Mathematical methods orthogonalization

Mathematical methods exist that guarantee an optimal placement of the collocation points. In orthogonal collocation (OC), the collocation points are equal to the zero points of the orthogonal polynomials. 

In our two-dimensional space, these two search directions are perpendicular to one another. Saying this in more general mathematical terms, the two search directions are orthogonal. This is not a coincidence that occurs just for the specific example we have defined it is a general property of steepest descent methods provided that the line search problem defined by Eq. (3.18) is solved optimally. 

There are many chemometric methods to build initial estimates some are particularly suitable when the data consists of the evolutionary profiles of a process, such as evolving factor analysis (see Figure 11.4b in Section 11.3) [27, 28, 51], whereas some others mathematically select the purest rows or the purest columns of the data matrix as initial profiles. Of the latter approach, key-set factor analysis (KSFA) [52] works in the FA abstract domain, and other procedures, such as the simple-to-use interactive self-modeling analysis (SIMPLISMA) [53] and the orthogonal projection approach (OPA) [54], work with the real variables in the data set to select rows of purest variables or columns of purest spectra, that are most dissimilar to each other. In these latter two methods, the profiles are selected sequentially so that any new profile included in the estimate is the most uncorrelated to all of the previously selected ones. 

Genetic programming [137] is an evolutionary technique which uses the concepts of Darwinian selection to generate and optimise a desired computational function or mathematical expression. It has been comprehensively studied theoretically over the past few years, but applications to real laboratory data as a practical modelling tool are still rather rare. Unlike many simpler modelling methods, GP model variations that require the interaction of several measured nonlinear variables, rather than requiring that these variables be orthogonal. 

Ideally, components that are not separated in the first separation step are resolved in the second. Peak capacity is the number of individual components that can be resolved by a separation method. A mathematical model shows that if the MD separations are orthogonal, then the total peak capacity is the product of the individual peak capacities of each dimension [14], Load capacity is defined as the maximum amount of material that can be run in a separation while maintaining chromatographic resolution. MD separations can be designed to significantly increase the load capacity in a first dimension to achieve enrichment of low-abundance or trace components in a peptide mixture, while the necessary peak capacity may be obtained in the second separation dimension [15]. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

It is customary, in the interconversion of these distribution functions, to assume that the particles are spherical this simplifies the mathematics, but is somewhat questionable physically. The method of measurement determines the nature of the reported radii of these hypothetical spheres e.g. in the case of microscopic sizing, the so-called surface radius is obtained, which is the radius of a circle having the same surface area as the orthogonal projection of the particle. 

Neuhaus JO, Wrigley C, The Quartimax method an analytic approach to orthogonal simple structure, British Journal of Mathematical and Statistical Psychology, 1954, 7, 81-91. 

The use of frozen orbitals, such as the bond orbitals connecting the quantum to the classical part of the system, can be extended to nonempirical quantum methods such as ab initio Hartree -Fock, post Hartree Fock, or DFT. In these cases, the overlap between atomic orbitals is taken into account and the orthogonality conditions are more difficult to fulfill. The mathematical formulation of the method has been developed in the original papers [26 28] and the process can be summarized as follows. 

Before we can discuss this method, however, we need certain mathematical results concerning the possibility of expanding arbitrary functions in infinite series of normalized orthogonal functions. These results, which are of great generality and widespread utility, we shall discuss in the next section without attempting any complete proof. 

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