Symmetrical rank

A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Math. Prog., 2, 177 (1991). Convergence of Quasi-Newton Matrices Generated by the Symmetric Rank One Update. 

According to the affine deformation postulate for the bulk deformation A, a symmetrical rank two tensor, the end-to-end distance vector deforms from Ro to... 

Try the form Hk+1 = Hk + a - uu1, noting that uu is a symmetric rank-one matrix. It turns out that the basic recursion requirement is satisfied by taking u = pk -Hkqk, and... 

For orbitally degenerate systems, in the presence of zero-field sphtting or strong spin-orbit coupling (Fe °, Co , Ln . ..), the expression of the magnetic susceptibility becomes a symmetric rank-2 tensor which can be typically defined empirically. In its principal axis system (PAS), i.e., the frame in which the tensor is diagonal (with Xzz > lyy > lxx ), X is defined by its isotropic value Xiso, its anisotropy Ax, and its asymmetry r ... 

If A is a symmetric positive definite matrix then we obtain that all eigenvalues are positive. As we have seen, this occurs when all columns (or rows) of the matrix A are linearly independent. Conversely, a linear dependence in the columns (or rows) of A will produce a zero eigenvalue. More generally, if A is symmetric and positive semi-definite of rank r[Pg.32]

Thus far we have considered the eigenvalue decomposition of a symmetric matrix which is of full rank, i.e. which is positive definite. In the more general case of a symmetric positive semi-definite pxp matrix A we will obtain r positive eigenvalues where r general case we obtain a pxr matrix of eigenvectors V such that ... 

The p-value for the sign test or Wilcoxon signed rank test can be found in the pValue variable in the pvalue data set. If the variable is from a symmetric distribution, you can get the p-value from the Wilcoxon signed rank test, where the Test variable in the pvalue data set is Signed Rank. If the variable is from a skewed distribution, you can get the p-value from the sign test, where the Test variable in the pvalue data set is Sign. ... 

This matrix describes the transformation from x y z to xyz as a rotation about the z axis over angle a, followed by a rotation about the new y" axis over angle /), followed by a final rotation over the new z " axis over angle y (Watanabe 1966 148). Formally, the low-symmetry situation is even a bit more complicated because the nondiagonal g-matrix in Equation 8.11 is not necessarily skew symmetric (gt] -g. Only the square g x g is symmetric and can be transformed into diagonal form by rotation. In mathematical terms, g x g is a second-rank tensor, and g is not. 

As the Chern classes of symmetric powers of vector bundles of rank 2 axe easy to compute, we know now the Chow ring of Hilbn(P( )/A). In particular we obtain ... 

Examples of force and stress acting on a parallelepipedic element is shown in Fig. 1. A rank 2, real and symmetric stress tensor may thus be defined as ... 

Coates cation [51], originally proposed to account for the exceptionally rapid stereospecific hydrolysis of the />-nitrobenzoate [59] (Coates and Kirkpatrick, 1970), joins the ranks of the well-established trishomoaro-matics. Proton and l3C NMR observations on [51] (and a methyl and phenyl derivative) supported the symmetrical delocalized nature of this ion (Coates... 

A computation of Raman intensities can be done precisely in the same way as for infrared intensities. One needs here, in addition to the wave functions of the initial and final state, the polarizability tensor a(r,0, < )). This is a symmetric tensor of rank 2 that in Cartesian coordinates can be written as... 

Although it is not related to the rest of this chapter, it is worth while to remark that the relationship between Chern classes/characters and symmetric functions. If i is a complex vector bundle of rank N, we can define its Ath Chern classes q. The total Chern class is its generating function ... 

Note 3 For mesophases eomposed of cylindrically symmetric molecules there is a precise relationship between the magnetic anisotropy, A, and the second-rank orientational parameter P ). 

A cumulant of rank s is a symmetric tensor with (s2 + 3s + 2)/2 unique elements for a three-dimensional distribution. Like the moments p and the quasimoments c, the cumulants are descriptors of the distribution. For a onedimensional distribution, the relations between the cumulants and the moments are defined by equating the two expansions ... 

The Hessian matrix H(r) is defined as the symmetric matrix of the nine second derivatives 82p/8xt dxj. The eigenvectors of H(r), obtained by diagonalization of the matrix, are the principal axes of the curvature at r. The rank w of the curvature at a critical point is equal to the number of nonzero eigenvalues the signature o is the algebraic sum of the signs of the eigenvalues. The critical point is classified as (w, cr). There are four possible types of critical points in a three-dimensional scalar distribution ... 

Symmetry restrictions for a number of crystal systems are summarized in Table B.l. The local symmetry restrictions for a site on a symmetry axis are the same as those for the crystal system defined by such an axis, and may thus be higher than those of the site. This is a result of the implicit mmm symmetry of a symmetric second-rank tensor property. For instance, for a site located on a mirror plane, the symmetry restrictions are those of the monoclinic crystal system. 

For binary diffusion in an isotropic medium, one diffusion coefficient describes the diffusion. For binary diffusion in an anisotropic medium, the diffusion coefficient is replaced by a diffusion tensor, denoted as D. The diffusion tensor is a second-rank symmetric tensor representable by a 3 x 3 matrix ... 

In an isotropic medium, D is a scalar, which may be constant or dependent on time, space coordinates, and/or concentration. In anisotropic media (such as crystals other than cubic symmetry, i.e., most minerals), however, diffusivity also depends on the diffusion direction. The diffusivity in an anisotropic medium is a second-rank symmetric tensor D that can be represented by a 3 x 3 matrix (Equation 3-25a). The tensor is called the diffusivity tensor. Diffusivity along any given direction can be calculated from the diffusivity tensor (Equation 3-25b). Each element in the tensor may be constant, or dependent on time, space coordinates and/or concentration. 

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