We present two versions of the maximum principle for nonsmooth hybrid optimal control problems, the first one of which requires differentiability along the reference trajectory and yields an adjoint equation of the usual kind, while the second one only requires approximability to first order by Lipschitz maps, and yields an adjoint differential inclusion involving a generalized gradient of the ...

We prove that each analytic set in R contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitzlike mappings of separable metric s...

We compare several notion of weak (modulus of) gradient in metric measure spaces and prove their equivalence. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independently of doubling and Poincaré assumptions on the metric measure space.

Let pn denote the nth prime, and consider function 1/n↦1/pn which maps reciprocals of positive integers bijectively to primes. We show that Hölder continuity this is equivalent a parametrised family Cramér type estimates on gaps between successive Here parametrisation comes from exponent. In particular, we Cramér’s conjecture map being Lipschitz. On other hand, inverse 1/pn↦1/n all orders but n...

Our argument relies on the well-known fact that Lipschitz maps from Euclidean space into metric spaces have metric derivatives almost everywhere, as proved by Kirchheim [Kir] and Korevaar-Schoen [KoSc] independently. We can combine the main result of [ChNi], a characterization of Gromov hyperbolicity via asymptotic cones [Dru], and Theorem 1.1 to obtain a partial generalization of Chatterji and...

1. Full proofs for every statement made in the main body. 2. Additional statements that introduce more details and more techniques for analyzing bi-Lipschitz learning settings. 3. Empirical results from the Machine Translation domain that indicate an underlying geometric structure in human and machine translation maps. The empirical results also show these properties for two " experts " : (1) G...

By the linearization property of Lipschitz-free spaces, any Lipschitz map $$f : M \rightarrow N$$ between two pointed metric spaces may be extended uniquely to a bounded linear operator $${\widehat{f}} {\mathcal {F}}(M) {F}}(N)$$ their corresponding spaces. In this note, we explore connections dynamics self-maps M$$ and extensions {F}}(M)$$ . This not only allows us relate topological dynamical...

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory