Covariant quantities

Covariant. We first present the transformation laws for covariant quantities. We want to transform an "old" set of quantities to a "new" set of quantities, due to a transformation (typically, rotation). Let the old unit cell be represented by the 1x3 row vector V0 = (a0b0c0), and the new unit cell be represented by the 1x3 row vector Vn = (anbncn). Then there exists a 3 x 3 transformation matrix P such that... 

This transformation applies to contravariant quantities such as zone axes. If, instead, one is transforming a unit cell Ua = (a0b0c0) into a new cell Un = (anb cn), it is really a covariant quantity, which should be represented as a 1 x 4 row vector it transforms using the matrix inverse to Q, namely P3 ... 

The updated quantities 0 and P represent our best estimates of the unknown parameters and their covariance matrix with information up to and including time tn. Matrix Pn represents an estimate of the parameter covariance matrix since,... 

In case of correlated parameters, the corresponding covariances have to be considered. For example, correlated quantities occur in regression and calibration (for the difference between them see Chap. 6), where the coefficients of the linear model y = a + b x show a negative mutual dependence. 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

Any set of quantities that transform according to this prescription are known as the covariant components of a vector, and represented by subscripted indices3. 

The quantities bij, 6U, and 6 are respectively called the components of covariant, contravariant or mixed tensors of the second order, if they transform according to the formulae... 

In a CFD calculation, one is usually interested in computing only the reacting-scalar means and (sometimes) the covariances. For binary mixing in the equilibrium-chemistry limit, these quantities are computed from (5.154) and (5.155), which contain the mixture-fraction PDF. However, since the presumed PDF is uniquely determined from the mixture-fraction mean and variance, (5.154) and (5.155) define mappings (or functions) from (I>- space ... 

The value of the proportionality constant /3 can be determined experimentally. In the absence of any other experimental data, an acceptable model would be based on any combination of the adjustable parameters C, K, and Ns that yields the correct value of /3, according to Equation 29. Since three adjustable parameters are available to define the value of one experimentally observable quantity, covariability among these parameters is expected. In reality an independent estimate of Ng might be available, and curvature of the <7q vs. log a + plot might reduce some of the covariability, but Equation 29 provides an initial step in understanding the relationship between covarying adjustable parameters. 

Following the common approach in relativistic field theory, which aims at a manifestly covariant representation of the dynamics inherent in the field operators, so far all quantities have been introduced in the Heisenberg picture. To develop the framework of relativistic DFT, however, it is common practice to transform to the Schrodinger picture, so that the relativistic theory can be formulated in close analogy to its nonrelativistic limit. As usual we choose the two pictures to coincide at = 0. Once the field operators in the Schroodinger-picture have been identified via j/5 (x) = tj/(x, = 0), etc, the Hamiltonians He,s, Hy s and are immediately obtained in terms of the Schrodinger-picture field operators. 

Further simplifications can be made by assuming the ofT-diagonal elements of Vi, which represent the covariance between the measured quantities, are zero this effectively means that there is no coupling between the different experimental determinations in the experimental procedure, which is reasonable. 

The vector F and the corresponding variance-covariance matrix My are the only parts of this equation that depend on the measurements the other quantities are derived from the observational equations. 

The terms variant and covariant refer to the transformation properties of the quantities. A transformation may be defined by the transformation matrix T operating on the direct space basis a, such that... 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

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 quantity c is very closely related to the odds ratio in fact c is the log of the OR, adjusted for the covariates. The anti-log of c (given by e"") gives the adjusted OR. Confidence intervals in relation to this OR can be constructed initially by obtaining a confidence interval for c itself and then taking the anti-log of the lower and upper confidence limits for c. 

We mentioned earlier, in Section 13.1, that if we did not have censoring then an analysis would probably proceed by taking the log of survival time and undertaking the unpaired t-test. The above model simply develops that idea by now incorporating covariates etc. through a standard analysis of covariance. If we assume that InT is also normally distributed then the coefficient c represents the (adjusted) difference in the mean (or median) survival times on the log scale. Note that for the normal distribution, the mean and the median are the same it is more convenient to think in terms of medians. To return to the original scale for survival time we then anti-log c, e, and this quantity is the ratio (active divided by control) of the median survival times. Confidence intervals can be obtained in a straightforward way for this ratio. 

All gauge theory depends on the rotation of an -component vector whose 4-derivative does not transform covariantly as shown in Eq. (18). The reason is that i(x) and i(x + dx) are measured in different coordinate systems the field t has different values at different points, but /(x) and /(x) + d f are measured with respect to different coordinate axes. The quantity d i carries information about the nature of the field / itself, but also about the rotation of the axes in the internal gauge space on moving from x + dx. This leads to the concept of parallel transport in the internal gauge space and the resulting vector [6] is denoted i(x) + d i. The notion of parallel transport is at the root of all gauge theory and implies the introduction of g, defined by... 

The last term in Equation C-2 reflects the fact that uncertainties in m and b are not independent of each other. The quantity smh is called the covariance and it can be positive or negative. 

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