Related papers: Hadamard triples generate self-affine spectral mea…

Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the…

Despite the moduli space of triangles being three dimensional, we prove the existence of two triangles which are not isometric to each other for which the first, second and fourth Dirichlet eigenvalues coincide, establishing a numerical…

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices $A_{n}$ and $B_{n}$ rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix $U_{n}$ (i.e.…

Let $ \Omega \subset R^d $ have finite positive Lebesgue measure, and let $ \mathcal{L}^{2}(\Omega) $ be the corresponding Hilbert space of $ \mathcal{L}^{2} $-functions on $ \Omega $. We shall consider the exponential functions $…

In this note we investigate some properties of equilibrium states of affine iterated function systems, sometimes known as K\"aenm\"aki measures. We give a simple sufficient condition for K\"aenm\"aki measures to have a gap between certain…

Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…

Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine…

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…

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…

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…

For singular numbers of integral operators of the form $u(x)\mapsto \int F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)\mu(dY),$ with measure $\mu$ singular with respect to the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the counting…

We consider spectral decomposition of the harmonic oscillator in $\mathbb R^n$ in terms of two different orthonormal bases in $L^2(\mathbb R^n)$ consisting of its eigenfunctions. Then, using purely functional analysis tools we provide…

An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension…

We construct spectral triples on a class of particular inductive limits of matrix-valued function algebras. In the special case of the Jiang-Su algebra we employ a particular $AF$-embedding.

We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting…

We investigate how spectral properties of a measure preserving system $(X,\mathcal{B},\mu,T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\to\mathbb{N}$ we provide natural…

In this paper we use the equivalence result originally proved by the author which relates a multi-resolution analysis (MRA) of $L^2(R)$ and an orthonormal set of single electron wave-functions in the lowest Landau level, to build up a…

We explore spectral duality in the context of measures in $\br^n$, starting with partial differential operators and Fuglede's question (1974) about the relationship between orthogonal bases of complex exponentials in $L^2(\Omega)$ and…

Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr\"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between…

In this paper we consider the (weighted) spectral measure $\mu_n$ of a $n\times n$ random matrix, distributed according to a classical Gaussian, Laguerre or Jacobi ensemble, and show a moderate deviation principle for the standardised…