INDEX scalar

The hardening index /I is a nondimensional scalar which has the same value... 

A few comments on the layout of the book. Definitions or common phrases are marked in italic, these can be found in the index. Underline is used for emphasizing important points. Operators, vectors and matrices are denoted in bold, scalars in normal text. Although I have tried to keep the notation as consistent as possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to switch notation between chapters. The second derivative of the energy, for example, is called the force constant k in force field theory, the corresponding matrix is denoted F when discussing vibrations, and called the Hessian H for optimization purposes. 

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

Generally speaking, neither dynamical, nor relaxation parts of (A7.20) are diagonal over index q. It is simpler to diagonalize the dynamical term, proceeding to the GF, where scalar product (j L) = (JzLq) =... 

If, in a vector space of an infinite number of dimensions the components Ai and Bi become continuously distributed and everywhere dense, i is no longer a denumerable index but a continuous variable (x) and the scalar product turns into an overlap integral f A(x)B(x)dx. If it is zero the functions A and B are said to be orthogonal. This type of function is more suitable for describing wave motion. 

We consider a radially symmetric structure as illustrated in Fig. 12.2. The guiding defect, consisting of a material of refractive index ndefect, is surrounded by distributed Bragg reflectors on both sides, where the reflectors layers are of refractive indices ni and n2. All the electromagnetic field components can be expressed in terms of the z-component of the electric and magnetic fields14. These components satisfy the scalar Helmholtz equation, which in cylindrical coordinates is given by ... 

A matrix is an array, either square or rectangular, of numbers. It will be noted subsequently with bold-face, upper-case symbols. The real matrix Am x has m rows and n columns. A matrix AmXn also represents either a set of n column-vectors in 91 ", or a set of m row-vectors in 91". A scalar element of the matrix A will be referred to by its row and column index and noted ay (ith row, jth column), so the array form of the matrix A is... 

Multivariate data are represented by one or several matrices. Variables (scalars, vectors, matrices) are written in italic characters scalars in lower or upper case (examples n, A), vectors in bold face lower case (example b). Vectors are always column vectors row vectors are written as transposed vectors (example bv). Matri ces are written in bold face upper case characters (example X). The first index of a matrix element denotes the row, the second the column. Examples x,- - or x(i, j) is an element of matrix X, located in row i and column / xj is the vector of row i xy is the vector of column j. 

The effort to solve Eqs.(l) evidently depends on the refractive index profile. For isotropic media in a one-dimensional refractive index profile the modes are either transversal-electric (TE) or transversal-magnetic (TM), thus the problem to be solved is a scalar one. If additionally the profile consists of individual layers with constant refractive index, Eq.(l) simplifies to the Flelmholtz-equation, and the solution functions are well known. Thus, by taking into account the relevant boundary conditions at interfaces, semi-analytical approaches like the Transfer-Matrix-Method (TMM) can be used. For two-dimensional refractive index profiles, different approaches can be... 

For our purpose, it is convenient to classify the measurements according to the format of the data produced. Sensors provide scalar valued quantities of the bulk fluid i. e. density p(t), refractive index n(t), viscosity dielectric constant e(t) and speed of sound Vj(t). Spectrometers provide vector valued quantities of the bulk fluid. Good examples include absorption spectra A t) associated with (1) far-, mid- and near-infrared FIR, MIR, NIR, (2) ultraviolet and visible UV-VIS, (3) nuclear magnetic resonance NMR, (4) electron paramagnetic resonance EPR, (5) vibrational circular dichroism VCD and (6) electronic circular dichroism ECD. Vector valued quantities are also obtained from fluorescence I t) and the Raman effect /(t). Some spectrometers produce matrix valued quantities M(t) of the bulk fluid. Here 2D-NMR spectra, 2D-EPR and 2D-flourescence spectra are noteworthy. A schematic representation of a very general experimental configuration is shown in Figure 4.1 where r is the recycle time for the system. 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

According to scalar diffraction theory (Section 4.4) the scattering amplitude in the forward direction is proportional to the cross-sectional area of the particle, regardless of its shape, and is independent of refractive index. To the extent that diffraction theory is a good approximation, therefore, the radius corresponding to the response of an instrument that collects light scattered near the forward direction by a nonspherical particle is that of a sphere with equal cross-sectional area. The larger the particle, however, the more the... 

Energy is a scalar and so does not carry an internal gauge index. There are three... 

A brief summary of the mathematical notation adopted throughout this text is in order. Scalar quantities, whether constants or variables, are represented by italic characters. Vectors and matrices are represented by boldface characters (individual matrix elements are scalar, however, and thus are represented by italic characters that are indexed by subscript(s) identifying the particular element). Quantum mechanical operators are represented by italic characters if diey have scalar expectation values and boldface characters if their expectation values are vectors or matrices (or if they are typically constructed as matrices for computational purposes). The only deliberate exception to the above rules is that quantities represented by Greek characters typically are made neither italic nor boldface, irrespective of their scalar or vector/matrix nature. 

Matrices are shown in bold capital letters e.g. R), column vectors in bold lowercase letters (e.g. c) (row vectors are transposed column vectors) and scalars in italic characters (e.g. q). True values are indicated by Greek characters or the subscript true . Calculated or measured values are indicated by Roman characters. The hat ( ), used in the literature to indicate calculated, has been dropped to simplify the notation whether the magnitude is measured or calculated can be deduced from the context. The running indexes in multivariate calibration are as follows k = 1 to Al analytes are present in z = 1 to I... 

Therefore R, is an antisymmetric Ricci tensor obtained from the index contraction from the Riemann curvature tensor. Further contraction of R leads to the scalar curvature R, which, for electromagnetism, is k2. The contraction must be... 

In this way, a complex function < )(r) can be interpreted as a map S3< S2. This is very important, since maps of this kind can be classified in homotopy classes labeled by a topological integer number called the Hopf index, so that the same topological property applies to any scalar field (provided that it is onevalued at infinity). 

A very important property is that the magnetic and electric lines of an electromagnetic knot are the level curves of the scalar fields 4>(r, t) and 0(r, f), respectively. Another is that the magnetic and the electric helicities are topological constants of the motion, equal to the common Hopf index of the corresponding pair of dual maps constant with dimensions of action times velocity. 

A scalar is a zero-rank tensor an ordinary vector is a first-rank tensor. A Cartesian tensor T consists of 3 quantities. If the index sets are denoted as... 

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