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Distances on the tropical line determined by two points
Let . Write if is a multiple of
. Two different points and in uniquely
determine a tropical line , passing through them, and stable under
small perturbations. This line is a balanced unrooted semi--labeled tree on
leaves. It is also a metric graph.
If some representatives and of and are the first and second
columns of some real normal idempotent order matrix , we prove that the
tree is described by a matrix , easily obtained from . We also
prove that is caterpillar. We prove that every vertex in
belongs to the tropical linear segment joining and . A vertex, denoted
, closest (w.r.t tropical distance) to exists in . Same for
. The distances between pairs of adjacent vertices in and the
distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the
matrix . In addition, if and are generic, then the tree
is trivalent. The entries of are differences (i.e., sum of principal
diagonal minus sum of secondary diagonal) of order 2 minors of the first two
columns of .Comment: New corrected version. 31 pages and 9 figures. The main result is
theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7
was obtained with A. Jim\'enez; see Arxiv 1205.416
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Modelling long-run trends and cycles in financial time series data
Copyright @ 2012 Wiley Publishing Ltd. This is the accepted version of the following article: "Modelling long-run trends and cycles in financial time series data", Journal of Time Series Analysis, 34(3), 405-421, 2013, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12010/abstract.This article proposes a general time series framework to capture the long-run behaviour of financial series. The suggested approach includes linear and segmented time trends, and stationary and non-stationary processes based on integer and/or fractional degrees of differentiation. Moreover, the spectrum is allowed to contain more than a single pole or singularity, occurring at both zero but non-zero (cyclical) frequencies. This framework is used to analyse five annual time series with a long span, namely dividends, earnings, interest rates, stock prices and long-term government bond yields. The results based on several likelihood criteria indicate that the five series exhibit fractional integration with one or two poles in the spectrum, and are quite stable over the sample period examined.Ministerio de Ciencia y Tecnologia and Jeronimo de Ayanz project of the Government of Navarra
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