Related papers: Structures and Numerical Ranges of Power Partial I…
We completely characterize the higher rank numerical range of the matrices of the form $J_n(\alpha)\oplus\beta I_m$, where $J_n(\alpha)$ is the $n\times n$ Jordan block with eigenvalue $\alpha$. Our characterization allows us to obtain…
Certain $*$-semigroups are associated with the universal $C^*$-algebra generated by a partial isometry, which is itself the universal $C^*$-algebra of a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is emphasized, and…
We show that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar Poincar\'e groups. Using these symmetries, we show that the intricate sums appearing in the…
A model is considered in which the scale of the heavy singlet neutrinos is a few orders of magnitude below the grand unification scale and where right-handed vector bosons play still a negligible role. In a basis with diagonal up-quark and…
A periodic structure sandwiched between two homogeneous media can support bound states in the continuum (BICs) that are valuable for many applications. It is known that generic BICs in periodic structures with an up-down mirror symmetry and…
We study representations of integral vectors in a number system with a matrix base $M$ and vector digits. We focus on the case when $M$ is similar to $J_n$, the Jordan block of $1$ of size $n$. If $M=J_2$, we classify digit sets of size 2…
Quasiperiodic systems offer an appealing intermediate between long-range ordered and genuine disordered systems, with unusual critical properties. One-dimensional models that break the so-called self-dual symmetry usually display a mobility…
Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of…
For a given set of $n\times n$ matrices $\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\mathcal F$, denoted by $W_C({\mathcal F})$. We obtain basic algebraic and topological properties of…
Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the…
We show that a metric Jordan curve $\Gamma$ is \emph{bounded turning} if and only if there exists a \emph{weak-quasisymmetric} homeomorphism $\phi\colon \mathsf{S}^1 \to \Gamma$.
The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show…
We present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of Wick contractions. The method gives exact results for monomials of low order. For higher--order monomials, it leads to an error…
One of the foundations of the Standard Model of Cosmology is statistical isotropy, which can be tested, among other probes, through the study of the Cosmic Microwave Background (CMB). However, a hemispherical power asymmetry on large scales…
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous…
The central elements of the algebra of monodromy matrices associated with the $\mathbb{Z}_n$ R-matrix are studied. When the crossing parameter $w$ takes a special rational value $w=\frac{n}{N}$, where $N$ and $n$ are positive coprime…
We consider a Weitzenb\"ock derivation $\Delta$ acting on a polynomial ring $R=K[\xi_1,\xi_2,...,\xi_m]$ over a field $K$ of characteristic 0. The $K$-algebra $R^\Delta = \{h \in R \mid \Delta(h) = 0\}$ is called the algebra of constants.…
We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(ln x)^2 potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have multifractal eigenstates. If the coefficient…
We study the boundary of the liquid region $\mathcal{L}$ in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line $(x,1)$, $x\in\mathbb{R}\equiv…
A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of R^m. Let l be any nonnegative integer. We prove…