Partial transposing

These rocks show clear cross-cutting relationships with the peridotite foliation and compositional layering (Shervais, 1979 Wilshire et al., 1980). They were generally deformed at high temperature, but only partially transposed into the peridotite... 

There is more than one criterium to determine whether a quantum state is entangled or not. One of those was proposed by Peres, in 1996 [4], In order to explain Peres criterium, we need to define the operation of partial transposing a density matrix. Let us write a density matrix p in the following way ... 

This means the matrix element Pikji becomes Pnjk- For instance, partial transposing the matrix representing the cat state. Equation (6.1.2) results in ... 

The partial transposing operation of this matrix can be easily calculated ... 

P6.4 - Apply the Peres criterium to the following two-qubit density matrices and determine the value of e for which the eigenvalues of the partially transposed matrices will be... 

On the other hand, under partial transposing, the second matrix becomes ... 

Carbon oxide species. Ionic cyanides are partially transposed to carbonates by... 

An excess of COs partially transposes the alkaline-earth oxalates, and vice versa, in accord with the Law of Mass Action ... 

To (hopefully) help keep things simple, we will organize all of our data into column-wise matrices. Later on, when we explore Partial Least-Squares (PLS), we will have to remember that the PLS convention expects data to be organized row-wise. This isn t a great problem since one convention is merely the matrix transpose of the other. Nonetheless, it is one more thing we have to remember. 

The vibrational kinetic energy can also be expressed in terms of the velocities in internal coordinates by taking the partial derivatives of Eq. (49). Thus, S = GP and, as G is square and nonsingular, P G lS and its transpose... 

In principle it is possible to write down the rate equation for any rate determining chemical step assuming any particular mechanism. To take a specific example, the overall rate may be controlled by the adsorption of A and the reaction may involve the dissociative adsorption of A, only half of which then reacts with adsorbed B by a Langmuir-Hinshelwood mechanism. The basic rate equation which represents such a process can be transposed into an equivalent expression in terms of partial... 

The properties of the partial molar volume which were studied in chapter I, 3 transpose immediately to the case of chemical potentials. If we consider a binary mixture and take the mole fraction Xg as independent variable we have, dropping the subscripts T and p for the rest of this paragraph ... 

The different terms that are added or subtracted in equation 4.8 are scalar quantities (a vector multiplied by its transpose, or vice versa, is a scalar, see appendix I). Y Y is the sum of squares of the responses. The so-called "least-squares" solution to this is the vector of estimators B which minimizes the sum of squares of the differences. The partial differential of the expression 4.8 with respect to the b must equal zero for b = B ... 

Here g is a transposed vector (gradient) containing all the partial first derivatives, and H is the (Hessian) matrix containing the partial second derivatives. In many cases, the expansion point Xo is a stationary point for the real function, making the first derivative disappear, and the zeroth-order term can be removed by a shift of the origin. 

Figure 10.2 Gas chromatogram obtained simultaneously with (a) katharometer and (b) flame ionisation detectors on a partially dried PVC powder. In (b) the dotted line represents the portion of the water peak as transposed from (a)...

An equivalent, but transposed, structure of an FIR hlter is shown in Fig. 8.98, which is much more modular and well suited to integrated circuit realization. Each module within the structure calculates a partial sum so that only a single addition is calculated at the output stage. 

The idea behind Peres separability criterium is the fact that the transposed of a density matrix is another density matrix that is, a positive operator with trace equal to one. Under partial transposition, this property is preserved for product states, but it fails for entangled states. 

Intuitively, one may expect that lumpability suggests that the system should have some degree of partial linearity, which is related to the Jacobian matrix J[c(t)] = df(c)/dc. Indeed, the system is exactly lumpable if and only if for any c in the composition space, the transpose of J[c(t)] has nontrivial fixed (i.e., c independent) invariant subspaces. It can be shown that the exactly lumped system is of the form... 

The objective of a principal component analysis (PCA) is to transform a number of correlated variables into a smaller set of new, uncorrelated variables (factors or latent variables). The first few factors should then explain most of the relevant variation in the data set. To allow this reduction in dimensionality, the variables are characterized by a partial correlation. The new variables can then be generated through a linear combination of the original variables, i.e. the original matrix X is then the product of a score matrix P and the transpose ( ) of the loading matrix A ... 

The second term is the creation energy of the induced multipoles. The + superscript indicates a matrix transpose. Vesely [55] has shown for a set of polarizable dipoles that if the induced dipoles are given by Eq. 9, it corresponds to a minimum of the total electrostatic energy, the partial derivative of E with respect to the induced multipole components being zero. 

Pavia, H., and Abrahamsson, K. (2012) Chemical images of marine bio-active compounds by surface enhanced Raman spectroscopy and transposed orthogonal partial least squares (T-OPLS). Anal. 

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