Resonances for Dirac operators on the half-line

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain.     

Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials, which can be considered as a generalization of the Stieltjes–Carlitz elliptic polynomials. Relations between characteristic (i.e., positive definite) functions, Toda chain, and orthogonal polynomials are developed in order to derive the main properties of these polynomials. Explicit expressions are found for the recurrence coefficients and the weight function for these polynomials. In the degenerate cases of the elliptic functions, the modified Meixner polynomials and the Krall–Laguerre polynomials appear.     

Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

We consider two models for directed polymers in space‐time independent random media (the O'Connell‐Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time τ, the probability distributions for the free energy fluctuations, when rescaled by τ^1/3, converges to the GUE Tracy‐Widom distribution. We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik–Ben Arous–Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm—the Kardar‐Parisi‐Zhang equation. The Laplace transform formula we prove can be inverted to give the one‐point probability distribution of the solution to these stochastic PDEs for the class of initial data. © 2014 Wiley Periodicals, Inc.     

1H NMR investigation of the hetero-association of phenanthridine dyes with Daunomycin: effect of substitution of amino with azido groups in the dye chromophore

¹H?NMR spectroscopy at 500?Mhz has been used to determine the structures and thermodynamics in aqueous salt solution of the hetero-association of Daunomycin (DAU) with a series of phenanthridine dyes having different numbers of amino/azido groups in the chromophore, together with the self-association of the phenthridine dyes under the same solution conditions (0.1?M phosphate buffer, pD 7.1, 298?K). The NMR measurements have been analyzed using statistical-thermodynamical models of both self-association and hetero- association in which no limitation is set on the size of molecular stacks. In this work the magnitudes of the self-association parameters of Ethidium Bromide (EB) and its azido-derivatives, 8-azido-Ethidium Bromide (EMB) and 3,8-diazido-Ethidium Chloride (EDC), show a successive decrease with mono- and di-substitition of the 3,8-amino groups of EB. A similar pattern is observed for the equilibrium constants for hetero-association of the phenanthridines with DAU. The thermodynamical and structural parameters of hetero-association of the phenanthridines with DAU are consistent with an intermolecular hydrogen bond between the 3,8 amino-groups of EB and the 9MeCO group of DAU contributing to the stability of the hetero-complex in aqueous solution.     

Two-dimensional liquid chromatography of diblock copolymers: Simulation at various adsorption interaction conditions

Diblock copolymers, which are heterogeneous in both molar mass and composition, can be fully characterized by using two-dimensional chromatography. Since the size-exclusion, the adsorption, and the critical interaction based modes of chromatography are possible for each of the polymers A and B, this leads to a variety of options for 2D-chromatography of copolymers AB. Using the theory of chromatography of block copolymers, 2D-chromatograms are simulated that correspond to the most interesting of these options. Orthogonal 2D-chromatograms are expected, if in the 1st dimension the critical condition is created for A, while in the 2nd dimension – for B. The situations, where A and B are both adsorbable, as well as those where the conditions of adsorption for A and SEC for B are created, are also considered. In particular, it is shown that the 2nd dimension combination of the critical condition for A and SEC – for B is preferable than that with SEC condition for both A and B. The simulated 2D-chromatograms of low- and high molar mass diblock copolymers, as well as of copolymers with one short block are compared with the reported real ones; it is concluded that the corresponding virtual and real 2D-chromatograms are qualitatively very similar.     

Anyons in an exactly solved model and beyond

A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z₂ gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.     

A reciprocal phase-perturbation theory for rough-surface scattering

We study the scattering of a scalar plane wave from a two-dimensional, randomly rough surface, on which the Dirichlet boundary condition is satisfied. The scattering amplitude is obtained in the form of the Fourier transform of an exponential, in which the exponent is written as an expansion in powers of the surface profile function. It is shown that the latter expansion can be written in such a way that the corresponding scattering matrix is manifestly reciprocal. Numerical results for the reflectivity, and for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered field, are obtained and compared with the predictions of small-amplitude perturbation theory and the Kirchhoff approximation. As the wavelength of the incident wave is varied continuously the results of the phase-perturbation theory change continuously from those of the small-amplitude perturbation theory to those of the Kirchhoff approximation.     

Enhanced coupling of subterahertz radiation with semiconductor periodic slot arrays

In this work, a theoretical study of the coupling of TM polarized subterahertz (THz) radiation with periodic semiconductor rectangular slot arrays was conducted, using InSb as an example. Simulation results showed that the structure with 4-12 microm thickness provides over a 20-30-fold increase in the electric field at slot edges in a nanosize region ( approximately 500 nm). The enhancement of the THz electromagnetic field extends across the slots and reaches peak values at the edges due to discontinuity effects. Because of the strong local electromagnetic field enhancement, the structure can potentially be used for the development of novel biophotonic sensors, leading to improved detection sensitivity.     

Cover Picture: Turning Spiroketals Inside Out: A Rearrangement Triggered by an Enol Ether Epoxidation (ChemistryOpen 5/2015)

Invited for this month''s cover picture is the group of Professor Mark Peczuh at the University of Connecticut. The cover picture compares the rearrangement of a small molecule to the process of turning a stuffed animal inside out. The recycled, inside-out stuffed animals are both artistic and philosophically provocative. They capture the essence of the rearrangement reaction because the compounds themselves turn inside out over the course of the reaction, extending the diversity of products that can arise from simple starting materials. Small molecules often have functional groups with latent reactivity; under the appropriate conditions, those groups can react with other compounds (e.g., reagents) and also with other groups in the same molecule in an intramolecular reaction. The research team found that the epoxidation of some highly functionalized spiroketal compounds promoted rearrangements of their structures that turned them inside out. Some of the features of the products led them to use X-ray crystallography or a combination of computer-assisted structure elucidation, computation, and a new version of the 1,1-ADEQUATE NMR experiment to determine their structures. For more details, see the Communication on p. 577 ff.     

Boundary Layers and KPP Fronts in a Cellular Flow

We study an eigenvalue problem associated with a reaction-diffusion-advection equation of the KPP type in a cellular flow. We obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude A ≪ 1. It follows that the minimal pulsating traveling front speed c _*(A) satisfies the upper and lower bounds C ₁ A ^1/4≦ c _*(A)≦ C ₂ A ^1/4. Physically, the speed enhancement is related to the boundary layer structure of the associated eigenfunction – accordingly, we establish an “averaging along the streamlines” principle for the unique positive eigenfunction.