Tensor eigenvalue

Because the choice of the particular decomposition method is non-unique (e.g., lost and Herrmann 1989 Julian et al. 1998), there is a growing interest in using graphical representations of the moment tensor eigenvalues (Hudson et al. 1989 Riedesel and Jordan 1989 Tape and Tape 2012). For example, the source-type diagram of Hudson et al. (1989) examines ratios of the three eigenvalues of the moment tensor decomposition to... 

Numerical tests performed with sets of 25-250 focal mechanisms are presented. The stress tensor is fixed for all datasets. The focal mechanisms are selected to satisfy the Mohr-Coulomb failure criterion (see Fig. 12a, b) and subsequently they are used for the calculation of moment tensors. The moment tensors were contaminated by uniform noise ranging from 0 to 50 % of the norm of the moment tensor (calculated as the maximum of absolute values of the moment tensor eigenvalues). The noisy moment tensors were decomposed back into strikes, dips and rakes of noisy focal mechanisms inverted for stress. The deviation between the true and noisy fault normals and slips attained values from 0° to 25°... 

Hudson et al. (1989) introduced two source-type plots a diamond x-k plot which is the diamond CL VD-ISO plot described in the previous section but with the opposite direction of the CLVD axis (Fig. 5a) and a skewed diamond u-v plot (Fig. 5b). The latter plot is introduced in order to conserve the uniform probability of moment tensor eigenvalues. If eigenvalues Mj, M2, and M3 have a uniform probability distribution between —1 and +1 and satisfy the ordering condition (9), then all points fill uniformly the skewed diamond plot. Axis u defines the deviatoric sources and axis V coimects the pure explosive and implosive sources. 

Fig. 2 Numeric simulations of the EPR spectra of nitroxide radicals ((A)-(Q) and Gd(iii) complexes ((D)-(F)). For both species the spectra were computed for the X-band detection frequency (9.5 GFIz, (A) and (D)) as well as for Q band (35 GFIz, (B) and (E)) and W band (95 GFIz, (C) and (F)). Spectra were simulated with EasySpin software (www. easyspin.org). Spectroscopic parameters for nitroxide radicals g-tensor eigenvalues -[2.0085 2.0061 2.0022], hyperfine tensor eigenvalues - [13 13 100] MFIz, FWFIM -[0.3 0.3] mT (mixed Lorentzian/Gaussian line shape). For nitroxide radicals subspectra corresponding to the spin projection of -Fl (left subspectrum), 0 (middle subspectrum), and -1 (right subspectrum) are plotted as dashed lines. Spectroscopic parameters for Gd(iii) centres isotropic g-value of 1.991 D-values normally distributed with =1500 MHz and a D)= /5 D/ values distributed, according to P(x)=x/3-2x /9 (see ref. 29 and 65).

FIG. 11 Eigenvalues of the radius of gyration tensor (dots largest, squares middle triangles smallest) of micelles vs aggregation number N in an oif-lattiee model of H2T2 surfaetants. The mieelle size distribution for this partieular system has a peak at 28. (From Viduna et al. [144].)... 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

Based on the foregoing discussion, a criterion for the separation of the fragments is easily obtained. If at least one eigenvalue of the tensor M is positive, the pair orients along the corresponding principal axis and the critical Fragmentation number for separation is given by... 

Fig. 2.10. Eigenvalues of the Fourier component of the dipole-dipole interaction tensor in two-dimensional infinite lattices. The solid lines are for a triangular lattice, the dashed lines are for an analytical approximation (2.2.9), and the dotted lines are for a square lattice.

The eigenvalues of tensor (k) at the symmetric points of the first Brillouin zone that are determined by formula (3.1.7) take the form ... 

Fig. 11. Tensor-valued elasticity parameters in a human breast in vivo. A dotted circle symbolizes a carcinoma previously localized using gadolinium-enhanced Ti-weighted imaging. Eigenvalues Ei, E2, and E3 of the elasticity tensor are shown in (a), (b), and (c) respectively. Also shown in (d) is the isotropic elasticity...

The second-order fluctuating rate-of-strain tensor is real and symmetric. Thus, its three eigenvalues are real and, due to continuity, sum to zero. The latter implies that one eigenvalue (a) is always positive, and one eigenvalue (y) is always negative. In the turbulence literature (Pope 2000), y is referred to as the most compressive strain rate. 

Odelius and co-workers reported some time ago an important study involving a combined quantum chemistry and molecular dynamics (MD) simulation of the ZFS fluctuations in aqueous Ni(II) (128). The ab initio calculations for hexa-aquo Ni(II) complex were used to generate an expression for the ZFS as a function of the distortions of the idealized 7), symmetry of the complex along the normal modes of Eg and T2s symmetries. An MD simulation provided a 200 ps trajectory of motion of a system consisting of a Ni(II) ion and 255 water molecules, which was analyzed in terms of the structure and dynamics of the first solvation shell of the ion. The fluctuations of the structure could be converted in the time variation of the ZFS. The distribution of eigenvalues of ZFS tensor was found to be consistent with the rhombic, rather than axial, symmetry of the tensor, which prompted the development of the analytical theory mentioned above (89). The time-correlation... 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

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