By Max Born, H. S. Green

This paper outlines a common idea whose item is to supply a foundation from which all of the equilibrium and dynamical houses of drinks may be investigated. a collection of multiform distribution features is outlined, and the generalized continuity equations happy by means of those services are derived. by means of introducing the equations of movement, a suite of relatives is bought from which the distribution features might be made up our minds. it really is proven that Boltzmann's equation within the kinetic thought of gases follows as a specific case, and that, in equilibrium stipulations, the idea offers effects in keeping with statistical mechanics. An essential equation for the radial distribution functionality is bought that is the usual generalization of 1 received by means of Kirkwood for 'rigid round molecules'. ultimately, it's indicated how the speculation should be utilized to resolve either equilibrium and dynamical difficulties of the liquid kingdom.

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These singular defect regions define topological obstructions; their representation in terms of homogeneous closed p-forms will be detailed below. Physically such p-forms can be related to "fixed" points of rotation or expansion, and tangential discontinuities such as found in hydrodynamic wakes, and propagating electromagnetic signals. Integrals of such singular p-forms around cycles lead to topological quantization. The values of such integrals are related to the integers. Relative to the Cartan topology [11], the "Pfaﬀ topological dimension" can be generated by each of the Pfaﬃan forms associated with each discipline.

However, not all topological properties fit this useful, but imprecise, description. As examples, note that the number of holes in a rubber sheet is a topological property, and is independent of the continuous deformation of the rubber sheet into diﬀerent sizes and shapes. The Planck black body radiation distribution of a hot body is a topological property, for the distribution of radiation frequencies (in first approximation) depends only on temperature, but not on the size and shape of the heated sample.

4 Holder Norms, Period Integrals and Topological Quantization Besides the invariant structures considered above, the Cartan methods may be used to generate other sets of topological invariants. Realize that over a domain of Pfaﬀ dimension n less than or equal to N, the Cartan criteria admits a submersive map to be made from N to a space of minimal dimension n. ˆ V n . , V n )Ωn . }m/p . )Ω = ρ{V 1 dV 2 ˆdV 3 ... − V 2 dV 1 ˆdV 3 ... + V 3 dV 1 ˆdV 2 ... }. 22) Theorem 1. The (n-1)-form J is closed, dJ = 0, for any choice of constants a,b,c..