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For n = 4, the Schur cover of the alternating group is given by SL(2, 3) → PSL(2, 3) ≅ A 4, which can also be thought of as the binary tetrahedral group covering the tetrahedral group. Similarly, GL(2, 3) → PGL(2, 3) ≅ S 4 is a Schur cover, but there is a second non-isomorphic Schur cover of S 4 contained in GL(2,9) – note that 9 = 3 ...
The compound polyhedron known as the stellated octahedron can be represented by a{4,3} (an altered cube), and , . The star polyhedron known as the small ditrigonal icosidodecahedron can be represented by a{5,3} (an altered dodecahedron), and , . Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to ...
A 4 is isomorphic to PSL 2 (3) [1] and the symmetry group of chiral tetrahedral symmetry. A 5 is isomorphic to PSL 2 (4), PSL 2 (5), and the symmetry group of chiral icosahedral symmetry. (See [1] for an indirect isomorphism of PSL 2 (F 5) → A 5 using a classification of simple groups of order 60, and here for a direct proof). A 6 is ...
Given a point A 0 in a Euclidean space and a translation S, define the point A i to be the point obtained from i applications of the translation S to A 0, so A i = S i (A 0).The set of vertices A i with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.
Some are structural, providing strength for a wall made with inferior stone or rubble, [2] while others merely add aesthetic detail to a corner. [3] According to one 19th-century encyclopedia, these imply strength, permanence, and expense, all reinforcing the onlooker's sense of a structure's presence.
The longest alternating subsequence problem has also been studied in the setting of online algorithms, in which the elements of are presented in an online fashion, and a decision maker needs to decide whether to include or exclude each element at the time it is first presented, without any knowledge of the elements that will be presented in the future, and without the possibility of recalling ...
All beta-barrels can be classified in terms of two integer parameters: the number of strands in the beta-sheet, n, and the "shear number", S, a measure of the stagger of the strands in the beta-sheet. [3] These two parameters (n and S) are related to the inclination angle of the beta strands relative to the axis of the barrel. [4] [5] [6]
A direct proof using classical geometry was developed by James Mercer in 1923. [2] This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles.