Related papers: Eigenvalue contour lines of Kac-Murdock-Szego matr…
We consider a family of $2 \times 2$ operator matrices ${\mathcal A}_\mu(k),$ $k \in {\Bbb T}^3:=(-\pi, \pi]^3,$ $\mu>0$, acting in the direct sum of zero- and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of…
We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only…
A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain…
A bordering of GUE matrices is considered, in which the bordered row consists of zero mean complex Gaussians N$[0,\sigma/2] + i {\rm N}[0,\sigma/2]$ off the diagonal, and the real Gaussian N$[\mu,\sigma/\sqrt{2}]$ on the diagonal. We…
In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among…
In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using…
In this paper, we consider rational cuspidal plane curves having exactly two cusps whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal…
We study the eigenvalue distributions for sums of independent rank-one $k$-fold tensor products of large $n$-dimensional vectors. Previous results in the literature assume that $k=o(n)$ and show that the eigenvalue distributions converge to…
A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical…
For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…
We explore the concept of eigenvalue avoidance, which is well understood for real symmetric and Hermitian matrices, for other classes of structured matrices. We adopt a differential geometric perspective and study the generic behaviour of…
This paper contains two main results. First, we provide combinatorial branching rules for $\text{GL}_n \downarrow \text{O}_n$ and $\text{GL}_{2n} \downarrow \text{Sp}_{2n}$ extending the Littlewood restriction rules. Second, we use these…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures-Wasserstein, Hellinger and Log-Determinant metrics. More precisely, we show that the matrix power means (Kubo-Ando and…
In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial…
Levels and sublevels in arrangements -- and, dually, $k$-sets and $(\leq k)$-sets -- are fundamental notions in discrete and computational geometry and natural generalizations of convex polytopes, which correspond to the $0$-level. A…
The still-unsolved problem of determining the set of eigenvalues realized by $n$-by-$n$ doubly stochastic matrices, those matrices with row sums and column sums equal to $1$, has attracted much attention in the last century. This problem is…
Many eigenvalue matrix models possess a peculiar basis of observables which have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb…
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…
Spectral properties of random matrices play an important role in statistics, machine learning, communications, and many other areas. Engaging results regarding the convergence of the empirical spectral distribution (ESD) and the…