Generation of Lie Incidence Geometries: A Survey

Author(s):  
Bruce N. Cooperstein
Keyword(s):  
2005 ◽  
Vol 5 (1) ◽  
Author(s):  
B. N. Cooperstein ◽  
Anna Kasikova ◽  
Ernest E. Shult
Keyword(s):  

2011 ◽  
Vol 64 (1-2) ◽  
pp. 105-128 ◽  
Author(s):  
Philippe Cara ◽  
Alice Devillers ◽  
Michael Giudici ◽  
Cheryl E. Praeger
Keyword(s):  

2014 ◽  
Vol 70 (a1) ◽  
pp. C66-C66
Author(s):  
Maria Elena Montero-Cabrera ◽  
Isai Castillo-Sandoval ◽  
Luis Fuentes-Cobas ◽  
Hilda Esparza-Ponce ◽  
Maria Elena Fuentes-Montero ◽  
...  

The Cave of Swords was discovered in 1910 at Naica mine, Chihuahua, Mexico. Its name refers to the look of the 1-2 m long crystals the cave had when it was discovered. Currently the crystals are 0.1-0.3 m long. The crystals surface is opaque and ocher. For over 100 years these crystals continue to amaze and give us clues about their formation. This work is part of a research aimed at the conservation of the Naica Giant Crystals. Thirteen samples from the Cave of Swords were analyzed by Scanning Electron Microscopy with Energy Dispersive Spectroscopy (SEM-EDS), Confocal Laser Microscopy with Differential Interference Contrast (LCM-DIM) and Transmission Electron Microscopy (TEM). X-Ray Fluorescence (μ-XRF) together with X-ray Absorption Near Edge Structure (μ-XANES) and X-ray Photoelectron Spectroscopy (XPS) were employed for elemental analysis. For phase analysis, X-ray diffraction (XRD) in both symmetric and grazing incidence geometries (GI-XRD) and Micro electron diffraction at TEM were used. Impurities on crystals surfaces show a heterogeneous distribution of the present elements. The thickness of impurities ranges from 120 nm to 150 μm. The phases identified were (see figure) gypsum (1, 2, 3, 6, 9, 10, 13), hematite (4, 7, 8), sphalerite (14), chalcopyrite (11), cuprite (15), galena (5), alabandite (12), halite, fluorite and amorphous Pb and Mn oxy-hydroxides. Al, C, Ca, Cl, Cu, F, Fe, Mg, Mn, Na, O, Pb, S, Si and Zn elements were identified. A model for the origin of impurities follows: Selenite stopped growing when the solution became sub-saturated. Then, hematite was deposited as the main phase, which was dissolved or suspended in the solution. Hematite matrix served for the adsorption of other crystalline and amorphous phases. We concluded that humans have not produced the impurities, which are witnesses of the gypsum crystals formation. Acknowledgment: Stanford Synchrotron Radiation Lightsource, Harvard Museum of Natural History and CONACYT CB-183706.


Author(s):  
D. J. A. Welsh

Matroid theory was first studied by Whitney (1) as an abstract theory of linear independence in vector spaces. Recently its importance in graph theory has been noticed by Tutte (2), Edmonds (3) and Nash-Williams (4,5). Less interest has been shown in the extremely close relationship between matroids and incidence geometries. In this note we develop the more geometrical aspects of matroid theory, paying particular attention to the fundamental role of the hyperplanes of a matroid in this theory.


1998 ◽  
pp. 39-54
Author(s):  
Francis Buekenhout ◽  
Michel Dehon ◽  
Dimitri Leemans

1998 ◽  
Vol 63 (2) ◽  
pp. 485-508 ◽  
Author(s):  
Mark J. Debonis ◽  
Ali Nesin

Generalized n-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).In a simplified world, every generalized n-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a dramatic way.Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.In this article we show that the same phenomenon occurs for the generalized n-gons if n ≥ 3 is an odd integer. For each such n we construct many nonisomorphic generalized n-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] for n = 3 carry over to an arbitrary positive odd integer n (which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphs K* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property.


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