A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

用非最大纠缠态实现三粒子纠缠态的几率隐形传输

提出一种利用三对非最大纠缠态粒子作为量子通道来传输三粒子纠缠态的方案。在此方案中发送者Alice作三次Bell态测量，并将测量结果通过经典通道告诉Bob，B。b根据接收到的信息即可通过相应的幺正变换来重建Alice要传输的量子态。     

三维裂纹体应力强度因子的计算

本文采用塌缩三棱柱形奇异单元的位移计算应力强度因子，给出了一个新的全三维外推公式，它是Ｃｈｅｎ和Ｋｕａｎｇ公式〔１３〕的全三维推广，特例证明，它的精度比Ｉｎ－ｇｒａｆｆｅａ和Ｍａｎｕ的公式〔８〕高一阶。数值计算表明，结果稳定和对单元尺寸改变不敏感     

三轴运动模拟转台的结构对称性与动力学计算

本文给出了三轴运动模拟转台动力学相互作用的计算公式，并讨论了它的结构对称性问题。     

三轴模拟台的交叉耦合及其控制方法的研究

本文主要从三轴转台的模型出发，研究了采用状态反馈的最优控制及状态观测器来提高控制系统的自适应能力，以解决三轴台特有的不确定非线性耦合问题和变结构解耦控制。     

等达因图象的自动采集与识别

文中提出了一种机机械扫描系统．小功率激光器、图象板、相移元件、微机和程序软件组成的等达因图象自动采集与处理系统，用灰度值比较的思想实现了等达因图象的跟踪采集。提出了四幅图象相移法实现等达因图象的自动识别。系统能快速、实时、精确和全自动地实现等达因图象的记录和分析。给出与物体内各点应力状态一一对应的位相图。并用来研究了其他实验方法困难的局部三维效应问题。     

侧柱对串列双柱脉动压力的干扰

研究了串列双圆柱旁加上一个等直径的圆柱，组成一个等边三角形排列的三圆柱的脉动压力分布．着重研究侧柱对串列双圆柱脉动压力分布的影响．研究的结果表明，绕流等边三角形排列的三圆柱，受影响最严重的是后柱．脉动压力分布出现了严重的不对称，外侧的压力脉动极其强烈，内侧的压力脉动较弱，与时均压力分布，很好的对应关系．另外，侧柱对于串列双圆柱是否达到超过临界间距的绕流流态，有很大的影响     

A three‐dimensional anionic metal–dicyanamide network in poly[ethyltriphenylphosphonium [tri‐μ2‐dicyanamidato‐cadmium(II)]]

The title compound, (C₂₀H₂₀P)[Cd(C₂N₃)₃], consists of ethyltriphenylphosphonium (EtPh₃P⁺) cations filling voids in a three‐dimensional anionic cadmium dicyanamide network. In the structure, each Cd^II atom is connected to six neighbouring Cd^II atoms through six separate dicyanamide ligands, forming cube‐shaped cages. The three‐dimensional anionic network encloses a solvent‐accessible void space of 1851 ų, amounting to 69.3% of the unit‐cell volume. Each cage accommodates only one EtPh₃P⁺ cation.     

A three‐dimensional coordination polymer based on the Zn4(μ3‐OH)2 unit: poly[[(μ4‐benzene‐1,4‐dicarboxylato)bis(μ3‐benzene‐1,4‐dicarboxylato)bis(μ3‐hydroxido)bis(1,10‐phenanthroline‐5,6‐dione)tetrazinc] dihydrate]

The title compound, {[Zn₄(C₈H₄O₄)₃(OH)₂(C₁₂H₆N₂O₂)₂]·2H₂O}_n, has been prepared hydrothermally by the reaction of Zn(NO₃)₂·6H₂O with benzene‐1,4‐dicarboxylic acid (H₂bdc) and 1,10‐phenanthroline‐5,6‐dione (pdon) in H₂O. In the crystal structure, a tetranuclear Zn₄(OH)₂ fragment is located on a crystallographic inversion centre which relates two subunits, each containing a [ZnN₂O₄] octahedron and a [ZnO₄] tetrahedron bridged by a μ₃‐OH group. The pdon ligand chelates to zinc through its two N atoms to form part of the [ZnN₂O₄] octahedron. The two crystallographically independent bdc²⁻ ligands are fully deprotonated and adopt μ₃‐κO:κO′:κO′′ and μ₄‐κO:κO′:κO′′:κO′′′ coordination modes, bridging three or four Zn^II cations, respectively, from two Zn₄(OH)₂ units. The Zn₄(OH)₂ fragment connects six neighbouring tetranuclear units through four μ₃‐bdc²⁻ and two μ₄‐bdc²⁻ ligands, forming a three‐dimensional framework with uninodal 6‐connected α‐Po topology, in which the tetranuclear Zn₄(OH)₂ units are considered as 6‐connected nodes and the bdc²⁻ ligands act as linkers. The uncoordinated water molecules are located on opposite sides of the Zn₄(OH)₂ unit and are connected to it through hydrogen‐bonding interactions involving hydroxide and carboxylate groups. The structure is further stabilized by extensive π–π interactions between the pdon and μ₄‐bdc²⁻ ligands.     

A three‐dimensional cadmium(II) coordination polymer with unequal homochiral double‐stranded concentric helical chains

A homochiral helical three‐dimensional coordination polymer, poly[[(μ₂‐acetato‐κ³O,O′:O)(hydroxido‐κO)(μ₄‐5‐nicotinamido‐1H‐1,2,3,4‐tetrazol‐1‐ido‐κ⁵N¹,O:N²:N⁴:N⁵)(μ₃‐5‐nicotinamido‐1H‐1,2,3,4‐tetrazol‐1‐ido‐κ⁴N¹,O:N²:N⁴:N⁵)dicadmium(II)] 0.75‐hydrate], {[Cd₂(C₇H₅N₆O)₂(CH₃COO)(OH)]·0.75H₂O}_n, was synthesized by the reaction of cadmium acetate, N‐(1H‐tetrazol‐5‐yl)isonicotinamide (H‐NTIA), ethanol and H₂O under hydrothermal conditions. The asymmetric unit contains two crystallographically independent Cd^II cations, two deprotonated 5‐nicotinamido‐1H‐1,2,3,4‐tetrazol‐1‐ide (NTIA⁻) ligands, one acetate anion, one hydroxide anion and three independent partially occupied water sites. The two Cd^II cations, with six‐coordinated octahedral and seven‐coordinated pentagonal bipyramidal geometries are located on general sites. The tetrazole group of one symmetry‐independent NTIA⁻ ligand links one of the independent Cd^II cations into 6₁ helical chains, while the other NTIA⁻ ligand links the other independent Cd^II cations into similar but unequal 6₁ helical chains. These chains, with a pitch of 24.937 (5) Å, intertwine into a double‐stranded helix. Each of the double‐stranded 6₁ helices is further connected to six adjacent helical chains through an acetate μ₂‐O atom and the tetrazole group of the NTIA⁻ ligand into a three‐dimensional framework. The helical channel is occupied by the isonicotinamide groups of NTIA⁻ ligands and two helices are connected to each other through the pyridine N and carbonyl O atoms of isonicotinamide groups. In addition, N—H...O and O—H...N hydrogen bonds exist in the complex.