Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q _y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.     

The ring lemma in three dimensions

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c _n (R ²) = (F _2n−3−1)⁻¹—where F _k denotes the kth Fibonacci number—and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration—thought to be extremal in three dimensions—is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n − 1.     

Far Red-Shifted CdTe Quantum Dots for Multicolour Stimulated Emission Depletion Nanoscopy

Stimulated emission depletion (STED) nanoscopy is a widely used nanoscopy technique. Two-colour STED imaging in fixed and living cells is standardised today utilising both fluorescent dyes and fluorescent proteins. Solutions to image additional colours have been demonstrated using spectral unmixing, photobleaching steps, or long-Stokes-shift dyes. However, these approaches often compromise speed, spatial resolution, and image quality, and increase complexity. Here, we present multicolour STED nanoscopy with far red-shifted semiconductor CdTe quantum dots (QDs). STED imaging of the QDs is optimized to minimize blinking effects and maximize the number of detected photons. The far-red and compact emission spectra of the investigated QDs free spectral space for the simultaneous use of fluorescent dyes, enabling straightforward three-colour STED imaging with a single depletion beam. We use our method to study the internalization of QDs in cells, opening up the way for future super-resolution studies of particle uptake and internalization.     

On Kovács–Newman ordered semigroups

Semigroup Forum - The irreducibility of pseudovarieties of semigroups is studied using many different approaches including a notion of Kovács–Newman semigroups. In this note, we answer a...     

Efficient and highly reproducible capillary electrophoresis-mass spectrometry of peptides using Polybrene-poly(vinyl sulfonate)-coated capillaries

The potential of capillaries noncovalently coated with a bilayer of oppositely charged polymers for the analysis of peptides by CE-MS was investigated. Bilayer coatings were produced by subsequently rinsing fused-silica capillaries with a solution of Polybrene (PB) and poly(vinyl sulfonate) (PVS). The PB-PVS coating showed to be fully compatible with MS detection causing no ionization suppression or background signals. The bilayer coating provided a considerable EOF at low pH, thereby facilitating the fast separation of peptides using a BGE of formic acid (pH 2.5). Under optimized CE-MS conditions, for enkephalin peptides high separation efficiencies were obtained with plate numbers in the range of 300,000-500,000. It is demonstrated that both the cancellation of the hydrodynamic capillary flow induced by the nebulizer gas and a sufficiently high-data acquisition rate are crucial for achieving these efficiencies. The overall performance of the CE-MS system using PB-PVS-coated capillaries was evaluated by the analysis of a tryptic digest of cytochrome c. The system provided an efficient separation of the peptide mixture, which could be effectively monitored by MS/MS detection allowing identification of at least 13 peptides within a time interval of 1.5 min. In addition, the PB-PVS coating proved to be very consistent yielding stable CE-MS patterns with highly favorable migration time reproducibilities (RSDs < 1% over a 3-day period).     

Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in [ 1 ]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.     

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability $$\mathbb{P}\left(\exists \text{no points in the Bessel process on}(0,x_1)\cup (x_2,x_3)\cup \cdots \cup (x_{2g},x_{2g+1})\right),$$ where and $g\ge 0$ is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a g-dimensional torus, whose integrand involves ratios of Riemann θ-functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain “good Diophantine properties”, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has “good Diophantine properties” (which is always the case for $g=1$, and “almost always” the case for $g\ge 2$), these results can be combined, yielding particularly precise and explicit large gap asymptotics.