Projections, vector

In the general case we use the symbols U and V to represent projection matrices in 5" and S , each containing r projection vectors, and the symbols S and L to represent their images in the dual space ... 

Eigenvector projections are those in which the projection vectors u and v are eigenvectors (or singular vectors) of the data matrix. They play an important role in multivariate data analysis, especially in the search for meaningful structures in patterns in low-dimensional space, as will be explained further in Chapters 31 and 32 on the analysis of measurement tables and general contingency tables. 

A rock is made of 0.45 moles Si02,0.10 moles CaO, and 0.45 moles MgO, which we will describe with the composition vector v (0.45,0.10,0.45). If we consider this rock as being made of virtual forsterite (Si02, 0, 2MgO), enstatite (2Si02, 0, 2MgO), and diopside (2Si02, CaO, MgO), what is the projection vector v of the rock composition onto the enstatite-diopside plane ... 

Li is the matrix representation of the lattice Liouvillian in the space of the basis operators, 1 is a unit (super)operator and Ci are projection vectors representing the operators of Eq. (32) in the same space. The... 

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

It is customary to think of the projective vector space of the spin-n/2 representation as instead of P . In a sense, this is a distinction without... 

As described above, time-delay analysis [389] of the energy derivative of the phase matrix 4> determines parametric functions that characterize the Breit-Wigner formula for the fixed-nuclei resonant / -matrix R[N(q e). The resonance energy eKS(q), the decay width y(q). and the channel-projection vector y(q) define R and its associated phase matrix such that tan = k(q)R , where... 

We shall organize the Chapman-Enskog reduction into four steps Step 1) Initial suggestion M w C Mi for the slow manifold Msiow C Mi is made. It is a manifold that has a one-to-one relation with the space M2. We can regard it as an imbedding of M2 in Mi. Step 2) The vector field v.f.) 1 is projected on M w (i.e., v.f. jf 1 denoting the vector field v. /.) 1 attached to a point of M, is projected on the tangent plane of M ow a at point). The projected vector field is denoted by the symbol... 

To test the correlation, a smoothing function is fitted to the plot of each c vector versus each r vector to minimize the squared error of c and the model s fit to c. The transformation of the projected vectors is performed by a... 

In a set of j projections with different projection vectors pi/, an Y-dimensional cross peak Q is projected orthogonally to the locations Qf Here,/is an arbitrary numeration of the set of j projections / = 1,. .., / In the 2D coordinate system of projection/ the projected cross peak has the position vector Ql = [v... 

For both planar and pyramidal mnpX systems, the letter subscript, in effect, defines the angle between the vector/projected vector and the XY-plane. For other orientations of Vy, one may replace the letter subscript with a numerical subscript for the value of the angle, e.g. v+45 is the orientation intermediate between 4a-3 and 4a-4, and v.450 corresponds to the intermediate between 2b-3 and 2b-4. 

We could say The four non-zero components are only determined within a gradient 91ogp/9x. These properties are characteristic of the four electromagnetic potentials. In other words The four electromagnetic potentials are the non-zero components of a projective vector whose null component is 1. 

The quantities 7 /3 and (pa represent a projective tensor, or rather a projective vector, both of index 0. is a scalar of index N. There are invariant... 

The projective tensor 0 0 contains, so to say, a projective scalar, a projective vector S lid an affine tensor gij. In fact... 

With the help of a projective connection we can construct, from the components of an arbitrary projective tensor, a new projective tensor, of higher rank, through the formulae of covariant diherentiation. For instance, if Ac is a covariant projective vector then... 

Z°- considered a function of q therefore is a contravariant projective vector with index -1 due to (16). 

A further affine invariant is established by the projective vector... 

Further we should stress that no use has been made of A further projective vector of index 0,... 

We have seen further that any projective vector X" corresponds through (1) to a point dx of the tangent space. We can also represent the relationship between the projective vectors and the tangent space in a somewhat different... 

Take as given a point x, a specific representation and five arbitrary numbers X°, X, ... X. We now combine the totality of all contravariant projective vectors of fixed index whose components assume the values X°, X, ... X in the given presentation, into a single geometrical objecf. The vectors of a... 

It follows immediately from the basic properties of a projective vector that the definition of associated spaces is independent of coordinate system. Hence we can in fact consider associated spaces as geometrical objects. 

Methods that use the direction of the function gradient as the projection vector on the active constraints are known as Gradient Projection Methods. 

Knowing all feasible localizations during assignment allows a small and fast optimization feedback loop in the assignment task itself, A tradeoff between the cost of non-matched throughput or a bad localization can then be made for a number of projection vector alternatives. 

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