The threaded BPM: A novel extension to the two-dimensional scalar beam propagation method

The two-dimensional scalar beam propagation method (BPM) is a widely used, computationally efficient tool for the analysis of planar optical waveguides and devices. The inherent paraxial limitations and rectilinear analysis grid limit its application to slightly curved structures and waveguides. In this novel extention to the BPM algorithm, the curvature restrictions are removed and in many cases the paraxial restrictions can be avoided, allowing for the first time, the efficient analysis of arbitrarily curved structures, such as S- or U-shaped bends, curved transitions of progressively varying curvature, and curved couplers. It can also handle concatenated devices and the curved interconnect sections between them. The process operates by the concatenation of micro-conformal maps, which progressively re-orientate the problem optimally towards a straight BPM analysis.     

Generalized Dirichlet tessellations

In this paper we study how to recognize when a dissection of the plane has been constructed in one of several natural ways each of which models some phenomena in the natural or social sciences. The prototypical case is the nearest-neighbor or Dirichlet tessellation.     

Very slowly varying functions

A real-valued functionf of a real variable is said to be?-slowly varying (?-s.v.) if lim _x→∞ ?(x) [f(x+α)?f(x)]=0 for each α. It is said to be uniformly?-slowly varying (u.?-s.v.) if lim _x→∞ sup _{α ∈ I} ?(x) |f(x+α)?f(x)|=0 for every bounded intervalI. It is supposed throughout that ? is positive and increasing. It is proved that if? increases rapidly enough, then every?-s.v. functionf must be u.?-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of?, a measurable functionf must be u.?-s.v. if it is?-s.v. Examples of pairs (?,f) are given that illustrate the necessity for the requirements on? andf in these results.     

Theoretical analysis of tautomerization of succinimide and analogous compounds: insights from DFT approach

Tautomerizations of biologically and medicinally important heterocyclic compound, 2,5-pyrrolidinedione, commonly known as succinimide and two of its analogous compounds, 2,4-pyrrolidinedione and 3,4-pyrrolidinedione have been investigated at the density functional theory (DFT)/M06-2X level in aqueous medium, implementing polarizable continuum model (PCM). We have extended our investigation of tautomerism to the sulfur analogues of the aforementioned compounds also, i.e., 2,5-pyrrolidinedithione, 2,4-pyrrolidinedithione, and 3,4-pyrrolidinedithione. Tautomerism observed in these compounds are mainly keto-enol, thio-thiol and amine-imine, but we have detected two new kind of tautomerization shown by some of the abovementioned compounds, named as keto-epoxy (for the oxygen analogue) and thio-thioepoxy (for the sulfur analogue) which have not yet been reported in the literature. Relative energies (E_r) and activation energies (E_a) have been calculated for all the tautomers and tautomerization processes. The potential energy surfaces (PESs) have been constructed using the M06-2X energy values. It has been observed that the energy difference found in the tautomers of sulfur analogues is relatively lower than that of the corresponding oxygen analogues, so does the activation energy barrier. As one-electron redox properties play an important role in biological systems, we have also explored the effect of conformational changes on the overall redox properties of the said compounds by calculating the adiabatic and vertical ionization potentials (AIPs and VIPs, respectively) and adiabatic electron affinities (AEAs).