On the average volume of sections of convex bodies

The average section functional as(K) of a star body in Rn is the average volume of its central hyperplane sections: \(as\left( k \right) = \int_{{S^{n - 1}}} {\left| {K \cap {\xi ^ \bot }} \right|} d\sigma \left( \xi \right)\). We study the question whether there exists an absolute constantC > 0 such that for every n, for every centered convex body K in R ⁿ and for every 1 ≤ k ≤ n ? 2,

$$as\left( K \right) \leqslant {C^k}{\left| K \right|^{\frac{k}{n}}}\mathop {\max }\limits_{|E \in G{r_{n - k}}} {\kern 1pt} as\left( {K \cap E} \right)$$

. We observe that the case k = 1 is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces C by CL _K orCd_ovr(K, BP _k ⁿ ), where L _K is the isotropic constant of K and dovr(K, BP _k ⁿ ) is the outer volume ratio distance of K to the class BP _k ⁿ of generalized k-intersection bodies. We also compare as(K) to the average of as(K ∩ E) over all k-codimensional sections of K. We examine separately the dependence of the constants on the dimension when K is in some classical position. Moreover, we study the natural lower dimensional analogue of the average section functional.

     

Normal modes of a model radiating system

In order to gain insight into normal modes of realistic radiating systems, we study the simple model problem of a finite string and a semi-infinite string coupled by a spring. As expected there is a family of modes which are basically the modes of the finite string slowly damped by the radiation of energy to infinity on the semi-infinite string. But we also study another family of modes, found by Dyson in a different model problem, which are strongly damped modes of the semi-infinite string itself. These may be analogous to the modes of black holes, and they are likely to be present in relativistic stars as well. The question of whether the instability in these modes which Dyson found is present in realistic stars remains open.     

Possibility of CP violation in the cascade non-leptonic decays

There is more than one parameter expressing CP violation in the non-leptonic hyperon decays. The recent experimental results provide us with indications that CP conservation may not be valid for the Ξ decays. Especially for the Ξ dacays this is not due to the deviations from the exact ΔI=1/2 rule.     

Extensional behavior influence on viscoelastic turbulent channel flow

In this work we use in the simulation of a viscoelastic turbulent channel flow a modification of the finitely extensible of non-linear elastic dumbbells with the Peterlin approximation (FENE-P) constitutive model for dilute polymer solutions, applicable to high extensional deformations. The new feature introduced by this modification is that the free energy of the polymer (since it is assumed to be entirely entropically driven) remains always bounded (FENE-PB). The characteristics of the model under steady shear flow, pure elongational flow and transient extensional behavior are presented. It is found that the FENE-PB model is more shear thinning than FENE-P. Most importantly, it also shows a higher extensional viscosity than the FENE-P model. Although the steady-state Trouton ratio asymptotically reaches at high extensional rates the same limit as the FENE-P model, the transition from the Newtonian value is sharper and faster. We use the FENE-PB model in direct numerical simulations (DNS) of viscoelastic turbulent channel flow using spectral approximations. The results for various statistics of the flow and the polymer conformation, when compared against those obtained with the original FENE-P model and the same rheological parameters, show an enhanced polymer-induced drag reduction effect and enhanced deformation of the polymer molecules. This indicates that it is not only the asymptotic but also details from the extensional rheological behavior that matter in quantitatively specifying turbulent viscoelastic flow behavior.     

Controlled Chemical Derivatisation of Carbon Nanotubes with Imaging,Targeting, and Therapeutic Capabilities

In drug delivery, carbon nanotubes (CNTs) hold a great potential as carriers because of their ability to easily cross biological barriers and be internalised into cells. Their high aspect ratio allows multi‐functionalisation and their development as a multimodal platform for targeted therapy. In this article, we report the controlled covalent derivatisation of triple‐functionalised CNTs with the anticancer drug gemcitabine, folic acid as a targeting ligand and fluorescein as a probe. The anticancer activity of gemcitabine was maintained after covalent grafting onto the CNTs. The functionalised nanotubes were internalised into both folate‐positive and negative cells, suggesting the passive diffusion of CNTs. Overall, our approach is versatile and offers a precise chemical control of the sidewall functionalisation of CNTs and the possibility to manoeuvre the types of functionalities required on the nanotubes for a multimodal therapeutic strategy.