The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szeg? weight functions consisting of any one of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}\,t^{2},\quad t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ? 1 and sum of semi-axes ρ > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99–127, 2006).     

Symmetry Reductions,Group‐Invariant Solutions,and Conservation Laws of a (2+1)‐Dimensional Nonlinear Schrödinger Equation in a Heisenberg Ferromagnetic Spin Chain

Nonlinear spin excitations in ferromagnetic spin chains are studied for spintronic and magnetic devices including magnetic‐field sensors and for high‐density data storage. Here, (2+1)‐dimensional nonlinear Schrödinger equation is investigated, which describes the nonlinear spin dynamics for a Heisenberg ferromagnetic spin chain. Lie point symmetry generators and Lie symmetry groups of that equation are derived. Lie symmetry groups are related to the time, space, scale, rotation transformations, and Galilean boosts of that equation. Certain solutions, which are associated with the known solutions, are constructed. Based on the Lie symmetry generators, the reduced systems of such an equation are obtained. Based on the polynomial expansion and through one of the reduced systems, group‐invariant solutions are constructed. Soliton‐type group‐invariant solutions are graphically investigated and effects of the magnetic coupling coefficients, that is, α₁, α₂, α₃, and α₄, on the soliton's amplitude, width, and velocity are discussed. It is seen that α₁, α₂, α₃, and α₄ have no influence on the soliton's amplitude, but can affect the soliton's velocity and width. Lax pair and conservation laws of such an equation are derived.     

Self-assembly of silanated poly(ethylene glycol) on silicon and glass surfaces for improved haemocompatibility

Surface immobilization of poly(ethylene glycol) (PEG) is an effective method to produce a material surface with protein repulsive property. This property could be made permanent by using covalent grafting of the PEG molecules onto material surfaces. In this study, self-assembled monolayers (SAMs) of PEG on silicon-containing materials (silicon chip and glassplate) were obtained through a one-step coating procedure of one kind of silanated PEG molecules made through the reaction between monomethoxy PEG and 3-isocyanatopropyltriethoxysilane. Atomic force microscopy (AFM) and water static contact angle measurement were employed to investigate the surface topography and wettability of the PEGylated material surfaces. The changes in the topography and the water contact angle of the surfaces with time of incubation in PBS solution were also measured. The results revealed that stable and uniform self-assembled monolayers of PEG could be formed on silicon or glass surfaces by simply soaking the substrates in the solution of silanated PEGs. The covalent coupling of PEGs to the substrates was also confirmed. In order to evaluate the stability of the SAMs, blood compatibility of the modified glassplate surface was evaluated by measuring full blood activated partial thromboplastin time (APTT), prothrombin time (PT), and thrombin time (TT), as well as by scanning electron microscopy (SEM) analysis of the appearance of adherence and denaturation of blood platelets onto the glassplate. The silanated PEGs were shown to have good effect on the protein-repulsion as well as haemocompatibility of the substrates.     

Congruences of multipartition functions modulo powers of primes

Let p _r (n) denote the number of r-component multipartitions of n, and let S _γ,λ be the space spanned by η(24z) ^γ ?(24z), where η(z) is the Dedekind’s eta function and ?(z) is a holomorphic modular form in \(M_{\lambda}(\mathrm{SL}_{2}(\mathbb{Z}))\) . In this paper, we show that the generating function of \(p_{r}(\frac{m^{k} n +r}{24})\) with respect to n is congruent to a function in the space S _γ,λ modulo m ^k . As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5,7,11 and Gandhi’s congruences of p ₂(n) modulo 5 and p ₈(n) modulo 11. Furthermore, using the invariance property of S _γ,λ under the Hecke operator \(T_{\ell^{2}}\) , we obtain two classes of congruences pertaining to the m ^k -adic property of p _r (n).     

两个非线性耦合网络间的自适应同步

本文研究了带非线性信号连接的两个复杂网络间的同步问题,引入非线性耦合参数α来调节两个复杂网络的同步．若耦合参数不能保证网络达到外部同步,这里我们提出了一种自适应同步方式,通过此方式可以使两个复杂网络达到同步,最后通过简单的数值算例来阐述得到的理论结果,包括网络具有相同和不相同的拓扑结构两种情形．     

Origin of thermally induced second harmonic generation in PbO-B2O3 glasses

Optical second harmonic generation (SHG) with second-order nonlinearity χ((2)) as high as 2.1 pm/V has been achieved in water quenched PbO-B(2)O(3) glasses. No nonlinear depletion layer or microcrystals were observed in these glasses and the mechanism for nonlinearity has been explored in this Letter. Our results show that the possible mechanism for SHG in these glasses can be attributed to their low thermal conductivity that led to a large surface stress gradient, which broke the inversion symmetry of the glasses and subsequently induced the nonlinear effect. These findings suggest that low thermal conductivity induced high stress gradients to lead to large SHG.     

形变量对MAE及BN的影响

研究了无取向硅钢Ｗ２３Ｇ，Ｗ１０，Ｗ１４和取向硅钢Ｑ８Ｇ４种材料在不同形变量下的磁声发射（ＭＡＥ）和巴克豪森噪音（ＢＮ），结果表明ＭＡＥ随形变量增大而减少，ＢＮ随形变量的增大而增大．测量了形变量为１０．４％的Ｗ１８和Ｗ２３Ｇ在１００，２００，３００，…，７００°Ｃ各退火１ｈ的ＭＡＥ与ＢＮ值．其中ＭＡＥ值随退火温度的升高逐渐增大，ＢＮ值在高于５００°Ｃ退火时明显下降．并对上述实验结果进行了详细讨论．     

钙钛矿型复合氧化物LaNi_(1-x)Mn_xO_3晶体结构与催化性能的关系

用XRD、IR和TPR等技术研究了钙钛矿型复合氧化物的晶体结构及其反应性能.整个组成范围内,LaNi_(i-x)Mn_xO_3(0.0≤x≤1.0)体系都生成单一钙钛矿相;x=0.0,0.6≤X≤1.0时为菱形晶系,0.2≤x≤0.5为立方晶系.IR和TPR研究表明,晶体结构对B—O键合作用存在较大影响.CO微反考察表明,本体系的催化活性与晶系存在密切关系,晶体对称性越高,反应活性越低.说明钙钛矿型复合氧化物的催化活性与体系活性中心的B—O键合作用能力有关.     

Determination and removal of sulfonamides and quinolones from environmental water samples using magnetic adsorbents

In this study, the magnetic materials known as polymerized ionic liquid@3‐(trimethoxysilyl)propyl methacrylate@Fe₃O₄ nanoparticles were synthesized and utilized as potential adsorbents. First, these nanoparticles were applied to the analysis of sulfonamides and quinolones present in different water samples using magnetic solid phase extraction and high‐performance liquid chromatography. Under optimized conditions, the developed method showed excellent detection sensitivity, with limits of detection (S/N = 3) and quantification limits (S/N = 10) within 0.2–1.0 and 0.8–3.4 μg/L, respectively. The spiked recoveries of the SAs and QNs in environmental water samples ranged from 83.5 to 103.0%, with RSDs of less than 4.5%. In addition, the adsorbents effectively removed sulfamethoxazole and ofloxacin present in existing aquatic environments. The adsorption kinetics and isotherms of sulfamethoxazole and ofloxacin on the magnetic adsorbents were studied to assess removal performance. The results indicate that the adsorption process follows a pseudo‐second‐order mechanism, which reveals that the sorption mechanism is the rate‐limiting step and produces high q_max values (sulfamethoxazole = 70.35 mg/g and ofloxacin = 48.95 mg/g), thus demonstrating the enormous adsorption capacity of these magnetic adsorbents.     

New benzoyl glucosides and cytotoxic pterosin sesquiterpenes from Pteris ensiformis Burm

Three new compounds: 2R,3R-pterosin L 3-O-beta-D-glucopyranoside (1), beta-D-xylopyranosyl(1-->2)-7-O-benzoyl-beta-D-glucopyranoside (2) and 4-O-benzoyl-beta-D-xylo-pyranosyl(1-->2)-7-O-benzoyl-beta-D-glucopyranoside (3), together with nine known compounds, were isolated from the ethyl acetate extract of Pteris ensiformis. 5-[2-Hydroxyethylidene]-2(5H)-furanone (4), which had been synthesized, was isolated from natural sources for the first time. The structures of all isolated compounds were determined on the basis of mass and spectroscopic evidence. Compound 1 and pterosin B (5) show cytotoxicity against HL 60 cells (human leukemia) with the IC(50) values of 3.7 and 8.7 microg/mL, respectively.