Theoretical papers, mathematical

Molecular diffusion was identifled as the important factor influencing overall design of the flow channel already in the first theoretical paper on FIA [20]. Since then its role in FIA processes has been clarified and mathematically described in a number of detailed papers. Works of Van-derslice et al. [1061], Wada [1065] and Hungerford [3.4] focused attention on molecular diffusion as the sole tool for redistribution of sample elements in radial direction. Their results, observations, and conclusions are reviewed in the next section. 

We have summarized two recent theoretical papers on the elasticity of polymer networks and chains confined within narrow slits[12, 13]. The neglect of polymer-polymer interactions (phantom chain assumption) allows an essentially exact — although sometimes complicated — mathematical description. The presentation has been didactic in spirit and we have strived to highlight the essential physics of these systems. Hopefully this will stimulate more theoretical, experimental and computational studies. 

The problem in interpretation of EPR spectra of textured paramagnetic complexes in liquid crystals, or radicals in prolated polymeric compounds was already discussed in literature. We can find some experimental and theoretical papers on this problem [1-3]. From the point of view of an experimental technique the EPR spectra acquisition and mathematical simulation algorithm for paramagnetic centers in a fiat surface film do not differ from centres in oriented liquid crystal solutions or inserted in prolated polymeric compounds. Solving the problem in a general form requires triple integration and is possible only by numerical methods. Attempts to solve this problem analytically were successful only in very particular cases. The techniques proposed in these papers based on theoretical tables and plots is not obvious enough. 

For Inter Molecular Perturbation Theory (IMPT) see Hayes, I. C. Stone, A. J. An intermolecular perturbation theory for the region of moderate overlap, Mol. Phys. 1984, 53, 83-105 papers of this kind, however, contain a large amount of theoretical and mathematical detail and are not transparent to the uninitiated. For Symmetry-Adapted Perturbation Theory (SAPT) see e.g. Bukowski, R. Szalewicz, K. Chabalovski, C. F. Ab initio interaction potentials for simulations of dinitramine solutions in supercritical carbon dioxide with cosolvents, J. Phys. Chem. 1999, A103, 7322-7340, and references therein. The Morokuma decomposition scheme is described in Kitaura, K. Morokuma, K. A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation, Int. J. Quantum Chem. 1976,10, 325-340. 

Adsorption of macromolecules has been widely investigated both theoretically [9—12] and experimentally [13 -17]. In this paper our purpose was to analyze the probable structures of polymeric stationary phases, so we shall not go into complicated mathematical models but instead consider the main features of the phenomenon. The current state of the art was comprehensively summarized by Fleer and Lyklema [18]. According to them, the reversible adsorption of macromolecules and the structure of adsorbed layers is governed by a subtle balance between energetic and entropic factors. For neutral polymers, the simplest situation, already four contributor factors must be distinguished ... 

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

Niels Bohr s 1913 hydrogen atom paper demonstrates the traditional interest of some physicists in placing the facts and laws of chemistry within a broader framework of foundational principles laid out by physicists. During the course of the next two decades, a number of physicists who became known as quantum physicists developed physical theories and mathematical techniques that they claimed would create a mathematical and theoretical chemistry. However, few of them had much chemical knowledge beyond a general understanding of the periodic table of the elements and familiarity with the Lewis-Langmuir theory of the electron duplet and octet. 

S.H. Bertz, A Mathematical Model of Molecular Complexity. A Collection of Papers from a Symposium Held at the University of Georgia, Athens, Georgia, U.S.A, 18-22 April 1983 (R.B. King, Editor) in "Studies in Physical and Theoretical Chemistry",Vol 28, Elsevier Science Publishers B.V., Amsterdam,... 

Since the Murday-Cotts paper in 1968 much progress has been made toward the theoretical description of the spin-echo intensity E g, A) for spins diffusing in well-defined geometries. Tanner and Stejskal derived already in 1968 the exact expression of ii(g. A) for spins diffusing in a rectangular box. The derivation of an exact expression for E g, A) for diffusion in a sphere with reflecting walls is not a trivial mathematical problem and it took between 1992 and 1994 when three expressions were published. All three expressions are only valid in the short-gradient-pulse approximation (see below). 

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

On the theoretical side, it was Hermann von Helmholtz (Sidebar 3.4) who first presented a clear and comprehensive mathematical formulation of energy conservation as a principle of universal validity, applicable to all natural phenomena. Helmholtz s landmark paper of 1847, Uber die Erhaltung der Kraft, reflected some lingering ambiguities of the force (Kraft) concept, but exhibited the deep integration of the first law into analytical dynamics in a clear and modem way. Helmholtz deserves to be counted the scientist most responsible for rigorous mathematical formulation of the first law. 

Fundamentals of the various methods of plastisol technology have been considered in a number of papers, nevertheless the theoretical analysis and the construction of mathematical models have not yet been completed so far. The dipping process has been studied in more detail cf.2 6 10). 

Yet who would have thought the old man to have had so much hlood in him This title, given by Prof. Rutherford Aris and his collaborator W.W. Farr to their recent paper [Chem. Eng. Sci., 41 (1986) 1385], is a phrase used by Lady Macbeth (Macbeth, V, 1, 42-44). Fierce, isn t it Apparently, they mean it to imply that traditional theoretical problems in the dynamics of chemical reactions, in particular the known problem of the dynamics of the continuous stirred tank reactor (CSTR), are far from being exhausted. Novel mathematical approaches provide new results oriented to physico-chemical comprehension. This current trend is confirmed by the present volume. 

The derivation of a consistent mixed quantum-classical dynamics discussed in this paper was first proposed in Ref. [15] and commented and clarified in Ref. [1], This derivation is based on a group-theoretical formulation of quantum and classical mechanics, which introduces a very elegant and formally rigorous mathematical apparatus and allows to directly obtain classical mechanics as the limit for h —> 0 of quantum mechanics, in the Heisenberg representation of quantum dynamics. 

In this paper we discuss a number of issues which manifest the theoretical particularity of quantum chemistry and which are usually not discussed in an explicit manner either in the historical or in the philosophical studies related to quantum chemistry. We shall focus on five issues the re-thinking of the problem of reductionism, the discourse of quantum chemistry as a confluence of the traditions of physics, chemistry, and mathematics, the role of textbooks in consolidating this discourse, the ontological status of resonance, and the more general problem of the status of the chemical bond. Finally, we shall briefly discuss the impact of large scale computing. 

Very few results are available to test this mathematical model. Circumstances have lead to two tests comparing a negative and a positive finding at 1A7 MHz with results at A50 MHz. One test was performed at 50 MHz (3.6A mW/cm2) to test a prediction of this model based on the result of 147 MHz (0.83 mW/cm2). All these tests have provided results that agree with the model. In this paper, we describe the results of two tests at 147 MHz which agree with the predictions, one positive and one negative, made from the 50 MHz data. Thus, all the results to date support the predictions of the theoretical analysis. Further support for the theory is the fact that no result is contradicted by a corresponding experiment at a different carrier frequency. A more complete test of the theory will be obtained when additional experiments are conducted at the untested values of P.. 

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