scholarly journals Fragility and excluded minors

2021 ◽  
Author(s):  
◽  
Ben Clark
Keyword(s):  
Representation Theory ◽  
Structural Characterisation ◽  
Excluded Minor ◽  
Excluded Minors ◽  
The Relationship ◽  
Parallel Extension

<p>Let Ɲ be a set of matroids. A matroid, M, is Ɲ -fragile, if for every element e, either M\e or M/e has no minor isomorphic to a member of Ɲ . This thesis gives new results in matroid representation theory that elucidate the relationship between Ɲ -fragile matroids and excluded minors.  Let ℙ be a partial field, and let Ɲ be a set of strong stabilizers for ℙ. The first main result of this thesis establishes a relationship between Ɲ -fragile matroids and excluded minors for the class of ℙ-representable matroids. We prove that if an excluded minor M for the class of ℙ-representable matroids has a pair of elements a,b such that M\a,b is 3-connected with an Ɲ -minor, then either M is close to an Ɲ -minor or M\a,b is Ɲ -fragile. The result motivates a study of the structure of ℙ-representable Ɲ -fragile matroids.  The matroids U₂,₅ and U₃,₅ are strong stabilizers for the U₂ and H₅ partial fields. The second main result of this thesis is a structural characterisation of the U₂- and H₅-representable {U₂,₅,U₃,₅}-fragile matroids. We prove that these matroids can be constructed from U₂,₅ and U₃,₅ by a sequence of moves, where, up to duality, each move consists of a parallel extension followed by a delta-wye or a generalised delta-wye exchange.  Finally, we obtain a bound on the size of an excluded minor M for the class of U₂- or H₅-representable matroids with the property that M has a pair of elements a,b such that M\a,b is 3-connected with a {U₂,₅,U₃,₅}-minor. Our proof uses the first and second main results of this thesis.</p>

scholarly journals Fragility and excluded minors

2021 ◽  
Author(s):  
◽  
Ben Clark
Keyword(s):  
Representation Theory ◽  
Structural Characterisation ◽  
Excluded Minor ◽  
Excluded Minors ◽  
The Relationship ◽  
Parallel Extension

<p>Let Ɲ be a set of matroids. A matroid, M, is Ɲ -fragile, if for every element e, either M\e or M/e has no minor isomorphic to a member of Ɲ . This thesis gives new results in matroid representation theory that elucidate the relationship between Ɲ -fragile matroids and excluded minors.  Let ℙ be a partial field, and let Ɲ be a set of strong stabilizers for ℙ. The first main result of this thesis establishes a relationship between Ɲ -fragile matroids and excluded minors for the class of ℙ-representable matroids. We prove that if an excluded minor M for the class of ℙ-representable matroids has a pair of elements a,b such that M\a,b is 3-connected with an Ɲ -minor, then either M is close to an Ɲ -minor or M\a,b is Ɲ -fragile. The result motivates a study of the structure of ℙ-representable Ɲ -fragile matroids.  The matroids U₂,₅ and U₃,₅ are strong stabilizers for the U₂ and H₅ partial fields. The second main result of this thesis is a structural characterisation of the U₂- and H₅-representable {U₂,₅,U₃,₅}-fragile matroids. We prove that these matroids can be constructed from U₂,₅ and U₃,₅ by a sequence of moves, where, up to duality, each move consists of a parallel extension followed by a delta-wye or a generalised delta-wye exchange.  Finally, we obtain a bound on the size of an excluded minor M for the class of U₂- or H₅-representable matroids with the property that M has a pair of elements a,b such that M\a,b is 3-connected with a {U₂,₅,U₃,₅}-minor. Our proof uses the first and second main results of this thesis.</p>

scholarly journals Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Thomas Bliem ◽  
Dido Salazar
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Ehrhart Polynomial ◽  
International Audience ◽  
Order Polytope ◽  
Finite Partially Ordered Set ◽  
The Relationship

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.

A Remark by Philip Hall

1958 ◽  
Vol 1 (1) ◽  
pp. 21-23 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Linear Transformations ◽  
Modern Physics ◽  
The Right ◽  
The Relationship

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

scholarly journals F--: A Parallel Extension to Cray Fortran

1997 ◽  
Vol 6 (3) ◽  
pp. 275-284 ◽  
Author(s):  
Robert W. Numrich
Keyword(s):  
Distributed Memory ◽  
Data Grid ◽  
Data Locality ◽  
Vertical Line ◽  
Data Layout ◽  
Local Data ◽  
Explicit Statement ◽  
Local Code ◽  
The Relationship ◽  
Parallel Extension

F--is a parallel extension to Cray Fortran 77 for distributed memory computers. It adds exactly one new symbol to the language, a vertical line that separates two sets of indices. The first set contains the coordinates for data in a local data grid. The second set contains the coordinates for processors in a global processor grid. A statement such as x(i,j)=y(i,j| p,q) generates a load from remote address y(i,j) in the data grid on processor (p,q) followed by a store to local address x(i,j) in the local data grid. F--syntax requires an explicit statement of the relationship between data layout and processor layout. It assumes that good performance on a distributed memory computer requires the programmer to understand and to exploit data locality. Programmers use the F--syntax only when it is needed. 0therwise all data are local and all code is local. Compiler and library developers concentrate on generating well-optimized local code.

scholarly journals The role of Heisenberg group in Harmonic analysis: دور زمرة هايزنبرغ في التحليل التوافقي

2019 ◽  
Vol 3 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Mathematical Concepts ◽  
Group Fourier Transform ◽  
The Relationship

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties. The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group. Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and  .

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>

Narrative Representation Theory: Identifying the human language with superstructure

2017 ◽  
Vol 19 (6) ◽  
pp. 648-672
Author(s):  
Hirokuni Masuda
Keyword(s):  
Biological Sciences ◽  
Mental Disorders ◽  
Representation Theory ◽  
Human Language ◽  
Computational System ◽  
Language Faculty ◽  
The Right ◽  
First Time ◽  
The Relationship ◽  
The Brain

Narrative Representation Theory (NRT), an evolved framework of Verse Analysis, has come into existence with the mission of explaining the operation of macro-systemic structure that could be hardwired in the brain. Based on the analyses of creoles or archetypal human languages, the theory puts forward the premise stating that the fundamental design of the human language faculty possesses the computational system for internalized discourse. The theory preserves the principles of Quint-patterning, Idea-formatting, N-ary-branching and X-numbering, complying respectively with the hierarchical orderings of constituency, the atomic elements of componentiality, the linear sequences of precedence and the specific measurement of terminal nodes. NRT tells that the macro-system of narrative superstructure must have emerged autonomously, yet links closely with the micro-system of phonology, morphology and syntax. This article explores for the first time scientific insights into the nature of human language, referring to recent research on the right cerebrum as well as on the prefrontal lobes of the brain, the relationship between mental disorders and their genetic deficiencies, and the investigations of human evolution during the period 200,000–40,000 years BP. All the converging evidence in biological sciences reinforces the hypothesis that the narrative superstructure of language faculty manifests as an inherent linguistic capacity in our mind.

scholarly journals Character theory of infinite wreath products

2005 ◽  
Vol 2005 (9) ◽  
pp. 1365-1379 ◽  
Author(s):  
Robert Boyer
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Finite Groups ◽  
Wreath Products ◽  
Character Theory ◽  
Group Algebras ◽  
Inductive Limits ◽  
Infinite Symmetric Group ◽  
The Relationship ◽  
Asymptotic Character

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” TheK0-invariant of the groupC∗-algebra is also determined.

scholarly journals A Riesz representation theory for completely regular Hausdorff spaces and its applications

2016 ◽  
Vol 14 (1) ◽  
pp. 474-496 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Representation Theory ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Weakly Compact ◽  
Completely Regular ◽  
Riesz Representation ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
The Relationship ◽  
Stieltjes Type

AbstractLet X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski.

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