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