Mathematical reasoning

Note that the upper limit of the second summation has been shifted from Nq to oo for mathematical reasons. The change is of little practical significance, since Eq. (5.24) drops off for very large values of n. To simplify the summation in Eq. (5.27) consider the following steps ... 

Eq. (7) is often referred to as the (N,6) Lennard-Jones potential. In particular, (12,6) is popular for mathematical reasons, despite the fact that an exponential form as in Eq. (6) usually describes the repulsive part of the potential better. The potentials shown in Fig. 6.2 form a good description for the physisorbed molecule, but they break down for small distances, where the attractive term in Eq. (6) starts to dominate in an unrealistic way, because for d 0 the repulsive part becomes constant (Vr Cr) while the Van der Waals part continuously goes towards infinity (Vvdw —> ) ... 

This paragraph is closely linked to the study of flashpoints. In order to be able to get an overall view on uses of lower explosive limits, one will need to refer to this paragraph and the next. To the user, LEL Is characterised by the high level of experimental error that goes with its measurement. But, for a purely mathematical reason, this level of error will not affect the use of this parameter greatly. It is thus undeniably useful and will be helpful in the difficult task of measuring flashpoints. 

Two comments can be made on the second point. For a simple mathematical reason mistakes made with the LEL value are of little consequence to the calculated value of flashpoint cc . Indeed, this mistake is not that significant since there Is a logarithm involved. Secondly, in theory no mistake is made with the stoichiometric concentration (except for nitrogenous compounds where there is an ambiguous aspect with regard to the nitrogen reaction). This second approach (with Cg) can thus provide preliminary control of the model parameters (S or the group) and there... 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

Introduction. The following two chapters are devoted to the evaluation of the orientation of structural entities in the studied material, not to the analysis of the inner structure (topology) of these entities. First discussions of the problem of orientation smearing go back to Kratky [248,249], Unfortunately, the corresponding mathematical concepts are quite involved, and a traceable presentation would require mathematical reasoning that is beyond the scope of this textbook. Thus only ideas, results and references are presented. 

There seems to be no mathematical reason to favor one formulation of the condensed Fukui function over the other. 

Whenever we see an author attempting to explain these hidden operations by invisible fluids, by aethers, by collisions, and vibrations, and particularly if we see him introducing mathematical reasonings into such explanationsthe best thing we can do is to shut the book, and take to some other subject.28... 

Consequently, it is important to note that the fluorescence intensity of a compound is proportional to the concentration over only a limited range of absorbances, due to this geometrical reason, in addition to the mathematical reason (as shown by Eq. 6.1). 

Consequently, discontinuities in certain correlation functions are not uncommon in the thermodynamic limit. Other examples are known. For example, Kirzhnits made a similar point concerning the static dielectric function [6]. The mathematical reason why such discontinuities are not prohibited is that the commutation rule, [JV, H] = 0, becomes meaningless in the thermodynamic limit. The reader is referred to the literature for additional discussion [7, 8]. 

One mathematical reason for reducing the number of columns in U can be seen upon examination of Equation 5-31. This equation shows that U is a function of S", which is calculated by taking the inverse of each of the diagonal elements of S. The calculation of S" is not stable if PCs with small diagonal dements (which likely describe only noise) are retained. Ttiis rank selection issue is discussed in much more depth in the examples below. 

Define y now to take the value one for a success and zero for a failure. For mathematical reasons, rather than modelling y as we did for continuous outcome variables, we now model the probability that y=l, written pr y= 1). 

It is nonetheless necessary to recognize that there are absolute mathematical reasons for demonstrating that the syntheses created by the Fourth Dimension are realized by 1 epanouissement [and] that, within these superior dimensions, beings and dominions will be of an ever more subtle fluidity, and that their surroundings will possess capacities ever more radiant. What horizons are now opened to scholars, to the occultists, to the poets ... 

Another atomist, prosecuted by the Italian church authorities, was Galileo Galilei (1564-1642 CE). He irritially used minimi to describe the smallest parts of substances but later applied the term to Epicurean atoms separated by a quantitatively infinite vacuum. The atomic structure of substances was necessary from mathematical reasoning, and the atom was indivisible without shape and dimensions. The qualities or properties (color, odor, taste, etc.) of atoms were not associated with atoms but with their serrsory detection by the observer (42). 

The mathematical reason is that the AOs of a given free atom form what is known as a complete set, that is any function can be produced... 

Problem-solving questions test your mathematical reasoning skills. This means that you will be required to apply several basic math techniques for each problem. Here are some helpful strategies to help you improve your math score on the problem-solving questions ... 

Five-choice questions test your mathematical reasoning skills. They require you to apply various math techniques for each problem. 

Consequently, there are four symmetry elements the n-fold axis of rotation, labeled C the plane of symmetry, labeled o the center of inversion, /, and the n-fold rotation-reflection axis, labeled S . Because of mathematical reasons, it is necessary to include the identity symmetry element, /. 

Gardner and Widstoe (1921) made attempts to develop a general equation. They assumed an ideal soil to be one in which the capillary potential was a linear function of the reciprocal of the moisture-content (after Buckingham, 1907), and that the inherent moisture conductivity is independent of the moisture-content. These assumptions are necessary for mathematical reasons. For downward flow through sand, whose surface is kept saturated, the equation connecting the time t and distance from the surface L was as follows ... 

Bundy, A. (1983) The Computer Modelling of Mathematical Reasoning, Academic Press, London. 

The exclusion of nonplanar helicenic and hollow coronoid species from the class of benzenoids was maybe not fully justified from a chemist s point of view, but there were good and convincing mathematical reasons for this anyway we use the term benzenoid in the same sense as in the book [3]. On the other hand, we find that it serves no purpose to strictly distinguish between benzenoid hydrocarbons (chemical objects) and benzenoid systems (mathematical objects), since this distinction is always obvious from the context. We note in passing that what we call benzenoid system is the same as hexagonal system or hexagonal animal in the mathematical literature. 

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop. 

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