Related papers: An introduction to higher cluster categories
Basic concepts of higher local fields and topologies on their additive and multiplicative groups are introduced.
Despite the inherent lack of a ground truth in clustering, a broad consensus is overall acknowledged in defining the concept of cluster in the continuous setting. Conversely, this remains controversial in the presence of categorical data.…
Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample patchworks' and…
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…
In [7], we introduced the category of chain bundles with examples and established its significance. Here we shall describe certain categories which we call the category of chains in the chain bundle category and discuss some interesting…
Cluster analysis requires many decisions: the clustering method and the implied reference model, the number of clusters and, often, several hyper-parameters and algorithms' tunings. In practice, one produces several partitions, and a final…
This paper proposes a new paradigm and computational framework for identification of correspondences between sub-structures of distinct composite systems. For this, we define and investigate a variant of traditional data clustering, termed…
The purpose of this article is to present the theory of higher order connections on vector bundles from a viewpoint inspired by projective differential geometry.
We present a comprehensive classification theory for saturated Fell bundles over locally compact groups, utilizing data associated with their base group and unit fiber. This framework offers a unified approach to understanding the structure…
This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.
There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and…
We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…
We give complete algorithms and source code for constructing (multilevel) statistical industry classifications, including methods for fixing the number of clusters at each level (and the number of levels). Under the hood there are…
We give a simple algebraic description of opetopes in terms of chain complexes, and we show how this description is related to combinatorial descriptions in terms of treelike structures. More generally, we show that the chain complexes…
The recent high level of interest in weighted complex networks gives rise to a need to develop new measures and to generalize existing ones to take the weights of links into account. Here we focus on various generalizations of the…
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…
In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing…
In this paper, we study the structure of a generalized near-group fusion category and classified it when it is slightly degenerate.
This chapter provides an overview of past and present techniques for optical detection of galaxy clusters. It follows the progression of cluster detection techniques through time, allowing readers to understand the development of the field…
Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be…