Lineability of non-differentiable Pettis primitives

Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0; 1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND

Rolewicz-type chaotic operators

In this article we introduce a new class of Rolewicz-type operators in l_p, $1 \le p < \infty$. We exhibit a collection F of cardinality continuum of operators of this type which are chaotic and remain so under almost all finite linear combinations, provided that the linear combination has sufficiently large norm. As a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm.Comment: 15 page

New insight into cataract formation -- enhanced stability through mutual attraction

Small-angle neutron scattering experiments and molecular dynamics simulations combined with an application of concepts from soft matter physics to complex protein mixtures provide new insight into the stability of eye lens protein mixtures. Exploring this colloid-protein analogy we demonstrate that weak attractions between unlike proteins help to maintain lens transparency in an extremely sensitive and non-monotonic manner. These results not only represent an important step towards a better understanding of protein condensation diseases such as cataract formation, but provide general guidelines for tuning the stability of colloid mixtures, a topic relevant for soft matter physics and industrial applications.Comment: 4 pages, 4 figures. Accepted for publication on Phys. Rev. Let

Eta and Rho invariants on manifolds with edges

We establish existence of eta-invariants as well as of the Atiyah–Patodi–Singer and the Cheeger–Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature and the spin Dirac operator. We derive an analogue of the Atiyah–Patodi–Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments are based on the microlocal analysis of the heat kernel asymptotics associated to the Dirac laplacian of an incomplete edge metric. As an application, we discuss stability results for the two rho-invariants we have defined

Signatures of Witt spaces with boundary

Let M‾ be a compact smoothly stratified pseudomanifold with boundary, satisfying the Witt assumption. In this paper we introduce the de Rham signature and the Hodge signature of M‾, and prove their equality. Next, building also on recent work of Albin and Gell-Redman, we extend the Atiyah-Patodi-Singer index theory established in our previous work under the hypothesis that M‾ has stratification depth 1 to the general case, establishing in particular a signature formula on Witt spaces with boundary. In a parallel way we also pass to the case of a Galois covering M‾Γ of M‾ with Galois group Γ. Employing von Neumann algebras we introduce the de Rham Γ-signature and the Hodge Γ-signature and prove their equality, thus extending to Witt spaces a result proved by Lück and Schick in the smooth case. Finally, extending work of Vaillant in the smooth case, we establish a formula for the Hodge Γ-signature. As a consequence we deduce the fundamental result that equates the Cheeger-Gromov rho-invariant of the boundary ∂M‾Γ with the difference of the signatures of and M‾ and M‾Γ: signdR(M‾,∂M‾)−signdRΓ(M‾Γ,∂M‾Γ)=ρΓ(∂M‾Γ). We end the paper with two geometric applications of our results

Modelling a Particle Detector in Field Theory

Particle detector models allow to give an operational definition to the particle content of a given quantum state of a field theory. The commonly adopted Unruh-DeWitt type of detector is known to undergo temporary transitions to excited states even when at rest and in the Minkowski vacuum. We argue that real detectors do not feature this property, as the configuration "detector in its ground state + vacuum of the field" is generally a stable bound state of the underlying fundamental theory (e.g. the ground state-hydrogen atom in a suitable QED with electrons and protons) in the non-accelerated case. As a concrete example, we study a local relativistic field theory where a stable particle can capture a light quantum and form a quasi-stable state. As expected, to such a stable particle correspond energy eigenstates of the full theory, as is shown explicitly by using a dressed particle formalism at first order in perturbation theory. We derive an effective model of detector (at rest) where the stable particle and the quasi-stable configurations correspond to the two internal levels, "ground" and "excited", of the detector.Comment: 13 pages, references added, final versio

Energy transfer in nonlinear network models of proteins

We investigate how nonlinearity and topological disorder affect the energy relaxation of local kicks in coarse-grained network models of proteins. We find that nonlinearity promotes long-range, coherent transfer of substantial energy to specific, functional sites, while depressing transfer to generic locations. Remarkably, transfer can be mediated by the self-localization of discrete breathers at distant locations from the kick, acting as efficient energy-accumulating centers.Comment: 4 pages, 3 figure

A new result on impulsive differential equations involving non-absolutely convergent integrals

AbstractIn this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations