Related papers: Index Theorems for Polynomial Pencils

We study the eigenvalues of a Laplace-Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. By assigning partitions with the restricted uniform measure,…

Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix…

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the…

Relative index theorems, which deal with what happens with the index of elliptic operators when cutting and pasting, are abundant in the literature. It is desirable to obtain similar theorems for other stable homotopy invariants, not the…

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating…

An eigenvalue $\lambda$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues…

This paper is devoted to the eigenvalue problem for the quantum Gaudin system. We prove the universal correspondence between eigenvalues of Gaudin Hamiltonians and the so-called G-opers without monodromy in general gl(n) case modulo a…

We prove the existence and uniqueness of a *projectively equivariant symbol map*, which is an isomorphism between the space of bidifferential operators acting on tensor densities over $R^n$ and that of their symbols, when both are…

The relation between the spectra of operator pencils with unbounded coefficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics.…

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained…

We present an elementary proof of a generalization of Kirchoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which only involves the…

The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the…

We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that…

For an $n \times n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of ${1,...,n}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\IR^{n}_{+}$. Motivated by this…

For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known…

In this paper, we describe the structure of the Laplace characteristic polynomial $\chi_n(\lambda)$ for the infinite family of graphs $H_n=H_n(G_1,\,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with…

The multidimensional chain rule formula for analytic functions and its generalisation to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to…

Let $K$ be an {\em arbitrary} field of characteristic $p>0$, let $A$ be one of the following algebras: $P_n:= K[x_1, ..., x_n]$ is a polynomial algebra, $\CD (P_n)$ is the ring of differential operators on $P_n$, $\CD (P_n)\t P_m$, the…

The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a…

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the…