Parameter equivalence for the Brooks-Corey and van Genuchten soil characteristics Preservin

a r X i v :0808.0806v 2 [c o n d -m a t .s u p r -c o n ] 7 A u g 2008Monotonic d-wave Superconducting Gap in Optimally-Doped Bi 2Sr 1.6La 0.4CuO 6Superconductor by Laser-Based Angle-Resolved Photoemission SpectroscopyJianqiao Meng 1,Wentao Zhang 1,Guodong Liu 1,Lin Zhao 1,Haiyun Liu 1,Xiaowen Jia 1,Wei Lu 1,Xiaoli Dong 1,Guiling Wang 2,Hongbo Zhang 2,Yong Zhou 2,Yong Zhu 3,Xiaoyang Wang 3,Zhongxian Zhao 1,Zuyan Xu 2,Chuangtian Chen 3,X.J.Zhou 1,∗1National Laboratory for Superconductivity,Beijing National Laboratory for Condensed Matter Physics,Institute of Physics,Chinese Academy of Sciences,Beijing 100190,China2Key Laboratory for Optics,Beijing National Laboratory for Condensed Matter Physics,Institute of Physics,Chinese Academy of Sciences,Beijing 100190,China3Technical Institute of Physics and Chemistry,Chinese Academy of Sciences,Beijing 100190,China(Dated:April 23,2008)The momentum and temperature dependence of the superconducting gap and pseudogap in optimally-doped Bi 2Sr 1.6La 0.4CuO 6superconductor is investigated by super-high resolution laser-based angle-resolved photoemission spectroscopy.The measured energy gap in the superconducting state exhibits a standard d -wave form.Pseudogap opens above T c over a large portion of the Fermi surface with a “Fermi arc”formed near the nodal region.In the region outside of the “Fermi arc”,the pseudogap has the similar magnitude and momentum dependence as the gap in the supercon-ducting state which changes little with temperature and shows no abrupt change across T c .These observations indicate that the pseudogap and superconducting gap are closely related and favor the picture that the pseudogap is a precursor to the superconducting gap.PACS numbers:74.25.Jb,71.18.+y,74.72.Dn,79.60.-iThe high temperature cuprate superconductors are characterized by their unusual superconducting state,manifested by the anisotropic superconducting gap with predominantly d -wave symmetry[1],as well as the anomalous normal state,exemplified by the existence of a pseudogap above the superconducting transition tem-perature (T c )[2].The origin of the pseudogap and its relation with the superconducting gap are critical is-sues in understanding the mechanism of superconduc-tivity and exotic normal state properties[3,4].It has been a long-standing debate on whether the pseudogap is intimately related to the superconducting gap like a precursor of pairing[5,6,7,8]or it originates from other competing orders that has no direct bearing on superconductivity[9,10,11,12].Angle-resolved photoemission spectroscopy (ARPES),as a powerful tool to directly measure the magni-tude of the energy gap,has provided key insights on the superconducting gap and pseudogap in cuprate superconductors[13].Recently,great effort has been fo-cused on investigating their relationship but the results are split in supporting two different pictures[8,11,14,15,16,17].In one class of ARPES experiments,dis-tinct doping and temperature dependence of the en-ergy gap between the nodal and antinodal regions are reported[11,15]which are used to support “two gap”picture where the pseudogap and the superconducting gap are loosely related or independent.Additional sup-port comes from the unusual gap form measured in the superconducting state[14,16].Its strong deviation from the standard d -wave form is interpreted as composing of “two components”:a “true”d-wave superconducting gapand the remanent pseudogap that is already present in the normal state[14,16].In another class of experiments that supports “one-gap”picture where the pseudogap is a precursor of the superconducting gap,the gap in the superconducting state is found to be consistent with a standard d -wave form[8,17].Slight deviation in the un-derdoped regime is interpreted as due to high-harmonic pairing terms[18].In light of the controversy surrounding the relationship between the pseudogap and superconducting gap and its importance in understanding high-T c superconductivity,we report in this paper detailed momentum and temper-ature dependence of the superconducting gap and pseu-dogap in Bi 2Sr 1.6La 0.4CuO 6(La-Bi2201)superconductor by super-high resolution laser-based ARPES measure-ments.In the superconducting state we have identified an anisotropic energy gap that is consistent with a stan-dard d -wave form.This is significantly different from the previous results on a similar superconductor[14].In the normal state,we have observed pseudogap opening with a small “Fermi arc”formed near the nodal region.Outside of the ”Fermi arc”,the pseudogap in the normal state has the similar magnitude and momentum dependence as the gap in the superconducting state:detailed tem-perature dependence shows that the pseudogap evolves smoothly into the superconducting gap with no abrupt change across T c .These results point to an intimate re-lationship between the pseudogap and the superconduct-ing gap which is in favor of the “one-gap”picture that pseudogap is a precursor to the superconducting gap.The ARPES measurements are carried out on our newly-developed Vacuum Ultraviolet(VUV)laser-based2E - EF (eV)E - EF (eV)1.00.50G (0,0)(p ,0)1510152025k xFIG.1:Fermi surface of the optimally-doped La-Bi2201(T c =32K)and corresponding photoemission spectra (EDCs)on the Fermi surface at various temperatures.(a).Spectral weight as a function of two-dimensional momentum (k x ,k y )integrated over [-5meV,5meV]energy window with respect to the Fermi level E F .The measured Fermi momenta are marked by red empty circles and labeled by numbers;(b).Original EDCs along the Fermi surface measured at 15K.The symmetrized EDCs along the Fermi surface are shown in (c)for 15K,(d)for 25K and (e and f)for 40K.The numbers on panels (b-f)corresponds to the Fermi momentum numbers in (a).angle-resolved photoemission system with advantages of super-high energy resolution,high momentum resolution,high photon flux and enhanced bulk sensitivity[19].The photon energy is 6.994eV with a bandwidth of 0.26meV and the energy resolution of the electron energy analyzer (Scienta R4000)was set at 0.5meV,giving rise to an overall energy resolution of 0.56meV.The angular res-olution is ∼0.3◦,corresponding to a momentum resolu-tion ∼0.004˚A −1at the photon energy of 6.994eV.The optimally doped Bi 2Sr 2−x La x CuO 6(La-Bi2201)(x=0.4,T c ∼32K,transition width ∼2K)single crystals were grown by the traveling solvent floating zone method[20].One advantage of choosing La-Bi2201system lies in its relatively low superconducting transition temperature that is desirable in investigating the normal state behav-ior with suppressed thermal broadening of photoemission spectra.The samples are cleaved in situ in vacuum with a base pressure better than 4×10−11Torr.Fig.1(a)shows the Fermi surface mapping of the op-timally doped La-Bi2201(T c =32K)measured at 15K.The low photon energy and high photon flux have made it possible to take dense sampling of the measurements in the momentum space.The photoemission spectra (En-ergy Distribution Curves,EDCs)along the Fermi surface are plotted in Fig.1(b).The EDCs near the nodal re-gion show sharp peaks that are similar to those observed in Bi2212[21].When the momentum moves away from the nodal region to the (0,π)antinodal region,the EDC peaks get weaker,but peak feature remains along the en-tire Fermi surface even for the one close to the antinodal region.The EDC peak position also shifts away from the Fermi level when the momentum moves from the nodal to the antinodal region,indicating a gap opening in the superconducting state.Note that the EDCs near the antinodal region do not show any feature near 40meV that was reported in a previous measurement[14].In order to extract the energy gap,we have sym-metrized the original EDCs with respect to the Fermi level,as shown in Fig.1c for the 15K measurements,and Fig.1d and Fig.1(e-f)for 25K and 40K,respec-tively.The symmetrization procedure not only provides an intuitive way in visualizing the energy gap,but also removes the effect of Fermi cutoffin photoemission spec-tra and provides a quantitative way in extracting the gap size[22].The symmetrized EDCs have been fitted using the general phenomenological form[22];the fitted curves are overlaid in Fig.1(c-f)and the extracted gap size is plotted in Fig.2.As shown in Fig.2,the gap in the superconducting state exhibits a clear anisotropic behavior that is consis-tent with a standard d -wave form ∆=∆0cos(2Φ)(or in a more strict sense,∆=∆0|cos (k x a )−cos (k y a )|/2form as shown in the inset of Fig.2)with a maximum energy gap ∆0=15.5meV.It is also interesting to note that the gap is nearly identical for the 15K and 25K measurements for such a T c =32K superconductor.These results are significantly different from a recent measurement where the gap in the superconducting state deviates strongly from the standard d -wave form with an antinodal gap at 40meV[14].An earlier measurement[23]gave an antin-3G a p S i z e (m e V )Angle F (degrees)FIG.2:Energy gap along the Fermi surface measured at 15K (solid circles),25K (empty circles)and 40K (empty squares)on the optimally-doped La-Bi2201(T c =32K).The solid red line is fitted from the measured data at 15K which gives ∆=15.5cos(2Φ).The Φangle is defined as shown in the bottom-right inset.The upper-right inset shows the gap size as a function of |cos (k x a )−cos (k y a )|/2at 15K and 25K.The pink line represents a fitted line with ∆=15.5|cos (k x a )−cos (k y a )|/2.odal gap at 10∼12meV which is close to our present mea-surement,but it also reported strong deviation from the standard d -wave form.While the non-d -wave energy gap can be interpreted ascomposed of two components in the previous measurement[14],our present results clearly in-dicate that the gap in the superconducting state is dom-inated by a d -wave component.In the normal state above T c =32K,the Fermi sur-face measured at 40K is still gapped over a large portion except for the section near the nodal region that shows a zero gap,as seen from the symmetrized EDCs (Fig.1e-f for 40K)and the extracted pseudo-gap (40K data in Fig.2).This is consistent with the “Fermi arc”picture observed in other high temperature superconductors[6,11,24].Note that the pseudogap out-side of the “Fermi arc”region shows similar magnitude and momentum dependence as the gap in the supercon-ducting state (Fig.2).Fig.3shows detailed temperature dependence of EDCs and the associated energy gap for two representa-tive momenta on the Fermi surface.Strong temperature dependence of the EDCs is observed for the Fermi mo-mentum A (Fig.3a).At high temperatures like 100K or above,the EDCs show a broad hump structure near -0.2eV with no observable peak near the Fermi level.Upon cooling,the high-energy -0.2eV broad hump shows little change with temperature,while a new structure emerges near the Fermi level and develops into a sharp “quasipar-ticle”peak in the superconducting state,giving rise to a peak-dip-hump structure in EDCs.This temperatureE - EF (eV)E - EF (eV)FIG.3:(a,b).Temperature dependence of representa-tive EDCs at two Fermi momenta on the Fermi surface in optimally-doped La-Bi2201.The location of the Fermi mo-menta is indicated in the inset.Detailed temperature depen-dence of the symmetrized EDCs for the Fermi momentum A are shown in (c)and for the Fermi momentum B in (d).The dashed lines in (c)and (d)serve as a guide to the eye.evolution and peak-dip-hump structure are reminiscent to that observed in other high temperature superconduc-tors like Bi2212[25].When moving towards the antin-odal region,as for the Fermi momentum B (Fig.3b),the EDCs qualitatively show similar behavior although the temperature effect gets much weaker.One can still see a weak peak developed at low temperatures,e.g.,13K,near the Fermi level.To examine the evolution of the energy gap with tem-perature,Fig.3c and 3d show symmetrized EDCs mea-sured at different temperatures for the Fermi momenta A and B,respectively.The gap size extracted by fit-ting the symmetrized EDCs with the general formula[22]are plotted in Fig. 4.For the Fermi momentum A,as seen from Fig.3c,signature of gap opening in the su-perconducting state persists above T c =32K,remaining obvious at 50K,getting less clear at 75K,and appear to disappear around 100K and above as evidenced by the appearance of a broad peak.The gap size below 50K (Fig.4)shows little change with temperature and no abrupt change is observed across T c .The data at 75K is hard to fit to get a reliable gap size,thus not included in Fig. 4.When the momentum moves closer to the antinodal region,as for the Fermi momentum B,simi-lar behaviors are observed,i.e.,below 50K,the gap size is nearly a constant without an abrupt change near T c .But in this case,different from the Fermi momentum A,4G a p S i z e (m e V )Temperature(K)FIG.4:Temperature dependence of the energy gap for two Fermi momenta A (empty squares)and B (empty circles)as indicated in insets of Fig.3(a)and (b),and also indicated in the up-right inset,for optimally-doped La-Bi2201.The dashed line indicates T c =32K.there is no broad peak recovered above 100K,probably indicating a higher pseudogap temperature.This is qual-itatively consistent with the transport[26]and NMR[27]measurements on the same material that give a pseudo-gap temperature between 100∼150K.From precise gap measurement,there are clear signa-tures that can distinct between “one-gap”and “two-gap”scenarios[4].In the “two-gap”picture where the pseudo-gap and superconducting gap are assumed independent,because the superconducting gap opens below T c in addi-tion to the pseudogap that already opens in the normal state and persists into the superconducting state,one would expect to observe two effects:(1).Deviation of the energy gap from a standard d -wave form in the super-conducting state with a possible break in the measured gap form[14];(2).Outside of the “Fermi arc”region,one should expect to see an increase in gap size in the superconducting state.Our observations of standard d -wave form in the superconducting state (Fig.2),similar magnitude and momentum dependence of the pseudogap and the gap in the superconducting state outside of the “Fermi arc”region (Fig.2),smooth evolution of the gap size across T c and no indication of gap size increase upon entering the superconducting state (Fig.4),are not com-patible with the expectations of the “two-gap”picture.They favor the “one-gap”picture where the pseudogap and superconducting gap are closely related and the pseu-dogap transforms into the superconducting gap across T c .Note that,although the region outside of the “Fermi arc”shows little change of the gap size with temperature (Fig.4),the EDCs exhibit strong temperature depen-dence with a “quasiparticle”peak developed in the su-perconducting state(Fig.3a and 3b)that can be related with the establishment of phase coherence[8,25].This suggests that the pseudogap region on the Fermi surface can sense the occurrence of superconductivity through acquiring phase coherence.In conclusion,from our precise measurements on the detailed momentum and temperature dependence of the energy gap in optimally doped La-Bi2201,we provide clear evidence to show that the pseudogap and super-conducting gap are intimately related.Our observations are in favor of the “one-gap”picture that the pseudogap is a precursor to the superconducting gap and supercon-ductivity is realized by establishing a phase coherence.We acknowledge helpful discussions with T.Xi-ang.This work is supported by the NSFC(10525417and 10734120),the MOST of China (973project No:2006CB601002,2006CB921302),and CAS (Projects IT-SNEM and 100-Talent).∗Corresponding author:XJZhou@[1]See,e.g.,C.C.Tsuei and J.R.Kirtley,Rev.Mod.Phys.72,969(2000).[2]T.Timusk and B.Statt,Rep.Prog.Phys.62,61(1999).[3]V.J.Emery and S.A.Kivelson,Nature (London)374,434(1995);X.G.Wen and P.A.Lee,Phys.Rev.Lett.76,503(1996);C.M.Varma,Phys.Rev.Lett.83,3538(1999);S.Chakravarty et al.,Phys.Rev.B 63,094503(2001);P.W.Anderson,Phys.Rev.Lett.96,017001(2006).[4]lis,Science 314,1888(2006).[5]Ch.Renner et al.,Phys.Rev.Lett.80,149(1998).[6]M.R.Norman et al.,Nature (London)392,157(1998).[7]Y.Y.Wang et al.,Phys.Rev.B 73,024510(2006).[8]A.Kanigel et al.,Phys.Rev.Lett.99,157001(2007).[9]G.Deytscher,Nature (London)397,410(1999).[10]M.Le.Tacon et al.,Nature Phys.2,537(2006).[11]K.Tanaka et al.,Scinece 314,1910(2006).[12]M.C.Boyer et al.,Nature Phys.3,802(2007).[13]A.Damascelli et al.,Rev.Mod.Phys.75,473(2003);J.C.Campuzano et al.,in The Physics of Superconductors,Vol.2,edited by K.H.Bennemann and J.B.Ketterson,(Springer,2004).[14]T.Kondo et al.,Phys.Rev.Lett.98,267004(2007).[15]W.S.Lee et al.,Nature (London)450,81(2007).[16]K.Terashima et al.,Phys.Rev.Lett.99,017003(2007).[17]M.Shi et al.,arXiv:cond-mat/0708.2333.[18]J.Mesot et al.,Phys.Rev.Lett.83,840(1999).[19]G.D Liu et al.,Rev.Sci.Instruments 79,023105(2008).[20]J.Q.Meng et al.,unpublished work.[21]W.T.Zhang et al.,arXiv:cond-mat/0801.2824.[22]M.R.Norman et al.,Phys.Rev.B 57,R11093(1998).[23]J.M.Harris et al.,Phys.Rev.Lett.79,143(1997).[24]A.Kanigel et al.,Nature Phys.2447(2006).[25]A.V.Fedorov et al.,Phys.Rev.Lett.82,2179(1999);D.L.Feng et al.,Science 289,277(2000);H.Ding et al.,Phys.Rev.Lett.87,227001(2001).[26]Y.Ando et al.,Phys.Rev.Lett.93,267001(2004).[27]G.-Q.Zheng et al.,Phys.Rev.Lett.94,047006(2005).。

偏微分方程数值解法英文英文回答：Numerical Solutions of Partial Differential Equations.Partial differential equations (PDEs) are a type of mathematical equation that describes how a quantity changes in relation to several independent variables. They are used to model a wide variety of physical phenomena, such asfluid flow, heat transfer, and electromagnetism.Analytical solutions to PDEs can be difficult or impossible to obtain, so numerical methods are often used to approximate solutions. Numerical methods discretize the PDE into a system of algebraic equations that can be solved by a computer.There are a variety of different numerical methods that can be used to solve PDEs, each with its own advantages and disadvantages. Some of the most common methods include:Finite difference methods discretize the PDE by replacing the derivatives with finite differences. This is a relatively simple method that can be used to solve a wide variety of PDEs.Finite element methods discretize the PDE by dividing the domain into a set of elements. This method is more flexible than finite difference methods and can be used to solve PDEs with complex geometries.Boundary element methods discretize the PDE by representing the solution in terms of the values on the boundary. This method is particularly well-suited for solving PDEs with infinite domains.The choice of which numerical method to use depends on the specific PDE being solved, as well as the available computational resources.中文回答：偏微分方程的数值解法。

Probabilistic Model Checking ofan Anonymity SystemVitaly ShmatikovSRI International333Ravenswood AvenueMenlo Park,CA94025U.S.A.shmat@AbstractWe use the probabilistic model checker PRISM to analyze the Crowds system for anonymous Web browsing.This case study demonstrates howprobabilistic model checking techniques can be used to formally analyze se-curity properties of a peer-to-peer group communication system based onrandom message routing among members.The behavior of group mem-bers and the adversary is modeled as a discrete-time Markov chain,and thedesired security properties are expressed as PCTL formulas.The PRISMmodel checker is used to perform automated analysis of the system and ver-ify anonymity guarantees it provides.Our main result is a demonstration ofhow certain forms of probabilistic anonymity degrade when group size in-creases or random routing paths are rebuilt,assuming that the corrupt groupmembers are able to identify and/or correlate multiple routing paths originat-ing from the same sender.1IntroductionFormal analysis of security protocols is a well-establishedfield.Model checking and theorem proving techniques[Low96,MMS97,Pau98,CJM00]have been ex-tensively used to analyze secrecy,authentication and other security properties ofprotocols and systems that employ cryptographic primitives such as public-key en-cryption,digital signatures,etc.Typically,the protocol is modeled at a highly ab-stract level and the underlying cryptographic primitives are treated as secure“black boxes”to simplify the model.This approach discovers attacks that would succeed even if all cryptographic functions were perfectly secure.Conventional formal analysis of security is mainly concerned with security against the so called Dolev-Yao attacks,following[DY83].A Dolev-Yao attacker is a non-deterministic process that has complete control over the communication net-work and can perform any combination of a given set of attacker operations,such as intercepting any message,splitting messages into parts,decrypting if it knows the correct decryption key,assembling fragments of messages into new messages and replaying them out of context,etc.Many proposed systems for anonymous communication aim to provide strong, non-probabilistic anonymity guarantees.This includes proxy-based approaches to anonymity such as the Anonymizer[Ano],which hide the sender’s identity for each message by forwarding all communication through a special server,and MIX-based anonymity systems[Cha81]that blend communication between dif-ferent senders and recipients,thus preventing a global eavesdropper from linking sender-recipient pairs.Non-probabilistic anonymity systems are amenable to for-mal analysis in the same non-deterministic Dolev-Yao model as used for verifica-tion of secrecy and authentication protocols.Existing techniques for the formal analysis of anonymity in the non-deterministic model include traditional process formalisms such as CSP[SS96]and a special-purpose logic of knowledge[SS99].In this paper,we use probabilistic model checking to analyze anonymity prop-erties of a gossip-based system.Such systems fundamentally rely on probabilistic message routing to guarantee anonymity.The main representative of this class of anonymity systems is Crowds[RR98].Instead of protecting the user’s identity against a global eavesdropper,Crowds provides protection against collaborating local eavesdroppers.All communication is routed randomly through a group of peers,so that even if some of the group members collaborate and share collected lo-cal information with the adversary,the latter is not likely to distinguish true senders of the observed messages from randomly selected forwarders.Conventional formal analysis techniques that assume a non-deterministic at-tacker in full control of the communication channels are not applicable in this case. Security properties of gossip-based systems depend solely on the probabilistic be-havior of protocol participants,and can be formally expressed only in terms of relative probabilities of certain observations by the adversary.The system must be modeled as a probabilistic process in order to capture its properties faithfully.Using the analysis technique developed in this paper—namely,formalization of the system as a discrete-time Markov chain and probabilistic model checking of2this chain with PRISM—we uncovered two subtle properties of Crowds that causedegradation of the level of anonymity provided by the system to the users.First,if corrupt group members are able to detect that messages along different routingpaths originate from the same(unknown)sender,the probability of identifyingthat sender increases as the number of observed paths grows(the number of pathsmust grow with time since paths are rebuilt when crowd membership changes).Second,the confidence of the corrupt members that they detected the correct senderincreases with the size of the group.Thefirstflaw was reported independently byMalkhi[Mal01]and Wright et al.[W ALS02],while the second,to the best ofour knowledge,was reported for thefirst time in the conference version of thispaper[Shm02].In contrast to the analysis by Wright et al.that relies on manualprobability calculations,we discovered both potential vulnerabilities of Crowds byautomated probabilistic model checking.Previous research on probabilistic formal models for security focused on(i)probabilistic characterization of non-interference[Gra92,SG95,VS98],and(ii)process formalisms that aim to faithfully model probabilistic properties of crypto-graphic primitives[LMMS99,Can00].This paper attempts to directly model andanalyze security properties based on discrete probabilities,as opposed to asymp-totic probabilities in the conventional cryptographic sense.Our analysis methodis applicable to other probabilistic anonymity systems such as Freenet[CSWH01]and onion routing[SGR97].Note that the potential vulnerabilities we discovered inthe formal model of Crowds may not manifest themselves in the implementationsof Crowds or other,similar systems that take measures to prevent corrupt routersfrom correlating multiple paths originating from the same sender.2Markov Chain Model CheckingWe model the probabilistic behavior of a peer-to-peer communication system as adiscrete-time Markov chain(DTMC),which is a standard approach in probabilisticverification[LS82,HS84,Var85,HJ94].Formally,a Markov chain can be definedas consisting in afinite set of states,the initial state,the transition relation such that,and a labeling functionfrom states to afinite set of propositions.In our model,the states of the Markov chain will represent different stages ofrouting path construction.As usual,a state is defined by the values of all systemvariables.For each state,the corresponding row of the transition matrix de-fines the probability distributions which govern the behavior of group members once the system reaches that state.32.1Overview of PCTLWe use the temporal probabilistic logic PCTL[HJ94]to formally specify properties of the system to be checked.PCTL can express properties of the form“under any scheduling of processes,the probability that event occurs is at least.”First,define state formulas inductively as follows:where atomic propositions are predicates over state variables.State formulas of the form are explained below.Define path formulas as follows:Unlike state formulas,which are simplyfirst-order propositions over a single state,path formulas represent properties of a chain of states(here path refers to a sequence of state space transitions rather than a routing path in the Crowds speci-fication).In particular,is true iff is true for every state in the chain;is true iff is true for all states in the chain until becomes true,and is true for all subsequent states;is true iff and there are no more than states before becomes true.For any state and path formula,is a state formula which is true iff state space paths starting from satisfy path formula with probability greater than.For the purposes of this paper,we will be interested in formulas of the form ,evaluated in the initial state.Here specifies a system con-figuration of interest,typically representing a particular observation by the adver-sary that satisfies the definition of a successful attack on the protocol.Property is a liveness property:it holds in iff will eventually hold with greater than probability.For instance,if is a state variable represent-ing the number of times one of the corrupt members received a message from the honest member no.,then holds in iff the prob-ability of corrupt members eventually observing member no.twice or more is greater than.Expressing properties of the system in PCTL allows us to reason formally about the probability of corrupt group members collecting enough evidence to success-fully attack anonymity.We use model checking techniques developed for verifica-tion of discrete-time Markov chains to compute this probability automatically.42.2PRISM model checkerThe automated analyses described in this paper were performed using PRISM,aprobabilistic model checker developed by Kwiatkowska et al.[KNP01].The toolsupports both discrete-and continuous-time Markov chains,and Markov decisionprocesses.As described in section4,we model probabilistic peer-to-peer com-munication systems such as Crowds simply as discrete-time Markov chains,andformalize their properties in PCTL.The behavior of the system processes is specified using a simple module-basedlanguage inspired by Reactive Modules[AH96].State variables are declared in thestandard way.For example,the following declarationdeliver:bool init false;declares a boolean state variable deliver,initialized to false,while the followingdeclarationconst TotalRuns=4;...observe1:[0..TotalRuns]init0;declares a constant TotalRuns equal to,and then an integer array of size,indexed from to TotalRuns,with all elements initialized to.State transition rules are specified using guarded commands of the form[]<guard>-><command>;where<guard>is a predicate over system variables,and<command>is the tran-sition executed by the system if the guard condition evaluates to mandoften has the form<expression>...<expression>, which means that in the next state(i.e.,that obtained after the transition has beenexecuted),state variable is assigned the result of evaluating arithmetic expres-sion<expression>If the transition must be chosen probabilistically,the discrete probability dis-tribution is specified as[]<guard>-><prob1>:<command1>+...+<probN>:<commandN>;Transition represented by command is executed with probability prob,and prob.Security properties to be checked are stated as PCTL formulas (see section2.1).5Given a formal system specification,PRISM constructs the Markov chain and determines the set of reachable states,using MTBDDs and BDDs,respectively. Model checking a PCTL formula reduces to a combination of reachability-based computation and solving a system of linear equations to determine the probability of satisfying the formula in each reachable state.The model checking algorithms employed by PRISM include[BdA95,BK98,Bai98].More details about the im-plementation and operation of PRISM can be found at http://www.cs.bham. /˜dxp/prism/and in[KNP01].Since PRISM only supports model checking offinite DTMC,in our case study of Crowds we only analyze anonymity properties offinite instances of the system. By changing parameters of the model,we demonstrate how anonymity properties evolve with changes in the system configuration.Wright et al.[W ALS02]investi-gated related properties of the Crowds system in the general case,but they do not rely on tool support and their analyses are manual rather than automated.3Crowds Anonymity SystemProviding an anonymous communication service on the Internet is a challenging task.While conventional security mechanisms such as encryption can be used to protect the content of messages and transactions,eavesdroppers can still observe the IP addresses of communicating computers,timing and frequency of communi-cation,etc.A Web server can trace the source of the incoming connection,further compromising anonymity.The Crowds system was developed by Reiter and Ru-bin[RR98]for protecting users’anonymity on the Web.The main idea behind gossip-based approaches to anonymity such as Crowds is to hide each user’s communications by routing them randomly within a crowd of similar users.Even if an eavesdropper observes a message being sent by a particular user,it can never be sure whether the user is the actual sender,or is simply routing another user’s message.3.1Path setup protocolA crowd is a collection of users,each of whom is running a special process called a jondo which acts as the user’s proxy.Some of the jondos may be corrupt and/or controlled by the adversary.Corrupt jondos may collaborate and share their obser-vations in an attempt to compromise the honest users’anonymity.Note,however, that all observations by corrupt group members are local.Each corrupt member may observe messages sent to it,but not messages transmitted on the links be-tween honest jondos.An honest crowd member has no way of determining whether6a particular jondo is honest or corrupt.The parameters of the system are the total number of members,the number of corrupt members,and the forwarding probability which is explained below.To participate in communication,all jondos must register with a special server which maintains membership information.Therefore,every member of the crowd knows identities of all other members.As part of the join procedure,the members establish pairwise encryption keys which are used to encrypt pairwise communi-cation,so the contents of the messages are secret from an external eavesdropper.Anonymity guarantees provided by Crowds are based on the path setup pro-tocol,which is described in the rest of this section.The path setup protocol is executed each time one of the crowd members wants to establish an anonymous connection to a Web server.Once a routing path through the crowd is established, all subsequent communication between the member and the Web server is routed along it.We will call one run of the path setup protocol a session.When crowd membership changes,the existing paths must be scrapped and a new protocol ses-sion must be executed in order to create a new random routing path through the crowd to the destination.Therefore,we’ll use terms path reformulation and proto-col session interchangeably.When a user wants to establish a connection with a Web server,its browser sends a request to the jondo running locally on her computer(we will call this jondo the initiator).Each request contains information about the intended desti-nation.Since the objective of Crowds is to protect the sender’s identity,it is not problematic that a corrupt router can learn the recipient’s identity.The initiator starts the process of creating a random path to the destination as follows: The initiator selects a crowd member at random(possibly itself),and for-wards the request to it,encrypted by the corresponding pairwise key.We’ll call the selected member the forwarder.The forwarderflips a biased coin.With probability,it delivers the request directly to the destination.With probability,it selects a crowd member at random(possibly itself)as the next forwarder in the path,and forwards the request to it,re-encrypted with the appropriate pairwise key.The next forwarder then repeats this step.Each forwarder maintains an identifier for the created path.If the same jondo appears in different positions on the same path,identifiers are different to avoid infinite loops.Each subsequent message from the initiator to the destination is routed along this path,i.e.,the paths are static—once established,they are not altered often.This is necessary to hinder corrupt members from linking multiple7paths originating from the same initiator,and using this information to compromise the initiator’s anonymity as described in section3.2.3.3.2Anonymity properties of CrowdsThe Crowds paper[RR98]describes several degrees of anonymity that may be provided by a communication system.Without using anonymizing techniques, none of the following properties are guaranteed on the Web since browser requests contain information about their source and destination in the clear.Beyond suspicion Even if the adversary can see evidence of a sent message,the real sender appears to be no more likely to have originated it than any other potential sender in the system.Probable innocence The real sender appears no more likely to be the originator of the message than to not be the originator,i.e.,the probability that the adversary observes the real sender as the source of the message is less thanupper bound on the probability of detection.If the sender is observed by the adversary,she can then plausibly argue that she has been routing someone else’s messages.The Crowds paper focuses on providing anonymity against local,possibly co-operating eavesdroppers,who can share their observations of communication in which they are involved as forwarders,but cannot observe communication involv-ing only honest members.We also limit our analysis to this case.3.2.1Anonymity for a single routeIt is proved in[RR98]that,for any given routing path,the path initiator in a crowd of members with forwarding probability has probable innocence against collaborating crowd members if the following inequality holds:(1)More formally,let be the event that at least one of the corrupt crowd members is selected for the path,and be the event that the path initiator appears in8the path immediately before a corrupt crowd member(i.e.,the adversary observes the real sender as the source of the messages routed along the path).Condition 1guarantees thatproving that,given multiple linked paths,the initiator appears more often as a sus-pect than a random crowd member.The automated analysis described in section6.1 confirms and quantifies this result.(The technical results of[Shm02]on which this paper is based had been developed independently of[Mal01]and[W ALS02],be-fore the latter was published).In general,[Mal01]and[W ALS02]conjecture that there can be no reliable anonymity method for peer-to-peer communication if in order to start a new communication session,the initiator must originate thefirst connection before any processing of the session commences.This implies that anonymity is impossible in a gossip-based system with corrupt routers in the ab-sence of decoy traffic.In section6.3,we show that,for any given number of observed paths,the adversary’s confidence in its observations increases with the size of the crowd.This result contradicts the intuitive notion that bigger crowds provide better anonymity guarantees.It was discovered by automated analysis.4Formal Model of CrowdsIn this section,we describe our probabilistic formal model of the Crowds system. Since there is no non-determinism in the protocol specification(see section3.1), the model is a simple discrete-time Markov chain as opposed to a Markov deci-sion process.In addition to modeling the behavior of the honest crowd members, we also formalize the adversary.The protocol does not aim to provide anonymity against global eavesdroppers.Therefore,it is sufficient to model the adversary as a coalition of corrupt crowd members who only have access to local communication channels,i.e.,they can only make observations about a path if one of them is se-lected as a forwarder.By the same token,it is not necessary to model cryptographic functions,since corrupt members know the keys used to encrypt peer-to-peer links in which they are one of the endpoints,and have no access to links that involve only honest members.The modeling technique presented in this section is applicable with minor mod-ifications to any probabilistic routing system.In each state of routing path construc-tion,the discrete probability distribution given by the protocol specification is used directly to define the probabilistic transition rule for choosing the next forwarder on the path,if any.If the protocol prescribes an upper bound on the length of the path(e.g.,Freenet[CSWH01]),the bound can be introduced as a system parameter as described in section4.2.3,with the corresponding increase in the size of the state space but no conceptual problems.Probabilistic model checking can then be used to check the validity of PCTL formulas representing properties of the system.In the general case,forwarder selection may be governed by non-deterministic10runCount goodbad lastSeen observelaunchnewstartrundeliver recordLast badObserve4.2Model of honest members4.2.1InitiationPath construction is initiated as follows(syntax of PRISM is described in section 2.2):[]launch->runCount’=TotalRuns&new’=true&launch’=false;[]new&(runCount>0)->(runCount’=runCount-1)&new’=false&start’=true;[]start->lastSeen’=0&deliver’=false&run’=true&start’=false;4.2.2Forwarder selectionThe initiator(i.e.,thefirst crowd member on the path,the one whose identity must be protected)randomly chooses thefirst forwarder from among all group mem-bers.We assume that all group members have an equal probability of being chosen, but the technique can support any discrete probability distribution for choosing for-warders.Forwarder selection is a single step of the protocol,but we model it as two probabilistic state transitions.Thefirst determines whether the selected forwarder is honest or corrupt,the second determines the forwarder’s identity.The randomly selected forwarder is corrupt with probability badCbe next on the path.Any of the honest crowd members can be selected as the forwarder with equal probability.To illustrate,for a crowd with10honest members,the following transition models the second step of forwarder selection: []recordLast&CrowdSize=10->0.1:lastSeen’=0&run’=true&recordLast’=false+0.1:lastSeen’=1&run’=true&recordLast’=false+...0.1:lastSeen’=9&run’=true&recordLast’=false;According to the protocol,each honest crowd member must decide whether to continue building the path byflipping a biased coin.With probability,the forwarder selection transition is enabled again and path construction continues, and with probability the path is terminated at the current forwarder,and all requests arriving from the initiator along the path will be delivered directly to the recipient.[](good&!deliver&run)->//Continue path constructionPF:good’=false+//Terminate path constructionnotPF:deliver’=true;The specification of the Crowds system imposes no upper bound on the length of the path.Moreover,the forwarders are not permitted to know their relative position on the path.Note,however,that the amount of information about the initiator that can be extracted by the adversary from any path,or anyfinite number of paths,isfinite(see sections4.3and4.5).In systems such as Freenet[CSWH01],requests have a hops-to-live counter to prevent infinite paths,except with very small probability.To model this counter,we may introduce an additional state variable pIndex that keeps track of the length of the path constructed so far.The path construction transition is then coded as follows://Example with Hops-To-Live//(NOT CROWDS)////Forward with prob.PF,else deliver13[](good&!deliver&run&pIndex<MaxPath)->PF:good’=false&pIndex’=pIndex+1+notPF:deliver’=true;//Terminate if reached MaxPath,//but sometimes not//(to confuse adversary)[](good&!deliver&run&pIndex=MaxPath)->smallP:good’=false+largeP:deliver’=true;Introduction of pIndex obviously results in exponential state space explosion, decreasing the maximum system size for which model checking is feasible.4.2.4Transition matrix for honest membersTo summarize the state space of the discrete-time Markov chain representing cor-rect behavior of protocol participants(i.e.,the state space induced by the abovetransitions),let be the state in which links of the th routing path from the initiator have already been constructed,and assume that are the honestforwarders selected for the path.Let be the state in which path constructionhas terminated with as thefinal path,and let be an auxiliary state. Then,given the set of honest crowd members s.t.,the transi-tion matrix is such that,,(see section4.2.2),i.e.,the probability of selecting the adversary is equal to the cumulative probability of selecting some corrupt member.14This abstraction does not limit the class of attacks that can be discovered using the approach proposed in this paper.Any attack found in the model where indi-vidual corrupt members are kept separate will be found in the model where their capabilities are combined in a single worst-case adversary.The reason for this is that every observation made by one of the corrupt members in the model with separate corrupt members will be made by the adversary in the model where their capabilities are combined.The amount of information available to the worst-case adversary and,consequently,the inferences that can be made from it are at least as large as those available to any individual corrupt member or a subset thereof.In the adversary model of[RR98],each corrupt member can only observe its local network.Therefore,it only learns the identity of the crowd member imme-diately preceding it on the path.We model this by having the corrupt member read the value of the lastSeen variable,and record its observations.This cor-responds to reading the source IP address of the messages arriving along the path. For example,for a crowd of size10,the transition is as follows:[]lastSeen=0&badObserve->observe0’=observe0+1&deliver’=true&run’=true&badObserve’=false;...[]lastSeen=9&badObserve->observe9’=observe9+1&deliver’=true&run’=true&badObserve’=false;The counters observe are persistent,i.e.,they are not reset for each session of the path setup protocol.This allows the adversary to accumulate observations over several path reformulations.We assume that the adversary can detect when two paths originate from the same member whose identity is unknown(see sec-tion3.2.2).The adversary is only interested in learning the identity of thefirst crowd mem-ber in the path.Continuing path construction after one of the corrupt members has been selected as a forwarder does not provide the adversary with any new infor-mation.This is a very important property since it helps keep the model of the adversaryfinite.Even though there is no bound on the length of the path,at most one observation per path is useful to the adversary.To simplify the model,we as-sume that the path terminates as soon as it reaches a corrupt member(modeled by deliver’=true in the transition above).This is done to shorten the average path length without decreasing the power of the adversary.15Each forwarder is supposed toflip a biased coin to decide whether to terminate the path,but the coinflips are local to the forwarder and cannot be observed by other members.Therefore,honest members cannot detect without cooperation that corrupt members always terminate paths.In any case,corrupt members can make their observable behavior indistinguishable from that of the honest members by continuing the path with probability as described in section4.2.3,even though this yields no additional information to the adversary.4.4Multiple pathsThe discrete-time Markov chain defined in sections4.2and4.3models construc-tion of a single path through the crowd.As explained in section3.2.2,paths have to be reformulated periodically.The decision to rebuild the path is typically made according to a pre-determined schedule,e.g.,hourly,daily,or once enough new members have asked to join the crowd.For the purposes of our analysis,we sim-ply assume that paths are reformulated somefinite number of times(determined by the system parameter=TotalRuns).We analyze anonymity properties provided by Crowds after successive path reformulations by considering the state space produced by successive execu-tions of the path construction protocol described in section4.2.As explained in section4.3,the adversary is permitted to combine its observations of some or all of the paths that have been constructed(the adversary only observes the paths for which some corrupt member was selected as one of the forwarders).The adversary may then use this information to infer the path initiator’s identity.Because for-warder selection is probabilistic,the adversary’s ability to collect enough informa-tion to successfully identify the initiator can only be characterized probabilistically, as explained in section5.4.5Finiteness of the adversary’s state spaceThe state space of the honest members defined by the transition matrix of sec-tion4.2.4is infinite since there is no a priori upper bound on the length of each path.Corrupt members,however,even if they collaborate,can make at most one observation per path,as explained in section4.3.As long as the number of path reformulations is bounded(see section4.4),only afinite number of paths will be constructed and the adversary will be able to make only afinite number of observa-tions.Therefore,the adversary only needsfinite memory and the adversary’s state space isfinite.In general,anonymity is violated if the adversary has a high probability of making a certain observation(see section5).Tofind out whether Crowds satisfies16。

概率统计中英术语对照表Probability Theory概率论Trial 试验intersection交union 并frequency 频率difference 差additivity 可加性complementation 对立contain 包含equivalent 等价mean 均值convolution [，kɔnvə'lu：ʃən］卷积variance 方差covariance 协方差correlated 相关standard deviation 标准差Random experiment 随机试验random event 随机事件certain event 必然事件impossible event 不可能事件elementary/fundamental event 基本事件the probability of event A 事件的概率sample point 样本点sample space 样本空间Classical probability 古典概型geometric probability 几何概型conditional probability 条件概型total probability 全概率formula of multiplication 乘法公式pair wise independence 两两相互独立Distribution function 分布函数discrete random variable 离散型随机变量two—point distribution （0-1)分布binomial distribution 二次分布Poisson distribution 泊松分布hyper geometric distribution 超几何分布Continuous random variable 连续型随机变量probability density function 概率密度函数uniform distribution 均匀分布Exponential distribution 指数分布standard normal distribution 标准正态分布Cauchy distribution 柯西分布n—dimensional random vector n维随机变量bivariate random variable [bai’vεəriət] 二维随机变量joint distribution function 联合分布函数bivariate discrete random variable 二维离散型随机变量joint distribution law 联合分布律bivariate continuous random variable 二维连续型随机变量joint probability density function 联合概率密度函数bivariate normal distribution 二维正态分布marginal distribution function 边缘分布函数marginal distribution law 边缘分布律marginal probability density function 边缘概率密度函数conditional distribution function 条件分布函数conditional probability density function 条件概率密度函数mathematical expectation 数学期望standard random variable 标准随机变量moment generating function 矩母函数characteristic function 特征函数positive correlated 正相关mixed moment 混合矩negative correlated 负相关mixed central moment 混合中心矩moment of order k about the origin 阶原点矩central moment of order k 阶中心矩covariance matrix 协方差矩阵convergence in probability 依概率收敛Bernouli large numbers law 伯努力大数定律Mathematical statistics数理统计individuality 个体population 总体sample size 样本大小simple random sample 随机样本efficiency有效statistic 统计量sample mean 样本均值sample variance样本方差sample standard deviation 样本标准差sample central moment of order k样本的阶中心矩skewness ［'skju：nis］偏度coefficient of variation 变异系数order statistics 次序统计量degrees or freedom 自由度sampling distribution 抽样分布parameter estimation 参数估计point estimation 点估计estimator 估计量estimate 估计值likelihood function 似然函数method of moment 矩估计法unbiased estimator 无偏估计量maximum likelihood estimate 最大似然估计system of likelihood equations似然方程组consistent estimator 一致估计量confidence level 置信水平confidence interval 置信区间upper confidence limit 置信上限parametric hypothesis 参数估计non-parametric hypothesis 非参数估计alternative hypothesis 备择假设null hypothesis 零假设Significance level 显著性水平rejection region 拒绝域acceptance region 接受域test for goodness of fit 拟和优度检验contingency table 列连表regression function 回归函数regression equation 回归方程linear regression model 线形回归模型regression coefficient 回归系数normal linear model 正态线形模型least squares estimate 最小二乘估计method of least squares 最小二乘法sum of squares of residual 残差平方和sum of squares of regression 回归平方和sum of residual 剩余平方和total sum of squares of deviations 总变差平方和coefficient of determination 判定系数point interval 点预测prediction interval 预测区间one—way analysis variance 单因素方差分析two—way analysis of variance 双因素方差分析interaction effect 交互效应。

On the modelling of multiphase turbulent flowsfor environmental and hydrodynamic applicationsDjamel LakehalInstitute of Energy Technology,ETH Zurich,ETH-Zentrum/CLT2,CH-8092Zurich,SwitzerlandReceived 1November 2000;received in revised form 24November 2001AbstractThe paper examines a selection of well-established prediction methods employed for the modelling of multiphase turbulent flows presented in typical environmental and hydrodynamic applications.The main objective is to provide a basic understanding of the subject with a deliberate intention to simplifying the presentation.Turbulence is approached on the basis of the conventional one-point closure context.The experience gathered by the author and by others with various predictive strategies all based on the Eule-rian–Eulerian (field description)and the Eulerian–Lagrangian methods are discussed and summarized;the goals,limitations,and required developments are described.Typical applications of each calculation method are presented,in which the interaction between the transported dispersed-phase and the field turbulence is treated on the basis of both one-way and two-way coupling.The case studies in question include aerosol production and transport over the oceans,pollutant dispersion in the atmospheric surface layer,hydrometeor impact on urban canopies,sedimentation of active sludge in secondary water clarifiers,and mixing and circulation within confined bubble plumes.Analysis of the various models reveals that for most of the reported applications the Reynolds averaged Navier–Stokes approach is inherently ill-posed and should be transcended by the promising large-eddy simulation concept.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Turbulent-Flow;Droplets;Dilute Suspensions;Bubbles;Particles1.IntroductionThis paper is written in the spirit of an overview of different multiphase modelling methods,with emphasis on the practical motivations for certain selected applications and theexpected International Journal of Multiphase Flow 28(2002)823–/locate/ijmulflowE-mail address:lakehal@iet.mavt.ethz.ch (kehal).0301-9322/02/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S0301-9322(01)00086-6824 kehal/International Journal of Multiphase Flow28(2002)823–863returns from computational analyses.The role of simulation strategies in the prediction and design processes is also discussed.The deliberate choice of applications is motivated by the variety of solution methods applied in each case;we aim at discussing them in the sections to follow.The methods are discussed in a comprehensive and simplistic way based on known ideas and prin-ciples.An overview of the state-of-the-art is presented in treating the various subjects using the Eulerian–Eulerian method in both the one-fluid and two-fluid(interpenetrating media)formu-lations,as well as by the Eulerian–Lagrangian variant.Since theflows considered herein are presently out of reach of direct and large-eddy simulation approaches(DNS and LES),we es-sentially focus on the implications of turbulence modelling(by reference to the Reynolds Aver-aged Navier–Stokes Equations,RANS)in the various computational frameworks discussed in this paper,and the way this conventional approach could be improved on by more elaborate ones. In support of this,practical case studies typical of environmental and hydrodynamic applications are presented.Apart from the applications with reference to hydrodynamic applications CS5and CS6,the interaction between the transported phase and thefield turbulence is treated in all other cases on the basis of one-way coupling.Note,too,that the paper does not deal with simplified simulation approaches,for example,with the so-called Gaussian models employed for atmospheric dispersion modelling(cf.Hangan,1999).Eulerian–Eulerian and Eulerian–Lagrangian methods have been extensively used to simulate particle dispersion.Depending on the nature of the case studies in question it is possible to employ a specific form of each of the two solution methods.But prior to that,it is worth highlighting the main differences between these two strategies,i.e.the Eulerian–Eulerian vs.the Eulerian–Lagrangian methods.The choice between these two procedures is in essence problem-dependent. The Eulerian orfield description methodology is commonly adopted for the prediction of in-terpenetrating media situations,including both highly particle loaded systems such asfluidized beds,dilute particle-ladenflows as in the case of dilute suspensions of aerosols,droplets and particles,and gas–liquid mixtures such as bubblyflows.This approach can be employed in two distinctive forms:The one-fluid formulation and the two-fluid approach.In thefirst approach, generally employed in the form of a one-field description of highly-loaded or dilute suspensions formed by concentrations of droplets and particles,the particle concentrations are assumed to have some characteristics of a continuous phase(e.g.the local concentration)and,when ap-propriate,some of a dispersed phase(e.g.the inertial slip).In other words,the method essentially consists in solving an extra conservation law for the concentration of particles or for their mean spatial density.Modifications of the transport equations are also needed to consider buoyancy forces whenever the two phases exhibit differences in density due to the presence of a heavier dispersed phase(e.g.sedimentation problems,snow avalanches,etc.),or when the carrier phase features thermal stratification as is often the case in geophysicalflows(e.g.thermal fronts,at-mospheric surface layer,etc.).In addition,the transport equations must include interfacial ex-change laws to account for mass transfer whenever the dispersed phase evaporates or condenses, e.g.evaporative marine droplets over the ocean.The combination of all these processes leads to a system of equations with a multitude of closure laws.In this respect,the closure relationships for the turbulent concentration or heatflux arising from Reynolds averaging conceptually follow the manner in which the mechanical turbulent stresses are approximated.This important issue is examined herein,too,in particular when the closure law for turbulence is a two-equation based approach,in which the buoyancy-induced contributions are represented in terms of additionalkehal/International Journal of Multiphase Flow28(2002)823–863825 source terms in the turbulence equations with some adjustable coefficients.In the second ap-proach,also known as the six equation model approach,the phases are treated as two inter-penetrating continua evolving within a single system:Each point in the mixture is occupied simultaneously(in variable proportions)by both phases.Each phase is then governed by its own conservation and constitutive equations;these are then coupled through interphase in-teraction properties.More precisely,in contrast to the one-fluid formulation,convective and diffusive processes are explicitly taken into account in each of the two phases.For example, mixtures of two immisciblefluids such as air bubbles in water cannot be considered as mixtures of dilute suspensions evolving within a liquid phase;they have to be simulated via the two-fluid approach.In the Lagrangian reference frame individual particles or clouds of particles are treated in a discrete way.The reference frame moves with the particles,and the instantaneous location of each particle is determined by reference to its origin and the time grangian methods em-ployed for particle tracking are conventionally based on the equation of motion for spherical particles at high-Reynolds numbers,as given by Clift et al.(1978),also known as Basset–Bous-sinesq–Oseen(BBO)equation(cf.Crowe et al.,1996).The dispersed phases are assumed to be heavy and smaller than the Kolmogorov microscales.As a prerequisite computational sequence theflowfield has to be known since tracking individual particles directly relies on its properties, i.e.velocityfield and turbulence statistics.In practical applications theflowfield is modelled by use of RANS,whereas the resort to DNS(Squires and Eaton,1990;Mosyak and Hetsroni,1999; Ahmed and Elghobashi,2000;Sawford and Yeung,2001)or LES(Yeh and Lei,1991;Wang and Squires,1996;Armenio et al.,1999;Boivin et al.,2000;Okong’o and Bellan,2000)is still confined to research studies dealing for example with turbulence–particle interactions.A variety of models accounting for the effects of turbulence on particle motion are available in the literature.A critical review of the variants employed for heavy particles in atmospheric tur-bulence is proposed by Wilson(2000).Another interesting review is that of Shirolkar et al.(1996) focusing on models used for dispersion in combustion problems.On the upper level of classifi-cation the models differ depending on whether they are applied to passive tracers(see,for ex-ample,Thomson,1987)or to inertial particles(IP).The present work places emphasis on the second class of models only.A subcategory of IP dispersion models is an approach based on a Markov chain process,which is afinite discrete form of the Langevin equation supposed to model thefluctuating particle velocities in a purely stochastic way.This equation wasfirst employed for the study of Brownian motion by Wang and Uhlenbeck(1945),and was only later applied to describe dispersion in homogeneous turbulence by Lin and Reid(1962).The other often employed random-flight algorithms treated in this paper are based on the generation of non-miscible(un-correlated)random eddies,in which particle trajectories are purely deterministic.These are known as eddy interaction models(EIM),perhaps initially proposed by Gossman and Ioannides(1981). Here it is assumed that individual particles are subject to a series of interactions with randomly sampled eddies;the particle velocity remains constant during each particle–eddy interaction time, during which the eddy velocity remains unchanged.The difference between the two methods is that Markov chain type models provide a continuousfluctuating velocityfield,whereas in EIMs thefluctuating velocity changes only when individual particles encounter a new eddy.This is the reason why MacInnes and Bracco(1992)refer to thefirst class as continuous random walk models and to the second as discontinuous random walk models.826 kehal/International Journal of Multiphase Flow28(2002)823–863The Eulerian–Lagrangian formalism thus amounts to the combination of two separate ap-proaches:The Eulerian part delivers theflowfield with its turbulent statistical properties,and the Lagrangian module employs these data to track individual particles.The parametrization of particle dispersion is therefore intimately tied to the dynamics offield turbulence.This is,of course,the case for dispersed phases smaller than the Kolmogorov micro-scales,whose inter-action with turbulence is commonly termed one-way coupling by reference to the weak effect of particle momentum on turbulence.These two methods are here discussed in their original modelling context and in the LES framework.Still,the Eulerian approach for simulating turbulent dispersion has its own advantages as compared to Lagrangian methods.Forflow laden with a large amount of particles the quanti-tative description of the variation in particle concentration is much simpler by means of the Eulerian method since,for the same purpose,statistical sampling is required with the Lagrangian grangian methods may also face problems whenever the cloud of particles tracked is larger than thefluid parcel over which volume averaging is performed.And apart from that,the Eulerian approach allows both phases to be computed over a single grid,whereas the Lagrangian methods require the interpolation of quantities between thefixed grid nodes and the local position of particles.However,treating particles via the Lagrangian formalism is in essence natural be-cause their motion is tracked as they move through theflowfield,which preserves their actual non-continuum behaviour and accounts for their history effects in a natural way.In addition,if attention is now redirected towards turbulence modelling,the Lagrangian approach holds a fundamental advantage over the Eulerian one in the sense that it does not require closure as-sumptions for turbulence correlations of tracer concentration and velocityfluctuations.More about the relative merits of these approaches is given by Durst et al.(1994)and Mostafa and Mongia(1987).The present paper is structured as follows:Selected applications arefirst introduced to grad-ually highlight the expected results of computational analyses.These selected case studies(see Section2)are referred to as CS1,CS2;...;CS6,respectively.Based on an extended literature survey the solution procedures employed so far in each case are introduced in Section3.Section4 is devoted to computational examples,where the solution methods are examined in the light of calculation results.Finally,key remarks are made in connection with computational strategies and turbulence models together with the presentation of an outlook on future developments. 2.Typical applications in environmental and hydrodynamic research2.1.Pollutant transport in the urban canopyThis type of study enters within the large framework of computational wind engineering(CWE), a discipline that has been progressing since the late1970s,boosted by its potential to overcome the limitations of earlier simplified physical models such as the Gaussian models evoked previously in Section1.Pollutant dispersion within the atmospheric surface layer encompasses a variety of aspects of vital interest that need to be explored:For example,predicting the transport of con-taminants from hazardous releases,analyzing the traffic-induced dispersion(Rafailidis,2000; Kastner-Klein et al.,1997;Meroney et al.,1999),and studying the effects of neighbouringkehal/International Journal of Multiphase Flow28(2002)823–863827 building topography on domestic gas-releases(Cowan et al.,1997;Delaunay et al.,1997;Hangan, 1999;Castro et al.,1999).Without considering the thorny question of predicting the behaviour of hazardous gas releases(Chernobyl type of tragedies)to the atmosphere,we could evoke a similar problem that draws less attention,but may nevertheless have an impact on daily life:The quality of air inside a single or a group of buildings and its relation to external aerodynamic conditions. Theseflow conditions can,for example,connect an external source of pollutants(chimneys re-leasing exhaust gases from centralized heating devices)with fresh-air admission(windows,etc.) which could in turn be contaminated.Although recent contributions to thefield have taken further steps by dealing with dispersion around complex(several buildings)configurations(e.g. Hangan,1999;Castro et al.,1999),the example selected here consists of the three-dimensional prediction of gas dispersion around an isolated,generic building model placed within a simulated urban canopy studied by Delaunay et al.(1997).The aim of this investigation was to provide architects and civil engineers with sufficient indications regarding theflow structure to help them design a group of buildings in which the recirculation of contaminants through fresh air admis-sions can be minimized.2.2.Car-induced pollution in urban areasCar-induced pollution in urban areas is a serious health concern,in particular within cities1 featuring many street canyons.Most often building aggregates placed within the atmospheric boundary layer may act as artificial obstacles to the wind and cause stagnant conditions.Ex-perimental and numerical studies of such problems aim in general at predicting the time evolution of pollutant concentrations and their implications for the comfort of pedestrians as a function of geometry and pollutant doses(Mestayer et al.,1993;Sini et al.,1996;Moussiopoulos et al.,1998; Rafailidis,2000).Previous studies showed the number and arrangement of vortex structures within the street canyon to strongly influence vertical exchange rates.It has also been shown that differential heating of street surfaces can grossly influence the capability of theflow to transport and exchange pollutants(Sini et al.,1996).In particular,differential heating could also shift the in-streetflow structure from a single-vortexflow to aflow with several counter-rotating vortices. We report here on the results of a recent simulation,conducted by Theodoridis and Moussio-poulos(2000),of theflow and contaminant transport within a typical street-canyon configuration studied experimentally by Rafailidis(2000).In contrast to earlier studies,a number of interesting and original issues typical for this type of problem have been dealt with by the authors,focusing for example on the determination of the subsequent production of NO x and ozone.2.3.Dispersion of marine dropletsThe fundamental issues of surface layer meteorology have been reviewed by many specialists, e.g.H€o gstro€m(1996).More specifically with respect to marine climatology Smith et al.(1996) have made available a complete overview leading to a better understanding of air–sea interaction.1Some Mediterranean cities suffer today because their developers opted in the past for street-canyon type conglomerations seeking for shadow(e.g.Medina and Casbah in North Africa).828 kehal/International Journal of Multiphase Flow28(2002)823–863The authors review the progress achieved in the study of air–sea interaction over the past three decades and its role in the modelling of the coupled system of ocean and atmosphere.Melville (1996)placed emphasis on the role of surface–wave breaking in air–sea interaction and the subsequent impact of aerosol production and transport.More precisely,it is the impact of marine droplets and aerosols on the heatflux balance that represents the key point in this branch,as discussed by Smith et al.(1996)and Fairall et al.(2000).Indeed,the evaporative droplets are known to distort the normal sensible/latent heatflux balance,whereas in their absence the entire surface moistureflux produces a latent heat loss by the ocean leading to an increase in the salinity at the surface.The central issue here is to understand the contribution of sea spray droplets to the transfer of moisture and latent heat from the sea to the atmosphere.The case study reported in the present review refers to the two-dimensional simulation of the turbulent transport and evapo-ration of droplets ejected by bursting bubbles within various simulated air–sea boundary layers (Edson and Fairall,1994;Edson et al.,1996).An integrated Eulerian–Lagrangian strategy was employed to compute theflow,temperature and moisturefields,and the trajectory of each ejected droplet;in particular,the particle trajectories were computed by means of a Markov chain based on the discretization of the Langevin equation for dispersed particles,modified to account for the effects of turbulence,gravity and inertia.This type of Lagrangian technique is presently being employed within the LES framework for other related subjects such as the prediction of pollution dispersion in the atmosphere(e.g.Sorbjan and Uliasz,1999).Studying the generation,transfer mechanisms and aerosol deposition over the ocean has also been migrating gradually from RANS (e.g.Ling et al.,1980;Burk,1984)to LES(e.g.Glendening and Burk,1992),although the Eulerian description is still preferred to the Lagrangian one.2.4.Impacting hydrometeors on buildingsThe deterioration experienced by buildings and monuments is caused in part by the direct impact of hydrometeors and subsequent deposition of moisture on the surface.In contrast to the effects caused by the spectacular impact of heavy hydrometeors such as hail,the more subtle degradations caused by moisture deposited by rain,snow and fog are less well assessed.In these instances,the deposited moisture can cause mechanical disruptions by freezing withinfissures or by actually dissolving the materials.In addition,atmospheric pollutants dissolved or suspended in water droplets can be carried to the surface.Once these pollutants have been deposited on the surface,capillarity can transport the moisture and pollutants into the interior of porous materials. This often results in chemical transformations and deterioration deep within these structures.For this class offlow the literature reports on a very limited number of computational investigations; the earlier ones have adopted simplified formulations relating the intensity of driving rains to the free-falling rain intensity and wind speed(cy,1977;Beguin,1985;Hilaire and Savina, 1989).More elaborate strategies based upon the Eulerian–Lagrangian approach appeared only recently(e.g.Choi,1994;Lakehal et al.,1995;Sankaran and Paterson,1997;Karagiozis et al., 1997).However,the only contribution in thisfield combining in a single model the effects of turbulence,gravity,and inertia is due to Lakehal et al.(1995).The example reported here(from these authors’work)centers around the prediction of wind-driven raindrop trajectories inside a two-dimensional street canyon;thefinal aim was to evaluate the impacting water rate on the facades.The solution procedure was again based on an integrated Eulerian–Lagrangian method,kehal/International Journal of Multiphase Flow28(2002)823–863829 and the particle trajectories were computed by means of a Markov chain modified to account for the effects of turbulence,gravity,and inertia.2.5.Sedimentation in water clarifiersSettling and sedimentation phenomena are complex processes,repeatedly evoked in hydro-dynamic applications;their presence within wastewater treatment plants as the most important unit operations is one example among various others.It is well known that gravity-induced sedimentation and the subsequent thickening process may be subdivided into four different types: Discrete particle settling,flocculent settling,hindered settling,and compression(see,for example, Karl and Wells(1999)for classification).The thickening process in water clarifiers occurs most often as a combination of the last three forms,which poses challenges to the modeler.The recent critical review of Parker et al.(2001)reports on the important design aspects properly applicable to clarifier technology that need to be observed.Investigating this type offlow is dictated by design interests:It is aimed at helping to design secondary clarifiers,whose efficiency is such that the overall performance of the entire wastewater treatment does not require post operations (Krebs et al.,1996).An intensive scientific effort has recently been made in order to understand this type offlow,and various numerical models have been developed for the purpose,most of which are based on two-equation turbulence models describing theflow pattern and sediment-induced density currents(Lyn et al.,1992;Zhou et al.,1992;Zhou and McCorquodale,1992; Szalai et al.,1994;Vitasovic et al.,1997;Armbruster et al.,2001).Apart from Lyn et al.(1992)the above cited works did not consider particle decompositions and were thereby based on the de-termination of an average settling velocity for suspended particles.Jin et al.(2000)have recently taken a step ahead by proposing a one-dimensional model for non-uniform sediment transport capable of handlingflocculation,coagulation,andfiltration.This type offlow raises additional complexities as compared to pollutant dispersion problems.Buoyancy effects may be more im-portant than those induced by turbulent stresses.The transported phase settles at a velocity strongly influenced by its concentration.Finally,the non-Newtonian behaviour of the activated sludge requires appropriate definition of its rheological properties.The results of modelling the sedimentation of a sludge blanket in a circular,center-fed secondary clarifier with inclined bottom and central withdrawal are presented.Axisymmetry is assumed and theflow and settling processes (with variable settling velocities)are computed in a radial section.The non-Newtonian behaviour of the sludge is also taken into account.2.6.Bubble plumesThree-dimensional mixing of multiphaseflows may occur in industrial applications as well as in environment protection processes.Industrial applications include gas stirring by liquid metal ladles in several metallurgical processes,or venting of vapour mixtures to liquid pools in chemical and nuclear reactors.Bubble plumes may also be involved in environment protection problems such as the aeration of lakes,mixing of stagnant water and,generally,de-stratification of water reservoirs.For all these applications the basic need is to determine the currents induced by the gaseous phase evolving in the surrounding liquid and thereby to establish the consequent mixing and partition of energy,or species concentration in the body of the liquid.Here the computationalmethodology to be followed is the two-fluid approach of Ishii (1975)evoked previously.However,more important is the fact that predicting bubbly flows cannot be achieved without suitable models capable of correctly representing interphase momentum transfer mechanisms and tur-bulence modulation induced by the bubbles.For the latter issue,various models have been published in the past,though all of them resort to a single-phase two-equation turbulence model modified to account for these exchange mechanisms (Malin and Spalding,1984).This includes the effect of bubble migration through the liquid (Simonin and Viollet,1988),and more often the interactions between the eddies and the dispersed phase via what is known as turbulent dispersion models (see,for example,Moraga et al.,2001,for a recent review).In practice the idea of tur-bulence dispersion induced by the dispersed phase has most often been reflected in terms of a superposition of the shear-induced and bubble-induced stress tensors in the equations for the liquid phase;the latter being constructed on the basis of scaling arguments.The example reported here consists of the prediction of a confined bubble plume studied experimentally by Anagbo and Brimacombe (1990).The numerical results reported here were obtained by Smith and Milelli (1998),who made a critical assessment of various models that have so far been advanced to support modelling of bubbly flows.3.Outline of the solution methods3.1.The Eulerian–Eulerian one-fluid approach3.1.1.BackgroundTo handle the transport of a dilute continuum acting as a passive scalar within a turbulent flow one generally resorts to the so-called Eulerian–Eulerian one-field formalism.In this approach the particle concentrations are assumed to have some characteristics of a continuous phase and some characteristics of a dispersed phase via the inertial slip,when appropriate (e.g.when the particles settle).An inherent concept in this formalism is the assumption that the transported (passive or active scalar)phase obeys the same Navier–Stokes equation governing the mean flow,since there is no interfacial or interphase exchange processes to account for.However,in general,the pres-ence of heavy particles with non-negligible inertia raises simulation problems not yet totally re-solved,such as the lack of appropriate boundary conditions.3.1.2.The transport equationsThe Eulerian–Eulerian approach is essentially based on the solution of the Reynolds Aver-aged 2Navier–Stokes equations (RANS)governing the motion of an incompressible carrier phase (cf.Hinze,1975),together with a transport equation for the dilute phase:@j U j ¼0;ð1ÞD t U i ¼À1=q w @i p þ@j r ij ÀÀs ij Áþ~f ;ð2Þ2Hereinafter each barred symbol represents an ensemble average,while primed letters denote the fluctuating counterparts.830 kehal /International Journal of Multiphase Flow 28(2002)823–863D t C¼D r2CÀ@j u0j c0:ð3ÞIn the above equations,D t¼@tþuÁr stands for the substantial derivative,r ij the viscous stress, q w the mean density of thefluid at a reference state,p the pressure,D the molecular diffusivity coefficient,and s ij u0i u0j the Reynolds stress tensor that requires a model.The buoyancy force term,~f,is effective only in thermally stratifiedflows and/or when the dispersed phase is appre-ciably heavier than the carrierfluid.For instance,case studies CS1,CS2and CS5were treated on the basis of Eqs.(1)–(3),with~f¼0for thefirst two examples.3.1.3.Thermally stratifiedflows with mass transferThe presence of an evaporative medium(e.g.marine droplets)within a thermally stratifiedflow (e.g.a marine sublayer)can be treated on the basis of the RANS equations(1)and(2),using theBoussinesq approximation,in which the buoyancy force term now reads~f¼Àg iðH vÀH rv Þ=H rv.This term is induced by the difference between the instantaneous virtual potential temperature3H v and that of the reference state H rv .According to Stull(1988),the presence of water dropletsrequires the virtual potential temperature to conform to the following relationH v¼H1½þ0:61q VÀq LÀq D ;ð5Þin which q L is the specific humidity of the liquid,and q D the contribution to the total specific humidity from the droplets to be determined by integrating the local droplet volume concen-tration.At high-Reynolds numbers the thermal and moisturefields are represented by the Rey-nolds averaged transport equations for the potential temperature H and the total specific humidity denoted:Q¼q Vþq LD t H¼À@j u0j hþLE S q=C p;ð6ÞD t Q¼À@j u0j qþS q:ð7ÞIn these equations C p stands for the specific heat at constant pressure,LE for the latent heat of vaporization,and S q for the total evaporation rate.The reader can refer to Pruppacher and Klett (1978)for more details on the modelling of the source term S q.Note that the molecular diffusion contributions in the above equations have been dropped,since only high-Re numberflows are of interest in these studies.The turbulentfluxes u0j q and u0j h appearing in Eqs.(6)and(7)need to be modelled,too,as will be discussed later.3.1.4.Density-induced stratification in non-newtonian mixturesIn certain class offlow the dense phase deposited over an impermeable surface forms a structure behaving like a non-Newtonian material.This is the case for biological material settling in water clarifiers.In a similar context,with use of the Boussinesq approximation the momentum equations take the form of Eqs.(1)and(2),but the buoyancy force is now driven by the difference 3The equation of state for‘‘humid air’’,a mixture of dry air and water vapor,is:p¼q a R a Hþq V R V H¼q a R a H v;H v¼H½1þ0:61q V ;ð4Þwhere H stands for the potential temperature,q V for the specific humidity of water vapor,and0.61is the explicit value ofðR VÀR aÞ=R a.kehal/International Journal of Multiphase Flow28(2002)823–863831。

Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOV A （analysis of variance）, 方差分析ANOV A Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 一般线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOV A , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性V ARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转V olume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换。

31. There are problems associated with providing a bad experience to a customer. The customercan turn on you. A term associated with this is:A) Defector B) T erroristC) Marginal D) Piranha39. If the project internal rate of return is estimated to be 11% andI) The company cost of capital is 10%II) The company cost of capital is 12%III) Funds are limited and another project will yield 14%IV) Funds are unlimited and another project will yield 14%A) I and III are true B) I and IV are trueC) II and III are true D) II and IV are true48. Advantages of computer software driven project management methods include:I) Ability to analyze "What-if" optionsII) Automatic calculation of the critical pathIII) The effects of actual results on the project are knownIV) Training requirements are minimalA) I, II and III only B) II, III and IV onlyC) II and III only D) I, II, III and IV75. In the six sigma define step, the critical to quality tree is used by the project team. the variouslevels of the tree are determined EXCEPT for:A) The exact metrics for the customerB) The needs of the customerC) The basic drivers for the customerD) The potential third level CTQ metrics79. If you are reading this question you are a customer of QCI, Identify QCI process outputelements from the list below:I) Binder suppliersII) Solution textsIII) AuthorsIV) Paper suppliersV) LibrariesVI) Question CDsVII) Study PrimersA) II, VI and VII only B) I and IV onlyC) III and V only D) II, III and VII only84. The probability of Steven passing his math course is 0.7, the probability of Steve passing hishistory class is 0.8. if the probability of Steve passing both course is 0.56, what is theprobability Steve will pass either math or history?A) 0 B) 1C) 0.99 D) 0.8485. The hypergeometric distribution should be used instead of the binomial distribution when:A) There are more than 2 outcomes on a single trialB) Each trial is independentC) Sampling does not involve replacementD) There is a fixed number of trials89. When conducting a process capability study consistent with PPAP requirements, which of thefollowing is mandatory?A) A submission of related control chart dataB) A selected characteristic that is controllableC) Data collected from a significant production run of 300 or more consecutive piecesD) A demonstrated 5 sigma capability90. For the Weibull distribution, as the scale parameter decreases:A) The Weibull is equivalent to the exponentialB) The location parameter approaches zeroC) The probability density function stretches to the rightD) The probability density function is compressed to the left101. Tremendous advances had been made in the quality of an electronic component, produced in quantities of one million units per year. last year only six defectives were discovered.A further improvement was mad. The plant manager asked the master black belt to run a100,000unit trial to determine with 95% confidence if the rate had been lowered by 2 DPMO.What was the master black belt response?A) It will take too much timeB) It can't be done with 100,000unitsC) It will only be proven if 0 defectives are foundD) We must look for a larger improvement for testing purposes103. In an experiment designed to compare two different ways of measuring a given quantity, it was desired to test the null hypothesis that the means were equal at the 0.05 level ofsignificance. A sample of five parts was measured by method I and a sample of seven parts with method II. A t-ratio of 2.179 was obtained. we should:A) Reject the null hypothesisB) Fail to reject the null hypothesisC) Conclude that X1 is significantly greater than X2D) Conclude that we must know the sample means in order to answer the question107. A study was made on the effects of several health additives for a number of elite runners.the results of a One-Way ANOVA are presented in the following table. how manysubjects(runners) were in the study?Source Df Sum of Squares MSBetween 3 55.00Within 15 450.50T otal 18 505.50A) 18 B) 19C) 3 D) 4110. Y ou have just conducted a designed experiment at three levels A, B, and C yielding the following "Coded" data:A B C6 5 33 9 45 1 2As a major step in your analysis, you calculate the degree of freedom for the "error" sum of squares to be:A) 7 B) 9C) 6 D) 3113. Which of the following nonparametric tests does NOT make a ranking evaluation by comparison with a critical value of chi-square?A) Mood's median testB) Spearman Rank correlation coefficientC) Kendall Coefficient of concordanceD) Kruskal-Wallis test115. When analyzing experimental data, which term describes the condition in which the error variance is inconsistent among observations?A) Stochastic variationB) HomoscedasticityC) HeterogeneousD) Heteroscedasticity128. If a sample space contains several unknown minima areas, then what can happen using steepest ascent methodology?A) Many tests may be requiredB) The yield contours must be ignoredC) The design area around point p must be expandedD) A wrong answer can result131. T aguchi methods uses a linear graph to help interpret the corresponding orthogonal array.For instance, for a L4 array, a linear graph with factors 1 and 2 at the endpoints, and factors3 at the midpoint indicates:A) Factor 3 is the interaction of factors 1 and 2B) That factor 4 is missing, since it is a L4C) The main factors (1 and 3) are interactionsD) Factor 2 will be the experimental result133. Plackett and Burman designs are used for screening experiments. There are geometric and non-geometric designs. it has been stated that runs of 12,20,24,28, and 36 runs arenon-geometric designs. This is because:A) The runs are in multiples of 4B) The non-geometric design has 2-factor interactions confounded with main effectsC) The geometric design runs are in powers of 2D) A PB design of 12 runs can have 11 factors covered137. A four factor, three level experiment must be conducted. What are the fewest number o of trials possible if all interactions are ignored?A) 9B) 18C) 27D) 81138. When selecting and scaling the process input variables for an experiment, what is NOT a desirable approach?A) Include as many important factors as possibleB) Set factor levels at practical or possible levelsC) Combine process measurement responses when possibleD) Be bold, but not foolish, in selecting high and low factor levels142. The most common subgrouping scheme for Xbar-R control charts is to separate the variation:A) Within stream versus stream to streamB) Within time versus time to timeC) Within piece versus piece to pieceD) Inherent process versus error of measurement143. If a process is out of control, the theoretical probability that a single point on the X bar chart will fall between plus one sigma and the upper control limits is:A) 0.2240B) 0.1587C) UnknownD) 0.3413144. A process is checked by inspection of random samples of four s hafts after a polishing operation. and Xbar and R charts are maintained. A person making a spot check picks out two shafts, measures them accurately, and plots the value of each on the chart. Both points fall just outside the control limits. He advises the department foreman to stop the process.This decision indicates that:A) The process levels is out of controlB) Both the level and dispersion are out of controlC) The process levels is out of control but not the dispersionD) The person is not using the chart correctly146. When using a pre-control chart, it's possible to have two consecutive samples outside of the target area but inside of the specification. What is the expectation that two consecutivesamples would both fall between the target area and the specification limit on the high side?A) 1/7B) 1/49C) 1/196D) 1/98147. The best chart for analyzing volatile data, like stock market averages or commodity prices, would be:A) EWMA 指数加权移动平均控制图B) CuSum累积和控制图C) Moving averageD) Short run148. What type of control chart employs a V-mask?A) EWMAB) Moving averageC) CuSumD) Short run150. If two-sigma limits are substituted for conventional three-sigma limits on a control chart, which of the following occurs?A) Decrease in alpha risk B) Increase in beta riskC) Increase in alpha risk D) Increase in sample size151. Which of the following types of control charts has the largest average run length for small shifts in the process mean?A) X bar B) Cumulative sumC) EWMA D) Dodge-Romig154. An operator is observed plotting nominal and target charts, what technique is being employed?A) Xbar-R charts B) Attribute chartsC) Short-run charts D) CuSum charts158. Y ou look at a process and note that the chart for averages has been in control. If the range suddenly and significantly increases, the mean will:A) Usually increaseB) Stay the sameC) Always decreaseD) Occasionally show out of control of either limit159. An Xbar and R chart was prepared for an operation using twenty samples with five pieces in each sample, Xbarbar was found to be 33.6 and Rbar was 6.20. During production, asample of five was taken and the pieces measured 36,43,37,25,and 38. At the time, this sample was taken:A) Both the average and range were within control limitsB) Neither the average nor range were within control limitsC) Only the average was outside control limitsD) Only the range was outside control limits172. The Shingo prize business model does NOT consider:A) The strategic planning processB) BechmarkingC) InnovationD) Community support175. Assume an operation speed rate of 80%. If 40 units are produced at 2 minutes/unit in two hours, what is the performance efficiency of the work unit?A) 0.800 B) 0.667C) 0.534 D) 0.435180. The Shingo prize business model does NOT consider:A) The strategic planning process B) BenchmarkingC) Innovation D) Community support181. A number of authors have recommended sequences by which the HOQ(QFD) can capture customer needs in the design. Please arrange the following design details inappropriate sequence from start to finish.I) Production requirementsII) Key process operationsIII) Parts characteristicsIV) Engineering characteristicsA) I, II, III, IVB) II, I, IV, IIC) IV, I I, III, ID) IV, III, II, I188. The design of a solution using a broad set of possible solutions, converging to a narrow set of possible, and then to a final solution, is referred to as:A) 20 questions approachB) Set-based designC) Systematic designD) Pugh method189. Review the following set of DFX statements and identify the single true description:A) DFSS is a subset of DFXB) The selection of DFX tools is relatively simpleC) DFX is a targeted development approachD) DFX was first created in the 1990's191. Cooper stresses that new products will have a greater chance of success if they have all of the following characteristics EXCEPT:A) Having an attractive marketB) Having a unique and superior productC) Being first to marketD) Having a good product launch194. TRIZ is a methodology for problem solving and is quite useful in the design phase of a product. Which of the following methods are employed in TRIZ?I) Trial and errorII) Reference to a trickIII) Use of physical effects (Physics)IV) Combination of tricks and physicsA) I and II onlyB) II and III onlyC) I, II and III onlyD) II, III and IV only198. In the design of many parts and products, it is best if the deviation from the target not exceed a certain amount. The best tolerance objective is termed:A) Nominal-is-bestB) Larger-is-betterC) Smaller-is-betterD) 6 sigma achievement199. The T aguchi loss function follows which type of relationship, as actual values deviate from the target?A) Reverse normal B) LinearC) Log normal D) Parabolic。

美国认可雅思成绩的大学和机构现在，美国越来越多的大学开始承认IELTS。

例如著名的加里福尼亚大学的各间分校，都接受IELTS成绩。

下面列出美国在IELTS中心注册的大学名单。

学校首字母查询 A B C D E F G H I J K L M N O P Q R S T U V W X Y ZAbilene Christian UniversityWeb: /Min IELTS Band Score:Undergraduate and Graduate AdmissionsAlbright CollegeWeb: /main.htmlMin IELTS Band Score: Undergraduate AdmissionsAllentown Business SchoolWeb: /index.aspAlliant International UniversityWeb: Min IELTS Band Score:Undergraduate and Graduate admissionsAmerican Association of Veterinary State Boards, Program for the Assessment of Veterinary Education Equivalence (PAVE)Web: American InterContinental University, BuckheadWeb: /Min IELTS Band Score: Undergraduate admissionsAmerican InterContinental University, DunwoodyWeb: /about.aspMin IELTS Band Score: Undergraduate and graduate admissions American InterContinental University, Ft. LauderdaleWeb: /Min IELTS Band Score: Undergraduate and graduate admissions American InterContinental University, HoustonAmerican InterContinental University, Los AngelesWeb: /Min IELTS Band Score: Undergraduate and graduate admissionsAmerican UniversityWeb: Min IELTS Band Score:Undergraduate admissionsAmerican University, Washington College of LawWeb: /Min IELTS Band Score: Graduate admissionsAmerican Veterinary Medical Association, Educational Commission for Foreign Veterinary Graduates (ECFVG)Web: Aquinas CollegeWeb: /Min IELTS Band Score: Undergraduate AdmissionsArdmore Higher Education ProgramWeb: Min IELTS Band Score:Undergraduate and graduate admissions for East Central University (Ada, OK), Murray State College (Tishomingo, OK), Southeastern Oklahoma State University (Durant, OK), and Oklahoma State University - Oklahoma City.Arizona State Board of NursingWeb: Arkansas State UniversityWeb: Min IELTS Band Score: Undergraduate AdmissionsArt Institute of AtlantaWeb: /programs.aspArt Institute of Boston at Lesley UniversityWeb: /aib/noflash_main.htmlMin IELTS Band Score:Undergraduate admissionsArt Institute of CaliforniaWeb: /Art Institute of CharlotteWeb: /Art Institute of ColoradoWeb: /index.aspArt Institute of DallasWeb: /Art Institute of Ft. LauderdaleWeb: /Art Institute of HoustonWeb: /Art Institute of Los AngelesWeb: /Art Institute of Los Angeles-Orange CountyWeb: /news_detail.asp?PressID=123Art Institute of MinnesotaWeb: /Art Institute of New York CityWeb: /Art Institute of PhiladelphiaWeb: /Art Institute of PhoenixWeb: /default.aspArt Institute of PittsburghWeb: /index2.aspArt Institute of SeattleWeb: /Art Institute of WashingtonWeb: /Art Institutes International at PortlandWeb: /index.aspAsbury Theological SeminaryWeb: /prospective/admiss_req.shtmlMin IELTS Band Score:Graduate admissionsAshland UniversityWeb: /Min IELTS Band Score: Undergraduate and graduate admissionsAssumption CollegeWeb: Min IELTS Band Score:Graduate admissionsAtlantic Culinary Academy at McIntosh CollegeWeb: /Bastyr UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsBellevue Community CollegeWeb: /Benedictine CollegeWeb: Min IELTS Band Score: Undergraduate AdmissionsBentley CollegeWeb: Min IELTS Band Score:Undergraduate admissionsBerea CollegeWeb: Min IELTS Band Score: Undergraduate AdmissionsBerklee College of MusicWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsBethany Lutheran CollegeWeb: /index.asp?bhcp=1Min IELTS Band Score: Undergraduate AdmissionsBoston UniversityWeb: /admissionsMin IELTS Band Score: Undergraduate and Graduate AdmissionsBoston University, School of Public HealthWeb: /sphMin IELTS Band Score: Graduate AdmissionsBradley UniversityWeb: /gradMin IELTS Band Score: Graduate AdmissionsBrandeis International Business SchoolWeb: /globalMin IELTS Band Score: Graduate AdmissionsBrandeis UniversityWeb: /admissions/Min IELTS Band Score: Undergraduate AdmissionsBrandeis University, Heller Graduate School, SID ProgramWeb: /sidMin IELTS Band Score:Graduate admissionsBrandeis University, SLIFKA Program in Intercommunal Coexistence Web: /programs/Slifka/Min IELTS Band Score:Graduate admissionsBriarcliffe College, BethpageWeb: /index.aspMin IELTS Band Score: Undergraduate AdmissionsBriarcliffe College, PatchogueWeb: /indexaspMin IELTS Band Score: Undergraduate AdmissionsBrigham Y oung University, HawaiiWeb: /Min IELTS Band Score: Undergraduate AdmissionsBrooks College, Long BeachWeb: /index.aspBrooks College, SunnyvaleWeb: /brookssj/index.jspBrooks Institute of PhotographyWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsBroward Community CollegeMin IELTS Band Score:Undergraduate admissionsBrown InstituteWeb: /brown/aboutus.htmlBrown UniversityWeb: /gsMin IELTS Band Score:Graduate admissionsBryn Mawr CollegeWeb: /admissionsMin IELTS Band Score: Undergraduate and Graduate AdmissionsButler UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia Culinary Academy - Le Gordon Bleu Hospitality & Restaurant Management Program Web: /California Institute of TechnologyWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia Lutheran UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia School of Culinary ArtsWeb: /California State University, ChicoWeb: Min IELTS Band Score: Undergraduate AdmissionsCalifornia State University, FullertonWeb: /Min IELTS Band Score:Undergraduate and Graduate admissionsCalifornia State University, HaywardWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia State University, Long BeachWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia State University, Los AngelesWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia State University, NorthridgeWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsCalifornia State University, SacramentoWeb: /Min IELTS Band Score:Undergraduate and Graduate admissionsCameron UniversityWeb: Min IELTS Band Score:Undergraduate and Graduate admissionsCarl Albert State College, PoteauWeb: /Carl Albert State College, SallisawWeb: /sequoyah_county/index.htmCarleton CollegeWeb: Min IELTS Band Score: Undergraduate AdmissionsCarnegie MellonWeb: /enrollment/admission/Min IELTS Band Score: Undergraduate AdmissionsCase Western Reserve UniversityWeb: Min IELTS Band Score: Undergraduate AdmissionsCatholic University of AmericaWeb: /Min IELTS Band Score: Undergraduate AdmissionsChandler-Gilbert Community CollegeChatham CollegeWeb: Min IELTS Band Score:Undergraduate admissionsCity University, BellevueWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsCity University, RentonWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsClaremont Graduate UniversityWeb: Min IELTS Band Score: Graduate AdmissionsClaremont Graduate University, The Peter F. Drucker and Masatoshi Ito Graduate School of ManagementWeb: /pages/1378.aspMin IELTS Band Score: Graduate AdmissionsClarion University of PennsylvaniaWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsClarkson UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate admissionsCleveland State UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsCollins CollegeWeb: /index.aspMin IELTS Band Score: Undergraduate AdmissionsColorado School of MinesWeb: /Academic/econbus/Min IELTS Band Score: Graduate AdmissionsColorado State UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsColumbia University Teachers' CollegeWeb: /academic/appliedlinguistics/default.htmMin IELTS Band Score: Graduate AdmissionsColumbia University, Mastr of Arts Program in Climate and SocietyWeb: /cu/climatesociety/admissions.htmlMin IELTS Band Score: Graduate admissionsColumbia University, School of Arts and SciencesWeb: /cu/gsas/Min IELTS Band Score:Graduate admissionsColumbia University, School of International and Public AffairsWeb: /Min IELTS Band Score:Graduate admissionsCommission on Graduates of Foreign Nursing SchoolsWeb: Min IELTS Band Score:Intl. Commission on Healthcare Professions; Credential Evaluation/Certification ProgramConcordia CollegeWeb: Min IELTS Band Score: Undergraduate AdmissionsConnecticut CollegeWeb: /admissions/visiting/international-info/admissionMin IELTS Band Score: Undergraduate AdmissionsConnors State College, MuskogeeConnors State College, WarnerContra Costa Community CollegeWeb: Min IELTS Band Score: Undergraduate admissionsCooking and Hospitality Institute of ChicagoWeb: /home.aspCornell University, Johnson School of Management (MBA program)Web: Min IELTS Band Score: Graduate admissionsDe Anza CollegeWeb: Min IELTS Band Score: Undergraduate admissionsDeV ry UniversityWeb: Min IELTS Band Score:Undergraduate and Graduate AdmissionsAll 29 CampusesDoane CollegeWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsDowntown College ConsortiumWeb: /Min IELTS Band Score:Undergraduate and graduate admissions for:Oklahoma City Community CollegeOklahoma State University-Oklahoma CityRedlands Community CollegeRose State CollegeUniversity of Central OklahomaDrury UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsDuke UniversityWeb: Min IELTS Band Score:Undergraduate and Graduate admissionsDuncan Higher Education CenterWeb: /duncan/Min IELTS Band Score:Undegraduate and graduate admissions for consortium of:Western Oklahoma State College (2 year college), Southwestern Oklahoma State University, East Central University, University of Sciences and Arts of Oklahoma and Cameron University.East Carolina UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsEast Central UniversityWeb: /Min IELTS Band Score:Undergraduate and Graduate admissionsEast Stroudsburg UniversityWeb: /servlet/RetrievePage?site=esu&page=home Min IELTS Band Score: Graduate AdmissionsEast Tennessee State UniversityWeb: /gradstudMin IELTS Band Score: Graduate AdmissionsEastern Michigan UniversityMin IELTS Band Score:Undergraduate and Graduate admissions Eastern Oklahoma State College, McAlesterEastern Oklahoma State College, WilburtonEastern Oregon UniversityWeb: /Min IELTS Band Score: Undergraduate AdmissionsEastern Washington UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsEckerd CollegeWeb: Min IELTS Band Score: Undergraduate AdmissionsEdmonds Community CollegeWeb: /Elon UniversityWeb: Min IELTS Band Score:Undergraduate and Graduate admissionsEmpire State College, State University of New Y orkWeb: /esconline/online2.nsf/eschome?openform Min IELTS Band Score: Undergraduate AdmissionsEstrella Mountain Community CollegeFairleigh Dickinson UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsFashion Institute of Design and MerchandisingWeb: /Florida Metropolitan UniversityWeb: Min IELTS Band Score: Undegraduate and Graduate admissionsFlorida National CollegeWeb: /Min IELTS Band Score: Undergraduate AdmissionsFoothill CollegeWeb: Min IELTS Band Score:Undergraduate admissionsFordham UniversityWeb: /prospective/Admissions/Graduate5142html Min IELTS Band Score: Graduate AdmissionsFranklin and Marshall CollegeWeb: /Min IELTS Band Score:Undergraduate admissionsFull Sail Real World EducationWeb: Min IELTS Band Score:Undergraduate admissionsGateway Community CollegeWeb: /Gemological Institute of AmericaWeb: /wd_3954htmMin IELTS Band Score: Undergraduate AdmissionsGeneva CollegeWeb: /Min IELTS Band Score: Undergraduate AdmissionsGeorge Mason UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsGeorge Washington UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsGeorgetown UniversityWeb: Min IELTS Band Score: Graduate admissionsGeorgia Institute of TechnologyWeb: /Min IELTS Band Score: Graduate AdmissionsGibbs College, MontclairWeb: /kgsnj/index.htmlGibbs College, NorwalkWeb: /kgsct/index.jspGibbs School, Washington DCWeb: /kgsdc/index.jspGlendale Community CollegeWeb: /index.htmlGlendale Community College, Maricopa CountyWeb: /Golden Gate UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsGoldey-Beacom CollegeWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsGraduate Institute of Applied LinguisticsWeb: Min IELTS Band Score: Graduate AdmissionsGrand V alley State UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsGreen River Community CollegeWeb: /Hamilton CollegeWeb: /Min IELTS Band Score: Undergraduate AdmissionsHamline UniversityWeb: /Harrington Institute of Interior DesignWeb: /Min IELTS Band Score:Undergraduate admissionsHarvard Business School (Doctoral Programs)Web: /doctoralMin IELTS Band Score: Graduate AdmissionsHarvard Business School MBA ProgramsWeb: Min IELTS Band Score:Graduate admissionsHarvard University, School of Public HealthWeb: /admissionsMin IELTS Band Score:Graduate admissionsHaverford CollegeWeb: /admissionMin IELTS Band Score: Undergraduate AdmissionsHawaii Pacific UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsHusson CollegeWeb: /Min IELTS Band Score: Undergraduate AdmissionsIdaho State UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsIllinois Institute of Art at Chicago-Schaumburg, TheWeb: /about_us_the_region.aspIndiana Institute of TechnologyWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsIndiana State UniversityWeb: /Min IELTS Band Score: Graduate AdmissionsIndiana UniversityWeb: /~iuadmit/graduate/index.shtmlMin IELTS Band Score: Graduate AdmissionsIndiana University Purdue UniversityWeb: /Min IELTS Band Score: Undergraduate AdmissionsIndiana University, School of LawWeb: /Min IELTS Band Score:Graduate admissionsIndiana University, School of MusicWeb: /Min IELTS Band Score: Undergraduate AdmissionsInternational Academy of Design and Technology, Chicago Web: /Min IELTS Band Score: Undergraduate AdmissionsInternational Academy of Design and Technology, DetroitMin IELTS Band Score: Undergraduate AdmissionsInternational Academy of Design and Technology, Fairmont Web: /Min IELTS Band Score: Undergraduate AdmissionsInternational Academy of Design and Technology, Orlando Web: /Min IELTS Band Score: Undergraduate AdmissionsInternational Academy of Design and Technology, PittsburghWeb: /Min IELTS Band Score: Undergraduate AdmissionsInternational Academy of Design and Technology, TampaWeb: /Min IELTS Band Score: Undergraduate AdmissionsInternational Commission on Healthcare Professions (ICHP)Web: International Culinary AcademyWeb: /ica/index.htmlInternational Monetary Fund (IMF) Institute, Washington DCWeb: Min IELTS Band Score: /external/np/ins/english/pdf/inst2004.pdfIowa State UniversityWeb: /Min IELTS Band Score: Undergraduate and Graduate AdmissionsJacksonville UniversityWeb: Min IELTS Band Score: Undergraduate and Graduate AdmissionsJohns Hopkins University, School of Advanced International StudiesWeb: http://www.jhubc.it/Min IELTS Band Score:Graduate admissionsJohnson and Wales University, DenverWeb: /denver/index.htmMin IELTS Band Score: Undergraduate and Graduate AdmissionsJohnson and Wales University, North MiamiWeb: /florida/index.htmMin IELTS Band Score: Undergraduate and Graduate AdmissionsJohnson and Wales University, ProvidenceWeb: /prov/index.htmMin IELTS Band Score: Undergraduate and Graduate AdmissionsKatharine Gibbs School, BostonWeb: /Katharine Gibbs School, MelvilleWeb: /index.aspKatharine Gibbs School, New Y orkWeb: /Katharine Gibbs School, PhiladelphiaWeb: /Katharine Gibbs School, PiscatawayWeb: /Katharine Gibbs School, ProvidenceWeb: /kgri/index.jspKentucky Board Of NursingWeb: /index-old.htmKirksville College of Osteopathic MedicineWeb: /Min IELTS Band Score:Graduate admissionsKnox CollegeWeb: /knox/Min IELTS Band Score: Undergraduate AdmissionsLane Community CollegeWeb: /Langston UniversityWeb: /Min IELTS Band Score:Undergraduate and graduate admissionsLangston University, Oklahoma CityWeb: /okcweb/Min IELTS Band Score:Undergraduate and graduate 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Controllability Improvement for LTI Systems*CAI Ning, XI Jianxiang, ZHONG Yi-ShengAutomation Department, Tsinghua University, Beijing 100084,P.R.ChinaE-mail: cain04@Abstract：For LTI systems represented by matrix pairs, the problem of configuration adjustment is dealt with to improve the controllability. As important concepts and theoretical foundation, the maximum controllability index of square matrix is defined and analyzed, and a generic controllability canonical form is introduced for single-input systems. Based on these concepts, approaches on adjusting the system configurations are presented respectively for two distinct scenarios.Key Words：LTI system, Matrix pair, Controllability1INTRODUCTIONIn engineering, one usually desires any application system to be controllable. Obviously, it is meaningful to consider the problem about how one can improve the controllability of an uncontrollable system by modifying its configuration. In this paper, the problem of controllability improvement for LTI systems will be dealt with.A notable background of this work is the control of composite systems. A composite system is a system composed of subsystems that may possess relatively independent configurations and functions. The information flow through the subsystems is regulated by certain interconnection configuration, which is often adjustable. The research of controllability for composite systems started long ago. For instance, Zazworsky and Knudsen [1] presented conditions for the controllability of compartmental models, focusing on the structure of interconnections. Recently, a few researchers have addressed the study of controllability problems for swarm systems or multi-agent systems. Ji and Egerstedt [2] gave sufficient conditions for controllability of first order swarm systems, based on the algebraic characteristics of certain matrices about the graphs. Tanner [3] postulated that more information exchange might be detrimental to controllability. Rahmani and Mesbahi [4] extended the work of Ji in [2], concentrating on the relationship between graph symmetry and controllability. Cai and Zhong [5] defined “formation controllability” and studied the condition for controllability of high order swarm systems. Liu et al. [6] discussed the controllability of discrete time systems with switching graph topologies. Ji et al. [7] presented some conditions for controllable graphs which are analogous to the results in [1].In most of the works mentioned in the last paragraph, the controllability of interconnection configuration ultimately amounted to the controllability of a pair of matrices. So the approaches about controllability improvement of matrix *This work is supported by National Nature Science Foundation under Grants 60736024 and 60674017. pairs by adjusting the elements in the matrices are the foundation for controllability improvement of composite systems.The main contribution of this paper is the presentation of approaches on controllability improvement for any typical LTI system which can be represented by a pair of matrices. There should be two considerations about the approaches: on the one hand, controllability is desired to be improved as much as possible; on the other hand, less change should be brought to the original system.Two concrete scenarios will be treated respectively in this paper:① the first matrix is fixed;② the second matrix is fixed. The assumption that one matrix is fixed while the other is adjustable is due to that the two matrices usually bear different mechanisms.As the theoretical foundation of this paper, a problem about square matrix will be first analyzed. That is: for a given square matrix A, how to find a vector b so that the controllability index of (,)A b can achieve its maximum possible value? On the basis of it, approaches on controllability improvement for both of the two scenarios① and② will be developed respectively. The main idea of the approach about scenario ①is directly based upon the theoretical analysis on the maximum possible controllability index. The main idea of the approach about scenario② is to set up a bridge to connect the uncontrollable subsystems with the controllable ones. For this end, a system considered should first be transformed into its generic controllability canonical form.The remaining part of this paper is organized as follows. In Section 2, the theoretical analysis about the maximum possible controllability index and some other relational mathematic results will be presented. In Sections 3 and 4, approaches of how to improve controllability of given LTI system will be discussed for the two different scenarios respectively. In Section 5, a numerical example will be shown to demonstrate the approach in Section 4. Notations:(,)A b denotes a pair of matrices and the system x Ax bu=+determined by them. ˆA denotes aProceedings of the 29th Chinese Control Conference July 29-31, 2010, Beijing, Chinasimilar matrix of A . SI is the abbreviation for “single-input”.2 THE MAXIMUM CONTROLLABILITYINDEX OF MATRIXSome concepts on square matrix which are closely relational to the controllability of LTI system will be introduced and analyzed in this section. They are fundamental for the discussions throughout this paper. The first important concept is the Maximum Controllability Index.Definition 1: (Maximum Controllability Index) Consideran LTI system: xAx bu =+ , where n x R ∈ is the state and the input. Suppose u R ∈n n A R ×∈ is given and can be freely selected in , then the maximum possible dimension for the controllable subspaces of systems among the set {(n b R ∈n R ,)|}n A b b R ∈ is defined as the Maximum Controllability Index of matrix A , denoted by ()A γ.Definition 2: (Nonderogatory Matrix [8]) A matrix is said to be nonderogatory if all its eigenvalues are of geometric multiplicity 1.There is a way to calculate the value of the maximum controllability index of a given matrix A and designate appropriate vector b s.t. the controllability index of (,)A b is maximum. The result is mainly stated as Theorem 1.Theorem 1: The maximum controllability index ()A γ equals to the degree of the minimum polynomial ()A φλ of matrix A .To demonstrate this theorem, several lemmas are required.Lemma 1: Suppose that matrix A has μ distincteigenvalues 12,,,μλλ λwith algebraic multiplicity i σ and geometric multiplicity i ς respectively. The Jordancanonical form of A is1J J μ⎡⎤⎢⎢⎢⎥⎣⎦⎥⎥， (1) where1i i i i J J J ς⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦(1,2,...,i μ=) ， (2) with each Jordan block (ik ikr r ik J R×∈1,2,...,i k ς=,1iiki k rςσ==∑). The degree of the minimum polynomial()A φλ of A is .11max {}iik k i r μς=∑≤≤It is simple to obtain Lemma 1 on the basis of the theoryin matrix analysis [8], so the detail of proof is omitted here.Lemma 2: 1) The maximum controllability index ()A γis equal to or greater than the degree of the minimumpolynomial ()A φλ of matrix A .2) Suppose that the given n n A R ×∈with Jordan canonical form 1ˆA PAP −= has μ distinct eigenvalues 12,,...,μλλλ. Let , where possesses at least 1ˆb P b−=ˆn bR ∈μ non-zero entries, each corresponding to the last row of the Jordan block that has the maximum dimension among the Jordan blocks of i λ (1,2,...,i μ=), then the dimension of controllable subspace of (,)A b is ()A γ.Proof: Consider the controllability of system ˆˆˆˆxAx bu =+ . Without loss of generality, let each 1i J (1,2,...,i μ=)respectively denote the specific Jordan block in ˆAwith the maximum dimension for each distinct eigenvalue i λ and let these 1i J (1,2,...,i μ=) be rearranged to be the first μ Jordan blocks.11111b u 12220ˆˆˆˆˆˆ0J x x J x x b μ⎡⎤⎢⎥⎢⎥⎡⎤⎡⎤⎢⎥⎡⎤=+⎢⎥⎢⎥⎢⎥⎢⎥×⎢⎥⎣⎦⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎢⎥×⎢⎥⎣⎦ . (3) The system is partitioned into two subsystems shown in(3), where ‘×’ denotes the other Jordan blocks except (1i J 1,2,...,i μ=). According to the criterion to check controllability, the first subsystem must be controllable ifsatisfies the condition of the statement 2) of this lemma, and the dimension of b 1ˆxequals to the degree of ()A φλ according to Lemma 2. Obviously, the controllability index of the system is equal to or greaterthan the dimension of 1ˆx. Lemma 3: The maximum controllability index ()A γ isequal to or less than the degree of the minimum polynomial ()A φλ of matrix n n A R ×∈.Proof: Let 01()s A s c c c λλ=+++ λ. If s φn =, the result of this lemma is obviously true. Assume that s n <. According to Cayley-Hamilton Theorem,010s s c c A c A +++= , Thus,010s s c c A c A +++= ,and evidently1()n rank bAb A b s −⎡⎤⎣⎦ ≤.Lemmas 2 and 3 can naturally lead to Theorem 1. Thestatement ② of Lemma 2 also provides a method to seekfeasible candidate of bto maximize the controllable dimension of (,)A b for any given A . Denote theindexes of the non-zero entries in such a bby ˆPb =12,,...,i i i μ. If all elements in P are real, simply, a b1−with the μ necessary nonzero elements being 1 and all the other elements being 0 is feasible. Otherwise, if Pand are complex valued, because , a feasibleshould not be orthogonal to all the 1P −ˆbPb =b μ rows in P indexed respectively by 12,,...,i i i μ. One will see that such a must exist in .b n R Theorem 2: For a given , almost any b makes the dimension of controllable subspace of (,n nA R ×∈nR ∈)A b be ()A γ.Proof: For convenience, if the dimension of the controllable subspace of some (,)A b is ()A γ, such a bis said to be feasible, otherwise this b is infeasible. Let() denote the rows in that may becomplex valued. If for any T i q 1,2,,i = n 1P −0T i q b =12{,,...,}i i i i μ∈, thenis infeasible. implies both b 0T i q b =Re()0T i q b = and, where Redenotes the real part and the imaginary part. Therefore, the domain in of infeasible is Im()0T i q b =()i Im()i n R b (Re())(Im())T Ti i q q ∩ (12,,...,i i i i μ=) where denotes the null space. For each() i 12{,,...,}i i i i μ∈, it is an intersection of two -dimensional hyperplanes and must be a subset of measure zero in . Apparently, the complete infeasibledomain of is(1)n −n R b 12{,,...,}((Re())(Im()))T T i i i i i i q q μ∈∪ ∩while any vector in the other area of must be feasible.Since this domain is the union of a finite number of measure zero subsets, its measure is still zero. Thus, almost any vector in as is feasible. □n R n R b It is a fact in matrix theory that for a given matrix , the degree of n n A R ×∈()A φλ is equal to if and only if n A is nonderogatory. The following statement is known in control theory and can be regarded as a corollary of Theorem 1.Corollary 1: For a given n n A R ×∈, there exists s.t. n b R ∈(,)A b is completely controllable if and only if A is nonderogatory.3 CONTROLLABILITY IMPROVEMENTOF MATRIX PAIRS AS THE FIRST MATRIX FIXEDSuppose that for an LTI system xAx Bu =+ , the coefficient matrix A represents some intrinsic property which is unalterable, while the coefficient matrix is to be designed to make the system controllable.B At first, there are two questions: Whether or not a single input variable u is sufficient to control the system? If a single input variable is sufficient, how can a proper be designed?R ∈n B R ∈Actually, the analysis in the last section has already answered the first two questions theoretically. Theorem 1shows that the maximum controllability index ()A γ equals to the degree of the minimal polynomial of A . Lemma 2 provides a method to calculate this value. Lemma 3 demonstrates a way to seek corresponding feasible n B R ∈. According to Corollary 1, if A is nonderogatory, i.e. ()A n γ=, then there exists somen B R ∈ to make the system completely controllable.If Ais derogatory, multiple input variables are necessary. There are also two questions about such asituation: What is the least number of input variablesrequired for controllability? If minput variables are sufficient, how can a proper be designed? Theorem 3 will answer these two questions theoretically. n m B R ×∈Theorem 3: If the matrix in system n n A R ×∈x Ax Bu =+ is derogatory, then at least ()m A α= input variables arerequired for complete controllability, where ()A αdenotes the maximum geometric multiplicity ofeigenvalues of A . Proof: Suppose that the Jordan canonical form of A is1ˆA PAP −=. Without loss of generality, assume that 1λ is the eigenvalue of A possessing the maximum geometricmultiplicity among the μ distinct eigenvalues.Thesubmatrix in ˆA associated with 1λ is 1J , comprising()A α Jordan blocks: 1111,()A J J J α⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦. Now consider the controllability of ˆˆ(,)AB , where is to be designed. According to the criterion of systemcontrollability based on Jordan canonical form, if ˆB ˆˆ(,)A B is controllable, then the ()A αrow vectors in respectively associated with the bottom rows inmust be linearly independent. This implicates that . Thus, the number of columns in should be at least ˆB 111,(),...,A J J αˆ()()rank BA α≥ˆB()A α. If ()ˆn A B R α×∈, the requirement of the criterion of system controllability for the rest of submatrices 23,,...,J J J μ can easily besatisfied by appropriately selected rows in because each geometric multiplicity of ˆB23,,...,μλλλ is less than orequal to ()A α. With such a , the physically realizable can be derived according to . ˆBB 1ˆB P B−=4 CONTROLLABILITY IMPROVEMENTOF MATRIX PAIRS AS THE SECOND MATRIX FIXEDConsider an SI LTI system xAx Bu =+ , where n B R ∈. In this section, suppose that the coefficient matrix B is fixed because it represents some intrinsic unalterable property. The question is: how can the coefficient matrix A be adjusted to notably improve the controllability ofthe system? If A could be arbitrarily redesigned, undoubtedly a controllable system can easily be derived so long as . However, what makes sense here is to make less change to the original configuration. In order to answer this question, the generic controllability canonical form of SI LTI system will be introduced as follows, which is one of the theoretical foundations of the discussions in this section.0B ≠Definition 4: (Generic Controllability Canonical Form) The generic controllability canonical form (,)M h of an SI LTI system of n th order that could be uncontrollable is321321001m O O O x x Θ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎢⎥u ⎥⎢⎥⎢⎥⎢⎥⎣⎦ +1×Θ=+⎢⎥×Θ⎢⎥⎢⎥×Θ⎣⎦ , (4) where () denote companion matrices [8]of the formi Θ1,,1i l = 01000***i ⎡⎤⎢⎥⎢⎥Θ=⎢⎥⎢⎥⎣⎦. with ‘*’ denoting any arbitrary element; ‘×’ denotes anyarbitrary submatrix; (i O 1,2,,i l = ) denote zero matrices.Remark 4.1: Let 0i η= () respectively denote the element at the left lower corner of O in (4). For instance,1,2,,i = l ⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦ i 0000i i O η== .These elements are very important to controllabilityimprovement.Theorem 4: Any th order SI system n xAx Bu =+ can be transformed into its generic controllability canonicalform ˆˆˆˆxAx Bu =+ by some similarity transformation , where n n P R ×∈ˆxPx =. Proof: If the system is controllable, it can be transformed into controllability canonical form, i.e. a specific case of generic controllability canonical form. Suppose the controllability index is k . The system can be decomposed into controllable/uncontrollable subsystems via some similarity transformation :n <1n n P R ×∈112122200c c c c x x A u x x A A B ⎡⎤⎡⎤⎡⎤⎡=+⎢⎥⎢⎥⎢⎥⎢⎣⎦⎣⎣⎦⎣⎦ ⎤⎥⎦. where k c x R ∈ and n k c x R −∈ represent the controllableand uncontrollable sub-state vectors respectively and thematrix pair (,222)A Bis controllable. The controllable subsystem can be transformed into controllabilitycanonical form via some similarity transformation 2k k P R ×∈:[]12222122,001TP A P P B −⎧=Θ⎪⎨=⎪⎩ . As to the subsystem 11c c xA x = , suppose the maximum controllability index of 11A is γ. Construct an auxiliary vector n k b R −∈according to the approach implied by Lemma 2, with the controllability index of 11(,)A b being γ. A virtual subsystem as follows can be obtained:11A bu ξξ=+ . (5) Obviously, (5) can still be decomposed into controllableand uncontrollable subsystems via some :()()3n k n k P R −×−∈(11)(1)(1)11(2)(21)(22)(2)(2)111100A u b A A ξξξξ⎡⎤⎡⎤⎡⎤⎡⎤=+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦ . where (,(22)(2)11A )b is controllable and can be transformed into controllability canonical form via some similarity transformation 4P R γγ×∈:[](22)141142(2)4,001TP A P P b −⎧=Θ⎪⎨=⎪⎩ .The entire similarity transformation so far is:3412n k k I P P P P I γ−−⎡⎤⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦⎢⎥⎣⎦P . (6) Implementing P on system, i.e. let ˆxPx =ˆ, yield the equation of the form:(11)112121ˆ01A O O x x ⎡⎤⎡⎤⎢⎥⎢⎥=×Θ⎢⎥⎢⎥⎢⎥⎢⎥×Θ⎣⎦⎣⎦ u +. (7) Obviously, the block can further be transformed by repeating the similar operations, until a generic controllability canonical form is finally achieved. □ (11)11A Actually, the elements i η () in the generic controllability canonical form are the keys for controllability improvement. It is easy to verify the following theorem and the detail is omitted here. 1,2,,i = l Theorem 5: If the values of 1η in the generic controllability canonical form (7) is altered from 0 to 1, then the dimension of controllable subspace of the system is increased from original k to k γ+. If all the i η (i 1,2,,l = ) are altered from 0 to 1, then the system is completely controllable.If certain i η is set to 1, an improved version of A can be retrieved via a series of reverse similarity transformations (6). i η (1,2,,i l = ) are the bridges that can connect the previously uncontrollable state variables with the input information. For instance, if 10η=, the first n k − state variables in (7) are separated from the input information;otherwise, the influence of input information could reach these variables. Actually, this effect of i η () is structural. 1,2,,i l = 1i η= is not necessary while any nonzero value could be effective for such a connection.The improvedA should be retrieved from ˆA by the relationship 1ˆA P AP −=. The adjustment forcontrollability improvement can be regarded as mountingan offset ˆAΔ upon ˆA . Accordingly, the new version of A is:1ˆˆ()P A A P A −+Δ=+ΔA , (8) 1ˆA P AP −Δ=Δ in (8) indicates the adjustment on theoriginal graph topology, which is expected to be less. Let (,) denote the matrix with a single nonzero element 1 at index (,. Suppose thatand n n i j E R ×∈)i j 1Tn P p p ⎡⎤=⎣⎦ T []11n P q −= q n , where () respectively denote the rows in,n i i p q R ∈1,2,...,i =P and columns in . If 1P −(,)ˆi j A E Δ=, then it is easy to verify that11(,)ˆT i j ijA P APP E P q p −−Δ=Δ==, (9) This implies that A Δ is equal to the outer product of theth column in and the i 1P −j th row in P , and it is independent of the other elements in the two matrices.The zero elements in A Δrepresent the unchanged elements in A after adjusting. The more zero elements in and 1P −P , the more zero elements there would be in A Δ.After controllability improvement, it is desired that there should be small change to the original system. On the one hand, it is better that the number of elements adjusted in A is less; on the other hand, it is better that the variations resulted from the adjustment are weaker. Next one will see that this requirement can be satisfied.Theorem 6: If an uncontrollable th order system n (,)A B is made controllable by adjusting A , some of the elements in this matrix can keep unchanged.Proof: Suppose the system is improved to be controllableby setting 12...1l ηηη==== in the generic controllability canonical form ˆˆ(,)AB . The elements in the set {} ( (,)i j 1,2,...,i n =1,...,j i =), i.e. the left lowerpart of ˆAincluding the diagonal, are free to be changed without affecting the controllability. Obviously, the amount of these free elements is . Now denote the variations on these free elements by 12 respectively, which are unknown variables to be determined. Then the variation on the original (1)/2r n n =+,,...,r y y y R ∈A is111r r l A y y Δ=Φ++Φ+Ψ++Ψ =. (10) where 1 respectively denote the variations aroused by 1 and ,...,n n r R ×ΦΦ∈1,...,1r y y =1,...,n n l R ×ΨΨ∈ the variations aroused by 11,...,1l ηη==.. The values ofmatrices 1 andcan be calculated according to (9).,...,r ΦΦ1,...,l ΨΨLet denote the column stacking of a matrix, then (10) is equivalent to()vec i 111()()()()()r r l vec A y vec y vec vec vec Δ=Φ++Φ +Ψ++Ψ (11)Suppose that some specific elements in A are designatedto be constant due to certain practical purpose, e.g. to keep the structural property. Without loss of generality, let the first ζ entries of be constant. For eachelement of ()vec A Ato keep constant, the corresponding element of A Δ with same index should be 0, accordingly, a scalar equation can be extracted from (11):(1)(1)(1)(1)111()()()()1110,0,r r l r r l y y y y ζζζϕϕψψϕϕψ⎧=+++++⎪⎨⎪=+++++⎩ζψ (12) where for instance, (1)1R ϕ∈ denotes the first entry in 1()vec Φ and (1)1R ψ∈ denotes the first element in 1()vec Ψ.If (12) is consistent, then it has solution for the unknown variables . If (12) covers all the entries in 12,,...,r y y y ()vec A Δ, i.e. nn ζ=, then it must be inconsistent. It is easy to derive the maximum number of consistent scalar equations by deleting the inconsistent ones from (12).□ Let r D R ζ×∈ denote the coefficient matrix in (12), i.e.(1)(1)1()()1r r D ζζϕϕϕϕ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦. Then the following corollary naturally arises according to the matrix theory and the proof of Theorem 6.Corollary 2: The maximum number of elements that can keep unchanged after adjustment for controllability is at least , with the row dimension of D being ()rank D nn ζ=.Suppose that (12) is consistent. If ()rank D r <, then the equation (12) has a solution set with infinite number of solutions. The less this rank, the wider the scope of the solution set is.Assume that the first ζ entries of are to keep constant. It is desired that the variations on the remaining entries are small. If (12) has a solution set Y , then an appropriate solution can be determined by solving such a problem:()vec A 1(,...,)min ()ry y Y vec A ∈Δ. (13)If ζ is small, then less number of elements in A keep constant. However, the scope of the solution set of equation (13) is possibly wider. Thus, a better result of ()vec A Δ may be derived for the optimal problem (13). A compromise is needed between the two aspects of requirements.5 NUMERICAL EXAMPLEConsider an uncontrollable system (,)A b :010101100111110011101011010001x x u ⎡⎤⎢⎥⎢⎥⎢⎥=+⎢⎥⎢⎥⎢⎥⎣⎦ ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦. The reason for that any element in the original matrices iseither 1 or 0 is to stress the structural property. Suppose that is fixed while b A is to be adjusted. In order to transform it into generic controllability canonical form, first decompose it into controllable/uncontrollable subsystems by the similarity transformation:110062110631006310163100311P −⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦. It follows that 11000010000002/31100013010110001c c c c x x u x x −⎡⎤⎡⎤⎢⎥⎢⎥⎡⎤⎢⎥+⎢⎥⎢⎥⎣⎦⎢⎥⎢⎥⎣⎦⎢⎥−⎢⎥⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦−⎢⎥⎢⎥−⎣⎦. (14) Construct a virtual dynamic system:110101u ξξ−⎡⎤⎡⎤=+⎢⎥⎢⎥−⎣⎦⎣⎦ . It is controllable and can be transformed into controllability canonical form. A new similarity transformation can be accordingly designed as21000011000001000011000411P ⎡⎤⎢⎥−⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦.which respectively transforms the two subsystems in (14)and results in the generic controllability canonical formˆˆ(,)Ab : 0100000011002/32/301001/31/300105/32/30311⎡⎤⎢⎥−−⎢⎥⎢⎥⎢⎥−−⎢⎥⎢⎥⎣⎦.The relationship between (,)A b and ˆˆ(,)Ab is: 1ˆAPAP −= and , where and ˆb Pb =21P P P =1011110021000211102100101P −⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥−⎣⎦.The index of 1η in ˆA is (2,3). Let the value of 1η be altered from 0 to 1, i.e. . It follows that,(2,3)ˆA E Δ=[][000102/301/301/3TA Δ=−−]0000000000002/301/301/300000⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥−−⎢⎥⎢⎥⎣⎦.Three elements in Aare adjusted while there is no structural adjustment.6 CONCLUSIONSThis paper presented approaches to improve controllabilityof LTI systems via adjusting their configurations. The models of these systems are usually represented by matrix pairs. As theoretical foundation, the concept of maximum controllability index for square matrices wasintroduced. 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