On the variance of the nodal volume of arithmetic random waves

Rudnick and Wigman (2008) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the d-dimensional torus is O ⁢ ( E / 𝒩 ) {O(E/\mathcal{N})} , as E → ∞ {E\to\infty} , where E is the energy and 𝒩 {\mathcal{N}} is the dimension of the eigenspace corresponding to E. Previous results have established this with stronger asymptotics when d = 2 {d=2} and d = 3 {d=3} . In this brief note we prove an upper bound of the form O ⁢ ( E / 𝒩 1 + α ⁢ ( d ) - ϵ ) {O(E/\mathcal{N}^{1+\alpha(d)-\epsilon})} , for any ϵ > 0 {\epsilon>0} and d ≥ 4 {d\geq 4} , where α ⁢ ( d ) {\alpha(d)} is positive and tends to zero with d. The power saving is the best possible with the current method (up to ϵ) when d ≥ 5 {d\geq 5} due to the proof of the ℓ 2 {\ell^{2}} -decoupling conjecture by Bourgain and Demeter.

Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

We investigate the following problem $$\begin{aligned} -\mathrm{div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \left( |x|^{-\theta }*\frac{|u|^{b}}{|x|^{\alpha }}\right) \frac{|u|^{b-2}}{|x|^{\alpha }}u+\lambda \left( |x|^{-\gamma }*\frac{|u|^{c}}{|x|^{\beta }}\right) \frac{|u|^{c-2}}{|x|^{\beta }}u \quad \text { in }{\mathbb {R}}^{N}, \end{aligned}$$ - div ( v ( x ) | ∇ u | m - 2 ∇ u ) + V ( x ) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where $$b, c, \alpha , \beta >0$$ b , c , α , β > 0 , $$\theta ,\gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$N\ge 3$$ N ≥ 3 , $$2\le m< \infty$$ 2 ≤ m < ∞ and $$\lambda \in {\mathbb {R}}$$ λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

On the number of excursion sets of planar Gaussian fields

The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

Topology of the Nodal Set of Random Equivariant Spherical Harmonics on 𝕊3

We show that real and imaginary parts of equivariant spherical harmonics on ${{\mathbb{S}}}^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left (\frac{N^2 - m^2}{2} + N\right ) $. Hence, if $\frac{m}{N}= c $ for fixed $0 &lt; c &lt; 1$, then the genus has order $N^3$.

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

In this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

Constructing a polynomial whose nodal set is any prescribed knot or link

We describe an algorithm that for every given braid [Formula: see text] explicitly constructs a function [Formula: see text] such that [Formula: see text] is a polynomial in [Formula: see text], [Formula: see text] and [Formula: see text] and the zero level set of [Formula: see text] on the unit three-sphere is the closure of [Formula: see text]. The nature of this construction allows us to prove certain properties of the constructed polynomials. In particular, we provide bounds on the degree of [Formula: see text] in terms of braid data.

Constructing links of isolated singularities of polynomials ℝ4 → ℝ2

We show that if a braid [Formula: see text] can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328 ) can be extended to a construction of a polynomial [Formula: see text] with the closure of [Formula: see text] as the link of an isolated singularity of [Formula: see text], showing that the closure of [Formula: see text] is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of [Formula: see text] over [Formula: see text].