Characterization of {(q + 1) + 2, 1;t, q}-min · hypers and {2(q + 1) + 2, 2; 2,q}-min · hypers in a Finite projective geometry

1989 ◽  
Vol 5 (1) ◽  
pp. 63-81 ◽  
Author(s):  
Noboru Hamada
Keyword(s):  
Projective Geometry ◽  
Finite Projective Geometry

Structures of the cubic and rhombohedral high-pressure modifications of silicon as packing of the rod-like substructures determined by the algebraic geometry

2008 ◽  
Vol 64 (1) ◽  
pp. 26-33 ◽  
Author(s):  
V. S. Kraposhin ◽  
A. L. Talis ◽  
V. G. Kosushkin ◽  
A. A. Ogneva ◽  
L. I. Zinober
Keyword(s):  
High Pressure ◽  
Algebraic Geometry ◽  
Projective Geometry ◽  
Symmetry Axis ◽  
Triangular Face ◽  
Threefold Symmetry ◽  
Finite Projective Geometry ◽  
The Common

Tessellations by generating clusters are proposed for the high-pressure phases BC8 and R8 of silicon. The structures of both high-pressure phases are represented by the parallel packings of rods. The latter are stacks of distorted icosahedra, joined by a common triangular face and flattened out along the threefold symmetry axis. Along the rod axis there is an alternation of empty and double-centered icosahedra. The empty icosahedra are additionally distorted in the cubic BC8 phase by antiparallel rotation about the rod axis, while the double-centered icosahedra are distorted by that rotation in the rhombohedral R8 phase. A possible mechanism of the reversible BC8 ↔ R8 transformation is proposed as the rotation about the rod axis of the common triangular face of the neighboring icosahedra, thus transforming between distorted and undistorted icosahedra. The graphs of the generating clusters for both the BC8 and R8 structures are determined by two subconfigurations of the same construction of the finite projective geometry.

Periodic Linear Transformations of Affine and Projective Geometries

1950 ◽  
Vol 2 ◽  
pp. 149-151 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Number Theory ◽  
Projective Geometry ◽  
Irreducible Polynomial ◽  
Galois Field ◽  
Linear Transformations ◽  
Finite Projective Geometry ◽  
Projective Geometries ◽  
Periodic Linear

Introduction. In a paper called “A Theorem in Finite Projective Geometry and some Applications to Number Theory” [Trans. Amer. Math. Soc, vol. 43 (1938), 377-385], J. Singer proved that the finite projective geometry PG(s — 1,pn), that is the projective geometry of dimension s — 1 whose coordinate field is the Galois field GF(pn), admits a collineation L of period q = (psn — 1)/ (pn — 1). Since this q is the number of points of PG(s — 1, pn), Singer's result states that the points of PG(s — 1, pn) are cyclically arranged. Singer's construction of L uses the notion of a “primitive irreducible polynomial of degree 5 belonging to a field GF(pn) which defines a PG(s — 1, pn).”

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