arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 163 records · Page 9Linked to original sources

arXiv2019

A Universality Law For Sign Correlations of Eigenfunctions of Differential Operators

We establish a universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a form of WKB approximation on compact intervals. This includes eigenfunctions of generic Schrödinger operators, as well as Laguerre and Chebyshev polynomials. Given two distinct points $x, y \in \mathbb{R}$, we ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all $x \neq y$, $$ \frac{1}{3} \leq \lim_{N \to \infty} \frac{1}{N} \# \left\{0 \leq n < N: \mathrm{sgn}(w_n(x)) = \mathrm{sgn}(w_n(y)) \right\} \leq \frac{2}{3}, $$ and that these bounds are optimal, and can be attained. Our methods extend to other questions of similar flavor and we also discuss a number of open problems.

math.CA↗

arXiv2019

A forgotten Theorem of Schoenberg on one-sided integral averages

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function for which we want to take local averages. Assuming we cannot look into the future, the 'average' at time $t$ can only use $f(s)$ for $s \leq t$. A natural way to do so is via a weight $ϕ$ and $$ g(t) = \int_{0}^{\infty}{f(t-s) ϕ(s) ds}.$$ We would like that (1) constant functions, $f(t) \equiv \mbox{const}$, are mapped to themselves and (2) $ϕ$ to be monotonically decreasing (the more recent past should weigh more heavily than the distant past). Moreover, we want that (3) if $f(t)$ crosses a certain threshold $n$ times, then $g(t)$ should not cross the same threshold more than $n$ times (if $f(t)$ is the outside wind speed and crosses the Tornado threshold at two points in time, we would like the averaged wind speed to cross the Tornado threshold at most twice). A Theorem implicit in the work of Schonberg is that these three conditions characterize a unique weight that is given by the exponential distribution $$ ϕ(s) = λ^{} e^{-λs} \qquad \mbox{for some} \quad λ> 0.$$

math.CA↗

arXiv2019

Leaky Roots and Stable Gauss-Lucas Theorems

Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the convex hull of its roots. A recent quantitative version Totik shows that if almost all roots are contained in a bounded convex domain $K \subset \mathbb{C}$, then almost all roots of the derivative $p'$ are in a $\varepsilon-$neighborhood $K_{\varepsilon}$ (in a precise sense). We prove another quantitative version: if a polynomial $p$ has $n$ roots in $K$ and $\lesssim c_{K, \varepsilon} (n/\log{n})$ roots outside of $K$, then $p'$ has at least $n-1$ roots in $K_{\varepsilon}$. This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture.

math.CV↗

arXiv2018

On algorithms to calculate integer complexity

We consider a problem first proposed by Mahler and Popken in 1953 and later developed by Coppersmith, Erdős, Guy, Isbell, Selfridge, and others. Let $f(n)$ be the complexity of $n \in \mathbb{Z^{+}}$, where $f(n)$ is defined as the least number of $1$'s needed to represent $n$ in conjunction with an arbitrary number of $+$'s, $*$'s, and parentheses. Several algorithms have been developed to calculate the complexity of all integers up to $n$. Currently, the fastest known algorithm runs in time $\mathcal{O}(n^{1.230175})$ and was given by J. Arias de Reyna and J. van de Lune in 2014. This algorithm makes use of a recursive definition given by Guy and iterates through products, $f(d) + f\left(\frac{n}{d}\right)$, for $d \ |\ n$, and sums, $f(a) + f(n - a)$, for $a$ up to some function of $n$. The rate-limiting factor is iterating through the sums. We discuss potential improvements to this algorithm via a method that provides a strong uniform bound on the number of summands that must be calculated for almost all $n$. We also develop code to run J. Arias de Reyna and J. van de Lune's analysis in higher bases and thus reduce their runtime of $\mathcal{O}(n^{1.230175})$ to $\mathcal{O}(n^{1.222911236})$. All of our code can be found online at: https://github.com/kcordwel/Integer-Complexity.

math.NT↗

arXiv2018

A Stability Version of the Gauss-Lucas Theorem and Applications

Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose $p$ has $n+m$ roots where $n$ are inside the unit disk, $$ \max_{1 \leq i \leq n}{|a_i|} \leq 1, \quad \mbox{and $m$ are outside} \quad \min_{n+1 \leq i \leq n+m}{ |a_i|} \geq d > 1 + \frac{2 m}{n},$$ then $p'$ has $n-1$ roots inside the unit disk and $m$ roots at distance at least $(dn - m)/(n+m) > 1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for $n$ sufficiently large each of the $m$ roots has a critical point at distance $\sim n^{-1}$.

math.CV↗

arXiv2018

On Pointwise Products of Elliptic Eigenfunctions

We consider eigenfunctions of Schrödinger operators on a $d-$dimensional bounded domain $Ω$ (or a $d-$dimensional compact manifold $Ω$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(ϕ_n)_{n \in \mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \mbox{span} \left\{ ϕ_i(x) ϕ_j(x): 1 \leq i,j \leq n\right\} \subseteq L^2(Ω).$$ Clearly, that vector space has dimension $\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $ϕ_i ϕ_j$ of eigenfunctions are simple in a certain sense: for any $\varepsilon > 0$, there exists a low-dimensional vector space $B_n$ that almost contains all products. More precisely, denoting the orthogonal projection $Π_{B_n}:L^2(Ω) \rightarrow B_n$, we have $$ \forall~1 \leq i,j \leq n~ \qquad \|ϕ_iϕ_j - Π_{B_n}( ϕ_i ϕ_j) \|_{L^2} \leq \varepsilon$$ and the size of the space $\mbox{dim}(B_n)$ is relatively small $$ \mbox{dim}(B_n) \lesssim \left( \frac{1}{\varepsilon} \max_{1 \leq i \leq n} \|ϕ_i\|_{L^{\infty}} \right)^d n.$$ In the generic delocalized setting, this bound grows linearly up to logarithmic factors: pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations.

math.SP↗

arXiv2018

Approximating Pointwise Products of Laplacian Eigenfunctions

We consider Laplacian eigenfunctions on a $d-$dimensional bounded domain $M$ (or a $d-$dimensional compact manifold $M$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(e_\ell)_{\ell \in \mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \mbox{span} \left\{ e_i(x) e_j(x): 1 \leq i,j \leq n\right\} \subseteq L^2(M).$$ Clearly, that vector space has dimension $\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $e_i e_j$ of eigenfunctions are simple in a certain sense: for any $\varepsilon > 0$, there exists a low-dimensional vector space $B_n$ that almost contains all products. More precisely, denoting the orthogonal projection $Π_{B_n}:L^2(M) \rightarrow B_n$, we have $$ \forall~1 \leq i,j \leq n~ \qquad \|e_ie_j - Π_{B_n}( e_i e_j) \|_{L^2} \leq \varepsilon$$ and the size of the space $\mbox{dim}(B_n)$ is relatively small: for every $δ> 0$, $$ \mbox{dim}(B_n) \lesssim_{M,δ} \varepsilon^{-δ} n^{1+δ}.$$ We obtain the same sort of bounds for products of arbitrary length, as well for approximation in $H^{-1}$ norm. Pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations.

math.AP↗

arXiv2018

A Compactness Principle for Maximizing Smooth Functions over Toroidal Geodesics

Let $f \in C^2(\mathbb{T}^2)$ have mean value 0 and consider $$ \sup_{γ~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|γ|} \left| \int_γ{ f ~~d\mathcal{H}^1}\right| },$$ where $γ$ ranges over all closed geodesics $γ:\mathbb{S}^1 \rightarrow \mathbb{T}^2$ and $|γ|$ denotes their length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $γ$ attaining the supremum in terms of \textit{smoothness} of the function: for all $s \geq 2$, $$ |γ|^{s} \lesssim_s \left( \max_{|α| = s}{ \| \partial_α f \|_{L^{1}(\mathbb{T}^2)}} \right) \| \nabla f \|_{L^2}^{} \|f\|_{L^2}^{-2}.$$ We also prove a sharp bound for trigonometric polynomials. This seems like an interesting phenomenon. We do not know at which level of generality it holds or whether versions or variants of it could be established in other settings (hyperbolic surfaces, groups,...).

math.CA↗

arXiv2018

The Hermite-Hadamard inequality in higher dimensions

The Hermite-Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions: let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a convex function satisfying $f \big|_{\partial Ω} \geq 0$, then $$ \frac{1}{|Ω|} \int_Ω{f ~d \mathcal{H}^n} \leq \frac{2 π^{-1/2} n^{n+1}}{|\partial Ω|} \int_{\partial Ω}{f~d \mathcal{H}^{n-1}}.$$ The constant $2 π^{-1/2} n^{n+1}$ is presumably far from optimal, however, it cannot be replaced by 1 in general. We prove slightly stronger estimates for the constant in two dimensions where we show that $9/8 \leq c_2 \leq 8$. We also show, for some universal constant $c>0$, if $Ω\subset \mathbb{R}^2$ is simply connected with smooth boundary, $f:Ω\rightarrow \mathbb{R}_{}$ is subharmonic, i.e. $Δf \geq 0$, and $f \big|_{\partial Ω} \geq 0$, then $$ \int_Ω{f~ d \mathcal{H}^2} \leq c \cdot \mbox{inradius}(Ω) \int_{\partial Ω}{ f ~d\mathcal{H}^{1}}.$$ We also prove that every domain $Ω\subset \mathbb{R}^n$ whose boundary is 'flat' at a certain scale $δ$ admits a Hermite-Hadamard inequality for all subharmonic functions with a constant depending only on the dimension, the measure $|Ω|$ and the scale $δ$.

math.CA↗

arXiv2018

Localization of Neumann Eigenfunctions near Irregular Boundaries

It has been empirically observed that eigenfunctions of Laplace's equation $-Δϕ= λϕ$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has some nontrivial implications in acoustics that has been put to real-life use (sound attenuation by noise-protective walls); this short paper describes the mathematical mechanism responsible for this and describes the quantitative strength of the phenomenon for some examples.

math.AP↗

arXiv2018

Ulam Sequences and Ulam Sets

The Ulam sequence is given by $a_1 =1, a_2 = 2$, and then, for $n \geq 3$, the element $a_n$ is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, \dots$, which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors $\left\{v_1, \dots, v_k\right\} \subset \mathbb{R}^n$, we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.

math.CO↗

arXiv2018

An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

The classical Alexandrov-Bakelman-Pucci estimate for the Laplacian states $$ \max_{x \in Ω}{ |u(x)|} \leq \max_{x \in \partial Ω}{|u(x)|} + c_{s,n} \mbox{diam}(Ω)^{2-\frac{n}{s}} \left\| Δu\right\|_{L^s(Ω)}$$ where $Ω\subset \mathbb{R}^n$, $u \in C^2(Ω) \cap C(\overlineΩ)$ and $s > n/2$. The inequality fails for $s = n/2$. A Sobolev embedding result of Milman & Pustylink, originally phrased in a slightly different context, implies an endpoint inequality: if $n \geq 3$ and $Ω\subset \mathbb{R}^n$ is bounded, then $$ \max_{x \in Ω}{ |u(x)|} \leq \max_{x \in \partial Ω}{|u(x)|} + c_n \left\| Δu\right\|_{L^{\frac{n}{2},1}(Ω)},$$ where $L^{p,q}$ is the Lorentz space refinement of $L^p$. This inequality fails for $n=2$ and we prove a sharp substitute result: there exists $c>0$ such that for all $Ω\subset \mathbb{R}^2$ with finite measure $$ \max_{x \in Ω}{ |u(x)|} \leq \max_{x \in \partial Ω}{|u(x)|} + c \max_{x \in Ω} \int_{y \in Ω}{ \max\left\{ 1, \log{\left(\frac{|Ω|}{\|x-y\|^2} \right)} \right\} \left| Δu(y)\right| dy}.$$ This is somewhat dual to the classical Trudinger-Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces, the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

math.AP↗

arXiv2018

Recovering Trees with Convex Clustering

Convex clustering refers, for given $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^p$, to the minimization of \begin{eqnarray*} u(γ) & = & \underset{u_1, \dots, u_n }{\arg\min}\;\sum_{i=1}^{n}{\lVert x_i - u_i \rVert^2} + γ\sum_{i,j=1}^{n}{w_{ij} \lVert u_i - u_j\rVert},\\ \end{eqnarray*} where $w_{ij} \geq 0$ is an affinity that quantifies the similarity between $x_i$ and $x_j$. We prove that if the affinities $w_{ij}$ reflect a tree structure in the $\left\{x_1, \dots, x_n\right\}$, then the convex clustering solution path reconstructs the tree exactly. The main technical ingredient implies the following combinatorial byproduct: for every set $\left\{x_1, \dots, x_n \right\} \subset \mathbb{R}^p$ of $n \geq 2$ distinct points, there exist at least $n/6$ points with the property that for any of these points $x$ there is a unit vector $v \in \mathbb{R}^p$ such that, when viewed from $x$, `most' points lie in the direction $v$ \begin{eqnarray*} \frac{1}{n-1}\sum_{i=1 \atop x_i \neq x}^{n}{ \left\langle \frac{x_i - x}{\lVert x_i - x \rVert}, v \right\rangle} & \geq & \frac{1}{4}. \end{eqnarray*}

stat.ML↗

arXiv2018

Electrostatic Interpretation of Zeros of Orthogonal Polynomials

We study the differential equation $ - (p(x) y')' + q(x) y' = λy,$ where $p(x)$ is a polynomial of degree at most 2 and $q(x)$ is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: the set of real numbers $\left\{x_1, \dots, x_n\right\}$ satisfies $$ p(x_i) \sum_{k = 1 \atop k \neq i}^{n}{\frac{2}{x_k - x_i}} = q(x_i) - p'(x_i) \qquad \mbox{for all}~ 1\leq i \leq n$$ if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on $p(x),q(x)$ whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.

math.CA↗

arXiv2018

Oscillatory functions vanish on a large set

Let $(M,g)$ be a $n-$dimensional, compact Riemannian manifold. We define the frequency scale $λ$ of a function $f \in C^{0}(M)$ as the largest number such that $\left\langle f, ϕ_k \right\rangle =0$ for all Laplacian eigenfunctions with eigenvalue $λ_k \leq λ$. If $λ$ is large, then the function $f$ has to vanish on a large set $$ \mathcal{H}^{n-1} \left\{x:f(x) =0\right\} \gtrsim_{} \left( \frac{ \|f\|_{L^1}}{\|f\|_{L^{\infty}}} \right)^{2 - \frac{1}{n}} \frac{ \sqrtλ}{(\logλ)^{n/2}}.$$ Trigonometric functions on the flat torus $\mathbb{T}^d$ show that the result is sharp up to a logarithm if $\|f\|_{L^1} \sim \|f\|_{L^{\infty}}$. We also obtain a stronger result conditioned on the geometric regularity of $\left\{x:f(x) = 0\right\}$. This may be understood as a very general higher-dimensional extension of the Sturm oscillation theorem.

math.CA↗

arXiv2018

On the Dual Geometry of Laplacian Eigenfunctions

We discuss the geometry of Laplacian eigenfunctions $-Δϕ= λϕ$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' $ α(ϕ_λ, ϕ_μ)$ between eigenfunctions $ϕ_λ$ and $ϕ_μ$ is given by a global average of local correlations $$ α(ϕ_λ, ϕ_μ)^2 = \| ϕ_λ ϕ_μ \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( ϕ_λ(y) - ϕ_λ(x))( ϕ_μ(y) - ϕ_μ(x)) dy} \right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t λ} + e^{-t μ} = 1$. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.

eess.SP↗

arXiv2018

Quantitative Projections in the Sturm Oscillation Theorem

There is $c_{} > 0$ such that for all $f \in C[0,π]$ with at most $d-1$ roots inside $(0,π)$ $$ \sum_{1 \leq n \leq d}{ \left| \left\langle f, \sin\left( n x\right) \right\rangle \right|} \geq κ^{-κ^2 \logκ}\|f\|_{L^2} \qquad \mbox{where} \quad κ= \frac{c_{} \| \nabla f\|_{L^2}}{\|f\|_{L^2}}.$$ This quantifies the Sturm-Hurwitz Theorem and connects a purely topological condition (number of roots) to the Fourier spectrum. It is also one of few estimates on Fourier coefficients from below. The result holds more generally for eigenfunctions of regular Sturm-Liouville problems $$ - (p(x) y'(x))' + q(x) y(x) = λw(x) y(x) \qquad \mbox{on}~(a,b).$$ Sturm-Liouville theory shows the existence of a sequence of solutions $(ϕ_n)_{n=1}^{\infty}$ that form an orthogonal basis of $L^2(a,b)$ with respect to $w(x)dx$. Sturm himself proved that if $f:(a,b) \rightarrow \mathbb{R}$ is a finite linear combinations of $ϕ_n$ having $d-1$ roots inside $(a,b)$, then $f$ cannot be orthogonal to $A = \mbox{span}\left\{ϕ_1, \dots, ϕ_{d}\right\}$. We prove a lower bound on the size of the projection $\| π_A f\|_{}$.

math.CA↗

arXiv2018

Numerical Integration on Graphs: where to sample and how to weigh

Let $G=(V,E,w)$ be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset $W \subset V$ of vertices and weights $a_w$ such that $$ \frac{1}{|V|}\sum_{v \in V}^{}{f(v)} \sim \sum_{w \in W}{a_w f(w)}$$ for functions $f:V \rightarrow \mathbb{R}$ that are `smooth' with respect to the geometry of the graph. The main application are problems where $f$ is known to somehow depend on the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem (`the optimal packing of heat balls'). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.

math.ST↗