220 research outputs found
A small frame and a certificate of its injectivity
We present a complex frame of eleven vectors in 4-space and prove that it
defines injective measurements. That is, any rank-one Hermitian
matrix is uniquely determined by its values as a Hermitian form on this
collection of eleven vectors. This disproves a recent conjecture of Bandeira,
Cahill, Mixon, and Nelson. We use algebraic computations and certificates in
order to prove injectivity.Comment: 4 pages, 3 figure
Semi-inverted linear spaces and an analogue of the broken circuit complex
The image of a linear space under inversion of some coordinates is an affine
variety whose structure is governed by an underlying hyperplane arrangement. In
this paper, we generalize work by Proudfoot and Speyer to show that circuit
polynomials form a universal Groebner basis for the ideal of polynomials
vanishing on this variety. The proof relies on degenerations to the
Stanley-Reisner ideal of a simplicial complex determined by the underlying
matroid. If the linear space is real, then the semi-inverted linear space is
also an example of a hyperbolic variety, meaning that all of its intersection
points with a large family of linear spaces are real.Comment: 16 pages, 1 figure, minor revisions and added connections to the
external activity complex of a matroi
MAXIMUM-LIKELIHOOD ESTIMATES OF RACEHORSE EARNINGS AND PROFITABILITY
Thoroughbred racehorses are commonly characterized as unprofitable investments. Previous studies, grouping all racehorses together, estimate that over 80% of all racehorses in training fail to earn enough to recover the variable costs of training. However, these studies are not truly representative, because they fail to account for a number of factors affecting profitability. This study estimates expected purse earnings and profitability of claiming horses in Kentucky. Maximum-likelihood estimates of probability distribution parameters show that expected purse earnings follow an exponential distribution with a mean of 4,824. Of the 305 claims analyzed for profitability, 61% were profitable. The results indicate substantial financial risk associated with claiming race horses, but conclude that there are positive economic returns on average.claiming horses, financial risk, maximum likelihood, probability, profitability, thoroughbred, Agribusiness,
Edges of the Barvinok-Novik orbitope
Here we study the k^th symmetric trigonometric moment curve and its convex
hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these
objects and show that there is some threshold so that for two points on S^1
with arclength below this threshold, the line segment between their lifts on
the curve form an edge on the Barvinok-Novik orbitope and for points with
arclenth above this threshold, their lifts do not form an edge. They also give
a lower bound for this threshold and conjecture that this bound is tight.
Results of Smilansky prove tightness for k=2. Here we prove this conjecture for
all k.Comment: 10 pages, 3 figures, corrected Lemma 4 and other minor revision
Gap distributions of Fourier quasicrystals via Lee-Yang polynomials
Recent work of Kurasov and Sarnak provides a method for constructing
one-dimensional Fourier quasicrystals (FQ) from the torus zero sets of a
special class of multivariate polynomials called Lee-Yang polynomials. In
particular, they provided a non-periodic FQ with unit coefficients and
uniformly discrete support, answering an open question posed by Meyer. Their
method was later shown to generate all one-dimensional Fourier quasicrystals
with -valued coefficients (-FQ).
In this paper, we characterize which Lee-Yang polynomials give rise to
non-periodic -FQs with unit coefficients and uniformly discrete
support, and show that this property is generic among Lee-Yang polynomials. We
also show that the infinite sequence of gaps between consecutive atoms of any
-FQ has a well-defined distribution, which, under mild conditions,
is absolutely continuous. This generalizes previously known results for the
spectra of quantum graphs to arbitrary -FQs. Two extreme examples
are presented: first, a sequence of -FQs whose gap distributions
converge to a Poisson distribution. Second, a sequence of random Lee-Yang
polynomials that results in random -FQs whose empirical gap
distributions converge to that of a random unitary matrix (CUE)
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