Related papers: The polynomial cluster value problem

Let $M$ be a manifold, $V$ be a vector field on $M$, and $B$ be a Banach space. For any fixed function $f:M\rightarrow B$ and any fixed complex number $\lambda$, we study Hyers-Ulam stability of the global differential equation $Vy=\lambda…

We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or…

Let $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ be a weighted projective line. We define the quantum cluster algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ and realize its specialized version as the subquotient of the…

The linear isometries between weighted Banach spaces of continuous functions are considered. Some of well known theorems on isometries between spaces of continuous functions are proved and stated, but all they are in an appropriate form. In…

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra $A$ and a compact space $X$, let…

This paper addresses the study and characterizations of variational convexity of extended-real-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and…

The notion of $p$-summing Bloch mapping from the complex unit open disc $\mathbb{D}$ into a complex Banach space $X$ is introduced for any $1\leq p\leq\infty$. It is shown that the linear space of such mappings, equipped with a natural…

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we can induce the…

We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study…

UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete…

In clustering problems, a central decision-maker is given a complete metric graph over vertices and must provide a clustering of vertices that minimizes some objective function. In fair clustering problems, vertices are endowed with a color…

We define and solve boundary value problems of Schwarz and Dirichlet type on the complex unit disk for bicomplex-valued functions.

Deep clustering has the potential to learn a strong representation and hence better clustering performance compared to traditional clustering methods such as $k$-means and spectral clustering. However, this strong representation learning…

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near…

In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to…

Motivated by the cluster structure of two-loop scattering amplitudes in N=4 Yang-Mills theory we define "cluster polylogarithm functions". We find that all such functions of weight 4 are made up of a single simple building block associated…

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…

Clustering with most objective functions is NP-Hard, even to approximate well in the worst case. Recently, there has been work on exploring different notions of stability which lend structure to the problem. The notion of stability,…

Cluster indices describe extremal behaviour of stationary time series. We consider runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain central limit theorems under conditions…

We address the classical problem of hierarchical clustering, but in a framework where one does not have access to a representation of the objects or their pairwise similarities. Instead, we assume that only a set of comparisons between…