Related papers: On the Random Wave Conjecture for Dihedral Maa{\ss…
We show algorithmic randomness versions of the two classical theorems on subsequences of normal numbers. One is Kamae-Weiss theorem (Kamae 1973) on normal numbers, which characterize the selection function that preserves normal numbers.…
Understanding the asymptotic behavior of the number of Galois orbits of newforms in $S_k(\Gamma_0(N), \Psi)$ as the weight increases is a central problem motivated by Maeda's conjecture. For trivial nebentypus, prior work of Dieulefait,…
Temporal and spatio-temporal (turbulence) distributed chaos in B\'{e}nard-Marangoni and Rayleigh-B\'{e}nard convection have been studied using results of laboratory experiments and direct numerical simulations in the terms of effective…
We consider a general model of Hamiltonian wave systems with triple resonances, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. In this asymptotic limit we show that the correct…
We use a third-order perturbation theory and Melnikov's method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points…
We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum $\sim 1/k$, where $k$ is the wave number. From very-high-resolution numerical simulations, in the…
We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2(…
Let $(\lambda_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta>0$, under the Ramanujan-Petersson conjecture for $\rm GL_2$ Maass forms,…
We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel…
We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over…
We resolve a long-standing riddle in quantum chaos, posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length. Depending on the boundary conditions for the quantum wave…
We study numerically the Cauchy problem for equivariant wave maps from 3+1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures.…
This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn's problem, the…
The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler…
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-sphere ($d\ge 2$). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and…
A refinement of so-called fast Johnson-Lindenstrauss transform, due to Ailon and Chazelle (2006), and Matou\v{s}ek (2008), is proposed. While it preserves the time efficiency and simplicity of implementation of the original construction, it…
We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of…
Mason's Conjecture asserts that for an $m$--element rank $r$ matroid $\M$ the sequence $(I_k/\binom{m}{k}: 0\leq k\leq r)$ is logarithmically concave, in which $I_k$ is the number of independent $k$--sets of $\M$. A related conjecture in…
We prove a joint value equidistribution statement for Hecke-Maa{\ss} cusp forms on the hyperbolic three-space $\mathbb{H}^3$. This supports the conjectural statistical independence of orthogonal cusp forms.
Multiple chaotic and turbulent regimes in Rayleigh-B\'{e}nard convection have been studied and classified from the onset of deterministic chaos to the fully developed turbulence using the distributed chaos approach supported by results of…