New method to solve electromagnetic parabolic equation

This paper puts forward a new method to solve the electromagnetic parabolic equation （EMPE） by taking the vertically-layered inhomogeneous characteristics of the atmospheric refractive index into account. First, the Fourier transform and the convo- lution theorem are employed, and the second-order partial differential equation, i.e., the EMPE, in the height space is transformed into first-order constant coefficient differential equations in the frequency space. Then, by use of the lower triangular characteristics of the coefficient matrix, the numerical solutions are designed. Through constructing ana- lytical solutions to the EMPE, the feasibility of the new method is validated. Finally, the numerical solutions to the new method are compared with those of the commonly used split-step Fourier algorithm.     

Uniform persistence for population interaction models with time delay

In this paper uniform persistence is establised for kolmogorov type Prey‐Predator and competion models with per capitals ent growth rates that are dependent on time‐lagged population densities.     

Siloxane polymers containing azo moieties synthesized by click chemistry for photo responsive and liquid crystalline applications

Three new types of siloxane‐based photoactive liquid crystalline polymers containing azo side groups were synthesized through the click chemistry route. The polymers having molecular weight range of 14,000–34,000 g mol^?1 were soluble in most of the polar solvents like chloroform, tetrahydrofuran, dimethylformamide, dimethyl sulfoxide, and dichloromethane. The photoresponsive trans–cis photoisomerization under UV radiation and cis–trans relaxation process in dark for the polymers were studied. The isomerization rate constants were found to be 0.01–0.04 sec^?1 and 1.16*10^?4–4.67*10^?4 sec^?1, respectively. It has been noted that the polymers showed high intensity absorption for n‐π* in chloroform. Both trans and cis forms of azide monomers having azo moiety exhibited molar extinction coefficient ( ? _max) in the range of 22,000–33,000 L mol^?1 cm^?1. The thermotropic behavior of the polymers was studied by polarizing optical microscope (POM) and differential scanning calorimetry (DSC) experiments. Polymer P1 showed liquid crystalline textures of nematic droplets, whereas P2 showed smectic focal conic texture and nematic droplets. Polymer P1 was also studied for photomechanical bending on exposure to UV radiation. The polymers showed initial degradation temperature in the range of 210–275°C. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Click chemistry‐based synthesis of azo polymers for second‐order nonlinear optics

Four linear polymers containing pendant azo moiety were synthesized through click chemistry for second‐order nonlinear optical study. The polymers were found soluble in most of the polar organic solvents such as tetrahydrofuran (THF), chloroform, and dimethyl formamide (DMF). The polymers showed thermal stability up to 300 °C and glass transition temperatures (T_g) in the range of 120–140 °C. The molecular weights (M_w) of these polymers (measured by gel permeation chromatography) were in the range 37,900–55,000 g/mol. The polymers were found to form optically transparent films by solution casting from THF solution. Order parameters were calculated from UV–vis absorption spectra. The morphology changes in the films after poling were characterized by atomic force microscopy. The angular dependence, temperature dependence, and time dependence of second harmonic generation (SHG) intensity were obtained by using 1064 nm Nd:YAG laser. The SHG intensity remained unchanged up to 95 °C. At room temperature, it remained stable up to 8 days after initial drop of about 14%. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Bio‐based comb‐like copolymers derived from alkyl 10‐undecenoates and maleic anhydride

In this article, 10‐undecenoic acid, based on castor oil, was used a raw material for the synthesis of alternating copolymers. ω‐Unsaturated fatty esters as alkyl 10‐undecenoates were prepared by the esterification reaction of 10‐undecenoic acid with alkyl alcohol. A series of comb‐like copolymers were synthesized by free radical polymerization from maleic anhydride and alkyl 10‐undecenoates copolymers with different length of alkyl side chains in a toluene solution. These copolymers were investigated by ¹H and ¹³C nuclear magnetic resonance, Fourier transform infrared spectroscopy, gel permeation chromatography, thermogravimetric analysis, differential scanning calorimetry (DSC). The copolymers were obtained in a low molecular weight in a range 3370–12,240 g mol⁻¹ and their structural characterization indicated the formation of alternating copolymers. DSC characterization revealed that these comb‐like copolymers showed amorphous to semicrystalline behavior by increasing the length of side chains. The bio‐based comb‐like copolymers allow for the development of new polymeric materials for several applications. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 1039–1045     

Beta-type means

Abstract

Based on the relationship between the Beta and the Euler Gamma function, beta-type functions of several variables, generated by a single variable function, are introduced. Applying the method of iterative functional equations, all means of beta-type are determined and some of their properties are discussed.     

On a problem of S.L. Sobolev

In his famous works of 1930 [1,2] Sergey L. Sobolev has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in Rş. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov (2002) and Smirnov (1964) to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.     

Ratio-conjugation tool and its application in linear optimization models†

The ratio-conjugation-tool (RC) is constructed systematically, and its applicability for analysis of economic models is demonstrated. One of the main results establishes a general formula for RC of aggregative function. This tool is applied to the analysis of linear economic dynamics models of Neumann–Gale type which can be solved by a dynamic programming method: for each direct problem we construct the corresponding dual RC-problem. Moreover, we consider three examples of the most important economic dynamics models and look at these now from a unified general position.