Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)
非线性流形降维方法结合近红外光谱技术快速鉴别不同海拔的茶叶
方面 LE 方法好于 LLE 方法。模型性能表明，LE_LSSVM 模型性能最佳，预测集总体判别率、Kappa 系数分
别为 100%和 1.00；相比于 PCA_LSSVM、KPCA_LSSVM 和 LLE_LSSVM，模型预测集总体判别率分别提高
1.7%、1.7%、3.3%；Kappa 系数分别提高 0.025、0.03、0.05。研究表明，LE 等非线性流形学习降维方法在近
作者简介：刘鹏，男，硕士研究生，主要从事农产品质量安全检测与模式识别方面的研究。*通信作者：aisrong@
716
茶叶科学
39 卷
altitude based on different dimensional reduction methods and least squares support vector machine (LSSVM) algorithm. Visualization of different dimensionality reduction results show that data processed by KPCA and PCA methods were more discrete. In particular, there were more overlaps between 400-800 m and 800-1 200 m samples. However, the same kind of sample points could be gathered well in three-dimensional space by the nonlinear manifold dimensionality reduction methods can. Tea at different altitude could be easily separated and the aggregation effect of the LE was better than that of the LLE. The results of models indicate the LE_LSSVM model had the best performance, with the prediction set accuracy and Kappa value of 100% and 1.00 respectively. Compared with PCA_LSSVM, KPCA_LSSVM and LLE_LSSVM models, the accuracy of prediction set was improved by 1.7%, 1.7%, 3.3% and Kappa values increased by 0.025, 0.03, and 0.05. The results show that LE and other nonlinear manifold dimensionality reduction methods were effective in reducing dimension of near infrared spectral data, simplifying model complexity, and improving model precision. The study provides a new means for rapid detecting for tea quality research. Keywords: tea, near infrared spectroscopy, nonlinear manifold dimensionality reduction methods, laplacian eigenmaps
基于谱聚类的流形学习算法研究
基于谱聚类的流形学习算法研究王洪波;罗贺【期刊名称】《中国科学技术大学学报》【年(卷),期】2013(043)001【摘要】传统流形学习算法虽然是一种常用的有效降维方法,但由于其自身计算结构的限制,往往存在数据分析不足和计算时间较长等问题.为此提出一种基于谱聚类的流形学习算法(spectral clustering locally linear embedding,SCLLE),并对其机理以及优点给予了实例证明.在UCI和NCBI数据集上的实验结果表明,该算法具有较好的识别效果和计算性能.%Although traditional manifold learning algorithms are common and effective dimension reduction methods, they still have calculating structure limits of their own, which lead to some problems such as inadequate data analyses and long calculation time. Therefore, on the basis of spectral clustering, a manifold learning algorithm named SCLLE (spectral clustering locally linear embedding) was proposed and its mechanisms as well as its advantages were demonstrated. Experiments with UCI and NCBI data sets show that the proposed algorithm has better recognition effect and computational performance.【总页数】8页(P79-86)【作者】王洪波;罗贺【作者单位】合肥工业大学管理学院,安徽合肥230009;过程优化与智能决策教育部重点实验室,安徽合肥230009;合肥工业大学管理学院,安徽合肥230009;过程优化与智能决策教育部重点实验室,安徽合肥230009【正文语种】中文【中图分类】TP301.6【相关文献】1.基于谱聚类和增量学习的运动目标物体检测算法研究 [J], 黄伟;杨文姬;曾璟;曾舒如;陈光2.基于流形学习的多光谱优化波段选择算法研究 [J], 顾章源;刘翔;苏枫;郑翰清;刘达;李建勋3.基于流形正则化极限学习机的文本分类算法研究 [J], 庞皓明;冀俊忠;刘金铎;姚垚4.基于主动学习的半监督谱聚类算法研究 [J], 刘晓丽;牟意红5.基于联邦学习的船舶AIS轨迹谱聚类算法研究 [J], 吕国华;胡学先;张启慧;魏江宏因版权原因，仅展示原文概要，查看原文内容请购买。
量子化学波谱计算基本流程
量子化学波谱计算基本流程英文回答：Quantum chemical spectroscopy is a powerful tool used to study the electronic structure and properties of molecules. It involves the calculation and analysis of various spectroscopic properties, such as electronic transitions, vibrational frequencies, and rotational constants.The basic workflow of quantum chemical spectroscopy can be divided into several steps. First, we need to choose an appropriate quantum chemical method to describe the electronic structure of the molecule. This can range from simple methods like the Hartree-Fock (HF) method to more sophisticated methods like density functional theory (DFT) or coupled cluster (CC) theory.Once the method is chosen, we need to specify the basis set, which is a set of functions used to represent themolecular orbitals. The choice of basis set cansignificantly affect the accuracy of the calculations. Commonly used basis sets include the Gaussian-type orbitals (GTOs) and the Slater-type orbitals (STOs).After setting up the method and basis set, we can perform the electronic structure calculation to obtain the molecular orbitals and their energies. This step is usually done using specialized quantum chemical software packages, such as Gaussian or NWChem.Next, we can calculate the spectroscopic properties of interest. For example, if we are interested in the UV-Vis absorption spectrum, we can calculate the excitation energies and oscillator strengths using methods like time-dependent density functional theory (TD-DFT). If we are interested in the infrared (IR) spectrum, we can calculate the vibrational frequencies and infrared intensities using methods like harmonic vibrational analysis.Once the spectroscopic properties are calculated, we can analyze and interpret the results. This involvescomparing the calculated values with experimental data, identifying the nature of the electronic transitions or vibrational modes, and understanding the underlying molecular interactions.It is important to note that quantum chemical spectroscopy calculations are not always perfect and can have limitations. The accuracy of the results depends on the choice of method, basis set, and other approximations made during the calculations. Therefore, it is always advisable to validate the results with experimental data and consider the uncertainties associated with the calculations.中文回答：量子化学波谱计算是一种用于研究分子电子结构和性质的强大工具。
基于Kelvin-Helmholtz不稳定性和界面剪切作用的扰动波高预测模型
化工进展Chemical Industry and Engineering Progress2024 年第 43 卷第 2 期基于Kelvin-Helmholtz 不稳定性和界面剪切作用的扰动波高预测模型孙宏军1，李腾1，李金霞2，丁红兵1（1 天津大学电气自动化与信息工程学院，天津 300072；2 中国民航大学电子信息与自动化学院，天津 300300）摘要：环雾状流广泛存在于天然气等工业环境中，深入探究扰动波特性对了解环雾状流的演化规律有着重要意义。
本文在内径为15mm 的垂直管路中进行了不同工况条件下的环雾状流实验，分别利用电导环传感器和液膜收集系统测量了液膜厚度和夹带率，并通过双阈值方法从液膜时序信号中提取了扰动波高数据。
探究了扰动波高和夹带率随气、液相流量和工况压力的变化规律，发现两者均随着液相流量的增大而增大，而随着气相流量和工况压力的增大，扰动波高呈下降趋势，夹带率呈上升趋势，验证了两者之间存在密切关联。
然后分析了影响扰动波高的尺度参数，建立了基于Kelvin-Helmholtz 不稳定性和界面剪切作用的扰动波高预测模型，相对均方根误差rRMSE 为4.05%，且98.7%的数据点都在±10.0%的误差以内，预测效果良好。
最后，将新模型与现有的扰动波高关系式进行比较，预测精度和可扩展性都有了较大提高。
关键词：气液两相流；液膜；Kelvin-Helmholtz 不稳定性；界面剪切作用；扰动波高中图分类号：TP212.9；TQ021 文献标志码：A 文章编号：1000-6613（2024）02-0609-10Disturbance wave height prediction model based on Kelvin-Helmholtzinstability and interfacial shearSUN Hongjun 1，LI Teng 1，LI Jinxia 2，DING Hongbing 1(1 School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China; 2 College ElectronicInformation and Automation, Civil Aviation University of China, Tianjin 300300, China)Abstract: Annular mist flow widely exists in natural gas and other industrial environments, in-depth exploration of the characteristics of disturbance waves is of significant importance for understanding the evolutionary patterns of annular mist flow. Experiments on gas-liquid two-phase flows were conducted in a vertical pipeline with an inner diameter of 15mm at different operating conditions. The liquid film thickness and droplet entrainment ratio were measured using a conductive ring sensor and a liquid film collection system respectively. The disturbance wave height data were extracted from the temporal signals of conductive ring sensor by using a dual-threshold method. The disturbance wave height and entrainment ratio with changes in gas-liquid flow rates and working pressure were investigated. It was found that both of them increased with increasing liquid flow rate, while they decreased with increasing gas phase flow rate and working pressure, indicating a close correlation between them. Then the scale parameters affecting the disturbance wave height were analyzed, and a disturbance wave height prediction model研究开发DOI ：10.16085/j.issn.1000-6613.2023-1071收稿日期：2023-06-28；修改稿日期：2023-08-17。
拉曼光谱分析法
3）环状化合物的对称呼吸振动常常是最强的拉曼谱带。
4）在拉曼光谱中，X=Y=Z，C=N=C，O=C=O-这类键 的对称伸缩振动是强谱带，反这类键的对称伸缩振 动是弱谱带。红外光谱与此相反。
5）C-C伸缩振动在拉曼光谱中是强谱带。
Infrared and Raman Spectra of Benzene
IR
Raman
拉曼光谱与红外光谱分析方法比较
拉曼光谱
红外光谱
光 谱 范 围 40-4000C m -1
光 谱 范 围 400-4000C m -1
水可作为溶剂
样品可盛于玻璃瓶，毛细管等容器 中直接测定
固体样品可直接测定
水不能作为溶剂 不能用玻璃容器测定 需要研磨制成 KBR 压片
• spectrum independent of excitation wavelength (488, 632.8, or 1064 nm)
Spectrum of CCl4, using an Ar+ laser at 488 nm.
Raman Spectroscopy
Another spectroscopic technique which probes the rovibrational structure of molecules. C.V. Raman discovered in 1928; received Nobel Prize in 1931. Can probe gases, liquids, and solids. Must use a laser source for excitation. Resurgence in recent years due to the development of new detectors with improved sensitivity. Shift back away from FT-Raman to dispersive Raman with multichannel detector systems.
新型不对称吡喃鎓方酸菁染料的晶体结构(英文)
新型不对称吡喃鎓方酸菁染料的晶体结构(英文)
陈建国;黄德音;黎源
【期刊名称】《感光科学与光化学》
【年(卷),期】2000(18)2
【摘要】通过缓慢挥发溶剂法得到一种不对称吡喃方酸菁染料的单晶 ,测定了其晶体结构 .其晶系为三斜晶系 ,空间群为P1 ,a =0 .92 2 8( 4 ) ,b =1 .41 2 2 ( 6) ,c=0 .61 2 4 ( 3)nm ,α =93.97( 4 ) ,β=98.1 4( 5) ,γ =71 .0 5( 4 ) ,V
=0 .7470 ( 6)nm3,Z =1 .用晶体结构数据解释了此不对称染料的1 H NMR谱 .【总页数】8页(P104-111)
【关键词】吡喃Weng方酸染料;菁染料;晶体结构;^1H-NMR谱
【作者】陈建国;黄德音;黎源
【作者单位】上海交通大学化学化工学院
【正文语种】中文
【中图分类】TQ614.4;O644
【相关文献】
1.新型不对称吡喃鎓方酸菁染料的晶体结构 [J], 陈建国;黄德音;黎源
2.新型吡喃鎓方酸菁染料溶剂效应、聚集行为研究 [J], 陈建国;黎源;黄德音
3.近红外水溶性不对称吲哚方酸菁染料的合成 [J], 宋波;于浩;刘翠翠;马文辉;彭孝军
4.一种新型含肉桂酸酯结构的螺吡喃光致变色染料的合成与性能研究 [J], 申凯华; 崔东熏
因版权原因，仅展示原文概要，查看原文内容请购买。
基于小波空间特征匹配及表面增强拉曼光谱技术快速检测混合物中的甲氨蝶呤和伏立康唑
基于小波空间特征匹配及表面增强拉曼光谱技术快速检测混合物中的甲氨蝶呤和伏立康唑刘察;臧颖超;曾惠桃;范夏琼;张志敏;卢红梅【摘要】为实现混合物中目标化合物的快速检测,合成了对甲氨蝶呤和伏立康唑具有普适性响应的金纳米复合(AuNPS-PDDA)基底,借助表面增强拉曼光谱技术,结合化学计量学方法,通过连续小波变换将甲氨蝶呤和伏立康唑的光谱信号转换到小波空间,依据小波空间特征匹配分析混合物的拉曼光谱,显著减轻信号中基线变化和随机噪声的影响,成功地识别出混合物中1.0×10-8mol/L甲氨蝶呤和2.86×10-4 mol/L伏立康唑.其中,甲氨蝶呤和伏立康唑的小波特征匹配系数均大于0.96,高于传统的命中质量系数,且采用非负最小二乘法成功实现了两种物质含量比例的确定,实验的组分比例预测值与真实值之间相关性大于0.98.实验结果表明,小波空间特征匹配结合表面增强拉曼光谱技术能有效地识别且半定量混合物中的目标化合物.【期刊名称】《分析测试学报》【年(卷),期】2019(038)006【总页数】7页(P668-674)【关键词】表面增强拉曼光谱技术;特征匹配;甲氨蝶呤;伏立康唑【作者】刘察;臧颖超;曾惠桃;范夏琼;张志敏;卢红梅【作者单位】中南大学化学化学工学院,湖南长沙410083;中南大学化学化学工学院,湖南长沙410083;中南大学化学化学工学院,湖南长沙410083;中南大学化学化学工学院,湖南长沙410083;中南大学化学化学工学院,湖南长沙410083;中南大学化学化学工学院,湖南长沙410083【正文语种】中文【中图分类】O657.3;TQ460.72拉曼光谱技术是一种重要的分析检测技术，广泛应用于各种领域，如古文物鉴定、生物蛋白质研究、医药物质检测、食物安全现场检测等[1]。
该技术无需复杂的样本预处理，几乎可以用于任何环境[2]。
拉曼光谱还包含了丰富的物质结构信息，被称为分子的“指纹图谱”[3]。
变时滞反馈控制的混合中立型随机延迟微分方程的指数稳定性
第50卷第6期2023年北京化工大学学报(自然科学版)Journal of Beijing University of Chemical Technology (Natural Science)Vol.50,No.62023引用格式:刘琪,兰光强.变时滞反馈控制的混合中立型随机延迟微分方程的指数稳定性[J].北京化工大学学报(自然科学版),2023,50(6):105-111.LIU Qi,LAN GuangQiang.Exponential stability of hybrid neutral stochastic differential delay equations with time⁃depend⁃ent delay feedback control[J].Journal of Beijing University of Chemical Technology (Natural Science),2023,50(6):105-111.变时滞反馈控制的混合中立型随机延迟微分方程的指数稳定性刘 琪 兰光强*(北京化工大学数理学院,北京 100029)摘 要:研究了变时滞反馈控制的混合中立型随机延迟微分方程(HNSDDEs)的指数稳定性㊂采用函数方法设置合适的变时滞反馈控制函数,得到了该系统的指数稳定性㊂对比已有的研究成果,本文的主要贡献是在变时滞反馈控制下对HNSDDEs 的指数稳定性作了进一步研究㊂最后,给出一个例子证明了结论的有效性㊂关键词:变时滞;混合中立型随机延迟微分方程(HNSDDEs);反馈控制;指数稳定性中图分类号:O211.6 DOI :10.13543/j.bhxbzr.2023.06.013收稿日期:2022-09-05基金项目:北京市自然科学基金(1192013)第一作者:女,1998年生,硕士生*通信联系人E⁃mail:langq@引 言带有变时滞反馈控制的混合中立型随机延迟微分方程(HNSDDEs)常被用于系统未来的建模,目前已经被广泛应用于种群生态㊁神经网络以及激光器动力学等领域㊂对于随机系统突然性的结构变化,常采用连续时间马氏链来描述,带有马氏链的随机延迟微分方程即为混合随机延迟微分方程㊂文献[1]具体研究了混合随机延迟微分方程,文献[2-4]则进一步考虑了其稳定性及有界性,文献[5-7]又扩展到了带中立项的混合随机延迟微分方程的稳定性研究㊂然而并非所有系统都是稳定的,因此设计一个合适的反馈控制使不稳定的系统变得稳定很有意义㊂相应地,文献[8-11]研究了系统稳定化问题㊂其中文献[8]研究了常时滞反馈控制的高阶非线性混合随机时滞微分方程的指数稳定性,文献[9]是在文献[10]的基础上进一步研究了变时滞反馈控制的HNSDDEs 的L p 渐进稳定性和H ∞稳定性㊂本文采用Lyapunov 函数方法,进一步研究了变时滞反馈控制下的HNSDDEs 的指数稳定性㊂文献[8]研究了常时滞反馈控制下的混合随机微分延迟方程的指数稳定性,其所涉及的时滞均为常量,本文进一步将常时滞推广到了函数时滞,并且将受控方程推广到了带有中立项的混合随机延迟微分方程,其难点在于找到时滞δ(t )的上界和利用引理2处理中立项㊂文献[9]研究了变时滞反馈控制的具有时变延迟的高度非线性HNSDDEs 的L p 渐近稳定性和H ∞稳定性,但缺少指数稳定性,本文则是通过进一步找到更合适的反馈函数确定了方程的收敛速度,即指数稳定性㊂1 基本假设与模型描述设(Ω,F ,{F t }t ≥0,P )是一个带有σ流(满足通常条件)的完备概率空间,{B (t )}t ≥0是定义在其上的m 维布朗运动,{r (t )}t ≥0是右连马氏链且独立于{B (t )}t ≥0,S ={1,2, ,N }是其状态空间,Γ=(γij )N ×N 是其生成算子㊂考虑变时滞反馈控制HNSDDEd ^x(t )=f (x (t ),x (t -τ(t )),t ,r (t ))d t +g (x (t ),x (t -τ(t )),t ,r (t ))d B (t ),t ≥0(1)其中^x(t )=x (t )-N (x (t -τ(t )),t ,r (t )),且初值满足{x(θ):-τ≤θ≤0}=φ∈C([-τ,0];n)r(0)=r0∈S(2)其中f,g,N均为Borel可测函数,并且满足f:n×n×+×S→ng:n×n×+×S→n×mN:n×+×S→n加上反馈控制函数u之后系统变为d^x(t)=[f(x(t),x(t-τ(t)),t,r(t))+u(x(t-δ(t)),t,r(t))]d t+g(x(t),x(t-τ(t)),t,r(t))㊃d B(t),t≥0(3)其中0≤δ(t)≤δ≤τ,0≤τ(t)≤τ㊂假设f(0,0,t,i)=N(0,t,i)≡0,g(0,0,t,i)≡0V(x,t,i)∈C2,1(n×+×S;+)为方便起见,简记^x=x-N(y,t,i)㊂对V(x,t,i)∈C2,1(n×+×S;+)定义如下算子LL V(x,y,t,i)=V t(^x,t,i)+V T x(^x,t,i)f(x,y,t, i)+12trace[g T(x,y,t,i)V xx(^x,t,i)g(x,y,t,i)]+∑j∈sγij V(^x,t,j)(4)为得到本文主要结论,提出以下假设㊂假设1 对任意l>0,存在K l>0,使得对任意i∈S,t∈+,且|x|∨|x|∨|y|∨|y|≤l,满足|f(x,y,t,i)-f(x,y,t,i)|∨|g(x,y,t,i)-g(x,y,t,i)|≤Kl(|x-x|+|y-y|)(5)假设2 存在K>0,m1>1,m2≥1,使得对∀x, y∈n,i∈S,t∈+,有|f(x,y,t,i)|≤K(|x|m1+|y|m1+1)|g(x,y,t,i)|≤K(|x|m2+|y|m2+1)(6)假设3 系统(3)中的时滞函数τ:+→[0,τ]满足τ′(t)=dτ(t)d t≤τ<1,t≥0(7)系统(3)反馈控制函数中的δ:+→[0,δ]满足δ′(t)=dδ(t)d t≤δ<1,t≥0(8)假设4 存在κ∈(0,1)使得对∀x,y∈n,i∈S,t∈+,有|N(x,t,i)-N(y,t,i)|≤κ(1-τ)|x-y|(9)并且N(0,t,i)≡0㊂假设5 存在常数c1,c2,c3,c4>0,c2>c3+c4和函数V∈C2,1(n×+×S;+),U1,U2∈C(×[-τ,+∞];+),使得对∀x,y∈n,i∈S,t∈+,有U1(x,t)≤V(x,t,i)≤U2(x,t)L V(x,y,t,i)+V x(x-N(y),t,i)u(z,t,i)≤c1-c2U2(x,t)+c3(1-τ)U2(y,t-τ(t))+c4(1-δ)U2(z,t-δ(t))(10)由文献[7]可得如下引理㊂引理1 设假设1~4成立,且假设5对于U1(x,t)=|x|w成立,那么系统(3)有唯一的全局解,并且满足sup-τ≤t<∞E|x(t)|w<∞,w≥2(m1∨m2)由文献[5]中引理2.2以及式(9)可得引理2 若p≥1,则[1-κ(1-τ)]p-1[|x|p-κ(1-τ)|y|p]≤|x-N(y,t,i)|p≤[1+κ(1-τ)]p-1[|x|p+κ(1-τ)|y|p](11) 2 主要结论与证明定义片段过程x(t)={x(t+s):-2τ≤s≤0,0≤t≤2τ}同理定义r(t),且令r(s)=r(0),s∈[-2τ,0)x(s)=φ(-τ),s∈[-2τ,-τ{)令U∈C2,1(n×+×S;+)且满足lim|x|→∞inf(t,i)∈+×SU(x,t,i[])=∞对于t∈+,定义V(x(t),t,r(t))=U(^x(t),t,r(t))+ρ∫0-δ∫t t+s J(v)㊃d v d s(12)其中ρ>0,且J(t):=δ|u(x(t-δ(t)),t,r(t))+f(x(t),x(t-τ(t)),t,r(t))|2+|g(x(t),x(t-τ(t)),t,r(t))|2对于x,y∈n,i∈S,s∈[-2τ,0),设f(x,y,s,i)≡f(x,y,0,i)g(x,y,s,i)≡g(x,y,0,i)u(z,s,i)≡u(z,0,i)由伊藤公式可得d U(^x(t),t,r(t))=[U t(^x(t),t,r(t))+ U T x(^x(t),t,r(t))(f(x(t),x(t-τ(t)),t,r(t))+ u(x(t-δ(t)),t,r(t)))+∑j∈Sγj,r(t)U(^x(t),t,j)+ 12trace[g T(x(t),x(t-τ(t)),t,r(t))U xx(^x(t),t,㊃601㊃北京化工大学学报(自然科学版) 2023年r(t))g(x(t),x(t-τ(t)),t,r(t))]d t+d B(t)(13)其中,B(t)是局部鞅,并且B(0)=0㊂整理式(13)得d U(^x(t),t,r(t))=l U(x(t),x(t-τ(t)),t, r(t))d t+U T x(^x(t),t,r(t))[u(x(t-δ(t)),t, r(t))-u(x(t),t,r(t))]d t+d B(t)其中,l U(x(t),x(t-τ(t)),t,r(t))=Ut(^x(t),t, r(t))+U T x(^x(t),t,r(t))[f(x(t),x(t-τ(t)),t, r(t))+u(x(t),t,r(t))]+∑j∈Sγj,r(t)U(^x(t),t,j)+ 12trace[g T(x(t),x(t-τ(t)),t,r(t))U xx(^x(t),t, r(t))g(x(t),x(t-τ(t)),t,r(t))]进而易得以下结论㊂引理3 V(x(t),t,r(t)),t≥0是伊藤过程,且有d V(x(t),t,r(t))=d B(t)+L V(x(t),t,r(t))㊃d t其中,L V(x(t),t,r(t))=l U(x(t),x(t-τ(t)),t, r(t))+ρδJ(t)-ρ∫t t-δJ(v)d v+U T x(^x(t),t,r(t))㊃[u(x(t-δ(t)),t,r(t))-u(x(t),t,r(t))](14)假设6 对于函数u:n×S×+→n,存在实数a i,a i,正数d i,d i和非负数b i,b i,e i,e i(i∈S),对于任意q1>1,p>2有x T[f(x,y,t,i)+u(x,t,i)]+12|g(x,y,t,i)|2≤a i|x|2+b i|y|2-d i|x|p+e i|y|px T[f(x,y,t,i)+u(x,t,i)]+q12|g(x,y,t,i)|2≤a i|x|2+b i|y|2-d i|x|p+e i|y|p且A1:=-2diag(a1,a2, ,a N)-ΓA2:=-(q1+1)diag(a1,a2, ,a N)-Γ是非奇异M矩阵(具体定义可参考文献[1]中的2.6部分),并有1>γ1,γ2>γ3,1>γ4,γ5>γ6(θ1,θ2, ,θN)T=A-11(1, ,1)T(θ1,θ2, ,θN)T=A-12(1, ,1)Tγ1=max i∈S2θi b i,γ2=min i∈S2θi d iγ3=max i∈S2θi e i,γ4=max i∈S(q1+1)θi b iγ5=min i∈S(q1+1)θi d i,γ6=max i∈S(q1+1)θi e i其中θi和θi是正数㊂需要注意的是,关于控制函数u的选取,考虑如下特殊情况x T f(x,y,t,i)+q-12|g(x,y,t,i)|2≤a(|x|2+ |y|2)-b|x|p+c|y|p其中a>0,b>c>0㊂由于|x|2,|y|2的系数均为正数,因此只能得到原方程的矩有界性,而得不到稳定性㊂此时可选取u(x,t,i)=Ax,其中矩阵A为实对称正定矩阵,且满足λmax(A)<-2a,从而x T[f(x,y,t,i)+u(x,t,i)]+q-12㊃|g(x,y,t,i)|2≤(λmax(A)+a)|x|2+a|y|2-b|x|p+c|y|p故加上控制项之后的系统指数稳定㊂假设7 存在U∈C2,1(n×+×S;+),H∈C(n;+),及常数0<α<1,0<β<λ,0<λ1,λ2,λ3,ρ1,ρ2,使得对任意的x,y∈n,i∈S,t∈+有l U(x,y,t,i)+λ1|U x(^x,t,i)|2+λ2㊃|f(x,y,t,i)|2+λ3|g(x,y,t,i)|2≤-λ|x|2+(1-τ)β|y|2-H(x)+(1-τ)αH(y)(15)其中,ρ1|x|p+q1-1≤H(x)≤ρ2(1+|x|p+q1-1)㊂假设8 存在λ4>0满足|u(x,t,i)-u(y,t,i)|≤λ4|x-y|(16)并且有u(0,t,i)=0㊂故有∀x∈n,u(x,t,i)≤λ4㊃|x|㊂定理1 令q∈[2,w),w≥2(m1∨m2)㊂若假设1~8成立,且常数满足κ(1-τ)<12δ≤λ1λ2(1-κ)(1-κ(1-τ))λ4∧2λ1λ3(1-κ)(1-κ(1-τ))λ24∧(λ-β)(1-δ)λ1(1-κ)(1-κ(1-τ))λ24则对任意初值,存在ε>0使得系统(3)的解满足lim t→∞sup1t ln(E|x(t)|q)≤-εw-q w-2(17)其中ε=ε1∧ε2∧ε3∧ε4,ε1,ε2,ε3,ε4分别是以下4个方程的根㊃701㊃第6期 刘 琪等:变时滞反馈控制的混合中立型随机延迟微分方程的指数稳定性εδ+2(1-κ)(1-κ(1-τ))=1[εh 3ρ-11(1+κ(1-τ))p +q 1-2](κe ετ+1)+e ετα=1ε(h 2+h 3)(1+κ(1-τ))(1+e ετκ)+βe ετ+2ρδ2λ24eεδ1-δ+λ4κ2(1-τ)e ετ(1-τ-δ+e εδ(1-τ ))λ1(1-δ-τ)=λ2e ετκ2(1-τ)2=1特别地,当q =2时有lim t →∞sup 1tln (E |x (t )|2)≤-ε(18)即满足均方指数稳定㊂证明:证明分为两步㊂1)第一步取k 0>0足够大使得‖φ‖:=sup -τ≤s ≤0φ(s )<k 0㊂定义σk =inf {t ≥0:|x (t )≥k |}(k ≥k 0),且inf ϕ=∞㊂由引理1和文献[7],当k →∞,则σk →∞,a.s.根据假设6再定义U (^x,i )=θi |^x |2+θi |^x |q 1+1(19)由伊藤公式有e εtEV (x (t ),t ,r (t ))=V (x (0),0,r (0))+∫te εs (εV (x (s ),s ,r (s ))+L V (x (s ),s ,r (s )))d s取h 1=min i ∈Sθi ,h 2=max i ∈S θi ,h 3=max i ∈Sθi ,结合式(12)可得h 1eε(t ∧σk )E |^x(t ∧σk )|2≤V (x (0),0,r (0))+∫t ∧σk0e εs E (L V (x (s ),s ,r (s )))d s +ερJ 1(t ∧σk )+∫t ∧σke εs (εh 2E |^x(s )|2+εh 3E |^x (s )|q 1+1)d s (20)其中,J 1(t ∧σk )=E ∫t ∧σke ε(s∫0-δ∫ss +uJ (v )d v d )u ㊃d s ㊂对于式(20)中的E |^x(t ∧σk )|2结合基本不等式可得到E |x (t ∧σk )|2≤2E |^x(t ∧σk )|2+2κ2(1-τ)2E |x (t ∧σk -τ(t ∧σk ))|2(21)对于式(20)中的L V (x (t ),t ,r (t ))结合式(14)和假设7有L V (x (t ),t ,r (t ))≤-λ|x (t )|2+(1-τ)β㊃|x (t -τ(t ))|2-H (x (t ))+(1-τ)αH (x (t -τ(t )))-λ1|U x (^x(t ),t ,r (t ))|2-λ2|f (x (t ),x (t -τ(t )),t ,r (t ))|2-λ3|g (x (t ),x (t -τ(t )),t ,r (t ))|2+ρδJ (t )-ρ∫tt-δJ (v )d v +U T x (^x (t ),t ,r (t ))㊃[u (x (t -δ(t )),t ,r (t ))-u (x (t ),t ,r (t ))]由假设8运用均值不等式可以得到U T x (^x (t ),t ,r (t ))[u (x (t -δ(t )),t ,r (t ))-u (x (t ),t ,r (t ))]≤λ1|U x (^x(t ),t ,r (t ))|2+λ244λ1㊃|x (t -δ(t ))-x (t )|2定义ρ=λ242λ1(1-κ)(1-κ(1-τ)),由定理1中δ满足的不等式知2ρδ2≤λ2,ρδ≤λ3㊂再由Hölder 不等式有E |x (t -δ(t ))-x (t )|2≤2E |^x(t )-^x (t -δ(t ))|2+2E |N (x (t -τ(t )),t ,r (t ))-N (x (t -τ(t )-δ(t ),t ,r (t ))|2≤4E∫tt-δ[δ|u (x (v -δ(v )),v ,r (v ))+f (x (v ),x (v -τ(v )),v ,r (v ))|2+|g (x (v ),x (v -τ(v )),v ,r (v ))|2]d v +2κ2(1-τ)2E |x (t -τ(t ))-x (t -τ(t )-δ(t ))|2所以有E L V (x (t ),t ,r (t ))≤-λE |x (t )|2+(1-τ)㊃βE |x (t -τ(t ))|2-EH (x (t ))+(1-τ)αEH (x (t -τ(t )))+2ρδ2λ24E |x (t -δ(t ))|2(+λ24λ1-)ρ㊃E∫t t -δJ (v )d v +λ4κ2(1-τ)22λ1E |x (t -τ(t ))-x (t -τ(t )-δ(t ))|2(22)对于式(20)中的E |^x(t )|q 1+1有以下关系式E |^x(t )|q 1+1≤E |^x (t )|2+E |^x (t )|p +q 1-1(23)又由假设7有|x (t )|p +q 1-1≤ρ-11H (x (t ))(24)所以结合式(20)~(23)有12h 1e ε(t ∧σk )E |x (t ∧σk )|2≤Π1+Π2+Π3+∫t ∧σke εs (εh 2E |^x(s )|2+εh 3E |^x (s )|2+εh 3㊃E |^x(s )|p +q 1-1)d s +∫t ∧σke εs E [-λ|x (s )|2+(1-τ)㊃β|x (s -τ(s ))|2-H (x (s ))+(1-τ)αH (x (s -τ(s )))+2ρδ2λ24|x (s -δ(s ))|2+λ4κ2(1-τ)22λ1㊃|x (s -τ(s ))-x (s -τ(s )-δ(s ))|2]d s(25)其中,Π1=h 1e ε(t ∧σk )κ2(1-τ)2E |x (t ∧σk -τ(t ∧σk ))|2Π2=V (x (0),0,r (0))㊃801㊃北京化工大学学报(自然科学版) 2023年Π3=ερJ 1(t ∧σk )(+λ24λ1-)ρJ 2(t ∧σk )J 2(t ∧σk )=E∫t ∧σke ε[s∫ss -δJ (v )d ]v d s易得J 1(t ∧σk )≤δJ 2(t ∧σk )㊂取ε1为ε1ρδ+λ24λ1-ρ=0的唯一解,则由ρ的定义知,对任意0<ε≤ε1,有Π3≤0㊂结合式(11),令k →∞,结合式(24),式(25)化为12h 1e εt E |x (t )|2≤Π1+Π2+Π4+Π5(26)其中,Π1=h 1e εt κ2(1-τ)2E |x (t -τ(t ))|2Π4=∫teεs{εh 3ρ-11[1+κ(1-τ)]p +q 1-2㊃[EH (x (s ))+κ(1-τ)EH (x (s -τ(s )))]-EH (x (s ))+(1-τ)αEH (x (s -τ(s )))}d sΠ5=∫te εs {ε(h 2+h 3)[1+κ(1-τ)]㊃[E |x (s )|2+κ(1-τ)E |x (s -τ(s ))|2]}d s +∫teε[s-λE |x (s )|2+(1-τ)βE |x (s -τ(s ))|2+2ρδ2λ24E |x (s -δ(s ))|2+λ4κ2(1-τ)22λ1E |x (s -τ(s ))-x (s -τ(s )-δ(s ))|]2d s对于Π2,由初值条件㊁假设2㊁假设8㊁引理2和式(12)得V (x (0),0,r (0))<∞,并且记为C 0,C 0为常数㊂对于Π4,根据假设3化简有Π4≤{[εh 3(1+κ(1-τ))p +q 1-2ρ-11](κe ετ+1)+e ετα-1}∫te εs E [H (x (s ))]d s +e ετ[εh 3(1+κ(1-τ))p +q 1-2ρ-11κ+α]∫-τe εs E [H (x (s ))]d s取ε2为[ε2h 3(1+κ(1-τ))p +q 1-2ρ-11](κe ε2τ+1)+e ε2τα-1=0的唯一解,则对任意0<ε≤ε2以及0<α<1即可满足Π4≤e ετ[εh 3(1+κ(1-τ))p +q 1-2ρ-11κ+α]㊃∫0-τe εs E [H (x (s ))]d s <∞(27)对于Π5,令ε3为ε3(h 2+h 3)(1+κ(1-τ))(1+e ε3τκ)+βe ε3τ+2ρδ2λ24eε3 δ1-δ+λ4κ2(1-τ)e ε3τ(1-τ-δ+e ε3δ(1-τ ))λ1(1-δ-τ)=λ的唯一解,对任意0<ε≤ε3,有Π5≤e [ετε(h 2+h 3)(1+κ(1-τ))κ+β+λ4κ2(1-τ)λ]1∫0-τe εs E |x (s )|2d s +2ρδ2λ24eεδ1-δ∫0-δe εs㊃E |x (s )|2d s +λ4κ2(1-τ)2e ε(τ+δ)λ1(1-δ-τ)∫-δ-τe εs E |x (s )|2d s [+ε(h 2+h 3)(1+κ-κτ)(1+e ετκ)+βe ετ+2ρδ2λ24eεδ1-δ+λ4κ2(1-τ)e ετ(1-τ-δ+e εδ(1-τ ))λ1(1-δ-τ)-]λ∫te εs E |x (s )|2d s ≤e [ετε(h 2+h 3)(1+κ(1-τ))κ+β+λ4κ2(1-τ)λ]1∫0-τe εs E |x (s )|2d s +2ρδ2λ24e εδ1-δ∫-δe εsE |x (s )|2d s +λ4κ2(1-τ)2e ε(τ+δ)λ1(1-δ-τ)㊃∫-δ-τe εs E |x (s )|2d s <∞(28)综上对任意0<ε≤ε1∧ε2∧ε3,可得12h 1e εt E |x (t )|2≤h 1e εt κ2(1-τ)2E |x (t -τ(t ))|2+C 1(29)其中C 1是一个常数㊂2)第二步式(29)经过整理可以得到e εt E |x (t )|2≤2e ετe ε(t -τ(t ))κ2(1-τ)2E |x (t -τ(t ))|2+2C 1h 1,故有sup 0≤s ≤t e εs E |x (s )|2≤2C 1h 1+2e ετκ2(1-τ)2sup 0≤s ≤t e εs ㊃E |x (s )|2+2κ2(1-τ)2e ετsup -τ≤s ≤0‖ϕ‖2由κ(1-τ)<12,令ε4为1-2e ε4τκ2(1-τ)2=0的唯一解,则对任意0<ε≤ε1∧ε2∧ε3∧ε4,有sup 0≤s ≤t e εs E |x (s )|2≤2C 1h 1+2κ2(1-τ)2e ετsup -τ≤s ≤0‖φ‖21-2κ2(1-τ)2e ετ:=C 2即当t ∈[0,∞)时,e εt E |x (t )|2≤C 2,即E |x (t )|2≤C 2e -εt ㊂对于任意的q ∈[2,w ),由Hölder 不等式得到㊃901㊃第6期 刘 琪等:变时滞反馈控制的混合中立型随机延迟微分方程的指数稳定性E |x (t )|q≤(E |x (t )|2)w - qw -2(E |x (t )|w)q -2w -2㊂由引理1知C 3:=E |x (t )|w <∞,故E |x (t )|q ≤C q -2w -23(C 2e -εt )w - qw -2≤C 4e -εt w - qw -2所以式(17)成立㊂特别地,当q =2时,有式(18)成立㊂3 例子考虑一维HNSDDEd[x (t )-N (x (t -τ(t )),t ,r (t ))]=f (x (t ),x (t -τ(t )),t ,r (t ))d t +g (x (t ),x (t -τ(t )),t ,r (t ))d B (t ),t ≥0(30)其中f (x ,y ,t ,1)=0.5x +y 3-6x 3f (x ,y ,t ,2)=x +y 3-4x3g (x ,y ,t ,1)=g (x ,y ,t ,2)=0.5y 2τ(t )=0.1(1-cos t ),N (y )=0.1y显然f ,g 不满足线性增长条件㊂令r (t )为一个连续的马氏链,状态空间S ={1,2},算子Γ=-22æèçöø÷1-1,B (t )为标准布朗运动且独立于r (t )㊂定义初值x (u )=0.2+cos u ,u ∈[-0.2,0],r (0)=2㊂由文献[10]可知系统(30)不稳定,以下将通过引入一个反馈控制函数使系统稳定㊂增加控制函数u (x ,t ,1)=-x ,u (x ,t ,2)=-2x ,增加控制函数后系统(3)的具体形式为 d[x (t )-0.1x (t -τ(t ))](=12x (t )+(x (t -τ(t )))3-6x (t )3-x (t - δ(t )))d t +12(x (t -τ(t )))2d B (t ),i (=1x (t )+(x (t -τ(t )))3-4x (t )3-2x (t - δ(t )))d t +12(x (t -τ(t )))2d B (t ),i ìîíïïïïïïïïïüþýïïïïïïïïï=2其中δ(t )=τ(t )㊂以下验证假设1~8㊂假设1显然成立㊂令m 1=3,m 2=2,可知假设2成立㊂令λ4=2,可知假设8成立㊂假设3对如下常数成立:δ=τ=0.2,δ=τ=0.1,且假设4对κ=19成立㊂取U 1(x ,t )=V (x ,i ,t )=|x |6,U 2(x ,t )=2.2x 6+x 8,由Young 不等式可得L V (x ,y ,t ,i )+V x (x -N (y ),t ,i )u (z ,t ,i )≤sup x ∈(43x 6-0.229x 8)-8×U 2(x ,t )+589×(1-τ)×U 2(y ,t -τ(t ))+109×(1-δ)×U 2(z ,t -δ(t ))故假设5对c 1=sup x ∈(43x 6-0.229x 8)<∞,c 2=8,c 3=589,c 4=109成立㊂取p =4,q 1=3,可知假设6成立㊂取U (x ,t ,i )=2x 2+x 4,i =1x 2+x 4,i ={2,再由Young 不等式,令λ1=0.05,λ2=0.1,λ3=4可得l U (x ,y ,t ,i )+λ1|U x (^x(t ),t ,i )|2+λ2㊃|f (x ,y ,t ,i )|2+λ3|g (x ,y ,t ,i )|2≤-1.845|x |2+0.369(1-τ)|y |2-6(x 4+x 6)+0.955×(1-τ)×6(y 4+y 6)若令H (x )=6(x 4+x 6),λ=1.845,β=0.369,α=0.955,则假设7成立㊂根据定理1条件发现κ,τ取值合理,进而可以得到δ≤0.0576时,定理1所有条件成立,故对∀w ≥6,∀q ∈[2,w ),存在ε>0使得lim t →∞sup1t ln (E |x (t )|q )≤-εw -qw -2特别地,q =2时有lim t →∞sup1tln (E |x (t )|2)≤-ε㊂4 结论本文采用函数方法,受文献[5]的启发在多项式增长的条件下讨论了变时滞反馈控制下的HNS⁃DDEs 的指数稳定性㊂最后,用一个例子证明了结论的有效性㊂参考文献:[1] MAO X R,YUAN C G.Stochastic differential equations with Markovian switching[M].London:Imperial CollegePress,2006.[2] FEI W Y,HU L J,MAO X R,et al.Delay dependentstability of highly nonlinear hybrid stochastic systems[J].Automatica,2017,82:165-170.[3] FEI C,SHEN M X,FEI W Y,et al.Stability of highlynonlinear hybrid stochastic integro⁃differential delay equa⁃tions[J].Nonlinear Analysis:Hybrid Systems,2019,31:180-199.㊃011㊃北京化工大学学报(自然科学版) 2023年[4] HU L J,MAO X R,SHEN Y.Stability and boundednessof nonlinear hybrid stochastic differential delay equations [J].Systems &Control Letters,2013,62:178-187.[5] WU A Q,YOU S R,MAO W,et al.On exponential sta⁃bility of hybrid neutral stochastic differential delay equa⁃tions with different structures [J].Nonlinear Analysis:Hybrid Systems,2021,39:100971.[6] SHEN M X,FEI W Y,MAO X R,et al.Stability ofhighly nonlinear neutral stochastic differential delay equa⁃tions[J].Systems &Control Letters,2018,115:1-8.[7] SHEN M X,FEI C,FEI W Y,et al.Boundedness andstability of highly nonlinear hybrid neutral stochastic sys⁃tems with multiple delays[J].Science China Information Sciences,2019,62:202205.[8] LI X Y,MAO X R.Stabilisation of highly nonlinear hy⁃brid stochastic differential delay equations by delay feed⁃back control[J].Automatica,2020,112:108657.[9] 周之薇,宋瑞丽.变时滞反馈控制的混合中立型随机延迟微分方程的稳定性[J].井冈山大学学报(自然科学版),2022,43(3):6-14.ZHOU Z W,SONG R L.Stabilization of the hybrid neu⁃tral stochastic differential equations controlled by thetime⁃varying delay feedback [J].Journal of Jinggangshan University (Natural Science),2022,43(3):6-14.(in Chinese)[10]SHEN M X,FEI C,FEI W Y,et al.Stabilisation by de⁃lay feedback control for highly nonlinear neutral stochasticdifferential equations [J ].Systems &Control Letters,2020,137:104645.[11]CHEN W M,XU S Y,ZOU Y.Stabilization of hybridneutral stochastic differential delay equations by delayfeedback control[J].Systems &Control Letters,2016,88:1-13.Exponential stability of hybrid neutral stochastic differential delay equations with time⁃dependent delay feedback controlLIU Qi LAN GuangQiang *(College of Mathematics and Physics,Beijing University of Chemical Technology,Beijing 100029,China)Abstract :The exponential stability of hybrid neutral stochastic differential delay equations (HNSDDEs)with time⁃dependent delay feedback control has been ing the Lyapunov function method,the exponential sta⁃bility of the system can be obtained by setting an appropriate feedback control function with a variable ⁃pared with the existing research results,the results of this work increase our understanding of the exponential stabil⁃ity of HNSDDEs under the influence of variable delay feedback.Finally,an example is given to prove the validity of the conclusions.Key words :time⁃dependent delay;hybrid neutral stochastic differential delay equations (HNSDDEs);feedbackcontrol;exponential stability(责任编辑:吴万玲)㊃111㊃第6期 刘 琪等:变时滞反馈控制的混合中立型随机延迟微分方程的指数稳定性。
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Contents 1. Introduction 2. The C ∗ -Algebra of SUq (2) 2.1. The q deformation picture of SUq (2) 2.2. SUq (2) as a graph algebra 3. The GNS representation for the Haar state 4. The Kasparov module 5. K-theory 6. The index pairing for the mapping cone 7. The semifinite spectral triple for SUq (2) 7.1. Semifinite spectral triples 7.2. Summability 7.3. The spectral flow formula 7.4. The SUq (2) spectral triple 7.5. The mapping cone pairing and spectral flow 8. The modular spectral triple for SUq (2) 8.1. Modular K -theory 8.2. The modular spectral triple and the index pairing 9. Concluding Remarks References 1 3 3 4 7 9 10 13 14 14 15 15 17 19 20 21 22 24 25
1. Introduction Motivated by our study of semifinite spectral triples and Kasparov modules for graph algebras, [CPR1, PR], we have found a new way of extracting invariants from algebras using non-tracial states. The basic constructions and first examples are in [CPR2], where we studied the Cuntz algebras using their unique KMS states for the canonical gauge action. However, we have found that this example, though illuminating, is not generic. Here we study a more generic situation, the example of SUq (2) using the Haar state, which is KMS for a certain circle action (which is not the gauge action).
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2
A. L. CAREY, A. RENNIE, AND K. TONG
The approach of [CPR2] yields a local index formula in twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the KMS state. The chief drawback of twisted cyclic theory, from the point of view of index theory, is that it does not pair with K -theory. However, in [CPR2] we showed that there was an abelian group, called modular K1 , which pairs with twisted cyclic cohomology. It seems that in general we must understand algebras that admit both traces (not necessarily faithful) and KMS states. The former situation, described in detail in Section 7 for SUq (2), exploits semifinite index theory, using untwisted cyclic cohomology. The latter, in Section 8, uses the Haar state and the associated twisted cyclic theory, and leads to a pairing with modular K1 . In both cases the pairing is given by computing spectral flow in the semifinite sense as described in [CP2]. This example points to the existence of a rich interplay between the tracial and KMS index theories. Many of the constructions of this paper mirror those of [CPR1, CPR2, PR], and we give precise references to those papers for more information about our constructions and, where necessary, for proofs. We focus here on the new aspects of these constructions that SUq (2) throws up. There are two motivations for this study. First there is the observation [CLM] that there is a way to associate directed graphs to Mumford curves and that from the corresponding graph C ∗ -algebras we might extract topological information about the curves. It eventuates that SUq (2) is an example of a graph algebra that shares with the Mumford curve graph algebras the property that it does not admit faithful traces but does admit faithful KMS states for nontrivial circle actions. If we are going to be able to exploit graph algebras to study invariants of Mumford curves then we need to demonstrate that it is possible to actually calculate numerical invariants explicitly. We find in Sections 7 and 8 that we are able to obtain not only abstract formulae but also the numbers produced by these formulae for particular unitaries in matrix algebras over SUq (2). The second motivation comes from the general formula obtained in the main result in Section 8, namely Theorem 8.2. This Theorem shows that there are two contributions to spectral flow in the twisted cocycle one of which comes from truncated eta type correction terms. This example points to the existence of a ‘twisted eta cocycle’, a matter we plan to investigate further in another place. The novel feature of our approach is to make use of the structure of SUq (2) as a graph C ∗ -algebra and a small part of its Hopf algebra structure via the Haar state. We find that this is best described by using an intermediate presentation of the algebra in terms of generators which are functions of the graph algebra generators. The Haar state provides us with a natural faithful KMS state on SUq (2) which we want to use because any trace on SUq (2) cannot be faithful. This has the consequence that any Dixmier type trace on SUq (2) will only see ‘part’ of the algebra. Nevertheless for non-faithful traces we can calculate what the odd semifinite local index formula in noncommutative geometry [CPRS2] tells us. More specifically we construct a particular (1, ∞) summable semifinite spectral triple for SUq (2) in Section 7. Then in subsection 7.5 we use ideas from [CPR1] to give some explicit computations which are actually the result of pairings with the K −theory of a mapping cone algebra constructed from SUq (2). These calculations use general formulae for spectral flow in von Neumann algebras found in [CP2, CPS2]. These pairings yield rational functions of the deformation parameter q , in fact q -numbers, which are naturally interpreted as q -winding numbers. Then, in Section 8, we turn to the question of what information we can extract from the Haar state or ‘twisted’ situation (where ‘twist’ refers to twisted cyclic cohomology). We constructed a twisted cocycle and the pairing with what we termed ‘modular K-theory’ in [CPR2]. When applied to particular unitaries in matrix algebras over SUq (2) the pairing of the twisted cocycle with modular K1 gives a spectral flow invariant that is a polynomial function of the deformation parameter q . These functions are distinct from those obtained in the tracial case although they depend on the same variables. We believe that the mapping cone plays a role here as well but much further investigation needs to be done to prove this. We remark that our aims are different from those of [ChP1, ChP2, DLSSV] where the quantum group structure of the algebra plays the main role through the construction of equivariant spectral