Related papers: Rational points in Cantor sets and spectral eigenv…

We show that the spectral measure of any non-commutative polynomial of a non-commutative $n$-tuple cannot have atoms if the free entropy dimension of that $n$-tuple is $n$ (see also work of Mai, Speicher, and Weber). Under stronger…

Let R be a real closed field and A=R[x_1,...,x_n]. Let sper A denote the real spectrum of A. There are two kinds of points in sper A : finite points (those for which all of |x_1|,...,|x_n| are bounded above by some constant in R) and points…

We study the spectral dimensions and spectral asymptotics of Krein-Feller operators for arbitrary finite Borel measures on $\left(0,1\right).$ Connections between the spectral dimension, the $L^{q}$-spectrum, the partition entropy and the…

Contour-integral-based rational filter leads to interior eigensolvers for non-Hermitian generalized eigenvalue problems. Based on Zolotarev's third problem, this paper proves the asymptotic optimality of the trapezoidal quadrature of the…

We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = \lambda K(x) |u|^{p-2} u \hspace{.1cm} \mbox {…

Let $\mu_{M,D}$ be the self-affine measure generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^n$. It is well known that the two measures $\mu_{M,D}$ and $\mu_{\tilde{M},\tilde{D}}$…

Let $S$ be a connected surface possibly with boundary, $\mu$ a finite Borel measure which is positive on open sets and $f:S\to S$ a homeomorphism preserving $\mu$. We prove that if $K$ is a compact connected subset of $S$ and $L$ is a…

The nonnegative inverse eigenvalue problem (NIEP) is to characterize the spectra of entrywise nonnegative matrices. A finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix. Monov…

We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce…

For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problems $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the…

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that…

This article introduces the $L_p$-Gauss dual curvature measure and proposes its related $L_p$-Gauss dual Minkowski problem as: for $p,q\in\mathbb{R}$, under what necessary and/or sufficient condition on a non-zero finite Borel measure $\mu$…

The joint spectral radius of a pair of 2x2 real matrices $(A_0,A_1)\in M_2(\mathbb{R})^2$ is defined to be $r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate of the norm of…

In this short technical note, we extend a recently published result [Liao2017] on the Perron root (or the spectral radius) of non-negative matrices to real-valued non-negative kernels on an arbitrary measurable space $(\mathrm{E},…

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under…

Given a closed subset $K$ in $\mathbb{R}$, the rational $K$-truncated moment problem ($K$-RTMP) asks to characterize the existence of a positive Borel measure $\mu$, supported on $K$, such that a linear functional $\mathcal{L}$, defined on…

We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kova\v{c}'s maximal restriction principle to show…

There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-Hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a…

By the m-spectrum of a real number q>1 we mean the set Y^m(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their…

Let $A$ be an $n \times n$ matrix with rational entries and let \[ \mathbb{Z}^n[A] := \bigcup_{k=1}^{\infty} \left( \mathbb{Z}^n + A\mathbb{Z}^n + \dots + A^{k-1}\mathbb{Z}^n\right) \] be the minimal $A$-invariant $\mathbb{Z}$-module…