Projected tensors inversion

The equivalence of Eqs. (2.133) and (2.136) for is a special case of a more general theorem relating inverses of projected tensors, which is stated and proved in the Appendix, Section B. Both Eqs. (2.133) and (2.136) yield tensors that satisfy Eq. (2.135), and that thus have vanishing hard components. The equivalence of the soft components of these tensors may be confirmed by substituting expansion (2.136) into the RHS of Eq. (2.132), expanding on the... 

In this section, we develop some useful relationships involving the determinants and inverses of projected tensors. Let S ap be the Riemannian representation of an arbitrary symmetric covariant tensor with a Cartesian representation S v We may write the Riemannian representation in block matrix form, using the indices a,b to denote blocks in which a or p mns over the soft coordinates and i,j to represent hard coordinates, as... 

We next present a relationship between the inverses of the projected tensors S and T within the soft and hard subspaces, respectively. By requiring that the matrix product of the block matrices (A.l) and (A.2) yield the identity tensor, it is straightforward to show that the elements of the matrix... 

We define Cartesian representations of these inverse projected tensors as sums ... 

Long-Period Moment-Tensor Inversion The Global CMT Project... 

Long-Period Moment-Tensor Inversion The Global CMT Project, Fig.l Kernel east-west seismograms for a hypothetical deep earthquake (/i = 500 km) in Bolivia recorded at a station in Los Angeles, California. Each trace corresponds to the motion associated with a single moment-tensor element/. The ground motion has been filtered between 30 and 300 s... 

Long-Period Moment-Tensor Inversion The Global analyses in 2012. Stations that contributed data for more CMT Project, Fig. 2 Map showing the locations of than 200 earthquakes ate shown with hexagons, other 206 stations that contributed seismograms to the GCMT stations are shown with squares... 

Long-Period Moment-Tensor Inversion The Global CMT Project, Fig. 3 A selection of observed black) and modeled red) waveforms for the May 20,2012, Mn = 6.1 Emilia-Romagna, Italy, earthquake (875 waveforms were used in the CMT inversirai). The station, network, and channel of motion are given to the right of each pair of... 

It is a basic quantity evaluated for earthquakes on all scales from acoustic emissions to large devastating earthquakes (see entries Long-Period Moment-Tensor Inversion The Global CMT Project Reliable Moment Tensor Inversion for Regional- to Local-Distance Earthquakes and Regional Moment Tensor Review An Example from the European-Mediterranean Region ). 

In the last few years, only GCMT solutions are computed inverting also for intermediate period surface waves, variation that allows a lowering of the magnitude threshold down to 5.0 (see entry on Long-Period Moment-Tensor Inversion The Global CMT Project ). 

Fixman has shown [2] that, for any covariant symmetric tensor S ap defined in the full space, with an inverse = (5 ) in the full space, the determinants S and f of the projections of S and T onto the soft and hard subspaces, respectively, are related by... 

The expression given by BCAH for elements of the constrained mobility within the internal subspace is based on inversion of the projection of the modified mobility within the internal subspace, rather than inversion of the projection (at of the mobility within the entire soft subspace. BCAH first define a tensor given by the projection of the modified friction tensor onto the internal subspace, which they denote by the symbol gat and refer to as a modified covariant metric tensor, which is equivalent to our CaT - They then define an inverse of this quantity within the subspace of internal coordinates, which they denote by g and refer to as a modified contravariant metric tensor, which is equivalent to our for afi = 1,..., / — 3. It is this last quantity that appears in their diffusion equation, given in Eq. (16.2-6) of Ref. 4, in place of our constrained mobility Within the space of internal coordinates, the two quantities are completely equivalent. 

The error concerned an explicit formula for the translational diffusion coefficient. Kirkwood calculated the diffusion tensor as the projection onto chain space of the inverse of the complete friction tensor he should have projected the friction tensor first, and then taken the inverse. This was pointed out by Y. Ikeda, Kobayashi Rigaku Kenkyushu Hokoku, 6, 44 (1956) and also by J. J. Erpenbeck and J. G. Kirkwood, J. Chem. Phys., 38, 1023 (1963). An example of the effects of the error was given by R. Zwanzig, J. Chem. Phys., 45, 1858 (1966). In the present article this question does not come up because we use the complete configuration space. 

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