1,845 research outputs found
Theory of valuations on manifolds, II
This article is the second part in the series of articles where we are
developing theory of valuations on manifolds. Roughly speaking valuations could
be thought as finitely additive measures on a class of nice subsets of a
manifold which satisfy some additional assumptions.
The goal of this article is to introduce a notion of a smooth valuation on an
arbitrary smooth manifold and establish some of the basic properties of it.Comment: 33 pages, minor correction
Grothendieck Rings of Theories of Modules
The model-theoretic Grothendieck ring of a first order structure, as defined
by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the
definable subsets of finite powers of the structure. In this paper we compute
the Grothendieck ring, , of a right -module , where
is any unital ring. As a corollary we prove a conjecture of Prest
that is non-trivial, whenever is non-zero. The main proof uses
various techniques from the homology theory of simplicial complexes.Comment: 42 Page
Intrinsic Volumes of Random Cubical Complexes
Intrinsic volumes, which generalize both Euler characteristic and Lebesgue
volume, are important properties of -dimensional sets. A random cubical
complex is a union of unit cubes, each with vertices on a regular cubic
lattice, constructed according to some probability model. We analyze and give
exact polynomial formulae, dependent on a probability, for the expected value
and variance of the intrinsic volumes of several models of random cubical
complexes. We then prove a central limit theorem for these intrinsic volumes.
For our primary model, we also prove an interleaving theorem for the zeros of
the expected-value polynomials. The intrinsic volumes of cubical complexes are
useful for understanding the shape of random -dimensional sets and for
characterizing noise in applications.Comment: 17 pages with 7 figures; this version includes a central limit
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