Mathematical progression

Cascade (dendritic) macromolecules are discrete, highly branched, monodisperse polymers that possess branching patterns described by a nonlinear mathematical progression. It has been demonstrated that although dendrimers have well-defined constitutions, their size in solution may vary drastically with conditions such as pH [10]. Therefore, shape-persistent... 

As the mysterious Stone engenders and multiplies itself in continuous mathematical progression the Illuminated One in his turn, transmits his own spiritual light to those who, intelligent and docile prima materia will themselves accept the need to die as lead in order to be better reborn as gold... 

The initial synthetic model was based on the architectural design for trees, thus the term arborol , in which a cascade approach, put forth by Professor Vdgtle (Scheme 1), would be based on a simple mathematical progression [ - 3->9->27->81 243-. ]. This is best demonstrated in our initial published example in 1985 (Scheme 2). The limiting feature is that a spacer moiety had to be utilized to circumvent the effects of the inherent neopentyl branching locus. 

In order to make mathematical progress with the problem of interest here, it is convenient to introduce dimensionless forms of the rate equations. For this we define dimensionless concentrations a and that are simply the actual concentrations divided by the initial concentration of A a = a/ao,P = b/ao. Thus both vary between 0 and 1, with a = 1 and / = 0 at the beginning of the reaction and a = 0, / = 1 for the final state. 

The letters s, p, and d originate in the names sharp, principal, and diffuse. These were the terms that were given to absorption and emission lines in the atomic spectra of alkali atoms on the basis of the appearance of those lines, usually on a photographic plate. Lines of similar type formed series, and it was learned that the transition frequencies measured for a series followed a simple mathematical progression. With quantum mechanics the progressions and spectral characteristics became understandable consequences of the allowed energy levels and transitions. From this association with types of lines come the orbital letters in use today. 

G. R. Keepin Delayed Neutrons Physics and Mathematics -Progress in Nuclear Energy, Series I (Pergamon Press,... 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

The logical order in which to present molecular orbital calculations is ab initio, with no approximations, through semiempirical calculations with a restricted number of approximations, to Huckel molecular orbital calculations in which the approximations are numerous and severe. Mathematically, however, the best order of presentation is just the reverse, with the progression from simple to difficult methods being from Huckel methods to ab initio calculations. We shall take this order in the following pages so that the mathematical steps can be presented in a graded way. 

All the isotherms give rise to BET plots which are linear over a limited range (e.g. for isotherm E, 001

apparent surface area A(app) from each isotherm. The increase in the value of c with the progressive removal of nonane (Table 4.6) is a mathematical consequence of the increasing contribution from the... 

Progress in modelling and analysis of the crack problem in solids as well as contact problems for elastic and elastoplastic plates and shells gives rise to new attempts in using modern approaches to boundary value problems. The novel viewpoint of traditional treatment to many such problems, like the crack theory, enlarges the range of questions which can be clarified by mathematical tools. 

The rate equation involves a mathematical expression describing the rate of progress of the reaction. To predict the size of the reactor required in achieving a given degree of conversion of reactants and a fixed output of the product, the following information is required ... 

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

It is no surprise that Mendeleev never gave precise mathematical expression to this periodic function . In fact, it would be impossible, we claim, to state at all precisely the content of Mendeleev s periodic law . (We are, of course, referring here to the law as articulated by Mendeleev himself and as understood by his contemporaries. There is no doubt that the subsequent development of chemistry has seen at least great progress toward the articulation of a precise version of the periodic law, based ultimately on quantum mechanics.24)... 

See Scerri (1998), where it is argued that, while still no one has succeeded in giving a mathematically precise version of the periodic law, and while the law has not exactly been reduced to quantum mechanics, sufficient progress has been made to suggest that a precise version of the law may eventually be possible (if only in the limit ). (See also Scerri, 1999, where further clarification is provided.)... 

Flade potential, 247 Flame-annealed gold surfaces and the work of Kolb, 81 Flat band potential, 483 Fluctuations asymmetrical and unstable systems, 255 controlling progress in pitting, 299 in pitting dissolution, 251 and corrosion processes, 217 during dissolution, 252 at electrodes, theory, 281 during film breakdown, 233 and mathematical expressions thereof, 276... 

Optional mathematical derivations. The How do we do that feature sets off derivations of key equations and encourages students to appreciate the power of mathematics by showing how it enables them to make progress and answer questions. All quantitative applications of calculus in the text are confined to this feature. The end-of-chapter exercises that make use of calculus are identified with a [cl... 

Before the advent of modem computer-aided mathematics, most mathematical models of real chemical processes were so idealized that they had severely limited utility— being reduced to one dimerrsion and a few variables, or Unearized, or limited to simplified variability of parameters. The increased availability of supercomputers along with progress in computational mathematics and numerical functional analysis is revolutionizing the way in which chemical engineers approach the theory and engineering of chemical processes. The means are at hand to model process physics and chenustry from the... 

The Michelson and Morley experiment shows the critical importance of intuitive concepts and understanding in the progress of Physics. It is essential that the abstract concepts and intuitive notions of a theory are accurate, precise and correct. A necessary condition is that they correspond exactly to the mathematical formulation of these concepts. In the case of MT during the last century and the beginning of this century, that correspondence was flawed. This example demonstrates the importance of teaching students both the concepts and the mathematics and to make sure that the relationship between the two is fully understood. 

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