Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.     

Stability in terms of two measures of hybrid systems with partially visible solutions

A smooth functioning system with some stability properties may break down abruptly due to unexpected disturbances and the entire mechanism may collapse. The system, in general, is made operable in a phased manner and after a finite time may again run smoothly, with some other stability properties. We explore this whole mechanism as a hybrid system and study the stability properties in terms of two measures so as to cater to a very general set up.     

Computing the spectral function for singular Sturm–Liouville problems

Algorithms for computing Sturm–Liouville spectral density functions are developed based on several mathematical characterizations. Convergence and error bounds are derived and methods are tested on several examples. The results are compared with those from the existing software package SLEDGE.     

Global solutions of a singular initial value problem to second order nonlinear delay differential equations

The paper is concerned with an initial value problem to second order nonlinear singular delay differential equations. By the use of the Schauder fixed point theorem, a result for the existence of global solutions is derived. Also, via the Banach contraction principle, another result concerning the existence and uniqueness of global solutions is established. Moreover, applications of these results to a particular case of second order nonlinear singular delay differential equations as well as to the special case of second order nonlinear singular ordinary differential equations are presented. Finally, some specific applications to certain equations and two examples are given to demonstrate the applicability of the results of the paper.     

Optimised synthesis of diamino-triazinylmethyl benzoates as inhibitors of Rad6B ubiquitin conjugating enzyme

Recently, we have identified the first inhibitors of Rad6B, an E2 enzyme essential for post-replication DNA repair and a potential new drug target for the treatment of breast cancer. We report two newly optimised synthetic routes to our [4-amino-6-(phenylamino)-1,3,5-triazin-2-yl]methyl 4-nitrobenzoate target compounds TZ8 and TZ9 with general applicability for further structure–activity relationship studies around this pharmacophore. The key step involved the condensation/cyclisation between phenylbiguanide and either ethyl bromoacetate or dimethyloxalate in good yield.     

Grain boundary,triple junction and quadruple point mobility controlled normal grain growth

Reduction in stored free energy provides the thermodynamic driving force for grain and bubble growth in polycrystals and foams. Evolution of polycrystalline networks exhibit the additional complication that grain growth may be controlled by several kinetic mechanisms through which the decrease in network energy occurs. Polyhedral boundaries, triple junctions (TJs), and quadruple points (QPs) are the geometrically distinct elements of three dimensional networks that follow Plateau’s rules, provided that grain growth is limited by diffusion through, and motion of, cell boundaries. Shvindlerman and co-workers have long recognized the kinetic influences on polycrystalline grain growth of network TJs and QPs. Moreover, the emergence of interesting polycrystalline nanomaterials underscored that TJs can indeed influence grain growth kinetics. Currently there exist few detailed studies concerned either with network distributions of grain size, number of faces per grain, or with ‘grain trajectories’, when grain growth is limited by the motion of its TJs or QPs. By contrast there exist abundant studies of classical grain growth limited by boundary mobility. This study is focused on a topological/geometrical representation of polycrystals to obtain statistical predictions of the grain size and face number distributions, as well as growth ‘trajectories’ during steady-state grain growth. Three limits to grain growth are considered, with grain growth kinetics controlled by boundary, TJ, and QP mobilities.     

Visible‐Light‐Responsive Reversible Photoacid Based on a Metastable Carbanion

A new photoacid that reversibly changes from a weak to a strong acid under visible light was designed and synthesized. Irradiation generated a metastable state with high C?H acidity due to high stability of a trifluoromethyl‐phenyl‐tricyano‐furan (CF₃PhTCF) carbanion. This long‐lived metastable state allows a large proton concentration to be reversibly produced with moderate light intensity. Reversible pH change of about one unit was demonstrated by using a 0.1 mM solution of the photoacid in 95 % ethanol. The quantum yield was calculated to be as high as 0.24. Kinetics of the reverse process can be fitted well to a second‐order‐rate equation with k=9.78×10² M ^?1 s^?1. Response to visible light, high quantum yield, good reversibility, large photoinduced proton concentration under moderate light intensity, and good compatibility with organic media make this photoacid a promising material for macroscopic control of proton‐transfer processes in organic systems.     

On the concept of solution for fractional differential equations with uncertainty

We consider a differential equation of fractional order with uncertainty and present the concept of solution. It extends, for example, the cases of first order ordinary differential equations and of differential equations with uncertainty. Some examples are presented.     

Three solutions of constant sign for a system of discrete equations

We consider the following system of discrete equations

Temporal reflectance from a light pulse irradiated medium embedded with highly scattering cores

This paper presents a new approach to utilize ultrashort pulsed laser for optical diagnostics with numerical simulations. The method is based on the use of ultrafast pulses with a pulsewidth selected according to the probed medium's radiative property and/or size. Our previous work in nonhomogeneous media has shown that the resulting time-resolved reflectance signal will have a unique characteristic: it will show a direct correlation of ballistic photon travel time and interface location, which is in between different layers or nonhomogeneous regions. The premise is based on utilizing the medium's structural information carried by the ballistic and snake photons without being masked by the diffuse photons. In this study, the space-time correlation is further explored in the case of minimally scattered photons from a large scattering coefficient core region embedded within a less-scattering medium. Time-resolved reflectance signals of the single scattering core and multiple scattering cores within a three-dimensional medium demonstrate the concept and illustrate the additional effect due to the scattered photons from the core region. A unique temporal signal profile's correlation at various detector positions with respect to the scattering core is explained in detail. The result has important implications. This approach will lead to a much simpler and more precise determination of the probed medium's composition or structure. Due to the large computational requirement to obtain the physical details of the light pulse propagation inside highly scattering multi-dimensional media, the reverse Monte-Carlo method is used. The potential applications of the method include non-destructive diagnostics, optical imaging, and remote sensing of underwater objects.