Related papers: Cluster Magnification, Root Capacity, Unique Chain…

In this article we establish certain variants of the Inverse Cluster Size problem. We introduce the notion of primitive extensions and establish the Primitive variant of the problem. Precisely, we prove the existence of primitive extensions…

We develop the theory of root clusters further in this article and give some applications. We introduce some new notions as well as recall earlier notions for field extensions over a perfect base field: root cluster size, its generalization…

The percolation properties of clustered networks are analyzed in detail. In the case of weak clustering, we present an analytical approach that allows to find the critical threshold and the size of the giant component. Numerical simulations…

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 consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail…

The determination of cluster centers generally depends on the scale that we use to analyze the data to be clustered. Inappropriate scale usually leads to unreasonable cluster centers and thus unreasonable results. In this study, we first…

We analyze a simple model for growing tree networks and find that although it never percolates, there is an anomalously large cluster at finite size. We study the growth of both the maximal cluster and the cluster containing the original…

Clustering is a well-known unsupervised machine learning approach capable of automatically grouping discrete sets of instances with similar characteristics. Constrained clustering is a semi-supervised extension to this process that can be…

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…

A cluster tree provides a highly-interpretable summary of a density function by representing the hierarchy of its high-density clusters. It is estimated using the empirical tree, which is the cluster tree constructed from a density…

In this paper we give an introduction on how one can extend a valuation from a field $K$ to the polynomial ring $K[x]$ in one variable over $K$. This follows a similar line as the one presented by the author in his talk at ALaNT 5. We will…

We study clustering on graphs with multiple edge types. Our main motivation is that similarities between objects can be measured in many different metrics. For instance similarity between two papers can be based on common authors, where…

The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety…

We study numerically a model of nonequilibrium networks where nodes and links are added at each time step with aging of nodes and connectivity- and age-dependent attachment of links. By varying the effects of age in the attachment…

A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous…

For a simple, normal and finite extension of a valued field, we prove that we can related the order of the ramification group of the field extension and the set of key polynomials associated to the extension of the valuation. More…

By virtue of their high galaxy space densities and their large spatial separations, clusters are efficient and accurate tracers of the large-scale density and velocity fields. Substantial progress has been made over the past decade in the…

We study the problem of explainability-first clustering where explainability becomes a first-class citizen for clustering. Previous clustering approaches use decision trees for explanation, but only after the clustering is completed. In…

For all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for…

Let us consider subcritical Bernoulli percolation on a connected, transitive, infinite and locally finite graph. In this paper, we propose a new (and short) proof of the exponential decay property for the volume of clusters. We do not rely…