Hamiltonian Structures of Multi-component Constrained KP Hierarchy

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Calcium Ions Are Involved in the Unusual Red Shift of the Light-harvesting 1 Qy Transition of the Co

Calcium Ions Are Involved in the Unusual Red Shift of the Light-harvesting 1Q y Transition of the Core Complex in Thermophilic Purple Sulfur Bacterium Thermochromatium tepidum *□SReceived for publication,January 10,2008,and in revised form,March 6,2008Published,JBC Papers in Press,March 10,2008,DOI 10.1074/jbc.M800256200Yukihiro Kimura ‡1,Yu Hirano ‡,Long-Jiang Yu ‡,Hiroaki Suzuki ‡,Masayuki Kobayashi §,and Zheng-Yu Wang ‡2From the ‡Faculty of Science,Ibaraki University,Mito 310-8512,Japan and §Ariake National College of Technology,Omuta,Fukuoka 836-8585,JapanThermophilic purple sulfur bacterium,Thermochromatium tepidum ,can grow at temperatures up to 58°C and exhibits an unusual Q y absorption at 915nm for the core light-harvesting complex (LH1),an ϳ35-nm red shift from those of its meso-philic counterparts.We demonstrate in this study,using a highly purified LH1-reaction center complex,that the LH1Q y transition is strongly dependent on metal cations and Ca 2؉is involved in the unusual red shift.Removal of the Ca 2؉resulted in formation of a species with the LH1Q y absorption at 880nm,and addition of the Ca 2؉to the 880-nm species recovered the native 915-nm form.Interchange between the two forms is fully reversible.Based on spectroscopic and isothermal titration cal-orimetry analyses,the Ca 2؉binding to the LH1complex was estimated to occur in a stoichiometric ratio of Ca 2؉/␣␤-sub-unit ؍1:1and the binding constant was in 105M ؊1order of magnitude,which is comparable with those for EF-hand Ca 2؉-binding proteins.Despite the high affinity,conformational changes in the LH1complex upon Ca 2؉binding were small and occurred slowly,with a typical time constant of ϳ6min.Replacement of the Ca 2؉with other metal cations caused blue shifts of the Q y bands depending on the property of the cations,indicating that the binding site is highly selective.Based on the amino acid sequences of the LH1complex,possible Ca 2؉-bind-ing sites are proposed that consist of several acidic amino acid residues near the membrane interfaces of the C-terminal region of the ␣-polypeptide and the N-terminal region of the ␤-polypeptide.Thermochromatium (Tch.)tepidum is a thermophilic purple sulfur photosynthetic bacterium that can grow at temperatures up to 58°C,the highest known for purple bacteria (1,2).Thecrystal structure of the reaction center (RC)3for this bacterium has been determined (3),revealing structural features similar to those of the RCs from other purple bacteria (4,5).The antenna system in Tch.tepidum contains two types of light-harvesting complex,LH1and LH2.The LH1complex has been shown to be tightly associated with RC to form the core complex (6,7),LH1-RC,and the LH2complex is supposed to be located in the periphery of the core complex (8,9).Although a number of high resolution structures are available for the LH2and LH3(a spec-troscopic variant of LH2)complexes (10–12),information on the molecular structure of the LH1complex has been lacking.This information is essential for studying its biolog-ical functions in terms of excitation energy transfer and reduced ubiquinone transport.The interaction mode be-tween LH1and RC complexes is also of particular interest because the RC has a structure of 2-fold symmetry (3,4,13),whereas low resolution structures of LH1show a pseudo 14-or 16-fold symmetry depending on the species (14–20).The sym-metry issue needs to be taken into account when addressing the possible contacting sites between the LH1and RC and the fac-tors determining the specific orientation of RC inside the LH1ring.The crystallographic structure of the LH1-RC complex from Rhodopseudomonas (Rps.)palustris at 4.8Åresolution shows a slightly elliptical LH1ring composed of 15pairs of the ␣␤-polypeptides with a gap that is associated with an unknown protein W and is proposed to be adjacent to the ubiquinone-binding site in RC (17).However,to gain insight into a more detailed picture of the configuration of LH1-RC,structural information at higher resolutions is required.Recently,the LH1-RC complex from Tch.tepidum has been crystallized (21).We have shown that the purified LH1-RC complex of Tch.tepidum is extremely stable (21,22).Besides the thermal stabil-ity,another striking feature is the unusual red shift of the Q y transition of the LH1complex.Although the core complex con-tains only a -type bacteriochlorophyll (BChl),the same as those found in most other purple bacteria,spectroscopically the LH1complex of Tch .tepidum exhibits a Q y absorption at 915nm,ϳ35nm red-shifted from those of its mesophilic counterparts.The reason for this red shift has been unclear (2).Two other LH1complexes from purple bacteria strain 970and Roseospi-*This work was supported by Grants-in-aid for Scientific Research on PriorityAreas “Structures of Biological Macromolecular Assemblies”from the Min-istry of Education,Culture,Sports,Science,and Technology of Japan.The costs of publication of this article were defrayed in part by the payment of page charges.This article must therefore be hereby marked “advertise-ment ”in accordance with 18U.S.C.Section 1734solely to indicate this fact.□SThe on-line version of this article (available at )contains supplemental Figs.S1–S4and supplemental data .1Present address:Organization of Advanced Science and Technology,Kobe University,1-1Rokkodai,Nada,Kobe 657-8501,Japan.2To whom correspondence should be addressed.Tel./Fax:81-29-228-8352;E-mail:wang@mx.ibaraki.ac.jp.3The abbreviations used are:RC,reaction center;BChl,bacteriochloro-phyll;LH,light-harvesting;ITC,isothermal titration calorimetry;OG,n -octyl-␤-D -glucopyranoside.THE JOURNAL OF BIOLOGICAL CHEMISTRY VOL.283,NO.20,pp.13867–13873,May 16,2008©2008by The American Society for Biochemistry and Molecular Biology,Inc.Printed in the U.S.A.by guest, on January 5, 2011 Downloaded from /content/suppl/2008/03/11/M800256200.DC1.htmlSupplemental Material can be found at:rillum parvum930I were reported to exhibit Qy absorptions at963nm(23)and909nm(24,25),respectively.The large red shifts were explained in terms of enhanced exciton interaction among the BChl a molecules and specific interactions between BChl a and LH1polypeptides.Recent studies on the LH com-plexes show that small differences in the primary structure of the apoproteins can alter the spectroscopic properties(12, 26–29).The hydrogen-bonding pattern and local environment were shown to have an effect on the conformation of the cou-pled BChl a molecules,and small changes in the conformationcould result in a significant shift of the Qy transition.In a preliminary study,we found that the LH1Qy absorptionof Tch.tepidum core complex appeared to be affected by vari-ous cations used in the experiment(6).However,the LH1-RC samples used in the previous work were partially damaged,and the LH2along with other two unknown proteins were not com-pletely removed.We demonstrate in the present work using highly purified samples that the calcium ions are stronglyinvolved in the unusual red shift of the LH1Qy transition.Aspecies with absorption maximum at880nm can be obtained from the LH1915-nm complex by removing the Ca2ϩ,and the native LH1915-nm form(B915)can be recovered from the880-nm species(B880)by addition of Ca2ϩ.Interchange between the two forms is reversible.Effects of the Ca2ϩon the properties of the LH1-RC complex from Tch.tepidum were analyzed by absorption,circular dichroism(CD)spectros-copies,isothermal titration calorimetry(ITC),and sucrose den-sity gradient ultracentrifugation analyses.The Ca2ϩbinding constants were estimated both spectroscopically and thermo-dynamically.The rates of conformational changes upon the Ca2ϩbinding were calculated from a pseudo first-order kinetics analysis.By comparison of the amino acid sequences of LH1complexes,possible Ca2ϩ-binding sites have been proposed.The Ca2ϩ-dependent spectral behavior of the LH1 complex has not been reported for other purple photosyn-thetic bacteria and is of great interest and significance in understanding the structural and functional roles of Ca2ϩin this thermophilic organism.MATERIALS AND METHODSSample Preparation—Cells from Tch.tepidum were cultured anaerobically at48°C for7days as described previously(7). Isolation and purification followed the same procedure described before(21)except that n-octyl-␤-D-glucopyranoside (OG)was used as detergent in solubilization and chromatogra-phy.The harvested cells were disrupted by sonication in20m M Tris-HCl buffer at pH8.5.Chromatophores were first treated with0.35%(w/v)lauryldimethylamine N-oxide at25°C for60 min followed by centrifugation at150,000ϫg for90min.The pellet was further treated with1.0%(w/v)OG and centrifuged at150,000ϫg for90min to extract the LH1-RC components. The supernatant was loaded onto a DEAE anion-exchange col-umn(Toyopearl650S;Tosoh Corp.)equilibrated at4°C with 20m M Tris-HCl buffer,pH7.5,containing0.8%(w/v)OG.The LH1-RC fraction was eluted by a linear gradient of NaCl from50to125m M,and peak fractions with a ratio of A915/A280over2.10were collected.The native LH1-RC complex was passed through a sizeexclusion column(Sephadex G25M PD10;GE Healthcare)to remove excess salts,followed by incubation at0°C for3h in darkness in the presence of1.25m M EDTA.The sample solu-tion was extensively washed with a buffer containing20m M Tris-HCl,pH7.5,and0.8%(w/v)OG to remove excess of EDTA.The final concentration of EDTA in the solution was estimated to be lower than1n M.The resulting core complex showed an LH1Qyabsorption maximumϳ880nm.To exam-ine the effects of metal cations,the B880sample was incubated at0°C overnight after adding various metal cations.Sucrose density gradient centrifugation analysis was conducted by ultracentrifugation at150,000ϫg and4°C for12h under a 10–40%(w/v)continuous gradient of sucrose concentration in a buffer containing20m M Tris-HCl,pH7.5,and0.7%(w/v)OG in the presence or absence of1m M EDTA.Spectroscopic Measurements—Absorption spectra were recorded on a Beckman DU-640spectrophotometer at room temperature.Kinetic analysis for the recovering process from B880to B915upon the addition of Ca2ϩwas performed by monitoring the increase of the absorbance at937nm.CD spec-tra were recorded on a Jasco J-720w spectropolarimeter in the ranges of200to300and400to1000nm.The measurements were conducted under the conditions of20nm/min of scan speed,1.0nm bandwidth,and2s of response time.For the CD measurement between200and300nm,data were averaged over10scans in order to improve the signal-to-noise ratio, while for the measurements between400and1000nm the spectra were obtained from one scan.The molar extinction coefficient of BChl a dimer in a native␣␤-subunit was estimated to be270 m MϪ1cmϪ1by extracting BChl a molecules from purified LH1-RC complexes with acetone/methanol(7/2,v/v)and using ⑀770ϭ76m MϪ1cmϪ1for the monomeric BChl a(30). ITC—ITC measurements were carried out on a MicroCal VP-ITC microcalorimeter at25°C.The LH1-RC samples were dissolved in20m M Tris-HCl,pH7.5,and0.7%OG at250␮M. Following thermal equilibration at25°C,a total of14injectionsof50m M CaCl2dissolved in the same buffer were made in1-␮l aliquots into the cell from the syringe using a300-rpm stirring speed.An injection delay of10min was utilized to allow for the baseline to return after each injection.The titration data were deconvoluted based on a binding model containing either one or two sets of noninteracting binding sites by a nonlinear least squares minimization method(31)performed using the Micro-Cal Origin ITC software.RESULTSFirst we confirmed the effect of various metal cations on the absorption spectrum using highly purified LH1-RC complex from the Tch.tepidum(supplemental Fig.S1).The native corecomplex(B915)exhibited LH1Qyabsorption at914nm asreported previously(21).The Qyband was retained at the orig-inal position only in the presence of Ca2ϩand blue-shifted by the addition of other cations.The results indicate that the native LH1form of the Tch.tepidum is sensitive to the coexist-ing metal cations.To investigate whether the B915species is bound to metal ions,metal chelators were employed.Fig.1shows an absorp-Ca2؉Binding to the LH1complex of Tch.tepidumby guest, on January 5, Downloaded fromtion spectrum obtained with the addition of EDTA to the B915species.The LH1Q y band was largely blue-shifted to 876nm with a spectral shape similar to those observed for the LH1complexes from other purple bacteria.This clearly indicated that metal ions are involved in the formation of the B915and removal of the cation by EDTA results in a conformational change of the BChl a molecules in the LH1complex,leading to a large blue shift of the LH1Q y band.Similar results were observed using different metal chelators,nitrilotriacetic acid and ethylene glycol-bis(␤-aminoethyl ether)-N ,N ,N Ј,N Ј-tet-raacetic acid (tetrasodium salt),suggesting that the blue shift was not caused by the specific interaction between EDTA and the core complex.We further investigated the effects of various cations on the EDTA-treated LH1-RC complex (B880).Typical results on the divalent cations are shown in Fig.1,and the results for monovalent cations are provided in supplemental Fig.S2.Red shifts with respect to the LH1Q y band (876nm)of the B880species were observed with additions of all cations.Among the divalent cations tested,only Ca 2ϩgave an absorp-tion spectrum closely resembling that of native B915species with the LH1Q y absorption maximum at 913.5nm.Addition of other cations resulted in small red shifts of the LH1Q y band to 885–889nm for the divalent cations and 879–884nm for the monovalent cations,respectively.The results strongly suggest that the B915species is induced from the B880species by the binding of Ca 2ϩto the LH1complex.The binding property of Ca 2ϩwas examined over a wide range of Ca 2ϩconcentrations.Fig.2shows the absorption spec-tra of the EDTA-treated core complex in the presence of Ca 2ϩat various concentrations.By increasing the Ca 2ϩconcentra-tion,the B915species was formed accompanied by a decrease of the B880species.There was an isosbestic point at 895nm,indi-cating that the B915recovered from the B880upon the addition of Ca 2ϩ.The Ca 2ϩ-dependent LH1Q y shifts were plotted in the inset as a function of the Ca 2ϩconcentration,revealing a sig-moid curve that saturated ϳ50␮M Ca 2ϩ.Kinetics of the Ca 2ϩbinding process at room temperature (25°C)was analyzed using the pseudo first-order model.Formation of the B915spe-cies was monitored at 937nm where we can observe the maxi-mum change in the intensity.Fig.3shows the time profile of the absorption change (⌬A )after addition of 5m M Ca 2ϩto the B880species of 7␮M LH1subunit.The absorbance at 937nm increased with time and reached maximum in ϳ30min.Because the concentration of Ca 2ϩwas much higher than that of the B880species,the binding process can be represented by a pseudo first-order kinetics model.The time-dependent absorp-tion change ⌬A (t )can be expressed by the first-order rate equa-tion as shown in Equation 1,⌬A ͑t ͒ϭ⌬A ϫ͓1Ϫexp ͑Ϫk ϫt ͔͒(Eq.1)where ⌬A is the final change of absorbance at 937nm,k is the rate constant for the conformational change of the LH1com-plex upon the Ca 2ϩbinding,and t is the time in minutes after the addition of Ca 2ϩ.By a least-square fitting of the data using Equation 1,the rise rate constant k of B915was determined to be 0.20min Ϫ1.This value did not change very much over aR e l a t i v e a b s o r b a n c eWavelength / nmFIGURE 1.Absorption spectra of the Ca 2؉-depleted LH1-RC complex (B880)from Tch.tepidum and the spectral changes induced by the addi-tion of various divalent cations (20m M )to the B880.Inset ,the expanded view of the LH1Q y region.B880(black ),Mg 2ϩ(blue ),Ca 2ϩ(red ),Sr 2ϩ(magenta ),Ba 2ϩ(green ),and Cd 2ϩ(purple ).R e l a t i v e A b s o r b a n c eWavelength / nmFIGURE 2.Spectral change of the absorptions after the addition of Ca 2؉at various concentrations to the B880complex of Tch.tepidum .All samples were incubated at 4°C overnight.The change of wavelength for the LH1Q y absorption peak is shown in the inset as a function of the Ca 2ϩconcentration.Concentration of the LH1subunit was 16␮M ,and residual EDTA in the solu-tion was lower than 1n M .0.150.100.050.00∆ A 937n m30252015105Time / min0.300.250.200.150.10R a t e c o n s t a n t k / m i n -1234561023456100[Ca 2+] / mMFIGURE 3.Time profile of the absorption change at 937nm after addition of 5m M Ca 2؉to the B880complex of Tch.tepidum .Concentration of the LH1subunit was 7␮M ,and the measurement was conducted at 25°C.The dotted line was obtained by pseudo first-order kinetic analysis using Equation 1.Inset ,a plot of apparent rate constants at different Ca 2ϩconcentrations.Ca 2؉Binding to the LH1complex of Tch.tepidumby guest, on January 5, 2011 Downloaded fromrange of2.5to100m M of the Ca2ϩconcentration(Fig.3,inset), suggesting that the pseudo first-order kinetic analysis is valid and the conformational changes from B880to B915upon the Ca2ϩbinding occur slowly under the experimental conditions. The binding affinity of Ca2ϩto B880was evaluated by a non-linear least-square fitting of the spectroscopic data.Fig.4shows the plots of⌬A at several wavelengths between930and960nm as a function of the Ca2ϩconcentration.These wavelengths were selected so as to minimize contribution from the RC that has absorption over750–950nm.Assuming that one LH1sub-unit of the B880species binds to one Ca2ϩto form one subunit of the B915species,the absorption changes⌬A observed can be expressed by Equation2(see supplemental data for details),⌬Aϭlϫ͑⑀915Ϫ⑀880͒ϫ͓͑1ϩC0KϩKx͒Ϫ͕͑1ϩC0KϩKx͒2Ϫ4C0K2x͖0.5͔/2K(Eq.2)where l is the light path length,⑀915and⑀880are molar extinc-tion coefficients of the LH1subunit in the B915and B880forms,respectively,C0is the initial concentration of subunit ofthe B880species,K is the Ca2ϩbinding constant,and x is the concentration of Ca2ϩ.By using the nonlinear least-square fit-ting of⌬A,the Ca2ϩbinding constant K was estimated to be6.5 (Ϯ0.3)ϫ105MϪ1.Fig.4shows a good agreement between the calculated result and the spectroscopic data.The value of the binding constant is comparable with those reported for typical EF-hand Ca2ϩ-binding proteins(32).The binding properties of Ca2ϩto the LH1complex were further examined by thermodynamic analysis.Fig.5A shows the ITC profile obtained with titration of Ca2ϩto the B880 rge exothermic change was observed for each injection,providing independent evidence for the strong Ca2ϩbinding.A remarkable feature in the thermogram is that each exothermic signal clearly revealed a two-phase change:a rapid return immediately after the injection and a slow recovery,as shown typically in the inset of Fig.5A.The two exothermic components can be interpreted in terms of a fast Ca2ϩbinding to the B880and a slow process of confor-mational rearrangement within the core complex,respec-tively.Based on a two-component analysis,the time con-stants were determined to be20and369s for the fast and slow processes,respectively.The latter is quite consistent with the rate constant observed from spectroscopic meas-urement(Fig.3).Fig.5B shows a plot of the integrated heat/ mol of injectant(Ca2ϩ)as a function of the molar ratio of Ca2ϩ/(B880subunit).The plot exhibited a sigmoid-like curve with a stoichiometrically equivalent pointϳ1.0,sug-gesting that the Ca2ϩbinding phenomenon can be described by a simple one-site model.From the model analysis,several thermodynamic parameters were obtained:the number of binding site/subunit nϭ0.97,the binding constant Kϭ9.6ϫ104MϪ1,and enthalpy change⌬HϭϪ4.0kcal molϪ1. Using these values,the changes of Gibbs free energy⌬G and entropy⌬S were evaluated to beϪ6.8kcal molϪ1and9.36cal molϪ1KϪ1,respectively,from Equation3,⌬Gϭ⌬HϪT⌬SϭϪnRT ln K(Eq.3) where T is the absolute temperature,n is the mole number,and R is the gas constant,8.315J molϪ1KϪ1.These thermodynamic parameters indicate that the Ca2ϩ-binding site has a high affin-ity with a1:1stoichiometric ratio and the Ca2ϩbinding is driven both enthalpically and entropically.∆A10101010[Ca2+] / Mparison between the measured and calculated absorption changes at selected wavelengths as a function of the Ca2؉concentra-tion.The experimental data were analyzed by the non-linear least-square fitting using Equation2.µcal/secTime (min)kcal/molofinjectant[Ca2+]/[αβ-subunit]FIGURE5.Isothermal titration calorimetric result of Ca2؉binding to the LH1complex.A,ITC titrations of50m M CaCl2into250␮M LH1B880␣␤-sub-units at25°C.B,binding isotherm of the titration as a function of Ca2ϩ/␣␤-subunit ratio.The solid line represents the best fit to a one-site model with the fitting parameters given under“Results.”The inset shows an expansion of the thermogram for the first injection.Ca2؉Binding to the LH1complex of Tch.tepidumby guest, on January 5, Downloaded fromStructural differences in conformation and assembly of the core complexes between the B915and B880species were inves-tigated by CD spectroscopy and sucrose density gradient cen-trifugation.Fig.6shows the CD spectra.As reported previously,the CD spectrum of B915was characterized by small and non-conservative signals in the Q y region (21),a typical feature for the core complexes from purple bacteria (33,34).Upon removal of Ca 2ϩ,the CD signals for the Q y and Q x bands were blue-shifted in accordance with the changes of absorption spec-tra,whereas other signals from the RC and carotenoids remained unchanged.These results support that Ca 2ϩ-dependent spectral changes originate from the change in con-figuration of the BChl a molecules in the LH1complex.We also examined the effect of Ca 2ϩon the secondary structure of the apoproteins of the LH1-RC complex by measuring the CD spectra in the far UV region (Fig.6,inset ).Spectra for both the B915and B880revealed almost the same shape and can be characterized by highly ␣-helical structures.This suggests that there was no change in the secondary structures of the polypep-tides.Sucrose density gradient ultracentrifugation exhibited a single band for both B915and B880at identical positions (sup-plemental Fig.S3),indicating that the B880is not a decomposed form of B915and the whole assembly of the core complex remains unchanged.All results described above were obtained using the puri-fied LH1-RC complex.In Fig.7,we show that similar phe-nomena can be observed for Tch.tepidum chromatophores.The chromatophore membrane contains a large amount of LH2complex,as revealed by the intensive absorption max-ima at 800and 850nm,and a small amount of LH1complex with the Q y transition at 915nm.In the presence of CaCl 2,there was no change in the absorption spectrum of the chro-matophore.However,upon the addition of EDTA or MgCl 2,the LH1915-nm band was shifted toward 880nm and merged with the LH2absorptions.The result implied that the Ca 2ϩ-binding site of the LH1complex exists in a hydro-philic part or at the membrane interface where metal cations and EDTA are readily accessible.DISCUSSIONThe present study demonstrated that the LH1Q y transition at 915nm of the core complex from Tch.tepidum is induced by the binding of Ca 2ϩto the LH1complex.Removal of the Ca 2ϩresulted in a blue shift of the LH1Q y transition to 876nm,the wavelength similar to those of other mesophilic purple bacteria.The B915native form can be recovered from the B880by the addition of Ca 2ϩ.The result indicates that Ca 2ϩis an indispen-sable cofactor for retaining the native structure of the Tch.tepi-dum LH1complex.The Ca 2ϩbinding property of the Tch.tepi-dum LH1complex is considered to be closely related to the living environment of this organism.Mammoth Hot Springs in Yellowstone National Park,from where the bacterium was col-lected (1),is known to contain rich mineral calcium carbonate deposited over millions of years to form thick layers of sedimen-tary limestone.This bacterium cannot grow in a Ca 2ϩ-depleted culture (data not shown).Through the spectroscopic and ITC analyses,binding of the Ca 2ϩto the LH1␣␤-subunit was found to occur in a one-to-one ratio with a binding constant of ϳ105order of magnitude.The high binding affinity is comparable with that for the well known EF-hand Ca 2ϩ-binding proteins (32,35).Generally,Ca 2ϩis classified as a hard acid and prefers to interact with a hard base.In the proteins with an EF-hand Ca 2ϩ-binding site,the ligands are mainly composed of oxygen atoms from acidic amino acid residues (Asp and Glu)as well as those from main chain car-bonyl groups and water molecules (35).Inspection of the pri-mary sequences of the Tch.tepidum LH1polypeptides revealed that three Asp in the ␣-polypeptide are located in the C-termi-nal region and are close to the BChl a -coordinating His residue (Fig.8)(6,7).Three similar residues (two Glu and one Asp)are found in the C terminus of the LH1␣-polypeptide from the mesophilic purple bacterium Allochromatium (Ach.)vinosum .However,a deletion exists in this region of the Tch.tepidum LH1␣-polypeptide that is thought to be essential for the for-mation of a Ca 2ϩ-binding site and the specific interaction between the polypeptide and pigment molecule.Positions of the deletion and the three Asp residues are estimated to be onC D / m d e gWavelength / nmFIGURE 6.CD spectra for the B915(broken line )and B880(solid line )com-plexes from Tch.tepidum in the 400–1000and 200–280nm regions (inset ).The spectra were normalized with respect to the absorption bands at 514and 280nm,respectively.R e l a t i v e a b s o r b a n c eWavelength / nmFIGURE 7.Effects of EDTA and metal cations on the absorption spectra of the chromatophores from Tch.tepidum .Inset ,an expanded view of the LH1Q y region:Native (solid line ),500m M Ca 2ϩ(dotted line )and Mg 2ϩ(dotted and dashed line ),and 20m M EDTA (dashed line ).Ca 2؉Binding to the LH1complex of Tch.tepidumby guest, on January 5, 2011 Downloaded fromthe membrane interface or in the hydrophilic part of the ␣-polypeptide.Similarly,several neighboring acidic residues are found in the N-terminal domain of the ␤-polypeptide,which might also serve as a potential binding site.Possibilities for other residues around the BChl a that are capable of hydro-gen bonding (27,29)and/or Ca 2ϩbinding need to be further investigated.Despite the high affinity between the Tch.tepidum LH1com-plex and Ca 2ϩ,the structural changes induced by the Ca 2ϩbinding occurred slowly as revealed from the spectroscopic and the ITC measurements.Therefore,the conformational change from B880to B915is the rate-limiting step under the experi-mental conditions.Previous study showed that the Tch.tepi-dum LH1-RC complex was purified in a monomeric form,with the LH1ring composed of 16␣␤-subunits (21).In the presentwork,the Ca 2ϩbinding to the LH1complex was estimated tooccur in a stoichiometric ratio of Ca 2ϩ/␣␤-subunit ϭ1:1.Thismeans that the B915form may not be completed from the B880form until all the 16binding sites are occupied by the Ca 2ϩions,because the Q y transition is a consequence of cooperative inter-action between the exciton states formed by the 32BChl a mol-ecules.In the case where there are less Ca 2ϩions than LH1-␣␤pairs,the original state of the exciton interaction in the LH1ring will be altered and become heterogeneous.As a result,a broadening in the Q y transition could be expected.Such broad-ening was actually observed at low Ca 2ϩconcentrations up to 3.2␮M (Fig.2and supplemental Fig.S4).At higher Ca 2ϩcon-centrations than 3.2␮M ,the full width at half maximum decreased rapidly with increasing Ca 2ϩconcentration.Taking into account the total molecular weight of ϳ330,000for the core complex,the slow structural change may be interpreted in terms of the large size of the membrane proteins associated with a considerable amount of the surrounded detergent mol-ecules.Other factors include the relatively long time required for the fine-tuning of the 16Ca 2ϩ-bound ␣␤-subunits within the LH1complex to form a stabilized conformation that can also contribute to the thermal stability of the Tch.tepidum .Generally,metal binding to proteins is known to be driven entropically because the dehydration enthalpy of divalent cat-ions is highly endothermic.However,overall enthalpy could become exothermic if the metal binding is thermodynamicallycoupled to the conformational change that eventually could result in a thermal stabilization of the protein (36).In calmodu-lins from bovine brain (37–39)and wheat germ (40),the Ca 2ϩbinding reactions were reported to be driven solely by a largefavorable entropy change despite unfavorable enthalpy change.In contrast,for the troponin C (41–43)and parvalbumins (32,44,45)the Ca 2ϩbinding reactions were mostly favorable in both entropy andenthalpy except for several specificsites where endothermic-andentropy-driven reactions occur in a similar way to the calmodulins (42–44).For the Tch.tepidum LH1-RC complex,the Ca 2ϩbinding is drivenboth entropically and enthalpically,similar to the cases of troponin C and parvalbumins.Therefore,the large exothermic changes of the Tch.tepidum core complex strongly support that the Ca 2ϩbinding to the LH1complex caused a conformational change that largely stabilized the structure of the LH1-RC complex.The Ca 2ϩbinding in calmodulins was reported to induce a conformational transition of random coils to ␣-helical struc-tures based on the results of CD measurement (46–48).For the Tch.tepidum core complex,the far UV-CD spectra did not show any change in the Ca 2ϩdepletion.This means that con-formational change of the LH1complex may occur mainly in the tertiary structure or the change in the secondary structure,if there is any,may be too small to be detected.Similar results were reported for troponin C from rabbit skeletal muscle (41).In EF-hand proteins (35),the Ca 2ϩ-binding sites are located in the loop of a helix-loop-helix motif and large conformationalchanges are accompanied with the binding of Ca 2ϩ.In contrast,the putative Ca 2ϩ-binding sites in the LH1complex are located at the terminal regions of the polypeptides.Furthermore,thenumber of acidic residues involved in the binding site and theirarrangement seem to be different from those of EF-hand Ca 2ϩ-binding proteins (32).As a result,the slight structural change in LH1can only be detected by its Q y transition that is extremely sensitive to the conformational change of the BChl a molecules.The changes in LH1Q y transition for different alkaline earthcations appeared to be partly related to the ionic radius,as the cations with larger (Sr 2ϩ,Ba 2ϩ)and smaller (Mg 2ϩ)ionic radiithan that of Ca2ϩshowed the cation-dependent Q y bands between 885and 889nm.However,substitution of Cd 2ϩthathas an ionic radius almost the same as that of Ca 2ϩalso induceda red shift to 887nm.The results indicate that the Ca 2ϩ-binding site in the LH1complex is highly selective and can distinguish the cations not only by the ionic radius but also by other prop-erties such as binding mode and coordination number (49)as well as the hard-soft nature.In this regard,the complex mech-anism of molecular recognition of the Tch.tepidum LH1is expected to be explored by detailed structural information,and the effort toward determining a high-resolution structure of the core complex is in progress (21).Acknowledgments—We thank Dr.J.Nakamura and Dr.Y.Sakaguchi of Nihon SiberHegner K.K.for the ITC measurements and M.Naka-mura and K.Horiguchi for technical assistance.MFTMNANLYKIWLILDPRRVLVSIVAFQIVLGLLI HMIVLST -D LNWL DD NIPVSYQALGKKMSPDLWKIWLLVDPRRILIAVFAFLTVLGLAIHMILLSTA E FNWL ED GVPAAAEQKSLTGLT DDE AK E FHAIFMQSMYAWFGLVVIAHLLAWLYRPWLNSSMTGLT E Q E AQ E FHGIFVQSMTAFFGIVVIAHILAWLWRPWL LH1-αTch. tepidum Ach. vinosum LH1-βTch. tepidum Ach. vinosum FIGURE parison of the amino acid sequences of LH1␣-and ␤-polypeptides from Tch.tepidum and Ach.vinosum (6,7).The sequences are aligned relative to the BChl a -coordinating histidine residues (shadowfonts ).Acidic amino acid residues in the C termini of ␣-polypeptides and N termini of ␤-polypeptides are indicated by bold fonts ,and a deletion in the Tch.tepidum ␣-polypeptide is indicated by an arrow .Underlined regions represent hydrophobic membrane-spanning domains predicted by SOSUI.Ca 2؉Binding to the LH1complex of Tch.tepidumby guest, on January 5, 2011 Downloaded from。

具有三个位势的广义耦合无色散可积方程的多孤子解(英文)

具有三个位势的广义耦合无色散可积方程的多孤子解（英文）2O11年5月第42巷第3期内蒙古大学(自然科学版)JournalofInnerMongoliaUniversity(NaturalScienceEdition)May2011V o1.42No.3ArticleID:1000—1638(2O11)03—0253-05 Multi—solitonSolutionsoftheGeneralized DispersionlessIntegrableEquationwithThreeCoupledPotentialsZhaqilao(C0llegeofMathematicsScience,InnerMongoliaNormalUniversity,Hohhot010022,Chin a)Abstract:AnewN—foldDarbouxtransformationofthegeneralizedcoupleddis—Dersionlessintegrableequationisderivedwiththeaidofthegaugetransforma—tionbetweencorresponding2×2matrixspectralproblemswiththreepoten—tla1s.AsanapplicationoftheDarbouxtransformation,N—solitonsolutionsof thegeneralizedcoupleddispersionlessintegrableequationareexplicitlygiven. Keywords:Darbouxtransformation;solitonsolution;coupleddispersionlessinte—grableequationCLCnumber:O175.2Documentcode:AIntr0ducti0n Thestudvofdispersionlesshierarchiesisoneofthemostprominentsubjectinthefieldofnon —linearscience,partlybecauseofdispersionlessequationsemergenceindiverseareasofmathe maticalandthe.retica1Dhysicssuchasquantumfieldtheory,conformalfieldtheory,stringtheory,sol itontheorv,etc.Inpast,thecoupleddispersionlessequationanditsgeneralizationhasalsoattracte da reatdealofinterestbecauseofitsniceintegrabilitystructureandsolitonsolutions. InRefs.[1—2],KonnoandKakuhataconsideredthegeneralformofthecoupleddispersionless integrableequation(CDIE)q+(rs)=0,,.一2r一0,s一25g一0(1) andobtainedthesingleanddoublesolitonsolutionsofCDIE(1)bytheinversescatteringtransf orm(IST).CDIE(1)hastwospecialcases:r=andr—.Theformercasewasfoundtobeequivalent tothesine—Gordone0uation[.andthelattertothePohlmeyer—gund--Reggeequation 们.Forr—s,CDIE(1)reducestothemodelwithtwopotentials.Inthepastyears,manyeffortshavebeen dedicatedtothereducedequationandrichfamiliesofexactanalyticalsolutionshavebeenobta inedeffectivemethods,suchasIST,Painlev6analysis∞,B/icklundtransformation(BT)"andDarbouxtransformation(DT)C783.However,toourknowledge,verylittleresearchhas beendoneonCDIE(1)viatheN—foldDTmethod. TheaimofthispaPeristoobtainN—solitonsolutionsofCDIE(1)withthehelpofspectral*Receiveddate:2011-05-06Foundationitem:SupportedbytheNaturalScienceFoundationofInnerMongolia(GrantNo 2009MS0108);the HighEducationScienceResearchofInnerMongoIiaAutonomousRegion(GrantNo?NJ10045)Biography:Zhaqila.(1971一),ma1e(Mongolian),anativeofOrdosofInnerMongolia,Viceprofesser,Ph?D?E_ mail:zhaqilao@.254内蒙古大学(自然科学版)2011矩problemsandtheDarbouxmatrixmethod(9-131.Tothisend,westartfromthecorresponding spec—tralproblemsofCDIE(1)andachieveanN—foldDTinsection1.ResortingtoDTinsection1.N—solitonsolutionsofCDIE(1)areobtainedinsection2.Mostimportantly,thedetailedstructure softhesinglesolitonanddoublesolitonsolutionsofCDIE(1)aregivenbothanalyticallyandgrap hical—ly.DarbouxtransformationInthissection,weshal1constructaThespectralproblemsofCDIE(1)whereU:=——iXnewN—foldDTforCDIE(1).aregivenas一U,≯===qfr5—q,V一(:--.r)十i(～0)(2)(3)Throughaa"lrectcalculation,thezerocurvatureequationUf—V+U,厂一【,『一0giverisetoCDIE(1).TheDarbouxtransformationisactuallyagau—ofspectralproblems(2).Itisrequiredthatalso一,一(+TU)T～,whereU一一iq尸sI—qlgetransformation(4)satisfiesthesamespectralproblemst—,V一(+)T-(5),V—O—+～o)Nowwediscussaconcretetransformation.LetmatrixTin(6)beintheformofT一丁c一A暑)whereN一1N～1N—lN--1A—一+∑A,B一∑B,c一∑CA一,D—一+∑D,0^=0=0k=OA,B,C^,Dkarefunctionsofzandt.A^,B,CandD^aregivenbyalinearalgebraicsystem N—l一一,∑(cI+3jD)—as117景0(6)(7)(8)怒蒋,,≤≤2N/1(9)1()+J1(J)'where一(l,2),一(1,)aretWObasicsolutionsofspectralproblems(2),and(≠J,:≠,asi≠)aresomeparameterssuitablychosensuchthatthedeterminantofcoefficientsfor (8)iSnonzero.Hence,ifwetakeDl,一--AN-一1.(10)therestA^,B^,CkandD^(1≤J≤N一1)areuniquelydeterminedby(8).Eq.(7)showsthatdetT()isa(一2N)th—orderpolynomialin,anddetT()=A()D()一B(,)C().Ontheotherhand,from(8)wehaveA()一一a,B(),CO,)一一D().,,,B+A,L∑h.代w第3期Zhaqila0Multi--solitonSolutionsoftheGeneralizedCoupled (255)Therefore,itholdsthatdet-/(,)一0whichimpliesthat,(1≤J≤2N)are2Nrootsofdet丁(),thatisdetT()===yⅡ(—)(11)whereyisindependentof.Inawaysimilartotheproofin[13],wecanverifythefollowingprop.sitions.Proposition1MatrixdeterminedbythesecondexpressionofEq.(5)hasthesameformasU.wherethetransformationformulafromtheoldpotentialsq,,-,intonewonesaregivenby ……一一邶一r+i一一瓮whereAN--1iscoefficientsof(8),i.e.AN一111…l__N÷1282),7….:LN+;;;:'.;j12N2N--N1….=L--2NN-.-2Ni.+(13)and△^…,△BareproducedfromGbyreplacingits(2N一1)th,2Nthcolumnwith(一,…, 一^.--2NN),respectively.△c1isproducedfrom△lbyreplacingits(2N一1)thcolumnwith(一,…,一2N).Accordingto(10),allJ(】≤≤2N)mustsatisfyconditions一一(一1,3,5,7,…).Proposition2MatrIxdefinedbythefourthexpressionofEq.(5)hasthesameformasV,m whchtheo1dpotentia1sg,r,saremappedintonewpotentials,r,,accordingtothesameDT (12).Propositions1and2showthatthetransformation(12)changesspectralproblems(2)intoan—otherspectra1problems(5),withU,Vand—U,一Vhavingthesameform.Thereforebothofthespec—tra1DroblemsleadtothesameCDIE(1),SO(12)istheDarbouxtransformationofCDIE(1)?F romPropositions1and2wehavethefollowingtheorem?Theorem1Let(q,r,s)beasolutionofCDIE(1).Thenfunction(,尸,)determinedbyDT (12)iSanewsolutionofCDIE(1).2SolitonsolutionsNowwecho.sethetrivialsolutionq.=fiz(|8isanarbitrarynon—zeroconstant)andro—s.一0,thecorrespondingcompatibIebasicsolutionofspectralproblems(2)canbewrittenas (.z,t,)一exp(一￡)00exp($i)(14)where￡一if1)!(一赢?Using(9),when,≠0andj≠0,wehave一exp(2岛),(:1,2)(15)Substituting(15)into(13),(12)denotesaunifiedandexplicitformulationofallk～solit.n.一lution(1≤是≤N),fr.mwhichitiseasytogetN—solitonsolutionsofCDIE(】)?Forsimplicity,wesha1】discl】sstwospectralcasesofN一1andN一2.256内蒙古大学(自然科学版)2011在Case1(N一1).Let,,,(一1,2)and2=::--A1.From(12),wehaveanexplicitsolutionof CDIE(1)asfollowing—fix-一一一一c16)'一一'where一lexp(2￡-i一蠢_1'2).If2一一1andl,2arepureimaginaryparameters,l,2arerealparameters,thesolutionde—terminedby(16)isthesinglesolitonsolutionq一—Itanhi,r一一sechi,s一一1sechi,一砉(￡一2Za12X)(17)Case2(N一2).Let,J(一1,2,3,4)and2一--;t1,4一--A3.From(12),wehaveanewso—lutionofCDIE(1)qz一卢z+i,r一i,s一i等c8whprA1一2j3 AA.△c1一一_『一一一一l一一3i一4l2iA3I23 AB.=一——.I_一where(一1,2,3,4)aredeterminedin(15).If,uz一--/zl,/14一一3and,(一1,2,3,4)arepureimaginaryparameters,J(一1,2,3,4)are realparameters,thesolutiondeterminedby(18)isthedoublesolitonsolution(seeFig.1).-4020(a】(b)(c)Fig.1Doublesolitonsolution(18)withr—s一0,一A1一2—5i,l一--p2—1,卢=0.O1,ps:一{一1,一A3=^=3iIteratingtheabovemethod,wecanobtainaseriesofmulti—solitonsolutionsofCDIE(1).3ConclusionByextendingelementsoftheDTmatrixintonegative--orderpolynomialsin,wedeveloped theDTmatrixmethod"todirectlyconstructanexplicitdeterminantformu1aforN—foldDTofCDIE(1).ThisN—foldDTformulacanbeinterpretedasanonlinearsuperpositionofasingleDT. Moreover,notonlytheconstructionofN—foldDTisverynaturalandmuchsimplerthanthatob一11111111乱1111融111l第3期ZhaqilaoMuIti—solitonSolutionsoftheGeneralizedCoupled (257)tainedintheusualwayEl-z],butalsotheN—foldDTisverysuitableforgeneratingaseriesofexplic—itsolutionsbysymboliccomputationonacomputer.References:[1][2][3][4][5]1-6][7][83[9][1o][11][12][13]KonnoK,KakuhataH.InteractionAmongGrowing,DecayingandStationarySolitonsforC oupledIntegrableDis—persionlessEquations[J].J.Phys.Soc.Jpn.,1995,64(8):2707-2709.KonnoK,KakuhataH.NovelSolitonicEvolutionsinaCoupledIntegrable,DispersionlessS ystem[J].J.Phys.Soc.Jpn.,1996,65(3):713—721.HirotaR,TsujimotoS.Noteon"NewCoupledIntegrableDispersionlessEquations"[J],J.Ph ys.Soc.Jpn.,I994,63(9):3533-3533.KotlyarovVP.OnEquationsGaugcEquivalenttOtheSine-GordonandPohlmeyer—Iund —ReggeEquations[J].J.Phys.Soc.Jpn.,1994,63:3535-3537.KonnoK,OonoH.NewCoupledIntegrableDispersionlessEquations[J].J.Phys.Soc.Jpn.,1 994,63:377—378.AlagesanT,ChungY,NakkeeranK.B~cklundtransformationandsolitonsolutionsfortheco upleddispersionlessequations[J].Chaos,SolitonsandFractals,2004,21:63—67.CbenAH,LiXM.Solitonsolutionsofthecoupleddispersionlessequation[J].Phys.Lett.A,2 007,370:281—286.HassanHJ.Darbouxtransformationofthegeneralizedcoupleddispersionlessintegrablesys tem[JⅢ.Phys.A:Math.Gen.,2009,42:l一11.NeugebaureG,MeineIR.GeneralN—so!itonsolutionoftheAKNSclassonarbitrarybackground[J].Phys.Lett.A,1984,100:467-470.MatveevVB,SalleMA.DarbouxTransformationsandSolitons[M].Berlin:Springer,1991. GuCH,HuHS,ZbouZx.DarbouxTransformationinSolitonTheoryandItsGeometricAppli cations[M].Shanghai:ShanghaiScienceandTechnologyPublishingHouse,2005.FanE(;.ComputerAlgebraandIntegrableSystems[M].Beijing:SciencePress,2004. Zhaqilao,ChenY,IiZB.Darbouxtransformationandmulti—solitonsolutionsforsomesolitonequations[J].Chaos,SolitonsandFractals.2009,41:661-670.(责任编委孙炯)具有三个位势的广义耦合无色散可积方程的多孤子解扎其劳(内蒙古师范大学数学科学学院,呼和浩特010022)摘要:利用具有三个位势的2×2矩阵谱问题的规范变换,给一个广义耦合无色散方程构造了一种新的N重达布变换.作为达布变换的应用,获得了该广义耦合无色散方程的N一孤子解.关键词:达布变换;孤子解;广义耦合无色散可积方程中图分类号:O175.2文献标志码:A收稿日期:2010—05-06基金项目:内蒙古自然科学基金项目(2009MS0108);内蒙古自治区高等学校科学研究项目(NJ10045)作者简介:扎其劳(1971一),男(蒙古族),内蒙古鄂尔多斯市人,副教授,博士.。

无穷维Hamilton算子谱理论研究综述

第50卷第1期2021年1月内蒙古师范大学学报(自然科学版)Journal of Inner Mongolia Normal University(Natural Science Edition)Vol.50No.1Jan.2021无穷维Hamilton算子谱理论研究综述阿拉坦仓】，吴德玉2（1.内蒙古师范大学数学科学学院，内蒙古呼和浩特010022；2.内蒙古大学数学科学学院，内蒙古呼和浩特010021）摘要：对无穷维Hamilton算子谱理论研究成果进行了梳理，包括无穷维Hamilton算子辛自伴性、无穷维Hamilton算子特征函数系的完备性，以及无穷维Hamilton算子数值域等方面的研究现状和展望。

提出一些关于无穷维Hamilton算子谱理论方面有待解决的问题。

关键词：无穷维Hamilton算子；自伴性；数值域；完备性中图分类号：0175.3文献标志码：A文章编号：1001—8735（2021）01—0001—06doi：10.3969/j.issn.1001—8735.2021.01.0010引言Hilbert空间中线性算子理论，尤其自伴算子谱理论是20世纪数学科学领域取得的最重要成果之一。

有界自伴算子谱理论的奠基人D.Hilbert,在1904年至1910年完成的关于积分方程的六篇论文中阐述了有界自伴算子谱理论的思想。

为研究量子力学问题，1927年至1929年J.von Neumann将有界自伴算子谱理论推广到了无界自伴算子领域。

20世纪30年代,F.Riesz和M.S.Stone等人进一步完善了无界自伴算子谱理论框架。

目前，自伴算子谱理论已经形成了比较完善的框架体系。

众所周知的求解偏微分方程的分离变量法，又称Fourier级数法，就是以自伴算子谱理论为基础的。

需要注意的是，当系统对应的算子是自伴算子时，该系统能量是守恒的，亦称封闭系统。

然而，实际问题中也存在诸多能量不守恒的开放系统，其状态算子为非自伴算子。

哈密顿拉格朗日多体系统动力学

哈密顿拉格朗日多体系统动力学（中英文实用版）Title: Hamilton and Lagrange Multibody System DynamicsHamilton"s mechanics and Lagrange"s equations are two essential frameworks in the field of multibody system dynamics.They provide a mathematical description of the motion of systems composed of multiple interacting particles or bodies.Hamilton"s mechanics, formulated by William Rowan Hamilton in the early 19th century, is based on the principle of least action.It provides a comprehensive framework for describing the dynamics of systems with a wide range of complexity, from simple mechanical systems to celestial mechanics and quantum mechanics.In Hamilton"s mechanics, the equations of motion are derived from the action principle, which states that the actual path of a system is the one that minimizes the action, a functional that depends on the configuration and time evolution of the system.In Chinese, Hamilton mechanics, formulated by William Rowan Hamilton in the early 19th century, is based on the principle of least action.It provides a comprehensive framework for describing the dynamics of systems with a wide range of complexity, from simple mechanical systems to celestial mechanics and quantum mechanics.In Hamilton"s mechanics, the equations of motion are derived from theaction principle, which states that the actual path of a system is the one that minimizes the action, a functional that depends on the configuration and time evolution of the system.Lagrange"s equations, on the other hand, were formulated by Joseph-Louis Lagrange in the mid-18th century.They provide an alternative approach to the study of dynamic systems, focusing on the conservation of grange"s equations are derived from the principle of virtual work, which states that the actual motion of a system is the one that minimizes the potential energy of the system.In Lagrange"s framework, the equations of motion are expressed in terms of generalized coordinates and their derivatives, which represent the configuration and time evolution of the system.In contrast to Hamilton"s mechanics, Lagrange"s equations focus on the conservation of energy.They were formulated by Joseph-Louis Lagrange in the mid-18th century.In Lagrange"s framework, the equations of motion are expressed in terms of generalized coordinates and their derivatives, which represent the configuration and time evolution of the system.The principle of virtual work underlies Lagrange"s equations, stating that the actual motion of a system is the one that minimizes the potential energy of the system.Both Hamilton"s and Lagrange"s frameworks are widely used in the study of multibody system dynamics.They provide powerful tools foranalyzing the motion of complex systems, such as robotic arms, vehicles, and biological organisms.By employing these frameworks, researchers and engineers can accurately predict the behavior of these systems under various conditions and design optimal control strategies for their operation.In summary, Hamilton"s and Lagrange"s mechanics are two complementary frameworks that play a crucial role in the analysis of multibody system dynamics.They provide a mathematical description of the motion of systems composed of multiple interacting particles or bodies, allowing for the study and optimization of complex dynamic systems.。

凝聚态物理相关诺贝尔奖(可编辑修改word版)

凝聚态物理相关诺贝尔化学奖1970-1985 年赫伯特·豪普特曼（美）杰罗姆·卡尔勒（美）在测定晶体结构的直接方法上的贡献2000 年艾伦·黑格（美）艾伦·麦克迪尔米德（美/新西兰）白川英树（日）对导电聚合物的研究2011 年丹·谢赫特曼（以）准晶的发现[5]凝聚态物理相关诺贝尔物理学奖1970-1972年约翰·巴丁美国“他们联合创立了超导微观理论，即常说的BCS 理论”"for their jointly developed theory ofsuperconductivity, usually called theBCS-theory"[73]利昂·库珀美国约翰·罗伯特·施里弗国美1973年江崎玲于奈日本“发现半导体和超导体的隧道效应”"for their experimental discoveriesregarding tunneling phenomena insemiconductors and superconductors,respectively"[74]伊瓦尔·贾埃弗挪威国英 “他理论上预测出通过隧道势垒的超电流的性质，特别是那些通常被称为约瑟夫森效应布赖的现象” 恩·戴"for his theoretical predictions of the 维·约瑟 properties of a supercurrent through a 夫森 tunnel barrier, in particular those phenomena which are generally known as the Josephson effect"[74]1977 年菲利普·沃伦·安德森国美“对磁性和无序体系电子结构的基础性理论研究”"for their fundamental theoretical investigations of the electronicstructure of magnetic and disorderedsystems"[78]内维尔·莫特国英约翰·凡扶累克国美年彼得·列昂尼多维奇·卡皮查苏联“低温物理领域的基本发明和发现” "for his basic inventions and discoveries in the area of low- temperature physics"[79]1982 年肯尼斯·威尔逊美国“对与相转变有关的临界现象理论的贡献”"for his theory for critical phenomena in connection with phase transitions"[83]1985 年克劳斯·冯·克利青德国“发现量子霍尔效应”"for the discovery of thequantized Hall effect"[86]特·鲁斯卡德国“电子光学的基础工作和设计了第一台显微镜"for his fundamental work in electron optics,electron microscope"格尔德·宾宁德国“研制"for their design of the scanningtunneling microscope"海因里希·罗雷尔瑞士约翰内斯·贝德诺尔茨德国“在发现"for their important break-through the discovery of superconductivity ceramic materials"卡尔·米勒瑞士皮埃尔吉勒纳法国“发现研究简单系统中有序现象的方法可以被推广到比较复杂的物质形式，特别是推广到"forfor studying order phenomena in simple systems can be generalized to morecomplexliquid crystals and polymers"1996年戴维·李美国“发现了在氦-3 里的超流动性”"for their discovery ofsuperfluidity inhelium-3"[97]道格拉斯·奥谢罗夫美国罗伯特·理查森美国2000年若雷斯·阿尔费罗夫俄罗斯“发展了用于高速电子学和光电子学的半导体异质结构”"for developing semiconductorheterostructures used in high-speed-and optoelectronics"[101]赫伯特·克勒默德国杰克·基尔比美国“在发明集成电路中所做的贡献”"for his part in the invention of theintegrated circuit"[101]2001年埃里克·康奈尔国美“在碱性原子稀薄气体的玻色-爱因斯坦凝聚态方面取得的成就，以及凝聚态物质属性质的早期基础性研究”"for the achievement of Bose-Einsteincondensation in dilute gases of alkaliatoms, and for early fundamental studiesof the properties of the condensates"[102]卡尔·威曼国美沃尔夫冈·克特勒国德2003年阿列克谢·阿布里科索夫美国俄罗斯“对超导体和超流体理论做出的先驱性贡献”"for pioneering contributions tothe theory of superconductors andsuperfluids"[104]维塔利·金俄罗斯兹堡安东尼·莱格特英国美国2007 年艾尔伯·费尔法国“发现巨磁阻效应”"for the discovery of giant magnetoresistance"[108]彼得·格林贝格德国2009 年高锟英国美国 [110] “在光学通信领域光在纤维中传输方面的突破性成就” "for groundbreaking achievementsconcerning the transmission of light infibers for optical communication"[111]威拉德·博伊尔美国“发明半导体成像器件电荷耦合器件” "for the invention of an imaging semiconductor circuit – the CCD sensor"[111]乔治·史密斯美国2010 年安德烈·海姆荷兰俄罗斯“在二维石墨烯材料的开创性实验”"for groundbreaking experimentsregarding the two-dimensionalmaterial graphene"[112]康斯坦丁·诺沃肖洛夫英国俄罗斯。

斯普林格数学研究生教材丛书

《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。

论文中参考文献和图、表的标准格式

论文中参考文献和图、表的标准格式1.参考文献的作用着录参考文献是科技论文写作的传统贯例和必要内容。

首先，它体现了科学的严肃性、继承性和对他人劳动的尊重。

其次，表明作者使用参考文献的起点和深度。

一份完整的参考文献，也是作者向读者提供的一份有价值的信息检索资料。

第三，为了保持篇章结构的简洁和紧凑，也不宜在正文中过多地赘述参考文献的内容，而善于利用参考文献，则可以适当节省篇幅，提高论文质量。

2.文献的着录规则参考文献的着录要求采用顺序编码标控制，即按引用文献出现的先后顺序，在文献的著者或成果叙述文字的右上角用方括号标注阿拉伯数字编排序号，在参考文献表中按此序号依次著录。

具体讲，有以下几种情况。

参考文献类型：专著［M］，论文集［C］，报纸文章［N］，期刊文章［J］，学位论文［D］，报告［R］，标准［S］，专利［P］，论文集中的析出文献［A］电子文献类型：数据库［DB］，计算机［CP］，电子公告［EB］电子文献的载体类型：互联网［OL］，光盘［CD］，磁带［MT］，磁盘［DK］（1）文内的标注视则①引文写出原著者时，序号应置于著者姓名的右上角。

如在谈到当前我国优质烤烟生产中存在的问题时，朱尊权[2]指出：……②不写出著者时，序号应置于引文之后右上角。

如“营养不良．发育不全，成熟不够．烘烤不当”[3]，这“四不”问题是符合我国烤烟生产实际情况的。

③当将参考文献序号作为文句的组成部分时，不用角码序号，而以正文形式用方括号序号列出。

如烟草增香剂的制各方法见文献[2]。

④当引用多篇文献时，只须将各篇文献的序号在一个角码内全部列出，各序号间用逗号隔开；如通连续序号，应在标注起止序号间加“～”连接。

如早期的研究结果[2，5，7～9]表明，……（2）文后的着录规则①书籍的基本著录项目和着录规则［序号］著者．书名．版本，出版地：出版者，出版年．页次．示例［1］管雨霖，张凤英，马超群，等．烟草病害诊断虫害识别及防治[M]．北京：农业出版社，1989．78-81．②期刊的基本著录项目和着录规则［序号］著者．题（篇）名．刊名，出版年，卷（期）号：页次．示例［1］朱尊权．论当前我国优质烤烟生产技术导向[J]．烟草科技，1994，（1）：2-3．［2］金龙林．关于在论文中标往参考文献顺序码的位置问题[J]．编辑学报，1994，6（4）：240-241．③专利文献的基本著录项目和着录规则［序号］专利申请者．专利题名．专利国别，专利文献种类，专利号．出版日期．示例［1］孙中培，刘清，段完晶．烟草薄片用烟杆纤维物质．中国专利，发明，CNA．1993-07-28．［2］牛聪阳，刘立全，荆海强．固体物料造粒机．中国专利，实用新型，ZL93 2 24039．9．1995-02-12．④报纸的基本著录项目和通用格式［序号］著者．题（篇）名．报纸名，出版年．月．日（版位）．示例［1］刘彩望．保护烟农利益稳定烟叶生产：河南省提高优质烟奖励标准．东方烟草报，1995．1．25（1）．A：专著、论文集、学位论文、报告［序号］主要责任者.文献题名［文献类型标识］.出版地：出版者，出版年.起止页码（可选）［1］刘国钧，陈绍业.图书馆目录［M］.北京：高等教育出版社，1957.15-18.B:期刊文章［序号］主要责任者.文献题名［J］.刊名，年，卷（期）：起止页码［1］何龄修.读南明史［J］.中国史研究，1998,(3):167-173.［2］OU J P，SOONG T T，et al.Recent advance in research on applications of passi ve energy dissipation systems［J］.Earthquack Eng,1997,38(3):358-361.C:论文集中的析出文献［序号］析出文献主要责任者.析出文献题名［A］.原文献主要责任者（可选）.原文献题名［C］.出版地：出版者，出版年.起止页码［7］钟文发.非线性规划在可燃毒物配置中的应用［A］.赵炜.运筹学的理论与应用——中国运筹学会第五届大会论文集［C］.西安：西安电子科技大学出版社，1996.468.D：报纸文章［序号］主要责任者.文献题名［N］.报纸名，出版日期（版次）［8］谢希德.创造学习的新思路［N］.人民日报，1998-12-25（10）.E：电子文献[文献类型/载体类型标识]：[J/OL]网上期刊、[EB/OL]网上电子公告、[M/CD]光盘图书、[DB/OL]网上数据库、[DB/MT]磁带数据库［序号］主要责任者.电子文献题名［电子文献及载体类型标识］.电子文献的出版或获得地址，发表更新日期/引用日期［12］王明亮.关于中国学术期刊标准化数据库系统工程的进展［EB/OL］.http://www.cajcd. /pub/wml.html，1998-08-16/1998-10-01.［8］万锦.中国大学学报文摘（1983-1993）.英文版［DB/CD］.北京：中国大百科全书出版社，1996.参考文献规范格式一、参考文献的类型参考文献（即引文出处）的类型以单字母方式标识，具体如下：M——专著C——论文集N——报纸文章J——期刊文章D——学位论文R——报告对于不属于上述的文献类型，采用字母“Z”标识。

几类新型无穷维矩阵Hamilton算子的一般化构造

几类新型无穷维矩阵Hamilton算子的一般化构造杜鹃;任文秀;霍晓霞【摘要】利用无穷维Hamilton算子的验证方法,推导出三种形式的矩阵Hamilton算子的判别条件,进而利用这些条件构造出六类新型Hamilton算子,同时说明了结论的正确性和一般性.最后,对于在形式化过程中未实现的算子做了分析,并对未来工作做了展望.【期刊名称】《内蒙古工业大学学报（自然科学版）》【年(卷),期】2018(037)001【总页数】7页(P7-13)【关键词】微分形式;判别条件;一般化;无穷维矩阵Hamilton算子【作者】杜鹃;任文秀;霍晓霞【作者单位】内蒙古工业大学理学院,呼和浩特 010051;内蒙古工业大学理学院,呼和浩特 010051;内蒙古工业大学理学院,呼和浩特 010051【正文语种】中文【中图分类】O1750 引言20世纪70年代,F.Magrin[1],Vinogradov[2]等人首次提出无穷维Hamilton系统的概念.后来,Gel’fand,Dorfman,Olver等人[3-5]做了进一步的探索.无穷维Hamilton系统在数学、物理、力学、控制、工程等领域都有广泛应用,因此将发展方程(组)平移到该体系下具有重要研究价值.而该系统的结构性质取决于其所对应的Hamilton算子.因此,无穷维Hamilton算子是一个重要的科研方向,可用于研究发展方程的对称、守恒律、双Hamilton系统等问题[6-9],而这些都离不开对Hamilton算子的验证.然而,该问题的难点在于雅克比恒等的证明,所以人们一般采用比较简单等价证明方法,即微分形式的语言:prvD θ(Θ)=0(1)由于Hamilton算子种类繁多,为了避免重复繁琐的证明过程,考虑将其形式化具有重要意义.例如,Cooke[10]用该方法构造出三阶和五阶标量Hamilton算子的一般式.贾[11]用同样的方法构造并分别证明了三种形式的矩阵Hamilton算子,但类型比较少,而且缺乏一般化.因此本文将对矩阵Hamilton算子做进一步探讨.主要构造了如下三种形式的候选矩阵Hamilton算子采用同样的方法分别得出使其成为Hamilton算子对应的判定条件,这是本文的关键.对于二阶算子D0,利用判定条件构造出四个类型的矩阵Hamilton算子,如分离变量型、指数型、幂型等,并用具体实例加以说明构造的意义.对于三阶算子K0,J0,推导出T,L满足的条件,从而构造出新型Hamilton算子,并对于在一般化过程中遇到的不能成为Hamilton算子的类型做了讨论,最后做了一些展望.1 本文主要结果本节首先给出与本文相关的公式及符号约定,然后分别将二、三阶矩阵Hamilton 算子形式化,并给出判别条件,从而构造出新型Hamilton算子,并对结论进行了验证.1.1 符号约定本节介绍的公式、符号类同文献[4],具体如下:第一,如果不做特殊声明,本文中出现的任何形如f(u,v)的函数都用f记;第二,延拓算子是对应于u=(u1,u2,…)T的单位向量;第三,标量Hamilton算子f Dx+Dx·f与f Dx·f具有等价性.1.2 二阶矩阵Hamilton算子的构造本小节主要将二阶矩阵Hamilton算子形式化.具体结论如下:命题1 若二阶矩阵微分算子具有以下形式且满足下面条件其中H=h1+h2,则D0一定是Hamilton算子.证明首先,由于算子D0满足下式故算子D0为斜伴随算子.其次,令θ=(θ,ξ)T是对应于变量u=(u,v)T的单位向量.算子D0对应的泛函双向量为通过计算D0θ并约定D0θ=(θ*,ξ*)T可得所以因为对于任意的函数M=M(u,v,ux,vx)其中,从而prvD0θ(Θ)=(fuH+2fvg-f Hu-HvH+Tuf+TvH)ξx∧θ∧θx+(HuH+Hvg-2guf-gvH+TuH+Tvg)ξx∧ξ∧θx}dx,其中T=h2-h1.令上式中三向量ξx∧θ∧θx,ξ∧θ∧θx,ξ∧θ∧ξx,ξx∧ξ∧θx的各系数为零,整理即得(a)-(d),从而prvD0θ(Θ)=0因此D0为Hamilton算子.命题得证.通过观察(a)-(d),由f,g,h1,h2的不同情形构造如下几个类型的Hamilton算子.型1 若算子D1具有以下形式且满足fu=hv,fv=hu则D1一定是Hamilton算子.证明在算子D0中,令f=g=f,h1=h2=h便得到算子D1.然后分别代入等式(a)-(d)的左边并整理得2h(fu-hv)+2f(fv-hu),h′(fu-hv)+f′(fv-hu),h′(hu-fv)+f′(hv-fu),2h(hu-fv)+2f(hv-fu),显然,条件(a)-(d)成立当且仅当fu=hv,fv=hu.由命题1可知D1为Hamilton算子. 特别地,由型1可得出如下结论:结论1 若算子D11具有以下形式其中fv=hu,则D11一定是Hamilton算子.结论2(分离变量型) 若算子D12具有以下形式其中,P=P(u),Q=Q(v),Pu=Qv,则D12一定是Hamilton算子.令结论2中P=u,Q=v,即为文[4]中当σ=1时,多变气体动力学方程所对应的一个Hamilton算子.根据命题1,还可以构造出如下类型的Hamilton算子,篇幅所限,这里不再给出证明. 型2(指数型) 若算子D2具有以下形式ki(i=1,2,3)为常数,则D2一定是Hamilton算子.特别地,令型2中的k1=k3=0,k2=1,即为文[12]中水动力系统所对应的一个Hamilton算子.型3(幂型) 若算子D3具有以下形式λ,ki(i=1,2,…,5)为常数,则D3一定是Hamilton算子.特别地,令型3中的k1=k3=1,λ=γ-2,其余常数为零,即为文[5]中气体动力学方程所对应的一个Hamilton算子.型4 若算子D4具有以下形式则D4一定是Hamilton算子.型4等价于文[11]中命题6的Hamilton算子,从而验证了该结论的正确性.1.3 三阶矩阵Hamilton算子的构造本小节将两种三阶矩阵Hamilton算子形式化,并给出如下具体结论:命题2 若三阶矩阵微分算子具有以下两种形式或其中T,L是斜伴随算子且分别满足则K0,J0一定是Hamilton算子.证明首先,故K0为斜伴随算子当且仅当T*=-T,即T是斜伴随算子.其次,令θ=(θ,ξ,τ)T是对应于变量u=(u,v,w)的单位向量.算子K0对应的泛函双向量为通过计算K0θ并约定K0θ=(θ*,ξ*,τ*)T可得故得从而因此,当时,K0为Hamilton算子.同理可验证,若L满足为Hamilton算子,证明略. 根据以上命题,通过对斜伴随算子T,L的分析,可构造出如下类型的Hamilton算子. 型1 三阶矩阵微分算子或一定是Hamilton算子.证明对于算子K1,这里T=f(w)Dx+Dx·f(w),由命题2,斜伴随性显然.其次,因此,K1为Hamilton算子.同理可证J1为Hamilton算子.型1中的J1是文[13]中非散射Boussinesq方程所对应的一个Hamilton算子的一般化.型2 三阶矩阵微分算子的形式为或其中,P=P(w),Q=Q(u),则K2,J2一定是Hamilton算子.证明对于算子K2,这里由命题2,斜伴随性显然.其次,因此,K2为Hamilton算子.同理可证J2为Hamilton算子.型2中J2与文[11]证明所得的Hamilton算子一致,再次验证了本文结论的正确性.2 结语本文由矩阵Hamilton算子一般化而得出的判别条件具有较重要意义.其一,可以构造出新型Hamilton算子;其二,验证其它同类型的Hamilton算子;其三,对于Hamilton算子对的证明也具有重要意义.原因在于如果算子D,K为Hamilton算子,二者为Hamilton算子对只需验证D+K为Hamilton算子即可.值得注意的是,命题1的判别条件仅仅是一个充分条件,对于一些具体的Hamilton算子并不满足这些条件,如文[12]中的二阶幂型Hamilton算子.本文在构造其它类型三阶Hamilton算子的过程中发现,验证条件(1)往往不能成立.比如下面三种形式对于E1,若斜伴随算子T,L为(P)型时(其中f,P为关于u,v,w的函数),其结果取决于Dx的系数,结果为零当且仅当Dx的系数为常数,即E1是平凡的,没有构造的必要.对于E2,特别地,若A是斜伴随算子,则A=B.相应地,若A也有T,L的如上两种形式,以上结果同样取决于Dx的系数,通过计算发现E2也是平凡的.同理,E3也是.值得一提的是还存在一类形如E=(Γ+ΓT)Dx+Γx的算子(DNLP型算子)[13],得出使其成为三阶矩阵Hamilton算子所满足的条件是构造的难点.今后值得进一步探讨的问题有：第一,其它类型二、三阶矩阵Hamilton算子的一般化问题;第二,高阶情形的一般化构造,如DNLP型;第三,本文所构造的新型矩阵Hamilton算子在Hamilton算子对、守恒律等相关问题上的应用.参考文献:[1] Magrin.A simple model of the integrable Hamiltonianequation[J].Journal of Mathematical Physics,1978,19(5):1156～1162. [2] Vinogradov.Hamiltonian Structures in Field Theory[J].Doklady Akademii Nauk Sssr,1978,19(1):790～794.[3] Gel'Fand I M,Dorfman I Y.Hamiltonian operators and algebraic structures related to them[J].Functional Analysis & ItsApplications,1979,13(4):248～262.[4] Olver P.J.Applications of Lie Groups to DifferentialEquations[M].Springer-Verlag,1999.[5] Olver P.J.Hamiltonian structures for systems of hyperbolic conservation laws[J].Journal of Mathematical Physics,1988,29(7):1610～1619.[6] 额布日力吐.无穷维Hamilton算子特征函数系的完备性及其在弹性力学中的应用[D].内蒙古大学,2011.[7] Wei F H,Alatancang,Huang J J.A Class of Infinite Dimensional Hamilton Operators with Finite Algebra Ascent[J].Journal of Inner Mongolia University,2012.[8] 闫利君,吴德玉,阿拉坦仓.一类无穷维Hamilton算子的谱[J].数学的实践与认识,2016,46(10):242～247.[9] 海国君,阿拉坦仓.某类无穷维Hamilton算子的Moore-Penrose可逆性[J].数学杂志,2017,37(2):358～364.[10] Cooke.Classification results and the Darboux theorem for low-order Hamiltonian operators[J].Journal of Mathematical Physics,1991,32(1):109～119.[11] 贾利东,任文秀,张育红.特殊无穷维矩阵Hamilton算子的探讨[J].内蒙古工业大学学报(自然科学版),2014,33(4):246～250.[12] H Gümral,Y Nutku.Multi-hamiltonian structure of equations of hydrodynamic type[J].Journal of Mathematical Physics,1990,31(11):2606～2611.[13] H Gumral,Y Nutku.Bi-Hamiltonian structures of d-Boussinesq and Benney-Lax equations[J].Journal of Physics A GeneralPhysics,1994,27(1):193.。

Inference of Population Structure Using Multilocus Genotype Data

Copyright©2000by the Genetics Society of AmericaInference of Population Structure Using Multilocus Genotype DataJonathan K.Pritchard,Matthew Stephens and Peter DonnellyDepartment of Statistics,University of Oxford,Oxford OX13TG,United KingdomManuscript received September23,1999Accepted for publication February18,2000ABSTRACTWe describe a model-based clustering method for using multilocus genotype data to infer populationstructure and assign individuals to populations.We assume a model in which there are K populations(where K may be unknown),each of which is characterized by a set of allele frequencies at each locus.Individuals in the sample are assigned(probabilistically)to populations,or jointly to two or more popula-tions if their genotypes indicate that they are admixed.Our model does not assume a particular mutationprocess,and it can be applied to most of the commonly used genetic markers,provided that they are notclosely linked.Applications of our method include demonstrating the presence of population structure,assigning individuals to populations,studying hybrid zones,and identifying migrants and admixed individu-als.We show that the method can produce highly accurate assignments using modest numbers of loci—e.g.,seven microsatellite loci in an example using genotype data from an endangered bird species.The softwareused for this article is available from /فpritch/home.html.I N applications of population genetics,it is often use-populations based on these subjective criteria representsa natural assignment in genetic terms,and it would beful to classify individuals in a sample into popula-tions.In one scenario,the investigator begins with a useful to be able to conﬁrm that subjective classiﬁcations sample of individuals and wants to say something aboutare consistent with genetic information and hence ap-the properties of populations.For example,in studies propriate for studying the questions of interest.Further, of human evolution,the population is often consideredthere are situations where one is interested in“cryptic”to be the unit of interest,and a great deal of work has population structure—i.e.,population structure that isdifﬁcult to detect using visible characters,but may be focused on learning about the evolutionary relation-ships of modern populations(e.g.,Cavalli et al.1994).signiﬁcant in genetic terms.For example,when associa-In a second scenario,the investigator begins with a settion mapping is used toﬁnd disease genes,the presence of predeﬁned populations and wishes to classify individ-of undetected population structure can lead to spurious uals of unknown origin.This type of problem arisesassociations and thus invalidate standard tests(Ewens in many contexts(reviewed by Davies et al.1999).A and Spielman1995).The problem of cryptic population standard approach involves sampling DNA from mem-structure also arises in the context of DNAﬁngerprint-bers of a number of potential source populations and ing for forensics,where it is important to assess thedegree of population structure to estimate the probabil-using these samples to estimate allele frequencies inity of false matches(Balding and Nichols1994,1995; each population at a series of unlinked ing theForeman et al.1997;Roeder et al.1998).estimated allele frequencies,it is then possible to com-Pritchard and Rosenberg(1999)considered how pute the likelihood that a given genotype originated ingenetic information might be used to detect the pres-each population.Individuals of unknown origin can beence of cryptic population structure in the association assigned to populations according to these likelihoodsmapping context.More generally,one would like to be Paetkau et al.1995;Rannala and Mountain1997).able to identify the actual subpopulations and assign In both situations described above,a crucialﬁrst stepindividuals(probabilistically)to these populations.In is to deﬁne a set of populations.The deﬁnition of popu-this article we use a Bayesian clustering approach to lations is typically subjective,based,for example,ontackle this problem.We assume a model in which there linguistic,cultural,or physical characters,as well as theare K populations(where K may be unknown),each of geographic location of sampled individuals.This subjec-which is characterized by a set of allele frequencies at tive approach is usually a sensible way of incorporatingeach locus.Our method attempts to assign individuals diverse types of information.However,it may be difﬁcultto populations on the basis of their genotypes,while to know whether a given assignment of individuals tosimultaneously estimating population allele frequen-cies.The method can be applied to various types ofmarkers[e.g.,microsatellites,restriction fragment Corresponding author:Jonathan Pritchard,Department of Statistics,length polymorphisms(RFLPs),or single nucleotide University of Oxford,1S.Parks Rd.,Oxford OX13TG,United King-dom.E-mail:pritch@ polymorphisms(SNPs)],but it assumes that the marker Genetics155:945–959(June2000)946J.K.Pritchard,M.Stephens and P.Donnellyloci are unlinked and at linkage equilibrium with one observations from each cluster are random draws another within populations.It also assumes Hardy-Wein-from some parametric model.Inference for the pa-berg equilibrium within populations.(We discuss these rameters corresponding to each cluster is then done assumptions further in background on clusteringjointly with inference for the cluster membership of methods and the discussion.)each individual,using standard statistical methods Our approach is reminiscent of that taken by Smouse(for example,maximum-likelihood or Bayesian et al.(1990),who used the EM algorithm to learn about methods).the contribution of different breeding populations to aDistance-based methods are usually easy to apply and sample of salmon collected in the open ocean.It is alsoare often visually appealing.In the genetics literature,it closely related to the methods of Foreman et al.(1997)has been common to adapt distance-based phylogenetic and Roeder et al.(1998),who were concerned withalgorithms,such as neighbor-joining,to clustering estimating the degree of cryptic population structuremultilocus genotype data(e.g.,Bowcock et al.1994). to assess the probability of obtaining a false match atHowever,these methods suffer from many disadvan-DNAﬁngerprint loci.Consequently they focused ontages:the clusters identiﬁed may be heavily dependent estimating the amount of genetic differentiation amongon both the distance measure and graphical representa-the unobserved populations.In contrast,our primarytion chosen;it is difﬁcult to assess how conﬁdent we interest lies in the assignment of individuals to popula-should be that the clusters obtained in this way are tions.Our approach also differs in that it allows for themeaningful;and it is difﬁcult to incorporate additional presence of admixed individuals in the sample,whoseinformation such as the geographic sampling locations genetic makeup is drawn from more than one of the Kof individuals.Distance-based methods are thus more populations.suited to exploratory data analysis than toﬁne statistical In the next section we provide a brief descriptioninference,and we have chosen to take a model-based of clustering methods in general and describe someapproach here.advantages of the model-based approach we take.TheTheﬁrst challenge when applying model-based meth-details of the models and algorithms used are given inods is to specify a suitable model for observations from models and methods.We illustrate our method witheach cluster.To make our discussion more concrete we several examples in applications to data:both onintroduce very brieﬂy some of our model and notation simulated data and on sets of genotype data from anhere;a fuller treatment is given later.Assume that each endangered bird species and from humans.incorpo-cluster(population)is modeled by a characteristic set rating population information describes how ourof allele frequencies.Let X denote the genotypes of the method can be extended to incorporate geographicsampled individuals,Z denote the(unknown)popula-information into the inference process.This may betions of origin of the individuals,and P denote the useful for testing whether particular individuals are mi-(unknown)allele frequencies in all populations.(Note grants or to assist in classifying individuals of unknownthat X,Z,and P actually represent multidimensional origin(as in Rannala and Mountain1997,for exam-vectors.)Our main modeling assumptions are Hardy-ple).Background on the computational methods usedWeinberg equilibrium within populations and complete in this article is provided in the appendix.linkage equilibrium between loci within populations.Under these assumptions each allele at each locus ineach genotype is an independent draw from the appro-BACKGROUND ON CLUSTERING METHODSpriate frequency distribution,and this completely speci-Consider a situation where we have genetic data fromﬁes the probability distribution Pr(X|Z,P)(given later a sample of individuals,each of whom is assumed toin Equation2).Loosely speaking,the idea here is that have originated from a single unknown population(nothe model accounts for the presence of Hardy-Weinberg admixture).Suppose we wish to cluster together individ-or linkage disequilibrium by introducing population uals who are genetically similar,identify distinct clusters,structure and attempts toﬁnd population groupings and perhaps see how these clusters relate to geographi-that(as far as possible)are not in disequilibrium.While cal or phenotypic data on the individuals.There areinference may depend heavily on these modeling as-broadly two types of clustering methods we might use:sumptions,we feel that it is easier to assess the validityof explicit modeling assumptions than to compare the 1.Distance-based methods.These proceed by calculatingrelative merits of more abstract quantities such as dis-a pairwise distance matrix,whose entries give thetance measures and graphical representations.In situa-distance(suitably deﬁned)between every pair of in-tions where these assumptions are deemed unreason-dividuals.This matrix may then be represented usingable then alternative models should be built.some convenient graphical representation(such as aHaving speciﬁed our model,we must decide how to tree or a multidimensional scaling plot)and clustersperform inference for the quantities of interest(Z and may be identiﬁed by eye.2.Model-based methods.These proceed by assuming that P).Here,we have chosen to adopt a Bayesian approach,947Inferring Population Structureby specifying models(priors)Pr(Z)and Pr(P),for both Assume that before observing the genotypes we haveZ and P.The Bayesian approach provides a coherent no information about the population of origin of eachframework for incorporating the inherent uncertainty individual and that the probability that individual i origi-of parameter estimates into the inference procedure nated in population k is the same for all k,and for evaluating the strength of evidence for the in-Pr(z(i)ϭk)ϭ1/K,(3) ferred clustering.It also eases the incorporation of vari-ous sorts of prior information that may be available,independently for all individuals.(In cases where somesuch as information about the geographic sampling lo-populations may be more heavily represented in thecation of individuals.sample than others,this assumption is inappropriate;itHaving observed the genotypes,X,our knowledge would be straightforward to extend our model to dealabout Z and P is then given by the posterior distribution with such situations.)We follow the suggestion of Balding and Nichols Pr(Z,P|X)ϰPr(Z)Pr(P)Pr(X|Z,P).(1)(1995)(see also Foreman et al.1997and Rannala While it is not usually possible to compute this distribu-and Mountain1997)in using the Dirichlet distri-tion exactly,it is possible to obtain an approximate bution to model the allele frequencies at each locus sample(Z(1),P(1)),(Z(2),P(2)),...,(Z(M),P(M))from Pr(Z,within each population.The Dirichlet distributionP|X)using Markov chain Monte Carlo(MCMC)meth-D(␭1,␭2,...,␭J)is a distribution on allele frequenciesods described below(see Gilks et al.1996b,for more pϭ(p1,p2,...,p J)with the property that these frequen-general background).Inference for Z and P may then cies sum to1.We use this distribution to specify the be based on summary statistics obtained from this sam-probability of a particular set of allele frequencies pkl·ple(see Inference for Z,P,and Q below).A brief introduc-for population k at locus l,tion to MCMC methods and Gibbs sampling may befound in the appendix.pkl·فD(␭1,␭2,...,␭J l),(4)independently for each k,l.The expected frequency of MODELS AND METHODS allele j is proportional to␭j,and the variance of thisfrequency decreases as the sum of the␭j increases.We We now provide a more detailed description of ourtake␭1ϭ␭2ϭ···ϭ␭J lϭ1.0,which gives a uniform modeling assumptions and the algorithms used to per-distribution on the allele frequencies;alternatives are form inference,beginning with the simpler case wherediscussed in the discussion.each individual is assumed to have originated in a singleMCMC algorithm(without admixture):Equations2, population(no admixture).3,and4deﬁne the quantities Pr(X|Z,P),Pr(Z),and The model without admixture:Suppose we genotypePr(P),respectively.By setting␪ϭ(␪1,␪2)ϭ(Z,P)and N diploid individuals at L loci.In the case without admix-letting␲(Z,P)ϭPr(Z,P|X)we can use the approach ture,each individual is assumed to originate in one ofoutlined in Algorithm A1to construct a Markov chain K populations,each with its own characteristic set ofwith stationary distribution Pr(Z,P|X)as follows: allele frequencies.Let the vector X denote the observedAlgorithm1:Starting with initial values Z(0)for Z(by genotypes,Z the(unknown)populations of origin ofdrawing Z(0)at random using(3)for example),iterate the the individuals,and P the(unknown)allele frequenciesfollowing steps for mϭ1,2,....in the populations.These vectors consist of the follow-ing elements,Step1.Sample P(m)from Pr(P|X,Z(mϪ1)).(x(i,1)l,x(i,2)l)ϭgenotype of the i th individual at the l th locus,Step2.Sample Z(m)from Pr(Z|X,P(m)).where iϭ1,2,...,N and lϭ1,2,...,L;z(i)ϭpopulation from which individual i originated;Informally,step1corresponds to estimating the allele p kljϭfrequency of allele j at locus l in population k,frequencies for each population assuming that the pop-where kϭ1,2,...,K and jϭ1,2,...,J l,ulation of origin of each individual is known;step2 where J l is the number of distinct alleles observed at corresponds to estimating the population of origin of locus l,and these alleles are labeled1,2,...,J l.each individual,assuming that the population allele fre-Given the population of origin of each individual,quencies are known.For sufﬁciently large m and c,(Z(m), the genotypes are assumed to be generated by drawing P(m)),(Z(mϩc),P(mϩc)),(Z(mϩ2c),P(mϩ2c)),...will be approxi-alleles independently from the appropriate population mately independent random samples from Pr(Z,P|X). frequency distributions,The distributions required to perform each step aregiven in the appendix.Pr(x(i,a)lϭj|Z,P)ϭp z(i)lj(2)The model with admixture:We now expand ourmodel to allow for admixed individuals by introducing independently for each x(i,a)l.(Note that p z(i)lj is the fre-a vector Q to denote the admixture proportions for each quency of allele j at locus l in the population of originof individual i.)individual.The elements of Q are948J.K.Pritchard,M.Stephens and P.Donnellyq (i )k ϭproportion of individual i ’s genome thattion of origin of each allele copy in each individual isknown;step 2corresponds to estimating the population originated from population k.of origin of each allele copy,assuming that the popula-It is also necessary to modify the vector Z to replace the tion allele frequencies and the admixture proportions assumption that each individual i originated in some are known.As before,for sufﬁciently large m and c ,unknown population z (i )with the assumption that each (Z (m ),P (m ),Q (m )),(Z (m ϩc ),P (m ϩc ),Q (m ϩc )),(Z (m ϩ2c ),P (m ϩ2c ),observed allele copy x (i ,a )l originated in some unknown Q (m ϩ2c )),...will be approximately independent random population z (i ,a )l :samples from Pr(Z ,P ,Q |X ).The distributions required to perform each step are given in the appendix.z (i ,a )l ϭpopulation of origin of allele copy x (i ,a )l .Inference:Inference for Z,P,and Q:We now discuss how We use the term “allele copy”to refer to an allele carried the MCMC output can be used to perform inference on at a particular locus by a particular individual.Z ,P ,and Q.For simplicity,we focus our attention on Q ;Our primary interest now lies in estimating Q.We inference for Z or P is similar.proceed in a manner similar to the case without admix-Having obtained a sample Q (1),...,Q (M )(using suitably ture,beginning by specifying a probability model for large burn-in m and thinning interval c )from the poste-(X ,Z ,P ,Q ).Analogues of (2)and (3)arerior distribution of Q ϭ(q 1,...,q N )given X using the MCMC method,it is desirable to summarize the Pr(x (i ,a )l ϭj |Z ,P ,Q )ϭp z (i ,a )l lj(5)information contained,perhaps by a point estimate of andQ.A seemingly obvious estimate is the posterior meanPr(z (i ,a )l ϭk |P ,Q )ϭq (i )k ,(6)E (q i |X )≈1M ͚M m ϭ1q (m )i .(8)with (4)being used to model P as before.To complete our model we need to specify a distribution for Q ,which However,the symmetry of our model implies that the in general will depend on the type and amount of admix-posterior mean of q i is (1/K ,1/K ,...,1/K )for all i ,ture we expect to see.Here we model the admixturewhatever the value of X.For example,suppose that there proportions q (i )ϭ(q (i )1,...,q (i )K )of individual i using are just two populations and 10individuals and that the the Dirichlet distributiongenotypes of these individuals contain strong informa-tion that the ﬁrst 5are in one population and the second q (i )فD (␣,␣,...,␣)(7)5are in the other population.Then eitherindependently for each individual.For large values of ␣(ӷ1),this models each individual as having allele q 1...q 5≈(1,0)and q 6...q 10≈(0,1)(9)copies originating from all K populations in equal pro-orportions.For very small values of ␣(Ӷ1),it models each individual as originating mostly from a single popu-q 1...q 5≈(0,1)and q 6...q 10≈(1,0),(10)lation,with each population being equally likely.As with these two “symmetric modes”being equally likely,␣→0this model becomes the same as our model leading to the expectation of any given q i being (0.5,without admixture (although the implementation of the 0.5).This is essentially a problem of nonidentiﬁability MCMC algorithm is somewhat different).We allow ␣caused by the symmetry of the model [see Stephens to range from 0.0to 10.0and attempt to learn about ␣(2000b)for more discussion].from the data (speciﬁcally we put a uniform prior on In general,if there are K populations then there will ␣෈[0,10]and use a Metropolis-Hastings update step be K !sets of symmetric modes.Typically,MCMC to integrate out our uncertainty in ␣).This model may schemes ﬁnd it rather difﬁcult to move between such be considered suitable for situations where little is modes,and the algorithms we describe will usually ex-known about admixture;alternatives are discussed in plore only one of the symmetric modes,even when run the discussion.for a very large number of iterations.Fortunately this MCMC algorithm (with admixture):The following does not bother us greatly,since from the point of algorithm may be used to sample from Pr(Z ,P ,Q |X ).view of clustering all the symmetric modes are the same Algorithm 2:Starting with initial values Z (0)for Z (by drawing Z (0)at random using (3)for example),iterate the [compare the clusterings corresponding to (9)and following steps for m ϭ1,2,....(10)].If our sampler explores only one symmetric mode then the sample means (8)will be very poor estimates Step 1.Sample P (m ),Q (m )from Pr(P ,Q |X ,Z (m Ϫ1)).of the posterior means for the q i ,but will be much better Step 2.Sample Z (m )from Pr(Z |X ,P (m ),Q (m )).estimates of the modes of the q i ,which in this case turn Step 3.Update ␣using a Metropolis-Hastings step.out to be a much better summary of the information in the data.Ironically then,the poor mixing of the Informally,step 1corresponds to estimating the allele MCMC sampler between the symmetric modes gives frequencies for each population and the admixture pro-portions of each individual,assuming that the popula-the asymptotically useless estimator (8)some practical949Inferring Population Structure value.Where the MCMC sampler succeeds in moving Simulated data:To test the performance of the clus-tering method in cases where the “answers”are known,between symmetric modes,or where it is desired to combine results from samples obtained using different we simulated data from three population models,using standard coalescent techniques (Hudson 1990).We as-starting points (which may involve combining results corresponding to different modes),more sophisticated sumed that sampled individuals were genotyped at a series of unlinked microsatellite loci.Data were simu-methods [such as those described by Stephens (2000b)]may be required.lated under the following models.Inference for the number of populations:The problem of Model 1:A single random-mating population of con-inferring the number of clusters,K ,present in a data stant size.set is notoriously difﬁcult.In the Bayesian paradigm the Model 2:Two random-mating populations of constant way to proceed is theoretically straightforward:place a effective population size 2N.These were assumed to prior distribution on K and base inference for K on the have split from a single ancestral population,also of posterior distributionsize 2N at a time N generations in the past,with no subsequent migration.Pr(K |X )ϰPr(X |K )Pr(K ).(11)Model 3:Admixture of populations.Two discrete popu-However,this posterior distribution can be peculiarly lations of equal size,related as in model 2,were fused dependent on the modeling assumptions made,even to produce a single random-mating population.Sam-where the posterior distributions of other quantities (Q ,ples were collected after two generations of random Z ,and P ,say)are relatively robust to these assumptions.mating in the merged population.Thus,individuals Moreover,there are typically severe computational chal-have i grandparents from population 1,and 4Ϫi lenges in estimating Pr(X |K ).We therefore describe an grandparents from population 2with probability alternative approach,which is motivated by approximat-(4i )/16,where i ෈{0,4}.All loci were simulated inde-ing (11)in an ad hoc and computationally convenient pendently.way.We present results from analyzing data sets simulated Arguments given in the appendix (Inference on K,the under each model.Data set 1was simulated under number of populations )suggest estimating Pr(X |K )usingmodel 1,with 5microsatellite loci.Data sets 2A and 2B Pr(X |K )≈exp(Ϫ␮ˆ/2Ϫ␴ˆ2/8),(12)were simulated under model 2,with 5and 15microsatel-lite loci,respectively.Data set 3was simulated under wheremodel 3,with 60loci (preliminary analyses with fewer loci showed this to be a much harder problem than ␮ˆϭ1M ͚M m ϭ1Ϫ2log Pr(X |Z (m ),P (m ),Q (m ))(13)models 1and 2).Microsatellite mutation was modeled by a simple stepwise mutation process,with the mutation andparameter 4N ␮set at 16.0per locus (i.e.,the expected variance in repeat scores within populations was 8.0).␴ˆ2ϭ1M ͚Mm ϭ1(Ϫ2log Pr(X |Z (m ),P (m ),Q (m ))Ϫ␮ˆ)2.We did not make use of the assumed mutation model in analyzing the simulated data.(14)Our analysis consists of two phases.First,we consider We use (12)to estimate Pr(X |K )for each K and substi-the issue of model choice—i.e.,how many populations tute these estimates into (11)to approximate the poste-are most appropriate for interpreting the data.Then,rior distribution Pr(K |X ).we examine the clustering of individuals for the inferred In fact,the assumptions underlying (12)are dubious number of populations.at best,and we do not claim (or believe)that our proce-Choice of K for simulated data:For each model,we dure provides a quantitatively accurate estimate of the ran a series of independent runs of the Gibbs sampler posterior distribution of K.We see it merely as an ad for each value of K (the number of populations)be-hoc guide to which models are most consistent with the tween 1and 5.The results presented are based on runs data,with the main justiﬁcation being that it seems of 106iterations or more,following a burn-in period of to give sensible answers in practice (see next section for at least 30,000iterations.To choose the length of the examples).Notwithstanding this,for convenience we burn-in period,we printed out log(Pr(X |P (m ),Q (m ))),and continue to refer to “estimating”Pr(K |X )and Pr(X |K ).several other summary statistics during the course of a series of trial runs,to estimate how long it took to reach (approximate)stationarity.To check for possible prob-APPLICATIONS TO DATAlems with mixing,we compared the estimates of P (X |K )and other summary statistics obtained over several inde-We now illustrate the performance of our method on both simulated data and real data (from an endangered pendent runs of the Gibbs sampler,starting from differ-ent initial points.In general,substantial differences be-bird species and from humans).The analyses make use of the methods described in The model with admixture.tween runs can indicate that either the runs should950J.K.Pritchard,M.Stephens and P.DonnellyTABLE 1Estimated posterior probabilities of K ,for simulated data sets 1,2A,2B,and 3(denoted X 1,X 2A ,X 2B ,and X 3,respectively)K log P (K |X 1)P (K |X 2A )P (K |X 2B )P (K |X 3)1ف1.0ف0.0ف0.0ف0.02ف0.00.210.999ف1.03ف0.00.580.0009ف0.04ف0.00.21ف0.0ف0.05ف0.0ف0.0ف0.0ف0.0The numbers should be regarded as a rough guide to which models are consistent with the data,rather than accurate esti-mates of posterior probabilities.Figure 1.—Summary of the clustering results for simulated data sets 2A and 2B,respectively.For each individual,webe longer to obtain more accurate estimates or that computed the mean value of q (i )1(the proportion of ancestry independent runs are getting stuck in different modes in population 1),over a single run of the Gibbs sampler.Thein the parameter space.(Here,we consider the K !dashed line is a histogram of mean values of q (i )1for individuals from population 0;the solid line is for individuals from popula-modes that arise from the nonidentiﬁability of the K tion 1.populations to be equivalent,since they arise from per-muting the K population labels.)We found that in most cases we obtained consistent and Q estimating the number of grandparents from estimates of P (X |K )across independent runs.However,each of the two original populations,for each individual.when analyzing data set 2A with K ϭ3,the Gibbs sampler Intuitively it seems that another plausible clustering found two different modes.This data set actually con-would be with K ϭ5,individuals being assigned to tains two populations,and when K is set to 3,one of clusters according to how many grandparents they have the populations expands to ﬁll two of the three clusters.from each population.In biological terms,the solution It is somewhat arbitrary which of the two populations with K ϭ2is more natural and is indeed the inferred expands to ﬁll the extra cluster:this leads to two modes value of K for this data set using our ad hoc guide [the of slightly different heights.The Gibbs sampler did not estimated value of Pr(X |K )was higher for K ϭ5than manage to move between the two modes in any of our for K ϭ3,4,or 6,but much lower than for K ϭ2].runs.However,this raises an important point:the inferred In Table 1we report estimates of the posterior proba-value of K may not always have a clear biological inter-bilities of values of K ,assuming a uniform prior on K pretation (an issue that we return to in the discussion ).between 1and 5,obtained as described in Inference for Clustering of simulated data:Having considered the the number of populations.We repeat the warning given problem of estimating the number of populations,we there that these numbers should be regarded as rough now examine the performance of the clustering algo-guides to which models are consistent with the data,rithm in assigning particular individuals to the appro-rather than accurate estimates of the posterior probabil-priate populations.In the case where the populations ities.In the case where we found two modes (data set are discrete,the clustering performs very well (Figure 2A,K ϭ3),we present results based on the mode that 1),even with just 5loci (data set 2A),and essentially gave the higher estimate of Pr(X |K ).perfectly with 15loci (data set 2B).With all four simulated data sets we were able to The case with admixture (Figure 2)appears to be correctly infer whether or not there was population more difﬁcult,even using many more loci.However,structure (K ϭ1for data set 1and K Ͼ1otherwise).the clustering algorithm did manage to identify the In the case of data set 2A,which consisted of just 5population structure appropriately and estimated the loci,there is not a clear estimate of K ,as the posterior ancestry of individuals with reasonable accuracy.Part probability is consistent with both the correct value,K ϭof the reason that this problem is difﬁcult is that it is 2,and also with K ϭ3or 4.However,when the number hard to estimate the original allele frequencies (before of loci was increased to 15(data set 2B),virtually all of admixture)when almost all the individuals (7/8)are the posterior probability was on the correct number of admixed.A more fundamental problem is that it is difﬁ-populations,K ϭ2.cult to get accurate estimates of q (i )for particular individ-Data set 3was simulated under a more complicated uals because (as can be seen from the y -axis of Figure model,where most individuals have mixed ancestry.In 2)for any given individual,the variance of how many this case,the population was formed by admixture of two populations,so the “true”clustering is with K ϭ2,of its alleles are actually derived from each population。

Structural Studies of the Ferroelectric Phase Transition in Bi4Ti3O12

Structural Studies of the Ferroelectric Phase Transitionin Bi4Ti3O12Qingdi Zhou and Brendan J.Kennedy*School of Chemistry,The University of Sydney,Sydney,NSW2006AustraliaChristopher J.HowardAustralian Nuclear Science and Technology Organization,Private Mail Bag1,Menai, NSW2234,Australia,and School of Physics,The University of Sydney,Sydney,NSW2006AustraliaReceived July3,2003.Revised Manuscript Received October20,2003A variable-temperature synchrotron X-ray diffraction study of the phase transitions in the ferroelectric n)3Aurivillius oxide Bi4Ti3O12is described.At room temperature the structure of Bi4Ti3O12is orthorhombic in space group B2eb and this continuously transforms to the high-temperature tetragonal I4/mmm structure via an intermediate orthorhombic phase.The possible space groups of this intermediate orthorhombic phase have been identified by using group theory.IntroductionThe Aurivillius oxides are represented by the general formula(Bi2O2)2+(A n-1B n O3n+1)2-where B is a diamag-netic transition metal such as Ti4+or Nb5+and A is an alkali or alkaline earth cation.1The structure of the Aurivillius oxides consists of arrays of Bi2O2and per-ovskite-like A n-1B n O3n+1layers.The ferroelectric prop-erties of such oxides have been known for around50 years,2,3yet the structural origins of their ferroelectricity have only recently been established.4,5Following the pioneering work of Scott and co-work-ers,6the possibility of employing these Aurivillius oxides in ferroelectric memory devices has been extensively studied.A serious barrier to their practical utilization is their poor thermal stability.7-10Formation of thin-film ferroelectric devices involves a sintering step at high temperatures and this invariably degrades the performance of the simpler Aurivillius oxides.Conse-quently,a number of studies of the high-temperature behavior and structures of Aurivillius oxides have been reported.11-14In comparison to the n)2oxides based on SrBi2Ta2O9very little is known about the high-temperature properties of the n)3oxides such as Bi4-Ti3O12,yet such oxides are reported to have ferroelectric properties superior to those of the better-studied n)2 oxides.15Although it has been reported5that Bi4Ti3O12is monoclinic at room temperature,very-high-resolution powder diffraction data suggest that powder samples of Bi4Ti3O12are actually orthorhombic at room temper-ature.16Heating Bi4Ti3O12above670°C is reported to result in an apparently first-order phase transition to a paraelectric tetragonal phase.16,17A significant volume change between the high-temperature and low-temper-ature phases is likely to be detrimental to the stability of any thin films annealed or sintered above the Curie temperature.Conversely,continuous transitions are less likely to adversely influence the properties of the thin films.Hervoches and Lightfoot have demonstrated, using powder neutron diffraction methods,that for Bi4-Ti3O12the high-temperature paraelectric phase is in space group I4/mmm and that the room-temperature orthorhombic phase is in B2eb.16The main structural basis for the ferroelectricity in Bi4Ti3O12is the displacement of the Bi atoms within the perovskite-like layers,along the crystallographic a-axis with respect to the chains of corner-sharing TiO6 octahedra.This corresponds to a[110]displacement referred to the parent I4/mmm.The TiO6octahedra are tilted relative to each other and the tilt system can be*To whom correspondence should be addressed.Phone:61-2-9351-2742.Fax:61-2-9351-3329.E-mail: b.kennedy@.au.(1)Aurivillius,B.Arkiv Kemi1949,1,463.(2)Subbarao,E.C.J.Phys.Chem.Solids1962,23,665.(3)Smolenskii,G.A.;Isupov,V.A.;Agranovskaya,A.I.Sov.Phys. Solid State1961,3,651.(4)Rae,A.D.;Thompson,J.G.;Withers,R.Acta Crystallogr.,Sect. B:Struct.Sci.1992,48,418.(5)Rae,A.D.;Thompson,J.G.;Withers,R.;Willis,A.C.Acta Crystallogr.,Sect.B:Struct.Sci.1990,46,474.(6)Paz de Araujo,C.A.;Cuchlaro,J.D.;McMillan,L.D.;Scott, M.;Scott,J.F.Nature(London)1995,374,627.(7)Shimakawa,Y.;Kubo,Y.;Tauchi,Y.;Kamiyama,T.;Asano,H.; Izumi,F.Appl.Phys.Lett.2000,77,2749.(8)Shimakawa,Y.;Kubo,Y.;Tauchi,Y.;Asano,H.;Kamiyama,T.; Izumi,F.;Hiroi,Z.Appl.Phys.Lett.2001,79,2791.(9)Boulle, A.;Legrand, C.;Guinebretie`re,R.;Mercurio,J.P.; Dauger,A.Thin Solid Films2002,391,42.(10)Aizawa,K.;Tokumitsu,E.;Okamoto,K.;Ishiwara,H.Appl. Phys.Lett.2000,79,2791.(11)Macquart,R.;Kennedy,B.J.;Hunter,B.A.;Howard,C.J.; Shimakawa,Y.Integr.Ferroelectr.2002,44,101-112.(12)Hervoches,C.H.;Irvine,J.T.S.;Lightfoot,P.Phys.Rev.B 2002,64,100102(R).(13)Liu,J.;Zou,G.;Yang,H.;Cui,Q.Solid State Commun.1994, 90,365.(14)Macquart,R.;Kennedy,B.J.;Vogt,T.;Howard,C.J.Phys. Rev B2002,66,212102.(15)Park,B.H.;Kang,B.S.;Bu,S.D.;Noh,T.W.;Lee,J.;Jo,W. Nature(London)1999,401,682.(16)Hervoches,C.H.;Lightfoot,P.Chem.Mater.1999,11,3359.(17)Hirata,T.;Yokokawa,T.Solid State Commun.1997,104,673.5025Chem.Mater.2003,15,5025-502810.1021/cm034580l CCC:$25.00©2003American Chemical SocietyPublished on Web11/21/2003described as a-a-c0in Glazer’s notation.18These two structural features act in concert to lower the symmetry from tetragonal to orthorhombic,however,they are not linked,and there is no reason to suppose that both modes will condense at precisely the same temperature. Rather,it is probable that these modes will condense successively and the two end member phases will be linked by an intermediate ing powder neutron diffraction methods,Macquart and Lightfoot have in-dependently shown that the A21am to I4/mmm transi-tion in the n)2Aurivillius phases SrBi2Ta2O911and Sr0.85Bi2.1Ta2O912proceeds via an intermediate paraelec-tric Amam phase in each case.This Amam phase is also seen in PbBi2M2O9(M)Nb,Ta).19That is,upon cooling from the I4/mmm phase,the tilting of the octahedra occurs before the cation displacement.The same se-quence is reported to occur in the n)4Aurivillius oxide SrBi4Ti4O15.20The aim of the present work is to establish whether the B2eb to I4/mmm transition in Bi4Ti3O12is first order as proposed by Hirata and Yokakowa17or if it actually occurs continuously via an intermediate phase as seen in SrBi2Ta2O9.11To establish this we have investigated the temperature dependence of the structure of Bi4-Ti3O12from room temperature to800°C using high-resolution synchrotron radiation.Near the Curie tem-perature fine temperature intervals have been used to detect the intermediate phase which exists over a very limited temperature range.Experimental SectionThe crystalline sample of Bi4Ti3O12was prepared by the solid-state reaction of stoichiometric quantities of Bi2O3 (99.999%,Aldrich)and TiO2(99.9%,Aldrich).The heating sequence used was700°C/24h and850°C/48h,with intermediate regrinding.The sample was slowly cooled to room temperature in the furnace.The sample purity was established by powder X-ray dif-fraction measurements using Cu K R radiation on a Shimadzu D-6000Diffractometer.Room-and variable-temperature syn-chrotron X-ray powder diffraction patterns were collected on a high-resolution Debye Scherrer diffractometer at beamline 20B,the Australian National Beamline Facility,at the Photon Factory,Japan.21The sample was finely ground and loaded into a0.3-mm quartz capillary that was rotated during the measurements.All measurements were performed under vacuum to minimize air scatter.Data were recorded using two Fuji image plates.Each image plate was20×40cm and each covered40°in2θ.A thin strip(ca.0.5cm wide)was used to record each diffraction pattern so that up to30patterns could be recorded before reading the image plates.The data were collected at a wavelength of0.75Å(calibrated with a NIST Si 640c standard)over the2θrange of5-75°with step size of 0.01°.The patterns were collected in the temperature range of100-800°C in25°C steps or600-803°C in7°C steps, and with30-min counting time at each temperature.Struc-tures were refined by the Rietveld method using the program Rietica.22The positions of the cations were well described in these analyses,however the estimated standard deviations (esds)of the Ti-O bond distances(typically around0.02Å) preclude any detailed discussion of the temperature depen-dence of the bond distances.Results and DiscussionThe published room-temperature structure for Bi4-Ti3O12was used as a starting model in our Rietveldrefinements,and the structural refinement proceededwithout event.The temperature dependence of thelattice parameters and volumes are illustrated in Figure1.All the lattice parameters show a smooth increasedue to thermal expansion as the sample is heated toca.500°C.Above this temperature the cell continuesto expand along both the b-and c-directions,howeverthe a-parameter is essentially constant.This behavioris very similar to that displayed by a number of simplerABO3perovskites23-25and is apparently related to thegradual reduction in distortion resulting from a reduc-tion in the magnitude of the tilting of the BO6octahedraas the temperature is increased.Near675°C there is arapid decrease in the a-parameter although as is clearlyevident from Figure1this is not a discontinuousdecrease but rather a rapid progressive drop.At thesame temperature the c-axis expands rapidly,Figure1.The transition to the tetragonal phase is clearlyevident in the200/020and317/137reflections(near2θ≈15.8and26.7°,respectively,whereλ)0.75Å).As illustrated in Figure2the diffraction pattern recordedat670°C shows obvious splitting of the200/020and317/137reflections that is clearly indicative of orthor-hombic symmetry.This splitting remains clearly visibleto the eye until677°C and can be discerned by profileanalysis at684°C.At691°C no splitting or diagnosticasymmetry of these or other peaks is apparent and itis concluded that the structure is tetragonal.That is,the transition to the tetragonal structure occurs near690°C,which is around15°above the reported ferro-electric Curie temperature for Bi4Ti3O12.(Some care isrequired when comparing the transition temperaturesreported in various studies because of both samplevariation(induced by the different heating regimesused)and possible variations in the high-temperaturethermometry.)Above691°C the structure has been refined in thetetragonal space group I4/mmm and is as described byHervoches and Lightfoot.16That the paraelectric phaseis tetragonal in I4/mmm well above the ferroelectricCurie temperature was confirmed from the high-resolu-tion powder neutron diffraction data by Hervoches andLightfoot from both the cell metric and the absence ofany superlattice reflections indicative of TiO6tilting. The thermal expansion of the c-axis both above and below the T c is relatively linear and can be well-fitted to a simple linear equation c)5.277×10-4T+32.725 for T<677°C,and c)5.116×10-4T+32.725for T> 691°C.Examining the temperature dependence of the long c-axis we observed a large difference between the(18)Glazer,A.M.Acta Crystallogr.B1972,28,3384.(19)Macquart,R.;Kennedy,B.J.;Hunter,B.A.;Howard,C.J.J. Phys.:Condens.Matter2002,14,7955.(20)Hervoches,C.H.;Snedden,A.;Riggs,R.;Kilcoyne,S.H.; Manuel,P.;Lightfoot,P.J.Solid State Chem.2002,164,280.(21)Sabine,T.M.;Kennedy,B.J.;Garrett,R.F.;Foran,G.J.; Cookson,D.J.J.Appl.Crystallogr.1995,28,513.(22)Howard C.J.;Hunter,B.A.A Computer Program for Rietveld Analysis of X-ray and Neutron Powder Diffraction Patterns;Lucas Heights Research Laboratories:New South Wales,Australia,1998; pp1-27.(23)Howard,C.J.;Knight,K.S.;Kisi,E.H.;Kennedy,B.J.J. Phys.:Condens.Matter2000,12,L677.(24)Kennedy,B.J.;Howard,C.J.;Thorogood,G.J.;Hester,J.R. J.Solid State Chem.2001,161,106.(25)Kennedy,B.J.;Howard,C.J.;Chakoumakos,B.C.J.Phys. C:Condens.Matter1999,11,1479.5026Chem.Mater.,Vol.15,No.26,2003Zhou et al.values of the low-temperature orthorhombic,ferroelec-tric phase and those for the high-temperature tetragonal and paraelectric phase.Clearly the value for the c -parameter at 684°C does not fall into either series.A similar conclusion can be made for both the a -and b -parameters.These temperature-dependent changes in the lattice parameters are more reminiscent of the continuous-phase transitions observed in oxides such as Bi 2PbNb 2O 919than of the more subtle changes observed in the high-temperature second-order incommensurate-to-commensurate phase transition observed in Bi 2-MoO 6.26A feature of many first-order phase transitions is the coexistence of a two phase region.Attempts to fit the pattern at 684°C to a two-phase orthorhombic/tetragonal model with lattice parameters obtained by appropriate linear extrapolation were unsuccessful.It was concluded that a single phase was present at this temperature and this was neither the orthorhombic in B 2eb nor the tetragonal I 4/mmm .(26)Buttrey,D.J.;Vogt,T.;White,B.D.J.Solid State Chem.2000,155,206.(27)/∼stokesh/isotropy.html.Figure 1.Temperature dependence of the lattice parameters and volume for Bi 4Ti 3O 12obtained from Rietveld analysis of variable-temperature synchrotron diffraction data.For ease of comparison the values in the 2× 2×1orthorhombic cells have been reduced to the equivalent tetragonal values.Figure 2.Portions of the Rietveld fits for Bi 4Ti 3O 12showing the temperature dependence of the splitting of the tetragonal 110reflection into the orthorhombic 200/020pair near 2θ)15.8°and of the 127reflection into the 317/137pair near 2θ)26.8°.The 0012reflection is apparent near 2θ)15.6°.Ferroelectric Phase Transition in Bi 4Ti 3O 12Chem.Mater.,Vol.15,No.26,20035027In summary,we observe an orthorhombic structure at 684°C whose lattice parameter clearly distinguishes it from the low-temperature orthorhombic and high-temperature tetragonal phases.(In fact,even if we could not distinguish this as a separate phase,the observed continuity of transition,together with the group theo-retical arguments to follow,would indicate that such an intermediate orthorhombic phase is involved in the transition.)If we assume that all three phases have commensurate structures then it should be possible to identify the space group of the intermediate phase from a group theoretical analysis.To do this we used the program ISOTROPY.27This analysis confirmed that a direct B 2eb to I 4/mmm transition could not be continu-ous.The two modes responsible for the transition were identified as Γ5-describing the cation displacement and X 3+associated with the tilting of the TiO 6octahedra.28Whichever of these modes condenses first,an interme-diate orthorhombic structure based on a 2× 2×1superstructure of the parent I 4/mmm structure is involved,as illustrated in Figure 3,and the successive phase transitions through either of these intermediates are allowed to be continuous.If the initial distortion is cation displacement via the Γ5-mode then a ferroelec-tric orthorhombic structure that lacks any tilting of the TiO 6octahedra is expected.The resulting structure in Fmm 2can continuously transform to the observed room-temperature orthorhombic B 2eb phase through tilting of the TiO 6octahedra via the X 3+mode.(This descrip-tion refers still to the parent structure in I 4/mmm .)Alternatively,the initial distortion may be the introduc-tion of tilting of the TiO 6octahedra resulting in a Cmca phase followed by cation displacement.The available synchrotron diffraction data do not allow us to unequivoc-ally distinguish between these two possibilities;how-ever,by analogy with SrBi 2Ta 2O 9,we favor the latter possibility.11The confirmation of this will require a high-resolution neutron diffraction study.It is illuminating to compare our results with those of Hirata and Yokokawa.17First,Figure 2of their report suggests they recorded very little data in the ferroelec-tric phase with only four temperature points obvious.A somewhat greater number of temperatures were examined (nine)in the paraelectric phase,apparently at 20°C intervals.Crucially,no data appear to have been collected between 475and 675°C,the latter corresponding to the reported ferroelectric Curie tem-perature for Bi 4Ti 3O 12.There is then a relatively large jump in the temperatures used by Hirata and Yokokawa to above 695°C.In our present study we have used relatively coarse temperature increments (25°C)to monitor the general form of the phase transition,but then we have used much finer intervals (7°C)to probe the nature (first order or continuous)of the ferroelectric to paraelectric transition.We conclude that as a result of the relatively coarse temperature intervals used by Hirata and Yokokawa it was not possible for those authors to establish whether the ferroelectric to paraelec-tric transition in Bi 4Ti 3O 12was first order or continuous.The temperature dependence of the lattice parameters established using powder neutron diffraction data de-scribed by Lightfoot and Hervoches 16,29is very similar to that observed here for the data collected in 25°C intervals.On the basis of a similar density of data,Lightfoot and Hervoches 29concluded that the transition is first order.In comparison,our data collected in 7°C intervals using high-resolution synchrotron X-ray meth-ods strongly suggests the transition is continuous,albeit involving an intermediate phase.In conclusion we have identified the existence of an intermediate orthorhombic phase in the solid-state phase transition of the ferroelectric n )3Aurivillius phases Bi 4Ti 3O 12.Two possible orthorhombic phases were iden-tified using group theory,Fmm 2and Cmca ,depending on the sequence in which the two modes responsible for the lowering of symmetry condense.By analogy with the n )2oxide SrBi 2Ta 2O 9,the most likely sequence of transitions is B 2cb 670°C fCmca 695°CfI 4/mmm .Clearly confirming the existence of and establishing the precise structure of the proposed intermediate phase is of considerable interest,and efforts aimed at this are in progress.Acknowledgment.This work performed at the Australian National Beamline Facility was supported by the Australian Synchrotron Research Program,which is funded by the Commonwealth of Australia under the Major National Research Facilities program.B.J.K.acknowledges the support of the Australian Research Council.The assistance of Dr.James Hester at the ANBF is gratefully acknowledged.We thank Dr.P.Lightfoot for bringing ref 29to our attention while this manuscript was under review.CM034580L(28)Miller,S.C.;Love,W.F.Tables of Irreducible Representations of Space Groups and Co-representations of Magnetic Space Groups ;Pruett Press:Boulder,CO,1967.(29)Hervoches,C.H.;Lightfoot,P.Proceedings of CIMTEC,10th International Ceramics Congress ,Florence,Italy,July 2002.Figure 3.Schematic diagram showing the group -subgroup relationships for the n )3Aurivillius oxides.The solid lines show the transitions that are allowed to be continuous.The tilt system for the intermediate phases is given.5028Chem.Mater.,Vol.15,No.26,2003Zhou etal.。

Parallel and Distributed Computing and Systems

Proceedings of the IASTED International ConferenceParallel and Distributed Computing and SystemsNovember3-6,1999,MIT,Boston,USAParallel Reﬁnement of Unstructured MeshesJos´e G.Casta˜n os and John E.SavageDepartment of Computer ScienceBrown UniversityE-mail:jgc,jes@AbstractIn this paper we describe a parallel-reﬁnement al-gorithm for unstructuredﬁnite element meshes based on the longest-edge bisection of triangles and tetrahedrons. This algorithm is implemented in P ARED,a system that supports the parallel adaptive solution of PDEs.We dis-cuss the design of such an algorithm for distributed mem-ory machines including the problem of propagating reﬁne-ment across processor boundaries to obtain meshes that are conforming and non-degenerate.We also demonstrate that the meshes obtained by this algorithm are equivalent to the ones obtained using the serial longest-edge reﬁne-ment method.Weﬁnally report on the performance of this reﬁnement algorithm on a network of workstations.Keywords:mesh reﬁnement,unstructured meshes,ﬁnite element methods,adaptation.1.IntroductionTheﬁnite element method(FEM)is a powerful and successful technique for the numerical solution of partial differential equations.When applied to problems that ex-hibit highly localized or moving physical phenomena,such as occurs on the study of turbulence inﬂuidﬂows,it is de-sirable to compute their solutions adaptively.In such cases, adaptive computation has the potential to signiﬁcantly im-prove the quality of the numerical simulations by focusing the available computational resources on regions of high relative error.Unfortunately,the complexity of algorithms and soft-ware for mesh adaptation in a parallel or distributed en-vironment is signiﬁcantly greater than that it is for non-adaptive computations.Because a portion of the given mesh and its corresponding equations and unknowns is as-signed to each processor,the reﬁnement(coarsening)of a mesh element might cause the reﬁnement(coarsening)of adjacent elements some of which might be in neighboring processors.To maintain approximately the same number of elements and vertices on every processor a mesh must be dynamically repartitioned after it is reﬁned and portions of the mesh migrated between processors to balance the work.In this paper we discuss a method for the paral-lel reﬁnement of two-and three-dimensional unstructured meshes.Our reﬁnement method is based on Rivara’s serial bisection algorithm[1,2,3]in which a triangle or tetrahe-dron is bisected by its longest edge.Alternative efforts to parallelize this algorithm for two-dimensional meshes by Jones and Plassman[4]use randomized heuristics to reﬁne adjacent elements located in different processors.The parallel mesh reﬁnement algorithm discussed in this paper has been implemented as part of P ARED[5,6,7], an object oriented system for the parallel adaptive solu-tion of partial differential equations that we have devel-oped.P ARED provides a variety of solvers,handles selec-tive mesh reﬁnement and coarsening,mesh repartitioning for load balancing,and interprocessor mesh migration.2.Adaptive Mesh ReﬁnementIn theﬁnite element method a given domain is di-vided into a set of non-overlapping elements such as tri-angles or quadrilaterals in2D and tetrahedrons or hexahe-drons in3D.The set of elements and its as-sociated vertices form a mesh.With theaddition of boundary conditions,a set of linear equations is then constructed and solved.In this paper we concentrate on the reﬁnement of conforming unstructured meshes com-posed of triangles or tetrahedrons.On unstructured meshes, a vertex can have a varying number of elements adjacent to it.Unstructured meshes are well suited to modeling do-mains that have complex geometry.A mesh is said to be conforming if the triangles and tetrahedrons intersect only at their shared vertices,edges or faces.The FEM can also be applied to non-conforming meshes,but conformality is a property that greatly simpliﬁes the method.It is also as-sumed to be a requirement in this paper.The rate of convergence and quality of the solutions provided by the FEM depends heavily on the number,size and shape of the mesh elements.The condition number(a)(b)(c)Figure1:The reﬁnement of the mesh in using a nested reﬁnement algorithm creates a forest of trees as shown in and.The dotted lines identify the leaf triangles.of the matrices used in the FEM and the approximation error are related to the minimum and maximum angle of all the elements in the mesh[8].In three dimensions,the solid angle of all tetrahedrons and their ratio of the radius of the circumsphere to the inscribed sphere(which implies a bounded minimum angle)are usually used as measures of the quality of the mesh[9,10].A mesh is non-degenerate if its interior angles are never too small or too large.For a given shape,the approximation error increases with ele-ment size(),which is usually measured by the length of the longest edge of an element.The goal of adaptive computation is to optimize the computational resources used in the simulation.This goal can be achieved by reﬁning a mesh to increase its resolution on regions of high relative error in static problems or by re-ﬁning and coarsening the mesh to follow physical anoma-lies in transient problems[11].The adaptation of the mesh can be performed by changing the order of the polynomi-als used in the approximation(-reﬁnement),by modifying the structure of the mesh(-reﬁnement),or a combination of both(-reﬁnement).Although it is possible to replace an old mesh with a new one with smaller elements,most -reﬁnement algorithms divide each element in a selected set of elements from the current mesh into two or more nested subelements.In P ARED,when an element is reﬁned,it does not get destroyed.Instead,the reﬁned element inserts itself into a tree,where the root of each tree is an element in the initial mesh and the leaves of the trees are the unreﬁned elements as illustrated in Figure1.Therefore,the reﬁned mesh forms a forest of reﬁnement trees.These trees are used in many of our algorithms.Error estimates are used to determine regions where adaptation is necessary.These estimates are obtained from previously computed solutions of the system of equations. After adaptation imbalances may result in the work as-signed to processors in a parallel or distributed environ-ment.Efﬁcient use of resources may require that elements and vertices be reassigned to processors at runtime.There-fore,any such system for the parallel adaptive solution of PDEs must integrate subsystems for solving equations,adapting a mesh,ﬁnding a good assignment of work to processors,migrating portions of a mesh according to anew assignment,and handling interprocessor communica-tion efﬁciently.3.P ARED:An OverviewP ARED is a system of the kind described in the lastparagraph.It provides a number of standard iterativesolvers such as Conjugate Gradient and GMRES and pre-conditioned versions thereof.It also provides both-and -reﬁnement of meshes,algorithms for adaptation,graph repartitioning using standard techniques[12]and our ownParallel Nested Repartitioning(PNR)[7,13],and work mi-gration.P ARED runs on distributed memory parallel comput-ers such as the IBM SP-2and networks of workstations.These machines consist of coarse-grained nodes connectedthrough a high to moderate latency network.Each nodecannot directly address a memory location in another node. In P ARED nodes exchange messages using MPI(Message Passing Interface)[14,15,16].Because each message has a high startup cost,efﬁcient message passing algorithms must minimize the number of messages delivered.Thus, it is better to send a few large messages rather than many small ones.This is a very important constraint and has a signiﬁcant impact on the design of message passing algo-rithms.P ARED can be run interactively(so that the user canvisualize the changes in the mesh that results from meshadaptation,partitioning and migration)or without directintervention from the user.The user controls the systemthrough a GUI in a distinguished node called the coordina-tor,.This node collects information from all the other processors(such as its elements and vertices).This tool uses OpenGL[17]to permit the user to view3D meshes from different angles.Through the coordinator,the user can also give instructions to all processors such as specify-ing when and how to adapt the mesh or which strategy to use when repartitioning the mesh.In our computation,we assume that an initial coarse mesh is given and that it is loaded into the coordinator.The initial mesh can then be partitioned using one of a num-ber of serial graph partitioning algorithms and distributed between the processors.P ARED then starts the simulation. Based on some adaptation criterion[18],P ARED adapts the mesh using the algorithms explained in Section5.Af-ter the adaptation phase,P ARED determines if a workload imbalance exists due to increases and decreases in the num-ber of mesh elements on individual processors.If so,it invokes a procedure to decide how to repartition mesh el-ements between processors;and then moves the elements and vertices.We have found that PNR gives partitions with a quality comparable to those provided by standard meth-ods such as Recursive Spectral Bisection[19]but which(b)(a)Figure2:Mesh representation in a distributed memory ma-chine using remote references.handles much larger problems than can be handled by stan-dard methods.3.1.Object-Oriented Mesh RepresentationsIn P ARED every element of the mesh is assigned to a unique processor.V ertices are shared between two or more processors if they lie on a boundary between parti-tions.Each of these processors has a copy of the shared vertices and vertices refer to each other using remote ref-erences,a concept used in object-oriented programming. This is illustrated in Figure2on which the remote refer-ences(marked with dashed arrows)are used to maintain the consistency of multiple copies of the same vertex in differ-ent processors.Remote references are functionally similar to standard C pointers but they address objects in a different address space.A processor can use remote references to invoke meth-ods on objects located in a different processor.In this case, the method invocations and arguments destined to remote processors are marshalled into messages that contain the memory addresses of the remote objects.In the destina-tion processors these addresses are converted to pointers to objects of the corresponding type through which the meth-ods are invoked.Because the different nodes are inher-ently trusted and MPI guarantees reliable communication, P ARED does not incur the overhead traditionally associated with distributed object systems.Another idea commonly found in object oriented pro-gramming and which is used in P ARED is that of smart pointers.An object can be destroyed when there are no more references to it.In P ARED vertices are shared be-tween several elements and each vertex counts the number of elements referring to it.When an element is created, the reference count of its vertices is incremented.Simi-larly,when the element is destroyed,the reference count of its vertices is decremented.When the reference count of a vertex reaches zero,the vertex is no longer attached to any element located in the processor and can be destroyed.If a vertex is shared,then some other processor might have a re-mote reference to it.In that case,before a copy of a shared vertex is destroyed,it informs the copies in other processors to delete their references to itself.This procedure insures that the shared vertex can then be safely destroyed without leaving dangerous dangling pointers referring to it in other processors.Smart pointers and remote references provide a simple replication mechanism that is tightly integrated with our mesh data structures.In adaptive computation,the struc-ture of the mesh evolves during the computation.During the adaptation phase,elements and vertices are created and destroyed.They may also be assigned to a different pro-cessor to rebalance the work.As explained above,remote references and smart pointers greatly simplify the task of creating dynamic meshes.4.Adaptation Using the Longest Edge Bisec-tion AlgorithmMany-reﬁnement techniques[20,21,22]have been proposed to serially reﬁne triangular and tetrahedral meshes.One widely used method is the longest-edge bisec-tion algorithm proposed by Rivara[1,2].This is a recursive procedure(see Figure3)that in two dimensions splits each triangle from a selected set of triangles by adding an edge between the midpoint of its longest side to the opposite vertex.In the case that makes a neighboring triangle,,non-conforming,then is reﬁned using the same algorithm.This may cause the reﬁnement to prop-agate throughout the mesh.Nevertheless,this procedure is guaranteed to terminate because the edges it bisects in-crease in length.Building on the work of Rosenberg and Stenger[23]on bisection of triangles,Rivara[1,2]shows that this reﬁnement procedure provably produces two di-mensional meshes in which the smallest angle of the re-ﬁned mesh is no less than half of the smallest angle of the original mesh.The longest-edge bisection algorithm can be general-ized to three dimensions[3]where a tetrahedron is bisected into two tetrahedrons by inserting a triangle between the midpoint of its longest edge and the two vertices not in-cluded in this edge.The reﬁnement propagates to neigh-boring tetrahedrons in a similar way.This procedure is also guaranteed to terminate,but unlike the two dimensional case,there is no known bound on the size of the small-est angle.Nevertheless,experiments conducted by Rivara [3]suggest that this method does not produce degenerate meshes.In two dimensions there are several variations on the algorithm.For example a triangle can initially be bisected by the longest edge,but then its children are bisected by the non-conforming edge,even if it is that is not their longest edge[1].In three dimensions,the bisection is always per-formed by the longest edge so that matching faces in neigh-boring tetrahedrons are always bisected by the same com-mon edge.Bisect()let,and be vertices of the trianglelet be the longest side of and let be the midpoint ofbisect by the edge,generating two new triangles andwhile is a non-conforming vertex doﬁnd the non-conforming triangle adjacent to the edgeBisect()end whileFigure3:Longest edge(Rivara)bisection algorithm for triangular meshes.Because in P ARED reﬁned elements are not destroyed in the reﬁnement tree,the mesh can be coarsened by replac-ing all the children of an element by their parent.If a parent element is selected for coarsening,it is important that all the elements that are adjacent to the longest edge of are also selected for coarsening.If neighbors are located in different processors then only a simple message exchange is necessary.This algorithm generates conforming meshes: a vertex is removed only if all the elements that contain that vertex are all coarsened.It does not propagate like the re-ﬁnement algorithm and it is much simpler to implement in parallel.For this reason,in the rest of the paper we will focus on the reﬁnement of meshes.5.Parallel Longest-Edge ReﬁnementThe longest-edge bisection algorithm and many other mesh reﬁnement algorithms that propagate the reﬁnement to guarantee conformality of the mesh are not local.The reﬁnement of one particular triangle or tetrahedron can propagate through the mesh and potentially cause changes in regions far removed from.If neighboring elements are located in different processors,it is necessary to prop-agate this reﬁnement across processor boundaries to main-tain the conformality of the mesh.In our parallel longest edge bisection algorithm each processor iterates between a serial phase,in which there is no communication,and a parallel phase,in which each processor sends and receives messages from other proces-sors.In the serial phase,processor selects a setof its elements for reﬁnement and reﬁnes them using the serial longest edge bisection algorithms outlined earlier. The reﬁnement often creates shared vertices in the bound-ary between adjacent processors.To minimize the number of messages exchanged between and,delays the propagation of reﬁnement to until has reﬁned all the elements in.The serial phase terminates when has no more elements to reﬁne.A processor informs an adjacent processor that some of its elements need to be reﬁned by sending a mes-sage from to containing the non-conforming edges and the vertices to be inserted at their midpoint.Each edge is identiﬁed by its endpoints and and its remote ref-erences(see Figure4).If and are sharedvertices,(a)(c)(b)Figure4:In the parallel longest edge bisection algo-rithm some elements(shaded)are initially selected for re-ﬁnement.If the reﬁnement creates a new(black)ver-tex on a processor boundary,the reﬁnement propagates to neighbors.Finally the references are updated accord-ingly.then has a remote reference to copies of and lo-cated in processor.These references are included in the message,so that can identify the non-conforming edge and insert the new vertex.A similar strategy can be used when the edge is reﬁned several times during the re-ﬁnement phase,but in this case,the vertex is not located at the midpoint of.Different processors can be in different phases during the reﬁnement.For example,at any given time a processor can be reﬁning some of its elements(serial phase)while neighboring processors have reﬁned all their elements and are waiting for propagation messages(parallel phase)from adjacent processors.waits until it has no elements to reﬁne before receiving a message from.For every non-conforming edge included in a message to,creates its shared copy of the midpoint(unless it already exists) and inserts the new non-conforming elements adjacent to into a new set of elements to be reﬁned.The copy of in must also have a remote reference to the copy of in.For this reason,when propagates the reﬁne-ment to it also includes in the message a reference to its copies of shared vertices.These steps are illustrated in Figure4.then enters the serial phase again,where the elements in are reﬁned.(c)(b)(a)Figure5:Both processors select(shaded)mesh el-ements for reﬁnement.The reﬁnement propagates to a neighboring processor resulting in more elements be-ing reﬁned.5.1.The Challenge of Reﬁning in ParallelThe description of the parallel reﬁnement algorithm is not complete because reﬁnement propagation across pro-cessor boundaries can create two synchronization prob-lems.Theﬁrst problem,adaptation collision,occurs when two(or more)processors decide to reﬁne adjacent elements (one in each processor)during the serial phase,creating two(or more)vertex copies over a shared edge,one in each processor.It is important that all copies refer to the same logical vertex because in a numerical simulation each ver-tex must include the contribution of all the elements around it(see Figure5).The second problem that arises,termination detection, is the determination that a reﬁnement phase is complete. The serial reﬁnement algorithm terminates when the pro-cessor has no more elements to reﬁne.In the parallel ver-sion termination is a global decision that cannot be deter-mined by an individual processor and requires a collabora-tive effort of all the processors involved in the reﬁnement. Although a processor may have adapted all of its mesh elements in,it cannot determine whether this condition holds for all other processors.For example,at any given time,no processor might have any more elements to re-ﬁne.Nevertheless,the reﬁnement cannot terminate because there might be some propagation messages in transit.The algorithm for detecting the termination of parallel reﬁnement is based on Dijkstra’s general distributed termi-nation algorithm[24,25].A global termination condition is reached when no element is selected for reﬁnement.Hence if is the set of all elements in the mesh currently marked for reﬁnement,then the algorithmﬁnishes when.The termination detection procedure uses message ac-knowledgments.For every propagation message that receives,it maintains the identity of its source()and to which processors it propagated reﬁnements.Each prop-agation message is acknowledged.acknowledges to after it has reﬁned all the non-conforming elements created by’s message and has also received acknowledgments from all the processors to which it propagated reﬁnements.A processor can be in two states:an inactive state is one in which has no elements to reﬁne(it cannot send new propagation messages to other processors)but can re-ceive messages.If receives a propagation message from a neighboring processor,it moves from an inactive state to an active state,selects the elements for reﬁnement as spec-iﬁed in the message and proceeds to reﬁne them.Let be the set of elements in needing reﬁnement.A processor becomes inactive when:has received an acknowledgment for every propa-gation message it has sent.has acknowledged every propagation message it has received..Using this deﬁnition,a processor might have no more elements to reﬁne()but it might still be in an active state waiting for acknowledgments from adjacent processors.When a processor becomes inactive,sends an acknowledgment to the processors whose propagation message caused to move from an inactive state to an active state.We assume that the reﬁnement is started by the coordi-nator processor,.At this stage,is in the active state while all the processors are in the inactive state.ini-tiates the reﬁnement by sending the appropriate messages to other processors.This message also speciﬁes the adapta-tion criterion to use to select the elements for reﬁnement in.When a processor receives a message from,it changes to an active state,selects some elements for reﬁne-ment either explicitly or by using the speciﬁed adaptation criterion,and then reﬁnes them using the serial bisection algorithm,keeping track of the vertices created over shared edges as described earlier.When itﬁnishes reﬁning its ele-ments,sends a message to each processor on whose shared edges created a shared vertex.then listens for messages.Only when has reﬁned all the elements speciﬁed by and is not waiting for any acknowledgment message from other processors does it sends an acknowledgment to .Global termination is detected when the coordinator becomes inactive.When receives an acknowledgment from every processor this implies that no processor is re-ﬁning an element and that no processor is waiting for an acknowledgment.Hence it is safe to terminate the reﬁne-ment.then broadcasts this fact to all the other proces-sors.6.Properties of Meshes Reﬁned in ParallelOur parallel reﬁnement algorithm is guaranteed to ter-minate.In every serial phase the longest edge bisectionLet be a set of elements to be reﬁnedwhile there is an element dobisect by its longest edgeinsert any non-conforming element intoend whileFigure6:General longest-edge bisection(GLB)algorithm.algorithm is used.In this algorithm the reﬁnement prop-agates towards progressively longer edges and will even-tually reach the longest edge in each processor.Between processors the reﬁnement also propagates towards longer edges.Global termination is detected by using the global termination detection procedure described in the previous section.The resulting mesh is conforming.Every time a new vertex is created over a shared edge,the reﬁnement propagates to adjacent processors.Because every element is always bisected by its longest edge,for triangular meshes the results by Rosenberg and Stenger on the size of the min-imum angle of two-dimensional meshes also hold.It is not immediately obvious if the resulting meshes obtained by the serial and parallel longest edge bisection al-gorithms are the same or if different partitions of the mesh generate the same reﬁned mesh.As we mentioned earlier, messages can arrive from different sources in different or-ders and elements may be selected for reﬁnement in differ-ent sequences.We now show that the meshes that result from reﬁning a set of elements from a given mesh using the serial and parallel algorithms described in Sections4and5,re-spectively,are the same.In this proof we use the general longest-edge bisection(GLB)algorithm outlined in Figure 6where the order in which elements are reﬁned is not spec-iﬁed.In a parallel environment,this order depends on the partition of the mesh between processors.After showing that the resulting reﬁned mesh is independent of the order in which the elements are reﬁned using the serial GLB al-gorithm,we show that every possible distribution of ele-ments between processors and every order of parallel re-ﬁnement yields the same mesh as would be produced by the serial algorithm.Theorem6.1The mesh that results from the reﬁnement of a selected set of elements of a given mesh using the GLB algorithm is independent of the order in which the elements are reﬁned.Proof:An element is reﬁned using the GLBalgorithm if it is in the initial set or reﬁnementpropagates to it.An element is reﬁnedif one of its neighbors creates a non-conformingvertex at the midpoint of one of its edges.Thereﬁnement of by its longest edge divides theelement into two nested subelements andcalled the children of.These children are inturn reﬁned by their longest edge if one of their edges is non-conforming.The reﬁnement proce-dure creates a forest of trees of nested elements where the root of each tree is an element in theinitial mesh and the leaves are unreﬁned ele-ments.For every element,let be the reﬁnement tree of nested elements rooted atwhen the reﬁnement procedure terminates. Using the GLB procedure elements can be se-lected for reﬁnement in different orders,creating possible different reﬁnement histories.To show that this cannot happen we assume the converse, namely,that two reﬁnement histories and generate different reﬁned meshes,and establish a contradiction.Thus,assume that there is an ele-ment such that the reﬁnement trees and,associated with the reﬁnement histories and of respectively,are different.Be-cause the root of and is the same in both reﬁnement histories,there is a place where both treesﬁrst differ.That is,starting at the root,there is an element that is common to both trees but for some reason,its children are different.Be-cause is always bisected by the longest edge, the children of are different only when is reﬁned in one reﬁnement history and it is not re-ﬁned in the other.In other words,in only one of the histories does have children.Because is reﬁned in only one reﬁnement his-tory,then,the initial set of elements to reﬁne.This implies that must have been reﬁned because one of its edges became non-conforming during one of the reﬁnement histo-ries.Let be the set of elements that are present in both reﬁnement histories,but are re-ﬁned in and not in.We deﬁne in a similar way.For each reﬁnement history,every time an ele-ment is reﬁned,it is assigned an increasing num-ber.Select an element from either or that has the lowest number.Assume that we choose from so that is reﬁned in but not in.In,is reﬁned because a neigh-boring element created a non-conforming ver-tex at the midpoint of their shared edge.There-fore is reﬁned in but not in because otherwise it would cause to be reﬁned in both sequences.This implies that is also in and has a lower reﬁnement number than con-。

数学专有名词英文

数学专业英语词汇英汉对照1 概率论与数理统计词汇英汉对照表Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test 柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数Mmain effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariate analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Qqadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ssample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数2. 最优化方法词汇英汉对照表Aactive constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子Bbandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界Ccoefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomial interpolation method 三次多项式插值法curve-fitting 曲线拟合Ddata-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search method 直接搜索法direction of search 搜索方向discontinuous 不连续Eeigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的Ffeasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶GGauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradient method 梯度法Hhandle 句柄Hessian matrix 海色矩阵Iindependent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代JJacobian 雅可比矩阵LLagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardt method 列文伯格-马夸尔特法line search 一维搜索linear 线性的linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解Mmedium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的norm 范数Oobjective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统Pparameter 参数partial derivatives 偏导数polynomial interpolation method 多项式插值法Qquadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划Rreal-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性Sscalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method 序列二次规划法simplex search method 单纯形法solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点stationary point 驻点step length 步长subspace trust region method 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵Ttermination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepest descent 最速下降法transpose 转置Uunconstrained 无约束的under-determined system 负定系统Vvariable 变量vector 矢量Wweighting matrix 加权矩阵3 样条词汇英汉对照表Aapproximation 逼近array 数组a spline in b-form/b-spline b样条a spline of polynomial piece /ppform spline 分段多项式样条Bbivariate spline function 二元样条函数break/breaks 断点Ccoefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation 三次样条插值/三次样条内插curve 曲线Ddegree of freedom 自由度dimension 维数end conditions 约束条件Iinput argument 输入参数interpolation 插值/内插interval 取值区间Kknot/knots 节点Lleast-squares approximation 最小二乘拟合Mmultiplicity 重次multivariate function 多元函数Ooptional argument 可选参数order 阶次output argument 输出参数Ppoint/points 数据点rational spline 有理样条rounding error 舍入误差（相对误差）Sscalar 标量sequence 数列（数组）spline 样条spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量weight/weights 权重4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图converge 收敛coordinate 坐标系Ddecomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditionsDirichlet边界条件eigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解Ggeneralized Neumann boundary condition 推广的Neumann边界条件geometry 几何形状geometry description matrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface（GUI）图形用户界面Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环Mmachine precision 机器精度mixed boundary condition 混合边界条件NNeuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量PParabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变plane stress 平面应力Poisson’s equation 泊松方程polygon 多边形positive definite 正定Qquality 质量Rrefined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数Ssingular 奇异的。

A thermal study on the structural changes of bimetallic ZrO2-modified TiO2 nanotubes synthesized

IOP P UBLISHING N ANOTECHNOLOGY Nanotechnology20(2009)195601(9pp)doi:10.1088/0957-4484/20/19/195601A thermal study on the structural changes of bimetallic ZrO2-modiﬁed TiO2 nanotubes synthesized using supercritical CO2R A Lucky and P A Charpentier1Department of Chemical and Biochemical Engineering,University of Western Ontario,London,ON,N6A5B9,CanadaE-mail:pcharpentier@eng.uwo.caReceived4November2008,inﬁnal form19February2009Published21April2009Online at /Nano/20/195601AbstractIn this study the thermal behavior of bimetallic ZrO2–TiO2(10/90mol/mol)nanotubes arediscussed which were synthesized via a sol–gel process in supercritical carbon dioxide(scCO2).The effects of calcination temperature on the morphology,phase structure,mean crystallite size,speciﬁc surface area and pore volume of the nanotubes were investigated by using a variety ofphysiochemical techniques.We report that SEM and TEM images showed that the nanotubularstructure was preserved at up to800◦C calcination temperature.When exposed to highertemperatures(900–1000◦C)the ZrO2–TiO2tubes deformed and the crystallites fused together,forming larger crystallites,and a bimetallic ZrTiO4species was detected.These results werefurther examined using TGA,FTIR,XRD and HRTEM analysis.The BET textural propertiesdemonstrated that the presence of a small amount of Zr in the TiO2matrix inhibited the graingrowth,stabilized the anatase phase and increased the thermal stability.(Someﬁgures in this article are in colour only in the electronic version)1.IntroductionConsiderable effort is being devoted to the preparation of one-dimensional(1D)oxide nanostructures due to their poten-tial applications in a diversity of technologies including catal-ysis,high efﬁciency solar cells,coatings and sensors[1–3].In particular,titania(TiO2)nanotubes are receiving considerable attention due to titania’s unique optoelectronic,photochemical and dielectric properties,along with being a low cost material for potential commercial employment[4–7].Having such unique properties,TiO2nanomaterials with deﬁned structures are highly desirable for electron-transport materials in dye-sensitized solar cells,as photocatalysts,photoconductive agents and nanoﬁllers in polymer composites[7,8].The performance of these titania nanomaterials widely relies on their crystallinity,crystallite size,crystal structure,speciﬁc surface area and thermal stability[9,10].As examples,the 1Author to whom any correspondence should be addressed.role of anatase and rutile crystal phases in TiO2nanostructures is still under active investigation for photocatalysis,where the interface between these nanostructures is believed to be the catalytic active site[11].The rutile crystal structure is important for stronger materials for use in orthopedic applications such as bone cements[12].For preparing these oxide nanostructures,the sol–gel method is becoming the standard method as it provides a uniform phase distribution,high purity,low temperature processing,and better size and morphology control[13]. However,the properties of sol–gel-synthesized binary metal oxides strongly depend on the synthesis conditions,such as the type of alkoxide(s),temperature,catalyst,solvent and solvent removal process[14,15].In the past decade,direct sol–gel reactions in supercritical carbon dioxide(scCO2)have attracted much attention for synthesizing oxide nanomaterials[16–19]. This approach has many advantages over the conventional sol–gel process operated in an organic solvent,as the resulting ma-terials maintain nanofeatures and a high surface area after CO2drying and venting[20].Low viscosity,‘zero’surface tension and high diffusivity of scCO2are favorable physical properties of the solvent for synthesizing potential superior ultraﬁne and uniform nanomaterials.As well,CO2is inexpensive, inﬂammable and considered a‘green’reaction medium[21].Previously,we discovered that bimetallic Zr-modiﬁed TiO2(Zr–TiO2)nanotubes could be synthesized in scCO2,and our preliminary results indicated that the nanotubes gave a high surface area,up to430m2g−1,as prepared[22].According to several studies,a small amount of transition metal doping has been found very effective to improve the thermal stability and activity of TiO2,particularly by using zirconia[23–26].In addition to the synthesis conditions,the calcination conditions are very important to the crystal structures of the metal oxide nanomaterials obtained,and subsequently,their potential end use applications.Spijksma et al[27]synthesized titania–zirconia microporous composite membranes using a1:1molar ratio by using the sol–gel process.The crystallization temperature for these materials was750◦C;however,after calcining at800◦C,an orthorhombic ZrTi2O6structure was formed,commonly known as srilankite.Whereas Zou et al [28]synthesized binary oxides by hydrolysis of titanium and zirconium nitrate solutions at various ratios.After calcining at 800◦C,the binary oxide showed the presence of the ZrTiO4 crystal phase and very low surface areas.Kitiyanan et al [23]synthesized5mol%zinconia-modiﬁed TiO2using a sol–gel process.They showed that this small amount of Zr stabilized the anatase phase up to800◦C,but that the anatase phase completely transformed into the rutile phase at 1000◦C.However,the materials calcined at these very high temperatures showed very low surface areas.As the nanotubular structure of the bimetallic ZrO2/TiO2 nanotubes has many potentially interesting applications, however,their structure and crystal morphology changes with calcination temperature have not been investigated.Hence, this study focused on the thermal behavior of the synthesized Zr–TiO2(10/90mol/mol)nanotubes prepared via an acetic acid modiﬁed sol–gel process in scCO2.The synthesized materials were calcined at different temperatures and the effects of calcination temperature on the morphology,phase structure,mean crystallite size,speciﬁc surface area and pore volume were investigated using a variety of physicochemical characterization techniques.2.Experimental details2.1.MaterialsReagent grade titanium(IV)isopropoxide(TIP,97%,Aldrich), zirconium(IV)propoxide(ZPO,70%,Aldrich),acetic acid(99.7%,Aldrich)and instrument grade carbon dioxide (99.99%,BOC)were used without further puriﬁcation.2.2.Nanotube synthesisThe procedure previously reported[22]was used for synthesis of10%ZrO2–90%TiO2nanotubes.2.3.CharacterizationScanning electron microscopy(SEM)measurements were usedto determine the size and morphology of nanomaterials usinga LEO1530scanning electron microscope.Transmissionelectron microscopy(TEM)and HRTEM images wereobtained using a Philips CM10and JEOL2010f.Thespecimens were dispersed in methanol and placed on a coppergrid covered with a holey carbonﬁlm.Thermo-gravimetricanalysis(TGA)was performed under nitrogen atmosphere ona TA Instruments TA-Q500at a heating rate of10◦C min−1 from room temperature to1000◦C.IR spectra were recordedon a Bruker IFS55Spectrometer in the range500–4000cm−1.Each spectrum was recorded at4cm−1resolution using500scans.Sample pellets were obtained from the powder calcinedat various temperatures by mixing with a small amount of KBr,then analyzing in transmission mode.Bulk composition wasdetermined using energy-dispersive x-ray spectroscopy(EDX)attached to a LEO1530scanning electron microscope(SEM).X-ray diffraction(XRD)was performed utilizing a Rigakuemploying Cu Kα1+Kα2=1.54184˚A radiation with a power of40kV–35mA for the crystalline analysis.The broad-scan analysis was typically conducted within the2θrangeof10◦–80◦.The samples were further analyzed using a RenishawModel2000Raman spectrometer equipped with a633nmlaser.The power at the sample varied between0.2and0.5mWwith the beam defocused to an area of approximately5–10μmin diameter.The textural characterization,such as surface area,pore volume and pore size distribution of the aerogels andthe oxides,was obtained by N2physisorption at77K usinga Micromeritics ASAP2010.Prior to the N2physisorption,the samples were degassed at200◦C under vacuum.Fromthe N2adsorption isotherms,the speciﬁc surface area wascalculated.The mesopore volume(V BJH),the average pore diameters(d p)and the pore size distributions were estimated by the Barret–Joyner–Halenda(BJH)method applied to the desorption branch of the isotherm.3.Results and discussion3.1.Electron microscopy(SEM/TEM)The effects of calcination temperature on the morphology of the Zr–TiO2nanotubes’shape and size were characterized by SEM and TEM analysis.In the SEM analysis for the as-prepared materials,it can be seen inﬁgure1(a)that the aerogel powders were composed of a one-dimensional structure,with the nanotubes having a diameter of50–140nm and a length of several micrometers.Throughout the course of heat treatment, phase changes(amorphous to anatase to rutile)and sintering phenomena of the nanotubes were revealed by SEM and TEM investigations.The SEM image inﬁgure1(b)shows that the material calcined at500◦C has a similar structure,although very small holes are visible in the TEM image(ﬁgure2(b))on the walls of the nanotubes.The morphology of the calcined nanotubes at800◦C is still preserved,as shown inﬁgure1(c). As the temperature was increased further to1000◦C,the initial nanotubes disappeared and were replaced by nanometer-sizedFigure1.SEM:Zr–TiO2nanotubes calcined at:(a)as-prepared,(b)500,(c)800and(d)1000◦C(bar represents200nm;all the samples were examined after platinum coating).aggregated particles in the50–100nm size range,as shown in ﬁgure1(d).Along with SEM,TEM images gave more detailed morphological information on the tubular structure.The TEM image inﬁgure2(a)indicates that the as-prepared nanotubes possessed uniform inner and outer diameters,having thicknesses approx.14–50nm along their length,depending on synthesis conditions.Upon heat treatment to500◦C, the chemically bonded organic layer was removed from the synthesized nanotubes,resulting in small holes on the tube wall,which was conﬁrmed by the TEM image in ﬁgure2(b),although the internal structure was still maintained at this temperature.The SEM images showed that the outer morphology was preserved at800◦C,although at900◦C the TEM images shown that the inner hole has almost vanished with this additional heat energy(ﬁgure2(c)).The TEM images along with SEM images reveal that the nanotubes were deformed and the crystallites were fused together when the calcination temperature was increased to1000◦C (ﬁgure2(d)).3.2.Decomposition behavior(TGA/FTIR)Thermo-gravimetric analysis(TG–DTG)analysis was carried out to study the thermal decomposition behavior of the synthesized Zr–TiO2nanotubes.Figure3shows three main peaks in the TG–DTG analysis,which are in the ranges of 20–120,120–250and250–500◦C.Theﬁrst stage with peak maxima at37◦C gave only6wt%loss,which we attribute to the removal of residual solvent present in the synthesized materials.The second peak with its maximum at200◦C is attributed to the removal of bounded water and chemically bonded organic material,with approx.19wt%lost at this stage. The third peak,with its maximum at341◦C,is broad and is attributed to the removal of any bonded/coordinated organic material and−OH groups,with approximately an additional 21wt%loss at this temperature.The weight loss over500◦C was extremely small(0.14%)and attributed to removal of bounded−OH groups.The total weight loss measured from the TG curve was46wt%.Elemental analysis(EDX)was also performed(see table1)to investigate the change of composition with calci-nation temperature.It showed that the as-prepared nanotubes contained approx.30%carbon,with this value decreasing upon increasing calcination temperature.At300◦C,the carbon content was about18%,while when the temperature was increased to500◦C,all carbon-containing organic material was removed,consistent with the TG–DTG results.Materials calcined at higher temperatures had only metal,oxygen and a ratio of oxygen to metal atom change with temperature. Surfaces of metal oxides consist of unsaturated metal and oxide ions,and are usually terminated by−OH groups.Figure 2.TEM:Zr–TiO 2nanotubes calcined at (a)as-prepared,(b)500,(c)900and (d)1000◦C.Figure 3.Weight loss of nanotubes as a function of temperature.The −OH groups are formed by dissociative adsorption of H 2O molecules to reduce the coordinative unsaturation of the surface sites.It is very difﬁcult to analyze the amount of oxygen bonded with metal atoms only by the EDX method,as the amount of H present in the materials cannot be determined by EDX due to the low atomic weight of H.Infrared spectroscopy is an excellent method to study the behavior and properties of metal oxides [28].The powder ATR-FTIR spectra of the Zr–TiO 2nanotubes calcined at different temperatures in air are given in ﬁgure 4.TheTable positional change with calcination temperature determined by EDX.Sample(Cal.temp ◦C)Carbon (at.%)Oxygen (at.%)Ti (at.%)Zr (at.%)As-prepared 30.6±258.5±19.9±0.51.3±0.2T-30018.5±362.7±116.5±12.3±0.5T-5000±379.6±317.9±0.52.5±0.5spectra (ﬁgure 4(a))for the as-prepared nanotubes shows a broad peak at 3400cm −1assigned to the −OH group of absorbed water [29].The peaks at 1548and 1452cm −1are due to symmetric and asymmetric stretching of the zirconium titanium acetate complex,respectively [30].This metal acetate complex conﬁrms that the acetic acid formed bridging complexes with the metal ions,helping to stabilize the structures during their synthesis and self-assembly into nanotubular structures in scCO 2.The −CH 3group contributes the small peak at 1343cm −1,while the two small peaks at 1037and 1024cm −1correspond to the ending and bridging −OPr groups,respectively [31],indicating that unhydrolyzed −OPr groups were present in the as-prepared materials [32].The oxo bonds can be observed by the bands present below 657cm −1[30].Calcination at 400◦C signiﬁcantly diminishes the intensity of the C–H stretching at 2800–3000cm −1and the zirconium titanium acetate complex band at 1548andFigure4.The powder ATR-FTIR spectra of Zr–TiO2nanotubes calcined at different temperatures.1452cm−1,spectra given inﬁgure4(b).This indicates thatthe calcination at400◦C removes any organic material presentin the as-prepared nanotubes.No trace of IR bands fromthe organic groups was detected upon further heat treatment(ﬁgure4(c))and the broad peak at3400cm−1signiﬁcantlydecreased.The nanotubes calcined at1000◦C showed onlya small band at3400cm−1(ﬁgure4(d)),indicating only asmall amount of−OH groups were still present at this hightemperature.These results are consistent with the TG–DTGmeasurements and the electron microscopy results.3.3.XRD and HRTEMIn order to examine the phase structure and crystallite size,XRD and HRTEM were used to investigate the effects of thecalcination temperature on the crystal size and phase structure.During heat treatment,the as-prepared materials transferredfrom the amorphous to anatase to rutile phases.The XRDpatterns(ﬁgure5)of all the calcined samples indicate that theZr–TiO2nanotubes consist of anatase crystal,with no rutile phase being present up to700◦C.The as-prepared materials were amorphous,while whenincreasing the calcination temperature up to400◦C,thematerial reorganized itself and the anatase particles beganto grow,resulting in crystalline material.There wasno distinct Zr peak,indicating no phase separation,andthat the Zr was integrated within the anatase crystalstructure for this composition.Previous experimentsshowed that increasing concentrations of Zr alkoxidewere incorporated homogeneously into the nanotubularstructure[22].As well,we previously prepared a crystalof Zr2Ti4(μ3–O)4(OPr)4(μ–OPr)2(μ–OAc)10using lower concentrations of acetic acid[33],showing that the Zr is partof the crystal hexamer structure.By increasing the calcinationtemperature from400to800◦C,the peak intensities increasedas well as the width of the peaks becoming narrower,indicatingan improvement of the anatase phase and simultaneously thegrowth of anatase crystallites.The XRD patterns for theobserved nanotubes indicate that no rutile phase appeared upto calcination temperatures of700◦C.It also shows thatheat Figure5.The powder XRD spectra of Zr–TiO2nanotubes calcined at different temperatures.(A–anatase,R–rutile).Table2.Crystal size and crystal structure at different calcination temperatures.Sample(Cal.temp◦C)Crystallite size(nm)Crystal structure As-prepared—AmorphousT-4009.8AnataseT-50012.5AnataseT-60015.9AnataseT-70019.5AnataseT-80021.8Anatase(88%)48.1Rutile(12%)T-90027.8Anatase(42%)69.5Rutile(58%)T-100028.3Anatase(6%)90.8Rutile(94%) treatment at800◦C forms a very small peak of the rutile phase, and further heat treatment increased the amount of rutile phase, and a new peak appeared at2θ=30.4◦,which is assigned to zirconium titanium oxide(ZrTiO4)[28].The crystallite sizes of the calcined samples are summarized in table2and were estimated from these XRD patterns using Scherrer’s equation(equation(1)):D=0.9λβcosθ(1) where D is the average nanocrystallite size(nm),λis the x-ray wavelength(1.541˚A),βis the full width at half-maximum intensity(in radians)andθis half of the diffraction peak angle.For the nanotubes calcined at400◦C,crystallite sizes of approx.9.8nm were calculated,while further heat treatment increased the crystallite size moderately.Nanotubes calcined at800◦C gave crystallite sizes up to21.8nm,resulting in smaller crystallite materials,indicating that a small amount of zirconia inhibited grain growth during heat treatment[26]. The rutile crystallite size was calculated by Scherrer’s equation using rutile(110).The obtained crystallite size was>90nm at1000◦C calcination temperature,smaller than the value reported in the literature for rutile crystallites[29],likely due to the constrained geometry of the nanotubular structure.The TEM images of the1000◦C calcined nanotubes previously showed that the crystallites were fused together,formingFigure6.HRTEM:Zr–TiO2nanotubes calcined at(a)as-prepared,(b)500and(c)1000◦C.Bar represents10nm. larger crystallites.It is known that rutile and anataseshare two and four polyhedra edges,respectively,althoughboth are tetragonal[34].Due to changes in the crystalstructure,the nanotubes’morphology deformed at highercalcination temperatures.The phase compositions of the calcined samples are alsoreported in table2and were calculated using the integratedintensities of anatase(101)and rutile(110)peaks by theequation developed by Spurr and Myers[35]:X rutile=11+K(I a/I r)(2)where I a and I r are the integrated peak intensities of the anatase and rutile phases,respectively,and the empirical constant K was taken as0.79according to Spurr and Myers.From table2 we see that the material calcined at800◦C contained only12% rutile.Further heat treatment caused a dramatic increase in both the composition and size of the rutile particles,where at 900◦C,58%of the material was converted into rutile,which increased to approx.94%at1000◦C.Rutile is the most stable crystalline phase of TiO2and the phase transformation(anatase to rutile)depends on both the size and dopant present in the system[26].Sui et al reported that anatase-type TiO2nanostructures transformed into rutile phase(56%)after calcination at600◦C[18],whereas only 12%of the nanomaterial was converted into rutile at800◦C in the present study.Hence,consistent with the literature for non-nanotubular structures[23],the anatase phase of the bimetallic nanotubes can be stabilized by modifying titania with a small amount of ZrO2.However,the mechanism by which the zirconia stabilizes the TiO2anatase phase at higher temperatures is unclear.As shown in the XRD data,no distinct zirconium peak was observed,indicating that zirconia was well integrated into the anatase structure.This suggests that particle agglomeration was not favored and the particles grew by the Oswald ripening process during heat treatment[26].Due to the presence of zirconia,the Oswald ripening process was restricted,reducing the crystal growth rate and increasing the phase transformation temperature.Once individual crystallites reach a threshold size,a spontaneous phase change can occur.The rapid growth of rutile particles formed during heat treatment suggests that the growth mechanism consists of particle agglomeration or grain coalescence by grain boundary diffusion[36].There may be a threshold size limit for this transformation,below which no transformation occurs,which was>27nm for this study, similar to that reported for SiO2and ZrO2doping in a titania matrix[26,36].For this reason,no rutile phase appeared up to 700◦C.When calcining at800◦C,the crystallites size became >27nm and the anatase phase started transforming rapidly into the rutile phase.In addition to the XRD data,detailed information on the structural transformations and crystal growth can be obtained using HRTEM.The HRTEM micrographs of the as-preparedFigure7.The Raman spectra of Zr–TiO2nanotubes calcined at: (a)500,(b)600,(c)700,(d)800,(e)900and(f)1000◦C.(A–anatase,R–rutile).nanotubes were amorphous(ﬁgure6(a))with no ordered structure.The lattice image of nanotubes calcined at500◦C is given inﬁgure6(b),showing a grain size of approx.12nm width with a d spacing0.35nm,very close to the lattice spacing of the(101)planes of the anatase phase.However, all grains were not the same in terms of size and shape. Some were long with signiﬁcant lattice mismatch and grain boundaries.All these defects prevent rapid grain growth.The HRTEM image for the material calcined at1000◦C is given inﬁgure6(c),where little amorphous phase is observed.The crystallites were very large having a d spacing of0.245nm, which value is very close to the rutile(110)plane of titania. These observations also support the XRD and TEM analysis.3.4.RamanTo further verify these results,Raman spectra for the bimetallic nanotube samples were measured for several different calcination temperatures,as shown inﬁgure7.The spectrum for the sample calcined at500◦C(ﬁgure7(a)) shows Raman peaks at142,395,517and639cm−1,which can be assigned to the E g,B1g,B1g/A1g and E g modes of the anatase phase of titania,respectively,which agrees with published values[37].With increasing calcination temperature inﬁgures7(b)–(d),the intensity of the anatase phase increased, indicating a larger particle size being present,with the anatase peak shifting to lower frequencies.Lottici et al[38]explained this effect as the size-induced pressure effect on the vibrational modes,with the smaller the crystallite size,the higher the pressure and Raman frequencies.After calcining at900◦C (ﬁgure7(e)),three new peaks at230,442and612cm−1 appeared,which match the literature values for the rutile phase[37,38].Upon calcining at1000◦C(ﬁgure7(f)), all anatase-related peaks vanished and only the rutile-related peaks remain,indicating complete anatase to rutile phase transformation.The Raman results agree with the previous characterizationresults.Figure8.Ln of anatase and rutile crystallite size in nm as a function of the reciprocal of absolute temperature according to equation(3). (—anatase,•/◦—rutile).3.5.Activation energy of phase transformationsThe activation energy(kJ mol−1)of phase transformations can be calculated from the slope of a plot of ln rutile weight fraction versus the reciprocal of annealing temperature from XRD spectra according to Burns et al[39].This relationship is given asE a=−∂ln(X r)∂(1/T)R(3) where T is the temperature in kelvin,R is the universal gas constant(8.314J mol−1K−1)and X r is the weight fraction of the rutile phase as determined using equation(2).Figure8 shows the plot of the logarithm of the average crystallite size versus the reciprocal of the calcination temperature(solid lines),according to equation(3).A linear relationship is observed and the activation energy for the crystal growth of the Zr–TiO2nanotubes was calculated as12.8kJ mol−1and 36.4kJ mol−1for anatase and rutile phases,respectively.The activation energy for the phase transformation from anatase to rutile was also calculated,as shown by the dashed line in ﬁgure8,with a value of171kJ mol−1.These values are higher than those reported for nanocrystalline pure titania[39,40], which is a beneﬁcial effect of doping.In explanation,during the heat treatment,there are two competitive processes:grain growth and A→R phase transformation in the nanocrystalline materials.Both processes are easier for small grain-sized materials because the activation energy for growth and phase transformation is lower[41].3.6.BET analysisThe textural properties,i.e.the surface area,pore volume and pore size distributions of the as-prepared and calcined bimetallic nanotubes were characterized by nitrogen adsorp-tion studies.Figure9shows the nitrogen adsorption isotherms for both the as-prepared and calcined materials,which exhibit H3hysteresis loops(to800◦C),typical for mesoporous materials.The isotherm for the bimetallic metal oxide nanotubes calcined at1000◦C changes to a type I isotherm, typical for a microporous material[42].The lower limit ofFigure9.N2adsorption/desorption isotherm of the Zr–TiO2 nanotubes calcined at differenttemperatures.Figure10.BJH pore size distribution of the Zr–TiO2nanotubes calcined at different temperatures.the relative pressure for the hysteresis loop is characteristic of a given adsorbate at a given temperature[43].It can be seen fromﬁgure9for both the as-prepared nanotubes,and for those calcined at300and400◦C that the lower pressure limit of the hysteresis loop is at P/P0=0.4.Calcinations at higher temperatures increase this value,e.g.at600◦C,P/P0increasesto approx.0.55,while after800◦C this value increases to0.6, indicating that the pores are becoming larger as the materials are calcined at higher temperatures.To evaluate the pore size distribution,the as-prepared and calcined materials were plotted as shown inﬁgure10. The average pore diameter for the as-prepared nanotubes is approx.3nm,whereas upon calcination the pore size became gradually larger and the pore size distribution shifted,forming larger pores at the expense of the smaller ones.At600◦C the pore size is approx.9nm while,when the nanomaterials were calcined at800◦C,the pore size became more than double at approx.19nm.Calcining the materials at1000◦C collapsed the small pores,resulting in only larger pores.The surface properties of the Zr–TiO2nanotubes calcined at different temperatures are summarized inﬁgure11.The as-prepared Zr–TiO2nanotubes,having a surface area of 430m2g−1,gradually decrease through calcination.Due to the sintering phenomena,the small pores collapse,reducingthe Figure11.Surface area and pore volume of the Zr–TiO2nanotubes calcined at different temperatures as a function of calcination temperature.pore volume and surface area.The transformations into anatase and rutile crystalline phases will also help to reduce the surface area.Interestingly,literature values show that pure titania has almost zero surface area at this high temperature[44,45]. Hence,the presence of10%zirconia increased the thermal stability and reduced grain growth rates during the course of heat treatment,resulting in moderate(23m2g−1)surface areas at very high temperature.4.ConclusionsThe thermal behavior of the Zr–TiO2nanotubes synthesized by an acid-modiﬁed sol–gel process in scCO2has been investigated in detail using SEM,TEM,TG,EDX,FTIR, XRD,HRTEM,Raman and BET analysis.SEM and TEM analysis conﬁrmed that the morphology of the nanotube structure was preserved at up to800◦C,whereas further heat treatment deformed the tubes.FTIR and EDX analysis showed that different organic residues were removed,depending on the calcination temperature.Along with HRTEM and Raman, XRD results showed anatase nanocrystallites were formed after calcining at400◦C,while no rutile phase appeared until calcining at700◦C,with further heat treatment resulting in a rutile phase transformation and ZrTiO4being formed. The activation energy for anatase and rutile crystal growth was calculated and the values were12.8and36.4kJ mol−1, respectively.The activation energy for phase transformation was determined to be171kJ mol−1,higher than that of pure titania nanomaterials.The as-prepared nanotubes had a430m2g−1speciﬁc surface area(SSA),whereas after calcining at1000◦C the SSA was reduced to23m2g−1. Hence,the ZrO2present in the titania matrix increased thermal stability,reducing grain growth resulting in smaller crystallites, and hence preserving the morphology and surface area at high temperatures.AcknowledgmentsThe authors would like to thank Nancy Bell from the UWO Nanotechnology Centre for the SEM analysis,Ronald。

N维Hamilton系统的Painleve分析及精确解

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N 维 Hamilton 系统的 Painleve 分析及精确解
utt − u xx = − sin u
上式左边与线性波动方程类似. 目前,孤立子的数学理论已初步形成比较完整的理论系统, 而这些孤立子方程的求解成为最受关注的问题,孤立子理论中已有一整套方法用于求解非线性偏微分方程的精确解,为非线性偏微分方程（NPDE）及非线性科学注入了新的活力. 例如 1967 年 GGKM 发现的用一维 Schrodinger 方程求解方程初值问题的反散射方法[5-8],在一定程度上可以看成非线性问题的傅立叶方法；利用多种变换和技巧求解非线性偏微分方程精确解的 Darboux 变换[9-11]、Backlund 变换[12-18]、 Hirota 双线性方法[19-22],Painleve 分析法[24-29]、齐次平衡法、非线性化方法、代数曲线方法等.这些方法涉及到经典分析、泛函分析、李群、李代数、无穷维李代数、微分几何、代数几何、拓扑学、动力系统和计算数学等多个数学分支. 它们的发现和应用,不仅使得过去难以求解的非线性偏微分方程得以成功求解,而且不断发现许多非线性方程的具有重要意义的新解. 给定一个非线性发展方程,如何判定它是否可用反散射方法（IST）求解？这是孤立子理论的基本问题,在这方面至今虽有大量的工作,但较有成效的还属 Wahlquist 和 Estabrook 的延拓结构法[23]以及 Ablowitz 等人的化为 Painleve 方程的方法[24-29]. Ablowitz 等人发现从 IST 可解的偏微分方程（PDE）,用相似约化方法得出常微分方程（ODE）,有非常简单的特征：都具有 Painleve 性质.比如大家熟知的 Boussinesq 方程 utt = u xx + ( u2 1 ) xx + u xxxx ,MKdV 方程 2 4

不带矩射的辛作用对应的约化的Hamilton函数

不带矩射的辛作用对应的约化的Hamilton函数
刘宝康
【期刊名称】《首都师范大学学报（自然科学版）》
【年(卷),期】2000(021)003
【摘要】There are reduced Hamiltonians on symplectic reduced spaces of symplectic actionrns without moments of Lie groups on symplectic manifolds.Two necessary and sufficrnient conditions for a poinf to be a relative equilibrium are also given in the prnaper.%Lie群G在辛流形(M，ω)上的辛作用(不带有矩射)的辛约化空间Nk=K＼Nx上具有约化的Hamilton 函数，并给出一点成为相对平衡点的两个充要条件.
【总页数】1页(P1)
【作者】刘宝康
【作者单位】首都师范大学数学系
【正文语种】中文
【中图分类】O-186.16
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2.Hamilton-Jacobi方程的广义条件对称约化 [J], 勾明;左苏丽
3.约化枚举及约化方程的Hamilton结构 [J], 王志宏
4.Hamilton算子特征函数系辛正交性的反问题 [J], 魏福红;阿拉坦仓;黄俊杰
5.弹性力学极坐标辛体系Hamilton函数的守恒律 [J], 朱炳麒;卓家寿;周建方因版权原因，仅展示原文概要，查看原文内容请购买。

本质集的邻域并和无爪图的哈密尔顿性

本质集的邻域并和无爪图的哈密尔顿性徐新萍【期刊名称】《东南大学学报（英文版）》【年(卷),期】2002(018)002【摘要】设G是一个图,G的独立集Y称为本质集,如果存在{y1,y2}Y,使得dist(y1,y2)=2.本文利用插点方法,给出了关于k或(k+1)连通(k≥2)无爪图G是哈密尔顿的或1哈密尔顿的统一的证明.2个结果的充分条件是关于∑ki=0N(Yi)与n(Y)的不等式,这里Y是图G的任一本质集,对于i∈{0,1,…,k},Yi={y1,yi-1,…,yi-(b-1)}Y(yj的下标将取模k+1);b是一个整数,且0＜b＜k+1;n(Y)={v∈V(G):dist (v,Y)≤2}.%Let G be a gra ph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y1,y2}Y such that dist (y1,y2)=2. In this paper, we will use the technique of the vertex insertion on l-connected (l=k or k+1,k≥2) claw-free graphs to provide a unified proof for G to be hamiltonian or 1-hamiltonian, the sufficient conditions are expressed by the inequality concerning ∑ki=0N(Yi) and n(Y) for each essential set Y={y0,y1,…,yk} of G, where Yi={yi,yi-1,…,yi-(b-1)}Y for i∈{0,1,…,k} (the subscriptions of yj's will be taken modulo k+1), b(0<b<k+1) is an integer, and n(Y)={v∈V(G): dist (v,Y)≤2}.【总页数】4页(P184-187)【作者】徐新萍【作者单位】南京师范大学数学与计算机科学学院,南京,210097,江苏省教育学院数学与计算机科学系,南京,210013【正文语种】中文【中图分类】O157.5因版权原因，仅展示原文概要，查看原文内容请购买。

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ider the Hamiltonian theory for the multicomponent KP hierarchy. We show that the second Hamiltonian structures constructed by Sidorenko and Strampp[J. Math. Phys. 34, 1429(1993)] are not Hamiltonian. A candidate for the second Hamiltonian Structures is proposed and is proved to lead to hereditary operators.
(2, 1) (2, 2)
where subscript + means the projection to the diﬀenertial part. Following the idea of Sidorenko and Strampp4 , we easily see that the systems(2.2) have the following presentation qtk = Bss0 where q = (q1 , q2 , r1 , r2 )T and
1
Mailing Address
1
1
INTRODUCTION
During last few years, the constrained KP hierarchy is studied intensively1−3,5 . This hierarchy is the result of generalizing Cao’s6 nonlinearization to the 2+1 dimensional case. Very interestingly, as the famous Gelfand-Dikii hierarchy7 , the constrained KP hierarchy is both mathematically and physically important. On the one hand, it contains physically applicable models, such as Yajima-Oikawa8 model and Melnikov9 model. On the other hand, the constrained KP hierarchy is Bi-Hamiltonian5 , has Darboux transformation10 , can be modiﬁed11 and is relevant to the theory of the W algebra12 . Very recently, the constrained KP is shown to be just a special case of a more general restriction of the KP hierarchy13 . Sidorenko and Strampp4 introduced multi-component KP hierarchy, which is a straightforward generalization of the scalar case. This is the hierarchy associated with the following Lax operator
m
Ln = ∂ + un−2 ∂
n
n−2
+ ... + u0 +
i=1
q i ∂ − 1 ri ,
(1, 1)
the corresponding ﬂows may be constructed by means of Fractional Power Method7 . For n = 1, one has multi-component AKNS, which includes the important coupled nonlinear Schr¨ odinger 14 equation as a special case. For the cases n = 2 and n = 3, one has the multi-component YajimaOikawa hierarchy and Melnikov hierarchy respectively. Sidorenko and Strampp4 further constructed recursion operators for the cases n = 2 and n = 3 by means of variational calculus developed in the Ref.15. They claimed that their recursion operators have implectic-symplectic factorizations in the sense of Fuchssteiner and Fokas17 . That is to say, they claimed that the Bi-Hamiltonian structures are found for the multi-component Yajima-Oikawa hierarchy and Melnikov hierarchy. Unfortunately, they did not prove their statement either directly or indirectly. The partial reason is that their candidates for the Hamiltonian structures are complicated nonlocal matrix operators and direct proof would be too tedious to do by ha The aim of the paper is to show that Sidorenko and Strampp’s claim is not correct. We will prove that their second Hamiltonian structures are not qualiﬁed as Hamiltonian at all. Furthermore, we will present alternative candidates for the second Hamiltonian structures and prove that it leads to hereditary operators. For simplicity, we concentrate on the simplest and non trivial case: twocomponent case. The generalization to the multi-component case will be commented in the due course. The paper is arranged as follows. The next section is on the two-component AKNS systems. Sidorenko-Strampp type operator is presented and shown to be not hereditary. A candidate for the second Hamiltonian operators is constructed. Also, we present a hereditary operator for this hierarchy. We do the same thing in the section three for the two-component Yajima-Oikawa hierarchy. Section four contains some comments on generalizations of the results of section two and section three and gives some outlines for the further study.

δHk+1 δHk = Bss1 , δq δq 0 0 0 0 −1 0 0 −1 1 0 0 0 0 1 0 0

(2, 3)
Bss0 =

, ,
(2, 4)
Bss1 =

2q1 ∂ −1 q1 2q1 ∂ −1 q2 ∂ − 2q1 ∂ −1 r1 −2q1 ∂ −1 r2 2q2 ∂ −1 q1 2q2 ∂ −1 q2 −2q2 ∂ −1 r1 ∂ − 2q2 ∂ −1 r2 − 1 − 1 ∂ − 2r1 ∂ q1 −2r1 ∂ q2 2r1 ∂ −1 r1 2r1 ∂ −1 r2 − 1 − 1 − 1 −2r2 ∂ q1 ∂ − 2r2 ∂ q2 2r2 ∂ r1 2r2 ∂ −1 r2 1 (Res(L1 )k ). k
We ﬁrst give Sidorenko-Strampp type operator and show it is not hereditary. The Lax operator is L1 = ∂ − q1 ∂ −1 r1 − q2 ∂ −1 r2 , the ﬂows are L1tk = [((L1 )k )+ , L1 ],
2
TWO-COMPONENT AKNS HIERARCHY
We consider the two-component AKNS system next. This hierarchy is known for long time. In fact, the so important coupled nonlinear Schr¨ odinger equation14 is a reduction of it. However, the explicit form of recursion operator is not written down to the best of my knowledge although it might be known to the specialists. For the motivation of the next section, we present it here. 2