Vector metric

Contravariant and covairant four vectors are connected through the metric = diag (1,-1,-1,-1) by... 

We must now mention, that traditionally it is the custom, especially in chemo-metrics, for outliers to have a different definition, and even a different interpretation. Suppose that we have a fc-dimensional characteristic vector, i.e., k different molecular descriptors are used. If we imagine a fe-dimensional hyperspace, then the dataset objects will find different places. Some of them will tend to group together, while others will be allocated to more remote regions. One can by convention define a margin beyond which there starts the realm of strong outliers. "Moderate outliers stay near this margin. 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

We shall denote the space time coordinates by a (which as a four-vector is denoted by a light face x) with x° — t, x1 = x, af = y, xz = z x — ai0,x. We shall use a metric tensor grMV = gliV with components... 

In effect the scalar product in (9-688), which makes the vector space into a Hilbert space, omits the factor ( —1) from the bilinear form (9-687). We shall always work with the indefinite bilinear form (9-687). Thus, for example, one verifies that with this indefinite metric... 

Note that in the Lorentz gauge we have to adopt the Gupta-Bleuler quantization scheme, with its indefinite metric in a vector space that contains, in addition to the physically realizable states, unphysical... 

It is possible to perform a systematic decoupling of this moment expansion using the superoperator formalism (34,35). An infinite dimensional operator vector space defined by a basis of field operators Xj which supports the scalar product (or metric)... 

Given a set of existing (x, y) data records, where x is a vector of operating or decision variables, which are believed to influence the values taken on by y, y is a performance metric, usually assumed to be a quality characteristic of the product or process under analysis ... 

This corresponds with a choice of factor scaling coefficients a = 1 and p = 0, as defined in Section 31.1.4. Note that classical PCA implicitly assumes a Euclidean metric as defined above. Let us consider the yth coordinate axis of column-space, which is defined by a p-vector of unit length of the form ... 

It has been shown in Chapter 29 that the set of vectors of the same dimension defines a multidimensional space S in which the vectors can be represented as points (or as directed line segments). If this space is equipped with a weighted metric defined by W, it will be denoted by the symbol S. The squared weighted distance between two points representing the vectors x and y in is defined by the weighted scalar product ... 

Distances in are different from those in the usual space S. A weighted space can be represented graphically by means of stretched coordinate axes [2]. The latter result when the basis vectors of the space are scaled by means of the corresponding quantities in Vw, where the vector w contains the main diagonal elements of W. Figure 32.3 shows that a circle is deformed into an ellipse if one passes from usual coordinate axes in the usual metric I to stretched coordinate axes in the weighted metric W. In this example, the horizontal axis in 5, is stretched by a factor. l-6 = 1.26 and the vertical axis is shrunk by a factor Vo.4 = 0.63. 

Generally, two vectors that are orthogonal in S will be oblique in 5 , unless the vectors are parallel to the coordinate axes. This is illustrated in Fig. 32.4. Furthermore, if X and y are orthogonal vectors in S, then the vectors W x and W" y are orthogonal in 5 ,. This follows from the definition of orthogonality in the metric W (eq. 32.12) ... 

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S" in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P". Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W. ... 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

In CFA we can derive biplots for each of the three types of transformed contingency tables which we have discussed in Section 32.3 (i.e., by means of row-, column- and double-closure). These three transformations produce, respectively, the deviations (from expected values) of the row-closed profiles F, of the column-closed profiles G and of the double-closed data Z. It should be reminded that each of these transformations is associated with a different metric as defined by W and W. Because of this, the generalized singular vectors A and B will be different also. The usual latent vectors U, V and the matrix of singular values A, however, are identical in all three cases, as will be shown below. Note that the usual singular vectors U and V are extracted from the matrix. ... 

From this point on, the analysis is identical to that of CFA. Briefly, this involves a generalized singular vector decomposition (SVD) of Z using the metrics W and Wp, such that ... 

U = metric tensor of the space nc = normal vector in gas-phase direction... 

The idea of a vector space is usefully extended to an infinite number of dimensions for continuous functions. Given a function /(e.g.,/ = sinx) and a definition domain (e.g., 0 to In), the coordinates of / = sin x will be the infinite number of values of the function over the definition domain. This definition is consistent with that of Euclidian spaces if a metric is defined. In about the same way as the squared norm of the n-vector x(xux2,. .., x ) is... 

Sequence comparison is a technique for comparing two strings, sequences, or vectors for the purpose of determining the distance between them, and consequently their relationship. The items making up the sequence could be numbers, symbols, tones, letters, or words. We could, for example, compute a metric distance between the names William and Victor. Recognizing this, it is a simple extension to understand that assertions concerning the causes of firm performance are sequences of words, and that sequence comparison methodology can be used to compute similarity measures between these assertions. 

Example 3.40. The moduli space of instantons on a 4-dimensional hyper-Kahler manifold X can be considered as a hyper-Kahler quotient. Let us take a smooth vector bundle E over X with a Hermitian metric. Let us denote the space of metric connections on E by A. Its tangent space at A G M can be identified with... 

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