369 research outputs found
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
Topics in algorithmic, enumerative and geometric combinatorics
This thesis presents five papers, studying enumerative and
extremal problems on combinatorial structures.
The first paper studies Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S_2xS_{n-2}-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties.
In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning
technique, the number of nonempty faces is counted, and in particular we confirm
Kalai's 3^d-conjecture for such polytopes. Furthermore, a characterization of
exactly which Hansen polytopes are also Hanner polytopes is given. We end by
constructing an interesting class of Hansen polytopes having very few faces and
yet not being Hanner.
The third paper studies the problem of packing a pattern as densely as possible into compositions. We are able to find the
packing density for some classes of generalized patterns, including all the three letter patterns.
In the fourth paper, we present combinatorial proofs of the enumeration of derangements with descents in prescribed positions. To this end, we
consider fixed point lambda-coloured permutations, which are easily
enumerated. Several formulae regarding these numbers are given, as
well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, the event that pi has descents in a set S of positions is positively correlated with the event that pi is a derangement, if pi is chosen uniformly in S_n.
The fifth paper solves a partially ordered generalization of the famous secretary problem. The elements of a finite nonempty partially ordered set are exposed in uniform random order to a selector who, at any given time, can see the relative order of the exposed elements. The selector's task is to choose online a maximal element. We describe a strategy for the general problem that achieves success probability at least 1/e for an arbitrary partial order, thus proving that the linearly ordered set is at least as difficult as any other instance of the problem. To this end, we define a probability measure on the maximal elements of an arbitrary partially ordered set, that may be interesting in its own right
An Implicitization Challenge for Binary Factor Analysis
We use tropical geometry to compute the multidegree and Newton polytope of
the hypersurface of a statistical model with two hidden and four observed
binary random variables, solving an open question stated by Drton, Sturmfels
and Sullivant in "Lectures on Algebraic Statistics" (Problem 7.7). The model is
obtained from the undirected graphical model of the complete bipartite graph
by marginalizing two of the six binary random variables. We present
algorithms for computing the Newton polytope of its defining equation by
parallel walks along the polytope and its normal fan. In this way we compute
vertices of the polytope. Finally, we also compute and certify its facets by
studying tangent cones of the polytope at the symmetry classes vertices. The
Newton polytope has 17214912 vertices in 44938 symmetry classes and 70646
facets in 246 symmetry classes.Comment: 25 pages, 5 figures, presented at Mega 09 (Barcelona, Spain
Equivariant Ehrhart theory
Motivated by representation theory and geometry, we introduce and develop an
equivariant generalization of Ehrhart theory, the study of lattice points in
dilations of lattice polytopes. We prove representation-theoretic analogues of
numerous classical results, and give applications to the Ehrhart theory of
rational polytopes and centrally symmetric polytopes. We also recover a
character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of
a Weyl group on the cohomology of a toric variety associated to a root system.Comment: 40 pages. Final version. To appear in Adv. Mat
Chiral extensions of chiral polytopes
Given a chiral d-polytope K with regular facets, we describe a construction
for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is
finite whenever K is finite. We provide explicit examples of chiral 4-polytopes
constructed in this way from chiral toroidal maps.Comment: 21 pages. [v2] includes several minor revisions for clarit
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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