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

Model-Independent Sum Rule Analysis Based on Limited-Range Spectral Data

Partial sum rules are widely used in physics to separate low- and high-energy degrees of freedom of complex dynamical systems. Their application, though, is challenged in practice by the always finite spectrometer bandwidth and is often performed using risky model-dependent extrapolations. We show that, given spectra of the real and imaginary parts of any causal frequency-dependent response function (for example, optical conductivity, magnetic susceptibility, acoustical impedance etc.) in a limited range, the sum-rule integral from zero to a certain cutoff frequency inside this range can be safely derived using only the Kramers-Kronig dispersion relations without any extra model assumptions. This implies that experimental techniques providing both active and reactive response components independently, such as spectroscopic ellipsometry in optics, allow an extrapolation-independent determination of spectral weight 'hidden' below the lowest accessible frequency.Comment: 5 pages, 3 figure

Magnification relations in gravitational lensing via multidimensional residue integrals

We investigate the so-called magnification relations of gravitational lensing models. We show that multidimensional residue integrals provide a simple explanation for the existence of these relations, and an effective method of computation. We illustrate the method with several examples, thereby deriving new magnification relations for galaxy lens models and microlensing (point mass lensing).Comment: 16 pages, uses revtex4, submitted to Journal of Mathematical Physic

Directed nucleation and growth by balancing local supersaturation and substrate/nucleus lattice mismatch

Controlling nucleation and growth is crucial in biological and artificial mineralization and self-assembly processes. The nucleation barrier is determined by the interfaces and local supersaturation. Although chemically tailored substrates and lattice mismatches are routinely used to modify various forms of energy contributions as resulted from the substrate/nucleus interface and thereby steer controlled heterogeneous nucleation, strategies to combine this with control over local supersaturations have remained virtually unexplored. Here we demonstrate simultaneous control over both parameters to direct the positioning and growth direction of mineralizing compounds on preselected polymorphic substrates. We exploit the polymorphic nature of calcium carbonate (CaCO3) to locally manipulate the carbonate concentration and lattice mismatch between the nucleus and substrate, such that barium carbonate (BaCO3) and strontium carbonate (SrCO3) nucleate only on specific CaCO3 polymorphs. Based on this approach we position different materials and shapes on predetermined CaCO3 polymorphs in sequential steps, and guide the growth direction using locally created supersaturations. These results shed light on nature’s remarkable mineralization capabilities and outline new fabrication strategies for advanced materials, such as ceramics, photonic structures and semiconductors.Chemistry and Chemical Biolog

Probe method and a Carleman function

A Carleman function is a special fundamental solution with a large parameter for the Laplace operator and gives a formula to calculate the value of the solution of the Cauchy problem in a domain for the Laplace equation. The probe method applied to an inverse boundary value problem for the Laplace equation in a bounded domain is based on the existence of a special sequence of harmonic functions which is called a {\it needle sequence}. The needle sequence blows up on a special curve which connects a given point inside the domain with a point on the boundary of the domain and is convergent locally outside the curve. The sequence yields a reconstruction formula of unknown discontinuity, such as cavity, inclusion in a given medium from the Dirichlet-to-Neumann map. In this paper, an explicit needle sequence in {\it three dimensions} is given in a closed form. It is an application of a Carleman function introduced by Yarmukhamedov. Furthermore, an explicit needle sequence in the probe method applied to the reduction of inverse obstacle scattering problems with an {\it arbitrary} fixed wave number to inverse boundary value problems for the Helmholtz equation is also given.Comment: 2 figures, final versio

Directional wetting in anisotropic inverse opals

Porous materials display interesting transport phenomena due to the restricted motion of fluids within the nano- to micro-scale voids. Here, we investigate how liquid wetting in highly ordered inverse opals is affected by anisotropy in pore geometry. We compare samples with different degrees of pore asphericity and find different wetting patterns depending on the pore shape. Highly anisotropic structures are infiltrated more easily than their isotropic counterparts. Further, the wetting of anisotropic inverse opals is directional, with liquids filling from the side more easily. This effect is supported by percolation simulations as well as direct observations of wetting using time-resolved optical microscopy

Free energy of colloidal particles at the surface of sessile drops

The influence of finite system size on the free energy of a spherical particle floating at the surface of a sessile droplet is studied both analytically and numerically. In the special case that the contact angle at the substrate equals $\pi/2$ a capillary analogue of the method of images is applied in order to calculate small deformations of the droplet shape if an external force is applied to the particle. The type of boundary conditions for the droplet shape at the substrate determines the sign of the capillary monopole associated with the image particle. Therefore, the free energy of the particle, which is proportional to the interaction energy of the original particle with its image, can be of either sign, too. The analytic solutions, given by the Green's function of the capillary equation, are constructed such that the condition of the forces acting on the droplet being balanced and of the volume constraint are fulfilled. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we observe a local free energy minimum for the particle being located at the drop apex or at an intermediate angle, respectively. This peculiarity can be traced back to a non-monotonic behavior of the Green's function, which reflects the interplay between the deformations of the droplet shape and the volume constraint.Comment: 24 pages, 19 figure

Tunable anisotropy in inverse opals and emerging optical properties

Using self-assembly, nanoscale materials can be fabricated from the bottom up. Opals and inverse opals are examples of self-assembled nanomaterials made from crystallizing colloidal particles. As self-assembly requires a high level of control, it is challenging to use building blocks with anisotropic geometry to form complex opals, which limits the realizable structures. Typically, spherical colloids are employed as building blocks, leading to symmetric, isotropic superstructures. However, a significantly richer palette of directionally dependent properties are expected if less symmetric, anisotropic structures can be created, especially originating from the assembly of regular, spherical particles. Here we show a simple method to introduce anisotropy into inverse opals by subjecting them to a post-assembly thermal treatment that results in directional shrinkage of the silica matrix caused by condensation of partially hydrated sol-gel silica structures. In this way, we can tailor the shape of the pores, and the anisotropy of the final inverse opal preserves the order and uniformity of the self-assembled structure, while completely avoiding the need to synthesize complex oval-shaped particles and crystallize them into such target geometries. Detailed X-ray photoelectron spectroscopy (XPS) and infrared (IR) spectroscopy studies clearly identify increasing degrees of sol-gel condensation in confinement as a mechanism for the structure change. A computer simulation of structure changes resulting from the condensation-induced shrinkage further confirmed this mechanism. As an example of property changes induced by the introduction of anisotropy, we characterized the optical spectra of the anisotropic inverse opals and found that the optical properties can be controlled in a precise way using calcination temperature

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