Related papers: M-systems and Cluster algebras

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose…

Despite the fundamental importance of clustering, to this day, much of the relevant research is still based on ambiguous foundations, leading to an unclear understanding of whether or how the various clustering methods are connected with…

We describe a modification of the stochastic coupled cluster algorithm that allows the use of multiple reference determinants. By considering the secondary references as excitations of the primary reference and using them to change the…

A natural way to characterize the cluster structure of a dataset is by finding regions containing a high density of data. This can be done in a nonparametric way with a kernel density estimate, whose modes and hence clusters can be found…

Clustering is a widely used technique in data mining applications for discovering patterns in underlying data. Most traditional clustering algorithms are limited to handling datasets that contain either numeric or categorical attributes.…

We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity…

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them…

In previous work "Betweenness algebras" we introduced and examined the class of betweenness algebras. In the current paper we study a larger class of algebras with binary operators of possibility and sufficiency, the weak mixed algebras.…

Convex clustering is a modern method with both hierarchical and $k$-means clustering characteristics. Although convex clustering can capture complex clustering structures hidden in data, the existing convex clustering algorithms are not…

We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.

This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…

We propose a new class of mathematical structures called (m,n)-semirings} (which generalize the usual semirings), and describe their basic properties. We also define partial ordering, and generalize the concepts of congruence, homomorphism,…

Cluster analysis is a fundamental research issue in statistics and machine learning. In many modern clustering methods, we need to determine whether two subsets of samples come from the same cluster. Since these subsets are usually…

Clustering algorithms partition a dataset into groups of similar points. The primary contribution of this article is the Multiscale Spatially-Regularized Diffusion Learning (M-SRDL) clustering algorithm, which uses spatially-regularized…

Higher homological algebra, basically done in the framework of an $n$-cluster tilting subcategory $\mathcal{M}$ of an abelian category $\mathcal{A}$, has been the topic of several recent researches. In this paper, we study a relative…

Reduced k-means clustering is a method for clustering objects in a low-dimensional subspace. The advantage of this method is that both clustering of objects and low-dimensional subspace reflecting the cluster structure are simultaneously…

This is a preliminary draft of Chapters 4-5 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. This installment contains: Chapter 4. New patterns from old Chapter 5. Finite…

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in…

Highly size-asymmetrical fluid mixtures arise in a variety of physical contexts, notably in suspensions of colloidal particles to which much smaller particles have been added in the form of polymers or nanoparticles. Conventional schemes…

The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…