532 research outputs found
Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings
The fundamental theorem of geometry of rectangular matrices describes the
general form of bijective maps on the space of all matrices over a
division ring which preserve adjacency in both directions. This
result proved by Hua in the nineteen forties has been recently improved in
several directions. One can study such maps without the bijectivity assumption
or one can try to get the same conclusion under the weaker assumption that
adjacency is preserved in one direction only. And the last possibility is to
study maps acting between matrix spaces of different sizes. The optimal result
would describe maps preserving adjacency in one direction only acting between
spaces of rectangular matrices of different sizes in the absence of any
regularity condition (injectivity or surjectivity).
A division ring is said to be EAS if it is not isomorphic to any proper
subring. It has been known before that it is possible to construct adjacency
preserving maps with wild behaviour on matrices over division rings that are
not EAS. For matrices over EAS division rings it has been recently proved that
adjacency preserving maps acting between matrix spaces of different sizes
satisfying a certain weak surjectivity condition are either degenerate or of
the expected simple standard form. We will remove this weak surjectivity
assumption, thus solving completely the long standing open problem of the
optimal version of Hua's theorem.Comment: 31 page
Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces
Convergence results are shown for full discretizations of quasilinear
parabolic partial differential equations on evolving surfaces. As a
semidiscretization in space the evolving surface finite element method is
considered, using a regularity result of a generalized Ritz map, optimal order
error estimates for the spatial discretization is shown. Combining this with
the stability results for Runge--Kutta and BDF time integrators, we obtain
convergence results for the fully discrete problems.Comment: -. arXiv admin note: text overlap with arXiv:1410.048
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