By Shapley H.
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Extra info for A Note on the Simple Device for Increasing a Photographic Power of Large Telescopes (1920)(en)(4s)
This makes the rotation group 0(2) for this space Abelian. This group is isomorphic with the one-dimensional infinite-connected group of unitary matrices. Group 14 (1) has infinitely many coverings and R. (the set of integers with summation as the group operation) constitutes its universal covering. If Z( x ) denotes a group generated by number x (Z x = : a = n • x,n E Z}), 9 0(3) is the set of all real unitary matrices 3 X 3 with determinant equal +I.. g. rotation on axis I at angles Tr and —7r) are treated as identical.
18). One can find the other presentation of the classic Artin homotopy group idly (Coxeter and Moser 1984). 15) the generators were related to the operations of exchanges of two neighbouring particles (generators ai). However, in this new presentation, two particles exchange only when they have indices 1 and 2 (the generator a). 2 Mill braid group for the sphere 82 Let S2 be a sphere in the three-dimensional Euclidean space. 23) ) v1(Q N( s2 ) ) = E N(s2 in other words, 7ri (QN (S2 )) is generated by operations of an exchange of neighbouring particles.
If Z( x ) denotes a group generated by number x (Z x = : a = n • x,n E Z}), 9 0(3) is the set of all real unitary matrices 3 X 3 with determinant equal +I.. g. rotation on axis I at angles Tr and —7r) are treated as identical. Points belonging to the sphere constructed in such a way form a double-connected space: all curves connecting pairs of points Ki and K2 of this sphere divide into two classes of homotopies. One class is represented by a curve connecting points K1 and K2 and consisting of points located in the distance shorter than 7T from thc sphere centre; the other class is represented by a curve connecting points K1 and K2 including some points from the surface of the sphere.