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

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition, the dimension equals to the…

In the present note we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs): from one side they could look standard, since they are, real, symmetric…

We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem: $$ (P_\la) \left\{ \begin{array}{lr}\ds \quad (-\Delta)^{s_1}u+\ba (-\Delta)^{s_2}_{p}u = \la a(x)|u|^{q-2}u+…

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $E(\Lambda,\varphi) = \{e^{2\pi i \lambda\cdot\varphi(x)}: \lambda\in\Lambda\}$ where the phase function $\varphi$ is a Borel measurable which…

Let $\mu$ be a measure on the real line $\mathbb{R}$ such that $\int_{\mathbb{R}}\frac{d\mu(t)}{1+t^2} < \infty$ and let $a>0$. Assume that the norms $\|f\|_{L^2(\mathbb{R})}$ and $\|f\|_{L^2(\mu)}$ are comparable for functions $f$ in the…

In a previous work by {\L}aba and Wang, it was proved that whenever there is a Hadamard triple $(N,{\mathcal D},{\mathcal L})$, then the associated one-dimensional self-similar measure $\mu_{N,{\mathcal D}}$ generated by maps $N^{-1}(x+d)$…

Let $b\ge 2$ be a given integer. In this paper, we show that there only finitely many positive integers $d$ which are not squares, such that the Pell equation $X^2-dY^2=1$ has two positive integer solutions $(X,Y)$ with the property that…

We study $L^q$-spectra of planar self-affine measures generated by diagonal systems with an emphasis on providing closed form expressions. We answer a question posed by Fraser in 2016 in the negative by proving that a certain natural closed…

For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) r(A D^2) >= r(AD)^2, leave open the question of what determines the order of r(A^2 D^2) with respect to…

Starting with a manifestly conformal ($O(d,2)$ invariant) mechanics model in $d$ space and 2 time dimensions, we derive the action for a massless spinning particle in $d$-dimensional anti-de Sitter space. The action obtained possesses both…

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients $\mathbf A, \mathbf b,\gamma$ in $L^\infty$ and symmetric and uniformly positive definite coefficient matrix $\mathbf A$, this paper…

In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures…

We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right.…

In this article we are interested for the numerical study of nonlinear eigenvalue problems. We begin with a review of theoretical results obtained by functional analysis methods, especially for the Schrodinger pencils. Some recall are given…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where $\Delta_p$ is the $p-$laplacian, with $1<p<n$. The…

Uniformly perfect measures are a common generalisation of Ahlfors regular measures, self-conformal measures on the line, and their push-forwards under sufficiently regular maps. We show that every uniformly perfect measure $\sigma$ on a…

Let $A$ be an expanding integer $n\times n$ matrix and $D$ be a finite subset of $Z^n$. The self-affine set $T=T(A,D)$ is the unique compact set satisfying the equality $A(T)=\cup_{d\in D} (T+d)$. We present an effective algorithm to…

We study two-to-two scattering amplitudes of a scalar particle of mass $m$. For simplicity, we assume the presence of $\mathbb{Z}_2$ symmetry and that the particle is $\mathbb{Z}_2$ odd. We consider two classes of amplitudes: the fully…

Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $\mu$ be the M\"obius function and $\mu_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=\mu^2g$ and $f=\mu_2^2g$ have unbounded partial sums. Our proofs…

We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $\Omega$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such…