In Congressus Numerantium 97 (1993), pp. 51--70. Perfect Dominating Sets on Cube-Connected

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Collective dynamics of 'small-world' networks

Nature © Macmillan Publishers Ltd 19988typically slower than ϳ1km s −1)might differ signiﬁcantly from what is assumed by current modelling efforts 27.The expected equation-of-state differences among small bodies (ice versus rock,for instance)presents another dimension of study;having recently adapted our code for massively parallel architectures (K.M.Olson and E.A,manuscript in preparation),we are now ready to perform a more comprehensive analysis.The exploratory simulations presented here suggest that when a young,non-porous asteroid (if such exist)suffers extensive impact damage,the resulting fracture pattern largely deﬁnes the asteroid’s response to future impacts.The stochastic nature of collisions implies that small asteroid interiors may be as diverse as their shapes and spin states.Detailed numerical simulations of impacts,using accurate shape models and rheologies,could shed light on how asteroid collisional response depends on internal conﬁguration and shape,and hence on how planetesimals evolve.Detailed simulations are also required before one can predict the quantitative effects of nuclear explosions on Earth-crossing comets and asteroids,either for hazard mitigation 28through disruption and deﬂection,or for resource exploitation 29.Such predictions would require detailed reconnaissance concerning the composition andinternal structure of the targeted object.ⅪReceived 4February;accepted 18March 1998.1.Asphaug,E.&Melosh,H.J.The Stickney impact of Phobos:A dynamical model.Icarus 101,144–164(1993).2.Asphaug,E.et al .Mechanical and geological effects of impact cratering on Ida.Icarus 120,158–184(1996).3.Nolan,M.C.,Asphaug,E.,Melosh,H.J.&Greenberg,R.Impact craters on asteroids:Does strength orgravity control their size?Icarus 124,359–371(1996).4.Love,S.J.&Ahrens,T.J.Catastrophic impacts on gravity dominated asteroids.Icarus 124,141–155(1996).5.Melosh,H.J.&Ryan,E.V.Asteroids:Shattered but not dispersed.Icarus 129,562–564(1997).6.Housen,K.R.,Schmidt,R.M.&Holsapple,K.A.Crater ejecta scaling laws:Fundamental forms basedon dimensional analysis.J.Geophys.Res.88,2485–2499(1983).7.Holsapple,K.A.&Schmidt,R.M.Point source solutions and coupling parameters in crateringmechanics.J.Geophys.Res.92,6350–6376(1987).8.Housen,K.R.&Holsapple,K.A.On the fragmentation of asteroids and planetary satellites.Icarus 84,226–253(1990).9.Benz,W.&Asphaug,E.Simulations of brittle solids using smooth particle put.mun.87,253–265(1995).10.Asphaug,E.et al .Mechanical and geological effects of impact cratering on Ida.Icarus 120,158–184(1996).11.Hudson,R.S.&Ostro,S.J.Shape of asteroid 4769Castalia (1989PB)from inversion of radar images.Science 263,940–943(1994).12.Ostro,S.J.et al .Asteroid radar astrometry.Astron.J.102,1490–1502(1991).13.Ahrens,T.J.&O’Keefe,J.D.in Impact and Explosion Cratering (eds Roddy,D.J.,Pepin,R.O.&Merrill,R.B.)639–656(Pergamon,New York,1977).14.Tillotson,J.H.Metallic equations of state for hypervelocity impact.(General Atomic Report GA-3216,San Diego,1962).15.Nakamura,A.&Fujiwara,A.Velocity distribution of fragments formed in a simulated collisionaldisruption.Icarus 92,132–146(1991).16.Benz,W.&Asphaug,E.Simulations of brittle solids using smooth particle put.mun.87,253–265(1995).17.Bottke,W.F.,Nolan,M.C.,Greenberg,R.&Kolvoord,R.A.Velocity distributions among collidingasteroids.Icarus 107,255–268(1994).18.Belton,M.J.S.et al .Galileo encounter with 951Gaspra—First pictures of an asteroid.Science 257,1647–1652(1992).19.Belton,M.J.S.et al .Galileo’s encounter with 243Ida:An overview of the imaging experiment.Icarus120,1–19(1996).20.Asphaug,E.&Melosh,H.J.The Stickney impact of Phobos:A dynamical model.Icarus 101,144–164(1993).21.Asphaug,E.et al .Mechanical and geological effects of impact cratering on Ida.Icarus 120,158–184(1996).22.Housen,K.R.,Schmidt,R.M.&Holsapple,K.A.Crater ejecta scaling laws:Fundamental forms basedon dimensional analysis.J.Geophys.Res.88,2485–2499(1983).23.Veverka,J.et al .NEAR’s ﬂyby of 253Mathilde:Images of a C asteroid.Science 278,2109–2112(1997).24.Asphaug,E.et al .Impact evolution of icy regoliths.Lunar Planet.Sci.Conf.(Abstr.)XXVIII,63–64(1997).25.Love,S.G.,Ho¨rz,F.&Brownlee,D.E.Target porosity effects in impact cratering and collisional disruption.Icarus 105,216–224(1993).26.Fujiwara,A.,Cerroni,P .,Davis,D.R.,Ryan,E.V.&DiMartino,M.in Asteroids II (eds Binzel,R.P .,Gehrels,T.&Matthews,A.S.)240–265(Univ.Arizona Press,Tucson,1989).27.Davis,D.R.&Farinella,P.Collisional evolution of Edgeworth-Kuiper Belt objects.Icarus 125,50–60(1997).28.Ahrens,T.J.&Harris,A.W.Deﬂection and fragmentation of near-Earth asteroids.Nature 360,429–433(1992).29.Resources of Near-Earth Space (eds Lewis,J.S.,Matthews,M.S.&Guerrieri,M.L.)(Univ.ArizonaPress,Tucson,1993).Acknowledgements.This work was supported by NASA’s Planetary Geology and Geophysics Program.Correspondence and requests for materials should be addressed to E.A.(e-mail:asphaug@).letters to nature440NATURE |VOL 393|4JUNE 1998Collective dynamics of ‘small-world’networksDuncan J.Watts *&Steven H.StrogatzDepartment of Theoretical and Applied Mechanics,Kimball Hall,Cornell University,Ithaca,New York 14853,USA.........................................................................................................................Networks of coupled dynamical systems have been used to model biological oscillators 1–4,Josephson junction arrays 5,6,excitable media 7,neural networks 8–10,spatial games 11,genetic control networks 12and many other self-organizing systems.Ordinarily,the connection topology is assumed to be either completely regular or completely random.But many biological,technological and social networks lie somewhere between these two extremes.Here we explore simple models of networks that can be tuned through this middle ground:regular networks ‘rewired’to intro-duce increasing amounts of disorder.We ﬁnd that these systems can be highly clustered,like regular lattices,yet have small characteristic path lengths,like random graphs.We call them ‘small-world’networks,by analogy with the small-world phenomenon 13,14(popularly known as six degrees of separation 15).The neural network of the worm Caenorhabditis elegans ,the power grid of the western United States,and the collaboration graph of ﬁlm actors are shown to be small-world networks.Models of dynamical systems with small-world coupling display enhanced signal-propagation speed,computational power,and synchronizability.In particular,infectious diseases spread more easily in small-world networks than in regular lattices.To interpolate between regular and random networks,we con-sider the following random rewiring procedure (Fig.1).Starting from a ring lattice with n vertices and k edges per vertex,we rewire each edge at random with probability p .This construction allows us to ‘tune’the graph between regularity (p ¼0)and disorder (p ¼1),and thereby to probe the intermediate region 0Ͻp Ͻ1,about which little is known.We quantify the structural properties of these graphs by their characteristic path length L (p )and clustering coefﬁcient C (p ),as deﬁned in Fig.2legend.Here L (p )measures the typical separation between two vertices in the graph (a global property),whereas C (p )measures the cliquishness of a typical neighbourhood (a local property).The networks of interest to us have many vertices with sparse connections,but not so sparse that the graph is in danger of becoming disconnected.Speciﬁcally,we require n q k q ln ðn Þq 1,where k q ln ðn Þguarantees that a random graph will be connected 16.In this regime,we ﬁnd that L ϳn =2k q 1and C ϳ3=4as p →0,while L ϷL random ϳln ðn Þ=ln ðk Þand C ϷC random ϳk =n p 1as p →1.Thus the regular lattice at p ¼0is a highly clustered,large world where L grows linearly with n ,whereas the random network at p ¼1is a poorly clustered,small world where L grows only logarithmically with n .These limiting cases might lead one to suspect that large C is always associated with large L ,and small C with small L .On the contrary,Fig.2reveals that there is a broad interval of p over which L (p )is almost as small as L random yet C ðp Þq C random .These small-world networks result from the immediate drop in L (p )caused by the introduction of a few long-range edges.Such ‘short cuts’connect vertices that would otherwise be much farther apart than L random .For small p ,each short cut has a highly nonlinear effect on L ,contracting the distance not just between the pair of vertices that it connects,but between their immediate neighbourhoods,neighbourhoods of neighbourhoods and so on.By contrast,an edge*Present address:Paul zarsfeld Center for the Social Sciences,Columbia University,812SIPA Building,420W118St,New York,New York 10027,USA.Nature © Macmillan Publishers Ltd 19988letters to natureNATURE |VOL 393|4JUNE 1998441removed from a clustered neighbourhood to make a short cut has,at most,a linear effect on C ;hence C (p )remains practically unchanged for small p even though L (p )drops rapidly.The important implica-tion here is that at the local level (as reﬂected by C (p )),the transition to a small world is almost undetectable.To check the robustness of these results,we have tested many different types of initial regular graphs,as well as different algorithms for random rewiring,and all give qualitatively similar results.The only requirement is that the rewired edges must typically connect vertices that would otherwise be much farther apart than L random .The idealized construction above reveals the key role of short cuts.It suggests that the small-world phenomenon might be common in sparse networks with many vertices,as even a tiny fraction of short cuts would sufﬁce.To test this idea,we have computed L and C for the collaboration graph of actors in feature ﬁlms (generated from data available at ),the electrical power grid of the western United States,and the neural network of the nematode worm C.elegans 17.All three graphs are of scientiﬁc interest.The graph of ﬁlm actors is a surrogate for a social network 18,with the advantage of being much more easily speciﬁed.It is also akin to the graph of mathematical collaborations centred,traditionally,on P.Erdo¨s (partial data available at /ϳgrossman/erdoshp.html).The graph of the power grid is relevant to the efﬁciency and robustness of power networks 19.And C.elegans is the sole example of a completely mapped neural network.Table 1shows that all three graphs are small-world networks.These examples were not hand-picked;they were chosen because of their inherent interest and because complete wiring diagrams were available.Thus the small-world phenomenon is not merely a curiosity of social networks 13,14nor an artefact of an idealizedmodel—it is probably generic for many large,sparse networks found in nature.We now investigate the functional signiﬁcance of small-world connectivity for dynamical systems.Our test case is a deliberately simpliﬁed model for the spread of an infectious disease.The population structure is modelled by the family of graphs described in Fig.1.At time t ¼0,a single infective individual is introduced into an otherwise healthy population.Infective individuals are removed permanently (by immunity or death)after a period of sickness that lasts one unit of dimensionless time.During this time,each infective individual can infect each of its healthy neighbours with probability r .On subsequent time steps,the disease spreads along the edges of the graph until it either infects the entire population,or it dies out,having infected some fraction of the population in theprocess.p = 0p = 1Regular Small-worldRandomFigure 1Random rewiring procedure for interpolating between a regular ring lattice and a random network,without altering the number of vertices or edges in the graph.We start with a ring of n vertices,each connected to its k nearest neighbours by undirected edges.(For clarity,n ¼20and k ¼4in the schematic examples shown here,but much larger n and k are used in the rest of this Letter.)We choose a vertex and the edge that connects it to its nearest neighbour in a clockwise sense.With probability p ,we reconnect this edge to a vertex chosen uniformly at random over the entire ring,with duplicate edges forbidden;other-wise we leave the edge in place.We repeat this process by moving clockwise around the ring,considering each vertex in turn until one lap is completed.Next,we consider the edges that connect vertices to their second-nearest neighbours clockwise.As before,we randomly rewire each of these edges with probability p ,and continue this process,circulating around the ring and proceeding outward to more distant neighbours after each lap,until each edge in the original lattice has been considered once.(As there are nk /2edges in the entire graph,the rewiring process stops after k /2laps.)Three realizations of this process are shown,for different values of p .For p ¼0,the original ring is unchanged;as p increases,the graph becomes increasingly disordered until for p ¼1,all edges are rewired randomly.One of our main results is that for intermediate values of p ,the graph is a small-world network:highly clustered like a regular graph,yet with small characteristic path length,like a random graph.(See Fig.2.)T able 1Empirical examples of small-world networksL actual L random C actual C random.............................................................................................................................................................................Film actors 3.65 2.990.790.00027Power grid 18.712.40.0800.005C.elegans 2.65 2.250.280.05.............................................................................................................................................................................Characteristic path length L and clustering coefﬁcient C for three real networks,compared to random graphs with the same number of vertices (n )and average number of edges per vertex (k ).(Actors:n ¼225;226,k ¼61.Power grid:n ¼4;941,k ¼2:67.C.elegans :n ¼282,k ¼14.)The graphs are deﬁned as follows.Two actors are joined by an edge if they have acted in a ﬁlm together.We restrict attention to the giant connected component 16of this graph,which includes ϳ90%of all actors listed in the Internet Movie Database (available at ),as of April 1997.For the power grid,vertices represent generators,transformers and substations,and edges represent high-voltage transmission lines between them.For C.elegans ,an edge joins two neurons if they are connected by either a synapse or a gap junction.We treat all edges as undirected and unweighted,and all vertices as identical,recognizing that these are crude approximations.All three networks show the small-world phenomenon:L ՌL random but C q C random.00.20.40.60.810.00010.0010.010.11pFigure 2Characteristic path length L (p )and clustering coefﬁcient C (p )for the family of randomly rewired graphs described in Fig.1.Here L is deﬁned as the number of edges in the shortest path between two vertices,averaged over all pairs of vertices.The clustering coefﬁcient C (p )is deﬁned as follows.Suppose that a vertex v has k v neighbours;then at most k v ðk v Ϫ1Þ=2edges can exist between them (this occurs when every neighbour of v is connected to every other neighbour of v ).Let C v denote the fraction of these allowable edges that actually exist.Deﬁne C as the average of C v over all v .For friendship networks,these statistics have intuitive meanings:L is the average number of friendships in the shortest chain connecting two people;C v reﬂects the extent to which friends of v are also friends of each other;and thus C measures the cliquishness of a typical friendship circle.The data shown in the ﬁgure are averages over 20random realizations of the rewiring process described in Fig.1,and have been normalized by the values L (0),C (0)for a regular lattice.All the graphs have n ¼1;000vertices and an average degree of k ¼10edges per vertex.We note that a logarithmic horizontal scale has been used to resolve the rapid drop in L (p ),corresponding to the onset of the small-world phenomenon.During this drop,C (p )remains almost constant at its value for the regular lattice,indicating that the transition to a small world is almost undetectable at the local level.Nature © Macmillan Publishers Ltd 19988letters to nature442NATURE |VOL 393|4JUNE 1998Two results emerge.First,the critical infectiousness r half ,at which the disease infects half the population,decreases rapidly for small p (Fig.3a).Second,for a disease that is sufﬁciently infectious to infect the entire population regardless of its structure,the time T (p )required for global infection resembles the L (p )curve (Fig.3b).Thus,infectious diseases are predicted to spread much more easily and quickly in a small world;the alarming and less obvious point is how few short cuts are needed to make the world small.Our model differs in some signiﬁcant ways from other network models of disease spreading 20–24.All the models indicate that net-work structure inﬂuences the speed and extent of disease transmis-sion,but our model illuminates the dynamics as an explicit function of structure (Fig.3),rather than for a few particular topologies,such as random graphs,stars and chains 20–23.In the work closest to ours,Kretschmar and Morris 24have shown that increases in the number of concurrent partnerships can signiﬁcantly accelerate the propaga-tion of a sexually-transmitted disease that spreads along the edges of a graph.All their graphs are disconnected because they ﬁx the average number of partners per person at k ¼1.An increase in the number of concurrent partnerships causes faster spreading by increasing the number of vertices in the graph’s largest connected component.In contrast,all our graphs are connected;hence the predicted changes in the spreading dynamics are due to more subtle structural features than changes in connectedness.Moreover,changes in the number of concurrent partners are obvious to an individual,whereas transitions leading to a smaller world are not.We have also examined the effect of small-world connectivity on three other dynamical systems.In each case,the elements were coupled according to the family of graphs described in Fig.1.(1)For cellular automata charged with the computational task of density classiﬁcation 25,we ﬁnd that a simple ‘majority-rule’running on a small-world graph can outperform all known human and genetic algorithm-generated rules running on a ring lattice.(2)For the iterated,multi-player ‘Prisoner’s dilemma’11played on a graph,we ﬁnd that as the fraction of short cuts increases,cooperation is less likely to emerge in a population of players using a generalized ‘tit-for-tat’26strategy.The likelihood of cooperative strategies evolving out of an initial cooperative/non-cooperative mix also decreases with increasing p .(3)Small-world networks of coupled phase oscillators synchronize almost as readily as in the mean-ﬁeld model 2,despite having orders of magnitude fewer edges.This result may be relevant to the observed synchronization of widely separated neurons in the visual cortex 27if,as seems plausible,the brain has a small-world architecture.We hope that our work will stimulate further studies of small-world networks.Their distinctive combination of high clustering with short characteristic path length cannot be captured by traditional approximations such as those based on regular lattices or random graphs.Although small-world architecture has not received much attention,we suggest that it will probably turn out to be widespread in biological,social and man-made systems,oftenwith important dynamical consequences.ⅪReceived 27November 1997;accepted 6April 1998.1.Winfree,A.T.The Geometry of Biological Time (Springer,New Y ork,1980).2.Kuramoto,Y.Chemical Oscillations,Waves,and Turbulence (Springer,Berlin,1984).3.Strogatz,S.H.&Stewart,I.Coupled oscillators and biological synchronization.Sci.Am.269(6),102–109(1993).4.Bressloff,P .C.,Coombes,S.&De Souza,B.Dynamics of a ring of pulse-coupled oscillators:a group theoretic approach.Phys.Rev.Lett.79,2791–2794(1997).5.Braiman,Y.,Lindner,J.F.&Ditto,W.L.Taming spatiotemporal chaos with disorder.Nature 378,465–467(1995).6.Wiesenfeld,K.New results on frequency-locking dynamics of disordered Josephson arrays.Physica B 222,315–319(1996).7.Gerhardt,M.,Schuster,H.&Tyson,J.J.A cellular automaton model of excitable media including curvature and dispersion.Science 247,1563–1566(1990).8.Collins,J.J.,Chow,C.C.&Imhoff,T.T.Stochastic resonance without tuning.Nature 376,236–238(1995).9.Hopﬁeld,J.J.&Herz,A.V.M.Rapid local synchronization of action potentials:Toward computation with coupled integrate-and-ﬁre neurons.Proc.Natl A 92,6655–6662(1995).10.Abbott,L.F.&van Vreeswijk,C.Asynchronous states in neural networks of pulse-coupled oscillators.Phys.Rev.E 48(2),1483–1490(1993).11.Nowak,M.A.&May,R.M.Evolutionary games and spatial chaos.Nature 359,826–829(1992).12.Kauffman,S.A.Metabolic stability and epigenesis in randomly constructed genetic nets.J.Theor.Biol.22,437–467(1969).gram,S.The small world problem.Psychol.Today 2,60–67(1967).14.Kochen,M.(ed.)The Small World (Ablex,Norwood,NJ,1989).15.Guare,J.Six Degrees of Separation:A Play (Vintage Books,New Y ork,1990).16.Bollaba´s,B.Random Graphs (Academic,London,1985).17.Achacoso,T.B.&Yamamoto,W.S.AY’s Neuroanatomy of C.elegans for Computation (CRC Press,BocaRaton,FL,1992).18.Wasserman,S.&Faust,K.Social Network Analysis:Methods and Applications (Cambridge Univ.Press,1994).19.Phadke,A.G.&Thorp,puter Relaying for Power Systems (Wiley,New Y ork,1988).20.Sattenspiel,L.&Simon,C.P .The spread and persistence of infectious diseases in structured populations.Math.Biosci.90,341–366(1988).21.Longini,I.M.Jr A mathematical model for predicting the geographic spread of new infectious agents.Math.Biosci.90,367–383(1988).22.Hess,G.Disease in metapopulation models:implications for conservation.Ecology 77,1617–1632(1996).23.Blythe,S.P .,Castillo-Chavez,C.&Palmer,J.S.T oward a uniﬁed theory of sexual mixing and pair formation.Math.Biosci.107,379–405(1991).24.Kretschmar,M.&Morris,M.Measures of concurrency in networks and the spread of infectious disease.Math.Biosci.133,165–195(1996).25.Das,R.,Mitchell,M.&Crutchﬁeld,J.P .in Parallel Problem Solving from Nature (eds Davido,Y.,Schwefel,H.-P.&Ma¨nner,R.)344–353(Lecture Notes in Computer Science 866,Springer,Berlin,1994).26.Axelrod,R.The Evolution of Cooperation (Basic Books,New Y ork,1984).27.Gray,C.M.,Ko¨nig,P .,Engel,A.K.&Singer,W.Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reﬂects global stimulus properties.Nature 338,334–337(1989).Acknowledgements.We thank B.Tjaden for providing the ﬁlm actor data,and J.Thorp and K.Bae for the Western States Power Grid data.This work was supported by the US National Science Foundation (Division of Mathematical Sciences).Correspondence and requests for materials should be addressed to D.J.W.(e-mail:djw24@).0.150.20.250.30.350.00010.0010.010.11rhalfpaFigure 3Simulation results for a simple model of disease spreading.The community structure is given by one realization of the family of randomly rewired graphs used in Fig.1.a ,Critical infectiousness r half ,at which the disease infects half the population,decreases with p .b ,The time T (p )required for a maximally infectious disease (r ¼1)to spread throughout the entire population has essen-tially the same functional form as the characteristic path length L (p ).Even if only a few per cent of the edges in the original lattice are randomly rewired,the time to global infection is nearly as short as for a random graph.0.20.40.60.810.00010.0010.010.11pb。

可持续性资本理论

B OSTON U NIVERSITY Center for Energy and Environmental Studies Working Papers SeriesNumber 9501 September 1995 THE CAPITAL THEORY APPROACH TO SUSTAINABILITY:A CRITICAL APPRAISALbyDavid Stern675 Commonwealth Avenue, Boston MA 02215Tel: (617) 353-3083Fax: (617) 353-5986E-Mail: dstern@WWW: /sterncv.htmlThe Capital Theory Approach to Sustainability:A Critical AppraisalDavid I. SternBoston UniversityNovember 1995______________________________________________________________________________ Center for Energy and Environmental Studies, Boston University, 675 Commonwealth Avenue, Boston MA 02215, USA. Tel: (617) 353 3083 Fax: (617) 353 5986, E-Mail: dstern@The Capital Theory Approach to Sustainability:A Critical Appraisal______________________________________________________________________________ SummaryThis paper examines critically some recent developments in the sustainability debate. The large number of definitions of sustainability proposed in the 1980's have been refined into a smaller number of positions on the relevant questions in the 1990's. The most prominent of these are based on the idea of maintaining a capital stock. I call this the capital theory approach (CTA). Though these concepts are beginning to inform policies there are a number of difficulties in applying this approach in a theoretically valid manner and a number of critics of the use of the CTA as a guide to policy. First, I examine the internal difficulties with the CTA and continue to review criticisms from outside the neoclassical normative framework. The accounting approach obscures the underlying assumptions used and gives undue authoritativeness to the results. No account is taken of the uncertainty involved in sustainability analysis of any sort. In addition, by focusing on a representative consumer and using market (or contingent market) valuations of environmental resources, the approach (in common with most normative neoclassical economics) does not take into account distributional issues or accommodate alternative views on environmental values. Finally, I examine alternative approaches to sustainability analysis and policy making. These approaches accept the open-ended and multi-dimensional nature of sustainability and explicitly open up to political debate the questions that are at risk of being hidden inside the black-box of seemingly objective accounting.I.INTRODUCTIONThe Brundtland Report (WCED, 1987) proposed that sustainable development is "development that meets the needs of the present generation while letting future generations meet their own needs". Economists initially had some difficulty with this concept, some dismissing it1 and others proliferating a vast number of alternative definitions and policy prescriptions (see surveys by: Pezzey, 1989; Pearce et al., 1989; Rees, 1990; Lélé, 1991).In recent years, economists have made some progress in articulating their conception of sustainability. The large number of definitions of sustainability proposed in the 1980's have been refined into a smaller number of positions on the relevant questions in the 1990's. There is agreement that sustainability implies that certain indicators of welfare or development are non-declining over the very long term, that is development is sustained (Pezzey, 1989). Sustainable development is a process of change in an economy that does not violate such a sustainability criterion. Beyond this, the dominant views are based on the idea of maintaining a capital stock as a prerequisite for sustainable development. Within this school of thought there are opposing camps which disagree on the empirical question of the degree to which various capital stocks can be substituted for each other, though there has been little actual empirical research on this question.There is a consensus among a large number of economists that the CTA is a useful means of addressing sustainability issues.2 Capital theory concepts are beginning to inform policy, as in the case of the UN recommendations on environmental accounting and the US response to them (Beardsley, 1994; Carson et al., 1994; Steer and Lutz, 1993). There are, however, a growing number of critics who question whether this is a useful way to address sustainability (eg. Norgaard, 1991; Amir, 1992; Common and Perrings, 1992; Karshenas, 1994; Pezzey, 1994; Common and Norton, 1994; Faucheux et al., 1994; Common, 1995). The literature on sustainable development and sustainability is vast and continually expanding. There are also a large number ofsurveys of that literature (eg. Tisdell, 1988; Pearce et al., 1989; Rees, 1990; Simonis, 1990; Lélé, 1991; Costanza and Daly, 1992; Pezzey, 1992; Toman et al., 1994). I do not intend to survey this literature.The aim of this paper is to present a critique of the capital theory approach to sustainability (CTA henceforth) as a basis for policy. This critique both outlines the difficulties in using and applying the CTA from a viewpoint internal to neoclassical economics and problems with this approach from a viewpoint external to neoclassical economics. I also suggest some alternative approaches to sustainability relevant analysis and policy. The neoclasscial sustainability literature generally ignores the international dimensions of the sustainability problem. I also ignore this dimension in this paper.The paper is structured as follows. In the second section, I discuss the background to the emergence of the capital theory approach, while the third section briefly outlines the basic features of the approach. The fourth section examines the limitations of the CTA from within the viewpoint of neoclassical economics and the debate between proponents of "weak sustainability" and "strong sustainability". The following sections examine the drawbacks of this paradigm from a viewpoint external to neoclassical economics and discuss alternative methods of analysis and decision-making for sustainability. The concluding section summarizes the principal points.SHIFTING DEBATE: EMERGENCE OF THE CAPITAL THEORY II. THEAPPROACHMuch of the literature on sustainable development published in the 1980's was vague (see Lélé, 1991; Rees, 1990; Simonis, 1990). There was a general lack of precision and agreement in defining sustainability, and outlining appropriate sustainability policies. This confusion stemmed in part from an imprecise demarcation between ends and means. By "ends" I mean the definition ofsustainability ie. what is to be sustained, while "means" are the methods to achieve sustainability or necessary and/or sufficient conditions that must be met in order to do the same. As the goal of policy must be a subjective choice, considerable debate surrounded and continues to surround the definition of sustainability (eg. Tisdell, 1988). As there is considerable scientific uncertainty regarding sustainability possibilities, considerable debate continues to surround policies to achieve any given goal.Sharachchandra Lélé (1991) stated that "sustainable development is in real danger of becoming a cliché like appropriate technology - a fashionable phrase that everyone pays homage to but nobody cares to define" (607). Lélé pointed out that different authors and speakers meant very different things by sustainability, and that even UNEP's and WCED's definitions of sustainable development were vague, and confused ends with means. Neither provided any scientific examination of whether their proposed policies would lead to increased sustainability. "Where the sustainable development movement has faltered is in its inability to develop a set of concepts, criteria and policies that are coherent or consistent - both externally (with physical and social reality) and internally (with each other)." (613). Judith Rees (1990) expressed extreme skepticism concerning both sustainable development and its proponents. “It is easy to see why the notion of sustainable development has become so popular ... No longer does environmental protection mean sacrifice and confrontation with dominant materialist values” (435). She also argued that sustainable development was just so much political rhetoric. A UNEP report stated: "The ratio of words to action is weighted too heavily towards the former" (quoted in Simonis, 1990, 35). In the early days of the sustainability debate, vagueness about the meaning of sustainability was advantageous in attracting the largest constituency possible, but in the longer run, greater clarity is essential for sustaining concern.In the 1990's many people have put forward much more precisely articulated definitions of sustainable development, conditions and policies required to achieve sustainability, and criteria toassess whether development is sustainable. This has coincided with a shift from a largely politically-driven dialogue to a more theory-driven dialogue. With this has come a clearer understanding of what kinds of policies would be required to move towards alternative sustainability goals, and what the limits of our knowledge are. There is a stronger awareness of the distinction between ends and means. Most, but not all (eg. Amir, 1992), analysts agree that sustainable development is a meaningful concept but that the claims of the Brundtland Report (WCED, 1987) that growth just had to change direction were far too simplistic.There is a general consensus, especially among economists, on the principal definition of sustainable development used by David Pearce et al. (1989, 1991): Non-declining average human welfare over time (Mäler, 1991; Pezzey, 1992; Toman et al., 1994).3 This definition of sustainability implies a departure from the strict principle of maximizing net present value in traditional cost benefit analysis (Pezzey, 1989), but otherwise it does not imply a large departure from conventional economics. John Pezzey (1989, 1994) suggests a rule of maximizing net present value subject to the sustainability constraint of non-declining mean welfare. It encompasses many but not all definitions of sustainability. For example, it excludes a definition of sustainability based on maintaining a set of ecosystem functions, which seems to be implied by the Holling-sustainability criterion (Common and Perrings, 1992; Holling, 1973, 1986) or on maintaining given stocks of natural assets irrespective of any contribution to human welfare. A sustainable ecosystem might not be an undesirable goal but it could be too strict a criterion for the goal of maintaining human welfare (Karshenas, 1994) and could in some circumstances lead to declining human welfare. Not all ecosystem functions and certainly not all natural assets may be necessary for human welfare. Some aspects of the natural world such as smallpox bacteria may be absolutely detrimental to people. In the context of the primary Pearce et al. definition, the Holling-sustainability criterion is a means not an end.The advantage of formalizing the concept of sustainability is that this renders it amenable to analysis by economic theory (eg. Barbier and Markandya, 1991; Victor, 1991; Common and Perrings, 1992; Pezzey, 1989, 1994; Asheim, 1994) and to quantitative investigations (eg. Repetto et al., 1989; Pearce and Atkinson, 1993; Proops and Atkinson, 1993; Stern, 1995). Given the above formal definition of sustainability, many economists have examined what the necessary or sufficient conditions for the achievement of sustainability might be. Out of this activity has come the CTA described in the next section. The great attractiveness of this new approach is that it suggests relatively simple rules to ensure sustainability and relatively simple indicators of sustainability. This situation has seemingly cleared away the vagueness that previously attended discussions of sustainability and prompted relatively fast action by governments and international organizations to embrace specific goals and programs aimed at achieving this notion of the necessary conditions for sustainability.III. THE ESSENCE OF THE CAPITAL THEORY APPROACHThe origins of the CTA are in the literature on economic growth and exhaustible resources that flourished in the 1970s, exemplified by the special issue of the Review of Economic Studies published in 1974 (Heal, 1974). Robert Solow (1986) built on this earlier literature and the work of John Hartwick (1977, 1978a, 1978b) to formalize the constant capital rule. In these early models there was a single non-renewable resource and a stock of manufactured capital goods. A production function produced a single output, which could be used for either consumption or investment using the two inputs. The elasticity of substitution between the two inputs was one which implied that natural resources were essential but that the average product of resources could rise without bound given sufficient manufactured capital.The models relate to the notion of sustainability as non-declining welfare through the assumption that welfare is a monotonically increasing function of consumption (eg. Mäler, 1991). The path ofconsumption over time (and therefore of the capital stock) in these model economies depends on the intertemporal optimization rule. Under the Rawlsian maxi-min condition consumption must be constant. No net saving is permissible as this is regarded as an unjust burden on the present generation. Under the Ramsey utilitarian approach with zero discounting consumption can increase without bound (Solow, 1974). Here the present generation may be forced to accept a subsistence standard of living if this can benefit the future generations however richer they might be. Paths that maximize net present value with positive discount rates typically peak and then decline so that they are not sustainable (Pezzey, 1994). Pezzey (1989) suggested a hybrid version which maximizes net present value subject to an intertemporal constraint that utility be non-declining. In this case utility will first increase until it reaches a maximum sustainable level. This has attracted consensus as the general optimizing criterion for sustainable development. Geir Asheim (1991) derives this condition more formally.Under the assumption that the elasticity of substitution is one, non-declining consumption depends on the maintenance of the aggregate capital stock ie. conventional capital plus natural resources, used to produce consumption (and investment) goods (Solow, 1986). Aggregate capital, W t,and the change in aggregate capital are defined by:W t=p Kt K t + p Rt S t (1)∆W t=p Kt∆K t + p Rt R t (2)where S is the stock of non-renewable resources and R the use per period. K is the manufactured capital stock and the p i are the relevant prices. In the absence of depreciation of manufactured capital, maintenance of the capital stock implies investment of the rents from the depletion of the natural resource in manufactured capital - the Hartwick rule (Hartwick 1977, 1978a, 1978b). Income is defined using the Hicksian notion (Hicks, 1946) that income is the maximum consumption in a period consistent with the maintenance of wealth. Sustainable income is,therefore, the maximum consumption in a period consistent with the maintenance of aggregate capital intact (Weitzman, 1976; Mäler, 1991) and for a flow of income to be sustainable, the stock of capital needs to be constant or increasing over time (Solow, 1986).The initial work can be extended in various ways. The definition of capital that satisfies these conditions can be extended to include a number of categories of "capital": natural, manufactured, human, and institutional.4 Natural capital is a term used by many authors (it seems Smith (1977) was the first) for the aggregate of natural resource stocks that produce inputs of services or commodities for the economy. Some of the components of natural capital may be renewable resources. Manufactured capital refers to the standard neoclassical definition of "a factor of production produced by the economic system" (Pearce, 1992). Human capital also follows the standard definition. Institutional capital includes the institutions and knowledge necessary for the organization and reproduction of the economic system. It includes the ethical or moral capital referred to by Fred Hirsch (1976) and the cultural capital referred to by Fikret Berkes and Carl Folke (1992). For convenience I give the name 'artificial capital' to the latter three categories jointly. None of these concepts is unproblematic and natural capital is perhaps the most problematic. Technical change and population growth can also be accommodated (see Solow, 1986).Empirical implementation of the CTA tends to focus on measurement of sustainable income (eg. El Serafy, 1989; Repetto, 1989) or net capital accumulation (eg. Pearce and Atkinson, 1993; Proops and Atkinson, 1993) rather than on direct estimation of the capital stock.5 The theoretical models that underpin the CTA typically assume a Cobb-Douglas production function with constant returns to scale, no population growth, and no technological change. Any indices of net capital accumulation which attempt to make even a first approximation to reality must take these variables into account. None of the recent empirical studies does so. For example, David Pearce and Giles Atkinson (1993) present data from eighteen countries on savings and depreciation of natural andmanufactured capital as a proportion of GNP. They demonstrate that only eight countries had non-declining stocks of total capital, measured at market prices, and thus passed a weak sustainability criterion of a constant aggregate capital stock, but their methodology ignores population growth, returns to scale or technological change.IV.INTERNAL APPRAISAL OF THE CAPITAL THEORY APPROACHIn this section, I take as given the basic assumptions and rationale of neoclassical economics and highlight some of the technical problems that are encountered in using the CTA as an operational guide to policy. From a neoclassical standpoint these might be seen as difficulties in the positive theory that may lead to difficulties in the normative theory of sustainability policy. In the following section, I take as given solutions to these technical difficulties and examine some of the problems inherent in the normative neoclassical approach to sustainability.a.Limits to Substitution in Production and "Strong Sustainability"Capital theorists are divided among proponents of weak sustainability and strong sustainability. This terminology is confusing as it suggests that the various writers have differing ideas of what sustainability is.6 In fact they agree on that issue, but differ on what is the minimum set of necessary conditions for achieving sustainability. The criterion that distinguishes the categories is the degree of substitutability believed to be possible between natural and artificial capital.7The weak sustainability viewpoint follows from the early literature and holds that the relevant capital stock is an aggregate stock of artificial and natural capital. Weak sustainability assumes that the elasticity of substitution between natural capital and artificial capital is one and therefore that there are no natural resources that contribute to human welfare that cannot be asymptotically replaced by other forms of capital. Reductions in natural capital may be offset by increases inartificial capital. It is sometimes implied that this might be not only a necessary condition but also a sufficient condition for achieving sustainability (eg. Solow, 1986, 1993).Proponents of the strong sustainability viewpoint such as Robert Costanza and Herman Daly (1992) argue that though this is a necessary condition for sustainability it cannot possibly be a sufficient condition. Instead, a minimum necessary condition is that separate stocks of aggregate natural capital and aggregate artificial capital must be maintained. Costanza and Daly (1992) state: "It is important for operational purposes to define sustainable development in terms of constant or nondeclining total natural capital, rather than in terms of nondeclining utility" (39).8 Other analysts such as members of the "London School" hold views between these two extremes (see Victor, 1991). They argue that though it is possible to substitute between natural and artificial capital there are certain stocks of "critical natural capital" for which no substitutes exist. A necessary condition for sustainability is that these individual stocks must be maintained in addition to the general aggregate capital stock.The weak sustainability condition violates the Second Law of Thermodynamics, as a minimum quantity of energy is required to transform matter into economically useful products (Hall et al., 1986) and energy cannot be produced inside the economic system.9 It also violates the First Law on the grounds of mass balance (Pezzey, 1994). Also ecological principles concerning the importance of diversity in system resilience (Common and Perrings, 1992) imply that minimum quantities of a large number of different capital stocks (eg. species) are required to maintain life support services. The London School view and strong sustainability accommodate these facts by assuming that there are lower bounds on the stocks of natural capital required to support the economy, in terms of the supply of materials and energy, and in terms of the assimilative capacity of the environment, and that certain categories of critical natural capital cannot be replaced by other forms of capital.Beyond this recognition it is an empirical question as to how far artificial capital can substitute for natural capital. There has been little work on this at scales relevant to sustainability. However, the econometric evidence from studies of manufacturing industry suggest on the whole that energy and capital are complements (Berndt and Wood, 1979).In some ways the concept of maintaining a constant stock of aggregate natural capital is even more bizarre than maintaining a non-declining stock of total capital. It seems more reasonable to suggest that artificial capital might replace some of the functions of natural capital than to suggest that in general various natural resources may be substitutes for each other. How can oil reserves substitute for clean air, or iron deposits for topsoil? Recognizing this, some of the strong sustainability proponents have dropped the idea of maintaining an aggregate natural capital stock as proposed by Costanza and Daly (1992) and instead argue that minimum stocks of all natural resources should be maintained (Faucheux and O'Connor, 1995). However, this can no longer really be considered an example of the CTA. Instead it is an approach that depends on the concept of safe minimum standards or the precautionary principle. The essence of the CTA is that some aggregation of resources using monetary valuations is proposed as an indicator for sustainability.The types of models which admit an index of aggregate capital, whether aggregate natural capital or aggregate total capital, is very limited. Construction of aggregate indices or subindices of inputs depend on the production function being weakly separable in those subgroups (Berndt and Christensen, 1973). For example it is only possible to construct an index of aggregate natural capital if the marginal rate of substitution between two forms of natural capital is independent of the quantities of labor or capital employed. This seems an unlikely proposition as the exploitation of many natural resources is impractical without large capital stocks. For example, in the production of caught fish, the marginal rate of substitution, and under perfect competition the price ratio, between stocks of fresh water fish and marine fish should be independent of the number of fishingboats available. This is clearly not the case. People are not likely to put a high value on the stock of deep sea fish when they do not have boats to catch them with.If substitution is limited, technological progress might reduce the quantity of natural resource inputs required per unit of output. However, there are arguments that indicate that technical progress itself is bounded (see Pezzey, 1994; Stern, 1994). One of these (Pezzey, 1994) is that, just as in the case of substitution, ultimately the laws of thermodynamics limit the minimization of resource inputs per unit output. Stern (1994) argues that unknown useful knowledge is itself a nonrenewable resource. Technological progress is the extraction of this knowledge from the environment and the investment of resources in this activity will eventually be subject to diminishing returns.Limits to substitution in production might be thought of in a much broader way to include nonlinearities and threshold effects. This view is sometimes described as the "ecological" viewpoint on sustainability (Common and Perrings, 1992; Common, 1995) or as the importance of maintaining the "resilience" of ecological systems rather than any specific stocks or species. This approach derives largely from the work of Holling (1973, 1986). In this view ecosystems are locally stable in the presence of small shocks or perturbations but may be irreversibly altered by large shocks. Structural changes in ecosystems such as those that come about through human interference and particularly simplification, may make these systems more susceptible to losing resilience and being permanently degraded. There is clearly some substitutability between species or inorganic elements in the role of maintaining ecosystem productivity, however, beyond a certain point this substitutability may suddenly fail to hold true. This approach also asks us to look at development paths as much less linear and predictable than is implied in the CTA literature.All things considered, what emerges is a quite different approach to sustainability policy. It is probable that substitution between natural and artificial capital is limited, as is ultimately technicalchange. Additionally the joint economy-ecosystem system may be subject to nonlinear dynamics. This implies that eventually the economy must approach a steady state where the volume of physical economic activity is dependent on the maximum economic and sustainable yield of renewable resources or face decline ie. profit (or utility) maximizing use of renewable resources subject to the sustainability constraint. As in Herman Daly's vision (Daly, 1977) qualitative change in the nature of economic output is still possible. Sustainability policy would require not just maintaining some stocks of renewable resources but also working to reduce "threats to sustainability" (Common, 1995) that might cause the system to pass over a threshold and reduce long-run productivity.The notion of Hicksian income originally applied to an individual price-taking firm (Faucheux and O'Connor, 1995). However, even here it is not apparent that the myopic policy of maintaining capital intact from year to year is the best or only way to ensure the sustainability of profits into the future. If a competing firm makes an innovation that renders the firm's capital stock obsolete, the latter's income may drop to zero. This is despite it previously following a policy of maintaining its capital intact. The firm's income measured up to this point is clearly seen to be unsustainable. In fact its policy has been shown to be irrelevant to long-run sustainability. In the real world firms will carry out activities that may not contribute to the year to year maintenance of capital and will reduce short-run profits such as research and development and attempts to gain market share.10 These activities make the firm more resilient against future shocks and hence enhance sustainability.b.Prices for AggregationSupposing that the necessary separability conditions are met so that aggregation of a capital stock is possible, analysts still have to obtain an appropriate set of prices so that the value of the capital stock is a sustainability relevant value. The CTA is more or less tautological if we use the "right" prices. However, these correct "sustainability prices" are unknown and unknowable. A number of。

完备基(perfect base)的存在性

完备基PB(v)的积构造蒋晓云1 冉育红 2 黄燕玲 3(1 桂林师范高等专科学校数学与计算机科学系广西桂林 541001； 2 广西师范大学数学与计算机科学学院广西桂林 541004 3 广西大学数学与计算机科学学院广西南宁 533XXX ）摘要：Etzion 提出了一个猜想，即对于正整数v ≡1(mod 6),存在完备基PB(v).本文给出了完备基PB(v)的一个积构造.利用该构造和已知PB(v)的存在结果，部分回答了Etzion 的猜想．关键词：完备基；广义斯坦纳系；光正交码中图分类号: O157.21 引言设G 是16+=t v 阶阿贝尔群，完备基PB(v)定义为集合}1|},,{{321t i x x x S i i i ≤≤=，, 1, 13ij x G i t j ∈≤≤≤≤,且满足如下性质：⑴x ±, 1≤i ≤t , 1≤j ≤3,互不相同;⑵ ),( ),( ),(312312i i i i i i x x x x x x -±-±-± 1≤i ≤t,互不相同.例1 取25Z G =，{{1, 2, 10},{3, 13, 17},{4, 6, 9},{5, 11, 18}},S =则S 即为PB （25）. 注：从PB(v)定义的条件可知，{ij x ±|1≤i ≤t , 1≤j ≤3}=G\{0}，{213213(), (), ()i i i i i i x x x x x x ±-±-±-|1≤i ≤t }= G\{0}. 在[1]中，Yin 引入PB(v)来构造光正交码，Yin 定义的完备基不限于v ≡1(mod 6). Etzion 在[2]中定义了 a good set of elements S(v)来构造广义斯坦纳系GS 4(2,4,v,2).当16+=t v 时，Yin 定义的完备基PB(v)和Etzion 定义的 S(v)是相同的，我们采用完备基这一名称来代替a good set of elements.关于光正交码和广义斯坦纳系的内容，可参看[1-5]，在此不做介绍.Etzion 在[2]中给出了如下猜想：猜想：对于任意正整数v ≡1（mod 6），存在PB(v)．以下引理见[2]．引理1.1 若v ≡1(mod 6)为质数幂，则存在PB(v)．在本文中，我们给出完备基的如下一个积构造：定理1.2 若存在PB(v 1)和PB(v 2),则存在PB(v 1 v 2)．利用定理1.2和引理1.1，可得到完备基PB(v)存在的更多结果，从而部分回答了Etzion 的猜想．在本文的第2部分，我们将给出定理1.2的证明，并对此结果做进一步的讨论，在第3部分我们将给出v<200时所有PB(v)的构造． ———————————基金项目：广西自然科学基金（桂科青0339021），广西教育厅科学基金作者简介：蒋晓云（1963—），广西全州人，桂林高等师范专科学校数学系副教授；冉育红（1981—），湖北公安人，广西师范大学数学与计算机科学学院；黄燕玲，（19XX —），广西XX 人，广西大学数学与计算机科学学院副教授．2 主要结果的证明定理1.2的证明设G 1和G 2分别是v 1和v 2阶的加法群，其中v 1≡1（mod 6）,v 2≡1（mod 6）, 11231(){{,,}|1}i i i PB v x x x i t =≤≤,6111-=v t ,11, 1, 13ij x G i t j ∈≤≤≤≤. 21232(){{,,}|1}j j j PB v y x x j t =≤≤, 6122-=v t ,22, 1, 13ij y G i t j ∈≤≤≤≤.令},|),{(2121G y G x y x G G G ∈∈=⊕=，对于任意),(y x ，),(y x ''G ∈,定义G 中的运算为 ),(y x +),(y x ''=),(y y x x '+'+，其中x x '+中的加法为G 1中的代数运算, y y '+中的加法为G 2中的代数运算．易知G 是一个21v v 阶的加法群．从G 中取出一些特殊元素作成三类三元集族321,,S S S 如下：} 1| )}0,(),0,(),0,{( {13211t i x x x S i i i ≤≤=， }1| )},0(),,0(),,0{( {23212t j y y y S j j j ≤≤=，31122331223311321321122331223311{ {(,),(,),(,)}, {(,),(,),(,)}, {(,),(,),(,)}, {(,),(,),(,)},{(,),(,),(,)}, {(,i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j S x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y =-------3213212),(,),(,)}|1,1}.i j i j x y x y i t j t --≤≤≤≤显然|S 1|= t 1 =611-v ，|S 2|= t 2 =612-v ,| S 3|=t 1×t 2×6=12(1)(1).6v v --由PB(v 1)，PB(v 2)的定义可知,10, 1, 13im x i t m ≠≤≤≤≤，20, 1, 13jn y j t n ≠≤≤≤≤,从而Φ=21S S ，Φ=32S S ，Φ=31S S .令321S S S S =，则||||321S S S S ==||||||321S S S ++=611-v +612-v +6)1)(1(21--v v =6121-v v .令}1| )0,(),0,(),0,( {13211t i x x x A i i i ≤≤±±±=,}1| ),0(),,0(),,0( {23211t j y y y B j j j ≤≤±±±=,11111222233331212232331311313212{ (,),(,),(,),(,),(,),(,), (,),(,),(,),(,),(,),(,), (,),(,),(,),(,i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i C x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y =----------------1323211112222333312122323313113),(,),(,), (,),(,),(,),(,),(,),(,), (,),(,),(,),(,),(,),(,), (,j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y ---------------1321213232),(,),(,),(,),(,),(,) }.i j i j i j i j i j x y x y x y x y x y -----由PB(v 1)，PB(v 2)的定义知1i x ±,2i x ±，3i x ±, 1≤i ≤1t ,互不相同, 1j y ±,2j y ±，3j y ±, 1≤j ≤2t ,互不相同．所以1A 中的元素互不相同，1B 中的元素互不相同，1C 中的元素也互不相同．由10i x ±≠,20i x ±≠，30i x ±≠, 1≤i ≤1t ，10 j y ±≠,20j y ±≠，30j y ±≠, 1≤j ≤2t ，知Φ=11B A ，Φ=11C A ，Φ=11C B ，从而111C B A 中的元素互不相同．令22132131{ (,0),(,0),(,0) |1}i i i i i i A x x x x x x i t =±-±-±-≤≤. }1| ),0(),,0(),,0( {23123122t j y y y y y y B j j j j j j ≤≤-±-±-±=.221213232131321323213132121133221133{ (,), (,), (,), (,), (,), (,), (,), (,), (,i i j j i i j j i i j j i i j j i i j j i i j j i i j j i i j j i i j C x x y y x x y y x x y y x x y y x x y y x x y y x x y y x x y y x x y =±--±--±--±--±--±--±--±--±--2211232231331212332311312213132121), (,), (,), (,), (,), (,), (,),(,), (,), (j i i j j i i j j i i j j i i j j i i j j i i j j i i j j i i j j i i y x x y y x x y y x x y y x x y y x x y y x x y y x x y y x x y y x x ±--±--±--±--±--±--±--±--±-32312,)|1i t , 1}j j y y j t -≤≤≤≤由PB(V 1)，PB(V 2)的定义知),( ),( ),(312312i i i i i i x x x x x x -±-±-±1≤i ≤1t ,互不相同,),( ),( ),(312312j j j j j j y y y y y y -±-±-±1≤j ≤2t , 互不相同．所以2A 中的元素互不相同，2B 中的元素互不相同，2C 中的元素也互不相同．由213213()0, ()0, ()0,i i i i i i x x x x x x ±-≠±-≠±-≠1≤i ≤1t ，21()0,j j y y ±-≠3213 ()0, ()0,j j j j y y y y ±-≠±-≠1≤j ≤2t ，知Φ=22B A ，Φ=22C A ，Φ=22C B ，从而222C B A 中的元素互不相同．综上所述，由完备基的定义知S 为G 中的一个完备基，即G 中存在PB(v 1 v 2).这样我们就证明了定理1.2．例2 设71Z G =，132Z G =，则1{{1, 2, 4}}S =为1G 中的PB( 7), 2{{1, 3, 9},{2, 5, 6}}S =为2G 中的PB(13).利用定理1.2可以构造出G 1⊕G 2上的PB （91）如下：S={{(1，0)，（2，0），（4，0）}，{（0，1），（0，3），（0，9）}，{（0，2），（0，5），（0，6）}，{(1，1)，（2，3），（4，9）}，{（1，3），（2，9），（4，1）}，{（1，9），（2，1），（4，3）}， {(1，2)，（2，5），（4，6）}，{（1，5），（2，6），（4，2）}， {(1，6)，（2，2），（4，5）}， {(1，12)，（2，10），（4，4）}，{（1，10），（2，4），（4，12）}，{（1，4），（2，12），（4，10）}，{(1，11)，（2，8），（4，7）}，{（1，8），（2，7），（4，11）}，{（1，7），（2，11），（4，8）} }．利用数学归纳法和定理1.2，很容易得到以下结果：推论2.1 若存在PB(v 1)，PB(v 2)，…,PB(v n ),则存在PB （v 1 v 2…v n ）．利用推论2.1和引理1.1，可得到以下结论：推论2.2 设v ≡1(mod 6)为正整数，若 ① v 不含模6余5的质因数;② v 的标准分解式为ts t s q q q p p p v βββααα 21212121=, 其中i p ≡1(mod 6), ,1 s i ≤≤j q ≡5(mod 6), t j ≤≤1, j β（t j ≤≤1）均为偶数．则存在PB(v)．利用推论2.1可知，若以下命题成立，则Etzion 的猜想成立，从而要证明Etzion 的猜想，只要证明以下命题即可：命题设1p ，2p 为两个质数，且1p ,2p ≡5(mod 6)，则存在PB （1p 2p ）.注：在第３部分中，我们将对某些质数1p ，2p ≡5(mod 6)，给出PB （1p 2p ）的构造．3 V<200时的PB(v)在本部分，对任意v ≡1(mod 6),v<200,我们将给出PB （v ）的构造，从而说明Etzion 的猜想对小于200的正整数成立．定理3.1 若v ≡1(mod 6),v<200, 则存在PB(v)．证明令V={v| v ≡1(mod 6)，v<200 }=E ∪F ∪H ，其中，E={7, 13, 19, 25, 31, 37, 43, 49, 61, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151,157, 163, 169, 181, 187,193,199}， F={91，133，175}，H={55，85，115，145，187}.E 中元素为模６余１的质数幂，由引理1.1，对v ∈E ，存在PB(v)．对Ｆ中元素，91=7×13，133=7×19，175=7×25，由定理1.2和引理1.1知，若v ∈F ，则存在PB(v)．对Ｈ中每个元素v ，利用计算机搜索，得到PB （v ）的构造如下：v=55S={{2, 7, 21}, {9, 26, 49}, {10, 19, 20}, {5, 33, 44}, {13, 31, 37}, {1,3, 23}, {4, 30, 38}, {8, 12, 15}, {14, 27, 39}}． v=85S={{4,9,36},{11,35,51},{14,22,23},{10,20,64},{12,41,58},{6,55,68},{3,25,67},{8,15,19},{1,52,72},{16,28,46},{5,38,53},{2,59,61},{7,42,45},{29,48,54}}．v=115S={{11,13,60},{17,53,73},{21,48,61},{10,38,41},{9,51,52},{28,36,58},{18,27,109},{25,35,46},{22,72,89},{14,30,49},{20,59,84},{1,7,70},{19,24,81},{16,39,113},{4,33,65},{5,12,83},{8,23,68},{3,15,29},{37,71,75}}． v=145S={{10,26,63},{12,76,105},{13,57,98},{24,29,115},{14,20,50},{6,37,46},{2,67,89},{18,21,32},{28,45,49},{19,58,86},{11,94,106},{7,79,81},{5,53,73}, {17,83,101},{16,43,118},{23,38,48},{22,112,120},{31,54,55},{1,35,70},{8,65,111},{3,52,85},{9,41,141},{15,71,109},{42,61,68}}．v=187S={{17,35,56},{38,61,133},{6,140,180},{44,115,135},{22,83,141},{33,74,108},{13, 110,127},{8,86,89},{12,114,125},{15,78,84},{2,123,153},{16,68,92},{37,59, 87},{9,137,142},{3,70,147},{4,42,91},{24,80,88},{10,58,146},{1,90,106},{21, 66,173},{19,48,105},{20,27,39},{18,102,111},{25,29,71},{5,30,31},{26,53, 63},{36,67,69},{28,43,75},{11,55,138},{32,94,164},{51,65,130}}.参考文献[1]Yin J. Some combinatorial constructions for optical orthogonal codes[J]. DiscreteMathematics, 1998, 185: 201-219.[2] Etzion T. Optimal constant weight codes over Z k and generalized designs[J]. DiscreteMathematics, 1997, 169: 55-82.[3] Brickell E. F and Wei V. K. Optical orthogonal codes and cyclic block designs[J]. CongressNumerantium, 1987, 58 : 175-192.[4] Wu D, Ge G and Zhu L. Generalized Steiner systems GS4(2,4,v,g) for g=2,3,6[J]. JCombinatorial Designs, 2001, 9: 401-423.[5] 吴佃华. 广义斯坦纳系GS k+1(2,k,v,g)的一个构造方法及简单应用[J]. 广西师范大学学报,2002, 4: 50-53.A Product Construction for Perfect Base PB(v)Jiang Xiao-yun1 Ran Y uhong 2Huang XXXXXX3（1 Department of Mathematics and Computer Science ,Guilin T eachers’ College, Guangxi 541001,China2 Department of Mathematics and Computer Science, Guangxi Normal University, Guilin 541004,China3 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX）Abstract: Etzion conjectured that there exists a perfect base PB(v) for any integer v≡1(mod 6). In this paper, a product construction for PB(v) is presented. By using this construction and the known results on PB(v)s, a partial answer to Etzion’s conjecture is obtained.Key Words:perfect base; generalized Steiner systems; optical orthogonal codes。

单圈图的边优美性

单圈图的边优美性陈淑贞;薛茗曦【摘要】边优美标号是图标号中的一种.文章主要研究了单圈图的边优美问题,定义了三类单圈图Ω1(m,2)、Ω2(m,2)和Ω(m,2),给出了这三类图在一定条件下的边优美标号,从而证明了它们是边优美图.【期刊名称】《海南师范大学学报（自然科学版）》【年(卷),期】2017(030)004【总页数】4页(P388-390,395)【关键词】单圈图;边优美图;边优美标号【作者】陈淑贞;薛茗曦【作者单位】海南师范大学数学与统计学院,海南海口571158;海南师范大学数学与统计学院,海南海口571158【正文语种】中文【中图分类】O157.5图标号问题由Ringel G和Rosa A在上世纪60年代中期提出并引起国内外许多学者的关注[1,2]，已成为图论中一个重要而活跃的研究分支.从图的优美标号被提出到现在已有三十来种图标号被定义[3].单圈图(只含一个圈的图)是一类重要的图形. 1984年，Truszczynski M关于单圈图的猜想: 除Cn (n≡1,2(mod 4))外，所有的单圈图都是优美的[4]. 从那时起，关于单圈图的研究一直未间断过，至今已有许多研究成果[3-8].1985年，Lo S引入边优美图的概念[9]，并给出边优美图的必要条件. 对图的边优美性的研究虽然已取得一些研究成果,但仍有许多问题尚待解决. 关于单圈图的边优美性有一个重要猜想[5]：奇阶单圈图是边优美的.本文研究了奇阶单圈图的边优美性问题，给出三类奇阶单圈图的边优美标号.1 相关定义定义1[3] 设G(V,E)是简单图，若存在双射f：E(G)→{1,2,…,|E(G)|}使得导出映射f*：V(G) →{0,1,2,…,|V(G)|-1}为双射，其中对任意的v∈V(G)，为顶点v的邻集，则称G为边优美图，f为G的边优美标号.以下所定义和研究的图均为简单图.定义2 在回路Cm相邻的两个顶点处分别粘接一条路所组成的单圈图称为靶图,记为Ω1(m,2).定义3 在回路Cm相距为2的两个顶点处分别粘接一条路所组成的单圈图称为定靶图,记为Ω2(m,2).定义4 在回路Cm的一个顶点处粘接两条路所组成的单圈图称为风筝图,记为Ω(m,2).2 主要结果定理1 奇阶靶图Ω1(m,2)是边优美图.证明设Ω1(m,2)的顶点数为2n+1(n>1)个，其中圈中有m个点，且有|V(Ω1)|= |E(Ω1)|= 2n+1,如图1所示，靶图Ω1(m,2)的顶点依次记为a1,a2,…，ai,…，a2n+1.定义其边标号f如下：f(akak+1)=k(k=1,2,…，2n),f(ai+1ai+m)=2n+1.显然，边标号f是E(Ω1)到{1,2,3,…,2n+1}的双射.由f导出的点标号f*为：f*(ak)=k-1+k=2k-1(k=1,2,…,n)，f*(an+k)=[2n+2k-1](mod|V(Ω1)|)=2k-2(k=1,2,…，n+1).特别地，顶点ai+1和ai+m有三条相关联的边，但多出的边ai+1ai+m满足f(ai+1ai+m)=2n+1,以上计算结果仍然成立.显然，f的导出映射f*是V(Ω1)到{0,1,2,…,2n}双射,所以靶图Ω1(m,2)是边优美图.图2给出了靶图Ω1(m,2)的边优美标号.图1 奇阶图Ω1(m,2)Fig. 1 The odd degree graph Ω1(m,2)图2 奇阶图Ω1(m,2)的边优美标号Fig.2 The edge-graceful labeling of odd degree graph Ω1(m,2)定理2 当回路上的点为偶数，且两条路的长度相差为1时，奇阶定靶图Ω2(m,2)是边优美图.证明设Ω2(m,2)的顶点数为2n+1(n>3)个，且有|V(Ω2)|= |E(Ω2)|= 2n+1.如图3所示，定靶图Ω2(m,2)的顶点依次记为a1,a2,…，ai,…，a2n+1，其中两条路的长分别为i+1和i,回路中的点数m为2n-2i.定义其边标号f如下：f(akak+1)=k(k=1,2,…,2i+3,2i+5,…,2n)，f(ai+2a2i+5)=2i+4，f(a2n+1ai+4)=2n+1.显然，边标号f是E(Ω2)到{1,2,3,…,2n+1}的双射.由f导出的点标号f*在取模|V(Ω2)|前为：f*(ak)=2k-1(k=1,2,…，i+1),f*(ai+2)=4i+7,f*(a2i+4)=2i+3,f*(ai+k)=2i+2k-1(k=3,4,5,…,i+3,i+5,…,2n+1-i).这2n+1个数恰好取遍集合{1,3,5,…,2i+1,2i+3, …,4i+5,4i+7,4i+9, …,4n-3,4n-1,4n+1}中的2n+1个数，显然取模|V(Ω2)|后点标号一一对应于{0,1,2,…,2n}. 所以f的导出映射f*是V(Ω2)到{0,1,2,…,2n}双射,于是定靶图Ω2(m,2)为边优美图.图4给出了定靶图Ω2(m,2)的边优美标号.图3 奇阶图Ω2(m,2)Fig.3 The odd degree graph Ω2(m,2)图4 奇阶图Ω2(m,2)的边优美标号Fig.4 The edge-graceful labeling of odd degree graph Ω2(m,2)定理3 一条路为n的2n+1阶风筝图Ω(m,2)是边优美图.证明Ω(m,2)的顶点数为2n+1(n>1)个，则有|V(Ω)|= |E(Ω)|= 2n+1.如图5所示，风筝图的顶点依次记为a1,a2,…，ai,…，a2n+1.定义其边标号f如下：f(akak+1)=k(k=1,2,…，n-1),f(akak+1)=k+1(k=n+1,n+2,…，2n),f(anan+m)=n,f(an+1an+m)=n+1.显然,边标号f是E(Ω)到{1,2,3,…,2n+1}的双射.由f导出的点标号f*为：f*(ak)=2k-1(k=1,2,…n),f*(a2n+1)=2n+1(mod|V(Ω)|)=0,f*(an+k)=[2n+2k+1](mod|V(Ω)| )=2k(k=1,2,…,n).特别地，顶点an+m有四条相关联的边，但多出的边an+1an+m和anan+m满足f(an+1an+m)+f(anan+m)=2n+1,以上计算结果仍然成立.显然，f的导出映射f*是V(Ω)到{0,1,2,…,2n}双射.于是风筝图Ω(m,2)是边优美图. 图6给出了风筝图Ω(m,2)的边优美标号.图5 奇阶图Ω(m,2)Fig.5 The odd degree graph Ω(m,2)图6 奇阶图Ω(m,2)的边优美标号Fig.6 The edge-graceful labeling of odd degree graph Ω(m,2)参考文献：[1] Ringel G. Problem 25 in theory of graphs and its application[C]// Proceedings of the Symposium Smolenice，1963. Prague Publ: House of Czcchoslovak Academy of Science, 1964.[2] Rosa A. On certain valuations of the vertices of a graph[C]// Theory of Graphs International Symposium, Rome, 1966. Newyork and Dunod Paris: Gordon and Breach, 1967: 349-355.[3]Gallian J A. A dynamic survey of graph labeling[J]. The Electronic Journal of Combinatorics, 2016, Dynamic Surveys 6: 1-408.[4]Truszczynski M. Graceful unicycle graphs[J]. Demonstration Math, 1984, 17: 377-387.[5]康庆德. 图标号问题[J]. 河北师范学院学报，1991(1):102-115.[6]郑学谦, 乔晓云. 图的边优美性[J].广西师范学院学报，2008,25(2):45-47.[7]陈淑贞, 王丽娜. Ω(2, k, n)型图的优美性[J]. 海南师范大学学报(自然科学版), 2008, 21(3)： 249-253.[8]郑学谦. 图Cn×K2的边优美标号的研究[J]. 太原师范学院学报，2012,11(4):12-13.[9]Lo S. On edge-graceful labelings of graphs[J]. Congressus Numerantium, 1985(50): 231-241.。

Quantum Computing for Computer Scientists

More informationQuantum Computing for Computer ScientistsThe multidisciplinaryﬁeld of quantum computing strives to exploit someof the uncanny aspects of quantum mechanics to expand our computa-tional horizons.Quantum Computing for Computer Scientists takes read-ers on a tour of this fascinating area of cutting-edge research.Writtenin an accessible yet rigorous fashion,this book employs ideas and tech-niques familiar to every student of computer science.The reader is notexpected to have any advanced mathematics or physics background.Af-ter presenting the necessary prerequisites,the material is organized tolook at different aspects of quantum computing from the speciﬁc stand-point of computer science.There are chapters on computer architecture,algorithms,programming languages,theoretical computer science,cryp-tography,information theory,and hardware.The text has step-by-stepexamples,more than two hundred exercises with solutions,and program-ming drills that bring the ideas of quantum computing alive for today’scomputer science students and researchers.Noson S.Yanofsky,PhD,is an Associate Professor in the Departmentof Computer and Information Science at Brooklyn College,City Univer-sity of New York and at the PhD Program in Computer Science at TheGraduate Center of CUNY.Mirco A.Mannucci,PhD,is the founder and CEO of HoloMathics,LLC,a research and development company with a focus on innovative mathe-matical modeling.He also serves as Adjunct Professor of Computer Sci-ence at George Mason University and the University of Maryland.QUANTUM COMPUTING FORCOMPUTER SCIENTISTSNoson S.YanofskyBrooklyn College,City University of New YorkandMirco A.MannucciHoloMathics,LLCMore informationMore informationcambridge university pressCambridge,New York,Melbourne,Madrid,Cape Town,Singapore,S˜ao Paulo,DelhiCambridge University Press32Avenue of the Americas,New York,NY10013-2473,USAInformation on this title:/9780521879965C Noson S.Yanofsky and Mirco A.Mannucci2008This publication is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2008Printed in the United States of AmericaA catalog record for this publication is available from the British Library.Library of Congress Cataloging in Publication dataYanofsky,Noson S.,1967–Quantum computing for computer scientists/Noson S.Yanofsky andMirco A.Mannucci.p.cm.Includes bibliographical references and index.ISBN978-0-521-87996-5(hardback)1.Quantum computers.I.Mannucci,Mirco A.,1960–II.Title.QA76.889.Y352008004.1–dc222008020507ISBN978-0-521-879965hardbackCambridge University Press has no responsibility forthe persistence or accuracy of URLs for external orthird-party Internet Web sites referred to in this publicationand does not guarantee that any content on suchWeb sites is,or will remain,accurate or appropriate.More informationDedicated toMoishe and Sharon Yanofskyandto the memory ofLuigi and Antonietta MannucciWisdom is one thing:to know the tho u ght by which all things are directed thro u gh allthings.˜Heraclitu s of Ephe s u s(535–475B C E)a s quoted in Dio g ene s Laertiu s’sLives and Opinions of Eminent PhilosophersBook IX,1. More informationMore informationContentsPreface xi1Complex Numbers71.1Basic Deﬁnitions81.2The Algebra of Complex Numbers101.3The Geometry of Complex Numbers152Complex Vector Spaces292.1C n as the Primary Example302.2Deﬁnitions,Properties,and Examples342.3Basis and Dimension452.4Inner Products and Hilbert Spaces532.5Eigenvalues and Eigenvectors602.6Hermitian and Unitary Matrices622.7Tensor Product of Vector Spaces663The Leap from Classical to Quantum743.1Classical Deterministic Systems743.2Probabilistic Systems793.3Quantum Systems883.4Assembling Systems974Basic Quantum Theory1034.1Quantum States1034.2Observables1154.3Measuring1264.4Dynamics1294.5Assembling Quantum Systems1325Architecture1385.1Bits and Qubits138viiMore informationviii Contents5.2Classical Gates1445.3Reversible Gates1515.4Quantum Gates1586Algorithms1706.1Deutsch’s Algorithm1716.2The Deutsch–Jozsa Algorithm1796.3Simon’s Periodicity Algorithm1876.4Grover’s Search Algorithm1956.5Shor’s Factoring Algorithm2047Programming Languages2207.1Programming in a Quantum World2207.2Quantum Assembly Programming2217.3Toward Higher-Level Quantum Programming2307.4Quantum Computation Before Quantum Computers2378Theoretical Computer Science2398.1Deterministic and Nondeterministic Computations2398.2Probabilistic Computations2468.3Quantum Computations2519Cryptography2629.1Classical Cryptography2629.2Quantum Key Exchange I:The BB84Protocol2689.3Quantum Key Exchange II:The B92Protocol2739.4Quantum Key Exchange III:The EPR Protocol2759.5Quantum Teleportation27710Information Theory28410.1Classical Information and Shannon Entropy28410.2Quantum Information and von Neumann Entropy28810.3Classical and Quantum Data Compression29510.4Error-Correcting Codes30211Hardware30511.1Quantum Hardware:Goals and Challenges30611.2Implementing a Quantum Computer I:Ion Traps31111.3Implementing a Quantum Computer II:Linear Optics31311.4Implementing a Quantum Computer III:NMRand Superconductors31511.5Future of Quantum Ware316Appendix A Historical Bibliography of Quantum Computing319 by Jill CirasellaA.1Reading Scientiﬁc Articles319A.2Models of Computation320More informationContents ixA.3Quantum Gates321A.4Quantum Algorithms and Implementations321A.5Quantum Cryptography323A.6Quantum Information323A.7More Milestones?324Appendix B Answers to Selected Exercises325Appendix C Quantum Computing Experiments with MATLAB351C.1Playing with Matlab351C.2Complex Numbers and Matrices351C.3Quantum Computations354Appendix D Keeping Abreast of Quantum News:QuantumComputing on the Web and in the Literature357by Jill CirasellaD.1Keeping Abreast of Popular News357D.2Keeping Abreast of Scientiﬁc Literature358D.3The Best Way to Stay Abreast?359Appendix E Selected Topics for Student Presentations360E.1Complex Numbers361E.2Complex Vector Spaces362E.3The Leap from Classical to Quantum363E.4Basic Quantum Theory364E.5Architecture365E.6Algorithms366E.7Programming Languages368E.8Theoretical Computer Science369E.9Cryptography370E.10Information Theory370E.11Hardware371Bibliography373Index381More informationPrefaceQuantum computing is a fascinating newﬁeld at the intersection of computer sci-ence,mathematics,and physics,which strives to harness some of the uncanny as-pects of quantum mechanics to broaden our computational horizons.This bookpresents some of the most exciting and interesting topics in quantum computing.Along the way,there will be some amazing facts about the universe in which we liveand about the very notions of information and computation.The text you hold in your hands has a distinctﬂavor from most of the other cur-rently available books on quantum computing.First and foremost,we do not assumethat our reader has much of a mathematics or physics background.This book shouldbe readable by anyone who is in or beyond their second year in a computer scienceprogram.We have written this book speciﬁcally with computer scientists in mind,and tailored it accordingly:we assume a bare minimum of mathematical sophistica-tion,aﬁrst course in discrete structures,and a healthy level of curiosity.Because thistext was written speciﬁcally for computer people,in addition to the many exercisesthroughout the text,we added many programming drills.These are a hands-on,funway of learning the material presented and getting a real feel for the subject.The calculus-phobic reader will be happy to learn that derivatives and integrals are virtually absent from our text.Quite simply,we avoid differentiation,integra-tion,and all higher mathematics by carefully selecting only those topics that arecritical to a basic introduction to quantum computing.Because we are focusing onthe fundamentals of quantum computing,we can restrict ourselves to theﬁnite-dimensional mathematics that is required.This turns out to be not much more thanmanipulating vectors and matrices with complex entries.Surprisingly enough,thelion’s share of quantum computing can be done without the intricacies of advancedmathematics.Nevertheless,we hasten to stress that this is a technical textbook.We are not writing a popular science book,nor do we substitute hand waving for rigor or math-ematical precision.Most other texts in theﬁeld present a primer on quantum mechanics in all its glory.Many assume some knowledge of classical mechanics.We do not make theseassumptions.We only discuss what is needed for a basic understanding of quantumxiMore informationxii Prefacecomputing as aﬁeld of research in its own right,although we cite sources for learningmore about advanced topics.There are some who consider quantum computing to be solely within the do-main of physics.Others think of the subject as purely mathematical.We stress thecomputer science aspect of quantum computing.It is not our intention for this book to be the deﬁnitive treatment of quantum computing.There are a few topics that we do not even touch,and there are severalothers that we approach brieﬂy,not exhaustively.As of this writing,the bible ofquantum computing is Nielsen and Chuang’s magniﬁcent Quantum Computing andQuantum Information(2000).Their book contains almost everything known aboutquantum computing at the time of its publication.We would like to think of ourbook as a usefulﬁrst step that can prepare the reader for that text.FEATURESThis book is almost entirely self-contained.We do not demand that the reader comearmed with a large toolbox of skills.Even the subject of complex numbers,which istaught in high school,is given a fairly comprehensive review.The book contains many solved problems and easy-to-understand descriptions.We do not merely present the theory;rather,we explain it and go through severalexamples.The book also contains many exercises,which we strongly recommendthe serious reader should attempt to solve.There is no substitute for rolling up one’ssleeves and doing some work!We have also incorporated plenty of programming drills throughout our text.These are hands-on exercises that can be carried out on your laptop to gain a betterunderstanding of the concepts presented here(they are also a great way of hav-ing fun).We hasten to point out that we are entirely language-agnostic.The stu-dent should write the programs in the language that feels most comfortable.Weare also paradigm-agnostic.If declarative programming is your favorite method,gofor it.If object-oriented programming is your game,use that.The programmingdrills build on one another.Functions created in one programming drill will be usedand modiﬁed in later drills.Furthermore,in Appendix C,we show how to makelittle quantum computing emulators with MATLAB or how to use a ready-madeone.(Our choice of MATLAB was dictated by the fact that it makes very easy-to-build,quick-and-dirty prototypes,thanks to its vast amount of built-in mathematicaltools.)This text appears to be theﬁrst to handle quantum programming languages in a signiﬁcant way.Until now,there have been only research papers and a few surveyson the topic.Chapter7describes the basics of this expandingﬁeld:perhaps some ofour readers will be inspired to contribute to quantum programming!This book also contains several appendices that are important for further study:Appendix A takes readers on a tour of major papers in quantum computing.This bibliographical essay was written by Jill Cirasella,Computational SciencesSpecialist at the Brooklyn College Library.In addition to having a master’s de-gree in library and information science,Jill has a master’s degree in logic,forwhich she wrote a thesis on classical and quantum graph algorithms.This dualbackground uniquely qualiﬁes her to suggest and describe further readings.More informationPreface xiii Appendix B contains the answers to some of the exercises in the text.Othersolutions will also be found on the book’s Web page.We strongly urge studentsto do the exercises on their own and then check their answers against ours.Appendix C uses MATLAB,the popular mathematical environment and an es-tablished industry standard,to show how to carry out most of the mathematicaloperations described in this book.MATLAB has scores of routines for manip-ulating complex matrices:we brieﬂy review the most useful ones and show howthe reader can quickly perform a few quantum computing experiments with al-most no effort,using the freely available MATLAB quantum emulator Quack.Appendix D,also by Jill Cirasella,describes how to use online resources to keepup with developments in quantum computing.Quantum computing is a fast-movingﬁeld,and this appendix offers guidelines and tips forﬁnding relevantarticles and announcements.Appendix E is a list of possible topics for student presentations.We give briefdescriptions of different topics that a student might present before a class of hispeers.We also provide some hints about where to start looking for materials topresent.ORGANIZATIONThe book begins with two chapters of mathematical preliminaries.Chapter1con-tains the basics of complex numbers,and Chapter2deals with complex vectorspaces.Although much of Chapter1is currently taught in high school,we feel thata review is in order.Much of Chapter2will be known by students who have had acourse in linear algebra.We deliberately did not relegate these chapters to an ap-pendix at the end of the book because the mathematics is necessary to understandwhat is really going on.A reader who knows the material can safely skip theﬁrsttwo chapters.She might want to skim over these chapters and then return to themas a reference,using the index and the table of contents toﬁnd speciﬁc topics.Chapter3is a gentle introduction to some of the ideas that will be encountered throughout the rest of the ing simple models and simple matrix multipli-cation,we demonstrate some of the fundamental concepts of quantum mechanics,which are then formally developed in Chapter4.From there,Chapter5presentssome of the basic architecture of quantum computing.Here one willﬁnd the notionsof a qubit(a quantum generalization of a bit)and the quantum analog of logic gates.Once Chapter5is understood,readers can safely proceed to their choice of Chapters6through11.Each chapter takes its title from a typical course offered in acomputer science department.The chapters look at that subﬁeld of quantum com-puting from the perspective of the given course.These chapters are almost totallyindependent of one another.We urge the readers to study the particular chapterthat corresponds to their favorite course.Learn topics that you likeﬁrst.From thereproceed to other chapters.Figure0.1summarizes the dependencies of the chapters.One of the hardest topics tackled in this text is that of considering two quan-tum systems and combining them,or“entangled”quantum systems.This is donemathematically in Section2.7.It is further motivated in Section3.4and formallypresented in Section4.5.The reader might want to look at these sections together.xivPrefaceFigure 0.1.Chapter dependencies.There are many ways this book can be used as a text for a course.We urge instructors to ﬁnd their own way.May we humbly suggest the following three plans of action:(1)A class that provides some depth might involve the following:Go through Chapters 1,2,3,4,and 5.Armed with that background,study the entirety of Chapter 6(“Algorithms”)in depth.One can spend at least a third of a semester on that chapter.After wrestling a bit with quantum algorithms,the student will get a good feel for the entire enterprise.(2)If breadth is preferred,pick and choose one or two sections from each of the advanced chapters.Such a course might look like this:(1),2,3,4.1,4.4,5,6.1,7.1,9.1,10.1,10.2,and 11.This will permit the student to see the broad outline of quantum computing and then pursue his or her own path.(3)For a more advanced class (a class in which linear algebra and some mathe-matical sophistication is assumed),we recommend that students be told to read Chapters 1,2,and 3on their own.A nice course can then commence with Chapter 4and plow through most of the remainder of the book.If this is being used as a text in a classroom setting,we strongly recommend that the students make presentations.There are selected topics mentioned in Appendix E.There is no substitute for student participation!Although we have tried to include many topics in this text,inevitably some oth-ers had to be left out.Here are a few that we omitted because of space considera-tions:many of the more complicated proofs in Chapter 8,results about oracle computation,the details of the (quantum)Fourier transforms,and the latest hardware implementations.We give references for further study on these,as well as other subjects,throughout the text.More informationMore informationPreface xvANCILLARIESWe are going to maintain a Web page for the text at/∼noson/qctext.html/The Web page will containperiodic updates to the book,links to interesting books and articles on quantum computing,some answers to certain exercises not solved in Appendix B,anderrata.The reader is encouraged to send any and all corrections tonoson@Help us make this textbook better!ACKNOLWEDGMENTSBoth of us had the great privilege of writing our doctoral theses under the gentleguidance of the recently deceased Alex Heller.Professor Heller wrote the follow-ing1about his teacher Samuel“Sammy”Eilenberg and Sammy’s mathematics:As I perceived it,then,Sammy considered that the highest value in mathematicswas to be found,not in specious depth nor in the overcoming of overwhelmingdifﬁculty,but rather in providing the deﬁnitive clarity that would illuminate itsunderlying order.This never-ending struggle to bring out the underlying order of mathematical structures was always Professor Heller’s everlasting goal,and he did his best to passit on to his students.We have gained greatly from his clarity of vision and his viewof mathematics,but we also saw,embodied in a man,the classical and sober ideal ofcontemplative life at its very best.We both remain eternally grateful to him.While at the City University of New York,we also had the privilege of inter-acting with one of the world’s foremost logicians,Professor Rohit Parikh,a manwhose seminal contributions to theﬁeld are only matched by his enduring com-mitment to promote younger researchers’work.Besides opening fascinating vis-tas to us,Professor Parikh encouraged us more than once to follow new directionsof thought.His continued professional and personal guidance are greatly appre-ciated.We both received our Ph.D.’s from the Department of Mathematics in The Graduate Center of the City University of New York.We thank them for providingus with a warm and friendly environment in which to study and learn real mathemat-ics.Theﬁrst author also thanks the entire Brooklyn College family and,in partic-ular,the Computer and Information Science Department for being supportive andvery helpful in this endeavor.1See page1349of Bass et al.(1998).More informationxvi PrefaceSeveral faculty members of Brooklyn College and The Graduate Center were kind enough to read and comment on parts of this book:Michael Anshel,DavidArnow,Jill Cirasella,Dayton Clark,Eva Cogan,Jim Cox,Scott Dexter,EdgarFeldman,Fred Gardiner,Murray Gross,Chaya Gurwitz,Keith Harrow,JunHu,Yedidyah Langsam,Peter Lesser,Philipp Rothmaler,Chris Steinsvold,AlexSverdlov,Aaron Tenenbaum,Micha Tomkiewicz,Al Vasquez,Gerald Weiss,andPaula Whitlock.Their comments have made this a better text.Thank you all!We were fortunate to have had many students of Brooklyn College and The Graduate Center read and comment on earlier drafts:Shira Abraham,RachelAdler,Ali Assarpour,Aleksander Barkan,Sayeef Bazli,Cheuk Man Chan,WeiChen,Evgenia Dandurova,Phillip Dreizen,C.S.Fahie,Miriam Gutherc,RaveHarpaz,David Herzog,Alex Hoffnung,Matthew P.Johnson,Joel Kammet,SerdarKara,Karen Kletter,Janusz Kusyk,Tiziana Ligorio,Matt Meyer,James Ng,SeverinNgnosse,Eric Pacuit,Jason Schanker,Roman Shenderovsky,Aleksandr Shnayder-man,Rose B.Sigler,Shai Silver,Justin Stallard,Justin Tojeira,John Ma Sang Tsang,Sadia Zahoor,Mark Zelcer,and Xiaowen Zhang.We are indebted to them.Many other people looked over parts or all of the text:Scott Aaronson,Ste-fano Bettelli,Adam Brandenburger,Juan B.Climent,Anita Colvard,Leon Ehren-preis,Michael Greenebaum,Miriam Klein,Eli Kravits,Raphael Magarik,JohnMaiorana,Domenico Napoletani,Vaughan Pratt,Suri Raber,Peter Selinger,EvanSiegel,Thomas Tradler,and Jennifer Whitehead.Their criticism and helpful ideasare deeply appreciated.Thanks to Peter Rohde for creating and making available to everyone his MAT-LAB q-emulator Quack and also for letting us use it in our appendix.We had a gooddeal of fun playing with it,and we hope our readers will too.Besides writing two wonderful appendices,our friendly neighborhood librar-ian,Jill Cirasella,was always just an e-mail away with helpful advice and support.Thanks,Jill!A very special thanks goes to our editor at Cambridge University Press,HeatherBergman,for believing in our project right from the start,for guiding us through thisbook,and for providing endless support in all matters.This book would not existwithout her.Thanks,Heather!We had the good fortune to have a truly stellar editor check much of the text many times.Karen Kletter is a great friend and did a magniﬁcent job.We also ap-preciate that she refrained from killing us every time we handed her altered draftsthat she had previously edited.But,of course,all errors are our own!This book could not have been written without the help of my daughter,Hadas-sah.She added meaning,purpose,and joy.N.S.Y.My dear wife,Rose,and our two wondrous and tireless cats,Ursula and Buster, contributed in no small measure to melting my stress away during the long andpainful hours of writing and editing:to them my gratitude and love.(Ursula is ascientist cat and will read this book.Buster will just shred it with his powerful claws.)M.A.M.。

新视野读写(第三版)(3)课本练习及答案Unit 7

8. Though she lost the title of female world reclaimed swimming champion last year, she _______ it this year when she gave it all she could. originated in the 9. This type of system which __________ United Kingdom has been widely used throughout Europe and is gaining in popularity in the United States of America and Japan.
Word Building The suffix -ion combines with verbs to form nouns. Nouns formed in this way refer to the state or process described by the verb, or to an instance of that process.
2. Their attempts to bring peace and stability to this civil war-torn region received suspension of peace another blow when ___________ talks was announced. 3. They went through the immigration and formality so quickly that they customs __________ were able to have a cup of coffee before I arrived to collect them.

Toni Morrison

Major Works
The Bluest Eye (1970)
Sula (1973) Song of Solomon (1977) Tar Baby (1981) Beloved (1987)
Song of Solomon (2部15章)
Brief introduction
Hero:Milkman Friend:Guitar Aunt:Pilate Why would he try to kill me before I got it or even found out what happened to it? (p835) And the skin of shame that he had rinsed away in the bathwater after having stolen from Pilate returned. But now it was as thick and as tight as a caul. (p848 para3)
1989 MLA Commonwealth Award in Literature 1993 Nobel Prize for Literature 1993 Commander of the Arts and Letters, Paris 1994 Condorcet Medal, Paris 1994 Pearl Buck Award 1994 Rhegium Julii Prize for Literature 1996 Jefferson Lecture 1996 National Book Foundation's Medal of Distinguished Contribution to American Letters 2000 National Humanities Medal 2002 100 Greatest African Americans, list by Molefi Kete Asante 2005 Honorary Doctorate of Letters from Oxford University 2009 Norman Mailer Prize, Lifetime Achievement 2010 Officier de la Lé gion d'Honneur 2011 Honorary Doctor of Letters at Rutgers University Graduation Commencement 2011 Honorary Doctorate of Letters from the University of Geneva 2012 Presidential Medal of Freedom 2013 The Nichols-Chancellor's Medal awarded by Vanderbilt University

铂类抗肿瘤药物作用机制研究进展_高传柱

0 引言
［1 ］自从Ｒosenberg 等人偶然发现顺铂的抗肿瘤活性后，铂类药物的研究得到迅速发展． 1978 年，第一，，代抗肿瘤铂类药物顺铂上市之后铂药被广泛应用于各种恶性肿瘤的临床治疗中并一直位列全球销售额领先的十大抗癌药物之中．目前已经上市的铂类药物均与顺铂具有相似的顺式平面结构，称为经典顺铂类抗肿瘤药物．经典顺铂类抗肿瘤药物广泛应用于各类实体瘤的化疗或者联合化疗方案中．据统计，目前临［2 ］ 70% ～ 80% 应用到顺铂类药物．顺铂对早期睾丸癌患者床化疗或者联合化疗治疗恶性肿瘤的方案中，［3 ］具有很好的疗效，治愈率甚至可达 100% ．但是，在铂类抗肿瘤药物的临床治疗过程中伴发严重的胃肠
Abstract： As one of the most important anticancer drugs with unique anticancer mechanism，high activity and broad anticancer spectrum，platinumbased drugs have been widely used in treatment of malignant tumors． based drugs has been greatly limited by its side effects，resistance However，the clinical application of platinumor crossresistance． Summarizing reported researches about anticancer mechanism of platinumbased drugs could deepen the understanding of the special anticancer mechanisms，provide directions in the clinical medication and give more information in designing and synthesizing novel platinumbased drugs with better efficiency，lower toxicity and less resistance． The research progress of platinum compounds for their anticancer mechanisms are therefore summarized in this paper． Key words： anticancer platinum drugs； antitumor mechanism； research progress

CollegeEnglish--我们唯一恐惧的就是恐惧本身[含翻译]

The Only Thing We Have to Fear Is Fear ItselfSpeaker: Franklin Roosevelt, January 30, 1882-April 12, 1945, the 32nd President of the United StatesDate & Place: March 4, 1933, Washington D.C., USABackground: First Inaugural Address with the nation in the grips of the Great Depression. The address is 1880 words and took 19 minutes to deliver, won 20 ovations, ranked3rd in The Top 100 American Speeches of the 20th Century.President Hoover, Mr. Chief Justice, my friends:This is a day of national consecration. And I am certain that on this day my fellow Americans expect that on my induction into the Presidency, I will address them with a candor and a decision which the present situation of our people impels. (Applause.)This is preeminently the time to speak the truth, the whole truth, frankly and boldly. Nor need we shrink from honestly facing conditions in our country today. This great Nation will endure, as it has endured, will revive and will prosper. (Applause.)So, first of all, let me assert my firm belief that the only thing we have to fear is fear itself- nameless, unreasoning, unjustified terror which paralyzes needed efforts to convert retreat into advance. In every dark hour of our national life, a leadership of frankness and of vigor has met with that understanding and support of the people themselves which is essential to victory. And I am convinced that you will again give that support to leadership in these critical days. (Applause.)In such a spirit on my part and on yours we face our common difficulties. They concern, thank God, only material things. Values have shrunk to fantastic levels; taxes have risen; our ability to pay has fallen; government of all kinds is faced by serious curtailment of income; the means of exchange are frozen in the currents of trade; the withered leaves of industrial enterprise lie on every side; farmers find no markets for their produce; and the savings of many years in thousands of families are gone. More important, a host of unemployed citizens face the grim problem of existence and an equally great number toil with little return. Only a foolish optimist can deny the dark realities of the moment.And yet our distress comes from no failure of substance. We are stricken by no plague of locusts. Compared with the perils which our forefathers conquered, because they believed and were not afraid, we have still much to be thankful for. Nature still offers her bounty and human efforts have multiplied it. Plenty is at our doorstep, but a generous use of it languishes in the very sight of the supply.Primarily, this is because the rulers of the exchange of mankind's goods have failed, through their own stubbornness and their own incompetence, have admitted their failure, and have abdicated. Practices of the unscrupulous money changers stand indicted in the court of public opinion, rejected by the hearts and minds of men.True, they have tried. But their efforts have been cast in the pattern of an out- worn tradition. Faced by failure of credit, they have proposed only the lending of more money. Stripped of the lure of profit by which to induce our people to follow their false leadership, they have resorted to exhortations, pleading tearfully for re- stored confidence. They only know the rules of a generation of self-seekers. They have no vision, and when there is no vision the people perish. (Applause.)Yes, the money changers have' fled from their high seats in the temple of our civilization. We may now restore that temple to the ancient truths. (Applause.) The measure of that restoration lies in the extent to which we apply social values more noble than mere monetary profit.Happiness lies not in the mere possession of money; it lies in the joy of achievement, in the thrill of creative effort. The joy, the moral stimulation of work no longer must be forgotten in the mad chase of evanescent profits. These dark days, my friends, will be worth all they cost us if they teach us that our true destiny is not to be ministered unto but to minister to ourselves, to our fellow men. (Applause.)Recognition of that falsity of material wealth as the standard of success goes hand in hand with theabandonment of the false belief that public office and high political position are to be valued only by the standards of pride of place and personal profit; and there must be an end to a conduct in banking and in business which too often has given to a sacred trust the likeness of callous and selfish wrong- doing. (Applause.) Small wonder that confidence languishes, for it thrives only on honesty, on honor, on the sacredness of obligations, on faithful protection, and on unselfish performance; without them it cannot live.Restoration calls, however, not for changes in ethics alone. This Nation is asking for action, and action now. (Applause.)Our greatest primary task is to put people to work. (Applause.) This is no unsolvable problem if we face it wisely and courageously. It can be accomplished in part by direct recruiting by the Government itself, treating the task as we would treat the emergency of a war, but at the same time, through this employment, accomplishing great -- greatly needed projects to stimulate and reorganize the use of our great natural resources.Hand in hand with that we must frankly recognize the overbalance of population in our industrial centers and, by engaging on a national scale in a redistribution, endeavor to provide a better use of the land for those best fitted for the land. (Applause.)Yes, the task can be helped by definite efforts to raise the values of agricultural products, and with this the power to purchase the output of our cities. It can be helped by preventing realistically the tragedy of the growing loss through foreclosure of our small homes and our farms. It can be helped by insistence that the Federal, the State, and the local governments act forthwith on the demand that their cost be drastically reduced. (Applause.) It can be helped by the unifying of relief activities which today are often scattered, uneconomical, unequal. It can be helped by nation- al planning for and supervision of all forms of transportation and of communications and other utilities that have a definitely public character. There are many ways in which it can be helped, but it can never be helped by merely talking about it.(Applause.) We must act. We must act quickly.And finally, in our progress towards a resumption of work, we require two safe- guards against a return of the evils of the old order. There must be a strict supervision of all banking and credits and investments. (Applause.) There must be an end to speculation with other people's money. (Applause.) And there must be provision for an adequate but sound currency. (Applause.)These, my friends, are the lines of attack. I shall presently urge upon a new Congress in special session detailed measures for their fulfillment, and I shall seek the immediate assistance of the 48 States. (Applause.) Through this program of action we address ourselves to putting our own national house in order and making income balance outgo. Our international trade relations, though vastly important, are in point of time, and necessity, secondary to the establishment of a sound national economy. (Applause.) I favor, as a practical policy, the putting of first things first. I shall spare no effort to restore world trade by international economic readjustment; but the emergency at home cannot wait on that accomplishment.The basic thought that guides these specific means of national recovery is not nationally -- narrowly nationalistic. It is the insistence, as a first consideration, up- on the interdependence of the various elements in and parts of the United States of America -- a recognition of the old and permanently important manifestation of the American spirit of the pioneer. It is the way to recovery. It is the immediate way. It is the strongest assurance that recovery will endure.In the field of world policy, I would dedicate this Nation to the policy of the good neighbor: the neighbor who resolutely respects himself and, because he does so, respects the rights of others; the neighbor who respects his obligations and respects the sanctity of his agreements in and with a world of neighbors. (Applause.) If I read the temper of our people correctly, we now realize, as we have neverrealized before, our interdependence on each other; that we can not merely take, but we must give as well; that if we are to go forward, we must move as a trained and loyal army willing to sacrifice for the good of a common discipline, because without such discipline no progress can be made, no leadership becomes effective.We are, I know, ready and willing to submit our lives and our property to such discipline, because it makes possible a leadership which aims at the larger good. This, I propose to offer, pledging that the larger purposes will bind upon us, bind upon us all as a sacred obligation with a unity of duty hitherto evoked only in times of armed strife. With this pledge taken, I assume unhesitatingly the leadership of this great army of our people dedicated to a disciplined attack upon our common problems.Action in this image, action to this end is feasible under the form of government which we have inherited from our ancestors. Our Constitution is so simple, so practical that it is possible always to meet extraordinary needs by changes in emphasis and arrangement without loss of essential form. That is why our constitutional system has proved itself the most superbly enduring political mechanism the modern world has ever seen. It has met every stress of vast expansion of territory, of foreign wars, of bitter internal strife, of world relations. And it is to be hoped that the normal balance of executive and legislative authority may be wholly equal, wholly adequate to meet the unprecedented task before us. But it may be that an unprecedented demand and need for undelayed action may call for temporary departure from that normal balance of public procedure.I am prepared under my constitutional duty to recommend the measures that a stricken nation in the midst of a stricken world may require. These measures, or such other measures as the Congress may build out of its experience and wisdom, I shall seek, within my constitutional authority, to bring to speedy adoption.But, in the event that the Congress shall fail to take one of these two courses, in the event that the national emergency is still critical, I shall not evade the clear course of duty that will then confront me. I shall ask the Congress for the one remaining instrument to meet the crisis broad Executive power to wage a war against the emergency, as great as the power that would be given to me if we were in fact invaded by a foreign foe. (Applause.)For the trust reposed in me, I will return the courage and the devotion that be- fit the time. I can do no less.We face the arduous days that lie before us in the warm courage of national unity; with the clear consciousness of seeking old and precious moral values; with the clean satisfaction that comes from the stern performance of duty by old and young alike. We aim at the assurance of a rounded, a permanent national life.We do not distrust the future of essential democracy. The people of the United States have not failed. In their need they have registered a mandate that they want direct, vigorous action. They have asked for discipline and direction under leadership. They have made me the present instrument of their wishes. In the spirit of the gift I take it.In this dedication. In this dedication of a Nation, we humbly ask the blessing of God. May He protect each and every one of us. May He guide me in the days to come. (Applause.)我们唯一恐惧的就是恐惧本身演讲者：富兰克林·罗斯福，1882.1.30—1945.4.12, 第32任美国总统时间地点：1933年3月4日，美国首都华盛顿演讲背景：美国处于历史上最严重的经济危机时期的首任总统就职演讲。

Vitali’s theorem and WWKL

VITALI’S THEOREM AND WWKLDOUGLAS K.BROWNMARIAGNESE GIUSTOSTEPHEN G.SIMPSONAbstract.Continuing the investigations of X.Yu and others,westudy the role of set existence axioms in classical Lebesgue mea-sure theory.We show that pairwise disjoint countable additivityfor open sets of reals is provable in RCA0.We show that sev-eral well-known measure-theoretic propositions including the VitaliCovering Theorem are equivalent to WWKL over RCA0.1.IntroductionThe purpose of Reverse Mathematics is to study the role of set ex-istence axioms,with an eye to determining which axioms are needed in order to prove speciﬁc mathematical theorems.In many cases,it is shown that a speciﬁc mathematical theorem is equivalent to the set existence axiom which is needed to prove it.Such equivalences are often proved in the weak base theory RCA0.RCA0may be viewed as a kind of formalized constructive or recursive mathematics,with full clas-sical logic but severely restricted comprehension and induction.The program of Reverse Mathematics has been developed in many publica-tions;see for instance[5,10,11,12,20].In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory.Historically,the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics.Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician.Although our program of Reverse Mathematics is quite diﬀerent from Bishop-style constructivism,we feel that Bishop’s remark implicitly raises an interesting question:Which nonconstructive set existence axioms are needed for measure theory?VITALI’S THEOREM AND WWKL 2This paper,together with earlier papers of Yu and others [21,22,23,24,25,26],constitute an answer to that question.The results of this paper build upon and clarify some early results of Yu and Simpson.The reader of this paper will ﬁnd that familiarity with Yu–Simpson [26]is desirable but not essential.We begin in section 2by exploring the extent to which measure theory can be developed in RCA 0.We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0.This is in contrast to a result of Yu–Simpson [26]:countable additivity for open sets of reals is equivalent over RCA 0to a nonconstructive set existence axiom known as Weak Weak K¨o nig’s Lemma (WWKL).We show in sections 3and 4that several other basic propositions of measure theory are also equivalent to WWKL over RCA 0.Finally in section 5we show that the Vitali Covering Theorem is likewise equivalent to WWKL over RCA 0.2.Measure Theory in RCA 0Recall that RCA 0is the subsystem of second order arithmetic with∆01comprehension and Σ01induction.The purpose of this section is toshow that some measure-theoretic results can be proved in RCA 0.Within RCA 0,let X be a compact separable metric space.We deﬁne C (X )= A,the completion of A ,where A is the vector space of rational “polynomials”over X under the sup-norm, f =sup x ∈X |f (x )|.For the precise deﬁnitions within RCA 0,see [26]and section III.E of Brown’s thesis [4].The construction of C (X )within RCA 0is inspired by the constructive Stone–Weierstrass theorem in section 4.5of Bishop and Bridges [2].It is provable in RCA 0that there is a natural one-to-one correspondence between points of C (X )and continuous functions f :X →R which are equipped with a modulus of uniform continuity ,that is to say,a function h :N →N such that for all n ∈N and x ,y ∈Xd (x,y )<12n .Within RCA 0we deﬁne a measure (more accurately,a nonnegative Borel probability measure)on X to be a nonnegative bounded linear functional µ:C (X )→R such that µ(1)=1.(Here µ(1)denotes µ(f ),f ∈C (X ),f (x )=1for all x ∈X .)For example,if X =[0,1],the unit interval,then there is an obvious measure µL :C ([0,1])→R given by µL (f )= 10f (x )dx ,the Riemann integral of f from 0to 1.We refer to µL as Lebesgue measure on [0,1].There is also the obvious generalization to Lebesgue measure µL on X =[0,1]n ,the n -cube.VITALI’S THEOREM AND WWKL 3Deﬁnition 2.1(measure of an open set).This deﬁnition is made in RCA 0.Let X be any compact separable metric space,and let µbe any measure on X .If U is an open set in X ,we deﬁneµ(U )=sup {µ(f )|f ∈C (X ),0≤f ≤1,f =0on X \U }.Within RCA 0this supremum need not exist as a real number.(Indeed,the existence of µ(U )for all open sets U is equivalent to ACA 0over RCA 0.)Therefore,when working within RCA 0,we interpret assertions about µ(U )in a “virtual”or comparative sense.For example,µ(U )≤µ(V )is taken to mean that for all >0and all f ∈C (X )with 0≤f ≤1and f =0on X \U ,there exists g ∈C (X )with 0≤g ≤1and g =0on X \V such that µ(f )≤µ(g )+ .See also [26].Some basic properties of Lebesgue measure are easily proved in RCA 0.For instance,it is straightforward to show that the Lebesgue measure of the union of a ﬁnite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.We deﬁne L 1(X,µ)to be the completion of C (X )under the L 1-norm given by f 1=µ(|f |).(For the precise deﬁnitions,see [5]and[26].)In RCA 0we see that L 1(X,µ)is a separable Banach space,but to assert within RCA 0that points of the Banach space L 1(X,µ)represent measurable functions f :X →R is problematic.We shall comment further on this question in section 4below.Lemma 2.2.The following is provable in RCA 0.If U n ,n ∈N ,is a sequence of open sets,then µ∞ n =0U n ≥lim k →∞µ k n =0U n .Proof.Trivial.Lemma 2.3.The following is provable in RCA 0.If U 0,U 1,...,U k is a ﬁnite,pairwise disjoint sequence of open sets,then µ k n =0U n ≥k n =0µ(U n ).Proof.Trivial.An open set is said to be connected if it is not the union of two disjoint nonempty open sets.Let us say that a compact separable metric space X is nice if for all suﬃciently small δ>0and all x ∈X ,the open ballB (x,δ)={y ∈X |d (x,y )<δ}VITALI’S THEOREM AND WWKL4 is connected.Such aδis called a modulus of niceness for X.For example,the unit interval[0,1]and the n-cube[0,1]n are nice, but the Cantor space2N is not nice.Theorem2.4(disjoint countable additivity).The following is prov-able in RCA0.Assume that X is nice.If U n,n∈N,is a pairwise disjoint sequence of open sets in X,thenµ∞n=0U n=∞n=0µ(U n).Proof.Put U= ∞n=0U n.Note that U is an open set.By Lemmas2.2and2.3,we have in RCA0thatµ(U)≥ ∞n=0µ(U n).It remainsto prove in RCA0thatµ(U)≤ ∞n=0µ(U n).Let f∈C(X)be suchthat0≤f≤1and f=0on X\U.It suﬃces to prove thatµ(f)≤∞n=0µ(U n).Claim1:There is a sequence of continuous functions f n:X→R, n∈N,deﬁned by f n(x)=f(x)for all x∈U n,f n(x)=0for all x∈X\U n.To prove this in RCA0,recall from[6]or[20]that a code for a continuous function g from X to Y is a collection G of quadruples (a,r,b,s)with certain properties,the idea being that d(a,x)<r im-plies d(b,g(x))≤s.Also,a code for an open set U is a collection of pairs(a,r)with certain properties,the idea being that d(a,x)<r im-plies x∈U.In this case we write(a,r)<U to mean that d(a,b)+r<s for some(b,s)belonging to the code of U.Now let F be a code for f:X→R.Deﬁne a sequence of codes F n,n∈N,by putting(a,r,b,s) into F n if and only if1.(a,r,b,s)belongs to F and(a,r)<U n,or2.(a,r,b,s)belongs to F and b−s≤0≤b+s,or3.b−s≤0≤b+s and(a,r)<U m for some m=n.It is straightforward to verify that F n is a code for f n as required by claim1.Claim2:The sequence f n,n∈N,is a sequence of elements of C(X). To prove this in RCA0,we must show that the sequence of f n’s has a sequence of moduli of uniform continuity.Let h:N→N be a modulus of uniform continuity for f,and let k be so large that1/2k is a modulus of niceness for X.We shall show that h :N→N deﬁned by h (m)=max(h(m),k)is a modulus of uniform continuity for all of the f n’s.Let x,y∈X and m∈N be such that d(x,y)<1/2h (m). To show that|f n(x)−f n(y)|<1/2m,we consider three cases.Case1:VITALI’S THEOREM AND WWKL5 x,y∈U n.In this case we have|f n(x)−f n(y)|=|f(x)−f(y)|<1VITALI’S THEOREM AND WWKL 6From (1)we see that for each >0there exists k such that µ(f )− ≤ kn =0µ(f n ).Thus we haveµ(f )− ≤kn =0µ(f n )≤k n =0µ(U n )≤∞ n =0µ(U n ).Since this holds for all >0,it follows that µ(f )≤ ∞n =0µ(U n ).Thus µ(U )≤ ∞n =0µ(U n )and the proof of Theorem 2.4is complete.Corollary 2.5.The following is provable in RCA 0.If (a n ,b n ),n ∈N is a sequence of pairwise disjoint open intervals,then µL ∞ n =0(a n ,b n ) =∞ n =0|a n −b n |.Proof.This is a special case of Theorem 2.4.Remark 2.6.Theorem 2.4fails if we drop the assumption that X is nice.Indeed,let µC be the familiar “fair coin”measure on the Cantor space X =2N ,given by µC ({x |x (n )=i })=1/2for all n ∈N and i ∈{0,1}.It can be shown that disjoint ﬁnite additivity for µC is equivalent to WWKL over RCA 0.(WWKL is deﬁned and discussed in the next section.)In particular,disjoint ﬁnite additivity for µC is not provable in RCA 0.3.Measure Theory in WWKL 0Yu and Simpson [26]introduced a subsystem of second order arith-metic known as WWKL 0,consisting of RCA 0plus the following axiom:if T is a subtree of 2<N with no inﬁnite path,thenlim n →∞|{σ∈T |length(σ)=n }|VITALI’S THEOREM AND WWKL 7see also Sieg [18].In this sense,every mathematical theorem provable in WKL 0or WWKL 0is ﬁnitistically reducible in the sense of Hilbert’s Program;see [19,6,20].Remark 3.2.The study of ω-models of WWKL 0is closely related to the theory of 1-random sequences,as initiated by Martin-L¨o f [16]and continued by Kuˇc era [7,13,14,15].At the time of writing of [26],Yu and Simpson were unaware of this work of Martin-L¨o f and Kuˇc era.The purpose of this section and the next is to review and extend the results of [26]and [21]concerning measure theory in WWKL 0.A measure µ:C (X )→R on a compact separable metric space X is said to be countably additive if µ∞ n =0U n =lim k →∞µ k n =0U n for any sequence of open sets U n ,n ∈N ,in X .The following theorem is implicit in [26]and [21].Theorem 3.3.The following assertions are pairwise equivalent over RCA 0.1.WWKL.2.(countable additivity)For any compact separable metric space Xand any measure µon X ,µis countably additive.3.For any covering of the closed unit interval [0,1]by a sequence of open intervals (a n ,b n ),n ∈N ,we have ∞n =0|a n −b n |≥1.Proof.That WWKL implies statement 2is proved in Theorem 1of [26].The implication 2→3is trivial.It remains to prove that statement 3implies WWKL.Reasoning in RCA 0,let T be a subtree of 2<N with no inﬁnite path.PutT ={σ i |σ∈T,σ i /∈T,i <2}.For σ∈2<N put lh(σ)=length of σanda σ=lh(σ)−1n =0σ(n )2lh(σ).Note that |a σ−b σ|=1/2lh(σ).Note also that σ,τ∈2<N are incompa-rable if and only if (a σ,b σ)∩(a τ,b τ)=∅.In particular,the intervals (a τ,b τ),τ∈ T,are pairwise disjoint and cover [0,1)except for some of the points a σ,σ∈2<N .Fix >0and put c σ=a σ− /4lh(σ),d σ=a σ+ /4lh(σ).Then the open intervals (a τ,b τ),τ∈ T,(c σ,d σ),VITALI’S THEOREM AND WWKL 8σ∈2<N and (1− ,1+ )form a covering of [0,1].Applying statement 3,we see that the sum of the lengths of these intervals is ≥1,i.e. τ∈ T12lh(τ)=1.From this,equation (2)follows easily.Thus we have proved that state-ment 3implies WWKL.This completes the proof of the theorem.It is possible to take a somewhat diﬀerent approach to measure the-ory in RCA 0.Note that the deﬁnition of µ(U )that we have given (Deﬁnition 2.1)is extensional in RCA 0.This means that if U and V contain the same points then µ(U )=µ(V ),provably in RCA 0.An alternative approach is the intensional one,embodied in Deﬁnition 3.4below.Recall that an open set U is given in RCA 0as a sequence of basic open sets.In the case of the real line,basic open sets are just intervals with rational endpoints.Deﬁnition 3.4(intensional Lebesgue measure).We make this deﬁni-tion in RCA 0.Let U = (a n ,b n ) n ∈N be an open set in the real line.The intensional Lebesgue measure of U is deﬁned by µI (U )=lim k →∞µL k n =0(a n ,b n ) .Theorem 3.5.It is provable in RCA 0that intensional Lebesgue mea-sure µI is countably additive on open sets.In other words,if U n ,n ∈N ,is a sequence of open sets,then µI∞ n =0U n =lim k →∞µI k n =0U n .Proof.This is immediate from the deﬁnitions,since ∞n =0U n is deﬁned as the union of the sequences of basic open intervals in U n ,n ∈N .Returning now to WWKL 0,we can prove that intensional Lebesgue measure concides with extensional Lebesgue measure.In fact,we have the following easy result.Theorem 3.6.The following assertions are pairwise equivalent over RCA 0.VITALI’S THEOREM AND WWKL91.WWKL.2.µI(U)=µL(U)for all open sets U⊆[0,1].3.µI is extensional on open sets.In other words,for all open setsU,V⊆[0,1],if∀x(x∈U↔x∈V)thenµI(U)=µI(V).4.For all open sets U⊇[0,1],we haveµI(U)≥1.Proof.This is immediate from Theorems3.3and3.5.4.More Measure Theory in WWKL0In this section we show that a good theory of measurable functions and measurable sets can be developed within WWKL0.Weﬁrst consider pointwise values of measurable functions.Our ap-proach is due to Yu[21,24].Let X be a compact separable metric space and letµ:C(X)→R be a positive Borel probability measure on X.Recall that L1(X,µ)is deﬁned within RCA0as the completion of C(X)under the L1-norm.In what sense or to what extent can we prove that a point of the Banach space L1(X,µ)gives rise to a function f:X→R?In order to answer this question,recall that f∈L1(X,µ)is given by a sequence f n∈C(X),n∈N,which converges to f in the L1-norm; more preciselyf n−f n+1 1≤12nfor all n,and|f m(x)−f m (x)|≤12k.VITALI’S THEOREM AND WWKL10 Then for x∈C fnand m ≥m≥n+2k+2we have|f m(x)−f m (x)|≤m −1i=m|f i(x)−f i+1(x)|≤∞i=n+2k+2|f i(x)−f i+1(x)|≤12k.We need a lemma:Lemma4.2.The following is provable in RCA0.For f∈C(X)and >0,we haveµ({x|f(x)> })≤ f 1/ .Proof.Put U={x|f(x)> }.Note that U is an open set.If g∈C(X),0≤g≤1,g=0on X\U,then we have g≤|f|, hence µ(g)=µ( g)≤µ(|f|)= f 1,henceµ(g)≤ f 1/ .Thus µ(U)≤ f 1/ and the lemma is proved.Using this lemma we haveµ(X\C fnk )=µx∞i=n+2k+2|f i(x)−f i+1(x)|>12i=1VITALI’S THEOREM AND WWKL 11hence by countable additivityµ(X \C f n )≤∞ k =0µ(X \C f nk )≤∞k =012n .This completes the proof of Proposition 4.1.Remark 4.3(Yu [21]).In accordance with Proposition 4.1,forf = f n n ∈N ∈L 1(X,µ)and x ∈ ∞n =0C f n ,we deﬁne f (x )=lim n →∞f n (x ).Thus we see thatf (x )is deﬁned on an F σset of measure 1.Moreover,if f =g in L 1(X,µ),i.e.if f −g 1=0,then f (x )=g (x )for all x in an F σset of measure 1.These facts are provable in WWKL 0.We now turn to a discussion of measurable sets within WWKL 0.We sketch two approaches to this topic.Our ﬁrst approach is to identify measurable sets with their characteristic functions in L 1(X,µ),accord-ing to the following deﬁnition.Deﬁnition 4.4.This deﬁnition is made within WWKL 0.We say that f ∈L 1(X,µ)is a measurable characteristic function if there exists a sequence of closed setsC 0⊆C 1⊆···⊆C n ⊆...,n ∈N ,such that µ(X \C n )≤1/2n for all n ,and f (x )∈{0,1}for all x ∈ ∞n =0C n .Here f (x )is as deﬁned in Remark 4.3.Our second approach is more direct,but in its present form it applies only to certain speciﬁc situations.For concreteness we consider only Lebesgue measure µL on the unit interval [0,1].Our discussion can easily be extended to Lebesgue measure on the n -cube [0,1]n ,the “fair coin”measure on the Cantor space 2N ,etc .Deﬁnition 4.5.The following deﬁnition is made within RCA 0.Let S be the Boolean algebra of ﬁnite unions of intervals in [0,1]with rational endpoints.For E 1,E 2∈S we deﬁne the distanced (E 1,E 2)=µL ((E 1\E 2)∪(E 2\E 1)),the Lebesgue measure of the symmetric diﬀerence of E 1and E 2.Thus d is a pseudometric on S ,and we deﬁne S to be the compact separable metric space which is the completion of S under d .A point E ∈ S is called a Lebesgue measurable set in [0,1].VITALI’S THEOREM AND WWKL 12We shall show that these two approaches to measurable sets (Deﬁ-nitions 4.4and 4.5)are equivalent in WWKL 0.Begin by deﬁning an isometry χ:S →L 1([0,1],µL )as follows.For 0≤a <b ≤1deﬁneχ([a,b ])= f n n ∈N ∈L 1([0,1],µL )where f n (0)=f n (a )=f n (b )=f n (1)=0and f n a +b −a 2n +1=1and f n ∈C ([0,1])is piecewise linear otherwise.Thus χ([a,b ])is a measurable characteristic function corresponding to the interval [a,b ].For 0≤a 1<b 1<···<a k <b k ≤1deﬁneχ([a 1,b 1]∪···∪[a k ,b k ])=χ([a 1,b 1])+···+χ([a k ,b k ]).It is straightforward to prove in RCA 0that χextends to an isometryχ: S→L 1([0,1],µL ).Proposition 4.6.The following is provable in WWKL 0.If E ∈ Sis a Lebesgue measurable set,then χ(E )is a measurable characteristic function in L 1([0,1],µL ).Conversely,given a measurable characteristic function f ∈L 1([0,1],µL ),we can ﬁnd E ∈ Ssuch that χ(E )=f in L 1([0,1],µL ).Proof.It is straightforward to prove in RCA 0that for all E ∈ S , χ(E )is a measurable characteristic function.For the converse,let f be a measurable characteristic function.By Deﬁnition 4.4we have that f (x )∈{0,1}for all x ∈ ∞n =0C n .ByProposition 4.1we have |f (x )−f 3n +3(x )|<1/2n for all x ∈C f n .Put U n ={x ||f 3n +3(x )−1|<1/2n }and V n ={x ||f 3n +3(x )|<1/2n }.Then for n ≥1,U n and V n are disjoint open sets.Moreover C n ∩C f n ⊆U n ∪V n ,hence µL (U n ∪V n )≥1−1/2n −1.By countable additivity(Theorem 3.3)we can eﬀectively ﬁnd E n ,F n ∈S such that E n ⊆U n and F n ⊆V n and µL (E n ∪F n )≥1−1/2n −2.Put E = E n +5 n ∈N .It is straightforward to show that E belongs to S and that χ(E )=f in L 1([0,1],µL ).This completes the proof.Remark 4.7.We have presented two notions of Lebesgue measurable set and shown that they are equivalent in WWKL 0.Our ﬁrst notion (Deﬁnition 4.4)has the advantage of generality in that it applies to any measure on a compact separable metric space.Our second no-tion (Deﬁnition 4.5)is advantageous in other ways,namely it is more straightforward and works well in RCA 0.It would be desirable to ﬁnd a single deﬁnition which combines all of these advantages.VITALI’S THEOREM AND WWKL 135.Vitali’s TheoremLet S be a collection of sets.A point x is said to be Vitali covered by S if for all >0there exists S ∈S such that x ∈S and the diameter of S is less than .The Vitali Covering Theorem in its simplest form says the following:if I is a sequence of intervals which Vitali covers an interval E in the real line,then I contains a countable,pairwise disjoint set of intervals I n ,n ∈N ,such that ∞n =0I n covers E except for a set of Lebesgue measure 0.The purpose of this section is to show that various forms of the Vitali Covering Theorem are provable in WWKL 0and in fact equivalent to WWKL over RCA 0.Throughout this section,we use µto denote Lebesgue measure.Lemma 5.1(Baby Vitali Lemma).The following is provable in RCA 0.Let I 0,...,I n be a ﬁnite sequence of intervals.Then we can ﬁnd a pair-wise disjoint subsequence I k 0,...,I k m such thatµ(I k 0∪···∪I k m )≥1VITALI’S THEOREM AND WWKL 14I =[2a −b,2b −a ].)Thusµ(I 0∪···∪I n )≤µ(I k 0∪···∪I k m )≤µ(I k 0)+···+µ(I k m )=3µ(I k 0)+···+3µ(I k m )=3µ(I k 0∪···∪I k m )and the lemma is proved.Lemma 5.2.The following is provable in WWKL 0.Let E be an in-terval,and let I n ,n ∈N ,be a sequence of intervals.If E ⊆ ∞n =0I n ,then µ(E )≤lim k →∞µ k n =0I n .Proof.If the intervals I n are open,then the desired conclusion follows immediately from countable additivity (Theorem 3.3).Otherwise,ﬁx >0and let I n be an open interval with the same midpoint as I n andµ(I n )=µ(I n )+µ(E \A ).(3)VITALI’S THEOREM AND WWKL 15To prove the claim,use Lemma 5.2and the Vitali property to ﬁnd a ﬁnite set of intervals J 1,...,J l ∈I such that J 1,...,J l ⊆E \A andµ(E \(A ∪J 1∪···∪J l ))<13µ(J 1∪···∪J l ).We then have µ(E \(A ∪I 1∪···∪I k ))<212µ(E \A )≤212µ(E \A )=34nµ(E ).Then by countable additivity we have µ E \∞ n =1A n =0and the lemma is proved.Remark 5.4.It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n -cube [0,1]n by closed balls or n -dimensional cubes.In the case of closed balls,the constant 3in the Baby Vitali Lemma 5.1is replaced by 3n .Theorem 5.5.The Vitali theorem for the interval [0,1](as stated in Lemma 5.3)is equivalent to WWKL over RCA 0.Proof.Lemma 5.3shows that,in RCA 0,WWKL implies the Vitali theorem for intervals.It remains to prove within RCA 0that the Vitali theorem for [0,1]implies WWKL.Instead of proving WWKL,we shall prove the equivalent statement 3.3.3.Reasoning in RCA 0,suppose thatVITALI’S THEOREM AND WWKL 16(a n ,b n ),n ∈N ,is a sequence of open intervals which covers [0,1].Let I be the countable set of intervals (a nki ,b nki )= a n +i k(b n −a n ) where i,k,n ∈N and 0≤i <k .Then I is a Vitali covering of [0,1].By the Vitali theorem for intervals,I contains a sequence of pairwise disjoint intervals I m ,m ∈N ,such that µ ∞ m =0I m ≥1.By disjoint countable additivity (Corollary 2.5),we have∞m =0µ(I m )≥1.From this it follows easily that∞n =0|a n −b n |≥1.Thus we have 3.3.3and our theorem is proved.We now turn to Vitali’s theorem for measurable sets.Recall our discussion of measurable sets in section 4.A sequence of intervals I is said to almost Vitali cover a Lebesgue measurable set E ⊆[0,1]if for all >0we have µL (E \O )=0,where O = {I |I ∈I ,diam(I )< }.Theorem 5.6.The following is provable in WWKL 0.Let E ⊆[0,1]be a Lebesgue measurable set with µ(E )>0.Let I be a sequence of intervals which almost Vitali covers E .Then I contains a pairwise disjoint sequence of intervals I n ,n ∈N ,such that µ E \∞ n =0I n =0.Proof.The proof of this theorem is similar to that of Lemma 5.3.The only modiﬁcation needed is in the proof of the claim.Recall from Deﬁnition 4.5that E =lim n →∞E n where each E n is a ﬁnite union of intervals in [0,1].Fix m so large thatµ((E \E m )∪(E m \E ))<1VITALI’S THEOREM AND WWKL 17andµ(E m \(A ∪J 1∪···∪J l ))<136µ(E \A )<236µ(E \A )≤236µ(E \A )<236µ(E \A )=3,The Baire category theorem in weak subsystems of second order arith-metic ,Journal of Symbolic Logic 58(1993),557–578.7.O.Demuth and A.Kuˇc era,Remarks on constructive mathematical analysis ,[3],1979,pp.81–129.8.H.-D.Ebbinghaus,G.H.M¨u ller,and G.E.Sacks (eds.),Recursion Theory Week ,Lecture Notes in Mathematics,no.1141,Springer-Verlag,1985,IX +418pages.VITALI’S THEOREM AND WWKL189.Harvey Friedman,unpublished communication to Leo Harrington,1977.10.Harvey Friedman,Stephen G.Simpson,and Rick L.Smith,Countable algebraand set existence axioms,Annals of Pure and Applied Logic25(1983),141–181.11.,Randomness and generalizations ofﬁxed point free functions,[1],1990, pp.245–254.15.,Subsystems of Second Order Arithmetic,Perspectives in Mathematical Logic,Springer-Verlag,1998,XIV+445pages.21.Xiaokang Yu,Measure Theory in Weak Subsystems of Second Order Arithmetic,Ph.D.thesis,Pennsylvania State University,1987,vii+73pages.22.,Riesz representation theorem,Borel measures,and subsystems of sec-ond order arithmetic,Annals of Pure and Applied Logic59(1993),65–78. 24.,A study of singular points and supports of measures in reverse mathe-matics,Annals of Pure and Applied Logic79(1996),211–219.26.Xiaokang Yu and Stephen G.Simpson,Measure theory and weak K¨o nig’slemma,Archive for Mathematical Logic30(1990),171–180.E-mail address:dkb5@,giusto@dm.unito.it,simpson@ The Pennsylvania State University。

Small ramsey numbers

Small Ramsey Numbers
*
Stanisław P. Radziszowski Department of Computer Science Rochester Institute of Technology Rochester, NY 14623, spr@ Submitted: June 11, 1994; Accepted: July 3, 1994 Current revision: August 28, 1995
2. Classical Two Color Ramsey Numbers We split the data into the table of values and a table with corresponding references. Known exact values appear as centered entries, lower bounds as top entries, and upper bounds as bottom entries. All the critical graphs for the numbers R (k , l ) (graphs on R (k , l ) − 1 vertices without Kk and without Kl in the complement) are known for k = 3 and l = 3, 4, 5, 6 [Ka2], 7 [RK3, MZ], and there are 1, 3, 1, 7 and 191 of them, respectively. There exists a unique critical graph for R (4,4) [Ka2]. There are 4 such graphs known for R (3,8) [RK2], 1 for R (3,9) [Ka2] and 350904 for R (4, 5) [MR5], but there might be more of them.

The Casino State and Crime - Macau_A Case Study

Proposed TitleThe Casino State and Crime - Macao: A Case StudyINTRODUCTIONIn this research, I would like to investigate the relationship between gambling and crime. Let me first define my research topic, a casino state means a state or city where gambling, such as blackjack and fan-tan (番摊), has been legalized by the government; by the word crime I mean violent crimes which include murder, aggravated assault and robbery. I have chosen Macao as a case study because Macao is properly known as a casino state in Asia and has been called the "Monte Carlo of the Orient". A vast amount of government revenue in Macao is collected from the casinos in the form of taxes. Since 1988 more than 30% of government's public revenue and expenditure has been derived from gambling taxes (Table 1). In addition, I also think that there is some specific link to Hong Kong. Firstly, many Hong Kong residents come to Macao to gamble since gambling is prohibited in Hong Kong; secondly, some triads in Hong Kong do fight for profits gained from the casinos in Macao. Moreover, the concerns about crime associated with the Macao casinos are a sensitive topic during the handover of Macao because of the need to demonstrate stability and order. In this study, I will focus on the link between gambling and crime in Macao. I will also investigate other issues such as the problems of the regulation of gambling and policing. The followings are some hypotheses (H) that I would like to investigate in this research:H1 Casinos enhance violent crime.H2 Less casino revenue and profits produce more violent crime.H3 More violent crime in a casino state, like Macao, reflects weak law enforcement.H1, H2 and H3 investigate the socio-economic (macro) aspects of gambling. In order to research H1, I will compare Macao with another casino state, such as Monte Carlo or Las Vegas. H2 reveals the idea of competing for scarcity. As there is less casino revenue, the intensity of crime will increase so that the social costs will increase. H3 has something to do with the reporting theory and policing.Macao has a special characteristic in which there is a strong community and people usually know each other very well, as a result, the people in Macao don't depend very much on the law enforcement system. The relationship between the police force and triads in Macao is perhaps similar to Hong Kong during the 60s and 70s.Another option is to investigate the psychological and sociological (micro) aspects of gambling and I have set the following hypothesis.H4 Gambling behavior (addiction) of adults produces more juvenile delinquency.In doing H4, I may use the method of self-reported crime in investigating the juvenile delinquency.I will also need to look at the cultural and sociological values.BRIEF INTRODUCTION OF MACAO AND STDMThe Portuguese came to Macao as early as 1535 and they traded in Macao, then finally China ceded Macao to Portugal in 1887 under some treaty terms. It is located on the western side of the Chu Chiang (Pearl River) Estuary, at the head of Kwang Chou (Canton) and stands opposite to Hong Kong. Together, they are called the Pearl River Delta. Macao has a total area of 23.5 square kilometers consistingof the Macao Peninsula, Taipa Island and Coloane Island. The total population today is about 650,000.Since gambling is prohibited in Hong Kong and China, Macao attracts a large number of tourists to its casinos, and is called the "Monte Carlo of the Orient". All gambling is controlled by a government-licensed syndicate, Sociedade De Turismo E Diversoes De Macao S.A.R.L. (STDM), which pays an annual tax of more than 4 billion dollars to the Macao government.The history of gambling in Macao goes back as early as 1875. However, gambling has not been legalized until 1937 and it is not franchised at that time. Only in 1962, the government has granted STDM with the Gambling Franchise. Sociedade De Turismo E Diversoes De Macao S.A.R.L (STDM), founded by Dr. Stanley Ho, plays a key role in the development, prosperity and growth of Macao. STDM is the largestcommercial employer in Macao providing jobs for more than 10,000 people. The company was granted the exclusive Gambling Franchise in Macao since 1962, which was extended in 1986 for another 15 years to 2001. The nine casinos operated by STDM have become the backbone of Macao tourism. The casinos offer games such as pai kau, fantan, blackjack and dice. There are also horse racing, dog racing and lottery. The annual income of STDM during 1978 and 1993 is given in Table 2.THE RECENT TREND OF CRIME IN MACAOsome crime statistics in Macao is given in Table 3. Over the last decade, the total number of crime in Macao has increased from 4,717 in 1987 to 8,576 in 1996, the increase is more than 80%. Crime rate per capita has increased from 14.11 per 1,000 people in 1987 to 20.62 per 1,000 people in 1996. The major problem is public order offense as the percentage increase is almost 6 times in 1996. This creates a serious social problem, as these crimes will directly affect the residents' daily life. They are worried about their life and property. The second highest percentage increase is robbery in buildings as it has increased for almost 3 times in 1996, then followed by other crimes which have increased for more than 2 times in 1996. The smallest increase is street robbery with only 8.39%.There is an upward trend in the number of crimes in Macao. The crime in Macao has becomemore publicized, serious, vice and organized. There are more juvenile delinquents and female criminals. In addition, the triads who are fighting for the profit of the casinos commit most of the serious crime (especially murder) in 1997 and 1998.However, the crime statistics may not be able to reflect the reality as some people may not report the crime because of the complicated procedures and time wasted; some victims may be tourists and they will just stay in Macao for a short period of time; some victims may be illegal immigrants so they don't report the crime.LITERATURE REVIEWSOCIO-ECONOMIC ASPECTS OF GAMBLINGThe literature review in this part is mainly about the socio-economic (macro) aspects of gambling and is related to the first three hypotheses that I have set.Some studies on "Casinos, crime, and real estate values: Do they relate?" were done in 1989 by Andrew J. Buck, Simon Hakim and Uriel Spiegel. The studies show that gambling can serve as the engine for growth in the Atlantic City region, but on the other hand, casinos will increase the level of crime significantly in that region. According to the studies, the sources of casino-related crime include the temporary visitors, criminals who realize the new crime opportunities offered by the casinos and the employees of the casinos. In addition, crime did not show a particular spatial pattern in the pre-casino years 1972 to 1977 but this emerged in the post-casino years 1978 to 1986. Andrew J. Buck has suggested in his article that some follow-up studies may lead to important policy implications and public security measures.Joseph Friedman, Simon Hakim and J. Weinblatt did another study in 1989 on "Casino gambling as a 'growth pole' strategy and its effect on crime." The studies show that crime in Atlantic City grew from 100.6 per 1000 people in 1977 to 353.7 in 1984. The highest percentage increase is in violent crimes, followed by burglaries and vehicle thefts. The lowest increase is in larcenies. In addition, the increase in crime was not limited to Atlantic City itself; there was also a significant crime increase in the neighboring communities. Joseph Friedman has raised an interesting question: Does crime increase due to normal growth or does the particular nature of the industry (casino gambling) cause additional crime? On the other hand, Friedman has suggested that casinos can serve as a "growth pole" device from an economic point of view. Between 1978 and 1984, the casinos attracted many visitors to Atlantic City. This had both direct and indirect income andemployment effects. Consumption by the tourists increases and the revenue of hotels also increased. The casino industry has invested more than $2.2 billion in property and equipment, and $199 million for economic redevelopment and housing investment in Atlantic City. Friedman has concluded that casinos can boost the economy but also increase the level of crime. He suggested that a social cost-benefit analysis on casinos and crime should be done and might lead to a different conclusion: that the social cost of crime may be high enough to outweigh the economicgain obtained from the casino. Besides, casinos may cause redistribution effects. Residents of localities far away from the casinos may suffer from crime spillover, while they may gain none or little development benefits generated by the casinos.These two studies support my first hypothesis: Casinos enhance violent crime.PSYCHOLOGICAL AND SOCIOLOGICAL (MICRO) ASPECTS OF GAMBLINGA study on "Adolescent machine gambling and crime" is done by Tim Yeoman and Mark Griffiths in 1990. This survey was done in Plymouth with the aim to find out whether there was any relationship between criminal activity (theft) and gambling machine use. The police officers collected information from 1851 juvenile offenders. The results show that there is little evidence showing that children and adolescents who gamble on fruit machines may engage themselves in stealing in order to fund their habit. The study also showed that many juveniles with heavy gaming machine use came from families who are less fortunate and they usually have poor relationships with their families.METHODOLOGYI intend to use secondary analysis and interviews for the data collection of my research. The reasons for that are direct observation will be dangerous and difficult because I may need to deal with the triad members who are in charge of the gambling tables, surveys will be quite difficult as well since it may involve doing surveys on the triad members who have dominated most of the casinos in Macao.SECONDARY ANAL YSISSince direct observation is not very accessible in the casinos, secondary analysis is more preferable. The sources that I would use include archives, newspaper, magazine articles, and law enforcement agency files. I will try to choose the sources which are more objective. I will adopt the econometric methods to find the correlation between crime and gambling revenue and profits in accordance to my second hypothesis.INTERVIEWSInterviews will enable me to do most of the qualitative part of my research, and the informationgained here is usually more realistic. I plan to interview some police officers, some owners of gambling tables in the casinos, some administrative employees of STDM, some reporters and journalists, some officers in the Autoridade Monetaria E Cambial De Macao (Gambling Authority), some prisoners who have some connections with the casinos, some gambling addictors in Macao and Hong Kong and if possible, the ex-triad members in Macao and Hong Kong.I have chosen to interview the above people that I have mentioned because of the following reasons:* the police officers can provide further data that I cannot get from the Macao Census and Statistics Department. They can also help me to examine the law enforcement system and give me some ideas on the organized crime of Macao.* the owners of gambling tables in the casinos and the administrative employees of STDM serve as the insight sources and enable me to understand how the casinos operate.* the reporters and journalists can give me some information from a different point of view since they represent the communication media.* the officers in the AMCM (Gambling Authority) can give me some ideas on the regulation of gambling and how the government manage the Gambling Franchise.* the prisoners and ex-triad members can give some information on the illegal activities associated with gambling and also enable me to understand more about organized crime in Macao, and maybe some of the secret rules of those triads. (Subject to permission to do so)* the gambling addictors can help me to understand more about the idea of addiction and morality of gambling and also the effects on the gamblers' family and society.DISCUSSIONSince the past studies have suggested that casino gambling has a positive effect on crime, especially on violent crimes so I would like to investigate this issue in Macao. In my opinion, although Macao is a casino state, it is not like other countries since the government, the police, the triads and Dr. Stanley Ho have some kind of hidden agreements so that Macao may appear to be more peaceful compared with Atlantic City.I think those news and issues about murdering committed by the triads in the casinos, and the war between the triads and the police officers in Macao are important for my study and the questions I raise. Another issue that interests me is the arresting of Wan Kuok Koi who has almost all the control of the gambling tables in the casinos in Macao.Macao begins to have published official statistics on crime only in 1987 and there has been very little research and analysis on crime in Macao. The crime statistics obtained from the Macao Census and Statistics Department are not detailed and are very limited. Therefore, I will try to contact other government departments such as the Police Department and the AMCM for their annual reports for additional data. If possible, I will try to go further back and find data in the 195 0s, 60s and 70s.But I believe that I would have an advantage of obtaining more insight information about STDM and the casinos because my mother has been working in the casinos for 27 years and my grandparents are also ex-employees of STDM.CONCLUSIONTo conclude, Macao is a casino state which is quite different from those in Australia and America because the government revenue in Macao depends very much on the gambling taxes and most of the infrastructure projects (international airport, deep-water port, new Macao-Taipa bridge and reclamation and development in Praia Grande Bay) are financed by STDM. As a result, the government may sometimes need to compromise with the casinos and triads so as to maintain its revenue. Besides, some police officers may also corrupt and profit from the casinos. Recently, the relationship between the triads and the police officers seems to be upset because of less revenue in the casinos and the handover of Macao in 1999. My research will mainly concentrate on the relationship of crime and gambling in Macao. I may also investigate other issues such as the problem of regulation of the Gambling Franchise and the policing in Macao.I have set three hypotheses on the macro aspects of gambling and also an option hypothesis on the micro aspects of gambling. Past studies have showed a positive relationship between casino gambling and crime and that casinos can stimulate economic growth. The methods that I plan to use are mainly secondary analysis and interviews. Some current events in Macao concerning the war between the police officers and the triads and the arresting of Wan Kuok Koi are also important for my studies because they may have hidden agreements in the past. The triads usually have some connections with the casinos or other associated illegal activities. Since not much research on crime is done in Macao, the major difficulty will be data collection.REFERENCESAndrew J. Buck, Joseph Deutsch, Simon Hakim, Uriel Spiegel and J. Weinblatt, "AV on Thunen Model of Crime, Casinos and Property Values in New Jersey", Urban Studies, Vol. 28, No. 5, 1991, p.673-686.Andrew J. Buck, Simon Hakim, Uriel Spiegel, "Casinos, Crime and Real Estate Values: Do they relate?", Journal of Research in Crime and Delinquency, Vol. 28, Aug1991, p.288-303.Joseph Friedman, Simon Hakim and J. Weinblatt, "Casinos Gambling as a 'Growth Pole' Strategy and Its Effect on Crime", Journal of Regional Science, V ol. 29, No.4, 1989, p.615-623.Kwong Che Leung and Kwan Fong, "Gambling in Macao: A Macroeconomic Analysis", Po pulation and Development in Macao (China Economic Research Centre, University of Macao), 1994, p.245-252.Penny Chan (1997) Notes for Presentation on "Sociological Analysis of the Current Problems of Public Security in Macao" (Unpublished Paper)Tim Yeoman and Mark Griffiths, "Adolescent Machine Gambling and Crime", Journalof Adolescence, Vol. 19, 1996, p.183-188.Encyclopedia Americana, Vol. 18, p.4Encyclopedia Britannica, V ol. 11, p.220-222Yearbook of Statistics, 1987-1996, Macao Census and Statistics DepartmentAutoridade Monetaria E Cambial De Macao (AMCM), Annual Report 1990Autoridade Monetaria E Cambial De Macao (AMCM), Report and Accounts 1985。

毕业论文参考题目及论著

汉语言文字‎学重要论文‎和著作现代汉语部‎分：著作类：理论语言学‎：《论人类语言‎结构的差异‎及其对人类‎精神发展的‎影响》，［德］威廉·冯·洪堡特，商务印书馆‎，1997年‎。

《普通语言学‎教程》，[瑞士]费尔德南·德·索绪尔，商务印书馆‎，2001。

《语言论》，[美]布龙菲尔德‎，1933/1980 ，商务印书馆‎Seman‎t ics,An intro‎d ucti‎o n to the scien‎c e of Meani‎n g , Ullma‎n,S , Oxfor‎d Oxfor‎d：Black‎well, 1962.Princ‎i ples‎o f seman‎t ics, Ullma‎n,S., Oxfor‎d Oxfor‎d：Black‎well ,1951.Words‎and their‎Use, Ullma‎n,S., N.Y Oxfor‎d Oxfor‎d：Black‎well, 1951,《历史语言学‎中的比较方‎法》，A·梅耶（法），岑麒祥译，科学出版社‎，1957。

《语言学概论‎》，L.R.帕尔默，李荣等译，商务印书馆‎，1983。

Langu‎a ge and Mind, Choms‎k y,N., New york : Harco‎u rt Brace‎Jovan‎o vi ch‎, 1972 .Essay‎s on Form and Inter‎p r eta‎t ion, Choms‎k y,N. , Londo‎n Fonta‎n a ,1977 .Rules‎and Repre‎s enta‎t ion, Choms‎k y,N, Oxfor‎d: Basil‎Black‎well ,1980.Langu‎a ge as Socia‎l Semio‎t ic,Halli‎d ay,M.A.K , Londo‎n: Edwar‎d Arnol‎d ,1978.Women‎,Fire,and Dange‎rous Thing‎s:What Categ‎o ries‎Revea‎l about‎ t he Mind. Lakof‎f,G.Chica‎g o: The Unive‎rsi ty‎of Chica‎g o Press‎.1987Histo‎ri cal‎ Lingu‎i stic‎s, R.L.Trask‎, 周流溪导读‎，外语教学与‎研究出版社‎北京，爱德华·阿诺德出版‎社，2000。

【能力】企业能力静态能力与动态能力理论界定及关系辨析

【关键字】能力企业能力：静态能力与动态能力理论界定及关系辨析黄培伦尚航标中国广州华南理工大学工商管理学院摘要：企业能力是企业的组织能力，按静态的观点称之为静态能力，按动态的观点称之为动态能力。

企业能力是静态能力与动态能力的统一，以静态能力为基础、动态能力为主导。

企业的静态能力集中表现为企业实力，企业的动态能力集中表现为企业活力。

静态能力和动态能力是企业能力的不同组成部分。

根据现有的文献分析，阐明企业能力、静态能力和动态能力的概念。

并且对静态能力与动态能力的关系进行辨析，认为动态能力是对静态能力的一种扬弃。

根据文中的研究，提出动态能力的研究模型。

此外，也探讨对未来研究的建议和在管理方面的含义。

关键词：企业能力；静态能力；动态能力引言学者对于“企业如何获取并保持其比赛优势”有许多不同的探讨，但大体上可以分为三类：比赛力量理论、战略冲突理论和资源基础理论。

这三个经典理论在基于静态的假设上解释企业是如何获取并保持比赛优势的(Teece, Pisano, and Shuen 1997)。

但，自上世纪九十年代以来，比赛的白炽化和环境的急剧动荡对这些忽视市场动态的理论提出挑战，使企业不断的否定其现有的资源和能力，所以要以动态的观点来审视企业的资源和能力，这促使动态能力理论的产生。

虽然对于动态能力的研究被视作最具发展前景的战略流派之一，但是由于动态能力的研究基础比较薄弱，在这个研究领域出现一些争论，甚至有学者提出“动态能力的概念和理论有价值么？”(鄢德春，2007)。

为什么会出现这种现象？显然这些学者把动态能力和企业静态能力对立起来。

本文的意图：(1)通过简要的描述关于企业能力的演化轨迹，明确企业静态能力与动态能力的理论界定；(2)明确企业静态能力、动态能力是企业能力的两个不同的层面，对于成功企业来说，二者缺一不可；(3)比较动态能力与静态能力的关系，并在此基础上提出一个包含动态能力逻辑推力的研究框架；(4)拟为下一步的研究提供理论支持和理论引导。

公共管理经典书目

公共管理经典书目（一）基础理论经典原著1、威尔逊：《行政学之研究》，《国外政治学》1987年第6期、1988年第1期。

2、古德诺：《政治与行政》，华夏出版社1987年版。

3、泰罗：《科学管理原理》，中国社会科学出版社1990年版。

4、法约尔：《工业管理与一般管理》，中国社会科学出版社1998年版。

5、马克斯·韦伯：《经济与社会》，商务印书馆1997年版。

6、怀特：《行政学概论》，上海商务印书馆1947年版。

7、西蒙：《管理行为》，北京经济学院出版社1988年版。

8、西蒙：《管理决策新科学》，中国社会科学出版社1982年版。

9、沃尔多：《行政国家：美国公共行政的政治理论研究》，纽约：罗纳德出版社1948年版。

10、林德布洛姆：《决策过程》，上海译文出版社1988年版。

11、德罗尔：《逆境中的政策制定》，上海远东出版社1996年版。

12、雷格斯：《行政生态学》，台湾商务印书馆1985年版。

13、弗雷德里克森：《新公共行政学》，美国亚拉巴马大学出版社1980版。

14、奥斯特罗姆：《美国公共行政的思想危机》上海三联书店1999年版。

15、尼斯坎南：《官僚制与公共经济学》，中国青年出版社2004年版。

16、詹姆斯·Q·威尔逊：《官僚机构：政府机构的作为及其原因》，三联书店2006年版。

17、奥斯特罗姆：《公共服务的制度建构》上海三联书店2000年版。

18、布坎南、塔洛克：《同意的计算：立宪民主的逻辑基础》，中国社会科学出版社2000年版。

19、缪勒：《公共选择理论》，中国社会科学出版社1999年版。

20、罗森布鲁姆：《公共行政学：管理、政治和法律的途径》，中国人民大学出版社2002年版。

21、R·J·斯蒂尔曼：《公共行政学：概念与案例》，中国人民大学出版社2004年版。

22、罗伯特•登哈特：《公共组织理论》，中国人民大学出版社2003年版。

23、沙夫里茨、海德：《公共行政学经典》，中国人民大学出版社2004年英文版。

【大学物理必备】量子阱与量子计算机

g+
g0
g- K 和-K ，瞄准原子照射，记原子与光相互作
用哈密顿量为 V，则
<g
，P k
V
e0 .P
eit 2
图 8-2 VSCPT 模型
应表示原子与光场复合系统能级差为，其中 2
是激光频率，Ω是原子拉比频率，我们发现存在一个 P=0 的相干迭加态，一
旦原子冷却处于该迭加态，便不再与光子发生作用。实际上，记此迭加态为
量子计算机除具备 0，1 二信息外，还有一个通常电子计算机不具备的信息位，就是原子可处在一半 0 位，一半 1 位的各占 5%几率的迭加态，称半翻转的量子位，利用它可以进行真正的随机运算，而电子计算机是没有这个功能的。
就可以达到囚禁原子的目的。
图 8-1 MOT 模型
实际上,当 B=0,即无外加磁场时,光子能量 E1 E0 E10 ,原子与光没有相互作用;当 B≠0 时,若 Z>0,ω与 S=1,ms=-1 的能级及 S=0 能级之差更为接近,所以,原子较多受到来自右方的光子的作用,t 完成从 S 0 态向
B=bz
b 是常数
则 S=1,并在空间有三个取向的三简并激发态将产生塞曼分裂。变成三个不同能
量的激发态 E ms B bms z
其中
eh zme
,e,me 分别是电子
的电量及静质量,能量分布图见
(ms 0,1)
图 8-1,如果用二束频率相同,分 ωh
σ-
ms 1
0
-1
σ+ B
别沿±Z 方向传播的激光 , 瞄准原子辐射,并冷冻原子及激光以减少噪声。
以以纯量子力学方式运行;此后,牛津大学数学所 David Deutseh 和美国、以色列