Related papers: A matricial view of the Karpelevi\v{c} Theorem
The statement of the Karpelevic theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevic region) is long and complicated and his proof methods are, at best, nebulous.…
The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of…
We consider the Karpelevi\v{c} region $\Theta_n \subset \mathbb{C}$ consisting of all eigenvalues of all stochastic matrices of order $n$. We provide an alternative characterisation of $\Theta_n$ that sharpens the original description given…
A celebrated result of Karpelevi\v c describes $\Theta_n,$ the collection of all eigenvalues arising from the stochastic matrices of order $n.$ The boundary of $\Theta_n$ consists of roots of certain one-parameter families of polynomials,…
Eigenvalues of stochastic matrices have been studied from two complementary perspectives. The individual eigenvalues are characterised through the well-established Karpelevich regions. The spectrum as a whole has also been analysed,…
Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the…
The problem of determining $DS_n$, the complex numbers that occur as an eigenvalue of an $n$-by-$n$ doubly stochastic matrix, has been a target of study for some time. The Perfect-Mirsky region, $PM_n$, is contained in $DS_n$, and is known…
In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(\Gamma)$ over Lyapunov curves have the shape of circular arcs. About 25…
The problem of finding "small" sets that meet every straight-line which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an {\em opaque set} or a {\em barrier} for that region. We consider the…
In 1933, Rado conjectured that for any positive integer n, there is always a linear homogeneous equation with degree of regularity n. In proving this conjecture, Alexeev and Tsimerman, and independently Golowich, found that some equations…
For a convex domain K in the complex plane C, the well-known general Markov inequality asserting that a polynomial p of degree n ||p'|| < c(K) n^2 ||p|| holds. On the other hand for polynomials in general, ||p'|| can be arbitrarily small as…
The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics…
In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlev\'e equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase…
Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using…
In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace…
Based on a result of Makeev, in 2012 Blagojevi\'c and Karasev proposed the following problem: given any positive integers $m$ and $1\leq \ell\leq k$, find the minimum dimension $d=\Delta(m;\ell/k)$ such that for any $m$ mass distributions…
In 1956, Dobrushin proved a definitive central limit theorem for non-homogeneous Markov chains. In this note, a shorter and different proof elucidating more the assumptions is given through martingale approximation.
The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on convex polygons. We prove that almost…
For an arrangement of $n$ pseudolines in the real projective plane let us denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a linear on $t_i$ inequality similar to the Hirzebruch one, but with an elementary proof. We…
Kohn introduced in 1979 the algorithm of multipliers to study the subelliptc estimate of the $\bar\partial$-Neumann problem for a smooth weakly pseudoconvex domain in a complex Euclidean space which satisfies D'Angelo's finite type…