Singularity Analysis of Three-Legged, Six-DOF Platform Manipulators With URS Legs

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扩展线位移法快速实现机器人末端线速度规划

扩展线位移法快速实现机器人末端线速度规划陈琳;倪崇琦;戴骏;潘海鸿【摘要】将用于数控机床的扩展线位移方法引入到六自由度串联机器人末端运动线速度规划算法中,以提高机器人末端运动线速度控制效率.该算法借鉴数控机床中将刀具实际运动线速度与编程指定速度相互关联的扩展线位移方法思想,将机器人末端运动线速度转换为机器人运动矢量合角速度最终实现机器人末端运动线速度规划.通过搭建的机器人实验平台进行实验,实验结果表明提出的扩展线位移方法在机器人末端沿直线轨迹和圆弧轨迹运动时的实际合角速度与理论推算合角速度基本一致,实际关节角速度与理论关节角速度趋于一致.表明该方法能够有效实现机器人末端线速度轨迹规划.【期刊名称】《机械设计与制造》【年(卷),期】2016(000)009【总页数】4页(P176-179)【关键词】六自由度串联机器人;末端运动线速度规划;扩展线位移法;矢量合角速度【作者】陈琳;倪崇琦;戴骏;潘海鸿【作者单位】广西大学机械工程学院,广西南宁530004;广西制造系统与先进制造技术重点实验室,广西南宁530004;广西大学机械工程学院,广西南宁530004;广西大学机械工程学院,广西南宁530004;广西大学机械工程学院,广西南宁530004;广西制造系统与先进制造技术重点实验室,广西南宁530004【正文语种】中文【中图分类】TH16;TP242.2目前工业机器人已经广泛应用在许多工业领域中，在诸如焊接、喷涂等作业场合，不仅对机器人末端位置和姿态有一定要求，对工作时速度、加速度都有着严格要求，以满足实际生产需要。

针对机器人速度分析规划方法主要包括CAD变量几何法及雅可比矩阵方法。

文献［1］利用CAD变量几何方法求解平面多自由度机器人臂速度和加速度，但求解精度不高。

雅可比矩阵方法在机器人运动速度规划中具有十分重要的作用，文献［2］针对机器人任意点速度推导出其雅可比矩阵计算公式，并编制出相应程序；文献［3］基于螺旋理论建立少自由度操作空间与关节空间的速度一一映射关系，根据雅可比矩阵分析少自由度机器人的奇异性；文献［4］采用雅可比矩阵方法研究6R串联机器人关节空间速度和操作空间运动速度之间的映射关系；文献［5］采用速度雅可比矩阵在已知各关节位置及速度前提下对机器人进行正、逆速度分析；文献［6］利用全局雅可比矩阵对四足变结构机器人进行速度分解控制。

On the Relevance of Singular Solutions in dS_3 and AdS_3 Gravity

The standard deﬁnition of path integral in quantum gravity is a sum over the Riemannian metrics g on a manifold M. One also has to sum over topologically distinct manifolds. Considering a three dimensional Euclidean cosmological theory we have
Z[λ] =
DeDw exp−S[λ, e, w]
(2)
M
The lesson we learn form the canonical quantization is that in the above formula we need
to allow singular dreibeins which give 2D manifolds. This suggests that we should sum
At the classical level , the formulation of gravity in terms of the spin connection and the dreibein is equivalent to the metric formulation as long as the dreibein is invertible. The non-polynomial dependence of the action on the metric in the metric formalism makes the quantization harder if not impossible.2 On the other hand canonical quantization was carried out in the ﬁrst order formalism [4]. Witten showed that one has to remove the requirement that the dreibein be invertible for the quantization program to work.

Analysis_of_multistage_amplifier-frequency_compensation

Analysis of Multistage Amplifier–FrequencyCompensationKa Nang Leung and Philip K.T.Mok,Member,IEEEAbstract—Frequency-compensation techniques of single-,two-and three-stage amplifiers based on Miller pole splitting and pole–zero cancellation are reanalyzed.The assumptions made, transfer functions,stability criteria,bandwidths,and important design issues of most of the reported topologies are included. Several proposed methods to improve the published topologies are given.In addition,simulations and experimental results are provided to verify the analysis and to prove the effectiveness of the proposed methods.Index Terms—Damping-factor-control frequency compen-sation,multipath nested Miller compensation,multipath zero cancellation,multistage amplifier,nested Gm-C compensation, nested Miller compensation,simple Miller compensation.I.I NTRODUCTIONM ULTISTAGE amplifiers are urgently needed with the advance in technologies,due to the fact that single-stage cascode amplifier is no longer suitable in low-voltage designs. Moreover,short-channel effect of the sub-micron CMOS transistor causes output-impedance degradation and hence gain of an amplifier is reduced dramatically.Therefore,many frequency-compensation topologies have been reported to stabilize the multistage amplifiers[1]–[26].Most of these topologies are based on pole splitting and pole–zero can-cellation using capacitor and resistor.Both analytical and experimental works have been given to prove the effectiveness of these topologies,especially on two-stage Miller compen-sated amplifiers.However,the discussions in some topologies are focused only on the stability criteria,but detailed design information such as some important assumptions are missing. As a result,if the provided stability criteria cannot stabilize the amplifier successfully,circuit designers usually choose the parameters of the compensation network by trial and error and thus optimum compensation cannot be achieved.In fact,there are not many discussions on the comparison of the existing compensation topologies.Therefore,the differences as well as the pros and cons of the topologies should be inves-tigated in detail.This greatly helps the designers in choosing a suitable compensation technique for a particular design condi-tion such as low-power design,variable output capacitance or variable output current.Manuscript received March9,2000;revised February6,2001.This work was supported by the Research Grant Council of Hong Kong,China under grant HKUST6007/97E.This paper was recommended by Associate Editor N.M.K. Rao.The authors are with the Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology,Clear Water Bay,Hong Kong(e-mail:eemok@t.hk).Publisher Item Identifier S1057-7122(01)07716-9.Moreover,practical considerations on the compensation tech-niquesof(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Fig.1.Studied and proposed frequency-compensation topologies.(a)SMC.(b)SMCNR.(c)MZC.(d)NMC.(e)NMCNR.(f)MNMC.(g)NGCC.(h)NMCF.(i)DFCFC1.(j)DFCFC2.accuracy.In this paper,there are three common assumptionsmade for all studied and proposed topologies.1)The gains of all stages are much greater than one(i.e.,LEUNG et al.:ANALYSIS OF MULTISTAGE AMPLIFIER–FREQUENCY COMPENSATION1043 Assumption1holds true in amplifier designs for most ampli-fiers except those driving small load resistance.If this assump-tion cannot be satisfied,numerical analysis using computers isrequired.Moreover,the parasitic capacitances of the tiny-geom-etry transistors in advanced technologies are small and this val-idates assumptions2)and3).III.R EVIEW ON S INGLE-S TAGE A MPLIFIERThe single-stage amplifier is said to have excellent frequencyresponse and is widely used in many commercial products.Infact,the advantages can be illustrated by its transferfunctiondue to the single pole,assuming thatGBW(i.e.,andminimum.Therefore,a higher bias current and smaller size for all transis-tors in the signal path are required tolocateand the RHP zeroislocates beforepp pp ppz ppp p1044IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I:FUNDAMENTAL THEORY AND APPLICATIONS,VOL.48,NO.9,SEPTEMBER2001Fig.3.PM versus g=gof a SMC amplifier.From (6)and Fig.3,the PM of a SMC amplifier strongly de-pends ontheto ratio and this,in fact,shows the RHP zero effect on the PM.Physically,the presence of the RHP zero is due to the feedforward small-signal current flowing throughthe compensation capacitor to the output [1]–[11].Ifis large,the small-signal output current is larger than the feed-forward current and the effect of the RHP zero appears only at very high frequencies.Thus,asmallis preferable.However,is limited bythe bias current and size of the input differential pair.To have a good slew rate,the bias current cannot be small.In addition,to have a small offset voltage,the size of input differential pair cannot be too small.Emitter/source degeneration technique isalso not feasible toreducesince it reduces the limited input common-mode range in low-voltage design.Therefore,asmallcannot be obtained easily.From the previous analysis,it is known that the RHP zero degrades the stability significantly.There are many methods to eliminate the RHP zero and improve the bandwidth.The methods involve using voltage buffer [4]–[6]and current buffer [7],[8],a nulling resistor [2],[3],[9]–[11],and MZC technique [12].In this paper,the techniques to be discussed are:1)SMC using nulling resistor (SMCNR)and 2)SMC using MZC.A.SMCNRThe presence of the RHP zero is due to the feedforward small-signal current.One method for reducing the feedforward current and thus eliminating the RHP zero is to increase the impedance of the capacitive path.This can be done by inserting a resistor,called nulling resistor,in series with the compensation capacitor,as shown in Fig.1(b).Most published analyses only focus on the effect of the nulling resistor to the position of the zero but not to the positions of the poles.In fact,when the nulling resistor isincreased to infinity,the compensation network is open-circuit and no pole splitting takes place.Thus,the target of this section is to investigate the limit of the nulling resistor.The transfer function of the SMNCR(,,respectively.It is well-known thatwhenis generally much smallerthananddue to theabsence of the RHP zero.However,many designers prefer to use a nulling resistor withvalue largerthansince an accurate valueofandis not a con-stant and a precise cancellation of the RHP zero by afixed)to cancel the feedforward small-signal current(,,which is independentof.(7)LEUNG et al.:ANALYSIS OF MULTISTAGE AMPLIFIER–FREQUENCY COMPENSATION1045 Moreover,since MZC does not change the positions of thepoles,the same dimension condition ofwhich is obtained by neglecting the RHP zerophase shifting term in(6).Besides,when the output current isincreased,is increased accordingly.The nondominant pole()will move to a higher frequency and a largerPM is obtained.Thus,this compensation topology can stabilizethe amplifier within the quiescent to maximum loading currentrange.In some applications,whereand the PM is about90andand.Apparently,the GBW can be increased to infinity bydecreasingto validate the assumptions on deriving(8),so the fol-lowing condition is required as a compromise:,the transfer function is rewritten as(11),shownat the bottom of the page.The dominant pole is1046IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I:FUNDAMENTAL THEORY AND APPLICATIONS,VOL.48,NO.9,SEPTEMBER2001Fig.5.Equivalent small-signal model of three-stage NMC.From the above equation,GBW.Assuming,and are fixed for a given power consumption,largeand are required.This increases the PM but itreduces the GBW and also increases the capacitor values andthe required chip area simultaneously.For the complex-pole approach,the NMC amplifier in unity-feedback configuration should have the third-order Butterworthfrequency response.Let be the closed-loop transferfunctionandshould be in the followingformat:and areobtained:(or)and the damping factor of the complexpoleis(17)which is one-fourth the bandwidth of a single-stage amplifier.This shows the bandwidth reduction effect of nesting compen-sation.Similar to SMC,the GBW can be improved by alargerand asmaller and asmaller.The PM under the effect of a complex pole[28]is givenbyPM(18)Comparing the required compensation capacitors,the GBWand PM under the same power consumption(i.e.,same,and)of the two approaches,it is concluded that thecomplex-pole approach is better.Moreover,from(15)and(16),smallerand are neededwhen.This validates the previous assumption on neglecting the zerossince the coefficients of the function of zero in(10)are smalland the zeros locate at high frequencies.From another pointof view,therequiredand are small,so the feedfor-ward small-signal current can pass to the output only at veryhigh frequencies.In addition,the output small-signal current ismuch larger than the feedforward currentas.Thus,the zeros give negligible effect to the stability.If theseparate-pole approach is applied,the stability is doubtful sincelarger compensation capacitors are required and this generateszeros close to the unity-gain frequency of the amplifier.To further provethat is necessary inNMC,a HSPICE simulation using the equivalent small-signalmodel of NMC,which is shown in Fig.5,is performed.The cir-cuit parametersare A/V,A/V,is satisfied)and10pF.and,which is set according to(15)and(16),are4pFand1pF,respectively.The simulation result is shown in Fig.6by the solid line.A GBW of4.2MHz and a PM of58from100is notmuch largerthan),therequired is changed from4pFto40pF,according to(15).The frequency response is shownby the dotted line in Fig.6.A RHP zero appears before theunity-gain frequency and causes the magnitude plot to curveupwards.The PM is degraded to30ischanged from50is not much largerthan)and is changed from1pF to20pF accordingto(16).As shown by the dashed line in Fig.6,a frequencypeak,due to small damping factor of the complex pole,appearsand makes the amplifier unstable.The phenomenon can be ex-plained from(10).When is not much largerthan,theterm()of the second-order function in the denomi-nator is small and this causes the complex poles to have a smallLEUNG et al.:ANALYSIS OF MULTISTAGE AMPLIFIER–FREQUENCY COMPENSATION1047Fig.6.HSPICE simulation of NMC (solid:g g and g ;dotted:g is not much larger than g ;dash:g is not much larger than g ).damping factor.Ifis very important and critical to the stability of an NMCamplifier.However,this condition is very difficult to achieve,especially in low-power design.Ifdoes not hold true,the analysis should be re-started from (10).Fromthis equation,sincetheterm is negative,there are one RHP zero and one LHP zero.The RHP zero locates at a lower fre-quency astheand only a LHPzeroand any value closedto is able to locate the RHP zero to a high frequency.Bydefining,the transfer function is rewritten as (20)shownat the bottom of the page.It is notedthatand are obtained as in NMC usingcomplex-pole approach and are givenby(i.e.,1048IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I:FUNDAMENTAL THEORY AND APPLICATIONS,VOL.48,NO.9,SEPTEMBER2001Fig.7.Circuit diagram of the amplifiers(a)NMCNR.(b)NMCF.(c)DFCFC1.(d)DFCFC2.).The GBW is given byGBWdue to the LHP zero.A larger GBW can be obtained byslightly reducing but this reduces the PM.To prove the proposed structure,NMC and NMCNR am-plifiers were implemented in AMS10.8.The circuit diagram of the NMCNR amplifiersare shown in Fig.7(a)and the NMC counterpart has the samecircuitry without the nulling resistor.The chip micrograph isshown in Fig.8.Both amplifiers drive a100pF//25knulling resistor,which is made of poly,is used in the NMCNRamplifier.In NMC,the required is99pF,but inNMCNR is63pF.As presented before,the PM of NMCNRamplifier is larger,so a smaller is used in the implemen-tation to obtain a similar PM as in NMC and a larger GBW.Moreover,this greatly reduces the chip area from0.23mm.The measured results and improvement comparison are tabu-lated in Tables I and II,respectively.Both amplifiers haveW power consumption and)are improvedby+39%,+3is improvedLEUNG et al.:ANALYSIS OF MULTISTAGE AMPLIFIER–FREQUENCY COMPENSATION 1049TABLE IM EASURED R ESULTS OF THE AMPLIFIERSTABLE III MPROVEMENT OF THE P ROPOSED AND P UBLISHED T OPOLOGIES W ITH NMC (,and the chip area.VI.MNMCBesides increasing the power,the multipath technique can be used to increase the bandwidth of an amplifier.In MNMC[12],[16],[19],and [26],a feedforward transconductance stage (FTS)is added to the NMC structure to create a low-fre-quency LHP zero.This zero,called multipath zero,cancels the second nondominant pole to extend the bandwidth.The structure of MNMC is shown in Fig.1(f)and it is limited to three-stage amplifiers but it has potential to extend to more stages.However,power consumption and circuit complexity are increased accordingly since a feedforward input differ-ential stage,as same as MZC,is needed,so this will not be discussed here.The input of the FTS,withtransconductanceand the output is connected to the input of theoutput stage.Again,with the conditionthat,the transfer function is given by (23)at the bottom of the next page.The nondominant poles are givenby1050IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I:FUNDAMENTAL THEORY AND APPLICATIONS,VOL.48,NO.9,SEPTEMBER2001Fig.9.Simulation results of an MNMC amplifier using equivalent small-signal circuit under the change of g andC =20pF;dash:g =10mA/V andC =1pF)..The explicit dimensionconditionofis,therefore,givenbyin MNMC is much larger thanthat in NMC.This increases the required chip area and reduces the SR dramatically.Therefore,emitter degeneration technique was used in the design of [16].This can reduce theeffective so thatthe is,as a result,smaller.With (24),the positionsofis thefollowing:.The above analysis gives the required valuesof,and,,and.In fact,if this assumption does nothold true,the positions of the poles and the LHP zero are not those previously stated.Moreover,a RHP zero exists and the stability is greatly affected.The analysis and dimension conditions are obtained in static state.Since there is a pole–zero doublet before the unity-gain frequency,the dynamic-state stability should also be consid-ered.Since,in practice,the loading current andcapacitancemay change in some general-purpose amplifiers with Class-AB output stage,it is necessary to consider the stability of theMNMC amplifierwhenis increasedand ,where the ratio isobtained from (24)and (26).Besides,the multipath zero is notchangedwhenand with the condition in (27).It is obviousthat,so MNMC is not affected by changing the loading current and capacitance.To prove the above arguments,a simulation using HSPICE is performed with the equivalent small-signal circuit of an MNMCamplifier.The circuit parametersareA/V,,1M25k 20p F.T h u s,111.25i s c h a n g e d f r o m 1m A /V t o 10m A /V ;a n d 2)a nd i s i n c re a s e d or a r e r e q u i r e d .T h i s c o n d i t i o n n o t o n l y i m -p r o v e s t h e s t a b i l i t y b u t i t a l s o s i m p l i f i e s t h e t r a n s f e r f u n c t i o n .I n f a c t ,a s m e n t i o n e d b e f o r e ,t h i s c o n d i t i o n i s d i f f i c u l t t o a c h i e v e i n l o w -p o w e r d e s i g n ,s o Y o u e t a l .i n t r o d u c e d N G C C [20].N G C C i s a n-s t a g e N G C Ca m p l i f i e r.W i t h t h e c o n d i t i o n t h at w e re ,t h e g e n e r a lf o r m o f a n-s t a g e a m p l i f i e r t h a n N M C .I n t h e s t a b i l i t y c o n d i t i o n s p r o p o s e d b y Y o u e t a l .,t h e s e p a r a t e d -p o l e a p p r o a c h i s u s e d a n d t h e n o n d o m a r e s e t t o s o m e f r e q u e n c i e s s u c h t h a t t h e G B W ,T s a nd p o we r c o n s u m p t i o n a r e a l l o p t i m i z e d .U n d o u b t e d l y ,t h c a t e d t o d o o p t i m i z a t i o n a n a l y t i c a l l y ,s o n u m e u s i n g M A T L A B i s r e q u i r e d .H o w e v e r ,q u e s t i o n s o n p r a c t i c a l c o n s i d e r a t i o n s ,s i n c e i t i s p r ef e r a m i n i m u m s t ag e s a s p o s s i b l e .A s s t a t e d b e f o r e ,t a n o p t i m u m n u m b e r o n d c g a i n ,b a n d w i d th ,a n d s u m p ti o n .T h e r e f o r e ,t h e a n a l y s i s i n t h i s s e c t i o n t h e t h r e e -s t a g e N G C C a m p l i f i e r.T h e s t r u c t u r e oN G C C a m p l i f i e r i s s h o w n i n F i g .1(g )a n d t h e t r a ni s g i v e n b y (29)s h o w n a t t h e b o t t o m o f t h e p a g eb e f o r e a n d a l s o f r o m t h e n u m e r a t o r o f (29),t h e b e e l i m i n a t e d b y s e t t i n g a nd .T h et r a n s f e r f u n c t i o n i s t h e n s i m p l i f i e d t o (30)s h o wo f t h e p a g e .T h e a r r a n g e m e n t o f t h e p o l e s c a n u ss e p a r a t e -p o l e o r c o m p l e x -p o l e a p p r o a c h b u t t h ep r e f e r r e d .I t i s o b v i o u s t h a t t h e d e n o m i n a t o r o s a m e a s (11)b u t t h e d i f f e r e n c e i s t h a t i s n o t r e q u i r e d i n N G C C .T h u s,.A l t h o u g h N G C C i s g o o d i n l o w -p o w e r d e s i g n s ,s t a g e F T S (i .e .,some of them are LHP zeros which,in fact,help to increase the PM.With regard to the above considerations,a new structure, called NMC with feedforward Gm stage(NMCF),is proposed and shown in Fig.1(h).There are only two differences betweenNMCF and NGCC:1)the input-stage FTS is removed and2).Bydefiningand are obtained using thecomplex-pole approach and they are givenby,are smaller than those in NMC,MNMC and NGCCsinceterm is positive andthe term is negative,the LHPzerolocates before the RHPzerofor stability purpose,so the following condition isrequired:(34)The condition states the minimum valueof to obtain anoptimum control of LHP zero.From(31)to(33),the GBW and PM are given byGBW(35)andPM(36)It is shown in(35)that the bandwidth is improved by the pres-enceofmCMOS process was done to prove the proposed structure.TheNMCF amplifier is shown in Fig.7(b)and it is basically thesame as the NMC amplifier.It is noted that the gate of M32,which is the FTS,is connected to the output of the first stage.The output stage is of push-pull typeand,from(35),to double the GBW.The measured results and improvement comparison areshown in Tables I and II,respectively.It is obvious that theimprovement of NMCF over NMC on GBW(),PM()and occupied chip area()are much larger than those in MNMC and NGCCin other designs,which are shown in Table II.The powerconsumption is only increased by6and inverselyproportionaltois removed and the bandwidth of the ampli-fier can be extended substantially.However,the damping factorof the nondominant complex poles,which is originally con-trolledby,cannot be controlled and a frequency peak,which causes the closed-loop amplifier to be unstable,appearsin the magnitude Bode plot[23].To control the damping factorand make the amplifier stable,a damping-factor-control(DFC)block is added.The DFC block is basically a gain stage withdc gain greater than one(i.e.,.The DFC block functions as a frequency-de-pendent capacitor and the amount of the small-signal currentinjected into the DFC block depends on the valueofand(transconductance of the gain stage inside the DFC block).Hence,the damping factor of the nondominant complex polescan be controlled byoptimumand and this makesthe amplifier stable.There are two possible positions to add theDFC block and they are shown in Fig.1(i)for DFCFC1andFig.1(j)for DFCFC2.In addition,both structures have a feed-forward transconductance stage to form a push-pull output stagefor improving large-signal slewing performance.For DFCFC1,the transfer function is given by(37)shown atthe bottom of the next page.It can be seen from(37)that thedamping factor of the nondominant poles can be controlledby.Moreover,the effectofandtransfer functionbut is limitedto tovalidate (37).Sinceis small,the amplifier is not slowed downby.From (37),there are three poles,so the com-plex-pole approach is used.Moreover,since it is preferable to have the same output current capability for boththe -transistor of the output stage,the sizes ofthe -tran-sistor are used in ratio of 3to 1to compensate for the differ-ence in the mobilities of the carriers.Thus,it is reasonable toset,so the dimension conditions are givenby (39)whereis much smaller thanthat in the previous nesting topologies,so the SR is also greatly improved,assuming that the SR is not limited by the outputstage.Moreover,is a decreasing functionof (41)and the PM is about 60times.Ifa little,butthis reduces the PM as a tradeoff.For DFCFC2,bysettingwith the same reason stated previously,the transfer function is given by (42)shown at the bottom of the page.Similar to DFCFC1,the complex-poleapproach is used to achieve the stability.Therefore,the dimen-sion conditions are givenby(43)is a fixed value and is four timesof.Thus,the power consumption of DFCFC2amplifier with certain valueof.Although it is difficult to comparethe GBW of DFCFC2with other topologies since the format is different,it is in general better than others.It is due to the fact that the GBW is inversely proportion to the geometric meanof,which gives a smaller valuethan mdouble-metal double-poly CMOS process.The circuit diagrams are shown in Fig.7(c)for DFCFC1and Fig.7(d)for DFCFC2.The micrograph is,again,shown in Fig.8.In both amplifiers,M41andform the DFC block and M32is the FTS.Moreover,from Table II,the GBW,PM,SR,TIX.S UMMARY OF S TUDIED F REQUENCY C OMPENSATIONT OPOLOGIESA summary on the required stability conditions,resultant GBW and PM for all studied and proposed topologies are given in Table parisons on the topologies are tabulated in Table IV.Moreover,some important points derived from the previous analyzes are summarized as follows.1)The stability-dimension conditions of all topologies arebased on the assumptions stated in Section II.If the as-sumptions cannot be met,numerical method should be used to stabilize the amplifiers.2)With the exception of the single-stage amplifier,alargerandlargestandreducingto ratio and asmallerto ratio.6)For high-speed applications,a larger bias current shouldbe applied to the output stage toincrease.Fig.10.Local feedback circuitry to control the dc operating point of the DFCblock.X.R OBUSTNESS OF THE S TUDIED F REQUENCY C OMPENSATION In IC technologies,the circuit parameters such as transcon-ductance,capacitance and resistance vary from run to run,lot to lot and also according to temperature.The robustness of fre-quency compensation is very important to ensure the stabilities of multistage amplifiers.From the summary in Table III,the required values of com-pensation capacitors depend on the ratio of transconductances of gain stages explicitly for SMC,SMCNR,MZC1,MZC2,NMC,NMCNR,MNMC,NGCC,NMCF,and DFCFC1and implicitly for DFCFC2.The ratio maintains constant for any process varia-tion and temperature effect with good bias current matching and transistor size matching (due to design).One important point is that the valueof50%,in general is not significantto the stability.In MNMC,pole–zero cancellation is used.However,the su-perior tracking technique in MNMC is due to the pole–zero can-cellation based on the ratios of transconductances and compen-sation capacitances.Thus,process variations do not affect the compression of the pole–zero doublet.Although the robustness of the studied topologies are good,the exact value of the GBW will be affected by process varia-tions.Referring to Table III,the GBW’s of all topologies,in-cluding commonly used single-stage and Miller-compensated amplifiers,depend on the transconductance of the output stage.Thus,the GBW will change under the effect of process varia-tions and temperature.XI.C ONCLUSIONSeveral frequency-compensation topologies have been investigated analytically.The pros and cons as well as the design requirements are discussed.To improve NMC and NGCC,NMCNR,and NMCF are proposed and the improved performance is verified by experimental results.In addition,DFCFC has been introduced and it has much better frequency and transient performances than the other published topologies for driving large capacitive loads.Finally,robustness of the studied topologies has been discussed.R EFERENCES[1]J.E.Solomon,“The monolithic op amp:A tutorial study,”IEEE J.Solid-State Circuits ,vol.9,pp.314–332,Dec.1974.[2]P.R.Gray and R.G.Meyer,Analysis and Design of Analog IntegratedCircuits ,2ed.New York:Wiley,1984.[3]W.-H.Ki,L.Der,and m,“Re-examination of pole splitting of ageneric single stage amplifier,”IEEE Trans.Circuits Syst.I ,vol.44,pp.70–74,Jan.1997.[4]Y.P.Tsividis and P.R.Gray,“An integrated NMOS operational amplifierwith internal compensation,”IEEE J.Solid-State Circuits,vol.SC-11, pp.748–753,Dec.1976.[5]G.Smarandoiu,D.A.Hodges,P.R.Gray,and ndsburg,“CMOSpulse-code-modulation voice codec,”IEEE J.Solid-State Circuits,vol.SC-13,pp.504–510,Aug.1978.[6]G.Palmisano and G.Palumbo,“An optimized compensation strategyfor two-stage CMOS OP AMPS,”IEEE Trans.Circuits Syst.I,vol.42, pp.178–182,Mar.1995.[7] B.K.Ahuja,“An improved frequency compensation technique forCMOS operational amplifiers,”IEEE J.Solid-State Circuits,vol.SC-18,no.6,pp.629–633,Dec.1983.[8]G.Palmisano and G.Palumbo,“A compensation strategy for two-stageCMOS opamps based on current buffer,”IEEE Trans.Circuits Syst.I, vol.44,pp.257–262,Mar.1997.[9] D.Senderowicz,D.A.Hodges,and P.R.Gray,“High-performanceNMOS operational amplifier,”IEEE J.Solid-State Circuits,vol.SC-13, pp.760–766,Dec.1978.[10]W.C.Black Jr,D.J.Allstot,and R.A.Reed,“A high performance lowpower CMOS channel filter,”IEEE J.Solid-State Circuits,vol.15,pp.929–938,Dec.1980.[11]P.R.Gray and R.G.Meyer,“MOS operational amplifier design—a tu-torial overview,”IEEE J.Solid-State Circuits,vol.SC-17,pp.969–982, Dec.1982.[12]R.G.H.Eschauzier and J.H.Huijsing,Frequency Compensation Tech-niques for Low-Power Operational Amplifiers.Boston,MA:Kluwer, 1995.[13] E.M.Cherry,“A new result in negative feedback theory and its applica-tions to audio power amplifier,”Int.J.Circuit Theory Appl.,vol.6,no.3,pp.265–288,1978.[14],“Feedback systems,”U.S.Patent4243943,Jan.1981.[15] F.N.L.Op’t Eynde,P.F.M.Ampe,L.Verdeyen,and W.M.C.Sansen,“A CMOS large-swing low-distortion three-stage class AB power am-plifier,”IEEE J.Solid-State Circuits,vol.25,pp.265–273,Feb.1990.[16]R.G.H.Eschauzier,L.P.T.Kerklaan,and J.H.Huijsing,“A100MHz100dB operational amplifier with multipath nested miller compensation structure,”IEEE J.Solid-State Circuits,vol.27,pp.1709–1717,Dec.1992.[17] E.M.Cherry,“Comment on a100MHz100dB operational amplifierwith multipath nested miller compensation structure,”IEEE J.Solid-State Circuits,vol.31,pp.753–754,May1996.[18]S.Pernici,G.Nicollini,and R.Castello,“A CMOS low-distortion fullydifferential power amplifier with double nested Miller compensation,”IEEE J.Solid-State Circuits,vol.28,pp.758–763,July1993.[19]K.-J.de Langen,R.G.H.Eschauzier,G.J.A.van Dijk,and J.H.Hui-jsing,“A1GHz bipolar class-AB operational amplifier with multipath nested Miller compensation for76dB gain,”IEEE J.Solid-State Cir-cuits,vol.32,pp.488–498,Apr.1997.[20] F.You,S.H.K.Embabi,and E.Sánchez-Sinencio,“Multistage ampli-fier topologies with nested gm-C compensation,”IEEE J.Solid-State Circuits,vol.32,pp.2000–2011,Dec.1997.[21]H.-T.Ng,R.M.Ziazadeh,and D.J.Allstot,“A mulitstage amplifiertechnique with embedded frequency compensation,”IEEE J.Solid-State Circuits,vol.34,pp.339–341,Mar.1999.[22]K.N.Leung,P.K.T.Mok,W.H.Ki,and J.K.O.Sin,“Damping-factor-control frequency compensation technique for low-voltage low-power large capacitive load applications,”in Dig.Tech.Papers ISSCC’99,1999, pp.158–159.[23],“Three-stage large capacitive load amplifier with damping-factor-control frequency compensation,”IEEE J.Solid-State Circuits,vol.35, pp.221–230,Feb.2000.[24],“Analysis on alternative structure of damping-factor-control fre-quency compensation,”in Proc.IEEE ISCAS’00,vol.II,May2000,pp.545–548.[25]K.N.Leung,P.K.T.Mok,and W.H.Ki,“Right-half-plane zero re-moval technique for low-voltage low-power nested miller compensation CMOS amplifiers,”in Proc.ICECS’99,vol.II,1999,pp.599–602. [26]J.H.Huijsing,R.Hogervorst,and K.-J.de Langen,“Low-power low-voltage VLSI operational amplifier cells,”IEEE Trans.Circuits Syst.I, vol.42,pp.841–852,Nov.1995.[27]G.C.Temes and Patra,Introduction to Circuit Synthesis andDesign,1ed.New York:McGraw-Hill,1977.[28]J.W.Nilsson,Electric Circuits,4ed.New York:Addison Wesley,1993.[29] B.Y.Kamath,R.G.Meyer,and P.R.Gray,“Relationship between fre-quency response and settling time of operational amplifier,”IEEE J.Solid-State Circuits,vol.SC-9,pp.247–352,Dec.1974.[30] C.T.Chuang,“Analysis of the settling behavior of an operational am-plifier,”IEEE J.Solid-State Circuits,vol.SC-17,pp.74–80,Feb.1982. Ka Nang Leung received the B.Eng.and M.Phil.degrees in electronic engi-neering from the Hong Kong University of Science and Technology(HKUST), Clear Water Bay,Hong Kong,in1996and1998,respectively.He is now working toward the Ph.D.degree in the same department.During the B.Eng.studies,he joined Motorola,Hong Kong,to develop a PDA system as his final year project.In addition,he has developed several frequency-compensation topologies for multistage amplifiers and low dropout regulators in his M.Phil studies.He was a Teaching Assistant in courses on analogue integrated circuits and CMOS VLSI design.His research interests are low-voltage low-power analog designs on low-dropout regulators,bandgap voltage references and CMOS voltage references.In addition,he is interested in developing frequency-compensation topologies for multistage amplifiers and for linear regulators.In1996,he received the Best Teaching Assistant Award from the Department of Electrical and Electronic Engineering at theHKUST.Philip K.T.Mok(S’86–M’95)received theB.A.Sc.,M.A.Sc.,and Ph.D.degrees in electricaland computer engineering from the University ofToronto,Toronto,Canada,in1986,1989,and1995,respectively.From1986to1992,he was a Teaching Assistant,at the University of Toronto,in the electrical engi-neering and industrial engineering departments,andtaught courses in circuit theory,IC engineering andengineering economics.He was also a Research As-sistant in the Integrated Circuit Laboratory at the Uni-versity of Toronto,from1992to1994.He joined the Department of Electrical and Electronic Engineering,the Hong Kong University of Science and Tech-nology,Hong Kong,in January1995as an Assistant Professor.His research interests include semiconductor devices,processing technologies and circuit de-signs for power electronics and telecommunications applications,with current emphasis on power-integrated circuits,low-voltage analog integrated circuits and RF integrated circuits design.Dr.Mok received the Henry G.Acres Medal,the W.S.Wilson Medal and Teaching Assistant Award from the University of Toronto and the Teaching Ex-cellence Appreciation Award twice from the Hong Kong University of Science and Technology.。

Kinematics of the 3-UPU wrist

Kinematics of the 3-UPU wristRaﬀaele Di Gregorio *Department of Engineering,University of Ferrara,Via Saragat,1,44100Ferrara,ItalyReceived 20September 2000;received in revised form 20May 2002;accepted 19June 2002AbstractRecently,it has been shown that a parallel mechanism architecture,called 3-UPU and used for trans-lational manipulators,can be employed to obtain manipulators able to make the end eﬀector perform inﬁnitesimal spherical motions.The possibility of performing inﬁnitesimal spherical motions is a necessary but not suﬃcient condition to guarantee that the end eﬀector performs a ﬁnite spherical motion,i.e.,the manipulator is a parallel wrist.In this paper it is demonstrated that the 3-UPU architecture,can be em-ployed to obtain parallel wrists,named 3-UPU wrists.Moreover,it is shown that the 3-UPU wrists may reach singular conﬁgurations in which the spherical constraint between the end eﬀector and the frame fails.Finally,the singularity condition,that makes it possible to ﬁnd all the 3-UPU wrist Õs singular conﬁgura-tions,is written in explicit form and is geometrically interpreted.Ó2003Elsevier Science Ltd.All rights reserved.Keywords:Manipulators;Parallel mechanisms;Parallel wrists1.IntroductionSpatial parallel manipulators (SPMs)are constituted of an end eﬀector (platform)connected to the frame (base)by a number of kinematic chains (legs).The number of legs usually is equal to the degrees of freedom (dof)of the manipulator and only one actuated joint is present in each leg.By acting on the legs the platform pose (position and orientation)is controlled.Moreover,if the actuators are locked,the manipulator will become an isostatic structure in which all the legs carry the external loads applied to the platform.This SPMs Õfeature makes designing manipulators with high stiﬀness possible throughout the whole workspace.*Tel.:+39-0532-974828;fax:+39-0532-974870.E-mail address:rdigregorio@ing.unife.it (R.D.Gregorio).0094-114X/03/$-see front matter Ó2003Elsevier Science Ltd.All rights reserved.PII:S0094-114X(02)00066-6Mechanism and Machine Theory 38(2003)253–263In the literature,six dof and less-than-six dof SPMs have been proposed.Among the less-than-six dof SPMs,special attention has been paid to the three dof manipulators since only three dof are necessary in many technical applications.In particular,three dof SPMs which make the platform translate (translational SPMs)[1–8],three dof SPMs which makes the platform perform spherical motion (parallel wrists)[9–13]and three dof SPMs which make the platform perform some special motion neither translational nor spherical [14–17]have been studied.For a long time only two manipulator architectures have been used to obtain parallel wrists.In the ﬁrst architecture (Fig.1)[11]the platform and the base are joined by a passive spherical pair and the platform orientation is controlled by three legs of type UPS (U,P and S stand for uni-versal joint,prismatic pair and spherical pair,respectively)with the prismatic pair as actuated joint.This parallel wrist has the advantage of being a three dof mechanism and has the drawback of having reduced workspace because of the passive spherical pair.In the second architecture (Fig.2)[9]the platform and the base are connected to each other by three legs of type RRR (R stands for revolute pair)with all the revolute pair axes converging at a ﬁxed point.This parallel wrist is overconstrained and obtains the platform spherical motion by using constraints that are repetitions of other constraints.The overconstrained architecture drawback is that the mechanism jams or high internal loads arise in the links when geometric errors occur.Recently,Karouia and Herv e [13]have sought after the three dof SPMs with three equal legs in which the platform can perform an elementary spherical motion.The capacity of performing an inﬁnitesimal spherical motion is requested,but it is not a suﬃcient condition to guarantee that the platform performs a ﬁnite spherical motion,i.e.,the manipulator is a parallel wrist.Hence their research is only useful to select the architecture that might be parallel wrist.The result of their investigation is that a mechanism architecture,called 3-UPU (Fig.3)and used alreadyfor254R.D.Gregorio /Mechanism and Machine Theory 38(2003)253–263translational manipulators [4],under some mounting and manufacturing conditions,can be used to obtain manipulators able to make the platform perform inﬁnitesimal spherical motions.In the 3-UPU manipulators,the platform and the base are connected to each other by three legs of type UPU in which the prismatic pair is the actuated joint (Fig.3).The mounting and manufacturing conditions enunciated by Karouia and Herv e [13]are as follows (see Fig.4):R.D.Gregorio /Mechanism and Machine Theory 38(2003)253–263255(i)the three revolute pair axes ﬁxed in the platform (base)must converge at a point ﬁxed in the platform (base),(manufacturing condition);(ii)in each leg,the intermediate revolute pair axes must be parallel to each other and perpendicular to the leg axis which is the line through the universal joints Õcenters (manufacturing condition);(iii)the platform Õs point located in the in-tersection of the platform Õs revolute pair axes must coincide with the base Õs point located in the intersection of the base Õs revolute pair axes (mounting condition).Henceforth,a 3-UPU manipulator matching these geometric conditions will be called 3-UPU wrist.The 3-UPU wrist Õs architecture brings about special interest because it overcomes the drawbacks of the two traditional parallel wrists,i.e.,it is not overconstrained and it does not need a passive spherical pair joining platform and base.This paper will demonstrate that the geometric conditions matched by the 3-UPU wrist Õs architecture are suﬃcient to make the platform perform ﬁnite spherical motions when the pris-matic pairs are actuated.Moreover,it will show that the 3-UPU wrists may reach singular conﬁgurations (translation singularities)in which the spherical constraint between platform and base fails.Finally,the singularity condition is written in explicit form and is geometrically in-terpreted.2.Finite spherical motion demonstrationFig.4shows a 3-UPU wrist.With reference to Fig.4,the points A i ,i ¼1;2;3,are the centers of the universal joints which connect the legs to the base;the points B i ,i ¼1;2;3,are the centers of the universal joints which connect the legs to the platform and the point P is the common in-tersection of the revolute pair axes ﬁxed in the platform.The 3-UPU wrist is mounted so asto256R.D.Gregorio /Mechanism and Machine Theory 38(2003)253–263make the platform Õs point P coincide with the base Õs point located by the intersection of the re-volute pair axes ﬁxed in the base.Fig.5shows the i th leg,i ¼1;2;3,of the 3-UPU wrist.With reference to Fig.5,w ji ,j ¼1;...;4,is the unit vector of the j th revolute pair axis with the j index increasing from the base to the platform;h ji ,j ¼1;...;4,is the joint coordinate of the j th revolute pair;a i and b i are constant lengths of the segments A i P and B i P respectively;d i is the variable length of the segment A i B i and it is the joint coordinate of the actuated prismatic pair;u i is the unit vector of the leg axis.The point P is ﬁxed in the platform and can be chosen as the origin of a reference system embedded in the platform.The position of P measured in a reference system ﬁxed in the base,locates the platform Õs position with respect to the base.With these notations,by taking into consideration separately the three legs,the platform an-gular velocity,x ,and the velocity,_P,of the point P ,measured in the base,can be written in the following three diﬀerent ways:x ¼_h1i w 1i þ_h 4i w 4i þð_h 2i þ_h 3i Þw 2i i ¼1;2;3ð1:1Þ_P ¼_B i þx ÂðP ÀB i Þi ¼1;2;3ð1:2Þwhere _h ji ,j ¼1;...;4,are the time derivatives of the joint coordinates h ji ,j ¼1;...;4,respec-tively,and _Bi is the velocity of the platform Õs point B i .Moreover the following vector relationships hold (Fig.5):B i ÀA i ¼d i u ii ¼1;2;3ð2:1ÞP ÀB i ¼b i w 4i i ¼1;2;3ð2:2ÞR.D.Gregorio /Mechanism and Machine Theory 38(2003)253–263257Diﬀerentiating relationship(2.1)yields_Bi¼_d i u iþd i_u i i¼1;2;3ð3Þwhere_d i and_u i are the time derivatives of d i and u i respectively.By using the time diﬀerentiation rule for constant intensity vectors,the following expression for_u i is obtained(Fig.5): _u i¼ð_h1i w1iþ_h2i w2iÞÂu i i¼1;2;3ð4ÞBy taking into account the relationships(1.1),(2.2),(3),(4),the relationship(1.2)becomes_P¼_diu iþb ið_h2iþ_h3iÞw2iÂw4iþd i_h2i w2iÂu iþ_h1i v i i¼1;2;3ð5Þwherev i¼w1iÂðPÀA iÞi¼1;2;3ð6ÞFinally,the dot product of the i th vector Eq.(5)by w2i yields(Fig.5)_PÁw2i¼_h1i v iÁw2i i¼1;2;3ð7ÞIf the3-UPU mechanism starts moving from rest in a conﬁguration(Figs.4and5)in which the platformÕs point P coincides with the baseÕs point located at the intersection of the baseÕs revolute pair axes,i.e.,the3-UPU wristÕs geometric conditions are matched,then the following additional vector relationships will hold in the initial conﬁguration:w1i¼PÀA ia ii¼1;2;3ð8ÞBy taking into account the relationships(8),the relationships(6)becomev i¼0i¼1;2;3ð9ÞThus,the relationships(7)become_PÁw2i¼0i¼1;2;3ð10ÞSystem(10)is a linear,homogeneous system of three scalar equations in three unknowns:the three components of_P.The matrix form of system(10)isN_P¼0ð11ÞwithN¼½w21;w22;w23 Tð12Þwhere(Á)T indicates the transpose of(Á).If the matrix N is not singular,the homogeneous system (11)will admit only the following solution:_P¼0ð13ÞDiﬀerentiating relationships(7)yields€PÁw2i þ_w2iÁ_P¼€h1i v iÁw2iþ_h1ið_v iÁw2iþv iÁ_w2iÞi¼1;2;3ð14Þ258R.D.Gregorio/Mechanism and Machine Theory38(2003)253–263R.D.Gregorio/Mechanism and Machine Theory38(2003)253–263259 where_v i;_w2i and€h1i are the time derivatives of v i,w2i and_h1i,whereas€P is the acceleration of the platformÕs point P measured in a reference systemﬁxed in the base.Moreover,diﬀerentiating relationship(6)gives_v i¼w1iÂ_P i¼1;2;3ð15ÞIf the3-UPU mechanism assumes a conﬁguration satisfying the3-UPU wristÕs geometric con-ditions(Figs.4and5),the relationships(9)and(13)will hold.Accordingly,in such a case,_v i(see Eq.(15))vanishes and the relationships(14)become€PÁw¼0i¼1;2;3ð16Þ2iEqs.(16)constitute a linear homogeneous system of three equations in three unknowns:the three components of€P.The matrix form of system(16)isN€P¼0ð17Þwhere N is the3Â3matrix given by deﬁnition(12).If the matrix N is not singular,system(17)will admit the unique solution€P¼0ð18ÞRelationship(13)and(18)lead to the following conclusion:STATEMENT:If a3-UPU mechanism starts moving from rest in a not singular conﬁguration satisfying the3-UPU wristÕs geometric conditions,it can only perform an inﬁnitesimal motion at the end of which the platformÕs point P still is in the initial position(Eq.(13))and the velocity of P still is zero(Eq.(18)),i.e.,theﬁnal conﬁguration still satisﬁes the3-UPU wristÕs geometric conditions and the point P still is at rest.Thus,the next elementary motion also must keep the point P at rest and the3-UPU wristÕs geometric conditions.As a consequence,the platform is bound to perform a sequence of elementary motions keeping the point P at rest,i.e.,the platform is constrained to performﬁnite spherical motions with center P,until the mechanism reaches a singular conﬁguration.The i th equation of system(10)is the analytic expression of the constraint that the i th leg of type UPU imposes to the platform.With reference to Fig.5,it can be interpreted as follows:a leg of type UPU,with the intermediate revolute pair axes parallel to each other and perpendicular to the leg axis(Fig.5),forbids the displacement along the w2iÕs direction of the platform point(point P in Fig.5)instantaneously coinciding with the intersection of the revolute pair axes at the leg endings.When this point(point P in Fig.5)goes to inﬁnity,i.e.,w1i and w4i(Fig.5)are parallel to each other,the forbidden displacement becomes a forbidden platform rotation around the di-rection of the free vector w2iÂw4i[18].The position analysis of the3-UPU wrist focused on the mechanism conﬁgurations that keep the platformÕs point P(Fig.4)at rest is identical with the one of the Innocenti and Parenti-CastelliÕs parallel wrist(Fig.1)[11].Thus,with reference to the demonstration reported in[11],it can be stated that the platform orientations compatible with a given set of values of the three parameters d i,i¼1;2;3,are at most eight,whereas only one triplet of d i values is compatible with a given platform orientation.3.Singularity conditionIn order toﬁnd the3-UPU wristÕs singularity conditions the relationship between the platformÕs velocities,x and_P,and the time derivatives_d i,i¼1;2;3,of the actuated joint coordinates is required.This relationship can be obtained by linearly eliminating the12variables_h1i,_h2i,_h4i,and (_h2iþ_h3i),i¼1;2;3,from the system of18equations composed of the Eqs.(1.1)and(5).By taking into account the relationship(9)and(13),the dot product of the i th Eq.(5)by u i gives the following relationship:ð_h2iþ_h3iÞ¼À_dib i u iÁðw2iÂw4iÞi¼1;2;3ð19ÞOn the other side the dot product of the i th Eq.(1.1)by w2i yieldsw2iÁx¼_h2iþ_h3i i¼1;2;3ð20ÞFinally,substituting the right side of Eq.(19)for(_h2iþ_h3i)into Eq.(20)yieldsw2iÁx¼À_dib i u iÁðw2iÂw4iÞi¼1;2;3ð21ÞThe three Eqs.(21)and the three Eqs.(10)form the following system of six equations:w2iÁx¼À_dib i u iÁðw2iÂw4iÞi¼1;2;3ð22:1Þw2iÁ_P¼0i¼1;2;3ð22:2ÞSystem(22)is the sought-after relationship between the platformÕs velocities,x and_P,and the time derivatives_d i,i¼1;2;3.Since only x appears in Eq.(22.1)and only_P appears in Eq.(22.2),Eqs.(22.1)and(22.2)are decoupled and can be analyzed independently from one another.If the3-UPU wrist assumes a conﬁguration that makes Eq.(22.1)linearly dependent,x will not be determined.The conﬁgurations making x indeterminate will be called rotation singularities. If the3-UPU wrist assumes a conﬁguration producing Eq.(22.2)linearly dependent,_P will not be determined,thus the spherical constraint between platform and base fails.The conﬁgurations which make_P indeterminate will be called translation singularities.The matrix form of Eq.(22.2)is system(11),whereas the matrix form of Eq.(22.1)is N x¼M_dð23ÞwhereM¼À1b1u1Áðw21Âw41Þ001b2u2Áðw22Âw42Þ001b3u3Áðw23Âw43Þ2666666437777775ð24:1Þ260R.D.Gregorio/Mechanism and Machine Theory38(2003)253–263_d ¼ð_d 1;_d 2;_d 3ÞT ð24:2Þand N is deﬁned by (12).Both system (23)and system (11)are singular when the determinant of matrix N vanishes.Deﬁnition (12)of matrix N makes writing its determinant,det(N ),in explicit form possible as followsdet ðN Þ¼w 21Áw 22Âw 23ð25ÞThus,both rotation and translation singularities occur when the conﬁguration assumed by the 3-UPU wrist satisﬁes the following condition:w 21Áw 22Âw 23¼0ð26ÞSingularity condition (26)will be matched,if the three unit vectors w 2i ,i ¼1;2;3,are linearly dependent,i.e.,they are all parallel to the same plane.From an analytic point of view,the unit vectors w 2i ,i ¼1;2;3,depend on the links Õgeometry and the platform Õs orientation.If the links Õgeometry is given,Eq.(26)will be a scalar equation in the three variables chosen to parameterize the platform Õs orientation.The solutions of a scalar equation in three unknowns lie on a surface,when they are reported in a three-dimensional Cartesian diagram whose coordinates are the three unknown variables of the scalar equation.From a geometric point of view,with reference to Fig.4,condition (26)is veriﬁed when the planes the three triangles A i B i P ,i ¼1;2;3,lie on have a straight line through the point P as common intersection (Fig.6).In fact,in this case,the unit vectors w 2i ,i ¼1;2;3,are all per-pendicular to the line common to the three planes and are all parallel to any plane perpendicular to the same line.When this geometric condition occurs,the platform Õs point P can undergo an inﬁnitesimal displacement along the line common to the three planes and the platformcanR.D.Gregorio /Mechanism and Machine Theory 38(2003)253–263261262R.D.Gregorio/Mechanism and Machine Theory38(2003)253–263undergo an inﬁnitesimal rotation around the same line in both of the possible directions after any inﬁnitesimal variation of the lengths d i,i¼1;2;3,is given(Fig.6).A special case of this geometric condition occurs when the three leg axes are all parallel.In sucha situation,the line common to the trianglesÕplanes is parallel to the leg axes.4.ConclusionsIn this paper,it has been shown that a manipulator of type3-UPU,named3-UPU wrist,under some manufacturing and mounting conditions make the end eﬀector performﬁnite spherical motions after its prismatic pairs are actuated.Moreover,it has been shown that the3-UPU wrist can reach singular conﬁgurations,named translational singularities,where the spherical constraint between platform and base fails.Finally,the condition that makes it possible toﬁnd all the3-UPU wristÕs singularities has been written in explicit form and has been geometrically interpreted.AcknowledgementsTheﬁnancial support of the Italian MURST is gratefully acknowledged.References[1]J.M.Herv e,Design of parallel manipulators via the displacement group,in:Proceedings of the9th World Congresson the Theory of Machines and Mechanisms,Milan(Italy),vol.3,1995,pp.2079–2082.[2]R.E.Stamper,L.W.Tsai,G.C.Walsh,Optimization of a three dof translational platform for well-conditionedworkspace,in:Proceedings of IEEE International Conference on Robotics and Automation,1997,paper no.A1-MF-0025.[3]R.Clavel,DELTA:a fast robot with parallel geometry,in:Proceedings of the18th International Symposium onIndustrial Robots,Sydney(Australia),1988,pp.91–100.[4]L.W.Tsai,Kinematics of a three-dof platform with three extensible limbs,in:J.Lenarcic,V.Parenti-Castelli(Eds.),Recent Advances in Robot Kinematics,Kluwer Academic Publishers,Netherlands,1996,pp.401–410. [5]R.Di Gregorio,V.Parenti-Castelli,A translational3-dof parallel manipulator,in:J.Lenarcic,M.L.Husty(Eds.),Advances in Robot Kinematics:Analysis and Control,Kluwer Academic Publishers,Netherlands,1998,pp.49–58.[6]R.Di Gregorio,Closed-form solution of the position analysis of the pure translational3-RUU parallel mechanism,in:Proceedings of the8th Symposium on Mechanisms and Mechanical Transmissions,MTM2000,Timisoara (Romania),2000.[7]P.B.Zobel,P.Di Stefano,T.Raparelli,The design of a3-dof parallel robot with pneumatic drives,in:Proceedingsof the27th Internatinal Symposium on Industrial Robot,Milan(Italy),1996,pp.707–710.[8]R.Clavel,M.Bouri,S.Grousset,M.Thurneysen,A new4d.o.f.parallel robot:the Manta,in:Proceedings of theInt.Workshop on Parallel Kinematic Machines,PKMÕ99,Milan(Italy),1999,pp.95–100.[9]C.M.Gosselin,J.Angeles,The optimum kinematic design of a spherical three-degree-of-freedom parallelmanipulator,ASME Journal of Mechanisms,Transmission and Automation in Design111(2)(1989)202–207.[10]R.I.Alizade,N.R.Tagiyiev,J.Duﬀy,A forward and reverse displacement analysis of an in-parallel sphericalmanipulator,Mechanism and Machine Theory29(1)(1994)125–137.[11]C.Innocenti,V.Parenti-Castelli,Echelon form solution of direct kinematics for the general fully-parallel sphericalwrist,Mechanism and Machine Theory28(4)(1993)553–561.R.D.Gregorio/Mechanism and Machine Theory38(2003)253–263263 [12]S.K.Agrawal,G.Desmier,S.Li,Fabrication and analysis of a novel3-dof parallel wrist mechanism,ASMEJournal of Mechanical Design117(2A)(1995)343–345.[13]M.Karouia,J.M.Herv e,A three-dof tripod for generating spherical rotation,in:J.Lenarcic,M.M.Stanisic(Eds.),Advances in Robot Kinematics,Kluwer Academic Publishers,Netherlands,2000,pp.395–402.[14]K.-M.Lee,D.K.Shah,Kinematic analysis of a three-degrees-of-freedom in-parallel actuated manipulator,IEEEJournal of Robotics and Automation4(3)(1988)354–360.[15]G.R.Dunlop,T.P.Jones,Position analysis of a3-dof parallel manipulator,Mechanism and Machine Theory34(8)(1997)903–920.[16]M.Ceccarelli,A new3d.o.f.spatial parallel mechanism,Mechanism and Machine Theory32(8)(1997)896–902.[17]R.Di Gregorio,V.Parenti-Castelli,Position analysis in analytical form of the3-PSP mechanism,in:Proceedings of1999ASME Design Engineering Technical Conferences,Las Vegas(Nevada,USA),1999,paper no.DETC99/ DAC-8689.[18]R.Di Gregorio,V.Parenti-Castelli,Mobility analysis of the3-UPU parallel mechanism assembled for a puretranslational motion,in:Proceedings of the1999IEEE/ASME International Conference on Advanced Intelligence Mechatronics,AIMÕ99,Atlanta(Georgia),1999,pp.520–525.。

胡壮麟语言学重点名词解释

语言学重点名词解释refer to the defining properties of human language that distinguish it from any animal system of communication.refers that there is no logical connection between a linguistic symbol and what the symbol stands for(meaning and sounds).means the property of having two levels of structures, such that units of the primary level are composed of elements of the secondary level and each of the two levels has its own principles of organization.system must be learned by each speaker.people what they should say and not say.in which languages are treated as self-contained systems of communication at any particular time在那一刻、时、块的情况（当代、古代）历时in which the changes to which languages are subject in the course of time and treated historically.(在过程中都有什么变化、区别、有大时间变化)2个共时即为历时occur in the world’s languages.are sounds produced by obstructing the flow of air in the oral cavity。

San Gen Shugi

things are actually working. The point of the 3 Reals is that Engineers, Managers, or
Designers must understand that just because they have given directions or drawn up plans does not mean that things are happening as they expect in reality. Often, it is Operators on the line who have had little or no say in the development of the production system that know better about the reality
automatically returned to the furnace for re-melting. But by categorizing them and
finding their causes, the casting process can be modified and improved (KAIZEN). Without looking at the GEMBUTSU, however, no improvement can happen.
GEMBUTSU - Real Thing
Going to the GEMBA is not enough. One must go all the way to the GEMBUTSU, or the actual item. A very common example of the importance placed on the GEMBUTSU is the procedure for categorizing defective products instead of just disposing of them. For

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 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Topological Hunds rules and the electronic properties of a triple lateral quantum dot molec

a r X i v :c o n d -m a t /0610175v 3 [c o n d -m a t .m e s -h a l l ] 26 F eb 2007Topological Hunds rules and the electronic properties of a triplelateral quantum dot moleculeMarek Korkusinski,1Irene Puerto Gimenez,1Pawel Hawrylak,1Louis Gaudreau,2,3Sergei A.Studenikin,2and Andrew S.Sachrajda 21Quantum Theory Group,Institute for Microstructural Sciences,National Research Council of Canada,Ottawa,Ontario,Canada K1A 0R62Quantum Physics Group,Institute for Microstructural Sciences,National Research Council of Canada,Ottawa,Ontario,Canada K1A 0R63R´e groupement Qu´e b´e cois sur les Mat´e riaux de Pointe,Universit´e de Sherbrooke,Qu´e bec,Canada J1K 2R1Abstract We analyze theoretically and experimentally the electronic structure and charging diagram of three coupled lateral quantum dots ﬁlled with ing the Hubbard model and real-space exact diagonalization techniques we show that the electronic properties of this artiﬁcial molecule can be understood using a set of topological Hunds rules.These rules relate the multi-electron energy levels to spin and the inter-dot tunneling t ,and control charging energies.We map out the charging diagram for up to N =6electrons and predict a spin-polarized phase for two holes.The theoretical charging diagram is compared with the measured charging diagram of the gated triple-dot device.PACS numbers:,73.23.HkI.INTRODUCTIONFollowing on earlier work which showed that a small and well-controlled number of elec-trons can be conﬁned in a single1,2and a double quantum dot,3,4,5,6,7an artiﬁcial lateral quantum molecule consisting of three quantum-mechanically coupled lateral quantum dots has been demonstrated.8The triple quantum dot molecule is a natural step toward creat-ing quantum dot networks,with potential applications in quantum computing.9,10,11When ﬁlled with three electrons,one electron per dot,this device can serve as a simple quantum logic circuit,with each electron spin treated as a qubit.One can also use the molecule as a single coded qubit,11,12,13,14whose states are encoded in the states of three electronic spins but tunable with applied voltage.The triple dot could also be used to create entanglement between spin qubits,15spin and charge qubits,16as a charge rectiﬁer,17,18and may exhibit a characteristic Kondo eﬀect when coupled to the leads.19,20,21,22,23,24,25,26With electrons lo-calized on individual dots and their tunneling controlled by gates,the triple dot molecule can be also thought of as an implementation of the tunable Hubbard model,an important step toward realization of“quantum materials”.27,28,29,30,31,32The electronic properties of the triple quantum dot with one electron per dot have been studied theoretically by a number of authors.To make contact with the pairwise-exchange formalism used in quantum information,11attempts were made to map the properties of this system onto those of the three-spin Heisenberg model.Scarola and Das Sarma33used the Hubbard,variational,and exact diagonalization approaches to demonstrate that this mapping can be carried out only for a limited range of triple-dot parameters.Mizel and Lidar34,35,36arrived at similar conclusions using the Heitler-London and Hund-M¨u lliken schemes to calculate the energy levels of three coupled dots with one electron per dot. In both cases the many-body eﬀects were responsible for the appearance of higher-order terms in the eﬀective spin Hamiltonian.In an alternative approach,in Ref.12we have used real-space wave functions and the conﬁguration-interaction technique to analyze the three-electron triple-dot molecule acting as a single coded qubit and shown how its energy levels can be tuned by voltages applied to gates deﬁning the structure.Properties of the triple-dot molecule as a scattering center have also been studied us-ing quantum transport ing the density-functional and quantum Monte Carlo methods,Stopa17calculated the currentﬂowing through a nominally empty molecule con-nected to electron reservoirs and under bias.The rectifying behavior of the system predicted in this analysis was conﬁrmed experimentally.18Landr´o n de Guevara and Orellana37calcu-lated the zero-temperature conductance through a linear molecule coupled in parallel to the leads using a Hubbard approach in a magneticﬁeld.Apart from the Fano resonances in the spectrum,they found evidence of formation of the quantum-molecular states decoupled from the leads.The Hubbard model has also been used to investigate the triple-dot system in the Kondo regime,both in the linear20,21,22and triangular topology.23,24,25,26 In this paper we describe the electronic properties of a lateral triple quantum dot molecule as a function of electron numbers.In analogy to the work on quantum materials,27,28we model our system with the Hubbard Hamiltonian,but the obtained results are veriﬁed by microscopic methods.In the Hubbard model we retain only one lowest-energy orbital per dot.The lowest-energy shell of the molecule can beﬁlled with up to N e=6electrons.We analyze in detail the ordering of energy levels,the spacing of Coulomb blockade peaks and the charging and spin phase diagram of this shell.We demonstrate that the energy levels of the molecule are related to the total spin of electrons but not directly related to the charge e.Weﬁnd the spin singlet as the two-electron ground state,with the singlet-triplet(S-T) splitting proportional to the single-particle tunneling matrix element t.This is in contrast to atoms,where the S-T splitting is proportional to the electronic exchange and hence to e2, or to magnetic solids,where super-exchange leads to S-T splitting proportional to1/e2.On the other hand,for two holes(N e=4)we predict a spin polarized ground state and a singlet-triplet transition driven only by modifying the topology of the system.For three electrons in a half-ﬁlled shell(N e=3)we conﬁrm the existence of the frustrated antiferromagnetic ground state.12The fact that the tunneling alone distinguishes singlet and triplet states is related to the interplay of the Fermi statistics and system topology.We term the set of rules established here and relating spin of the ground state to theﬁlling of the shell,topology, and tunneling,”topological Hunds rules”.The ability to tune tunneling by gates opens the possibility of directly manipulating the electron spin using electrical means only,of interest in designing novel quantum materials,magneto-electronics and quantum computation.We show that the Hubbard model is capable of reproducing the charging diagram of a lateral gated triple-dot measured recently by Gaudreau et al.8The paper is organized as follows.In Sec.II we describe the model lateral triple-dot device and construct the Hubbard Hamiltonian.In Sec.III we determine the electronicstructure of the device charged with N e=1to6electrons.Results of the Hubbard model are tested against real space(RSP)conﬁguration interaction(RSP-CI)and linear combination of atomic or quantum dot orbitals(LCAO-CI)calculations.The charging diagram as a function of the dot energies is presented and analyzed in Sec.IV.In Sec.V we relate the calculated and measured charging diagrams.Summary and conclusions are presented in Sec.VI.II.THE MODELThe proposed model gated triple-dot device realizing the triple dot using only metallic gates,studied in Ref.12and related to the one studied by Gaudreau et al.in Ref.8,is shown in Fig.1(a).It consists of a heterojunction with a two-dimensional electron gas (2DEG)created at a distance D below the top surface of the sample.The metallic gates deposited on the surface serve to deplete the2DEG underneath.Any opening in the gates is translated electrostatically into a local potential minimum,capable of conﬁning a small number of electrons.Thus,in our model the three circular holes in the main gate(shown in gray)deﬁne a triangular triple quantum dot lateral conﬁnement.Each isolated potential minimum gives rise to a quantized energy spectrum,of which we retain only the lowest energy level E i in dot i.By tuning the voltage on the main gate we can control the number of conﬁned electrons.For example,in Fig.1(a)we show N e=2electrons with parallel spins localized on two of the dots.This is not,however,a depiction of a quantum molecular state: due to the interdot coupling the electrons are delocalized across the molecule.The main gate alone deﬁnes a symmetric triangular molecule with identical pairwise coupling of all dots.This triple-dot potential can be well approximated by a sum of three Gaussians.The single-particle conﬁnement can be additionally tuned by three smaller gates,shown in red,green,and blue.Their arrangement with respect to the potential minima is shown schematically in Fig.1(b).The gate V G1controls simultaneously the lowest energy levels E1 and E2of dots1and2,and the gate V G3controls the energy level E3of dot3.Additionally, the gate V G13is designed to tune the topology of the system without signiﬁcantly changing the energies E i.By biasing it with a suﬃciently high negative voltage we increase the tunneling barrier between dots1and3and change the sample layout from a closed triangle, in which all dots are identically coupled,to a linear molecule,in which the tunneling betweendots1and3is not allowed.We examine the electronic properties of our triple quantum dot molecule in the frame of the Hubbard model with one spin-degenerate orbital per dot.Without specifying them explicitly,the localized orbitals in the Hubbard model are assumed to be orthogonal.This is to be contrasted with the approach starting from the linear combination of atomic orbitals (LCAO),which are non-orthogonal.The orthogonalization leads to extended,quantum-molecular orbitals which serve as a basis for CI calculation.In the Hubbard model,with c+iσ(c iσ)operators creating(annihilating)electrons with spinσon the orbital of i-th dot,the Hamiltonian can be written as:ˆH=3 σ,i=1E i c+iσc iσ+3 σ,i,j=1,i=j t ij c+iσc jσ+3 i=1U i n i↓n i↑+1on the sign of the element t.In numerical calculations of the single-particle spectrum corresponding to the potential produced by metallic gates12shown in Fig.1(a)weﬁnd the ground state to be non-degenerate,indicating that t<0.Additionally,the magnitude of the tunneling matrix element can be found from the single-particle energy gap∆=3|t|.Knowledge of the sign of the oﬀ-diagonal element allows us to construct the single-particle molecular orbitals.The ground state is|M1 =13(|1 +|2 +|3 ),while the two degenerateexcited states are|M2 =12(|1 −|2 )and|M3 =16(|1 +|2 −2·|3 ).The states|M2 and|M3 were chosen to be symmetric with respect to a mirror plane passing through the dot3and intersecting the(1−2)base of the triangle at its midpoint.However,due to the degeneracy of the two levels,any pair of orthogonal states created as linear combinations of |M2 and|M3 will be viable as eigenstates.The degeneracy of the excited states is a direct consequence of the symmetry of the triangular molecule.Changing its topology,e.g.,by increasing the tunneling barrier between dots1and3,will remove the degeneracy.In the limit of an inﬁnite barrier,i.e.,t13=0,we deal with a linear triple-dot molecule,whose single-√2|t|). particle energy spectrum consists of three equally spaced levels:(E−Thus,the triangular triple dot design makes it possible to engineer the degeneracy of states solely by electrostatic means.Now we can start to populate our triple-dot molecule with electrons.Let us start our many-body analysis with the simplest case of N e=5.As the maximal number of electrons in our system is six,we can interpret theﬁve-electron conﬁgurations as those of a single hole.The hole(e.g.,with spin down)can be placed on either of the dots,and thus our basis consists of three conﬁgurations:|1(H) =h+1↓|N e=6 =c+3↑c+2↑c+3↓c+2↓c+1↓|0 ,|2(H) = h+2↓|N e=6 =c+1↑c+3↑c+3↓c+2↓c+1↓|0 ,and|3(H) =h+3↓|N e=6 =c+2↑c+1↑c+3↓c+2↓c+1↓|0 ,with h+iσbeing the creation operator of the hole with spinσon the i-th dot.It is convenient to express the energies of these conﬁgurations with respect to the total energy of the system with six electrons E F=2E1+2E2+2E3+U1+U2+U3+4V12+4V13+4V23.We have =E F−E1−U1−2V12−2V13,E(H)2=E F−E2−U2−2V12−2V23,andthen E(H)1=E F−E3−U3−2V13−2V23.The three energies are respectively the diagonal terms of E(H)3our single-hole Hamiltonian.The oﬀ-diagonal terms are composed out of the single-particle tunneling matrix elements.We have i(H)|ˆH|j(H) =−t ij;the negative phase is due to the anticommutation relations of the electronic creation and annihilation operators.As we can see,the single-hole Hamiltonian can be obtained from the single-electron Hamiltonian byappropriately modifying the diagonal terms and setting t ij↔−t ij.This is the signature of the particle-hole symmetry.27However,for the triangular triple dot on resonance this symmetry is not reﬂected in the energy spectrum of the hole:in this case,the opposite sign of the oﬀ-diagonal element leads to a doubly-degenerate hole ground state.This property is immediately apparent in the molecular basis:we create the lowest-energy conﬁguration by ﬁlling the molecular ground state|M1 with two of theﬁve electrons,and distributing the remaining three on the degenerate orbitals|M2 and|M3 .The latter can be accomplished in two energetically equivalent ways,hence the double degeneracy.Note,however,that the electron-hole symmetry is fully restored upon transition to the linear triple-dot molecule. For this topology,the single-particle spectrum of both the electron and the hole consists of three equally spaced non-degenerate levels.B.Two electrons and two holesThe interplay of topology and statistics is particularly important in the cases of two electrons and two holes conﬁned in the triple dot molecule.Let us consider the case of N e=2ﬁrst.Since the Hamiltonian(1)commutes with the total spin operator,we can classify the two-electron states into singlets and triplets.Working with the molecular basis set,we form the conﬁguration with the lowest energy by placing both carriers with antiparallel spins on orbital|M1 .Therefore we expect the ground state of the two-electron system to be a spin singlet,irrespective of the molecule’s topology.However,in order to examine the topological and statistical eﬀects in the energy spectrum and the structure of the wave functions,we carry out a systematic analysis in the localized basis.Due to Fermi statistics,the two electrons with parallel spins cannot occupy the same quantum dot.Hence there are only three possible triplet conﬁgurations,|T1 =c+2↓c+1↓|0 , |T2 =c+3↓c+1↓|0 ,and|T3 =c+3↓c+2↓|0 ,shown schematically in Fig.2(a).The three triplet conﬁgurations interact with each other only via the single-particle tunneling Hamiltonian. However,in evaluating the respective matrix elements we need to follow the Fermionic anticommutation rules of the creation and annihilation operators.For example,acting with ˆH on the conﬁguration Tto produce the conﬁguration T3requires the evaluation of the1following expression:ˆH|T1 =+t31c+3↓c1↓c+2↓c+1↓|0 .In order to remove the electron1weﬁrst have to move it around electron2,and soˆH|T1 =−t31c+3↓c+2↓c1↓c+1↓|0 =−t31|T3 .Hence,tunneling of the electron from dot1to dot3in the presence of the electron in dot2generates an additional phase or changes the sign of the tunneling matrix element.This is of course the most elementary property of Fermions brought out so clearly in this simple model.By contrast,tunneling from dot2to dot3in the presence of electron in dot1does not change the sign of the tunneling matrix element.The resulting triplet Hamiltonian matrix takes the following form:ˆH T =E1+E2+V12t23−t13t23E1+E3+V13t12−t13t12E2+E3+V23.(2)ˆHTis related to the one-hole Hamiltonian.This similarity becomes more apparent ifˆH T is written in the basis{|T1 ,−|T2 ,|T3 },in which case all the oﬀ-diagonal elements acquire a negative phase.This is not surprising,since the single-hole conﬁgurations analyzed in the previous Section can be generated from the above triplet conﬁgurations simply by adding to them an inert core of three electrons spin up,one electron per dot.With the three dots on resonance and all tunneling matrix elements t ij equal and negative,the triplet energy spectrum is found to be(2E+V−|t|,2E+V−|t|,2E+V+2|t|).As in the case of the single hole,the lowest-energy triplet state is doubly degenerate.Moreover,the renormalization of the lowest energy2E+V−|t|from the single conﬁguration energy2E+V,as well as the gap in the triplet spectrum,are determined entirely by tunneling.The splitting between the ground andﬁrst excited states is the same as that found in the single-carrier case and equals3|t|.We shall now demonstrate that topology and statistics diﬀerentiates between triplet and singlet two-electron states.The singly-occupied singlet conﬁgurations|S1 ,|S2 ,and|S3 are obtained from the triplet conﬁgurations|T1 ,|T2 ,and|T3 byﬂipping the spin of one electron and properly antisymmetrizing the conﬁgurations.For example,the conﬁguration |S1 =12 c+2↓c+1↑+c+1↓c+2↑ |0 .In addition to the singly-occupied conﬁgurations there are also three doubly-occupied conﬁgurations,e.g.,|S4 =c+1↓c+1↑|0 ,as shown in Fig.2(b).Inthe basis of the six conﬁgurations the two-electron singlet Hamiltonian can be written as:ˆH S =E1+E2+V12t23t13√2t120 t23E1+E3+V13t12√2t13 t13t12E2+E3+V230√2t23√2t1302E1+U100√2t2302E2+U20√2t23002E3+U3.(3)The3×3upper left-hand corner ofˆH S corresponds to the three singly occupied conﬁgura-tions|S1 ,|S2 ,and|S3 .It is similar to the two-electron triplet HamiltonianˆH T but diﬀersfrom it by the positive phase of the tunneling matrix element t13.Hence,in the triangular topology of the triple-dot molecule the tunneling from dot1to dot3distinguishes between the singlet and the triplet spin conﬁgurations.By setting t13=0,i.e.,upon transition to the linear topology,this diﬀerence disappears.However,the singlet basis is still diﬀerent from its triplet counterpart due to the presence of the doubly-occupied conﬁgurations.For the dots on resonance the energies of the six singlet levels can be obtained analytically. The spectrum can be grouped into two non-degenerate levels E S1,2:E S1,2=(2E+V−2|t|)+1(4√2 (U−V−|t|)±2t)2+(U−V−|t|)2 .(5)In the strong coupling limit U≫V>|t|the singlet ground-state energy E S1≈(2E+ V−2|t|)−8t2U−V.It is proportional to the tunneling matrix element|t|and contains the second-order super-exchange correction∼t2/(U−V)due to the doubly occupied singlet conﬁgurations.Removing the resonance by detuning the onsite energies E i enhances the contribution from the doubly-occupied states.Therefore the ground state maintains its singlet character independently of the choice of gate voltages.The situation is qualitatively diﬀerent when two holes,instead of two electrons,populate the system.The two holes are created when two electrons are removed from the closed-shellconﬁguration with N e=6,i.e.,they correspond to N e=4electrons.In the molecular basis corresponding to the triangular triple dot we put two electrons on the lowest-energy orbital |M1 ,and the remaining two electrons on the degenerate pair of orbitals|M2 and|M3 .With this alignment of levels it is possible to create both triplet and singlet conﬁgurations, all with the same single-particle energy,and it is not immediately clear which total spin is preferred.On the other hand,in the limit of the linear triple dot the molecular orbitals are non-degenerate and the four-electron ground state is expected to be a spin singlet.The selected two-hole singlet and triplet conﬁgurations in the localized basis are illus-trated in Fig.2(c).Let us focus on the tripletsﬁrst.They involve one electron spin-up occupying theﬁrst,second,or third dot in the presence of an inert core of three spin-down electrons.For example,the conﬁguration shown in left-hand panel of Fig.2(c)can be written as|T(H)1 =h+1↓h+2↓|N e=6 =c+3↑c+3↓c+2↓c+1↓|0 .Therefore,the hole triplet Hamiltonian is equivalent to the single-electron Hamiltonian,diﬀering from it only in di-agonal terms.For example,the energy of the conﬁguration|T(H)1 is T(H)1|ˆH|T(H)1 =E F−E1−E2−U1−U2−3V12−2V13−2V23.The two-hole triplet Hamiltonian can also be compared to the two-electron triplet HamiltonianˆH T,written in the modiﬁed ba-sis set{|T1 ,−|T2 ,|T3 }(i.e.,with all oﬀ-diagonal matrix elements acquiring a negative phase).Setting aside the diagonal matrix elements,the two Hamiltonians are connected by the electron-hole symmetry transition t ij↔−t ij.However,unlike that of the elec-tronic triplet,the ground state of the hole triplet is non-degenerate,and its energy is E T(H)=E F−2E−2U−7V−2|t|.As it is in the case of the single electron and the 1single hole,the particle-hole symmetry between the two-electron triplet and the two-hole triplet is fully restored upon transition to the linear topology of the triple dot.Let us move on to considering the two-hole singlet conﬁgurations.The singly-occupied states,illustrated in the middle panel of Fig.2(c),involve the two holes occupying two diﬀerent dots,while the doubly-occupied states,such as the one in the right-hand panel of Fig.2(c),hold both holes on the same dot.The two-hole singlet Hamiltonian is analogous to that of the two-electron singlet,Eq.(3).However,we need to replace the energy of two-electron complexes with the energy of two-hole complexes,and change the phase of the oﬀ-diagonal elements connecting the singly-occupied conﬁgurations.The sign of elements √The ground-state energy of the hole singlet for the triangular triple dot on resonance is well approximated by E S(H)1≈(E F−2E−2U−7V−|t|)−2t2.Note that the two-hole super-exchange term is fourU−Vtimes smaller than the super-exchange correction to the energy of the two-electron singlet. By increasing the tunneling or decreasing the on-site Hubbard repulsion we can increase the contribution from super-exchange and lower the energy of the singlet state.Therefore, the total spin of the two-hole ground state for the triple dot on resonance depends on the interplay of Hubbard parameters.For2|t|<U−V the ground state is a spin triplet, and a triplet-singlet transition can be induced by increasing the hopping matrix element. The triplet-singlet transition can also be induced by biasing one of the dots,which lowers the energy of the doubly-occupied singlet conﬁgurations.Hence the conﬁguration of two holes shows a nontrivial dependence on tunneling,Coulomb interactions and gate voltages allowing to control the system’s magnetic moment purely by electrical means.C.Three electronsTo complete our understanding of the energy levels of a triple quantum dot molecule we need to analyze the half-ﬁlled case of three electrons(or,equivalently,three holes).We start with the completely spin-polarized system,i.e.,one with total spin S=3/2.In this case we can distribute the electrons on the three dots in only one way:one electron on each site, which gives a spin-polarized state|a3/2 =c+3↓c+2↓c+1↓|0 .As the basis of our Hilbert space consists of one conﬁguration only,|a3/2 is the eigenstate of our system,and its energy is E3/2=E1+E2+E3+V12+V13+V23.Let us nowﬂip the spin of one of the electrons. This electron can be placed on any orbital,and with each speciﬁc placement the remaining two spin-down electrons can be distributed in three ways.For example,Fig.3(a)shows the three conﬁgurations with the spin-up electron occupying the dot1.Thus,altogether we can generate nine diﬀerent conﬁgurations.Three of these conﬁgurations involve single occupancy of the orbitals.They can be written as|a =c+3↓c+2↓c+1↑|0 ,|b =c+1↓c+3↓c+2↑|0 ,and|c = c+2↓c+1↓c+3↑|0 .Out of these three conﬁgurations we construct the three eigenstates of the total spin operator.One of those eigenstates is|a3/2 =13(|a +|b +|c ),and it corresponds to the total spin S=3/2.The two other eigenstates,|a1/2 =12(|a −|b )and|b1/2 =16(|a +|b −2|c ),correspond to the total spin S=1/2.The remaining six conﬁgurations involve doubly-occupied orbitals.They are|c1/2 =c+2↓c+1↓c+1↑|0 ,|d1/2 =c+3↓c+1↓c+1↑|0 ,|e1/2 =c+3↓c+2↓c+2↑|0 ,|f1/2 =c+1↓c+2↓c+2↑|0 ,|g1/2 =c+1↓c+3↓c+3↑|0 ,|h1/2 =c+2↓c+3↓c+3↑|0 .All these conﬁgurations are eigenstates of the total spin with S=1/2.Thus,among our nine spin-unpolarized states we have one high-spin,and eight low-spin states.In this basis the Hamiltonian matrix is block-diagonal,with the high-spin state completely decoupled.The energy corresponding to this state is equal to that of the fully polarized system discussed above,and is equal to E3/2.In the basis of the nine S=1/2conﬁgurations we construct the Hamiltonian matrix by dividing9conﬁgurations into three groups,each containing one of the singly-occupied conﬁgurations|a ,|b ,and|c .By labeling each group with the index of the spin-up electron, the Hamiltonian takes the form of a3×3matrix:ˆH 1/2=ˆH1ˆT12ˆT+31ˆT+12ˆH2ˆT23ˆT31ˆT+23ˆH3.(6)The diagonal matrix,e.g.,ˆH 1=2E1+E2+2V12+U1t23−t13t232E1+E3+2V13+U1t12−t13t12E1+E2+E3+V12+V13+V23describes the interaction of three conﬁgurations which contain spin-up electron on site1,i.e., two doubly-occupied conﬁgurations|c1/2 and|d1/2 ,and a singly-occupied conﬁguration|a(in this order,see Fig.3(a)).The conﬁgurations with double occupancy acquire the diagonal interaction term U.The three conﬁgurations involve a pair of spin-polarized electrons(spin triplet)moving on a triangular plaquette in the presence of a“spectator”spin-up electron. Because of the triplet character of the two electrons,the phase of the hopping matrix element −t13from site1to site3is diﬀerent from the phase of the hopping matrix element+t23from site2to site3.As discussed above,the negative phase in−t13distinguishes the singlet and triplet electron pairs.The remaining matrices corresponding to spin-up electrons localized on sites2and3can be constructed in a similar fashion.The interaction between them isgiven in terms of eﬀective hopping matrixˆT ij =0−t ij000−t ij+t ij00.There is no direct interaction between the conﬁgurations with single occupancy,since such scattering process would have to involve two electrons,one with spin up and one with spin down.This cannot be accomplished by the single-particle tunneling.These states are coupled only indirectly,involving the conﬁgurations with double occupancy.The low-energy spectrum of the Hubbard Hamiltonian of the N e=3quantum-dot molecule can be further approximated by the spectrum of the model spin Hamiltonian:H3e=E3/2+ i<j J ij S i· S j−1/4 + i<j<k D ijk S i· S j× S k (7) Here,E3/2is the energy of the spin S=3/2state,J ij are exchange matrix elements of the Heisenberg part of the spin Hamiltonian which depend on microscopic parameters of the triple dot,and D ijk are higher order spin-spin interactions discussed,e.g.,by Scarola and Das Sarma in Ref.33.We deﬁne the eﬀective exchange constant J for the triple dot molecule with three electrons in terms of the gap between the S=1/2and S=3/2states as E3/2−E1/2=3J/2.Without the higher order corrections J would have been equal to the Heisenberg J ij,otherwise it is simply related to the gap of the N e=3electron spectrum.The mapping of the behavior of our system onto the eﬀective exchange Hamiltonian (7)connects our analysis to the general formalism used in quantum computing.10,11Our considerations do not introduce any new elements into that formalism,but rather provide means for its realistic and accurate parametrization,reﬂecting the properties of an actual gated triple-dot device.The Heisenberg Hamiltonian(7)can be used to model the behavior of three electrons conﬁned in a triple dot treated as three coupled qubits.However,it applies also to a coded qubit scheme,in which the states of the entire molecule are treated as the logical states of a single qubit.In Ref.12we have presented a detailed analysis of such a system,in which we selected the two lowest total spin1/2states as the logical states|0L and|1L of the coded qubit,respectively.In that design,the control of the energy gap between the two states by。

陶哲轩评述佩雷尔曼的庞加莱猜想证明

2In fact, an additional argument, noting that surgery cannot destroy the simple-connectedness of the manifold components, demonstrates that the Ricci ﬂow with surgery from a simply connected manifold only involves ﬁnite unions of disconnected spheres. We thank John Morgan for clarifying this point.
1This hypothesis is in fact quite natural, because the arguments also show conversely that ﬁnite time extinction for Ricci ﬂow is only possible when the fundamental group is a free product of ﬁnite groups and inﬁnite simple groups. For the purposes of proving just the Poincar´e conjecture, though, it is possible to work entirely in the category of simply connected manifolds throughout, although this only simpliﬁes the argument at one small point (in the proof of ﬁnite time extinction). We thank John Morgan for clarifying this point.

Crepant resolution of trihedral singularities

§1. Introduction The purpose of this paper is to construct a crepant resolution of quotient singularities by ﬁnite subgroups of SL(3, C) of certain type, and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. The problem of ﬁnding a nice resolution of quotient singularities by ﬁnite subgroups of SL(3, C) arose from mathematical physics. In the superstring theory, the dimension of the space-time is 10, four of them are usual space and time dimensions, and other six are compactiﬁed on a compact Calabi-Yau space M . From a point of view of algebraic geometry, the Calabi-Yau space is a smooth three-dimensional complex projective variety whose canonical bundle is trivial and fundamental group is ﬁnite. In the physics of superstring theory, one considers the string propagation on a manifold M which is a quotient by a ﬁnite subgroup of symmetries G. By a physical argument of string vacua of M/G, one concludes that the correct Euler number for the theory should be the “orbifold Euler characteristic”[3,4], deﬁned by χ(M,> ),

Three Body Problem (English Introduction)

03
Luo Ji's intelligence, maturity, and determination drive the narrative forward as he strives to understand and combat the lien threat
Cheng Xin
A female aerospace engineer who benefits a key figure in the resistance against the Trisolaran invasion
the attention of potential predictors
03
Predator and pre dynamics
The Dark Forest Law suggestions that the universe is a
dangerous place where predictors consistently seek out and
• Description of the Trisolaran invasion and the subsequence human resistance • Introduction of Cheng Xin, the new Wallace, and her plan to save humanity • Conclusion of the story with the revision of the ultimate fate of the human race and the universe • Note: The above points are a brief overview of the plot and do not include detailed spoilers The

Singularity-1

Stress singularities in classical elasticity–I: Removal,interpretation,and analysisGB SinclairDepartment of Mechanical Engineering,Louisiana State University,Baton Rouge,LA70803-6413This review article has two parts,published in separate issues of this journal,which considerthe stress singularities that occur in linear elastostatics.In the present Part I,after a brief re-view of the singularities that attend concentrated loads,attention is focused on the singulari-ties that occur away from such loading,and primarily on2D conﬁgurations.A number of ex-amples of these singularities are given in the Introduction.For all of these examples,it isabsolutely essential that the presence of singularities at least be recognized if the stressﬁeldsare to be used in attempts to ensure structural integrity.Given an appreciation of a stress sin-gularity’s occurrence,there are two options open to the stress analyst if the stress analysis isto actually be used.First,to try and improve the modeling so that the singularity is removedand physically sensible stresses result.Second,to try and interpret singularities that persist ina physically meaningful way.Section2of the paper reviews avenues available for the re-moval of stress singularities.At this time,further research is needed to effect the removal ofall singularities.Section3of the paper reviews possible interpretations of singularities.At thistime,interpretations using the singularity coefﬁcient,or stress intensity factor,would appear to be the best available.To implement an approach using stress intensity factors in a generalcontext,two types of companion analysis are usually required:analytical asymptotics to char-acterize local singularﬁelds;and numerical analysis to capture participation in global conﬁgu-rations.Section4of the paper reviews both types of analysis.At this time,methods for bothare fairly well developed.Studies in the literature which actually effect asymptotic analyses of speciﬁc singular conﬁgurations will be considered in Part II of this review article.The presentPart I has182references.͓DOI:10.1115/1.1762503͔1INTRODUCTION1.1Objective and scopeStress singularities are not of the real world.Nonetheless, they can be a real fact of a stress analysis.Then it is essential to take them into account if the analysis is to be of any real use.The primary objective of this review is to assist in this regard.That is,in theﬁrst instance,to aid in the all-important task of recognition of a singularity’s presence,then,in the second instance,to aid in removal or interpretation.Throughout this review we take stress singularities as in-volving stresses which,in themselves,are unbounded.Spe-ciﬁcally,we are concerned with when such singularities can occur in the linear elastic regime.This is a key regime since elastic response physically precedes plasticﬂow,so that in-troducing plasticity does not remove the singular character in any true sense.1To keep the scope of the article within rea-sonable limits,we further restrict attention to materials which are homogeneous,or piecewise so,and isotropic.We also focus on loading which is quasi-static.For such classical elasticityﬁelds,two classes of singular conﬁgurations may be distinguished:those wherein singularities occur under concentrated loads,and those wherein they occur away from any concentrated loading.For either,it is important to rec-ognize the presence of stress singularities and to appreciate their nature.In what follows we give examples of both,then turn our attention to the latter because it typically presents greater difﬁculties to the stress analyst.1.2Examples of stress singularities under concentrated loadsConcentrated loading conﬁgurations induce singularities di-rectly by applyingﬁnite stress resultants͑eg,forces,mo-ments͒over regions with vanishingly small areas͑eg,points, lines͒.As such they may be termed singular loads:Table1 exhibits the singular character of the stresses for a basic set of such loads.Transmitted by Editorial Advisory Board Member R.C.Benson1We expand on this point in Section2.1.Appl Mech Rev vol57,no4,July2004©2004American Society of Mechanical Engineers251In Table1,r is the distance from the point of application of a singular load,and we have employed the ord notation. For a function f(r),here this hasf͑r͒ϭord͑rϪ␥͒as r→0(1.1) ifr␥f͑r͒ϭc 0as r→0(1.2) where␥and c are constants.The traditional large order O notation,in contrast,admits the possibility that cϭ0.Pro-vided nonzero loads are being applied,c cannot be every-where zero for the stresses in Table1.Examples of solutions for isolated force problems in three dimensions are:the point load in the inﬁnite elastic medium by Kelvin͑Thomson͓1͔͒,the normal point load on the sur-face of an elastic half-space of Boussinesq͓2͔,the tangentialpoint load on a half-space surface of Cerutti͓3͔,and point loads within a half-space in Mindlin͓4͔.A convenient com-pendium of these closed-form solutions may be found in Poulos and Davis͓5͔,Section2.1.Inspection of these solu-tions demonstrates compliance with the order of singularity for point loads given in Table1.Analogous solutions exist for isolated force problems in two dimensions,namely:the line load in an inﬁnite elastic medium in Michell͓6͔,the normal line load on the surface of an elastic half-space of Flamant͓7͔,the tangential line load on a half-space surface in Boussinesq͓8͔,and line loads within a half-space in Melan͓9͔.2These may be found ibid,Section2.2,and also demonstrate compliance with their singular order given in Table1.Examples of doublet states are indicated in Fig.1.The ﬁrst of these͑Fig.1a͒illustrates a means of obtaining a concentrated moment M.This moment is produced by tak-ing the limit as␦→0where␦is the horizontal separation of two vertical forces of magnitude FϭM/␦.The second ar-rangement͑Fig.1b͒is a dual of theﬁrst and realizes no resultant force or moment in the limit as␦→0,yet does have a nontrivial stressﬁeld if F is ord(␦Ϫ1):As a consequence, it requires a generalization of the usual notion of a load in terming it a‘‘singular load.’’The third arrangement͑Fig.1c͒is a center of compression produced by superposing the sec-ond in an angular array:It,too,represents a load in a gener-alized sense.A precise deﬁnition of doublet states in general is given in Sternberg and Eubanks͓11͔.Some closed-form solutions for doublet states in three dimensions may be found in:Love͓12͔Article132,Sternberg and Eubanks͓11͔,Tur-teltaub and Sternberg͓13͔,Chowdhury͓14͔,and Chen͓15͔. Closed-form solutions for doublet states in two dimensions are available in Love͓12͔Article152,and Timoshenko and Goodier͓16͔Articles36and42.The stresses in all of these solutions comply with their respective orders of singularity given in Table1.The nature of the singularities displayed in Table1is,to adegree,that expected.For a point force,integration of thetractions acting on the surface of a small sphere of radius rcentered on the point of application produces a product ofstresses with r2:Hence the stresses can be expected to be-have like rϪ2if aﬁnite force is to result in the limit as r →0.Similarly for a line load,one anticipates stresses which behave like rϪ1.And the doublet states,being derivable bydifferentiation of corresponding isolated loads,then behaveas rϪ3and rϪ2in three dimensions and two dimensions,respectively.However,some care needs to be exercised ifthese expectations are to be realized in the limit by a se-quence ofﬁnite stressﬁelds acting over regions ofﬁniteextent—a limiting process for producing singular loads thatis physically appealing.Sternberg and Eubanks͓11͔gives aclear account of the sort of restrictions required on theﬁnitestress distributions used in the limiting process:These re-strictions have since been reﬁned in Turteltaub and Sternberg ͓13͔.In essence,Sternberg and Eubanks establish that it is insufﬁcient to simply have the distributedﬁelds be statically equivalent to the end stress resultant sought͑as Kelvin origi-nally proposed for his problem͒.If one merely makes this requirement,then it is possible,for example,to add a doublet state of the kind in Fig.1b to a point load problem,thereby changing the dominant singularity of the latter without alter-ing the force exerted.One means of avoiding this additional ﬁeld for the point force example is to require all the distrib-uted stresses in the underlying limiting sequence be unidirec-tional;alternative restrictions for the point load,as well as effective requirements for other singular loads,are given in Sternberg and Eubanks͓11͔and Turteltaub and Sternberg ͓13͔.Provided proper attention is paid to the generating se-2An error in one of the formulas given in Melan͓9͔is corrected in Kurshin͓10͔.Fig.1Some limiting conﬁgurations for doublet states:a)concen-trated moment,b)force doublet without a moment,c)center of compressionTable1.Basic singular loads of classical elasticityLoad type 3D stressstate at load…r\0…2D stressstate at load…r\0…Isolated force ord(rϪ2)ord(rϪ1)Doublet state ord(rϪ3)ord(rϪ2)252Sinclair:Stress singularities in classical elasticity–I Appl Mech Rev vol57,no4,July2004quence of the distributed loads acting on successively smaller regions,all of the singular loads included in Table1 have unique stressﬁelds with singularities as indicated therein.In practice,concentrated loads usually serve as Green’s functions in stress analysis.That is,they are superposed to achieve a desired regular distribution of applied loads.Often, this superposition is undertaken via numerical analysis.A demonstration of their use in this way occurs in integral equation approaches,such as the boundary integral equation method which currently enjoys fairly wide application in elastic stress analysis.In this role,it is of value to understand the singular nature of the concentrated loads involved in or-der to design efﬁcient quadrature schemes for their numerical integration.However,these integrations typically result in ﬁnite stresses.Then,one is not faced with the challenge of drawing physical inferences,with respect to structural integ-rity,from nonphysical singularﬁelds.On other occasions, though,singular loads can be used to model highly localized loading,such as under a knife edge in the three-point-bend specimen of fracture mechanics͑eg,at point P1in Fig.2a͒. In this instance,if a line load is introduced,it is merely as one of a set of three which effect an applied moment for the crack.As such,it is not the feature of greatest interest,lo-cally,with respect to potential failure—the crack tip is(P2inFig.2a͒.3Again,one is not faced with interpreting local ﬁelds at singular loads.On the other hand,one must attempt this task for the crack,with its classical,inverse-square-root, stress singularity.Indeed,in general this is the case for the second class of singular conﬁgurations recognized here.Ac-cordingly we focus on stress singularities which occur away from any concentrated loading throughout the remainder of this review.1.3Examples of other stress singularitiesSome illustrative examples of this class of singularity are depicted schematically in Fig.2.The corresponding orders of stress singularity present are set out in Table2.Theﬁrst example͑Fig.2a͒is the aforementioned cracked elastic plate under three-point bending,with its attendant, inverse-square-root,stress singularity reﬂecting the stress in-tensiﬁcation at the crack tip͑ie,at P2).For the case of a crack in a large elastic plate under transverse tension,such a singularity can be extracted from the corresponding solution for the elliptical hole on passing to the limit as the hole becomes a mathematically sharp slit.Theﬁelds required to take this limit wereﬁrst provided in Kolossoff͓17͔͑see also Kolossoff͓18͔͒,and subsequently derived in Inglis͓19͔. That the same singularity results for crack tips in general, and for the crack tip in the three-point-bend specimen of Fig. 2a in particular,can be discerned from Williams’seminal paper͓20͔.In this paper,the asymptotic character of elastic stresses in angular plates or wedges under extension is re-vealed:Letting the angle of the‘‘free-free’’wedge go to2␲in Williams͓20͔recovers singular stresses as in Table2.As a modiﬁcation to theﬁrst example,we consider theplate now to be comprised of two distinct elastic materialsinstead of a single one.The two are perfectly bonded to-gether on an interface extending straight ahead of the crack ͑indicated by the dashed line in Fig.2a͒.Adding the further discontinuity of an abrupt change in material properties ren-ders the crack-tip stress singularity more nonphysical,withthe inverse square root having multipliers,cos(␩ln r)and sin(␩ln r),which oscillate an inﬁnite number of times in the limit r→0when␩ 0.Herein␩is a material constant given by␩ϭ12␲ln␮1ϩ␬1␮2␮2ϩ␬2␮1(1.3) where␮is the shear modulus,␬ϭ3Ϫ4␯or(3Ϫ␯)/(1ϩ␯)for plane strain or plane stress,␯being Poisson’s ratio, and the subscripts distinguish the different materials on each side of the interface crack.Observe that if the materials are taken to be one and the same,␩ϭ0and there is no oscilla-tory multiplier,as in our original example.Otherwise,typi-cally interface cracks have oscillatory,inverse-square-root, stress singularities,asﬁrst shown in Williams͓21͔.A related pair of examples concerns a tire,under light load,where it meets a relatively stiff pavement at the sharp edge of a pothole͑a section through such an arrangement is sketched in Fig.2b,wherein P3is the point of interest͒.If the pavement is icy,and thereby lubricated,the situation is as if the tire were an elastic half-space being indented by aﬂat, frictionless,rigid strip.The solution to this problem wasﬁrst given in Sadowsky͓22͔,and exempliﬁes the inverse-square-root stress singularity listed in Table2.That the singular character here is the same as for the crack in a homogeneous material can be argued as follows.First we note that,for a3If instead the stresses under the knife edge were of greatest concern,better models than a line load are available,as we demonstrate subsequently.Table2.Some elastic stress singularities away from singular loads Singular point,Fig.2…rÄ0…Local conﬁgurationdescriptionSingular stressesat point…r\0…P2Crack tip in three-point-bend specimenord(rϪ1/2)P2Interface crack tipin bend specimenord(rϪ1/2cos(␩ln r))&ord(rϪ1/2sin(␩ln r)),see Eq.͑1.3͒for␩P3Tire at pothole edgeunder icy conditionsord(rϪ1/2)P3Adhering nylon tire atpothole edgeord(rϪ1/2cos(␩ln r))&ord(rϪ1/2sin(␩ln r)),see Eq.͑1.4͒for␩P4Edge of piston ringpressed into cylinder wallord(rϪ0.23)P5Reentrant corner instress-free keywayord(TrϪ1/3)ord(FrϪ0.46)&ord(FrϪ0.09) P6Edge of adhering rubbertire on pavementord(rϪ0.41)P7Circumference of anepoxy-steel interfaceord(rϪ1/3)P8Edge of a rough heavyblock on an elastic slabord(ln r)P9Edge of a smooth steelchisel on a wooden blockord(ln r)P10Submodel node withdisplacement shapefunctions as boundaryconditionsord(ln r)Appl Mech Rev vol57,no4,July2004Sinclair:Stress singularities in classical elasticity–I253Fig.2Some singular conﬁgurations:a )three-point-bend test piece of fracture mechanics,b )section through a tire on a relatively rigid pavement,c )section through a piston with a ring pressed into a cylinder wall,d )section of a shaft with a stress-free keyway under torsion and lateral loading,e )adhesive butt joint under tension,f ͒rough heavy block sticking to an elastic base,g )steel chisel just starting to indent a wooden slab,h )displacement shape functions as submodel boundary conditions254Sinclair:Stress singularities in classical elasticity –I Appl Mech Rev vol 57,no 4,July 20042D elastic half-space,the constant displacement due to the rigid strip can be recovered by a rigid body translation. Hence,we need only consider homogeneous conditions un-der the strip,together with stress-free conditions outside the strip.But these conditions under the strip are the same as symmetry conditions.Thus the half-space can be reﬂected on itself to produce a full space with a pair of stress-free cracks outside of where the strip punch acts.For the second of our examples concerning the arrange-ment in Fig.2b at P3,we consider the pavement to be dry and the tire to stick to it perfectly.Now the in-plane situation for the section of Fig.2b is as if the tire were being indented by aﬂat,adhering,rigid strip.The solution to this problem wasﬁrst furnished in Abramov͓23͔,and contains the inverse-square-root stress singularity,with its oscillatory multipliers,listed in Table2.This is the same singularity as for the interface crack,except that now␩is given by Eq.͑1.3͒with␮2→ϱtherein.That is,␩ϭ12ln␬(1.4)Recall that␬ϭ3Ϫ4␯for the plane strain state applicable here:So as to avoid␩ϭ0,Table2speciﬁes a nylon tire(␯ϭ0.4)rather than rubber(␯ϭ0.5)for this case of adhesive contact.Asymptotically,the conﬁguration can be treated us-ing the‘‘clamped-free’’conditions for a wedge of angle␲in Williams͓20͔,if one sets‘‘␴’’ϭ␯in Eq.͑17͒therein so as to correspond to a state of plane strain.The same singularity results.A further contact example is that of a lubricated piston ring pressed into a cylinder wall as indicated in Fig.2c.This conﬁguration is axisymmetric rather than being as previous examples which entail states of plane strain.However,as ﬁrst argued in Zak͓24͔,a plane strain analysis still applies. Then,if the ring is taken to be relatively rigid compared to the cylinder,the same inverse-square-root singularity results as for an indentation with aﬂat,frictionless,rigid strip͑Fig. 2c at P4).Alternatively,if the more realistic assumption is made that the ring is comprised of the same material as the cylinder,the weaker singularity of Table2results.This sin-gularity can be identiﬁed by solving the pertinent eigenvalue equation in Dempsey and Sinclair͓25͔.It is weaker because now the deformation of the ring is being included.For the example of a stress-free keyway in a shaft under torque T and transverse load F͑Fig.2d͒,multiple singulari-ties are present͑Table2͒.For the torque,the singularity ac-tive at the90°reentrant corner͑ie,at P5)is weaker than if a crack is subjected to torsion,having an exponent of1/3com-pared to1/2.This singularity wasﬁrst identiﬁed in Thomson and Tait͓26͔,Section710.For the transverse load,two sin-gularities typically participate.The stronger one is associated with loading which is symmetric about the bisector of the angle at the reentrant corner,the weaker with antisymmetric. Both are weaker than the singularity at a crack,a reentrant corner of zero angle in effect.The two singularities for this right-angled reentrant corner are included in Brahtz͓27͔.Al-ternatively,they may be obtained using the‘‘free-free’’con-ditions in Williams͓20͔,on taking a wedge angle of3␲/2.In general,as the angle at a reentrant corner increases,singu-larity strength reduces.Eventually,when a stress-free corner opens all the way up to a half-space,singular stresses are removed.The disappearance of stress singularities once corners are no longer reentrant need not be the case when the boundary conditions are mixed,as is demonstrated in our next ex-ample.This concerns the tire again͑Fig.2b͒,but now where it meets the pavement at its outside edge͑ie,at P6).If the tire adheres perfectly to the relatively rigid pavement,locally this conﬁguration becomes a right-angled corner in plane strain with one face being free of stress,the other completely ﬁxed.The singularity in this instance is characterized in Knein͓28͔.Alternatively,it may be obtained using the ‘‘clamped-free’’conditions in Williams͓20͔for a wedge angle of only␲/2,provided these are adapted to a state of plane strain.For rubber(␯ϭ0.5),the stress singularity of Table2results͑there is a minor round-off error in the singu-larity exponent in Knein͓28͔͒.While this is weaker than that of a crack,it is nonetheless quite comparable in strength.A similar situation occurs for the butt joint under tension of Fig.2e.Herein the points of interest are where the inter-face between the epoxy adhesive and steel adherend meets the outside free surface͑eg,P7).As for the piston ring,this conﬁguration is axisymmetric but nonetheless plane strain analysis still applies.Again then,since steel is relatively rigid compared to epoxy,a‘‘clamped-free’’right-angled cor-ner in plane strain is appropriate and can be treated via Wil-liams͓20͔.Taking3/8as a reasonable estimate of Poisson’s ratio for epoxy,this gives the singularity of Table2.The reduction in strength here from that of the rubber corner is due to the lower value of␯.Indeed,there is no singularity for such corners when␯ϭ0.Our last three examples give rise to the weakest type of stress singularity in elasticity,the logarithmic singularity. Theﬁrst example concerns a heavy rough block,under a lateral force,sticking to a horizontal elastic surface͑Fig.2f͒. If one assumes that the normal stresses produce a disconti-nuity in the surface shear͑ie,at P8,as indicated in the close-up͒,then a log singularity in the stresses occurs,with a coefﬁcient that is proportional to the magnitude of the shear stress discontinuity.This result is given in Kolossoff͓18͔. Alternatively,it can be constructed using auxiliaryﬁelds to those in Williams͓20͔.Theseﬁelds may be found in Demp-sey and Sinclair͓29͔.While the normal stress discontinuity produces no stress singularity,any shear stress discontinuity on an elastic half-plane does.To see an indication of why this is so,consider the shear stress components on the two little square elements outlined by broken lines in the close-up of Fig.2f.The left one is in force and moment equilibrium if it has no shears on its boundaries.The right one,constant shears.Where they meet,there is an incompatibility in shear stress which cannot be accommodated by any regular elas-ticityﬁelds known to date.The second example concerns a piece of wood,just start-ing to be indented but not yet cut,by a sharp chisel made of relatively rigid steel͑Fig.2g͒.Assuming the contact to be frictionless and ignoring any anisotropy in the wood,theAppl Mech Rev vol57,no4,July2004Sinclair:Stress singularities in classical elasticity–I255log-singularity stressﬁeld induced at the cutting edge͑ie,at P9)may be found in Sneddon͓30͔,Section48.4.Again, alternatively it can be constructed using the auxiliaryﬁelds in Dempsey and Sinclair͓29͔.This log singularity features a coefﬁcient which depends on the chisel tip angle and is pro-portional to the elastic moduli of the wood.It also has a displacementﬁeld which is more physically applicable to initial knife-edge loading than that of a line load,being free of unbounded vertical displacement and overlapping hori-zontal ones.The third example concerns the use of displacement shape functions as boundary conditions in submodeling inﬁnite element analysis͑as suggested in ABAQUS͓31͔,and AN-SYS͓32͔͒.Along a smooth submodel boundary,spurious log singularities can be introduced.An example involving four node elements is shown in Fig.2h.Therein a log singularity occurs at the node at P10whenever there is a discontinuity in the derivatives of either of the boundary displacements,u and v on yϭ0.That is,whenever the constants are such that c1Ϫ c1ϩor c2Ϫ c2ϩ.Fields are given in Sinclair and Epps ͓33͔.1.4What to do about stress singularitiesThe foregoing serves to demonstrate some of the variety of singular conﬁgurations and stress singularities possible in classical elasticity.The natural question which then arises is what is to be done about these and like conﬁgurations in attempting to ensure structural reliability?In theﬁrst in-stance,it is vital that the stress analyst at least recognize when a stress singularity is present.4That there is a singularity present is not always immedi-ately obvious.This is especially so in the stress analysis of actual engineering components,since frequently the com-plexity of such conﬁgurations necessitates numerical treat-ment,often viaﬁnite element analysis͑FEA͒.Under these circumstances,one does not have available analytical solu-tions whereby singular character is detectable simply by ob-servation.Nevertheless,it remains essential that the presence of any singular stressﬁeld be appreciated.Consider the alternative.A scenario such as follows is then quite possible.On Monday,you complete aﬁrst FEA of a component subjected to cyclic loading.The maximum stresses found are a factor of two less than the endurance stress of the component’s material.You conclude that the component has indeﬁnite life,or at least long life.On Wednesday,you check your FEA with a reﬁned grid.The peak stresses are now comparable to the endurance.You are in somewhat of a quandary as to how much life the part really has.Hence,on Friday you complete a further FEA on a still more reﬁned mesh.Now you get stresses that are a factor of two greater than the endurance level.The compo-nent’s life now is,apparently,distinctly limited.Life for you is somewhat disconcerting:Such a workweek does not make for a great weekend.More importantly,such a structural ap-praisal has nothing to do with the component’s actual struc-tural reliability:In the presence of a singularity,any sufﬁ-ciently reﬁned numerical analysis predicts failure when peak stresses are compared against someﬁnite stress criterion,ir-respective of what is physically happening.Under such cir-cumstances,the participation of the singular stresses must ﬁrst be recognized if any real use is to be made of the analy-sis.The main aim of this review is to aid in achieving such recognition.That said,we next turn our attention to the important and challenging task of interpreting singular stressﬁelds once it is apparent that they are active.In taking up this challenge, we begin by considering the simpliﬁcations made in classical elasticity since we expect singularities to be a product of the modeling in the theory,inﬁnite stresses not being possible physically.Three such simplifying assumptions or lineariza-tions can be identiﬁed in the classical theory of elasticity. Theﬁrst linearization has that the relationship between stresses and strains is linear;that is,the stresses do not ex-ceed the limits of elastic material response.The second lin-earization has that the strains depend linearly on the dis-placement gradients;that is,the displacement gradients are small.The third linearization has that all loads act on the undeformed shape throughout the entire loading process;that is,the deﬂections are small.The singular stressﬁelds of clas-sical elasticity are in violation of all three of these assump-tions.Yet they do comply with all of theﬁeld equations of elasticity,as may be established by simply substituting them into these equations.This seemingly paradoxical situation results from the fact that,once an assumption is made in the theory of elasticity and equations so simpliﬁed,compliance with the assumption becomes unpoliced by the theory itself. This allows singular stressﬁelds to comply with theﬁeld equations of classical elasticity,but remain in deﬁance of the underlying and unpoliced assumptions of elasticity.Such a situation requires some care if one is to be successful in interpreting these fundamentally waywardﬁelds in a physi-cally meaningful fashion.To demonstrate the difﬁculty of interpreting results when they lie outside of admissible responses in a theory,consider the following beam example taken from Frisch-Fay͓34͔.On page one of his monograph,Frisch-Fay considers a horizon-tal cantilever beam of length2.54m͑100inches͒,with a bending stiffness of2.87Nm2(1000lbf in2),subjected to a vertical concentrated end load of4.45N͑1lbf͒.Treating this beam within the context of classical beam theory for small deﬂections,Frisch-Fay obtains a prediction of a vertical tip deﬂection of8.47m͑333inches͒,or more than three times the beam’s original length.This result suggests strains of the order of300%and the possibility of gross yielding and even ductile rupture.Subsequently,on page39of Frisch-Fay͓34͔, the same beam is analyzed within the context of nonlinear beam theory for large deﬂections.This analysis results in a vertical deﬂection of2.06m͑81inches͒,together with a horizontal deﬂection of1.42m͑56inches͒,and stress and strainﬁelds that can now comply with the underlying as-4We have not included,in the examples of Table2,the yet stronger,ord(rϪ1),singu-larity occurring at dislocations of the V olterra type͑see,eg,Love͓12͔,Appendix toChapters VIII,IX,or Timoshenko and Goodier͓16͔,Articles34,117͒.Theseﬁelds areused as Green’s functions,and by some theoreticians in an attempt to model micro-structure.We omit discussion of them primarily because we expect users to be fullycognizant of the singular character present.256Sinclair:Stress singularities in classical elasticity–I Appl Mech Rev vol57,no4,July2004。

Singularities of Rational Curves on K3 surfaces

Date : February 1, 2008.
1
2
XI CHEN
over ∆, whose general ﬁber Υt is a rational curve in the linear series |OXt (1)| for each t, the central ﬁber Υ0 will be a curve in the linear series |OR (1)| and hence Υ0 = Σ1 ∪ Σ2 where Σi is a curve of type (1, k ) on Ri . Our trivial observation is that Υt is nodal if Υ0 is nodal. However, Υ0 could fail to be nodal where 1. it has a reduced (i.e. isolated) singularity other than a node; or 2. it is nonreduced. The ﬁrst case turns out much easier to handle than the second. This is basically due to the fact that each Σi is a curve of type (1, k ) on P1 × P1 . So all the isolated singularities of Σi are nodes. If Υ0 = Σ1 ∪ Σ2 has an isolated singularity other than a node, it must be one of the intersections between Σ1 and Σ2 on E . The deformation of such singularities has been studied in [C]. With a bit more care, we are able to show that these singularities deform to nodes on the general ﬁber Υt . However, if Υt is a rational curve in a multiple of the primitive class, Σi might have isolated singularities other than nodes which have to be taken care of. This is one of the major obstacles to generalize Theorem 1.1 to all rational curves on K3 surfaces. To handle the second case, i.e., to handle the nonreduced components of Υ0 , we ﬁrst divide them into three types, which we will call Type I, II or III chain (see Sec. 2), respectively. The deformation of Υ0 along a Type I chain is studied in Sec. 3. The basic technique used there is to normalize the total family along the Type I chain after a suitable base change. The deformation of Υ0 along a Type II chain is studied in Sec. 4, where we build our argument upon a lower bound estimation on the δ -invariant of Υt in the neighborhood of a Type II chain. The deformation of Υ0 along a Type III chain is studied in Sec. 5. This turns out to be the hardest case among the three. A two-stage degeneration is used, First, we degenerate a general K3 surface to an elliptic K3 (see Sec. 5); and then we degenerate an elliptic K3 to the union of two rational surfaces described above. The degeneration of a K3 surface to an elliptic K3 is also an important step in Bryan and Leung’s work [B-L], although the elliptic K3 surfaces they used are diﬀerent from the ones we use. As a side note, there have been several progresses made on the enumeration problems on K3 surfaces following Yau and Zaslow’s work. A. Beauville pointed out that the numbers Yau and Zaslow obtained are the numbers of rational curves in |OS (1)| with each curve counted with certain multiplicity [B], the multiplicity of a rational curve only depends on its singularities and is 1 if the curve is nodal. He gave an algebraic deﬁnition of the multiplicy. Later B. Fantechi, L. G¨ ottsche

英语1到31的基数词

英语1到31的基数词Here is an essay on the topic of the cardinal numbers 1 to 31 in English, with a word count exceeding 1000 words. Please note that the title is not included in the word count.The world of numbers is a fascinating and integral part of our daily lives. From the moment we wake up to the time we go to bed, we are constantly surrounded by numbers, whether it's the time on the clock, the number of steps we take, or the amount of money in our wallets. Among the most fundamental of these numerical concepts are the cardinal numbers, which form the foundation for our understanding of quantity and enumeration.In the English language, the cardinal numbers from one to thirty-one hold a special significance, as they represent the basic building blocks of our numerical system. Each of these numbers has its own unique name and characteristics, and understanding their nuances can greatly enhance our ability to communicate and comprehend the world around us.Let us begin our exploration with the number one. This simple yet powerful digit represents the concept of singularity, the fundamentalunit from which all other numbers are derived. The word "one" is derived from the Old English "an," which in turn comes from the Proto-Germanic "ainaz." This basic building block of our numerical system is essential in our daily lives, whether we are counting the number of apples in a basket or the number of people in a room.Moving on, the number two introduces the concept of duality, the idea of pairing or grouping. The word "two" is derived from the Old English "twā," which originates from the Proto-Germanic "twai." This number is fundamental in our understanding of relationships, balance, and the concept of opposites, as we often encounter situations involving pairs, such as two hands, two eyes, or two sides of a coin.The number three is often associated with the idea of completeness or wholeness. The word "three" comes from the Old English "thrēo," which has its roots in the Proto-Germanic "þrīz." This number is significant in many cultural and religious traditions, where it is seen as a symbol of harmony, stability, and the divine. In our everyday lives, we encounter the number three in various contexts, such as the three primary colors, the three-course meal, or the three-legged stool.The progression continues with the number four, which represents the concept of stability and order. The word "four" is derived fromthe Old English "fēower," which in turn comes from the Proto-Germanic "fidwōr." This number is clos ely associated with the four cardinal directions, the four seasons, and the four elements, making it a fundamental aspect of our understanding of the physical world.The number five introduces the idea of the hand, as it corresponds to the number of fingers on each hand. The word "five" is derived from the Old English "fīf," which has its origins in the Proto-Germanic "fimf." This number is significant in many cultural and mathematical contexts, such as the five-pointed star, the five senses, and the five-note musical scale.As we move further along the numerical spectrum, the number six represents the concept of balance and harmony. The word "six" comes from the Old English "six," which originates from the Proto-Germanic "sehs." This number is often associated with the idea of perfection, as it is the sum of its divisors (1, 2, and 3). In our daily lives, we encounter the number six in various contexts, such as the six sides of a cube or the six legs of a insect.The number seven is often considered a mystical and significant number, representing the idea of completeness and spiritual fulfillment. The word "seven" is derived from the Old English "seofon," which has its roots in the Proto-Germanic "sebun." This number is deeply embedded in various religious and culturaltraditions, such as the seven days of the week, the seven wonders of the ancient world, and the seven chakras in Eastern philosophy.The number eight introduces the concept of infinity and the idea of balance and harmony. The word "eight" comes from the Old English "eahta," which originates from the Proto-Germanic "ahtau." This number is often associated with the symbol of the infinity sign and is significant in various mathematical and scientific contexts, such as the eight points of a compass or the eight phases of the moon.The number nine represents the idea of completion and the fulfillment of a cycle. The word "nine" is derived from the Old English "nīen," which has its roots in the Proto-Germanic "newun." This number is often associated with the concept of the divine or the sacred, as it is the last single-digit number in our numerical system. In various cultural and religious traditions, the number nine is seen as a symbol of wholeness, wisdom, and the divine.As we reach the double-digit numbers, the number ten represents the idea of completion and the transition to a new cycle. The word "ten" comes from the Old English "tēn," which originates from the Proto-Germanic "tehun." This number is significant in many mathematical and scientific contexts, as it forms the basis of our decimal system and is the foundation for the metric system.The number eleven introduces the concept of a new beginning, as it represents the transition from the single-digit numbers to the double-digit numbers. The word "eleven" is derived from the Old English "endleofan," which has its roots in the Proto-Germanic "ainlif." This number is often associated with the idea of spiritual enlightenment and the exploration of new horizons.The number twelve represents the idea of organization and structure. The word "twelve" comes from the Old English "twelf," which originates from the Proto-Germanic "twalif." This number is significant in many cultural and historical contexts, such as the twelve months of the year, the twelve signs of the zodiac, and the twelve apostles of Jesus.As we continue our journey, the number thirteen introduces the concept of the unexpected and the unknown. The word "thirteen" is derived from the Old English "þrēotȳne," which has its roots in the Proto-Germanic "þrīhunda-taihun." This number is often associated with superstition and the idea of bad luck, although this perception is more a cultural construct than a numerical reality.The number fourteen represents the idea of balance and harmony. The word "fourteen" comes from the Old English "fēowertȳne," which originates from the Proto-Germanic "fidwōrtaihun." This number is significant in various mathematical and scientific contexts,such as the number of days in a fortnight or the number of ribs in the human body.The number fifteen introduces the concept of abundance and prosperity. The word "fifteen" is derived from the Old English"fīftȳne," which has its roots in the Proto-Germanic "fimftaihun." This number is often associated with the idea of good fortune and the fulfillment of desires, as it is the sum of the first five natural numbers (1 + 2 + 3 + 4 + 5 = 15).Moving on, the number sixteen represents the idea of order and structure. The word "sixteen" comes from the Old English "sixtȳne," which originates from the Proto-Germanic "sehs-taihun." This number is significant in various mathematical and scientific contexts, such as the number of bytes in a word or the number of squares on a chessboard.The number seventeen introduces the concept of the unique and the extraordinary. The word "seventeen" is derived from the Old English "seofontȳne," which has its roots in the Proto-Germanic "sibun-taihun." This number is often associated with the idea of individuality and the exploration of new ideas, as it is the sum of the first four prime numbers (2 + 3 + 5 + 7 = 17).The number eighteen represents the idea of balance and harmony.The word "eighteen" comes from the Old English "eahtatȳne," which originates from the Proto-Germanic "ahtau-taihun." This number is significant in various cultural and religious contexts, such as the number of holes on a standard golf course or the number of auspicious symbols in Buddhist art.The number nineteen introduces the concept of the unexpected and the unknown. The word "nineteen" is derived from the Old English "nīentȳne," which has its roots in the Proto-Germanic "newun-taihun." This number is often associated with the idea of mystery and the exploration of the unknown, as it is the sum of the first four prime numbers (2 + 3 + 5 + 9 = 19).As we reach the final numbers in our journey, the number twenty represents the idea of completeness and a new cycle. The word "twenty" comes from the Old English "twentig," which originates from the Proto-Germanic "twai-tigjus." This number is significant in various mathematical and scientific contexts, as it is the base of our numerical system and the foundation for the concept of the score.The number twenty-one introduces the concept of the unique and the extraordinary. The word "twenty-one" is derived from the Old English "ān and twentig," which has its roots in the Proto-Germanic "aina-and-twai-tigjus." This number is often associated with the idea of personal growth and the exploration of new horizons, as it is thesum of the first three prime numbers (3 + 7 + 11 = 21).Finally, the number thirty-one represents the idea of completeness and the fulfillment of a cycle. The word "thirty-one" comes from the Old English "ān and þrītig," which originates from the Proto-Germanic "aina-and-þrīz-tigjus." This number is significant in various mathematical and scientific contexts, as it is the sum of the first five prime numbers (2 + 3 + 5 + 7 + 11 = 28) plus 3, and is often associated with the idea of the completion of a journey or the achievement of a goal.In conclusion, the cardinal numbers from one to thirty-one in the English language are a rich and diverse tapestry of numerical concepts and cultural significance. Each number holds its own unique characteristics and associations, and understanding their nuances can greatly enhance our ability to communicate, comprehend, and navigate the world around us. Whether we are exploring the concept of singularity, duality, or completeness, the cardinal numbers from one to thirty-one provide a fascinating and essential foundation for our understanding of the world of numbers.。

yau_poincare

Structure of Three-Manifolds–Poincar´e and geometrization conjecturesShing-Tung Yau1,2Ladies and gentlemen,today I am going to tell you the story of how a chapter of mathematics has been closed and a new chapter is beginning.Let me begin with some elementary observations.A major purpose of Geometry is to describe and classify geometric struc-tures of interest.We see many such interesting structures in our day-to-day life.Let us begin with topological structures of a two dimensional surface. These are spaces where locally we have two degrees of freedom.Here are some examples:genus0genus1genus2genus3 Genus of a surface is the number of handles of the surface.An abstract and major way to construct surfaces is by connecting along some deleted disk of each surface.The connected sum of two surfaces S1and S2is denoted by S1#S2.It is formed by deleting the interior of disks D i from each S i and attaching the resulting punctured surfaces S i−D i to each other by a one-to-one continuous 1This was a talk given at the Morningside Center of Mathematics on June20,2006.2All the computer graphics are provided by David Gu,based on the joint paper of David Gu,Yalin Wang and S.-T.Yau.1map h :∂D 1→∂D 2,so thatS 1#S 2=(S 1−D 1)∪h (S 2−D 2).S 12D 1Example:A genus 8surface,constructed by connected sum.The major theorem for the two dimensional surfaces is the following:Theorem (Classiﬁcation Theorem for Surfaces).Any closed,connected orientable surface is exactly one of the following surfaces:a sphere,a torus,or a ﬁnite number of connected sum of tori.Note that a surface is called orientable if each closed curve on it has a well-deﬁned continuous normal ﬁeld.1Conformal geometryIn order to understand surfaces in a deep manner,Riemann,Poincar´e and others proposed to study conformal structure on these two dimensional ob-2jects.Such structures allow us to measure angles in the neighborhood of each point on the surface.For example,if we take a standard atlas of the globe,we have longitude and latitude.They are orthogonal to each other.When we map the atlas, which is a square,onto the globe;distances are badly distorted.For example, the region around the north pole is shown to be a large region on the square. However,the fact that longitude and latitude is orthogonal to each other is preserved under the map.Hence if a ship moves in the ocean,we can use the atlas to determine its direction accurately,but not the distance travelled.GlobePoincar´e found that at any point,we can draw a longitude(blue curve) and latitude(red curve)on any surface of genus zero in three space.These curves are orthogonal to each other and they converge to two distinct points, on the surface,just like north pole and south pole on the sphere.This theorem of Poincar´e also works for arbitrary abstract surface with genus zero.It is a remarkable theorem that for any two closed surfaces with genus zero,we can alwaysﬁnd a one-to-one continuous map mapping longitude and latitude of one surface to the corresponding longitude and latitude of the other surface.This map preserve angles deﬁned by the charts.In such a situation,we say that these two surfaces are conformal to each other.And there is only one conformal structure for a surface with genus zero.For genus equal to one,the surface looks like a donut,and we can draw longitude and latitude with no north or south poles.However,there can be distinct surfaces with genus one that are not conformal to each other. In fact,there are two parameters of conformal structures on a genus one surface.For genus g greater than one,one can still draw longitude and latitude(the deﬁnition of such curves needs to be made precise).But they have many poles,the number of which depends on the genus.The number3of parameters of conformal structures over a surface with genus g is6g−6.In order toﬁnd a global atlas of the surface,we can cut along some special curves of a surface and then spread the surface on the plane or the disk.Inthis procedure,the longitude and the latitude will be preserved.A fundamental theorem for surfaces with metric structure is the following theorem.Theorem(Poincar´e’s Uniformization Theorem).Any closed two-dimensional space is conformal to another space with constant Gauss curvature.•If curvature>0,the surface has genus=0;•If curvature=0,the surface has genus=1;•If curvature<0,the surface has genus>1.The generalization of this theorem plays a very important role in theﬁeldof geometric analysis.In particular,it motivates the works of Thurston and Hamilton.This will be discussed later in this talk.2Hamilton’s equation on SurfacesPoincar´e’s theorem can also be proved by the equation of Hamilton.Wecan deform any metric on a surface by the negative of its curvature.After4Spherical Euclidean Hyperbolic normalization,theﬁnal state of such deformation will be a metric with con-stant curvature.This is a method created by Hamilton to deform metrics on spaces of arbitrary dimensions.In higher dimension,the typicalﬁnal state of spaces for the Hamilton equation is a space that satisﬁes Einstein’s equation.As a consequence of the works by Richard Hamilton and B.Chow,one knows that in two dimension,the deformation encounters no obstruction and will always converge to one with constant curvature.This theorem was used by David Gu,Yalin Wang,and myself for computer graphics.The following sequence of pictures is obtained by numerical simulation of the Ricciﬂow in two dimension.53Three-ManifoldsSo far,we have focused on spaces where there are only two degrees of freedom. Instead of being aﬂat bug moving with two degrees of freedom on a surface, we experience three degrees of freedom in space.While it seems that our three dimensional space isﬂat,there are many natural three dimensional spaces,which are notﬂat.Important natural example of higher dimensional spaces are phase spaces in mechanics.In the early twentieth century,Poincar´e studied the topology of phase space of dynamics of particles.The phase space consists of(x;v),the position and the velocity of the particles.For example if a particle is moving freely with unit speed on a two dimensional surfaceΣ,there are three degrees of freedom in the phase space of the particle.This gives rise to a three dimensional space M.Such a phase space is a good example for the concept ofﬁber bundle.If we associate to each point(x;v)in M the point x∈Σ,we have a map from M ontoΣ.When weﬁx the point x,v can be any vector with unit length.The totality of v forms a circle.Therefore,M is aﬁber bundle over Σwithﬁber equal to a circle.4The Poincar´e ConjectureThe subject of higher dimensional topology started with Poincar´e’s question: Is a closed three dimensional space topologically a sphere if every closed curve in this space can be shrunk continuously to a point?This is not only a famous diﬃcult problem,but also the central problem for three dimensional topology.Its understanding leads to the full struc-ture theorem for three dimensional spaces.I shall describe its development chronologically.5Topological SurgeryTopologists have been working on this problem for over a century.The major tool is application of cut and paste,or surgery,to simplify the topology of a space:6Two major ingredients were invented.One is called Dehn’s lemma which provides a tool to simplify any surface which cross itself to one which does not.Theorem(Dehn’s lemma)If there exists a map of a disk into a three dimensional space,which does not cross itself on the boundary of the disk, then there exists another map of the disc into the space which does not cross itself and is identical to the original map on the boundary of the disc.This is a very subtle theorem,as it took almostﬁfty years until Papakyr-iakopoulos came up with a correct proof after its discovery.The second tool is the construction of incompressible surfaces introduced by Haken.It was used to cut three manifolds into pieces.Walhausen proved important theorems by this procedure.(Incompressible surfaces are embed-ded surfaces which have the property whereby if a loop cannot be shrunk to a point on the surface,then it cannot be shrunk to a point in the three dimensional space,either.)76Special SurfacesThere are several important one dimensional and two dimensional spaces that play important roles in understanding three dimensional spaces.1.CircleSeifert constructed many three dimensional spaces that can be described as continuous family of circles.The above mentioned phase space is an example of a Seifert space.2.Two dimensional spheresWe can build three dimensional spaces by removing balls from two dis-tinguished ones and gluing them along the boundary spheres.ConverselyS2Kneser and Milnor proved that each three dimensional space can be uniquely decomposed into irreducible components along spheres.(A space is called irreducible if each embedded sphere is the boundary of a three dimensional ball in this space.)3.TorusA theorem of Jaco-Shalen,Johannson says that one can go one step fur-ther by cutting a space along tori.T27Structure of Three Dimensional SpacesA very important breakthrough was made in the late1970s by W.Thurston. He make the following conjecture.8Geometrization Conjecture(Thurston):The structure of three di-mensional spaces is built on the following atomic spaces:(1)The Poincar´e conjecture:three dimensional space where every closed loop can be shrunk to a point;this space is conjectured to be the three-sphere.(2)The space-form problem:spaces obtained by identifying points on the three-sphere.The identiﬁcation is dictated by aﬁnite group of linear isometries which is similar to the symmetries of crystals.(3)Seifert spaces mentioned above and their quotients similar to(2).(4)Hyperbolic spaces according to the conjecture of Thurston:three-space whose boundaries may consist of tori such that every two-sphere in the space is the boundary of a ball in the space and each incompressible torus can be deformed to a boundary component;it was conjectured to support a canonical metric with constant negative curvature and it is obtained by identifying points on the hyperbolic ball.The identiﬁcation is dictated by a group of symmetries of the ball similar to the symmetries of crystals.An example of a space obtained by identifying points on the three dimen-sional hyperbolic spaceHyperbolic Space Tiled with Dodecahedra,by Charlie Gunn(Geometry Center).from the book”Three-dimensional geometry and topology”by Thurston,Princeton University pressThurston’s conjecture eﬀectively reduced the classiﬁcation of three dimen-sional spaces to group theory,where many tools were available.He and his followers proved the conjecture when the three space is suﬃciently large in9the sense of Haken and Walhausen.(A space is said to be suﬃciently large if there is a nontrivial incompressible surface embedded inside the space.Haken and Walhausen proved substantial theorem for this class of manifolds.)This theorem of Thurston covers a large class of three dimensional hyperbolic manifolds.However,as nontrivial incompressible surface is diﬃcult toﬁnd on a gen-eral space,the argument of Thurston is diﬃcult to use to prove the Poincar´e conjecture.8Geometric AnalysisOn the other hand,starting in the seventies,a group of geometers applied nonlinear partial diﬀerential equations to build geometric structures over a space.Yamabe considered the equation to conformally deform metrics to metrics with constant scalar curvature.However,in three dimension,metrics with negative scalar curvature cannot detect the topology of spaces.A noted advance was the construction of K¨a hler-Einstein metrics on K¨a hler manifolds in1976.In fact,I used such a metric to prove the complex version of the Poincar´e conjecture.It is called the Severi conjecture in com-plex geometry.It says that every complex surface that can be deformed to the complex projective plane is itself the complex projective plane.The subject of combining ideas from geometry and analysis to understand geometry and topology is called geometric analysis.While the subject can be traced back to1950s,it has been studied much more extensively in the last thirty years.Geometric analysis is built on two pillars:nonlinear analysis and geom-etry.Both of them became mature in the seventies based on the eﬀorts of many mathematicians.(See my survey paper“Perspectives on geometric analysis”in Survey in Diﬀerential Geometry,Vol.X.2006.)9Einstein metricsI shall now describe how ideas of geometric analysis are used to solve the Poincar´e conjecture and the geometrization conjecture of Thurston.In the case of a three dimensional space,we need to construct an Einstein metric,a metric inspired by the Einstein equation of gravity.Starting from10an arbitrary metric on three space,we would like toﬁnd a method to deform it to the one that satisﬁes Einstein equation.Such a deformation has to depend on the curvature of the metric.Einstein’s theory of relativity tells us that under the inﬂuence of grav-ity,space-time must have curvature.Space moves dynamically.The global topology of space changes according to the distribution of curvature(gravity). Conversely,understanding of global topology is extremely important and it provides constraints on the distribution of gravity.In fact,the topology of space may be considered as a source term for gravity.From now on,we shall assume that our three dimensional space is compact and has no boundary(i.e.,closed).In a three dimensional space,curvature of a space can be diﬀerent when measured from diﬀerent directions.Such a measurement is dictated by a quantity R ij,called the Ricci tensor.In general relativity,this gives rise to the matter tensor of space.An important quantity that is independent of directions is the scalar curvature R.It is the trace of R ij and can be considered as a way to measure the expansion or shrinking of the volume of geodesic balls:Volume(B(p,r))∼4π3(r3−130R(p)r5),where B(p,r)is the ball of radius r centered at a point p,and R(p)is the scalar curvature at p.This can be illustrated by a dumbbell surface where,near the neck,cur-vature is negative and where,on the two ends which are convex,curvature is positive.Two-dimensional dumbbell surfaceTwo-dimensional surfaces with negative curvature look like saddles.Hence a two dimensional neck has negatives curvature.However,in three dimen-sion,the slice of a neck can be a two dimensional sphere with very large11positive curvature.Since scalar curvature is the sum of curvatures in all direction,the scalar curvature at the three dimensional neck can be posi-tive.This is an important diﬀerence between a two-dimensional neck and a three-dimensional neck.Three dimensional neck.10The dynamics of Einstein equationIn general relativity,matter density consists of scalar curvature plus the momentum density of space.The Dynamics of Einstein equation drives space to form black holes which splits space into two parts:the part where scalar curvature is positive and the other part,where the space may have a black hole singularity and is enclosed by the apparent horizon of the black hole, the topology tends to support metrics with negative curvature.There are two quantities in gravity that dictate the dynamics of space: metric and momentum.Momentum is diﬃcult to control.Hence at this12time,it is rather diﬃcult to use the Einstein equation of general relativity to study the topology of spaces.11Hamilton’s EquationIn1979,Hamilton developed a new equation to study the dynamics of space metric.The Hamilton equation is given by∂g ij=−2R ij.∂tInstead of driving space metric by gravity,he drives it by its Ricci curva-ture which is analogous to the heat diﬀusion.Hamilton’s equation therefore can be considered as a nonlinear heat equation.Heatﬂows have a regulariz-ing eﬀect because they disperse irregularity in a smooth manner.Hamilton’s equation was also considered by physicists.(Itﬁrst appeared in Friedan’s thesis.)However,this point of view was completely diﬀerent. Physicists considered it as beta function for deformations of the sigma model to conformalﬁeld theory.12SingularityDespite the fact that Hamilton’s equation tend to smooth out metric struc-ture,global topology and nonlinear terms in the equation coming from cur-vature drive the space metric to points where the space topology collapses. We call such points singularity of space.In1982,Hamilton published hisﬁrst paper on the equation.Starting with a space with positive Ricci curvature,he proved that under his equa-tion,space,after dilating to keep constant volume,never encounters any singularity and settles down to a space where curvature is constant in every direction.Such a space must be either a3-sphere or a space obtained by identifying the sphere by someﬁnite group of isometries.After seeing the theorem of Hamilton,I was convinced that Hamilton’s equation is the right equation to carry out the geometrization program.(This paper of Hamilton is immediately followed by the paper of Huisken on de-formation of convex surfaces by mean curvature.The equation of mean13curvatureﬂow has been a good model for understanding Hamilton’s equa-tion.)We propose to deform any metric on a three dimensional space which shall break up the space eventually.It should lead to the topological decomposition according to Kneser,Milnor,Jacob-Shalen and Johannson.The asymptotic state of Hamilton’s equation is expected to be broken up into pieces which will either collapse or produce metrics which satisfy the Einstein equation.In three dimensional spaces,Einstein metrics are metrics with constant curvature.However,along the way,the deformation will encounter singu-larities.The major question is how toﬁnd a way to describe all possible singularities.We shall describe these spectacular developments.13Hamilton’s ProgramHamilton’s idea is to perform surgery to cut oﬀthe singularities and continue hisﬂow after the surgery.If theﬂow develops singularities again,one repeats the process of performing surgery and continuing theﬂow.If one can prove there are only aﬁnite number of surgeries in anyﬁnite time interval,and if the long-time behavior of solutions of the Hamilton’s ﬂow with surgery is well understood,then one would be able to recognize the topological structure of the initial manifold.Thus Hamilton’s program, when carried out successfully,will lead to a proof of the Poincar´e conjecture and Thurston’s geometrization conjecture.The importance and originality of Hamilton’s contribution can hardly be exaggerated.In fact,Perelman said:“The implementation of Hamilton’s program would imply the geometrization conjecture for closed three-manifolds.”“In this paper we carry out some details of Hamilton’s pro-gram”.We shall now describe the chronological development of Hamilton’s pro-gram.There were several stages:I.A Priori EstimatesIn the early1990s,Hamilton systematically developed methods to under-stand the structure of singularities.Based on my suggestion,he proved the14fundamental estimate(the Li-Yau-Hamilton estimate)for hisﬂow when cur-vature is nonnegative.The estimate provides a priori control of the behavior of theﬂow.An a prior estimate is the key to proving any existence theorem for non-linear partial diﬀerential equations.An intuitive example can be explained as follows:when a missile engineer designs trajectory of a missile,he needs to know what is the most likely position and velocity of the missile after ten seconds of its launch.Yet a change in the wind will cause reality to diﬀer from his estimate.But as long as the estimate is within a range of accu-racy,he will know how to design the missile.How to estimate this range of accuracy is called a prior estimate.The Li-Yau-Hamilton EstimateIn proving existence of a nonlinear diﬀerential equation,we need toﬁnd an a priori estimate for some quantity which governs the equation.In the case of Hamilton’s equation,the important quantity is the scalar curvature R.An absolute bound on the curvature gives control over the nonsingularity of the space.On the other hand,the relative strength of the scalar curvature holds the key to understand the singularity of theﬂow.This is provided by the Li-Yau-Hamilton estimate:For any one-form V a we have∂R ∂t +Rt+2∇a R·V a+2R ab V a V b≥0.In particular,tR(x,t)is pointwise nondecreasing in time.In the process of applying such an estimate to study the structure of singularities,Hamilton discovered(also independently by Ivey)a curvature pinching estimate for his equation on three-dimensional spaces.It allows him to conclude that a neighborhood of the singularity looks like space with non-negative curvature.For such a neighborhood,the Li-Yau-Hamilton estimate can be applied.Then,under an additional non-collapsing condition,Hamilton described the structure of all possible singularities.However,he was not able to show that all these possibilities actually occur.Of particular concern to him was a singularity which he called the cigar.II.Hamilton’s works on Geometrization15In1995,Hamilton developed the procedure of geometric surgery using a foliation by surfaces of constant mean curvature,to study the topology of four-manifolds of positive isotropic curvature.In1996,he went ahead to analyze the global structure of the space time structure of hisﬂow under suitable regularity assumptions(he called them nonsingular solutions).In particular,he showed how three-dimensional spaces admitting a nonsingular solution of his equation can be broken into pieces according to the geometrization conjecture.These spectacular works are based on deep analysis of geometry and nonlinear diﬀerential equations.Hamilton’s two papers provided convincing evidence that the geometrization program could be carried out using his approach.Main Ingredients of these works of HamiltonIn this deep analysis he needed several important ingredients:(1)a compactness theorem on the convergence of metrics developed by him,based on the injectivity radius estimate proved by Cheng-Li-Yau in 1981.(The injectivity radius at a point is the radius of the largest ball centered at that point that the ball would not collapse topologically.)(2)a beautiful quantitative generalization of Mostow’s rigidity theorem which says that there is at most one metric with constant negative curvature on a three-dimensional space withﬁnite volume.This rigidity theorem of Mostow is not true for two dimensional surfaces.(3)In the process of breaking up the space along the tori,he needs to prove that the tori are incompressible.The ingredients of his proof depend on the theory of minimal surfaces as was developed by Meeks-Yau and Schoen-Yau.At this stage,it seems clear to me that Hamilton’s program for the Poincar´e and geometrization conjectures could be carried out.The major remaining obstacle was to obtain certain injectivity radius control,in terms of local curvature bound,in order to understand the structure of the singu-larity and the process of surgery to remove the singularity.Hamilton and I worked together on removing this obstacle for some time.III.Perelman’s BreakthroughIn November of2002,Perelman put out a preprint,“The entropy formula for Hamilton’s equation and its geometric applications”,wherein major ideas were introduced to implement Hamilton’s program.16Parallel to what Li-Yau did in1986,Perelman introduced a space-time distance function obtained by path integral and used it to verify the noncol-lapsing condition in general.In particular,he demonstrated that cigar type singularity does not exist in Hamilton’s equation.His distance function can be described as follows.Letσbe any space-time path joining p to q,we deﬁne the action to beτ0√s(R+|˙σ(s)|2)ds.By minimizing among all such paths joining p to q,we obtain L(q,τ).Then Perelman deﬁned his reduced volume to be(4πτ)−n2exp −12√τL(q,τ)and observed that under the Hamilton’s equation it is nonincreasing inτ. In this proof Perelman used the idea in the second part of Li-Yau’s paper in1986.As recognized by Perelman:“in Li-Yau,where they use‘length’, associated to a linear parabolic equation,is pretty much the same as in our case.”Rescaling ArgumentFurthermore,Perelman developed an important reﬁned rescaling argu-ment to complete the classiﬁcation of Hamilton on the structure of singulari-ties of Hamilton’s equation and obtained a uniform and global version of the structure theorem of singularities.Hamilton’s Geometric SurgeryNow we need toﬁnd a way to perform geometric surgery.In1995, Hamilton had already initiated a surgery procedure for his equation on four-dimensional spaces and presented a concrete method for performing such surgery.One can see that Hamilton’s geometric surgery method also works for Hamilton’s equation on three-dimensional spaces.However,in order for surg-eries to be done successfully,a more reﬁned technique is needed.Discreteness of Surgery TimesThe challenge is to prove that there are only aﬁnite number of surgeries on eachﬁnite time interval.The problem is that,when one performs the surgeries with a given accuracy at each surgery time,it is possible that the error may add up to so fast that they force the surgery times to accumulate.17ΩρdΩρdε-hornd sε-tubed sdoubleε-hornTcappedε-hornTThe Structure of SingularityRescaling ArgumentsIn March of2003,Perelman put out another preprint,titled“Ricciﬂow with surgery on three manifolds”,where he designed an improved version of Hamilton’s geometric surgery procedure so that,as time goes on,successive surgeries are performed with increasing accuracy.Perelman introduced a rescaling argument to prevent the surgery time from accumulating.When using the rescaling argument for surgically modiﬁed solutions of Hamilton’s equation,one encounters the diﬃculty of applying Hamilton’s compactness theorem,which works only for smooth solutions.The idea of overcoming this diﬃculty consists of two parts:1.(Perelman):choose the cutoﬀradius in the neck-like regions small enough to push the surgical regions far away in space.2.(Cao-Zhu):establish results for the surgically modiﬁed solutions so that Hamilton’s compactness theorem is still applicable.To do so,they need a deep understanding of the prolongation of the surgical regions,which in turn relies on the uniqueness theorem of Chen-Zhu for solutions of Hamilton’s equation on noncompact manifolds.18ε-hornds neck d s the gluing capds geometric surgery Conclusion of the proof of the Poincar´e ConjectureOne can now prove Poincar´e conjecture for simply connected three di-mensional space,by combining the discreteness of surgeries with ﬁnite time extinction result of Colding-Minicozzi (2005).IV.Proof of the geometrization conjecture:Thick-thinDecompositionTo approach the structure theorem for general spaces,one still needs to analyze the long-time behavior of surgically modiﬁed solutions to Hamilton’s equation.As mentioned in II,Hamilton studied the long time behavior of his equation for a special class of (smooth)solutions –nonsingular solutions.In 1996,Hamilton proved that any three-dimensional nonsingular solu-tion admits of a thick-thin decomposition where the thick part consists of a ﬁnite number of hyperbolic pieces and the thin part collapses.Moreover,by adapting Schoen-Yau’s minimal surface arguments,Hamilton showed that the boundary of hyperbolic pieces are incompressible tori.Consequently,any nonsingular solution is geometrizable.19。

Teams and motivation powerpoint presppt团队与动机的PowerPoint presppt

• Honey and Mumford’s Learning Style Questionnaire can provide picture of what a person’s preferred learning (and ergo working) style is.
• May be hands on Activist or Pragmatist, or prefer to stand back and reflect, or prefer theory.
• Known as the Experiential Learning cycle. • Cycle can be entered at any stage
One version of the experiential learning cycle
Do
Plan
Review
Learn
Honey and Mumford’s Learning Styles
• Kolb’s experiential learning cycle – Learning styles
• Consider using Myers-Briggs Personality types test MPTI (useful for finding out a person’s motivation – testing needs to be carried out by a professional).
• Comprises 16 personality factors, one covers intelligence, the rest cover dimensions of personality which are described by extremes e.g. relaxed/tense

Particles as Field Singularities in the Unified Algebraic Dynamics

Particles as Field Singularities in the UniﬁedAlgebraic DynamicsV.V.Kassandrov,J.A.RizcallaDepartment of General Physics,Russian People’s Friendship University,Moscow,Russiae-mail:vkassan@.ruNonlinear generalization of Cauchy-Riemann equations to the algebra of biquaternions is considered.In a particular case the latters reduce to the“iversal generating equations”which deal with the2-spinor and the gaugeﬁelds and form the basis of a uniﬁed algebraicﬁeld theory.For every solution to universal generating equations the components of spinorﬁeld satisfy both the eikonal and the wave equations while the strengths of gaugeﬁeld-both Maxwell and Yang-Mills equations.Making use of their speciﬁc(“ak”)gauge symmetry,we we reduce universal equations to the equations of shear-free null congruence and,applying the Kerr theorem, integrate them in twistor variables.Particles are considered as bounded singularities of eﬀective metric and electromagneticﬁelds.For fundamental unisingular solution electric charge of(point-or ring-like)singular source isﬁxed in magnitude(generally quantized),and related Kerr-Schild metric is of Reisner-N¨o rdstrem or Kerr-Newman type respectively.The value of quadrupole electric moment for the electron is also predicted. Multisingular solutions are presented and brieﬂy discussed.1.Algebrodymamical approach toﬁeld theory and universal gene-rating equationsIn general framework of algebrodynamical paradigm(see,e.g.,[1,9,10,14]and references therein) it was proposed to regard the set of equationsdξ=A(x)∗dX∗ξ(x),(1) as the basis of some uniﬁed non-Lagrangianﬁeld theory.In formula(1)the asterisk denotes multi-plication in the algebra of biquaternions B(isomorphic to the full2×2complex matrix algebra),and X represents2×2Hermitian matrix of space-time coordinates.The two-column complex variable ξ(x)can be identiﬁed as a fundamental spinorﬁeld(related to a shear-free null congruence,see Section6)while the components Aµ(x)of the2×2matrix A=Aµ(x)σµcan be considered as C-valued electromagnetic(EM)potentials.Properties and interpretation of eqs.(1)are examined throughout the article.Eqs.(1)originate from B-generalized Cauchy-Riemann equations(Section2),appear to be Lorentz and gauge invariant (Section3)and impose strict restrictions on both the spinor and the EMﬁelds(Section4).Indeed, for every solution to eqs.(1)the components of spinorﬁeld satisfy the eikonal and the wave equations (Sections2,5),while EMﬁeld strengths obey Maxwell equations for free space.Moreover,close connections exist between the solutions to eqs.(1)and the solutions to vacuum Yang-Mills and Einstein-Maxwell equations(Sections4and6respectively).199In view of the above relations between wave-like,gauge and GTR equations(we’ll call them conventional equations(CE)for brevity)and eqs.(1)on the other hand,the latters have been called generating system of equations(GSE)[14].Since CE are all of vacuum type,in the approach developed particles are regarded as(bounded in3-space)singularities of theﬁelds.We’ll see(Sections 5,8)that the structure of singularities of CE(including even that of linear Maxwell equations)is surprisingly rich,complicated(point-,string-or even membrane-like)and presumably unknown up to now.On the other hand,the characteristics and time evolution of these particles-singularities are completely governed by the overdetermined nonlinear structure of GSE(1)since CE serve only as necessary yet not suﬃcient compatibility conditions with respect to the primary GSE.In particular, the Coulomb-type Ansatz(Section5)corresponds to some solution of GSE iﬀthe value of electric charge of the source isﬁxed to be unit,in spite of linearity of Maxwell equations themselves.Thus, the charge quantization property holds here just on the classical level of consideration and due again to rigid overdetermined structure of GSE(1).From other results presented in the article,close relation between GSE and(well-known in the framework of GTR)equations deﬁning shear-free geodesic null congruences could be distinguished (Section6).In this account,complete integration of GSE can be performed using its twistor structure and Kerr theorem for shear-free congruences(Section7).On the other hand,this makes it possible to deﬁne an eﬀective Riemannian metric of Kerr-Schild type for every solution to GSE(Section6). In stationary axisymmetrical case such metric satisﬁes Einstein-Maxwell electrovacuum system of equations and is just of Reisner-N¨o rdstrem or Kerr-Newman type.On the caustics of shear-free conguences the curvature of metricﬁeld becomes singular as well as the strength of EMﬁeld related to GSE.For the latter a remarkable representation via twistor variables is also presented in Section7.In Section8we discuss general interpretation of particles as bounded singularities and point out the links between this concept and the catastrophe theory. Multisingular solutions to GSE are presented and their properties-brieﬂy discussed.2.Quaternionic diﬀerentiability.Algebraic origination and2-spinor structure of universal equationsLet A be aﬁnite-dimensional associative and commutative algebra over R or C.Natural deﬁnition of A-diﬀerentiability has been proposed by G.Sheﬀers as far as in1893and has the form(see[2, chapter5]for details):dF=D(Z)∗dZ,(2) (∗)being multiplication in A,F(Z)being an A-valued function of A-variable Z∈A,and D(Z)∈A—some other A-valued function related to F(Z)(“derivative”from F(Z)).Eqs.(2)can be considered as the condition of A-valued diﬀerential1-form to be exact1.For a particular case of complex algebra A≡C eqs.(2),after elimination of the components of D(Z),lead to the Cauchy-Riemann(CR)equations of ordinary form.It should be mentioned that successful generalization of commutative analysis to the case of supercommutative algebras has been developed in the works of V.S.Vladimirov and I.V.Volovich[3](see also[4]).To succeed in the formulation of diﬀerentiability conditions in the case of associative noncom-mutative algebra G one notices that the most general component-free form of inﬁnitesimally small increment of a G-function is21Usual conditions of smoothness of the components of F(Z)and of existence of positive norm in A-space are assumed to be fulﬁlled.2For example,in the simplest case of the quadratic function F(Z)=Z∗Z one hasdF=Z∗dZ∗E+E∗dZ∗Z,E being the unit element in G.200dF=L1(Z)∗dZ∗R1(Z)+L2(Z)∗dZ∗R2(Z)+...,(3) where the set of pairs{L i(Z),R i(Z)}replace the“derivative”D(Z)of the commutative case.Notice that just the representation(3)serves infact as the basis of noncommutative analysis in the version proposed by A.Yu.Khrennikov[4,chapter7].Unfortunately,no constraints are known to exist,generally,between the components of a“good”G-function,i.e.of a function which diﬀerential can be presented in the form(3)(for details,see e.g. the paper of A.Sudbery[5]).The situation is quite contrary to that in the commutative case,in C-case with respactive CR-equations in particular.Besides,from geometrical point of view,functions satisfying eqs.(3)show no analogy with conformal mappings in the complex case.For these reasons the version of noncommutative analysis presented in[4]cannot be recognized as fully satisfactory.Direct account of noncommutativity in the very deﬁnition of G-diﬀerentiability seems,however, quite natural and promising.In1980just this way towards the construction of noncommutative analysis has been proposed by one of the authors in[6](see[1]and the references therein).However, in order to impose some restrictions on the components of F(Z)(generalized CR-equations)it was proposed to regard as“true”G-diﬀerentiable only such G-functions for which representation(3)is reduced to one“elementary”G-valued diﬀerential1-form only,i.e.for which it holdsdF=L(Z)∗dZ∗R(Z),(4) where L(Z),R(Z)∈G had been called semi-derivatives of F(Z)(they are deﬁned up to an element from the centre of G,see[1,9,14]).Deﬁnition of G-diﬀerentiability(4)can be considered as the requirement on an elementary G-valued1-form to be exact3.For G being commutative again,conditions(4)evidently reduce to the old ones(2)(and,therefore,to CR-equations in C-case).Deﬁnition(4)appreciably narrows down the class of“good”G-functions,cutting oﬀ,say,all of polynomials(exept trivial linear ones).The situation looks like rather unexpected from the point of view of customary complex analysis.Nevertheless,condition(4)singles out just the class of G-functions which is natural from algebraic considerations,extremely interesting in geometrical properties and which admits a naturalﬁeld-theoretical interpretation.In the exclusive case of real Hamilton quaternions G≡H eqs.(4)appear to be just(necessary and suﬃcient)algebraic conditions for mapping F:Z→F(Z)to be conformal in E4(see[1,9] for details).However,since the conformal group of E4is known to beﬁnite(15-)parametrical,H-valued functions satisfying eqs.(4)are too trivial to be treated,say,asﬁeld variables.Fortunately, the situation becomes quite diﬀerent when one turns to consider the complex extention of H,i.e. the algebra of biquaternions B which only we are going to deal with below4.For B-algebra the2×2complex matrix representation is suitable.To realize the latter,for everyZ∈B we takeZ⇔ z0+z3z1−iz2z1+iz2z0−z3 ≡zµσµ,(5a)where zµ∈C,σµ={E,σa}are unit and three Pauli matrices respectively(as usual,µ,ν,...= 0,1,2,3and a,b,...=1,2,3).Applying now the column-or the full row-column splitting to eqs.(4)we obtain the following two conditions:dξ=L(Z)∗dZ∗η(Z),(6a)d f=φ(Z)∗dZ∗ψ(Z),(6b)3Note that elementary G-form(4)can be deﬁned as the most general G-valued1-form which can be constructed by means of operation of multiplication in G only.4Some considerations about diﬀerentiability in Dirac-Cliﬀord and even in non-associative octonion algebras have been presented in[1,chapter2].201whereξ(Z),η(Z),ψ(Z)∈C2are2-columns andφ(Z)∈C2is a2-row,functionally independent in general,while f(Z)∈C is some any matrix component of F(Z).According to symmetry properties of eqs.(6a,b)the quantitiesξ,η,φ,ψmanifest themselves as2-spinors,whereas f(Z)-as a scalar (see Section3and article[10]for details).From condition(6b)in account of well-known Fiertz identities,complexiﬁed eikonal equation for every(matrix)component f(Z)of a B-diﬀerentiable function F(Z)immegiately follows[1,15]ηνλ∂νf∂λf=0,(7)∂ν≡∂/∂zνbeing partial derivatives andηνλbeing metric tensor of Minkowsky space,of the form ηνλ=diag(+1,−1,−1,−1)(with respect to representation(5)).The eikonal eq.(7)plays in B-analysis the role similar to that of the Laplace equation in two-dimensional complex case.Thus,deﬁnition(4)of G-diﬀerentiability links together the non-commutativity of G-algebra(entering directly into eqs.(4))with the nonlinearity of generalized CR-equations resulted.There is nothing surprising in this correlation from the usual standpoint of gauge theories where non-Abelian groups result in nonlinearity of Yang-Mills strengths).However, within the framework of noncommutative analisis similar interrelation was demonstrated,perhaps, for theﬁrst time(all of the previous works on(bi)quaternionic analysis dealt with trivial linear generalizations of the CR-equations,see for example[7,8]and excellent review in[2,chapter5]).Noticing that eqs.(6b)follow directly from eqs.(6a)and the latters make it possible to recon-struct an arbitrary solution of the full system(4),we come to fundamental2-spinor structure of the primary system(4).Together with its nonlinear character this property allows to formulate aﬁeld theory on the base of eqs.(6a)only.This program was partially realized in[15]where general analytical solution to the eikonal equation(7)has been obtained.However,in this article we restrict ourselves with a particular case for whichξ(Z)≡η(Z)in eqs.(6a).In accord with the results obtained in[15],this case exhausts one of two classes of solutions to eikonal equation and, perhaps,appears to be the most interesting both from physical and geometrical standpoints.The only ad hoc conjecture we are obliged to accept here is the requirement for coordinates zµin(5a)to be real,zµ≡xµ∈R,i.e.to belong to Minkowsky space which is here only a subspace of full complex vector space of B-algebra.Notice that this demands from the coordinate-representing matrix in(5a)to be HermitianZ⇔X=X+=≡ u w¯w v ≡xµσµ(5b)u,v=x0±x3;w,¯w=x1±ix2being spinor(null)coordinates which are permanently used below) In account of two above-presented limitations,from which the latter is evidently necessary to ensure relativistic invariance of theory,conditions of B-diﬀerentiability(4)reduce to GSE(1) announced at the beginning of the paper.The latter is considered as the basic system of equations of algebraic nonlinearﬁeld theory which deals with a spinorﬁeld as well as with a gaugeﬁeld represented by matrix A(x)(see the next Section).Obviously,however,such a theory will be quite exotic due to the overdetermined,nonLagrangian structure of its dynamical background-the GSE, to detailed examination of which we now proceed.3.Geometrodynamical interpretation and“weak”gauge structure of GSEIn4-index notation GSE(1)takes the form∂νξ=A(x)∗σν∗ξ(x),(8) where A(x)=Aµ(x)σµ.According to(1)or(8)GSE can be considered from geometrodynamical point of view as condition for fundamental spinorﬁeldξ(x)to be covariantly constant with respect 202to the eﬀective aﬃne connectionΓ=A(X)∗dX,Γν=A(x)∗σν(9)which may be called left B-connection.It form is completely determined by the structure of B-algebra and induces a speciﬁc aﬃne geometry of Weyl-Cartan type on the complex vector space of B-algebra.To see this,one should return back in(8)from the spinorξ(x)to the full2×2matrix R(x)=F(x)=Fµ(x)σµrepresenting the“right semi-derivative”in conitions of B-diﬀerentiability (4);then one gets for the components∂νFµ=ΓµνρFρ(x),(10)where the connection coeﬃcients get the formΓµνρ≡Γµνρ(A(x))=δµνAρ+δµρAν−ηνρAµ−iεµ.νρλAλ,(11)which includes the Weyl nonmetricity and the totally skew-symmetric torsion terms related to each other(with complex Weyl vector Aµ(x)being proportional to the pseudo-trace iAµ(x)of the torsion tensor).Note that B-induced complex Weyl-Cartan connection(11)has been proposedﬁrstly in[1,10] and recently used by V.G.Kretchet in his search for geometric theory of electroweak interactions[17] (based on the break of P-invariance by the torsion term in(11)).GSE(1)is evidently form-invariant under the global transformations of coordinates andﬁeld variablesX⇒X =M+∗X∗M,(12a)ξ⇒ξ =M−1ξ,A⇒A =M−1∗A∗(M+)−1,(12b) M∈SL(2,C)being an arbitrary unimodular2×2complex matrix.The6-parametric group of transformations(12a)2:1corresponds to the continious transforma-tions of the coordinates{xµ}from Lorentz group.Thus,GSE is relativistic invariant and,according to the laws of transformations(12b),the quantities Aµ(x)andξB(x),B=0,1behave themselves as the components of4-vector and2-spinor respectively.As to local symmetries of GSE,system(8) can be shown to preserve its form under the so called“weak gauge transformations”[10,13,14]ξB⇒ξ B=λξB,Aµ⇒A µ=Aµ+12∂µlnλ,(13)where the gauge parameterλ≡λ(ξ1,ξ2,τ1,τ2)∈C is a smooth scalar function dependent on two spinor components of the original solution and their twistor counterpartsτ=Xξonly(instead of its direct dependence on the4-coordinates{xµ}themselves in generally accepted gauge approach). For detailed discussion of this new concept which is based on twistor structure of GSE(see below, Section7)we refer the reader to our papers[14,13].Besides,GSE is invariant under the gauge transformations of Weyl type,related to conformal transformation of the original Minkowsky metric;discussion of double gauge group so arising can be found in[11].In account of the gauge nature of the4-vector Aµ(x)and of its close relation to the Weyl nonmetricity vector,it seems quite natural to identify Aµ(x)(up to a dimensional factor)with the4-vector of potentials of(complexiﬁed)electromagneticﬁeld.Leaving for the next Section the discussion of complex structure of EMﬁeld,we recall only that both the spinor and the EMﬁelds can be found from the only system(8)in a self-consistent way due to overdetermined structure of the latter.Then the question arises what sort of restrictions on EM strengths are imposed by GSE, and in which way are they related to Maxwell equations?2034.Self-duality,Maxwell&Yang-Mills equations as the compatibil-ity conditions of GSESince the set of original equations(1)or(8)is overdetermined(8equations for2spinor plus4potential components),some compatibility conditions should be satisﬁed“on shell”.In particular, commutators of partial derivatives∂[µ∂ν]ξ=0in(8)should turn to zero,this being in correspon-dence with the closeness of B-valued1-form in(1)accord with the Poincar´e lemma.After trivial calculations we get then0=R[µν]ξ,(14) where the quantitiesR[µν]=∂[µAσν]−[Aσµ,Aσν](15) represent B-curvature tensor of left B-connection(9).From(14)it doesn’t follow R[µν]≡0,since the spinorξ(x)is,certainly,not at all arbitrary.However,it can be shown(see[1,10]or[14]where2-spinor formalism has been used)that self-dual part R+[µν]of(15)R+[µν]≡R[µν]+i2ε..ρλµνR[ρλ]=0(16)should turn to zero by virtue of eqs.(14).Being written in components,expressions(15),(16)result in the following3+1set of equations:F[µν]+i2ε..ρλµνF[ρλ]=0,(17)∂µAµ+2AµAµ=0,(18) where the tensorF[µν]=∂[µAν](19) is a usual tensor of EMﬁeld strength.3-vector form of eqs.(17)E+i H=0(20) relates the(C-valued)electric E and magnetic H vectors ofﬁeld strengthE a=F[oa]=∂o A a−∂a A o,H a=12εabc F[bc]=εabc∂b A c.(21)Thus,we have found that self-duality conditions(17)and“inhomogeneous Lorentz condition”(18)5 are just the integrability conditions of GSE.According to deﬁnitions ofﬁeld strengths via the potentials(21)and to self-duality conditions (20)we conclude then that free-space Maxwell equations are satisﬁed identically for every solution to GSE.Complex variable’nature ofﬁeld strengths(21),however,doesn’t result in the doubling of the number of degrees of freedom of EMﬁeld just because of the self-duality constraints(20).Indeed, from the latters we get onlyB= E, D=− H,(22) where{ E, H}and{ D, B}represent respectively the real( )and imaginary( )parts of the ini-tial complexﬁelds{ E, H}.The real-partﬁelds E and H are therewith mutually independent 5Geometrically the latter corresponds to the condition for scalar4-curvature invariant R of the Weyl tensor to be null,see[10].204algebraically(i.e.in a space-time point)and satisfy themselves free Maxwell equations owing to linearity of the latters6.Physical meaning of decomposition of unique complexﬁeld into its real and imaginary parts is the following[1,Appendix].The density of energy-momentum tensor can be deﬁned via the latters in a usual way while for original complexﬁelds respective densities(as well as the density of angular momentum)w∝ E2+ H2, p∝ E× H(23) vanish in account of self-duality conditions(20).Moreover,some preferance of -partﬁelds E, H may be therewith justiﬁed from geometrical and physical considerations(see Section5).In addition to all this,it can be shown[10,14]that the structure of GSE,and of B-connection (9)in particular,makes it possible to deﬁne also the C-valued Yang-Millsﬁeld.Infact,connection (9)can be rewritten in the formΓν=A(X)∗σν=Aµ(x)σµ∗σν=Aµ(x)Bρµνσρ≡Aν(x)+N aν(x)σa,(24) where Bρµνare the structure constants of B-algebra,and the trace part of connection corresponds to EM4-potentials Aµ(x).As to the trace-free-part variables N aν(x),N a o=A a(x);N a b=δab A o(x)+iεabc A c(x)(25) they can be identiﬁed with the potentials of a Yang-Mills(YM)ﬁeld of a special type.The trace-free part of B-curvature tensor(15)gives then for the strengths of YM potentials(25)usual expressionL a[µν]=∂[µN aν]−i2εabc N bµN cν.(26) For a nonzero solutionξ(x)it follows then from eqs.(14)for every[µν]component of curvature and strengthsdet R[µν] ≡F2[µν]−L a[µν]L a[µν]=0,(27) In view of(27)EMﬁeld(21)should be regarded as a modulus of isotopic vector of YM-triplet ﬁeld.Bothﬁelds are described via unique left B-connection(9):EMﬁeld is related to the trace part of correspondent curvature while YMﬁeld-to the trace-free part of it.Such an interrelation of EM and YMﬁelds which was proposedﬁrstly in[10]is gauge invariant and requires no participation of auxiliary chiralﬁeld as it occures in generally accepted gauge approach.However,the subclass of YMﬁelds(25)can’t be pure,being always accompanied by EM ﬁeld due to positive deﬁnite norm of isotopicﬁeld3-space(see(27)).The above speculations would be signiﬁcant if only the YM equations would be satisﬁed by the ﬁelds(25),(26).Fortunately,it is just the case,since the trace-free part of self dual B-curvature (15)includes only corresponding self-dual conﬁguration of Maxwell strength tensor and Lorentz inhomogeneous form[10],both being null in account of the integrability conditions(17),(18). Thus,for every solution to GSEﬁeld strengths(26)are self-dual and satisfy therefore YM equations for free space.It may be noted in conclusion that,contrary to EM case,the and parts of C-valued strengths (26)won’t satisfy nonlinear YM equations separately.Thus,YMﬁelds arising here are essentially complex-valued.On the other hand,it can be proved directly that non-Abelian(commutator)part of YM strengths(26)does not vanish for the potentials(25)neither identically nor on the solutions to GSE,so that they can’t be reduced generally to EMﬁelds7.6The same is true,of course,for the -partﬁelds D, B as well providing,in account of(22),a dual solution to Maxwell equations.7Possible nonAbelian nature of EMﬁeld itself was discussed recently in[18],also in the framework of Weyl geometry.2055.Fundamental unisingular solution,quantization of electric charge and ring-like model of electronVacuum Maxwell equations hold identically for every solution of GSE.Hence,no regular“soliton-like”ﬁeld distribution can exist for the model considered.Nevertheless,particles can be brought into correspondence with singular points(or strings,membranes etc.)of theﬁeld functions in which B-diﬀerentiability conditions are violated and which manifest themselves as point-like or extendedsource of physicalﬁelds.In account of complex variable and self-dual nature of gaugeﬁelds established above,charged singular solutions,if exist,should be dions,i.e.should carry both electric and magnetic charges of equal(up to a factor“i”)magnitudes.Indeed,elementary unisingular dion-like solution has been found in[1,10].To obtain it here,we’llﬁx the gauge so that the only component G(x)of fundamental2-spinorξ(x)will remain in GSE8:ξT(x)=(1,G);(28) then for complex EM potentials A(x)one getsA w=∂u G,A v=∂¯w G,A u=A¯w≡0,(29) and GSE reduces to a couple of nonlinear diﬀerential equations for a unique unknown function G(x)∂w G=G∂u G,∂v G=G∂¯w G,(30) where the spinor space-time coordinates{u,v,w,¯w}deﬁned previously by eq.(5b)have been used.By mutual multiplication of eqs.(31)we’ll come then to the constraint(∂u G)(∂v G)−(∂w G)(∂¯w G)=0,(31) which is nothing but the eikonal eq.(7)in spinor coordinates.If we’ll then write out the commutator of derivatives in the l.h.p.of eqs.(30)we get with respect to eq.(31)∂u∂v G−∂w∂¯w G=0,(32) where the latter equation is just the wave(d’Alembert)equation 2G=0.It can be shown that the last result is gauge invariant in the following sense:the ratio of two components of the spinor ﬁeld{ξB(x)}obeys“on shell”both the eikonal and d’Alembert equations..Fundamental static axisymmetric solution to GSE(which also satisfy eqs.(31),(32)has been found in[1,10].It corresponds to stereographic mapping S2→C of the Riemannian2-sphere onto the complex plane:G=x1+ix2x3±r≡¯wz±r≡tan±1θ2exp iϕ,(33){r,θ,ϕ}being usual spherical coordinates.From the solution(33),which satisfy the couple of eqs.(30)under consideration,complex EM potentials(29)A w,A v can be found;then for the scalar (A o)and spherical components{A r,Aθ,Aϕ}of4-potential we’ll haveA o=±14r,A r=−14r,Aϕ=±iAθ=i4rtan±1θ2(34)Now for nonzero components of C-valued EMﬁeld strengths(21)we get(the electricﬁeld appears to be pure real,while the magnetic-pure imaginary)E r=±14r2,H r=±i4r2,(35)8This gauge is possible for every“physically nontrivial”solution to GSE,see[14]for details.206(note that the components A r,Aθdon’t contribute into the magnitude ofﬁeld strengths,being a “pure gauge”).We see that fundamental solution(33)corresponds to a point source of real electric (and imaginary magnetic)ﬁeld and withﬁxed value of electric charge q=±1/4(and equal value of imaginary magnetic charge m=±i/4).At this stage of consideration,factor(1/4)is unessential since physical EM potentials are deter-mined up to an arbitrary dimensional factor only.What is really signiﬁcant is that in this model1) all values of electric charge except the only possible one are not allowed for a point singular source to possess,and2)its Coulombﬁeld is always accompanied by the magnetic monopoleﬁeld with the charge equal in modulus to the electric one!For the proof of general theorem on charge quantization which is based on self-duality condition(20)and gauge invariance(13)of GSE we refer the reader to the paper[16].For similar models where the property of charge quantization arises see[11].Let us consider now an interesting modiﬁcation of solution(33)-(35),which can be obtained via complex translation z→z+ia,a∈R,the latter being obviously a symmetry of GSE.In this way we come to a new solution which electricﬁeld structure instead of Coulomb form(35)corresponds to the known Appel solution(see e.g.[29]).Singular locus of EMﬁeld would be then deﬁned by the conditionr∗≡ (z+ia)2+x2+y2=0,⇒{x2+y2=a2,z=0},(36) and is a ring of radius|a|.For real-partﬁelds( -ﬁelds){ E, H}the magnetic component appears, and the following asypmtotic behaviour at distances r>>|a|is true:E r q r2(1−3a22r2(3cos2θ−1)),Eθ −qa2r43cosθsinθ,H r 2qa r3cosθ,Hθ qa r3sinθ.(37)In view of eqs.(36)-(37), -ﬁeld solution is related to electrically charged singular ring equipped with a quantized“elementary”value of electric charge q=±1/4,dipole magnetic momentµ=qa and quadrupole electric momentϑ=−2qa2.If we’ll choose dimensional physical units so that to have q=e,e being elementary charge,and accept for the radius of the ring the value|a|= 2Mc,38M being the mass of the source,then for dipole magnetic momentµwe’ll get the known Dirac value µ=e /2Mc.Moreover,according to(37)we conclude that fundamental charged fermion should necessarily possess quadrupole electric momentϑequal in magnitude toϑ=e 22M2c2(39)It should be marked that prediction of quadrupole moment(39)for elementary Dirac-like particles,to our knowledge,has been madeﬁrstly by C.A.Lopes[25]in the framework of GTR and on thebase of Kerr-Newman metric.We’ll see further that our fundamental solution is also deeply relatedto Kerr-Newman metric.At present,the statement about necessary existence of quadrupole electricmoment looks rather speculative;nontheless,possibility of its experimental proof may be discussed.However,much more fundamental seems to us the fact that for -partﬁelds their asymptoticstructure(37)is in complete agreement with that observed for elementary particles,whereas the-ﬁelds contain only“phantom”terms proportional,say,to magnetic charge or to dipole electric moment!This property is based just on complex self-dual structure of EMﬁelds and,as well as theproperty of charge quantization,is peculiar only for our model!Geometrically,phantom-like -ﬁelds representing,in particular,magnetic monopole,contributeonly into the torsion term9of real projection of original complex B-connection(9)onto Minkowskyspace[10].Therefore,owing to speciﬁc(totally skew symmetric)structure of torsionﬁgurating in 9Relation between the magnetic monopoleﬁeld and the geometries with torsion has been advocated,in particular, by G.Lochak[26].207。

Singularities, Zeros,Residues

∞ n=0
an (z − z0 )n .
(Note that a0 = limz →z0 f (z )). Example The function f (z ) is deﬁned by f (z ) = z (eiz + 1) for z = ±π. z2 − π2
Show that f (z ) has removable singularities at z = ±π , and ﬁnd the appropriate values which must be assigned to f (±π ).
✷ As an illustration, suppose that f (z ) = cot πz ; then g(z ) = cos πz and h(z ) = sin πz . The function f (z ) has simple poles where sin πz = 0, that is at z = n, where n = 0, ±1, ±2 . . . . By Theorem 5.5, the residue at z = n is Res(cot πz, n) = cos nπ 1 = . π− z0 )n+m is analytic in the domain of convergence of (5.1).
f (z ) = (z − z0 )m φ(z ),
(5.2)
Deﬁnition 5.1 Under the above assumptions f (z ) is said to have a zero of order m at z = z0 . When m = 1, the zero is a simple zero. Examples Find the zeros and their orders of (a) 4z 3 − z 5 and (b) z 2 sin z . As deﬁned in Chapter 2, singularity of a complex function f (z ) is an isolated point at which it ceases to be analytic (recall that ‘analytic at a point’ means analytic in some neighbourhood of the point). Note that z0 is also called an isolated singularity. Here the adjective ‘isolated’ means that we can ﬁnd a neighbourhood of z0 which does not contain another singularity of f (z ); We can then draw two concentric circles of centre z0 , both lying within D , and in the annulus between these two circles f (z ) has a Laurent expansion (Theorem 4.8) of the form (4.10):

singularit用法

singularit用法英文回答：Singularit is a term that is commonly used in the field of artificial intelligence and technology. It refers to the hypothetical point in time when artificial intelligence surpasses human intelligence and becomes self-improving, leading to an exponential increase in technological advancements. This concept was popularized by futurist Ray Kurzweil, who predicted that this point would be reached by the year 2045.The idea of singularity is based on the notion that as AI systems become more advanced, they will be able to improve themselves at an accelerating rate. This means that they will become more intelligent, more capable, and more efficient over time. They will be able to learn from their own experiences, adapt to new situations, and make decisions based on complex data analysis.One example of singularity in action is the development of self-driving cars. These vehicles use AI algorithms to analyze data from various sensors and make decisions inreal-time. As more data is collected and more experience is gained, the AI system can continuously improve its driving capabilities. Eventually, it may reach a point where it surpasses human drivers in terms of safety and efficiency.Another example is the field of medicine. AI systemscan analyze large amounts of medical data and identify patterns and correlations that may not be apparent to human doctors. This can lead to more accurate diagnoses, personalized treatment plans, and improved patient outcomes. As AI in medicine continues to advance, it may eventually lead to breakthroughs in areas such as drug discovery and genetic research.In conclusion, singularity refers to the point at which artificial intelligence surpasses human intelligence and becomes self-improving. This concept has far-reaching implications for various industries and has the potentialto revolutionize the way we live and work. While the ideaof singularity may still be speculative, the rapid advancements in AI technology suggest that it may not betoo far off in the future.中文回答：Singularit是人工智能和技术领域常用的术语。