Dynamical Properties of a Haldane Gap Antiferromagnet

a r X i v :0806.1256v 1 [p h y s i c s .s o c -p h ] 7 J u n 2008Understanding individual human mobility patternsMarta C.Gonz´a lez,1,2C´e sar A.Hidalgo,1and Albert-L´a szl´o Barab´a si 1,2,31Center for Complex Network Research and Department of Physics and Computer Science,University of Notre Dame,Notre Dame IN 46556.2Center for Complex Network Research and Department of Physics,Biology and Computer Science,Northeastern University,Boston MA 02115.3Center for Cancer Systems Biology,Dana Farber Cancer Institute,Boston,MA 02115.(Dated:June 7,2008)Despite their importance for urban planning [1],traffic forecasting [2],and the spread of biological [3,4,5]and mobile viruses [6],our understanding of the basic laws govern-ing human motion remains limited thanks to the lack of tools to monitor the time resolved location of individuals.Here we study the trajectory of 100,000anonymized mobile phone users whose position is tracked for a six month period.We find that in contrast with the random trajectories predicted by the prevailing L´e vy flight and random walk models [7],human trajectories show a high degree of temporal and spatial regularity,each individual being characterized by a time independent characteristic length scale and a significant prob-ability to return to a few highly frequented locations.After correcting for differences in travel distances and the inherent anisotropy of each trajectory,the individual travel patterns collapse into a single spatial probability distribution,indicating that despite the diversity of their travel history,humans follow simple reproducible patterns.This inherent similarity in travel patterns could impact all phenomena driven by human mobility,from epidemic prevention to emergency response,urban planning and agent based modeling.Given the many unknown factors that influence a population’s mobility patterns,ranging from means of transportation to job and family imposed restrictions and priorities,human trajectories are often approximated with various random walk or diffusion models [7,8].Indeed,early mea-surements on albatrosses,bumblebees,deer and monkeys [9,10]and more recent ones on marine predators [11]suggested that animal trajectory is approximated by a L´e vy flight [12,13],a random walk whose step size ∆r follows a power-law distribution P (∆r )∼∆r −(1+β)with β<2.While the L´e vy statistics for some animals require further study [14],Brockmann et al.[7]generalized this finding to humans,documenting that the distribution of distances between consecutive sight-ings of nearly half-million bank notes is fat tailed.Given that money is carried by individuals, bank note dispersal is a proxy for human movement,suggesting that human trajectories are best modeled as a continuous time random walk with fat tailed displacements and waiting time dis-tributions[7].A particle following a L´e vyflight has a significant probability to travel very long distances in a single step[12,13],which appears to be consistent with human travel patterns:most of the time we travel only over short distances,between home and work,while occasionally we take longer trips.Each consecutive sightings of a bank note reflects the composite motion of two or more indi-viduals,who owned the bill between two reported sightings.Thus it is not clear if the observed distribution reflects the motion of individual users,or some hitero unknown convolution between population based heterogeneities and individual human trajectories.Contrary to bank notes,mo-bile phones are carried by the same individual during his/her daily routine,offering the best proxy to capture individual human trajectories[15,16,17,18,19].We used two data sets to explore the mobility pattern of individuals.Thefirst(D1)consists of the mobility patterns recorded over a six month period for100,000individuals selected randomly from a sample of over6million anonymized mobile phone users.Each time a user initiates or receives a call or SMS,the location of the tower routing the communication is recorded,allowing us to reconstruct the user’s time resolved trajectory(Figs.1a and b).The time between consecutive calls follows a bursty pattern[20](see Fig.S1in the SM),indicating that while most consecutive calls are placed soon after a previous call,occasionally there are long periods without any call activity.To make sure that the obtained results are not affected by the irregular call pattern,we also study a data set(D2)that captures the location of206mobile phone users,recorded every two hours for an entire week.In both datasets the spatial resolution is determined by the local density of the more than104mobile towers,registering movement only when the user moves between areas serviced by different towers.The average service area of each tower is approximately3km2 and over30%of the towers cover an area of1km2or less.To explore the statistical properties of the population’s mobility patterns we measured the dis-tance between user’s positions at consecutive calls,capturing16,264,308displacements for the D1and10,407displacements for the D2datasets.Wefind that the distribution of displacements over all users is well approximated by a truncated power-lawP(∆r)=(∆r+∆r0)−βexp(−∆r/κ),(1)withβ=1.75±0.15,∆r0=1.5km and cutoff valuesκ|D1=400km,andκ|D2=80km(Fig.1c,see the SM for statistical validation).Note that the observed scaling exponent is not far fromβB=1.59observed in Ref.[7]for bank note dispersal,suggesting that the two distributions may capture the same fundamental mechanism driving human mobility patterns.Equation(1)suggests that human motion follows a truncated L´e vyflight[7].Yet,the observed shape of P(∆r)could be explained by three distinct hypotheses:A.Each individual follows a L´e vy trajectory with jump size distribution given by(1).B.The observed distribution captures a population based heterogeneity,corresponding to the inherent differences between individuals.C.A population based heterogeneity coexists with individual L´e vy trajectories,hence(1)represents a convolution of hypothesis A and B.To distinguish between hypotheses A,B and C we calculated the radius of gyration for each user(see Methods),interpreted as the typical distance traveled by user a when observed up to time t(Fig.1b).Next,we determined the radius of gyration distribution P(r g)by calculating r g for all users in samples D1and D2,finding that they also can be approximated with a truncated power-lawP(r g)=(r g+r0g)−βr exp(−r g/κ),(2) with r0g=5.8km,βr=1.65±0.15andκ=350km(Fig.1d,see SM for statistical validation). L´e vyflights are characterized by a high degree of intrinsic heterogeneity,raising the possibility that(2)could emerge from an ensemble of identical agents,each following a L´e vy trajectory. Therefore,we determined P(r g)for an ensemble of agents following a Random Walk(RW), L´e vy-Flight(LF)or Truncated L´e vy-Flight(T LF)(Figure1d)[8,12,13].Wefind that an en-semble of L´e vy agents display a significant degree of heterogeneity in r g,yet is not sufficient to explain the truncated power law distribution P(r g)exhibited by the mobile phone users.Taken together,Figs.1c and d suggest that the difference in the range of typical mobility patterns of indi-viduals(r g)has a strong impact on the truncated L´e vy behavior seen in(1),ruling out hypothesis A.If individual trajectories are described by a LF or T LF,then the radius of gyration should increase in time as r g(t)∼t3/(2+β)[21,22]while for a RW r g(t)∼t1/2.That is,the longer we observe a user,the higher the chances that she/he will travel to areas not visited before.To check the validity of these predictions we measured the time dependence of the radius of gyration for users whose gyration radius would be considered small(r g(T)≤3km),medium(20<r g(T)≤30km)or large(r g(T)>100km)at the end of our observation period(T=6months).Theresults indicate that the time dependence of the average radius of gyration of mobile phone users is better approximated by a logarithmic increase,not only a manifestly slower dependence than the one predicted by a power law,but one that may appear similar to a saturation process(Fig.2a and Fig.S4).In Fig.2b,we have chosen users with similar asymptotic r g(T)after T=6months,and measured the jump size distribution P(∆r|r g)for each group.As the inset of Fig.2b shows,users with small r g travel mostly over small distances,whereas those with large r g tend to display a combination of many small and a few larger jump sizes.Once we rescale the distributions with r g(Fig.2b),wefind that the data collapses into a single curve,suggesting that a single jump size distribution characterizes all users,independent of their r g.This indicates that P(∆r|r g)∼r−αg F(∆r/r g),whereα≈1.2±0.1and F(x)is an r g independent function with asymptotic behavior F(x<1)∼x−αand rapidly decreasing for x≫1.Therefore the travel patterns of individual users may be approximated by a L´e vyflight up to a distance characterized by r g. Most important,however,is the fact that the individual trajectories are bounded beyond r g,thus large displacements which are the source of the distinct and anomalous nature of L´e vyflights, are statistically absent.To understand the relationship between the different exponents,we note that the measured probability distributions are related by P(∆r)= ∞0P(∆r|r g)P(r g)dr g,whichsuggests(see SM)that up to the leading order we haveβ=βr+α−1,consistent,within error bars, with the measured exponents.This indicates that the observed jump size distribution P(∆r)is in fact the convolution between the statistics of individual trajectories P(∆r g|r g)and the population heterogeneity P(r g),consistent with hypothesis C.To uncover the mechanism stabilizing r g we measured the return probability for each indi-vidual F pt(t)[22],defined as the probability that a user returns to the position where it was first observed after t hours(Fig.2c).For a two dimensional random walk F pt(t)should follow ∼1/(t ln(t)2)[22].In contrast,wefind that the return probability is characterized by several peaks at24h,48h,and72h,capturing a strong tendency of humans to return to locations they visited before,describing the recurrence and temporal periodicity inherent to human mobility[23,24].To explore if individuals return to the same location over and over,we ranked each location based on the number of times an individual was recorded in its vicinity,such that a location with L=3represents the third most visited location for the selected individual.Wefind that the probability offinding a user at a location with a given rank L is well approximated by P(L)∼1/L, independent of the number of locations visited by the user(Fig.2d).Therefore people devote mostof their time to a few locations,while spending their remaining time in5to50places,visited with diminished regularity.Therefore,the observed logarithmic saturation of r g(t)is rooted in the high degree of regularity in their daily travel patterns,captured by the high return probabilities(Fig.2b) to a few highly frequented locations(Fig.2d).An important quantity for modeling human mobility patterns is the probabilityΦa(x,y)tofind an individual a in a given position(x,y).As it is evident from Fig.1b,individuals live and travel in different regions,yet each user can be assigned to a well defined area,defined by home and workplace,where she or he can be found most of the time.We can compare the trajectories of different users by diagonalizing each trajectory’s inertia tensor,providing the probability offinding a user in a given position(see Fig.3a)in the user’s intrinsic reference frame(see SM for the details).A striking feature ofΦ(x,y)is its prominent spatial anisotropy in this intrinsic reference frame(note the different scales in Fig3a),and wefind that the larger an individual’s r g the more pronounced is this anisotropy.To quantify this effect we defined the anisotropy ratio S≡σy/σx, whereσx andσy represent the standard deviation of the trajectory measured in the user’s intrinsic reference frame(see SM).Wefind that S decreases monotonically with r g(Fig.3c),being well approximated with S∼r−ηg,forη≈0.12.Given the small value of the scaling exponent,other functional forms may offer an equally goodfit,thus mechanistic models are required to identify if this represents a true scaling law,or only a reasonable approximation to the data.To compare the trajectories of different users we remove the individual anisotropies,rescal-ing each user trajectory with its respectiveσx andσy.The rescaled˜Φ(x/σx,y/σy)distribution (Fig.3b)is similar for groups of users with considerably different r g,i.e.,after the anisotropy and the r g dependence is removed all individuals appear to follow the same universal˜Φ(˜x,˜y)prob-ability distribution.This is particularly evident in Fig.3d,where we show the cross section of ˜Φ(x/σ,0)for the three groups of users,finding that apart from the noise in the data the curves xare indistinguishable.Taken together,our results suggest that the L´e vy statistics observed in bank note measurements capture a convolution of the population heterogeneity(2)and the motion of individual users.Indi-viduals display significant regularity,as they return to a few highly frequented locations,like home or work.This regularity does not apply to the bank notes:a bill always follows the trajectory of its current owner,i.e.dollar bills diffuse,but humans do not.The fact that individual trajectories are characterized by the same r g-independent two dimen-sional probability distribution˜Φ(x/σx,y/σy)suggests that key statistical characteristics of indi-vidual trajectories are largely indistinguishable after rescaling.Therefore,our results establish the basic ingredients of realistic agent based models,requiring us to place users in number propor-tional with the population density of a given region and assign each user an r g taken from the observed P(r g)ing the predicted anisotropic rescaling,combined with the density function˜Φ(x,y),whose shape is provided as Table1in the SM,we can obtain the likelihood offinding a user in any location.Given the known correlations between spatial proximity and social links,our results could help quantify the role of space in network development and evolu-tion[25,26,27,28,29]and improve our understanding of diffusion processes[8,30].We thank D.Brockmann,T.Geisel,J.Park,S.Redner,Z.Toroczkai and P.Wang for discus-sions and comments on the manuscript.This work was supported by the James S.McDonnell Foundation21st Century Initiative in Studying Complex Systems,the National Science Founda-tion within the DDDAS(CNS-0540348),ITR(DMR-0426737)and IIS-0513650programs,and the U.S.Office of Naval Research Award N00014-07-C.Data analysis was performed on the Notre Dame Biocomplexity Cluster supported in part by NSF MRI Grant No.DBI-0420980.C.A.Hi-dalgo acknowledges support from the Kellogg Institute at Notre Dame.Supplementary 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[15]Sohn,T.,Varshavsky,A.,LaMarca,A.,Chen,M.Y.,Choudhury,T.,Smith,I.,Consolvo,S.,High-tower,J.,Griswold,W.G.&de Lara,E.Lecture Notes in Computer Sciences:Proc.8th International Conference UbiComp2006.(Springer,Berlin,2006).[16]Onnela,J.-P.,Saram¨a ki,J.,Hyv¨o nen,J.,Szab´o,G.,Lazer,D.,Kaski,K.,Kert´e sz,K.&Barab´a si A.L.Structure and tie strengths in mobile communication networks.Proceedings of the National Academy of Sciences of the United States of America104,7332-7336(2007).[17]Gonz´a lez,M.C.&Barab´a si,plex networks:From data to models.Nature Physics3,224-225(2007).[18]Palla,G.,Barab´a si,A.-L.&Vicsek,T.Quantifying social group evolution.Nature446,664-667(2007).[19]Hidalgo C.A.&Rodriguez-Sickert C.The dynamics of a mobile phone network.Physica A387,3017-30224.[20]Barab´a si,A.-L.The origin of bursts and heavy tails in human dynamics.Nature435,207-211(2005).[21]Hughes,B.D.Random Walks and Random Environments.(Oxford University Press,USA,1995).[22]Redner,S.A Guide to First-Passage Processes.(Cambridge University Press,UK,2001).[23]Schlich,R.&Axhausen,K.W.Habitual travel behaviour:Evidence from a six-week travel diary.Transportation30,13-36(2003).[24]Eagle,N.&Pentland,A.Eigenbehaviours:Identifying Structure in Routine.submitted to BehavioralEcology and Sociobiology(2007).[25]Yook,S.-H.,Jeong,H.&Barab´a si A.L.Modeling the Internet’s large-scale topology.Proceedings ofthe Nat’l Academy of Sciences99,13382-13386(2002).[26]Caldarelli,G.Scale-Free Networks:Complex Webs in Nature and Technology.(Oxford UniversityPress,USA,2007).[27]Dorogovtsev,S.N.&Mendes,J.F.F.Evolution of Networks:From Biological Nets to the Internet andWWW.(Oxford University Press,USA,2003).[28]Song C.M.,Havlin S.&Makse H.A.Self-similarity of complex networks.Nature433,392-395(2005).[29]Gonz´a lez,M.C.,Lind,P.G.&Herrmann,H.J.A system of mobile agents to model social networks.Physical Review Letters96,088702(2006).[30]Cecconi,F.,Marsili,M.,Banavar,J.R.&Maritan,A.Diffusion,peer pressure,and tailed distributions.Physical Review Letters89,088102(2002).FIG.1:Basic human mobility patterns.a,Week-long trajectory of40mobile phone users indicate that most individuals travel only over short distances,but a few regularly move over hundreds of kilometers. Panel b,displays the detailed trajectory of a single user.The different phone towers are shown as green dots,and the V oronoi lattice in grey marks the approximate reception area of each tower.The dataset studied by us records only the identity of the closest tower to a mobile user,thus we can not identify the position of a user within a V oronoi cell.The trajectory of the user shown in b is constructed from186 two hourly reports,during which the user visited a total of12different locations(tower vicinities).Among these,the user is found96and67occasions in the two most preferred locations,the frequency of visits for each location being shown as a vertical bar.The circle represents the radius of gyration centered in the trajectory’s center of mass.c,Probability density function P(∆r)of travel distances obtained for the two studied datasets D1and D2.The solid line indicates a truncated power law whose parameters are provided in the text(see Eq.1).d,The distribution P(r g)of the radius of gyration measured for the users, where r g(T)was measured after T=6months of observation.The solid line represent a similar truncated power lawfit(see Eq.2).The dotted,dashed and dot-dashed curves show P(r g)obtained from the standard null models(RW,LF and T LF),where for the T LF we used the same step size distribution as the onemeasured for the mobile phone users.FIG.2:The bounded nature of human trajectories.a,Radius of gyration, r g(t) vs time for mobile phone users separated in three groups according to theirfinal r g(T),where T=6months.The black curves correspond to the analytical predictions for the random walk models,increasing in time as r g(t) |LF,T LF∼t3/2+β(solid),and r g(t) |RW∼t0.5(dotted).The dashed curves corresponding to a logarithmicfit of the form A+B ln(t),where A and B depend on r g.b,Probability density function of individual travel distances P(∆r|r g)for users with r g=4,10,40,100and200km.As the inset shows,each group displays a quite different P(∆r|r g)distribution.After rescaling the distance and the distribution with r g(main panel),the different curves collapse.The solid line(power law)is shown as a guide to the eye.c,Return probability distribution,F pt(t).The prominent peaks capture the tendency of humans to regularly return to the locations they visited before,in contrast with the smooth asymptotic behavior∼1/(t ln(t)2)(solid line)predicted for random walks.d,A Zipf plot showing the frequency of visiting different locations.The symbols correspond to users that have been observed to visit n L=5,10,30,and50different locations.Denoting with(L)the rank of the location listed in the order of the visit frequency,the data is well approximated by R(L)∼L−1. The inset is the same plot in linear scale,illustrating that40%of the time individuals are found at theirfirsttwo preferred locations.FIG.3:The shape of human trajectories.a,The probability density functionΦ(x,y)offinding a mobile phone user in a location(x,y)in the user’s intrinsic reference frame(see SM for details).The three plots, from left to right,were generated for10,000users with:r g≤3,20<r g≤30and r g>100km.The trajectories become more anisotropic as r g increases.b,After scaling each position withσx andσy theresulting˜Φ(x/σx,y/σy)has approximately the same shape for each group.c,The change in the shape of Φ(x,y)can be quantified calculating the isotropy ratio S≡σy/σx as a function of r g,which decreases as S∼r−0.12(solid line).Error bars represent the standard error.d,˜Φ(x/σx,0)representing the x-axis cross gsection of the rescaled distribution˜Φ(x/σx,y/σy)shown in b.。

翻译部分Research on support bearing and top coal stability of fully mechanized caving face in deep mineABSTRACT: On the basis of geological and production conditions of 34223 fullymechanized caving face in Quantaicoal mine, the distribution of displacement field of topcoal deformation is analyzed by means of UDEC3.0 numerical calculating method, which indicates that the top coal deformation shows a characteristic offront-to-back dynamicunstable areas. And the impact of support rigidity and rotation angle of main roof onsupport bearing and on top coal deformation is investigated. Last, the feasibility of light-duty support used in fully mechanized caving face is also analyzed. introductionAlong with the wide application of top coal caving technology in fullymechanized mining face, the powered support for the top coal caving shows a varietyof development. And compared with the high-resistance powered support for top coal caving, the light-duty support for top coal caving is in common use due to suchadvantages as the lower working resistance, lower cost, light weight, and theconvenient operation. With the increase of mining depth, the face condition of “isolated island” formed by the mining sequence and the high stress ca used by deep mining results in a superimposition of rock stress in working face, which has a greatimpact on the rock and coal control and on the safetyin production. Under this condition, whether the light-duty support for top coal caving can be used successfully or not to realize a safe mining of a thick coal seam has become a hot topic. In this paper, on the basis of geological and production conditions of fully mechanized caving face with deep high stress in Quantai colliery, UDEC3.0 numerical calculating method is used to analyze and discuss the relations of the main roof movement and top coal deformation with the support bearing in order to provide some references for rational lectotype of the powered support in fully mechanized caving face in deep mining of thick coal seam.SIMULATED GEOLOGICAL AND PRODUCTIVE CONDITIONS No.3 coal seam is now exploited by 34223 working face of Quantai mine, the thickness of coal seam is 4.5 m, the dip angle is 2 ~ 14°, averaged by 7°. Its Protodyakonov coefficient f is 1.0, being soft in hardness and simple in structure, with a buried depth of 800 m. The inclined length of the working face is 140 m and the strike length is 708 m. Since the upper and low adjacent coal seams of this working face have already been mined out, during the face mining, the working face will become an “isolated island” form, being pendent in three sides. The immediate roof consists of sandy mudstone with a thickness of 3.4 m, and the main roof consists of sandstone with a thickness of 4.6 m. The immediate floor is composed of sandy mudstone with a thickness of 1.7 m. The geological structure of the working face is quite simple. And ZFZ2600-16/24 powered support for lower top coal caving is used as the face support. The setting load of the support is 1950 kN, with a workingresistance of 2600 kN and a supporting strength of 0.45 MPa. ESTABLISHMENT OF NUMERICAL CALCULATING MODEL The stability of top coal is influenced by both the support rigidity and the movement of main roof, but it can impact on the support bearing as well. In order to analyze the impact of the support rigidity and main roof movement on the stability of top coal and the impact of the stability of top coal on the working resistance of support, the numerical calculation model is established according to the geological condition of 34223 working face (Fig. 1). The model is 150min length and 30min height, with asimulating mining depth of800m. The gravity stress of the upper strata is exerted on the upper boundary of the model.The immediate roof and the top coal in the range of the roof-controlled area of the working face are considered as an emphasis to be studied.. In the calculation model, the excavation of the coal seam starts from the left boundary to make the rotation of main roof form a certain rotation angle. But the rotation angle of the main roof is relevant to the backfilling degree of the goaf, and in the simulation, the rotation angles of the main roof are determined to be 4.98°, 5.81°, and 8.16°, respectively. The simulated support is replaced by the rod element with certain rigidities, such as 40 kN/mm, 85 kN/mm, and 120 kN/mm, respectively. The mechanical parameters of various strata in the calculating model are listed in Table 1 and Table 2.Table 1 Simulating mechanical parameters of strataTable 2 Mechanical parameters of joint surface in simulating strataANALYSIS AND NUMERICALCALCULATING RESULTDeformation characteristic of top coalIf the support rigidity is 80 kN/mm and the rotation angle of the main roof is 5.81°,the displacement vector distribution of the top coal is shown in Figure 2. From Figure 2, it can be seen that the deformation of the top coal has mainly two areas. The first is the deformation nearby the goaf-side which causes the support to move toward the goaf direction, resulting in the transverse instability of support. And the softer the top coal is, the larger this area will be. The second is the deformation at the end face nearby the rib, which results in a serious deformation of the top coal at the end face and an enlargement of fall area.Therefore, according to the deformationImpact of rotation of main roof on top coal stabilityThe rotation of the main roof is an important factor influencing on the rock deformation, and its rotation angle is relevant to the backfilling degree of the rock fall in the goaf.It can be seen that the vertical and horizontal displacements of the top coal increase with the increase of the rotation angle. But if the rotation angle of main roof is small, the displacement is not obvious. The obvious impact of main roof rotation on the top coal deformation is shown in the vertical displacement of upper top coal and the horizontal displacement of lower top coalImpact of support rigidity on stability of top coalThe support rigidity is an important parameter to reflect the supporting performance of the support. Usually, to increase the support rigidity is favorable for controlling the roof. But as for the top coal caving, the impact degree of the support rigidity on top coal deformation is different under the influence of the mechanical characteristics of the top coal. .It can be seen that the increase of support rigidity can reduce the vertical displacement of top coal, and finally the vertical displacements of both upper and lower top coals tend to be the same. Meanwhile, the horizontal displacement wouldincrease with the increase of the support rigidity. So, if the support rigidity is raised properly and the horizontal component of the support toward the rib direction is kept unchanged, it should be favorable for the roof stability of the end face and for the stability of the rock-support system.From the above analysis, the support bearing is the comprehensive result of the support rigidity, the deformation and breakage degree of the top coal, and the rotation degree of the main roof. The increase of mining depth will increase the breakage degree of the top coal. The deformation and breakage characteristic of top coal determines that the support bearing is lowered with the increase of the support rigidity. Therefore, under mining condition of the fully mechanized caving face, if the main roof can form a relatively stable st ructure of “voussior beam”, the working resistance of support will be properly decreased and the light-duty support of top coal caving can be used.CONCLUSIONS1) The deformation of the top coal in roof-controlled area can be divided into two dynamic instable areas: the front one and the back one. And the size of the dynamic instable area is relevant to the hardness of the top coal. The joint of both areas may result in an instability of rock-support system. And under the condition of the soft coal, the occurrence of the front instable area should be avoided and the back instable area should be also reduced. But, in the condition of the hard coal, the back instable area should be properly enlarged to make it be favorable for the top coal caving.2) To increase the rotation angle of main roof will increase both the vertical displacement of upper top coal and the horizontal displacement of lower top coal. And to increase the support rigidity will decrease the vertical displacement of the top coal but increase the horizontal displacement, with the lower top coal in particular.3) The support bearing is the comprehensive result of the support rigidity, the deformation degree of the top coal, and the rotation degree of the main roof. As for fully mechanized caving face, if the main roof can form a relatively stable structure of “voussior beam”, the working resistance of support should be properly lowered, and then the light-duty support of top coal caving used is feasible.REFERENCES[1] Liu, C.Y., Cao, S.G. & Fang, X.Q. 2003. Relation between rock and support instope and its monitoring. Xuzhou: Press of China University of Mining & Technology.[2] Liu, C.Y., Cao, S.G. & Yang, P.J. 1999. Research on bearing characteristics of immediate roof in stope. Journal of Rock Pressure and Ground Control (3 - 4): 35 - 39.中文译文：深部综放开采顶煤稳定性与支架承载研究摘要:依据权台煤矿34223综放工作面的地质及生产条件，采用UDEC3.0数值计算方法，分析了顶煤变形的位移场分布，得出了顶煤变形呈前后动态失稳区特征，分析了老顶回转角、支架刚度对顶煤变形以及支架承载的影响规律，分析了综放开采采用轻型支架的可行性。

a r X i v :0712.1575v 1 [g r -q c ] 10 D e c 2007Fundamental properties and applications of quasi-local black hole horizonsBadri Krishnan Max Planck Institut f¨u r Gravitationsphysik,Am M¨u hlenberg 1,D-14476Golm,Germany E-mail:badri.krishnan@aei.mpg.de Abstract.The traditional description of black holes in terms of event horizons is inadequate for many physical applications,especially when studying black holes in non-stationary spacetimes.In these cases,it is often more useful to use the quasi-local notions of trapped and marginally trapped surfaces,which lead naturally to the framework of trapping,isolated,and dynamical horizons.This framework allows us to analyze diverse facets of black holes in a unified manner and to significantly generalize several results in black hole physics.It also leads to a number of applications in mathematical general relativity,numerical relativity,astrophysics,and quantum gravity.In this short review,I will discuss the basic ideas and recent developments in this framework,and summarize some of its applications with an emphasis on numerical relativity.1.Introduction The surface of a black hole has traditionally been defined using event horizons.Event horizons play a fundamental role in many seminal investigations in black hole physics.This includes Hawking’s area increase theorem,black hole thermodynamics,the uniqueness theorems,black hole perturbation theory and the topological censorship results.Moreover,the most important family of black holes for many purposes are the Kerr-Newman black holes.Similarly for almost all astrophysical purposes,most studies are carried our using Kerr black holes.Given this list of successful results and applications,is there any real need to go beyond event horizons and Kerr black holes?There are indeed some situations where event horizons are not sufficient,and most of these have to do with the global nature of event horizons;we need to know the entire history of the spacetime in order to locate them.This leads to a practical problem for numerical relativity simulations.There is no way to locate event horizons using only Cauchy data at a given time without actually performing the simulation and constructing the full spacetime.Moreover,even after the event horizon is located,using it to calculate the physical parameters is fraught with difficulties.In particular,the Hamiltonian methods used to define the black hole parameters as generators of symmetries are not well adapted to the event horizon.All these problems are resolvedin the case when the spacetime is stationary.However we would like to go beyond stationarity,and even for black holes in equilibrium,it should not be necessary to require the entire spacetime to be stationary.One of the classic results of crucial importance to black holes which does not use event horizons are the singularity theorems of Penrose and Hawking[1,2].The presence of a closed trapped surface implies geodesic incompleteness in the future.Thefirst singularity theorem was proved by Penrose in1965[1],and this paper also introduced the notion of a trapped surface.We shall use Penrose’s trapped surfaces to study black holes quasi-locally,without relying on global properties of the spacetime.The rest of this review is organized as follows.Following a discussion of basic notions and definitions,we discuss the existence and non-uniqueness of quasi-local horizons,and the time evolution of marginally trapped surfaces in Sec.2.Sec.3discusses the black hole area increase law.Finally Sec.4describes some applications in numerical relativity. The reader should beware that this is a biased review of quasi-local horizons with a focus on numerical relativity applications.There are a number of other interesting mathematical and physical aspects of quasi-local horizons which we shall not have time to discuss.The reader is referred to[3,4,5]for more complete reviews and references. Trapped surfaces and the trapping regionThe expansion of a congruence of null geodesics is defined as the rate of increase of an infinitesimal transverse2-dimensional cross-section areaδA carried along with the geodesics:Θ=1dt.(1)The definitions of the shearσab and twistωab are also based on deformations of the cross-section.The particular geodesic congruence we consider are the ones orthogonal to a2-surface S.Let us denote the in-going and out-going null normals to S byℓa and n a respectively,and letΘ(ℓ)andΘ(n)be their respective expansions.For a sphere in flat space,the out-going light rays are diverging and the ingoing ones are converging, i.e.Θ(ℓ)>0andΘ(n)<0.S is said to be a trapped surface if both sets of null-normals are converging:Θ(ℓ)<0andΘ(n)<0.A marginally trapped surface(MTS)is one for whichΘ(ℓ)=0andΘ(n)<0.As shown by the singularity theorems,the presence of such surfaces is the signature of a spacetime containing a black hole.Note however that this is not necessarily a signature of strong gravitationalfield;they are present even for large black holes which have correspondingly small tidal forces at the horizon.It can be shown that trapped surfaces must lie inside the event horizon,and that cross-sections of the event horizon for stationary black holes are MTSs.The spacetime region T containing trapped surfaces is called the trapped region. Similarly,if we restrict our attention to a initial-data surfaceΣ,and to trapped surfaces lying onΣ,we can similarly define the trapped region TΣ⊂Σwhich is,by definition,a subset of the full four-dimensional trapped region.An apparent horizon is the outermost2.Fundamental properties of black hole horizonsLet us outline some basic definitions and properties of quasi-local horizons.The starting point for most of these constructions is the notion of a marginally trapped tube(MTT) defined to be a three-surface of topology S2×R foliated by MTSs.It is useful to think of a MTT as being obtained by the time evolution of a MTS.The MTT is thus constructed by stacking up MTSs found at different times.The various kinds of quasi-local horizons are MTTs with additional conditions on whether it is a spacelike,timelike or null surface, and additional geometric requirements onΘ(n).Name Additional conditions onΘ(n)andΘ(ℓ)NullSpacelikeTimelikeNo restrictionAn MTS S on a spatial sliceΣis said to be strictly-stably-outermost if there exists an infinitesimalfirst order outward deformation which makes S strictly untrapped. Explicitly,if r is the unit spacelike normal to S onΣ,then we consider displacements of S(and geometricfields on S)along f r for some function f;outward deformations have f≥0.Then S is strictly-stably-outermost if thefirst order variation of the expansion Θ(ℓ)is positive:δf rΘ(ℓ)>0for f≥0.A crucial tool in these results is the stability operator L r[f]:=δf rΘ(ℓ)which turns out to be an elliptic operator,and the stability condition can be recast as a condition on the principal eigenvalue of L;see also If this stability condition is satisfied,then the MTT produced by the time evolution of S exists at least for a sufficiently short duration,and it continues to exist as long as this stability condition holds.Furthermore,the MTT in the neighborhood of S is either null or spacelike.It is spacelike of the matterflux T abℓaℓb is non-vanishing somewhere on S.The elliptic nature of L ensures that the MTT is spacelike everywhere in a neighborhood of S if theflux T abℓaℓb is non-zero even in a very small region on S.It should be emphasized that these results do not imply that the time development of S is unique.It in fact implies quite the opposite:for every choice of time evolution by a foliation by spacelike surfaces(i.e.for every choice of lapse and shift functions)there exists an MTT and the different MTTs constructed from the different gauge choices are,in general,distinct from each other.It is also worth noting that not all MTSs will satisfy the stability condition;the unstable MTSs will be inner horizons and actually occur quite frequently in numerical simulations[28](though they are usually not looked for).However,even the unstable MTSs always seem to evolve smoothly as far as the numerical simulations are concerned.This leads us to believe that the existence result might be of more general validity and it might be possible to extend the above techniques to prove this.See[33,34,35]for further results on trapped surfaces and quasi-local horizons using similar techniques.See also[36]for interesting numerical results on the behavior of MTTs in binary black hole spacetimes.Complementary to these existence results,there are other important results on dynamical horizons worth mentioning.In[37]it is proved that the foliation of a dynamical horizon H by MTSs is unique.This,together with the existence results above implies that for a given MTSs on an initial sliceΣ,the MTTs corresponding different time developments must really be distinct as3-manifolds.There are also some restrictions on the location of the various dynamical horizons.For example,it is shown in[37]that for a given dynamical horizon H,there cannot be any closed MTSs(and thus no other DH)lying in the past domain of dependence of H.Thus, while DHs are far from unique,there are some restrictions on where they can occur.We briefly mention results regarding the(non-)existence of dynamical horizons in spacetimes with symmetries[37,38,39].For example,it is shown in[38]that strictly stationary spacetime regions cannot contain trapped or marginally trapped surfaces,and thus no quasi-local horizons as well.Finally,a different approach to studying dyamical horizons is presented in[40]which considers the conditions on the Cauchy data on H that must be satisfied if H is a dynamical horizon.This can be studied in spherical symmetry,and it leads to necessary conditions for the spacetime to contain a dynamical horizon; in this regard,see also[41].3.The second lawIn this section,we outline the second law for dynamical/trapping horizons and its ramifications.But before doing so,it is worth mentioning the significant amount of work devoted to understanding thefirst law for quasi-local horizons.Thefist law connects variations in the mass M between two nearby black hole solutions to the surface gravity κ,area A,angular velocityΩand angular momentum J(the presence of other conserved charges is easy to incorporate):δM=κdt2−κdAThe right hand side of this equation contains the source terms from the infalling matter/radiation which causes the black hole to grow,and all terms in thus equation are quasi-local.The sign ofκis however a problem;it will generically lead to an exponential divergence if we attempt to solve(4)as an initial value problem.In fact,we need to impose dA/dt→0as t→∞to getfinite solutions.Can we reformulate the second law for quasi-local horizons?In fact,∆A>0is a simple consequence ofΘ(ℓ)=0andΘ(n)<0for dynamical horizons.We can do better. Performing the decomposition of all geometricfields on a dynamical horizon,and using the constraint equations on a dynamical horizon,it can be shown thatR22G =F(R)m+F(R)g(5)whereF(R)g :=1dt2+κ′dAThey have so far been primarily used to extract gauge invariant information about the black hole such as its mass,angular momentum etc.,and this is what we shall mostly focus on in this section.However,there are also other important applications which we will not be able to discuss.This includes the construction of initial data with trapped and marginally trapped surfaces as the inner boundary[44,45,46,47,48,49],and the possibility of using fully constrained evolution schemes with a dynamical horizon as the inner boundary[50].There has also been interest in clarifying the definition of surface gravity pf a quasi-local horizon,and the closely related notion of extremality[51,52,53]. This could prove to be useful for mathematical and astrophysical applications.Mass and angular momentumNumerical simulations are based on the initial value formulation of general relativity. Thus we are given an initial data set(Σ,h ab,K ab)whereΣis a3-manifold embedded in the full spacetime,h ab a Riemannian metric onΣ,and K ab the second fundamental form describing howΣis embedded in the spacetime.Such an initial data set is evolved in time to construct the full spacetime.There are various formalisms for performing these evolutions and there are different choices of the variables that can be evolved,but these issues are not of much concern for our purposes.We instead pose a straightforward question.AssumingΣto be some slice of the Kerr spacetime,and assuming thatΣintersects the event horizon in a complete sphere‡,how can we determine its parameters, i.e.its mass and angular momentum?Depending on the choice ofΣand the choice of coordinates onΣ,the shape of the apparent horizon may turn out to be quite complicated.It may not seem axisymmetric and it may even be difficult to say whether we have a stationary black hole.Let us reformulate this question in a much more general context.Consider a quasi-local horizon H(i.e.an isolated,dynamical or trapping horizon)and let us assume that Σintersects H in a cross-section S which is a marginally trapped surface.We assume thatΣis an asymptoticallyflat slice with S as its inner boundary(the generalization to multiple black holes,i.e.when S consists of several disconnected components is straightforward).Let us assume that H is axisymmetric,i.e.it has a rotational vectorϕa which has closed orbits,vanishes at two points on each cross-section of the MTSs which foliate H,and which is a symmetry of the geometricalfields on H.In particularϕa is a symmetry of the two metric q ab on every cross-section S of H.Note that we only asked forϕa to exist on H,and not in the full spacetime and not even in a neighborhood of H.of the horizon.Just like in It is then possible to associate an angular momentum J(ϕ)Sclassical mechanics where conserved quantities are defined as generators of symmetries, this calculation is based on a Hamiltonian formalism.We calculate the generator of diffeomorphisms along a rotational vectorfieldφa which coincides withϕa on H and with an asymptotic rotational symmetry at infinity[14,15,16,17,18](analogous calculations also work in2+1[19]and higher[20,21]dimensions).It is then easy to identify the ‡And also assuming that we have an efficient way of locating marginally trapped surfaces onΣ.contribution of the horizon to the angular momentum,and it turns out to be given by a surface integral over S[54]:J(ϕ) S =1A S/4π,the horizon mass M(ϕ)SisM(ϕ)S =1R4S+4(J(ϕ)S)2.(9)This will give the correct answer for Kerr,and it is in fact also the result of a Hamiltonian calculation in the more general case of an axisymmetric quasi-local horizon.Apart from getting the magnitude of the spin,it is also possible to estimate the direction of the angular momentum vector.The basic idea is to use the poles ofϕa to define the axis of rotation.While the poles themselves are well defined on a given quasi-local horizon,the procedure of assigning a vector is not as clear cut.For example, it is not clear how the spin direction thus obtained can be compared with the spin direction calculated at spatial infinity.Nevertheless,this method has been applied and preliminary results are promising[57].Equation(8)is now being used fairly widely in numerical relativity,though there is possibly room for improvement in the calculation of the symmetry vectorfieldϕa along the lines of[55],for having a better conceptual understanding of the meaning of J(ϕ)Swhenϕa is only an approximate symmetry vector(which is invariably the case in numerical simulations),and also a better understanding of the spin direction which is important for astrophysical applications.§This may seem surprising because this is certainly not the case for a normal S2×R cylinder in Euclidean space.If there is a symmetry vectorϕa on the cylinder,it need not project to a symmetry vector on a given cross-section S of the cylinder.It is nevertheless true for an isolated horizon because ifϕa is a symmetry,then so isϕa+fℓa for any function f and null generatorℓa;projectingϕa to S is equivalent to a particular choice of f.Higher multipole momentsApart from calculating the angular momentum and mass,it turns out that it is also possible to meaningfully define the higher multipole moments of a quasi-local horizon, at least in the axi-symmetric case.This construction wasfirst carried out by[58]for isolated horizons,and subsequently applied by[3,28]to the general case of a dynamical horizon.The construction starts with a coordinate system built using the given axial vectorϕa.φ∈[0,2π)is affine parameter alongϕa andζ=cosθdefined by dζ∝⋆ϕ(the proportionality factor is chosen by requiring Sζd2V=0).Then we use the spherical harmonics in these(θ,φ)coordinates to define the mass and current multipole moments:M(ϕ)n=R n S M(ϕ)S8πSP′n(ζ)¯K abϕa d2S b(10)where P n is the n th order Legendre polynomial and P′n its derivative.These equations provides the source multipole moments of black hole,which are in general distinct from thefield multipole moments defined at infinity.For isolated horizons,it can be shown[58]that the intrinsic horizon geometry is completely characterized by these multipole moments,i.e.any two isolated horizons with the same multipole moments are diffeomorphic to each other.J0vanishes by absence of monopole (NUT)charges,M0is mass and J1is angular momentum.In Kerr,M0and J1determine all higher moments.In Schwarzschild,only M0=0.The higher moments provide a convenient way of quantifying the deviation from Kerr.These multipole moments were applied in some example numerical simulations in[28]where it was shown that the black holes do indeed converge to Kerr very quickly after merger.However,the simulations were unfortunately not accurate enough to extract the late time decay rates of the multipole moments which would be the analog of Price’s law for dynamical horizons. Hopefully this can be measured in the future using long duration accurate simulations. Quasi-local linear momentumThe calculation of black hole linear momentum is of astrophysical importance in the context of the recoil velocity produced during the merger of two black holes.The reason for the recoil is the anisotropic emission of gravitational radiation.It has been found that certain initial spin configurations lead to a much larger than expected value of this recoil velocity and the largest contribution turns out to be from the merger phase.This result is especially interesting for the case of super-massive black holes;if the recoil is large enough,the remnant black hole may be kicked out of the host galaxy and this has important astrophysical implications.Most calculations of the recoil velocity are based on the gravitational waveform extracted far away from the black hole which measures the center-of-mass momentum of the system.It is thus natural to ask whether one can measure the momentum quasi-locally for the two individual black holes.This would be a useful consistency check and it could also give us more detailed information about the dynamics of the merger.The possibility of measuring the linear momentum,andmore generally the quasi-local energy-momentum four-vector,is also of mathematical interest.So far,we have justified the equations for angular momentum,mass and energy by Hamiltonian methods.For the angular momentum we assumed the existence of a rotational symmetry and for energy and mass we need to pick out a preferred time evolution vectorfield at the horizon.Following the same line of reasoning,one might think of defining linear momentum by assuming the existence of a translational symmetry in a neighborhood of the horizon,or at least some preferred translational vectorfield.While it might be possible to do this in special cases,for example when the data is conformallyflat,it is clearly not something we can assume generally.Unlike for angular momentum where there are interesting regimes where approximate axisymmetry is a valid assumption,the basis for carrying over this approach to linear momentum is much less secure.Let us then try a more heuristic approach.Just as the formula for horizon angular momentum is analogous to the angular momentum at spatial infinity,let us apply the formula for linear momentum at infinity to the horizon.At spatial infinity,for a given asymptotic translational Killing vectorfieldξa,the momentum isP ADM ξ=18πS(K ab−Kh ab)ξa dS b(12)whereξa is some translational vector at the ing the constraint equations of Σ,it is easy to showP ADM ξ−P(S)ξ=15060708090100110120t/M −80−60−40−20020406080P y /M (k m s −1)M/32M/48M/646575859510521252933Figure 2.Linear momentum of the remnant black hole produced by the headonmerger of black holes with anti-aligned spins for different resolutions.behaved as one might have thought based on intuition about apparent horizons.It is possible to prove useful and interesting mathematical results about them and they can be used to study black hole physics.They 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宽禁带半导体ZnS物性的第一性原理研究摘要硫化锌（ZnS）是一种新型的II-VI族宽禁带电子过剩本征半导体材料，其禁带宽度为3.67eV,具有良好的光致发光性能和电致发光性能。

在常温下禁带宽度是3.7eV，具有光传导性好，在可见光和红外范围分散度低等优点。

ZnS和基于ZnS的合金在半导体研究领域己经得到了越来越广泛的关注。

由于它们具有较宽的直接带隙和很大的激子结合能，在光电器件中具有很好的应用前景。

本文介绍了宽禁带半导体ZnS目前国内外的研究现状及其结构性质和技术上的应用。

阐述了密度泛函理论的基本原理，对第一性原理计算的理论基础作了详细的总结，并采用密度泛函理论的广义梯度近似（GGA）下的平面波贋势法，利用Castep软件计算了闪锌矿结构ZnS晶体的电子结构和光学性质。

电子结构如闪锌矿ZnS晶体的能带结构，态密度。

光学性质如反射率，吸收光谱，复数折射率，介电函数，光电导谱和损失函数谱。

通过对其能带及结构的研究，可知闪锌矿硫化锌为直接带隙半导体，通过一系列对光学图的分析，可以对闪锌矿ZnS的进一步研究做很好的预测。

关键词ZnS;宽禁带半导体；第一性原理；闪锌矿结构-I -First-principles Research on Physical Properties of Wide Bandgap Semiconductor ZnSAbstractZinc sulfide (ZnS) is a new family of ll-VI wide band gap electronic excess in tri nsic semic on ductor material with good photolu min esce nee properties and electroluminescent properties. At room temperature band gap is 3.7 eV, and there is good optical transmission in the visible and infrared range and low dispersi on. ZnS and Zn S-based alloy in the field of semic on ductor research has bee n paid more and more atte nti on. Because of their wide direct ban dgap and large excit on binding en ergy, the photovoltaic device has a good prospect.This thesis describes the current research status and structure of nature and tech ni cal applicati ons on wide band gap semic on ductor ZnS. Described the basic principles density functional theory, make a detailed summary for the basis of first principles theoretical calculations, using the density functional theory gen eralized gradie nt approximati on (GGA) un der the pla ne wave pseudopote ntial method, calculated using Castep software sphalerite ZnS crystal structure of electronic structure and optical properties. Electronic structures, such as sphalerite ZnS crystal band structure, density of states. Optical properties such as reflecta nee, absorpti on spectra, complex refractive in dex, dielectric function, optical conductivity spectrum and the loss function spectrum. Band and the structure through its research known as zin cble nde ZnS direct band gap semic on ductor,Through a series of optical map an alysis, can make a good predicti on for further study on zin cble nde ZnS.Keywords ZnS; Wide ban dgap semic on ductor; First-pri nciples; Zin cble nde structure-2-目录摘要 (I)Abstract (II)第1章绪论 ........................................................... 1..1.1 ZnS半导体材料的研究背景 ....................................... 1.1.2 ZnS的基本性质和应用........................................... 1.1.3 ZnS材料的研究方向和进展 (3)1.4 ZnS的晶体...................................................... 4.1.4.1 ZnS晶体结构 ............................................... 4.1.4.2 ZnS的能带结构............................................. 5.1.5 ZnS的发光机理................................................. 6.1.6研究目的和主要内容.............................................. 7.2.1相关理论......................................................... 9.2.1.1密度泛函理论 (9)2.1.2交换关联函数近似........................................... 1.12.2总能量的计算 (13)2.2.1势平面波方法 (14)2.2.2结构优化 (16)2.3 CASTEP软件包功能特点........................................ 1.8第3章ZnS晶体电子结构和光学性质.................................... 1.93.1闪锌矿结构ZnS的电子结构 (19)3.1.1晶格结构 (19)3.1.2能带结构 (20)3.1.3态密度 (21)3.2闪锌矿ZnS晶体的光学性质 (24)结论 (30)致谢 (31)参考文献 (32)附录A (33)附录B (45)-ill -第1章绪论1.1 ZnS半导体材料的研究背景Si是应用最为广泛的半导体材料，现代的大规模集成电路之所以成功推广应用，关键就在于Si半导体在电子器件方面的突破。

The Formation of Coral Reefs and Atolls Coral reefs and atolls are some of the most fascinating and diverse ecosystems on the planet. They are formed from the accumulation of calcium carbonate exoskeletons of coral polyps, which are tiny, invertebrate animals that live in colonies. These structures provide a habitat for a wide variety of marine life, including fish, invertebrates, and algae. The formation of coral reefs and atolls is a complex process that involves geological, biological, and environmental factors.One of the key factors in the formation of coral reefs and atolls is the presence of suitable substrate for coral growth. Coral polyps require a hard surface on which to attach and grow, and they typically thrive in warm, shallow, clear waters. As the polyps grow and reproduce, they form colonies that eventually build up into large, solid structures. Over time, the accumulation of coral skeletons creates the framework for a reef or atoll.The geological processes that contribute to the formation of coral reefs and atolls are also important to consider. Reefs often form along the edges of continents or around volcanic islands, where the ocean floor is relatively shallow. As the Earth's tectonic plates shift and move, these areas can experience changes in sea level and land elevation, which can impact the growth and development of coral reefs. Additionally, the presence of ocean currents and wave action can influence the shape and structure of reefs, as well as the distribution of coral species.In addition to geological factors, biological processes play a crucial role in the formation of coral reefs and atolls. Coral polyps rely on a symbiotic relationship with photosynthetic algae called zooxanthellae, which live within their tissues and provide them with essential nutrients. This relationship allows corals to thrive in nutrient-poor waters, but it also makes them sensitive to environmental stressors such as changes in temperature and water quality. When corals are stressed, they can expel their zooxanthellae, a process known as coral bleaching, which can have devastating effects on reef ecosystems.The environmental conditions in which coral reefs and atolls form and grow are also significant. These ecosystems are typically found in tropical and subtropical regions, wherethe water is warm and clear. The availability of sunlight is crucial for the photosynthetic processes of both corals and their symbiotic algae, so reefs are most commonly found in shallow waters where light can penetrate. Additionally, the stability of water temperature and chemistry is important for the health and growth of coral reefs, as they are sensitive to changes in conditions such as ocean acidification and pollution.Human activities also play a role in the formation and degradation of coral reefs and atolls. Overfishing, destructive fishing practices, coastal development, and pollution can all have negative impacts on reef ecosystems. Additionally, climate change is a major threat to coral reefs, as rising sea temperatures and ocean acidification are causing widespread coral bleaching and mortality. Conservation efforts, such as marine protected areas and sustainable fishing practices, are essential for the preservation of these valuable and vulnerable ecosystems.In conclusion, the formation of coral reefs and atolls is a complex and dynamic process that involves a combination of geological, biological, and environmental factors. These ecosystems are not only incredibly diverse and beautiful, but they also provide important ecological and economic benefits. Understanding the processes that shape coral reefs and atolls is crucial for their conservation and management, especially in the face of growing threats from human activities and climate change. As stewards of the natural world, it is our responsibility to protect and preserve these irreplaceable ecosystems for future generations.。

dynamic kinetic resolutionDynamic kinetic resolution (DKR) is an innovative process developed to achieve asymmetric synthesis in complex molecules. It is a type of chemical reaction which helps to obtain a target product with enhanced asymmetric selectivity by using a dynamic kinetic process. It is an environmentally friendly process which is cost-effective and offers shorter reaction timelines as compared to conventional methods.Dynamic kinetic resolution (DKR) consists of two parts, i.e., a kinetic resolution part and a dynamic resolution part. During the kinetic resolution part, a catalyst is used to promote a reaction between two similar molecules. The dynamicresolution part, on the other hand, involves a complex set of reactions in which two different components of a mixture are separated into two different fractions. The reaction is carried out at a temperature and pressure over a certain period of time such that the components are rearranged in accordance with the desired target product.The main advantage of dynamic kinetic resolution (DKR) is that it allows for the production of a target product with high chiral selectivity. This feature is essential for the production of enhanced enantioselective products. It also offers other advantages such as shorter reaction times, cost savings, and reduction of the use of hazardous chemicals. Moreover, it is a selective and targeted process which avoids the use of toxic and hazardous solvents and catalysts. Moreover, itis capable of producing chiral compounds in yields that exceed 50%.Dynamic kinetic resolution (DKR) is most commonly employed in the synthesis of chiral compounds such as natural products, pharmaceuticals, fine chemicals, and agrochemicals. This process is also used in the development of new drugs, fragrances, and optical brighteners. It is also used for the production of complex compounds and the separation of optically active compounds from racemic mixtures. DKR is a versatile and powerful method which provides high yields of the desired product with excellent selectivity.In conclusion, dynamic kinetic resolution (DKR) is a powerful and selective process which offers a number of advantages. It has many applications in the production of chiral compounds and the separation of optically active compoundsfrom racemic mixtures. It is also cost-effective, environmentally friendly and offers shorter reaction times as compared to conventional methods. Thus, DKR is an important tool for the production of complex molecules in organic chemistry.。

a r X i v :m a t h /0501424v 1 [m a t h .O A ] 24 J a n 2005KMS STATES AND COMPLEX MULTIPLICATIONALAIN CONNES,MATILDE MARCOLLI,AND NIRANJAN RAMACHANDRAN1.Introduction Several results point to deep relations between noncommutative geometry and class field theory ([2],[9],[18],[20]).In [2]a quantum statistical mechanical system is exhibited,with partition function the Riemann zeta function,and whose arithmetic properties are related to the Galois theory of the maximal abelian extension of Q .In [9],this system is reinterpreted in terms of the geometry of commensurable 1-dimensional Q -lattices,and a generalization is constructed for 2-dimensional Q -lattices.The arithmetic properties of this GL 2-system and its KMS states at zero temperature related to the Galois theory of the modular field.The ground states and the Galois properties are analyzed in [9]for the generic case of elliptic curves with transcendental j -invariant.As the results of [9]show,one of the main new features of the GL 2-system is the presence of symmetries by endomorphism ,through which the full Galois group of the modular field appears as symmetries acting on the KMS equilibrium states of the system.In both the original BC system and in the GL 2-system,the arithmetic properties of zero temperature KMS states rely on an underlying result of compatibility between ad`e lic groups of symmetries and Galois groups.This correspondence between ad`e lic and Galois groups naturally arises within the context of Shimura varieties.In fact,a Shimura variety is a pro-variety defined over Q ,with a rich ad`e lic group of symmetries.In that context,the compatibility of the Galois action and the automorphisms is at the heart of Langlands program.This leads us to give a reinterpretation of the BC and the GL 2systems in the language of Shimura varieties,with the BC system corresponding to the simplest (zero dimensional)Shimura variety.In the case of the GL 2system,we show how the data of 2-dimensional Q -lattices and commensurability can be also described in terms of elliptic curves together with a pair of points in the total Tate module,and the system is related to the Shimura variety of GL2.This viewpoint suggests considering our systems as noncommutative pro-varieties defined over Q ,more specifically as noncommutative Shimura varieties.We then present our main result,which is the construction of a new system,whose arithmetic prop-erties fully incorporate the explicit class field theory for an imaginary quadratic field K ,and whose partition function is the Dedekind zeta function of K .The underlying geometric structure is given by commensurability of 1-dimensional K -lattices.This new system can be regarded in two different ways.On the one hand,it is a generalization ofthe BC system of [2],when changing the field from Q to K ,and is in fact Morita equivalent to the one considered in [18],but with no restriction on the class number.On the other hand,it is also a specialization of the GL 2-system of [9]to elliptic curves with complex multiplication by K .In this case the ground states can be related to the non-generic ground states of the GL 2-system,associated to points τ∈H with complex multiplication,and the group of symmetries is the Galois group of the maximal abelian extension of K .Here also we show that symmetries by endomorphisms play a crucial role,as they allow for the action of the class group Cl(O ),so that our results hold for any class field.Since this complex multiplication (CM)case can be realized as a subgroupoid of the GL 2-system,it has a natural choice of a rational subalgebra (an arithmetic structure)inherited from that of the GL 2-system.This is crucial,in order to obtain the intertwining of Galois action on the values of extremal states and action of symmetries of the system.12CONNES,MARCOLLI,AND RAMACHANDRANWe summarize and compare the main properties of the three systems(BC,GL2,and CM)in the following table.GL1CMpartition functionζ(β)ζ(β−1)A∗/Q∗A∗K,f/K∗fautomorphisms GL2(ˆZ)Cl(O)Galois group Aut(F)Sh(GL1,±1)A∗K,f/K∗The paper consists of two parts,with sections2and3centered on the relation of the BC and GL2 system to the arithmetic of Shimura varieties,and sections4and5dedicated to the construction of the CM system and its relation to the explicit classfield theory for imaginary quadraticfields.The two parts are closely interrelated,but can also be read independently.2.Quantum Statistical Mechanics and Explicit Class Field TheoryThe BC quantum statistical mechanical system[1,2]exhibits generators of the maximal abelian extension of Q,parameterizing ground states(i.e.at zero temperature).Moreover,the system has the remarkable property that these ground states take algebraic values,when evaluated on a rational subalgebra of the C∗-algebra of observables.The action on these values of the absolute Galois group factors through the abelianization Gal(Q ab/Q)and is implemented by the action of the id`e le class group as symmetries of the system,via the classfield theory isomorphism.This suggests the intriguing possibility of using the setting of quantum statistical mechanics to address the problem of explicit classfield theory for other numberfields.In this section we recall some basic notions of quantum statistical mechanics and of classfield theory, which will be used throughout the paper.We also formulate a general conjectural relation between quantum statistical mechanics and the explicit classfield theory problem for numberfields.Quantum Statistical Mechanics.A quantum statistical mechanical system consists of an algebra of observables,given by a unital C∗-algebra A,together with a time evolution,consisting of a1-parameter group of automorphismsσt, (t∈R),whose infinitesimal generator is the Hamiltonian of the system.The analog of a probability measure,assigning to every observable a certain average,is given by a state,namely a continuous linear functionalϕ:A→C satisfying positivity,ϕ(x∗x)≥0,for all x∈A,and normalization,ϕ(1)=1.In the quantum mechanical framework,the analog of the classical Gibbs measure is given by states satisfying the KMS condition(cf.[13]).Definition2.1.A triple(A,σt,ϕ)satisfies the Kubo-Martin-Schwinger(KMS)condition at inverse temperature0≤β<∞,if,for all x,y∈A,there exists a bounded holomorphic function F x,y(z)on the strip0<Im(z)<β,continuous on the boundary of the strip,such that(2.1)F x,y(t)=ϕ(xσt(y))and F x,y(t+iβ)=ϕ(σt(y)x),∀t∈R.KMS AND CM3 We also say thatϕis a KMSβstate for(A,σt).The set Kβof KMSβstates is a compact convex Choquet simplex[3,II§5]whose set of extreme points Eβconsists of the factor states.At0temperature(β=∞)the KMS condition(2.1)says that,for all x,y∈A,the function(2.2)F x,y(t)=ϕ(xσt(y))extends to a bounded holomorphic function in the upper half plane H.This implies that,in the Hilbert space of the GNS representation ofϕ(i.e.the completion of A in the inner productϕ(x∗y)), the generator H of the one-parameter groupσt is a positive operator(positive energy condition). However,this notion of0-temperature KMS states is in general too weak,hence the notion of KMS∞states that we shall consider is the following.Definition2.2.A stateϕis a KMS∞state for(A,σt)if it is a weak limit ofβ-KMS states for β→∞.One can easily see the difference between these two notions in the case of the trivial evolutionσt= id,∀t∈R,where any state has the property that(2.2)extends to the upper half plane(as a constant),while weak limits ofβ-KMS states are automatically tracial states.With Definition2.2we still obtain a weakly compact convex setΣ∞and we can consider the set E∞of its extremal points.The typical framework for spontaneous symmetry breaking in a system with a unique phase transition (cf.[12])is that the simplexΣβconsists of a single point forβ≦βc i.e.when the temperature is larger than the critical temperature T c,and is non-trivial(of some higher dimension in general)when the temperature lowers.A(compact)group of automorphisms G⊂Aut(A)commuting with the time evolution,(2.3)σtαg=αgσt∀g∈G,t∈R,is a symmetry group of the system.Such G acts onΣβfor anyβ,hence on the extreme points E(Σβ)=Eβ.The choice of an equilibrium stateϕ∈Eβmay break this symmetry to a smaller subgroup given by the isotropy group Gϕ={g∈G,gϕ=ϕ}.The unitary group U of thefixed point algebra ofσt acts by inner automorphisms of the dynamical system(A,σt),by(2.4)(Ad u)(a):=u a u∗,∀a∈A,for all u∈U.One can define an action modulo inner of a group G on the system(A,σt)as a map α:G→Aut(A,σt)fulfilling the condition(2.5)α(gh)α(h)−1α(g)−1∈Inn(A,σt),∀g,h∈G,i.e.,as a homomorphism of G to Aut(A,σt)/U.The KMSβcondition shows that the inner automor-phisms Inn(A,σt)act trivially on KMSβstates,hence(2.5)induces an action of the group G on the setΣβof KMSβstates,for0<β≤∞.More generally,one can consider actions by endomorphisms(cf.[9]),where an endomorphismρof the dynamical system(A,σt)is a∗-homomorphismρ:A→A commuting with the evolutionσt.There is an induced action ofρon KMSβstates,for0<β<∞,given by(2.6)ρ∗(ϕ):=Z−1ϕ◦ρ,Z=ϕ(e),provided thatϕ(e)=0,where e=ρ(1)is an idempotentfixed byσt.An isometry u∈A,u∗u=1,satisfyingσt(u)=λit u for all t∈R and for someλ∈R∗+,defines an inner endomorphism Ad u of the dynamical system(A,σt),again of the form(2.4).The KMSβcondition shows that the induced action of Ad u onΣβis trivial,cf.[9].The induced action(modulo inner)of a semigroup of endomorphisms of(A,σt)on the KMSβstates in general may not extend directly to KMS∞states(in a nontrivial way),but it may be defined on E∞by“warming up and4CONNES,MARCOLLI,AND RAMACHANDRANcooling down”(cf.[9]),provided the“warming up”map Wβ:E∞→Eβis a bijection between KMS∞states(in the sense of Definition2.2)and KMSβstates,for sufficiently largeβ.The map is given byTr(πϕ(a)e−βH)(2.7)Wβ(ϕ)(a)=−d),for some positive integer d>1.We denote byˆZ the profinite completion of Z and by A f=ˆZ⊗Q the ring offinite adeles of Q.For any abelian group G,we denote by G tors the subgroup of elements offinite order.For any ring R,we write R∗for the group of invertible elements,while R×denotes the set of nonzero elements of R,which is a semigroup if R is an integral domain.We write O for the ring of algebraic integers of K.We setˆO:=(O⊗ˆZ)and write A K,f=A f⊗Q K and I K=A∗K,f=GL1(A K,f). Note that K∗embeds diagonally into I K.The classfield theory isomorphism provides the canonical identification(2.8)θ:I K/K∗∼−→Gal(K ab/K),with K∗replaced by Q∗+when K=Q.KMS AND CM5Fabulous states for numberfields.The connection between classfield theory and quantum statistical mechanics can be formulated as the problem of constructing a class of quantum statistical mechanical systems,whose set of ground states E∞has special arithmetic properties,because of which we refer to such states as“fabulous states”. Given a numberfield K,with a choice of an embedding K⊂C,the“problem of fabulous states”consists of constructing a C∗-dynamical system(A,σt),with an arithmetic subalgebra A Q of A,with the following properties:(1)The quotient group G=C K/D K acts on A as symmetries compatible withσt.(2)The statesϕ∈E∞,evaluated on elements of the arithmetic subalgebra A Q,satisfy:•ϕ(a)∈6CONNES,MARCOLLI,AND RAMACHANDRANaction(by S=R∗+in the1-dimensional case,or by S=C∗in the2-dimensional case),the algebra of coordinates associated to the quotient R/S is obtained by restricting the convolution product of thealgebra of R to weight zero functions with S-compact support.The algebra obtained this way,whichis unital in the1-dimensional case,but not in the2-dimensional case,has a natural time evolution given by the ratio of the covolumes of a pair of commensurable lattices.Every unit y∈R(0)of Rdefines a representationπy by left convolution of the algebra of R on the Hilbert space H y=ℓ2(R y),where R y is the set of elements with source y.This construction passes to the quotient by the scaling action of S.Representations corresponding to points that acquire a nontrivial automorphism groupwill no longer be irreducible.If the unit y∈R(0)corresponds to an invertible Q-lattice,thenπy is apositive energy representation.In both the1-dimensional and the2-dimensional case,the set of extremal KMS states at low tem-perature is given by a classical ad`e lic quotient,namely,by the Shimura varieties for GL1and GL2,respectively,hence we argue here that the noncommutative space describing commensurability classes of Q-lattices up to scale can be thought of as a noncommutative Shimura variety,whose set of classicalpoints is the corresponding classical Shimura variety.In both cases,a crucial step for the arithmetic properties of the action of symmetries on extremal KMS states at zero temperature is the choice of an arithmetic subalgebra of the system,on whichthe ground states are evaluated.Such choice gives the underlying noncommutative space a more rigidstructure,of“noncommutative arithmetic variety”.Tower Power.If V is an algebraic variety–or a scheme or a stack–over afield k,a“tower”T over V is a family V i(i∈I)offinite(possibly branched)covers of V such that for any i,j∈I,there is a l∈I with V l a cover of V i and V j.Thus,I is a partially ordered set.In case of a tower over a pointed variety(V,v),onefixes a point v i over v in each V i.Even though V i may not be irreducible,we shall allow ourselves to loosely refer to V i as a variety.It is convenient to view a“tower”T as a category C with objects(V i→V)and morphisms Hom(V i,V j)being maps of covers of V.One has the group Aut T(V i)of invertible self-maps of V i over V(the group of deck transformations);the deck transformations are not required to preserve the points v i.There is a(profinite)group of symmetries associated to a tower,namely(3.2)G:=lim←−i Aut T(V i).The simplest example of a tower is the“fundamental group”tower associated with a(smooth con-nected)complex algebraic variety(V,v)and its universal covering(˜V,˜v).Let C be the category of all finite´e tale(unbranched)intermediate covers˜V→W→V of V.In this case,the symmetry group G of(3.2)is the algebraic fundamental group of V;it is also the profinite completion of the(topological) fundamental groupπ1(V,v).Simple variants of this example include allowing controlled ramification. Other examples of towers are those defined by iteration of self maps of algebraic varieties.For us,the most important examples of“towers”will be the cyclotomic tower and the modular tower.Another very interesting case of towers is that of more general Shimura varieties.These,however,will not be treated in this paper.(For a more general treatment of noncommutative Shimura varieties see [11].)The cyclotomic tower and the BC system.In the case of Q,an explicit description of Q ab is provided by the Kronecker–Weber theorem.Thisshows that thefield Q ab is equal to Q cyc,thefield obtained by attaching all roots of unity to Q. Namely,Q ab is obtained by attaching the values of the exponential function exp(2πiz)at the torsion points of the circle group R/ing the isomorphism of abelian groups¯Q∗tors∼=Q/Z and the identification Aut(Q/Z)=GL1(ˆZ)=ˆZ∗,the restriction to¯Q∗tors of the natural action of Gal(¯Q/Q) on¯Q∗factors asGal(¯Q/Q)→Gal(¯Q/Q)ab=Gal(Q ab/Q)∼−→ˆZ∗.KMS AND CM7 Geometrically,the above setting can be understood in terms of the cyclotomic tower.This has base Spec Z=V1.The family is Spec Z[ζn]=V n whereζn is a primitive n-th root of unity(n∈N∗). The set Hom(V m→V n),non-trivial for n|m,corresponds to the map Z[ζn]֒→Z[ζm]of rings.Thegroup Aut(V n)=GL1(Z/n Z)is the Galois group Gal(Q(ζn)/Q).The group of symmetries(3.2)of the tower is then(3.3)G=lim←−n GL1(Z/n Z)=GL1(ˆZ),which is isomorphic to the Galois group Gal(Q ab/Q)of the maximal abelian extension of Q.The classical object that we consider,associated to the cyclotomic tower,is the Shimura variety given by the ad`e lic quotient(3.4)Sh(GL1,{±1})=GL1(Q)\(GL1(A f)×{±1})=A∗f/Q∗+.Now we consider the space of1-dimensional Q-lattices up to scaling modulo commensurability.This can be described as follows([9]).In one dimension,every Q-lattice is of the form(3.5)(Λ,φ)=(λZ,λρ),for someλ>0and someρ∈Hom(Q/Z,Q/Z).Since we can identify Hom(Q/Z,Q/Z)endowed with the topology of pointwise convergence with(3.6)Hom(Q/Z,Q/Z)=lim←−n Z/n Z=ˆZ,we obtain that the algebra C(ˆZ)is the algebra of coordinates of the space of1-dimensional Q-lattices up to scaling.The groupˆZ is the Pontrjagin dual of Q/Z,hence we also have an identification C(ˆZ)=C∗(Q/Z).The group of deck transformations G=ˆZ∗of the cyclotomic tower acts by automorphisms on the algebra of coordinates C(ˆZ).In addition to this action,there is a semigroup action of N×=Z>0 implementing the commensurability relation.This is given by endomorphisms that move vertically across the levels of the cyclotomic tower.They are given by(3.7)αn(f)(ρ)=f(n−1ρ),∀ρ∈nˆZ.Namely,αn is the isomorphism of C(ˆZ)with the reduced algebra C(ˆZ)πby the projectionπn givennby the characteristic function of nˆZ⊂ˆZ.Notice that the action(3.7)cannot be restricted to the set of invertible Q-lattices,since the range ofπn is disjoint from them.The algebra of coordinates A1on the noncommutative space of equivalence classes of1-dimensional Q-lattices modulo scaling,with respect to the equivalence relation of commensurability,is given then by the semigroup crossed product(3.8)A=C(ˆZ)⋊αN×.Equivalently,we are considering the convolution algebra of the groupoid R1/R∗+given by the quotient by scaling of the groupoid of the equivalence relation of commensurability on1-dimensional Q-lattices, namely,R1/R∗+has as algebra of coordinates the functions f(r,ρ),forρ∈ˆZ and r∈Q∗such that rρ∈ˆZ,with the convolution product(3.9)f1∗f2(r,ρ)= f1(rs−1,sρ)f2(s,ρ),and the adjoint f∗(r,ρ)=8CONNES,MARCOLLI,AND RAMACHANDRANAs a set,the space of commensurability classes of1-dimensional Q-lattices up to scaling can also be described by the quotient(3.10)GL1(Q)\A·/R∗+=GL1(Q)\(A f×{±1}),where A·:=A f×R∗is the set of ad`e les with nonzero archimedean component.Rather than considering this quotient set theoretically,we regard it as a noncommutative space,so as to be able to extend to it the ordinary tools of geometry that can be applied to the“good”quotient(3.4).The noncommutative algebra of coordinates of(3.10)is the crossed product(3.11)C0(A f)⋊Q∗+.This is Morita equivalent to the algebra(3.8).In fact,(3.8)is obtained as a full corner of(3.11),C(ˆZ)⋊N×= C0(A f)⋊Q∗+ π,by compression with the projectionπgiven by the characteristic function ofˆZ⊂A f(cf.[17]).The quotient(3.10)with its noncommutative algebra of coordinates(3.11)can then be thought of as the noncommutative Shimura variety(3.12)Sh(nc)(GL1,{±1}):=GL1(Q)\(A f×{±1})=GL1(Q)\A·/R∗+,whose set of classical points is the well behaved quotient(3.4).This has a compactification,obtained by replacing A·by A,as in[7],(3.13)−1)c(Z)=1.One considers the rational subalgebra A1,Q of(3.8)generated by the functions e1,a(r,ρ):=e1,a(ρ)and by the functionsµn(r,ρ)=1for r=n and zero otherwise,that implement the semigroup action of N×in(3.8).As proved in[9],the algebra A1,Q is the same as the rational subalgebra considered in[2],generated over Q by theµn and the exponential functions(3.17)e(r)(ρ):=exp(2πiρ(r)),forρ∈Hom(Q/Z,Q/Z),and r∈Q/Z,KMS AND CM9 with relations e(r+s)=e(r)e(s),e(0)=1,e(r)∗=e(−r),µ∗nµn=1,µkµn=µkn,and(3.18)µn e(r)µ∗n=1ζ(β)∞n=1n−βρ(ζk r).•The group GL1(ˆZ)acts by automorphisms of the system.The induced action of GL1(ˆZ)on the set of extreme KMS states below critical temperature is free and transitive.•The vacuum states(β=∞)are fabulous states for thefield K=Q,namelyϕ(A1,Q)⊂Q cycl and the classfield theory isomorphismθ:Gal(Q cycl/Q)∼=→ˆZ∗intertwines the Galois action on values with the action ofˆZ∗by symmetries,(3.21)γϕ(x)=ϕ(θ(γ)x),for allϕ∈E∞,for allγ∈Gal(Q cycl/Q)and for all x∈A1,Q.The modular tower and the GL2-system.Modular curves arise as moduli spaces of elliptic curves endowed with additional level structure.Every congruence subgroupΓ′ofΓ=SL2(Z)defines a modular curve YΓ′;we denote by XΓ′the smooth compactification of the affine curve YΓ′obtained by adding cusp points.Especially important among these are the modular curves Y(n)and X(n)corresponding to the principal congruence subgroups Γ(n)for n∈N∗.Any XΓ′is dominated by an X(n).We refer to[15,29]for more details.We have the following descriptions of the modular tower.10CONNES,MARCOLLI,AND RAMACHANDRANCompact version:The base is V=P1over Q.The family is given by the modular curves X(n),consid-ered over the cyclotomicfield Q(ζn)[23].We note that GL2(Z/n Z)/±1is the group of automorphisms of the projection V n=X(n)→X(1)=V1=V.Thus,we have(3.22)G=GL2(ˆZ)/±1=lim←−n GL2(Z/n Z)/{±1}.Non-compact version:The open modular curves Y(n)form a tower with base the j-line Spec Q[j]= A1=V1−{∞}.The ring of modular functions is the union of the rings of functions of the Y(n),with coefficients in Q(ζn)[15].This shows how the modular tower is a natural geometric way of passing from GL1(ˆZ)to GL2(ˆZ). The formulation that is most convenient in our setting is the one given in terms of Shimura varieties. In fact,rather than the modular tower defined by the projective limit(3.23)Y=lim←−n Y(n)of the modular curves Y(n),it is better for our purposes to consider the Shimura variety(3.24)Sh(H±,GL2)=GL2(Q)\(GL2(A f)×H±)=GL2(Q)\GL2(A)/C∗,of which(3.23)is a connected component.In fact,it is well known that,for arithmetic purposes,it is always better to work with nonconnected rather than with connected Shimura varieties(cf.e.g.[23]). The simple reason why it is necessary to pass to the nonconnected case is the following.The varieties in the tower are arithmetic varieties defined over numberfields.However,the numberfield typically changes along the levels of the tower(Y(n)is defined over the cyclotomicfield Q(ζn)).Passing to nonconnected Shimura varieties allows precisely for the definition of a canonical model where the whole tower is defined over the same numberfield.This distinction is important to our viewpoint,since we want to work with noncommutative spaces endowed with an arithmetic structure,specified by the choice of an arithmetic subalgebra.Every2-dimensional Q-lattice can be described by data(3.25)(Λ,φ)=(λ(Z+Z z),λα),for someλ∈C∗,some z∈H,andα∈M2(ˆZ)(using the basis(1,−z)of Z+Z z as in(87)[9]to view αas a mapφ).The diagonal action ofΓ=SL2(Z)yields isomorphic Q-lattices,and(cf.(87)[9])the space of2-dimensional Q-lattice up to scaling can be identified with the quotient(3.26)Γ\(M2(ˆZ)×H).The relation of commensurability is implemented by the partially defined action of GL+2(Q)on(3.26). The groupoid R2of the commensurability relation on2-dimensional Q-lattices not up to scaling(i.e. the dual space)has as algebra of coordinates the convolution algebra ofΓ×Γ-invariant functions on (3.27)˜U={(g,α,u)∈GL+2(Q)×M2(ˆZ)×GL+2(R)|gα∈M2(ˆZ)}.Up to Morita equivalence,this can also be described as the crossed product(3.28)C0(M2(A f)×GL2(R))⋊GL2(Q).When we pass to Q-lattices up to scaling,we take the quotient R2/C∗.If(Λk,φk)k=1,2are a pair of commensurable2-dimensional Q-lattices,then for anyλ∈C∗,the Q-lattices(λΛk,λφk)are also commensurable,withr(g,α,uλ)=λ−1r(g,α,u).However,the action of C∗on Q-lattices is not free due to the presence of lattices L=(0,z),where z∈Γ\H has nontrivial automorphisms.Thus,the quotient Z=R2/C∗is no longer a groupoid.This can be seen in the following sim-ple example.Consider the two Q-lattices(α1,z1)=(0,2i)and(α2,z2)=(0,i).The composite ((α1,z1),(α2,z2))◦((α2,z2),(α1,z1))is equal to the identity((α1,z1),(α1,z1)).We can also considerthe composition (i (α1,z 1),i (α2,z 2))◦((α2,z 2),(α1,z 1)),where i (α2,z 2)=(α2,z 2),but this is not the identity,since i (α1,z 1)=(α1,z 1).However,we can still consider the convolution algebra of Z ,by restricting the convolution product of R 2to homogeneous functions of weight zero with C ∗-compact support,where a function f has weight k if it satisfiesf (g,α,uλ)=λk f (g,α,u ),∀λ∈C ∗.This is the analog of the description (3.8)for the 1-dimensional case.The noncommutative algebra of coordinates A 2is thus given by a Hecke algebra of functions on(3.29)U ={(g,α,z )∈GL +2(Q )×M 2(ˆZ )×H ,gα∈M 2(ˆZ )}invariant under the Γ×Γaction(3.30)(g,α,z )→(γ1gγ−12,γ2α,γ2(z )),with convolution(3.31)(f 1∗f 2)(g,α,z )= s ∈Γ\GL +2(Q ),sα∈M 2(ˆZ )f 1(gs −1,sα,s (z ))f 2(s,α,z )and adjoint f ∗(g,α,z )=Another reformulation uses the Pontrjagin duality between profinite abelian groups and discrete tor-sion abelian groups given by Hom(−,Q/Z).This reformulates the datumφof a Q-lattice as aˆZ-linear map Hom(QΛ/Λ,Q/Z)→ˆZ⊕ˆZ,which is identified withΛ⊗ˆZ→ˆZ⊕ˆZ.Here we use the fact that ΛandΛ⊗ˆZ∼=H1(E,ˆZ)are both self-dual(Poincar´e duality of E).In this dual formulation com-mensurability means that the two maps agree on the intersection of the two commensurable lattices,(Λ1∩Λ2)⊗ˆZ.With the formulation of Proposition3.2,can then give a new interpretation of the result of Proposition 43of[9],which shows that the space of commensurability classes of2-dimensional Q-lattices up to scaling is described by the quotient(3.36)Sh(nc)(H±,GL2):=GL2(Q)\(M2(A f)×H±).In fact,the data(Λ,φ)of a Q-lattice in C are equivalent to data(E,η)of an elliptic curve E=C/Λand an A f-homomorphism(3.37)η:Q2⊗A f→Λ⊗A f,withΛ⊗A f=(Λ⊗ˆZ)⊗Q,where we can identifyΛ⊗ˆZ with the total Tate module of E,as in(3.35). Since the Q-lattice need not be invertible,we do not require thatηbe an A f-isomorphism(cf.[23]). The commensurability relation between Q-lattices corresponds to the equivalence(E,η)∼(E′,η′) given by an isogeny g:E→E′andη′=(g⊗1)◦η.Namely,the equivalence classes can be identified with the quotient of M2(A f)×H±by the action of GL2(Q),(ρ,z)→(gρ,g(z)).Thus,(3.36)describes a noncommutative Shimura variety which has the Shimura variety(3.24)as the set of its classical points.The results of[9]show that,as in the case of the BC system,one recovers the classical points from the low temperature extremal KMS states.We shall return to this in the next section.In this case,the“compactification”,analogous to passing from(3.10)to(3.14),corresponds to the replacing(3.36)by the noncommutative space(3.38)in[31].This suggests that modular functions should appear naturally in the arithmetic subalgebra A2,Q of the GL2-system,but that requires working with unbounded multipliers.This is indeed the case for the arithmetic subalgebra A2,Q defined in[9],which we now recall.Let F be the modularfield,namely thefield of modular functions over Q ab(cf.e.g.[19]).This is the union of thefields F N of modular functions of level N rational over the cyclotomicfield Q(ζN),that is,such that the q-expansion at a cusp has coefficients in the cyclotomicfield Q(ζN).The action of the Galois groupˆZ∗≃Gal(Q ab/Q)on the coefficients of the q-expansion determines a homomorphism(3.42)cycl:ˆZ∗→Aut(F).If f is a continuous functions on Z=R2/C∗,we writef(g,α)(z)=f(g,α,z)so that f(g,α)∈C(H).For p N:M2(ˆZ)→M2(Z/N Z)the canonical projection,we say that f is of level N iff(g,α)=f(g,p(α))∀(g,α).NThen f is completely determined by the functionsf(g,m)∈C(H),for m∈M2(Z/N Z).Notice that the invariance f(gγ,α,z)=f(g,γα,γ(z)),for allγ∈Γand for all(g,α,z)∈U,implies that f(g,m)|γ=f(g,m),for allγ∈Γ(N)∩g−1Γg,i.e.f is invariant under a congruence subgroup. The arithmetic algebra A2,Q defined in[9]is a subalgebra of continuous functions on Z=R2/C∗with the convolutions product(3.31)and with the properties:•The support of f inΓ\GL+2(Q)isfinite.•The function f is offinite level withf(g,m)∈F∀(g,m).•The function f satisfies the cyclotomic condition:f(g,α(u)m)=cycl(u)f(g,m),for all g∈GL+2(Q)diagonal and all u∈ˆZ∗,withα(u)= u001and cycl as in(3.42).Here F is the modularfield,namely thefield of modular functions over Q ab.This is the union of the fields F N of modular functions of level N rational over the cyclotomicfield Q(ζN).The cycloomic condition is a consistency condition on the roots of unity that appear in the coefficients of the q-series,which allows for the existence of“fabulous states”(cf.[9]).Forα∈M2(ˆZ),let Gα⊂GL+2(Q)be the set ofGα={g∈GL+2(Q):gα∈M2(ˆZ)}.Then,as shown in[9],an element y=(α,z)∈M2(ˆZ)×H determines a unitary representation of the Hecke algebra A on the Hilbert spaceℓ2(Γ\Gα),(3.43)((πy f)ξ)(g):= s∈Γ\Gαf(gs−1,sα,s(z))ξ(s),∀g∈Gαfor f∈A andξ∈ℓ2(Γ\Gα).Invertible Q-lattices determine positive energy representations,due to the fact that the condition gα∈M2(ˆZ)for g∈GL+2(Q)andα∈GL2(ˆZ)(invertible case)implies g∈M2(Z)+,hence the time evolution(3.32)is implemented by the positive Hamiltonian with spectrum{log det(m)}⊂[0,∞)for。