Vector, defined

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... 

Rather than a single objective, y, as in Section IV, we now have a total of P distinct categorical performance variables, y, i = 1,..., P, associated with an equivalent number of objectives. Consequently, each data record is now composed of a (x, y) pair, where y is a performance vector defined by... 

Where a is the unit vector defined by the local frame defining the axial quadupole, r is the vector oriented towards r from the monopole to the axial quadupole. Qa is the corresponding matrix element. Following the modification of the charge-dipole interaction, we introduced a modified mono-quad interaction, namely I -mono-quad1 ... 

The aggregation-induced changes of absorption spectra result from intermolecular interactions between closely spaced carotenoid molecules. For two molecules whose transition dipole moment vectors, p, are located at places characterized by position vectors /- and r2, with the relative position vector defined as R=r1-r2, Figure 8.3, the interaction energy is expressed as (van Amerongen et al. 2000)... 

The Rh-Ir vector defines the z-axis, and the NN bridges define the x- and y-axes. The N and P atoms are solely electron donors, but the CO ligand involves a lot of re-back-bonding. Thus the quadrupole matrix, which has large components if there is an electric field gradient at the Ir nucleus, is rotated 45° compared with the g-matrix principal axes. 

Fig. 4. Schematic showing the SAXS measurement on the Siemens D5000 diffractometer. The wave-vector, k, is determined as (27t/X)(s-s0), where s and s0 are the umt vectors defining the directions of the scattered and incident radiation respectively.

In the course of this work there will be occasion to use vectors defined in more than three dimensions. Although an n-dimensional vector is difficult to visualize geometrically the algebraic definition is no more complicated than for three-dimensional vectors. 

Any three-dimensional orthogonal coordinate system may be specified in terms of the three coordinates q, q2 and q3. Because of the orthogonality of the coordinate surfaces, it is possible to set up, at any point, an orthogonal set of three unit vectors ex, e2, e3, in the directions of increasing qx, q2, q3, respectively. It is important to select the qt such that the unit vectors define a right-handed system of axes. The set of three unit vectors defines a Cartesian coordinate system that coincides with the curvilinear system in... 

In an n-dimensional Hilbert space, Ln the set of n independent vectors define a complete set in Ln. This set is called a basis, and the vectors are called the basis vectors. Basis vectors can be chosen in an infinite number... 

The equations satisfied by the fields and potentials may be reduced to a more compact form by the construction of a complex vector whose real and imaginary parts are formed from the vectors defining the magnetic and electric fields. 

The generation of invariants in the Lorentz transformation of four-vectors has been interpreted to mean that the transformation is equivalent to a rotation. The most general rotation of a four-vector, defined as the quaternion q = w + ix + jy + kz is given by [39]... 

Together these unit vectors define an n-dimensional vector space and they are said to constitute an orthonormal set. State vectors of this type are... 

The graphic demonstration thereof is self-explanatory. The linear(resultant) vector defines a fluctuating electromagnetic field. Each of the rotating vectors describes a helix around the time axis. 

Let u(t) be an n-vector with components (u1,u2,..u ) depending on a single scalar variable t. The derivative of the vector u with respect to the scalar t is the n-vector defined as... 

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