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Various notions of stable ranks are studied for topological algebras. Some partial answers to R.G. Swan's problem (Have two Banach or good Fr\'echet algebras as in the density theorem in K-theory the same stable rank ?) are obtained. For…
Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial…
We classify possible supersymmetry-preserving relevant, marginal, and irrelevant deformations of unitary superconformal theories in $d \geq 3$ dimensions. Our method only relies on symmetries and unitarity. Hence, the results are model…
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…
We use recent results on the boundary behavior of Cauchy integrals to study the Krein spectral shift of a rank one perturbation problem for self-adjoint operators. As an application, we prove that all self-adjoint rank one perturbations of…
We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ which is accumulated by a sequence of elements of the complement $M\setminus L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\setminus L$ is not a…
We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schr\"odinger operator on a star-graph with a finite number of…
An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra…
A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where…
In this paper we deal with branched coverings over the complement to finitely many exceptional points on the Riemann sphere having the property that the local monodromy around each of the branching points is of finite order. To such a…
In this paper we study random matrix models where the matrices in question contain infinitely many spikes. Recent work has characterized the possible outliers in the spectrum of large deformed unitarily invariant models when the number of…
We study the problem of maximizing a spectral risk measure of a given output function which depends on several underlying variables, whose individual distributions are known but whose joint distribution is not. We establish and exploit an…
We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties:…
We prove various notions of uniform continuity for compact-quantum-group representations on Hilbert or Banach spaces equivalent to having finite spectrum, i.e. finitely many isotypic components. This generalizes the classical analogue for…
We give a brief account of the interplay between spectral morphisms, K-theory, and stable ranks in the context of Banach algebras.
In this short survey article, we try to list maximum number of known results on class preserving automorphisms of finite $p$-groups. We conclude the survey with some interesting (at least for the author) open problems on this topic.
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar…
Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic…
This paper is dedicated to the unique continuation properties of the solutions to nonlinear variational problems. Our analysis covers the case of nonlinear autonomous functionals depending on the gradient, as well as more general double…
We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The notion of point degree spectrum…