2,454 research outputs found
HodgeRank is the limit of Perron Rank
We study the map which takes an elementwise positive matrix to the k-th root
of the principal eigenvector of its k-th Hadamard power. We show that as
tends to 0 one recovers the row geometric mean vector and discuss the geometric
significance of this convergence. In the context of pairwise comparison
ranking, our result states that HodgeRank is the limit of Perron Rank, thereby
providing a novel mathematical link between two important pairwise ranking
methods
Enumerating Polytropes
Polytropes are both ordinary and tropical polytopes. We show that tropical
types of polytropes in are in bijection with cones of a
certain Gr\"{o}bner fan in restricted
to a small cone called the polytrope region. These in turn are indexed by
compatible sets of bipartite and triangle binomials. Geometrically, on the
polytrope region, is the refinement of two fans: the fan of
linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite
binomial fan. This gives two algorithms for enumerating tropical types of
polytropes: one via a general Gr\"obner fan software such as \textsf{gfan}, and
another via checking compatibility of systems of bipartite and triangle
binomials. We use these algorithms to compute types of full-dimensional
polytropes for , and maximal polytropes for .Comment: Improved exposition, fixed error in reporting the number maximal
polytropes for , fixed error in definition of bipartite binomial
Computation of transient viscous flows using indirect radial basis function networks
In this paper, an indirect/integrated radial-basis-function network (IRBFN) method is further developed to solve transient partial differential equations (PDEs) governing fluid flow problems. Spatial derivatives are discretized using one- and two-dimensional IRBFN
interpolation schemes, whereas temporal derivatives are
approximated using a method of lines and a finite-difference technique. In the case of moving interface problems, the IRBFN method is combined with the level set method to capture the evolution of the interface. The accuracy of the method is investigated by considering several benchmark test problems, including the classical lid-driven cavity flow. Very accurate results are achieved using relatively low numbers of data points
Antibiotics Time Machine is NP-hard
The antibiotics time machine is an optimization question posed by Mira
\latin{et al.} on the design of antibiotic treatment plans to minimize
antibiotic resistance. The problem is a variation of the Markov decision
process. These authors asked if the problem can be solved efficiently. In this
paper, we show that this problem is NP-hard in general.Comment: 5 page
Zeros of random tropical polynomials, random polytopes and stick-breaking
For , let be independent and identically
distributed random variables with distribution with support .
The number of zeros of the random tropical polynomials is also the number of faces of the lower convex
hull of the random points in . We show that this
number, , satisfies a central limit theorem when has polynomial decay
near . Specifically, if near behaves like a
distribution for some , then has the same asymptotics as the
number of renewals on the interval of a renewal process with
inter-arrival distribution . Our proof draws on connections
between random partitions, renewal theory and random polytopes. In particular,
we obtain generalizations and simple proofs of the central limit theorem for
the number of vertices of the convex hull of uniform random points in a
square. Our work leads to many open problems in stochastic tropical geometry,
the study of functionals and intersections of random tropical varieties.Comment: 22 pages, 5 figure
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