AFM, SEM and GIXRD studies of thin films of red polycarbazolyldiacetylenes

In this paper we report the results of a morphological and structural investigation on film properties of a soluble polydiacetylene, the poly[1,6-bis(3,6-dihexadecyl-N-carbazolyl)-2,4-hexadiyne] (polyDCHD-HS). The red films of this polymer, prepared by standard spin-coating techniques, revealed absence of linear dichroism and birefringence in contrast with the ordered mesophases detected by powder X-ray studies. In order to interpret the optical behavior of this polymer, we performed AFM and SEM studies of polyDCHD-HS films spun on hydrophylic and hydrophobic glass substrates. We found the presence of surfaces organized in rod-like particles, more regularly oriented on the hydrophylic substrate. GIXRD studies, carried out on films sufficiently thick to allow the observation of the diffraction pattern, reveled the presence of a lamellar structure with a spacing of 3.22 nm. The low intensity of the diffraction peaks and the isotropic linear optical properties of the films show that the lamellar mesophases are not extended over large areas. These findings were compared with the data obtained from AFM and SEM studies on films of two other polydiacetylenes, the poly[1-(3,6-dihexadexyl-N-carbazolyl)-6-(N-carbazolyl)-2,4-hexadyine] (polya-DCHD) and the poly[1,6-bis(3,6-dipalmitoyl-N-carbazolyl)-2,4-hexadyine] (polyDPCHD), spun on hydrophylic glass substrate. The results confirmed the presence of nodular morphologies which seem to be a general characteristic of this class of materials. The particles organization appears instead related to the chemical nature of the substituents on the carbazolyl rings.     

Differentiators and the geometry of polynomials

In 1959, Davis introduced the concept of a differentiator of an operator on a finite-dimensional Hilbert space. We prove that every such operator possesses a differentiator. We also use the theory of differentiators to solve several problems in the geometry of polynomials. For instance, we answer in the affirmative a twenty year old unsolved conjecture of Schoenberg, a related conjecture of Katsoprinakis and a fifty year old unsolved conjecture of De Bruijn and Springer.     

On the successive supersymmetric rank‐1 decomposition of higher‐order supersymmetric tensors

In this paper, a successive supersymmetric rank‐1 decomposition of a real higher‐order supersymmetric tensor is considered. To obtain such a decomposition, we design a greedy method based on iteratively computing the best supersymmetric rank‐1 approximation of the residual tensors. We further show that a supersymmetric canonical decomposition could be obtained when the method is applied to an orthogonally diagonalizable supersymmetric tensor, and in particular, when the order is 2, this method generates the eigenvalue decomposition for symmetric matrices. Details of the algorithm designed and the numerical results are reported in this paper. Copyright © 2007 John Wiley & Sons, Ltd.     

Monotonic solutions of a quadratic integral equation of fractional order

In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.     

Dual effects of fullerene doped to holographic polymer dispersed liquid crystals

Doping of conductive fullerene particles to the formulation of conventional holographic polymer dispersed liquid crystal‐induced dual effects of reducing both droplet coalescence and operating voltage. Fullerene induced an induction period which otherwise does not exist, followed by a gradual increase of diffraction efficiency to a saturation value being increased with increasing fullerene content. The increased diffraction efficiency was caused by the decreased droplet coalescence which was due to the hindered migration of LC by the fullerene particles. On the other hand, doped fullerene particles augmented the conductivity of polymer phase and hence the local electrical field imposed on LC droplet, which overcome the threshold for driving and reduced operating voltage and response times. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 5590–5596, 2007     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.     

弱混强平稳序列随机容量、随即秩次序统计量的极限分布律

设W(1n)W2(n)…W(nn)是严平稳随即序列{Xn,n 1}的前几个变量的顺序统计量,Nn Mn是两正整值r.v,n 1.在较弱一些的混合条件下,本文给出了W(NMn)n标准化后的极限分布,从而使一些提法更趋一般化,主要结果推进了Rao(1984)和张国胜(1993)的工作.由于混合条件的减弱,使得对m-相关和正态平稳过程的具体应用讨论更加充分.     

具非线性中立项的二阶时滞微分方程的振动性质

By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.     

Algebraic structure of association schemes of prime order

Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.     

一个新的最优模糊存储模型

提出一个新的具有积压定单的关于模糊订购量的模糊存储模型.在模糊函数原理下,给出了模糊总存储成本.为了寻找最优解,把最优模糊存储模型转化为双目标最优化模型,利用L ingo8.0求解不等式约束问题,我们发现最优解都是确定的实数.此外,当模糊订购量和模糊总需求都是三角形(或权重均为1/2梯形)模糊数时,我们提出模型的最优解与经典的具有积压定单存储模型具有相同的结果.