321 research outputs found
On Partitioning Colored Points
P. Kirchberger proved that, for a finite subset of such
that each point in is painted with one of two colors, if every or
fewer points in can be separated along the colors, then all the points in
can be separated along the colors. In this paper, we show a more colorful
theorem
Shape of matchbox manifolds
In this work, we develop shape expansions of minimal matchbox manifolds
without holonomy, in terms of branched manifolds formed from their leaves. Our
approach is based on the method of coding the holonomy groups for the foliated
spaces, to define leafwise regions which are transversely stable and are
adapted to the foliation dynamics. Approximations are obtained by collapsing
appropriately chosen neighborhoods onto these regions along a "transverse
Cantor foliation". The existence of the "transverse Cantor foliation" allows us
to generalize standard techniques known for Euclidean and fibered cases to
arbitrary matchbox manifolds with Riemannian leaf geometry and without
holonomy. The transverse Cantor foliations used here are constructed by purely
intrinsic and topological means, as we do not assume that our matchbox
manifolds are embedded into a smooth foliated manifold, or a smooth manifold.Comment: 36 pages. Revision of the earlier version: introduction is rewritten.
Accepted to a special issue of Indagationes Mathematica
On the combinatorics of plethysm
AbstractWe construct three (large, reduced) incidence algebras whose semigroups of multiplicative functions, under convolution, are anti-isomorphic, respectively, to the semigroups of what we call partitional, permutational and exponential formal power series without constant term, in infinitely many variables x = (x1, x2,…), under plethysm. We compute the Möbius function in each case. These three incidence algebras are the linear duals of incidence bialgebras arising, respectively, from the classes of transversals of partitions (with an order that we define), partitions compatible with permutations (with the usual refinement order), and linear transversals of linear partitions (with the order induced by that on transversals). We define notions of morphisms between partitions, permutations and linear partitions, respectively, whose kernels are defined to be, in each case, transversals, compatible partitions and linear transversals. We introduce, in each case, a pair of sequences of polynomials in x of binomial type, counting morphisms and monomorphisms, and obtain expressions for their connection constants, by summation and Möbius inversion over the corresponding posets of kernels
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