4,142 research outputs found
On the variety of four dimensional lie algebras
Lie algebras of dimension are defined by their structure constants ,
which can be seen as sets of scalars (if we take into
account the skew-symmetry condition) to which the Jacobi identity imposes
certain quadratic conditions. Up to rescaling, we can consider such a set as a
point in the projective space . Suppose , hence . Take
a random subspace of dimension in , over the complex numbers. We
prove that this subspace will contain exactly points giving the
structure constants of some four dimensional Lie algebras. Among those,
will be isomorphic to , will be the sum of two copies of the Lie
algebra of one dimensional affine transformations, will have an abelian,
three-dimensional derived algebra, and will have for derived algebra the
three dimensional Heisenberg algebra. This answers a question of Kirillov and
Neretin.Comment: To appear in Journal of Lie Theor
Typical Geometry, Second-Order Properties and Central Limit Theory for Iteration Stable Tessellations
Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. The purpose of this paper is to
describe large scale asymptotic geometry of STIT tessellations in
and more generally that of non-stationary iteration infinitely
divisible tessellations. We study several aspects of the typical first-order
geometry of such tessellations resorting to martingale techniques as providing
a direct link between the typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations. Further, we
also consider second-order properties of STIT and iteration infinitely
divisible tessellations, such as the variance of the total surface area of cell
boundaries inside a convex observation window. Our techniques, relying on
martingale theory and tools from integral geometry, allow us to give explicit
and asymptotic formulae. Based on these results, we establish a functional
central limit theorem for the length/surface increment processes induced by
STIT tessellations. We conclude a central limit theorem for total edge
length/facet surface, with normal limit distribution in the planar case and
non-normal ones in all higher dimensions.Comment: 51 page
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