Completion of continuity spaces with uniformly vanishing asymmetry

The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in question. For qausi-metric spaces and quasi-uniform spaces various non-equivalent completions exist, often defined on a certain subcategory of spaces that satisfy a key property required for the particular completion to exist. The classical filter completion of a uniform space can be adapted to yield a filter completion of a metric space. We show that this completion by filters generalizes to continuity spaces that satisfy a form of symmetry which we call uniformly vanishing asymmetry

Global Flows with Invariant Measures for the Inviscid Modified SQG Equations

We consider the family known as modified or generalized surface quasi-geostrophic equations (mSQG) consisting of the classical inviscid surface quasi-geostrophic (SQG) equation together with a family of regularized active scalars given by introducing a smoothing operator of nonzero but possibly arbitrarily small degree. This family naturally interpolates between the 2D Euler equation and the SQG equation. For this family of equations we construct an invariant measure on a rough $L^2$-based Sobolev space and establish the existence of solutions of arbitrarily large lifespan for initial data in a set of full measure in the rough Sobolev space.Comment: 18 page

Entanglement rates and the stability of the area law for the entanglement entropy

We prove a conjecture by Bravyi on an upper bound on entanglement rates of local Hamiltonians. We then use this bound to prove the stability of the area law for the entanglement entropy of quantum spin systems under adiabatic and quasi-adiabatic evolutions

Completeness of Randomized Kinodynamic Planners with State-based Steering

Probabilistic completeness is an important property in motion planning. Although it has been established with clear assumptions for geometric planners, the panorama of completeness results for kinodynamic planners is still incomplete, as most existing proofs rely on strong assumptions that are difficult, if not impossible, to verify on practical systems. In this paper, we focus on an important class of kinodynamic planners, namely those that interpolate trajectories in the state space. We provide a proof of probabilistic completeness for these planners under assumptions that can be readily verified from the system's equations of motion and the user-defined interpolation function. Our proof relies crucially on a property of interpolated trajectories, termed second-order continuity (SOC), which we show is tightly related to the ability of a planner to benefit from denser sampling. We analyze the impact of this property in simulations on a low-torque pendulum. Our results show that a simple RRT using a second-order continuous interpolation swiftly finds solution, while it is impossible for the same planner using standard Bezier curves (which are not SOC) to find any solution.Comment: 21 pages, 5 figure

Orthonormal bases of regular wavelets in spaces of homogeneous type

Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have H\"older regularity. This is used to build an orthonormal basis of H\"older-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calder\'on reproducing formula to study and develop function space theory and singular integrals. We discuss the examples of $L^p$ spaces, BMO and apply this to a proof of the T(1) theorem. As no extra condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the space of homogeneous type is required, our results extend a long line of works on the subject.Comment: We have made improvements to section 2 following the referees suggestions. In particular, it now contains full proof of formerly Theorem 2.7 instead of sending back to earlier works, which makes the construction of splines self-contained. One reference adde