Impossibility theory

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Another phenomenon is so called two-side filling of one-side closed conical capillaries with liquid [5]. On the one hand the more penetrant is trapped by the defect the wider indication will appear. Contrariwise it is almost impossible to extract a penetrant from the completely filled surface defects by dry developer [6]. In this study we propose the theory of the phenomenon. Besides experimental results of the investigation of two-side filling with various penetrants of conical capillaries are presented. Practical recommendations to optimize liquid penetrant testing process are proposed. 

The miderstanding of molecular motions is necessarily based on quaiitum mechanics, the theory of microscopic physical behaviour worked out in the first quarter of the 20th century. This is because molecules are microscopic systems in which it is impossible—or at least very dangerous —to ignore the dual wave-particle nature of matter first recognized in quaiitum theory by Einstein (in the case of classical waves) and de Broglie (in the case of classical particles). 

Simulations act as a bridge in another sense between theory and experiment (see figure B3.3.2. We can test a theory using idealized models, conduct thought experiments , and clarify what we measure in the laboratory. We may also carry out simulations on the computer tliat are difficult or impossible in the laboratory (for example, working at extremes of temperature or pressure). 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

In general, the foUowing steps can occur in an overall Hquid—soHd extraction process solvent transfer from the bulk of the solution to the surface of the soHd penetration or diffusion of the solvent into the pores of the soHd dissolution of the solvent into the solute solute diffusion to the surface of the particle and solute transfer to the bulk of the solution. The various fundamental mechanisms and processes involved in these steps make it impracticable or impossible to describe leaching by any rigorous theory. 

However, theories that are based on a basis set expansion do have a serious limitation with respect to the number of electrons. Even if one considers the rapid development of computer technology, it will be virtually impossible to treat by the MO method a small system of a size typical of classical molecular simulation, say 1000 water molecules. A logical solution to such a problem would be to employ a hybrid approach in which a chemical species of interest is handled by quantum chemistry while the solvent is treated classically. 

It is worthwhile to present this episode in eonsiderable detail, beeause it eneapsulates very elearly what was new in physieal metallurgy in the middle of the eentury. The elements are an aecurate theory of the effects in question, preferably without disposable parameters and, to check the theory, the use of a technique of measurement (the Snoek pendulum) which is simple in the extreme in construction and use but subtle in its quantitative interpretation, so that theory ineluctably comes into the measurement itself. It is impossible that any handwaver could ever have conceived the use of a pendulum to measure dissolved carbon concentrations ... 

Although it is often possible to predict the effect of the solvent on retention, due to the unique interactive character of both the solvents and the enantiomers, it is virtually impossible to predict the subtle differences that control the separation ratio from present knowledge. Nevertheless, some accurate retention data, taken at different solvent compositions, can allow the retention and separation ratios to be calculated over a wide range of concentrations using the procedure outlined above. From such data the phase system and the column can be optimized to provide the separation in the minimum time, a subject that will be discussed later in the treatment of chromatography theory. 

The blowdown valves on the boilers were operated by a special key, which had a lug on it so that it could not be removed when the valve was open. It was therefore impossible, in theory, for two blowdown valves to be open at the same time. However, the boiler fitter kept and jealously guarded a private key w ithout a lug and had used this one to open the blowdown valve on the boiler that was under repair. He forgot to tell the process foreman what he had done or to close the valve. The presence of this key would appear to have been of little moment as long as the correct procedure of complete isolation was maintained, but as soon as it was departed from, the additional key became a menace, which eventually enabled the present tragedy to occur, the accident report said. 

Although Eqs. (33), (34), and especially (35), are useful they have a problem. They all predict that the hard sphere system is a fluid until = 1. This is beyond close packing and quite impossible. In fact, hard spheres undergo a first order phase transition to a solid phase at around pd 0.9. This has been estabhshed by simulations [3-5]. To a point, the BGY approximation has the advantage here. As is seen in Fig. 1, the BGY equation does predict that dp dp)j = 0 at high densities. However, the location of the transition is quite wrong. Another problem with the PY theory is that it can lead to negative values of g(r). This is a result of the linearization of y(r) - 1 that... 

Adsorption of hard sphere fluid mixtures in disordered hard sphere matrices has not been studied profoundly and the accuracy of the ROZ-type theory in the description of the structure and thermodynamics of simple mixtures is difficult to discuss. Adsorption of mixtures consisting of argon with ethane and methane in a matrix mimicking silica xerogel has been simulated by Kaminsky and Monson [42,43] in the framework of the Lennard-Jones model. A comparison with experimentally measured properties has also been performed. However, we are not aware of similar studies for simpler hard sphere mixtures, but the work from our laboratory has focused on a two-dimensional partly quenched model of hard discs [44]. That makes it impossible to judge the accuracy of theoretical approaches even for simple binary mixtures in disordered microporous media. 

Ideally, the sample should be injected onto the column as an infinitely thin disc, which covers the total cross section of the column. Because this is impossible, PSS has injected finite volumes onto the columns. In theory, these injection volumes should be as low as possible. In order to be able to detect the sample with significance, a certain (high) concentration of the sample has to be injected. This concept works well for low molar mass compounds, which do not generate much sample viscosity. However, when working with samples... 

Clearly, proximity and orientation play a role in enzyme catalysis, but there is a problem with each of the above comparisons. In both cases, it is impossible to separate true proximity and orientation effects from the effects of entropy loss when molecules are brought together (described the Section 16.4). The actual rate accelerations afforded by proximity and orientation effects in Figures 16.14 and 16.15, respectively, are much smaller than the values given in these figures. Simple theories based on probability and nearest-neighbor models, for example, predict that proximity effects may actually provide rate increases of only 5- to 10-fold. For any real case of enzymatic catalysis, it is nonetheless important to remember that proximity and orientation effects are significant. 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

Newtonian inochaiiics. Since it is impossible to completely specify the initial state of such a system, kinetic theory (contents itself with describing a smoothed version of the system. The smoothed version is simply one whore all exact state-information below some characteristic length and time is replaced by a probabilistic description. 

Having thus established at least a formal equivalency between a discretized field theory on a lattice and CA, Svozil invokes the so-called no-go theorem to show that field theory cannot be discretized in this simple fashion. The no-go theorem (see [karstSl] and [nielSl]) states essentially that under a set of only mild assumptions it is impossible to formulate a local, unitary, charge conserving lattice held theory without effectively doubling the size of the predicted fermion population (i.e. species doubling see discussion box). 

An interesting and practically valuable result was obtained in [21] for PE + N2 melts, and in [43] for PS + N2 melts. The authors classified upper critical volumetric flow rate and pressure with reference to channel dimensions x Pfrerim y Qf"im-Depending on volume gas content

channel entrance (pressure of 1 stm., experimental temperature), x and y fall, in accordance with Eq. (24), to tp 0.85. At cp 0.80, in a very narrow interval of gas concentrations, x and y fall by several orders. The area of bubble flow is removed entirely. It appears that at this concentration of free gas, a phase reversal takes place as the polymer melt ceases to be a continuous phase (fails to form a continuous cluster , in flow theory terminology). The theoretical value of the critical concentration at which the continuous cluster is formed equals 16 vol. % (cf., for instance, Table 9.1 in [79] and [80]). An important practical conclusion ensues it is impossible to obtain extrudate with over 80 % of cells without special techniques. In other words, technology should be based on a volume con-... 

Even in the well-known domain of nearly linear phenomena it would be impossible to make predictions accurately on a purely theoretical basis historically the progress was just opposite, going from the experimental evidence to the theory and not in the opposite direction. 

Nonanalytic Nonlinearities.—A somewhat different kind of nonlinearity has been recognized in recent years, as the result of observations on the behavior of control systems. It was observed long ago that control systems that appear to be reasonably linear, if considered from the point of view of their differential equations, often exhibit self-excited oscillations, a fact that is at variance with the classical theory asserting that in linear systems self-excited oscillations are impossible. Thus, for instance, in the van der Pol equation... 

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