Uniqueness, mathematical

Finally, it is important to mention another class of methods. In many cases it is not possible to obtain a unique mathematical solution to the multivariate resolution of complex mixtures, and the problem of embedded peaks without selectivity, which may occur, for example, in impurity monitoring, causes difficulties when using many conventional approaches. 

As yet, no analytical and readily integrated unique mathematical function of the type - dCjdt — Cx) exists for describing the kinetics, where (C ) is the resin phase concentration of the counter-ion A initially in the exchanger, B the ion in solution, and t the elapsed time. However, analytical solutions of the rate equations are available which account for the observed rate behaviour under specified circumstances or boundary conditions. 

US stat ent can be validated both through model experiments and with rigoroi mathematical analyses of the properties of adsorption isotherm equations. However, the point that these equations are not unique mathematical descriptions of surface reactions in soils can be made effectively by a simple counterexample. Consider a typical batch sorption experiment in which o-phosphate is reacted with soil material and suppose that in this reaction an amorphous aluminum phosphate phase is formed. 

The result is used to determine regions of maximum purity. These approximate to the elution profiles or spectra (in coupled chromatography) of each component in a mixture that can then be employed in factor analysis as some information on each pure component is known. Sometimes there are embedded peaks, for which there is no pure (or selective or composition 1) region. The eigenvalue plots can still provide valuable information as to where each component elutes but sometimes it is hard to obtain unique mathematical solutions to the determination of information on each compound. 

Feature diagrams are a formal representation of product data with a precise semantics. Consequently, they define a unique mathematical model of the product data that consists of the hierarchical structure and additional logical formulas that specify the properties and links to aU product parts. The analyzing procedures presented here operate on purely logical representations of the product data. As a consequence, the structure and content of a feature diagram have to be transformed into an equivalent representation in logic. 

The elements of this determinant are the coefficients of the unknown amplitudes Ai in the set of equations (10). When a fixed value of X, say k, is chosen so as to cause the determinant to vanish, the coefficients of the unknown Ai in (10) become fixed, and it is then possible to obtain a solution, Aik, for which the additional subscript k will be used to indicate the correspondence with the particular value of X -. Such a system of equations does not determine the Aik uniquely, but gives only their ratios an arbitrary set A. may be obtained by putting, 4u = 1-A convenient and unique mathematical solution may be designated by the quantities which are defined in terms of an arbitrary solution, by the formula ... 

Mandelbrot also pointed out that these unique mathematical concepts are now the fundamental tools defining natural phenomena in the world around us. He referenced particular mathematicians who made singular contributions to the theory of fractals their theories and the objects named after them are standards of the canon of fractal geometry. 

One of the most important elements in nondisplay applications of liquid crystals is the positive focal length lens and its relationship to the Fourier transform. It is a Fourier transform and its unique mathematical properties that allow us to perform many... 

However, by constructing a nested sequence of successively larger discrete spaces and approximations therein we hope to end up with some approximation of a unique invariant measure, which is then implicitly defined via the constructed sequence of subspaces. An expression of this mathematical consideration is the multilevel structure of the suggested algorithm - details see below (Section 3.2). In physical terms, we hope that the perturbations introduced by discretization induce a unique and smooth invariant measure but are so weak that they do not destroy the essential physical structure of the problem. 

The mathematical requirements for unique determination of the two slopes mi and ni2 are satisfied by these two measurements, provided that the second equation is not a linear combination of the first. In practice, however, because of experimental error, this is a minimum requirement and may be expected to yield the least reliable solution set for the system, just as establishing the slope of a straight line through the origin by one experimental point may be expected to yield the least reliable slope, inferior in this respect to the slope obtained from 2, 3, or p experimental points. In univariate problems, accepted practice dictates that we... 

For practical reasons, the blast furnace hearth is divided into two principal zones the bottom and the sidewalls. Each of these zones exhibits unique problems and wear mechanisms. The largest refractory mass is contained within the hearth bottom. The outside diameters of these bottoms can exceed 16 or 17 m and their depth is dependent on whether underhearth cooling is utilized. When cooling is not employed, this refractory depth usually is determined by mathematical models these predict a stabilization isotherm location which defines the limit of dissolution of the carbon by iron. Often, this depth exceeds 3 m of carbon. However, because the stabilization isotherm location is also a function of furnace diameter, often times thermal equiHbrium caimot be achieved without some form of underhearth cooling. 

In 1893, it was shown that corresponding states are not unique to van der Waals equation of state (73). Rather, for any equation of state having not more than three constants, corresponding states are only a mathematical consequence. 

Model discrimination is a procedure for developing a suitable description of the unit performance. The techniques are drawn from the mathematics hterature where the goodness-of-fit of various proposed models are compared. Unfortunately, the various proposed models will usually describe a unit s performance equally well. Model discrimination is better accomplished when raw or adjusted measurements from many, unique operating conditions provide the foundation for the comparisons. 

The model-dependent aspect of ellipsometric analysis makes it a difficult technique. Several different models fit to one set of data may produce equivalendy low MSEs. The user must integrate and evaluate all available information about the sample to develop a physically realistic model. Another problem in applying ellip-sometry is determining when the parameters of the model are mathematically correlated for example, a thicker film but lower index of refraaion might give the same MSE as some other combinations of index and thickness. That is, the answer is not always unique. 

Beeler defined the broad scope of computer experiments as follows Any conceptual model whose definition can be represented as a unique branching sequence of arithmetical and logical decision steps can be analysed in a computer experiment... The utility of the computer... springs mainly from its computational speed. But that utility goes further as Beeler says, conventional analytical treatments of many-body aspects of materials problems run into awkward mathematical problems computer experiments bypass these problems. 

The above discussion leads to the conclusion that time-related and demand-related failures for a piece of equipment cannot be equated through a general mathematical relationship. These issues are better dealt with in a data base taxonomy (classification scheme) for equipment reliability data by defining a unique application through equipment description, service description, and failure description. 

After Hertz s death Helmholtz paid tribute to his former student as a consummate physicist, who uniquely combined mathematical ability, theoretical insight, and experimental skill. These qualities enabled Hertz to make many important contributions to physics, of which only those relating more directly to energy are outlined below. 

Space. Some fixed reference system in which the position of a body can be uniquely defined. The concept of space is generally handled by imposition of a coordinate system, such as the Cartesian system, in which the position of a body can be stated mathematically. 

A mathematical technique used to convert a time-domain plot into its unique frequency components. 

Mathematical models are the link between what is observed experimentally and what is thought to occur at the molecular level. In physical sciences, such as chemistry, there is a direct correspondence between the experimental observation and the molecular world (i.e., a nuclear magnetic resonance spectrum directly reflects the interaction of hydrogen atoms on a molecule). In pharmacology the observations are much more indirect, leaving a much wider gap between the physical chemistry involved in drug-receptor interaction and what the cell does in response to those interactions (through the cellular veil ). Hence, models become uniquely important. 

Our first task is to knock this barrier down to size. Instead of facing a brick wall, we should simply become aware that we are about to cross over a threshold, and a low threshold, at that. This author feels uniquely qualified to guide you over this threshold, having, himself, become a chemist because of a pronounced aversion to heavy mathematics. 

Interest in developing and refining the mathematical methods of operations research has become intensified and sophisticated. Attention is generally given to a priori upper bounds on the number of solutions of a problem, the existence and uniqueness of solutions,... 

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