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For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general… (More)

In this paper, we study the regularity of solutions to the p-Poisson equation for all 1 < p < ∞. In particular, we are interested in smoothness estimates in the adaptivity scale B σ τ (L τ (Ω)), 1/τ = σ/d + 1/p, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear… (More)

- Benjamin Scharf
- 2009

The first part of this paper deals with the topic of finding equivalent norms and characterizations for vector-valued Besov and Triebel-Lizorkin spaces Bs p,q(E) and F s p,q(E). We will deduce general criteria by transferring and extending a theorem of Bui, Paluszyński and Taibleson from the scalar to the vector-valued case. By using special norms and… (More)

- Benjamin Scharf
- 2012

In Chapter 4 of [28] Triebel proved two theorems concerning pointwise multipliers and diffeomorphisms in function spaces Bp,q(R n) and Fs p,q(R n). In each case he presented two approaches, one via atoms and one via local means. While the approach via atoms was very satisfactory concerning the length and simplicity, only the rather technical approach via… (More)

The aim of the paper is to characterize the trace space of vector-valued Sobolev spaces W p (R , E) , where E is an arbitrary Banach space. In particular, we do not assume that the underlying Banach space E has the UMD property. Vector-valued Sobolev and Besov spaces are widely used in abstract evolution equations, cf. e.g. Amann [1, 2, 4], Veraar and Weis… (More)

- Benjamin Scharf
- 2010

The rst part of this diploma thesis deals with the topic of nding equivalent norms and characterizations for vector-valued Besov and Triebel-Lizorkin spaces Bs p,q(E) and F s p,q(E). We will deduce general criteria by transferring and extending a theorem of Bui, Paluszy«ski and Taibleson from the scalar to the vector-valued case. By using special norms and… (More)

- Benedetto Piccoli, Nastassia Pouradier, Benjamin Scharf
- 2015

Collective migration of animals in a cohesive group is rendered possible by a strategic distribution of tasks among members: some track the travel route, which is time and energy-consuming, while the others follow the group by interacting among themselves. In this paper, we study a social dynamics system modeling collective migration. We consider a group of… (More)

- BENJAMIN SCHARF
- 2013

A rather tricky question is the construction of wavelet bases on domains for suitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In his monograph from 2008, Triebel presented an approach how to construct wavelet (Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellular domains , in particular on the cube. However, he had… (More)

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