1,092 research outputs found
Distributions of points in the unit square and large k-gons
AbstractWe consider a generalization of Heilbronn’s triangle problem by asking, given any integers n≥k, for the supremum Δk(n) of the minimum area determined by the convex hull of some k of n points in the unit square [0,1]2, where the supremum is taken over all distributions of n points in [0,1]2. Improving the lower bound Δk(n)=Ω(1/n(k−1)/(k−2)) from [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, SIAM Journal on Computing 30 (2000) 383–390] and from [W.M. Schmidt, On a problem of Heilbronn, Journal of the London Mathematical Society (2) 4 (1972) 545–550] for k=4, we show that Δk(n)=Ω((logn)1/(k−2)/n(k−1)/(k−2)) for fixed integers k≥3 as asked for in [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, SIAM Journal on Computing 30 (2000) 383–390]. Moreover, we provide a deterministic polynomial time algorithm which finds n points in [0,1]2, which achieve this lower bound on Δk(n)
Probability Theory of Random Polygons from the Quaternionic Viewpoint
We build a new probability measure on closed space and plane polygons. The
key construction is a map, given by Knutson and Hausmann using the Hopf map on
quaternions, from the complex Stiefel manifold of 2-frames in n-space to the
space of closed n-gons in 3-space of total length 2. Our probability measure on
polygon space is defined by pushing forward Haar measure on the Stiefel
manifold by this map. A similar construction yields a probability measure on
plane polygons which comes from a real Stiefel manifold.
The edgelengths of polygons sampled according to our measures obey beta
distributions. This makes our polygon measures different from those usually
studied, which have Gaussian or fixed edgelengths. One advantage of our
measures is that we can explicitly compute expectations and moments for
chordlengths and radii of gyration. Another is that direct sampling according
to our measures is fast (linear in the number of edges) and easy to code.
Some of our methods will be of independent interest in studying other
probability measures on polygon spaces. We define an edge set ensemble (ESE) to
be the set of polygons created by rearranging a given set of n edges. A key
theorem gives a formula for the average over an ESE of the squared lengths of
chords skipping k vertices in terms of k, n, and the edgelengths of the
ensemble. This allows one to easily compute expected values of squared
chordlengths and radii of gyration for any probability measure on polygon space
invariant under rearrangements of edges.Comment: Some small typos fixed, added a calculation for the covariance of
edgelengths, added pseudocode for the random polygon sampling algorithm. To
appear in Communications on Pure and Applied Mathematics (CPAM
Local geometry of random geodesics on negatively curved surfaces
It is shown that the tessellation of a compact, negatively curved surface
induced by a typical long geodesic segment, when properly scaled, looks locally
like a Poisson line process. This implies that the global statistics of the
tessellation -- for instance, the fraction of triangles -- approach those of
the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces
with possibly variable negative curvatur
Resolvents of R-Diagonal Operators
We consider the resolvent of any -diagonal operator
in a -factor. Our main theorem gives a universal asymptotic
formula for the norm of such a resolvent. En route to its proof, we calculate
the -transform of the operator where is Voiculescu's
circular operator, and give an asymptotic formula for the negative moments of
for any -diagonal . We use a mixture of complex analytic
and combinatorial techniques, each giving finer information where the other can
give only coarse detail. In particular, we introduce {\em partition structure
diagrams}, a new combinatorial structure arising in free probability.Comment: 29 pages, 12 figures, used gastex.st
Geometry of Log-Concave Density Estimation
Shape-constrained density estimation is an important topic in mathematical
statistics. We focus on densities on that are log-concave, and
we study geometric properties of the maximum likelihood estimator (MLE) for
weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the
optimal log-concave density is piecewise linear and supported on a regular
subdivision of the samples. This defines a map from the space of weights to the
set of regular subdivisions of the samples, i.e. the face poset of their
secondary polytope. We prove that this map is surjective. In fact, every
regular subdivision arises in the MLE for some set of weights with positive
probability, but coarser subdivisions appear to be more likely to arise than
finer ones. To quantify these results, we introduce a continuous version of the
secondary polytope, whose dual we name the Samworth body. This article
establishes a new link between geometric combinatorics and nonparametric
statistics, and it suggests numerous open problems.Comment: 22 pages, 3 figure
Transition from damage to fragmentation in collision of solids
We investigate fracture and fragmentation of solids due to impact at low
energies using a two-dimensional dynamical model of granular solids. Simulating
collisions of two solid discs we show that, depending on the initial energy,
the outcome of a collision process can be classified into two states: a damaged
and a fragmented state with a sharp transition in between. We give numerical
evidence that the transition point between the two states behaves as a critical
point, and we discuss the possible mechanism of the transition.Comment: Revtex, 12 figures included. accepted by Phys. Rev.
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