micrographs shown in Figure 19 Illustrates a probable transition from a nucleation and growth phase separation mechanism to a spinodal decomposition. The 7% PS sample shows more or less spherical PS domains, characteristics of nucleation and growth. However, a few domains appear to have some elliptical characteristics. When the PS content increases up to midrange, the shape of the PS domains becomes more obviously elliptical, suggestive of cross-sections of cylinders or other worm-like structures. As the PS content increases further, the... 

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