INDEX orthogonal

Molecular connectivity terms can also be orthogonalized, generating orthogonal molecular connectivity terms that (1) reduce the inter cor relation among the X terms, (2) improve u), (3) generate coefficients that are stable upon introduction of a new orthogonal index, and (4) detect dominant descriptors whenever X indices are poor descriptors [55-57]. When properties with negative and antithetical values are considered, hke the specific rotations... 

We shall meet examples later where the basis is not denumerable, but has a continuous index, say g>, where the orthogonality and closure conditions, analogs of Eqs. (8-7) and (8-13), are respectively... 

Here the indices a and b stand for the valence orbitals on the two atoms as before, n is a number operator, c+ and c are creation and annihilation operators, and cr is the spin index. The third and fourth terms in the parentheses effect electron exchange and are responsible for the bonding between the two atoms, while the last two terms stand for the Coulomb repulsion between electrons of opposite spin on the same orbital. As is common in tight binding theory, we assume that the two orbitals a and b are orthogonal we shall correct for this neglect of overlap later. The coupling Vab can be taken as real we set Vab = P < 0. 

The orthogonal projection of the epitaxial poly(DMDA) could not be indexed using the unit cell data for the bulk polymerized crystal (8). However, poly(DMDA) cannot usually be polymerized to completion or to high crystallinity in the bulk due to cross-linking. The use of an epitaxial substrate may have controlled the polymerization process that led to oriented single crystals. 

Each irreducible representation of a group consists of a set of square matrices of order lt. The set of matrix elements with the same index, grouped together, one from each matrix in the set, constitutes a vector in -dimensional space. The great orthogonality theorem (16) states that all these vectors are mutually orthogonal and that each of them is normalized so that the square of its length is equal to g/li. This interpretation becomes more obvious when (16) is unpacked into separate expressions ... 

If, in a vector space of an infinite number of dimensions the components Ai and Bi become continuously distributed and everywhere dense, i is no longer a denumerable index but a continuous variable (x) and the scalar product turns into an overlap integral f A(x)B(x)dx. If it is zero the functions A and B are said to be orthogonal. This type of function is more suitable for describing wave motion. 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

It is well known that for lossless media, all squared effective indexes are real, and for any transversally limited structure they form a discrete nongrowing sequence. The field distributions / (O and /i (0 of the m-th mode are mutually orthogonal, have the same phase at each point of the cross-section, and the set of functions corresponding to all modes is complete. The orthogonality relations can be taken in the form... 

For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

Index 1 problems two-point orthogonal collocation (4-stable) ... 

As in solid-phase synthesis programs, one of the most challenging problems with solution-phase synthesis is to separate products obtained in the reaction mixtures and identify the compound or compounds with biological potency or some other desirable property. An especially effective technique that has been developed to deal with this problem is called indexed libraries, also known as orthogonal libraries. In this process, each product compound is prepared twice. Analysis then permits identification of the positive part of the compound and the negative part of the compound with the greatest potency. Once these have been determined, it is possible to identify the specific compound most active in the mixture. 

Note 2 In the case of a short pitch, when P is less than the wavelength X, the macroscopic extraordinary axis for the refractive index is orthogonal to the director. 

Orthogonal crystalline powder refractive index (for montroydite) 2.37 density 11.14 g/cm3 Moh s hardness 2.5 insoluble in water and ethanol soluble in dilute acids and aqueous solutions of alkali iodides and cyanides. 

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