Mathematical constructs

John von Neuman, one of the greatest mathematicians of the twentieth century, believed that the sciences, in essence, do not try to explain, they hardly even try to interpret they mainly make models. By a model he meant a mathematical construct that, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work. Stephen Hawking also believes that physical theories are just mathematical models we construct and that it is meaningless to ask whether they correspond to reality, just as it is to ask whether they predict observations. 

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

The previous section used a mathematical construct called a ray to predict behavior of light in an optical system. We should emphasize that rays are purely a mathematical construct, not a physical reality. Rays work well to describe the behavior of light in cases where we can ignore its wave-like behavior. These situations are ones in which the angular size of the point-spread-function is much greater than A/d, where A is the wavelength of light and d is the diameter of the optical system. 

The Heisenberg uncertainty principle is a consequence of the stipulation that a quantum particle is a wave packet. The mathematical construction of a wave packet from plane waves of varying wave numbers dictates the relation (1.44). It is not the situation that while the position and the momentum of the particle are well-defined, they cannot be measured simultaneously to any desired degree of accuracy. The position and momentum are, in fact, not simultaneously precisely defined. The more precisely one is defined, the less precisely is the other, in accordance with equation (1.44). This situation is in contrast to classical-mechanical behavior, where both the position and the momentum can, in principle, be specified simultaneously as precisely as one wishes. 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Using regression analysis on a data set of about 50 different molecules, it was found that a. = —4.4,8 = —0.5, Df = 12 cm2/s, and =2.5x 10 5 cm2/s [192], A graphic representation of the effect of relative molecular mass (Mr) and distribution coefficient on corneal permeability is shown in Fig. 13. One observes a rapid reduction in permeability coefficient with decreasing P and increasing Mr. The addition of pores to the model, a mathematical construct, is necessary to account for permeability of polar molecules, such as mannitol and cromolyn. These would also be required for correlating effects of compounds, such as benzalkonium chloride, which may compromise the... 

Another method of predicting human pharmacokinetics is physiologically based pharmacokinetics (PB-PK). The normal pharmacokinetic approach is to try to fit the plasma concentration-time curve to a mathematical function with one, two or three compartments, which are really mathematical constructs necessary for curve fitting, and do not necessarily have any physiological correlates. In PB-PK, the model consists of a series of compartments that are taken to actually represent different tissues [75-77] (Fig. 6.3). In order to build the model it is necessary to know the size and perfusion rate of each tissue, the partition coefficient of the compound between each tissue and blood, and the rate of clearance of the compound in each tissue. Although different sources of errors in the models have been... 

Optical designers and specialists in heat transfer calculations in the chemical engineering and mechanical engineering sciences are familiar with the mathematical construct known as The Equation of Radiative Transfer, although most chemists and spectroscopists are not. The Equation of Radiative Transfer states that, disregarding absorbance and scattering, in a lossless optical system... 

Whereas the wave function is a mathematical construct, with interpre-tational rules for obtaining the answers to physical problems, the physical... 

It is worth remarking that a gas sensor array is a mere mathematical construction where the sensor outputs are arranged as components of a vector. Arrays can also be utilized to investigate the properties of chemical sensors, or even better, the peculiar behaviour of a sensor as a component of an array. In this chapter, the more common sensor array methodologies are critically reviewed, including the most general steps of a multivariate data analysis. The application of such methods to the study of sensor properties is also illustrated through a practical example. 

In their theoretical work,43 the various self-affine fractal interfaces were mathematically constructed employing the Weierstrass function /ws(x), 151>152... 

The total ion current (TIC) can either be measured by a hardware TIC monitor before mass analysis, or it can be reconstructed by the data system from the spectra after mass analysis. [27] Thus, the TIC represents a measure of the overall intensity of ion production or of mass spectral output as a function of time, respectively. The TIC obtained by means of data reduction, [28] i.e., by mathematical construction from the mass spectra as successively acquired while the sample evaporates, is also termed total ion chromatogram (TIC). For this purpose, the sum of all ion intensities belonging to each of the spectra is plotted as a function of time or scan number, respectively. 

In the last chapter, the full formalism of Hartree-Fock theory was developed. While this theory is impressive as a physical and mathematical construct, it has several limitations in a practical sense. Particularly during the early days of computational chemistry, when computational power was minimal, carrying out HF calculations without any further approximations, even for small systems with small basis sets, was a challenging task. 

The fact that the velocity of a fluid changes from layer to layer is evidence of a kind of friction between these layers. The layers are mathematical constructs, but the velocity gradient is real and a characteristic of the fluid. The property of a fluid that describes the internal friction or resistance to flow is the viscosity of the material. Chapter 4 is devoted to a discussion of the measurement and interpretation of viscosity. For now, it is enough for us to recall that this property is quantified by the coefficient of viscosity 77 of a material. The coefficient of viscosity has dimensions of mass length-1 time-1, kg m-ls-1 in SI units. In actual practice, the cgs unit of viscosity, the poise (P), is widely used. Note that pure water at 20°C has a viscosity of about 0.01 P = 10-3kgm-ls-1... 

Mathematical constructs of physical/ chemical processes that predict fixed outputs for a fixed set of inputs... 

Mathematical construction of physical/ chemical processes that predict the range and probability density distribution of an exposure model outcome (e.g. predicted distribution of personal exposures within a study population)... 

J. P. Vigier, Explicit mathematical construction of relativistic non-linear de Broglie waves described by three-dimensional (wave and electromagnetic) solitons piloted (controlled) by corresponding solutions of associated linear Klein-Gordon and Schrodinger equations, Found. Phys. 21(2) (1991). 

The hole theory was perceived as a Active mathematical construction and was initially rejected by prominent contemporary physicists such as Pauli and Bohr. The physical reality of antiparticles was not taken seriously even by Dirac himself. In 1931 he wrote about his anti-electron we should not expect to find it in Nature [2]. Surprisingly, the first anti-electrons were discovered already in 1932 by Anderson, who studied cosmic rays in Caltech s magnet cloud chamber. Anderson noticed abnormally bending trajectories indicating the presence of light positively charged particles and, as related by Fowler [3], "could not resist the devastating conclusion that they are caused by positive electrons The first piece of antimatter, a positron, made its physical appearance. 

This kind of change can be accompanied by new ways of perceiving space. Lines may become curved instead of straight, for example. Some people report perceiving four or more dimensions in d-ASCs, not as a mathematical construct but as an experiential reality. The difficulties of expressing this in a language evolved from external adaptation to... 

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