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We consider the problem of the reconstruction of a Schwarz matrix from exactly one given eigenvalue. This inverse eigenvalue problem leads to the Jacobi orthogonal polynomials~$\{P_k^{(-n,n)}\}_{k=0}^{n-1}$ that can be treated as a discrete finite analogue of Bessel polynomials.

We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and they can be viewed as such for other rank $2$ Lie algebras.

Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to each directed edge, and a linear map to each directed edge. Two representations A and A' of G are called isomorphic if there is a system of linear bijections between the vector spaces corresponding to the same vertices that transforms A to A'. We prove that if two representations are isomorphic and close to each other, then their isomorphism can be chosen close to the identity., Comment: 12 pages

Based on a less-known result, we prove a recent conjecture concerning the determinant of a certain Sylvester-Kac type matrix and consider an extension of it.

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$$ as the $\mu$-permanent of $A$, where $\ell(\sigma)$ is the number of inversions of the permutation $\sigma$ in the symmetric group $S_n$. In this note, we review several less known results of the $\mu$-permanent, recalling some of its interesting properties. Some determinantal conjectures are considered and extended to that polynomial. A correction to a previous note is presented as well., Comment: This manuscript is largely based on the talk "On a generalization of the determinant of a matrix" that the author gave in Washington \& Lee University, Lexington, VA, USA, on July 19, 2012. It was submitted to Linear and Multilinear Algebra on September 20, 2015. A reduced form will be published as a Letter to the Editor in the same journal

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph $G$, the ABC index is defined as $\sum_{uv\in E}\sqrt{\frac{d(u) +d(v)-2}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ denotes the set of edges of $G$. In this paper we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a $B_4$-branch and $B_1$ or $B_2$-branches.

Let $\cal A$ and $\cal B$ be two systems consisting of the same vector spaces $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$ and bilinear or sesquilinear forms $A_i,B_i:\mathbb C^{n_{k(i)}}\times\mathbb C^{n_{l(i)}}\to\mathbb C$, for $i=1,\dots,s$. We prove that $\cal A$ is transformed to $\cal B$ by homeomorphisms within $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$ if and only if $\cal A$ is transformed to $\cal B$ by linear bijections within $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$., Comment: 5 pages

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$$ as the $\mu$-permanent of $A$, where $\ell(\sigma)$ is the number of inversions of the permutation $\sigma$ in the symmetric group $S_n$. In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $\mu$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided., Comment: The example for a $\mu$-labeling on page 3 has been corrected

Two sesquilinear forms $\Phi:\mathbb C^m\times\mathbb C^m\to \mathbb C$ and $\Psi:\mathbb C^n\times\mathbb C^n\to \mathbb C$ are called topologically equivalent if there exists a homeomorphism $\varphi :\mathbb C^m\to \mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $\Phi(x,y)=\Psi(\varphi (x),\varphi (y))$ for all $x,y\in \mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix $A$; that is, a direct sum $SAS^*=R\oplus J_{n_1}\oplus\dots\oplus J_{n_p}$, in which $S$ and $R$ are nonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ singular Jordan block. In this paper, we prove that $\Phi$ and $\Psi$ are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands $J_{n_i}$ and replacement of $R\in\mathbb C^{r\times r}$ by a nonsingular matrix $R'\in\mathbb C^{r\times r}$ such that $R$ and $R'$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained., Comment: 11 pages

We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$ S_{m,v}=\left(\frac \pi{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{k\pi}{2m}\right)\, , $$ We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$ S_{m,v}=\left(\frac \pi{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{k\pi}{2m}\right)\, , $$ where $m$ and $v$ are positive integers. This method not only confirms the results obtained earlier by an empirical method, but it is also much more expedient from a computational point of view. By comparing the formulas from both methods, we derive several new interesting number theoretic results involving symmetric polynomials over the set of quadratic powers up to $(v-1)^2$ and the generalized cosecant numbers. The method is then extended to other related trigonometric power sums including the untwisted Dowker sum. By comparing both forms for this important sum, we derive new formulas for specific values of the N\"{o}rlund polynomials. Finally, by using the results appearing in the tables, we consider more advanced sums involving the product of powers of cotangent and tangent with powers of cosecant and secant respectively.

We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.