Related papers: Resonance between planar self-affine measures

Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined…

We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also…

Poincare-type series, such as Selberg's, are known to produce automorphic functions, in the hyperbolic half-plane, the decompositions of which into eigenfunctions (genuine or generalized) of the automorphic Laplacian contain all modular…

We establish a complete algebraic characterization of self-similar iterated function systems $\Phi$ on $\mathbb{R}^{d}$, for which there exists a positive probability vector $p$ so that the Fourier transform of the self-similar measure…

Single-particle resonance parameters and wave functions in spherical and deformed nuclei are determined through analytic continuation in the potential strength. In this method, the analyticity of the eigenvalues and eigenfunctions of the…

The Haar functional on the quantum $SU(2)$ group is the analogue of invariant integration on the group $SU(2)$. If restricted to a subalgebra generated by a self-adjoint element the Haar functional can be expressed as an integral with a…

Experiments have confirmed the presence of a mass gap between the Standard Model and potential New Physics. Consequently, the exploration of effective field theories to detect signals indicative of Physics Beyond the Standard Model is of…

We study a hard regime of stimulation of two-frequency oscillations in the main resonance equation with a fast oscillating external force: $ \ve i \psi' + |\psi|^2\psi = \exp\big(it^2/ (2\ve)\big), 0<\ve\ll1$. This phenomenon is caused by…

Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is…

We show that if a compact connected $n$-dimensional manifold $M$ has a conformal class containing two non-homothetic metrics $g$ and $\tilde g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite covering, either $n=4$…

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsed\`a and Misiurewicz and contributing to a question of Navas on the…

In this paper, we consider a product of a symmetric stable process in $\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that…

In this paper, we establish the following result: Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, let $Y$ be a reflexive real Banach space, and let $\varphi, \psi:Y\to {\bf R}$ be two sequentially weakly lower semicontinuous…

Let $\mu$ be a planar Mandelbrot measure and $\pi_*\mu$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of $\pi_*\mu$. We first show that $\pi_*\mu$ is exactly dimensional, with…

In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa\{\vphi_i\}_{i \in…

The autocorrelation function, A(t), measures the overlap (in Hilbert space) of a time-dependent quantum mechanical wave function, psi(x,t), with its initial value, psi(x,0). It finds extensive use in the theoretical analysis and…

We are concerned with an harmonic analysis in Hilbert spaces $L^2(\mu)$, where $\mu$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_\lambda(x) = \exp(2\pi i…

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice…

We present a theoretical and experimental study of electrodynamics of a planar spiral superconducting resonator of a finite length. The resonator is made in the form of a monofilar Archimedean spiral. By making use of a general model of…