4,412 research outputs found
Projective geometry of Wachspress coordinates
We show that there is a unique hypersurface of minimal degree passing through
the non-faces of a polytope which is defined by a simple hyperplane
arrangement. This generalizes the construction of the adjoint curve of a
polygon by Wachspress in 1975. The defining polynomial of our adjoint
hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a
key ingredient for the definition of Wachspress coordinates, which are
barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial
also appears both in algebraic statistics, when studying the moments of uniform
probability distributions on polytopes, and in intersection theory, when
computing Segre classes of monomial schemes. We describe the Wachspress map,
the rational map defined by the Wachspress coordinates, and the Wachspress
variety, the image of this map. The inverse of the Wachspress map is the
projection from the linear span of the image of the adjoint hypersurface. To
relate adjoints of polytopes to classical adjoints of divisors in algebraic
geometry, we study irreducible hypersurfaces that have the same degree and
multiplicity along the non-faces of a polytope as its defining hyperplane
arrangement. We list all finitely many combinatorial types of polytopes in
dimensions two and three for which such irreducible hypersurfaces exist. In the
case of polygons, the general such curves< are elliptic. In the
three-dimensional case, the general such surfaces are either K3 or elliptic
Finite length spectra of random surfaces and their dependence on genus
The main goal of this article is to understand how the length spectrum of a
random surface depends on its genus. Here a random surface means a surface
obtained by randomly gluing together an even number of triangles carrying a
fixed metric.
Given suitable restrictions on the genus of the surface, we consider the
number of appearances of fixed finite sets of combinatorial types of curves. Of
any such set we determine the asymptotics of the probability distribution. It
turns out that these distributions are independent of the genus in an
appropriate sense.
As an application of our results we study the probability distribution of the
systole of random surfaces in a hyperbolic and a more general Riemannian
setting. In the hyperbolic setting we are able to determine the limit of the
probability distribution for the number of triangles tending to infinity and in
the Riemannian setting we derive bounds.Comment: 30 pages, 6 figure
Total embedding distributions of Ringel ladders
The total embedding distributions of a graph is consisted of the orientable
embeddings and non- orientable embeddings and have been know for few classes of
graphs. The genus distribution of Ringel ladders is determined in [Discrete
Mathematics 216 (2000) 235-252] by E.H. Tesar. In this paper, the explicit
formula for non-orientable embeddings of Ringel ladders is obtained
Minkowski Functional Description of Microwave Background Gaussianity
A Gaussian distribution of cosmic microwave background temperature
fluctuations is a generic prediction of inflation. Upcoming high-resolution
maps of the microwave background will allow detailed tests of Gaussianity down
to small angular scales, providing a crucial test of inflation. We propose
Minkowski functionals as a calculational tool for testing Gaussianity and
characterizing deviations from it. We review the mathematical formalism of
Minkowski functionals of random fields; for Gaussian fields the functionals can
be calculated exactly. We then apply the results to pixelized maps, giving
explicit expressions for calculating the functionals from maps as well as the
Gaussian predictions, including corrections for map boundaries, pixel noise,
and pixel size and shape. Variances of the functionals for Gaussian
distributions are derived in terms of the map correlation function.
Applications to microwave background maps are discussed.Comment: 24 pages with 2 figures. Submitted to New Astronom
Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves
We describe an algorithm for computing certain quaternionic quotients of the
Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to
obtain (conjectural) equations for the canonical embedding of Shimura curves.Comment: Accepted for publication in LMS Journal of Computation and
Mathematic
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Methods for Computing Genus Distribution Using Double-Rooted Graphs
This thesis develops general methods for computing the genus distribution of various types of graph families, using the concept of double-rooted graphs, which are defined to be graphs with two vertices designated as roots (the methods developed in this dissertation are limited to the cases where one of the two roots is restricted to be of valence two). I define partials and productions, and I use these as follows: (i) to compute the genus distribution of a graph obtained through the vertex amalgamation of a double-rooted graph with a single-rooted graph, and to show how these can be used to obtain recurrences for the genus distribution of iteratively growing infinite graph families. (ii) to compute the genus distribution of a graph obtained (a) through the operation of self-vertex-amalgamation on a double-rooted graph, and (b) through the operation of edge-addition on a double-rooted graph, and finally (iii) to develop a method to compute the recurrences for the genus distribution of the graph family generated by the Cartesian product of P3 and Pn
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