scholarly journals The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

2009 ◽  
Vol 30 (8) ◽  
pp. 1868-1886 ◽  
Author(s):  
Emeric Gioan ◽  
Michel Las Vergnas
Keyword(s):  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Bounded Region ◽  
Optimal Basis

scholarly journals Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

2021 ◽  
Author(s):  
Michael Cuntz ◽  
Sophia Elia ◽  
Jean-Philippe Labbé
Keyword(s):  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Weak Order ◽  
Polyhedral Cones ◽  
Normal Lattice ◽  
Poset Of Regions

AbstractA catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.

scholarly journals Tropical hyperplane arrangements and oriented matroids

2008 ◽  
Vol 262 (4) ◽  
pp. 795-816 ◽  
Author(s):  
Federico Ardila ◽  
Mike Develin
Keyword(s):  
Hyperplane Arrangements ◽  
Oriented Matroids

scholarly journals Equivariant Log Concavity and Representation Stability

2021 ◽  
Author(s):  
Jacob P Matherne ◽  
Dane Miyata ◽  
Nicholas Proudfoot ◽  
Eric Ramos
Keyword(s):  
Lie Algebra ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Computer Assisted ◽  
Representation Stability ◽  
Low Degree ◽  
Coxeter Arrangement

Abstract We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{s}\mathfrak{l}_n$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree.

scholarly journals Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

2021 ◽  
Vol 274 (1345) ◽  
Author(s):  
Stuart Margolis ◽  
Franco Saliola ◽  
Benjamin Steinberg
Keyword(s):  
Representation Theory ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Regular Band ◽  
Left Regular ◽  
The Face ◽  
Left Regular Band ◽  
Poset Topology ◽  
Left Regular Bands

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

scholarly journals Diameters of Cocircuit Graphs of Oriented Matroids: An Update

10.37236/9653 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ilan Adler ◽  
Jesús A. De Loera ◽  
Steven Klee ◽  
Zhenyang Zhang
Keyword(s):  
Directed Graphs ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Low Rank ◽  
Oriented Matroid ◽  
Computer Search ◽  
Linear Programs ◽  
Large Computer ◽  
Positive Side ◽  
Point Configurations

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key  role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof).  For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false. 

Prevention and Treatment of Early Childhood Caries (ECC)

2016 ◽  
Vol 1 (3) ◽  
Author(s):  
Jeffrey .
Keyword(s):  
Early Childhood ◽  
Health Problem ◽  
Early Childhood Caries ◽  
Health Providers ◽  
Missing Teeth ◽  
Optimal Basis ◽  
Major Health Problem ◽  
Major Health ◽  
Childhood Caries

Early Childhood Caries (ECC) is a chronic disease that can be prevented. It commonlyaffects children involving in one or more decayed (with lesions or not) teeth, missing teeth (dueto caries), or teeth with fillings in children aged under 71 months. The disease is sometimesoverlooked, but this condition usually affects the general health of children. Early detection ofEarly Childhood Caries (ECC) can prevent problems which are harmful to children. Therefore,the ECC must be prevented and for teeth that have had dental caries they should be givenproper treatment so as not to worsen and affect the quality of life in children. Prevention of thisdisease is a significant component in any health program to prepare for the optimal basis forthe oral health of children. This condition will become a serious health problem if not handledproperly, and it is a major health problem for health providers throughout the world.Primarypreventive must be initiated since a woman getting pregnant.Keywords: Early Childhood Caries (ECC), prevention, treatment

A study on appropriate investment of pipeline rehabilitation for water distribution network

2005 ◽  
Vol 5 (2) ◽  
pp. 31-38
Author(s):  
A. Asakura ◽  
A. Koizumi ◽  
O. Odanagi ◽  
H. Watanabe ◽  
T. Inakazu
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Water Supply ◽  
Water Distribution ◽  
Ad Hoc ◽  
Distribution Network ◽  
Water Distribution Network ◽  
Distribution Networks ◽  
Optimal Basis ◽  
Model Case

In Japan most of the water distribution networks were constructed during the 1960s to 1970s. Since these pipelines were used for a long period, pipeline rehabilitation is necessary to maintain water supply. Although investment for pipeline rehabilitation has to be planned in terms of cost-effectiveness, no standard method has been established because pipelines were replaced on emergency and ad hoc basis in the past. In this paper, a method to determine the maintenance of the water supply on an optimal basis with a fixed budget for a water distribution network is proposed. Firstly, a method to quantify the benefits of pipeline rehabilitation is examined. Secondly, two models using Integer Programming and Monte Carlo simulation to maximize the benefits of pipeline rehabilitation with limited budget were considered, and they are applied to a model case and a case study. Based on these studies, it is concluded that the Monte Carlo simulation model to calculate the appropriate investment for the pipeline rehabilitation planning is both convenient and practical.

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