Pure mathematical modeling

If data analysis with a single exponential decay is not satisfactory, a double exponential can be used, but such a decay must be considered as a purely mathematical model. 

The new theory developed from two independent publications - a purely mathematical model and the Schrodinger alternative with a clear physical foundation. The latter was immediately branded as a futile attempt to revive the concepts of classical physics, already refuted by the new paradigm. 

Purely mathematical models can be criticized as curve fitting exercises with very little appeal to chemical intuition and very little transferability. But models that seem reasonable chemically can be inaccurate, and maybe they will not be transferable either. To set some stakes on this rugged landscape, let us first examine the atomic multipole model, In this model each atom is a multipolar center which makes a contribution to the electric potential. 

With the advent of the computer, numerous simulated unique patterns such as the rich Julia and Mandelbrot sets have been created this is computer graphical generation by pure mathematical models. However, these unique patterns were formed only in the computer by the mathematical means. The twin problems of how to realize these patterns in a real physical system and how to bridge the gap between the real physical system and the pure mathematical model have stimulated the interest of many natural scientists. Since the discovery of buckminsterfullerene its physical properties and interactions with atoms, molecules, polymers, and crystalline surfaces have also been the subject of intensive investigations. Moreover, fullerene-doped polymers show particular promise as new materials with novel electrical, optical, and/or optoelectrical properties. The present study focuses on the striking morphological properties of fullerene-TCNQ multilayered thin films formed under proper growth conditions and explores the relationship between the real physical system and the pure mathematical model. 

Factor analysis. This method is used to interpret underlying factors responsible for data and is one of the most versatile chemometric methods. Factor analysis provides a purely mathematical model prepared from abstract values, which are related to a data matrix as follows,... 

The last approach, named pure mathematical modeling, consists of the correlation between the in vitro results and the plasmatic concentration reached in the in vivo assays. These correlations are commonly called in vitro/ in vivo correlations (IVIVC). The IVIVC can be classified into four levels level A, level B, level C and multiple level C correlations [201]. Only the IVIVC of level A is accepted by regulatory boards (FDA, USP) for scale-up and postapproval changes (SUPAC), which can be justified without the need for additional in vivo studies. This level A IVIVC comprises a point to point relationship between the in vitro release profile and the in vivo plasmatic concentration of the drug. One procedure to achieve this goal consists of the... 

Models are the handiwork of theoreticians and may be mechanical or electrical analogs, pictorial representations, or purely mathematical constructs. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

The problem of how to fit a random process model to a given physical situation, i.e., what values to assign to the time averages, is not a purely mathematical problem, but one involving a skillful combination of both empirical and theoretical results, as well as a great deal of judgement based on practical experience. Because of their involved nature, we shall not consider such problems (called problems in statistics to distinguish them from the purely mathematical problems of the theory of random processes) in detail here, but instead, refer the reader to the literature. ... 

Atmospheric aerosols have a direct impact on earth s radiation balance, fog formation and cloud physics, and visibility degradation as well as human health effect[l]. Both natural and anthropogenic sources contribute to the formation of ambient aerosol, which are composed mostly of sulfates, nitrates and ammoniums in either pure or mixed forms[2]. These inorganic salt aerosols are hygroscopic by nature and exhibit the properties of deliquescence and efflorescence in humid air. That is, relative humidity(RH) history and chemical composition determine whether atmospheric aerosols are liquid or solid. Aerosol physical state affects climate and environmental phenomena such as radiative transfer, visibility, and heterogeneous chemistry. Here we present a mathematical model that considers the relative humidity history and chemical composition dependence of deliquescence and efflorescence for describing the dynamic and transport behavior of ambient aerosols[3]. 

There are two basic classes of mathematical models (see Fig. 5.3-18) (1) purely empirical models, and (2) models based on physicochemical principles. 

A deliquescent material takes up moisture freely in an atmosphere with a relative humidity above a specific, well-defined critical point. That point for a given substance is defined as the critical relative humidity (RH0). Relative humidity (RH) is defined as the ratio of water vapor pressure in the atmosphere divided by water vapor pressure over pure water times 100% [RH = (PJP0) X 100%]. Once moisture is taken up by the material, a concentrated aqueous solution of the deliquescent solute is formed. The mathematical models used to describe the rate of moisture uptake involve both heat and mass transport. 

Berggren, K.-F., and A.F. Sadreev. Chaos in quantum billiards and similarities with pure-tone random models in acoustics, microwave cavities and electric networks. Mathematical modelling in physics, engineering and cognitive sciences. Proc. of the conf. Mathematical Modelling of Wave Phenomena , 7 229, 2002. 

This chapter is concerned with the design and improvement of chemically-active ship bottom paints known as antifouling paints. The aims have been to illustrate the challenges involved in working with such multi-component, functional products and to show which scientific and engineering tools are available. The research in this field includes both purely empirical formulation and test methods and advanced tools including mathematical modelling of paint behaviour. 

Quantum wave mechanics gave chemistry a new "understanding," but it was an understanding absolutely dependent on purely chemical facts already known. What enabled the theoretician to get the right answer the first time, in a set of calculations, was the experimental facts of chemistry, which, Coulson wrote, "imply certain properties of the solution of the wave equation, so that chemistry could be said to be solving the mathematicians problems and not the other way around."36 So complex are the possible interactions among valence electrons that one must either use an exact mathematical model of a... 

In many cases of practical interest, no theoretically based mathematical equations exist for the relationships between x and y we sometimes know but often only assume that relationships exist. Examples are for instance modeling of the boiling point or the toxicity of chemical compounds by variables derived from the chemical structure (molecular descriptors). Investigation of quantitative structure-property or structure-activity relationships (QSPR/QSAR) by this approach requires multivariate calibration methods. For such purely empirical models—often with many variables—the... 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

Assuming each reaction is linearily dependent on the concentrations of each reacr-tant, derive a dynamic mathematical model of the system. There are two feed streams, one pure benzene and one concentrated nitric acid (98 wt %). Assume constant densities and complete miscibility. 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I< 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

Calculated descriptors have generally fallen into two broad categories those that seek to model an experimentally determined or physical descriptor (such as ClogP or CpKJ and those that are purely mathematical [such as the Kier and Hall connectivity indices (4)]. Not surprisingly, the latter category has been heavily populated over the years, so much so that QSAR/QSPR practitioners have had to rely on model validation procedures (such as leave-k-out cross-validation) to avoid models built upon chance correlation. Of course, such procedures are far less critical when very few descriptors are used (such as with the Hansch, Leo, and Abraham descriptors) it can even be argued that they are unnecessary. 

Gray fit Yang (Ref 1), a mathematical model was proposed to unity the chain and thermal mechanisms of explosion. It was shown that the trajectories in the phase plane of the coupled energy and radical concentration equations of an explosive system will oive the time-dependent behavior of the system when the initial temperature and radical concentration are given. In the 2nd paper of the same investigators (Ref 2), a general equation for explosion limits (P—T relation) is derived from a unified thermal and chain theory and from chis equation, the criteria of explosion limits for either the pure chain or pure thermal theory can be deduced. For detailed discussion see Refs... 

We generally distinguish between two methods when the determination of the composition of the equilibrium phases is taking place. In the first method, known amounts of the pure substances are introduced into the cell, so that the overall composition of the mixture contained in the cell is known. The compositions of the co-existing equilibrium phases may be recalculated by an iterative procedure from the predetermined overall composition, and equilibrium temperature and pressure data It is necessary to know the pressure volume temperature (PVT) behaviour, for all the phases present at the experimental conditions, as a function of the composition in the form of a mathematical model (EOS) with a sufficient accuracy. This is very difficult to achieve when dealing with systems at high pressures. Here, the need arises for additional experimentally determined information. One possibility involves the determination of the bubble- or dew point, either optically or by studying the pressure volume relationships of the system. The main problem associated with this method is the preparation of the mixture of known composition in the cell. 

The Sharpless epoxidation is sensitive to preexisting chirality in selected substrate positions, so epoxidation in the absence or presence of molecular sieves allows easy kinetic resolution of open-chain, flexible allylic alcohols (Scheme 26) (52, 61). The relative rates, kf/ks, range from 16 to 700. The lower side-chain units of prostaglandins can be prepared in high ee and in reasonable yields (62). A doubly allylic alcohol with a meso structure can be converted to highly enantiomerically pure monoepoxy alcohol by using double asymmetric induction in the kinetic resolution (Scheme 26) (63). A mathematical model has been proposed to estimate the degree of the selectivity enhancement. 

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