The clustering instability of inertial particles spatial distribution in turbulent flows

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摘要摘要基于植保无人机的低空药液喷施技术具有安全、高效的作业优势，已广泛应用于北方大田种植环境下的农作物精准施药。

南方农作物种植环境多以丘陵山地为主，主要特征是农作物种植不规整、随意性大，不适合直接采用大田种植环境下的均匀药液喷施模式。

为了提高山地果园种植环境中植保无人机药液喷施的作业效果，以柑橘果树为研究对象，提出一种复杂果园环境下基于单目视觉的果树检测方法，实现无人机在果园环境下获取果树大小、位置等基本的作物信息，可以为农药的精准对靶施药提供技术支持；为检验果园的施药效果，提出一种基于地—空信息融合的柑橘产量估计方法，实现单棵果树的产量估计。

本文主要研究内容和结论如下：1、基于单目机器视觉的果园环境下柑橘果树检测。

获取了山地柑橘果园环境下果树分布的低空RGB (Red/green/blue)图像数据，为了减少亮度对果树检测的干扰，采用选择性直方图改善果树区域的亮度，由于果树与杂草区域存在较为显著的视觉差异性，通过色差图方法实现潜在果树区域(Regions of Interest, RoI)的分割；利用色彩空间变换技术(Color spaces transformation technology, CST)提取了表征果树颜色信息的14种颜色特征，分别采用灰度共生矩阵(Gray-level co-occurrence matrix, GLCM)和局部二值模式(Local binary pattern, LBP)获取了RoIs的6种纹理特征，并建立支持向量机(Support vector machine, SVM)模型实现果树分割和半径检测。

结果表明，本文方法可在维持图像色彩信息不变的情况下实现果树亮度的自适应调整，提高了前背景间的对比度，确保了更准确的果树区域分割，平均分割准确率为83.09%；基于果树外观特征和SVM的柑橘果树检测模型能进一步抑制复杂果园环境中干扰场景参与物，检测准确率达到了85.27%。

a r X i v :0802.1452v 1 [a s t r o -p h ] 11 F eb 2008accepted in ApJPreprint typeset using L A T E X style emulateapj v.04/21/05THE SHAPE OF THE INITIAL CLUSTER MASS FUNCTION:WHAT IT TELLS US ABOUT THE LOCAL STAR FORMATION EFFICIENCYG.Parmentier 1,2,4,S.P.Goodwin 3,P.Kroupa 1and H.Baumgardt 1accepted in ApJABSTRACTWe explore how the expulsion of gas from star-cluster forming cloud-cores due to supernova explo-sions affects the shape of the initial cluster mass function,that is,the mass function of star clusters when effects of gas expulsion are over.We demonstrate that if the radii of cluster-forming gas cores are roughly constant over the core mass range,as supported by observations,then more massive cores undergo slower gas expulsion.Therefore,for a given star formation efficiency,more massive cores retain a larger fraction of stars after gas expulsion.The initial cluster mass function may thus dif-fer from the core mass function substantially,with the final shape depending on the star formation efficiency.A mass-independent star formation efficiency of about 20per cent turns a power-law core mass function into a bell-shaped initial cluster mass function,while mass-independent efficiencies of order 40per cent preserve the shape of the core mass function.Subject headings:galaxies:star clusters —stars:formation —stellar dynamics1.INTRODUCTIONOne of the greatest discoveries made by the Hubble Space Telescope is that the formation of star clusters with mass and compactness comparable to those of old globular clusters is still ongoing in violent star forming environments,such as starbursts and galaxy mergers.The young cluster mass function (CMF:the number of objects per logarithmic cluster mass interval,d N/dlog m )is often reported to be a power-law of spectral index α=−2(i.e.equivalent to d N ∝m −2d m )down to the detection limit (Fall &Zhang 2001;Bik et al.2003;Hunter et al.2003).Similar results are obtained for the cluster luminosity function (e.g.,Miller et al.1997;Whitmore et al.1999).However,old globular cluster systems are found to have a CMF which is Gaussian-like with a peak mass of ∼105M ⊙(Kavelaars &Hanes 1997).These very different CMFs between young and old systems present the greatest dissimilarity between the two populations,and perhaps the greatest barrier to identifying young massive clusters as analogues of young globular clusters.Yet,it is worth keeping in mind that the luminos-ity/mass distribution of some young star cluster systems does not obey a power-law.Based on high resolution Very Large Array observations,Johnson (2008)rules out a power-law cluster luminosity function for the young star clusters formed in the starburst galaxy Henize 2-10.In NGC 5253,an irregular dwarf galaxy in the Cen-taurus Group,Cresci et al.(2005)detect a turnover at 5×104M ⊙in the mass function of the young massive star clusters in the central st but not least,the case of the Antennae merger of galaxies NGC 4038/39remains1Argelander-Institut f¨u r Astronomie,University of Bonn,Auf dem H¨u gel 71,D-53121Bonn,Germany2Institute of Astrophysics &Geophysics,University of Li`e ge,All´e e du 6Aoˆu t 17,B-4000Li`e ge,Belgium3Department of Physics and Astronomy,University of Sheffield,Hicks Building,Hounsfield Road,Sheffield S37RH,United King-dom4Alexander von Humboldt Fellow and Research Fellow of Bel-gian Science Policyheavily debated.While Whitmore et al.(1999)infer from their HST/WFPC2imaging data a power-law clus-ter luminosity function with a spectral index α≃−2.1,Anders et al.(2007)report the first statistically robust detection of a turnover at M V ≃−8.5mag,correspond-ing to a mass of ≃16×103M ⊙.The origin of that discrepancy likely resides in differences in the data re-duction and the statistical analysis aimed at rejecting bright-star contamination and disentangling complete-ness effects from intrinsic cluster luminosity function sub-structures.These findings raise the highly interesting prospect that the shape of the CMF at young ages does vary from one galaxy to another.If it were conclusively proven that the shape of the initial cluster mass function (ICMF)is not universal among galaxies but,instead,depends on environmental conditions and/or on the star formation process,consequences would be far reaching since this would imply that the young CMF may help us decipher physical conditions prevailing in external galaxies.In this contribution,we report the first results of a project investigating the evolution of the mass function of star forming cores into that of bound gas-free star clusters which have survived expulsion of their resid-ual star forming gas.A similar study was carried out by Parmentier &Gilmore (2007)in which they showed that protoglobular cloud mass functions characterized by a mass-scale of ≃106M ⊙evolve into the observed globular cluster mass function.They propose that the origin of the universal Gaussian mass function of old globular clusters thus likely resides in a protoglobular cloud mass-scale common among galaxies and imposed in the protogalactic era.Present-day star forming cores,however,do not show any characteristic mass-scale as their mass function is a power-law down to the detec-tion limit.Previously,Kroupa &Boily (2002)showed that a power-law embedded-cluster mass function can evolve into a mass function with a turnover near 105M ⊙if the physics of gas removal is taken into account.Tak-ing advantage of the grid of N -body models recently compiled by Baumgardt &Kroupa (2007),we explore2Parmentier et al.in greater detail how a featureless core mass function evolves into the ICMF as a result of gas removal,which we model as the supernova-driven expansion of a su-pershell of gas.We also explore how the ICMF re-sponds to model parameter variations so as to under-stand what may make the mass function of young star clusters vary from one galaxy to another.In a companion paper (Baumgardt,Kroupa &Parmentier 2008),we ex-plore this issue from a different angle,that is,by defining an energy criterion,and we apply that alternative model to the mass function of the Milky Way old globular clus-ters.2.NOTION OF INITIAL CLUSTER MASSFUNCTIONBefore proceeding any further,it is worth defining the notion of the ICMF properly.Following the onset of massive star activity at an age of 0.5to 5Myr,a gas-embedded cluster expels its residual star forming gas.As a result,the potential in which the stars reside changes rapidly causing the cluster to violently relax in an attempt to reach a new equilibrium.This violent relaxation takes 10–50Myr during which time the cluster loses stars (”infant weight-loss”)and may be completely destroyed (”infant-mortality”)(this process has been studied in detail by Hills (1980);Mathieu (1983);Elmegreen (1983);Lada,Margulis &Dearborn (1984);Elmegreen &Pinto (1985);Pinto (1987);Verschueren &David (1989);Goodwin (1997a);Goodwin (1997b);Kroupa,Aarseth &Hurley (2001);Geyer &Burkert (2001);Boily &Kroupa (2003a);Boily &Kroupa (2003b);Goodwin &Bastian (2006);Baumgardt &Kroupa (2007)).After ∼50Myr the evolution of clusters is driven by internal two-body relaxation processes and tidal interactions with the host galaxy (see Spitzer 1987).Following Kroupa &Boily (2002)we define the ICMF as the mass function of clusters at the end of the violent relaxation phase ,that is,when their age is ≃50Myr.The effect of gas expulsion depends strongly on the virial ratio of the cluster immediately before the gas is removed.If the stars and gas are in virial equilibrium before gas expulsion the virial ratio Q ⋆of the stars de-pends entirely on the star formation efficiency (SFE,or ε)such that Q ⋆=1/(2ε)(where we define virial equi-librium to be Q =1/2)(Goodwin 2008).Formally,εis the effective SFE (eSFE:see Verschueren (1990);Goodwin &Bastian (2006)),that is,a measure of how far from virial equilibrium the cluster is at the onset of gas expulsion,rather than the actual efficiency with which the gas has formed stars.We note that even quite small deviations from virial equilibrium can equate to large differences between the eSFE and the SFE.How-ever,in this paper we will make the assumption that the stars have had sufficient time to come into virial equi-librium with the gas potential (ie.gas expulsion occurs after a few crossing times),and so the eSFE very closely matches the SFE.We discuss this assumption further in the conclusions.In this case,the radius of the embedded cluster is the same as that of the core and the three-dimensional ve-locity dispersion of the stars in the embedded cluster is dictated by thedepth of the core gravitational poten-tial,namely:σ=(Gm c /r c )1/2,where m c and r c are themass and radius of the core,respectively,and G is the gravitational constant.A star-cluster forming ‘core’of mass m c (ie.the region of a molecular cloud that forms a cluster)with a SFE of εwill form a cluster of mass ε×m c .However,gas expulsion will unbind a fraction (1−F bound )of the stars in that cluster such that the mass of the cluster at ∼50Myr will bem init =F bound ×ε×m c .(1)(Note that when F bound ∼zero the cluster is destroyed.)It is important to keep in mind that,for clusters whose age does not exceed,say,10-20Myr,the mass frac-tion of stars that have escaped after gas removal is still small,as most stars do not have a high enough veloc-ity to have become spatially dissociated from the cluster (see fig.1in Goodwin &Bastian (2006)and fig.1in Kroupa,Aarseth &Hurley (2001)).The instantaneous mass of these young clusters thus follows:m cl ε×m c and,in the absence of any explicit dependence of εon m c ,the CMF at such a young age mirrors the core mass func-tion.Any meaningful comparison of our model ICMFs therefore has to be limited to populations of clusters which have either re-virialised or been destroyed.In our Monte-Carlo simulations,we assume that the SFE of a core is independent of its mass.The core mass function is assumed to follow a power-law of spectral in-dex α=−2and we describe the probability distribution function of the SFE as a Gaussian of mean ¯εand stan-dard deviation σε,which we denote P (ε)=G (¯ε,σε).Clearly,if the bound fraction of stars after gas ex-pulsion,F bound ,depends upon the core mass,then the shape of the ICMF will differ from the (power-law)core mass function,an effect whose study was pioneered by Kroupa &Boily (2002).We now take this point fur-ther by combining the detailed N -body model grid of Baumgardt &Kroupa (2007),which provides the bound fraction F bound as a function of the SFE εand of the ratio between the gas removal time-scale τGR and the proto-cluster crossing-time τcross .3.THE GAS EXPULSION TIME-SCALEWe model the gas expulsion from an (embedded)clus-ter as an outwardly expanding supershell whose radius r s varies with time t as (Castor,McCray &Weaver 1975):r s (t )=125ρg 1/5t 3/5.(2)Here ˙E0is the energy input rate from Type II supernovae and ρg is the mass density of the residual star forming gas in the core.We assume the mass density profiles of the core and of the newly formed stars to be alike,that is,ρg =(1−ε)ρc .The energy input rate from massive stars expelling the gas is˙E0=N GR ×E SNInitial cluster mass function3˙E 0=N SN×E SN1253N SN E SN(1−ε)m c 1/3.(5)While gas-embedded cluster masses vary by many or-ders of magnitude,their radii are remarkably constant, always of order1pc(see table1in Kroupa(2005)).As N SN∝m c,it follows from eq.5that,for a givenε,the gas removal time-scaleτGR is independent of the core mass m c.However,the impact of gas expulsion upon a cluster is not governed by the absolute value ofτGR,but by its ratio with the core crossing timeτcross,ie.F bound is a function ofτGR/τcross.The dynamical crossing-time of the star forming core is given byτcross=(15/πGρc)0.5 which,combined with eq.5,gives:τGRE511−ε1M⊙ 1/2r c4Parmentier et al.100101102103104105106100101102103104105106l o g ( d N c l / d l o g m c l )10101102103104105106log( m cl . M o -1)Fig.2.—How the form of the ICMF produced from a −2power-law cluster mass function responds to variations in the SFE Gaus-sian distribution G(¯ε,σε),where ¯εand σεare the mean SFE and the standard deviation,respectively.Solid lines depict the original core mass function.These different behaviours stem from how large the variations of the bound fraction F bound with the core mass m c are.In order to illustrate this,let us focus on the top panel of Fig.2for which σε=0.A SFE as high as ε=0.4guarantees that F bound is only a weak function of the core mass m c as 0.35 F bound 0.95over the core mass range 103M ⊙ m c 108M ⊙(see Fig.1).As a result,the shape of the ICMF mirrors that of the core mass function and is close to a power-law of spectral index α=−2.In the case of a SFE of ¯ε=0.33,the same core mass range corresponds to an increase of the bound fraction with the core mass of almost two orders of magnitude (i.e.0.01 F bound 0.90),making the ICMF significantly shallower than the core mass function.It is worth keep-ing in mind that the detailed form of this ICMF is uncer-tain,as the determination of the bound fraction F bound depends on the fine details of the N -body modelling ofgas expulsion when F bound 0.1.As for ε=0.33,F bound ≃0.01−0.04when log(τGR /τcross ) −0.5(see Fig.1).If the bound fraction were larger over that range of gas removal time-scale to crossing time ratio,the shape of the ICMF would be closer to that of the core mass function,while if F bound were zero before increasing at,say,m core ≃105−3×105M ⊙,this would result in a bell-shaped ICMF (see the case of ¯ε=0.20below).The core mass function is most affected when ¯ε=0.20.Fig.1shows that the formation of a bound gas-free star cluster requires its parent core to be more massive than 106M ⊙,since the bound fraction is zero otherwise (note that the top x -axis is rightward-shifted by ≃0.1in log(τGR /τcross )when ε=0.20,see eq.6).Owing to the wide variations of F bound over the core mass range (0 F bound 0.85),the transformation of the core mass function into the ICMF is,in this case,heavily core mass dependent:the featureless power-law core mass func-tion evolves into a bell-shaped ICMF.The time-scale for this evolution is that required for the exposed cluster to get back into virial equilibrium following gas expul-sion,that is,50Myr.Examination of the model grid computed by Baumgardt &Kroupa (2007)for the SFE and the gas removal time-scale of relevance (ε=0.20and τGR τcross )shows that the end of the violent re-laxation phase occurs at about that age and the cluster mass is then as defined in eq.1.As indicated by the arrows in Fig.1and in the top panel of Fig.2,the turnover location is determined by the core mass at which the bound fraction is a few per cent.For instance,a core mass of 2×106M ⊙with ε=0.20leads to log(τGR /τcross )=0.13and F bound =0.06.The corresponding initial cluster mass is thus m init =(0.06)×(0.20)×(2×106M ⊙)≃2.4×104M ⊙,roughly the mass of the ICMF turnover.That ε=0.20leads to a bell-shaped ICMF directly re-sults from the zero bound fraction of stars for core masses less than ≃106M ⊙.A SFE of ε=0.25would result in the same effect (see Fig.1).This is equivalent to creating a truncated power-law for the cluster forming cores,even though the star forming core mass function is a feature-less power-law down to the low-mass regime.That result is reminiscent of the detailed investigation performed by Parmentier &Gilmore (2007)who found that a system of cluster parent clouds devoid of low-mass objects,such as a power-law mass function truncated at low-mass,re-sults in a bell-shaped ICMF.We emphasize that,if these clusters were observed at an age of 50Myr,their observed mass would closely match the mass predicted by our model.Actually,all clusters of the bell-shaped ICMF,irrespective of their mass,stem from gaseous progenitors more massive than 106M ⊙.This implies that unbound stars formed in the core leave the exposed cluster with velocities of order 60km.s −1and do not linger as part of a halo around the bound part of the cluster.As a related consequence,note that if the very early Milky Way disc went through a violent star formation phase producing such massive clusters,then the corresponding high veloc-ity dispersions lead naturally to the creation of a thick disc component (Kroupa 2002).In the middle and bottom panels of Fig.2we show the ICMF that results from the core mass function for SFE distributions with variance σε=0.03and 0.05re-spectively.The general features of the panels are theInitial cluster mass function5100 200 300 400 500 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5d N c l / d l o g m c llog( m cl . M o -1)Fig. 3.—How the shape and position of the ICMF produced by a −2power-law core mass function with a SFE of ¯ε=0.20,responds to variations in the width of the SFE distribution,and in the core radiussame as for zero variance,however there are increasing differences,especially at the low-mass end of the ICMF.These differences are due to the effect of the SFE not being symmetric around the mean ¯ε.For ¯ε=0.2,low-mass gas-embedded clusters with ε=0.2are completely destroyed (F bound ∼0)and this is the case for any clus-ter selected below the mean.However,a low-mass gas-embedded cluster can be drawn with ε=0.3–0.4,in which case it will be able to survive and retain a sig-nificant fraction of its mass.For this reason,the ICMF flattens rather than turning-over in the bottom panel of Fig.2.A key assumption underlying the above analysis is the hypothesis of a constant radius for all star forming cores,regardless of their mass.In Fig.3,we examine the effect of the core radius on the form of the ICMF produced from a core mass function with ε=0.20(ie.a bell-shaped ICMF).We explore how the cluster mass at the turnover responds to core radius variations.A linear scaling for the y -axis is adopted as it enables us to pinpoint the turnover location more accurately.For the sake of com-parison,the ICMFs obtained with σε=0.00and 0.03and with r c =1pc (dotted curves with crosses in top and middle panels of Fig.2,respectively)are shown as solid lines.ICMFs depicted with open symbols correspond to constant core radius of r c =0.5pc and r c =2pc.A larger core radius implies smaller core densities,thereby increasing the disruptive effect of gas expulsion and low-ering the number of bound gas-free star clusters.That the turnover is at higher cluster mass arises from the leftward-shift of the mass-scaling of the top x -axis of Fig.1when the core radius is larger (see eq.6).As stated in Section 3,observations support the as-sumption of an almost constant core radius.This,how-ever,is at variance with theoretical expectations that themass-radius relation of virialised cores obeys r c ∝m 1/2c (e.g.Harris &Pudritz 1994).ICMFs obtained with r c [pc]=10−3(m c /1M ⊙)1/2are shown as lines with filled symbols in Fig.3.The normalisation is such that a core of mass 106M ⊙has a radius of 1pc.Higher normal-isations,corresponding to less dense cores,lead to the disruption of practically the whole original population ofembedded clusters.While,for a constant core radius of r c =1pc,the bell-shape of the ICMF is preserved when σεis increased from 0.00to 0.03,the virial mass-radius relation leads to a power-law ICMF for σε=0.03.That is,the sensitivity of the ICMF shape to the width of the distribution function for the SFE is greater than in case of a constant core radius.5.SUMMARYThe shape of the mass function of young star clusters remains much debated.Observational results are sorted into two distinct,often opposed,categories:power-law or bell-shaped mass functions.We have shown in this con-tribution that both shapes can arise from the same phys-ical process,namely,the supernova-driven gas expulsion from newly formed star clusters on a core mass dependent time-scale.Specifically,more massive cores have a higher gas removal time-scale to dynamical crossing-time ratio by virtue of their deeper potential wells (assuming the near constancy of core radii over the core mass range).For a given SFE,this results in larger bound fractions F bound of stars at the end of the violent relaxation which follows gas expulsion.The shape of the ICMF is chiefly governed by the probability distribution of the SFE ε.Bell-shaped ICMFs arise when most star forming cores have ε 0.25.In that case,however,bound gas-free star clusters only form out of cores more massive than 106M ⊙.This implies that bell-shaped ICMFs are also associated with gas-rich environments so as to guarantee that the core mass function is sampled to such a high mass.In contrast,power-law ICMFs mirroring the core mass function arise when ε∼0.4for most star forming cores.This implies that the shape of the ICMF also de-pends on the width σεof the SFE distribution.When ¯ε 0.25,an increasing σεenhances the contribution of cores with ε≃0.40and turns a bell-shaped ICMF into a power-law.The amplitude of that evolution is greater when core radii are assumed to follow the mass-radius relation of virialised cores than when the core radius is assumed to be constant.As illustrated by Figs.2and 3,that the mass function of young clusters observed in a variety of environments appears not to be universal is not surprising .Vesperini (1998)pointed out that a power-law ICMF of spectral index (-2)evolves to a bell-shaped mass func-tion the turnover of which is located at a significantly lower cluster mass than what is observed for the uni-versal globular cluster mass function.For such models,Parmentier &Gilmore (2005)showed that a power-law ICMF with α=−2also leads to a radial gradient in the peak of the evolved mass function contrary to what is observed.Baumgardt,Kroupa &Parmentier (2008)demonstrated that these two issues do not arise if the ICMF is depleted in low-mass clusters compared to a power-law of spectral index (-2)as a result of gas expul-sion.In that case,it evolves within 13Gyr into a cluster mass function similar to that of old globular clusters both in the inner and outer Galactic halo (see their fig.4).While a mean SFE of ¯ε 0.40preserves the shape of the core mass function,a mean SFE not exceeding ¯ε≃0.25leads to a low-mass cluster depleted ICMF (see Fig.2).The universality of the globular cluster mass function may thus stem from the existence of an upper limit of,say,25per cent on the mean SFE in all protogalaxies.6Parmentier et al.The origin of that constraint on the mean SFE in the protogalactic era remains to be explained,however.In this preliminary study,the energy input rate from massive stars encompasses the contribution made by su-pernovae only,i.e.that of stellar winds has been ne-glected.Consequently,the above results apply to the sole case of metal-poor environments.As noted in Section2, we have assumed that the effect of gas expulsion depends on the actual star formation efficiency-i.e.we assume that the stars and gas(potential)are in virial equilib-rium at the onset of gas expulsion.This assumption is reasonable if(the stellar component of)the cluster has had time to relax.A cluster will have been able to relax if the onset of gas expulsion occurs after a few crossing times.In a low-metallicity regime such as we are con-sidering in this paper,gas expulsion will be driven by supernovae rather than stellar winds and so will not be-gin until a few Myr after the stars have formed.Such a time-scale is several crossing times,and so we could ex-pect the eSFE to closely match the SFE.We note how-ever,that there may well be a mass-eSFE dependence in higher metallicity cluster populations that may alter our conclusions(which would be applicable to populations such as those in the Antennae merger of disc galaxies). In a follow-up paper,we will consider the additional impact of stellar winds and the case of star clusters form-ing out of dense environments where strong tidalfields prevail.GP acknowledges support from the Alexander von Humboldt Foundation in the form of a Research Fellow-ship and from the Belgian Science Policy Office in the form of a Return Grant.GP and SG are grateful for re-search support and hospitality at the International Space Science Institute in Bern(Switzerland),as part of an In-ternational Team Programme.We acknowledge partial financial support from the UK’s Royal Society through an International Joint Project grant aimed at facilitating networking activities between the universities of Sheffield and Bonn.REFERENCESAnders,P.,Bissantz,N.,Boysen,L.,de Grijs,R.,Fritze-v.Alvensleben,U.2007,MNRAS,377,91Baumgardt,H.,Kroupa,P.2007,MNRAS,380,1589 Baumgardt,H.,Kroupa,P.,Parmentier,G.,accepted in MNRAS, arXiv:0712.1591Bik,A.,Lamers,H.J.G.L.M.,Bastian,N.,Panagia,N.,Romaniello, M.2003,A&A,397,473Boily,C.M.&Kroupa,P.2003a,MNRAS,338,643Boily,C.M.&Kroupa,P.2003b,MNRAS,338,673Castor,J.,McCray,R.,Weaver,R.1975ApJ,200,107Cresci,G.,Vanzi,L.,Sauvage,M.2005,A&A,433,447 Elmegreen,B.G.1983,MNRAS,203,1011Elmegreen,B.G.&Clemens,C.1985,ApJ,294,523Fall,S.M.,Zhang,Q.2001,ApJ,561,751Geyer,M.P.,Burkert,A.2001,MNRAS,323,988Goodwin,S.P.1997a,MNRAS,284,785Goodwin,S.P.1997b,MNRAS,286,669Goodwin,S.P.,Bastian,N.2006,MNRAS,373,752Goodwin,S.P.2008,in:Young massive star clusters:Initial conditions and environments,Perez, E.,de Grijs R.,and Gonzalez Delgado,R.(eds),Springer,in pressHarris W.E.,Pudritz R.E.1994,ApJ,429,177Hills,J.,G.1980,ApJ,235,986Hunter,D.A.,Elmegreen,B.G.,Dupuy,T.J.,Mortonson,M. 2003,AJ,126,1836Johnson,K.E.2008,in:Young massive star clusters:Initial conditions and environments,Perez, E.,de Grijs R.,and Gonzalez Delgado,R.(eds),Springer,in press Kavelaars,J;J.,Hanes D.A.1997,MNRAS,285,L31Kroupa P.2001,MNRAS,322,231Kroupa,P.,Aarseth,S.,Hurley,J.,2001,MNRAS,321,699 Kroupa P.2002,MNRAS,330,707Kroupa,P.,Boily,C.M.2002,MNRAS,336,1188Kroupa,P.2005,In:Proceedings of”The Three-Dimensional Universe with Gaia”(ESA SP-576)C.Turon,K.S.O’Flaherty, M.A.C.Perryman(eds),p.629Lada,C.J.,Margulis,M.,Dearborn,D.1984,ApJ,285,141 Mathieu,R.D.1983,ApJ,267,97Miller,B..,Whitmore,B.C.,Schweizer,F.,Fall,S.M.1997,AJ, 114,2381Parmentier,G.,Gilmore,G.F.2005,MNRAS,363,326 Parmentier,G.,Gilmore,G.F.2007,MNRAS,377,352Pinto,F.1987,PASP,99,1161Spitzer,L.,1987,in:Dynamical evolution of globular clusters, Princeton University PressVerschueren,W.&David,M.1989,A&A,219,105 Verschueren,W.1990,A&A,234,156Vesperini E.1998,MNRAS,299,1019Weidner,C.,Kroupa,P.2004,MNRAS,348,187Whitmore,B.C.,Zhang Q.,Leitherer,C.,Fall,S.M.,Schweizer, F.,;Miller,B.W.,1999,AJ,118,1551。

a rXiv:as tr o-ph/98528v115May1998The evolution of clustering and bias in the galaxy distribution B y J.A.Peacock Institute for Astronomy,Royal Observatory,Edinburgh EH93HJ,UK This paper reviews the measurements of galaxy correlations at high redshifts,and discusses how these may be understood in models of hierarchical gravita-tional collapse.The clustering of galaxies at redshift one is much weaker than at present,and this is consistent with the rate of growth of structure expected in an open universe.If Ω=1,this observation would imply that bias increases at high redshift,in conflict with observed M/L values for known high-z clusters.At redshift 3,the population of Lyman-limit galaxies displays clustering which is of similar amplitude to that seen today.This is most naturally understood if the Lyman-limit population is a set of rare recently-formed objects.Knowing both the clustering and the abundance of these objects,it is possible to deduce em-pirically the fluctuation spectrum required on scales which cannot be measured today owing to gravitational nonlinearities.Of existing physical models for the fluctuation spectrum,the results are most closely matched by a low-density spa-tially flat universe.This conclusion is reinforced by an empirical analysis of CMB anisotropies,in which the present-day fluctuation spectrum is forced to have the observed form.Open models are strongly disfavoured,leaving ΛCDM as the most successful simple model for structure formation.2J.A.Peacockcommon parameterization for the correlation function in comoving coordinates:ξ(r,z)=[r/r0]−γ(1+z)−(3−γ+ǫ),(1.2) whereǫ=0is stable clustering;ǫ=γ−3is constant comoving clustering;ǫ=γ−1isΩ=1linear-theory evolution.Although this equation is frequently encountered,it is probably not appli-cable to the real world,because most data inhabit the intermediate regime of 1<∼ξ<∼100.Peacock(1997)showed that the expected evolution in this quasilin-ear regime is significantly more rapid:up toǫ≃3.(b)General aspects of biasOf course,there are good reasons to expect that the galaxy distribution will not follow that of the dark matter.The main empirical argument in this direction comes from the masses of rich clusters of galaxies.It has long been known that attempts to‘weigh’the universe by multiplying the overall luminosity density by cluster M/L ratios give apparent density parameters in the rangeΩ≃0.2to0.3 (e.g.Carlberg et al.1996).An alternative argument is to use the abundance of rich clusters of galaxies in order to infer the rms fractional density contrast in spheres of radius8h−1Mpc. This calculation has been carried out several different ways,with general agree-ment on afigure close to(1.3)σ8≃0.57Ω−0.56m(White,Efstathiou&Frenk1993;Eke,Cole&Frenk1996;Viana&Liddle1996). The observed apparent value ofσ8in,for example,APM galaxies(Maddox,Efs-tathiou&Sutherland1996)is about0.95(ignoring nonlinear corrections,which are small in practice,although this is not obvious in advance).This says that Ω=1needs substantial positive bias,but thatΩ<∼0.4needs anti bias.Although this cluster normalization argument depends on the assumption that the density field obeys Gaussian statistics,the result is in reasonable agreement with what is inferred from cluster M/L ratios.What effect does bias have on common statistical measures of clustering such as correlation functions?We could be perverse and assume that the mass and lightfields are completely unrelated.If however we are prepared to make the more sensible assumption that the light density is a nonlinear but local function of the mass density,then there is a very nice result due to Coles(1993):the bias is a monotonic function of scale.Explicitly,if scale-dependent bias is defined asb(r)≡[ξgalaxy(r)/ξmass(r)]1/2,(1.4) then b(r)varies monotonically with scale under rather general assumptions about the densityfield.Furthermore,at large r,the bias will tend to a constant value which is the linear response of the galaxy-formation process.There is certainly empirical evidence that bias in the real universe does work this way.Consider Fig.1,taken from Peacock(1997).This compares dimen-sionless power spectra(∆2(k)=dσ2/d ln k)for IRAS and APM galaxies.The comparison is made in real space,so as to avoid distortions due to peculiar veloc-ities.For IRAS galaxies,the real-space power was obtained from the the projectedThe evolution of galaxy clustering and bias3Figure1.The real-space power spectra of optically-selected APM galaxies(solid circles)and IRAS galaxies(open circles),taken from Peacock(1997).IRAS galaxies show weaker clustering, consistent with their suppression in high-density regions relative to optical galaxies.The relative bias is a monotonic but slowly-varying function of scale.correlation function:Ξ(r)= ∞−∞ξ[(r2+x2)1/2]dx.(1.5)Saunders,Rowan-Robinson&Lawrence(1992)describe how this statistic can be converted to other measures of real-space correlation.For the APM galaxies, Baugh&Efstathiou(1993;1994)deprojected Limber’s equation for the angular correlation function w(θ)(discussed below).These different methods yield rather similar power spectra,with a relative bias that is perhaps only about1.2on large scale,increasing to about1.5on small scales.The power-law portion for k>∼0.2h Mpc−1is the clear signature of nonlinear gravitational evolution,and the slow scale-dependence of bias gives encouragement that the galaxy correlations give a good measure of the shape of the underlying massfluctuation spectrum.2.Observations of high-redshift clustering(a)Clustering at redshift1At z=0,there is a degeneracy betweenΩand the true normalization of the spectrum.Since the evolution of clustering with redshift depends onΩ,studies at higher redshifts should be capable of breaking this degeneracy.This can be done without using a complete faint redshift survey,by using the angular clustering of aflux-limited survey.If the form of the redshift distribution is known,the projection effects can be disentangled in order to estimate the3D clustering at the average redshift of the sample.For small angles,and where the redshift shell being studied is thicker than the scale of any clustering,the spatial and angular4J.A.Peacockcorrelation functions are related by Limber’s equation(e.g.Peebles1980): w(θ)= ∞0y4φ2(y)C(y)dy ∞−∞ξ([x2+y2θ2]1/2,z)dx,(2.1)where y is dimensionless comoving distance(transverse part of the FRW metric is[R(t)y dθ]2),and C(y)=[1−ky2]−1/2;the selection function for radius y is normalized so that y2φ(y)C(y)dy=1.Less well known,but simpler,is the Fourier analogue of this relation:π∆2θ(K)=The evolution of galaxy clustering and bias5 ever,the M/L argument is more powerful since only a single cluster is required, and a complete survey is not necessary.Two particularly good candidates at z≃0.8are described by Clowe et al.(1998);these are clusters where significant weak gravitational-lensing distortions are seen,allowing a robust determination of the total cluster mass.The mean V-band M/L in these clusters is230Solar units,which is close to typical values in z=0clusters.However,the comoving V-band luminosity density of the universe is higher at early times than at present by about a factor(1+z)2.5(Lilly et al.1996),so this is equivalent to M/L≃1000, implying an apparent‘Ω’of close to unity.In summary,the known degree of bias today coupled with the moderate evolution in correlation function back to z=1 implies that,forΩ=1,the galaxy distribution at this time would have to consist very nearly of a‘painted-on’pattern that is not accompanied by significant mass fluctuations.Such a picture cannot be reconciled with the healthy M/L ratios that are observed in real clusters at these redshifts,and this seems to be a strong argument that we do not live in an Einstein-de Sitter universe.(b)Clustering of Lyman-limit galaxies at redshift3The most exciting recent development in observational studies of galaxy clus-tering is the detection by Steidel et al.(1997)of strong clustering in the popula-tion of Lyman-limit galaxies at z≃3.The evidence takes the form of a redshift histogram binned at∆z=0.04resolution over afield8.7′×17.6′in extent.For Ω=1and z=3,this probes the densityfield using a cell with dimensionscell=15.4×7.6×15.0[h−1Mpc]3.(2.3) Conveniently,this has a volume equivalent to a sphere of radius7.5h−1Mpc,so it is easy to measure the bias directly by reference to the known value ofσ8.Since the degree of bias is large,redshift-space distortions from coherent infall are small; the cell is also large enough that the distortions of small-scale random velocities at the few hundred km s−1level are also ing the model of equation (11)of Peacock(1997)for the anisotropic redshift-space power spectrum and integrating over the exact anisotropic window function,the above simple volume argument is found to be accurate to a few per cent for reasonable power spectra:σcell≃b(z=3)σ7.5(z=3),(2.4) defining the bias factor at this scale.The results of section1(see also Mo& White1996)suggest that the scale-dependence of bias should be weak.In order to estimateσcell,simulations of synthetic redshift histograms were made,using the method of Poisson-sampled lognormal realizations described by Broadhurst,Taylor&Peacock(1995):using aχ2statistic to quantify the nonuni-formity of the redshift histogram,it appears thatσcell≃0.9is required in order for thefield of Steidel et al.(1997)to be typical.It is then straightforward to ob-tain the bias parameter since,for a present-day correlation functionξ(r)∝r−1.8,σ7.5(z=3)=σ8×[8/7.5]1.8/2×1/4≃0.146,(2.5) implyingb(z=3|Ω=1)≃0.9/0.146≃6.2.(2.6) Steidel et al.(1997)use a rather different analysis which concentrates on the highest peak alone,and obtain a minimum bias of6,with a preferred value of8.6J.A.PeacockThey use the Eke et al.(1996)value ofσ8=0.52,which is on the low side of the published range of ingσ8=0.55would lower their preferred b to 7.6.Note that,with both these methods,it is much easier to rule out a low value of b than a high one;given a singlefield,it is possible that a relatively‘quiet’region of space has been sampled,and that much larger spikes remain to be found elsewhere.A more detailed analysis of several furtherfields by Adelberger et al. (1998)in fact yields a biasfigure very close to that given above,so thefirstfield was apparently not unrepresentative.Having arrived at afigure for bias ifΩ=1,it is easy to translate to other models,sinceσcell is observed,independent of cosmology.For lowΩmodels, the cell volume will increase by a factor[S2k(r)dr]/[S2k(r1)dr1];comparing with present-dayfluctuations on this larger scale will tend to increase the bias.How-ever,for lowΩ,two other effects increase the predicted densityfluctuation at z=3:the cluster constraint increases the present-dayfluctuation by a factor Ω−0.56,and the growth between redshift3and the present will be less than a factor of4.Applying these corrections givesb(z=3|Ω=0.3)The evolution of galaxy clustering and bias7 87GB survey(Loan,Lahav&Wall1997),but these were of only bare significance (although,in retrospect,the level of clustering in87GB is consistent with the FIRST measurement).Discussion of the87GB and FIRST results in terms of Limber’s equation has tended to focus on values ofǫin the region of0.Cress et al.(1996)concluded that the w(θ)results were consistent with the PN91 value of r0≃10h−1Mpc(although they were not very specific aboutǫ).Loan et al.(1997)measured w(1◦)≃0.005for a5-GHz limit of50mJy,and inferred r0≃12h−1Mpc forǫ=0,falling to r0≃9h−1Mpc forǫ=−1.The reason for this strong degeneracy between r0andǫis that r0parame-terizes the z=0clustering,whereas the observations refer to a typical redshift of around unity.This means that r0(z=1)can be inferred quite robustly to be about7.5h−1Mpc,without much dependence on the rate of evolution.Since the strength of clustering for optical galaxies at z=1is known to correspond to the much smaller number of r0≃2h−1Mpc(e.g.Le F`e vre et al.1996),we see that radio galaxies at this redshift have a relative bias parameter of close to 3.The explanation for this high degree of bias is probably similar to that which applies in the case of QSOs:in both cases we are dealing with AGN hosted by rare massive galaxies.3.Formation and bias of high-redshift galaxiesThe challenge now is to ask how these results can be understood in cur-rent models for cosmological structure formation.It is widely believed that the sequence of cosmological structure formation was hierarchical,originating in a density power spectrum with increasingfluctuations on small scales.The large-wavelength portion of this spectrum is accessible to observation today through studies of galaxy clustering in the linear and quasilinear regimes.However,non-linear evolution has effectively erased any information on the initial spectrum for wavelengths below about1Mpc.The most sensitive way of measuring the spectrum on smaller scales is via the abundances of high-redshift objects;the amplitude offluctuations on scales of individual galaxies governs the redshift at which these objectsfirst undergo gravitational collapse.The small-scale am-plitude also influences clustering,since rare early-forming objects are strongly correlated,asfirst realized by Kaiser(1984).It is therefore possible to use obser-vations of the abundances and clustering of high-redshift galaxies to estimate the power spectrum on small scales,and the following section summarizes the results of this exercise,as given by Peacock et al.(1998).(a)Press-Schechter apparatusThe standard framework for interpreting the abundances of high-redshift objects in terms of structure-formation models,was outlined by Efstathiou& Rees(1988).The formalism of Press&Schechter(1974)gives a way of calculating the fraction F c of the mass in the universe which has collapsed into objects more massive than some limit M:F c(>M,z)=1−erf δc2σ(M) .(3.1)8J.A.PeacockHere,σ(M)is the rms fractional density contrast obtained byfiltering the linear-theory densityfield on the required scale.In practice,thisfiltering is usually performed with a spherical‘top hat’filter of radius R,with a corresponding mass of4πρb R3/3,whereρb is the background density.The numberδc is the linear-theory critical overdensity,which for a‘top-hat’overdensity undergoing spherical collapse is1.686–virtually independent ofΩ.This form describes numerical simulations very well(see e.g.Ma&Bertschinger1994).The main assumption is that the densityfield obeys Gaussian statistics,which is true in most inflationary models.Given some estimate of F c,the numberσ(R)can then be inferred.Note that for rare objects this is a pleasingly robust process:a large error in F c will give only a small error inσ(R),because the abundance is exponentially sensitive toσ.Total masses are of course ill-defined,and a better quantity to use is the velocity dispersion.Virial equilibrium for a halo of mass M and proper radius r demands a circular orbital velocity ofV2c=GMΩ1/2m(1+z c)1/2f 1/6c.(3.3)Here,z c is the redshift of virialization;Ωm is the present value of the matter density parameter;f c is the density contrast at virialization of the newly-collapsed object relative to the background,which is adequately approximated byf c=178/Ω0.6m(z c),(3.4) with only a slight sensitivity to whetherΛis non-zero(Eke,Cole&Frenk1996).For isothermal-sphere haloes,the velocity dispersion isσv=V c/√The evolution of galaxy clustering and bias9 and the more recent estimate of0.025from Tytler et al.(1996),thenΩHIF c=2for the dark halo.A more recent measurement of the velocity width of the Hαemission line in one of these objects gives a dispersion of closer to100km s−1(Pettini,private communication),consistent with the median velocity width for Lyαof140km s−1 measured in similar galaxies in the HDF(Lowenthal et al.1997).Of course,these figures could underestimate the total velocity dispersion,since they are dominated by emission from the central regions only.For the present,the range of values σv=100to320km s−1will be adopted,and the sensitivity to the assumed velocity will be indicated.In practice,this uncertainty in the velocity does not produce an important uncertainty in the conclusions.(3)Red radio galaxies An especially interesting set of objects are the reddest optical identifications of1-mJy radio galaxies,for which deep absorption-line spectroscopy has proved that the red colours result from a well-evolved stellar population,with a minimum stellar age of3.5Gyr for53W091at z=1.55(Dun-10J.A.Peacocklop et al.1996;Spinrad et al.1997),and4.0Gyr for53W069at z=1.43(Dunlop 1998;Dey et al.1998).Such ages push the formation era for these galaxies back to extremely high redshifts,and it is of interest to ask what level of small-scale power is needed in order to allow this early formation.Two extremely red galaxies were found at z=1.43and1.55,over an area 1.68×10−3sr,so a minimal comoving density is from one galaxy in this redshift range:N(Ω=1)>∼10−5.87(h−1Mpc)−3.(3.9) Thisfigure is comparable to the density of the richest Abell clusters,and is thus in reasonable agreement with the discovery that rich high-redshift clusters appear to contain radio-quiet examples of similarly red galaxies(Dickinson1995).Since the velocity dispersions of these galaxies are not observed,they must be inferred indirectly.This is possible because of the known present-day Faber-Jackson relation for ellipticals.For53W091,the large-aperture absolute magni-tude isM V(z=1.55|Ω=1)≃−21.62−5log10h(3.10) (measured direct in the rest frame).According to Solar-metallicity spectral syn-thesis models,this would be expected to fade by about0.9mag.between z=1.55 and the present,for anΩ=1model of present age14Gyr(note that Bender et al.1996have observed a shift in the zero-point of the M−σv relation out to z=0.37of a consistent size).If we compare these numbers with theσv–M V relation for Coma(m−M=34.3for h=1)taken from Dressler(1984),this predicts velocity dispersions in the rangeσv=222to292km s−1.(3.11) This is a very reasonable range for a giant elliptical,and it adopted in the following analysis.Having established an abundance and an equivalent circular velocity for these galaxies,the treatment of them will differ in one critical way from the Lyman-αand Lyman-limit galaxies.For these,the normal Press-Schechter approach as-sumes the systems under study to be newly born.For the Lyman-αand Lyman-limit galaxies,this may not be a bad approximation,since they are evolving rapidly and/or display high levels of star-formation activity.For the radio galax-ies,conversely,their inactivity suggests that they may have existed as discrete systems at redshifts much higher than z≃1.5.The strategy will therefore be to apply the Press-Schechter machinery at some unknown formation redshift,and see what range of redshift gives a consistent degree of inhomogeneity.4.The small-scalefluctuation spectrum(a)The empirical spectrumFig.2shows theσ(R)data which result from the Press-Schechter analysis, for three cosmologies.Theσ(R)numbers measured at various high redshifts have been translated to z=0using the appropriate linear growth law for density perturbations.The open symbols give the results for the Lyman-limit(largest R)and Lyman-α(smallest R)systems.The approximately horizontal error bars showThe evolution of galaxy clustering and bias11Figure2.Theradius R.Thecircles)Theredshifts2,4,...The horizontal errors correspond to different choices for the circular velocities of the dark-matter haloes that host the galaxies.The shaded region at large R gives the results inferred from galaxy clustering.The lines show CDM and MDM predictions,with a large-scale normalization ofσ8=0.55forΩ=1orσ8=1for the low-density models.the effect of the quoted range of velocity dispersions for afixed abundance;the vertical errors show the effect of changing the abundance by a factor2atfixed velocity dispersion.The locus implied by the red radio galaxies sits in between. The different points show the effects of varying collapse redshift:z c=2,4,...,12 [lowest redshift gives lowestσ(R)].Clearly,collapse redshifts of6–8are favoured12J.A.Peacockfor consistency with the other data on high-redshift galaxies,independent of the-oretical preconceptions and independent of the age of these galaxies.This level of power(σ[R]≃2for R≃1h−1Mpc)is also in very close agreement with the level of power required to produce the observed structure in the Lyman alpha forest(Croft et al.1998),so there is a good case to be made that thefluctu-ation spectrum has now been measured in a consistent fashion down to below R≃1h−1Mpc.The shaded region at larger R shows the results deduced from clustering data (Peacock1997).It is clear anΩ=1universe requires the power spectrum at small scales to be higher than would be expected on the basis of an extrapolation from the large-scale spectrum.Depending on assumptions about the scale-dependence of bias,such a‘feature’in the linear spectrum may also be required in order to satisfy the small-scale present-day nonlinear galaxy clustering(Peacock1997). Conversely,for low-density models,the empirical small-scale spectrum appears to match reasonably smoothly onto the large-scale data.Fig.2also compares the empirical data with various physical power spectra.A CDM model(using the transfer function of Bardeen et al.1986)with shape parameterΓ=Ωh=0.25is shown as a reference for all models.This appears to have approximately the correct shape,although it overpredicts the level of small-scale power somewhat in the low-density cases.A better empirical shape is given by MDM withΩh≃0.4andΩν≃0.3.However,this model only makes physical sense in a universe with highΩ,and so it is only shown as the lowest curve in Fig.2c,reproduced from thefitting formula of Pogosyan&Starobinsky(1995; see also Ma1996).This curve fails to supply the required small-scale power,by about a factor3inσ;loweringΩνto0.2still leaves a very large discrepancy. This conclusion is in agreement with e.g.Mo&Miralda-Escud´e(1994),Ma& Bertschinger(1994),Ma et al.(1997)and Gardner et al.(1997).All the models in Fig.2assume n=1;in fact,consistency with the COBE results for this choice ofσ8andΩh requires a significant tilt forflat low-density CDM models,n≃0.9(whereas open CDM models require n substantially above unity).Over the range of scales probed by LSS,changes in n are largely degenerate with changes inΩh,but the small-scale power is more sensitive to tilt than to Ωh.Tilting theΩ=1models is not attractive,since it increases the tendency for model predictions to lie below the data.However,a tilted low-Ωflat CDM model would agree moderately well with the data on all scales,with the exception of the ‘bump’around R≃30h−1Mpc.Testing the reality of this feature will therefore be an important task for future generations of redshift survey.(b)Collapse redshifts and ages for red radio galaxiesAre the collapse redshifts inferred above consistent with the age data on the red radio galaxies?First bear in mind that in a hierarchy some of the stars in a galaxy will inevitably form in sub-units before the epoch of collapse.At the time offinal collapse,the typical stellar age will be some fractionαof the age of the universe at that time:age=t(z obs)−t(z c)+αt(z c).(4.1) We can rule outα=1(i.e.all stars forming in small subunits just after the big bang).For present-day ellipticals,the tight colour-magnitude relation only allows an approximate doubling of the mass through mergers since the termination ofThe evolution of galaxy clustering and bias13Figure3.The age of a galaxy at z=1.5,as a function of its collapse redshift(assuming an instantaneous burst of star formation).The various lines showΩ=1[solid];openΩ=0.3 [dotted];flatΩ=0.3[dashed].In all cases,the present age of the universe is forced to be14 Gyr.star formation(Bower at al.1992).This corresponds toα≃0.3(Peacock1991).A non-zeroαjust corresponds to scaling the collapse redshift asapparent(1+z c)∝(1−α)−2/3,(4.2) since t∝(1+z)−3/2at high redshifts for all cosmologies.For example,a galaxy which collapsed at z=6would have an apparent age corresponding to a collapse redshift of7.9forα=0.3.Converting the ages for the galaxies to an apparent collapse redshift depends on the cosmological model,but particularly on H0.Some of this uncertainty may be circumvented byfixing the age of the universe.After all,it is of no interest to ask about formation redshifts in a model with e.g.Ω=1,h=0.7when the whole universe then has an age of only9.5Gyr.IfΩ=1is to be tenable then either h<0.5against all the evidence or there must be an error in the stellar evolution timescale.If the stellar timescales are wrong by afixed factor,then these two possibilities are degenerate.It therefore makes sense to measure galaxy ages only in units of the age of the universe–or,equivalently,to choose freely an apparent Hubble constant which gives the universe an age comparable to that inferred for globular clusters.In this spirit,Fig.3gives apparent ages as a function of effective collapse redshift for models in which the age of the universe is forced to be14 Gyr(e.g.Jimenez et al.1996).This plot shows that the ages of the red radio galaxies are not permitted very much freedom.Formation redshifts in the range6to8predict an age of close to 3.0Gyr forΩ=1,or3.7Gyr for low-density models,irrespective of whetherΛis nonzero.The age-z c relation is ratherflat,and this gives a robust estimate of age once we have some idea of z c through the abundance arguments.It is therefore14J.A.Peacockrather satisfying that the ages inferred from matching the rest-frame UV spectra of these galaxies are close to the abovefigures.(c)The global picture of galaxy formationIt is interesting to note that it has been possible to construct a consistent picture which incorporates both the large numbers of star-forming galaxies at z<∼3and the existence of old systems which must have formed at very much larger redshifts.A recent conclusion from the numbers of Lyman-limit galaxies and the star-formation rates seen at z≃1has been that the global history of star formation peaked at z≃2(Madau et al.1996).This leaves open two possibilities for the very old systems:either they are the rare precursors of this process,and form unusually early,or they are a relic of a second peak in activity at higher redshift,such as is commonly invoked for the origin of all spheroidal components. While such a bimodal history of star formation cannot be rejected,the rareness of the red radio galaxies indicates that there is no difficulty with the former picture. This can be demonstrated quantitatively by integrating the total amount of star formation at high redshift.According to Madau et al.,The star-formation rate at z=4is˙ρ∗≃107.3h M⊙Gyr−1Mpc−3,(4.3) declining roughly as(1+z)−4.This is probably a underestimate by a factor of at least3,as indicated by suggestions of dust in the Lyman-limit galaxies(Pettini et al.1997),and by the prediction of Pei&Fall(1995),based on high-z element abundances.If we scale by a factor3,and integrate tofind the total density in stars produced at z>6,this yieldsρ∗(z f>6)≃106.2M⊙Mpc−3.(4.4) Since the red mJy galaxies have a density of10−5.87h3Mpc−3and stellar masses of order1011M⊙,there is clearly no conflict with the idea that these galaxies are thefirst stellar systems of L∗size which form en route to the general era of star and galaxy formation.(d)Predictions for biased clustering at high redshiftsAn interesting aspect of these results is that the level of power on1-Mpc scales is only moderate:σ(1h−1Mpc)≃2.At z≃3,the correspondingfigure would have been much lower,making systems like the Lyman-limit galaxies rather rare.For Gaussianfluctuations,as assumed in the Press-Schechter analysis,such systems will be expected to display spatial correlations which are strongly biased with respect to the underlying mass.The linear bias parameter depends on the rareness of thefluctuation and the rms of the underlyingfield asb=1+ν2−1δc(4.5)(Kaiser1984;Cole&Kaiser1989;Mo&White1996),whereν=δc/σ,andσ2is the fractional mass variance at the redshift of interest.In this analysis,δc=1.686is assumed.Variations in this number of order10 per cent have been suggested by authors who have studied thefit of the Press-Schechter model to numerical data.These changes would merely scale b−1by a small amount;the key parameter isν,which is set entirely by the collapsed。

华南农业大学学报 Journal of South China Agricultural University 2023, 44(4): 628-637DOI: 10.7671/j.issn.1001-411X.202208008赵润茂, 朱政, 陈建能, 等. 3D LiDAR感知的植物行信息提取方法与试验[J]. 华南农业大学学报, 2023, 44(4): 628-637.ZHAO Runmao, ZHU Zheng, CHEN Jianneng, et al. 3D LiDAR sensing method and experiment of plant row information extraction[J]. Journal of South China Agricultural University, 2023, 44(4): 628-637.3D LiDAR 感知的植物行信息提取方法与试验赵润茂1,2 ，朱　政1，陈建能1,2 ，范国帅1，王麒程1，黄培奎3(1 浙江理工大学 机械与自动控制学院, 浙江 杭州 310018； 2 浙江省种植装备技术重点实验室,浙江 杭州 310018； 3 华南农业大学 工程学院, 广东 广州 510642)摘要: 【目的】针对林间或冠层下等卫星信号严重遮挡的区域，提出一种面向农业机器人导航环境感知的低成本3D激光雷达(LiDAR)点云信息处理与植物行估计方法。

【方法】利用直通滤波器滤除感兴趣区域外的目标无关点；提出均值漂移聚类、扫描区域自适应的方法分割每棵植物主干，垂直投影主干点云估算中心点；利用最小二乘法拟合主干中心，估计植物行。

分别在开阔地的仿真果园与水杉树林进行模拟试验与田间试验，以植物行向量与正东方夹角为指标，计算本研究提出的方法识别的植物行信息与GNSS卫星天线定位测得的植物行真值间的角度误差。

【结果】采用提出的3D LiDAR点云信息处理与植物行估计方法，模拟试验和田间试验对植物行识别误差平均值分别为0.79°和1.48°，最小值分别为0.12°和0.88°，最大值分别为1.49°和2.33°。

8 Cluster Analysis:Basic Concepts andAlgorithmsCluster analysis divides data into groups(clusters)that are meaningful,useful, or both.If meaningful groups are the goal,then the clusters should capture the natural structure of the data.In some cases,however,cluster analysis is only a useful starting point for other purposes,such as data summarization.Whether for understanding or utility,cluster analysis has long played an important role in a wide variety offields:psychology and other social sciences,biology, statistics,pattern recognition,information retrieval,machine learning,and data mining.There have been many applications of cluster analysis to practical prob-lems.We provide some specific examples,organized by whether the purpose of the clustering is understanding or utility.Clustering for Understanding Classes,or conceptually meaningful groups of objects that share common characteristics,play an important role in how people analyze and describe the world.Indeed,human beings are skilled at dividing objects into groups(clustering)and assigning particular objects to these groups(classification).For example,even relatively young children can quickly label the objects in a photograph as buildings,vehicles,people,ani-mals,plants,etc.In the context of understanding data,clusters are potential classes and cluster analysis is the study of techniques for automaticallyfinding classes.The following are some examples:488Chapter8Cluster Analysis:Basic Concepts and Algorithms •Biology.Biologists have spent many years creating a taxonomy(hi-erarchical classification)of all living things:kingdom,phylum,class, order,family,genus,and species.Thus,it is perhaps not surprising that much of the early work in cluster analysis sought to create a discipline of mathematical taxonomy that could automaticallyfind such classifi-cation structures.More recently,biologists have applied clustering to analyze the large amounts of genetic information that are now available.For example,clustering has been used tofind groups of genes that have similar functions.•Information Retrieval.The World Wide Web consists of billions of Web pages,and the results of a query to a search engine can return thousands of pages.Clustering can be used to group these search re-sults into a small number of clusters,each of which captures a particular aspect of the query.For instance,a query of“movie”might return Web pages grouped into categories such as reviews,trailers,stars,and theaters.Each category(cluster)can be broken into subcategories(sub-clusters),producing a hierarchical structure that further assists a user’s exploration of the query results.•Climate.Understanding the Earth’s climate requiresfinding patterns in the atmosphere and ocean.To that end,cluster analysis has been applied tofind patterns in the atmospheric pressure of polar regions and areas of the ocean that have a significant impact on land climate.•Psychology and Medicine.An illness or condition frequently has a number of variations,and cluster analysis can be used to identify these different subcategories.For example,clustering has been used to identify different types of depression.Cluster analysis can also be used to detect patterns in the spatial or temporal distribution of a disease.•Business.Businesses collect large amounts of information on current and potential customers.Clustering can be used to segment customers into a small number of groups for additional analysis and marketing activities.Clustering for Utility Cluster analysis provides an abstraction from in-dividual data objects to the clusters in which those data objects reside.Ad-ditionally,some clustering techniques characterize each cluster in terms of a cluster prototype;i.e.,a data object that is representative of the other ob-jects in the cluster.These cluster prototypes can be used as the basis for a489 number of data analysis or data processing techniques.Therefore,in the con-text of utility,cluster analysis is the study of techniques forfinding the most representative cluster prototypes.•Summarization.Many data analysis techniques,such as regression or PCA,have a time or space complexity of O(m2)or higher(where m is the number of objects),and thus,are not practical for large data sets.However,instead of applying the algorithm to the entire data set,it can be applied to a reduced data set consisting only of cluster prototypes.Depending on the type of analysis,the number of prototypes,and the accuracy with which the prototypes represent the data,the results can be comparable to those that would have been obtained if all the data could have been used.•Compression.Cluster prototypes can also be used for data compres-sion.In particular,a table is created that consists of the prototypes for each cluster;i.e.,each prototype is assigned an integer value that is its position(index)in the table.Each object is represented by the index of the prototype associated with its cluster.This type of compression is known as vector quantization and is often applied to image,sound, and video data,where(1)many of the data objects are highly similar to one another,(2)some loss of information is acceptable,and(3)a substantial reduction in the data size is desired.•Efficiently Finding Nearest Neighbors.Finding nearest neighbors can require computing the pairwise distance between all points.Often clusters and their cluster prototypes can be found much more efficiently.If objects are relatively close to the prototype of their cluster,then we can use the prototypes to reduce the number of distance computations that are necessary tofind the nearest neighbors of an object.Intuitively,if two cluster prototypes are far apart,then the objects in the corresponding clusters cannot be nearest neighbors of each other.Consequently,to find an object’s nearest neighbors it is only necessary to compute the distance to objects in nearby clusters,where the nearness of two clusters is measured by the distance between their prototypes.This idea is made more precise in Exercise25on page94.This chapter provides an introduction to cluster analysis.We begin with a high-level overview of clustering,including a discussion of the various ap-proaches to dividing objects into sets of clusters and the different types of clusters.We then describe three specific clustering techniques that represent490Chapter8Cluster Analysis:Basic Concepts and Algorithms broad categories of algorithms and illustrate a variety of concepts:K-means, agglomerative hierarchical clustering,and DBSCAN.Thefinal section of this chapter is devoted to cluster validity—methods for evaluating the goodness of the clusters produced by a clustering algorithm.More advanced clustering concepts and algorithms will be discussed in Chapter9.Whenever possible, we discuss the strengths and weaknesses of different schemes.In addition, the bibliographic notes provide references to relevant books and papers that explore cluster analysis in greater depth.8.1OverviewBefore discussing specific clustering techniques,we provide some necessary background.First,we further define cluster analysis,illustrating why it is difficult and explaining its relationship to other techniques that group data. Then we explore two important topics:(1)different ways to group a set of objects into a set of clusters,and(2)types of clusters.8.1.1What Is Cluster Analysis?Cluster analysis groups data objects based only on information found in the data that describes the objects and their relationships.The goal is that the objects within a group be similar(or related)to one another and different from (or unrelated to)the objects in other groups.The greater the similarity(or homogeneity)within a group and the greater the difference between groups, the better or more distinct the clustering.In many applications,the notion of a cluster is not well defined.To better understand the difficulty of deciding what constitutes a cluster,consider Figure 8.1,which shows twenty points and three different ways of dividing them into clusters.The shapes of the markers indicate cluster membership.Figures 8.1(b)and8.1(d)divide the data into two and six parts,respectively.However, the apparent division of each of the two larger clusters into three subclusters may simply be an artifact of the human visual system.Also,it may not be unreasonable to say that the points form four clusters,as shown in Figure 8.1(c).Thisfigure illustrates that the definition of a cluster is imprecise and that the best definition depends on the nature of data and the desired results.Cluster analysis is related to other techniques that are used to divide data objects into groups.For instance,clustering can be regarded as a form of classification in that it creates a labeling of objects with class(cluster)labels. However,it derives these labels only from the data.In contrast,classification8.1Overview491(a)Original points.(b)Two clusters.(c)Four clusters.(d)Six clusters.Figure8.1.Different ways of clustering the same set of points.in the sense of Chapter4is supervised classification;i.e.,new,unlabeled objects are assigned a class label using a model developed from objects with known class labels.For this reason,cluster analysis is sometimes referred to as unsupervised classification.When the term classification is used without any qualification within data mining,it typically refers to supervised classification.Also,while the terms segmentation and partitioning are sometimes used as synonyms for clustering,these terms are frequently used for approaches outside the traditional bounds of cluster analysis.For example,the term partitioning is often used in connection with techniques that divide graphs into subgraphs and that are not strongly connected to clustering.Segmentation often refers to the division of data into groups using simple techniques;e.g., an image can be split into segments based only on pixel intensity and color,or people can be divided into groups based on their income.Nonetheless,some work in graph partitioning and in image and market segmentation is related to cluster analysis.8.1.2Different Types of ClusteringsAn entire collection of clusters is commonly referred to as a clustering,and in this section,we distinguish various types of clusterings:hierarchical(nested) versus partitional(unnested),exclusive versus overlapping versus fuzzy,and complete versus partial.Hierarchical versus Partitional The most commonly discussed distinc-tion among different types of clusterings is whether the set of clusters is nested492Chapter8Cluster Analysis:Basic Concepts and Algorithmsor unnested,or in more traditional terminology,hierarchical or partitional.A partitional clustering is simply a division of the set of data objects into non-overlapping subsets(clusters)such that each data object is in exactly one subset.Taken individually,each collection of clusters in Figures8.1(b–d)is a partitional clustering.If we permit clusters to have subclusters,then we obtain a hierarchical clustering,which is a set of nested clusters that are organized as a tree.Each node(cluster)in the tree(except for the leaf nodes)is the union of its children (subclusters),and the root of the tree is the cluster containing all the objects. Often,but not always,the leaves of the tree are singleton clusters of individual data objects.If we allow clusters to be nested,then one interpretation of Figure8.1(a)is that it has two subclusters(Figure8.1(b)),each of which,in turn,has three subclusters(Figure8.1(d)).The clusters shown in Figures8.1 (a–d),when taken in that order,also form a hierarchical(nested)clustering with,respectively,1,2,4,and6clusters on each level.Finally,note that a hierarchical clustering can be viewed as a sequence of partitional clusterings and a partitional clustering can be obtained by taking any member of that sequence;i.e.,by cutting the hierarchical tree at a particular level. Exclusive versus Overlapping versus Fuzzy The clusterings shown in Figure8.1are all exclusive,as they assign each object to a single cluster. There are many situations in which a point could reasonably be placed in more than one cluster,and these situations are better addressed by non-exclusive clustering.In the most general sense,an overlapping or non-exclusive clustering is used to reflect the fact that an object can simultaneously belong to more than one group(class).For instance,a person at a university can be both an enrolled student and an employee of the university.A non-exclusive clustering is also often used when,for example,an object is“between”two or more clusters and could reasonably be assigned to any of these clusters. Imagine a point halfway between two of the clusters of Figure8.1.Rather than make a somewhat arbitrary assignment of the object to a single cluster, it is placed in all of the“equally good”clusters.In a fuzzy clustering,every object belongs to every cluster with a mem-bership weight that is between0(absolutely doesn’t belong)and1(absolutely belongs).In other words,clusters are treated as fuzzy sets.(Mathematically, a fuzzy set is one in which an object belongs to any set with a weight that is between0and1.In fuzzy clustering,we often impose the additional con-straint that the sum of the weights for each object must equal1.)Similarly, probabilistic clustering techniques compute the probability with which each8.1Overview493 point belongs to each cluster,and these probabilities must also sum to1.Be-cause the membership weights or probabilities for any object sum to1,a fuzzy or probabilistic clustering does not address true multiclass situations,such as the case of a student employee,where an object belongs to multiple classes. Instead,these approaches are most appropriate for avoiding the arbitrariness of assigning an object to only one cluster when it may be close to several.In practice,a fuzzy or probabilistic clustering is often converted to an exclusive clustering by assigning each object to the cluster in which its membership weight or probability is highest.Complete versus Partial A complete clustering assigns every object to a cluster,whereas a partial clustering does not.The motivation for a partial clustering is that some objects in a data set may not belong to well-defined groups.Many times objects in the data set may represent noise,outliers,or “uninteresting background.”For example,some newspaper stories may share a common theme,such as global warming,while other stories are more generic or one-of-a-kind.Thus,tofind the important topics in last month’s stories,we may want to search only for clusters of documents that are tightly related by a common theme.In other cases,a complete clustering of the objects is desired. For example,an application that uses clustering to organize documents for browsing needs to guarantee that all documents can be browsed.8.1.3Different Types of ClustersClustering aims tofind useful groups of objects(clusters),where usefulness is defined by the goals of the data analysis.Not surprisingly,there are several different notions of a cluster that prove useful in practice.In order to visually illustrate the differences among these types of clusters,we use two-dimensional points,as shown in Figure8.2,as our data objects.We stress,however,that the types of clusters described here are equally valid for other kinds of data. Well-Separated A cluster is a set of objects in which each object is closer (or more similar)to every other object in the cluster than to any object not in the cluster.Sometimes a threshold is used to specify that all the objects in a cluster must be sufficiently close(or similar)to one another.This idealistic definition of a cluster is satisfied only when the data contains natural clusters that are quite far from each other.Figure8.2(a)gives an example of well-separated clusters that consists of two groups of points in a two-dimensional space.The distance between any two points in different groups is larger than494Chapter8Cluster Analysis:Basic Concepts and Algorithmsthe distance between any two points within a group.Well-separated clusters do not need to be globular,but can have any shape.Prototype-Based A cluster is a set of objects in which each object is closer (more similar)to the prototype that defines the cluster than to the prototype of any other cluster.For data with continuous attributes,the prototype of a cluster is often a centroid,i.e.,the average(mean)of all the points in the clus-ter.When a centroid is not meaningful,such as when the data has categorical attributes,the prototype is often a medoid,i.e.,the most representative point of a cluster.For many types of data,the prototype can be regarded as the most central point,and in such instances,we commonly refer to prototype-based clusters as center-based clusters.Not surprisingly,such clusters tend to be globular.Figure8.2(b)shows an example of center-based clusters. Graph-Based If the data is represented as a graph,where the nodes are objects and the links represent connections among objects(see Section2.1.2), then a cluster can be defined as a connected component;i.e.,a group of objects that are connected to one another,but that have no connection to objects outside the group.An important example of graph-based clusters are contiguity-based clusters,where two objects are connected only if they are within a specified distance of each other.This implies that each object in a contiguity-based cluster is closer to some other object in the cluster than to any point in a different cluster.Figure8.2(c)shows an example of such clusters for two-dimensional points.This definition of a cluster is useful when clusters are irregular or intertwined,but can have trouble when noise is present since, as illustrated by the two spherical clusters of Figure8.2(c),a small bridge of points can merge two distinct clusters.Other types of graph-based clusters are also possible.One such approach (Section8.3.2)defines a cluster as a clique;i.e.,a set of nodes in a graph that are completely connected to each other.Specifically,if we add connections between objects in the order of their distance from one another,a cluster is formed when a set of objects forms a clique.Like prototype-based clusters, such clusters tend to be globular.Density-Based A cluster is a dense region of objects that is surrounded by a region of low density.Figure8.2(d)shows some density-based clusters for data created by adding noise to the data of Figure8.2(c).The two circular clusters are not merged,as in Figure8.2(c),because the bridge between them fades into the noise.Likewise,the curve that is present in Figure8.2(c)also8.1Overview495 fades into the noise and does not form a cluster in Figure8.2(d).A density-based definition of a cluster is often employed when the clusters are irregular or intertwined,and when noise and outliers are present.By contrast,a contiguity-based definition of a cluster would not work well for the data of Figure8.2(d) since the noise would tend to form bridges between clusters.Shared-Property(Conceptual Clusters)More generally,we can define a cluster as a set of objects that share some property.This definition encom-passes all the previous definitions of a cluster;e.g.,objects in a center-based cluster share the property that they are all closest to the same centroid or medoid.However,the shared-property approach also includes new types of clusters.Consider the clusters shown in Figure8.2(e).A triangular area (cluster)is adjacent to a rectangular one,and there are two intertwined circles (clusters).In both cases,a clustering algorithm would need a very specific concept of a cluster to successfully detect these clusters.The process offind-ing such clusters is called conceptual clustering.However,too sophisticated a notion of a cluster would take us into the area of pattern recognition,and thus,we only consider simpler types of clusters in this book.Road MapIn this chapter,we use the following three simple,but important techniques to introduce many of the concepts involved in cluster analysis.•K-means.This is a prototype-based,partitional clustering technique that attempts tofind a user-specified number of clusters(K),which are represented by their centroids.•Agglomerative Hierarchical Clustering.This clustering approach refers to a collection of closely related clustering techniques that producea hierarchical clustering by starting with each point as a singleton clusterand then repeatedly merging the two closest clusters until a single,all-encompassing cluster remains.Some of these techniques have a natural interpretation in terms of graph-based clustering,while others have an interpretation in terms of a prototype-based approach.•DBSCAN.This is a density-based clustering algorithm that producesa partitional clustering,in which the number of clusters is automaticallydetermined by the algorithm.Points in low-density regions are classi-fied as noise and omitted;thus,DBSCAN does not produce a complete clustering.Chapter 8Cluster Analysis:Basic Concepts and Algorithms (a)Well-separated clusters.Eachpoint is closer to all of the points in itscluster than to any point in anothercluster.(b)Center-based clusters.Each point is closer to the center of its cluster than to the center of any other cluster.(c)Contiguity-based clusters.Eachpoint is closer to at least one pointin its cluster than to any point inanother cluster.(d)Density-based clusters.Clus-ters are regions of high density sep-arated by regions of low density.(e)Conceptual clusters.Points in a cluster share some generalproperty that derives from the entire set of points.(Points in theintersection of the circles belong to both.)Figure 8.2.Different types of clusters as illustrated by sets of two-dimensional points.8.2K-meansPrototype-based clustering techniques create a one-level partitioning of the data objects.There are a number of such techniques,but two of the most prominent are K-means and K-medoid.K-means defines a prototype in terms of a centroid,which is usually the mean of a group of points,and is typically8.2K-means497 applied to objects in a continuous n-dimensional space.K-medoid defines a prototype in terms of a medoid,which is the most representative point for a group of points,and can be applied to a wide range of data since it requires only a proximity measure for a pair of objects.While a centroid almost never corresponds to an actual data point,a medoid,by its definition,must be an actual data point.In this section,we will focus solely on K-means,which is one of the oldest and most widely used clustering algorithms.8.2.1The Basic K-means AlgorithmThe K-means clustering technique is simple,and we begin with a description of the basic algorithm.Wefirst choose K initial centroids,where K is a user-specified parameter,namely,the number of clusters desired.Each point is then assigned to the closest centroid,and each collection of points assigned to a centroid is a cluster.The centroid of each cluster is then updated based on the points assigned to the cluster.We repeat the assignment and update steps until no point changes clusters,or equivalently,until the centroids remain the same.K-means is formally described by Algorithm8.1.The operation of K-means is illustrated in Figure8.3,which shows how,starting from three centroids,the final clusters are found in four assignment-update steps.In these and other figures displaying K-means clustering,each subfigure shows(1)the centroids at the start of the iteration and(2)the assignment of the points to those centroids.The centroids are indicated by the“+”symbol;all points belonging to the same cluster have the same marker shape.1:Select K points as initial centroids.2:repeat3:Form K clusters by assigning each point to its closest centroid.4:Recompute the centroid of each cluster.5:until Centroids do not change.In thefirst step,shown in Figure8.3(a),points are assigned to the initial centroids,which are all in the larger group of points.For this example,we use the mean as the centroid.After points are assigned to a centroid,the centroid is then updated.Again,thefigure for each step shows the centroid at the beginning of the step and the assignment of points to those centroids.In the second step,points are assigned to the updated centroids,and the centroids498Chapter8Cluster Analysis:Basic Concepts and Algorithms(a)Iteration1.(b)Iteration2.(c)Iteration3.(d)Iteration4.ing the K-means algorithm tofind three clusters in sample data.are updated again.In steps2,3,and4,which are shown in Figures8.3(b), (c),and(d),respectively,two of the centroids move to the two small groups of points at the bottom of thefigures.When the K-means algorithm terminates in Figure8.3(d),because no more changes occur,the centroids have identified the natural groupings of points.For some combinations of proximity functions and types of centroids,K-means always converges to a solution;i.e.,K-means reaches a state in which no points are shifting from one cluster to another,and hence,the centroids don’t change.Because most of the convergence occurs in the early steps,however, the condition on line5of Algorithm8.1is often replaced by a weaker condition, e.g.,repeat until only1%of the points change clusters.We consider each of the steps in the basic K-means algorithm in more detail and then provide an analysis of the algorithm’s space and time complexity. Assigning Points to the Closest CentroidTo assign a point to the closest centroid,we need a proximity measure that quantifies the notion of“closest”for the specific data under consideration. Euclidean(L2)distance is often used for data points in Euclidean space,while cosine similarity is more appropriate for documents.However,there may be several types of proximity measures that are appropriate for a given type of data.For example,Manhattan(L1)distance can be used for Euclidean data, while the Jaccard measure is often employed for documents.Usually,the similarity measures used for K-means are relatively simple since the algorithm repeatedly calculates the similarity of each point to each centroid.In some cases,however,such as when the data is in low-dimensional8.2K-means499Table8.1.Table of notation.Symbol Descriptionx An object.C i The i th cluster.c i The centroid of cluster C i.c The centroid of all points.m i The number of objects in the i th cluster.m The number of objects in the data set.K The number of clusters.Euclidean space,it is possible to avoid computing many of the similarities, thus significantly speeding up the K-means algorithm.Bisecting K-means (described in Section8.2.3)is another approach that speeds up K-means by reducing the number of similarities computed.Centroids and Objective FunctionsStep4of the K-means algorithm was stated rather generally as“recompute the centroid of each cluster,”since the centroid can vary,depending on the proximity measure for the data and the goal of the clustering.The goal of the clustering is typically expressed by an objective function that depends on the proximities of the points to one another or to the cluster centroids;e.g., minimize the squared distance of each point to its closest centroid.We illus-trate this with two examples.However,the key point is this:once we have specified a proximity measure and an objective function,the centroid that we should choose can often be determined mathematically.We provide mathe-matical details in Section8.2.6,and provide a non-mathematical discussion of this observation here.Data in Euclidean Space Consider data whose proximity measure is Eu-clidean distance.For our objective function,which measures the quality of a clustering,we use the sum of the squared error(SSE),which is also known as scatter.In other words,we calculate the error of each data point,i.e.,its Euclidean distance to the closest centroid,and then compute the total sum of the squared errors.Given two different sets of clusters that are produced by two different runs of K-means,we prefer the one with the smallest squared error since this means that the prototypes(centroids)of this clustering are a better representation of the points in their ing the notation in Table8.1,the SSE is formally defined as follows:。

数据库英⽂版第六版课后答案（32）C H A P T E R11Indexing and HashingThis chapter covers indexing techniques ranging from the most basic one tohighly specialized ones.Due to the extensive use of indices in database systems,this chapter constitutes an important part of a database course.A class that has already had a course on data-structures would likely be famil-iar with hashing and perhaps even B+-trees.However,this chapter is necessaryreading even for those students since data structures courses typically cover in-dexing in main memory.Although the concepts carry over to database accessmethods,the details(e.g.,block-sized nodes),will be new to such students.The sections on B-trees(Sections11.4.5)and bitmap indexing(Section11.9) may be omitted if desired. Exercises11.15When is it preferable to use a dense index rather than a sparse index?Explain your answer.Answer:It is preferable to use a dense index instead of a sparse indexwhen the?le is not sorted on the indexed?eld(such as when the indexis a secondary index)or when the index?le is small compared to the sizeof memory.11.16What is the difference between a clustering index and a secondary index?Answer:The clustering index is on the?eld which speci?es the sequentialorder of the?le.There can be only one clustering index while there canbe many secondary indices.11.17For each B+-tree of Practice Exercise11.3,show the steps involved in thefollowing queries:a.Find records with a search-key value of11.b.Find records with a search-key value between7and17,inclusive.Answer:With the structure provided by the solution to Practice Exer-cise11.3a:9798Chapter11Indexing and Hashinga.Find records with a value of11i.Search the?rst level index;follow the?rst pointer.ii.Search next level;follow the third pointer.iii.Search leaf node;follow?rst pointer to records with key value11.b.Find records with value between7and17(inclusive)i.Search top index;follow?rst pointer.ii.Search next level;follow second pointer.iii.Search third level;follow second pointer to records with keyvalue7,and after accessing them,return to leaf node.iv.Follow fourth pointer to next leaf block in the chain.v.Follow?rst pointer to records with key value11,then return.vi.Follow second pointer to records with with key value17.With the structure provided by the solution to Practice Exercise12.3b:a.Find records with a value of11i.Search top level;follow second pointer.ii.Search next level;follow second pointer to records with key value11.b.Find records with value between7and17(inclusive)i.Search top level;follow second pointer.ii.Search next level;follow?rst pointer to records with key value7,then return.iii.Follow second pointer to records with key value11,then return.iv.Follow third pointer to records with key value17.With the structure provided by the solution to Practice Exercise12.3c:a.Find records with a value of11i.Search top level;follow second pointer.ii.Search next level;follow?rst pointer to records with key value11.b.Find records with value between7and17(inclusive)i.Search top level;follow?rst pointer.ii.Search next level;follow fourth pointer to records with key value7,then return.iii.Follow eighth pointer to next leaf block in chain.iv.Follow?rst pointer to records with key value11,then return.v.Follow second pointer to records with key value17.Exercises99 11.18The solution presented in Section11.3.4to deal with nonunique searchkeys added an extra attribute to the search key.What effect could this change have on the height of the B+-tree?Answer:The resultant B-tree’s extended search key is unique.This results in more number of nodes.A single node(which points to mutiple records with the same key)in the original tree may correspond to multiple nodes in the result tree.Dependingon how they are organized the height of the tree may increase;it might be more than that of the original tree.11.19Explain the distinction between closed and open hashing.Discuss therelative merits of each technique in database applications.Answer:Open hashing may place keys with the same hash function value in different buckets.Closed hashing always places such keys together in the same bucket.Thus in this case,different buckets can be of different sizes,though the implementation may be by linking together?xed size buckets using over?ow chains.Deletion is dif?cult with open hashing as all the buckets may have to inspected before we can ascertain that a key value has been deleted,whereas in closed hashing only that bucket whose address is obtained by hashing the key value need be inspected.Deletions are more common in databases and hence closed hashing is more appropriate for them.For a small,static set of data lookups may be more ef?cient using open hashing.The symbol table of a compiler would be a good example.11.20What are the causes of bucket over?ow in a hash?le organization?Whatcan be done to reduce the occurrence of bucket over?ows?Answer:The causes of bucket over?ow are:-a.Our estimate of the number of records that the relation will have wastoo low,and hence the number of buckets allotted was not suf?cient.b.Skew in the distribution of records to buckets.This may happeneither because there are many records with the same search keyvalue,or because the the hash function chosen did not have thedesirable properties of uniformity and randomness.To reduce the occurrence of over?ows,we can:-a.Choose the hash function more carefully,and make better estimatesof the relation size.b.If the estimated size of the relation is n r and number of records perblock is f r,allocate(n r/f r)?(1+d)buckets instead of(n r/f r)buckets.Here d is a fudge factor,typically around0.2.Some space is wasted:About20percent of the space in the buckets will be empty.But thebene?t is that some of the skew is handled and the probability ofover?ow is reduced.11.21Why is a hash structure not the best choice for a search key on whichrange queries are likely?100Chapter11Indexing and HashingAnswer:A range query cannot be answered ef?ciently using a hashindex,we will have to read all the buckets.This is because key valuesin the range do not occupy consecutive locations in the buckets,they aredistributed uniformly and randomly throughout all the buckets.11.22Suppose there is a relation r(A,B,C),with a B+-tree index with searchkey(A,B).a.What is the worst-case cost of?nding records satisfying10using this index,in terms of the number of records retrieved n1and the height h of the tree?b.What is the worst-case cost of?nding records satisfying1050∧5n2that satisfy this selection,as well as n1and h de?ned above?c.Under what conditions on n1and n2would the index be an ef?cient way of?nding records satisfying10Answer:a.What is the worst case cost of?nding records satisfying10using this index,in terms of the number of records retrieved n1and the height h of the tree?This query does not correspond to a range query on the search key as the condition on the?rst attribute if the search key is a comparison condition.It looks up records which have the value of A between10and50.However,each record is likely to be in a different block, because of the ordering of records in the?le,leading to many I/O operation.In the worst case,for each record,it needs to traverse the whole tree(cost is h),so the total cost is n1?h.b.What is the worst case cost of?nding records satisfying1050∧5n2that satisfy this selection,as well as n1and h de?ned above.This query can be answered by using an ordered index on the search key(A,B).For each value of A this is between10and50,the system located records with B value between5and10.However,each record could is likely to be in a different disk block.This amounts to exe-cuting the query based on the condition on A,this costs n1?h.Then these records are checked to see if the condition on B is satis?ed.So, the total cost in the worst case is n1?h.c.Under what conditions on n1and n2would the index be an ef?cient way of?nding records satisfying10n1records satisfy the?rst condition and n2records satisfy the second condition.When both the conditions are queried,n1records are output in the?rst stage.So,in the case where n1=n2,no extra records are output in the furst stage.Otherwise,the records whichExercises101 dont satisfy the second condition are also output with an additionalcost of h each(worst case).11.23Suppose you have to create a B+-tree index on a large number of names,where the maximum size of a name may be quite large(say40characters) and the average name is itselflarge(say10characters).Explain how pre?x compression can be used to maximize the average fanout of nonleaf nodes. Answer:There arises2problems in the given scenario.The?rst prob-lem is names can be of variable length.The second problem is names can be long(maximum is40characters),leading to a low fanout and a correspondingly increased tree height.With variable-length search keys, different nodes can have different fanouts even if they are full.The fanout of nodes can be increased bu using a technique called pre?x compres-sion.With pre?x compression,the entire search key value is not stored at internal nodes.Only a pre?x of each search key which is suf?ceint to distinguish between the key values in the subtrees that it seperates.The full name can be stored in the leaf nodes,this way we dont lose any information and also maximize the average fanout of internal nodes. 11.24Suppose a relation is stored in a B+-tree?le organization.Suppose sec-ondary indices stored record identi?ers that are pointers to records on disk.a.What would be the effect on the secondary indices if a node splithappens in the?le organization?b.What would be the cost of updating all affected records in a sec-ondary index?c.How does using the search key of the?le organization as a logicalrecord identi?er solve this problem?d.What is the extra cost due to the use of such logical record identi?ers?Answer:a.When a leaf page is split in a B+-tree?le organization,a numberof records are moved to a new page.In such cases,all secondaryindices that store pointers to the relocated records would have tobe updated,even though the values in the records may not havechanged.b.Each leaf page may contain a fairly large number of records,andeach of them may be in different locations on each secondary index.Thus,a leaf-page split may require tens or even hundreds of I/Ooperations to update all affected secondary indices,making it a veryexpensive operation.c.One solution is to store the values of the primary-index search keyattributes in secondary indices,in place of pointers to the indexed102Chapter11Indexing and Hashingrecords.Relocation of records because of leaf-page splits then doesnot require any update on any secondary index.d.Locating a record using the secondary index now requires two steps:First we use the secondary index to?nd the primary index search-key values,and then we use the primary index to?nd the corre-sponding records.This approach reduces the cost of index updatedue to?le reorganization,although it increases the cost of accesingdata using a secondary index.11.25Show how to compute existence bitmaps from other bitmaps.Make sure that your technique works even in the presence of null values,by using a bitmap for the value null.Answer:The existence bitmap for a relation can be calculated by takingthe union(logical-or)of all the bitmaps on that attribute,including thebitmap for value null.11.26How does data encryption affect index schemes?In particular,how might it affect schemes that attempt to store data in sorted order?Answer:Note that indices must operate on the encrypted data orsomeone could gain access to the index to interpret the data.Otherwise,the index would have to be restricted so that only certain users could access it.To keep the data in sorted order,the index scheme would haveto decrypt the data at each level in a tree.Note that hash systems wouldnot be affected.11.27Our description of static hashing assumes that a large contiguous stretch of disk blocks can be allocated to a static hash table.Suppose you can allocate only C contiguous blocks.Suggest how to implement the hash table,if it can be much larger than C blocks.Access to a block should stillbe ef?cient.Answer:A separate list/table as shown below can be created.Starting address of?rst set of C blocksCStarting address of next set of C blocks2Cand so onDesired block address=Starting adress(from the table depending onthe block number)+blocksize*(blocknumber%C)For each set of C blocks,a single entry is added to the table.In thiscase,locating a block requires2steps:First we use the block number tond the actual block address,and then we can access the desired block.。

a rX iv:c ond-ma t/37186v1[c ond-m at.str-el]9J ul23Charge stripes due to electron correlations in the two-dimensional spinless Falicov-Kimball model R.Lema´n ski ∗Institute of Low Temperature and Structure Research,Polish Academy of Sciences,Wroc l aw,Poland J.K.Freericks †Department of Physics,Georgetown University,Washington,DC 20057G.Banach Daresbury Laboratory,Cheshire WA44AD,Daresbury,United Kingdom (Dated:February 2,2008)Abstract We calculate the restricted phase diagram for the Falicov-Kimball model on a two-dimensional square lattice.We consider the limit where the conduction electron density is equal to the localized electron density,which is the limit related to the S z =0states of the Hubbard model.After considering over 20,000different candidate phases (with a unit cell of 16sites or less)and their thermodynamic mixtures,we find only about 100stable phases in the ground-state phase diagram.We analyze these phases to describe where stripe phases occur and relate these discoveries to the physics behind stripe formation in the Hubbard model.Keywords:charge-stripes–Falicov-Kimball–Hubbard–phase-diagramI.INTRODUCTIONWefind itfitting to write a paper on the spinless Falicov-Kimball(FK) model[1]to celebrate Elliott Lieb’s seventieth birthday.Elliott,and his collaborators,provided two seminal results on this model:(i)thefirst,with Tom Kennedy,proved that there was afinite temperature phase transition to a checkerboard charge-density-wave(CDW)phase in two or more di-mensions for the symmetric half-filled case[2,3],and(ii)the second,with Daniel Ueltschi and Jim Freericks,proved that the segregation principle holds for all dimensions[4,5](which states that if the total particle density is less than one,then the ground state is phase separated if the interac-tion strength is large enough[6]).The Kennedy-Lieb result(along with an independent Brandt-Schmidt paper[7,8])inspired dozens of follow-up papers by researchers across the world.The Freericks-Lieb-Ueltschi paper generalized Lemberger’s proof[9]from one dimension to all dimensions, whichfinally proved the decade old Freericks-Falicov conjecture[6].Both papers are important,because they are the only examples where long-range order and phase separation can be proved to occur in a correlated electronic system.The Falicov-Kimball model has an interesting history too.Leo Falicov and John Kimball invented the spin-one-half version of the model in1969to describe metal-insulator transitions of rare-earth compounds[1].It turns out that John Hubbard actually“discovered”the spinless version of the FK model four years earlier in1965[10],when he developed the alloy-analogy solution to the Hubbard model[11](the so-called Hubbard III solution). This latter version was rediscovered by Kennedy and Lieb in1986[2]when they formulated it as a simple model for how crystallization can be driven by the Pauli exclusion principle.In this contribution,we focus on another problem that can be analyzed in the FK model—the problem of stripe formation in two dimensions.The question of the relation between charge stripes,correlated electrons,and high-temperature superconductivity has been asked ever since static stripes werefirst seen in the nickelate[12,13,14,15]and cuprate[16,17,18,19] materials starting in1993.Two schools of thought emerged to describe the theoretical basis for stripe formation in the Hubbard model.The Kivelson-Emery scenario[20,21,22,23]says that at large U the Hubbard model is close to a phase separation instability but the long-range Coulomb force restricts the phase separation on the nanoscale;a compromise results in static stripe-like order.The Scalapino-White scenario[24,25,26,27]says that stripes can form due to a subtle balance between kinetic-energy effects and potential-energy effects,mediated by spinfluctuations.No long range Coulomb interaction or phase separation is needed to form these stripes. There are numerous numerical studies that have tried to shed light onto this problem.Unfortunately,they have conflicting results.High temperature2series expansions on the related t−J model[28,29]show that phase separation exists,but only when J is large enough,so it is not present in the large-correlation-strength limit of the Hubbard model(where J→0). Monte Carlo calculations[30,31,32]and exact diagonalization studies[33, 34]give different results:some calculations predict the stripe formation to occur,others show a linkage between the stripe formation and the boundary conditions selected for the problem.Mean-field-theory analyses[35,36] seem to predict stripe formation without any phase separation.One way to make sense of these disparate results is that both the energy of the intrinsic stripe phases and the energy for phase separation are quite close to one another,so any small change(induced byfinite-size effects,statistical errors,effects of correlations not included in the perturbative expansions,or due to terms dropped or added to the Hamiltonian)can have a large effect on the phase diagram by producing a small relative change in the energetics of the different many-body states(because of their near degeneracy).We take an alternate point of view here.We choose to examine a model that can be analyzed rigorously,and can be continuously connected to the Hubbard model.We choose the regime that connects directly with the S z=0states of the Hubbard model.The model we analyze is the spinless Falicov-Kimball model on a square latticeH=−t ij (c†i c j+c†j c i)+U|Λ| i=1w i c†i c i,(1) where c†i(c i)creates(destroys)a spinless conduction electron at site i, t is the hopping matrix element( ij denotes a summation over nearest-neighbor pairs on a square lattice),w i=0or1is a classical variable denot-ing the localized electron number at site i,and U is the on-site Coulomb interaction energy.The Fermionic operators satisfy anticommutation re-lations(c†i,c†j)+=0,(c i,c j)+=0,and(c†i,c j)+=δij.The symbol|Λ| denotes the total number of lattice sites in the square latticeΛ.We will always be dealing with periodic configurations of localized electrons,which means we can always consider our lattice to have a large butfinite number of lattice sites and periodic boundary conditions.A short presentation of these results has already appeared[37].The Falicov-Kimball model can be viewed as a Fermionic quantum ana-logue of the Ising model,while the Hubbard model can be viewed as the Fermionic quantum analogue of the Heisenberg model(indeed in the large-U limit at halffilling,the Falicov-Kimball model maps onto an effective nearest-neighbor Ising model,while the Hubbard model maps onto an ef-fective nearest-neighbor Heisenberg model).The way to link the Falicov-Kimball model to the Hubbard model is to imagine a generalization of the Hubbard model where the down-spin hopping matrix element differs from the up-spin hopping matrix element.Then as t↓→0,the down spins3become heavy and are localized on the lattice;the quantum-mechanical ground state is determined by the configuration of down-spin electrons that minimizes the energy of the up-spin electrons.This is precisely the Falicov-Kimball model!In order to maintain the connection to the Hubbard model in zero magneticfield,we must choose the conduction electron densityρe= |Λ|i=1 c†i c i /|Λ|to be equal to the localized electron densityρf= |Λ|i=1w i/|Λ|,which we do here.We study the evolution from the checker-board phase at halffilling(ρe=ρf=1/2)to the segregated phase,which appears whenρe=ρf is small enough.Since these two phases are dras-tically different from each other,the transition is likely to include many different intermediate phases.Indeed,the ground state phase diagram of the Falicov-Kimball model can be quite complex.There are many different periodic phases that can be stabilized for different values of U orρe=ρf. As U becomes large though,the phase diagram simplifies,as the segregated phase becomes the ground state for wider and wider ranges of the electron densities.II.FORMALISMOur strategy to examine the FK model is a brute-force approach which is straightforward to describe,but tedious to carry out.We employ the so-called restricted phase diagram approach,where we consider the grand-canonical thermodynamic potential of the system for all possible periodic phases of the localized electrons,selected from afinite set of candidate phases.In this work,we consider23,755phases,which corresponds to the set of all inequivalent phases with a unit cell that includes16or fewer lattice sites.In order to calculate the thermodynamic potential,wefirst must determine the electronic band structure for the conduction electrons for each candidate periodic phase.We employ a Brillouin-zone grid of 110×110momentum points for each bandstructure.This requires us to diagonalize up to16×16matrices at each discrete momentum point in the Brillouin zone and results in at most16different energy bands.Hence, our calculations can be viewed asfinite-size cluster calculations with cluster sizes ranging from110×110×1up to110×110×16depending on the number of atoms in the unit cell.An example of such a bandstructure is shown in Fig.1.The eigenvalues of the band structure are summed to determine the ground-state energy for each number of conduction electrons.The Gibbs thermodynamic potential is then calculated for all possible values of the chemical potentials of the conduction and localized electrons through the4formula1G({w i})=We separate the different stable phases into10different groups.Every stable phase is labeled by a number and depicted in Fig.3.The small dots indicate the absence of a localized electron,while the large dots indicate the positions of the localized electrons.In the lower left corner,we shade in the unit cell of the configuration and we show with the two solid lines the translation vectors of the unit cell that allow the square lattice to be tiled by the unit cell.The different families of configurations are as follows: (i)the empty lattice(ρe=0andρf=0)denoted E which contains no lo-calized electrons(configuration1);(ii)the full lattice(ρe=0andρf=1) denoted F which contains a localized electron at each site(configuration 2);(iii)the checkerboard phase(ρe=ρf=1/2)denoted Ch which has the localized electrons occupying the A sublattice only of the square lattice in a checkerboard arrangement(configuration3);(iv)diagonal non-neutral stripe phases(ρe=1−ρf)denoted DS which consist of diagonal checker-board phases separated by empty diagonal stripes of slope1(configuration 4);(v)axial non-neutral checkerboard stripes(ρe=1−ρf)denoted AChS which consist of checkerboard regions arranged in stripes oriented parallel to the x-axis and separated by empty stripes with slope0(configurations 5–10);(vi)diagonal neutral stripe phases(ρe=1−ρf)denoted DNS which consist of localized electrons arranged in the checkerboard phase separated by fully occupied striped regions of slope1,or equivalently,checkerboard phases with diagonal antiphase boundaries(configurations11–19);(vii)ax-ial non-neutral stripe phases(ρe=1−ρf)denoted AS which consist of fully occupied vertical(or horizontal)stripes separated by empty stripes, which are translationally invariant in the vertical(or horizontal)direction (configurations20–54);(viii)neutral phases(ρe=1−ρf)denoted N which consist of neutral phases in an arrangement that does not look like any simple stripe phase(some neutral phases can be described in a stripe pic-ture,such as configuration61which has a slope1/3empty lattice stripe, but we prefer to refer to them as non-stripe phases)(configurations55–70); (ix)four-molecule phases(ρe=1−ρf)denoted4M which can be described as a“bound”four-molecule square of empty sites tiled inside an occupied lattice framework(configurations71–74);(x)two-dimensional non-neutral phases(ρe=1−ρf)denoted2D which consist of phases with the local-ized electrons arranged in a fashion that is not stripe-like and requires a two-dimensional unit cell to describe them(once again,some phases like configuration75could be described as a slope3/2stripe,but appears to us more like a2D phase)(configurations75–111).Generically,wefind the canonical phase diagram does not contain pure phases from one of the111stable phases,but rather forms mixtures of two or three periodic phases,or one or two periodic phases and the empty lattice(which is often needed to get the conduction-electronfilling correct in the mixture).When we are doped sufficiently far from halffilling,we are in the segregated phase,which is a mixture of the E and F phases. We consider5different values of U in our computations:U=1,2,4,6,6and8.The phase diagram is quite complex,with many of the different111 phases appearing for different values of U.We summarize which phases appear in Table I.We begin our discussion with the weak-coupling value U=1where50 phases appear.The phase diagram is summarized in Fig.4.We use a solid line to indicate the region of the particle density where a particular phase appears in the ground state(either as a pure phase or as a mixture). The phases that appear in a mixture at a given density are found by de-termining the solid vertical lines that intersect a horizontal line drawn to pass through the given particle density.The phase diagram has shading included to separate the regions of the different categories of phases.The numeric labels are shown to make it easier to determine the actual phases present in the diagram.We plot similar phase diagrams for U=2(38 phases),4(42phases),6(30phases),and8(25phases)in Figs.5–8,re-spectively.A schematic phase diagram that illustrates the generic features of the phase diagram in the electron density,interaction-strength plane appears in Fig.9.As can be seen from thesefigures,the generic phase diagram is quite complex,and by looking at the different phases in Fig.3,many of the phases have stripe-like structures to them.To begin our discussion of these results,we mustfirst recall the rigorous results known for this model.When ρe=ρf=1/2,the ground state is the checkerboard phase(configuration 3)for all U.This can be seen in all of the phase diagrams plotted.When ρe=ρf=1/2,the ground state becomes the phase separated segregated phase when U is large enough.So there is a simplification in the phase diagram as we increase U,and the most complex phase diagram appears in the U→0limit.That limit is also the most difficult computationally, because the differences in the energies between different configurations also becomes small for small U,and the numerical accuracy must be huge in order to achieve trustworthy results.This is why we do not report any phase diagrams with U<1here.Looking at the U=8case shown in Fig.8,we see that as we move away from halffilling,we initiallyfind mixtures between the checkerboard phase,other diagonal stripe phases,and the empty lattice.When we ex-amine the structure factors associated with the diagonal stripe phases,we find that they tend to have more weight along the Brillouin zone diagonal than elsewhere.Hence,these diagonal stripe phases are being stabilized by a“near-nesting”instability of the noninteracting Fermi surface,and the overall mixtures are required to maintain the averagefillings of the conduction and localized electrons.As we move farther from halffilling, the checkerboard phase disappears from the mixtures,and then a series of neutral phases enter the mix which retain some appearance of diago-nal stripes,but with more and more“defects”to the stripes that make them look more two-dimensional.Wefind the localized electron density of these phases increases as we reduce the totalfilling,which is what we7expect as we move toward the segregated phase which involves a mixture of the E and F configurations.Note that the formation of many different stripe phases,occurs without needing the long-range Coulomb interaction to oppose the tendency towards phase separation,when we are close to half filling.Indeed,the ground state is often a phase separated mixture,but it is a mixture of stripe-like phases,which occur automatically,without the need to add any other physics to the system.This regime,is the closest to the Kivelson-Emery picture,but we see it has more complex behavior than what they envisioned when they examined the Hubbard model.Moving on to the U=6case in Fig.7,wefind a significant change in the phase diagram.The grouping of diagonal stripes near halffilling disappears and we insteadfind the ground state to initially be a mixture between the checkerboard phase,a truly two-dimensional screen-like phase (configuration108)and the empty lattice.Here,if we include a long-range Coulomb interaction,we would likely form diagonal stripes,but the mix-ture would be more complicated because it would include this screen-like structure as well.As we dope further away,we see a smaller number of the neutral phases,which look somewhat like diagonal stripes with a large number of defects in them,and then we go to a very different class of mixtures,dominated by the presence of the axial stripe phase in configu-ration33.As that phase becomes destabilized,wefind a cascade of many other axial stripes entering,before the segregated phase takes over.This transition from diagonal stripes to axial stripes as a function of the elec-tronfilling,also occurs because of a“near-nesting”effect.The structure factors of the axial stripe phases are peaked predominantly along the zone edge,and as we dope further from halffilling,this is where nesting is more likely to occur.The cascade of stable phases that enter after configura-tion33is destabilized,have a progression of the peaks in their structure factor moving towards the zone center,which is also expected,since they are progressively heading towards the segregated phase.A similar kind of transition from diagonal stripes to axial stripes is seen in the Hubbard model studies,with the critical density lying near0.375,as we see here too. By the time we decrease to U=4shown in Fig.6,wefind even more interesting behavior.Now,when we are near halffilling,wefind two more configurations,a nonneutral phase(configuration59)and a two-dimensional phase(configuration109)joining with the checkerboard phase and configuration108in the initial mixtures.Each of these phases looks like a“square-lattice screen”with differing size“holes”in the screen.These two-dimensional structures are not stripe-like and it would be interesting to see if they could appear in the Hubbard model.As we dope further away,we enter the axial stripe region,now dominated by configuration20first,then there is a cascade to configuration33,then a cascade to the seg-regated phase.This value of U is a truly intermediate value,where many different mechanisms for ordering are present and the system can change very rapidly in response to a modification in the density.8As U=2(Fig.5),we see more modifications in the phase diagram.Now we see other diagonal stripe phases mixing with the checkerboard phase near halffilling.This region would correspond to the Scalapino-White regime,where the stripe formation is driven more by interplays between the kinetic and potential energies and nesting effects(driven by charge fluctuations in the FK model and spinfluctuations in the Hubbard model). In addition,a much larger number of the2D phases enter also close to half filling,illustrating the prevalence of these“screen-like”phases as well.The axial stripes also enter as we dope further away from halffilling,but the configurations20and33are not nearly as stable as they are for slightly larger U.Here,we see the four-molecule phases being stabilized just before the system phase separates into the segregated phase.Finally,for U=1,shown in Fig.4,the predominance of the diagonal stripes,near halffilling increases now supplemented by the axial checker-board stripes,but then there is a plethora of different2D phases that also enter as the system is doped somewhat farther from halffilling,then we see a similar evolution,first to AS and then to4M phases before the segre-gated phase.Here there is a tremendous complexity to the phase diagram, with many different mixtures being present due to the competition be-tween kinetic energy and potential energy minimization brought about by the many-body aspects of the problem.The general picture,illustrated schematically in Fig.9,now emerges: near halffilling,we oftenfind diagonal stripes and screen-like two-dimensional phases,then a rapid transition to the segregated phase for large U.As U is reduced,we can dope farther away from halffilling before segregating,which allows many other phases to enter.In particular,there is a large region of stability for axial stripes,and as U is reduced further,we see the emergence of axial checkerboard stripes close to halffilling,near the diagonal stripes,and four-molecule phases appearing near the segregation boundary.IV.CONCLUSIONSIn this manuscript we have numerically studied how the FK model makes the transition from the checkerboard phase at halffilling to the segregated phase as the density is lowered.Since these two phases are very different from one another,there are many different pathways that one might imag-ine the system to take in making this crossover.Indeed wefind that the pathway varies dramatically as a function of U.For large U,we have a relatively simple transition between diagonal stripe-like phases which be-come more two-dimensional as the localized electron density increases,until the system gives way to the segregated phase.As U is lowered,wefirst see two-dimensional-“screen”-like phases enter,then we see axial stripes9emerge,followed by four-molecule phases and axial checkerboard stripes. The complexity of the phase diagram greatly increases as the interaction strength decreases.It is interesting to ask how we might expect these results to change if we allowed more configurations into our restricted phase diagram.We don’t know this answer in particular,but we do know,that of the23,755 candidate phases only a small fraction(111or0.5%)appear in the phase diagram,so we don’t expect too many additional phases to appear. Another interesting question to ask is how do these results for the FK model shed light on the stripe-formation problem in the Hubbard model. By continuity,we expect these results not to change too dramatically as we turn on a small hopping for the localized electrons(although now we must summarize our results in terms of correlation functions for the two kinds of electrons,since both are now mobile).But we also know for many fillings,there will be a“phase transition”as a function of the hopping, since the ground state of the Hubbard model is not ferromagnetic for all fillings and large U(which is what the segregated phase maps to in the Hubbard model).The results are likely to be closer to what happens in the Hubbard model close to halffilling,because the analogue of the anti-ferromagnetic phase is the checkerboard phase,and that is present for all U in the Hubbard model at T=0.In general,we also feel that the FK model phase diagram must be more complicated than the Hubbard model phase diagram because of the mobility of both electrons in the latter.We feel one of the most important results of this work is that there may be two-dimensional phases that are not stripe like that form ground-state con-figurations for some values of thefilling in the Hubbard model,and such configurations will be worthwhile to investigate with the numerical tech-niques that currently exist.In conclusion,we are delighted to be able to shed some light on the inter-esting question for the FK model of how one makes a transition from the checkerboard phase at halffilling to the segregated phase away from half filling.Since Elliott Lieb has had an important impact in proving the sta-bilization of these two phases,wefind itfitting to ask the questions about how the two phases inter-relate.Perhaps these numerical calculations can further inspire new rigorous work that helps to identify the pathway be-tween these two phases.V.ACKNOWLEDGMENTSR.L.and G.B.acknowledge support from the Polish State Committee for Scientific Research(KBN)under Grant No.2P03B13119and J.K.F. acknowledges support from the NSF under grant No.DMR-0210717.We also acknowledge support from Georgetown University for a travel grant in10the fall of 2001.[1] L. 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Reading Material 12Principles of Momentum transfer1. IntroductionThe flow and behavior of fluid is important in many of the unit operations in process engineering. A fluid may be defined as a substance that does not permanently resist distortion and, hence, will change its shape. In this text gases, liquids, and vapors are considered to have the characteristics of fluids and to obey many of the same laws.2. Fluid FlowThe principles of the statics of fluids are almost an exact science. On the other hand, the principles of the motions of fluids are quite complex. The basic relations describing the motions of a fluid are the equations for the overall balances of mass, energy, and momentum, which will be covered in the following sections.The study of momentum transfer, or fluids mechanics as it is often called, can be divided into tow branches: fluid statics, or fluid at rest, and fluid dynamics, or fluids in motion. In other sections we treat fluid statics; in the remaining sections, fluid dynamics. Since in fluid dynamics momentum is being transferred, the term “momentum transfer” or “transfer” is usually used.In momentum transfer we treat the fluid as a continuous distribution of matter or as a “continuum”. This treatment as a continuum is valid when the smallest volume of fluid contains a large enough number of molecules so that a statistical average is meaningful and the macroscopic properties of the fluid such as density, pressure, and so on, vary smoothly or continuously from point to point.Like all physical matter, a fluid is composed of an extremely large number of molecules per unit volume. A theory such as the kinetic theory of gases or statistical mechanics treats the motions of molecules in terms of statistical groups and not in terms of individual molecules. In engineering we are mainly concerned with the bulk or macroscopic behavior of a fluid rather than with the individual molecular or microscopic behavior.In the process industries, many of the materials are in fluid form and must be stored, handled, pumped, and processed, so it is necessary that we become familiar with the principles that govern the flow of fluids and also with the equipment used. Typical fluids encountered include water, air, CO2, oil, slurries, and thick syrups.If a fluid is inappreciably affected by change in pressure, it is said to be incompressible. Most liquids are incompressible. Gases are considered to be compressible fluids. However, if gases are subjected to small percentage changes in pressure and temperature, their density changes will be small and they can be considered to be incompressible.3. Laminar and Turbulent FlowThe type of flow occurring in a fluid in a channel is important in fluid dynamics problems. When fluids move through a closed channel of any cross section, either of tow distinct types of flow can be observed according to the conditions present.These two types of flow can be commonly seen in a flowing open stream or river. When the velocity of flow is slow, the flow patterns are smooth. However, when the velocity is quite high, an unstable pattern is observed in which eddies or small packets of fluid particles are present moving in all directions and at all angles to the normal line of flow.The first type of flow at low velocities where the layers of fluid seem to slide by one another without eddies or swirls being present is called laminar flow and Newton ’s law of viscosity holds. The second type of flow at higher velocities where eddies are present giving the fluid a fluctuating nature is called turbulent flow. The existence of laminar and turbulent flow is most easily visualized by the experiments of Reynolds. Water was allowed to flow at steady state through a transparent pipe with the flow rate controlled by a valve at the end of the pipe.A fine steady stream of dye-colored water was introduced from a fine jet as shown and its flow pattern observed. At low rates of water flow, the dye pattern was regular and formed a single line or stream similar to a thread. There was no lateral of the fluid, and it flowed in streamlines down the tube. By putting in additional jets at other points in the pipe cross section, it was shown that there was no mixing in any parts of the tube and the fluid flowed in straight parallel lines. This type of flow is called laminar or viscous flow.As the velocity was increased, it was found that at a define velocity the thread of dye become dispersed and the pattern was very erratic. This type of flow is known as turbulent flow. The velocity at which the flow changes is known as the critical velocity.4. Reynolds NumberStudies have shown that the transition from laminar to turbulent flow in tubes is not only a function of velocity but also of density and viscosity of the fluid and the tube diameter. These variables are combined into the Reynolds number, which is dimensionless.R e =μρv DWhere Re is the Reynolds number, D the diameter in m, ρ the fluid density in kg/m3,u the fluid viscosity in Pa*s, and ν the average velocity of the fluid in m/s .(where average velocity is defined as the volumetric rate of flow divided by the cross sectional area of the pipe.)The instability of the flow that leads to disturbed or turbulent flow is determined by the ratio of the kinetic or inertial forces to the viscous forces in the fluid stream.The inertial forces are proportional to ρν2 and the viscous forces to D μν,and the ratio )(2D yv pv is the Reynolds number y vp DFor a straight circular pipe when the value of the Reynolds number is less than 2100, the flow is always laminar. When the value is over 4000, the flow will be turbulent, except in very special cases. In between, which is called the transition region, the flow can be viscous or turbulent, depending upon the apparatus details, whichcannot be predicted.5. Simple Mass BalancesIn fluid dynamics fluids are in motion. Generally, they are moved from place to place by means of mechanical devices such as pumps or blowers, by gravity head, or by pressure, and flow through systems of piping and/or process equipment .The first step in the solution of flow problem is generally to apply the conservation of mass to the whole system or to any part of the system. We will consider an elementary balance on a simple geometry. Simple mass or material balances were followed. Input=output + accumulationsince, in fluid flow, we are usually working with rates of flow and usually at steady state, the rate of accumulation is zero and we obtainrate of input = rate of outputWhen making overall balances on mass, energy, and momentum we are not interested in the details of what occurs inside the enclosure. For example, in an overall balance average inlet and outlet velocities are considered. However ,in a differential balance the velocity distribution inside an enclosure can be obtained with the use of Newton’s law of viscosity.These overall or macroscopic balances will be applied to a finite enclosure or control volume fixed in space. We use the term “overall” because we wish to describe these balances from outside the enclosure. The changes inside the enclosure are determined in terms of the properties of the streams entering and leaving and the exchanges of energy between the enclosure and its surroundings.阅读材料 12动量交换定理1. 介绍在许多工程的单元操作中流体的流量和状态是很重要的。