相关论文: Cluster algebras and triangulated surfaces. Part I…
Given a finite dimensional algebra $C$ (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension $C\ltimes \Ext_C^2(DC,C)$ of $C$ by the $C$-$C$-bimodule…
We express cluster variables of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$. Then we associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$.…
Usual formulations of the clustering coefficient can be shown to be insufficient in the task of describing the local topology of very simple networks. Motivated by this, we review some alternatives in order to present an extension, the…
After generalizing the concept of clusters to incorporate clusters that are linked to other clusters through some relatively narrow bridges, an approach for detecting patches of separation between these clusters is developed based on an…
We study cluster algebras with principal and arbitrary 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…
This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables…
We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the…
Clustering is an unsupervised learning problem that aims to partition unlabelled data points into groups with similar features. Traditional clustering algorithms provide limited insight into the groups they find as their main focus is…
For the cluster category of a hereditary or a canonical algebra, equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an…
We study quantum cluster algebras from marked surfaces without punctures. We express the quantum cluster variables in terms of the canonical submodules. As a byproduct, we obtain the positivity for this class of quantum cluster algebra.
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This…
We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effect of clustering on the…
We generalize to the non simply-laced case results of Gei\ss, Leclerc and Schr\"oer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non…
Clustered solutions in oscillator networks provide an important insight into how a system might diversify from a synchronous solution into spatiotemporal complex solutions. They can therefore form a link between fully synchronized and…
For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum…
In this paper, we provide novel definitions of clustering coefficient for weighted and directed multilayer networks. We extend in the multilayer theoretical context the clustering coefficients proposed in the literature for weighted…
We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several…
A hypergraph is a useful combinatorial object to model ternary or higher-order relations among entities. Clustering hypergraphs is a fundamental task in network analysis. In this study, we develop two clustering algorithms based on…
In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of…
In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster…