一类非线性发展方程的精确解

利用hirota双线性法,得到(3+1)维孤子方程、(3+1)维KP-Boussinesq方程、(2+1)维修正Caudrey-Dodd-Gibbon-Kotera-S awada方程、Hirota-Satsuma浅水波方程的精确解,并做出一部分解的图形,进一步研究解的结构和性质.     

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

两个基于2,5-二甲氧基对苯二甲酸和咪唑衍生物的超分子钴配位聚合物的结构及光催化反应

本文以咪唑衍生物为配体,通过水热合成法与钴离子制备出两个配位聚合物:{[Co(DTA)(1,4-DIB)(H₂O)]·H₂O}_n(1)和[Co(DTA)(1,3-BMIB)]_n(2)(1,4-DIB=1,4-二(1H-咪唑-1-基)苯; 1,3-BMIB=1,3-二(4-甲基-1H-咪唑-1-基)苯;H₂DTA=2,5-二甲氧基对苯二甲酸)。利用X射线单晶衍射、粉末衍射、热失重、元素分析、红外光谱以及固体紫外-可见光谱等对两个配合物进行了表征。结构分析证实配合物1和2是通过二维结构堆积成的三维超分子化合物。粉末衍射测试则显示两个配合物在水中有很好的稳定性。固体紫外-可见光谱显示两个配合物属半导体材料,对紫外-可见光有很强的吸收作用。在光催化实验中,配合物1和2可加快亚甲基蓝的降解速度。     

Synthesis and optical properties of new chalcones containing 4-[bis(2-hydroxyethyl)amino]phenyl fragment

New chalcones with 4-[bis(2-hydroxyethyl)amino] phenyl fragment were obtained from 4-[bis(2-acetoxyethyl) amino]- benzaldehyde by the Claisen–Schmidt reaction. From their UV-VIS absorption and emission spectra, optical band gap values were calculated based on the Stokes shifts as well as the molar absorption coefficients and fluorescence quantum yields were estimated. The dependence of the absorption and emission maxima on solvent polarity and pH was evaluated.     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

Highly sensitive perylene tetra-(alkoxycarbonyl) based colorimetric and ratiometric probes for fluorine detection

Fluoride anion (F^?) affects environmental, biological, and chemical processes significantly. Therefore, its detection has received increasing attention, and sensitive, effective, and convenient probes for F^? detection need to be developed urgently. In this work, two perylene tetra-(alkoxycarbonyl) (PTAC) based colorimetric and ratiometric probes, P1 and P2, were developed for the detection of F^?. The interactions between F^? and these two probes were investigated by absorption, electrochemistry, ¹H NMR, and density functional methods. Both the two probes were complexed with F^? with a ratio of 1:1. The detection limits of P1 and P2 were 0.22 μM and 0.87 μM, respectively. It was worth noting that the absorption peak of P1 showed a 190 nm red shift when sensing F^?, and P1 is the largest red shift value reported in F^? probes based on PTAC derivatives. This phenomenon was resulted from the unique configuration and deprotonation of P1 that can promote the intramolecular charge transfer (ICT). This strategy provides an example for the development of other ion probes based on D-A type ICT mechanism.     

Thermoelastic solutions for thermal distributions moving over thin slim rod under memory-dependent three-phase lag magneto-thermoelasticity

Abstract

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of three-phase lag (TPL) theory of thermoelasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effect of moving heat source speed on temperature, stress, and temperature is studied. It is found from the distributions that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. For the better understanding of the effect of moving heat source on all the distributions, three animations are added.     

Sharper uncertainty principles associated with Lp-norm

Motivated by the well-established phase derivative embedded technique, this study devotes to sharper uncertainty principles related to the L^p-norm type of uncertainty product, giving rise to two kinds of uncertainty inequalities that improve the classical result through providing tighter lower bounds. The conditions that truly reach these better estimates are obtained. Examples and simulations are carried out to verify the correctness of the derived results, and finally, possible applications in time-frequency analysis are also given.     

Hardy和Littlewood的一个三角和

本文对Hardy和Littlewood考虑的一个有限三角和做了进一步地研究.通过充分运用Chebyshev多项式和M?bius函数的性质,建立了该有限三角和的一个有趣的恒等式,并得到了一个精确的渐近公式.