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 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 Immunocytochemical detection of regional protein changes in rat brain sections using computer-assisted image analysis.

1996 ◽  
Vol 44 (9) ◽  
pp. 981-987 ◽  
Author(s):  
X Huang ◽  
S Chen ◽  
E I Tietz
Keyword(s):  
Image Analysis ◽  
Rat Brain ◽  
Immunohistochemical Analysis ◽  
Brain Homogenate ◽  
Model System ◽  
Computer Assisted ◽  
Antigen Concentration ◽  
Immunocytochemical Detection ◽  
Low Degree ◽  
Computer Assisted Image

We used several approaches to assess the reliability and sensitivity of computer-assisted densitometry to detect regional changes in tissue antigen content as a function of immunohistochemical staining density. We designed a model system to mimic variations in antigen concentration in postfixed, slide-mounted rat brain sections by varying the ratios of conjugated (biotinylated) to unconjugated secondary antibody. Antigen concentration was also varied in tissue discs made from mixing rat brain homogenate with increasing amounts of tissue embedding compound. The monoclonal antibody bd-17 to the beta2/3 subunit of the GABAA receptor was used as the primary antibody. Immunostaining density was visualized with diaminobenzidine (DAB). There was a significant, positive linear relationship (r = 0.97-0.99) between immunostaining intensity and antigen concentration. With this approach, changes in antigen content of less than 10%, as reflected in immunostaining intensity, were detectable in brain sections. The low degree of variability in measures of regional variation in immunostaining in sections from naive rats (n = 7) suggested that the method was suitable for quantitative analysis and indicated the reliability of the method. This systematic study of the utility of computer-assisted image analysis for semiquantitative immunohistochemical analysis found the method to be both reliable and sensitive.

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. 

scholarly journals The Varchenko determinant of a Coxeter arrangement

2018 ◽  
Vol 21 (4) ◽  
pp. 651-665 ◽  
Author(s):  
Götz Pfeiffer ◽  
Hery Randriamaro
Keyword(s):  
Explicit Formula ◽  
Coxeter Groups ◽  
Hyperplane Arrangements ◽  
Arrangement Of Hyperplanes ◽  
Coxeter Arrangement

AbstractThe Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula for this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their associated determinants.

3-D organization of fibrinogen receptors using computer reconstruction and whole-mount microscopy

1988 ◽  
Vol 46 ◽  
pp. 30-31
Author(s):  
E. T. O'Toole ◽  
R. R. Hantgan ◽  
J. C. Lewis
Keyword(s):  
Whole Blood ◽  
Colloidal Gold ◽  
Blood Platelets ◽  
Cell Contact ◽  
Computer Assisted ◽  
Adherent Cells ◽  
Differential Centrifugation ◽  
Computer Reconstruction ◽  
Sds Page ◽  
Whole Mount

Thrombocytes (TC), the avian equivalent of blood platelets, support hemostasis by aggregating at sites of injury. Studies in our lab suggested that fibrinogen (fib) is a requisite cofactor for TC aggregation but operates by an undefined mechanism. To study the interaction of fib with TC and to identify fib receptors on cells, fib was purified from pigeon plasma, conjugated to colloidal gold and used both to facilitate aggregation and as a receptor probe. Described is the application of computer assisted reconstruction and stereo whole mount microscopy to visualize the 3-D organization of fib receptors at sites of cell contact in TC aggregates and on adherent cells.Pigeon TC were obtained from citrated whole blood by differential centrifugation, washed with Ca++ free Hank's balanced salts containing 0.3% EDTA (pH 6.5) and resuspended in Ca++ free Hank's. Pigeon fib was isolated by precipitation with PEG-1000 and the purity assessed by SDS-PAGE. Fib was conjugated to 25nm colloidal gold by vortexing and the conjugates used as the ligand to identify fib receptors.

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