scholarly journals On the Excluded Minors for Matroids of Branch-Width Three

10.37236/1648 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Petr Hliněný
Keyword(s):  
Computer Assisted ◽  
Binary Matroids ◽  
Alternative Characterization ◽  
Excluded Minor ◽  
Practical Algorithm ◽  
Branch Width ◽  
A Minor ◽  
Excluded Minors

Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if $M$ is an excluded minor for matroids of branch-width $3$, then $ M$ has at most $14$ elements. We show that there are exactly $10$ such binary matroids $M$ (7 of which are regular), proving a conjecture formulated by Dharmatilake in 1994. We also construct numbers of such ternary and quaternary matroids $ M$, and provide a simple practical algorithm for finding a width-$3$ branch-decomposition of a matroid. The arguments in our paper are computer-assisted — we use a program $MACEK$ [P. Hliněný, The MACEK Program, http://www.mcs.vuw.ac.nz/research/macek, 2002] for structural computations with represented matroids. Unfortunately, it seems to be infeasible to search through all matroids on at most $14$ elements.

scholarly journals K-Regular Matroids

2021 ◽  
Author(s):  
◽  
Charles A Semple
Keyword(s):  
Finite Number ◽  
Prime Power ◽  
Negative Integer ◽  
Regular Matroid ◽  
Excluded Minor ◽  
Excluded Minors

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

scholarly journals K-Regular Matroids

2021 ◽  
Author(s):  
◽  
Charles A Semple
Keyword(s):  
Finite Number ◽  
Prime Power ◽  
Negative Integer ◽  
Regular Matroid ◽  
Excluded Minor ◽  
Excluded Minors

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

scholarly journals Excluding Kuratowski graphs and their duals from binary matroids

2020 ◽  
Author(s):  
Dillon Mayhew ◽  
G Royle ◽  
Geoffrey Whittle
Keyword(s):  
Growth Rate ◽  
Critical Exponent ◽  
Binary Matroid ◽  
Binary Matroids ◽  
Practical Algorithm ◽  
A Minor

© 2017 Elsevier Inc. We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in M, where M is a subset of {M(K3,3),M⁎(K3,3),M(K5),M⁎(K5)} that contains either M(K3,3) or M⁎(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in M. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.

scholarly journals Excluding Kuratowski graphs and their duals from binary matroids

2020 ◽  
Author(s):  
Dillon Mayhew ◽  
G Royle ◽  
Geoffrey Whittle
Keyword(s):  
Growth Rate ◽  
Critical Exponent ◽  
Binary Matroid ◽  
Binary Matroids ◽  
Practical Algorithm ◽  
A Minor

© 2017 Elsevier Inc. We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in M, where M is a subset of {M(K3,3),M⁎(K3,3),M(K5),M⁎(K5)} that contains either M(K3,3) or M⁎(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in M. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.

scholarly journals Correction to: An extension and an alternative characterization of May’s theorem

2021 ◽  
Author(s):  
Josep Freixas ◽  
Montserrat Pons
Keyword(s):  
Alternative Characterization

scholarly journals Occurrence of fatty acid epoxide hydrolases in soybean (Glycine max). Purification and characterization of the soluble form

1992 ◽  
Vol 282 (3) ◽  
pp. 711-714 ◽  
Author(s):  
E Blée ◽  
F Schuber
Keyword(s):  
Glycine Max ◽  
Gel Filtration ◽  
Soluble Form ◽  
Soluble Enzyme ◽  
Purification And Characterization ◽  
Epoxide Hydrolases ◽  
Major Activity ◽  
Apparent Homogeneity ◽  
A Minor

Epoxide hydrolases catalysing the hydration of cis-9,10-epoxystearate into threo-9,10-dihydroxystearate have been detected in soybean (Glycine max) seedlings. The major activity was found in the cytosol, a minor fraction being strongly associated with microsomes. The soluble enzyme, which was purified to apparent homogeneity by (NH4)2SO4 fractionation, hydrophobic, DEAE- and gel-filtration chromatographies, has a molecular mass of 64 kDa and a pI of 5.4.

scholarly journals Demonstration of an integrated nanophotonic chip-scale alkali vapor magnetometer using inverse design

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Yoel Sebbag ◽  
Eliran Talker ◽  
Alex Naiman ◽  
Yefim Barash ◽  
Uriel Levy
Keyword(s):  
Spatial Resolution ◽  
Near Field ◽  
Computer Assisted ◽  
Experimental Characterization ◽  
Optical Isolation ◽  
New Concepts ◽  
On Chip ◽  
The Cost ◽  
Micrometer Scale

AbstractRecently, there has been growing interest in the miniaturization and integration of atomic-based quantum technologies. In addition to the obvious advantages brought by such integration in facilitating mass production, reducing the footprint, and reducing the cost, the flexibility offered by on-chip integration enables the development of new concepts and capabilities. In particular, recent advanced techniques based on computer-assisted optimization algorithms enable the development of newly engineered photonic structures with unconventional functionalities. Taking this concept further, we hereby demonstrate the design, fabrication, and experimental characterization of an integrated nanophotonic-atomic chip magnetometer based on alkali vapor with a micrometer-scale spatial resolution and a magnetic sensitivity of 700 pT/√Hz. The presented platform paves the way for future applications using integrated photonic–atomic chips, including high-spatial-resolution magnetometry, near-field vectorial imaging, magnetically induced switching, and optical isolation.

scholarly journals A characterization of a class of non-binary matroids

1990 ◽  
Vol 49 (2) ◽  
pp. 181-189 ◽  
Author(s):  
James G Oxley
Keyword(s):  
Binary Matroids

Detection and characterization of a factor which rescues spliceosome assembly from a heat-inactivated HeLa cell nuclear extract

1991 ◽  
Vol 11 (7) ◽  
pp. 3425-3431
Author(s):  
P Delannoy ◽  
M H Caruthers
Keyword(s):  
Heat Treatment ◽  
Hela Cell ◽  
Nuclear Extract ◽  
Mrna Splicing ◽  
Splicing Factor ◽  
Nuclear Extracts ◽  
Small Nuclear Rnas ◽  
A Minor ◽  
Cell Nuclear Extract

Mild heat treatment of HeLa cell nuclear extracts (NE) selectively inhibits pre-mRNA splicing. Heat-inactivated extracts can be complemented by a small amount of untreated NE. Utilizing this complementation assay and a combination of ion-exchange, affinity, and hydrophobic chromatography, a heat reversal factor (HRF) was purified from NE that is required to rescue pre-mRNA splicing from a heat-inactivated extract. This activity in its most purified form consistently copurified in a fraction containing two 70-kDa proteins and a minor polypeptide of approximately 100 kDa. It was free of the major small nuclear RNAs, sensitive to protease, and required to rescue spliceosome formation from a heat-inactivated nuclear extract. These results suggest that this factor is a protein that may be an important component in pre-mRNA splicing, or alternatively, it may be involved in renaturation of a heat-sensitive splicing factor.

Functional Characterization of GABAA Receptors in Neonatal Hypothalamic Brain Slice

2002 ◽  
Vol 88 (4) ◽  
pp. 1655-1663 ◽  
Author(s):  
Ren-Qi Huang ◽  
Glenn H. Dillon
Keyword(s):  
Functional Characterization ◽  
Minor Component ◽  
Dehydroepiandrosterone Sulfate ◽  
Receptor Subtypes ◽  
Hypothalamic Neurons ◽  
Inhibitory Neurotransmitter ◽  
Old Rats ◽  
A Minor ◽  
Whole Cell Recordings

The hypothalamus influences a number of autonomic functions. The activity of hypothalamic neurons is modulated in part by release of the inhibitory neurotransmitter GABA onto these neurons. GABAA receptors are formed from a number of distinct subunits, designated α, β, γ, δ, ε, and θ, many of which have multiple isoforms. Little data exist, however, on the functional characteristics of the GABAA receptors present on hypothalamic neurons. To gain insight into which GABAA receptor subunits are functionally expressed in the hypothalamus, we used an array of pharmacologic assessments. Whole cell recordings were made from thin hypothalamic slices obtained from 1- to 14-day-old rats. GABAA receptor-mediated currents were detected in all neurons tested and had an average EC50 of 20 ± 1.6 μM. Hypothalamic GABAA receptors were modulated by diazepam (EC50 = 0.060 μM), zolpidem (EC50 = 0.19 μM), loreclezole (EC50 = 4.4 μM), methyl-6,7-dimethoxy-4-ethyl-β-carboline (EC50= 7.7 μM), and 5α-pregnan-3α-hydroxy-20-one (3α-OH-DHP). Conversely, these receptors were inhibited by Zn2+ (IC50 = 70.5 μM), dehydroepiandrosterone sulfate (IC50 = 16.7 μM), and picrotoxin (IC50 = 2.6 μM). The α4/6-selective antagonist furosemide (10–1,000 μM) was ineffective in all hypothalamic neurons tested. The results of our pharmacological analysis suggest that hypothalamic neurons express functional GABAA receptor subtypes that incorporate α1 and/or α2 subunits, β2 and/or β3 subunits, and the γ2 subunit. Our results suggest receptors expressing α3–α6, β1, γ1, and δ, if present, represent a minor component of functional hypothalamic GABAA receptors.

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