Related papers: The Geometry of Nodal Sets and Outlier Detection

We study the growth of Laplacian eigenfunctions $ -\Delta \phi_k = \lambda_k \phi_k$ on compact manifolds $(M,g)$. H\"ormander proved sharp polynomial bounds on $\| \phi_k\|_{L^{\infty}}$ which are attained on the sphere. On a `generic'…

Given a real-valued function $f$ defined over a manifold $M$ embedded in $\mathbb{R}^d$, we are interested in recovering structural information about $f$ from the sole information of its values on a finite sample $P$. Existing methods…

We consider a Laplace eigenfunction $\varphi_\lambda$ on a smooth closed Riemannian manifold, that is, satisfying $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$. We introduce several observations about the geometry of its vanishing…

It is an open problem in general to prove that there exists a sequence of $\Delta_g$-eigenfunctions $\phi_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(\phi_{j_k}) $ of nodal domains tends to infinity with the eigenvalue.…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the…

Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure:…

It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the…

We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$ and investigate a density property of their zero sets. More precisely, let $f=\sum_{k=1}^m a_k…

We study the size of nodal sets of Laplacian eigenfunctions on compact Riemannian manifolds without boundary and recover the currently optimal lower bound by comparing the heat flow of the eigenfunction with that of an artifically…

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…

We look at the average sum of the Euler's phi function $\phi{(n)}$ and it's relation with the visibility of a point from the origin.We show that $\forall{\hspace{0.05in}{k} \ge{1}},k\in\mathbb{N},\exists$ a $k$$\times$$k$ grid in the 2D…

We consider a family of domains $(\Omega_N)_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set $\Omega_0 = \{(x,y): 0<y<1, -\phi(y)<x<0\}$, for a Lipschitz function $\phi\geq 0$. We derive full asymptotic expansions, as…

Let $\mathcal{F}$ be a family of subsets of a ground set $\{1,\ldots,n\}$ with $|\mathcal{F}|=m$, and let $\mathcal{F}^{\updownarrow}$ denote the family of all subsets of $\{1,\ldots,n\}$ that are subsets or supersets of sets in…

Let $({\mathcal{X}},g)$ be a closed Riemmanian manifold of dimension $n>0$. Let $\Delta$ be the Laplacian on ${\mathcal{X}}$, and let $(e\_k)\_k$ be an $L^2$-orthonormal and dense family of Laplace eigenfunctions with respective eigenvalues…

We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the function $f:\mathbb{T} \rightarrow \mathbb{R}$ satisfy $\hat{f}(k) \geq c|k|^{-2}$ and define a…

Recently, W. M. Schmidt and L. Summerer introduced a new theory which allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and to discover new ones. They…

We investigate the distance function $\boldsymbol{\delta}_{K}^{\phi}$ from an arbitrary closed subset $ K $ of a~finite-dimensional Banach space $ (\mathbf{R}^{n}, \phi) $, equipped with a uniformly convex $\mathcal{C}^{2}$-norm $ \phi $.…

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the…

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in…

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional…