Convolution factorability of bilinear maps and integral representations

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators.     

A new method for pyrrole peri-annulation: synthesis of 1H-1,5,7-triazacyclopenta[c,d]phenalenes from 1H-perimidines

New methods for the synthesis of 1H-1,5,7-triazacyclopenta[c,d]phenalenes have been developed based on a sequence involving Schmidt reaction of keto-perimidines and acylation of the intermediate amides with 1,3,5-triazines or carboxylic acids. The same synthetic sequence starting from the corresponding alkylperimidine includes acylation with acetic acid as the first step.     

Exact classical solutions of nonlinear sigma models on supermanifolds

We study two-dimensional nonlinear sigma models with target spaces being the complex super-Grassmannian manifolds, that is, coset supermanifolds G(m,p|n,q)≅U(m|n)/[U(p|q)⊗U(m−p|n−q)]$G(m,p|n,q)\cong U(m|n)/[U(p|q)\otimes U(m-p|n-q)]$ for 0?p?m$0?p?m$, 0?q?n$0?q?n$ and 1?p+q$1?p+q$. The projective superspace CPm−1|n${\mathbf{CP}}^{m-1|n}$ is a special case of p=1$p=1$, q=0$q=0$. For the two-dimensional Euclidean base space, a wide class of exact classical solutions (or harmonic maps) are constructed explicitly and elementarily in terms of Gramm–Schmidt orthonormalisation procedure starting from holomorphic bosonic and fermionic supervector input functions. The construction is a generalisation of the non-super-case published more than twenty years ago by one of the present authors.     

Intramolecular and Intermolecular Schmidt Reactions of Alkyl Azides with Aldehydes

Despite recent advances in the use of alkyl azides in ring expansion reactions of ketones, there has been little work done on the corresponding chemistry of aldehydes. In the present study, the Lewis acid-promoted reactions of alkyl azides with aldehydes were studied in both intermolecular and intramolecular contexts. The intramolecular reactions of azidoalkyl aldehydes in which the azide and carbonyl groups were separated by 2-5 carbons were examined. Although the examples having the shortest tether failed (3-azidopropanals), each of the other systems gave good yields of either NH-substituted lactams (resulting from hydride migration in the initially formed azidohydrin adduct) or formamides (alkyl migration). The product formed was dependent on the chain length of the starting azido aldehyde. The intermolecular reactions were less efficient, requiring TiCl₄ promotion for even moderate yields, and in each case gave mixtures of products resulting from hydride and alkyl migration.     

Schmidt representation of bilinear operators on Hilbert spaces

Current work defines Schmidt representation of a bilinear operator $T:H_1\times H_2\to K$, where $H_1,H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.     

Stability and bifurcation in a two-species reaction–diffusion–advection competition model with time delay

In this paper, we are concerned with the dynamics of a class of two-species reaction–diffusion–advection competition models with time delay subject to the homogeneous Dirichlet boundary condition or no-flux boundary condition in a bounded domain. The existence of steady state solution is investigated by means of the Lyapunov–Schmidt reduction method. The stability and Hopf bifurcation at the spatially nonhomogeneous steady-state are obtained by analyzing the distribution of the associated eigenvalues. Finally, the effect of advection on Hopf bifurcation is explored, which shows that with the increase of convection rate, the Hopf bifurcation phenomenon is more likely to emerge.     

基于波像差函数建立大口径施密特校正板方程

施密特光学系统由施密特校正板和球面反射镜组成,校正板设置在球面反射镜的球心处,系统的焦点不一定和反射镜的焦点重合。为了得到精确的校正板面形初始结构参数,基于波像差函数建立了带有离焦量的大口径施密特校正板的数学模型,同时校正了系统的三级和五级球差。利用光学设计软件对校正板口径为1000mm、主镜的曲率半径为2000mm、F数为1的系统进行了设计和分析,来验证校正板面形数学模型的正确性。结果显示此校正板的数学模型与优化后结果吻合得较好,校正板面形初始结构参数的精度得到了极大的提高,这为大口径、大相对孔径的施密特光学系统的设计提供了理论基础。     

The behavior of a crack in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading

Summary In this paper, the behavior of a crack in functionally graded piezoelectric/piezomagnetic materials subjected to an anti-plane shear loading is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using a Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. These equations are solved using the Schmidt method. The relations among the electric displacement, the magnetic flux and the stress field near the crack tips are obtained. Numerical examples are provided to show the effect of the functionally graded parameter on the stress intensity factors of the crack.The authors are grateful for financial support from the Natural Science Foundation of Hei Long Jiang Province (A0301), the National Natural Science Foundation of China (50232030, 10172030), the Natural Science Foundation with Excellent Young Investigators of Hei Long Jiang Province(JC04-08) and the National Science Foundation with Excellent Young Investigators (10325208).     

偏振和衍射双重效应影响的Schmidt棱镜特性

一束偏振光经过Schmidt棱镜的两个不同路径,成为两个不同的偏振状态,使得出射光束的偏振态呈现非均匀分布.为了探索偏振态非均匀分布对Schmidt棱镜传光质量的影响机理,将两个路径对应的光波函数引入屋脊衍射积分方程,得到了偏振效应影响的屋脊衍射场强分布.场强分布的数字计算表明:在偏振效应和衍射效应的双重影响下,经Schmidt棱镜出射的光场分布出现了严重变形;对应同一入射线偏振光出现的P、S两分量位相差的差异,使得P、S分量的屋脊衍射光强分布IP、IS有很大差异,这种差异在入射线偏振光方位角为0°和90°时达到最大;而合光波的衍射光强IP+IS是分裂为有一定空间间距的多峰分布,但多峰分布随入射线偏振光方位角的变化比较小.实验拍摄了He-Ne激光经Schmidt棱镜衍射后出射光强分布图.实验结果和理论分析一致性表明:Schmidt棱镜中的偏振效应和屋脊衍射效应导致了一束入射线偏振光分裂为有一定空间间距的多峰光束,严重破坏了Schmidt棱镜的传光特性.