Related papers: Lipschitz Flow-box Theorem

In the present paper we prove that a necessary condition for a Banach space $X$ to admit a generating compact Lipschitz retract $K$, which satisfies an additional mild assumption on its shape, is that $X$ enjoys the Bounded Approximation…

We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal…

This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show…

In this paper, we shall establish Banach-Stone type theorems on spaces of uniformly continuous and lipschitz continuous pseudometrics.

These notes have the intent to introduce the study of the nonlinear aspects of operator space theory. We investigate some results on the nonlinear theory of Banach spaces which remain valid in the noncommutative case. In particular, we show…

Helmholtz theorem states that, in ideal fluid, vortex lines move with the fluid. Another Helmholtz theorem adds that strength of a vortex tube is constant along the tube. The lines may be regarded as integral surfaces of a 1-dimensional…

We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently…

We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive…

Flow Matching, a promising approach in generative modeling, has recently gained popularity. Relying on ordinary differential equations, it offers a simple and flexible alternative to diffusion models, which are currently the…

The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is…

For a Banach space $V$ we define its Lipschitz extension constant, $\cL\cE(V)$, to be the infimum of the constants $c$ such that for every metric space $(Z,\rho)$, every $X \subset Z$, and every $f: X \to V$, there is an extension, $g$, of…

In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz…

In a recent paper [5], the global well-posedness of the two-dimensional Euler equation with vorticity in \mbox{$L^1\cap LBMO$} was proved, where $ LBMO$ is a Banach space which is strictly imbricated between \mbox{$L^\infty$} and $BMO$. In…

In 1948, Schwinger developed a local Lorentz covariant formulation of relativistic quantum electrodynamics in space-time which is fundamentally inconsistent with any delocalized interpretation of quantum mechanics. An interpretation…

We prove a fixed-point theorem that generalises and simplifies a number of results in the theory of $F$-contractions. We show that all of the previously imposed conditions on the operator can be either omitted or relaxed. Furthermore, our…

Relativistic invariance of the vacuum is (or follows from) one of the Wightman axioms which is commonly believed to be true. Without these axioms, here we present a direct and general proof of continuous relativistic invariance of all…

Let X and Y be Banach spaces and F a subset of B_{Y^*}. Endow Y with the topology \tau_F of pointwise convergence on F. Let T: X^* \to Y be a bounded linear operator which is (w^*, \tau_F) continuous. Assume that every vector in the range…

Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-dimensional domain, and suppose that the set has finite $n$-dimensional Hausdorff measure and the complement is not connected. Starting from this given…

In this article, we prove a general viability theorem for continuity inclusions in Wasserstein spaces, and provide an application thereof to the existence of exponentially stable trajectories obtained via the second method of Lyapunov.

The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which…