Scalar, mathematical concept

The mathematical concept of an operator needs to be introduced before launching into quantum mechanics. We do so in the briefest possible manner. An operator when applied to some function gives a scalar number multiplied by the function. Mathematically, oG = cG, when o is some operator, G is a function, and c is a scalar. A simple example would be the partial derivative, 8/8x(e") = a e ). When there are only certain allowable solutions to such an equation, it is called an eigenvalue equation. The solutions are called eigenfunctions or eigenvectors, and the scalars are called eigenvalues. 

The mathematical conception of an independent definition of geometric subjects (as reaction paths) in the configuration space starts with the idea of an analogous transformation of the coordinates as well as the angle relations in the new system. The distortion of equipotential lines in the new system should be compensated by an inverse distortion of the scalar product defining the angles ... 

The definition of a mathematical space begins with the set of objects X, Y, Z,. .. that occupy the space (an intrinsically empty space being a physically problematic concept). Among the simplest algebraic structures that can characterize such objects is that of a linear manifold, also called a linear vector space, affine space, etc. By definition, such a manifold has only two operations— addition (X + Y) and multiplication by a scalar (AX)— resulting in each case in another element of the manifold. These operations have the usual distributive,... 

Before we can demonstrate the connection between process control and Eq. (A.20), we need to introduce the concept of Lyapunov functions (Schultz and Melsa. 1967). Lyapunov functions wnre originally designed to study the stability of dynamic systems. A Lyapunov function is a positive scalar that depends upon the system s state. In addition, a Lyapunov function has a negative time derivative indicative of the system s drive toward its stable operating point where the Lyapunov function becomes zero. Mathematically we can describe these conditions as... 

What can a scalar time series tell us about the multivariate state (or phase) space buried in the observations The so-called embedology attributed to Whitney [75] and Takens [76] provides us with an essential clue to the answer of such a question. A detailed description of the mathematical proof is beyond the scope of this review, and here we focus on describing the brief concept and methodology. 

In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

In solid-state NMR, a very important concept is that the resonance frequency of a given nucleus within a particular crystallite depends on the orientation of the crystallite [3—5]. Considering the example of the CSA of a nucleus in a carboxyl group, Fig. 9.1 illustrates how the resonance frequency varies for three particular orientations of the molecule with respect to the static magnetic field, Bq. At this point, we note that the orientation dependence of the CSA, dipolar, and first-order quadrupo-lar interactions can all be represented by what are referred to as second-rank tensors. This simply means that the interaction can be described mathematically in Cartesian space by a 3 X 3 matrix (this is to be compared with scalar and vector quantities, which are actually zero- and first- rank tensors, and are specified by a single element and a 3 X 1 matrix, respectively). For such a second-rank tensor, there exists a principal axes system (PAS) in which only the diagonal elements of the matrix are non-zero. Indeed, the orientations illustrated in Fig. 9.1 correspond to the orientation of the three principal axes of the chemical shift tensor with respect to the axis defined by Bq. 

The self-sensing actuator concept requires the powerful mathematical machinery of complex hysteresis operators - first for reconstructing the mechanical quantities by means of the measured values of electrical quantities and second for compensating the hysteretic nonlinearities and the load dependency. Whereas robust software tools exist for modeling, identifying and compensating scalar complex hysteretic nonlinearities in practical applications, a considerable amount of research activities is necessary in the field of vectorial hysteresis phenomena to obtain a similar status. 

First the concept of Lyapunov functions will be introduced. A Lyapunov function, Y x t)), is a positive scalar that depends on the system s state. By definition, the time derivative of a Lyapunov function is non-positive. Mathematically these conditions can be described by ... 

This chapter explores how the directed nature of chemical bonding affects molecular collisions and chemical reactions. Whereas the concept of size or cross-section leads to numbers (scalar quantities), the concept of chemical shape leads to vectors (numbers tied to directions). Consequently, this topic is more mathematically challenging, but its study yields important insights into the nature of chemical transformations. As expected, just as the size of a molecule depends on die probe selected to measure this quantity, the chemical shape of a molecule also depends sensitively on the probe chosen for the measurement. It is important to stress that the physical shape of a molecule is most commonly determined from molecular spectroscopy it is usually expressed in terms of bond angles and bond lengths. The chemical shape of a molecule refers to the apparent size and shape of a molecule as experienced by another atom or molecule that collides with it. 

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