Vector covariant component

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

Although it is customary to refer to covariant and contravariant vectors, this may be misleading. Any vector can be described in terms of its contravariant or its covariant components with equal validity. There is no reason other than numerical simplicity for the preference of one set of components over the other. 

Requiring these order parameters to transform in a Lorentz-covariant way, we are led to a particular basis of 4 x 4 matrices , which was recently derived in detail (Capelle and Gross 1999a). The resulting order parameters represent a Lorentz scalar (one component), a four vector (four components), a pseudo scalar (one component), an axial four vector (four components), and an antisymmetric tensor of rank two (six independent components). This set of 4 x 4 matrices is different from the usual Dirac y matrices. The latter only lead to a Lorentz scalar, a four vector, etc., when combined with one creation and one annihilation operator, whereas the order parameter consists of two annihilation operators. 

According to this theoretical framework, one can easily constmct the three fundamental vector operators. So, the gradient of a scalar quantity

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Assume the transverse cut of the curvilinear boundary of Figure 4.9, appearing in the secondary mesh, with n = hu, hv, T a normal unit vector in the (u, v, w) general coordinate system. The covariant components of E 1 = and fields... 

Magnitude of local acceleration of gravity vector g (295) determinant of fundamental tensor (318)-(319) Function of C (25)-(26) Covariant component of metric tensor (318)... 

In Seet. 4.2, we need veetor spaee with abasis whieh is formed by A linear independent vectors gp p =, ..., k) which are not generally perpendicular or of unit length [12, 18, 19]. Sueh nonorthogonal basis, we eall a contravariant one. Covariant components of the so called metric tensor are defined by... 

The quantities labeled with subscript indices are called the covariant components of the 4-vector X and are given by... 

IS. Note that the velocity n is a 3-vector and therefore its components are labeled with subscript indices, e.g., Vi = Vx. This must not be confused with the covariant components of a 4-vector. The velocity vector v may point in any direction, but the coordinate axes of IS and IS must be parallel to each other, i.e., there is no constant rotation between the axes of the two systems. For a particle at rest in IS (for example at the origin r = 0), we have dr = 0 and can therefore immediately write down the Lorentz transformation given by Eq. (3.13) as... 

Similarly to the nonrelativistic situation [cf. Eq. (2.29)], the components of the Lorentz transformation matrix A may be expressed as derivatives of the new coordinates with respect to the old ones or vice versa. However, since we have to distinguish contra- and covariant components of vectors in the relativistic framework, there are now four different possibilities to express these derivatives ... 

The weights for the state and covariance components of the sigma point vector are calculated as... 

Let n be a vector whose components are random variables, not necessarily independent. Then, the covariance matrix of v, cov(n), has elements... 

Similar to vectors, based on the transfomiation properties of the second tensors the following three types of covariant, contravariant and mixed components are defined... 

Principal component analysis (PCA) takes the m-coordinate vectors q associated with the conformation sample and calculates the square m X m matrix, reflecting the relationships between the coordinates. This matrix, also known as the covariance matrix C, is defined as... 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Ab. [In the manner of Eq. (94), this can be decomposed into components Ab, in which the superscripta labels the matrices in the theory). Next, we define the field intensity tensor through a covariant curl by... 

The products of the components of two covariant vectors, taken in pairs, form the components of a covariant tensor. If the vectors are contravariant,... 

The fractional coordinates in crystallography, referred to direct and reciprocal cells respectively, are examples of covariant and contravariant vector components. 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

Principal component analysis (PCA) is aimed at explaining the covariance structure of multivariate data through a reduction of the whole data set to a smaller number of independent variables. We assume that an m-point sample is represented by the nxm matrix X which collects i=l,...,m observations (measurements) xt of a column-vector x with j=, ...,n elements (e.g., the measurements of n=10 oxide weight percents in m = 50 rocks). Let x be the mean vector and Sx the nxn covariance matrix of this sample... 

PLS with a matrix Y instead of a vector y is called PLS2. The purpose of data evaluation can still be to create calibration models for a prediction of the y-variables from the x-variables in PLS2 the models for the various y-variables are connected. In a geometric interpretation (Figure 4.25), the m-dimensional x-space is projected on to a small number of PLS-x-components (summarizing the x-variables), and the -dimensional y-space is projected on to a small number of PLS-y-components (summarizing the y-variables). The x- and the y-components are related pairwise by maximum covariance of the scores, and represent a part of the relationship between X and Y. Scatter plots with the x-scores or the y-scores are projections of... 

FIGURE 4.25 PLS2 works with X- and K-matrix in this scheme both have three dimensions. t and u are linear latent variables with maximum covariance of the scores (inner relation) the corresponding loading vectors are p und q. The second pair of x- and y-components is not shown. A PLS2 calibration model allows a joint prediction of all y-variables from the x-variables via x- and y-scores. 

Technique 2 Elgenanalysls. It Is well known that the structure of a data set can be uncovered by performing an elgenanalysls of Its covariance matrix.(14) This Is often called principal component analysis. That Is, we arrange the M measurement made on each of N objects as a column vector and combine them to form an M x N matrix, A. A matrix B, resembling the covariance matrix of this data set, Is an M x M matrix AA whose elements are given by... 

Any 37/-dimensional Cartesian vector that is associated with a point on the constraint surface may be divided into a soft component, which is locally tangent to the constraint surface and a hard component, which is perpendicular to this surface. The soft subspace is the /-dimensional vector space that contains aU 3N dimensional Cartesian vectors that are locally tangent to the constraint surface. It is spanned by / covariant tangent basis vectors... 

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... 

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