3,399 research outputs found
The complexity of normal form rewrite sequences for Associativity
The complexity of a particular term-rewrite system is considered: the rule of
associativity (x*y)*z --> x*(y*z). Algorithms and exact calculations are given
for the longest and shortest sequences of applications of --> that result in
normal form (NF). The shortest NF sequence for a term x is always n-drm(x),
where n is the number of occurrences of * in x and drm(x) is the depth of the
rightmost leaf of x. The longest NF sequence for any term is of length
n(n-1)/2.Comment: 5 page
Characteristic polynomials of supertropical matrices
Supertropical matrix theory was investigated in [6], whose terminology we
follow. In this work we investigate eigenvalues, characteristic polynomials and
coefficients of characteristic polynomials of supertropical matrices and their
powers, and obtain the analog to the basic property of matrices that any power
of an eigenvalue of a matrix is an eigenvalue of the corresponding power of the
matrix.Comment: Department of Mathematics, Bar-Ilan University, Ramat Gan 52900,
Israel. Email: [email protected]. This paper is part of the author's
Ph.D thesis, which was written at Bar-Ilan University under the supervision
of Prof. L. H. Rowe
Tropical totally positive matrices
We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of
FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author
is sported by the French Chateaubriand grant and INRIA postdoctoral
fellowshi
Seed mass diversity along resource gradients: the role of allometric growth rate and size-asymmetric competition
The large variation in seed mass among species inspired a vast array of
theoretical and empirical research attempting to explain this variation. So
far, seed mass variation was investigated by two classes of studies: one class
focuses on species varying in seed mass within communities, while the second
focuses on variation between communities, most often with respect to resource
gradients. Here, we develop a model capable of simultaneously explaining
variation in seed mass within and between communities. The model describes
resource competition (for both soil and light resources) in annual communities
and incorporates two fundamental aspects: light asymmetry (higher light
acquisition per unit biomass for larger individuals) and growth allometry
(negative dependency of relative growth rate on plant biomass). Results show
that both factors are critical in determining patterns of seed mass variation.
In general, growth allometry increases the reproductive success of small-seeded
species while light asymmetry increases the reproductive success of
large-seeded species. Increasing availability of soil resources increases light
competition, thereby increasing the reproductive success of large-seeded
species and ultimately the community (weighted) mean seed mass. An unexpected
prediction of the model is that maximum variation in community seed mass (a
measure of functional diversity) occurs under intermediate levels of soil
resources. Extensions of the model incorporating size-dependent seed survival
and disturbance also show patterns consistent with empirical observations.
These overall results suggest that the mechanisms captured by the model are
important in determining patterns of species and functional diversity
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