Related papers: Stein-Malliavin Approximations for Nonlinear Funct…
We consider sequences of needlet random fields defined as weighted averaged forms of spherical Gaussian eigenfunctions. Our main result is a Central Limit Theorem in the high energy setting, for the boundary lengths of their excursion sets.…
By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$, then the sequence of random variables…
We recently introduced a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics…
In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that the eigenfunction cannot be a polynomial on any open set, which may be reviewed as a refinement of the classic unique…
We provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of $\ell^2(\mathbb{N}^*)$, the space of real valued sequences with finite $\ell^2$ norm. We use this bound to estimate the…
A new Berry-Esseen bound for non-linear functionals of non-symmetric and non-homogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin-Stein method and an analysis of the discrete…
We analyse some PT-symmetric oscillators with $T_{d}$ symmetry that depend on a potential parameter $g$. We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of $g$. Pairs of…
We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the…
We consider nonlinear functionals of discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{2}$, including i.i.d. supercritical percolation clusters, where the conductances are possibly…
We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for several years -- shows that the so-called…
By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in…
The approximation of integral type functionals is studied for discrete observations of a continuous It\^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions…
For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}( A(\cdot/\varepsilon) \nabla) $ with tiny $\varepsilon>0$, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions…
This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting…
We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an…
This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner.…
We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a…
This paper derives noncentral limit theorems (NCLTs) for suitable scaling of functionals of spatially homogeneous and isotropic, and stationary in time, LRD Gaussian subordinated Spatiotemporal Random Fields (STRFs) with Hermite rank equal…
Quantitative multivariate central limit theorems for general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal…
This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are…