Related papers: Cluster Theories and Cluster Structures of Type A
The analysis of the presence of substructures in 16 well-sampled clusters of galaxies suggests a stimulating hypothesis: Clusters could be classified as unimodal or bimodal, on the basis of to the sub-clump distribution in the {\em 3-D}…
Recent critiques of the semantic conception of scientific theories suggest that a theory is not best formulated as a collection of models satisfying some set of kinematical or dynamical conditions. Thus it has been argued that additional…
This is an expanded version of the notes of our lectures given at the conference "Current Developments in Mathematics 2003" held at Harvard University on November 21--22, 2003. We present an overview of the main definitions, results and…
Using a measure of clustering derived from the nearest neighbour distribution and the void probability function we are able to distinguish between regular and clustered structures. With an example we show that regularity is a property of a…
Following our previous work [18], we introduce the notions of partial seed homomorphisms and partial ideal rooted cluster morphisms. Related to the theory of Green's equivalences, the isomorphism classes of sub-rooted cluster algebras of a…
Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network…
Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified…
We give a geometric realization of cluster categories of type $D_n$ using a polygon with $n$ vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain…
We consider triangulated orbit categories, with the motivating example of cluster categories, in their usual context of algebraic triangulated categories, then present them from another perspective in the framework of topological…
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…
Clustering is an essential data mining tool that aims to discover inherent cluster structure in data. For most applications, applying clustering is only appropriate when cluster structure is present. As such, the study of clusterability,…
Clustering is one of the most common unsupervised learning tasks in machine learning and data mining. Clustering algorithms have been used in a plethora of applications across several scientific fields. However, there has been limited…
In the framework of the Time Dependent Scattering Theory we discuss three forms of Cluster Separability as well as the conditions for the representation of the scattering system dynamics implied by their respective use.
Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories are actually just quotients of cluster categories. The other…
We study the version of the C-Planarity problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the Strip Planarity problem. We give algorithms to decide several families of instances for…
Abstract separation systems provide a simple general framework in which both tree-shape and high cohesion of many combinatorial structures can be expressed, and their duality proved. Applications range from tangle-type duality and tree…
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric…
We review clustering as an analysis tool and the underlying concepts from an introductory perspective. What is clustering and how can clusterings be realised programmatically? How can data be represented and prepared for a clustering task?…
We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a…
We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality. The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second…