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Matrix nonstationary semigroups: Semigroup property and relativity. (English) Zbl 0887.60091
Bainov, Drumi (ed.) et al., Proceedings of the 5th international colloquium on differential equations, Plovdiv, Bulgaria, August 18–23, 1994. Invited lectures and short communications. Vol. 1. Singapore: SCT Publishing, 54-66 (1995).
This is a draught concerning the matrix nonstationary semigroups, a structure generalizing the notion of matrix exponential. On open time intervals of regularity, of which the union is everywhere dense in the parameter time set, we expand the reduction (in its locally independent constituents) of such a matrix semigroup into a sum generalizing a Taylor series. These expansions are expressed in terms of diffused and atomic measures involved by the reduction itself on every interval of regularity. On these intervals, we also give the Fokker-Planck equations corresponding to this reduction. In the last section, we use these results to show that the propagator of a (regularized) monochromatic wave is necessarily based on the combinatorial numbers $${n\choose k}$$ only, which so appear to be the basic invariants in the relativistic formalism when seen from a semigroup point of view, giving the subsequent classical invariants, to wit the quadratic form and the phase increment. In such an approach, the space-time structure arises from the monochromatic waves, that is, from an ergodism where inertia is involved, and the relativistic formalism arises from the propagators of these waves, according to Huygens’ principle, that is, from the semigroup property expressing Huygens’ spherelets in terms of discrete combinatorial distributions or, in the limit, in terms of Gaussian distributions (see also the author [in: Proceedings of the fourth international colloquium on differential equations, 71-82 (1994; Zbl 0843.60100)], of which the general structure of matrix semigroups and this last section sketches the main lines of the reciprocal part).
For the entire collection see [Zbl 0870.00028].

##### MSC:
 60K40 Other physical applications of random processes 20M99 Semigroups 26A21 Classification of real functions; Baire classification of sets and functions 60J27 Continuous-time Markov processes on discrete state spaces 60J75 Jump processes (MSC2010) 83A05 Special relativity 81P20 Stochastic mechanics (including stochastic electrodynamics)