Transfinite interpolation on the medians of a triangle and best L 1-approximation

Let be a triangle in and let be the set of its three medians. We construct interpolants to smooth functions using transfinite (or blending) interpolation on The interpolants are of type f(₁)+g(₂)+h(₃), where (₁,₂,₃) are the barycentric coordinates with respect to the vertices of . Based on an error representation formula, we prove that the interpolant is the unique best L¹-approximant by functions of this type subject the function to be approximated is from a certain convexity cone in C³().Received: 17 December 2003     

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

Repeated passing principle for propagation in optical resonators

In this paper we make comparison between a well-known theoretical model of light propagation through racetrack resonator and experimentally obtained results. Observed differences are studied and some original modifications are made in the existing model so as to achieve better alignment with experiment. The influence of several geometric parameters on racetrack’s response is used for further adjustments to be performed. This procedure opens up the possibility to estimate the free spectral range and resonant wavelength for different geometric parameters and consequently to predict resonator functionality and working conditions, as well as functionality of complex photonic devices based on resonant structures.     

Effects of dielectric barrier discharge in air on morphological and electrical properties of graphene nanoplatelets and multi-walled carbon nanotubes

Multi-walled carbon nanotubes (MWCNTs) and graphene nanoplatelets (GNPs) have been functionalized by dielectric barrier discharge (DBD) in air. The extent of functionalization of MWCNTs and GNPs reaches a maximum at the delivered discharge energy of 720 and 240 J mg⁻¹, respectively. Further exposure to plasma leads to reduction of functional groups from the surface of the treated nanomaterials. It has also been demonstrated that DBD plasma does not produce dramatic structural changes in MWCNTs, while flakes of the treated GNPs become thinner and smaller in the lateral size. Conductive thin films, obtained by drop casting a solution of the treated nanomaterials in N-methyl-1-pyrrolidone on poly(methyl methacrylate) substrate, show significantly lower sheet resistance.     

Engineering hairy cellulose nanocrystals for chemotherapy drug capture

Cancer is one of the leading causes of death worldwide, affecting millions of people every year. Although chemotherapy remains one of the most common cancer treatments in the world, the severe side effects of chemotherapy drugs impose serious concerns to cancer patients. In many cases, the chemotherapy can be localized to maximize the drug effects; however, the drug systemic circulation induces undesirable side effects. Here, we have developed a highly efficient cellulose-based nanoadsorbent that can capture more than 6,000 milligrams of doxorubicin (DOX), one of the most widely used chemotherapy drugs, per gram of the adsorbent at physiological conditions. Such drug capture capacity is more than 3,200% higher than other nanoadsorbents, such as DNA-based platforms. We show how anionic hairy cellulose nanocrystals, also known as electrosterically stabilized nanocrystalline cellulose (ENCC), bind to positively charged drugs in human serum and capture DOX immediately without imposing any cytotoxicity and hemolytic effects. We elucidate how ENCC provides a remarkable platform for biodetoxification at varying pH, ionic strength, ion type, and protein concentration. The outcome of this research may pave the way for developing the next-generation in vitro and in vivo drug capture additives and devices.