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Elementary holograms and 3-orbifolds. (English) Zbl 0646.43005

This article summarizes results on holograms which are obtained by harmonic analysis on the 3-dimensional real Heisenberg group A(\({\mathbb{R}})\). First of all the holographic plane is identified with A(\({\mathbb{R}})/Z\), Z the 1-dimensional center of A(\({\mathbb{R}})\), and the points of the holographic plane are represented by the equivalence class of \( \begin{pmatrix} 1&x&0 \\ 0&1&y \\ 0&0&1 \end{pmatrix} \). Elementary holograms are the functions on the holographic plane defined by the coefficients of the irreducible Schrödinger representation of A(\({\mathbb{R}})\). With the identification \(z=x+iy\) one can transform the Gaussian integers to the holographic plane forming a holographic grid. The Euclidean orientable 3- orbifolds of planar holographic grids are enumerated with the help of the crystallographic groups. Finally applications of holographic grids in neurophysiology are listed.
For complete proofs the reader is referred to [Harmonic analysis on the Hsure on a finite time-interval avoidance motions of a nonlinear conflict-controlled system from a closed target set given in the space of continuous functions under the presence of phase restrictions. A description of solving strategies of both players is given with the aid of the method of program iterations developed by A. G. Chentsov [Dokl. Akad. Nauk SSSR 226, 73-76 (1976; Zbl 0395.90105); and Izv. Akad. Nauk SSSR, Ser. Mat. 42, 455-467, 471 (1978; Zbl 0423.90096)] and its generalization to games with information memory obtained by A. G. Chentsov and the author [Kibernetika 1985, No.3, 95-97 (1985; Zbl 0625.90102)]. An alternative separation of the space of initial histories of the game is found.
Reviewer: E.Al’brekht

MSC:

43A80 Analysis on other specific Lie groups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
78A70 Biological applications of optics and electromagnetic theory
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