Commensurate relationship

Until the parallel between number and cosmos is demonstrated to be an illusion we shall use this idea to model the universe. The power of this approach lies therein that all regularities in the physical world can be reduced to the same mathematical rules as the commensurable relationships in the solar system. The same mathematics that optimizes the distribution of matter in spiral galaxies and solar systems, shapes the growth of nautilus shells and sunflower heads. This ubiquitous symmetry, known as self-similarity is... 

In the conclusion, the relationship of disciplines to schools and traditions is discussed, both in the particular instances of the Paris and London-Manchester schools and in the more general history of Continental and Anglo-American science, using elements of the disciplinary identity schema developed at the beginning of the book. I end with an inquiry into the views of mid-twentieth-century theoretical chemists and chemical physicists on the commensurability of their disciplines, the reducibility of chemical explanation to physical principles and laws, and the distinctiveness of chemistry from physics. 

The types of harm that chemicals have the capacity to cause (in other words their hazards) are qualitatively different and not commensurate with each other. At one extreme are effects that require reasonably high concentrations of the substance, are immediate and localized (for example flammability, explosivity, corrosiveness and acute toxicity). The causal relationship between a particular chemical exposure and such effects is usually obvious, or at least easy to demonstrate. Those affected generally include those who deal directly with the chemical and it is often possible to handle and store the chemical in such a way that harm is avoided. The existing European Union (EU) regulatory system, dating from the 1967 dangerous substances directive (Council Directive 67/548/EEC) was clearly set up with... 

The commensurable kinetics of an electrode reaction bring about a hindrance or resistance to the transfer of charge at the interface and the current—potential relationship for the interface is more complex than the one predicted by Ohm s law for conducting phases. 

In this addendum, we will derive the spectral function from Weyl s theory and in particular demonstrate the relationship between the imaginary part of the Weyl-Titchmarsh m-function, mi, and the concept of spectral concentration. For simplicity we will restrict the discussion to the spherical symmetric case with the radial coordinate defined on the real half-line. Remember that m could be defined via the Sturm-Liouville problem on the radial interval [0,b] (if zero is a singular point, the interval [a,b], b > a > 0), and the boundary condition at the left boundary is given by [commensurate with Eq. (5)]... 

Among other predictions, the integrated model reveals that as work rate is varied, commensurate increases in the rate of mitochondrial ATP synthesis are effected by changes in concentrations of available ADP and inorganic phosphate. In other words, mitochondrial respiratory control is achieved in vivo by substrate feedback control. The predicted relationship between substrates and work rate is plotted in Figure 7.14. Model predictions are compared to data obtained from NMR spectroscopy of exercising flexor forearm muscle in healthy human subjects [106],... 

A myriad of minor moons and rings have the same disposition with respect to the parent planet that the asteroid belt and the inner planets have with respect to the srm. Closer examination of the orbital relationship within these groups, and also of the accumulation of the asteroids with respect to their mean orbit, demonstrate the same commensuration pattern as the planets around the srm, only on a different scale. 

Conventionally, the ratio cjc-a is used as a basis for dividing these systems into two categories—commensurate and incommensurate. The ciassical definition assigns the inclusion compound as commensurate if cjc is a rational number and as incommensurate if Cg/ci, is an irrational number. However, because experimental measurements of and c can never be made with absolute precision, it is more practical to define an inclusion compound as commensurate if (and only if) Cg/cj, is sufficiently close to a rational number with low denominator. This is equivalent to stating that, for a commensurate system, small integers p and q can be found that satisfy the relationship pcy, = qc,. If no sufficiently small integers p and q can be found, the inclusion compound is incommensurate. 

An important consequence of an incommensurate relationship between the host and guest substructures is that, in principle, each guest molecule within a given tunnel samples a slightly different environment with respect to the host structure. For a commensurate material, on the other hand, the guest molecules in the tunnel sample only one or a small number of different local environments within the host structure. Clearly, several chemical and spectroscopic properties of the guest... 

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