The tropical double description method

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characterization of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redundant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on Theoretical Aspects of Computer Science, 2010, Nancy, Franc

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

A-Tint: A polymake extension for algorithmic tropical intersection theory

In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.Comment: 32 pages, 5 figures, 4 tables. Second version: Revised version, to be published in European Journal of Combinatoric

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.Comment: 27 pages, 6 figures, revised versio

Influence of Doubled CO2 on Ozone via Changes in the Brewer–Dobson Circulation

In this short note, the effect of enhanced circulation due to doubling CO2 on ozone is investigated. The difference of Brewer–Dobson circulation (BDC) between the doubled CO2 and control run from an idealized atmospheric general circulation model is added to the BDC climatology derived from National Centers for Environmental Prediction—Department of Energy Reanalysis 2 (NCEP2) from 1979 to 2002. Then it is used to drive the California Institute of Technology/Jet Propulsion Laboratory (Caltech/JPL) two-dimensional chemistry and transport model. The results reveal that the total ozone increases by 7 and 3.5 Dobson units (DU) in the high latitudes of the Northern and Southern Hemispheres, respectively, and decreases by 4 DU in the Tropics as a result of the increase in BDC associated with doubled CO2. If the change of eddy mixing coefficients after doubling CO2 is also considered, the total ozone will increase by 6.5 and 3 DU in the high latitudes of the Northern and Southern Hemispheres after combining both effects from the change in BDC and eddy mixing coefficients

Studies on Barbula consanguinea (Thw. & Mitt.) Jaeg. sensu Eddy : a pan-tropical species

Barbula consanguinea (Thw. & Mitt.) Jaeg. sensu Eddy is considered as a pan-tropical species. This taxon is reported new for several central African countries, the Arabian peninsula, and Middle America. This variable species is shortly discussed and compared with (most) related taxa. Hymenostylium crispulum Broth. & Par. and Barbula obscura Sull. (= Barbula wrightii Sauerb.) are considered as synonyms of Barbula consanguinea

A seed-diffusion model for tropical tree diversity patterns

Diversity patterns of tree species in a tropical forest community are approached by a simple lattice model and investigated by Monte Carlo simulations using a backtracking method. Our spatially explicit neutral model is based on a simple statistical physics process, namely the diffusion of seeds. The model has three parameters: the speciation rate, the size of the meta-community in which the studied tree-community is embedded, and the average surviving time of the seeds. By extensive computer simulations we aim the reproduction of relevant statistical measures derived from the experimental data of the Barro Colorado Island tree census in year 1995. The first two parameters of the model are fixed to known values, characteristic of the studied community, thus obtaining a model with only one freely adjustable parameter. As a result of this, the average number of species in the considered territory, the relative species abundance distribution, the species-area relationship and the spatial auto-correlation function of the individuals in abundant species are simultaneously fitted with only one parameter which is the average surviving time of the seeds.Comment: 12 pages, 5 figure

Logarithmic Gromov-Witten invariants

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li, and completes a program first proposed by the second named author in 2002. One considers target spaces X carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on X is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that X is log smooth, one obtains a theory of log Gromov-Witten invariants.Comment: 58 pages, 5 figure