In this paper we construct global dispersive solutions to the space inhomogeneous kinetic wave equation (KWE) which propagate L^1_xv —moments and conserve mass, momentum and energy. We prove that they scatter, and that the wave operators mapping the initial data to the scattering states are 1–1, onto and continuous in a suitable topology. This is the first global existence result for strong solutions for KWE. This contrasts with prior global existence results for mild solutions, which satisfy a transported version of the equation but do not solve the equation itself. Our proof is carried out entirely in physical space and combines dispersive estimates for the free transport with new trilinear bounds for the gain and loss operators of the KWE on weighted Lebesgue spaces. The main difficulty is the fast growth of the hard-sphere kernel. Our fundamental tool to handle it is a novel collisional averaging estimate. We also show that the nonlinear evolution preserves positivity forward in time. For this, we use the Kaniel–Shinbrot iteration scheme (Kaniel and Shinbrot in Commun Math Phys 58:65–84, 1978), properly initialized to ensure successive approximations are dispersive.
We introduce a notion of \emph{efficient stability} for finite presentations of groups. Informally, a finite presentation using generators $S$ and relations $R$ is \emph{stable} if any map from $S$ to unitaries that approximately satisfies the relations (in the tracial norm) is close to the restriction of a representation of $G$ to the subset $S$. This notion and variants thereof have been extensively studied in recent years, in part motivated by connections to property testing in computer science. The novelty in our work is the focus on \emph{efficiency}, which, informally, places an onus on small presentations -- in the sense of encoding length. The goal in this setup is to achieve non-trivial tradeoffs between the presentation length and its modulus of stability. With this goal in mind we analyze various natural examples of presentations. We provide a general method for constructing presentations of $\mathbb{Z}_2^k$ from linear error-correcting codes. We observe that the resulting presentation has a weak form of stability exactly when the code is \emph{testable}. This raises the question of whether testable codes give rise to genuinely stable presentations using this method. While we cannot show that this is the case in general, we leverage recent results in the study of non-local games in quantum information theory (Ji et al., Discrete Analysis 2021) to show that a specific instantiation of our construction, based on the Reed-Muller family of codes, leads to a stable presentation of $\mathbb{Z}_2^k$ of size polylog$(k)$ only. As an application, we combine this result with recent work of de la Salle (arXiv:2204.07084) to re-derive the quantum low-degree test of Natarajan and Vidick (IEEE FOCS'18), which is a key building block in the recent refutation of Connes' Embedding Problem via complexity theory (Ji et al., arXiv:2001.04383).
Dedekind's problem, dating back to 1897, asks for the total number ψ(n) of antichains contained in the Boolean lattice B_n on n elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in B_n. We obtain detailed estimates for both ψ(n) and the number of antichains of size βn⌊ n/2 ⌋ for any fixed β>0. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size m is contained in a middle layer of B_n.
We consider the following question arising in the theory of differential inclusions: Given an elliptic set Gamma and a Sobolev map u whose gradient lies in the quasiconformal envelope of Gamma and touches Gamma on a set of positive measure, must u be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where Gamma subset of R(2 & times;2 i)s an elliptic, smooth, closed curve. More precisely, we prove that the distance of Du to Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge-Amp & egrave;re equation: concerning the latter, we obtain a new proof of the W-2,W-1+epsilon-regularity for two-dimensional solutions.
We study the fundamental problem of learning a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonlinear dynamical system that has finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This significantly generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.
