566 research outputs found
Intrinsic Volumes of the Maximal Polytope Process in Higher Dimensional STIT Tessellations
Stationary and isotropic iteration stable random tessellations are
considered, which can be constructed by a random process of cell division. The
collection of maximal polytopes at a fixed time within a convex window
is regarded and formulas for mean values, variances, as
well as a characterization of certain covariance measures are proved. The focus
is on the case , which is different from the planar one, treated
separately in \cite{ST2}. Moreover, a multivariate limit theorem for the vector
of suitably rescaled intrinsic volumes is established, leading in each
component -- in sharp contrast to the situation in the plane -- to a
non-Gaussian limit.Comment: 27 page
Simplicial Ricci Flow
We construct a discrete form of Hamilton's Ricci flow (RF) equations for a
d-dimensional piecewise flat simplicial geometry, S. These new algebraic
equations are derived using the discrete formulation of Einstein's theory of
general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation
is naturally associated to each edge, L, of a simplicial lattice. In defining
this equation, we find it convenient to utilize both the simplicial lattice, S,
and its circumcentric dual lattice, S*. In particular, the RRF equation
associated to L is naturally defined on a d-dimensional hybrid block connecting
with its (d-1)-dimensional circumcentric dual cell, L*. We show that
this equation is expressed as the proportionality between (1) the simplicial
Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume
weighted average of the fractional rate of change of the edges, lambda in L*,
of the circumcentric dual lattice, S*, that are in the dual of L. The inherent
orthogonality between elements of S and their duals in S* provide a simple
geometric representation of Hamilton's RF equations. In this paper we utilize
the well established theories of Regge calculus, or equivalently discrete
exterior calculus, to construct these equations. We solve these equations for a
few illustrative examples.Comment: 34 pages, 10 figures, minor revisions, DOI included: Commun. Math.
Phy
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