X, and*, vector

X and vectors are the actual and the running coordinates over the planar contact area respectively. The boundary conditions are ... 

In this chapter several equations wiU be shown explaining the main mathematical interpretation of three-way models. Scalars will be indicated with lowercase italics (e.g. x) and vectors with bold lowercase characters (e.g. y). Ordinary two-way arrays (matrices) will be denoted with bold uppercase (e.g. X), whereas higher-order arrays will be indicated with underscored bold uppercase (e.g. X). The ijkth element of a three-way array X will be where... 

X and y are vectors containing the liquid mole fractions and vapor mole fractions respectively. 

Let us consider that the niunber of echoes M and the incident wavelet (/) (e.g., a normalized comer echo) are known. Least Squares approach for estimating parameter vectors x and requires the solution to the nonlinear least squares problem ... 

Figure 5, Sketch of a conical intersection. The vectors x and X2 are the GD and DC respectively, that lift the degeneracy of the two adiabatic surfaces, The plane containing these vectors is known as the branching space.

The coordinate vector is X and V is the velocity part. At is expected to be small. 

Linear regression models a linear relationship between two variables or vectors, x and y Thus, in two dimensions this relationship can be described by a straight line given by tJic equation y = ax + b, where a is the slope of tJie line and b is the intercept of the line on the y-axis. 

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

For a multidimensional function, the variable x is replaced by the vector x and matrices are used for the various derivatives. Thus if the potential energy is a function of 3N... 

The Euler angles are often used to describe the orientations of a molecule. There are thre Euler cmgles d and ip. is a rotation about the Cartesian z axis this has the effec of moving the x and y axes, d is a rotation about the new x axis. Finally, ip is rotation about the new z axis (Figure 8.4). If the Euler angles are randomly changed b small amounts S[Pg.437]

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

To set up the problem for a microcomputer or Mathcad, one need only enter the input matrix with a 1.0 as each element of the 0th or leftmost column. Suitable modifications must be made in matrix and vector dimensions to accommodate matrices larger in one dimension than the X matrix of input data (3-56), and output vectors must be modified to contain one more minimization parameter than before, the intercept otq. 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

A vector is said to be indifferent if it does not change magnitude or direction under (A.50). For example, the vector a = x — y where x and y are positions in space, under the transformation (A.50), becomes... 

---