Degenerate cascade fluorescence: Optical spectral-line narrowing via a single microwave cavity

For a three-level atom, two nondegenerate(even microwave and optical) electric dipole transitions are usually allowed; for either of these, the fluorescence spectra are well-described in terms of spontaneous transitions from a triplet of dressed sublevels to an adjacent lower-lying triplet. When the three dressed sublevels are equally spaced from each other, a remarkable feature known as degenerate cascade fluorescence takes place, which displays a five-peaked structure. We show that a single cavity can make all the spectral lines extremely narrow, whether they arise from cavity-coupled or cavity-free transitions. This effect is based on intrinsic cascade lasing feedback and makes it possible to use a single microwave cavity(even a bad cavity) to narrow the spectral lines in the optical frequency regime.     

Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

Synthesis,structural characterization and in vitro cytotoxic activity of novel polymeric triorganotin(IV) complexes of urocanic acid

Two novel triorganotin carboxylate complexes of the biologically active urocanic acid have been synthesized and characterized. Elemental analysis, melting point, spectroscopic techniques – IR, ¹H, ¹³C and ¹¹⁹Sn NMR – mass spectrometry and X‐ray diffraction studies have been used for structural characterization. Crystal structures of the tin(IV) derivatives show that urocanic acid acts as a bridging bidentate ligand through its imidazole nitrogen atom and its carboxylic group, producing a polymeric one‐dimensional chain. The molecular structures of the complexes, catena‐poly‐tri(n‐butyl)tin(IV) 3‐(3H‐imidazol‐4‐yl)prop‐2‐enoate (1) and catena‐poly‐triphenyltin(IV) 3‐(3H‐imidazol‐4‐yl)prop‐2‐enoate (2), present a distorted trigonal–bipyramidal configuration. This is further confirmed by ¹¹⁹Sn NMR in the solid state. The tin(IV) derivatives form double‐stranded ribbons via N―H^…O―H bonds. Nevertheless, the compounds are essentially monomeric in solution, with a tetrahedral configuration as observed by ¹¹⁹Sn NMR in solution. The cytotoxic activity of the titled compounds has been tested against six human cell lines and the corresponding IC₅₀ values are reported. Both tin(IV) compounds have a high to very high in vitro cytotoxic activity against the tumor cell lines K562, HCT‐15 and MCF‐7. Compound 1 is 86 times more active than cisplatin in the HTC‐15 cell line. Copyright © 2012 John Wiley & Sons, Ltd.     

Bounds for the fundamental solutions of elliptic and parabolic equations

It is shown that any elloptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.     

Numerical solution for the anisotropic Willmore flow of graphs

The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge–Kutta–Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.     

Formations of n-order two-soliton bound states in Bose–Einstein condensates with spatiotemporally modulated nonlinearities

The formations of n-order two-soliton bound states (TSBSs) in the Bose–Einstein condensates with spatiotemporally modulated nonlinearities are studied. Exact analytical expressions of the n-order TSBSs are derived by means of the similarity transformations. Further, the numerical simulations are carried out, consistent with the analytical results very well. The stability analysis shows that the solutions can be stable. Our results indicate that the attractive spatiotemporal inhomogeneous nonlinearities can support n-order TSBSs, which has the potential applications to the generation of matter-wave bright solitons in harmonic traps.     

A note on general solution of Stokes equations

Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).     

Nonlinear resonant tunneling diode (RTD) circuits for microwave A/D conversion

Short-duration electrical pulses play important roles in ultrafast time-domain metrology: they are used to sample rapidly varying signals or as probe signals in ranging radars, time-domain reflectometry and in communication. In this work, we design a nonlinear transmission, which is loaded with resonant tunneling diode to be suitable for microwave A/D conversion. A resonant tunneling diode (RTD) has a negative differential resistance that means when the voltage increases the current decreases. The equivalent circuit of monostable line is given. The simulation is performed by using OrCad program. Results show that a spike is produced and after a charging time constant, another switching occurs. Hence – similar to a relaxation oscillator – the spiking period is determined by the amplitude and frequency of the input current. The transmission line itself ensures the generation and propagation of identical spikes, such as solitons formed after few diodes.