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微凸体干涉机制英语Microasperity interference is a fascinating phenomenon that occurs when two surfaces with tiny protrusions come into contact. These protrusions, or microasperities, can cause all sorts of interesting effects, from increased friction to changes in surface adhesion.In the world of tribology, microasperity interference is often seen as a nuisance, as it can lead to wear and tear. But when viewed through the lens of creativity, these tiny imperfections actually have the potential to open up new areas of research. For instance, engineers are exploring ways to utilize these microasperities to create unique surface textures that enhance lubrication or reduce noise.The study of microasperity interference is not just about physics and mechanics. It's also about understanding how materials interact at the nanoscale. This requires a multidisciplinary approach, blending knowledge from fieldslike materials science, surface engineering, and tribology.One fascinating aspect of microasperity interference is how it can affect the performance of mechanical components. Even the smallest imperfections can have a significant impact on friction, wear, and the overall lifespan of a component. By understanding and controlling these microasperities, engineers can design more efficient and durable systems.In the future, as we continue to push the boundaries of technology, microasperity interference will become an even more important topic of research. As we build smaller and more complex devices, the role of these tiny imperfections will become increasingly significant. Who knows, maybe one day we'll even be able to harness the power of microasperity interference to create entirely new types of materials and devices.。
肿瘤坏死因子受体超级家族（tumor necrosis fac⁃tor receptor superfamily，TNFRSF）的死亡受体（death receptor）以及它们的配体在胚胎正常发育及机体免疫和炎症反应过程中扮演了重要角色。
外胚层发育不良受体（ectodysplasin A2receptor，EDA2R）是一个在20年前被鉴定出来的TNFRSF成员（TNFRSF27）［1］，在肿瘤发生、雄激素性脱发等过程中起到重要的作用，但对于该受体作系统性介绍的综述文章尚未见报道。
本文就该受体的研究进展作一系统性的综述，旨在为相关研究提供新的思路。
1EDA2R的蛋白结构和配体1.1EDA2R的蛋白结构EDA2R基因位于人类染色体Xq12，全长约43kb，有6个外显子（GenBank登录号：NG_013271），外胚层发育不良受体EDA2R的研究进展蓝希钳1，2，肖海婷1，2，罗怀容1，2，陈建宁1，2（西南医科大学药学院：1衰老与再生医学实验室，2药理学教研室，四川泸州646000）【摘要】外胚层发育不良受体EDA2R（ectodysplasin A2receptor）是肿瘤坏死因子受体超级家族（tumor necrosis factor recep⁃tor superfamily，TNFRSF）中的一个较新的成员，在发育中的胚胎里有很高的表达，在成年人和动物的多个器官组织中也有表达。
与其它TNFRSF成员不同，尽管EDA2R蛋白在胞内没有死亡结构域（death domain，DD），但它仍可激活NF-κB和JNK通路，并介导细胞的凋亡。
本文广泛回顾了近年来与EDA2R有关的文献，就该蛋白分子的相关研究进展进行综述，以期为与该蛋白相关的分子功能或其介导的相关疾病的研究提供新的思路。
【关键词】EDA2R受体肿瘤坏死因子受体超级家族死亡结构域凋亡【中图分类号】R34文献标志码A doi：10.3969/j.issn.2096-3351.2021.03.018Research progress of ectodysplasin A2receptorLAN Xi-qian1，2，XIAO Hai-ting1，2，LUO Huai-rong1，2，CHEN Jian-ning1，2 1Key Laboratory for Aging and Regenerative Medicine；2Department of Pharmacology，School of Pharmac，South⁃west Medical University，Luzhou646000，Sichuan，China【Abstract】Ectodysplasin A2receptor（EDA2R）is a relatively new member of the tumor necrosis factor re⁃ceptor superfamily（TNFRSF），and it is highly expressed in developing embryos and is also expressed in multiple organs and tissues of adult human and animals.Different from other TNFRSF members，EDA2R protein does not contain the death domain in the intracellular region，but it can still activate the NF-κB and JNK pathways and medi⁃ate cell apoptosis.This article reviews related articles on EDA2R in recent years and related research advances in this protein，in order to provide new ideas for research on molecular functions associated with EDA2R or related dis⁃eases mediated by EDA2R.【Key words】Ectodysplasin A2receptor Tumor necrosis factor receptor superfamily Death domain Apoptosis基金项目：泸州市科技局－西南医科大学联合项目（2018LZXNYD-ZK12）；西南医科大学-泸州市中医医院基地项目（2019-LH005）第一作者简介：蓝希钳，博士。
水溶液中几种芳香族氨基酸π-π自堆叠作用胡新根1,*朱玉青1余生1张贺娟1刘飞1于丽2(1温州大学化学与材料科学学院,浙江温州325027;2山东大学胶体与界面化学教育部重点实验室,济南250100)摘要：利用精密的流动混合微量热法测定了298.15K 时D/L -色氨酸、L -色氨酸、L -组氨酸和L -苯丙氨酸四种天然芳香族氨基酸水溶液的稀释焓,根据所建立的拟等步自堆叠作用的化学模型对实验数据进行了处理,计算得到模型参数K ΔH m .该化学作用参数与McMillan -Mayer 理论模型中的焓对作用系数具有高度一致性,即h xx =K ΔH m .结合文献报道的结果,认为芳核π-π自堆叠作用在本质上是一种特殊的疏水-疏水作用,一般表现为吸热效应;取代基空间位阻、芳核以外部分的静电、氢键和手性选择性作用等对芳核π-π自堆叠作用有显著影响;组合参数K ΔH m 实际上描述了芳核π-π自堆叠作用平衡及焓变的综合效应.关键词：芳香族氨基酸;π-π自堆叠;稀释焓;微量热法;化学作用模型中图分类号：O642Aromatic π-πSelf -Stacking of Some Aromatic Amino Acids inAqueous SolutionsHU Xin -Gen 1,*ZHU Yu -Qing 1YU Sheng 1ZHANG He -Juan 1LIU Fei 1YU Li 2(1College of Chemistry and Materials Engineering,Wenzhou University,Wenzhou 325027,Zhejiang Province,P.R.China ;2Key Laboratory for Colloid and Interface Chemistry of the Ministry of Education,Shandong University,Jinan 250100,P.R.China )Abstract ：Dilution enthalpies of some aromatic amino acids such as D/L -α-tryptophan,L -α-tryptophan,L -α-tyrosine and L -α-phenylalanine in aqueous solutions at 298.15K were determined by sensitive mixing -flow microcalorimetry.A chemical interaction model for quasi -isodemic self -stacking was proposed and used to process the calorimetric data from which the model parameter K ΔH m was calculated.The chemical interaction parameter (K ΔH m )agrees well with and provides good insights into the pairwise enthalpic interaction coefficient (h xx )in the McMillan -Mayer approach for the existence of the equation K ΔH m =h xx .Combined with results from literature we considered that aromatic π-πself -stacking is essentially a kind of special hydrophobic interaction manifesting commonly as an endothermic effect.Noteworthy effects arising from substituent hindrance,electrostatic interaction,hydrogen bonding and chiral recognition which are directed away from the aromatic core exert on aromatic π-πself -stacking.In nature,the composite parameter K ΔH m describes a complex effect between the equilibrium and an enthalpic change of aromatic π-πself -stacking.Key Words ：Aromatic amino acid;π-πself -stacking;Dilution enthalpy;Microcalorimetry;Chemicalinteraction model[Article]物理化学学报(Wuli Huaxue Xuebao )Acta Phys.-Chim.Sin .,2009,25(4)：729-734Received:October 17,2008;Revised:January 9,2009;Published on Web:February 18,2009.*Corresponding author.Email:hxgwzu@.国家自然科学基金(20673077)、胶体与界面化学教育部重点实验室(山东大学)开放课题(200506)资助项目鬁Editorial office of Acta Physico -Chimica Sinica芳香族分子间的相互作用(π-π,OH-π,NH-π和阳离子-π)在化学和生物学的许多领域都非常重要,这些相互作用控制了芳香族分子的晶体结构、生物系统的稳定性以及分子识别过程[1-4].在蛋白质折叠过程中,氨基酸残基之间的相互作用是一个决定性的因素[5],对认识蛋白质的高级结构非常关键.在各种氨基酸残基中,极性基团间的氢键作用和芳香性残基间可能的π-π相互作用对决定蛋白质的April 729Acta Phys.-Chim.Sin.,2009Vol.25结构起着重要影响[6,7].Burley等[8]在对一组34个蛋白质的研究中发现,大约60%的芳香性侧链涉及了π-π相互作用,而80%的这种作用通过联系蛋白质二级结构的不同要素对稳定其三级结构有贡献.另外,π-π相互作用在与化学和生化过程相关的质子偶合电子转移反应(PCET)中也普遍存在[9].这类弱相互作用的大小和物理起源对理解各种分子聚集体的结构与性质是必需的,同时对发展材料和药物设计策略也相当重要的.在理论上进行模拟计算是研究芳香族分子上述弱相互作用的一种重要途径[5],而在实验上人们往往利用合适的分子模型系统进行研究[10,11].近年来,对生物体系中弱的非键相互作用的能量学(energetics)研究引起了人们的重视[12-17].作为研究工作的继续[18],本文建立了一种描述水溶液中芳香族分子拟等步的π-π自堆叠作用的化学模型,用于处理D/L-色氨酸、L-色氨酸、L-组氨酸和L-苯丙氨酸水溶液的稀释焓实验数据,并结合文献报道的结果,对芳核π-π自堆叠作用的焓效应及其影响因素作了讨论.1芳核π-π自堆叠作用的化学模型对于芳香族溶质M的π-π自堆叠作用,假设可视同如下化学反应[18]:M i-1+M=M i,i叟2(1)K i=b(M i)b(M i-1)b(M)=b(M i)(K2K3…K i-1)[b(M)]i,i叟2(2)式中b(M i)为物种M i的质量摩尔浓度.设反应的摩尔焓变为ΔH m,i,平衡常数为K i,并假定溶液的稀释过程热效应主要是缔合物种M i的解堆叠作用产生的,则对于一定物质的量n的溶质M在质量为m s 的溶剂中无限稀释的稀释焓(以每kg溶剂计)为Δdil H∞=nm sΔdil H∞m=-[b(M2)ΔH m,2+b(M3)(ΔH m,2+ΔH m,3)+…]=-{K2[b(M)]2ΔH m,2+K2K3[b(M)]3(ΔH m,2+ΔH m,3)+…}(3)式中,Δdil H∞m为每摩尔溶质无限稀释的稀释焓.溶液的总质量摩尔浓度b t为b t=n/m s=b(M)+2b(M2)+3b(M3)+…=b(M)+2K2[b(M)]2+3K2K3[b(M)]3+ (4)=b(M){1+2K2[b(M)]+3K2K3[b(M)]2+…}定义物种M i的配分函数(权重分数)为单体:α=b(M)b t=11+2K2[b(M)]+3K2K3[b(M)]2+…(5a) i聚体:αi=ib(M i)b t=i(K2K3…K i-1K i)[b(M)]i-11+2K2[b(M)]+3K2K3[b(M)]2+…,i叟2(5b)假定M分子的二聚作用是主要的,其平衡常数K2较大;但在二聚之后,每一步π-π自堆叠作用的模式和强度都相同,可看作是拟等步堆叠(quasi-isodemic packing),即假定每一步堆叠作用的平衡常数和摩尔反应焓变都相等:K=K2/β=K3=K4=…=K i,i叟3(6)ΔH m=ΔH2=ΔH3=…=ΔH i,i叟3(7)式中β表示二聚和三聚作用的强度比.则式(3)和(4)分别可变为Δdil H∞m=-βKΔH m[b(M)]2{1+2Kb(M)+3K2[b(M)]2+…}/b t(8)b t=b(M){1+2βKb(M)+3βK2[b(M)]2+…}(9)若以无限稀释溶液为参考态,则溶液的相对表观摩尔焓LΦ为LΦ=-Δdil H∞m=βKΔH m b(M)·(1+2Kb(M)+3K2[b(M)]2+…)/(1+2βKb(M)+3βK2[b(M)]2+…)(10)在稀溶液中,若Kb(M)垲1,βKb(M)垲1,则如下展开式成立:[1-Kb(M)]-2=1+2Kb(M)+3K2[b(M)]2+…(11a)β[1-Kb(M)]-2-β+1=1+2βKb(M)+3βK2[b(M)]2+…(11b)故由式(9)得b t=b(M){β[1-Kb(M)]-2-β+1}(12)由式(10)得LΦ=βKΔH m[b(M)]2b t[1-Kb(M)]2=KΔH m{1-βK[1-βKb(M)]2b t}2b t(13)式(13)展开后得到LΦ=KΔH m b t[1-2βKb t+5βK2b2t+…](14)对于稀溶液(b t垲1mol·kg-1)或较弱的自堆叠作用(K垲1kg·mol-1),式(13)中右边的高次项可以忽略.则对任意的稀释过程(b i→b f),其摩尔稀释焓为Δdil H m(b i→b f)=LΦ(b f)-LΦ(b i)=KΔH m(b f-b i)-2βK2ΔH m(b2f-b2i)+5βK3ΔH m(b3f-b3i)+ (15)由实验测得的摩尔稀释焓Δdil H m(b i→b f),根据式(15)进行多元线性回归分析,可以求得KΔH m的值.730No.4胡新根等：水溶液中几种芳香族氨基酸π-π自堆叠作用K ΔH m 称为对分子化学作用参数(pairwise molecular chemical interaction parameter).2实验部分2.1仪器与试剂采用LKB -2277BioActivity Monitor 精密微热量计的流动混合检测系统测量稀释焓.待测溶液和纯溶剂分别由两台微蠕动泵(LKB BROMMA 2132)输入热量计的混合池,在池中迅速混合、反应,然后流出,在流动过程中,反应的热功率被检测,经放大、显示、记录.仪器的温度控制精度为0.0005K,热功率测量精度优于0.2%,检测极限为2μW.实验所用D/L -色氨酸、L -色氨酸、L -组氨酸和L -苯丙氨酸均为Sigma 试剂,使用前用甲醇-水混合溶剂重结晶处理并真空干燥至少48h.L -酪氨酸因为在中性条件下在水中溶解度太小,稀释热很小,所以没有进行测定.溶液配制用水为二次重蒸水.2.2方法与步骤当热量计恒温系统及检测系统达到热平衡后,用溶剂水设定基线,进行电标定.然后,溶液和溶剂分别通过一对蠕动泵以一定体积流速比(v 1∶v 2)泵入热量计的混合池.当稀释过程产生的热功率(P )达到稳定值后记录P 值.质量流速f 1和f 2则以称重法进行标定,流速精度优于0.1%.总体积流速(v 1+v 2)对电标定值有较大影响,在不同总流速下测定时必须重新进行电标定.实验发现,总体积流速不变,改变进样流速比,热标定值基本不变.通过调整两台蠕动泵的体积流速比(v 1∶v 2)就可以方便地改变稀释比,而不必每次进行电标定.另外,总流速在40mL ·h -1以下时,测得KCl 水溶液的稀释焓与总体积流速呈线性关系,可以直线外推至零,说明在这一流速以下两种试液能充分混合稀释.实验中控制体积流速比(v 1∶v 2)为20∶10,15∶15,10∶20,总流速为30,单位为mL ·h -1.稀释焓的计算公式为Δdil H m =P /(c i f 2)(16)稀释后溶液的浓度为b f =b i f 2/[f 1(b i M 2+1)+f 2](17)式中,f 2和f 1为溶液和溶剂的质量流速(mg ·s -1),b i 和b f 为溶液稀释前和稀释后的质量摩尔浓度(mol ·kg -1),c i 为以每kg 溶液计的溶液的稀释前浓度(mol ·kg -1),M 2为溶质的摩尔质量(kg ·mol -1).每个浓度下的稀释热功率P 都平行测定三次,取其平均值根据式(16)计算摩尔稀释焓.b i and b f are the initial and final molalities of solutes.The values in parentheses are the experimental errors.表1298.15K 时D/L -色氨酸、L -色氨酸、L -组氨酸和L -苯丙氨酸水溶液的摩尔稀释焓Table 1Molar dilution enthalpies of aqueous solutions of D/L -α-tryptophan,L -α-tryptophan L -α-histidine andL -α-phenylalanine at 298.18Kb i (mol ·kg -1)b f(mol ·kg -1)Δdil H m,exp b i(mol·kg -1)b f D/L -α-tryptophan L -α-tryptophan0.05110.033422.97(0.19)0.02110.01380.05110.023236.18(0.32)0.02110.00960.05110.017343.85(0.39)0.02110.00720.04520.029420.24(0.13)0.01820.01190.04520.020431.87(0.28)0.01820.00830.04520.015238.63(0.24)0.01820.00620.04050.026218.05(0.11)0.01650.01080.04050.018228.43(0.22)0.01650.00750.04050.013634.46(0.27)0.01650.00560.03550.022915.78(0.09)0.01440.00940.03550.015924.85(0.21)0.01440.00650.03550.011930.12(0.25)0.01440.00490.02990.019513.46(0.07)0.01210.00790.02990.013621.20(0.18)0.01210.00550.02990.010125.69(0.13)0.01210.00410.02540.016411.28(0.05)0.01070.00720.02540.011417.76(0.07)0.01070.00490.02540.008521.52(0.12)0.01070.00360.02030.01319.06(0.04)0.00890.00580.02030.009114.26(0.09)0.00890.00430.02030.006817.29(0.11)0.00890.0034Δdil H m,exp(kJ ·mol -1)-21.09(0.13)-33.17(0.21)-40.21(0.32)-18.41(0.14)-28.68(0.17)-34.76(0.25)-16.72(0.08)-26.26(0.19)-31.31(0.24)-14.61(0.06)-22.69(0.16)-27.58(0.19)-12.31(0.11)-19.17(0.09)-23.11(0.14)-10.61(0.09)-16.86(0.08)-20.33(0.13)-8.95(0.07)-14.02(0.11)-17.31(0.13)b i(mol ·kg -1)L -α-histidine 0.13970.13970.13970.11130.11130.11130.09520.09520.09520.08010.08010.08010.06920.06920.06920.05210.05210.05210.04560.04560.0456b f(mol·kg -1)0.09450.06500.04750.07530.05180.03790.06440.04430.03240.05420.03730.02720.04680.03220.02350.03530.02420.01770.03090.02120.0155Δdil H m,exp(kJ ·mol -1)28.25(0.21)46.85(0.33)58.11(0.45)22.57(0.12)37.14(0.26)46.05(0.36)19.39(0.07)32.15(0.22)40.02(0.18)16.24(0.11)26.86(0.14)33.34(0.23)14.03(0.08)23.21(0.12)28.27(0.21)10.56(0.08)17.47(0.11)21.56(0.16)9.21(0.07)15.29(0.11)19.05(0.13)b i (mol ·kg -1)L -α-phenylalanine 0.15010.15010.15010.11130.11130.11130.07270.07270.07270.05210.05210.05210.04880.04880.04880.04520.04520.04520.04090.04090.0409b f (mol ·kg -1)0.10210.06610.04780.07550.04890.03530.04940.03250.02310.03520.02280.01650.03320.02150.01550.03040.01970.01420.02730.01770.0128Δdil H m,exp(kJ ·mol -1)-61.06(0.41)-104.50(0.82)-125.80(0.97)-43.28(0.34)-74.43(0.55)-89.86(0.49)-27.17(0.12)-46.98(0.21)-56.89(0.33)-18.87(0.12)-32.73(0.24)-39.77(0.18)-17.75(0.07)-30.81(0.26)-37.38(0.27)-16.21(0.06)-28.13(0.17)-34.13(0.15)-14.46(0.07)-25.12(0.09)-30.55(0.16)731Acta Phys.-Chim.Sin.,2009Vol.25 3结果与讨论D/L-色氨酸、L-色氨酸、L-组氨酸和L-苯丙氨酸在298.15K时不同质量摩尔浓度的摩尔稀释焓数据见表1.根据式(15)进行多元线性回归分析,求得KΔH m的值,见表2.根据McMillan-Mayer统计热力学理论[19],非电解质水溶液的稀释焓常用以下方程进行处理:Δdil H m=h xx(b f-b i)+h xxx(b2f-b2i)+ (18)对比方程(18)和(15),得到h xx=KΔH m,说明本文的化学作用参数KΔH m与McMillan-Mayer理论中的焓对作用系数h xx之间具有高度的一致性.表2中还列举了根据文献报道的一些芳香族非电解质在水中的稀释焓计算得到的KΔH m,部分直接引用了h xx的值.在McMillan-Mayer理论中,h xx被认为是溶剂介入的溶质-溶质相互作用(solvent-mediated solute-solute interaction)在焓效应上的一种度量:h xx=鄣(g xx/T)鄣(1/T)鄣鄣p=g xx+Ts xx=u xx+λRT g xx-RTM1-(E0Φ)21鄣鄣(19)式(19)中,u xx是超额内能的第二维里系数,λ是溶剂的热膨胀系数,E0Φ是溶质的标准偏摩尔膨胀率, M1和V01分别是溶剂的摩尔质量和体积.在理论上,焓对作用系数h xx和自由能对作用系数g xx可通过对两个溶剂化溶质分子的平均力势(potential of mean force,PMF)[19]积分求得;在混合溶剂中平均力势不仅与两个溶质分子的取向有关,而且还与所有水分子和共溶剂分子的取向有关.可见,h xx不仅反映了溶质-溶质直接作用,而且反映了无限稀释时由于溶质-溶剂、溶质-共溶剂、溶剂-共溶剂相互作用的重新分布引起的影响,因此h xx实际上是综合反映了溶质-溶质、溶质-溶剂各种相互作用的复杂效应.尽管McMillan-Mayer理论在很多情况下运用是成功的,但在h xx的理论解释和理论计算上还存在相当多的困难.根据本文提出的化学作用模型,由于h xx=KΔH m,因此h xx实际上既包含了对分子作用过程(pairwise molecular interaction,PMI)中的化学平衡因素(ΔG m=-RT ln K),又包含了热效应因素(ΔH m).因此借助化学作用模型可以更好地理解McMillan-Mayer理论中h xx的物理意义,尤其对于存在较强非键作用的分子体系,如疏水缔合、氢键缔合和π-π堆叠等.根据h xx=KΔH m,由于平衡常数K>0,因此h xx 值的符号由热效应ΔH m来决定.当ΔH m>0时,h xx> 0,例如在疏水-疏水作用或亲水-疏水作用占优势的情况下;当ΔH m<0时,h xx<0,例如在亲水-亲水作用占优势的情况下.另外值得注意的是,如果体系中存在其它较强的影响化学作用平衡(K)的复杂因表2298.15K时一些芳香族非电解质在水中的化学作用参数(KΔH m)Table2Chemical interaction parameters(KΔH m)of some aromatic nonelectrolytes in water at298.15K Aromatic compound h xx=KΔH m/(J·kg·mol-2)L or D/L-α-Tryptophan D-L:-1282(8)[this work]-1368(25)[24]L-L:2864(24)[this work]2900(170)[25]L-α-histidine-634(5)[this work]-631(7)[24]-620(10)[25]L-phenylalanine1074(10)[this work]1229(29)[24]1140(30)[25]L-phenylalanine(in acidic solution)2750(238)[12]L-tyrosine(in acidic solution)563(75)[12]D-p-hydroxyphenylglycine-726(8)[16]phenol816(8)[12]4-hydroxyphenylacetic acid-2032(134)[12] pyridine1192[14]o-methylpyridine(α-picoline)598[14]m-methylpyridine(β-picoline)2828[14]p-methylpyridine(γ-picoline)3577[14]The values in parentheses are the experimental errors.732No.4胡新根等：水溶液中几种芳香族氨基酸π-π自堆叠作用素,例如静电作用、氢键、取代基位阻、手性识别等,就会显著影响h xx的绝对值大小.芳核的π-π堆叠作用在本质上包含了van der Waals、疏水和静电作用[20].每一种成分的相对贡献和大小,以及芳核采取何种几何模式进行堆叠与芳核本身的结构与性质有关[10].由于包含了疏水成分,因此它是一种非常令人感兴趣的分子识别要素;同时如果其中的静电成分比较显著,这种作用还具有选择性.Hunter等[21]在这种相互作用的静电成分基础上提出了几种几何堆叠构型(图1),认为静电成分是芳核的四极矩引起的.比如苯环,虽然没有净的偶极,但有非偶的电荷分布,在其环面上电子密度较大,而在其周边上较小,从而产生四极矩.三种可能的堆叠几何构型分别是边-面型(edge-to-face)、偏置型(offset)和面-面型(face-to-face),环表面的覆盖程度依次增大.当一个或两个环面上的电子密度较低时,第二种模式就比较常见.第三种模式常见于供体-受体作用,以及具有相反四极矩的化合物,此时环面是相互吸引的.例如苯-六氟苯的相互作用,根据计算具有-15.5kJ·mol-1的稳定化能[22],其中静电成分有着不可忽视的贡献[23].本文认为,以芳环(包括芳杂环)为中心的π-π堆叠作用,虽然其成分比较复杂,但仍然可以看作是一种特殊的疏水-疏水作用,其内在的van der Waals 力和静电作用成分对疏水作用具有协同效应.因此芳核的π-π堆叠作用一般都表现为吸热效应(KΔH m>0),除非对分子作用过程的疏水协同性遭到严重的取代基空间阻碍,或者遭受芳核以外部分的静电、氢键等作用的破坏,或者由于芳核以外部分存在手性中心而受到手性选择作用的影响.如表2所示,纯水中L-苯丙氨酸的KΔH m值是一个较大的正值.其对分子作用过程,一方面因为分子具有较大的疏水区域(C6H5CH2—),所以疏水-疏水作用占绝对优势;而另一方面,由于氨基酸两性离子所带的正、负电荷头基之间静电作用的存在,因此也可以大大降低同手性分子间(L-L)疏水作用的协同性.与在中性溶液中相比,L-苯丙氨酸在酸性溶液中随着羧基的质子化,所带负电荷消失,对分子作用过程中疏水协同性反而增强,表现为较大的吸热效应,因此其KΔH m是一个更大的正值.同在酸性溶液中, L-酪氨酸因为比苯丙氨酸在芳环上多了一个—OH 官能团,与水分子之间可以形成氢键,使两性离子在对分子作用过程中的疏水协同性进一步降低,吸热效应更小,所以KΔH m值也较小.尤其有意思的是,D/L-色氨酸与L-色氨酸的KΔH m值出现了反号.在同手性情况下(L-L),KΔH m是一个很大的正值((2864±24)J·kg·mol-2);而在消旋情况下(D-L), KΔH m是一个较大的负值(-(1282±8)J·kg·mol-2).与其它氨基酸不同的是,色氨酸的芳香结构部分是由一个苯环和一个吲哚环连接而成的.苯环是疏水性的,吲哚环则具有一定的亲水性.吲哚环与两性离子头基扩展形成了一个较大的水结构破坏区域,结果使得同手性分子(L-L)在发生π-π堆叠作用时,可以采取更为合适的构型,疏水成分得到显著增强,因此表现出很大的吸热效应(KΔH m>0).而对于消旋体系(D-L),由于堆叠构型不合适(如采取edge-to-face的构型)疏水成分不强,分子的亲水部分在对分子作用过程中起决定作用,因此表现出较大的放热效应(KΔH m<0).L-组氨酸分子中带有一个咪唑环,与吲哚环类似具有亲水性,因此在发生π-π堆叠作用时疏水成分较少,而主要是与两性离子头基一起扩展而成的结构破坏区域在起决定作用,因此它的KΔH m是负值.表2中苯酚的KΔH m=816J·kg·mol-2,是一个较大正值,说明苯酚分子在水中具有较大的疏水能力,其对分子作用过程中疏水-疏水作用占优势,表现为吸热效应;与之相比,对羟基苯乙酸尽管在分子结构上增加了一个疏水性的—CH2—基,分子的(a)edge-to-face(b)offset(c)face-to-face图1芳核π-π堆叠作用的几何构型[21]Fig.1Geometric configurations of aromaticπ-πstacking [21]733Acta Phys.-Chim.Sin.,2009Vol.25疏水性得到增强,但因为同时又多了一个亲水性较强的—COOH官能团,尤其—COOH的存在造成对分子作用过程中分子疏水区域并排重叠(juxtaposition)的严重空间阻碍,反而使得亲水-亲水作用占据了绝对优势,因此KΔH m是一个很大的负值(KΔH m=(-2032±134)J·kg·mol-2).而具有手性中心的D-对羟苯基甘氨酸,在分子结构上比对羟基苯乙酸多了一个亲水性的—NH2基团,按理应该具有更负的KΔH m,但实际是KΔH m=(-726±8)J·kg·mol-2.说明芳核外部的手性中心在起识别作用,使得π-π堆叠时环表面覆盖度增大,疏水成分得到增强,削弱了对分子作用过程的亲水-亲水效应.对于表2中属芳杂环的吡啶及其甲基取代衍生物,Yu等[14]报道了它们的稀释焓,并从溶质-溶剂相互作用角度对甲基的位置异构取代产生的焓对作用作了初步探讨.如表2所示,这些化合物在水溶液中KΔH m值的递变如下:o-methylpyridine＜pyridine＜m-methylpyridine＜p-methylpyridine,而且均为正值.说明在吡啶环上甲基的引入和甲基在环上的取代位置直接影响了π-π堆叠作用的几何构型,使得疏水成分发生不同程度的变化.邻位取代时吡啶环的疏水成分最小,对位取代时最大.UV 光谱等研究也表明[26,27],吡啶及其衍生物在水溶液中存在自缔合现象,吡啶环上取代基的种类、数目和取代位置,以及分子不同的质子解离形态对自缔合作用有重要影响;在大多数情况下光谱效应表现为减色效应(hypochromic effects),因此认为其自缔合模式是吡啶环的垂直π-π堆叠.在少数情况下,例如吡啶-2,6-二羧酸的单质子解离形态,则表现为增色效应(hyperchromic effects),其自缔合模式是吡啶环之间通过氢键的共线连接.4结论综上所述,微量热法是研究溶液中分子间非键相互作用的一种可靠方法,稀释焓可以用拟等步自堆叠作用的化学模型进行处理,模型参数KΔH m描述了对分子作用过程(PMI)中平衡和焓变的综合效应.所研究的四种芳香族氨基酸及其它芳香族化合物的π-π堆叠作用都可以看作是一种特殊的疏水-疏水作用,静电作用、取代基空间位阻、氢键和手性选择性作用等对疏水协同性都具有重要影响.References1Tsuzuki,S.Structure&bonding,intermolecular forces and clusters 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于德涵，黎莉，苏适. 天然低共熔溶剂提取黄酮类化合物的研究进展[J]. 食品工业科技，2023，44（24）：367−375. doi:10.13386/j.issn1002-0306.2023020204YU Dehan, LI Li, SU Shi. Research Progress on Extraction of Flavonoids Using Natural Deep Eutectic Solvents[J]. Science and Technology of Food Industry, 2023, 44(24): 367−375. (in Chinese with English abstract). doi: 10.13386/j.issn1002-0306.2023020204· 专题综述 ·天然低共熔溶剂提取黄酮类化合物的研究进展于德涵*，黎 莉，苏 适（绥化学院食品与制药工程学院，黑龙江绥化 152061）摘 要：天然低共熔溶剂是一种新型绿色溶剂，有望替代传统有机溶剂实现对黄酮等天然产物的高效提取。
为了阐明天然低共熔溶剂在黄酮化合物萃取方面的应用，本文对近5年发表的相关研究论文进行了整理和分析，综述了天然低共熔溶剂提取黄酮的研究现状，并详细讨论了影响提取率的各种因素。
天然低共熔溶剂在黄酮、黄酮醇、二氢黄酮、花色素、异黄酮等多类天然黄酮产物的提取方面表现良好，其萃取率普遍优于甲醇、乙醇等传统溶剂，且萃取产物活性更高；低共熔溶剂的组成、摩尔比、含水量和温度等条件会显著影响其对黄酮化合物的萃取。
文章还对天然低共熔溶剂在未来的发展趋势作出展望，希望能为黄酮化合物的高效、绿色提取提供有益参考。
关键词：低共熔溶剂，黄酮类化合物，绿色溶剂，提取本文网刊: 中图分类号：TQ28、TS201 文献标识码：A 文章编号：1002−0306（2023）24−0367−09DOI: 10.13386/j.issn1002-0306.2023020204Research Progress on Extraction of Flavonoids Using Natural DeepEutectic SolventsYU Dehan *，LI Li ，SU Shi（Food and Pharmaceutical Engineering Department, Suihua University, Suihua 152061, China ）Abstract ：The natural deep eutectic solvent is a new type of green solvent that is expected to replace traditional organic solv-ents for efficient extraction of natural products such as flavonoids. In order to clarify the application of natural deep eutectic solvents in the extraction of flavonoids, the author summarizes and analyzes relevant research papers published in the past 5years. This article provides a review of the current research status of natural deep eutectic solvents for extracting flavonoids,and discuss in detail the various factors that affect the extraction rate. The natural deep eutectic solvents perform well in the extraction of various natural flavonoid products such as flavonoids, flavonols, flavonones, anthocyans, and isoflavones.Their extraction rates are generally better than traditional solvents such as methanol and ethanol, and the extracted products have higher activity. The composition, molar ratio, water content, and temperature of deep eutectic solvents signi-ficantly affect their extraction of flavonoids. Finally, the development trend of natural eutectic solvents in the future is prospected. This paper aims to provide reference for the efficient and green extraction of flavonoids.Key words ：deep eutectic solvents ；flavonoids ；green solvent ；extraction黄酮是植物细胞中一种重要的次级代谢产物，能够消除人体内自由基，有较强抗氧化、抗衰老的功能[1]，在抗菌、抗病毒、抗炎、降血糖、降血脂等方面也颇有功效[2]。
黑龙江省哈尔滨师范大学附属中学2024-2025学年高三上学期10月月考英语试题一、听力选择题1．How many of the dresses does the woman have?A．One.B．Two.C．Three.2．How does the man feel about the shoes?A．Satisfied.B．Embarrassed.C．Dissatisfied.3．Where are the speakers probably?A．In a store.B．In an office.C．In a classroom.4．What is the relationship between the speakers?A．Strangers.B．Friends.C．Husband and wife. 5．What is the weather like now?A．Cloudy.B．Sunny.C．Rainy.听下面一段较长对话，回答以下小题。
6．What do we know about the woman?A．She likes the outdoors.B．She tripped up on a rock.C．She never camped in the woods.7．What is hard in the dark according to the man?A．Setting up a tent.B．Avoiding rocks.C．Building a fire.听下面一段较长对话，回答以下小题。
8．What did the man do yesterday?A．He called his friends.B．He visited the gallery.C．He made a reservation. 9．What is the man’s problem?A．He found the gallery was full of people.B．He didn’t know where to pick up the tickets.C．His name is not on the list.10．What will the woman most likely do next?A．Give some tickets to the man.B．Close the gallery.C．Contact a lady.听下面一段较长对话，回答以下小题。
专利名称：一种从硬骨鱼中鉴定新的广谱抗菌肽的基因组途径专利类型：发明专利
发明人：苏珊·道格拉斯,杰弗里·加朗特,亚历山大·帕特里凯特申请号：CN03819924.6
申请日：20030822
公开号：CN1678632A
公开日：
20051005
专利内容由知识产权出版社提供
摘要：本发明提供了一种鉴定编码抗菌肽的候选核酸序列的方法。
该方法包括：鉴定感兴趣的起始肽；鉴定编码所述起始肽的基因组DNA；鉴定位于所述起始肽每一侧的侧翼序列；获得与所述侧翼序列互补的引物；及大范围筛选核酸序列来鉴定能够使用步骤e)的引物来扩增的候选序列。
在一些例子中，所述抗菌肽为hepcidin或pleurocidin。
申请人：加拿大国家研究所
地址：加拿大安大略省
国籍：CA
代理机构：北京英赛嘉华知识产权代理有限责任公司
更多信息请下载全文后查看。
PHYSICAL REVIEW E 86,046305(2012)Shear viscosity of dilute suspensions of ellipsoidal particles with a lattice Boltzmann methodHaibo Huang,YanFeng Wu,and Xiyun LuDepartment of Modern Mechanics,University of Science and Technology of China,Hefei,Anhui 230026,China(Received 21March 2012;published 8October 2012)The intrinsic viscosities for prolate and oblate spheroidal suspensions in a dilute Newtonian fluid are studied using a three-dimensional lattice Boltzmann method.Through directly calculated viscous dissipation,the minimum and maximum intrinsic viscosities and the period of the tumbling state all agree well with the analytical solution for particles with different aspect ratios.This numerical test verifies the analysis on maximum and minimum intrinsic viscosities.Different behavior patterns of transient intrinsic viscosity in a period are analyzed in detail.A phase lag between the transient intrinsic viscosity and the orientation of the particle at finite Reynolds number (Re)is found and attributed to fluid and particle inertia.At lower Re,the phase lag increases with Re.There exists a critical Reynolds number Re a at which the phase lag begins to decrease with Re.The Re a depends on the aspect ratio of the particle.We found that both the intrinsic viscosity and the period change linearly with Re when Re <Re a (low-Re regime)and nonlinearly when Re >Re a (high-Re regime).In the high-Re regime,the dependence of the period on Re is consistent with a scaling law,and the dependence of the intrinsic viscosity on Re is well described by second-degree polynomial fits.DOI:10.1103/PhysRevE.86.046305PACS number(s):47.55.Kf,47.11.−jI.INTRODUCTIONSuspended particles in flows occur in many applications and play an important role in industry.For example,the behavior of suspended particles may affect the quality of paper [1].Shear viscosity of dilute suspensions of spheres has been studied analytically by Einstein [2].According to the theory,the relative viscosity ¯π in dilute suspensions of spheres is [2] ¯π =1+φ ¯η ,where φis the solid volume fraction and¯η =52is the intrinsic viscosity.The relative viscosity ¯π is defined as the ratio of the effective suspension viscosity νs tothe corresponding viscosity of the pure fluid νf ,i.e., ¯π ≡νsνf.The intrinsic viscosity is defined as ¯η = ¯π −1φ.Later,Krieger and Dougherty [3]extended Einstein’s formula to a larger φregime semiempirically.Recently,Lishchuk et al.[4]studied the shear viscosity of bulk suspensions of spherical particles at a low Reynolds number (Re)using a lattice Boltzmann (LB)method.In their result,they included a correction for the effective hydrodynamic radius of particles due to the simple bounce-back scheme they used [4].In the present study,such corrections are not necessary.For a dilute suspension of nonspherical ellipsoidal particles,the variation of intrinsic viscosity would be more complex.The rotational behavior of prolate or oblate spheroids at very low Reynolds numbers has been studied theoretically for a long time.Jeffery investigated the motion of a single ellipsoid in shear flow while completely neglecting inertial effects [5].He concluded that the final rotational state of an ellipsoid cannot be determined because it depends on initial conditions.To definitively determine the final rotational state,Jeffery [5]hypothesized that,“The particle will tend to adopt that motion which,of all the motions possible under the approximated equations,corresponds to the least dissipation of energy.”Extensive analytical investigations [6]studied the inertial effect at Re <1using perturbation theory.However,their analysis is not applicable to large-Re cases.Leal [7]reviewed most of the previous relevant theoretical studies.There are also some relevant experimental works in the literature.Taylor [8]confirmed Jeffery’s hypothesis by investigating the orbit of a prolate or oblate spheroid in a Couette flow at a very low Reynolds number.However,Karnis et al.[9]found that even when the inertial effect is very small [e.g.,at ReO(10−3)],nonspherical particles may adopt a motion that is different from Jeffery’s hypothesis.For suspensions of many spherical particles,states of maximum dissipation have been observed experimentally [10].Different numerical methods have also been used to study the motion of particles in flows.However,some methods such as Stokesian dynamics [11]are only applicable to spherical particles and they neglect the inertial term,which may have a significant influence on the motion of particles.For finite-Reynolds-number flows,the Navier-Stokes equations have to be solved.Feng et al.[12]simulated the motion of a single ellipse in two-dimensional (2D)creeping flows using a finite element approach.They confirmed Jeffery’s hypothesis at Re ≈1.In the past 30years,the lattice Boltzmann methods (LBMs)have been developed into an efficient numerical tool to study particulate suspensions [1,4,13–15].Qi and Luo studied the energy dissipation of a prolate spheroid at Re =0.1and 18using a LBM [1].According to their study,the numerical results did not support Jeffery’s hypothesis.However,the calculated intrinsic viscosities in their study [1]seem not to be consistent with Jeffery’s study [5].That will be illustrated in Sec.III in detail.Here we demonstrate that our results are much more reliable than those of Qi and Luo [1]and further confirm that Jeffery’s hypothesis may be incorrect.Some numerical studies investigated prolate and oblate spheroid suspensions in 3D Couette flows for Re up to approximately several hundred [1,15,16].They focused on the critical transition Re for different rotational modes,which may depend on the initial orientation [15].There are also some studies that focused on the critical Re at which the particle would stop rotating [14].Lin et al.[17]studied the inertial effects on the suspension of a rigid sphere in a simple shear flow.They proposed a formula for the suspension viscosity:νs =νf [1+φ(52+1.34Re 1.5)].However,for suspensions ofHAIBO HUANG,Y ANFENG WU,AND XIYUN LU PHYSICAL REVIEW E86,046305(2012) elliptical spheroids,no such formula is proposed.Here,fromour numerical results,we propose to separate the dependencesof the intrinsic viscosity and the period of the tumbling stateon Re into two regimes.In the low-Re and high-Re regimes,they change linearly and nonlinearly with Re,respectively.The phase lag between the transient intrinsic viscosity and theorientation of the particle atfinite Reynolds number is alsoanalyzed in detail.In this study,we focus on the intrinsic viscosity of spheroidsin a3D Couetteflow for Re up to about200.The numericalmethod used in our study is based on the LBM[13,14]withimprovements in the collision model[18]and curved-wallboundary treatment[18].For the collision model,the multi-relaxation-time(MRT)model[18]is used because it has betternumerical stability.For the curved-wall boundary condition,anaccurate momentum-exchange-based scheme[18]is applied.The translational and orientational motions of the spheroid aremodeled by the Newtonian and Euler equations,respectively.Our scheme for calculating relative viscosity was validated inRef.[15].The present work is intended to provide a betterunderstanding of the intrinsic viscosity of dilute suspensionsof nonspherical particles.In Sec.II,the LBM and the basicequations for the motion of a solid particle are introducedbriefly.Results and discussion on the shear viscosity ofdilute suspensions of ellipsoidal particles with different Reand aspect ratios are described in Sec.III.Conclusions arepresented in Sec.IV.II.NUMERICAL METHODIn our study,thefluidflow is solved by the MRT-LBM[18].The following MRT lattice Boltzmann equation(LBE)[19]is employed to solve the incompressible Navier-Stokesequations:|f(x+e iδt,t+δt) −|f(x,t)=−M−1ˆS[|m(x,t) −|m(eq)(x,t) ],(1)where the Dirac notation of ket|· vectors symbolizes thecolumn vectors.The particle distribution function|f(x,t)has19components f i with i=0,1,2,3,...,18in our3Dsimulations because the D3Q19velocity model is used.Thecollision matrixˆS=MSM−1is diagonal withˆS[19],ˆS≡diag(0,s1,s2,0,s4,0,s4,0,s4,s9,s10,s9,s10,s13,s13,s13,s16,s16,s16).|m(eq) is the equilibrium value of the moment|m .The19×19matrix M[19]is a linear transformation which is usedto map a vector|f in discrete velocity space to a vector|m inmoment space,i.e.,|m =M|f ,|f =M−1|m .In the aboveequation,e i are the discrete velocities of the D3Q19model.The lattice speed is defined as c=δx,whereδx andδt are thelattice spacing and time step,respectively,in LB simulations.The matrix M,the discrete velocities of the D3Q19model,and|m(eq) are all identical to those in Refs.[15,19].The macrovariable densityρand momentum jζare ob-tained fromρ=i f i,jζ=if i e iζ,(2)FIG.1.(Color online)Schematic diagram of a spheroid withits symmetry axis in the x direction in a Couetteflow.Line OMrepresents the intersection of the(x,y)and the(x ,y )coordinateplanes.The two walls at y=−L and L move in opposite directions.Periodic boundary conditions are applied in the x and z directions.whereζdenotes x,y,or z coordinates.Here the collisionprocess is executed in moment space[19].In our simulations,the parameters are chosen as s1=1.19,s2=s10=1.4,s4=1.2,s9=1τ,s13=s9,and s16=1.98.The parameterτisrelevant to the kinematic viscosity of thefluid withνf=c2s(τ−0.5)δt and c s=c√3.The pressure in theflowfield canbe obtained from the density via the equation of state p=c2sρ.The particle’s movement and rotation are updated at eachtime step through Newton’s law and Euler equations.Theprolate or oblate spheroid is described byx 2a2+y2b2+z2c2=1,(3)where(x ,y ,z )represents the body-fixed coordinate systemand a,b,and c are the lengths of three semiprincipal axes of aspheroid in the x ,y ,and z directions,respectively.The spatialorientation of any body-fixed frame(coordinate system)can beobtained by a composition of rotations around the z -x -z axiswith Euler angles(ϕ,θ,ψ)from the space-fixed frame(x,y,z)that initially overlaps the body-fixed frame.The compositionof rotations is illustrated in Fig.1.Noted that the symmetryaxis of the spheroid is in the x direction.The translational velocity U(t)of the solid particle isdetermined by solving Newton’s equations[15].The rotationof the spheroid is determined by Euler equations,which arewritten asI·d (t)dt+ (t)×[I· (t)]=T(t),(4)where I is the inertial tensor.Note that in the body-fixedcoordinate system[coordinates(x ,y ,z )in Fig.1],thetensor is diagonal and the principal moments of inertiaare I x x =m b2+c25,I y y =m a2+c25,and I z z =m a2+b25,wherem=ρ043πabc is the mass of the suspended particle.Therepresent angular velocities and T is the torque exerted onthe solid particle in the same coordinate system.Here fourquaternion parameters[15]are used as generalized coordinatesto solve the corresponding system of equations.With theSHEAR VISCOSITY OF DILUTE SUSPENSIONS OF...PHYSICAL REVIEW E86,046305(2012) quaternion formulation, in[Eq.(4)]can be solved using afourth-order accurate Runge-Kutta integration procedure[15].Thefluid-solid coupling in our study is based on theschemes in Refs.[13]and[18].The force on solid boundarynodes is calculated through the momentum exchange scheme[18,20],and the force due to thefluid particle entering andleaving the solid region[13]is also considered.A.Rotational modes and parameters in simulationsIn the space-fixed coordinates,the streamwise directionof the Couetteflow is along the z direction.The velocitygradient and the vorticity are oriented in the y and x directions,respectively.The tumbling mode indicates a particle rotationabout its y or z axis with the axis parallel to the vorticitydirection.The log rolling mode indicates the spheroid spinaround the evolution axis(the x axis),which overlaps the xaxis.To describe the geometry of the spheroid,“ellipticity”is defined here to be the difference of the greatest andleast diameters divided by the greatest,i.e.,ε=a−ba for aprolate spheroid andε=b−ab for an oblate spheroid[5].Thecomputational domain in all of our following simulations has dimension2L in the x,y,and z directions.Two walls located at y=−L and L move in opposite directions with speed U, as shown in Fig.1.Periodic boundary conditions are applied in both the x and z directions.To calculate the force acting on the spheroid more accurately,grid refinement is used near the particle.Both thefine grid and the particle are located in the center of the domain,and the particle is immersed in the fine grid.We use1l.u.and1t.s.to denote1 x f and1 t f, respectively.Thefine grid has dimension L in three directions with lattice spacing1l.u.The rest of the computational domain isfilled with coarse mesh and the lattice spacing x c=2l.u. and t c=2t.s.Thefine and coarse grids are coupled using the scheme proposed by Filippova et al.[21].The particle Reynolds number is defined to be Re=4Gd2νf,where G=UL and d is the length of the semimajor axis(i.e.,d=a for the prolate spheroid and d=b=c for the oblatespheroid).In all of our cases,the confinement ratio La >5andthe volume fraction is less than0.7%.Hence,the moving wall effect can be neglected and the suspensions of particles can be regarded as dilute suspensions.In our simulations,theflow is intended to be incompress-ible.To ensure the incompressibility condition,the maximum velocity in theflowfield should not exceed0.1l.u./t.s.[15].On the other hand,to ensure that the bounce-back boundary condition correctly mimics the nonslip boundary condition on the wall,τused in the simulations should not be too large.τf andτc are the relaxation times in thefine mesh and coarse mesh,respectively.They satisfy the formula1 3(τf−0.5) t f=13(τc−0.5) t c.In our simulations,weadoptτf<1.3;hence the bounce back can correctly recover the nonslip physical boundary condition.Considering the above two effects,the parameters are chosen in the following way.For the low Reynolds number cases,U is calculated from the definition of Re.For example,in the case of Re=0.5, L=192l.u.,a=24l.u.,b=c=12l.u.,τf is set to be1.2, and the calculated velocity of the wall is0.00486l.u./t.s.For larger Reynolds number cases,U is kept at0.1l.u./t.s.and τf is calculated from the definition of Re.For example,in the case of a prolate spheroid with Re=30,the parameters are L=192l.u.,a=24l.u.,b=c=12l.u.,and the calculated τf andτc are0.74and0.62,respectively.In all of our numerical tests,the velocityfield was initialized as a Couetteflow with a uniform pressurefield(p0=c2sρ0). The particle is released at the center of the computational domain with zero velocity.Although the translational motion of the particle is not constrained,the spheroid center is not found to depart from the center of the computational domain in all simulated cases.To make the particle enter log rolling mode and tumbling mode quickly,it was excited by setting the initial orientation as(ϕ0,θ0,ψ0)=(0◦,0◦,0◦)and(ϕ0,θ0,ψ0)= (90◦,90◦,90◦),respectively.B.Calculation of the intrinsic viscosityFrom Fig.1,we can see that the streamlines of the Couette flow are in the z direction and that the velocity gradient is in the y direction.The shear stress at afluid node x can be obtained throughσ(x)=ρνf(∂y w+∂z v),where v and w are the velocity components in the y and z directions at x,respectively.We note that in the MRT LBM,the second-order moments of the distribution function are given by p yz= i e iy e iz f i=ρvw−τc2sρ(∂y w+∂z v).Therefore, at eachfluid node the shear stress is obtained throughσ(x)=ρνf(∂y w+∂z v)=νfτc2s(ρvw−p yz).(5)In the lattice Bhatnagar-Gross-Krook(BGK)method,at eachfluid node the shear stress can be obtained byσ(x)=ρνf(∂y w+∂z v)=−(1−12τ)f n eqie iy e iz,wheref n eqi= f i−f eq i.The energy dissipation represented by the relative viscosity ¯π of theflow system[1]is given by¯π =¯σρνf G,(6)where ¯σ is the temporally and spatially averaged shear stress and G is the shear rate of the Couetteflow without particles. The transient intrinsic viscosity is defined to beη(t) =π(t) −1φ,(7)where π(t) = σ(t)ρνf Gand σ(t) is the spatially averaged shear stress at time t.To calculate transient intrinsic viscosity η ,first the drag force acting on theflat wall is obtained through integrating the shear stressσ(x)on moving wall nodes,which is equal to that acting on thefluid nodes that are nearest to the wall nodes. Then the spatially averaged shear stress σ is equal to the drag force divided by the area of the movingflat wall.After η is further averaged in time, ¯η is obtained.Here we can see that σ can be regarded as theρνf G w ,where G w is the strength of the area-averaged shear rate near the wall.Hence, G w is directly related to η .,046305(2012)¯πtheToisfor suspensionsWe found thatspheroids,analyticalfor the prolateand tumblingpredicted,for the oblatemode have therespectively.for the prolate[15],whichFor the oblate[15],whichresults argueassuming theparticleThat iset al.,whichof manywith differentare alsoThe analyticalperiods in ourones.Hereand itnote for clarityby1G.viscositiestheir calculatedanalyticalIn Ref.[1],thel.u.,b=c=×64l.u.×=43π×12×viscosities arethe tumbling¯η = ¯π −1φareHowever,FIG.3.(Color online)Intrinsic viscosities for dilute suspensions of(a)prolate spheroids and(b)oblate spheroids at Re=0.5for different ellipticities.PHYSICAL REVIEW E 86,046305(2012)FIG.4.(Color online)Period T for the prolate and oblate spheroidal particles at Re =0.5for different ellipticity.according to Jeffery’s study,the minimum and maximum ¯η should be 2.174and 2.819,respectively [5].Hence,the numerical results for ε=0.5in the study of Qi and Luo [1]have large discrepancies with Jeffery’s study [5].It is worth noting that although the results of Ref.[1]are somewhat different from Jeffery’s solution,the ratio of the minimum and maximum values is consistent with that solution.It is possible that the present interpretation of the results in Ref.[1]is not completely correct,perhaps due to missing or incorrect information in Ref.[1].B.Transient propertyFor the log rolling mode,the intrinsic viscosity would reach a constant value when the flow becomes steady,i.e.,the angular velocity of the spheroid becomes constant.However,for the tumbling mode,the transient intrinsic viscosity η is not a constant in a period of rotating and it varies with the particle’s orientation.The orientation is described by angles α,β,and γ,which denote the angles between the x axis and the space-fixed x ,y ,and z axes,respectively (cos 2α+cos 2β+cos 2γ=1).The typical variations of the transient intrinsic viscosity as a function of cos βfor the dilute suspension of prolate spheroids at Re ∼O(1)and Re ∼O(10)are shown in Fig.5.The arrows in the figure demonstrate the chronological order.Here we can see that there are two peaks and two valleys in a period when the Reynolds number is low [Fig.5(a)]while there is only one maximum and one minimum in a period at Re ∼O(10).For dilute suspension of oblate spheroids,similar behavior is also observed.In the following,the mechanics for such different behaviors at low and larger Re will be explored.First,we would like to discuss the case of low Re.The transient intrinsic viscosity, η ,the angular velocity in the x direction,ω,and the torque and orientation as functions of t for the suspension of prolate spheroids with Re =0.1and ε=0.5are shown in Fig.6.Note that the angular velocity is normalized by G and the torque is normalized by m (Ga )2.In the figure,to elucidate more clearly,the times A to E are labeled in chronological order.The time from A to E is a half-period of the tumbling state.In the discussion,we will focus on the orientation and the torque (or the distance-averaged force).The two features may affect the intrinsic viscosity significantly.From the figure,we can see that when the major axis (x axis)of the spheroid is parallel to the z axis, η reaches a valley with a value of about 2.51at t =12.3(point A).At this orientation,the particle has a minimum effect on G w ,and hence η ,compared to the other orientations.The particle rotates counterclockwise (refer to Fig.1,view in the −X direction).When the x axis approaches the y axis, η increases to a peak value about 3.37at t =15.4(point B).At this orientation,the torque increases to a maximum value.From A to B,the torque is positive,which means that it would push the particle to rotate counterclockwise.At point B,the particle strongly hinders the Couette flow and the shear rate near the moving walls is the strongest.After point B,the torque decreases to a negative value,which means the hindering effect becomes weak.Consequently, η becomes smaller and reaches its second valley with η =2.65at t =16.4(point C).After point C,the torque acting on the particle increases and hence η increases.On the other hand,the x axis becomes closer to the z axis.That means the effect of the suspended particle on the shear rate near the moving walls,i.e., G w,FIG.5.Transient intrinsic viscosity η as a function of cos βfor a dilute suspension of prolate spheroids,(a)a case of low Reynolds number (Re =0.5,ε=0.5)and (b)a case of larger Reynolds number (Re =20with ε=0.7).HAIBO HUANG,Y ANFENG WU,AND XIYUN LUPHYSICAL REVIEW E 86,046305(2012)ωFIG.6.(Color online)(a)Transient intrinsic viscosity η ,(b)angular velocity ω,(c)torque,and (d)orientation as functions of time for a dilute suspension of prolate spheroids at Re =0.1with ε=0.5.becomes smaller.Due to the combination of the above two features, η reaches a second maximum of 3.37at point D and later η decreases to the minimum value when the x axis is parallel to the z axis.Figure 7shows typical variations of the intrinsic viscosity,angular velocity,torque,and orientation as functions of time when Re =20and ε=0.7.In a half-period,there is only one maximum value.The variation of η becomes simpler than that in the low-Re case.From the figure,we can see that at t ≈49,cos β=0and the x axis is parallel to the z axis.In this orientation, η should reach its minimum value because the particle has a minimum effect on the intrinsic viscosity if the inertia of the fluid is neglected.However,it takes until t ≈55before η reaches its minimum value.Hence,the change of η is not synchronous with the change of the orientation of the particle.We also found that the phase lag between η min and orientation depends on Re.Typical curves of phase lag are shown in Fig.8.From the figure,we can see that the phase lag χ/πof prolate spheroids with ε=0.7,0.8,and 0.9increases with Re until Re ≈Re a and then it decreases.Re a depends on ε.For ε=0.7,0.8,and 0.9,the Re a ’s are approximately 100,60,and 30,respectively.The dashed line in the figure is a line connecting the Re a for different ε.There are two factors that may affect the phase lag between η min and cos β=0:the particle inertia and the fluid inertia.Note that even when the inertia of the fluid is small,i.e.,when Re is small,the inertia of the particle exits.We notethat the nondimensional maximum torque acting ontheωFIG.7.(Color online)(a)Transient intrinsic viscosity η ,(b)angular velocity ω,(c)torque,and (d)orientation as functions of time for a dilute suspension of prolate spheroids at Re =20with ε=0.7.particle decreases with Re (not shown here).This means the inertia of the particle decreases with Re.On the other hand,the inertia of the fluid increases with Re.The competition between the two factors results in the behavior of the phase lag.In the next section,we show that when Re <Re a ,both the intrinsic viscosity and the period change linearly with Re,while for Re >Re a ,they change nonlinearly withRe.ReFIG.8.(Color online)Phase lag between the minimum intrinsic viscosity η min and cos β=0for dilute suspensions of prolate spheroids (ε=0.7,0.8,0.9)as functions of Re.SHEAR VISCOSITY OF DILUTE SUSPENSIONS OF...ReFIG.9.(Color online)Intrinsic viscosity ¯η and angular velocityωfor dilute suspensions of prolate spheroids(ε=0.5)as functionsof Re when the spheroid is constrained into the log rolling mode.C.Re effect on intrinsic viscosity and periodIn the study of Huang et al.[15],it is found that for a givendynamical mode,the relative viscosity increases with Re whenRe<18.Here we extend the study to higher Re and proposea scaling law for Re∼O(10).For the tentatively constrained spheroid,the x axis alwaysoverlaps the x axis and the intrinsic viscosity and equilibriumangular velocity would change with Re.That is illustratedin Fig.9.When Re increases from0.5to90,The angularvelocityωwould decrease from0.5to about0.44for the prolatespheroid withε=0.5.That may be due to the particle andfluid inertia.We note that the angular velocityω=0.5is alsoobserved in the study of a spherical particle in shearflows byNirschl et al.[22].Here we found that ¯η would increase from2.17to3.09.The possible reason is that the shearflow nearthe particle would be retarded when the equilibrium angularvelocity becomes smaller at large Re.That may increase theshear rate near the walls.For the oblate spheroid,a similarvariation with Re was also observed.For the tumbling state,the scaling of the period of rotationis[13]T=C(Re c−Re)−12,where Re c is the critical Re whenthe spheroid would stop rotating.However,we found that Re cdepends slightly on the confinement ratio Lb.In the followingsimulation for an oblate spheroid withε=0.5,Re c is about140.The intrinsic viscosity and period as functions of Re areshown in Fig.10.We can see that the period increases withRe.For the larger Re near to Re c,the period seems close tothe scaling law found by Aidun et al.[13].Here we found thatReFIG.10.(Color online)The intrinsic viscosity ¯η and period Tas functions of Re for the dilute suspension of the tumbling oblatespheroid(ε=0.5).T=220(Re c−Re)−12fits the high-Re section well.However,in the lower Re section,T increases linearly with Re.For the prolate spheroid,the situation is similar to that ofthe oblate spheroid.Figure11shows the intrinsic viscosityand period as functions of Re for the prolate spheroid.Both¯η and T change almost linearly with Re when Re<Re a.When Re>Re a,both ¯η and T change nonlinearly with Re.From Fig.11,we can see that forε=0.7,0.8,and0.9,thereis a critical Reynolds number Re c,beyond which the periodof rotation would become infinite,i.e.,the particle would stopand remain stationary in theflow.The corresponding Re c’sare approximately167,120,and75,respectively.Hence,Re cdecreases with the ellipticityε.Forε=0.5,due to numericalinstability,we have not been able to carry out simulations ofcases with Re>500.At Re≈500,we did not observe thestationary state for cases ofε=0.5.It is believed that forε=0.5,they have similar behavior asε=0.7,0.8,and0.9.The difference is that Re a and Re c are much larger than thoseofε=0.7,0.8,and0.9.Here only cases Re<200forε=0.5are illustrated.First we analyze the linear dependence of ¯η and T on Re(Re<Re a).In Fig.11,the solid lines are the linearfits.FromFig.11(a),we can see that forε=0.5and0.7, ¯η increaseswith Re.However,forε=0.8and0.9, ¯η decrease with Re.However,in Fig.11(b),for allε,the period increases with Re.For Re<Re a,the scaling law is¯η = ¯η0 +κ1Re,T=T0+κ2Re,(8) TABLE I.Slopes of the linearfit for the ¯η -Re and T-Re curves(dilute suspensions of tumbling prolate spheroids). Ellipticityε ¯η0 κ1(slope of the ¯η -Re curve)T0κ2(slope of the T-Re curve)0.0 2.50.025312.570.0720.3 2.6410.019413.690.0960.5 2.7380.013017.060.1170.7 3.1160.0028722.490.2320.8 3.443−0.0033731.720.4170.9 4.482−0.020158.83 1.069。
Research on Ellipsoidal Intersection Fusion Method with Unknown CorrelationTao-Tao Wu, Jin An, Chun-Shan Ding and Shuang-Xi LuoJiang-Su Automation Research InstituteLian-Yun-Gang, Chinattwu@, anjin308@, dingcs2009@, luosxjary@Abstract—This paper reviews the advantages and shortages of the covariance intersection (CI) and ellipsoidal intersection (EI) methods for decentralized state fusion with unknown correlation, and makes some progress on both of them. New results are: a). For CI method, the convexity property is proved for the two classical cost functions (i.e., trace and natural logarithm of determinant of the fused covariance), and a simple form of the optimization conditions is derived for the latter one. Furthermore, a fast 2-sensor CI algorithm is proposed by expressing the cost function in scalar form. b). For the 2-sensor EI algorithm which minimized the natural logarithm of determinant of the mutual covariance, a new proof for its optimality is presented, which partly makes up the gap in [17]. Simulation results show the efficiency for both the 2-sensor CI and EI algorithms.Keywords-Covariance intersection; cost function; restricted convex programming; algorithm; Newton iteration; ellipsoidal intersection.I.I NTRODUCTIONIt’s well known that Kalman Filter (KF) estimate is optimal in the sense of minimizing the mean square error for the linear system when the noises of system dynamics and measurement are Gussian white noises, and it’s also the optimal linear estimate when the noises are not Gussian but its second order statics is known. In both cases, fusion based on KF will result in consistent estimate for system state.Practically, it’s difficult to get the accurate variance, especially the covariance of data from sensors of different types. Thus, fusion system usually adopt the de-coupling method to deal with each state component by ignoring their covariance. There are two reasons for this. The first one is we do not understand the real system well due to restriction of impersonal condition, and the complexity of realization for fusion algorithm is the other.To overcome the divegence of filter, two classes of fusion methods are developed. The key idea of the former is to dynamically adjust the system noise factitiously. The latter could be named the Ellipsodial Intersection Fusion Methods (EIFM), which mainly includes the Covariance Intersection (CI) method, Largest Ellipsoid (LE) method, Internal Ellipsoidal Approximation (IEA) method and Ellipsoidal Intersection (EI) method. The disadvantage of the first one is its strong dependance on the empirical knowledge, in other words, special analysis for the particular real system and a compromise on the fusion performance are needed. Luckily, the second class will avoid this. Actually, the CI method provided an efficient way to get consistent estimate by fusion with unknown correlation and it is capable of fusing any probablity density function, not just Gaussian distributions. But CI method is pessimistic with the ellipse being larger than it needs to be. LE and IEA avoid this by providing new ways to find tighter estimates. However, they have a common big shortcoming, that is, accurate estimates are treated with an equal importance as inaccurate ones, which results in an estimate with bigger error than it needs to be. This motivates the appearance of EI method, which seperates the recieved information as mutual part and exclusive part and only uses the latter to update the estimate.The above methods, i.e., CI, LE, IEA and EI methods, form the mainstream of currect decentralized fusion algorithms, and each of them has its own disadvantages and shortcomings, which are discribed as follows.A.CI methodSince CI method appeared, much progress was made on the optimization conditions and improved algorithms, which focus on two aspects:a) Fusion of two local estimates. When the error ellipse of one estimate contains the one of the other, the fused estimate given by standard CI method degrades as the local estimate which has a smaller error ellipse,and thus the advantage of fusion was not demonstrated since it doesn’t consider the directions of local estimates.b) CI algorithm is highly nonlinear optimization problem, which always take the trace or determinant/its variant of fused variance as its cost function. Thus many improved algorithms aimed at finding some inferior-optimal and fast approximation methods, e.g., directly compute the convex combination coefficients by using the variance matrices of local estimates ([6~9]).Much attention is paid on CI method and its applications. It is indicated in [1] that the Newton-Raphson algorithm and many convex optimization techniques could be used to determine the coefficients of convex combination. However, the realization of CI algorithm depends on the existing low-efficient optimization algorithms, which neither proved theconvexity of the cost function nor made use of the structureproperty of CI algorithm sufficiently.B.LE and IEA methodsCurrently, many fusion algorithms are based on CI method.However, recent researches showed its deficiency andpresented some new methods which will result in tighterestimate. For details, it is pointed in [6] and [13] that CI ispessimistic in the sense that it overestimates the error ellipse.To find tighter estimate, [13] proposed the LE method to findfused estimate of two local estimates with unknown correlation.The idea of LE is creating the biggest ellipse that will fit withinthe intersection of the covariances. Unfornately, it is indicatedin [18] that the LE algorithm did not derive the computation ofestimated fusion correctly. To fill in the gap, [18] presentedthe IEA method based on the internal approximation of the intersection region of the covariance matrices, wherein somesimple simultion results showed that the fusion accuracy byIEA upgrades at about 2.13 percent than that by CI method.Although a little improvement than CI method, IEAmethod is only limited to the case of 2 sensors.C.2-sensor EI methodAs indicated before, both CI and IEA methods treated theaccurate and inaccurate estimates equally. When the mutualinformation is treated as exclusive one, the estimates become “over-confident”and incorrect. Thus, how to cope with thecorrelation of two local estimates is a main difficult ofdecentralized state-estimate algorithms. To overcome it, [17]proposed a novel fusion method, EI method, which containstwo steps. First, it derived a fusion estimate which isrepresented in a mutual mean and covariance of the mutualinformation. Secondly, solve the optimal mutual mean andcovariance by maximizing the effect of the mutual information.The key difference between EI and CI/LEA methods lieson that, the former uses only the received exclusiveinformation to update the current estimate while the latter resembles to an agreement, i.e., a combination of two local estimates.Apparently, the 2-sensor EI method is better than 2-sensorCI and other methods. However, the critical proof in [17] is notstrict, which aims at proving the equivalence of the twooptimization problems which minimizing the trace and natural logarithm of determinant of the mutual covariance respectively.D.Focus of this paperFrom above analysis, LEA and EI methods will get estimates with better accuracy than CI method does in 2-sensor case. But CI method is capable of fusing any probablity density functions, while EI method is restricted for only Gaussian distributions. Furthermore, CI method can deal with the n-case while LEA and EI methods couldn’t.To sum up, EI method is the best algorithm for 2-sensor case, and CI method is the basic algorithm for general case. How to overcome the shortcomings for CI and EI adressed above is a meaningful topic, which is the focus of this paper.II.MAIN RESULTS-P ART 1: FOR CI METHODA.Convexity of Cost Funtion in CI algorithmIn the case of known correlation, if the local estimates are consistent, the CI-based fusion algorithm is given by11111ˆˆ,,m mi i i i ii ix P P x P Pωω---====∑∑ (1) where ˆ,i ix P denote the i-th estimate and its covariance, ˆ,x P denote the fused estimate and fused covariance, andiωis the i-th weight satisfying 1,0 1.i iiωω=≤≤∑Consider the optimization problem:12min().:(,,,)Tm mfs tωωωωω=∈Ω, (2)where{}1|1,0,1~mmm i iiu R u u i m+=Ω∈=≥=∑.The cost function always takes the following forms:()()f tr Pω= (3)()ln(det())f Pω= (4) Next analyze the property of CI algorithm. From (1), we get111()0,ii iPP PPωω---∂∂==∂∂(5) Expand the first equation of (5) to result in1iiPPP Pω-∂=-∂(6)1()()iitr Ptr PP Pω-∂=-∂,1~i m= (7)21111()(())j i i ji jtr Ptr P P PP P PP Pωω----∂=+∂∂(8)Lemma 1. Suppose1()(,,)n nij nA a a a R⨯==∈, 1(,,)ndiagλλΛ= , then 221()nTi iitr A A aλ=Λ=∑ . (9)The proof of Lemma 1 is easy and we omit it.From Lemma 1, we can show the convexity property of CI algorithm.Theorem 2.For CI algorithm (1), the cost functions defined by (3) and (4) are both convex function with respect toω, i.e.22()0i jffωωω⎡⎤∂∇≥⎢⎥∂∂⎢⎥⎣⎦. (10) The proof is presented in Appendix A.From Theorem 2, when ()()f tr P ω=, the minimizing problem (2) is a classical restricted convex optimization problem, and its optimality condition is in such a simple form:1()()i tr PP P tr P -=, if 0i w >, 1()()i tr PP P tr P -≥, if 0i w =.Thus optimization problem (2) could be solved by high-efficiency algorithms such as Newton and quasi-Newtoniteration, and also by optimization software like SeDuMi ([15]), YALMIP ([16]), etc. B. 2-sensor CI AlgorithmBefore giving a novel 2-sensor CI algorithm, we first rewrite the fusion covariance in a simple form when 2m =.Theorem 3.Denote 1TP LL=,12T T L P L QDQ -=, where L is a lower triangular matrix, Q is a orthogonal matrixand 1(,,)n D diag d d = is a diagonal matrix. Let()ij A LQ a ==, 21,1,,ni ki k c a i n ===∑ . Then, we have1121(),()TTP AD A PP P AD A ωω---==, (11)1112()()T PP P AD DD A ωω---=, (12)where ()(1)D I D ωωω=+-.The proof is presented in Appendix B. Corollary 4. If ()()f tr P ω=, then 1()(1)njj jc f dωωω==+-∑,21(1)()((1))nj j j jc d f d ωωω=-'=-+-∑, 2312(1)()((1))n j j j j c d f d ωωω=-''=+-∑ (13) Corollary 5. If ()ln(det())f P ω=, then1()2ln det()ln (1)nj j f A d ωωω=⎡⎤=-+-⎣⎦∑ (14a)1(1)()((1))nj j j d f d ωωω=-'=-+-∑, 221(1)()((1))nj j jd f d ωωω=-''=+-∑ (14b)Denote min 1min j j nd d ≤≤=, max 1min j j nd d ≤≤=. Further analysis willgive the following theorem without proof.Theorem 6. For cost function (3)~(4), the minimizingproblems (2) have the following common properties: (a) If max 1d ≤, then 21P P ≥,()0f ω'≤, and ()f ω achieves its minimum at 1ω=, and 1P P =; (b) If min 1d ≥, then 21P P ≤,()0f ω'≥, and ()f ω achieves its minimum at 0ω=, and 2P P =;(c) If min max 1d d <<, then ()f ω' has at most one zero point, and ()f ω'has a unique zero point if and only if (0)0f '≤and (1)0f '≥.Combine Theorem 3 and 6 to get the following high efficient algorithm for cost function (3) and (4):2-sensor CI algorithmStep 1. Compute Cholesky decomposition 1TP LL=; Step 2. Do orthogonal decomposition 112T T L P L QD Q ---= to get min max ,,,j Q d d d . If min 1d ≥, output 20,opt P P ω==, stop; if max 1d ≤, output 11,opt P P ω== , stop; otherwise, go to step 3;Step 3. Compute21,,(0),(1)nj kj k A LQ c a f f =''==∑.If (0)0f '≥, output 20,opt P P ω== , stop; if (1)0f '≤,output 11,opt P P ω==, stop; otherwise, go to step 4; Step 4. Select an initial 0ω, compute the zero point of ()f ω' by Newton method.Step 5. Calculate P by (11), output ,opt P ω, stop.The highlight of the 2-sensor CI algorithm lies on that one can express the cost function and its first and second derivatives analytically in scalar form with respect to the weight omega, which enables us to find the optimal solution by using Newton, quasi-Newton or other high-efficient algorithm.III. MAIN RESULTS -P ART 2: FOR EI METHODAs indicated above, the 2-sensor EI method proposed in [17] contains two steps. We will review the first step, and for the second step we will give a new proof for the 2-sensor EI algorithm ’s optimality when cost function is the natural logarithm of determinant of the mutual covariance.A. Fusion in terms of mutual mean and covarianceSuppose all the probability density function (PDF) estimates are Guassian with mean []i x k and covariance []i P k , that is, ([])(,[],[])i i i p x k N x x k P k =, and ([])(,[],[])f f f p x k N x x k P k =. The estimates arebased on information which could be (partly) mutual. We denote the mutual information by d γ and their own excusive information by i d and j d . i p and j p could be rewritten by applying the Bayes ’ rule as follows:(|)(|)()(|,)(|)(|)i i i iRp x d p d x p x p x d d p x d p dx dxγγγ==⎰, (15a)(|)(|)()(|,)(|)(|)j j j jRp x d p d x p x p x d d p x d p dx dxγγγ==⎰. (15b)The fused PDF becomes (|,)(|)()(|)(|,,)(|,)(|)()(|)i j i j f i j iji j RRp x d d p d x p x p d x p p x d d d p x d d p dx p x p d x γγγ===⎰⎰. (16) Assume (|)(,,)j j j p d x N x U μ=,(|)(,,)p x d N x γγ=Γ, then we have11111(),()j j j i j j P U x P U μγ-----=+Γ=+Γ , (17) 11111(),()f i jf f i i jj P P U x P P x U μ-----=+=+ , (18)which yields111111,j j j j j j U P U P x μγ------=-Γ=-Γ (19) 1111111(),()f i jf f i i jj P P P x P P x P x γ-------=+-Γ=+-Γ (20)B. Determine mutual covariance Γand mean γThe criterion considered in this paper is to maximize the mutual information using cost function (4), i.e.,10,0,arg min ln .:1,i jTP P s t x x x εε-ΓΓ≤∀∈⋃ , (21)where ()(){}1,()|1TP x x x P x μεμμ---≤ denotes the sub-level-set of Gaussian PDF (,,)G x P μ.Lemma 7. Suppose ,A B are symmetric and positive, then (a)A B ≥if and only if 11A B --≤;(b) 0,0,A B εε⊇ if and only if A B ≥.The proof is easy and we omit it.According to Lemma 7, problem (21) becomesarg min ln ,.:,i j s t P P ΓΓ≥Γ≥ (22) Now we present the main results for EI algorithm. Theorem 8.Decompose matrices ,i j P P as T i i i iP L D L =, 1/211/2T T i i j i i j j jD L P L D Q D Q ----=, where i L is a unit lower triangular matrix, j Q is orthogonal and ,i j D D are diagonal. Denote 12(,,)j n D diag d d d = ,()1,2,,max(,1)k k ndiag d =Λ= ,Then 1/21/2T T i ij ji iL D Q Q D L Γ=Λ is the optimal solution of (21).The proof is presented in Appendix C.Corollary 9. When i L is a orthogonal matrix, then1/21/2T Ti i j j i iL D Q Q D L Γ=Λ is still the optimal solution of (21). It worth noting that the EI algorithm is consistent, i.e.,f i P P ≤and f j P P ≤. We omit the calculation for mutual mean γ, see [17] for details.IV. S IMULATION RESULTSA. 2-sensor CI algorithmFive examples are designed to verify the efficiency of the 2-sensor CI algorithm. Simulation results are obtained for both cases, i.e.,()ln(det())f P ω= and ()()f trace P ω=, whose curves are shown in Fig. 1~ 5 respectively for the purpose of comparison and verification. Example 1. 12101000,010001P P ⎡⎤⎡⎤==⎢⎥⎢⎥⎣⎦⎣⎦. Obviously,we have121210,,1,100,0.01,100.010L Q I c c d d ⎡⎤======⎢⎥⎣⎦For both cases, 1.980200.5,0 1.9802opt CI P ω⎡⎤==⎢⎥⎣⎦，andln(det())1.3664,() 3.9604CI CI P trace P ==. Example 2.1210050364110,50250110775P P ⎡⎤⎡⎤==⎢⎥⎢⎥⎣⎦⎣⎦. Obviously,12P P ≤, thus for both cases 11,opt CI P P ω==, and ln(det())10.0213,()350.CI CI P trace P ==Example 3.12100509370.5,.5025070.5154.25P P ⎡⎤⎡⎤==⎢⎥⎢⎥⎣⎦⎣⎦Obviously, we have12720.5020.595.75P P -⎡⎤-=≥⎢⎥-⎣⎦. For both cases, the results are 20,,opt CI P P ω==andln(det())9.1458,()247.25CI CI P trace P ==.Figure 1. Curves of two cost functions for Example 1.Figure 2. Curves of two cost functions for Example 2.Figure 3. Curves of two cost functions for Example 3.Figure 4. Curves of two cost functions for Example 4.Figure 5. Curves of two cost functions for Example 5.Example 4.12125509040,.5010040150P P ⎡⎤⎡⎤==⎢⎥⎢⎥⎣⎦⎣⎦When ()ln(det())f P ω=, since (1)0f '≤, we have11,,ln(det())9.2103.opt CI CI P P P ω=== When ()()f trace P ω=, the result is110.9162 45.47980.6747,,()221.3583. 45.4798 110.4422opt CI CI P trace P ω⎡⎤===⎢⎥⎣⎦Example 5.123020.8,.00.40.81P P -⎡⎤⎡⎤==⎢⎥⎢⎥-⎣⎦⎣⎦When ()ln(det())f P ω=, we have 2.09900.26210.5644,,ln(det())0.0228.0.26210.5201opt CI CI P P ω-⎡⎤===⎢⎥-⎣⎦When ()()f trace P ω=, the results areTABLE 1. (N EWTON ) ITERATION NUMBER WHEN ()ln(det())f P ω= Iteration number Example 1Example 5ω()f ω' ω()f ω' 0 0.2000 -3.5252 0.0000 -1.1667 1 0.3458 -1.3044 0.3404 -0.3627 2 0.4742 -0.1990 0.5477 -0.0262 3 0.4999 -0.0010 0.5645 -0.0000 40.5000 -0.00001.9693 -0.36620.4037,,ln(det())2.5676.-0.3662 0.5983optCI CI P P ω⎡⎤===⎢⎥⎣⎦According to our algorithm, one can judge the optimal solution without iteration when (0)0f '≥ or (1)0f '≤ . This is verified by the above examples. For the case (0)0(1)f f ''<<, it is seen from Table 1 that it only needs 2~4 iterations to get a high-accuracy solution. B. 2-sensor EI algorithmOur 2-sensor EI algorithm is applied to all the five examples when ()ln(det())f P ω=, and only show the 5-th due to the length limitation, with initial states 1(1,2)T x =- and2(2,1)T x =--. The results by EI and CI are3.5737 -0.5991,(-0.18,-0.7676),-0.5991 1.0257T γ⎡⎤Γ==⎢⎥⎣⎦1.4263 -0.2009(-0.82,-2.2324),,-0.2009 0.3743T EI EI x P ⎡⎤==⎢⎥⎣⎦and (-0.4453,-1.8819),0.5644,T CI opt x ω==2.09900.2621,ln(det())0.02280.26210.5201CI CI P P -⎡⎤==⎢⎥-⎣⎦. Fig.6 and 7 show the difference of results of CI and EI algorithms. Obviously, the 2-sensor EI algorithm gives a tighter estimate than the 2-sensor CI algorithm. It should be remarked that CI algorithm may cost much more computation than EI since CI keeps on searching the optimal solution while EI doesn ’t.Figure 6. The two initial, fused results according to CI and EI.Figure 7.The ellipses in Fig. 6 when centered at origin.V.C ONCLUSIONSThis paper makes some progress on CI and EI algorithms. For CI algorithm, we give two explicit expressions for the first and second derivatives of the trace, which enables us to use high-efficient algorithm (such as Newton or quasi-Newton algorithm) but not simple linear search algorithm, to find the optimal solution. Further, we present a fast 2-sensor CI algorithm by expressing the cost functions and their first and second derivatives in scalar form analytically. For 2-sensor EI algorithm, we give a new proof for its optimality, which partly fills in the gap in [17]. Simulation results show that both the 2-sensor CI and EI algorithm are efficient and the latter gives a tighter estimate.It is worth nothing that most applications of EI algorithm limit to the 2-sensor case ([17]). How to extend the 2-sensor EI algorithm to n-sensor case is still an open problem. 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Bosch, “State fusion with unknowncorrelation: Ellipsodial intersection,” Proceedings of the 2010 American Control Conference, Baltimore, Maryland, USA, pp. 3992-3997, 2010.[18]Yan Zhou, Jianxun Li, “Data fusion of unknown correlations usinginternal ellipsodial approximation,”Proceedings of the 17th World Congress, The International Federation of Automatic Control, Seoul, Korea, July 6-11, pp. 2856-2860, 2008.A PPENDIXA.Proof of Theorem 2We first prove (10) for()()f tr Pω=.,0mz R z∀∈≠, ithas2()2()Tz f z tr PGPGP∇=, where()11mj jjG z P-=∑. Note that G is symmetric and P is symmetric and positive, let 1(,,)TnP Qdiag Qλλ= , where Q is orthogonaland0,1~ii nλ>=. Denote1(,,)nA PGQ a a=, it follows from Lemma 1 that2121()((,,))nTn i iitr PGPGP tr Adiag A aλλλ===∑, which results in 2()0Tz f z∇≥. The equality holds if and only if 0,1~ia i n==, ,which is equivalent to G=0 by definition of A, that is 11mj jjz P-==∑.Next we prove the convexity of ln(det())P in two steps. Step 1. Prove ln(det())X is concave with respect to nX S++∈, where nS++denotes the set of symmetric and positive n n⨯matrix. In fact, for,nA B S++∈,[0,1]λ∀∈, let TA LL=, 1T TL BL QDQ--=, where L is a low triangular matrix and Q isorthogonal,1(,,)nD diag d d= with 0id> for all i.Notice that ln()⋅ is concave, so we have11ln det((1))ln det()ln((1))ln det()(ln1(1)ln )ln det()(1)ln det()ni i mii A B A d A d A B λλλλλλλλ==+-=++-≥++-=+-∑∑Step 2. Prove ln(det(()))P w is concave with respect to ω. ,,[0,1]m u v λ∀∈Ω∈, let (1)u v ωλλ=+-.By substituting1u A P-= and1v B P-=in Step 1 to get1111ln det()ln det((1))ln det()(1)ln det()ln det()(1)ln det()u v u vu v P P P P P P P ωλλλλλλ----=-+-≤---=+-,which completes the proof. B. Proof of Theorem 3Since A LQ =, it has 11()T T T A Q L L Q ---== and11112(1)()T TP L L P A D A ωωω------=+-=,i.e.,1()TP AD A ω-=. Similarly,121()TPP P AD A ω--=, and1112()()T PP P AD DD A ωω---=, which completes the proof.C. Proof of Theorem 8Consider reversible linear transformation T:1/2i i j x L D Q z =. Simple calculation yields 111,TTTTi j j x P x z z x P x z D z ---==, which means 0,0,0,0,~,~ij j P I P D εεεε. It holds that11T T x x z B z --Γ=, where11/211/21/21/2,T T T Tj i i i i j i i j j i i B Q D L L D Q L D Q BQ D L ------=ΓΓ= .Thus11/211/21ln ln ln ln ln ln T Tj i i i ij i iB Q D L L D Q D D -------=Γ=+Γ=Γ-,which lead problem (21) to bearg min ln ,.:,jB s t B I B D ≥≥ (23)Now we prove ()max(,1)k diag d Λ= is the optimalsolution of (23), which means 1/21/2T Ti i j j i iL D Q Q D L Γ=Λ is the solution of original problem (21). The proof is divided into three cases below.(a) if max 1k kd ≤, then 0,0,j D I εε⊆, (23) becomesarg min ln ,.:B s t B I ≥.Obviously,I Λ= is the optimal solution of (23).(b) if min 1k kd ≥, then 0,0,j D I εε⊇and j D Λ=, (23) becomesarg min ln ,.:j B s t B D ≥.Since j C B D - is semi-positive, it has1/21/21/21/21/21/2ln ln ln ()ln ln ln j j j j jj jj jB DCD I D CD D D I D CDD ----=+=+=++≥,Thus j D Λ= is the optimal solution of (23). (c) if min 1max k k d d <<, then ,j jD I I DE E E E ⊄⊄.Note that (max(,1))k diag d Λ=. Without loss of generality,Assume()j k D diag d =with 1,1~k d k m>=and1,1,,k d k m n ≤=+ , and ()T s Qdiag Q λΓ=with12n λλλ≥≥ and ()1n Q q q = is orthogonal. Let s edenote the vector whose s-th element is 1 and others are 0. Since T I Q Q =, it has ,j Tk k k k D e Q q d e E =∈,2111()()T T Tksk k k k k ksk k s sq d e d e d e Qdiag Q e d λλ--≥Γ==∑,which means ()12k kss s d q λ-≤∑. Notice that ln x -isconvex and21ks sq =∑, we have 221~ln ln ln ks kks s s k m ss q d q λλ=⎛⎫≤-≤ ⎪⎝⎭∑∑∑, which yields1~1~221~1~ln ln max(,1)ln ln ln ln ln kkk nk mks s ks sk m s s k m s sddq q λλλ====Λ==⎛⎫≤= ⎪⎝⎭≤=Γ∑∑∑∑∑∑∑. The proof is completed.。
