Related papers: Sequences of analytic disks
We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For…
We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…
We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.
We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials…
In this paper, we introduce and investigate two new subclasses of analytic functions in the open unit disk in the complex plane. Several interesting properties of the functions belonging to these classes are examined. Here, sufficient, and…
This paper studies the computational difficulty of clustering problems that are defined directly on a continuous probability density. Rather than working with finite samples, we assume the density is given as a polynomial and ask whether it…
This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We first consider two finite type cases: $B_n$ and $C_n$, completing a classification by Grabowski for…
Based on the competition between members of a hierarchy of length scales in complex multi-scale systems, it is shown how clustering of active quantities into concentrated sets, like bubbles in a Swiss cheese, is a generic property that…
Cluster analysis methods seek to partition a data set into homogeneous subgroups. It is useful in a wide variety of applications, including document processing and modern genetics. Conventional clustering methods are unsupervised, meaning…
For any closed analytic set X in C^2 there exists a proper holomorphic embedding of the unit disk into C^2 such that the image avoids X.
The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster…
In cluster analysis, a common first step is to scale the data aiming to better partition them into clusters. Even though many different techniques have throughout many years been introduced to this end, it is probably fair to say that 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…
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional…
We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras,…
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out…
Cluster analysis plays an indispensable role in machine learning and data mining. Learning a good data representation is crucial for clustering algorithms. Recently, deep clustering, which can learn clustering-friendly representations using…
We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the…
For cluster algebras from surfaces, there is a known formula for cluster variables and F-polynomials in terms of the perfect matchings of snake graphs. If the cluster algebra has trivial coefficients, there is also a known formula for…