scholarly journals Matching Complexes of $3 \times n$ Grid Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Shuchita Goyal ◽  
Samir Shukla ◽  
Anurag Singh
Keyword(s):  
Simplicial Complex ◽  
Homotopy Equivalent ◽  
Grid Graphs ◽  
Comprehensive List ◽  
Matching Complex

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed  that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.  

scholarly journals Cosheaf representations of relations and Dowker complexes

2021 ◽  
Author(s):  
Michael Robinson
Keyword(s):  
Weight Function ◽  
Simplicial Complex ◽  
Binary Relation ◽  
Homotopy Equivalent ◽  
Covariant Functor ◽  
The Matrix ◽  
Abstract Simplicial Complex ◽  
Integer Weight

AbstractThe Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

scholarly journals 3-torsion in the Homology of Complexes of Graphs of Bounded Degree

2013 ◽  
Vol 65 (4) ◽  
pp. 843-862
Author(s):  
Jakob Jonsson
Keyword(s):  
Boundary Element ◽  
Simplicial Complex ◽  
Homology Class ◽  
Chain Complex ◽  
Semigroup Algebras ◽  
Bounded Degree ◽  
Matching Complex ◽  
Image Position

AbstractFor δ ≥ 1 and n ≥ 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When δ = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n − 4)/3 ≤ d ≤ (n − 6)/2. This paper establishes similar bounds for δ ≥ 2. Specifically, there is 3-torsion in degree d wheneverThe procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3zis a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.

scholarly journals Matching Complexes of Small Grids

10.37236/8480 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Takahiro Matsushita
Keyword(s):  
Simplicial Complex ◽  
Line Graph ◽  
Simple Graph ◽  
Grid Graph ◽  
Matching Complex ◽  
Independence Complex

The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$.  Braun and Hough introduced a family of graphs $\Delta^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $\Delta^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.

scholarly journals Directed Subgraph Complexes

10.37236/1828 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Axel Hultman
Keyword(s):  
Simplicial Complex ◽  
Directed Graph ◽  
Homotopy Type ◽  
Disjoint Union ◽  
Undirected Graph ◽  
Hyperplane Arrangements ◽  
Homotopy Equivalent ◽  
Dimensional Sphere ◽  
Strongly Connected ◽  
Connected Subgraphs

Let $G$ be a directed graph, and let $\Delta^{ACY}_G$ be the simplicial complex whose simplices are the edge sets of acyclic subgraphs of $G$. Similarly, we define $\Delta^{NSC}_G$ to be the simplicial complex with the edge sets of not strongly connected subgraphs of $G$ as simplices. We show that $\Delta^{ACY}_G$ is homotopy equivalent to the $(n-1-k)$-dimensional sphere if $G$ is a disjoint union of $k$ strongly connected graphs. Otherwise, it is contractible. If $G$ belongs to a certain class of graphs, the homotopy type of $\Delta^{NSC}_G$ is shown to be a wedge of $(2n-4)$-dimensional spheres. The number of spheres can easily be read off the chromatic polynomial of a certain associated undirected graph. We also consider some consequences related to finite topologies and hyperplane arrangements.

Beetles of the family Cryptophagidae (Coleoptera) in the collection of Zoological Museum of Taras Shevchenko National University of Kyiv

2019 ◽  
Vol 17 (2) ◽  
pp. 10-24 ◽  
Author(s):  
K. Ocheretna
Keyword(s):  
Species Identification ◽  
Broad Sense ◽  
Zoological Museum ◽  
World War ◽  
Identification Number ◽  
Comprehensive List ◽  
Carpathian Region ◽  
The Family ◽  
National University ◽  
Collection Sites

The Cryptophagidae collection (Coleoptera: Cucujoidea) deposited at the Zoological Museum of the Taras Shevchenko National University of Kyiv (ZMKU) is described. The main authors of the collection are well-known researchers from the 1910–1930s, Orest Marcu and Karl Penecke. This is the largest collection of cryptophagids among the natural museums of Ukraine containing 304 specimens belonging to 85 species of 13 genera. In addition, 15 specimens of 5 species belonging to the families Erotylidae, Biphyllidae and Languriidae were among Cryptophagidae specimens. The collection, according to information available in the ZMKU, came to the museum not earlier than 1947 as the indemnity for the results of the II World War, most likely from Chernivtsi, where Marcu and Penecke worked. The vast majority of specimens is collected in the territory of modern Romania and Ukraine, and many specimens came from Chernivtsi. A table with an overview of all key details of the specimens is given, in which there are 6 fields: the name of the species on the label, details on the species identification, number of specimens, collection locality with the name of collector and remarks on the specimen, in particular, the instructions for decoding collection sites from the original labels. Annotations are made on the amount of the collection and the most important specimens and re-identification for each of the 13 genera. Some specimens are lost, probably during numerous collection migrations. In particular, some species (Cryptophagus simplex, C. lapidicola, C. nitidulus, Caenoscelis subdeplanata, Atomaria grandicollis, A. peltata, etc.) are represented in the collection only by the labels. The collection is important for the analysis of the composition of the fauna of the Carpathian region in the broad sense, since some species are encountered in the collection rarely; therefore it is important to clarify their locations to form the most comprehensive list of species of the Cryptophagids in the region. Several species of the family were included on the actual list of the fauna of the region on the basis of the study of this collection, in particular: Atomaria linearis, A. analis, A. apicalis, A. gravidula, Cryptophagus fasciatus, C. setulosus, etc.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

scholarly journals What Do Meditators Do When They Meditate? Proposing a Novel Basis for Future Meditation Research

Mindfulness ◽  
2021 ◽  
Author(s):  
Karin Matko ◽  
Ulrich Ott ◽  
Peter Sedlmeier
Keyword(s):  
Qualitative Study ◽  
Quantitative Study ◽  
Literature Search ◽  
Comprehensive Overview ◽  
Contemplative Practices ◽  
Comprehensive List ◽  
Scientific Investigations ◽  
Wide Range ◽  
Vast Range ◽  
Selection Of

Abstract Objectives Meditation is an umbrella term for a vast range of contemplative practices. Former proposals have struggled to do justice to this variety. To our knowledge, there is to date no comprehensive overview of meditation techniques spanning all major traditions. The present studies aimed at providing such a comprehensive list of meditation techniques. Methods In a qualitative study, we compiled a collection of 309 meditation techniques through a literature search and interviews with 20 expert meditators. Then, we reduced this collection to 50 basic meditation techniques. In a second, quantitative study, 635 experienced meditators from a wide range of meditative backgrounds indicated how much experience they had with each of these 50 meditation techniques. Results Meditators’ responses indicated that our choice of techniques had been adequate and only two techniques had to be added. Our additional statistical and cluster analyses illustrated preferences for specific techniques across and within diverse traditions as well as sets of techniques commonly practiced together. Body-centered techniques stood out in being of exceptional importance to all meditators. Conclusions In conclusion, we found an amazing variety of meditation techniques, which considerably surpasses previous collections. Our selection of basic meditation techniques might be of value for future scientific investigations and we encourage researchers to use this set.

scholarly journals Robo Ludens

2021 ◽  
Vol 10 (4) ◽  
pp. 1-28
Author(s):  
John Edison MUñOZ ◽  
Kerstin Dautenhahn
Keyword(s):  
Human Behavior ◽  
Scoping Review ◽  
Game Design ◽  
Research Community ◽  
Human Robot Interaction ◽  
Multiplayer Games ◽  
Robot Interaction ◽  
Design Elements ◽  
Comprehensive List ◽  
Suitable Solution

The use of games as vehicles to study human-robot interaction (HRI) has been established as a suitable solution to create more realistic and naturalistic opportunities to investigate human behavior. In particular, multiplayer games that involve at least two human players and one or more robots have raised the attention of the research community. This article proposes a scoping review to qualitatively examine the literature on the use of multiplayer games in HRI scenarios employing embodied robots aiming to find experimental patterns and common game design elements. We find that researchers have been using multiplayer games in a wide variety of applications in HRI, including training, entertainment and education, allowing robots to take different roles. Moreover, robots have included different capabilities and sensing technologies, and elements such as external screens or motion controllers were used to foster gameplay. Based on our findings, we propose a design taxonomy called Robo Ludens, which identifies HRI elements and game design fundamentals and classifies important components used in multiplayer HRI scenarios. The Robo Ludens taxonomy covers considerations from a robot-oriented perspective as well as game design aspects to provide a comprehensive list of elements that can foster gameplay and bring enjoyable experiences in HRI scenarios.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

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