Mathematical concept

Numerical optimization is quite a broad general concept for finding optima, given an objective Junction of your problem at hand. In general, an objective function has a number of input variables (multivariate) and a real-valued output. For the sake of simplicity, we wiU assume that the objective function should be minimized. 

In general, linear functions and correspondingly linear optimization methods can be distinguished from nonlinear optimization problems. The former, being in itself the wide field of linear programming with the predominant Simplex algorithm for routine solution [75] shall be excluded here. 

The area of nonlinear optimization can be subdivided further into various classes of problems. Firstly, there is constrained versus unconstrained optimization. Normally, the inclusion of constraints generates only a special case for the sibling, unconstrained 

and most general, is the case of an objective function that may or may not be smooth and may or may not allow for the computation of a gradient at every point. The nonlinear Simplex method [77] (not to be confused with the Simplex algorithm for linear programming) performs a pattern search on the basis of only function values, not derivatives. Because it makes little use ofthe objective function characteristics, it typically requires a great many iterations to find a solution that is even close to an optimum. 

Some of the most important variations are the so-called Quasi-Newton Methods, which update the Hessian progressively and therefore economize compute requirements considerably. The most successful scheme for that purpose is the so-called BFGS update. For a detailed overview of the mathematical concepts, see [78, 79] an excellent account of optimization methods in chemistry can be found in [80]. 

Consider a physical property (such as the total Gibbs free energy G) of a continuous mixture, the value of which depends on the composition of the mixture. Because the latter is a function of, say, the mole distribution n(x), one has a mapping from a function to (in this case) a scalar quantity G, which is expressed by saying that G is given by afunctional of n(x). [One could equally well consider the mass distribution function m(x), and consequently one would have partial mass properties rather than partial molar ones.] We use z for the label x when in- 

One now needs to extend the concept of a derivative from ordinary functions to functionals. This is done by writing, with s z) being a displacement function. 

A Euclidean norm declares two functions that differ only on a countable infinity of isolated points as being close. This is not too much of a difficulty for the problems we consider, but there is another difficulty. If we want to consider distributions that include one or more discrete components (a semicontinuous distribution), s(x) may well contain some delta functions. This implies, first, that all integrals have to be interpreted as a Stjieltjies ones but even so one has a problem with the right-hand side of Eq. (177), because the delta function is not Stjieltjies square-integrable. One could be a bit cavalier here and say that we agree that 5 (j ) = 5(x), but it is perhaps preferable to keep continuous and discrete components separate. Let, for instance, the mole distribution be i, ri2,.. ., /v, n(x) in a mixture with N discrete components and a distributed spectrum. One can now define the scalar product as the ordinary one over the discrete components, plus 

Because 6G is linear in its second argument, it must be expressible as a weighted integral of s(z) the weighting function, however, may still depend on the particular point (z) in Hilbert space where the Frechet differential is evaluated. Hence, the weighting function is given by a functional G [ ] of n z), which depends parametrically on x G [ ] is called the functional derivative of G[ ]. One has 

This formalism can now be applied, as an example, to our specific physical example where G is the Gibbs free energy and n(x) is the mole distribution. The usual statement in thermodynamics that G is an extensive property can be formalized by requiring the functional G[ ] to be homogeneous of the first degree. Say for any positive scalar O one has 

There are various series expansions that are useful for approximating functions. Particularly important is the Taylor series if/(x) is a continuous, single-valued function of x with continuous derivatives (x),then we can expand the function about a point Xq as follows  

Taylor series are often truncated after the term involving the second derivative, which makes the function vary in a quadratic fashion This is a common assumption in many of the minimisation algorithms that we will discuss in Chapter 5 

A Maclaurin series is a specific form of the Taylor series for which Xq = 0. Some standard expansions in Taylor series form are  

Some of the common manipulations that are performed with vectors include the scalar product, vector product and scalar triple product, which we will illustrate using vectors ri, T2 and r3 that are defined in a rectangular Cartesian coordinate system  

V = (yiZ2 - Ziy2)i + ZlX2 - XiZ2)j - - (Xiy2 - yiX2)k (1.9) 

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To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for... this uncanny usefulness of mathematical concepts... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

The HETP equation is not simply a mathematical concept of little practical use, but a tool by which the function of the column can be understood, the best operating conditions deduced and, if required, the optimum column to give the minimum analysis time calculated. Assuming that appropriate values of (u) and (Dm) and (Ds)... 

Naturally, fibers and whiskers are of little use unless they are bonded together to take the form of a structural element that can carry loads. The binder material is usually called a matrix (not to be confused with the mathematical concept of a matrix). The purpose of the matrix is manifold support of the fibers or whiskers, protection of the fibers or whiskers, stress transfer between broken fibers or whiskers, etc. Typically, the matrix is of considerably lower density, stiffness, and strength than the fibers or whiskers. However, the combination of fibers or whiskers and a matrix can have very high strength and stiffness, yet still have low density. Matrix materials can be polymers, metals, ceramics, or carbon. The cost of each matrix escalates in that order as does the temperature resistance. 

Although other descriptions are possible, the mathematical concept that matches more closely the intuitive notion of smoothness is the frequency content of the function. Smooth functions are sluggish and coarse and characterized by very gradual changes on the value of the output as we scan the input space. This, in a Fourier analysis of the function, corresponds to high content of low frequencies. Furthermore, we expect the frequency content of the approximating function to vary with the position in the input space. Many functions contain high-frequency features dispersed in the input space that are very important to capture. The tool used to describe the function will have to support local features of multiple resolutions (variable frequencies) within the input space. 

It is clear that 6 q) tends to infinity as h, and we deal with the second mathematical concept of a mass. Respectively, the expression for the attraction field... 

Note, that the flux of the attraction field is a purely mathematical concept it does not have any physical meaning. Next, we evaluate the flux through the surface S,... 

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. 

The problem we all have is that we all want answers to be in clear, unambiguous terms yes/no, black/white, is/isn t linear, and so on while Statistics deals in probabilities. It is certainly true that there is no single statistic not SEE, not R2, not DW, nor any other that is going to answer the question of whether a given set of data, or residuals, has a linear relation. If we wanted to be REALLY ornery, we could even argue that linearity is, as with most mathematical concepts, an idealization of a property that... 

This set of terms is a supplement to the text. Many of these terms are included to clarify issues discussed in the text. We refer to the text index for more detailed coverage of the statistics and chemometrics terms. Many of these terms refer to the measuring instrument or the process of making a measurement rather than to mathematical concepts. 

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. 

An important objective has been a book that is sufficiently self-contained to allow first reading without consulting too many primary sources. The introductory material should elucidate most mathematical concepts and applications not familiar to the reader. For more detailed assistance one should refer to specialized volumes treating of mathematical methods for scientists, e.g. [5, 6, 7, 8, 9]. It may be unavoidable to consult appropriate texts in pure mathematics, of which liberal use has been made here without reference. 

Crystal structures and crystal lattices are different, although these terms are frequently (and incorrectly) used as synonyms. A crystal structure is built of atoms. A crystal lattice is an infinite pattern of points, each of which must have the same surroundings in the same orientation. A lattice is a mathematical concept. 

In those moments, when he could escape from the grind of daily life, having access to the facilities of the Nottingham Subscription Library in Bromley House enabled him to become acquainted with the advanced mathematical concepts embodied in the works of the French school of analytical physics, which was only then being established at Cambridge. In 1828, at the age of thirty-five, he published by subscription the paper that immortalized his name. It was entitled, An Essay on the Application of Mathematical... 

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... 

In literature, PLS is often introduced and explained as a numerical algorithm that maximizes an objective function under certain constraints. The objective function is the covariance between x- and y-scores, and the constraint is usually the orthogonality of the scores. Since different algorithms have been proposed so far, a natural question is whether they all maximize the same objective function and whether their results lead to comparable solutions. In this section, we try to answer such questions by making the mathematical concepts behind PLS and its main algorithms more transparent. The main properties of PLS have already been summarized in the previous section. 

This calculation of the standard cell potential for the Daniell cell used the mathematical concept that the subtraction of a negative number is equivalent to the addition of its positive value. You saw that 0.342 V - (-0.762 V) = 0.342 V + 0.762 V... 

In this chapter we will deal with those parts of acoustic wave theory which are relevant to chemists in the understanding of how they may best apply ultrasound to their reaction system. Such discussions tvill of necessity involve the use of mathematical concepts to support the qualitative arguments. Wherever possible the rigour necessary for the derivation of the basic mathematical equations has been kept to a minimum within the text. An expanded treatment of some of the derivations of key equations is provided in the appendices. For those readers who would like to delve more deeply into the physics and mathematics of acoustic cavitation numerous texts are available dealing with bubble dynamics [1-3]. Others have combined an extensive treatment of theory with the chemical and physical effects of cavitation [4-6]. 

In the MPC theory, the problem is not even posed. One starts defining the purely mathematical concept of dynamical system without any reference to a representation of reality. (The baker s transformation or the Bernoulli shift are obvious examples.) From here on, one proves mathematically the existence of a class of abstract dynamical systems (K-flows) that are intrinsically stochastic —that is, that possess precise mathematical properties (including a temporal symmetry breaking that can be revealed by a change of representation). 

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