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Let $\Omega$ be a bounded Lipshcitz domain in $\mathbb{R}^n$ and we study boundary behaviors of solutions to the Laplacian eigenvalue equation with constant Neumann data. \begin{align} \label{cequation0} \begin{cases} -\Delta u=cu\quad…

We prove a Schwarz lemma for a domain E in 3-dimensional complex space that arises in connection with a problem in H infinity control theory. We describe a class of automorphisms of E and determine the distinguished boundary of E. We obtain…

Let $\Gamma_n$, $n \geq 2$, denote the symmetrized polydisc in $\mathbb{C}^n$, and $\Gamma_1$ be the closed unit disc in $\mathbb{C}$. We provide some characterizations of elements in $\Gamma_n$. In particular, an element $(s_1, \ldots,…

It was shown recently that associated with a pair of real sequences $\{\{c_{n}\}_{n=1}^{\infty}, \{d_{n}\}_{n=1}^{\infty}\}$, with $\{d_{n}\}_{n=1}^{\infty}$ a positive chain sequence, there exists a unique nontrivial probability measure…

We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$.…

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions $p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres…

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction…

Extended particles are considered in terms of the fields on the Poincar\'{e} group. Dirac like wave equations for extended particles of any spin are defined on the various homogeneous spaces of the Poincar\'{e} group. Free fields of the…

There is a natural conjugation action on the set of endomorphism of $\P^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $\P^N$, denoted $\mathcal{M}_d^N$. We construct invariant…

We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some rank one matrix T. A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such…

With $<\cdot>$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,\mu) := < \prod_{l=1}^N \chi_{(0,\infty)\backslash I}^{(l)} (\lambda - \lambda_l)^\mu>$…

We give a solution to Pick's interpolation problem on the unit polydisc in $\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite kernels…

In this study, we present a novel family of Meixner-type $d$-orthogonal polynomials, which are distinguished as a particular subset of multiple orthogonal polynomials. We demonstrate their connection to the Lie algebra $\mathfrak{su}(1,1)$…

We consider local semitransparent Neumann boundary conditions for a quantum scalar field as imposed by a quadratic coupling to a source localized on a flat codimension-one surface. Upon a proper regularization to give meaning to the…

We derive the multiplicative duality "q<->1/q" and other typical mathematical structures as the special cases of the (mu,nu,q)-relation behind Tsallis statistics by means of the (mu,nu)-multinomial coefficient. Recently the additive duality…

We give a characterization for the existence of a holomorphic interpolant on the unit polydisc $\mathbb{D}^n,$ $n\geq 2,$ for prescribed three-point Pick--Nevanlinna data. One of the key steps is a characterization for the existence of an…

We study supersymmetric domain walls of four dimensional $SU(N)$ SQCD with $N$ and $N+1$ flavors. In $4d$ we analyze the BPS differential equations numerically. In $3d$ we propose the $\mathcal{N}=1$ Chern-Simons-Matter gauge theories…

We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${\rm U}(n)$ part are considered…

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} \Delta + V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported…

This paper is an extension of the author's lecture "Unique Determination of Polyhedral Domains and $p$-Moduli of Path Families" given at the International Conference "Metric Geometry of Surfaces and Polyhedra" dedicated to the 100th…