Related papers: Sequences of analytic disks

Despite the widespread use of Clustering, there is distressingly little general theory of clustering available. Questions like "What distinguishes a clustering of data from other data partitioning?", "Are there any principles governing all…

What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number…

We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…

We prove the existence of cluster characters for Hom-infinite cluster categories. For this purpose, we introduce a suitable mutation-invariant subcategory of the cluster category. We sketch how to apply our results in order to categorify…

The rapid development of high-throughput sequencing technologies has led to an explosive increase in biological sequence data, making sequence clustering a fundamental task in large-scale bioinformatics analyses. Unlike traditional…

Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering…

We initiate a systematic study of the deep points of a cluster algebra; that is, the points in the associated variety which are not in any cluster torus. We describe the deep points of cluster algebras of type A, rank 2, Markov, and…

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point…

We study the notion of positive and negative complexity of pairs of objects in cluster categories. The first main result shows that the maximal complexity occurring is either one, two or infinite, depending on the representation type of the…

Typically clustering algorithms provide clustering solutions with prespecified number of clusters. The lack of a priori knowledge on the true number of underlying clusters in the dataset makes it important to have a metric to compare the…

Let $X,X_1,X_2,\ldots$ be i.i.d. mean zero random vectors with values in a separable Banach space $B$, $S_n=X_1+\cdots+X_n$ for $n\ge1$, and assume $\{c_n:n\ge1\}$ is a suitably regular sequence of constants. Furthermore, let…

Taking a representation-theoretic viewpoint, we construct a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. We show that our associahedron shares important properties with the…

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert…

In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare…

Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby discover, relate, and structure types: of behaviour,…

We study supervised learning problems using clustering constraints to impose structure on either features or samples, seeking to help both prediction and interpretation. The problem of clustering features arises naturally in text…

In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster…

The paper concerns clustering with respect to the shape and size of 2D contours that are boundaries of cross-sections of 3D objects of revolution. We propose a number of similarity measures based on combined disparate Procrustes analysis…

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,…