Mathematical concept definition

While it is desirable to formulate the theories of physical sciences in terms of the most lucid and simple language, this language often turns out to be mathematics. An equation with its economy of symbols and power to avoid misinterpretation, communicates concepts and ideas more precisely and better than words, provided an agreed mathematical vocabulary exists. In the spirit of this observation, the purpose of this introductory chapter is to review the interpretation of mathematical concepts that feature in the definition of important chemical theories. It is not a substitute for mathematical studies and does not strive to achieve mathematical rigour. It is assumed that the reader is already familiar with algebra, geometry, trigonometry and calculus, but not necessarily with their use in science. 

Let us first introduce some important definitions with the help of some simple mathematical concepts. Critical aspects of the evolution of a geological system, e.g., the mantle, the ocean, the Phanerozoic clastic sediments,..., can often be adequately described with a limited set of geochemical variables. These variables, which are typically concentrations, concentration ratios and isotope compositions, evolve in response to change in some parameters, such as the volume of continental crust or the release of carbon dioxide in the atmosphere. We assume that one such variable, which we label/ is a function of time and other geochemical parameters. The rate of change in / per unit time can be written... 

As in chapter 8, we will refer to contributions from the electron density centered on the nucleus as central contributions, and to the remainder as peripheral contributions. In the spectroscopic literature, the latter are commonly referred to as lattice contributions, a term we will avoid as it conflicts with the common definition of the lattice as a mathematical concept. 

Mathematicians have termed a set of elements in a poset that are all mutually incomparable an anti-chain. (See chapter by Briiggemann and Carlsen, p. 61 and for more detailed mathematics and definitions see Combinatorics and Partially Ordered Sets Dimension Theory by Trotter (Trotter 1992)). If we consider all anti-chains that contain a partition [A] as an element, the complexity of [a] is the number of elements in those antichains (i.e. the cardinality or size of the anti-chains) that have the maximum number of elements, maximum anti-chains. Clearly, this concept can be generalized to any poset, though, as we have seen, the case of the YDL is of particular interest and relevance to physics and chemistry. 

The definition given in Eqs. (2.29) for the partial pressures of the gases in a mixture is a purely mathematical one we now ask whether or not this mathematical concept of partial pressure has any physical significance. The results of two experiments, illustrated in Figs. 2.8 and 2.9, provide the answer to this question. First consider the experiment shown in Fig. 2.8. A container. Fig. 2.8(a), is partitioned into two compartments of equal volume... 

The following formal definition of the space Y refers to a given choice of reference states %p) and fp) [as in Eq. (1)] that are proper eigenstates of a given Fock-space Hamiltonian H. The mathematical concepts used in this formal chapter can be found in common textbooks on functional anetlysis,... 

The concept of a spherical world is attributed to the Greek philosopher Pythagoras in the sixth century B.C. (see also Chapter 1.3.1). Based on this mathematical-astronomical definition, the ancient geographers, namely Ptolemy, introduced 24 climata, being zones between two parallel circles, for which the length of the longest... 

The definition of crystal as a periodic array of identical motifs corresponds to the mathematical concept of lattice. Under this assumption, crystallography has developed powerful methods for allowing crystal structure determination of different materials, from simple metals (where the asymmetric imit may be the single atom) to protein crystals (containing thousands of atoms per cell) [62]. 

The book is composed of 11 chapters. Chapter 1 presents the various introductory aspects of patient safety including patient safety-related facts and figures, terms and definitions, and sources for obtaining useful information on patient safety. Chapter 2 reviews mathematical concepts considered useful to understand subsequent chapters and covers topics such as mode, median, mean deviation. Boolean algebra laws, probability definition and properties, Laplace transforms, and probability distributions. 

One must be aware that nowadays the mathematical concept of distance has evolved into an intricate labyrinth of alternative definitions and variants however, one can safely rely on the classic Euclidean concepts for practical QS purposes. From the QS point of view, any DF can be studied as a function belonging to a vector semispace. " Furthermore, DF can be seen as vectors belonging to infinite-dimensional Hilbert semispaces and thus can be also subject to comparative measures of distances and angles between the two of them. A pair of DF may, in this way, be considered as vectors subtending an angle a and situated in a plane... 

Appendix 19 Rigorous Definitions and Descriptions of a Selection of Mathematical Concepts of Discrete Mathematics... 

In this appendix, we have collected more rigorous definitions and descriptions of a selection of mathematical concepts of discrete mathematics. 

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 ... 

Mathematical concepts, which are fundamental for the understanding of physical or chemical definitions and derivations in the text, but which due to their length would make it harder to get an overview of the text (Linear regression. Exact differential, etc.). 

This chapter focuses on types of models used to describe the functioning of biogeochemical cycles, i.e., reservoir or box models. Certain fundamental concepts are introduced and some examples are given of applications to biogeochemical cycles. Further examples can be found in the chapters devoted to the various cycles. The chapter also contains a brief discussion of the nature and mathematical description of exchange and transport processes that occur in the oceans and in the atmosphere. This chapter assumes familiarity with the definitions and basic concepts listed in Section 1.5 of the introduction such as reservoir, flux, cycle, etc. 

When economical schemes for multidimensional problems in mathematical physics are developed in Chapter 9, we shall need a revised concept of approximation error, thereby changing the definition of scheme. The notion of summed (in t) approximation in Section 3 of Chapter 9 is of a constructive nature, making it possible to produce economical schemes for various problems. 

Time is a fundamental property of the physical world. Because time encompasses the antinomic qualities of transience and duration, the definition of time poses a dilemma for the formulation of a comprehensive physical theory. The partial elimination of time is a common solution to this dilemma. In his mechanical philosophy, Newton appears to resort to the elimination of the transient quality of time by identifying time with duration. It is suggested, however, that the transient quality of time may be identified as the active component of the Newtonian concept of inertia, a quasi occult quality of matter that is correlated with change, and that is essential to defining duration. The assignment of the transient quality of time to matter is a necessary consequence of Newton s attempt to render a world system of divine mathematical order. Newton s interest in alchemy reflects this view that matter is active and mutable in nature... 

It is this ordering that gave the concept a theoretical bent as real separations are not ordered the retention times in most separation techniques appear almost random across a range of separation time. The mathematical definition of peak capacity, nc, for an isocratic separation is given as (Grushka, 1970)... 

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