Related papers: Mysterious points in keys but not trees
Clustering is an unsupervised machine learning methodology where unlabeled elements/objects are grouped together aiming to the construction of well-established clusters that their elements are classified according to their similarity. The…
The explosion in the amount of data available for analysis often necessitates a transition from batch to incremental clustering methods, which process one element at a time and typically store only a small subset of the data. In this paper,…
We discuss basic structural properties of finite black box groups. A special emphasis is made on the use of centralisers of involutions in probabilistic recognition of black box groups. In particular, we suggest an algorithm for finding the…
Recent work on explainable clustering allows describing clusters when the features are interpretable. However, much modern machine learning focuses on complex data such as images, text, and graphs where deep learning is used but the raw…
The community structure of complex networks reveals both their organization and hidden relationships among their constituents. Most community detection methods currently available are not deterministic, and their results typically depend on…
We find a duality between two well-known trees, the Calkin-Wilf tree and the Stern-Brocot tree, derived from cluster algebra theory. The vertex sets of these trees are the set of rational numbers, and they have cluster structures induced by…
For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting objects in the m-cluster category. There…
Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and…
We consider the general notion of coloured quiver mutation and show that the mutation class of a coloured quiver $Q$, arising from an $m$-cluster tilting object associated with $H$, is finite if and only if $H$ is of finite or tame…
To cluster data is to separate samples into distinctive groups that should ideally have some cohesive properties. Today, numerous clustering algorithms exist, and their differences lie essentially in what can be perceived as ``cohesive…
Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…
In this paper, we define weakly coherent rings, and examine the transfer of these rings to homomorphic image, trivial ring extension, localization, and direct product. These results provide examples of weakly coherent rings that are not…
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…
It is explained that, for black holes much heavier than the Planck mass, black hole microstates can be well understood without string theory. It is essential to understand the antipodal identification at the horizon. We show why the…
In general a contractible complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one "gets stuck", that is one can choose the collapses in such a way that one…
Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters…
Developing a better understanding of surprising or counterintuitive phenomena has constituted a significant portion of deep learning research in recent years. These include double descent, grokking, and the lottery ticket hypothesis --…
This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct…
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study…
We complete the discrete cluster categories of type $\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a…