On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius

On the Classification of Residues of the Grassmannian

We study leading singularities of scattering amplitudes which are obtained as residues of an integral over a Grassmannian manifold. We recursively do the transformation from twistors to momentum twistors and obtain an iterative formula for Yangian invariants that involves a succession of dualized twistor variables. This turns out to be useful in addressing the problem of classifying the residues of the Grassmannian. The iterative formula leads naturally to new coordinates on the Grassmannian in terms of which both composite and non-composite residues appear on an equal footing. We write down residue theorems in these new variables and classify the independent residues for some simple examples. These variables also explicitly exhibit the distinct solutions one expects to find for a given set of vanishing minors from Schubert calculus.Comment: 20 page

Recent Developments in Modeling Heteroepitaxy/Heterogeneous Nucleation by Dynamical Density Functional Theory

Crystallization of supersaturated liquids usually starts by epitaxial growth or by heterogeneous nucleation on foreign surfaces. Herein, we review recent advances made in modeling heteroepitaxy and heterogeneous nucleation on flat/modulated surfaces and nanoparticles within the framework of a simple dynamical density functional theory, known as the phase-field crystal model. It will be shown that the contact angle and the nucleation barrier are nonmonotonous functions of the lattice mismatch between the substrate and the crystalline phase. In continuous cooling studies for substrates with lattice mismatch, we recover qualitatively the Matthews–Blakeslee mechanism of stress release via the misfit dislocations. The simulations performed for particle-induced freezing will be confronted with recent analytical results, exploring thus the validity range of the latter. It will be demonstrated that time-dependent studies are essential, as investigations based on equilibrium properties often cannot identify the preferred nucleation pathways. Modeling of these phenomena is essential for designing materials on the basis of controlled nucleation and/or nano-patterning

Soft network composite materials with deterministic and bio-inspired designs

Hard and soft structural composites found in biology provide inspiration for the design of advanced synthetic materials. Many examples of bio-inspired hard materials can be found in the literature; far less attention has been devoted to soft systems. Here we introduce deterministic routes to low-modulus thin film materials with stress/strain responses that can be tailored precisely to match the non-linear properties of biological tissues, with application opportunities that range from soft biomedical devices to constructs for tissue engineering. The approach combines a low-modulus matrix with an open, stretchable network as a structural reinforcement that can yield classes of composites with a wide range of desired mechanical responses, including anisotropic, spatially heterogeneous, hierarchical and self-similar designs. Demonstrative application examples in thin, skin-mounted electrophysiological sensors with mechanics precisely matched to the human epidermis and in soft, hydrogel-based vehicles for triggered drug release suggest their broad potential uses in biomedical devices. &amp;#169; 2015 Macmillan Publishers Limited. All rights reservedopen7

Putative Nanobacteria Represent Physiological Remnants and Culture By-Products of Normal Calcium Homeostasis

Putative living entities called nanobacteria (NB) are unusual for their small sizes (50–500 nm), pleomorphic nature, and accumulation of hydroxyapatite (HAP), and have been implicated in numerous diseases involving extraskeletal calcification. By adding precipitating ions to cell culture medium containing serum, mineral nanoparticles are generated that are morphologically and chemically identical to the so-called NB. These nanoparticles are shown here to be formed of amorphous mineral complexes containing calcium as well as other ions like carbonate, which then rapidly acquire phosphate, forming HAP. The main constituent proteins of serum-derived NB are albumin, fetuin-A, and apolipoprotein A1, but their involvement appears circumstantial since so-called NB from different body fluids harbor other proteins. Accordingly, by passage through various culture media, the protein composition of these particles can be modulated. Immunoblotting experiments reveal that antibodies deemed specific for NB react in fact with either albumin, fetuin-A, or both, indicating that previous studies using these reagents may have detected these serum proteins from the same as well as different species, with human tissue nanoparticles presumably absorbing bovine serum antigens from the culture medium. Both fetal bovine serum and human serum, used earlier by other investigators as sources of NB, paradoxically inhibit the formation of these entities, and this inhibition is trypsin-sensitive, indicating a role for proteins in this inhibitory process. Fetuin-A, and to a lesser degree albumin, inhibit nanoparticle formation, an inhibition that is overcome with time, ending with formation of the so-called NB. Together, these data demonstrate that NB are most likely formed by calcium or apatite crystallization inhibitors that are somehow overwhelmed by excess calcium or calcium phosphate found in culture medium or in body fluids, thereby becoming seeds for calcification. The structures described earlier as NB may thus represent remnants and by-products of physiological mechanisms used for calcium homeostasis, a concept which explains the vast body of NB literature as well as explains the true origin of NB as lifeless protein-mineralo entities with questionable role in pathogenesis

Improved functionalization of oleic acid-coated iron oxide nanoparticles for biomedical applications

Superparamagnetic iron oxide nanoparticles can providemultiple benefits for biomedical applications in aqueous environments such asmagnetic separation or magnetic resonance imaging. To increase the colloidal stability and allow subsequent reactions, the introduction of hydrophilic functional groups onto the particles’ surface is essential. During this process, the original coating is exchanged by preferably covalently bonded ligands such as trialkoxysilanes. The duration of the silane exchange reaction, which commonly takes more than 24 h, is an important drawback for this approach. In this paper, we present a novel method, which introduces ultrasonication as an energy source to dramatically accelerate this process, resulting in high-quality waterdispersible nanoparticles around 10 nmin size. To prove the generic character, different functional groups were introduced on the surface including polyethylene glycol chains, carboxylic acid, amine, and thiol groups. Their colloidal stability in various aqueous buffer solutions as well as human plasma and serum was investigated to allow implementation in biomedical and sensing applications.status: publishe