Related papers: Morse inequalities for ordered eigenvalues of gene…
Motivated by the nodal distribution universality conjecture for discrete operators on graphs and by the spectral analysis of their maximal abelian covers, we consider a family of Hermitian matrices $h_{\alpha}$ obtained by varying the…
In this paper we consider generalized eigenvalue problems for a family of operators with a polynomial dependence on a complex parameter. This problem is equivalent to a genuine non self-adjoint operator. We discuss here existence of non…
The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…
We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these…
We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which…
We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree $d+1$. The polynomials that we investigate are…
The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. We define these manifolds as submanifolds of $\R^n$ with a finite number of conical singularities. To formulate a good Morse theory we must use an…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
We provide a new approach to studying the moduli space of curves via Morse theory and hyperbolic geometry, by introducing a family of Morse functions on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$…
The problem of membrane topology in the matrix model of M-theory is considered. The matrix regularization procedure, which makes a correspondence between finite-sized matrices and functions defined on a two-dimensional base space, is…
We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship between the index defined in [44] and [45]. Then we generalize the method of saddle point…
In this work we consider generic coalescing of eigenvalues of smooth complex valued matrix functions depending on 2 parameters. We call generic cuspidal points the parameter values where eigenvalues coalesce and we discuss the relation…
A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any other eigenvalue $\mu$. Nonnegative matrices and symmetric ones are typical examples of this…
In this work we present a framework for studying the eigenvalues of a family of matrices with a particular displacement structure. The family admits a specific decomposition as the product of an upper and a lower triangular matrices having…
Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix…
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient…
To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There…
Symplectic geometry plays an increasingly important role in mathematics, physics and applications, and naturally gives rise to interesting matrix families and properties. One of these is the notion of symplectic eigenvalues, whose existence…
Along with the partition of a planar bounded domain $\Omega$ by the nodal set of a fixed eigenfunction of the Laplace operator in $\Omega$, one can consider another natural partition of $\Omega$ by, roughly speaking, gradient flow lines of…
This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we…