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In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in…

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an…

We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform…

We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded…

We consider Gabor frames $\{e^{2\pi i bm \cdot} g(\cdot-ak)\}_{m,k \in \mathbb{Z}}$ with translation parameter $a=L/2$, modulation parameter $b \in (0,2/L)$ and a window function $g \in C^n(\mathbb{R})$ supported on $[x_0,x_0+L]$ and…

We study the determination of a holomorphic function from its absolute value. Given a parameter $\theta \in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $\Lambda \subseteq \mathbb{R}$: if…

A non perturbative computation of the evolution of singlet parton densities without gauge--fixing requires a gauge invariant gluon source operator. Within the Schr\"odinger Functional scheme (SF), such a source can be defined in terms of…

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an…

We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the…

We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast…

Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $\Delta$. It is well-known that every complex-valued harmonic function in the unit disc $\Delta$ can be uniquely…

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,\Omega\right)=\int_{\Omega}\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,\xi)$…

In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and…

We discuss in this paper the lattice discretizations of all topological defect lines (TDLs) for diagonal, minimal CFTs, using integrable restricted solid-on-solid (RSOS) models. For these CFTs, the TDLs can be labeled by the Kac labels. In…

In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] the instabilities of the Gabor phase retrieval problem, i.e., the problem of…

Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any entire solution of the fourth order…

We study existence of minimizers of the least gradient problem \[\inf_{v \in BV_g} \int_{\Omega}\varphi(x, Dv),\] where $BV_g=\{v \in BV(\Omega): \int_{\partial \Omega}gv=1\}$, $\varphi(x,p): \Omega\times \R^n \rightarrow \R$ is a convex,…

We consider the following problem: given a set $\Lambda \subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 \pi isx}\}$, $(t,s) \in \Lambda$,…

In this paper we solve an open problem concerning the characterization of those measurable sets $\Omega\subset \mathbb{R}^{2d}$ that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in…

Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this…