The Exact S-Matrix for an osp(22) Disordered System
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。
Autodesk Nastran 2023 参考手册说明书
FILESPEC ............................................................................................................................................................ 13
DISPFILE ............................................................................................................................................................. 11
File Management Directives – Output File Specifications: .............................................................................. 5
BULKDATAFILE .................................................................................................................................................... 7
算法导论 第三版 第22章 答案 英
Michelle Bodnar, Andrew Lohr April 12, 2016
Exercise 22.1-1 Since it seems as though the list for the neighbors of each vertex v is just an undecorated list, to find the length of each would take time O(out − degree(v )). So, the total cost will be v∈V O(outdegree(v )) = O(|E | + |V |). Note that the |V | showing up in the asymptotics is necessary, because it still takes a constant amount of time to know that a list is empty. This time could be reduced to O(|V |) if for each list in the adjacency list representation, we just also stored its length. To compute the in degree of each vertex, we will have to scan through all of the adjacency lists and keep counters for how many times each vertex has appeared. As in the previous case, the time to scan through all of the adjacency lists takes time O(|E | + |V |). Exercise 22.1-2 The adjacency list representation: 1 : 2, 3 2 : 1, 4, 5 3 : 1, 6, 7 4:2 5:5 6:3 7 : 3.
Twisted Quantum Affine Superalgebra $U_q[sl(22)^{(2)}]$, $U_q[osp(22)]$ Invariant R-matrice
![Twisted Quantum Affine Superalgebra $U_q[sl(22)^{(2)}]$, $U_q[osp(22)]$ Invariant R-matrice](https://img.taocdn.com/s3/m/429506202f60ddccda38a0cd.png)
becomes sl(2|2) invariant. Using this R-matrix, we will derive a new Uq [osp(2|2)] invariant
affine superalgebra sl(2|2)(2) and its quantized version Uq [sl(2|2)(2) ], respectively. The
Abstract We describe the twisted affine superalgebra sl(2|2)(2) and its quantized version Uq [sl(2|2)(2) ].
We investigate the tensor product representation of the 4-dimensional grade star represen-
model of strongly correlated electrons which is integrable on a one dimension lattice. This model has different interaction terms from the ones in the models [3, 4, 5]. This paper is organized as follows. In section 2 and section 3, we study the twisted tensor product representation of the 4-dimensional grade star representation for the fixed
subsuperalgebra Uq [osp(2|2)] is also investigated in details, and basis and its dual for this
分块下三角阵 英语
分块下三角阵英语Partitioned Lower Triangular MatricesMatrices are fundamental mathematical objects that find applications in various fields, including linear algebra, computer science, physics, and engineering. Among the different types of matrices, lower triangular matrices hold a special place due to their unique structure and properties. In this essay, we will delve into the concept of partitioned lower triangular matrices, exploring their characteristics, applications, and the underlying mathematical principles.A lower triangular matrix is a square matrix in which all the elements above the main diagonal are zero. In other words, the non-zero elements are concentrated in the lower triangular portion of the matrix. Formally, a matrix A = [a_ij] is said to be lower triangular if a_ij = 0 for all i < j, where i and j represent the row and column indices, respectively.Partitioned lower triangular matrices are a generalization of thisconcept, where the matrix is divided into smaller submatrices, each of which is also lower triangular. This partitioning can be achieved by grouping the rows and columns of the original matrix into blocks or submatrices. The resulting partitioned matrix maintains the overall lower triangular structure, with the submatrices satisfying the same property.One of the key advantages of partitioned lower triangular matrices is their ability to simplify complex computations and facilitate efficient matrix operations. By exploiting the lower triangular structure, various algorithms and numerical methods can be optimized, leading to improved computational performance and reduced memory requirements.For instance, in the context of solving systems of linear equations, partitioned lower triangular matrices can be used to implement efficient algorithms, such as the Gaussian elimination method or the LU decomposition. These techniques leverage the lower triangular structure to reduce the computational complexity and improve the numerical stability of the solution process.Another important application of partitioned lower triangular matrices lies in the field of control theory and systems engineering. In the analysis and design of linear dynamical systems, the state-space representation of the system often involves matrices with apartitioned lower triangular structure. This structure can be exploited to study the stability, controllability, and observability of the system, as well as to design optimal control strategies.Furthermore, partitioned lower triangular matrices find applications in the analysis of graph-theoretic structures, such as social networks, transportation networks, and communication networks. The lower triangular structure can be used to model and analyze the relationships and dependencies between different entities or nodes in these networks, enabling the study of network dynamics, centrality measures, and community detection.In the realm of numerical linear algebra, partitioned lower triangular matrices play a crucial role in the development and implementation of efficient numerical algorithms. For instance, in the context of iterative methods for solving large-scale linear systems, the lower triangular structure can be exploited to design preconditioners that accelerate the convergence of the iterative process.The mathematical properties of partitioned lower triangular matrices are also of great interest. These matrices exhibit specific characteristics that can be leveraged in various theoretical and computational contexts. For example, the product of two partitioned lower triangular matrices is also a partitioned lower triangular matrix, which is a fundamental property that underpins many algorithmsand matrix operations.Additionally, the eigenvalues of a partitioned lower triangular matrix are closely related to the eigenvalues of its diagonal submatrices. This property can be utilized in the analysis of the spectral properties of these matrices, with applications in areas such as numerical linear algebra, control theory, and graph theory.In conclusion, partitioned lower triangular matrices are a fascinating and versatile concept in linear algebra, with a wide range of applications across various scientific and engineering disciplines. Their unique structure and properties enable the development of efficient computational algorithms, the analysis of complex systems, and the exploration of intricate mathematical relationships. As the field of matrix computations continues to evolve, the study of partitioned lower triangular matrices will undoubtedly remain an active and important area of research.。
皮肤表面pH值及其临床意义
OSHA现场作业手册说明书
DIRECTIVE NUMBER: CPL 02-00-150 EFFECTIVE DATE: April 22, 2011 SUBJECT: Field Operations Manual (FOM)ABSTRACTPurpose: This instruction cancels and replaces OSHA Instruction CPL 02-00-148,Field Operations Manual (FOM), issued November 9, 2009, whichreplaced the September 26, 1994 Instruction that implemented the FieldInspection Reference Manual (FIRM). The FOM is a revision of OSHA’senforcement policies and procedures manual that provides the field officesa reference document for identifying the responsibilities associated withthe majority of their inspection duties. This Instruction also cancels OSHAInstruction FAP 01-00-003 Federal Agency Safety and Health Programs,May 17, 1996 and Chapter 13 of OSHA Instruction CPL 02-00-045,Revised Field Operations Manual, June 15, 1989.Scope: OSHA-wide.References: Title 29 Code of Federal Regulations §1903.6, Advance Notice ofInspections; 29 Code of Federal Regulations §1903.14, Policy RegardingEmployee Rescue Activities; 29 Code of Federal Regulations §1903.19,Abatement Verification; 29 Code of Federal Regulations §1904.39,Reporting Fatalities and Multiple Hospitalizations to OSHA; and Housingfor Agricultural Workers: Final Rule, Federal Register, March 4, 1980 (45FR 14180).Cancellations: OSHA Instruction CPL 02-00-148, Field Operations Manual, November9, 2009.OSHA Instruction FAP 01-00-003, Federal Agency Safety and HealthPrograms, May 17, 1996.Chapter 13 of OSHA Instruction CPL 02-00-045, Revised FieldOperations Manual, June 15, 1989.State Impact: Notice of Intent and Adoption required. See paragraph VI.Action Offices: National, Regional, and Area OfficesOriginating Office: Directorate of Enforcement Programs Contact: Directorate of Enforcement ProgramsOffice of General Industry Enforcement200 Constitution Avenue, NW, N3 119Washington, DC 20210202-693-1850By and Under the Authority ofDavid Michaels, PhD, MPHAssistant SecretaryExecutive SummaryThis instruction cancels and replaces OSHA Instruction CPL 02-00-148, Field Operations Manual (FOM), issued November 9, 2009. The one remaining part of the prior Field Operations Manual, the chapter on Disclosure, will be added at a later date. This Instruction also cancels OSHA Instruction FAP 01-00-003 Federal Agency Safety and Health Programs, May 17, 1996 and Chapter 13 of OSHA Instruction CPL 02-00-045, Revised Field Operations Manual, June 15, 1989. This Instruction constitutes OSHA’s general enforcement policies and procedures manual for use by the field offices in conducting inspections, issuing citations and proposing penalties.Significant Changes∙A new Table of Contents for the entire FOM is added.∙ A new References section for the entire FOM is added∙ A new Cancellations section for the entire FOM is added.∙Adds a Maritime Industry Sector to Section III of Chapter 10, Industry Sectors.∙Revises sections referring to the Enhanced Enforcement Program (EEP) replacing the information with the Severe Violator Enforcement Program (SVEP).∙Adds Chapter 13, Federal Agency Field Activities.∙Cancels OSHA Instruction FAP 01-00-003, Federal Agency Safety and Health Programs, May 17, 1996.DisclaimerThis manual is intended to provide instruction regarding some of the internal operations of the Occupational Safety and Health Administration (OSHA), and is solely for the benefit of the Government. No duties, rights, or benefits, substantive or procedural, are created or implied by this manual. The contents of this manual are not enforceable by any person or entity against the Department of Labor or the United States. Statements which reflect current Occupational Safety and Health Review Commission or court precedents do not necessarily indicate acquiescence with those precedents.Table of ContentsCHAPTER 1INTRODUCTIONI.PURPOSE. ........................................................................................................... 1-1 II.SCOPE. ................................................................................................................ 1-1 III.REFERENCES .................................................................................................... 1-1 IV.CANCELLATIONS............................................................................................. 1-8 V. ACTION INFORMATION ................................................................................. 1-8A.R ESPONSIBLE O FFICE.......................................................................................................................................... 1-8B.A CTION O FFICES. .................................................................................................................... 1-8C. I NFORMATION O FFICES............................................................................................................ 1-8 VI. STATE IMPACT. ................................................................................................ 1-8 VII.SIGNIFICANT CHANGES. ............................................................................... 1-9 VIII.BACKGROUND. ................................................................................................. 1-9 IX. DEFINITIONS AND TERMINOLOGY. ........................................................ 1-10A.T HE A CT................................................................................................................................................................. 1-10B. C OMPLIANCE S AFETY AND H EALTH O FFICER (CSHO). ...........................................................1-10B.H E/S HE AND H IS/H ERS ..................................................................................................................................... 1-10C.P ROFESSIONAL J UDGMENT............................................................................................................................... 1-10E. W ORKPLACE AND W ORKSITE ......................................................................................................................... 1-10CHAPTER 2PROGRAM PLANNINGI.INTRODUCTION ............................................................................................... 2-1 II.AREA OFFICE RESPONSIBILITIES. .............................................................. 2-1A.P ROVIDING A SSISTANCE TO S MALL E MPLOYERS. ...................................................................................... 2-1B.A REA O FFICE O UTREACH P ROGRAM. ............................................................................................................. 2-1C. R ESPONDING TO R EQUESTS FOR A SSISTANCE. ............................................................................................ 2-2 III. OSHA COOPERATIVE PROGRAMS OVERVIEW. ...................................... 2-2A.V OLUNTARY P ROTECTION P ROGRAM (VPP). ........................................................................... 2-2B.O NSITE C ONSULTATION P ROGRAM. ................................................................................................................ 2-2C.S TRATEGIC P ARTNERSHIPS................................................................................................................................. 2-3D.A LLIANCE P ROGRAM ........................................................................................................................................... 2-3 IV. ENFORCEMENT PROGRAM SCHEDULING. ................................................ 2-4A.G ENERAL ................................................................................................................................................................. 2-4B.I NSPECTION P RIORITY C RITERIA. ..................................................................................................................... 2-4C.E FFECT OF C ONTEST ............................................................................................................................................ 2-5D.E NFORCEMENT E XEMPTIONS AND L IMITATIONS. ....................................................................................... 2-6E.P REEMPTION BY A NOTHER F EDERAL A GENCY ........................................................................................... 2-6F.U NITED S TATES P OSTAL S ERVICE. .................................................................................................................. 2-7G.H OME-B ASED W ORKSITES. ................................................................................................................................ 2-8H.I NSPECTION/I NVESTIGATION T YPES. ............................................................................................................... 2-8 V.UNPROGRAMMED ACTIVITY – HAZARD EVALUATION AND INSPECTION SCHEDULING ............................................................................ 2-9 VI.PROGRAMMED INSPECTIONS. ................................................................... 2-10A.S ITE-S PECIFIC T ARGETING (SST) P ROGRAM. ............................................................................................. 2-10B.S CHEDULING FOR C ONSTRUCTION I NSPECTIONS. ..................................................................................... 2-10C.S CHEDULING FOR M ARITIME I NSPECTIONS. ............................................................................. 2-11D.S PECIAL E MPHASIS P ROGRAMS (SEP S). ................................................................................... 2-12E.N ATIONAL E MPHASIS P ROGRAMS (NEP S) ............................................................................... 2-13F.L OCAL E MPHASIS P ROGRAMS (LEP S) AND R EGIONAL E MPHASIS P ROGRAMS (REP S) ............ 2-13G.O THER S PECIAL P ROGRAMS. ............................................................................................................................ 2-13H.I NSPECTION S CHEDULING AND I NTERFACE WITH C OOPERATIVE P ROGRAM P ARTICIPANTS ....... 2-13CHAPTER 3INSPECTION PROCEDURESI.INSPECTION PREPARATION. .......................................................................... 3-1 II.INSPECTION PLANNING. .................................................................................. 3-1A.R EVIEW OF I NSPECTION H ISTORY .................................................................................................................... 3-1B.R EVIEW OF C OOPERATIVE P ROGRAM P ARTICIPATION .............................................................................. 3-1C.OSHA D ATA I NITIATIVE (ODI) D ATA R EVIEW .......................................................................................... 3-2D.S AFETY AND H EALTH I SSUES R ELATING TO CSHO S.................................................................. 3-2E.A DVANCE N OTICE. ................................................................................................................................................ 3-3F.P RE-I NSPECTION C OMPULSORY P ROCESS ...................................................................................................... 3-5G.P ERSONAL S ECURITY C LEARANCE. ................................................................................................................. 3-5H.E XPERT A SSISTANCE. ........................................................................................................................................... 3-5 III. INSPECTION SCOPE. ......................................................................................... 3-6A.C OMPREHENSIVE ................................................................................................................................................... 3-6B.P ARTIAL. ................................................................................................................................................................... 3-6 IV. CONDUCT OF INSPECTION .............................................................................. 3-6A.T IME OF I NSPECTION............................................................................................................................................. 3-6B.P RESENTING C REDENTIALS. ............................................................................................................................... 3-6C.R EFUSAL TO P ERMIT I NSPECTION AND I NTERFERENCE ............................................................................. 3-7D.E MPLOYEE P ARTICIPATION. ............................................................................................................................... 3-9E.R ELEASE FOR E NTRY ............................................................................................................................................ 3-9F.B ANKRUPT OR O UT OF B USINESS. .................................................................................................................... 3-9G.E MPLOYEE R ESPONSIBILITIES. ................................................................................................. 3-10H.S TRIKE OR L ABOR D ISPUTE ............................................................................................................................. 3-10I. V ARIANCES. .......................................................................................................................................................... 3-11 V. OPENING CONFERENCE. ................................................................................ 3-11A.G ENERAL ................................................................................................................................................................ 3-11B.R EVIEW OF A PPROPRIATION A CT E XEMPTIONS AND L IMITATION. ..................................................... 3-13C.R EVIEW S CREENING FOR P ROCESS S AFETY M ANAGEMENT (PSM) C OVERAGE............................. 3-13D.R EVIEW OF V OLUNTARY C OMPLIANCE P ROGRAMS. ................................................................................ 3-14E.D ISRUPTIVE C ONDUCT. ...................................................................................................................................... 3-15F.C LASSIFIED A REAS ............................................................................................................................................. 3-16VI. REVIEW OF RECORDS. ................................................................................... 3-16A.I NJURY AND I LLNESS R ECORDS...................................................................................................................... 3-16B.R ECORDING C RITERIA. ...................................................................................................................................... 3-18C. R ECORDKEEPING D EFICIENCIES. .................................................................................................................. 3-18 VII. WALKAROUND INSPECTION. ....................................................................... 3-19A.W ALKAROUND R EPRESENTATIVES ............................................................................................................... 3-19B.E VALUATION OF S AFETY AND H EALTH M ANAGEMENT S YSTEM. ....................................................... 3-20C.R ECORD A LL F ACTS P ERTINENT TO A V IOLATION. ................................................................................. 3-20D.T ESTIFYING IN H EARINGS ................................................................................................................................ 3-21E.T RADE S ECRETS. ................................................................................................................................................. 3-21F.C OLLECTING S AMPLES. ..................................................................................................................................... 3-22G.P HOTOGRAPHS AND V IDEOTAPES.................................................................................................................. 3-22H.V IOLATIONS OF O THER L AWS. ....................................................................................................................... 3-23I.I NTERVIEWS OF N ON-M ANAGERIAL E MPLOYEES .................................................................................... 3-23J.M ULTI-E MPLOYER W ORKSITES ..................................................................................................................... 3-27 K.A DMINISTRATIVE S UBPOENA.......................................................................................................................... 3-27 L.E MPLOYER A BATEMENT A SSISTANCE. ........................................................................................................ 3-27 VIII. CLOSING CONFERENCE. .............................................................................. 3-28A.P ARTICIPANTS. ..................................................................................................................................................... 3-28B.D ISCUSSION I TEMS. ............................................................................................................................................ 3-28C.A DVICE TO A TTENDEES .................................................................................................................................... 3-29D.P ENALTIES............................................................................................................................................................. 3-30E.F EASIBLE A DMINISTRATIVE, W ORK P RACTICE AND E NGINEERING C ONTROLS. ............................ 3-30F.R EDUCING E MPLOYEE E XPOSURE. ................................................................................................................ 3-32G.A BATEMENT V ERIFICATION. ........................................................................................................................... 3-32H.E MPLOYEE D ISCRIMINATION .......................................................................................................................... 3-33 IX. SPECIAL INSPECTION PROCEDURES. ...................................................... 3-33A.F OLLOW-UP AND M ONITORING I NSPECTIONS............................................................................................ 3-33B.C ONSTRUCTION I NSPECTIONS ......................................................................................................................... 3-34C. F EDERAL A GENCY I NSPECTIONS. ................................................................................................................. 3-35CHAPTER 4VIOLATIONSI. BASIS OF VIOLATIONS ..................................................................................... 4-1A.S TANDARDS AND R EGULATIONS. .................................................................................................................... 4-1B.E MPLOYEE E XPOSURE. ........................................................................................................................................ 4-3C.R EGULATORY R EQUIREMENTS. ........................................................................................................................ 4-6D.H AZARD C OMMUNICATION. .............................................................................................................................. 4-6E. E MPLOYER/E MPLOYEE R ESPONSIBILITIES ................................................................................................... 4-6 II. SERIOUS VIOLATIONS. .................................................................................... 4-8A.S ECTION 17(K). ......................................................................................................................... 4-8B.E STABLISHING S ERIOUS V IOLATIONS ............................................................................................................ 4-8C. F OUR S TEPS TO BE D OCUMENTED. ................................................................................................................... 4-8 III. GENERAL DUTY REQUIREMENTS ............................................................. 4-14A.E VALUATION OF G ENERAL D UTY R EQUIREMENTS ................................................................................. 4-14B.E LEMENTS OF A G ENERAL D UTY R EQUIREMENT V IOLATION.............................................................. 4-14C. U SE OF THE G ENERAL D UTY C LAUSE ........................................................................................................ 4-23D.L IMITATIONS OF U SE OF THE G ENERAL D UTY C LAUSE. ..............................................................E.C LASSIFICATION OF V IOLATIONS C ITED U NDER THE G ENERAL D UTY C LAUSE. ..................F. P ROCEDURES FOR I MPLEMENTATION OF S ECTION 5(A)(1) E NFORCEMENT ............................ 4-25 4-27 4-27IV.OTHER-THAN-SERIOUS VIOLATIONS ............................................... 4-28 V.WILLFUL VIOLATIONS. ......................................................................... 4-28A.I NTENTIONAL D ISREGARD V IOLATIONS. ..........................................................................................4-28B.P LAIN I NDIFFERENCE V IOLATIONS. ...................................................................................................4-29 VI. CRIMINAL/WILLFUL VIOLATIONS. ................................................... 4-30A.A REA D IRECTOR C OORDINATION ....................................................................................................... 4-31B.C RITERIA FOR I NVESTIGATING P OSSIBLE C RIMINAL/W ILLFUL V IOLATIONS ........................ 4-31C. W ILLFUL V IOLATIONS R ELATED TO A F ATALITY .......................................................................... 4-32 VII. REPEATED VIOLATIONS. ...................................................................... 4-32A.F EDERAL AND S TATE P LAN V IOLATIONS. ........................................................................................4-32B.I DENTICAL S TANDARDS. .......................................................................................................................4-32C.D IFFERENT S TANDARDS. .......................................................................................................................4-33D.O BTAINING I NSPECTION H ISTORY. .....................................................................................................4-33E.T IME L IMITATIONS..................................................................................................................................4-34F.R EPEATED V. F AILURE TO A BATE....................................................................................................... 4-34G. A REA D IRECTOR R ESPONSIBILITIES. .............................................................................. 4-35 VIII. DE MINIMIS CONDITIONS. ................................................................... 4-36A.C RITERIA ................................................................................................................................................... 4-36B.P ROFESSIONAL J UDGMENT. ..................................................................................................................4-37C. A REA D IRECTOR R ESPONSIBILITIES. .............................................................................. 4-37 IX. CITING IN THE ALTERNATIVE ............................................................ 4-37 X. COMBINING AND GROUPING VIOLATIONS. ................................... 4-37A.C OMBINING. ..............................................................................................................................................4-37B.G ROUPING. ................................................................................................................................................4-38C. W HEN N OT TO G ROUP OR C OMBINE. ................................................................................................4-38 XI. HEALTH STANDARD VIOLATIONS ....................................................... 4-39A.C ITATION OF V ENTILATION S TANDARDS ......................................................................................... 4-39B.V IOLATIONS OF THE N OISE S TANDARD. ...........................................................................................4-40 XII. VIOLATIONS OF THE RESPIRATORY PROTECTION STANDARD(§1910.134). ....................................................................................................... XIII. VIOLATIONS OF AIR CONTAMINANT STANDARDS (§1910.1000) ... 4-43 4-43A.R EQUIREMENTS UNDER THE STANDARD: .................................................................................................. 4-43B.C LASSIFICATION OF V IOLATIONS OF A IR C ONTAMINANT S TANDARDS. ......................................... 4-43 XIV. CITING IMPROPER PERSONAL HYGIENE PRACTICES. ................... 4-45A.I NGESTION H AZARDS. .................................................................................................................................... 4-45B.A BSORPTION H AZARDS. ................................................................................................................................ 4-46C.W IPE S AMPLING. ............................................................................................................................................. 4-46D.C ITATION P OLICY ............................................................................................................................................ 4-46 XV. BIOLOGICAL MONITORING. ...................................................................... 4-47CHAPTER 5CASE FILE PREPARATION AND DOCUMENTATIONI.INTRODUCTION ............................................................................................... 5-1 II.INSPECTION CONDUCTED, CITATIONS BEING ISSUED. .................... 5-1A.OSHA-1 ................................................................................................................................... 5-1B.OSHA-1A. ............................................................................................................................... 5-1C. OSHA-1B. ................................................................................................................................ 5-2 III.INSPECTION CONDUCTED BUT NO CITATIONS ISSUED .................... 5-5 IV.NO INSPECTION ............................................................................................... 5-5 V. HEALTH INSPECTIONS. ................................................................................. 5-6A.D OCUMENT P OTENTIAL E XPOSURE. ............................................................................................................... 5-6B.E MPLOYER’S O CCUPATIONAL S AFETY AND H EALTH S YSTEM. ............................................................. 5-6 VI. AFFIRMATIVE DEFENSES............................................................................. 5-8A.B URDEN OF P ROOF. .............................................................................................................................................. 5-8B.E XPLANATIONS. ..................................................................................................................................................... 5-8 VII. INTERVIEW STATEMENTS. ........................................................................ 5-10A.G ENERALLY. ......................................................................................................................................................... 5-10B.CSHO S SHALL OBTAIN WRITTEN STATEMENTS WHEN: .......................................................................... 5-10C.L ANGUAGE AND W ORDING OF S TATEMENT. ............................................................................................. 5-11D.R EFUSAL TO S IGN S TATEMENT ...................................................................................................................... 5-11E.V IDEO AND A UDIOTAPED S TATEMENTS. ..................................................................................................... 5-11F.A DMINISTRATIVE D EPOSITIONS. .............................................................................................5-11 VIII. PAPERWORK AND WRITTEN PROGRAM REQUIREMENTS. .......... 5-12 IX.GUIDELINES FOR CASE FILE DOCUMENTATION FOR USE WITH VIDEOTAPES AND AUDIOTAPES .............................................................. 5-12 X.CASE FILE ACTIVITY DIARY SHEET. ..................................................... 5-12 XI. CITATIONS. ..................................................................................................... 5-12A.S TATUTE OF L IMITATIONS. .............................................................................................................................. 5-13B.I SSUING C ITATIONS. ........................................................................................................................................... 5-13C.A MENDING/W ITHDRAWING C ITATIONS AND N OTIFICATION OF P ENALTIES. .................................. 5-13D.P ROCEDURES FOR A MENDING OR W ITHDRAWING C ITATIONS ............................................................ 5-14 XII. INSPECTION RECORDS. ............................................................................... 5-15A.G ENERALLY. ......................................................................................................................................................... 5-15B.R ELEASE OF I NSPECTION I NFORMATION ..................................................................................................... 5-15C. C LASSIFIED AND T RADE S ECRET I NFORMATION ...................................................................................... 5-16。
口服固体制剂 GMP 实施指南
目录
目录
1 简介 .............................................................................................................................................. 1 2 质量管理....................................................................................................................................... 3 2.1 概论 ................................................................................................................................... 3 2.2 风险管理............................................................................................................................ 4 2.3 产品质量回顾.................................................................................................................... 9 2.4 自检 ............................................................
Matrix包用户指南说明书
2nd Introduction to the Matrix packageMartin Maechler and Douglas BatesR Core Development Team******************.ethz.ch,*******************September2006(typeset on November29,2023)AbstractLinear algebra is at the core of many areas of statistical computing and from its inception the S lan-guage has supported numerical linear algebra via a matrix data type and several functions and operators,such as%*%,qr,chol,and solve.However,these data types and functions do not provide direct accessto all of the facilities for efficient manipulation of dense matrices,as provided by the Lapack subroutines,and they do not provide for manipulation of sparse matrices.The Matrix package provides a set of S4classes for dense and sparse matrices that extend the basic matrix data type.Methods for a wide variety of functions and operators applied to objects from theseclasses provide efficient access to BLAS(Basic Linear Algebra Subroutines),Lapack(dense matrix),CHOLMOD including AMD and COLAMD and Csparse(sparse matrix)routines.One notable char-acteristic of the package is that whenever a matrix is factored,the factorization is stored as part of theoriginal matrix so that further operations on the matrix can reuse this factorization.1IntroductionThe most automatic way to use the Matrix package is via the Matrix()function which is very similar to the standard R function matrix(),>library(Matrix)>M<-Matrix(10+1:28,4,7)>M4x7Matrix of class"dgeMatrix"[,1][,2][,3][,4][,5][,6][,7][1,]11151923273135[2,]12162024283236[3,]13172125293337[4,]14182226303438>tM<-t(M)Such a matrix can be appended to(using cbind()or rbind())or indexed,>(M2<-cbind(-1,M))4x8Matrix of class"dgeMatrix"[,1][,2][,3][,4][,5][,6][,7][,8][1,]-111151923273135[2,]-112162024283236[3,]-113172125293337[4,]-1141822263034381>M[2,1][1]12>M[4,][1]14182226303438where the last two statements show customary matrix indexing,returning a simple numeric vector each1. We assign0to some columns and rows to“sparsify”it,and some NA s(typically“missing values”in data analysis)in order to demonstrate how they are dealt with;note how we can“subassign”as usual,for classical R matrices(i.e.,single entries or whole slices at once),>M2[,c(2,4:6)]<-0>M2[2,]<-0>M2<-rbind(0,M2,0)>M2[1:2,2]<-M2[3,4:5]<-NAand then coerce it to a sparse matrix,>sM<-as(M2,"sparseMatrix")>10*sM6x8sparse Matrix of class"dgCMatrix"[1,].NA......[2,]-10NA150 (310350)[3,]...NA NA...[4,]-10.170 (330370)[5,]-10.180 (340380)[6,]........>identical(sM*2,sM+sM)[1]TRUE>is(sM/10+M2%/%2,"sparseMatrix")[1]TRUEwhere the last three calls show that multiplication by a scalar keeps sparcity,as does other arithmetic, but addition to a“dense”object does not,as you might have expected after some thought about“sensible”behavior:>sM+106x8Matrix of class"dgeMatrix"[,1][,2][,3][,4][,5][,6][,7][,8][1,]10NA101010101010[2,]9NA251010104145[3,]101010NA NA101010[4,]910271010104347[5,]910281010104448[6,]10101010101010101because there’s an additional default argument to indexing,drop=TRUE.If you add“,drop=FALSE”you will get submatrices instead of simple vectors.2Operations on our classed matrices include(componentwise)arithmetic(+,−,∗,/,etc)as partly seen above,comparison(>,≤,etc),e.g.,>Mg2<-(sM>2)>Mg26x8sparse Matrix of class"lgCMatrix"[1,].N......[2,]:N|...||[3,]...N N...[4,]:.|...||[5,]:.|...||[6,]........returning a logical sparse matrix.When interested in the internal str ucture,str()comes handy,and we have been using it ourselves more regulary than print()ing(or show()ing as it happens)our matrices; alternatively,summary()gives output similar to Matlab’s printing of sparse matrices.>str(Mg2)Formal class'lgCMatrix'[package"Matrix"]with6slots..@i:int[1:16]1340113422.....@p:int[1:9]0358910101316..@Dim:int[1:2]68..@Dimnames:List of2....$:NULL....$:NULL..@x:logi[1:16]FALSE FALSE FALSE NA NA TRUE.....@factors:list()>summary(Mg2)6x8sparse Matrix of class"lgCMatrix",with16entriesi j x121FALSE241FALSE351FALSE412NA522NA623TRUE743TRUE853TRUE934NA1035NA1127TRUE1247TRUE1357TRUE1428TRUE1548TRUE1658TRUEAs you see from both of these,Mg2contains“extra zero”(here FALSE)entries;such sparse matrices may be created for different reasons,and you can use drop0()to remove(“drop”)these extra zeros.This should never matter for functionality,and does not even show differently for logical sparse matrices,but the internal structure is more compact:3>Mg2<-drop0(Mg2)>str(Mg2@x)#length 13,was 16logi [1:13]NA NA TRUE TRUE TRUE NA ...For large sparse matrices,visualization (of the sparsity pattern)is important,and we provide image()methods for that,e.g.,>data(CAex,package ="Matrix")>print(image(CAex,main ="image(CAex)"))#print(.)needed for Sweaveimage(CAex)Dimensions: 72 x 72Column R o w204060204060−0.4−0.20.00.20.40.60.81.0Further,i.e.,in addition to the above implicitly mentioned "Ops"operators (+,*,...,<=,>,...,&which all work with our matrices,notably in conjunction with scalars and traditional matrices),the "Math"-operations (such as exp(),sin()or gamma())and "Math2"(round()etc)and the "Summary"group of functions,min(),range(),sum(),all work on our matrices as they should.Note that all these are implemented via so called group methods ,see e.g.,?Arith in R .The intention is that sparse matrices remain sparse whenever sensible,given the matrix classes and operators involved,but not content specifically. E.g.,<sparse>+<dense>gives <dense>even for the rare cases where it would be advantageous to get a <sparse>result.These classed matrices can be “indexed”(more technically “subset”)as traditional S language (and hence R )matrices,as partly seen above.This also includes the idiom M [M ⟨op ⟩⟨num ⟩]which returns simple vectors,>sM[sM >2][1]NA NA 151718NA NA 313334353738>sml <-sM[sM <=2]>sml [1]0-10-1-10NA NA 000000000NA[24]NA 0000000and “subassign”ment similarly works in the same generality as for traditional S language matrices.41.1Matrix package for numerical linear algebraLinear algebra is at the core of many statistical computing techniques and,from its inception,the S language has supported numerical linear algebra via a matrix data type and several functions and operators,such as %*%,qr,chol,and solve.Initially the numerical linear algebra functions in R called underlying Fortran routines from the Linpack(Dongarra et al.,1979)and Eispack(Smith et al.,1976)libraries but over the years most of these functions have been switched to use routines from the Lapack(Anderson et al.,1999) library which is the state-of-the-art implementation of numerical dense linear algebra.Furthermore,R can be configured to use accelerated BLAS(Basic Linear Algebra Subroutines),such as those from the Atlas(Whaley et al.,2001)project or other ones,see the R manual“Installation and Administration”.Lapack provides routines for operating on several special forms of matrices,such as triangular matrices and symmetric matrices.Furthermore,matrix decompositions like the QR decompositions produce multiple output components that should be regarded as parts of a single object.There is some support in R for operations on special forms of matrices(e.g.the backsolve,forwardsolve and chol2inv functions)and for special structures(e.g.a QR structure is implicitly defined as a list by the qr,qr.qy,qr.qty,and related functions)but it is not as fully developed as it could be.Also there is no direct support for sparse matrices in R although Koenker and Ng(2003)have developed the SparseM package for sparse matrices based on SparseKit.The Matrix package provides S4classes and methods for dense and sparse matrices.The methods for dense matrices use Lapack and BLAS.The sparse matrix methods use CHOLMOD(Davis,2005a), CSparse(Davis,2005b)and other parts(AMD,COLAMD)of Tim Davis’“SuiteSparse”collection of sparse matrix libraries,many of which also use BLAS.Todo:triu(),tril(),diag(),...and as(.,.),but of course only when they’ve seen a few different ones.Todo:matrix operators include%*%,crossprod(),tcrossprod(),solve()Todo:expm()is the matrix exponential......Todo:symmpart()and skewpart()compute the symmetric part,(x+t(x))/2and the skew-symmetric part,(x-t(x))/2of a matrix x.Todo:factorizations include Cholesky()(or chol()),lu(),qr()(not yet for dense)Todo:Although generally the result of an operation on dense matrices is a dgeMatrix,certain operations return matrices of special types.Todo: E.g.show the distinction between t(mm)%*%mm and crossprod(mm).2Matrix ClassesThe Matrix package provides classes for real(stored as double precision),logical and so-called“pattern”(binary)dense and sparse matrices.There are provisions to also provide integer and complex(stored as double precision complex)matrices.Note that in R,logical means entries TRUE,FALSE,or NA.To store just the non-zero pattern for typical sparse matrix algorithms,the pattern matrices are binary,i.e.,conceptually just TRUE or FALSE.In Matrix, the pattern matrices all have class names starting with"n"(patter n).2.1Classes for dense matricesFor the sake of brevity,we restrict ourselves to the real(d ouble)classes,but they are paralleled by l ogical and patter n matrices for all but the positive definite ones.dgeMatrix Real matrices in general storage modedsyMatrix Symmetric real matrices in non-packed storagedspMatrix Symmetric real matrices in packed storage(one triangle only)5dtrMatrix Triangular real matrices in non-packed storagedtpMatrix Triangular real matrices in packed storage(triangle only)dpoMatrix Positive semi-definite symmetric real matrices in non-packed storagedppMatrix ditto in packed storageMethods for these classes include coercion between these classes,when appropriate,and coercion to the matrix class;methods for matrix multiplication(%*%);cross products(crossprod),matrix norm(norm); reciprocal condition number(rcond);LU factorization(lu)or,for the poMatrix class,the Cholesky decom-position(chol);and solutions of linear systems of equations(solve).Whenever a factorization or a decomposition is calculated it is preserved as a(list)element in the factors slot of the original object.In this way a sequence of operations,such as determining the condition number of a matrix then solving a linear system based on the matrix,do not require multiple factorizations of the same matrix nor do they require the user to store the intermediate results.2.2Classes for sparse matricesUsed for large matrices in which most of the elements are known to be zero(or FALSE for logical and binary (“pattern”)matrices).Sparse matrices are automatically built from Matrix()whenever the majority of entries is zero(or FALSE respectively).Alternatively,sparseMatrix()builds sparse matrices from their non-zero entries and is typically recommended to construct large sparse matrices,rather than direct calls of new().Todo: E.g.model matrices created from factors with a large number of levelsTodo:or from spline basis functions(e.g.COBS,package cobs),etc.Todo:Other uses include representations of graphs.indeed;good you mentioned it!particularly since we still have the interface to the graph package.I think I’d like to draw one graph in that article—maybe the undirected graph corresponding to a crossprod()result of dimension ca.502Todo:Specialized algorithms can give substantial savings in amount of storage used and execution time of operations.Todo:Our implementation is based on the CHOLMOD and CSparse libraries by Tim Davis.2.3Representations of sparse matrices2.3.1Triplet representation(TsparseMatrix)Conceptually,the simplest representation of a sparse matrix is as a triplet of an integer vector i giving the row numbers,an integer vector j giving the column numbers,and a numeric vector x giving the non-zero values in the matrix.2In Matrix,the TsparseMatrix class is the virtual class of all sparse matrices in triplet representation.Its main use is for easy input or transfer to other classes.As for the dense matrices,the class of the x slot may vary,and the subclasses may be triangular, symmetric or unspecified(“general”),such that the TsparseMatrix class has several3‘actual”subclasses,the most typical(numeric,general)is dgTMatrix:>getClass("TsparseMatrix")#(i,j,Dim,Dimnames)slots are common to allVirtual Class"TsparseMatrix"[package"Matrix"]Slots:2For efficiency reasons,we use“zero-based”indexing in the Matrix package,i.e.,the row indices i are in0:(nrow(.)-1)and the column indices j accordingly.3the3×3actual subclasses of TsparseMatrix are the three structural kinds,namely t riangular,s ymmetric and g eneral, times three entry classes,d ouble,l ogical,and patter n.6Name:ijDim Dimnames Class:integer integer integer listExtends:Class "sparseMatrix",directlyClass "Matrix",by class "sparseMatrix",distance 2Class "replValueSp",by class "Matrix",distance 3Known Subclasses:"ngTMatrix","ntTMatrix","nsTMatrix","lgTMatrix","ltTMatrix","lsTMatrix","dgTMatrix","dtTMatrix","dsTMatrix">getClass("dgTMatrix")Class "dgTMatrix"[package "Matrix"]Slots:Name:ijDim Dimnames xfactorsClass:integerintegerinteger listnumericlistExtends:Class "TsparseMatrix",directly Class "dsparseMatrix",directly Class "generalMatrix",directlyClass "dMatrix",by class "dsparseMatrix",distance 2Class "sparseMatrix",by class "dsparseMatrix",distance 2Class "compMatrix",by class "generalMatrix",distance 2Class "Matrix",by class "TsparseMatrix",distance 3Class "replValueSp",by class "Matrix",distance 4Note that the order of the entries in the (i,j,x)vectors does not matter;consequently,such matrices are not unique in their representation.42.3.2Compressed representations:CsparseMatrix and RsparseMatrixFor most sparse operations we use the compressed column-oriented representation (virtual class CsparseMatrix )(also known as “csc”,“compressed sparse column”).Here,instead of storing all column indices j ,only the start index of every column is stored.Analogously,there is also a compressed sparse row (csr)representation,which e.g.is used in in the SparseM package,and we provide the RsparseMatrix for compatibility and completeness purposes,in ad-dition to basic coercion ((as(.,<cl>)between the classes.These compressed representations remove the redundant row (column)indices and provide faster access to a given location in the matrix because you only need to check one row (column).There are certain advantages 5to csc in systems like R ,Octave and Matlab where dense matrices are stored in column-major order,therefore it is used in sparse matrix libraries such as CHOLMOD or CSparse of which we make use.For this reason,the CsparseMatrix class and subclasses are the principal classes for sparse matrices in the Matrix package.The Matrix package provides the following classes for sparse matrices ...FIXMEmany more —maybe ex plain namingscheme?...4Furthermore,there can be repeated (i,j)entries with the customary convention that the corresponding x entries are addedto form the matrix element m ij .5routines can make use of high-level (“level-3”)BLAS in certain sparse matrix computations 7dgTMatrix general,numeric,sparse matrices in(a possibly redundant)triplet form.This can be a conve-nient form in which to construct sparse matrices.dgCMatrix general,numeric,sparse matrices in the(sorted)compressed sparse column format.dsCMatrix symmetric,real,sparse matrices in the(sorted)compressed sparse column format.Only the upper or the lower triangle is stored.Although there is provision for both forms,the lower triangle form works best with TAUCS.dtCMatrix triangular,real,sparse matrices in the(sorted)compressed sparse column format.Todo:Can also read and write the Matrix Market and read the Harwell-Boeing representations.Todo:Can convert from a dense matrix to a sparse matrix(or use the Matrix function)but going through an intermediate dense matrix may cause problems with the amount of memory required.Todo:similar range of operations as for the dense matrix classes.3More detailed examples of“Matrix”operationsHave seen drop0()above,showe a nice double example(where you see“.”and“0”).Show the use of dim<-for resizing a(sparse)matrix.Maybe mention nearPD().Todo:Solve a sparse least squares problem and demonstrate memory/speed gainTodo:mention lme4and lmer(),maybe use one example to show the matrix sizes.4Notes about S4classes and methods implementationMaybe we could give some glimpses of implementations at least on the R level ones?Todo:The class hierarchy:a non-trivial tree where only the leaves are“actual”classes.Todo:The main advantage of the multi-level hierarchy is that methods can often be defined on a higher (virtual class)level which ensures consistency[and saves from“cut&paste”and forgetting things]Todo:Using Group Methods5Session Info>toLatex(sessionInfo())•R version4.3.2Patched(2023-11-24r85645),x86_64-pc-linux-gnu•Locale:LC_CTYPE=de_CH.UTF-8,LC_NUMERIC=C,LC_TIME=en_US.UTF-8,LC_COLLATE=C,LC_MONETARY=en_US.UTF-8,LC_MESSAGES=de_CH.UTF-8,LC_PAPER=de_CH.UTF-8,LC_NAME=C,LC_ADDRESS=C,LC_TELEPHONE=C,LC_MEASUREMENT=de_CH.UTF-8,LC_IDENTIFICATION=C•Time zone:Europe/Zurich•TZcode source:system(glibc)•Running under:Fedora Linux38(Thirty Eight)•Matrix products:default•BLAS:/u/maechler/R/D/r-patched/F38-64-inst/lib/libRblas.so•LAPACK:/usr/lib64/liblapack.so.3.11.08•Base packages:base,datasets,grDevices,graphics,methods,stats,utils•Other packages:Matrix1.6-4•Loaded via a namespace(and not attached):compiler4.3.2,grid4.3.2,lattice0.22-5,tools4.3.2 ReferencesE.Anderson,Z.Bai,C.Bischof,S.Blackford,J.Demmel,J.Dongarra,J.Du Croz,A.Greenbaum,S.Ham-marling,A.McKenney,and PACK Users’Guide.SIAM,Philadelphia,PA,3rd edition, 1999.Tim Davis.CHOLMOD:sparse supernodal Cholesky factorization and update/downdate. http://www.cise.ufl.edu/research/sparse/cholmod,2005a.Tim Davis.CSparse:a concise sparse matrix package.http://www.cise.ufl.edu/research/sparse/CSparse, 2005b.Jack Dongarra,Cleve Moler,Bunch,and G.W.Stewart.Linpack Users’Guide.SIAM,1979.Roger Koenker and Pin Ng.SparseM:A sparse matrix package for R.J.of Statistical Software,8(6),2003.B.T.Smith,J.M.Boyle,J.J.Dongarra,B.S.Garbow,Y.Ikebe,V.C.Klema,and C.B.Moler.Matrix Eigensystem Routines.EISPACK Guide,volume6of Lecture Notes in Computer Science.Springer-Verlag, New York,1976.R.Clint Whaley,Antoine Petitet,and Jack J.Dongarra.Automated empirical optimization of software and the ATLAS project.Parallel Computing,27(1–2):3–35,2001.Also available as University of Tennessee LAPACK Working Note#147,UT-CS-00-448,2000(/lapack/lawns/lawn147.ps).9。
试题英文数理统计
一、填空（一）各章节的introduction1、Continuous variables or interval data can assume any value in some interval of real numbers.连续变量或间隔数据可以假设在某个实数间隔中的任意值。
（measurement）Discrete variables assume only isolated values.离散变量只假定孤立的值。
(counting)11、The lower or first quartile is the 25th percentile and the upper or third quartile is the 75th percentile.12、The fist qurtile Q1 is the median of the observations falling below the median of the entire sample and the third quartile Q3 is the median of the observations falling above the median of the entire sample.The interquartile range is defined as IQR=Q3-Q1.第一个四分位数Q1是低于整个样本中位数的观测值的中位数，第三个四分位数Q3是高于整个样本中位数的观测值的中位数。
四分位数范围定义为IQR=Q3-Q1。
2、Statistics applied to the life sciences in often called biostatistics or biometry.统计学应用于生命科学，通常称为生物统计学或生物计量学。
3、A descriptive measure associated with a random variable when it is considered over the entire population is called a parameter.当在整个总体中考虑一个随机变量时，与它相关的描述性度量称为参数4、One is forced to examine a subset or sample of the population and make inferences about the entire variable of a sample is called a statistic.人们被迫检查总体中的一个子集或样本，并对样本中的整个变量做出推断，这被称为统计量。
FE-matrix of elliptic operators with L ∞-coefficients
and bi =
1
ϕµ (x)vi (x)dx
µ∈Iω2
.
Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators
3
or close to corners of the boundary ∂ . In the case of cij ∈ L∞ ( ) the theorem of De Giorgi (1957; see [8, page 200]) guarantees only local H¨ older continuity of G. In this article we consider the extreme case when cij ∈ L∞ ( ) and ⊂ Rd is a bounded Lipschitz domain, and prove that nevertheless B as well as the inverse FE stiffness matrix A−1 are well approximated by H-matrices. On the other hand, we require no smoothness of the functions ui , vi in (1.2). Usually, iterative methods are applied for the efficient numerical solution of elliptic partial differential equations, a prominent example are multigrid methods (cf. [13], [14, Chapter 10]). The first aim of iterative methods is a convergence rate independent3 of the dimension of the problem (“optimality”). However, the influence of other problem parameters may still deteriorate the method and is not so easy to cure (“robustness”). Jumping coefficients and oscillatory coefficients (as it may happen for cij ∈ L∞ ( )) are two examples of this kind. A weak point of traditional iterative methods is the treatment of arising Schur complements, since its explicit calculation is avoided but nevertheless a good preconditioning is required. This is hard to achieve for real life problems involving difficult problem parameters. The concept of H-matrices allows to compute the Schur complement since the class of H-matrices provides both efficient storage and efficient arithmetic of the matrix algebra. Consequently, this article is designed to lay ground to future efficient and easy to implement algorithms for the solution of elliptic partial differential equations with extremely general coefficients. The efficient treatment of the inverse of the stiffness matrix might be used for (a) the direct solution of FEM systems, (b) for preconditioning another iterative method or (c) for the calculation of a Schur complement. It is interesting to remark that the easily available inverse enables also the calculation of matrix functions (e.g., exp(−tA); cf. [6]) or the solution of matrix equations (e.g., the Riccati equation; cf. [9]). Since we emphasise the rather weak conditions cij ∈ L∞ ( ) on the coefficients and “ bounded Lipschitz” on the domain, we simplify other aspects in order not the distract the attention of the reader by other complications. These simplifications are listed below. 1. We consider L to be an differential operator (1.3) consisting only of the principal part. Lower order terms cause no problem as long as we can guarantee L−1 to exist. A dominant low order term changes the situation, since we obtain a singularly perturbed problem. 2. We consider a second order differential operator L as in (1.3). 3. L is assumed to be a scalar operator, systems are not considered.
TIMIT
5 Output from T¯ I MIT 5.1 Hydrostatic Quantities . . . . . . . . . . . . . . . 5.1.1 The format of the hydrostatic output . . 5.2 Time Domain Hydrodynamic Quantities . . . . . 5.2.1 Format of the time domain hydrodynamic 5.3 Frequency Domain Hydrodynamic Quantities . . 5.3.1 Format of the frequency domain output .
T¯ I MIT
A panel-method program for transient wave-body interactions.
VERSION 4.0: For zero and forward speed analysis of a single body with any number of waterlines, arbitrary wave heading, generalized modes, and infinite or finite depth.
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The development of T¯ I MIT has been supported by the Office of Naval Reseach, the Joint Industry Project “Wave Effects on Offshore Structures”, the Consortium for Numerical Analysis of Wave Effects on Offshore Structures, and the Naval Ship Warfare Center.
Multivariable link invariants arising from Lie superalgebras of type I
MULTIVARIABLE LINK INVARIANTS ARISING FROM LIE SUPERALGEBRAS OF TYPE I
NATHAN GEER AND BERTRAND PATUREAU-MIRAND Abstract. In this paper we construct new links invariants from a type I basic Lie superalgebra g. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module, by non-trivial “fake quantum dimensions.” Using this, we get a multivariable link invariant associated to any one parameter family of irreducible g-modules.
Date : February 2, 2008.
1
2
NATHAN GEER AND BERTRANDe non-zero and lead to non-trivial link invariants. The first of these examples recover the hierarchy of invariants defined by Akutsu, Deguchi and Ohtsuki [1], using a regularize of the Markov trace and nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev’s quantum dilogarithm invariants of knot (see [16]). The second example, is the invariants defined in this paper. The definition of the “fake quantum dimensions” given in [9] is abstract where the analogous definition in this paper is given by explicit formulas. One can use general theory to show that these definitions are equivalent. The explicit formulas given in this paper are useful when one wants to compute the invariant or compare it to other invariants. • In the second subsequent paper the authors will use the explicit formulas for the “fake quantum dimensions” to define “fake superdimensions” of typical representations of the Lie superalgebra g. These “fake superdimensions” are non-zero and lead to a kind of supertrace on the category representations of g which is non-trivial and invariant. These statements are completely classical statements. However, the only proof we know of uses the quantum algebra and low-dimensional topology developed in this paper. • We will now discuss the final subsequent paper in relation with the multivariable invariants defined in this paper. In Section 3 we will show that for c ∈ Nr−1 the pair (g, c) gives rise to c a multivariable link invariant Mg . These invariants associate a variable to each component of the link. There are only a handful of such invariants including the multivariable Alexander polynomial and the ones defined in [1]. All of these invariants are related to the invariants defined in this paper. Let us now explain these relationships. First, in [8] we plan (0,...,0) on showing that the invariant Msl(m|1) specializes to the multivariable Alexander polynomial. Second, in order to define their link invariants the authors of [1] regularize the Markov trace. Although using different methods, the invariants of this paper have a similar regularization. In both case, the standard method using ribbon categories or the Markov trace is trivial. Moreover, both families of invariants are generalization of the multivariable Alexander polynomial. In [8], we plan on conjec(0,...,0) turing that the invariants Msl(m|1) , for m ∈ N, specialize to the hierarchy of invariants defined in [1]. (Note this specialization
Structure of neutrino mass matrix
(3)
(we use the LMA MSW solution of the solar neutrino problem). The absolute mass scale and the three CP violating phases are not constrained by oscillation data. In order to study the dominant structure of the mass matrix, we will neglect O(s13 ) terms with respect to O(1) terms. Notice, however, that some matrix elements can be of order s13 . Therefore, the subdominant structure of the matrix, formed by small elements, cannot be studied in this approximation. A more detailed analysis can be found in a previous work 3 . Using Eqs.(1) and (2) and defining k ≡ m1 /m2 , r ≡ m3 /m2 , for s13 = 0 we get: z c23 y s23 y m −2iσx | s c | − x + re−2iσx | , 2 (4) = . . . |c2 23 23 23 x + s23 re m2 2 re−2iσx | x + c ... ... |s2 23 23
Ref.SISSA 32/2002/EP
STRUCTURE OF NEUTRINO MASS MATRIX
Unstable particle mixing and CP violation in weak decays
Abstract
We discuss unstable particle mixing in CP-violating weak decays. It is shown that for a completely degenerate system unstable particle mixing does not introduce a CP-violating partial rate difference, and that when the mixings are small only the off-diagonal mixings are relevant. Also, in the absence of mixing, unstable particle wave function renormalization does not introduce any additional effect. An illustrative example is given to heavy scalar decays with arbitrary mixing.
b,c −1 Tf b Vbc Vca e−iωcT
b,c
¯ −1 V ¯ca e−iωc t , Tf∗b V bc
¯ are the mixing matrices where V and V |φa = Vca |φ′c , ¯a = V ¯′ , ¯ca |φ |φ c (4) (5)
1
Introduction
The smallness of KL − KS mass difference allows us to have an access to rare processes such as CP violation. Up to now the only established experimental evidence of CP violation ¯ 0 [1] . comes from the mixing of the unstable particles K 0 and K Earlier studies of unstable particle mixing followed two physically equivalent paths. One is due to Weisskopf and Wigner [2], in which one introduces an effective complex mass matrix. The evolution of the system is determined by the standard time-dependent Hamiltonian formalism [3]. The other is due to Sachs [4], in which one studies the dynamics of the complex pole of the kaon field propagator. The Hamiltonian method is expressed directly in terms of the measured quantities and is therefore more transparent from a phenomenological viewpoint. On the other hand, the propagator method arises naturally in the context of quantum field theory, and hence is more easily adapted to fundamental gauge theories of weak interactions. Both approaches are phenomenological, having difficulties handling ultraviolet divergences arising from higher order corrections. In spite of these fundamental difficulties, the phenomenological formalisms have been very successful. They provide the standard descriptions for the study of unstable particle mixing. The advent of renormalizable gauge theory provides a connection between the parameters of a phenomenological formalism and the parameters of a given fundamental theory. In this paper we would like to study these connections for unstable particle mixing in some detail, focusing on CP-violating processes. We will adopt an approach that combines the two methods mentioned above. Instead of introducing a complete renormalization prescription, our immediate goal is more modest. In the next section we discuss some general properties of S -matrix elements in the presence of unstable particle mixing. The results of this analysis turn out to be very useful for simplifying Feynman diagram calculations. In section 3 we study the relationship between the unstable particle mixing and antiparticle mixing. For simplicity, we only focus on scalars. A simple formula valid for small mixings is derived for CP-violating partial rate differences. The formalisms developed in section 2 and section 3 are applied to a simple example of baryogenesis by heavy scalar decay. The results are shown to agree with the published results obtained directly from Feynman diagram calculations. This part is presented in section 4, followed by a discussion in section 5 of large mixing and renormalization. Our conclusion is presented in section 6. We give two appendices to present some technical details: one discusses the renormalization of unstable particle mixing and the other shows how to diagonalize an arbitrary n × n complex matrix.
微电子工艺习题参考解答
CRYSTAL GROWTH AND EXPITAXY1．画出一50cm 长的单晶硅锭距离籽晶10cm 、20cm 、30cm 、40cm 、45cm 时砷的掺杂分布。
(单晶硅锭从融体中拉出时,初始的掺杂浓度为1017cm —3） 2．硅的晶格常数为5.43Å．假设为一硬球模型： （a ）计算硅原子的半径。
(b ）确定硅原子的浓度为多少（单位为cm —3）?(c ）利用阿伏伽德罗（Avogadro)常数求出硅的密度。
3．假设有一l0kg 的纯硅融体，当硼掺杂的单晶硅锭生长到一半时，希望得到0。
01 Ω·cm 的电阻率，则需要加总量是多少的硼去掺杂？4．一直径200mm 、厚1mm 的硅晶片，含有5。
41mg 的硼均匀分布在替代位置上,求： (a ）硼的浓度为多少？(b ）硼原子间的平均距离。
5．用于柴可拉斯基法的籽晶，通常先拉成一小直径（5。
5mm ）的狭窄颈以作为无位错生长的开始。
如果硅的临界屈服强度为2×106g/cm2，试计算此籽晶可以支撑的200mm 直径单晶硅锭的最大长度。
6．在利用柴可拉斯基法所生长的晶体中掺入硼原子，为何在尾端的硼原子浓度会比籽晶端的浓度高?7．为何晶片中心的杂质浓度会比晶片周围的大？8．对柴可拉斯基技术,在k 0=0。
05时，画出C s /C 0值的曲线。
9．利用悬浮区熔工艺来提纯一含有镓且浓度为5×1016cm —3的单晶硅锭。
一次悬浮区熔通过,熔融带长度为2cm,则在离多远处镓的浓度会低于5×1015cm —3？10．从式L kx s e k C C /0)1(1/---=，假设k e =0。
3，求在x/L=1和2时，C s /C 0的值。
11．如果用如右图所示的硅材料制造p +—n 突变结二极管，试求用传统的方法掺杂和用中子辐照硅的击穿电压改变的百分比。
12．由图10.10,若C m =20％，在T b 时,还剩下多少比例的液体?13．用图10。
DISTA, Universita del Piemonte Orientale,
The Scale Factor:A New Degree of Freedom in Phase Type ApproximationAndrea BobbioDISTA,Universit`a del Piemonte Orientale, Alessandria,Italy,bobbio@unipmn.itAndr´a s Horv´a th,Mikl´o s TelekDept.of Telecommunications, Budapest University of Technology and Economics, Hungary,horvath,telek@webspn.hit.bme.huAbstractThis paper introduces a unified approach to phase-type approximation in which the discrete and the continuous phase-type models form a common model set.The models of this common set are assigned with a non-negative real parameter,the scale factor.The case when the scale factor is strictly positive results in Discrete phase-type distribu-tions and the scale factor represents the time elapsed in one step.If the scale factor is0,the resulting class is the class of Continuous phase-type distributions.Applying the above view,it is shown that there is no qualitative difference be-tween the discrete and the continuous phase-type models.Based on this unified view of phase-type models one can choose the best phase-type approximation of a stochastic model by optimizing the scale factor.Keywords:Discrete and Continuous Phase type distri-butions,Phase type expansion,approximate analysis.1IntroductionThis paper presents new comparative results on the use of Discrete Phase Type(DPH)distributions[11]and of Continuous Phase Type(CPH)distributions[12]in applied stochastic modeling.DPH distributions of order are defined as the time to absorption in a Discrete-State Discrete-Time Markov Chain (DTMC)with transient states and one absorbing state. CPH distributions of order are defined,similarly,as the distribution of the time to absorption in a Discrete-State Continuous-Time Markov Chain(CTMC)with transient states and one absorbing state.The above definition im-plies that the properties of a DPH distribution are computed over the set of the natural numbers while the properties of a CPH distribution are defined as a function of a continuous time variable.When DPH distributions are used to model timed activities,the set of the natural numbers must be re-lated to a time measure.Hence,a new parameter need to be introduced that represents the time span associated to each step.This new parameter is the scale factor of the DPH dis-tribution,and can be viewed as a new degree of freedom, since its choice largely impacts the shape and properties of a DPH distribution over the continuous time axes.When DPH distributions are used to approximate a given continu-ous distribution,the scale factor affects the goodness of the fit.The paper starts discussing to what extent DPH or CPH distributions can be utilized tofit a given continuous distri-bution.It is shown that a DPH distribution of any order con-verges to a CPH distribution of the same order as the scale factor goes to zero.Even so,the DPH class contains dis-tributions whose behavior differs substantially from the one of the corresponding distributions in the CPH class.Two main peculiar points differentiate the DPH class from the CPH class.Thefirst point concerns the coefficient of varia-tion:indeed,while in the continuous case the minimum co-efficient of variation is a function of the order only and its lower bound is given by the well known theorem of Aldous and Shepp[1],in the discrete case the minimum coefficient of variation is proved to depend both on the order and on the mean(and hence on the scale factor)[13].Furthermore, it is easy to see that for any order,there exist members of the DPH class that represent a deterministic value with a coefficient of variation equal to zero.Hence,for any order (greater than1),the coefficient of variation of the DPH class spans from zero to infinity.The second peculiar point that differentiate the DPH class is the support of the distributions.While a CPH dis-tribution(of any order)has always an infinite support,there exist members of the DPH class of any order that have a finite support(between a minimum non-negative value and a maximum)or have a mass equal to one concentrated in a single value(deterministic distribution).It turns out that the possibility oftuning the scale factor to optimize the goodness of the fit,having distributions with coefficient of variation span-ning from0to infinity,representing deterministic values exactly,coping withfinite support distributions,makes the DPH class a very interesting and challenging class of distributions to be explored in applied stochastic models.The purpose of this paper is to show how these fa-vorable properties can be exploited in practice,and to pro-vide guidelines to the modeler to a reasonably good choice of the distributions to be used.Indeed,since a DPH dis-tribution tends to a CPH distribution as the scale factor ap-proaches zero,considering the scale factor as a new decision variable in afitting experiment,andfinding the value of the optimal scale factor(with respect to some error measure) provides a valuable tool to decide whether to use a discrete or a continuous approximation to the given problem.Thefitting problem for the CPH class has been exten-sively studied and reported in the literature by resorting to a variety of structures and numerical techniques(see[10]for a survey).Conversely,thefitting problem for the DPH class has received very little attention[4].In recent years,a considerable effort has been devoted to define models with generally distributed timings and to merge in the same model random variables and determin-istic duration.Analytical solutions are possible in special cases,and the approximation of the original problems by means of CPH distributions is a rather well known tech-nique[7].This paper is aimed at emphasizing that DPH approximation may provide a more convenient alternative with respect to CPH approximation,and also to provide a way to quantitatively support this choice.Furthermore,the use of DPH approximation can be extended from stochas-tic models to functional analysis where time intervals with nondeterministic choice are considered[3].Finally,dis-cretization techniques for continuous problems[8]can be restated in terms of DPH approximations.The rest of the paper is organized as follows.After defin-ing the notation to be used in the paper in Section2,Section 3discusses the peculiar properties of the DPH class with re-spect to the CPH class.Some guidelines for bounding the parameters of interest and extensive numerical experiments to show how the goodness of thefit is influenced by the op-timal choice of the scale factor are reported in Section4. Section5discusses the quality of the approximation when passing from the analysis of a single distribution to the anal-ysis of performance measures in complete non-Markovian stochastic models.The paper is concluded in Section6.2Definition and NotationA DPH distribution[11,12]is the distribution of the time to absorption in a DTMC with transient states,and one absorbing state numbered.The one-step transition probability matrix of the corresponding DTMC can be par-titioned as:(1)where is the matrix collecting the transi-tion probabilities among the transient states,is the column vector of length grouping the probabilities from any state to the absorbing one,and is the zero vector.The initial probability vectoris of length,with.In the present paper,we consider only the class of DPH distribu-tions for which,but the extension to the case when is straightforward.The tuple is called the representation of the DPH distribution,and the order.Similarly,a CPH distribution[12]is the distribution of the time to absorption in a CTMC with transient states, and one absorbing state numbered.The infinites-imal generator of the CTMC can be partitioned in the following way:(2) where,is a matrix that describes the tran-sient behavior of the CTMC and is the column vector grouping the transition rates to the absorbing state.Letbe the initial probability(row) vector with.The tuple is called the representation of the CPH distribution,and the order.It has been shown in[4]for the discrete case and in[6] for the continuous case that the representations in(1)and (2),because of their too many free parameters,do not pro-vide a convenient form for running afitting algorithm.In-stead,resorting to acyclic phase-type distributions,the num-ber of free parameters is reduced significantly since both in the discrete and the continuous case a canonical form can be used.The canonical form and its constraints for the discrete case[4]is depicted in Figure1.Figure2gives the canonical form and associated constraints for the continuous case.In both cases the canonical form corresponds to a mixture of Hypo-exponential distributions.Afitting algorithm that provides acyclic CPH,acyclic DPH distributions has been provided in[2]and[4],respec-tively.Experiments suggests(an exhaustive comparison of fitting algorithms can be found in[10])that,from the point of view of applications,the Acyclic phase-type class is as flexible as the whole phase-type class.3Comparing properties of CPH and DPH distributionsCTMC are defined as a function of a continuous time variable,while DTMC are defined over the set of the nat-ural numbers.In order to relate the number of jumps in a DTMC with a time measure,a time span must be assigned to each step.Let be(in some arbitrary units)the scaleFigure2.Canonical representation of acyclicCPH distributions and its constraintsfactor,i.e.the time span assigned to each step.The valueof establishes an equivalence between the sentence”prob-ability at the-th step”and”probability at time”,and hence,defines the time scale on which the properties of theDTMC are measured.The consideration of the scale factor introduces a new parameter,and consequently a new de-gree of freedom,in the DPH class with respect to the CPHclass.In the following,we discuss how this new degree of freedom impacts the properties of the DPH class and how it can be exploited in practice.Let be an”unscaled”DPH distributed random variable (r.v.)of order with representation,defined over the set of the non-negative natural numbers.Let us consider a scale factor;the scaled r.v.is defined over the dis-crete set of time points,being a non-negative natural number.For the unscaled and the scaled DPH r.v.the following equations hold.(3)where is the column vector of ones,and is the -th moment calculated from the factorial moments of:.It is evident from(3)that the mean of the scaled r.v.is times the mean of the unscaled r.v..While is an invariant of the representation,is a free parame-ter;adjusting,the scaled r.v.can assume any mean value .On the other hand,one can easily infer from(3) that the coefficients of variation of and are equal.A consequence of the above properties is that one can easily provide a scaled DPH of order with arbitrary mean and arbitrary coefficient of variation with an appropriate scale factor.Or more formally:the unscaled DPH r.v.of any order can exhibit a coefficient of variation between .For the coefficient of variation ranges between.As mentioned earlier,an important property of the DPH class with respect to the CPH class is the possibility of exactly representing a deterministic delay.A determinis-tic distribution with value can be realized by means of a scaled DPH distribution with phases with scale factor if is integer.In this case,the structure of the DPH distribution is such that phase is connected with probabil-ity1only to phase(),and with an initial probability concentrated in state1.If is not inte-ger for the given,the deterministic behavior can only be approximated.3.1First order discrete approximation of CTMCsGiven a CTMC with infinitesimal generator,the tran-sition probability matrix over an interval of length can be written as:hence thefirst order approximation of is matrix.is a proper stochastic matrix if,where.is the exact transition probability matrix of the CTMC assumed that at most one transition occurs in the interval of length.We can approximate the behavior of the CTMC at timeusing the DTMC with transition probability matrix.The approximate transition prob-ability matrix at time is:.Since matrices and commute we can obtain the matrix version of the same expression as followsAn obvious consequence of Theorem1for PH distribu-tions is given in the following corollary.Corollary1Given a scaled DPH distribution of order, representation and scale factor,the limiting behavior as is the CPH distribution of order with representation.3.2The minimum coefficient of variationIt is known that one of the main limitation in approx-imating a given distribution by a PH one is the attainable minimal coefficient of variation,.In order to discuss this point,we recall two theorems that state the for the class of CPH and DPH distributions.Theorem2(Aldous and Shepp[1])The of a CPH distributed r.v.of order is and is attained by the Erlang()distribution independent of its mean or of its parameter.The corresponding theorem for the unscaled DPH class has been proved in[13].In the following,denotes the integer part and denotes the fractional part of. Theorem3The of an unscaled DPH r.v.of order and mean is:In this particular case,when the structure of the bestfit-ting scaled DPH and CPH distributions are known,we can show that the distribution of the bestfitting scaled DPH dis-tribution converges to the distribution of the bestfitting CPH distribution when.Unfortunately,the same conver-gence property cannot be proved in general,since the struc-tural properties of the bestfitting PH distributions are not known and they depend on the chosen(arbitrary)optimiza-tion criterion.Instead,in Section4we provide an extensive experimental study on the behavior of the bestfitting scaled DPH and CPH distributions as a function of the scale factor .3.4DPH distributions withfinite supportAnother peculiar characteristic of the DPH class is to contain distributions withfinite support.A DPH distribu-tion hasfinite support if its structure does not contain cycles and self-loops(any cycle or self loop implies an infinite sup-port).Let be thefinite support of a given distribution,with and(when thefinite support distri-bution reduces to a deterministic distribution with mass1at ).If and are both integers,it is possible to construct a scaled DPH of order for which the probabil-ity mass function has non-zero elements only for the values.As an example,the discrete uniform distribution between and is reported in Figure 5,for scale factor.0.10.20.30.40.50.60.70.80.910.40.60.811.2 1.41.61.82c d fxOriginalScale factor: 0.01Scale factor: 0.06Scale factor: 0.1CPH 00.511.522.50.40.60.81 1.2 1.4 1.6 1.82p d fxOriginalScale factor: 0.01Scale factor: 0.06Scale factor: 0.1CPH Figure 6.Approximating the L3distribution with -phase PH approximationsWhen is less than its lower bound the required can-not be attained;when becomes too large the wide separa-tion of the discrete steps increases the approximation error;when is in the proper range (e.g.)a reasonably good fit is achieved.This example also suggests that an optimal value of exists that minimizes the chosen distance measure in (6).In order to display the goodness of fit for the L3distribu-tion,Figure 7shows the distance measure as a function of for various values of the order .A minimum value of is attained in the range where the parameters fit the bounds of Table 1.Notice also that,as increases,the advantage of having more phases disappears,according to Theorem 3.The circles in the left part of this figure (as well as in all the successive figures)indicate the corresponding distance measure obtained from CPH fitting.The figure (and the subsequent ones as well)suggests that the distance measure obtained from DPH fitting converges to the distance mea-sure obtained by the CPH approximation as tends to .upper bound of equation (7)0.20920.07920.04250.021700.010.020.030.040.050.060.070.080.090.100.050.10.150.20.250.3d i s t a n ce m e a s u r escale factor2 phases 4 phases 6 phases 8 phases 10 phases 12 phasesFigure 9.Distance measure as the function of the scale factor for Uniform(1,2)(U2)00.0020.0040.0060.0080.010.0120.0140.01600.050.10.150.20.250.3d i s t a n ce m e a s u r escale factor2 phases 4 phases 6 phases 8 phases 10 phases 12 phasesFigure 10.Distance measure as the function of the scale factor for Uniform(0,1)(U1)be stressed that the chosen distance measure in (6)can be considered as not completely appropriate in the case of finite support,since it does not force the approximating PH to have its mass confined in the finite support and 0outside.Let be a uniform r.v.over the interval ,withand (this is the distributionU2taken from the benchmark in [5,4]).Figure 9shows the distance measure as a function of for various orders .It is evident that,for each ,a minimal value of is obtained,that provides the best approximation according to the chosen distance measure.As a second example,let be a uniform r.v.over theinterval,with and (this is the distribution U1taken from the benchmark in [5,4]).Figure 10shows the distance measure as a function of forvarious orders .Since,in this example,an order is large enough for a CPH to attain the coefficient of variation of the distribution.Nevertheless,the optimal in Figure (10),which minimizes the distance mea-sure for high order PH (),ranges between and ,thus leading to the conclusion that a DPH provides a better fit.This example evidences that the coef-ficient of variation is not the only factor which influences the optimal value.The shape of the distribution plays an essential role as well.Our experiments show that a discon-00.20.40.60.810.20.40.60.811.21.4c d fxOriginalScale factor: 0.03Scale factor: 0.1CPH00.20.40.60.811.21.41.600.20.40.60.811.21.4p d fxOriginalScale factor: 0.03Scale factor: 0.1CPHFigure 11.Approximating the Uniform ()distribution (U1)tinuity in the pdf (or in the cdf)is hard to approximate with CPH,hence in the majority of these cases DPH provides a better approximation.Figure 11shows the cdf and the pdf of the U1distribu-tion,compared with the best fit PH approximations of order,and various scale factors .In the case of DPH ap-proximation,the values are calculated as in (9).With respect to the chosen distance measure,the best approxi-mation is obtained for,which corresponds to a DPH distribution with infinite support .Whenthe approximate distribution has a finite support.Hence,the value (for )provides a DPH able to rep-resent the logical property that the random variable is less than .Another fitting criterion may,of course,stress this property.5Approximating non-Markovian modelsSection 4has explored the problem of how to find the best fit among either a DPH or a CPH distribution by tuning the scale factor .When dealing with a stochastic model of a system that incorporates non exponential distributions,a well know solution technique consists in a markovianiza-tion of the underlying non-Markovian process by substi-tuting the non exponential distribution with a best fit PH distribution,and then expanding the state space.A natural question arises also in this case,on how to decide among a discrete (using DPH)or a continuous (using CPH)approx-imation,in order to minimize the error in the performances2s4s3Figure12.The state space of the consideredM/G/1/2/2queuemeasures we are interested in for the overall model.One possible way to handle this problem could consist infinding the best PHfits for any single distribution and to plug them in the model.In the present paper,we only consider the case where the PH distributions are either all discrete(and with the same scale factor)or they are all continuous.Various embedding techniques have been ex-plored in the literature for mixing DPH(with different scale factors)and CPH([8,9]),but these techniques are out of the scope of the paper.In order to quantitatively evaluate the influence of the scale factor on some performance measures defined at the system level,we have considered a preemptive M/G/1/2/2 queue with two classes of customers.We have chosen this example because accurate analytical solutions are available both in transient condition and in steady-state using the methods presented in e.g.[8].The general distribution is taken from the set of distributions(L1,L3,U1,U2)already considered in the previous section.Customers arrive at the queue with rate in both classes.The service time of a higher priority job is exponen-tially distributed with parameter.The service time distribution of the lower priority job is either L1,L3,U1 or U2.Arrival of a higher priority job preempts the lower priority one.The policy associated to the preemption of the lower priority job is preemptive repeat different(prd),i.e. after the departure of the higher priority customer the ser-vice of the low priority customer starts from the beginning with a new service time sample.The system has4states(Figure12):in state s1the server is empty,in state s2a higher priority customer is under ser-vice with no lower priority customer in the system,in state s3a higher priority customer is under service with a lower priority customer waiting,in state s4a lower priority job is under service(in this case there cannot be a higher priority job).Let denote the steady state probability of the M/G/1/2/2queue obtained from an exact analytical solution.In order to evaluate the correctness of the PH approxima-0.020.040.060.080.10.120.140.1600.020.040.060.080.10.120.140.160.180.2sumoferrorsscale factor2 phases4 phases6 phases8 phases10 phases12 phasesFigure13.with scale factor and distri-bution L30.010.020.030.040.050.0600.020.040.060.080.10.120.140.160.180.2sumoferrorsscale factor2 phases4 phases6 phases8 phases10 phases12 phasesFigure14.with scale factor and distri-bution L3tion we have solved the model by substituting the original general distribution(either L1,L3,U1or U2)with approx-imating DPH or CPH distributions.Letdenote the steady state probability of the M/PH/1/2/2queue with the PH approximation.The overall approximation error is measured in terms of the difference between the exact steady state probabilities and the approximate steady state probabilities.Two error measures are defined:andThe evaluated numerical values for and are reported in Figures13and14for the distribution L3.Since the behavior of is very similar to the behavior of in all the cases,for the other distributions we reportonly(Figures15,16,17).Thefigures,which re-fer to the error measure in a performance index of a global stochastic model,show a behavior similar to the one ob-tained for a single distributionfitting.Depending on the coefficient of variation and on the shape of the considered non-exponential distributions an optimal value of is found which minimizes the approximation error.In this example, the optimal value of is close to the one obtained for the single distributionfitting.Based on our experiments,we guess that the observed0.050.10.150.20.2500.020.040.060.080.10.120.140.160.180.2s u m o f e r r o r sscale factor2 phases 4 phases 6 phases 8 phases 10 phases 12 phasesFigure 15.with scale factorand distri-bution L100.020.040.060.080.10.120.140.1600.050.10.150.20.250.3s u m o f e r r o r sscale factor2 phases 4 phases 6 phases 8 phases 10 phases 12 phasesFigure 16.with scale factorand distri-bution U1property is rather general.If the stochastic model under study contains a single non-exponential distribution,then the approximation error in the evaluation of the perfor-mance indices of the global model can be minimized by re-sorting to a PH type approximation (and subsequent DTMC or CTMC expansion)with the optimal of the single distri-bution.The same should be true if the stochastic model under study contains more than one general distribution,whose best PH fit provides the same optimal .In order to investigate the approximation error in the transient behavior,we have considered distribution U2for the service time and we have computed the transient proba-bility of state with two different initial conditions.Figure 18depicts the transient probability of state with initial state .Figure 19depicts the transient probability of the same state,,when the service of a lower priority job starts at time 0(the initial state is ).All approximations are with DPH distributions of order .Only the DPH ap-proximations are depicted because the CPH approximationis very similar to the DPH one with scale factor.In the first case,(Figure 18),the scale factor,which was the optimal one from the point of view of fitting the single distribution in isolation,provides the most accu-rate results for the transient analysis as well.Instead,in the second case,the approximation with a scale factor0.020.040.060.080.10.120.140.160.180.200.050.10.150.20.250.3s u m o f e r r o r sscale factor2 phases 4 phases 6 phases 8 phases 10 phases 12 phasesFigure 17.with scale factorand distri-bution U20.10.20.30.40.50.60.70.80.91012345t r a n s i e n t p r o b a b i l i t ytimeTransient behaviour Scale factor: 0.03Scale factor: 0.1Scale factor: 0.2Figure 18.Approximating transient probabili-tiescaptures better the sharp change in the transient probability.Moreover,this value of is the only one among the values reported in the figure that results in 0probability for time points smaller than 1.In other words,the second example depicts the advantage given by DPH distributions to model durations with finite support.This example suggests also that DPH approximation can be of importance when pre-serving reachability properties is crucial (like in modeling time-critical systems)and,hence,DPH approximation can be seen as a bridge between the world of stochastic model-ing and the world of functional analysis and model checking [3].6Concluding remarksThe main result of this paper has been to show that the DPH and CPH classes of distributions of the same order can be considered a single model set as a function of a scalefactor .The optimal value of ,,determines the best distribution in a fitting experiment.When the bestchoice is a CPH distribution,while whenthe best choice is a DPH distribution.This paper has also shown that the transition from DPH class to CPH class is continu-ous with respect to several properties,like the distance (de-noted by in 6)between the original and the approximate0.050.10.150.20.250.3012345t r a n s i e n t p r o b a b i l i t ytimeTransient behaviour Scale factor: 0.03Scale factor: 0.1Scale factor: 0.2Figure 19.Approximating transient probabili-tiesdistributions.The paper presents limit theorems for special cases;however,extensive numerical experiments show that the limiting behavior is far more general than the special cases considered in the theorems.The numerical examples have also evidenced that for very small values of ,the diagonal elements of the tran-sition probability matrix become very close to ,rendering numerically unstable the DPH fitting procedure.A deep analytical and numerical sensitivity analysis is required to draw more general conclusions for the model level “optimal value”and its dependence on the consid-ered performance measure than the ones presented in this work.It is definitely a field of further research.Finally,we summarize the advantages and the disadvan-tages of applying approximate DPH models (even with op-timal value)with respect to using CPH approximations.Advantages of using DPH:An obvious advantage of the ap-plication of DPH distributions is that one can have a closer approximate of distributions with low coefficient of varia-tion.An other important quantitative property of the DPH class is that it can capture distributions with finite support and deterministic values.This property allows to capture the periodic behavior of a complex stochastic model,while any CPH based approximation of the same model tends to a steady state.Numerical experiments have also shown that DPH can better approximate distributions with some abrupt or sharp changes in the CDF or in the PDF.Disadvantages of using DPH:There is a definite disad-vantage of discrete time approximation of continuous time models.In the case of CPH approximation,coincident events do not have to be considered (they have zero proba-bility of occurrence).Instead,when applying DPH approxi-mation coincident events have to be handled,and their con-sideration may burden significantly the complexity of the analysis.AcknowledgmentsThis work has been performed under the Italian-Hungarian R&D program supported by the Italian Ministry of Foreign Affairs and the Hungarian Ministry of Education.A.Bob-bio was partially supported by the MURST Under Grant ISIDE;M.Telek was partially supported by Hungarian Sci-entific Research Fund (OTKA)under Grant No.T-34972.References[1] D.Aldous and L.Shepp.The least variable phase type dis-tribution is Erlang.Stochastic Models ,3:467–473,1987.[2] A.Bobbio and A.Cumani.ML estimation of the param-eters of a PH distribution in triangular canonical form.In G.Balbo and G.Serazzi,editors,Computer Performance Evaluation ,pages 33–46.Elsevier Science Publishers,1992.[3] 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Non-Markovian Stochastic Petri Nets .John Wiley and Sons,2000.[9]R.Jones and G.Ciardo.On phased delay stochastic Petrinets:Definition and application.In Proceedings 9th Inter-national Workshop on Petri Nets and Performance Models -PNPM01.IEEE Computer Society,2001.[10] ng and J.L.Arthur.Parameter approximation forphase-type distributions.In Matrix-analytic methods in stochastic models ,Lecture notes in pure and applied mathe-matics,pages 151–206.Marcel Dekker,Inc.,1996.[11]M.Neuts.Probability distributions of phase type.In LiberAmicorum Prof.Emeritus H.Florin ,pages 173–206.Uni-versity of Louvain,1975.[12]M.Neuts.Matrix Geometric Solutions in Stochastic Models .Johns Hopkins University Press,Baltimore,1981.[13]M.Telek.Minimal coefficient of variation of discretephase type distributions.In 3rd International Conference on Matrix-Analitic Methods in Stochastic models,MAM3,pages 391–400,Leuven,Belgium,2000.Notable Publica-tions Inc.。
disturbance has an elliptical distribution
The mean mean squared error of the instrumental variables estimator when thedisturbance has an elliptical distribution¤Fernanda P.M.PeixeUniversity of Evora yAlastair R.HallNorth Carolina State University zFebruary15,2002AbstractThis paper generalizes Nagar’s(1959)approximation to the…nite sam-ple mean squared error(MSE)of the instrumental variables estimator,assuming that the errors possess an elliptical distribution whose momentsexist up to in…nite order.This allows for types of excess kurtosis exhib-ited by some…nancial data series.We compare the results with Knight’s(1985)approach for deriving exact moments under non-normality.Weuse the results to explore two questions on instrument selection.First,wecomplement Buse’s(1992)analysis by considering the impact of additionalinstruments on both bias and MSE.Second,we evaluate the properties ofAndrews’(1999a)selection method in terms of the bias and MSE of theresulting IV estimator.¤This work was begun while Peixe was a graduate student and Hall was a Senior Research Fellow at the Department of Economics,University of Birmingham,UK,and this support is gratefully acknowledged.Peixe also gratefully acknowledges…nancial support from FCT under grant PRAXIS XXI/BD/13453/97.This work represents part of Peixe’s PhD dissertation at the University of Birmingham.We are grateful for the comments of Peter Burridge and Chris Orme.y Departamento de Economia,Universidade de Evora,Apartado94,7002-554Evora,Por-tugal.Email:fmp@uevora.ptz Department of Economics,Box8110,North Carolina State University,Raleigh NC27695-8110,USA.Email:alastair_hall@1IntroductionGeneralized instrumental variables(GIV)estimation(Hansen and Singleton, 1982)has been widely applied in economics and…nance.The method rests cru-cially on the speci…cation of a vector of valid instruments.In practice,there is typically a large set of possible valid instruments and so it is important to understand how the choice between them may e¤ect subsequent inference.From the perspective of asymptotic theory,the issue is clear cut.The estimator is consistent and asymptotically normal for any choice of valid instrument,z t say. Furthermore,there are no costs to augmenting z t with additional valid instru-ments because such augmentation can never increase the asymptotic variance of the estimator.1However,the issue is far less clear in…nite samples.Simulation studies indicate that the behaviour of the estimator is not always well approx-imated by asymptotic theory in the sample sizes encountered in economics2. These studies also reveal that the relationship between the instrument vector and the properties of the estimator is far more complex than would be an-ticipated from asymptotic theory.These…ndings provide a strong motivation for the derivation of analytic expressions for the…nite sample distribution or moments of the estimator.To date,it has proved impossible to develop a…nite sample distribution theory for GIV in nonlinear models.However,such an analysis becomes feasible if attention is restricted to the instrumental variables(IV)estimator in linear simultaneous equations models.Phillips(1980)derives the exact density of the IV estimator in the case where the error distribution is normal.3This result is useful because it facilitates the computation of exact probability statements about the estimator.However,it is less convenient for the characterization of particular features of the distribution,such as bias or mean squared error, which are often of interest.Hillier,Kinal,and Srivastava(1984)derive exact formulae for the moments of the IV estimator under normality.These formulae can be used to calculate the bias and mean squared error but are su¢ciently complicated to be uninterpretable.In this regard,the approach taken by Nagar (1959)is more convenient.He developed approximations to the…nite sample bias and mean squared error of the two stage least squares(2SLS)estimator4 which are analytically simpler than the exact distribution but more accurate than asymptotic distribution theory.All these studies provide useful insights into…nite sample behaviour but they all su¤er from one weakness,that is the assumption of normality.There is considerable evidence that…nancial time series possess fatter tails than the normal,and so it is desirable to derive results which allow for this type of behaviour.5Buse(1992)provides a very valuable 1See White(1984,p.81–82).2E.g.see Tauchen(1986),Kocherlakota(1990),Hansen,Heaton,and Yaron(1996).3See Phillips(1983)for a review of…nite sample distribution theory in linear simultaneous equations models.4Alternatively such expressions may be interpreted as the moments of an approximating distribution,e.g.see Linton(1995)and Linton(1996).5See Bollerslev,Chou,and Kroner(1992)and Bollerslev,Engle,and Nelson(1994)for a discussion of the distributional properties of…nancial series.1…rst step in this direction.He extends Nagar’s(1959)expression for the bias to the class of IV estimators in linear simultaneous equations models in which the error is assumed only to be independently and identically distributed.He also derives conditions under which the inclusion of an additional instrument leads to an increase or decrease of the…nite sample bias.Buse’s(1992)analysis is very instructive but only provides a partial assessment of the impact of instrument selection.In many cases,researchers may be prepared to accept an increase in bias if it is outweighed by a decrease in variance.To explore the nature of this trade-o¤,it is necessary to analyze the impact of instrument selection on the mean squared error of the estimator.Furthermore,as argued above,it is desirable for this analysis to allow for the possibility of fat tailed distributions. Knight(1985)derives exact formulae for the moments of the2SLS estimator when the errors follow an Edgeworth type distribution.Although the resulting formulae is extremely complicated,the results of his simulations provide a basis of comparison for the e¤ects of departures from normality.In this paper,we generalize Nagar’s(1959)approximation for the second moment matrix to IV estimators in linear simultaneous equation models.In-stead of assuming that the disturbance is normally distributed,we consider the class of elliptically symmetric distributions,which contains the Normal but also fat-tailed distributions like the Student’s t and some mixtures of normals. This approach is compared with Knight’s(1985)approach for obtaining exact moments with non-normal errors.We use the results to explore two speci…c questions relating to instrument selection.First,we complement Buse’s(1992) analysis by considering the impact of additional instruments on both bias and mean squared error.Second,we use the results to examine the performance of Andrews’s(1999)method for instrument selection.An outline of the paper is as follows.Section2brie‡y reviews Buse’s(1992) results for the bias.Section3presents our main results.Section4reports various numerical calculations designed to illuminate how instrument selection e¤ects the bias and MSE of the IV estimator.Some concluding remarks are o¤ered in Section5.All derivations are relegated to a mathematical appendix. 2The Bias of the IV EstimatorThere has been some misunderstanding in the literature about the impact of the instruments on the…nite-sample properties of the IV estimator.Phillips (1980)concluded that there is a trade-o¤between bias and e¢ciency of the IV estimator:as the number of included“excess”instruments increases,the bias is supposed to increase and the dispersion is supposed to decrease.Buse(1992) showed however that this statement is not true in general.Speci…cally,Buse showed that Phillips’statement will be valid only under the condition that the additional instrumental variables play no role in the determination of the en-dogenous regressor in the reduced form.Furthermore Buse extended the result of Nagar(1959)for the bias of the2SLS estimator to an IV context without2recourse to the assumption of normality in order to establish an unambiguous relationship between bias and the number of instrumental variables.In this section we summarize Buse’s(1992)result.Consider the G linear simultaneous equations modelY B+N¡=U(1)where Y is the T£G matrix of endogenous variables,N is the T£K matrix of exogenous variables and the rows of U are G£1vectors of independent disturbances with zero mean and covariance matrix§=[¾ij],with reduced formY=N¦+VWithout loss of generality,we focus attention on the…rst equation of the system which can be written asy1=Y1¯+N1°+u1(2)whereY1=N¦1+V1=£N1N2¤·¦12¦22¸+V1and Y1is T£G1,N1is T£K1,N2is T£K2,K1+K2=K.It is assumed that rank(¦22)=G1to ensure identi…cation.At times,it is useful to refer to the following more compact version of(2),y1=X1µ+u1(3) where X1=[Y1;N1]andµ0=£¯0;°0¤is a p£1vector,p=G1+K1.The relationship between the errors in the structural and reduced form is given byV1=UB1(4)B1being the relevant G£G1submatrix of B¡1.The observations on the instruments are contained in the T£q matrix Z such thatZ=[N1N2S]=[NS1NS2]=NS zwhere K¸q¸p,and S,S i are selection matrices.The IV estimator ofµis given by^µ=(X01P z X1)¡1X01P z y1(5)zwhere P z=Z(Z0Z)¡1Z0,with main diagonal elements denoted by f p tt g.The bias of^µz is given by E(b z)whereb z=^µz¡µ=(X01P z X1)¡1X01P z u1(6)3If we let X=[N¦1;N1]and V x=[V1;0]and substitute X1=X+V x into(6) then we obtainb z=(I+Q z¢)¡1Q z(X0+V0x)P z u1(7) where Q z=(X0P z X)¡1and¢=X0P z V x+V0x P z X+V0x P z V x.If we take a series expansion of(I+Q z¢)¡1and drop terms of smaller order than O p¡T¡1¢, then we obtainb z=Q z X0P z u1+Q z V0x P z u1¡Q z V0x P z XQ z X0P z u1¡Q z X0P z V x Q z X0P z u1 Buse(1992)provides the following approximation to the bias of the IV estimator.Lemma1(Buse,1982)The bias of the IV estimator(5),to the order of T¡1, isE[b z]=(L¡1)Q z s(8) where L=q¡p,s is de…ned to be:s=·B01¾10¸and¾1is the…rst column of§.From(8)it can be seen that the inclusion of an additional instrument e¤ects both L and Q z and so the overall impact on the bias is unclear.However,Buse (1992)derives an inequality which delineates the circumstances under which the bias increases.To illustrate,consider the case in which G=2(i.e.there is just one endogenous right-hand side variable),and consider the bias associated with two choices of instrument Z1and Z2such that Z1½Z2.If we let b z(Z)denote the bias when the instrument vector is Z,then Buse(1992)shows:b z(Z2) z1R1as L2¡11R Y01P z2Y1¡Y01P n1Y101z1101n11=R22¡R20212(9)where R2i=Y01P zi Y1=Y01Y1(i=1;2),P z i=NS z i(S0zi N0NS z i)¡1S0ziN0,S ziisa K£q i selection matrix(i=1;2),L i=q i¡p,P n1=N1(N01N1)¡1N01,and R20=Y01P n1Y1=Y01Y1.Thus the estimated bias increases with the number of excess instrumental variables only if the proportional increase in the instruments is faster than the rate of increase in R2measured relative to the…t of Y1on N1.This inequality indicates that whether or not bias increases depends on the properties of both the additional instruments and also those already included.Buse(1992,p.178)concludes that“whether or not there is a trade-o¤between bias and e¢ciency depends therefore on the instrumental variable se-lection sequence”.However,his analysis only looks explicitly at one part of this trade-o¤,that is the bias.To develop a more complete understanding,it is4necessary to consider how the variance changes in response to the introduction of additional instruments.In practice,the trade-o¤between bias and e¢ciency is often assessed using the mean squared error(MSE)which is just the sum of the variances of the elements of^µz plus the bias squared.Therefore,we focus directly on the MSE.Nagar(1959)provides an approximation to the MSE un-der the assumption that the errors have a normal distribution.As remarked in the introduction,this distributional assumption does not cover many cases of interest particularly in…nance.It is therefore of interest to extend Nagar’s (1959)MSE result to allow for the types of error distributions encountered in practice.This is the topic of the next section.3The Moment Matrix of the IV EstimatorIn this section we derive the second moment matrix of the IV estimator,as-suming that the errors are drawn from an elliptical distribution.This class of symmetric distributions,whose main properties are presented below,includes members with di¤erent degrees of fatness of tails.The purpose of our analysis is to obtain an expression for the MSE of the IV estimator in…nite samples, analogous to the one for the bias in the previous section,so that we can check its determinants and in particular use it to simulate the impact of an extra instrument.3.1Elliptical DistributionsThe series expansion for the second moment matrix depends on the third and fourth moments of the errors.The resulting expression is very complicated in the general case,and so to facilitate the analysis it is assumed that U is drawn from a multivariate symmetric distribution.This class of distribution is particularly attractive here because it both contains the normal as a special case but also includes distributions with fatter tails than the normal.In univariate distributions the property of symmetry implies E(u3t)=0,but the generalization to a multivariate setting is not so obvious.According to Fang, Kotz,and Ng(1990),one of the ways of de…ning symmetry in a multivariate distribution is looking at the shape of the density.This property implies that the contours of surfaces of equal density have the same shape in a given class, giving rise to the name of elliptically contoured distributions.These are an extension of spherical distributions to which we now turn.If a random vector x possesses a density of the form f(x0x),for some function f,we say that x has a spherical distribution,denoted x»S n(Á).It is not necessary however that a spherical distribution possesses a density,so we can de…ne sphericity through the characteristic function of x,which is of the form Á(t0t).Examples of spherical distributions are a random vector u(n)distributed uniformly on the unit surface in R n,a random vector x distributed uniformly inside the unit sphere in R n,or x distributed as N n(0;I).5A spherical distribution is a special case of an elliptical distribution,or the latter can be constructed from the former by introducing some additional parameters.Based on Fang,Kotz,and Ng(1990)and Traat(1990)we present the following de…nition and lemma.De…nition1The n£1random vector y is said to have an elliptically sym-metric,elliptically contoured or simply elliptical distribution,denoted y»E n(¹;¤),with parameters¹(n£1)and¤(n£n),rank(¤)=k,if:(i)The ran-dom vector y has the same distribution as¹+z0x,where x»S k(Á),and z(k£n) is such that z0z=¤;(ii)The characteristic function of y is of the form:ª(t)=e it0¹Á(t0¤t)for some scalar functionÁ.(iii)The density function of y is of the form:f(y)=a n j¤j¡12g£(y¡¹)0¤¡1(y¡¹)¤for some function g,being a n a normalizing constant.Examples of elliptical distributions are the multinormal,N n(¹;§),the mul-tivariate t distribution,t n(º;¹;¤),and some mixtures of normal distributions. Fang,Kotz,and Ng(1990)discuss the properties of elliptical distributions,such as:¤,Á,and z are not unique unless we impose the condition j¤j=1;any linear combination of elliptically distributed variables is still elliptical.When the vector y is normally distributed we can use standard statistics such as the t-ratio and the F-test to make inference about the parameters. Fang,Kotz,and Ng(1990)show that these statistics,being invariant under scalar multiplication,are robust in the class of spherical distributions.Also the normal theory for inference on the parameters of a regression model with spherical disturbances remains valid,and large sample inference based on an i.i.d.elliptical sample is possible.The following lemma shows the expressions for the moments of the elliptical distributions up to the fourth order:Lemma2The mean and covariance matrix of y»E n(¹;¤)are given by:E(y)=¹V ar(y)=§=c2¤(10)where c2=¡2Á0(0):Denoting u=y¡¹,the mixed central moments of y of order3and4are:E(u i u j u k)=0(11)andE(u i u j u k u l)=c4[E(u i u j)E(u k u l)+E(u i u k)E(u j u l)+E(u i u l)E(u j u k)] =c4(¾ij¾kl+¾ik¾jl+¾il¾jk)(12) where c4=Á00(0),and i;j;k;l=1;2;:::;n.026The result(11)comes directly from the symmetry property,see Roomeldi (1992).The kurtosis parameter is given by:·=Á00(0)£Á0(0)¤¡1=c4¡1(13)We use Lemma2to evaluate the moments of the disturbances that arise from the series expansion for the moment matrix,assuming that u»E G(0;¤),where u is any of the rows of U.Below we use our formula to calculate the bias and MSE for two speci…c members of the family of elliptical distributions:the normal and the mixture of normals.For these two distributions c2and c4are as follows.If u»N G(0;¤)thenc2=c4=1(14)If u»²N G(0;¤)+(1¡²)N G¡0;h2¤¢thenc2=²+h2(1¡²)and c4=²+h4(1¡²)22(15)The results stated above will be used to evaluate the expectations of the mo-ments of the disturbances,which we need to work out the approximate expres-sion for the second moment matrix of the IV estimator.3.2Mean Squared ErrorThe matrix that contains the second moments of the IV estimator,^µz,around the true parameter valueµ,is de…ned by E³^µz¡µ´³^µz¡µ´0=E(b z b0z).The diagonal elements of that matrix give us the MSE of the corresponding elements of the parameter vector.In order to…nd an approximate expression for the moment matrix,we begin by de…ning for simpli…cation purposes=(X0+V0x)P z u1,and writing(7)asb z=(I+Q z¢)¡1Q zand then construct the productb z b0z=(I+Q z¢)¡1Q z P Q z(I+¢Q z)¡1so that E(b z b0z)will give us the second moment matrix of the estimator;Q z,¢and P=0are symmetric matrices.The orders in probability of these matrices areQ z=O p¡T¡1¢,P=O p(T)and¢=O p³T1=2´,and they will be used to calculate the order in probability of each term in the expansion that follows.7Let us now derive the moment matrix to the order of T¡2.For that we expand(I+Q z¢)¡1and its transpose,considering the…rst three terms only since the others are of lower order in probability,which gives:b z b0z=h I¡Q z¢+(Q z¢)2i Q z PQ z h I¡¢Q z+(¢Q z)2i=Q z P Q z¡Q z P Q z¢Q z+Q z PQ z¢Q z¢Q z¡Q z¢Q z P Q z+Q z¢Q z P Q z¢Q z+Q z¢Q z¢Q z P Q z¡O p(T¡5=2)(16) or,equivalently,we can writeQ¡1z b z b0z Q¡1z=P¡P Q z¢+¢Q z P+P Q z¢Q z¢+¢Q z P Q z¢+¢Q z¢Q z P¡O p(T¡1=2)(17) which is O p(1).In order to obtain the moment matrix E(b z b0z),we take ex-pectations of the somewhat simpler expression(17)and pre and post-multiply afterwards by Q z.Those calculations are shown in the appendix.At this point it is useful to introduce the following notation:A=D z¦01N0(P z¡P n1)(18)D z=[¦01N0(P z¡P n1)N¦1]¡1(19)C=P z XQ z X0P z(with the main diagonal element,c tt)(20)¹4=£¹ij¤=¾11¹4¹ij=E¡u21t u it u jt¢¾1¾01=[¾1i¾1j]=¾11¹§i;j=1;:::;G:H¤¹=B01¹4B1H¤§=B01§B1=1E(V1V01)H¤¾=B01¹§B1Bordering the(G1£G1)matrices H¤¹,H¤§and H¤¾(which are scalars in the special case of one right-hand endogenous variable)with K1rows and K1 columns of zeros,we obtainH¹=·H¤¹000¸H§=·H¤§000¸H¾=·H¤¾000¸=1¾11ss0The matrices H¤§,H§and H¤¾,H¾correspond to the matrices C¤,C and C¤1, C1in Nagar(1959),respectively.Finally,we de…nei0=£I0¤H=H¹¡H§¡2H¾H¤=H¤¹¡H¤§¡2H¤¾=(c4¡1)B01¡§+2¹§¢B1(21) The second equality in(21)is valid for elliptical variables,and was obtained by applying(12)to¹4.We are now ready to state our main result.8Theorem1If(i)the rows of U in(1)are G£1vectors of independent and elliptically distributed disturbances with zero mean,covariance matrix§=[¾ij], and…nite higher order moments;(ii)E(b z b0z)exists;then the moment matrix, to the order of T¡2,of the IV estimator^µz around the parameter valueµis given by:E(b z b0z)=¾11Q z(I p+A¤)(22) whereA¤=[¡(2L¡2)tr(Q z H¾)+tr(Q z H¾)]¢I p(23) +£¡L2¡3L+4¢H¾¡(L¡2)H§¤Q z+³X p2tt+X c2tt¡2X p tt c tt´HQ z+3X0P z diag(A0H¤A)P z XQ z+3X0P z diag(C¡P z)A0H¤i0Q z+3iH¤A diag(C¡P z)P z XQ z+2tr(Q z H)X0P z diag(C¡P z)P z XQ z We notice from Theorem1that the moment matrix of the IV estimator depends on three types of variables:L,the number of excess instruments;H¾and H§matrices,based on the moments of the errors“…ltered”by the struc-tural parameters of the endogenous variables;in particular H and H¤contain the“di¤erence”between the considered distribution and the Normal;and…-nally other matrices(X,Q z,P z,C,A)based on the exogenous variables,their moments and the reduced form parameters.Corollary1If the rows of U follow a Normal(0;§)distribution,the moment matrix,to the order of T¡2,of the IV estimator^µz around the parameter value µis given by(22)withA¤=[¡(2L¡2)tr(Q z H¾)+tr(Q z H§)]¢I p(24)+£¡L2¡3L+4¢H¾¡(L¡2)H§¤Q zWe note that Corollary1agrees with Nagar(1959,p.579),with Q z= Q2SLS,H¾=C1and H§=C.This means that our result,stated in Theorem 1,while valid for the class of elliptical distributions(so accommodating di¤erent degrees of kurtosis)and for the general IV estimator,specializes for the2SLS estimator with normally distributed errors.The result expressed in Theorem1 will be used in the next section to simulate the e¤ect of adding extra instruments on the MSE of the IV estimator.3.3Exact momentsIn this section we compare the expressions for the moments of the IV estimator in a non-normality context using two approaches:our extension of Nagar’s (1959)technique of asymptotic expansions,which is presented in the previous section,with the exact moments approach of Knight(1985).This is at present work in progress.94Simulation ResultsThe expression for the approximated MSE(and bias)of the IV estimator that we have derived in the previous sections will now be used in a simulation study. The purpose is to investigate the e¤ect on the MSE of adding extra instruments, and whether(and in which direction)that e¤ect changes with the sample size and with the degree of kurtosis of the error distribution.The simulations are based on the modely1=y2¯+n1°11+u1(25)y2=N°2+u2(26) where y1and y2are(T£1)vectors of endogenous variables,¯and°11are scalar parameters and°2is a(9£1)vector of parameters,N=[n1:::n9]is a (T£9)matrix of exogenous variables and£u1u2¤=U forms the matrix of disturbances for the model;the transpose of a row of U is u t=(u1t;u2t)0and is elliptically distributed with zero mean and covariance matrix§=·1:8:81¸The columns of N are generated from a joint standard normal distribution, ensuring that they are…xed in repeated samples and that N0N=T¢I9.The vector u t comes from a bivariate mixture of two Normal distributions; with probability²=0:8the process is realized from a N(0;¤)and with prob-ability(1¡²)from N¡0;h2¤¢,where the parameter that in‡ates the variance takes values h2f1;2;3;4;6;8;10g.Those values of h correspond to a kurtosis of·2f0;0:6;1:5;2:3;3:1;3:4;3:6g,as given by(13),(14)and(15). The…rst case h=1,·=0characterizes the Normal distribution.As h grows (keeping²constant),the kurtosis becomes larger as compared to the Normal. We control for the values of¤so that the variance of u t remains unchanged as we vary the h parameter.That ensures that any changes in the MSE of the IV estimator that we may…nd are due to the kurtosis of the distribution only,via changes in the fourth moment.We set the coe¢cients¯=°11=1,°21=:::°25=:03,°26=:::°29=:33; this implies that the…rst…ve columns of N have only a marginal contribution to the explanation of y2,whereas the last four variables together lead us to expect a theoretical R2of around30%in the second equation,calculated as:plim R2=plim ^°02N0y2022=°02°2022(27)Note that in this simple model the second equation is already in the reduced form,so the simultaneity comes only from the correlation between u1and u2.The parameter vector to be estimated by IV isµ=(¯;°11)0.The matrix of instruments is Z=NS z where S z is a(9£q)selection matrix,q=5;¢¢¢;9, composed by zeros and ones according to the columns of Z that we want to10select.We have always z1=n1,and therefore have eight possible instruments for y2,out of which only four(n6;:::;n9)have a signi…cant contribution to the determination of y2in the reduced form,but all of them are orthogonal to the error.We shall refer for simplicity to this set as“good”instruments and to(n2;:::;n5)as“bad”instruments for y2.We consider nineteen di¤erent selection matrices,the…rst one selecting only bad instruments and the last selecting all instruments,with di¤erent combinations in between.For each combination we calculate the bias and the second moment matrix of^µz according to(8)and(22)respectively.A priori we expect the best choice in terms of bias and MSE to be the selection of only good instruments(plus n1).We consider three sample sizes,T=30,60and90and di¤erent degrees of kurtosis as mentioned.Table1reports the e¤ects on the bias of adding an instrument,according to its quality.The bias is computed for…ve di¤erent choices of instrument vector at sample sizes T=30,60and90respectively.All choices of instrument vector include n1but di¤er in the composition of the remaining four variables. The columns are ordered from left to right according to the number of bad instruments with the left-hand column containing the results for the case with four bad instruments.The third(+1G%)and fourth(+1B%)rows of the tables show the percentage change in the bias when we add a good(n9)and a bad(n5) instrument respectively to the combination written in the…rst row.Tables2-4 report analogous computations for the square root of the MSE(RMSE)of^¯, but the results in this case di¤er according to the degree of kurtosis considered.Comparing the rows showing the“bias”for the di¤erent sample sizes,in Table1,we see that the bias decreases as T increases as would be expected. The relative e¤ect of adding an instrument is,however,roughly constant with the sample size.The bias increases by50%when a poor instrument is added (rows labelled+1B%).Analyzing now the e¤ect of adding a good instrument (+1G%)we note that the…rst and second good instruments included cause a clear reduction in the bias,whereas continuing adding a third one does not have much a¤ect,and a fourth one makes the bias increase by12%.This result agrees with Buse(1992),in the sense that the existence of a bias/variance trade-o¤depends on the instrument selection sequence,namely on the quantity and quality of the other selected instruments.Tables2-4report the RMSE of the IV estimator calculated using the formula in Theorem1.These results indicate that the MSE decreases when we add a good quality instrument and increases when we add a bad quality one.The …rst e¤ect results from a decrease in the variance,together with a decrease in the bias for the…rst two instruments added.The second e¤ect shows that even when a trade-o¤is to be expected,the increase in the bias when we add a poor instrument can o¤set the decrease in the variance of the estimator,so making the MSE increase,which leads to a worse estimator by this criterion.The absolute value of the change is nevertheless much greater in magnitude when a good instrument is added.Moreover,both e¤ects show a tendency to increase with h(and thus with the degree of kurtosis),which is clearer in the small sample.With the other11。
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CLNS 99/1646; hep-th Introduction
Non-perturbative techniques have proven to be very useful in the investigation of interacting field theories. This is especially true in two-dimensions, where rich symmetry structures have allowed the calculation of fundamental quantities, such as exact S -matrices and correlation functions, for many important models. In recent times, this progress has been further advanced by a better understanding of the mathematical framework underlying the symmetry structures. Perhaps this is most apparent in the construction of exact S matrices for integrable 2D systems. Here a knowledge of the symmetry algebra and its representation theory is essential to building a S -matrix satisfying the Yang-Baxter constraint [1–6]. Well known examples of models with exact S -matrices include the sine-Gordon/massive Thirring model [7–9], Gross-Neveu models [10–12], principal chiral and sigma-type models [13–16], various statistical systems, such as RSOS systems [17,18], and Toda theories based on Lie algebras [19–25]. A relatively new class of models that have been investigated are based on Lie superalgebras. Supersymmetric Toda models belong to this category and their exact S -matrices have been calculated [26, 27]. In this paper we use the S -matrix approach to study a model based on the Lie superalgebra osp(2|2), which is closely related to a disordered system introduced to describe the integer quantum Hall transition [28]. As shown by Bernard [29], the model arises after disorder averaging over a random scalar potential and consists of a free fermionic and bosonic piece, combined with a marginal osp(2|2) current-current perturbation. The bosonic part is the result of rewriting the fermion partition function as a path integral over complex bosonic variables. This pairs the fermions and bosons thus making the action “supersymmetric”. The model is integrable with factorized scattering and Yangian symmetry. By introducing an anisotropy, which allows us to flow between a relevant and marginal perturbation, we construct non-local charges for half the current-current operators. The remaining current-current operators are generated under renormalization at the marginal point. The non-local charges are shown to satisfy the Uq [osp(2|2)(1) ] quantum superalgebra and to be conserved to lowest order in conformal perturbation theory. Requiring the theory to have Uq [osp(2|2)(1) ] quantum group symmetry, we calculate the S -matrix, or more appropriately the R -matrix, for the fundamental vector representation. The physical S -matrix is then obtained, up to CDD factors, by imposing the unitarity and crossing constraints. We propose that in the marginal limit this S -matrix is the exact S -matrix (in the fundamental representation) with Yangian symmetry for the osp(2|2) current-current model. The particle spectrum of the model and the corresponding S -matrix are massive. In particular, this means that all states are Anderson localized. The quantum Hall disordered system discussed in [28] contains three types of randomness. Though the generic case, with all types of randomness present, is not believed to be integrable, it is certainly possible that on some submainifold in the three coupling parameter space the model can be exactly solved. Various such subspaces have already been investigated [28–30]. An interesting subset was recently studied in [31] (see also [32]), where supersymmetric disorder averaging led to a gl(N |N ) current-current type model for which exact correlation functions were computed. Our S -matrix analysis considers the situation where there is only one specific type of disorder, namely a random scalar potential. We present our results as follows. In section 2 we write down the models and show how the full current algebra is generated under renormalization. The non-local charges are constructed in section 3. From the quantum group structure we build the S -matrix for the fundamental representation in section 4. The pole structure is briefly discussed in section 5. Lastly we conclude with a summary and comment on open questions for further study. An appendix reviews the osp(2|2) algebras.
The Exact S -Matrix for an osp(2|2) Disordered System
Zorawar S. Bassi∗ Andr´ e LeClair
arXiv:hep-th/9911105v2 11 Dec 1999
Newman Laboratory Cornell University Ithaca, NY 14853, USA December 1999