We use the simple, but little-known, result that a uniformly continuous function on a convex set is -Lipschitz (as defined below) to extend Piyavskii’s algorithm for Lipschitz global optimization to the larger domain of continuous (not-necessarilyLipschitz) global optimization.

where φ is the angle between the unit vortex line tangent vectors ξ(x−y, t) and ξ(x, t). Some degree of smoothness of the bundle of vortex lines near a potential singularity may result in averting blowup [3]. For simplicity, we’ll discuss Lipschitz continuous cases, although Hölder continuous cases may be analyzed in a similar fashion. We distinguish between the sine-Lipschitz case (i.e. sinφ i...

We prove that a Lipschitz (or uniformly continuous) mapping f : X → Y can be approximated by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a C(K) space (resp. super-reflexive space). As a corollary we obtain also smooth approximation of C1-smooth mappings together with their first derivatives.

Consider an o-minimal expansion of the real field. We show that definable Lipschitz continuous maps can be definably fine approximated by definable continuously differentiable Lipschitz maps whose Lipschitz constant is close to that of the original map.

In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α : H → Rm is injective, with (α(x))k = |〈x, fk〉|, where {f1, · · · , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω : Rm → H that is Li...

It is well known that the absolute value map on the self-adjoint operators on an infinite dimensional Hilbert spaces is not Lipschitz continuous, although Lipschitz continuity holds on certain subsets of operators. In this note, we provide an elementary proof that the absolute value map is Lipschitz continuous on the set of all operators which are bounded below in norm by any fixed positive con...