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Tropical polar cones, hypergraph transversals, and mean payoff games

Xavier Allamigeon, Stéphane Gaubert, Ricardo D. Katz
2011 Linear Algebra and its Applications  
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.  ...  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).  ... 
doi:10.1016/j.laa.2011.02.004 fatcat:jnmingfkdbhjbj6okzxzqmtpzq

Dynamic resource location with tropical algebra

D. Jacob Wildstrom
2011 Linear Algebra and its Applications  
The traditional dynamic resource location problem attempts to 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  ... 
doi:10.1016/j.laa.2010.03.028 fatcat:lh345ko6ijczpnaydbcdjxtsyi

Enumeration of Real Conics and Maximal Configurations [article]

Erwan Brugallé, Nicolas Puignau
2011 arXiv   pre-print
We use floor decompositions of 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.  ... 
arXiv:1102.1834v1 fatcat:oajvynusfvhlrbmgmvchhs2fv4

Enumeration of real conics and maximal configurations

Erwan Brugallé, Nicolas Puignau
2013 Journal of the European Mathematical Society (Print)  
We use floor decompositions of 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.  ... 
doi:10.4171/jems/418 fatcat:j26cpnvgtzdkhjsx2swvpilft4

A Tropical Approach to Neural Networks with Piecewise Linear Activations [article]

Vasileios Charisopoulos, Petros Maragos
2019 arXiv   pre-print
We treat neural network layers with piecewise linear activations as 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.  ... 
arXiv:1805.08749v2 fatcat:u3d642mdcndwnaabyxsdiiozh4

Tropical Open Hurwitz Numbers

Benoît Bertrand, Erwan Brugallé, Grigory Mikhalkin
2011 Rendiconti del Seminario Matematico della Universita di Padova  
We give a 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  ... 
doi:10.4171/rsmup/125-10 fatcat:epuhusitrfbgxdw3mmv72gwhpe

Parametric shortest-path algorithms via tropical geometry [article]

Michael Joswig, Benjamin Schröter
2021 arXiv   pre-print
This is most easily expressed in terms of 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.  ... 
arXiv:1904.01082v4 fatcat:mwt4xy4dkfd6lcmao7mf4hmun4

A Novel Electronic Data Collection System for Large-Scale Surveys of Neglected Tropical Diseases

Jonathan D. King, Joy Buolamwini, Elizabeth A. Cromwell, Andrew Panfel, Tesfaye Teferi, Mulat Zerihun, Berhanu Melak, Jessica Watson, Zerihun Tadesse, Danielle Vienneau, Jeremiah Ngondi, Jürg Utzinger (+3 others)
2013 PLoS ONE  
Large cross-sectional household surveys are common for measuring indicators of neglected 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.  ... 
doi:10.1371/journal.pone.0074570 pmid:24066147 pmcid:PMC3774718 fatcat:7ol3hxrosrhrllye5wkha4wsme

Counting Algebraic Curves with Tropical Geometry [article]

Florian Block
2012 arXiv   pre-print
In this survey, we give an introduction to 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 .  ... 
arXiv:1206.1925v1 fatcat:5bxic37phzatrllicqf3caaglq

Enumeration of rational curves with cross-ratio constraints

Ilya Tyomkin
2017 Advances in Mathematics  
In this paper we prove the algebraic-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.  ... 
doi:10.1016/j.aim.2016.10.010 fatcat:wmq6trk4qvg2navifw2ujitdki

Tropical and Ordinary Convexity Combined [article]

Michael Joswig, Katja Kulas
2010 arXiv   pre-print
A d-dimensional polytrope turns out to be a 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.  ... 
arXiv:0801.4835v3 fatcat:t4dtl4p3orbdnom2xwtbvryhgq

Implementation of digital technology solutions for a lung health trial in rural Malawi

Blessings Chisunkha, Hastings Banda, Rachael Thomson, S. Bertel Squire, Kevin Mortimer
2016 European Respiratory Journal  
Acknowledgements: Sincere gratitude to the entire project team from both REACH Trust (Malawi) and Liverpool School of 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.  ... 
doi:10.1183/13993003.00045-2016 pmid:27076597 pmcid:PMC4887431 fatcat:y5ixrmdzfrb7tergbryf3ocbfq

Computing the Vertices of Tropical Polyhedra Using Directed Hypergraphs

Xavier Allamigeon, Stéphane Gaubert, Éric Goubault
2012 Discrete & Computational Geometry  
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  ...  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.  ... 
doi:10.1007/s00454-012-9469-6 fatcat:ky2bcc2ogvfzjgbkl562cakhxa

Algorithms for tight spans and tropical linear spaces

Simon Hampe, Michael Joswig, Benjamin Schröter
2018 Journal of symbolic computation  
We describe a new method for computing 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.  ... 
doi:10.1016/j.jsc.2018.06.016 fatcat:jonawvdfjbanbgmfilp5jfyzje

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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