Bounding the number of k-faces in arrangements of hyperplanes

Fukuda, Komei; Saito, Shigemasa; Tamura, Akihisa; Tokuyama, Takeshi

doi:10.1016/0166-218x(91)90067-7

Cited by 15 publications

(14 citation statements)

References 12 publications

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“…(This is given in terms of hyperplane arrangements and oriented matroids in Fukuda, Tamura & Tokuyama [214,213].) 7.19 For any vector configuration V = {v 1 , .…”

Section: 18mentioning

confidence: 99%

Lectures on Polytopes

Ziegler¹

1995

Graduate Texts in Mathematics

2,192

1,950

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“…(This is given in terms of hyperplane arrangements and oriented matroids in Fukuda, Tamura & Tokuyama [214,213].) 7.19 For any vector configuration V = {v 1 , .…”

Section: 18mentioning

confidence: 99%

Lectures on Polytopes

Ziegler¹

1995

Graduate Texts in Mathematics

2,192

1,950

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“…In fact, in the GSMM of B.cuenoti , every single FT has more EFMs than the whole network has FTs. This is in contrast to general hyperplane arrangements, in which there are least as many topes (sign vectors with maximal support) as vertices (sign vectors with minimal support) (Fukuda et al , 1991). We conjecture that the lower number of FTs compared to EFMs is a typical feature of GSMMs; a detailed comparison will be the scope of further work.…”

Section: Discussionmentioning

confidence: 88%

Flux tope analysis: studying the coordination of reaction directions in metabolic networks

et al. 2018

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MotivationElementary flux mode (EFM) analysis allows an unbiased description of metabolic networks in terms of minimal pathways (involving a minimal set of reactions). To date, the enumeration of EFMs is impracticable in genome-scale metabolic models. In a complementary approach, we introduce the concept of a flux tope (FT), involving a maximal set of reactions (with fixed directions), which allows one to study the coordination of reaction directions in metabolic networks and opens a new way for EFM enumeration.ResultsA FT is a (nontrivial) subset of the flux cone specified by fixing the directions of all reversible reactions. In a consistent metabolic network (without unused reactions), every FT contains a ‘maximal pathway’, carrying flux in all reactions. This decomposition of the flux cone into FTs allows the enumeration of EFMs (of individual FTs) without increasing the problem dimension by reaction splitting. To develop a mathematical framework for FT analysis, we build on the concepts of sign vectors and hyperplane arrangements. Thereby, we observe that FT analysis can be applied also to flux optimization problems involving additional (inhomogeneous) linear constraints. For the enumeration of FTs, we adapt the reverse search algorithm and provide an efficient implementation. We demonstrate that (biomass-optimal) FTs can be enumerated in genome-scale metabolic models of B.cuenoti and E.coli, and we use FTs to enumerate EFMs in models of M.genitalium and B.cuenoti.Availability and implementationThe source code is freely available at https://github.com/mpgerstl/FTA.Supplementary information Supplementary data are available at Bioinformatics online.

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“…For results concerning inequalities and unimodality of the f -vector of zonotopes, see [17,18,19]. The minimal and maximal number of connected components of arrangements has been studied by Shnurnikov [30] in the cases of Euclidean, projective and Lobachevskiȋ spaces.…”

Section: Discussionmentioning

confidence: 99%