Burnside-type problems in discrete geometry

2019 ◽  
Vol 29 (6) ◽  
pp. 357-362
Author(s):  
Leonid V. Kuz’min

Abstract The paper is concerned with systems of incidence involving a space of points X and lines consisting of q points each. A free space X is defined. For a space X an analogue of the Burnside problem (solved in the negative) and an analogue of the weakened Burnside problem are formulated. In the case q = 3 the positive answer to the analogue of the weakened Burnside problem is equivalent to the existence of a universal finite geometry.

1997 ◽  
Vol 161 ◽  
pp. 203-218 ◽  
Author(s):  
Tobias C. Owen

AbstractThe clear evidence of water erosion on the surface of Mars suggests an early climate much more clement than the present one. Using a model for the origin of inner planet atmospheres by icy planetesimal impact, it is possible to reconstruct the original volatile inventory on Mars, starting from the thin atmosphere we observe today. Evidence for cometary impact can be found in the present abundances and isotope ratios of gases in the atmosphere and in SNC meteorites. If we invoke impact erosion to account for the present excess of129Xe, we predict an early inventory equivalent to at least 7.5 bars of CO2. This reservoir of volatiles is adequate to produce a substantial greenhouse effect, provided there is some small addition of SO2(volcanoes) or reduced gases (cometary impact). Thus it seems likely that conditions on early Mars were suitable for the origin of life – biogenic elements and liquid water were present at favorable conditions of pressure and temperature. Whether life began on Mars remains an open question, receiving hints of a positive answer from recent work on one of the Martian meteorites. The implications for habitable zones around other stars include the need to have rocky planets with sufficient mass to preserve atmospheres in the face of intensive early bombardment.


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