Useful Mathematical Definitions

This section presents a number of mathematical definitions that are considered to be useful in performing various types of safety and reliability studies concerned with systems used in the oil and gas industry. 

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The mathematical definition of the Born-Oppenheimer approximation implies following adiabatic surfaces. However, software algorithms using this approximation do not necessarily do so. The approximation does not reflect physical reality when the molecule undergoes nonradiative transitions or two... 

The complete mathematical definition of a particle size distribution is often cumbersome and it is more convenient to use one or two single numbers representing say the mean and spread of the distribution. The mean particle size thus enables a distribution to be represented by a single dimension while its standard deviation indicates its spread about the mean. There are two classes of means ... 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

Prove the following relationships using only definitions and mathematical principles ... 

The strict mathematical definition of a functional derivative is slightly more subtle than the more familiar definition of a function s derivative, but conceptually you can think of this just as a regular derivative. The functional derivative is written using 8 rather than d to emphasize that it not quite identical to a normal derivative. 

The key to use of any control system is to select the correct values for K, Tj and T,. The optimum control is a somewhat subjective term that is made clear by the presence of different mathematical definitions ... 

In 1997, D.L. Massart suggested the following definition "Chemometrics is a chemical discipline that uses mathematics, statistics and formal logic (a) to design or select optimal experimental procedures (b) to provide maximum relevant chemical information by analyzing chemical data and (c) to obtain knowledge about chemical systems" (Massart et ah, 1997). 

Physicists often refer to complex scalar product spaces as Hilbert spaces. The formal mathematical definition of a Hilbert space requires more than just the existence of a complex scalar product the space must be closed a.k.a. complete in a certain technical sense. Because every scalar product space is a subset of some Hilbert space, the discrepancy in terminology between mathematicians and physicists does not have dire consequences. However, in this text, to avoid discrepancies with other mathematics textbooks, we will use complex scalar product. ... 

Many theories developed in this book are expressed by equations or results involving continuous functions for example, the spatially variable concentration c(r). Materials systems are fundamentally discrete and do not have an inherent continuous structure from which continuous functions can be constructed. Whereas the composition at a particular point can be understood both intuitively and as an abstract quantity, a rigorous mathematical definition of a suitable composition function is not straightforward. Moreover, using a continuous position vector f in conjunction with a crystalline system having discrete atomic positions may lead to confusion. 

Matrix elements of the operators of the interaction energy between two shells of equivalent electrons may be expressed, with the aid of the CFP, in terms of the corresponding two-electron quantities. Substituting in such formula the explicit expression for the two-electron matrix element, after a number of mathematical manipulations and using the definition of submatrix elements of operators composed of unit tensors, we get convenient expressions for the matrix elements in the case of two shells of equivalent electrons. The corresponding details may be found in [14], here we present only final results. 

Naturally this coincidence does not mean that the geometric optics added to the classical physics could be used for the exact description of the light propagation since the Michelson-Morley experiment refuted its validity forever. It is evident that there are possible new mathematical definitions for c+ and c instead of the ordinary speed addition mle of the classical physics seen in Eqs. (9) and (11). These can be compatible with the experimental results as well. 

The International Union of Pure and Applied Chemistry (IUPAC, 10) proposed a definition of computational chemistry about 15 years after the term was already well established in the lexicon of chemists A discipline using mathematical methods for the calculation of molecular properties or for the simulation of molecular behavior. It also includes, e.g., synthesis planning, database searching, combinatorial library manipulation. The open-ended second part of this statement reflects the disparate opinions of the IUPAC committee and others who contributed comments. 

Line helices (chordal helices) are composed of dimensionless points. A chordal helix is uniform if all points are equivalent such a helix will have a circular cross-section and could be inscribed on a right circular cylinder. A nonuniform chordal helix is regular if it contains motifs repeated in a definite pattern and irregular if the motifs are not so repeated. For our purposes, chordal helices are useful mathematical abstractions. 

This characterization suffice to determine Clar structure if there is but one such structure (the cases illustrated in Fig. 1 and Fig. 2). In order to define Clar structure in a general case we have to find which Kekule valence structure are used more than once in a superposition. The concept of k-sextet structure considered in the previous section allow us to arrive at the mathematical definition of Clar s structure shown below which we are comparing with the geometrical definition of Clar s structure ... 

When only one factor is involved in the experiment, the predictive ability is often visualised by confidence bands. The size of these confidence bands depends on the magnitude of the experimental error. The shape , however, depends on the experimental design, and can be obtained from the design matrix (Section 2.2.3) and is influenced by the arrangement of experiments, replication procedure and mathematical model. The concept of leverage is used as a measure of such confidence. The mathematical definition is given by... 

Mathematical definitions of the various parameters and of the temperature groups used in this study are given in Tables I and II. 

Before going into a detailed description, a mathematical definition of the problem is useful. If we have conserved quantities, like the total angular momentum J and its projection J, the eigenvalues of the linearized motion matrix M has now the following eigenvalue structure. Let us have, as usual, n degrees of freedom and k < n —2 conserved quantities. Then we have... 

The mean partide size measured by these various methods is different depending on the method used. The mathematical definition of the various mean sizes, with their mathematical definitions, follow [1]. Number, length mean diameter or arithimatic diameter ... 

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