Augmented vector

In certain circumstances, the model equations may not have an explicit expression for the measured variables. Namely, the model can only be represented implicitly. In such cases, the distinction between dependent and independent variables becomes rather fuzzy, particularly when all the variables are subject to experimental error. As a result, it is preferable to consider an augmented vector of measured variables, y, that contains both regressor and response variables (Box. 

It is easy to verify that the product of the augmented matrix A and the augmented vector V results in the vector V, which is the same as the vector X (Eqs. 1.38 and 1.39) plus additional 1 as the fourth element of the vector as shown in Eq. 1.48. Therefore, instead of specifying rotational and translational parts separately, they can be combined into a single matrix ... 

We now consider how the two interacting symmetry operations produce a third symmetry operation, similar to how it was described in section 1.6 but now in terms of their algebraic representation. Assume that two symmetry operations, which are given by the two augmented matrices A and A, are applied in sequence to a point, coordinates of which are represented by the augmented vector V. Taking into account Eq. 1.48, but written in a short form, the first symmetry operation will result in the vector V given as... 

Gauss-Jordan elimination is a variation of the Gauss elimination scheme. Instead of obtaining the triangular matrix at the end of the elimination, the Gauss-Jordan has one extra step to reduce the matrix A to an identity matrix. In this way the augmented vector b is simply the solution vector x. 

The augmented vector lead aVR should be checked after a recording is made. PQRST waves in lead aVR are almost always negatively deflected (point downwards below the baseline). A positive deflection should alert the recorder to the possibility that the cables/leads attached to the limb electrodes could have been applied the wrong way round. This can mimic a condition called dextrocardia and is known as technical dextrocardia . 

The function rjk is used to select the secondary particles disappearing due to coalescence in the I -th section, while is a sparse coalescence matrix whose nonzero elements represent the successful coalescence events between any pairs of secondary particle with locations and weights that are given by the augmented vectors 6 and w. ... 

Assuming the T-matrix equations s = Te and s = Te our task is to express the transition matrix in the global coordinate system T in terms of the transition matrix in the particle coordinate system T. Defining the augmented vectors of spherical wave functions in each coordinate sjretem... 

For an incident wave propagating along the Z-axis, the augmented vector of spherical harmonics can be computed by using (1.121). Choosing the... 

When developing the dusty gas model flux relations in Chapter 3, the thermal diffusion contributions to the flux vectors, defined by equations (3.2), were omitted. The effect of retaining these terms is to augment the final flux relations (5.4) by terms proportional to the temperature gradient. Specifically, equations (5.4) are replaced by the following generalization... 

Step 1 - the n x n coefficient matrix is augmented with the load vector on the right-hand side to form an n x n + ) matrix. 

Case 1. A bias present in the measurement of /2 was identified by the sequential processing of the measurements (see Example 7.2). We augment, in consequence, the vector of parameters of the original problem by adding an additional component to represent the uncertain parameter (bias term). 

In Equation 6.19, which can be called TD Kohn-Sham-type equation, the effective scalar and vector potentials vcff(r, f) and Aeff(r, t), respectively, consist of contributions from the external potentials augmented by internal contributions determined by the density variables and can be expressed as... 

If complete spectra at nl wavelengths are measured for the above ns solutions, there are nsx.nl equations of the kind (3.8). Again, matrix notation is much more convenient and easier to understand. The vectors y, a and r are augmented by the wavelength dimension and become matrices Y, A and R. 

These vectors allow us to separate the coordinates ri and r j from the coordinates over which the integration is performed (v) in the density calculation. Likewise, the vector of the shifts needs to be augmented to match the dimension of r" " ... 

X is the matrix of the predictors augmented with a column of 1, necessary for fhe esfimafion of fhe intercept values, b is the column vector of fhe regression coefficienfs. The regression coefficients are estimated by... 

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