INDEX triangularity

Provided that the m variances Vj - sj are roughly equal (Bartlett s test, see Section 1.7.3), the m means are ordered (cf. subroutine SORT, Table 5.17). The smallest mean has index 1, the largest has index m. A triangular matrix (see Tables 4.9, 4.10) is then printed that gives the m (m- l)/2 differences AXmean.wi = Xmean.M Xmean.i for all possible pairings. Every element of the... 

This index is redundant due to the symmetry of the adjacency matrix. It can be reduced to the sum A of the upper triangular adjacency submatrix, which is called 11 the total adjacency of a molecule ... 

For real symmetric matrices of dimension n, a triangular pattern (referred to as T) is used with the location of i,j computed as L 3 i+j(j-l)/2 for ij]. The Cl hamiltonian matrix is a large real symmetric matrix with mostly zero entries (provided orthonormal configurations constructed from orthonormal orbitals are used). If more than half the entries are zero it is more efficient to omit zero entries and include the index as a label (if the word length is long and the matrix is small enough, this label may be packed into the insignificant bits of the matrix element). 

Elastic neutron diffraction was first performed (analyzer in Fig. 1 set to zero energy transfer) to establish the structure of the monolayer at low temperature. Three Bragg reflections were observed which could be indexed by a triangular lattice having a nearest-neighbor distance about 10% smaller than required for a 3 X /3 R30° commensurate structure (every third carbon hexagon in the graphite basal plane occupied). 

Figure 1. Triangulation numbers T = (fp+hk + k2) represented on an equilateral triangular net. An icosadeltahedron (see figure 2) with a five-fold vertex at the origin of this net and a neighbouring five-fold vertex at a lattice point of index h, k will have A = 207 triangular facets, V6 = lOfT1— 1) six-connected vertices, and V5 = 12 five-vertices.

Analysis of ternary mixtures of certain drugs including salicylamide were studied by plotting the refractive index isotherms of the fused mass and the isotherms of the primary crystals on triangular concentration diagrams. The influence of each factor upon the quantitative determination could be found. In each case there was an area unsuitable for measurement where the isotherms were practically parallel when this was not the case, quantitative determination was possible. Precaution should be taken to prevent loss of highly volatile components (7l). 

A liquid prism was created on a PDMS chip for detection based on absorption and refractive index shift. The liquid prism was formed by filling a hollow triangular-shaped chamber with a liquid sample. Excitation and emission were arranged at the minimum deviation configuration. At a low concentration of fluorescein (< 100 pM), excitation light from an optical fiber was absorbed by the molecule, but there was no shift in the excitation maximum. In this absorption-only mode, the LOD of fluorescein was 6 pM. At higher concentration (i.e., 100-1000 pM), there is an additional shift in the excitation maximum This leads to a much sharper decrease in the measured intensity, which is more than can be accounted for simply by the absorption effect [714]. 

The matrix (Aqte)is lower-triangular by construction. For the index range j solution matrix elements are determined by... 

Multilin utilizes transpositional symmetry for any pairs of indices. Thus, if A is declared symmetric, antisymmetric, Hermi-tean, or anti-Hermitean in any index pair, an appropriate (multiply) triangular storage and addressing scheme is invoked. 

Corollary. Let S be any solvable matrix group over an algebraically closed field. Then S has a normal subgroup of finite index which can be put in triangular form. 

Fig. 21 Comparison of refractive index sensitivity for Ag nanoparticles with different geometries. The spherical nanoparticle ( ) has a sensitivity of 161 nm per RIU, the triangular nanoparticle (A) has a sensitivity of 197 nm per RIU, and the rodlike nanoparticle ( ) has a sensitivity of 235 nm per RIU. Reproduced with permission from [41]. Copyright 2003 American Chemical Society...

CoO(lll) and MnO(lll) polished single crystals of purity 99.9% were provided by Crystal GmBH (Berlin, Germany). The samples were cut and aligned to better than 0.1° prior polishing. After polishing, samples with a surface mosaic spread of about 0.8° for CoO and 1.6° for MnO were obtained. A triangular unit cell was chosen to describe the 3-fold symmetry of the (111) surface plane. The unit vectors are related to the cubic ones by as=[il0]/2, bs=[ll0]/2, Cs=[lll]. The h and k indexes describe the in-plane momentum... 

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