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Tropical polar cones, hypergraph transversals, and mean payoff games

2011
*
Linear Algebra and its Applications
*

We characterize the extreme rays of the polar in terms of certain

doi:10.1016/j.laa.2011.02.004
fatcat:jnmingfkdbhjbj6okzxzqmtpzq
*minimal**set*covers which may be thought of as weighted generalizations of*minimal*transversals in hypergraphs. ... In particular, the polar of a*tropical*polyhedral cone represents the*set*of linear inequalities that its elements satisfy. ... It follows that*enumerating*the extreme rays of the polar is at least as hard as the well known problem of*enumerating*the*minimal*transversals of an hypergraph (Corollary 7 and Proposition 2). ...##
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Dynamic resource location with tropical algebra

2011
*
Linear Algebra and its Applications
*

The traditional dynamic resource location problem attempts to

doi:10.1016/j.laa.2010.03.028
fatcat:lh345ko6ijczpnaydbcdjxtsyi
*minimize*the cost of servicing a number of sequential requests, given foreknowledge of a limited number of requests. ... This paper presents an algebraic framework for addressing this question in general, and relates he algebraic properties of a generating*set*to questions in long-term optimizability, addressing the two ... Note that while membership of a i in the*minimal*row-*set*of A is necessary for a i ⊗ B to be in the*minimal*row-*set*of B, it is not sufficient, so the cardinality of the*minimal*row-*set*of B is less than ...##
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Enumeration of Real Conics and Maximal Configurations
[article]

2011
*
arXiv
*
pre-print

We use floor decompositions of

arXiv:1102.1834v1
fatcat:oajvynusfvhlrbmgmvchhs2fv4
*tropical*curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in ^n is maximal. ... Given a*minimal**tropical*morphism f : C → R n , the morphism π • f : C → R n−1 is not*minimal*in general. ... Then we define TC(ω) as the*set*of all*minimal*rational morphisms f : C → R n of degree d such that f (C) intersects all*tropical*linear spaces in ω. ...##
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Enumeration of real conics and maximal configurations

2013
*
Journal of the European Mathematical Society (Print)
*

We use floor decompositions of

doi:10.4171/jems/418
fatcat:j26cpnvgtzdkhjsx2swvpilft4
*tropical*curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in RP n is maximal. ...*Enumeration*of real conics and maximal configurations ... Given a*minimal**tropical*morphism f : C → R n , the morphism π • f : C → R n−1 is not*minimal*in general. ...##
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A Tropical Approach to Neural Networks with Piecewise Linear Activations
[article]

2019
*
arXiv
*
pre-print

We treat neural network layers with piecewise linear activations as

arXiv:1805.08749v2
fatcat:u3d642mdcndwnaabyxsdiiozh4
*tropical*polynomials, which generalize polynomials in the so-called (, +) or*tropical*algebra, with possibly real-valued exponents. ... Our work follows a novel path, exclusively under the lens of*tropical*geometry, which is independent of the improvements reported in (arXiv:1611.01491, arXiv:1711.02114). ...*connections*between popular neural network models and*tropical*geometry. ...##
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Tropical Open Hurwitz Numbers

2011
*
Rendiconti del Seminario Matematico della Universita di Padova
*

We give a

doi:10.4171/rsmup/125-10
fatcat:epuhusitrfbgxdw3mmv72gwhpe
*tropical*interpretation of Hurwitz numbers extending the one discovered in [CJM]. ... Hurwitz numbers are defined as the (weighted) number of ramified coverings of a compact closed oriented surface S of a given genus having a given*set*of critical values with given ramification profiles ... We denote by T the*set*of all*minimal**tropical*morphisms h : C 1 3 C such that C 1 is a*tropical*curve with boundary; h(@C 1 ) & ; h is unramified over C n ; h jh À1 (C H ) has degree d(C H ) for each ...##
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Parametric shortest-path algorithms via tropical geometry
[article]

2021
*
arXiv
*
pre-print

This is most easily expressed in terms of

arXiv:1904.01082v4
fatcat:mwt4xy4dkfd6lcmao7mf4hmun4
*tropical*geometry. Applications include shortest paths in traffic networks with variable link travel times. ... The graph G(D) contains the dual graph of the polyhedral subdivision S as a*connected*subgraph. A graph traversal*enumerates*all nodes in the*connected*component of some first node. ... The*tropical*hypersurface T (f ) is the locus where at least two terms of f are*minimal*; i.e., T (f ) "lies between" pairs of regions. ...##
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A Novel Electronic Data Collection System for Large-Scale Surveys of Neglected Tropical Diseases

2013
*
PLoS ONE
*

Large cross-sectional household surveys are common for measuring indicators of neglected

doi:10.1371/journal.pone.0074570
pmid:24066147
pmcid:PMC3774718
fatcat:7ol3hxrosrhrllye5wkha4wsme
*tropical*disease control programs. ... Data recorders felt a lack of*connection*with the interviewee during the first days using electronic devices, but preferred to collect data electronically in future surveys. ... Unique identification numbers were generated based on survey preferences*set*on the tablet in the home screen of the application and from*minimal*input of the data recorder. ...##
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Counting Algebraic Curves with Tropical Geometry
[article]

2012
*
arXiv
*
pre-print

In this survey, we give an introduction to

arXiv:1206.1925v1
fatcat:5bxic37phzatrllicqf3caaglq
*tropical*geometry techniques for algebraic curve counting problems. ... This paper is based on the author's lecture at the Workshop on*Tropical*Geometry and Integrable Systems in Glasgow, July 2011. ... Let C trop be the*set*of irreducible*tropical*plane curves of degree d and genus g through the points P R . ...##
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Enumeration of rational curves with cross-ratio constraints

2017
*
Advances in Mathematics
*

In this paper we prove the algebraic-

doi:10.1016/j.aim.2016.10.010
fatcat:wmq6trk4qvg2navifw2ujitdki
*tropical*correspondence for stable maps of rational curves with marked points to toric varieties such that the marked points are mapped to given orbits in the big torus ... This research was initiated while the author was visiting the Centre Interfacultaire Bernoulli in 2014 in the framework of the special program on*Tropical*geometry in its complex and symplectic aspects ...*Set*v := v r , and let a, b ∈ I v be the*minimal*and the maximal indices. ...##
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Tropical and Ordinary Convexity Combined
[article]

2010
*
arXiv
*
pre-print

A d-dimensional polytrope turns out to be a

arXiv:0801.4835v3
fatcat:t4dtl4p3orbdnom2xwtbvryhgq
*tropical*simplex, that is, it is the*tropical*convex hull of d+1 points. ... A polytrope is a*tropical*polytope which at the same time is convex in the ordinary sense. ... Let P be a d-polytrope in TA d with*tropical*vertices v 1 , . . . , v n . We have to show that n = d + 1. ...##
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Implementation of digital technology solutions for a lung health trial in rural Malawi

2016
*
European Respiratory Journal
*

Acknowledgements: Sincere gratitude to the entire project team from both REACH Trust (Malawi) and Liverpool School of

doi:10.1183/13993003.00045-2016
pmid:27076597
pmcid:PMC4887431
fatcat:y5ixrmdzfrb7tergbryf3ocbfq
*Tropical*Medicine (UK), with expertise in Health and Medicine, Data Management, Computer ... The digital methodology adopted was not without challenges, although these were*minimal*and readily overcome as described. ... Data were transferred to the database using a wireless internet*connection*at our site office which enabled the data management team to monitor progress, and view and map data in real time. ...##
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Computing the Vertices of Tropical Polyhedra Using Directed Hypergraphs

2012
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Discrete & Computational Geometry
*

This allows us to develop a

doi:10.1007/s00454-012-9469-6
fatcat:ky2bcc2ogvfzjgbkl562cakhxa
*tropical*analogue of the classical double description method, which computes a*minimal*internal representation (in terms of vertices) of a polyhedron defined externally (by ... We establish a characterization of the vertices of a*tropical*polyhedron defined as the intersection of finitely many half-spaces. ... The*tropical*method computes a*minimal*generating*set*of a polyhedral cone, starting from a system of*tropically*linear inequalities defining it. ...##
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Algorithms for tight spans and tropical linear spaces

2018
*
Journal of symbolic computation
*

We describe a new method for computing

doi:10.1016/j.jsc.2018.06.016
fatcat:jonawvdfjbanbgmfilp5jfyzje
*tropical*linear spaces and more general duals of polyhedral subdivisions. It is based on Ganter's algorithm (1984) for finite closure systems. ... For a vector p ∈ R n , we define the*set*(5) M p := B ∈ M v(B) − i∈B p i is*minimal*. ... The closed*sets*of a matroid are called flats. Remark 2.5. For matroids it is not necessary to check for the*minimality*of the closed*sets*N i in Algorithm 1. ...##
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Page 7355 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

Among others, we show that the l-skeleton of a k-

*set*polytope restricted to vertices corresponding to the affine k-*sets*is not always*connected*.” ... Summary: “We present two versions of an algorithm based on the reverse search technique for*enumerating*all k-*sets*of a point*set*in R’. ...
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