scholarly journals The Discrete Fundamental Group of the Associahedron

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Christopher Severs ◽  
Jacob White
Keyword(s):  
Fundamental Group ◽  
Lattice Theory ◽  
Homotopy Theory ◽  
Equivalence Classes ◽  
Point Of View ◽  
Cluster Algebras ◽  
Type B ◽  
Simple Description ◽  
Generating Set ◽  
International Audience

International audience The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \par L'associahèdre est un objet bien etudié que l'on retrouve dans plusieurs contextes. Par exemple, il est associé à la théorie des opérades, à l'étude des partitions non-croisées, à la théorie des treillis et plus récemment aux algèbres dámas. Nous étudions cet objet par le biais de la théorie des homotopies discretes. En bref cette théorie signifie qu'un cycle de longueur 5 (sur le squelette de l'associahèdre) est considéré comme étant le bord d'un trou combinatoire, alors qu'un cycle de longueur 4 peut être contracté sans problème. Les classes d'homotopies discrètes sont donc des classes d'équivalence de cycles de longueurs 5. Nous donnons une description simple de ces classes d'équivalence et identifions un ensemble de générateurs du groupe correspondant (abélien) d'homotopies discrètes. Nous d'ecrivons également les liens entre notre construction et les algèbres d'amas.

scholarly journals THE DISCRETE FUNDAMENTAL GROUP OF THE ASSOCIAHEDRON, AND THE EXCHANGE MODULE

2013 ◽  
Vol 23 (04) ◽  
pp. 745-762
Author(s):  
HÉLÈNE BARCELO ◽  
CHRISTOPHER SEVERS ◽  
JACOB A. WHITE
Keyword(s):  
Fundamental Group ◽  
Lattice Theory ◽  
Homotopy Theory ◽  
Cluster Algebra ◽  
Type A ◽  
Point Of View ◽  
Cluster Algebras ◽  
Combinatorial Description

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank [Formula: see text]. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type An cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank [Formula: see text].

scholarly journals Divisors on graphs, Connected flags, and Syzygies

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Farbod Shokrieh
Keyword(s):  
Gröbner Basis ◽  
Betti Numbers ◽  
Point Of View ◽  
Grobner Basis ◽  
Generating Set ◽  
Term Order ◽  
Syzygy Module ◽  
International Audience ◽  
The Given ◽  
Minimal Generating Set

International audience We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.

WIGNER MEASURES AND MOLECULAR PROPAGATION THROUGH GENERIC ENERGY LEVEL CROSSINGS

2003 ◽  
Vol 15 (10) ◽  
pp. 1285-1317 ◽  
Author(s):  
CLOTILDE FERMANIAN KAMMERER
Keyword(s):  
Energy Transfer ◽  
Normal Form ◽  
Energy Level ◽  
Point Of View ◽  
Time Dependent ◽  
Type B ◽  
Unified Framework ◽  
Level Crossings ◽  
Normal Form Theorem ◽  
Uniformly Bounded

We study the time-dependent Schrödinger equation with matrix-valued potential presenting a generic crossing of type B, I, J or K in Hagedorn's classification. We use two-scale Wigner measures for describing the Landau–Zener energy transfer which occurs at the crossing. In particular, in the case of multiplicity 2 eigenvalues, we calculate precisely the change of polarization at the crossing. Our method provides a unified framework in which codimension 2, 3 or 5 crossings can be discussed. We recover Hagedorn's result for wave packets, from Wigner measure point of view, and extend them to any data uniformly bounded in L2. The proof is based on a normal form theorem which reduces the problem to an operator-valued Landau–Zener formula.

scholarly journals The Fundamental Group of Balanced Simplicial Complexes and Posets

10.37236/73 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Steven Klee
Keyword(s):  
Fundamental Group ◽  
Upper Bound ◽  
Large Family ◽  
Simplicial Complexes ◽  
Generating Set ◽  
Minimal Generating Set

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

Variations in Course and Branching of Sciatic Nerve and it’s Relation to Pyriformis Syndrome

2021 ◽  
Vol 9 (4) ◽  
pp. 8156-8159
Author(s):  
Patel Dinesh K ◽  
◽  
Shinde Amol A ◽  
Keyword(s):  
Sciatic Nerve ◽  
Hip Replacement ◽  
Popliteal Fossa ◽  
Point Of View ◽  
Lower Limbs ◽  
Type B ◽  
Sacral Plexus ◽  
Type D ◽  
Variant Anatomy ◽  
Anatomy Teaching

Background: Sciatic nerve is a branch of sacral plexus. It passes below the pyriformis and divides in the popliteal fossa. Higher division and relation of sciatic nerve to pyriformis have been documented. Beaton and Anson have classified relation of sciatic nerve to pyriformis. The aim of this study is to find incidence of variant anatomy of sciatic nerve as per Beaton and Anson classification. Materials and methods: 48 formalin embalmed lower limbs used for regular anatomy teaching were used. Branching and course of sciatic nerve was observed in gluteal region,thigh and popliteal fossa. Observations: As per Beaton and Anson classification, we found 81.2% showed type A or normal arrangement. Type B variation was seen in 14.6% while 4.2% showed type D variation. Conclusion: Variations in branching of sciatic nerve and it’s relation to pyriformis muscle are important from point of view of Surgeons and Anaesthetists. Knowledge of these variations will help reducing block failures in cases of sciatica, pyriformis syndrome and hip replacement surgeries. KEY WORDS: Sciatic nerve, Sacral plexus, Pyriformis Syndrome, Hip replacement.

scholarly journals The Pantelides algorithm for delay differential-algebraic equations

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Ines Ahrens ◽  
Benjamin Unger
Keyword(s):  
Differential Algebraic Equations ◽  
Equivalence Classes ◽  
Point Of View ◽  
Algebraic Equations ◽  
Sufficient Criterion ◽  
Differential Algebraic Equation ◽  
Graph Theoretical Approach ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Structural Point

Abstract We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

scholarly journals CROSSED MODULE BUNDLE GERBES; CLASSIFICATION, STRING GROUP AND DIFFERENTIAL GEOMETRY

2011 ◽  
Vol 08 (05) ◽  
pp. 1079-1095 ◽  
Author(s):  
BRANISLAV JURČO
Keyword(s):  
Differential Geometry ◽  
Geometric Realization ◽  
Equivalence Classes ◽  
Point Of View ◽  
Crossed Module ◽  
Simplicial Group ◽  
Stable Equivalence ◽  
Bundle Gerbe ◽  
Bundle Gerbes ◽  
Simplicial Methods

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to a (Lie) crossed module (H → D) there is a simplicial group [Formula: see text], the nerve of the groupoid [Formula: see text] defined by the crossed module, and its geometric realization, the topological group [Formula: see text]. We introduce crossed module bundle gerbes so that their (stable) equivalence classes are in a bijection with equivalence classes of principal [Formula: see text]-bundles. We discuss the string group and string structures from this point of view. Also, we give a simplicial interpretation to the bundle gerbe connection and bundle gerbe B-field.

scholarly journals Cluster algebras of unpunctured surfaces and snake graphs

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Gregg Musiker ◽  
Ralf Schiffler
Keyword(s):  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Polynomial Expansion ◽  
International Audience ◽  
Polynomial Expansions

International audience We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ . Nous étudions des algèbres amassées avec coefficients principaux associées aux surfaces. Nous présentons une formule directe pour les développements de Laurent des variables amassées dans ces algèbres en terme de couplages parfaits d'un certain graphe $G_{T,\gamma}$ que l'on construit a partir de la surface en recollant des pièces élémentaires que l'on appelle carreaux. Nous donnons aussi une seconde formule pour ces développements en termes de sous-graphes de $G_{T,\gamma}$ .

scholarly journals Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Soojin Cho ◽  
Kyoungsuk Park
Keyword(s):  
Bruhat Order ◽  
Type B ◽  
Coxeter System ◽  
Covering Relation ◽  
Signed Permutations ◽  
Weak Bruhat Order ◽  
International Audience

International audience Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$. De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$.

Sign in / Sign up

Export Citation Format

Share Document