114 research outputs found
Generalization of the Apollonius theorem for simplices and related problems
The Apollonius theorem gives the length of a median of a triangle in terms of
the lengths of its sides. The straightforward generalization of this theorem
obtained for m-simplices in the n-dimensional Euclidean space for n greater
than or equal to m is given. Based on this, generalizations of properties
related to the medians of a triangle are presented. In addition, applications
of the generalized Apollonius' theorem and the related to the medians results,
are given for obtaining: (a) the minimal spherical surface that encloses a
given simplex or a given bounded set, (b) the thickness of a simplex that it
provides a measure for the quality or how well shaped a simplex is, and (c) the
convergence and error estimates of the root-finding bisection method applied on
simplices
The hamburger theorem
We generalize the ham sandwich theorem to measures in as
follows. Let be absolutely continuous finite
Borel measures on . Let for , and assume that . Assume that for every . Then there
exists a hyperplane such that each open halfspace defined by
satisfies for every
and . As a
consequence we obtain that every -colored set of points in
such that no color is used for more than points can be
partitioned into disjoint rainbow -dimensional simplices.Comment: 11 pages, 2 figures; a new proof of Theorem 8, extended concluding
remark
Jensen's inequality for the Tukey median
--Shape analysis,spherical harmonic descriptors,optimal designs,mean square error,3D-image data,minimax optimal designs,robust designs,dependent data
The space of ultrametric phylogenetic trees
The reliability of a phylogenetic inference method from genomic sequence data
is ensured by its statistical consistency. Bayesian inference methods produce a
sample of phylogenetic trees from the posterior distribution given sequence
data. Hence the question of statistical consistency of such methods is
equivalent to the consistency of the summary of the sample. More generally,
statistical consistency is ensured by the tree space used to analyse the
sample.
In this paper, we consider two standard parameterisations of phylogenetic
time-trees used in evolutionary models: inter-coalescent interval lengths and
absolute times of divergence events. For each of these parameterisations we
introduce a natural metric space on ultrametric phylogenetic trees. We compare
the introduced spaces with existing models of tree space and formulate several
formal requirements that a metric space on phylogenetic trees must possess in
order to be a satisfactory space for statistical analysis, and justify them. We
show that only a few known constructions of the space of phylogenetic trees
satisfy these requirements. However, our results suggest that these basic
requirements are not enough to distinguish between the two metric spaces we
introduce and that the choice between metric spaces requires additional
properties to be considered. Particularly, that the summary tree minimising the
square distance to the trees from the sample might be different for different
parameterisations. This suggests that further fundamental insight is needed
into the problem of statistical consistency of phylogenetic inference methods.Comment: Minor changes. This version has been published in JTB. 27 pages, 9
figure
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Jensen's inequality for the Tukey median
Jensen's inequality states for a random variable X with values in Rd and existing
expectation and for any convex function f : R^d -> R, that f(E(X)) <= E(f(X)).
We prove an analogous inequality, where the expectation operator is replaced by
the halfspace-median-operator (or Tukey-median-operator)
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