Related papers: Non-spectral problem for the planar self-affine me…

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the…

In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates…

We consider the mixed Dirichlet-conormal problem on irregular domains in $\mathbb{R}^d$. Two types of regularity results will be discussed: the $W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain is assumed to be…

The spectral and scattering properties of non-selfadjoint problems pose a mathematical challenge. Apart from exceptional cases, the well-developed methods used to examine the spectrum of selfadjoint problems are not applicable. One of the…

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, the reduced free product von Neumann algebra of $M_2(\cc)$ with $M_2(\cc)$. Examples include $AB$ and $A+B$, where A and…

$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation.…

Strong evidence indicates that the spectrum of planar anomalous dimensions of N=4 super Yang-Mills theory is given asymptotically by Bethe equations. A curious observation is that the Bethe equations for the psu(1,1|2) subsector lead to…

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite…

We solve the Neumann problem in the half space $\mathbb{R}^{n+1}_+$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in the negative smoothness space $\dot…

Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where $D$ and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \times N$ and $n \times n$, and where $X$ is a random matrix with…

Nonsmooth nonconvex optimization problems involving the $\ell^p$ quasi-norm, $p \in (0, 1]$, of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary…

We consider the nonlinear Neumann eigenvalue problem in outward cuspidal domains with a weighted measure. Using composition operators on Sobolev spaces, we establish embeddings of Sobolev spaces into weighted Lebesgue spaces. These…

Given a domain $\Omega\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $\Lambda\subset \Bbb R^d$, we say that $(\Omega, \Lambda)$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal…

This paper is concerned with a shape optimization problem governed by a non-smooth PDE, i.e., the nonlinearity in the state equation is not necessarily differentiable. We follow the functional variational approach of [40] where the set of…

We consider the problem -{\epsilon}^2\Delta_gu+u = |u|^{p-2}u in M, where (M,g) is a symmetric Riemannian manifold. We give a multiplicity result for antisymmetric changing sign solutions.

We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting…

Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set-valued equation…

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main…

For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x \quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when $b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point, but…

The paper studies integral functionals with non-smooth functions from L_2 defined on solutions of ODEs. Some regularity is obtained in the form of estimates of L_2-norm for these functionals. This result is used for regularization of…