Unitary Triangularization of a Nonsymmetric Matrix

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证明开普勒猜想的原文(英文)并附译后汉语

Thus it can seen, the general volume of two such rectangular parallelepipeds is just equal to the general volume of aforementioned the annular solid, for either general volume is equal to (√2-1)L3.
Thus it can computed, the ratio of the volume of the internal tangent sphere of cube N to the volume of cube K is equal to √2π/6, i.e. π /√18. This ratio is just that ratio which the Kepler’s conjecture mentions.
Abstract
Heap together equivalent spheres into a cube up to most possible, then variant general volumes of equivalent spheres inside the cube depend on variant arrangements of equivalent spheres fundamentally. This π/√18 which the Kepler’s conjecture mentions is the ratio of the general volume of equivalent spheres under the maximum to the volume of the cube. We will do a closer arrangement of equivalent spheres inside a cube. Further let a general volume of equivalent spheres to getting greater and greater, up to tend upwards the super-limit, in pace with which each of equivalent spheres is getting smaller and smaller, and their amount is getting more and more. We will prove the Kepler’s conjecture by such a way in this article.

On distinguishing quotients of symmetric groups

The notation used is fairly standard. We use κ, λ, µ, and ν to stand for cardinals (usually inﬁnite), and |X | for the cardinality of the set X . If Ω is any set we write Sym(Ω) for the group of all permutations of Ω (1–1 maps from Ω onto itself), with permutations acting on the right, and we write S (µ) for Sym(µ) for any cardinal µ. For g ∈ Sym(Ω) we let supp g be the support of g . If we are working in Sλ (µ)/Sκ (µ) (where Sλ (µ), Sκ (µ) are as introduced above) then we refer to sets of cardinality less than κ as small. We use overlines such as x to stand for ﬁnite sequences (‘tuples’) (x1 , x2 , . . . , xn ). By a permutation representation or action of a group G we understand a homomorphism θ from G into Sym(X ) for some set X . The representation is faithful if θ is 1–1, it is transitive if for any x, y ∈ X there is g ∈ G such that x(gθ) = y , and it is trivial if its image is the trivial group. If X is a subset (or sequence of elements) of a group G, we let X denote the subgroup generated by X . If g, h ∈ G we write g h for the conjugate h−1 gh of g by h. If g is a sequence of members of G and h ∈ G, we write g h for the h sequence whose ith entry is gi , and if g, h are sequences of members of G of the same length, we let g ∗ h be the sequence whose ith entry is gi hi . If gh 1 = g 2 for some h, g1 and g2 are said to be conjugate. If N ≤ G and f = (f1 , . . . , fn ) ∈ Gn we let N.f = (N f1 , . . . , N fn ). We write P (X ) for the power set of the set X , and Pκ (X ) for the set of subsets of X of cardinality less than κ. Then P (X ) is a boolean algebra, and each Pκ (X ) for κ inﬁnite is a ring of sets. Moreover, if ℵ0 ≤ κ < λ ≤ |X |+ , Pκ (X ) is an ideal of Pλ (X ), so we may study the quotient ring Pλ (X )/Pκ (X ), which is a boolean algebra just in the case where λ = |X |+ (that is, where Pλ (X ) = P (X )). In the remainder of this introductory section we give an outline of the main arguments of the paper. Our analysis of the quotient groups Sλ (µ)/Sκ (µ) is carried out using certain 2 many sorted structures Mκλµ and Nκλµ . (There is also a simpler version M∗ κλµ of Mκλµ applicable just in the case cf (κ) > 2ℵ0 .) These structures are devised with the object of describing the permutation action of tuples of elements of Sλ (µ), modulo small sets. The essential properties of such an n-tuple g = (g1 , g2 , . . . , gn ) are described by its action on the orbits of the subgroup g . In fact, if g1 and g 2 are n-tuples of elements of Sλ (µ) then g1 and g2 are conjugate if and only if the orbits of g 1 and g 2 can be put into 1–1 correspondence in such a way that the action of g1 on each orbit of g1 is isomorphic to that of g 2 on the corresponding orbit of g2 . Similar remarks apply in the quotient group, except that we have to allow fewer than κ ‘mistakes’ (by passing to equivalence classes of a suitable equivalence relation). These considerations lead us to observe that what should represent g in Mκλµ is a list of how many g -orbits there are of the various possible isomorphism types, where by ‘isomorphic’ here we mean ‘under the action of g’. Included among the sorts of Mκλµ are therefore, for each positive integer n, the family ISn of isomorphism types of pairs (A, f ), where f is an n-tuple of 3

Non-Dimensionalization

Non-dimensionalization of T umor-Immune ODESystem∗L.G.de Pillis and A.E.RadunskayaAugust22,2002Non-DimensionalizationRescaling and Dimensionless VariablesWhy is it important to rescale and non-dimensionalize?•T o SIMPLIFY the equations by reducing the number of variables.•T o analyze the behavior of the system,regardless of the UNITS used to measure the variables.Example:The Reynolds number of aﬂuid or T umorgrowth/normal growth.•T o rescale the parameters and variables so that all computed quantities are of relatively similar MAGNITUDES.2Non-DimensionalizationNotes for Rescaling and Dimensionless Variables slide:Answers:(1)simplify(2)units Note:The Reynolds number is the ratio of intertial force to the viscous force, and is deﬁned as:ULR=References:See[LS74],[Ban98]An Additional Example:The following example is included in case this material is unfamiliar,but may be omitted.It might also be given as an exercise,providing a review of simple initial value problems.A more involved problem is given in the exercises.2-2Non-DimensionalizationA Simple Preliminary ExampleSuppose that we want to test an exponential model of tumor growth against a listof data collected in an oncologist’s ofﬁce.We are ignoring the effect of theimmune system,as well as the competition between tumor cells for nutrients.Let T denote the number of tumor cells,and r the growth rate per cell.Thus,the differential equation describing growth is:dTNon-DimensionalizationA Simple Preliminary Example(continued)Suppose the detection threshold for tumors is approximately105cells.Therefore, the number of tumor cells present at the previous visit is T0<105cells.If T1is the number of tumor cells at the time of detection,andτis the time between visits,we canﬁnd the MINIMUM rate of growth by solving the initial value problem:dTNon-DimensionalizationPreliminary Example:Dimensionless Variables Introduce the dimensionless variable:Y=T/T0=TD t,our new‘time’,whereD is the doubling time of normal cells(≈10days).Using the chain rule,the differential equation in terms of the dimensionless variables is:dYln2r5Non-DimensionalizationNotes for Preliminary Example Dimensionless Variables slide:Answers:(1)T/(105cells)Note:This rescaling has the effect of reducing the order of magni-tude of this variable.(2)dimensionless Note:This may seem an odd choice of rescaling,but it simpliﬁes the end result.The factor of ln2comes in because the rate,r,is in the exponent,and measurements are usually made in terms of doublings,i.e.powers of2,rather than powers of e.(3)Dds =dYdtdtdTdtdt=1ln2rT=1ln2rY T0=D ds=kY has the solution Y0e ks.5-1Non-DimensionalizationPreliminary Example:Non-dimensionlized SolutionLet sτbe the time-data,to the time between doctor visits,(τ)expressed in termsof the variable s,i.e.sτ=τln2sτ .What have we gained by this procedure?•We can normalize a long list of data all at once,leaving us with one less parameter.•k is in terms relative to the growth rate of normal cells,(k=1⇒tumor cell doubling time=D(Normal Cell Doubling Time).•The values of Y are O(1),as opposed to O(105).6Non-DimensionalizationNotes for Preliminary Example Non-dimensionalized Solution slide:Answers:ln2(1)τsτ.Details:Previously,k was D105=Y(sτ)(where e ksτis the solution at the time of detection and Y(sτ)=T1)105.sτ(5)D Note:That is,when k=1,the tumor cell doubling equals the normal cell dou-bling time.Details:We know that Y(s)=e ks since Y0=Y(0)=1.T oﬁnd the doubling time,Y(s D)=e ks D=2Y0=2,which gives that s D=ln2s D=ing s D= ln2dt=s+pETdt=aT(1−bT)−nETWrite these differential equations in terms of the new variables:E∗=ˆEE T∗=ˆT T t∗=ˆt twhere the unit-carrying quantitiesˆE,ˆT andˆt will be chosen ing the chain rule:dE∗ˆt +pgˆT+T∗−mE∗T∗ˆtE∗dT∗ˆt T∗(1−bˆtˆEE∗T∗.7Notes for Step1Deﬁne Dimensionless Variables of the T umor-Immune System slide: Answers:(1)sˆEˆtE∗T∗ˆTˆt−dˆtT∗(1−bˆtˆEE∗T∗Details:Apply the chain rule,to get:dE∗dEdtdt=ˆEdtand a similar equation for dT∗/dt∗.After substituting the expressions for the starred variables on the right hand sides,rearrange to get the new equations.Note:“unit-carrying”means,for example,ˆE andˆT will be some quantity of cells,andˆt is some number of days.7-1Non-DimensionalizationStep2:Choose the Non-dimensionalizing ConstantsWe have3choices to make,and our goal is to reduce the number of PARAMETERS while rescaling very large and very small quantities.One option(among many):•Letˆt=a,i.e.scale time relative to THE TUMOR GROWTH RATE.•LetˆE=n/ˆt=n/a,i.e.scale the immune population relative to THE NUMBER OF TUMOR CELLS INACTIVATED BY EACH IMMUNE CELL(PER UNIT OF RESCALED TIME).•LetˆT=b,i.e.scale the tumor population relative to THE MAXIMUM TUMOR POPULATION(TUMOR CARRYING CAPACITY).8Notes for Choose the Non-dimensionalizing Constants slide:Answers:(1)three(ˆE,ˆT,andˆt).(2)parameters Note:There are many ways to choose these constants.The class might come up with different choices,which could then be compared.Our choice of constants differs from that of Kuznetsov[KMTP94].We comment on this further in a later slide.(3)the tumor growth rate(4)the number of tumor cells inactivated by each immune cell per unit of rescaled time(5)the maximum tumor population the tumor carrying capacityQuestion:What are the units of each of these quantities?Answer:ˆE andˆT are in units of cells−1,whileˆt is in units of days−1.8-1Non-DimensionalizationStep3:Deﬁne New(Dimensionless)ParametersDeﬁne new system parameters in order to simplify the equations:σ=sˆEa2,ρ=pab,δ=dNotes for Deﬁne New(Dimensionless)Parameters slide:Answers:(1).04414(2).6917(3).04038(4).9506(5).2289Question:What are the dimensions of each new parameter?Answer:None of the parameters(or variables)have dimension.(Verify this.)Note:In the paper[KMTP94]a different non-dimensionalization is used.The choice of non-dimensionalizing constants presented in these slides,in contrast,results in sim-pler non-dimensional equations than those in[KMTP94].This choice also has the ad-vantage of highlighting the relevant relationships between the original parameters.For example,the parameterδnow represents the death rate of the immune cells,(original parameter:d),relative to the birth rate of the tumor cells(original parameter:a).When the equations are in this form,it is clearer that it is the ratio of these two quantities which9-1will determine the fate of the system,rather than the values themselves.Since we have reduced the number of parameters as much as possible,and we are left withﬁve,we say that this system has“ﬁve degrees of freedom”.See[EK88,chapter4.4,p.126].9-2Non-DimensionalizationThe Equations in Final Form Step4:Write the non-dimensionalized equations:dE∗η+T∗−µET−δE∗dT∗η+T∗−µET−δE∗(2)T∗(1−T∗)−T∗E∗Note for bifurcation analysis:Our choice of parameters leaves the equation for the evolution of the tumor cell population invariant under parameter changes.This has the effect ofﬁxing the tumor-nullcline.The effector-nullcline,in contrast,will shift as parameters shift.We note that with Kuznetsov’s choice of non-dimensionalization,both the tumor-nullcline and the effector-nullcline are parameter dependent.The effect of parameter changing on Kuznetsov’s non-dimensionalized equations is discussed in the section on bifurcation analysis.We point out that either choice of non-dimensionalization is valid,since it is always the relative orientation of the null-clines which is important.Suggested Exercise:As an exercise,have the students go through Kuznetsov’s [KMTP94]non-dimensionalization,in preparation for the section on bifurcation analysis in which we return to using Kuznetsov’s equations.10-1References[Ban98]Robert Banks.T owing Icebergs,Falling Dominoes,and Other Adventures in Mathematics.Princeton University Press,1998.[EK88]Leah Edelstein-Keshet.Mathematical Models in Biology.Random House/Birkhauser,1988.[KMTP94]Vladmir A.Kuznetsov,Iliya A.Makalkin,Mark A.T aylor,and Alan S.Perel-son.Nonlinear dynamics of immunogenic tumors:Parameter estimation andglobal bifurcation analysis.Bulletin of Mathematical Biology,56(2),1994.[LS74] C.C.Lin and L.A.Segel.Mathematics Applied to Deterministic Problems in the Natural Sciences.Macmillan Publishing Co.,Inc.,1974.10-2。

慢加急性肝衰竭

GUIDELINESAcute-on-chronic liver failure:consensus recommendationsof the Asian Paciﬁc Association for the Study of the Liver (APASL)2014Shiv Kumar Sarin•Chandan Kumar Kedarisetty•Zaigham Abbas•Deepak Amarapurkar•Chhagan Bihari•Albert C.Chan•Yogesh Kumar Chawla•A.Kadir Dokmeci•Hitendra Garg•Hasmik Ghazinyan•Saeed Hamid•Dong Joon Kim•Piyawat Komolmit•Suman Lata•Guan Huei Lee•Laurentius A.Lesmana•Mamun Mahtab•Rakhi Maiwall•Richard Moreau•Qin Ning•Viniyendra Pamecha•Diana Alcantara Payawal•Archana Rastogi•Salimur Rahman•Mohamed Rela•Anoop Saraya•Didier Samuel•Vivek Saraswat•Samir Shah•Gamal Shiha•Brajesh Chander Sharma•Manoj Kumar Sharma•Kapil Sharma•Amna Subhan Butt•Soek Siam Tan•Chitranshu Vashishtha•Zeeshan Ahmed Wani•Man-Fung Yuen•Osamu Yokosuka•the APASL ACLF Working PartyReceived:4April2014/Accepted:25August2014/Published online:26September2014ÓAsian Paciﬁc Association for the Study of the Liver2014Abstract Theﬁrst consensus report of the working party of the Asian Paciﬁc Association for the Study of the Liver (APASL)set up in2004on acute-on-chronic liver failure (ACLF)was published in2009.Due to the rapid advancements in the knowledge and available information, a consortium of members from countries across Asia Paciﬁc,‘‘APASL ACLF Research Consortium(AARC),’’was formed in2012.A large cohort of retrospective and prospective data of ACLF patients was collated and fol-lowed up in this data base.The current ACLF deﬁnition was reassessed based on the new AARC data base.These initiatives were concluded on a2-day meeting in February 2014at New Delhi and led to the development of theﬁnal AARC consensus.Only those statements which were based on the evidence and were unanimously recommended were accepted.These statements were circulated again to all the experts and subsequently presented at the annual confer-ence of the APASL at Brisbane,on March14,2014.The suggestions from the delegates were analyzed by the expert panel,and the modiﬁcations in the consensus were made. Theﬁnal consensus and guidelines document was pre-pared.After detailed deliberations and data analysis,theS.K.Sarin(&)ÁC.K.KedarisettyÁH.GargÁR.MaiwallÁM.K.SharmaÁK.SharmaÁC.VashishthaÁZ.A.Wani Department of Hepatology,Institute of Liver and Biliary Sciences,New Delhi110070,Indiae-mail:shivsarin@Z.AbbasDepartment of Hepatogastroenterology,Sindh Institute of Urology and Transplantation,Karachi,PakistanD.AmarapurkarDepartment of Gastroenterology and Hepatology,Bombay Hospital and Medical Research,Mumbai,IndiaC.BihariÁA.RastogiDepartment of Pathology,Institute of Liver and Biliary Sciences, New Delhi110070,IndiaA.C.ChanDivision of Hepatobiliary and Pancreatic Surgery,and Liver Transplantation,Department of Surgery,The University of Hong Kong,Hong Kong,China Y.K.ChawlaDepartment of Hepatology,Post Graduate Institute of Medical Education and Research,Chandigarh,IndiaA.K.DokmeciDepartment of Gastroenterology,Ankara University School of Medicine,Ankara,TurkeyH.GhazinyanDepartment of Hepatology,Nork Clinical Hospital of Infectious Diseases,Yerevan,ArmeniaS.HamidÁA.S.ButtDepartment of Medicine,Aga Khan University Hospital, Karachi,PakistanD.J.KimCenter for Liver and Digestive Diseases,Hallym University Chuncheon Sacred Heart Hospital,Chuncheon,Gangwon-Do, Republic of KoreaHepatol Int(2014)8:453–471 DOI10.1007/s12072-014-9580-2original proposed deﬁnition was found to withstand the test of time and identify a homogenous group of patients pre-senting with liver failure.Based on the AARC data,liver failure grading,and its impact on the‘‘Golden therapeutic Window,’’extra-hepatic organ failure and development of sepsis were analyzed.New management options including the algorithms for the management of coagulation disor-ders,renal replacement therapy,sepsis,variceal bleed, antivirals,and criteria for liver transplantation for ACLF patients were proposed.Theﬁnal consensus statements along with the relevant background information are pre-sented here.Keywords Liver failureÁChronic liver diseaseÁCirrhosisÁAscitesÁAcute liver failure and Scute liver failureIntroductionLiver failure is a common medical ailment,and its inci-dence is increasing with the use of alcohol and growing epidemic of obesity and diabetes.It can present as acute liver failure(ALF)(in the absence of any preexisting liver disease),acute-on-chronic liver failure(ACLF)(an acute deterioration of known or unknown chronic liver disease), or an acute decompensation of an end-stage liver disease. Each of these is a well-deﬁned disease entity with a homogenous population of patients with expected out-comes.Due to an overlap and lack of clarity of deﬁnitions and outcomes,entities like late-onset liver failure and subacute hepatic failure have become less relevant and are not often used.The growing interest in ACLF after theﬁrst consensus deﬁnition of ACLF from APASL[1]is evident by the fact that more than200publications as full paper have been published and the trend is surely increasing.A seminal paper from the EASL-CLIF consortium on the deﬁnition and out-come of ACLF has since appeared[2]based on the work of experts from several European and Western countries.The group of investigators working on liver failure in the Asia–Paciﬁc region working for the past decade carefully analyzed the patient characteristics,natural history,and outcome over the years.The group met on yearly basis and collated data on Web site(www.aclf.in)since2009.The data were analyzed at meeting in China and Dhaka in2012,with the setting up of the APASL ACLF Research Consortium(AARC).The ret-rospective and prospective data of patients from different centers were analyzed,and the completed patient records were utilized for deﬁning predictors of mortality and grades of liver failure and incidence of other organ failures.Experts from all over the globe,especially from the Asia–Paciﬁc region,and members of theﬁrst consensus group were requested to identify pertinent and contentious issues in ACLF.Six major contentious issues and unmet needs in the management of ACLF were approached for the update:(1)what constitutes an acute insult;(2)whether chronic liver disease should be included or only cirrhosis of the liver in deﬁning underlying liver disease;(3)the role of SIRS and sepsis as a cause or consequence of liver failure;(4)the incidence and impact of non-hepatic organ failures;(5)the relevance and grades of liver failure,the urgency, and outcome of liver transplant;and(6)an AARC pre-diction model of outcome of ACLF.The process for the development of the recommendations and guidelines included review of all available published literature onP.KomolmitDivision of Gastroenterology and Hepatology,Department of Medicine,Chulalongkorn University,Bangkok,ThailandtaDepartment of Nephrology,Institute of Liver and Biliary Sciences,New Delhi110070,IndiaG.H.LeeDepartment of Gastroenterology and Hepatology,National University Health System,Singapore,SingaporeL.A.LesmanaDivision of Hepatology,University of Indonesia,Jakarta, IndonesiaM.MahtabÁS.RahmanDepartment of Hepatology,Bangabandhu Sheikh Mujib Medical University,Dhaka,BangladeshR.MoreauInserm,U1149,Centre de recherche sur l’Inﬂammation(CRI), Paris,France R.MoreauUMR_S1149,Labex INFLAMEX,Universite´Paris Diderot Paris7,Paris,FranceR.MoreauDe´partement Hospitalo-Universitaire(DHU)UNITY,Service d’He´patologie,Hoˆpital Beaujon,APHP,Clichy,FranceQ.NingDepartment of Infectious Disease,Tongji Hospital of Tongji Medical College,Huazhong University of Science and Technology,Wuhan,ChinaV.PamechaDepartment of Hepatobiliary Surgery,Institute of Liver and Biliary Sciences,New Delhi110070,IndiaD.A.PayawalDepartment of Hepatology,Cardinal Santos Medical Center, Manila,PhilippinesACLF by individual and group of experts;preparation of a review manuscript and consensus statements based on Oxford system of evidence-based approach[3]for devel-oping the consensus statements,circulation of all consen-sus statements to all experts,an effort to deﬁne the acute hepatic insults;the underlying chronic liver disease,a survey of the current approaches for the diagnosis and management of ACLF;discussion on contentious issues; and deliberations to prepare the consensus statement by the experts of the working party.A2-day meeting was held on February22–23,2014,at New Delhi,India,to discuss and ﬁnalize the recommendations and guidelines.These state-ments were circulated to all the experts,posted on the AARC Web site(www.aclf.in),and subsequentlyﬁnalized. These consensus statements and guidelines for the man-agement of such patients are included in this review.A brief background is included providing the available data and published information on each issue.Statements from theﬁrst consensus have been reproduced at places to give a background and continuity.The concept of ACLF and need for a deﬁnitionAcute liver failure is a well-deﬁned medical emergency which is deﬁned as a severe liver injury,leading to coag-ulation abnormality usually with an INR C1.5,and any degree of mental alteration(encephalopathy)in a patient without pre-existing liver disease and with an illness of up to4weeks duration[4].A proportion of patients who present with features mimicking ALF,however,have an underlying chronic liver disease or cirrhosis of the liver. These patients grouped together as acute-on-chronic liver failure(ACLF)also have a poor outcome.These patients are distinctly different from a group of cirrhotic patients who are already decompensated and have a sudden worsening of their condition due to an acute event as liver failure is central.The ACLF is a clinical syndrome manifesting as acute and severe hepatic derangements resulting from varied insults.This term wasﬁrst used in1995to describe a condition in which two insults to liver are operating simultaneously,one of them being ongoing and chronic, and the other acute[5].Over the years,nearly thirteen different deﬁnitions have been proposed,creating confu-sion in theﬁeld[6].Any patient who has an underlying chronic liver disease with superimposed acute insult is being labeled as having ACLF,irrespective of evidence of liver failure per se or evidence of preexisting cirrhotic decompensation.Several investigators were concerned that this would lead to overlap with decompensated liver dis-ease.The main emphasis of the third consensus meeting of the APASL working party was to identify from this large group of patients,a subset of patients who have a relatively homogenous presentation and potentially similar outcome and restrict the use of the term‘‘acute-on-chronic liver failure’’to this subset.The2009APASL deﬁnition had provided a basis to select patients presenting with a distinct syndrome.To cover the entire spectrum of these patients, from mild to most severe,patients with chronic liver dis-ease with or without cirrhosis of the liver were included and carefully analyzed.It is understandable,though not well deﬁned,that the nature and degree acute insult and the status of the underlying chronic liver disease would determine the outcome of the patient(Fig.1).To give clarity to the primary event,a hepatic insult, jaundice and coagulopathy,which deﬁned liver failure was considered essential.In acute liver failure,though hepatic encephalopathy is part of the deﬁnition,it follows liver failure.Encephalopathy in the absence of overt jaundice or liver failure is separately categorized as due to by-pass[7]. Should one wait for deﬁning the outcome of‘‘liver failure’’M.RelaInstitute of Liver Diseases and Transplantation,Global Health City,Chennai,IndiaA.SarayaDepartment of Gastroenterology,All India Institute of Medical Sciences,New Delhi,IndiaD.SamuelINSERM,Centre He´patobiliarie,Hoˆpital Paul Brousse,Villejuif, FranceV.SaraswatDepartment of Gastroenterology,Sanjay Gandhi Post Graduate Institute of Medical Sciences,Lucknow,IndiaS.ShahDepartment of Gastroenterology and Hepatology,Global Hospitals,Mumbai,India G.ShihaDepartment of Internal Medicine,Egyptian Liver Research Institute and Hospital,Cairo,EgyptB.C.SharmaDepartment of Gastroenterology,GB Pant Hospital,New Delhi, IndiaS.S.TanDepartment of Gastroenterology and Hepatology,Selayang Hospital,Kepong,MalaysiaM.-F.YuenDepartment of Medicine,The University of Hong Kong, Hong Kong,ChinaO.YokosukaDepartment of Gastroenterology and Nephrology,Graduate School of Medicine,Chiba University,Chiba,Japantill the time extra-hepatic organ failures set in or not remain contentious.For deﬁnition,the event must be universally present in all patients.From the point of view of intensi-vists,it is well known that with increasing number of organ dysfunction or failure,the mortality would cumulatively increase.Undoubtedly,these events are predictive of the outcome,the basis of SOFA score.It is therefore not sur-prising;the same has been reported in the CANONIC study [2].However,should organ failure be included in deﬁning the clinical syndrome of liver failure needs a thorough analysis.As a corollary,despite decades of extensive experience,renal or circulatory dysfunction has not been included in the deﬁnition of ALF.The issue whether sepsis per se could lead to liver failure or is a result of liver failure had been debated for many years and was again revisited. However,sepsis is an integral part of development of multi-organ failure in any patient,be it of renal,pancreatic, or cardiac origin.The differences between the current deﬁnitions of CLIF consortium and APASL have been recently published[8].While theﬁrst APASL consensus was based on the data of only about200patients,the data of1700patients are now available from14countries.Records of1,363ACLF patients were analyzed.This formed the basis of re-eval-uating the validity of the APASL2009consensus.It was decided that,like in other studies,the analysis of the original data should be sent for separate publications and only the conclusions and recommendations based on these data can be used for the purpose of the consensus.To improve our understanding of the West,Prof Richard Moreau,theﬁrst author of the CANONIC study,kindly consented to join the consensus meeting.The6major issues as mentioned above,and28sub-issues,were deﬁned,and systematic reviews were made available to all participants.These were addressed at length in the meeting.What constitutes an acute insultThis issue was divided into two parts:ﬁrst,what is the time frame for the term‘‘acute,’’and second,what are the cri-teria to deﬁne the nature of an‘‘insult.’’A review of the different published deﬁnitions of acute liver failure and ACLF was done,and the current APASL deﬁnition of ACLF was reassessed.It was clear that the event must be new and acute,and its impact on the patient’s condition should be observable as liver failure within a given time frame.The EASL-AASLD consortium had initially kept the assessment of outcomes at3months[9],but subse-quently revised it to28days in the recent CANONIC study [2].The AARC data were carefully analyzed,and themortality rates were at different time points.A mortality rate of more than33%at4weeks was considered to be signiﬁcant allowing recovery to less than two-thirds of the ing these criteria,the data showed that more than50%patients of ACLF die by week 4.It was, therefore,unanimously agreed that the4-week(28days) period should be maintained as per the initial deﬁnition for deﬁning the impact of an acute event.Efforts were made in light of all the available data on deﬁning the nature of acute event.The acute insult could vary depending on the geographic region and the popu-lation under study.These include both infectious and non-infectious causes.These were well characterized in the past.While Hepatitis B reactivation remains the pre-dominant cause of acute hepatic insult in the East,from the global perspective,the major etiologic agent was alcohol,both in the West and the East.This was a bit unexpected for the Asian countries where alcoholic hep-atitis is emerging as a major acute insult and shows the growing westernization of Asia.The predominant causes of acute hepatic insults are shown in Fig.2.A review of the recent CANONIC study data showed that in the West, the term precipitating event is generally used and proba-bly details of events such as Hepatitis B or superadded Hepatitis A and E are rarely encountered or recorded[2]. However,it was a bit surprising that active alcohol abuse and alcoholic hepatitis were also not the predominant causes.A plausible explanation could be that since the CANONIC study only recorded the acute decompensation of cirrhosis and not the hepatic insults,the major events recorded were only non-hepatic,such as bacterial infec-tions or sepsis.Acute decompensation of cirrhosis is a different entity than ACLF.As the core premise of ACLF is presented as liver failure,the acute insults should be hepatic insults.Both,hepatotropic or non-hepatotropic insults,should manifest in the patientﬁrst with liver failure.Acute hepatic insults of infectious etiology included reactivation of Hepatitis B virus(HBV)as the leading cause of ACLF in the Asian region[10–19].Reactivation of HBV could be either spontaneous or due to intensive chemotherapy or immunosuppressive therapy[10,11], immune restoration after highly active antiretroviral ther-apy for HIV[12,13],treatment-related[14],or reactivation of the occult HBV infection by rituximab(anti-CD20)-based chemotherapy[15–17].Similarly,reactivation of Hepatitis C virus infection has also been reported,espe-cially after immune suppressive therapy[18,19].The other very important infectious etiology of the acute event is super-infection with Hepatitis E virus,predominantly in patients in the Indian subcontinent[20–23].Various bac-terial,parasitic,and fungal infections may affect the liver.Spirochetal,protozoal,helminthic,or fungal organisms may directly infect the liver,whereas bacterial or parasitic infection may spread to the liver from other sites[24]. These infections may lead to liver failure in patients with underlying chronic liver disease.Among the non-infectious etiologies,alcoholic hepatitis is the major cause of acute deterioration in stable known or unknown chronic liver diseases,more often in the western countries[25,26].It was not clear what should be the interval from the last alcoholic drink to be included as acute insult.Since,after the direct hepatic injury,the immunologic injury starts to decline[27],a period of28days was considered adequate for inclusion as the last drink.The issue which remains to be addressed was of binge drinking in patients with ACLF due to recent alcohol intake.It was appreciated that a prospective data collection including the drinking behavior especially in the past6months would help decide the inﬂuence of drinking behavior on the clinical outcome and help in deﬁning the time frame of what should be consid-ered as an acute insult.Hepatotoxic drugs and complimentary and alternative medicines(CAM)are important causes for acute and acute-on-chronic liver failure in the Asia–Paciﬁc region[28]. Hepatitis following the use of anti-tubercular drugs was considered to be an important cause of acute insult leading to ACLF.In a proportion of patients,despite a history of use of CAM,the precise nature and injurious inﬂuence of the agent cannot be determined.The need for further data on the hepatic injury caused by different herbal prepara-tions needs to be studied.Acute variceal bleeding has been included as one of the events to deﬁne hepatic decompensation in the natural history of cirrhosis[29].Variceal bleeding has also been taken as an acute insult for ACLF in some western trials of ACLF.It was extensively debated whether to consider variceal bleed as an acute event of ACLF.Since the deﬁ-nition of ACLF includes liver failure,jaundice,and coag-ulopathy,the variceal bleed should result in liver failure. The liver failure in such patients is mainly due to hepatic ischemia[30]and subsequent bacterial infections[31].It was discussed that for a patient with portal hypertension and cirrhosis of the liver who presents for theﬁrst time with variceal bleed without any previous or present signs or symptoms of chronic liver disease,it would not constitute an acute insult.This is especially relevant if such a patient does not develop any jaundice.The experts discussed the stratiﬁcation of patients based on the stage of underlying liver disease and the severity of variceal bleed.However, since patients with ACLF never decompensated before and are distinct from patients with decompensated cirrhosis,it is unlikely that a variceal bleed would per se lead to sig-niﬁcant liver failure manifesting as jaundice and coagu-lopathy.Based on the data,it was unanimously agreed that acute variceal bleeding is not an acute hepatic insult unless in the patients where it produces jaundice and coagulopa-thy deﬁning ACLF.A scenario may exist that a patient who has already fulﬁlled the criteria of ACLF,and has been diagnosed ACLF,develops a variceal bleed.In such a patient,variceal bleed would be considered as a complication in the natural history of ACLF.The issue of other non-hepatotropic insults which have been considered in other studies such as surgery,trauma, insertion of transjugular intrahepatic porto-systemic shunt, transartrial chemoembolization,or radiofrequency ablation for hepatocellular carcinoma was discussed in detail.While there is an indirect connection with each of these,it was debated that a patient who already has cirrhosis with HCC or a cirrhotic who undergoes surgery,and separate risk scores are already in practice and being utilized.The likely potential for hepatic decompensation would vary depend-ing on the nature of intervention and underlying hepatic reserve.It was agreed that non-hepatotropic insults pro-ducing direct hepatic insult and ACLF in an otherwise compensated liver disease could be considered as acute hepatic insults(2b,C).In a proportion of patients in Asia or even in the west,the precise agent(s)leading to acute hepatic insult is not well recognized on routine assessment. In such patients,this should be recorded as such. RecommendationsDeﬁning the acute event in ACLFThe ACLF can develop from one or more clearly deﬁned acute hepatic insults,which can be due to hepatotropic or non-hepatotropic agents/causes.Acute insults vary depending on the geographic region and the population under study.Major etiologic agents responsible for pre-cipitating ACLF are as follows:1.1Hepatotropic viral infections(1a,A).1.1.1Among these,reactivation of Hepatitis Bvirus(HBV)infection and super-infectionwith HEV are the major causes of acuteinsult in ACLF(1a,A).1.1.2Among the non-infectious causes,activealcohol consumption(within the last28days)remains the commonest cause(1a,A).1.1.3Drug-induced liver injury,consumption ofcomplimentary and alternative medicines(CAM),severe autoimmune hepatitis,andﬂare of Wilson’s disease are other causesof acute insult in ACLF(1a,A).1.2Non-hepatotropic insults like surgery,trauma,andviral infections if producing direct hepatic insultcould lead to ACLF(2b,C).1.3Variceal bleed per se may not qualify as an acuteinsult for ACLF,and we need more data toascertain this(5,D).1.4In a proportion of patients,the acute hepatic insultmay not be identiﬁable by the current routineassessment(5,D).Deﬁning the underlying chronic liver diseaseTwo aspects were carefully analyzed,what constitutes chronic liver disease,cirrhosis alone or non-cirrhotic chronic liver diseases,and the etiology of the chronic liver disease.The degree of hepaticﬁbrosis and the functional hepa-tocellular mass remains heterogeneous in patients with the chronic hepatitis[32,33].Even in patients with stage IV disease,critical mass varies according to the parenchymal reserves.Modiﬁed Laennec Scoring System divides stage IV further,according to the thickness of septa into three, ending up in six stages altogether[34,35].Moreover, ACLF is not equivalent to the acute decompensation of cirrhosis,which is the result of parenchymal extinction. Majority of the ACLF patients present with liver failure without any previous assessment of liver disease.It is not possible to distinguish accurately patients with different degree ofﬁbrosis at this point in time.The liver with any signiﬁcant degree ofﬁbrosis,with activated stellate cells, and inﬁltrated by the inﬂammatory cells,is expected to respond in a different way to the acute insult compared to the liver without inﬂammatory inﬁltrate[36].The NAFLD is the leading cause of donor rejection in liver transplantation[37].Experience from liver trans-plantation centers shows that steatosis[30%in the donor liver is associated with a higher risk of primary non-function and graft initial poor function as compared to grafts with no or\30%steatosis[38].Patients with met-abolic syndrome and fatty liver,diabetics,male patients of age[45–50years,patients with obesity,and dyslipidemia have the increased risk ofﬁbrosis[39].While cirrhosis could be a late event,a large proportion of them may have stage2or3ﬁbrosis.Hence,NASH is indeed an important cause of chronic liver disease[40].Furthermore,in the East,a large proportion of patients do have reactivation of chronic Hepatitis B.In these patients,while liver failure and ACLF-like presentation does develop,cirrhosis is not necessarily present.The AARC data,based on the liver biopsy studies,corroborated the facts that a fair proportion of patients with ACLF do not have underlying cirrhosis, but still carry a poor prognosis,with mortality above33% at4weeks.Based on the available data,the published literature and the validity of the2009consensus on including the non-cirrhotic chronic liver disease were reafﬁrmed.Accurate and reliable assessment of underlying CLD in the setting of ACLF is important for the subsequent man-agement and need for liver transplant in these patients. Diagnosis of chronic liver disease in the setting of ACLF is made by history,physical examination,and previously available or recent laboratory,endoscopic or radiologic investigations[41].Ultrasound and CT abdomen may pick up CLD.However,to assess the degree ofﬁbrosis in an un-shrunken liver would require other radiologic modalities. The current noninvasive tests cannot clearly diagnose the presence of chronic liver disease in the presence of inﬂammation and liver failure.Hence,liver biopsy through the transjugular route remains an important tool to conﬁrm the stage ofﬁbrosis and presence of cirrhotic or non-cir-rhotic liver disease.A liver biopsy through the transjugular route may be of help when the presence of already underlying CLD and the cause of liver disease are not clear.The liver biopsy may highlight the etiology,stage ofﬁbrosis and prognosis,and outcome in patients with ACLF[42].In addition,transju-gular access directly into the hepatic vein allows the hepatic venous pressure gradient to be measured(HVPG). There is a risk of bleeding leading to hemobilia,hemo-peritoneum,and hepatic hematoma in the setting of the deranged clotting proﬁle[43].The need of liver biopsy in ACLF should therefore be individualized.Standardization of liver biopsy assessment would help a uniform approach to the diagnosis and treatment for CLD and the acute insult.There is a need to have reliable noninvasive tools to assess the severity ofﬁbrosis in a previously undiagnosed CLD.Ultrasound and CT abdomen may pick up CLD. However,to assess the degree ofﬁbrosis in an un-shrunken liver would require other radiologic modalities.Transient elastography(ﬁbroscan)is a good modality to detect ﬁbrosis radiologically[44].However,the liver tissue stiffness may also increase with hepatitis,steatosis,and inﬂammation present in the ACLF setting[45].The second issue was about the etiology of chronic liver disease and cirrhosis in the Asian–Paciﬁc region.Experts reviewed the data from the AARC,and the etiologic proﬁle of cirrhosis in ACLF was found to be similar to the etiol-ogy of cirrhosis in general in the respective countries[26, 46,47].With the rising incidence of obesity and NAFLD, proportion of burnt-out NASH presenting as cryptogenic cirrhosis is also increasing[48–50].Viral serology and nucleic acid testing are required to identify viral etiology.Specialized tests to rule to diagnose metabolic and autoimmune diseases would be needed as well. The presence of stigmata of liver disease on clinical exami-nation,low platelets,evidence of synthetic dysfunction in。

Analogue of Weil representation for abelian schemes

a r X i v :a l g -g e o m /9712021v 1 19 D e c 1997ANALOGUE OF WEIL REPRESENTATION FOR ABELIANSCHEMESA.POLISHCHUK This paper is devoted to the construction of projective actions of certain arithmetic groups on the derived categories of coherent sheaves on abelian schemes over a normal base S .These actions are constructed by mimicking the construction of Weil in [27]of a projective representation of the symplectic group Sp(V ∗⊕V )on the space of smooth functions on the lagrangian subspace V .Namely,we replace the vector space V by an abelian scheme A/S ,the dual vector space V ∗by the dual abelian scheme ˆA ,and the space of functions on V by the (bounded)derived category of coherent sheaves on A which we denote by D b (A ).The role of the standard symplectic form on V ⊕V ∗is played by the line bundle B =p ∗14P ⊗p ∗23P −1on (ˆA ×S A )2where P is the normalized Poincar´e bundle on ˆA ×A .Thus,the symplectic group Sp(V ∗⊕V )is replaced by the group of automorphisms of ˆA ×S A preserving B .We denote the latter group by SL 2(A )(in [20]the same group is denoted by Sp(ˆA ×S A )).We construct an action of a central extension of certain ”congruenz-subgroup”Γ(A,2)⊂SL 2(A )on D b (A ).More precisely,if we write an element of SL 2(A )in the block form a 11a 12a 21a 22 then the subgroup Γ(A,2)is distinguished by the condition that elements a 12∈Hom(A,ˆA )and a 21∈Hom(ˆA,A )are divisible by 2.We construct autoequivalences F g of D b (A )corresponding to elements g ∈Γ(A,2),such the composition F g ◦F g ′diﬀers from F gg ′by tensoring with a line bundle on S and a shift in the derived category.Thus,we get an embedding of the central extension of Γ(A,2)by Z ×Pic(S )into the group of autoequivalences of D b (A ).The 2-cocycle of this central extension is described by structures similar to the Maslov index of a triple of lagrangian subspaces in a symplectic vector space.However,the situation here is more complicated since the construction of the functor F g requires a choice of a Schr¨o dinger representation of certain Heisenberg group G g associated with g .The latter is a central extension of a ﬁnite group scheme K g over S by G m such that the commutation pairing K g ×K g →G m is non-degenerate.If the order of K g is d 2then a Schr¨o dinger representation of G g is a representation of G g in a vector bundle of rank d over S such that G m ⊂G g acts naturally.Any two such representations diﬀer by tensoring with a line bundle on S .The classical example due to D.Mumford arises in the situation when there is1a relatively ample line bundle L on an abelian schemeπ:A→S.Then the vector bundleπ∗L on S is a Schr¨o dinger representation of the Mumford group G(L)attached to L,this group is a central extension of theﬁnite group scheme K(L)by G m where K(L)is the kernel of the symmetric homomorphismφL:A→ˆA associated with L. Our Heisenberg groups G g are subgroups in some Mumford groups and there is no canonical choice of a Schr¨o dinger representation for them in general(this ambiguity is responsible for the Pic(S)-part of our central extension ofΓ(A,2)).Moreover, unless S is a spectrum of an algebraically closedﬁeld,it is not even obvious that a Schr¨o dinger representation for G g exists.Our main technical result that deals with this problem is the existence of a”Schr¨o dinger representation”forﬁnite symmetric Heisenberg group schemes of odd order established in section2.Further we observe that the obstacle to existence of such a representation is an elementδ(G g)in the Brauer group of S,and that the map g→δ(G g)is a homomorphism.This allows us to use the theory of arithmetic groups to prove the vanishing ofδ(G g).When an abelian scheme A is equipped with some additional structure(such as a symmetric line bundle)one can sometimes extend the above action of a central exten-sion ofΓ(A,2)on D b(A)to an action of a bigger group.The following two cases seem to be particularly interesting.Firstly,assume that a pair of line bundles on A andˆA is given such that the composition of the corresponding isogenies between A andˆA is the morphism of multiplication by some integer N>0.Then we can construct an action of a central extension of the principal congruenz-subgroupΓ0(N)⊂SL2(Z) on D b(A)(note that in this situation there is a natural embedding ofΓ0(N)into SL2(A)but the image is not necessarily contained inΓ(A,2)).Secondly,assume that we have a symmetric line bundle L on A giving rise to a principal polarization.Then there is a natural embedding of Sp2n(Z)into SL2(A n),where A n denotes the n-fold ﬁbered product over S,which is an isomorphism when End(A)=Z.In this case we construct an action of a central extension of Sp2n(Z)on D b(A n).The main point in both cases is to show the existence of relevant Schr¨o dinger representations.Both these situations admit natural generalizations to abelian schemes with real multipli-cation treated in the same way.For example,we consider the case of an abelian scheme A with real multiplication by a ring of integers R in a totally real number ﬁeld,equipped with a symmetric line bundle L such thatφL:A →ˆA is an R-linear principal polarization.Then there is an action of a central extension of Sp2n(R)by Z×Pic(S)on D b(A n).When R=Z we determine this central extension explicitly using a presentation of Sp2n(Z)by generators and relations.It turns out that the Pic(S)-part of this extension is induced by a non-trivial central extension of Sp2n(Z) by Z/2Z via the embedding Z/2Z֒→Pic(S)given by an element(π∗L)⊗4⊗ωA is the restriction of the relative canonical bundle of A/S to the zero section.Also we show that the restriction of this central extension to certain congruenz-subgroup Γ1,2⊂Sp2n(Z)splits.In the case when S is the spectrum of the algebraically closedﬁeld the constructions2of this paper were developed in[22]and[20].In the latter paper the Z-part of the above central extension is described in the analytic situation.Also in[22]we have shown that the above action of an arithmetic group on D b(A)can be used to construct an action of the corresponding algebraic group over Q on the(ungraded)Chow motive of A.In the present paper we extend this to the case of abelian schemes and their relative Chow motives.Under the conjectural equivalence of D b(A n)with the Fukaya category of the mir-ror dual symplectic torus(see[12])the above projective action of Sp2n(Z)should correspond to a natural geometric action on the Fukaya category.The central exten-sion by Z appears in the latter situation due to the fact that objects are lagrangian subvarieties together with some additional data which form a Z-torsor(see[12]). The paper is organized as follows.In theﬁrst two sections we studyﬁnite Heisen-berg group schemes(non-degenerate theta groups in the terminology of[14])and their representations.In particular,we establish a key result(Theorem2.3)on the existence of a Schr¨o dinger representation for a symmetric Heisenberg group scheme of odd order.In section3we consider another analogue of Heisenberg group:the central extension H(ˆA×A)ofˆA×S A by the Picard groupoid of line bundles on S.We develop an analogue of the classical representation theory of real Heisenberg groups for H(ˆA×A).Schr¨o dinger representations forﬁnite Heisenberg groups enter into this theory as a key ingredient for the construction of intertwining operators.In section4we construct a projective action ofΓ(A,2)on D b(A),in section5—the corresponding action of an algebraic group over Q on the relative Chow motive of A. In section6we study the group SL2(A),the extension of SL2(A)which acts on the Heisenberg groupoid.In section7we extend the action ofΓ(A,2)to that of a bigger group in the situation of abelian schemes with real multiplication.In section8we study the corresponding central extension of Sp2n(Z).All the schemes in this paper are assumed to be noetherian.The base scheme S is always assumed to be connected.We denote by D b(X)the bounded derived category of coherent sheaves on a scheme X.For a morphism of schemes f:X→Y ofﬁnite cohomological dimension we denote by f∗:D b(X)→D b(Y)(resp.f∗:D b(Y)→D b(X))the derived functor of direct(resp.inverse)image.For any abelian scheme A over S we denote by e:S→A the zero section.For an abelian scheme A(resp. morphism of abelian schemes f)we denote byˆA(resp.ˆf)the dual abelian scheme (resp.dual morphism).For every line bundle L on A we denote byφL:A→ˆA the corresponding morphism of abelian schemes(see[17]).When this is reasonable a line bundle on an abelian scheme is tacitly assumed to be rigidiﬁed along the zero section(one exception is provided by line bundles pulled back from the base).For every integer n and a commutative group scheme G we denote by[n]=[n]G:G→G the multiplication by n on G,and by G n⊂G its kernel.We use freely the notational analogy between sheaves and functions writing in particular F x= Y G y,x dy,where x∈X,y∈Y,F∈D b(X),G∈D b(Y×X),instead of F=p2∗(G).31.Heisenberg group schemesLet K be aﬁniteﬂat group scheme over a base scheme S.Aﬁnite Heisenberg group scheme is a central extension of group schemes(1.1)0→G m→G p→K→0such that the corresponding commutator form e:K×K→G m is a perfect pairing. Let A be an abelian scheme over S,L be a line bundle on A trivialized along the zero section.Then the group scheme K(L)={x∈A|t∗x L≃L}has a canonical central extension G(L)by G m(see[17]).When K(L)isﬁnite,G(L)is aﬁnite Heisenberg group scheme.A symmetric Heisenberg group scheme is an extension0→G m→G→K→0 as above together with an isomorphism of central extensions G →[−1]∗G(identical on G m),where[−1]∗G is the pull-back of G with respect to the inversion morphism [−1]:K→K.For example,if L is a symmetric line bundle on an abelian scheme A (i.e.[−1]∗L≃L)with a symmetric trivialization along the zero section then G(L)is a symmetric Heisenberg group scheme.For any integer n we denote by G n the push-forward of G with respect to the morphism[n]:G m→G m.For any pair of central extensions(G1,G2)of the same group K we denote by G1⊗G2their sum(given by the sum of the corresponding G m-torsors).Thus,G n≃G⊗n.Note that we have a canonical isomorphism of central extensions(1.2)G−1≃[−1]∗G opwhere[−1]∗G op is the pull-back of the opposite group to G by the inversion morphism [−1]:K→K.In particular,a symmetric extension G is commutative if and only if G2is trivial.Lemma1.1.For any integer n there is a canonical isomorphism of central exten-sions[n]∗G≃G n(n+1)2where[n]∗G is the pull-back of G with respect to the multiplication by n morphism [n]:K→K.In particular,if G is symmetric then[n]∗G≃G n2.Proof.The structure of the central extension G of K by G m is equivalent to the following data(see e.g.[3]):a cube structure on G m-torsor G over K and a trivial-ization of the corresponding biextensionΛ(G)=(p1+p2)∗G⊗p∗1G−1⊗p∗2G−1of K2. Now for any cube structure there is a canonical isomorphism(see[3])[n]∗G≃G n(n+1)2which is compatible with the natural isomorphism of biextensions([n]×[n])∗Λ(G)≃Λ(G)n2≃Λ(G)n(n+1)2.4The latter isomorphism is compatible with the trivializations of both sides when G arises from a central extension.Remark.Locally one can choose a splitting K→G so that the central extension is given by a2-cocycle f:K×K→G m.The previous lemma says that for any 2-cocycle f the functions f(nk,nk′)and f(k,k′)n(n+1)2diﬀer by a canonical coboundary.In fact this coboundary can be written explicitly in terms of the functions f(mk,k)for various m∈Z.Proposition1.2.Assume that K is annihilated by an integer N.If N is odd then for any Heisenberg group G→K the central extension G N is canonically trivial, otherwise G2N is trivial.If G is symmetric and N is odd then G N(resp.G2N if N is even)is trivial as a symmetric extension.bining the previous lemma with(1.2)we get the following isomorphism: [n]∗G≃G n(n+1)2≃G n⊗(G⊗G op−1)n(n−1)2=1for n=2N(resp.for n=N if N is odd).Hence, the triviality of G n in these cases.Corollary1.3.Let G→K be a symmetric Heisenberg group such that the order of K over S is odd.Then the G m-torsor over K underlying G is trivial.Proof.The isomorphism(1.2)implies that the G m-torsor over K underlying G2is trivial.Together with the previous proposition this gives the result.If G→K is a(symmetric)Heisenberg group scheme,such that K is annihilated by an integer N,n is an integer prime to N then G n is also a(symmetric)Heisenberg group.When N is odd this group depends only on the residue of n modulo N(due to the triviality of G N).We call aﬂat subgroup scheme I⊂K G-isotropic if the central extension(1.1) splits over I(in particular,e|I×I=1).Ifσ:I→G is the corresponding lifting,then we have the reduced Heisenberg group scheme0→G m→p−1(I⊥)/σ(I)→I⊥/I→0where I⊥⊂K is the orthogonal complement to I with respect to e.If G is a symmetric Heisneberg group,then I⊂K is called symmetrically G-isotropic if the restriction of the central extension(1.1)to I can be trivialized as a symmetric ex-tension.Ifσ:I→G is the corresponding symmetric lifting them the reduced Heisenberg group p−1(I⊥)/σ(I)is also symmetric.5Let us deﬁne the Witt group WH sym (S )as the group of isomorphism classes of ﬁnite symmetric Heisenberg groups over S modulo the equivalence relation generated by[G ]∼[p −1(I ⊥)/σ(I )]for a symmetrically G -isotropic subgroup scheme I ⊂K .The (commutative)addition in WH sym (S )is deﬁned as follows:if G i →K i (i =1,2)are Heisenberg groups with commutator forms e i then their sum is the central extension0→G m →G 1×G m G 2→K 1×K 2→0so that the corresponding commutator form on K 1×K 2is e 1⊕e 2.The neutral element is the class of G m considered as an extension of the trivial group.The inverse element to [G ]is [G −1].Indeed,there is a canonical splitting of G ×G m G −1→K ×K over the diagonal K ⊂K ×K ,hence the triviality of [G ]+[G −1].We deﬁne the order of a ﬁnite Heisenberg group scheme G →K over S to be the order of K over S (specializing to a geometric point of S one can see easily that this number has form d 2).Let us denote by WH ′sym (S )the analogous Witt group of ﬁnite Heisenberg group schemes Gover S of odd order.Let also WH(S )and WH ′(S )be the analogous groups deﬁned for all (not necessarily symmetric)ﬁnite Heisenberg groups over S (with equivalence relation given by G -isotropic subgroups).Remark.Let us denote by W(S )the Witt group of ﬁnite ﬂat group schemes over S with non-degenerate symplectic G m -valued forms (modulo the equivalence relation given by global isotropic ﬂat subgroup schemes).Let also W ′(S )be the analogous group for group schemes of odd order.Then we have a natural homomorphism WH(S )→W(S )and one can show that the induced map WH ′sym →W ′(S )is anisomorphism.This follows essentially from the fact that a ﬁnite symmetric Heisen-berg group of odd order is determined up to an isomorphism by the corresponding commutator form,also if G →K is a symmetric ﬁnite Heisenberg group with the commutator form e ,I ⊂K is an isotropic ﬂat subgroup scheme of odd order,then there is a unique symmetric lifting I →G .Theorem 1.4.The group WH sym (S )(resp.WH ′sym (S ))is annihilated by 8(resp.4).Proof .Let G →K be a symmetric ﬁnite Heisenberg group.Assume ﬁrst that the order N of G is odd.Then we can ﬁnd integers m and n such that m 2+n 2≡−1mod(N ).Let αbe an automorphism of K ×K given by a matrix m −n n m .Let G 1=G ×G m G be a Heisenberg extension of K ×K representing the class 2[G ]∈WH ′sym (S ).Then from Lemma 1.1and Proposition 1.2we get α∗G 1≃G −11,hence 2[G ]=−2[G ],i.e.4[G ]=0in WH ′(S ).If N is even we can apply the similar argument to the 4-th cartesian power of G and the automorphism of K 4given by an integer 4×4-matrix Z such that Z t Z =6(2N−1)id.Such a matrix can be found by considering the left multiplication by a quaternion a+bi+cj+dk where a2+b2+c2+d2=2N−1.2.Schr¨o dinger representationsLet G be aﬁnite Heisenberg group scheme of order d2over S.A representation of G of weight1is a locally free O S-module together with the action of G such that G m⊂G acts by the identity character.We refer to chapter V of[14]for basic facts about such representations.In this section we study the problem of existence of a Schr¨o dinger representation for G,i.e.a weight-1representation of G of rank d(the minimal possible rank).It is well known that such a representation exists if S is the spectrum of an algebraically closedﬁeld(see e.g.[14],V,2.5.5). Another example is the following.As we already mentioned one can associate a ﬁnite Heisenberg group scheme G(L)(called the Mumford group)to a line bundle L on an abelian schemeπ:A→S such that K(L)isﬁnite.Assume that the base scheme S is connected.Then R iπ∗(L)=0for i=i(L)for some integer i(L) (called the index of L)and R i(L)π∗(L)is a Schr¨o dinger representation for G(L)(this follows from[17]III,16and[16],prop.1.7).In general,L.Moret-Bailly showed in [14]that a Schr¨o dinger representation exists after some smooth base change.The main result of this section is that for symmetric Heisenberg group schemes of odd order a Schr¨o dinger representation always exists.Let G be a symmetricﬁnite Heisenberg group scheme of order d2over S.Then lo-cally(in fppf topology)we can choose a Schr¨o dinger representation V of G.According to Theorem V,2.4.2of[14]for any weight-1representation W of G there is a canon-ical isomorphism V⊗HomG (V,V)≃O.Choose an open covering U i such that there existSchr¨o dinger representations V i for G over U i.For a suﬃcentlyﬁne covering we have G-isomorphismsφij:V i→V j on the intersections U i∩U j,andφjkφij=αijkφik on the triple intersections U i∩U j∩U k for some functionsαijk∈O∗(U i∩U j∩U k).Then(αijk) is a Cech2-cocycle with values in G m whose cohomology class e(G)∈H2(S,G m) doesn’t depend on the choices made.Furthermore,by deﬁnition e(G)is trivial if and only if there exists a global weight-1representation we are looking for.Using the language of gerbs(see e.g.[8])we can rephrase the construction above withoutﬁxing an open ly,to eachﬁnite Heisenberg group G we can associate the G m-gerb Schr G on S such that Schr G(U)for an open set U⊂S is the category of Schr¨o dinger representations for G over U.Then Schr G represents the cohomology class e(G)∈H2(S,G m).Notice that the class e(G)is actually represented by an Azumaya algebra A(G) which is deﬁned as follows.Locally,we can choose a Schr´o dinger representation V for G and put A(G)=End(V)≃Endisomorphism f:V→V′(since any other G-isomorphism diﬀers from f by a scalar), hence these local algebras glue together into a global Azumaya algebra A(G)of rank d2.In particular,d·e(G)=0(see e.g.[9],prop.1.4).Now let W be a global weight-1representation of G which is locally free of rank l·d over S.Then we claim that EndG (W)≃Mat l(O).Now we claim that there is a global algebra isomorphismA(G)⊗End(W).Indeed,we have canonical isomorphism of G-modules of weight1(resp.−1)V⊗Hom G(V∗,W∗) →W∗).Hence,we have a sequence of natural morphismsEnd G(V∗,W∗)⊗Hom(V)⊗Hom G×G(V∗⊗V,W∗⊗W)→End G(W)—the latter map is obtained by taking the image of the identity section id∈V∗⊗V under a G×G-morphism V∗⊗V→W∗⊗W.It is easy to see that the composition morphism gives the required isomorphism.This leads to the following statement. Proposition2.1.For anyﬁnite Heisenberg group scheme G over S a canonical element e(G)∈Br(S)is deﬁned such that e(G)is trivial if and only if a Schr¨o dinger representation for G exists.Furthermore,d·e(G)=0where the order of G is d2, and if there exists a weight-1G representation which is locally free of rank l·d over S then l·e(G)=0.Proposition2.2.The map[G]→e(G)deﬁnes a homomorphism WH(S)→Br(S). Proof.First we have to check that if I⊂K is a G-isotropic subgroup, I⊂G its lifting, and G)=e(G).Indeed,there is a canonical equivalence of G m-gerbs Schr G→SchrProof.Let[G]∈WH′sym(S)be a class of G in the Witt group.Then4[G]=0by Theorem1.4,hence4e(G)=0by Proposition2.2.On the other hand,d·e(G)=0 by Proposition2.1where d is odd,therefore,e(G)=0.Let us give an example of a symmetricﬁnite Heisenberg group scheme of even order without a Schr¨o dinger representation.First let us recall the construction from [23]which associates to a group scheme G over S which is a central extension of aﬁnite commutative group scheme K by G m,and a K-torsor E over S a class e(G,E)∈H2(S,G m).Morally,the mapH1(S,K)→H2(E,G m):E→e(G,E)is the boundary homomorphism corresponding to the exact sequence0→G m→G→K→0.To deﬁne it consider the category C of liftings of E to to a G-torsor.Locally such a lifting always exists and any two such liftings diﬀer by a G m-torsor.Thus,C is a G2-gerb over S,and by deﬁnition e(G,E)is the class of C in H2(S,G m)Note that e(G,E)=0if and only if there exists a G-equivariant line bundle L over E,such that G m⊂G acts on L via the identity character.A K-torsor E deﬁnes a commutative group extension G E of K by G m as follows. Choose local trivializations of E over some covering(U i)and letαij∈K(U i∩U j) be the corresponding1-cocycle with values in K.Now we glue G E from the trivial extensions G m×K over U i by the following transition isomorphisms over U i∩U j:f ij:G m×K→G m×K:(λ,x)→(λe(x,αij),x)where e:K×K→G m is the commutator form corresponding to G.It is easy to see that G E doesn’t depend on a choice of trivializations.Now we claim that if G isa Heisenberg group then(2.1)e(G,E)=e(G⊗G E)−e(G).This is checked by a direct computation with Cech cocycles.Notice that if E2is a trivial K-torsor then G2E is a trivial central extension of K,hence G E is a symmetric extension.Thus,if G is a symmetric Heisenberg group,then G⊗G E is also symmetric. As was shown in[23]the left hand side of(2.1)can be ly,consider the case when S=A is a principally polarized abelian variety over an algebraically closedﬁeld k of characteristic=2.Let K=A2×A considered as a(constant)ﬁnite group scheme over A.Then we can consider E=A as a K-torsor over A via the morphism[2]:A→A.Now if G→A2is a Heisenberg extension of A2(deﬁned over k)then we can consider G as a constant group scheme over A and the class e(G,E) is trivial if and only if G embeds into the Mumford group G(L)of some line bundle L over A(this embedding should be the identity on G m).When NS(A)=Z this means, in particular,that the commutator form A2×A2→G m induced by G is proportional9to the symplectic form given by the principal polarization.When dim A≥2there is a plenty of other symplectic forms on A2,hence,e(G,E)can be non-trivial.Now we are going to show that one can replace A by its general point in this example.In other words,we consider the base S=Spec(k(A))where k(A)is the ﬁeld of rational functions on A.Then E gets replaced by Spec(k(A))considered asa A2-torsor over itself corresponding to the Galois extension[2]∗:k(A)→k(A):f→f(2·)with the Galois group A2.Note that the class e(G,E)for any Heisenberg extension G of A2by k∗is annihilated by the pull-back to E,hence,it is represented by the class of Galois cohomology H2(A2,k(A)∗)⊂Br(k(A))where A2acts on k(A) by translation of argument.It is easy to see that this class is the image of the class e G∈H2(A2,k∗)of the central extension G under the natural homomorphism H2(A2,k∗)→H2(A2,k(A)∗).From the exact sequence of groups0→k∗→k(A)∗→k(A)∗/k∗→0we get the exact sequence of cohomologies0→H1(A2,k(A)∗/k∗)→H2(A2,k∗)→H2(A2,k(A)∗)(note that H1(A2,k(A)∗)=0by Hilbert theorem90).It follows that central exten-sions G of A2by k∗with trivial e(G,E)are classiﬁed by elements of H1(A2,k(A)∗/k∗).Lemma2.4.Let A be a principally polarized abelian variety over an algebraically closedﬁeld k of characteristic=2.Assume that NS(A)=Z.Then H1(A2,k(A)∗/k∗)= Z/2Z.Proof.Interpreting k(A)∗/k∗as the group of divisors linearly equivalent to zero we obtain the exact sequence0→k(A)∗/k∗→Div(A)→Pic(A)→0,where Div(A)is the group of all divisors on A.Note that as A2-module Div(A) is decomposed into a direct sum of modules of the form Z A2/H where H⊂A2is a subgroup.Now by Shapiro lemma we have H1(A2,Z A2/H)≃H1(H,Z),and the latter group is zero since H is a torsion group.Hence,H1(A2,Div(A))=0.Thus, from the above exact sequence we get the identiﬁcationH1(A2,k(A)∗/k∗)≃coker(Div(A)A2→Pic(A)A2).Now we use the exact sequence0→Pic0(A)→Pic(A)→NS(A)→0,10where Pic0(A)=ˆA(k).Since the actions of A2on Pic0(A)and NS(A)are trivial we have the induced exact sequence0→Pic0(A)→Pic(A)A2→NS(A).The image of the right arrow is the subgroup2NS(A)⊂NS(A).Note that Pic0(A)= [2]∗Pic0(A),hence this subgroup belongs to the image of[2]∗Div(A)⊂Div(A)A2. Thus,we deduce thatH1(A2,k(A)∗/k∗)≃coker(Div(A)A2→2NS(A)).Let[L]⊂NS(A)be the generator corresponding to a line bundle L of degree1on A. Then L4=[2]∗L,hence4·[L]=[L4]belongs to the image of Div(A)A2.On the other hand,it is easy to see that there is no A2-invariant divisor representing[L2],henceH1(A2,k(A)∗/k∗)≃Z/2Z.It follows that under the assumptions of this lemma there is a unique Heisenberg extensions G of A2by k∗with the trivial class e(G,E)(the Mumford extension corresponding to L2,where L is a line bundle of degree1on A).Hence,for g≥2 there exists a Heisenberg extension with a non-trivial class e(G,E)∈Br(k(A)).3.Representations of the Heisenberg groupoidRecall that the Heisenberg group H(W)associated with a symplectic vector space W is a central extension0→T→H(W)→W→0of W by the1-dimensional torus T with the commutator form exp(B(·,·))where B is the symplectic form.In this section we consider an analogue of this extension in the context of abelian schemes(see[22],sect.7,[23]).Namely,we replace a vector space W by an abelian scheme X/S.Bilinear forms on W get replaced by biextensions of X2.Recall that a biextension of X2is a line bundle L on X2together with isomorphismsL x+x′,y≃L x,y⊗L x′,y,L x,y+y′≃L x,y⊗L x,y′—this is a symbolic notation for isomorphisms(p1+p2,p3)∗L≃p∗13L⊗p∗23L and (p1,p2+p3)∗L≃p∗12L⊗p∗13L on X3,satisfying some natural cocycle conditions (see e.g.[3]).The parallel notion to the skew-symmetric form on W is that of a skew-symmetric biextension of X2which is a biextension L of X2together with an isomorphism of biextensionsφ:σ∗L →L−1,whereσ:X2→X2is the permutation of11factors,and a trivialization∆∗L≃O X of L over the diagonal∆:X→X2compati-ble withφ.A skew-symmetric biextension L is called symplectic if the corresponding homomorphismψL:X→ˆX(whereˆX is the dual abelian scheme)is an isomor-phism.An isotropic subscheme(with respect to L)is an abelian subscheme Y⊂X such that there is an isomorphism of skew-symmetric biextensions L|Y×Y≃O Y×Y. This is equivalent to the condition that the composition Y i→XψL→ˆXˆi→ˆY is zero. An isotropic subscheme Y⊂X is called lagrangian if the morphism Y→ker(ˆi) induced byψL is an isomorphism.In particular,for such a subscheme the quotient X/Y exists and is isomorphic toˆY.Note that to deﬁne the Heisenberg group extension it is not suﬃcient to have a symplectic form B on W:one needs a bilinear form B1such that B(x,y)=B1(x,y)−B1(y,x).In the case of the real symplectic space one can just take B1=B/2,however in our situation we have to simply add necessary data.An enhanced symplectic biextension(X,B)is a biextension B of X2such that L:=B⊗σ∗B−1is a symplectic biextension.The standard enhanced symplectic biextension for X=ˆA×A,where A is any abelian scheme,is obtained by settingB=p∗14P∈Pic(ˆA×A×ˆA×A),where P is the normalized Poincar´e line bundle on A×ˆA.Given an enhanced symplectic biextension(X,B)one deﬁnes the Heisenberg groupoid H(X)=H(X,B)as the stack of monoidal groupoids such that H(X)(S′)for an S-scheme S′is the monoidal groupoid generated by the central subgroupoid P ic(S′)of G m-torsors on S′and the symbols T x,x∈X(S′)with the composition lawT x◦T x′=B x,x′T x+x′.The Heisenberg groupoid is a central extension of X by the stack of line bundles on S in the sense of Deligne[4].In[22]we considered the action of H(ˆA×A)on D b(A)which is similar to the stan-dard representation of the Heisenberg group H(W)on functions on a lagrangian sub-space of W.Below we construct similar representations of the Heisenberg groupoid H(X)associated with lagrangian subschemes in X.Further,we construct inter-twining functors for two such representations corresponding to a pair of lagrangian subschemes,and consider the analogue of Maslov index for a triple of lagrangian subschemes that arises when composing these intertwining functors.To deﬁne an action of H(X)associated with a lagrangian subscheme one needs some auxilary data described as follows.An enhanced lagrangian subscheme(with respect to B)is a pair(Y,α)where Y⊂X is a lagrangian subscheme with respect to X,αis a line bundle on Y with a rigidiﬁcation along the zero section such that an isomorphism of symmetric biextensionsΛ(α)≃B|Y×Y is given,whereΛ(α)=12。

笛卡尔的本体论之争

笛卡尔的本体论之争首先周一公布2001年6月18日;实质性修改太阳2006年10月15日笛卡尔的本体论（或先验）的论点，既是哲学的一个最迷人，他的理解方面的不足。

论据与魅力源于努力证明神的存在，从简单的处所，但功能强大。

存在是产生立即从清晰和明确的想法是一个无比完美。

讽刺的是，简单的说法也产生了一些误读，加剧了部分由笛卡尔没有一套单一版本。

该声明的论点主要出现在第五沉思。

这种说法因果来得早在接踵而至的一个神的存在，沉思在第三，不同的证据提出问题的两项之间的秩序和关系。

重复笛卡尔哲学原理，包括本体论争论的几个文本等中央。

他还辩解首先由一些主要的知识分子，他在一天，严厉打击反对第二次回复，和第五。

笛卡尔不是第一位哲学家，制订一个本体论的论点。

一个早期版本的说法已大力安瑟伦辩护圣在11世纪，然后圣托马斯阿奎那批评由当代），后来被命名为Gaunilo和尚（安瑟伦（尽管他的言论是针对然而，另一个版本参数）。

阿奎那的批评被视为如此具有破坏性，本体论的争论了数百年死亡。

它的出现，作为一个同时代的惊喜笛卡尔，他应该试图复活它。

虽然他声称没有被证明的熟悉安瑟伦的版本，笛卡尔似乎他自己的工艺参数，以阻止传统的反对。

尽管相似之处，笛卡尔的论点的版本不同于安瑟伦方式在重要的。

后者的版本被认为要从定义这个词的含义“上帝”，上帝是一个被一大于不能设想。

笛卡尔的观点相反，中，主要是基于两个他的哲学的中心原则-天生的思想理论和学说明确的印象和独特的。

他声称不依赖于上帝的任意定义，而是一种天生的想法，其内容是“的。

” 笛卡尔的版本也非常简单。

神的存在是直接从推断的事实，有必要存在的想法是包含在一个清晰而鲜明的超级完美的存在。

事实上，在一些场合，他建议，所谓的本体论“的论调”是不是一个正式的哲学偏见的证据，而是在所有不言而喻的公理直观地掌握了一个心灵的自由。

笛卡尔的本体论的争论相比往往以几何论证，认为有必要存在的想法不能排除再从神比事实平等的角度，其角度，例如两权，可以被排除在一个三角形的想法。

Some aspects of normal development in the colonial ciliate zoothamnium alternans(生活史重要文献)

optical sections of the barely immersed colonies were not clear, the
water level in the Petri dish was adjusted by a slight turn of the dish on the slanted microscope stage until the surface film flattened the
INTRODUCTION
This is the first of a series of inquiries into some of the problems of organization and regulation in a colonial protozoan of a rather special
FIG.
1.
A
relatively
mature
colony
of
ZoÃ¶thamnium
alternans
showing
the
characteristic arrangement of the zoOids and branches. Axial microzoÃ¶ids on branches E, G, L, and N are differentiating into ciliospores; one of them at E is in an advanced stage of metamorphosis, almost ready to break away from the parent colony. Approximately X 100. GENERAL FORM OF THE MATURE COLONY

非酒精性脂肪性肝病代谢组学研究进展

机制尚未完全明确，１９９８年Ｄａｙ等［１２］提出“二次打击”学说。开。同时ＮＡＦＬＤ肝硬化患者与酒精性肝硬化患者也可有效区
随后Ｔｉｌｇ等［１３－１４］提出“多重平行打击”理论，包括遗传因素、分开（ＡＵＣ＝０．８３）。他们认为此方法可作为区分ＮＡＦＬＤ纤维
ＩＲ、氧化应激、脂毒性、慢性炎症、纤维化、免疫和肠道菌群等，化程度及诊断的无创生物标志物，且可以显著减少对肝活检的
黄酯和１３－ｃｉｓＲＡ呈正相关。他们在人类组织中首次检测到验证；单不饱和ＴＡＧ的增加可能是ＮＡＦＬＤ和ＣＨＢ患者ＮＡＳＨ
ａｔＲＡ的活性代谢物４－ｏｘｏ－ａｔＲＡ，表明这种类维生素Ａ可能的特异性标志物。
有助于人体类维生素Ａ的信号传导。肝脏维生素Ａ的稳态平２．３代谢组学对ＮＡＦＬＤ药物作用与疗效研究的推动作用
录组学、蛋白质组学为代表的系统生物学技术提供了新的技术展的新学科，代谢组学较为全面的展示了机体的代谢结果，为
与思路。区别于其他组学技术，以内源性小分子代谢物为研究临床医学提供了新的技术和方法。
对象的代谢组学可以很好的揭示机体变化的最终代谢结果。因２非酒精性脂肪性肝病（ＮＡＦＬＤ）
收基作ＤＯ稿金者Ｉ：日项简１０期目介．３：：：９１２上栾研６７０９）２海究雨／０ｊ．中婷－ｉｓ医１ｓ（ｎ１药．１－１９大０９０００６学１—；修－附）５回，属２女５日第６，．期七主２０：人２要２１０民．２从００医４事－．院０１慢４２人７－性才１肝７培病养计的划基（础ＸＸ与２０临１９床－通信作者：顼志兵，ｘｚｂ６１６０＠１６３．ｃｏｍ
和遗传易感密切相关的代谢应激性肝损伤，包括非酒精性单纯１代谢组学概述
性肝脂肪变（ＮＡＦＬ）、非酒精性脂肪性肝炎（ＮＡＳＨ）、肝硬化和１．１代谢组学含义代谢组学最初于１９９９年由Ｎｉｃｈｏｌｓｏｎ

黎曼猜想英语

黎曼猜想英语The Riemann Hypothesis, named after the 19th-century mathematician Bernhard Riemann, is one of the most profound and consequential conjectures in mathematics. It is concerned with the distribution of the zeros of the Riemann zeta function, a complex function denoted as $$\zeta(s)$$, where $$s$$ is a complex number. The hypothesis posits that all non-trivial zeros of this analytical function have their real parts equal to $$\frac{1}{2}$$.To understand the significance of this conjecture, one must delve into the realm of number theory and the distribution of prime numbers. Prime numbers are the building blocks of arithmetic, as every natural number greater than 1 is either a prime or can be factored into primes. The distribution of these primes, however, has puzzled mathematicians for centuries. The Riemann zeta function encodes information about the distribution of primes through its zeros, and thus, the Riemann Hypothesis is directly linked to understanding this distribution.The zeta function is defined for all complex numbers except for $$s = 1$$, where it has a simple pole. For values of $$s$$ with a real part greater than 1, it converges to a sum over the positive integers, as shown in the following equation:$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$。

Symmetric invariant manifolds in the Fermi Pasta Ulam chain

Abstract
1 Introduction
The Fermi Pasta Ulam chain or FPU chain is a discrete model for a continuous nonlinear string, introduced by E. Fermi, J. Pasta and S. Ulam 4]. This string is modeled by a nite number of point masses which represent the material elements of the string. Each of the point masses interacts with its nearest neighbors only.
1
conditions. We speak of xed boundary conditions if the rst and the last particle do not move, meaning that we have q0 = qn = 0 for all time. The FPU chain with xed boundary conditions models a string with Dirichlet boundary conditions. It is also possible to choose periodic boundary conditions, in which case the rst and the last particle are identi ed, that is q0 = qn for all time. The FPU chain with periodic boundary conditions models a circular string. Both types of boundary conditions occur very often in the literature. In this paper we shall only consider chains with periodic boundary conditions, as it will become clear that each chain with xed boundary conditions is naturally embedded as an invariant manifold of an appropriate periodic chain. The particles of the periodic chain are labeled by elements of the cyclic group Z=nZ The Hamiltonian equations of motion for the FPU chain are derived as follows. . The space of positions q = (q1; : : : ; qn ) of the particles in the chain is Rn. The space of positions and conjugate momenta is the cotangent bundle T Rn of Rn, the elements of which are denoted (q; p) = (q1; : : : ; qn ; p1; : :P pn ). T Rn is a symplectic :; manifold, endowed with the symplectic form dq ^ dp = n=1 dqj ^ dpj . Any smooth j function H : T Rn ! R now induces the Hamiltonian vector eld XH given by the de ning relation (dq ^ dp)(XH ; ) = dH . In other words, we have the system of ordinary @H _ @H di erential equations q_j = @pj ; pj = ? @qj . The Hamiltonian function for the FPU chain with periodic boundary conditions and n particles consists of a kinetic energy and a potential energy. The potential energy is assumed to depend only on the vertical distance between pairs of neighboring particles. Hence the Hamiltonian is X 12 p + W (qj+1 ? qj ) ; (1.1) H= 2 j j 2Z nZ = in which W : R ! R is a Lennard-Jones potential energy density function of the form 1 (1.2) W (x) = 2! x2 + 3! x3 + 4! x4 : The ; are real parameters measuring the nonlinearity in the forces between the particles in the chain. We e chain consists of a nite number n 2 N particles. De ne qj 2 R the vertical position of the j -th particle. We distinguish two di erent types of boundary

Symmetry classes of disordered fermions

a r X i v :m a t h -p h /0411040v 1 10 N o v 2004Abstract .—Building upon Dyson’s fundamental 1962article known in random-matrix theory as the threefold way ,we classify disordered fermion systems with quadratic Hamil-tonians by their unitary and antiunitary symmetries.Important physical examples are af-forded by noninteracting quasiparticles in disordered metals and superconductors,and by relativistic fermions in random gauge ﬁeld backgrounds.The primary data of the classiﬁcation are a Nambu space of fermionic ﬁeld opera-tors which carry a representation of some symmetry group.Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms.After reduction,the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V ,a space of en-domorphisms in End (V ⊕V ∗),a bilinear form on V ⊕V ∗which is either symmetric or alternating,and one or two antiunitary symmetries that may mix V with V ∗.Every such set of block data is shown to determine an irreducible classical compact symmetric space.Conversely,every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces con-jectured some time ago.Keywords :disordered electron systems,random Dirac fermions,quantum chaos;repre-sentation theory,symmetric spaces1.IntroductionIn a famous and inﬂuential paper published in 1962(“The threefold way:algebraic structure of symmetry groups and ensembles in quantum mechanics”[D ]),Freeman J.Dyson classiﬁed matrix ensembles by a scheme that became fundamental to several areas of theoretical physics,including the statistical theory of complex many-body systems,mesoscopic physics,disordered electron systems,and the area of quantum chaos.Being set in the context of standard quantum mechanics,Dyson’s classiﬁcation asserted that “the most general matrix ensemble,deﬁned with a symmetry group that may be completely arbitrary,reduces to a direct product of independent irreducible2P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERensembles each of which belongs to one of three known types.”These three ensembles, or rather their underlying matrix spaces,are nowadays known as the Wigner-Dyson symmetry classes of orthogonal,unitary,and symplectic symmetry.Over the last ten years,various matrix spaces beyond Dyson’s threefold way have come to the fore in random-matrix physics and mathematics.On the physics side,such spaces arise in problems of disordered or chaotic fermions;among these are the Eu-clidean Dirac operator in a stochastic gaugeﬁeld background[V2],and quasiparticle excitations in disordered superconductors or metals in proximity to a superconductor [A2].In the mathematical research area of number theory,the study of statistical cor-relations in the values of the Riemann zeta function,and more generally of families of L-functions,has prompted some of the same extensions[K].A brief account of why new structures emerge on the physics side is as follows. When Diracﬁrst wrote down his famous equation in1928,he assumed that he was writing an equation for the wavefunction of the ter,because of the insta-bility caused by negative-energy solutions,the Dirac equation was reinterpreted(via second quantization)as an equation for the fermionicﬁeld operators of a quantum ﬁeld theory.A similar change of viewpoint is carried out in reverse in the Hartree-Fock-Bogoliubov mean-ﬁeld description of quasiparticle excitations in superconduc-tors.There,one starts from the equations of motion for linear superpositions of the electron creation and annihilation operators,and reinterprets them as a unitary quan-tum dynamics for what might be called the quasiparticle‘wavefunction’.In both cases–the Dirac equation and the quasiparticle dynamics of a superconduc-tor–there enters a structure not present in the standard quantum mechanics underlying Dyson’s classiﬁcation:the fermionicﬁeld operators are subject to a set of conditions known as the canonical anticommutation relations,and these are preserved by the quantum dynamics.Therefore,whenever second quantization is undone(assuming it can be undone)to return fromﬁeld operators to wavefunctions,the wavefunction dy-namics is required to preserve some extra structure.This puts a linear constraint on the allowed Hamiltonians.A good viewpoint to adopt is to attribute the extra invariant structure to the Hilbert space,thereby turning it into a Nambu space.It was conjectured some time ago[A2]that extending Dyson’s classiﬁcation to the Nambu space setting,the relevant objects one is led to consider are large families of symmetric spaces of compact type.Past understanding of the systematic nature of the extended classiﬁcation scheme relied on the mapping of disordered fermion problems toﬁeld theories with supersymmetric target spaces[Z]in combination with renormalization group ideas and the classiﬁcation theory of Lie superalgebras.An extensive review of the mathematics and physics of symmetric spaces,covering the wide range from the basic deﬁnitions to various random-matrix applications,has recently been given in[C].That work,however,offers no answers to the question as to why symmetric spaces are relevant for symmetry classiﬁcation,and under what assumptions the classiﬁcation by symmetric spaces is complete.SYMMETRY CLASSES OF DISORDERED FERMIONS3 In the present paper,we get to the bottom of the subject and,using a minimal set of tools from linear algebra,give a rigorous answer to the classiﬁcation problem for disor-dered fermions.The rest of this introduction gives an overview over the mathematical model to be studied and a statement of our main result.We begin with aﬁnite-or inﬁnite-dimensional Hilbert space V carrying a unitary representation of some compact Lie group G0–this is the group of unitary symmetries of the disordered fermion system.We emphasize that G0need not be connected;in fact,it might be just aﬁnite group.Let W=V⊕V∗,called the Nambu space of fermionicﬁeld operators,be equipped with the induced G0-representation.This means that V is equipped with the given representation,and g(f):=f◦g−1for f∈V∗,g∈G0.Let C:W→W be the C-antilinear involution determined by the Hermitian scalar product , V on V.In physics this operator is called particle-hole conjugation.Another canonical structure on W is the symmetric complex bilinear form b:W×W→C deﬁned byb(v1+f1,v2+f2):=f1(v2)+f2(v1).It encodes the canonical anticommutation relations for fermions,and is related to the unitary structure , of W by b(w1,w2)= Cw1,w2 for all w1,w2∈W.It is assumed that G0is contained in a group G–the total symmetry group of the fermion system–which is acting on W by transformations that are either unitary or antiunitary.An element g∈G either stabilizes V or exchanges V and V∗.In the latter case we say that g∈G mixes,and in the former case we say that it is nonmixing. The group G is generated by G0and distinguished elements g T which act as anti-unitary operators T:W→W.These are referred to as distinguished‘time-reversal’symmetries,or T-symmetries for short.The squares of the g T lie in the center of the abstract group G;we therefore require that the antiunitary operators T representing them satisfy T2=±Id.The subgroup G0is deﬁned as the set of all elements of G which are represented as unitary,nonmixing operators on W.If T and T1are distinguished time-reversal operators,then P:=T T1is a unitary sym-metry.P may be mixing or nonmixing.In the latter case,P is in G0.Therefore,modulo G0,there exist at most two different T-symmetries.If there are exactly two such sym-metries,we adopt the convention that T is mixing and T1is nonmixing.Furthermore, it is assumed that T and T1either commute or anticommute,i.e.,T1T=±T T1.As explained throughout this article,all of these situations are well motivated by physical considerations and examples.We note that time-reversal symmetry(and all other T-symmetries)of the disordered fermion system may also be broken;in this case T and T1are eliminated from the mathematical model and G0=G.Given W and the representation of G on it,the object of interest is the real vector space H of C-linear operators in End(W)that preserve the canonical structures b and , of W and commute with the G-action.Physically speaking,H is the space of ‘good’Hamiltonians:theﬁeld operator dynamics generated by H∈H preserves both4P.HEINZNER,A.HUCKLEBERRY &M.R.ZIRNBAUERthe canonical anticommutation relations and the probability in Nambu space,and is compatible with the prescribed symmetry group G .When unitary symmetries are present,the space H decomposes by blocks associ-ated with isomorphism classes of G 0-subrepresentations occurring in W .To formal-ize this,recall that two unitary representations ρi :G 0→U (V i ),i =1,2,are equiv-alent if and only if there exists a unitary C -linear isomorphism ϕ:V 1→V 2so that ρ2(g )(ϕ(v ))=ϕ(ρ1(g )(v ))for all v ∈V 1and for all g ∈G 0.Let ˆG 0denote the space of equivalence classes of irreducible unitary representations of G 0.An element λ∈ˆG 0is called an isomorphism class for short.By standard facts (recall that every represen-tation of a compact group is completely reducible)the unitary G 0-representation on V decomposes as an orthogonal sum over isomorphism classes:V =⊕λV λ.The subspaces V λare called the G 0-isotypic components of V .Some of them may be zero.(Some of the isomorphism classes of G 0may just not be realized in V .)For simplicity suppose now that there is only one distinguished time-reversal sym-metry T ,and for any ﬁxed λ∈ˆG0with V λ=0,consider the vector space T (V λ).If T is nonmixing,i.e.,T :V →V ,then T (V λ)⊂V must coincide with the isotypic component for the same or some other isomorphism class.(Since conjugation by g T is an automorphism of G 0,the decomposition into G 0-isotypic components is preserved by T .)If T is mixing,i.e.,T :V →V ∗,then T (V λ)=V ∗λ′,still with some λ′∈ˆG 0.Now deﬁne the block B λto be the smallest G -invariant space containing V λ⊕V ∗λ.Note that if we are in the situation of nonmixing and T (V λ)=V λ,thenB λ= V λ⊕T (V λ) ⊕ V λ⊕T (V λ) ∗.On the other hand,if we are in the situation of mixing and T (V λ)=V ∗λ,thenB λ= V λ⊕T (V ∗λ) ⊕ V ∗λ⊕T (V λ) .The block B λis halved if T (V λ)=V λresp.T (V λ)=V ∗λ.Note that if there are two distinguished T -symmetries,the above discussion is only slightly more complicated.In any case we now have the basic G -invariant blocks B λ.Because different blocks are built from representations of different isomorphism classes,the good Hamiltonians do not mix blocks.Thus every H ∈H is a direct sum over blocks,and the structure analysis of H can be carried out for each block B λseparately.If V λis inﬁnite-dimensional,then to have good mathematical control we truncate to a ﬁnite-dimensional space V λ⊂V λand form the associated block B λ⊂W .The truncation is done in such a way that B λis a G -representation space and is Nambu.The goal now is to compute the space of Hermitian operators on B λwhich commute with the G -action and respect the canonical symmetric C -bilinear form b induced from that on V ⊕V ∗;such a space of operators realizes what is called a symmetry class .SYMMETRY CLASSES OF DISORDERED FERMIONS5 For this,certain spaces of G0-equivariant homomorphisms play an essential role, i.e.,linear maps S:V1→V2between G0-representation spaces which satisfyρ2(g)◦S=S◦ρ1(g)for all g∈G0,whereρi:G0→U(V i),i=1,2,are the respective representations.If it is clear which representations are at hand,we often simply write g◦S=S◦g or S=gSg−1.Thus we regard the space Hom G0(V1,V2)of equivariant homomorphisms as thespace of G0-ﬁxed vectors in the space Hom(V1,V2)of all linear maps.If V1=V2=V,then these spaces are denoted by End G0(V)and End(V)respectively.Roughly speaking,there are two steps for computing the relevant spaces of Hermi-tian operators.First,the block Bλis replaced by an analogous block Hλof G0-equi-variant homomorphisms from aﬁxed representation space Rλof isomorphism classλand/or its dual R∗λto Bλ.The space Hλcarries a canonical form(called either s or a)which is induced from b.As the notation indicates,although the original bilinear form on Bλis symmetric,this induced form is either symmetric or alternating.Change of parity occurs in the most interesting case when there is a G0-equivariantisomorphismψ:Rλ→R∗λ.In that case there exists a bilinear form Fψ:Rλ×Rλ→Cdeﬁned by Fψ(r,t)=ψ(r)(t),which is either symmetric or alternating.In a certain sense the form b is a product of Fψand a canonical form on Hλ.Thus,if Fψis alternating,then the canonical form on Hλmust also be alternating.After transferring to the space Hλ,in addition to the canonical bilinear form s or a we have a unitary structure and conjugation by one or two distinguished time-reversal symmetries.Such a symmetry T may be mixing or not,and both T2=Id and T2=−Id are possible.The second main step of our work is to understand these various cases, each of which is directly related to a classical symmetric space of compact type.Such are given by a classical Lie algebra g which is either su n,usp2n,or so n(R).In the notation of symmetric spaces we have the following situation.Let g be the Lie algebra of antihermitian endomorphisms of Hλwhich are isometries(in the sense of Lie algebra elements)of the induced complex bilinear form b=s or b=a.This is of compact type,because it is the intersection of the Lie algebra of the unitary group of Hλand the complex Lie algebra of the group of isometries of b.Conjugation by the antiunitary mapping T deﬁnes an involutionθ:g→g.The good Hamiltonians(restricted to the reduced block Hλ)are the Hermitian oper-ators h∈i g such that at the level of group action the one-parameter groups e−i th satisfy T e−i th=e+i th T,i.e.,i h∈g must anticommute with T.Equivalently,if g=k⊕p is the decomposition of g intoθ-eigenspaces,the space of operators which is to be computed is the(−1)-eigenspace p.The space of good Hamiltonians restricted to Hλthen is i p. Since the appropriate action of the Lie group K(with Lie algebra k)on this space is just conjugation,one identiﬁes i p with the tangent space g/k of an associated symmetric space G/K of compact type.It should be underlined that there is more than one symmetric space associated to a Cartan decomposition g=k⊕p.We are most interested in the one consisting of the6P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERphysical time-evolution operators e−i th;if G(not to be confused with the symmetry group G)is the semisimple and simply connected Lie group with Lie algebra g,this is given as the image of the compact symmetric space G/K under the Cartan embedding into G deﬁned by g K→gθ(g)−1,whereθ:G→G is the induced group involution. The following mathematical result is a conseqence of the detailed classiﬁcation work in Sects.3,4and5.Theorem1.1.—The symmetric spaces which occur under these assumptions are ir-reducible classical symmetric spaces g/k of compact type.Conversely,every irre-ducible classical symmetric space of compact type occurs in this way.We emphasize that here the notion symmetric space is appliedﬂexibly in the sense that depending on the circumstances it could mean either the inﬁnitesimal model g/k or the Cartan-embedded compact symmetric space G/K.Theorem1.1settles the question of symmetry classes in disordered fermion systems; in fact every physics example is handled by one of the situations above.The paper is organized as follows.In Sect.2,starting from physical considerations we motivate and develop the model that serves as the basis for subsequent mathemati-cal work.Sect.3proves a number of results which are used to eliminate the group of unitary symmetries G0.The main work of classiﬁcation is given in Sect.4and Sect.5. In Sect.4we handle the case where at most one distinguished time-reversal operator is present,and in Sect.5the case where there are two.There are numerous situations that must be considered,and in each case we precisely describe the symmetric space which occurs.Various examples taken from the physics literature are listed in Sect.6, illustrating the general classiﬁcation theory.2.Disordered fermions with symmetries‘Fermions’is the physics name for the elementary particles which all matter is made of.The goal of the present article is to establish a symmetry classiﬁcation of Hamilto-nians which are quadratic in the fermion creation and annihilation operators.To mo-tivate this restriction,note that any Hamiltonian for fermions at the fundamental level is of Dirac type;thus it is always quadratic in the fermion operators,albeit with time-dependent coefﬁcients that are themselves operators.At the nonrelativistic or effective level,quadratic Hamiltonians arise in the Hartree-Fock mean-ﬁeld approximation for metals and the Hartree-Fock-Bogoliubov approximation for superconductors.By the Landau-Fermi liquid principle,such mean-ﬁeld or noninteracting Hamiltonians give an adequate description of physical reality at very low temperatures.In the present section,starting from a physical framework,we develop the appropri-ate model that will serve as the basis for the mathematical work done later on.Please be advised that disorder,though advertised in the title of the section and in the title of paper,will play no explicit role here.Nevertheless,disorder(and/or chaos)are theSYMMETRY CLASSES OF DISORDERED FERMIONS7 indispensable agents that must be present in order to remove speciﬁc and nongeneric features from the physical system and make a classiﬁcation by basic symmetries mean-ingful.In other words,what we carry out in this paper is theﬁrst step of a two-step program.Thisﬁrst step is to identify in the total space of Hamiltonians some lin-ear subspaces that are relevant(in Dyson’s sense)from a symmetry perspective.The second step is to put probability measures on these spaces and work out the disor-der averages and statistical correlation functions of interest.It is this latter step that ultimately justiﬁes theﬁrst one and thus determines the name of the game.2.1.The Nambu space model for fermions.—The starting point for our considera-tions is the formalism of second quantization.Its relevant aspects will now be reviewed so as to introduce the key physical notions as well as the proper mathematical language. Let i=1,2,bel an orthonormal set of quantum states for a single fermion. Second quantizing the many-fermion system means to associate with each i a pair of operators c†i and c i,which are called fermion creation and annihilation operators, respectively,and are related to each other by an operation of Hermitian conjugation †:c i→c†i.They are subject to the canonical anticommutation relationsc†i c†j+c†j c†i=0,c i c j+c j c i=0,c†i c j+c j c†i=δi j,(2.1) for all i,j.They act in a Fock space,i.e.,in a vector space with a distinguished vec-tor,called the‘vacuum’,which is annihilated by all of the operators c i(i=1,2,...). Applying n creation operators to the vacuum one gets a state vector for n fermions.A ﬁeld operatorψis a linear combination of creation and annihilation operators,ψ=∑i v i c†i+f i c i ,with complex coefﬁcients v i and f i.To put this in mathematical terms,let V be the complex Hilbert space of single-fermion states.(We do not worry here about complications due to the dimension of V being inﬁter rigorous work will be carried out in theﬁnite-dimensional setting.) Fock space then is the exterior algebra∧V=C⊕V⊕∧2V⊕...,with the vacuum being the one-dimensional subspace of constants.Creating a single fermion amounts to exterior multiplication by a vector v∈V and is denoted byε(v):∧n V→∧n+1V.To annihilate a fermion,one contracts with an element f of the dual space V∗.In other words,one applies the antiderivationι(f):∧n V→∧n−1V given byι(f)·1=0,ι(f)v=f(v),ι(f)(v1∧v2)=f(v1)v2−f(v2)v1,etc.In that mathematical framework the canonical anticommutation relations readε(v)ε(˜v)+ε(˜v)ε(v)=0,ι(f)ι(˜f)+ι(˜f)ι(f)=0,(2.2)ι(f)ε(v)+ε(v)ι(f)=f(v).8P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERThey can be viewed as the deﬁning relations of an associative algebra,the so-called Clifford algebra C(W),which is generated by the vector space W:=V⊕V∗over C. This vector space W is sometimes referred to as Nambu space in physics.Since we only consider Hamiltonians that are quadratic in the creation and annihi-lation operators,we will be able to reduce the second-quantized formulation on∧V to standard single-particle quantum mechanics,albeit on the Nambu space W carrying some extra structure.Note that W is isomorphic to the space ofﬁeld operatorsψ.On W=V⊕V∗there exists a canonical symmetric C-bilinear form b deﬁned by b(v+f,˜v+˜f)=f(˜v)+˜f(v)=∑i(f i˜v i+˜f i v i).The signiﬁcance of this bilinear form in the present context lies in the fact that it en-codes on W the canonical anticommutation relations(2.1),or(2.2).Indeed,we can view aﬁeld operatorψ=∑i(v i c†i+f i c i)either as a vectorψ=v+f∈V⊕V∗,or equivalently as a degree-one operatorψ=ε(v)+ι(f)in the Clifford algebra acting on∧V.Adopting the operator perspective,we get from(2.2)thatψ˜ψ+˜ψψ=f(˜v)+˜f(v)=∑i f i˜v i+˜f i v i .Switching to the vector perspective we have the same answer from b(ψ,˜ψ).Thusψ˜ψ+˜ψψ=b(ψ,˜ψ).Deﬁnition2.1.—In the Nambu space model for fermions one identiﬁes the space ofﬁeld operatorsψwith the complex vector space W=V⊕V∗equipped with its canonical unitary structure , and canonical symmetric complex bilinear form b. Remark.—Having already expounded the physical origin of the symmetric bilinear form b,let us now specify the canonical unitary structure of W.The complex vector space V,being isomorphic to the Hilbert space of single-particle states,comes with a Hermitian scalar product(or unitary structure) , V.Given , V deﬁne a C-antilinear bijection C:V→V∗byCv= v,· V,and extend this to an antilinear transformation C:W→W by the requirement C2=Id. Thus C|V∗=(C|V)−1.The operator C is called particle-hole conjugation in physics. Using C,transfer the unitary structure from V to V∗in the natural way:f,˜f V∗:=SYMMETRY CLASSES OF DISORDERED FERMIONS9Proof .—Given an orthonormal basis c †1,c 1,c †2,c 2,...this is immediate from C ∑i (v i c †i +f i c i )=∑i (¯v i c i +¯f i c †i )and the expressions for , and b in components.Returning to the physics way of telling the story,consider the most general Hamilto-nian H which is quadratic in the single-fermion creation and annihilation operators.Assuming H to be Hermitian,using the canonical anticommutation relations (2.1),and omitting an additive constant (which is of no consequence in physics)this has the form H =12∑i jB i j c †i c †j +¯B i j c j c i ,where A i j =¯Aji (from H =H †)and B i j =−B ji (from c i c j =−c j c i ).The Hamiltonians H act on the ﬁeld operators ψby the commutator,ψ→[H ,ψ]≡H ψ−ψH ,and the time evolution is determined by the Heisenberg equation of motion,−i ¯h d ψ¯h A B −¯B−¯A.To recast all this in concise terms,we need some further mathematical background.Notwithstanding the fact that in practice we always consider the Fock space represen-tation C (W )→End (∧V )by w =v +f →ε(v )+ι(f ),it should be stated that the primary (or universal)deﬁnition of the Clifford algebra C (W )is as the associative algebra generated by W ⊕C with relationsw 1w 2+w 2w 1=b (w 1,w 2)×Id(w 1,w 2∈W ).(2.3)The Clifford algebra is graded byC (W )=C 0(W )⊕C 1(W )⊕C 2(W )⊕...,where C 0(W )≡C ,C 1(W )∼=W ,and C n (W )for n ≥2is the linear space of skew-symmetrized degree-n monomials in the elements of W .In particular,C 2(W )is the linear space of skew-symmetric quadratic monomials w 1w 2−w 2w 1(w 1,w 2∈W ).From the Clifford algebra perspective,a quadratic Hamiltonian H is viewed as an operator in the degree-two component C 2(W ).Let us therefore gather some standard facts about C 2(W ).First among these is that C 2(W )is a complex Lie algebra with10P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERthe commutator playing the role of the Lie bracket(an exposition of this fact for the case of a Clifford algebra over R is found in[B3];the complex case is no different). Second,in addition to acting on itself by the commutator,the Lie algebra C2(W) acts(still by the commutator)on all of the components C k(W)of degree k≥1of the Clifford algebra C(W).In particular,C2(W)acts on C1(W).Third,C2(W)turns out to be canonically isomorphic to the complex orthogonal Lie algebra so(W,b)which is associated with the vector space W=V⊕V∗and its canonical symmetric complex bilinear form b;this Lie algebra so(W,b)is deﬁned to be the subspace of elements E∈End(W)satisfying the conditionb(Ew1,w2)+b(w1,Ew2)=0(for all w1,w2∈W).The canonical isomorphism C2(W)→so(W,b)is given by the commutator action of C2(W)on C1(W)∼=W,i.e.,by sending a∈C2(W)to[a,·]=E∈End(W);the latter indeed lies in so(W,b)as follows from the expression for b(Ew1,w2)+b(w1,Ew2) given by the canonical anticommutation relations(2.3),from the Jacobi identity [a,w1]w2+w2[a,w1]+w1[a,w2]+[a,w2]w1=[a,w1w2+w2w1],and from the fact that w1w2+w2w1lies in the center of the Clifford algebra.To describe so(W,b)explicitly,decompose the endomorphisms E∈End(V⊕V∗) into blocks asE= A B C D ,where A∈End(V),B∈Hom(V∗,V),C∈Hom(V,V∗)and D∈End(V∗).Let the adjoint(or transpose)of A∈End(V)be denoted by A t∈End(V∗),and call an element C in Hom(V,V∗)skew if C t=−C,i.e.,if(C v1)(v2)=−(C v2)(v1).Proposition2.3.—An endomorphism E= A B C D ∈End(V⊕V∗)lies in the com-plex orthogonal Lie algebra so(V⊕V∗,b)if and only if B,C are skew and D=−A t. Proof.—Considerﬁrst the case B=C=0,and let D=−A t.Thenb E(v+f),˜v+˜f =b(A v−A t f,˜v+˜f)=˜f(A v)−A t f(˜v)=A t˜f(v)−f(A˜v)=−b(v+f,A˜v−A t˜f)=−b v+f,E(˜v+˜f) .Using B t=−B and C t=−C,a similar calculation for the case A=0givesb E(v+f),˜v+˜f =b(B f+C v,˜v+˜f)=C v(˜v)+˜f(B f)=−f(B˜f)−C˜v(v)=−b(v+f,B˜f+C˜v)=−b v+f,E(˜v+˜f) . Since these two cases complement each other,we see that the stated conditions on E∈End(W)are sufﬁcient in order for E to be in so(W,b).The calculation can equally well be read backwards;thus the conditions are both sufﬁcient and necessary.SYMMETRY CLASSES OF DISORDERED FERMIONS11 Let us now make the connection to physics,where C(W)is represented on Fock space and the elements v+f=w∈W becomeﬁeld operatorsψ=ε(v)+ι(f).Fixing orthonormal bases c†1,c†2,...of V and c1,c2,...of V∗as before,we assign matrices with matrix elements A i j,B i j,C i j to the linear operators A,B,C.A straightforward computation using the canonical anticommutation relations then yields: Proposition2.4.—The inverse of the Lie algebra automorphism C2(W)→so(W,b) is the C-linear mapping given byA B C−A t →12∑i j(B i j c†i c†j+C i j c i c j).Now recall that C2(W)acts on the degree-one component C1(W)by the commuta-tor.By the isomorphisms C2(W)∼=so(W,b)and C1(W)∼=W,this action coincides with the fundamental representation of so(W,b)on its deﬁning vector space W.In other words,taking the commutator of the Hamiltonian H∈C2(W)with aﬁeld op-eratorψ∈C1(W)yields the same answer as viewing H as an element of so(W,b), then applying H= A B C−A t to the vectorψ=v+f∈W byH·(v+f)=(A v+B f)+(C v−A t f),andﬁnally reinterpreting the result as aﬁeld operator in C1(W).The closure relation[C2(W),C1(W)]⊂C1(W)and the isomorphism C1(W)∼= W make it possible to reduce the dynamics ofﬁeld operators to a dynamics on the Nambu space W.After reduction,as we have seen,the generators X∈End(V⊕V∗) of time evolutions of the physical system are of the special formiX=。

基于偏序集的连续映射

第38卷第1期2024年2月南华大学学报(自然科学版)Journal of University of South China(Science and Technology)Vol.38No.1Feb.2024收稿日期:2023-10-18基金项目:湖南省教育厅科研基金项目(22B0447)作者简介:杨大子(1998 ),男,硕士研究生,主要从事偏序集理论方面的研究㊂E-mail:ydz1344939@㊂∗通信作者:邹志伟(1983 ),男,副教授,博士,主要从事Domain 理论方面的研究㊂E-mail:zouzhiwei1983@DOI :10.19431/ki.1673-0062.2024.01.008基于偏序集的连续映射杨大子,邹志伟∗(南华大学数理学院,湖南,衡阳,421001)摘㊀要:基于实数集R 的经典邻域的概念,引入了一个新的基于偏序集P 的邻域的概念,随后定义了一个从P 到R 的映射的极限和连续的新概念,并在偏序集的背景下验证了连续函数的各种经典结论,例如有界性㊁介值定理等等㊂关键词:偏序集;邻域;极限;连续性中图分类号:O153.1文献标志码:A 文章编号:1673-0062(2024)01-0060-07Continuous Mapping Based on PosetYANG Dazi ,ZOU Zhiwei ∗(School of Mathematics and Physics,University of South China,Hengyang,Hunan 421001,China)Abstract :Different from the classical neighborhood based on real number set R ,this paper introduces we introduce a new notion of neighborhood based on a poset P .Subsequently,the novel notions of limit and continuity of a mapping from P to R are also defined.Various classical results about a continuous functions are verified in the context of poset inthis paper,such as boundness,intermediate value theorem and etc.key words :poset;neighborhood;limit;continuity0㊀引㊀言偏序集作为近代数学三大母结构之一,在数学中是广泛存在的㊂偏序集(P ,ɤ)由一个非空集合P 和满足自反性㊁反对称性和传递性的二元关系 ɤ 组成㊂如果偏序集中任意两个元素都存在上下确界则称为格㊂格论起源于G.Birkhoff在1940年出版的巨著Lattice Theory [1]㊂经过这些年的发展,格论在拓扑学㊁模糊数学㊁组合数学等都得到了广泛的应用[2-8]㊂在微积分中,如果对于定义域中的任一点x ,U (x )⊆R 且f (x )在像集中,存在f (U (x ))⊆U (f (x )),则称函数f 连续㊂连续这一概念是微积分的基础,在微积分理论的发展过程中起到不可第38卷第1期杨大子等:基于偏序集的连续映射2024年2月或缺的作用㊂同时连续性本身也带来了分析中的许多关键性质和定理,如闭区间上的有界性㊁零点存在定理,介值定理和不动点理论[9]等等㊂函数是一个从实数集R 的子集到实数集R 的映射,类似地在偏序集理论中也有从偏序集到偏序集的映射㊂考虑到实数集可以看作是一个特殊的偏序集,那么如何定义和研究从偏序集到实数集的连续映射,推广连续函数的性质成为有意义的?函数的极限和连续性的概念是基于邻域的概念,因此可以首先在偏序集上定义邻域的概念;其次,定义从P 到R 的映射上的极限的概念并推广相关性质;然后给出从P 到R 的映射的连续的概念并着手对其性质进行研究㊂考虑闭区间上连续函数的性质,需要将它们推广到从P 到R 的映射上,由于一般偏序集不具有实数集的完备性,所以不能使这些性质成立,因此,需要完备格来代替一般偏序集㊂最后,才能在此背景下验证数学分析中的各种重要定理㊂1㊀预备知识下文将使用以下基本定义和命题㊂定义1[10]㊀设P 为偏序集,称P 有一个底元,如果存在ʅɪP 且对任意的x ɪP 有ʅɤx ㊂对偶地,P 有一个顶元,如果存在ʅɪP 使得对于任意的x ɪP 有x ɤʅ㊂对于a ,b ɪP ,如果它们满足a ɤb 且a ʂb ,则记为a <b ㊂定义2[10]㊀设P 为偏序集且x ,y ɪP ,称x 被y 覆盖(或y 覆盖x ),并且写作x ≼y 或y ≽x ,如果x <y 且x ɤz <y 蕴含z =x ㊂后一个条件要求不存在z ɪP 使得x <z <y ㊂定义3[10]㊀假设P 和Q 是(不相交的)偏序集㊂P 和Q 的不交并集PQ 是偏序集,在PQ中定义x ɤy 当且仅当x ,y ɪP 且在P 中x ɤy 或者x ,y ɪQ 且在Q 中x ɤy ㊂设P 为偏序集且取Q ⊆P ,那么称a ɪQ 是Q 的一个极大元,如果a ɤx 且x ɪQ 蕴含a =x ,记Q 的极大元的集合为Max Q ㊂如果Q (继承P 中的序)有顶元ʅQ ,那么Max Q ={ʅQ },这种情况下ʅQ 称为Q 中的最大元,写作max Q =ʅQ ;极小元㊁极小元的集合Min Q 和最小元min Q 的定义与上对偶㊂定义4[10]㊀设P 是一个非空偏序集,如果对于所有的S ⊆P ,ᶱS 和ɡS 都存在,那么P 被称为完备格㊂其中用x ᶱy 代替sup{x ,y },x ɡy 代替inf{x ,y },类似地用ᶱS 和ɡS 代替sup S 和inf S (如果它们存在)㊂命题1[11]㊀设P 是一个偏序集,若任意并存在(或者任意交存在),则P 为完备格㊂2㊀主要结论2.1㊀邻域和极限设P 为偏序集且在P 中有a <b ㊂称子集{x ɪP a ɤx ɤb }为闭区间,记为[a ,b ];称子集{x ɪP a <x <b }为开区间,记为(a ,b )㊂定义5㊀设P 为偏序集,对于P 中的点x (1)若存在a ɪP 使得a <x ,则称[a ,x ]为x的一个左邻域,记作U -(x ,a ),简记为U -(x );称[a ,x )为x 的一个去心左邻域,记作U ㊂-(x ,a ),简记为U ㊂-(x );(2)若存在b ɪP 使得x <b ,则称[x ,b ]为x 的一个右邻域,记作U +(x ,b ),简记为U +(x );称(x ,b ]为x 的一个去心右邻域,记作U ㊂+(x ,b ),简记为U ㊂+(x );(3)若存在a ,b ɪP 使得a <x 0和a <x 0㊂存在U ㊂-(x 0)⊆[a ,x 0),使得对于任意的x ɪU ㊂-(x 0)有f (x )-A <ε,则称A 为f 在x 0的左极限,记为lim x ңx -0f (x )=A ;(2)对任意的ε>0和x 00和a <x 0<b ㊂存在U ㊂(x 0)⊆U ㊂(x 0,a ,b ),使得对于任意的x ɪU ㊂(x 0)有f (x )-A <ε,则称A 为f 在x 0的极限,记为lim x ңx 0f (x )=A ㊂注1:如果x 0ɪP 没有左(右)邻域但是映射f在x 0处有右(左)极限时,亦称f 在点x 0处有极限㊂命题2㊀假设A 与B 都是f 在点x 0的极限,第38卷第1期南华大学学报(自然科学版)2024年2月则A =B ㊂证明:由极限的定义,可知对任意的ε>0和P 中任意的a <x 0<b ,存在U ㊂1(x 0)⊆U ㊂(x 0,a ,b ),使得对任意的x ɪU ㊂1(x 0),f (x )-A <ε2成立㊂对U ㊂1(x 0)中任意的c <x 0<d ,存在U ㊂2(x 0)⊆U ㊂(x 0,c ,d ),使得对任意的x ɪU ㊂2(x 0),f (x )-B <ε2成立㊂则对于任意的x ɪU ㊂2(x 0),使得|A -B |ɤf (x )-A +f (x )-B <ε由于ε可以无限接近于0,可知A =B ㊂证毕㊂命题3㊀若lim x ңx 0f (x )=A ,lim x ңx 0g (x )=B ,且A >B ,则存在U ㊂(x 0),使得对于任意x ɪU ㊂(x 0),有f (x )>g (x )㊂证明:取ε0=A -B2>0㊂由lim x ңx 0f (x )=A ,对于P中任意的a <x 0<b ,存在U ㊂1(x 0)⊆U ㊂(x 0,a ,b ),使得对任意的x ɪU ㊂1(x 0),有f (x )-A <ε0,从而A +B2<f (x );由lim x ңx 0g (x )=B ,对U ㊂1(x 0)中任意的c <x 0<d ,存在U ㊂2(x 0)⊆U ㊂(x 0,c ,d ),使得对任意的x ɪU ㊂2(x 0),有|g (x )-B |<ε0,从而g (x )<A +B2㊂则对于任意的x ɪU ㊂2(x 0),有g (x )<A +B2<f (x )㊂证毕㊂推论1㊀若lim x ңx 0f (x )=A >0,对P 中任意的a <x 0A2>0成立㊂证明:令g (x )=A2,由命题3可知存在U ㊂(x 0)⊆P ,使得对于任意的x ɪU ㊂(x 0)有f (x )>A2>0㊂证毕㊂对偶地,当A <0时,有f (x )<A2<0㊂推论2㊀如果lim x ңx 0f (x )=A >0,lim x ңx 0g (x )=B ,那么对P 中任意的a <x 0A ,则由命题3,对U ㊂(x 0)中任意的c <x 0<d ,存在U ㊂1(x 0)⊆U ㊂(x 0,c ,d ),使得对任意的x ɪU ㊂1(x 0)有g (x )>f (x )㊂可知对于任意的x ɪU ㊂1(x 0)既有g (x )ɤf (x )又有g (x )>f (x ),从而产生矛盾㊂证毕㊂推论3㊀令lim x ңx 0f (x )=A >0,对P 中任意的a <x 0<b ,存在U ㊂(x 0)⊆U ㊂(x 0,a ,b ),使得f (x )在U ㊂(x 0)中有界㊂证明:取常数M 和m ,满足m <A <M ,并且令g (x )=m ,h (x )=M 为两个常值映射㊂由命题3可知对P 中任意的a <x 0<b ,存在U ㊂(x 0)⊆U ㊂(x 0,a ,b ),使得对任意的x ɪU ㊂(x 0)有m <f (x )<M ㊂证毕㊂如果f 在x 0有定义,取G =max{|m |,|M |,|f (x )|},则对任意x ɪU (x 0)有f (x )ɤG ㊂命题4㊀如果对P 中任意的a <x 00和U ㊂(x 0)中任意的a 1<x 0<b 1㊂由lim x ңx 0h (x )=A 可知存在U ㊂1(x 0)⊆U ㊂(x 0,a 1,b 1),使得对于任意的x ɪU ㊂1(x 0)有|h (x )-A |<ε,那么h (x )<A +ε㊂由lim x ңx 0g (x )=A ,对U ㊂1(x 0)中任意的a 2<x 0<b 2可知存在U ㊂2(x 0)⊆U ㊂(x 0,a 2,b 2),使得对于任意的x ɪU ㊂2(x 0)有|g (x )-A |<ε,那么A -ε<g (x )㊂对任意的x ɪU ㊂2(x 0)有A -ε<g (x )ɤf (x )ɤh (x )<A +ε即lim x ңx 0f (x )=A ㊂证毕㊂2.2㊀连续性定义7㊀设P 为偏序集,x 0ɪP 且f :P ңR ㊂(1)若lim x ңx -0f (x )=f (x 0)或者x 0没有左邻域,称f 在x 0处左连续;(2)若lim x ңx +0f (x )=f (x 0)或者x 0没有右邻域,称f 在x 0处右连续;(3)如果f 在x 0处既左连续又右连续,那么f 在x 0处连续;(4)对任意的x ɪP ,如果f 在x 处连续,那么f 在P 上连续㊂第38卷第1期杨大子等:基于偏序集的连续映射2024年2月以lim x ңx -0f (x )=f (x 0)为例㊂由极限的定义,左极限存在且等于f (x 0)当且仅当对于任意的ε>0和a <x 0,存在U ㊂-(x 0)⊆[a ,x 0),使得对于任意的x ɪU ㊂-(x 0)有f (x )-f (x 0)<ε㊂然后称使得f (x )-f (x 0)<ε成立的所有U ㊂-(x 0)的集族为x 0的连续左邻域系并记作U P -(x 0,ε)㊂类似地,连续右邻域系U P +(x 0,ε)和连续邻域系U P (x 0,ε)㊂若P 为一个反链,则映射f :P ңR 连续㊂证明:因为P 是一个反链,所以任意的x ɪP 都没有左右邻域,所以f 在任意x ɪP 上连续,所以f 在P 上连续㊂证毕㊂定义8㊀设P 为偏序集,对于任意的a ,b ɪP ,如果P 中存在一列有限个元素即x 1,x 2, ,x n ,使得序列a ,x 1,x 2, ,x n ,b 任意相邻两个元素之间存在覆盖关系,则称a 与b 有关系,记为a ~b ,反之则称a 与b 没有关系㊂定理1㊀设P 为有限偏序集,f :P ңR 为一个映射,则以下条件等价:(1)f 在P 上连续;(2)∀a ,b ɪP ,若a ≼b ,则f (a )=f (b );(3)∀a ,b ɪP ,若a ~b ,则f (a )=f (b )㊂证明:(1)⇒(2)假设存在a ,b ɪP 且a ≼b ,使得f (a )ʂf (b ),那么对于点a ,存在ε=f (a )-f (b )2>0,有f (a )-f (b )>ε,则映射f 在点a 处不连续,与条件(1)矛盾;(2)⇒(3)由定义8可知对a ~b ,则P 中存在一列有限个元素x 1,x 2, ,x n ,使得序列a ,x 1,x 2, ,x n ,b 任意相邻两个元素之间存在覆盖关系,由条件(2)可知任意两个元素之间如果存在覆盖关系则他们在映射的作用下的像相等,此即f (a )=f (x 1)=f (x 2)= =f (x n )=f (b );(3)⇒(1)由条件(3)可知对任意的a ɪP ,若存在b ɪP ,使得a ~b ,则f (a )=f (b )㊂此即对任意的ε>0,存在U ㊂-(a )ʂØ或U ㊂+(a )ʂØ,使得对任意x ɪU ㊂-(a )或任意x ɪU ㊂+(a )时,有f (x )-f (a )<ε㊂对任意a ɪP ,如果任意U ㊂(a )=Ø,那么映射f 在点a 处连续㊂因为点a 是P 中任意点,所以f 在P 上连续㊂证毕㊂设P 为偏序集,P 1⊆P 且f 是从P 到R 的映射㊂构造一个映射f P 1:P 1ңR 且其满足f P 1(x )=f (x )㊂那么称fP 1是f 的投影㊂定理2㊀假设f 在P 上连续,取P 1⊆P ,映射fP 1在P 1上连续,如果P 1满足以下条件:(1)对任意的a ,x 0ɪP 1且a <x 0,有{x ɪP 1a ɤx <x 0}={x ɪP a ɤx <x 0};(2)对任意的b ,x 0ɪP 1且x 00存在U P -(x 0,ε)㊂由条件(1)对P 1中任意的a <x 0,有{x ɪP 1a ɤx <x 0}={x ɪP a ɤx <x 0},易知存在U ㊂-(x 0)⊆{x ɪP 1a ɤx <x 0},使得U ㊂-(x 0)ɪU P -(x 0,ε)㊂那么f P 1在x 0处左连续㊂对偶地,fP 1在x 0处右连续㊂这能推出fP 1在P 1上连续㊂证毕㊂定理3㊀设P 1,P 2是两个不相交的偏序集,且令P =P 1P 2,则以下条件等价:(1)映射f 在P 上连续;(2)映射f P i 在P i (i =1,2)上连续㊂证明:(1)⇒(2)只需证明fP 1在P 1上连续㊂由条件(1),f 在任意x ɪP 处连续㊂取任意x 0ɪP 1,如果x 0在P 1上有左邻域,因为P 1,P 2是两个不相交的偏序集,所以对P 1中任意的a <x 0,有{x ɪP 1a ɤx <x 0}={x ɪP a ɤx <x 0}㊂对偶地,对P 1中任意的x 0<b ,有{x ɪP 1x 0<x ɤb }={x ɪP x 0<x ɤb }㊂由定理2可知映射f P 1在P 1上连续㊂类似地f P 2在P 2上连续㊂(2)⇒(1)因为fP i在P i (i =1,2)上连续,对任意x ɪP ,x ɪP 1或者x ɪP 2,所以映射f 在P 上连续㊂证毕㊂注2:若存在一族两两不相交的偏序集{P γ}γɪΓ,令P =γɪΓP γ,映射f 在P 上连续当且仅当映射f P γ在任意P γ上连续㊂证明过程与定理3一致㊂2.3㊀连续映射的性质设P 为偏序集㊂对于P 中任意a ɤb ,[a ,b ]是P 上的闭区间㊂定理4㊀设P 为偏序集,f :P ңR 为一个映射㊂对P 中任意的a ɤb ,如果映射f 在P 上连续,那么在f 闭区间[a ,b ]⊆P 上连续㊂证明:若a ≼b ,则f (a )=f (b ),即f 在闭区间[a ,b ]上连续㊂如果(a ,b )ʂØ,对任意的x 0ɪ[a ,b ]㊂容易知道对于[a ,b ]中任意的y <x 0,有{x ɪ[a ,b ]y ɤx <x 0}={x ɪP y ɤx <x 0}㊂对偶的对于[a ,b ]中任意的x 0<z ,有{x ɪ[a ,b ]x 0<第38卷第1期南华大学学报(自然科学版)2024年2月x ɤz }={x ɪP x 0<x ɤz }㊂然后根据定理2,f 在[a ,b ]上连续㊂证毕㊂找到闭区间之后,发现上面的这些性质在偏序集上不一定成立㊂下面是两个反例㊂例2㊀设P 为(R \{0},ɤ),对任意的x ɪP有f (x )=1|x |㊂以闭区间[-1,1]⊆P 为例㊂由定理4,f 在[-1,1]上连续,容易发现lim x ң0f (x )=+ɕ,于是映射f 无界并且没有最大值㊂例3㊀设P 为(R \{0},ɤ),对任意的x ɪP 有f (x )=x ㊂以闭区间[-1,1]⊆P 为例㊂由定理4,f 在[-1,1]上连续,f (-1)=-1<0且f (1)=1>0,但是0在P 中没有原像,所以零点存在定理和介值定理在偏序集上的闭区间上也不一定成立㊂从数学分析中可以看出,如果没有实数集的完备性,就不能建立闭区间上连续函数的性质㊂从以上两个例子可以看出,与实数集相比,一般偏序集缺乏完备性㊂为此,选择了一个特殊的偏序集完备格,并考虑它的完备性㊂类似于实数理论,首先定义序列及其极限的概念㊂设P 为偏序集,P 中的一列由正整数编号的元素x 1,x 2, ,x n 被称为一个序列,记作{x n }㊂定义9㊀设P 为偏序集,{x n }为P 中的序列且x 0ɪP ㊂如果P 中有一条升链{a n }和一条降链{b n }并且sup {a n }=inf {b n }=x 0㊂对于任意的m ɪN +,存在N >0使得对于任意的n >N 有x n ɪU (x 0,a m ,b m )㊂那么序列{x n }趋近于x 0㊂称x 0为{x n }的极限并记作lim n ң+ɕx n =x 0㊂命题5㊀如果f :P ңR 连续且lim n ң+ɕx n =x 0,则f (lim n ң+ɕx n )=lim n ң+ɕf (x n )=f (x 0)㊂证明:由lim n ң+ɕx n =x 0可知f (lim n ң+ɕx n )=f (x 0)㊂对任意的ε>0,又因为lim n ң+ɕx n =x 0可知P 中有一条升链{a n }和一条降链{b n }并且sup{a n }=inf{b n }=x 0㊂对于任意的m ɪN +,存在N >0使得对于任意的n >N 有x n ɪU (x 0,a m ,b m )㊂因为f 在P 上连续,对a 1<x 00使得对任意的x ɪU ㊂(x 0,a m ,b m )有f (x )-f (x 0)<ε㊂因为x n ɪU (x 0,a m ,b m ),所以对于任意n >N 有f (x n )-f (x 0)<ε㊂所以lim n ң+ɕf (x n )=f (x 0)㊂证毕㊂设P 为完备格,S 为P 任意非空子集㊂由完备格的定义可知sup S 和inf S 在P 中存在㊂然后通过实数理论研究完备格的一些完备性质㊂实数理论的确界原理在完备格上自然成立㊂然后我们由实数理论来考虑完备格的完备性质㊂命题6㊀设P 为偏序集,对P 1⊆P ,X ⊆P 1,若sup PX ɪP 1,则sup P 1X 存在且sup P 1X =sup PX ㊂证明:令a =sup PX ɪP 1,首先对于任意的x ɪX ,有x ɤa ㊂若存在y ɪP 1,使得对任意的x ɪX 有x ɤy ,则y 在P 中为X 的上界,所以a ɤy ,即a 为X 在P 1的上确界,即sup P 1X =sup PX ㊂证毕㊂命题7㊀设P 为完备格,对于P 中任意的a ɤb ,[a ,b ]也为完备格㊂证明:对任意的X ⊆[a ,b ]⊆P ,sup P X 存在㊂因为对任意的x ɪX 有a ɤx ɤb ,所以sup PX ɪ[a ,b ]㊂由命题6可知,sup [a ,b ]X 存在且sup [a ,b ]X =sup PX ,由命题1,[a ,b ]为完备格㊂证毕㊂注3:由完备格定义可知,完备格自身也为闭区间形如[ʅ,ʅ]㊂如果没有特殊说明,以下的P 为完备格㊂命题8㊀设{x n }为P 中的序列,若{x n }单调递增且有上界,则序列{x n }有极限且极限为上确界㊂证明:设x 0=sup{x n },构造集合E ={x n n =1,2, }⊆P ,所以ᶱE 存在且ᶱE =x 0㊂取{a n }={x n }且取{b n }={x 0,x 0, },则对任意的m ɪN +,存在N >0,使得对任意的n >N ,有x n ɪ[a m ,x 0]ɣ[x 0,b m ]=[a m ,x 0],则lim n ң+ɕx n =x 0㊂证毕㊂对偶地,当{x n }单调递减且有下界时lim n ң+ɕx n =inf{x n }㊂命题9㊀有界序列有收敛子列㊂证明:设{x n }为P 上有界序列㊂首先证明单调子列的存在性㊂如果{x n }中存在单调增子列{x n k },那么证明完毕㊂如果{x n }中不存在单调增子列,那么存在n 1>n 总有x n 1ɤx n ㊂类似地,{x n }(n >n 1)中也不存在单调增子列,存在n 2>n 1总有x n 2ɤx n 1ɤx n ㊂如此无限下去,可以得到一列单调递减子列{x n k }㊂那么有界序列存在单调子列{x n k },又因为它有界由命题8可知{x n k }收敛㊂证毕㊂定理5㊀若映射f :P ңR 在闭区间[a ,b ]⊆P 上连续,则它在[a ,b ]上有界㊂证明:假设f 在[a ,b ]上无界,则对于每个正实数A ,都存在x ɪ[a ,b ],使得f (x )ȡA ;所以对每个n ɪN +,存在x ɪ[a ,b ],使得f (x )ȡn ㊂由命题9,可以找到一个序列{x n }⊆[a ,b ](n ɪN +),使得f (x n )ȡn ㊂由命题9可知{x n }(n ɪ第38卷第1期杨大子等:基于偏序集的连续映射2024年2月N +)有收敛子列{x n k }(k ɪN +)㊂令x 0=lim k ң+ɕx n k 并且构造集合E ={x n k k =1,2, }⊆[a ,b ],那么x 0ɪ[a ,b ]㊂又因为f 在[a ,b ]上连续,由命题5易知f (lim n ң+ɕx n )=lim n ң+ɕf (x n )=f (x 0)且f (x 0)<+ɕ㊂所以lim k ң+ɕf (x n k )<+ɕ㊂对于f (x n k )ȡn k ȡk ,lim k ң+ɕf (x n k )=+ɕ,产生矛盾㊂所以f 在[a ,b ]上有界㊂证毕㊂为了验证最值定理,本文引入了复合映射的概念㊂设映射f :P ңR 和函数g :R ңR ,称g f :P ңR 为复合映射㊂定理6㊀如果u =f (x )在点x 0处连续且u 0=f (x 0)并且y =g (u )在点u 0处连续,则g f :P ңR 在点x 0处连续㊂证明:对任意ε>0,由于lim u ңu 0g (u )=g (u 0),存在U ㊂(u 0)⊂R ,使得对于任意u ɪU ㊂(u 0),有g (u )-g (u 0)<ε㊂因为lim x ңx 0f (x )=f (x 0)=u 0,存在U ㊂(x 0)⊂P ,使得对于任意x ɪU ㊂(x 0),有f (x )ɪU ㊂(u 0)㊂那么对于任意的x ɪU ㊂(x 0)有g f (x )-g f (x 0)=g f (x )-g (u 0)<ε,此即lim x ңx 0g f (x )=g f (x 0)㊂证毕㊂由定理6可知当映射f :P ңR 在P 上连续,函数g :R ңR 在R f 上连续时,复合映射g f :P ңR 在P 上连续㊂定理7㊀如果映射f 在闭区间[a ,b ]⊆P 上连续,则它在[a ,b ]上能取到最大值和最小值,即存在A ,B ɪ[a ,b ],对任意x ɪ[a ,b ]有f (A )ɤf (x )ɤf (B )㊂证明:由定理5,集合R f ={f (x )x ɪ[a ,b ]}是一个有界实数集,所以必有上下确界,记m =inf R f ,M =sup R f ㊂接着需要证明存在B ɪ[a ,b ],使得f (B )=M ㊂假设对于任意x ɪ[a ,b ]有f (x )<M ㊂令g (x )=1M -f (x ),x ɪ[a ,b ]㊂由定理6,复合映射g f :P ңR 在[a ,b ]上连续且值为正的㊂那么g 在[a ,b ]上有上确界并记为G ㊂则有0<g (x )=1M -f (x )ɤG ,x ɪ[a ,b ]㊂从而推得f (x )ɤM -1G,x ɪ[a ,b ]㊂这与M 为f ([a ,b ])的上确界相矛盾,所以必有B ɪ[a ,b ],使得f (B )=M ,即f 在[a ,b ]上有最大值㊂同理可证f 在[a ,b ]上有最小值㊂证毕㊂以下是数学分析中零点存在定理的推广㊂定理8㊀如果映射f 在闭区间[a ,b ]⊆P 上连续且f (a )㊃f (b )<0,则存在c ɪ(a ,b ),使得f (c )=0㊂证明:不失一般性,不妨设f (a )<0,f (b )>0,集合E ={x ɪ[a ,b ]f (x )<0},显然E 有底元a ,且存在极大元集合Max E ⊆E ,因为f (a )<0,所以Max E 和E 非空㊂对任意的c ɪMax E ,c ɪ[a ,b ]㊂假设f (c )ʂ0㊂如果f (c )>0,由推论1,存在U ㊂-(c )⊆[a ,b ],使得对于任意的x ɪU ㊂-(c )有f (x )>0,那么U ㊂-(c )ɘE ʂØ,这与c ɪMax E 矛盾㊂如果f (c )<0,对于f (b )>0且c <b ,由命题3,存在U ㊂+(c )⊆[a ,b ],使得对任意的x ɪU ㊂+(c )有f (x )<0,这与c ɪMax E 矛盾㊂所以f (c )=0㊂证毕㊂定理9㊀如果映射f 在闭区间[a ,b ]⊆P 上连续,存在A ,B ɪ[a ,b ],使得m =f (A )=inf R f ,M =f (B )=sup R f ㊂那么映射f 在[a ,b ]上能取到[m ,M ]中的任意值㊂证明:对于A ,B ɪ[a ,b ]⊆P ,A 和B 不一定比,所以取两个闭区间[a ,A ]和[a ,B ],因为对任意的x ɪ[a ,b ]有f (A )ɤf (x )ɤf (a )或者f (a )ɤf (x )ɤf (B )㊂取任意y ɪ(m ,f (a )),令g (x )=f (x )-y ,易知g (A )<0且g (a )>0,由定理8,存在c ɪ(a ,A ),使得g (c )=0㊂类似地,对于任意地y ɪ(f (a ),M ),存在c ɪ(a ,B ),使得g (c )=0㊂证毕㊂推论4㊀如果映射f 在闭区间[a ,b ]⊆P 上连续,m 是最小值,M 是最大值,则映射的值域是闭区间R f =[m ,M ]㊂参考文献:[1]BIRKHOFF ttice theory[M].Providence:American Mathematical Society,1940:1-20.[2]JIN Q,LI L Q,MA Z M,et al.A note on the relationshipsbetween generalized rough sets and topologies[J].Inter-national journal of approximate reasoning,2021,130(1):292-296.[3]PEI Z,PEI D W,ZHENG L.Topology vs generalizedrough sets[J].International journal of approximate rea-soning,2011,52(2):231-239.(下转第89页)第38卷第1期屈㊀星等:一种考虑光伏发电系统的广义综合模型2024年2月(5):312-318.[15]李培强,曾小军,黄际元,等.面向综合负荷的并网光伏发电系统等效建模[J].电力系统自动化,2016,40(8):43-50.[16]盛四清,关皓闻,雷业涛,等.基于混沌海鸥优化算法的含光伏发电系统负荷模型参数辨识[J].太阳能学报,2022,43(7):64-72.[17]李善寿,张兴.改进的光伏组件工程数学模型建模方法[J].电力自动化设备,2015,35(9):108-112. [18]周德佳,赵争鸣,袁立强,等.具有改进最大功率跟踪算法的光伏并网控制系统及其实现[J].中国电机工程学报,2008,28(31):94-100.[19]SON S E,LEE S H,CHOI D H,et al.Improvement ofcomposite load modeling based on parameter sensitivityand dependency analyses[J].IEEE transactions onpower systems,2014,29(1):242-250. [20]屈星,李欣然,宋军英,等.遗传算子自适应设计及其在负荷建模中的应用[J].电力系统及其自动化学报,2018,30(7):65-72.(上接第65页)[4]ZOU Z W,LI Q G,HO W K.Domains via approximation operators[J].Logical methods in computer science,2018, 14(2):1-17.[5]LEI Y B,LUO M K.Rough concept lattices and domains [J].Annals of pure and applied logic,2009,159(3): 333-340.[6]YAO Y Y.A comparative study of formal concept analysis and Rough Set Theory in data analysis[J].Lecture notes in computer science,2004,3066:59-68.[7]GIERZ G,LAWSON J,STRALKA A.Quasicontinuous posets[J].Houston journal of mathematics,1983,9:2.[8]KESZEGH B,LEMONS N,MARTIN R R,et al.Induced and non-induced poset saturation problems[J].Journal of com-binatorial theory,series A,2021,184:105497. [9]JAMES S.Calculus[M].7th ed.Boston:Cengage Learn-ing,2014:108-130.[10]DAVEY B A,PRIESTLEY H A.Introduction to latticesand order[M].2nd ed.Cambridge:Cambridge UniversityPress,2002:1-64.[11]GIERZ G,HOFMANN K H,KEIMEL K,et al.Continu-ous lattices and domains[M].Cambridge:CambridgeUniversity Press,2003:1-47.。

Triangulations of Seifert Fibred Manifolds

a r X i v :m a t h /0301246v 1 [m a t h .G T ] 22 J a n 2003TRIANGULATIONS OF SEIFERT FIBRED MANIFOLDSALEKSANDAR MIJATOVI ´CAbstract .It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists.In this paper we describe a procedure yielding an explicit formula for such a function if the 3-manifold in question is a Seifert ﬁbred space.1INTRODUCTION There is a natural way of modifying a triangulation T of an n -manifold.Suppose D is a combinatorial n -disc which is a subcomplex both in this triangulation and in the boundary of the (n +1)-simplex ∆n +1.We can change T by removing D and inserting ∆n +1−int(D ).What we’ve just described is called a Pachner move .In dimension 3there are four possible moves (see ﬁgure 1).Note that we can deﬁne Pachner moves even if the triangulation T is non-combinatorial (i.e.simplices of T are not uniquely determined by their vertices).Since our aim is to deal with the triangulations of the manifolds that are not necessarily closed,we need to allow for some additional moves that will modify the simplicial structure on the boundary (throughout this paper we will be using the term simplicial structure as a synonym for a possibly non-combinatorial triangulation).The deﬁnition of a Pachner move readily generalises to this setting.Changing the triangulation of the boundary by an (n −1)-dimensional Pachner move amounts to gluing onto (or removing form)our manifold an n -simplex ∆n ,which must exist by the deﬁnition of the move.So in dimension 3we have to use the three two-dimensional moves (usually referred to as (2−2),(1−3)and (3−1))that can be implemented by gluing on or shelling a tetrahedron.It was proved by Pachner (see [9])that any two triangulations of the same PL n -manifold are related by a ﬁnite sequence of Pachner moves and simplicial isomorphisms.It is well known (see proposition 1.3in [8])that a computable function bounding the length of the sequence from Pach-ner’s theorem in terms of the number of tetrahedra for a ﬁxed 3-manifold M ,gives an algorithm for recognising M among all 3-manifolds.The following theorem gives an explicit formula for such a bound in case M is a Seifert ﬁbred space with a ﬁxed triangulation on its boundary.Theorem 1.1Let M →B be a Seifert ﬁbred space with non-empty boundary.Let P and Q be two triangulations of M that coincide on ∂M and contain p and q tetrahedra respectively.Then there exists a sequence of Pachner moves of length at most e 6(10p )+e 6(10q )which transforms P into a triangulation isomorphic to Q .The homeomorphism of M that realizes the simplicial isomorphismis,when restricted to ∂M ,equal to the identity on the boundary of M .The exponent in the above expression containing the exponential function e (x )=2x stands for the composition of the function with itself rather than for multiplication.The shameful enormity of the bound can be curbed by a more careful choice of subdivisions.The height of the tower of exponents can be reduced from 6to 3,but the complexity of the constructions involved grows tenfold.Since we are mainly interested in the existence of an explicit formula,we shall not strive to get the best numbers possible.The bound in theorem 1.1is clearly computable and it hence gives a conceptually simple recognition algorithm for every bounded Seifert ﬁbred space.It is true that the topology of1(2-3)(1-4)(4-1)Figure1:Three dimensional Pachner moves.the Seifertﬁbred manifolds is not terribly exciting.They are however ubiquitous pieces in JSJ-decompositions of more interesting3-manifolds.It is there that our theorem willﬁnd its best application.2SOME NORMAL SURF ACE THEORY AND SOME TOPOLOGY The pivotal tool for probing the triangulations of our3-manifolds is normal surface theory. Many good accounts of it appeared in the literature(see for example[2],[6]or[1]).In this section we will state the basic deﬁnitions that will allow us to give some known properties of normal surface.We will then go on to explore how normal surfaces interact with boundary patterns.The section will be concluded with a discussion of some well known features of incompressible surfaces in Seifertﬁbred manifolds.We are assuming throughout that all3-manifolds we are dealing with are orientable.Let T be a triangulation of a3-manifold M.An arc in a2-simplex of T is normal if its ends lie in diﬀerent sides of a2-simplex.A simple closed curve in the2-skeleton of T is a normal curve if it intersects each2-simplex of T in normal arcs.A properly embedded surface F in M is in normal form with respect to T if it intersects each tetrahedron in T in a collection of discs all of whose boundaries are normal curves consisting of3or4normal arcs,i.e.triangles and quadrilaterals.A normal discs is a triangle or a quadrilateral.There are precisely seven normal disc types in any tetrahedron of T.An isotopy of M is called a normal isotopy with respect to T if it leaves all simplices of T invariant.In particular this implies that it isﬁxed on the vertices of T.A normal surface is determined,up to normal isotopy,by the number of normal disc types in which it meets the tetrahedra of T.It therefore deﬁnes a vector with7t coordinates.Each coordinate represents the number of copies of a normal disc type that are contained in the surface with t being the number of tetrahedra in T.It turns out that there is a certain restricted linear system that such a vector is a solution of.Moreover there is a one to one correspondence between the solutions of that restricted linear system and normal surface in M.If the sum of two vector solutions of this system satisﬁes the restrictions on the system,then it represents a normal surface in M.On the other hand there is a geometric process called regular alteration(seeﬁgure2in[1]) which can be carried out on the normal surfaces representing the summands and which yields the2normal surface corresponding to the sum.It follows directly from the deﬁnition of regular alteration that the Euler characteristic is additive over normal addition.We can deﬁne the weight w(F)of a surface F,which is transverse to the1-skeleton of T,to be the number of points of intersection between the surface and the1-skeleton.Since regular alteration only changes the surfaces involved away from the1-skeleton,the weight too is additive over normal addition.A normal surface is called fundamental if the vector corresponding to it is not a sum of two integral solutions of the linear system.The solution space of the linear system projects down to a compact convex linear cell which is called the projective solution space.A vertex surface is a connected two-sided normal surface that projects onto a vertex of the projective solution space (see[6]for a more detailed description).The next proposition is proved in[2].It will be important for us that its proof does not de-pend on the number of equations in the linear system arising from the triangulation of the manifold. Proposition2.1Let M be a compact triangulated3-manifold containing t tetrahedra.Then each normal coordinate of a vertex surface in M is bounded above by27t.If the normal surface is fundamental,then7t27t puts an upper bound on all of its normal coordinates.Recall that a properly embedded surface F in a3-manifold M is injective if the homomorphism π1(F)→π1(M),induced by the inclusion of F into M,is a monomorphism.A surface F is said to be incompressible if it satisﬁes the following conditions:•The surface F does not contain2-spheres that bound3-balls nor does it contain discs which are isotopic rel boundary to discs in∂M and•for every disc D in M with D∩S=∂D there is a disc D′in S with∂D=∂D′.A horizontal boundary of an I-bundle over a surface is a part of the boundary corresponding to the∂I-bundle.The vertical boundary is a complement of the horizontal boundary and consists of annuli thatﬁbre over the bounding circles of the base surface.It is a well-known fact that a properly embedded one-sided surface in M is injective if and only if the horizontal boundary of its regular neighbourhood is incompressible.An embedded torus in M that is incompressible and is not boundary parallel is sometimes referred to as an essential torus.There is also a relative notion of incompressibility which we will need to consider.A surface F is∂-incompressible if for each disc D in M such that∂D splits into two arcsαandβmeeting only at their common endpoints,with D∩F=αand D∩∂M=βthere is a disc D′in F with α⊂∂D′and∂D′−α⊂∂F.Such a disc D is called a∂-compression disc for F.Note that if the manifold M is irreducible and has incompressible boundary,then we can isotope F relαso that the disc D′becomes the∂-compression disc D.A properly embedded annulus in M that is both incompressible and∂-incompressible is called an essential annulus.Before we can state the main technical results from normal surface theory,we need to deﬁne a very useful concept.Deﬁnition.A boundary pattern P in a compact3-manifold M is a(possibly empty)collection of disjoint simple closed curves and trivalent graphs embedded in∂M such that the surface∂M−P is incompressible in M.Boundary patterns usually appear in the context of hierarchies.In this paper however we will mainly be concerned with a special case when the pattern P consists of simple closed curves only.Therefore we will not investigate the correspondence between hierarchies and patterns any further.Notice also that it follows from the deﬁnition of the pattern that any simple closed curve component of P is homotopically non-trivial in∂M.Let M be a3-manifold with non-empty boundary that contains a boundary pattern P.It follows from the deﬁnition that the pattern P can be empty if and only if the manifold M has incompressible boundary.Assume now that the pattern P is not empty.A subset of M is pure if it has empty intersection with the pattern P.Most concepts from general3-manifolds carry over to3-manifolds with pattern in a very natural way.For example,a properly embedded surface F in3-manifold M with pattern P is P-boundary incompressible if for any pure disc D in M,such that D∩(∂M∪F)=∂D and D∩F is a single arc in F,the arc D∩F cuts oﬀa pure disc from F.3Notice that for P=∅this notion reduces to∂-incompressibility.Also this notion is well deﬁned only up to an isotopy of M that is invariant on the pattern P.In other words we can have two isotopic surfaces in M,out of which only one is P-boundary incompressible.Let T be some triangulation of a3-manifold M with pattern P.Throughout this paper we will be assuming that the pattern P lies in the1-skeleton of T.This assumption immediately implies that any incompressible P-boundary incompressible surface in M can be isotoped into normal form. Also any normal surface F in M has a well deﬁned intersection numberι(F),equal to the number of points in∂F∩P.Moreover this intersection number is additive over geometric sums of normal surfaces.We say that a surface F in a3-manifold M with pattern P has minimal weight if it can not be isotoped to a surface with lower weight by an isotopy that is invariant on the pattern.In case of P=∅this reduces to the usual deﬁnition of a minimal weight.An incompressible P-boundary incompressible surface in M of minimal weight has to intersect each triangle in the2-skeleton of T in normal arcs and possibly some simple closed curves.If M is irreducible we can remove these simple closed curves by isotopies in the usual way.The isotoped surface is then in normal form.The sum F=F1+F2is in reduced form if the number of components of F1∩F2is minimal among all normal surfaces F′1and F′2isotopic rel P to F1and F2respectively such that F=F′1+F′2.If the pattern P is empty we get the familiar notion of reduced form which was used in[1].We are now going to deﬁne a concept which will be of some signiﬁcance to all that follows.Deﬁnition.A patch for the normal sum F=F1+F2is a component of F1−int(N(F1∩F2)) or F2−int(N(F1∩F2)).A trivial patch is a pure patch which is topologically a disc and whose boundary intersects only one component of the1-manifold F1∩F2.This means that a boundary of a trivial patch is either a single simple closed curve in F1∩F2or it consists of two arcs:one in(∂M)−P and the other in F1∩F2.Theﬁrst possibility coincides with what was called a disc-patch in[1].In the absence of pattern the notion of the trivial patch coincides with what is called a disc-patch in[6].The reason for our slightly non-standard terminology is to avoid the confusion arising from the patches which are discs but are not disc-patches.There are several ways in which a disc patch(i.e.a patch which is a disc)can fail to be trivial,if∂M is not empty.Clearly a disc patch which intersects the pattern P is non-trivial.A pure disc patch will also be non-trivial if it intersects∂M in more than one arc.It is not hard to prove that a trivial patch can not have zero weight.The argument of lemma 3.3in[1]gives it to us for all trivial patches bounded by simple closed curve components of F1∩F2. In case when our disc patch is bounded by two arcs,we can use a simple doubling trick and then apply lemma3.3from[1]to obtain the desired conclusion.However there are patches that are topologically discs and have zero weight.But they must contain more than one component of F1∩F2in their boundaries.Now we areﬁnally in the position to state the following lemma. Lemma2.2Let M be an irreducible3-manifold with a(possibly empty)boundary pattern P.Let F be a minimal weight incompressible P-boundary incompressible normal surface.If the sum F= F1+F2is in reduced form then each patch is both incompressible and P-boundary incompressible and no patch is trivial.Furthermore if F is injective,then each patch has to be injective.This lemma is a mild generalisation of both lemma3.6in[1]and lemma6.6in[6].Even though patches are not properly embedded surfaces in M the notions of P-boundary incompressibility and injectivity can be naturally extended to this setting.Note also that if P=∅,the manifold M has incompressible boundary,the surface F is boundary incompressible and so are the patches of F=F1+F2.Proof.We start by reducing the lemma to the statement that no patch of F1+F2is trivial.In case we have a patch which is either compressible or not injective,we can argue in precisely the same way as in the proof of lemma3.6of[1]to obtain a disc patch whose boundary is a single simple closed curve from F1∩F2and is therefore trivial.If there is a patch R of F1+F2which has a pure boundary compressing disc D,then we can assume without loss of generality that D is also a boundary compressing disc for the surface F.4Then the unique arc D∩R=D∩F cuts oﬀa pure disc D′in F.If D′contains a simple closed curve of F1∩F2,then we canﬁnd a compressible patch of F1+F2and we are in the previous case. If there are no simple closed curves of F1∩F2in D′,then the edge most arc from F1∩F2in D′cuts oﬀa pure disc patch which is clearly trivial.So it is enough to prove that no trivial patch exists.If there exists a disc patch in F1+F2bounded by a single simple closed curve from F1∩F2,we can use an identical argument to the one in the proof of lemma3.6in[1]to construct two normal surfaces F′and S such that F=F′+S.F′is isotopic to F and S is a closed normal surface withχ(S)=w(S)=0.This is clearly a contradiction because no normal surface can miss the 1-skeleton.Also by our hypothesis the surface F might be only incompressible and not necessarily injective.So in order to use the argument from[1]which shows that every simple closed curve in F1∩F2,bounding a trivial patch,has to be two-sided in both F1and F2,we need to note that when M equals R P3,every embedded projective plane in M is actually injective.So now we can assume that F1+F2contains no disc patch disjoint from∂M.Let D be a trivial patch,lying in F1say,which has least weight among all trivial patches in F1+F2.The intersection D∩F2consists of a unique arcαthat is a component of F1∩F2.After regular alterationαproduces two properly embedded arcs in F one of which cuts oﬀa pure disc D′from F.This is because F is P-boundary incompressible.Disc D′is distinct from but might contain the trivial patch D.We must have w(D′)=w(D).Otherwise we could isotope F,by an isotopy invariant on the pattern,so that its weight is decreased.Since D minimises the weight of all trivial patches(which is strictly positive)and D′must contain at least one such,there is exactly one trivial patch in D′and its weight is equal to w(D).Every other patch in D′is topologically a disc whose boundary consists of4arcs.Two of them are in∂M and the other two are components of F1∩F2.If D′was itself a patch then we could isotope F1and F2,using the deﬁnition of the pattern P and the irreducibility of M,to obtain normal surfaces F′1and F′2still summing up to F but having fewer components of intersection.This contradicts our assumption on the reduced form of F1+F2.So D′is not itself a trivial patch,but it has to contain one.Now we can imitate the argument in the proof of lemma3.6from[1]to obtain a normal sum F=A+F′where F′is a normal surface isotopic to F and A is a pure normal annulus of zero weight.Since no normal surface can live in the complement of the1-skeleton,this gives aﬁnal contradiction.2 Another crucial fact from normal surface theory,tying up normal addition with the topological properties of surfaces involved,is contained in the next theorem.It appeared several times in the literature in slightly diﬀerent forms.The version that is of interest to us is the following.Theorem2.3Let M be an irreducible3-manifold with a possibly empty boundary pattern P.Let F be a least weight normal surface properly embedded in M.Assume also that F is two-sided incompressible P-boundary incompressible and F=F1+F2.Then F1and F2are incompressible and P-boundary incompressible.The proof of this theorem can be obtained by using lemma2.2and following(verbatim)the proof of theorem6.5in[6].Before we state a further consequence of theorem2.3,we need the following simple fact from topology.Lemma2.4Let M be an irreducible3-manifold with incompressible boundary.Assume also that M is neither homeomorphic to the product S1×S1×I,nor to an I-bundle over a Klein bottle. Let S be a toral boundary component of M.If A and B are two properly embedded incompressible ∂-incompressible annuli in M such that at least one boundary component of both annuli lies in S, then these bounding simple closed curves must be isotopic in S,i.e.they determine the same slope in S.Proof.We start by isotoping the annuli A and B so that their intersection is minimal.If A∩B is either empty or it consists only of essential simple closed curves,then the boundary curves must5be parallel in S.So we can assume that∂A∩∂B is non-empty.This implies that the boundary slopes of∂A and∂B in S are distinct.Therefore the complement S−(∂A∪∂B)is a disjoint union of disc.Since both A and B are incompressible and∂-incompressible,the intersection contains neither contractible simple closed curves nor boundary parallel arcs in either of the two annuli.In other words A∩B consists of spanning arcs in both annuli.So an I-bundle structure extends from A∪B to the the regular neighbourhood N(A∪B).If the bounding circles of A lie in distinct components of∂M,then the manifold M has to be homeomorphic to S1×S1×I.This is because each annulus V in the vertical boundary of the I-bundle N(A∪B)cuts oﬀfrom M a3-ball of the form D×I where D is one of the discs in S−(∂A∪∂B).This3-ball can not contain A∪B since both annuli are incompressible.We can therefore extend the product structure over this3-ball,thus obtaining an I-bundle over the torus S.If both components of∂A live in S,then there are two possibilities for the compressible annulus V.The dichotomy comes from the discs D1and D2in the surface S,bounded by the circles of∂V. They can either be nested,say D1⊂int(D2),or disjoint.It is clear that the annulus A is disjoint from∂V=∂D1∪∂D2.The horizontal boundary of the regular neighbourhood N(A∪B)contains an embedded arc,running from∂D1to∂A,which is disjoint from∂D2.If theﬁrst of the two cases were true,this would imply that at least one of the boundary components of A is contained in D2,which is clearly a contradiction.So we must have an embedded2-sphere D1∪D2∪V which bounds a3-ball D1×I,disjoint from A∪B,like before.Adjoining all these solid cylinders to N(A∪B)makes our manifold M into an I-bundle with a single toral boundary component.But this has to be an I-bundle over a closed non-orientable surface of Euler characteristic zero,i.e.a Klein bottle.This concludes the proof.2A very useful consequence of theorem2.3that deals with orientable surfaces in M with zero Euler characteristic is contained in proposition2.5.Most of it is a direct consequence of corollary 6.8in[6].Proposition2.5Let M be an irreducible3-manifold with incompressible boundary that contains either an essential torus or an essential annulus.Let T be the triangulation of M.Then the following holds:(a)If A is a least weight normal representative(with respect to T)in an isotopy class of anessential torus in M,then every vertex surface in the face of the projective solution space that carries A is an essential torus.(b)Let A be a least weight essential annulus in M that is normal with respect to the triangula-tion T.Then there exists a vertex surface in M that is an essential annulus.If the boundary of M consists of tori only and if M is neither S1×S1×I nor an I-bundle over a Klein bottle,then the boundary of the vertex annulus is parallel to the boundary of A in∂M. Proof.Since vertex surfaces are two-sided by deﬁnition,(a)is just restating corollary6.8in[6].In (b)at least one of the vertex surfaces in the face of projective solution space has to be an annulus. In fact,by theorem2.3,it has to be an essential annulus.Assume now that M has toral boundary.All vertex annuli in the face of the projective solution space that carries A must have their respective boundaries lying in precisely the components of ∂M that contain∂A.Since all these vertex annuli are essential we can apply lemma2.4andﬁnish the proof.2 We will conclude this section by a short discussion of incompressible surfaces in Seifertﬁbred manifolds.A surface in a Seifertﬁbred space M→B is called vertical if it can be expressed as a union of regularﬁbres.So the only possibilities are a torus,a Klein bottle or an annulus.Moebius band does not come into the picture because the generator of its fundamental group can not be a regularﬁbre in M.On the other hand the surface that is transverse to allﬁbres in M is called horizontal.The following proposition says roughly that every essential surface in M→B has to be either vertical or horizontal.6Proposition2.6Let S be an incompressible∂-incompressible surface in a3-manifold M.Then the following holds:(a)Assume further that M→B is an irreducible Seifertﬁbred space,possibly containing nosingularﬁbres,over a(perhaps non-orientable)compact surface B.If S is two-sided and if it contains no singularﬁbres of M,then it is isotopic to either a vertical surface or a horizontal surface.(b)Let M→B be an S1-bundle over a(perhaps non-orientable)compact bounded surface B.If S is a one-sided,i.e.non-orientable,surface in M,then it is isotopic to either a vertical or a horizontal surface.It should be noted that(b)of proposition2.6actually fails if the base surface B is closed.It is not hard to see that the lens space L(2n,1),whichﬁbres as an S1-bundle over S2with Euler number2n,contains an incompressible surface homeomorphic to a connected sum of n projective planes.Such a surface can not be vertical because of its Euler characteristic,but it can also not be horizontal since the Euler number of our S1-bundle does not vanish.Also notice that the statement (b)for two-sided surfaces is already contained in(a).Proof.If the surface S contained a disc component,then,by the deﬁnition of incompressibility,∂M would have to compress.Since M is irreducible,this would make it into a solid torus.The surface S is then a disjoint collection of compression discs which is horizontal in anyﬁbration of the solid torus.So from now on we can assume that S contains no disc components.Part(a)of the proposition is a well-known fact about two-sided incompressible∂-incompressible surfaces(which contain no disc components)in Seifertﬁbred spaces.Its proof can be found in[5] (theorem VI.34).We shall now prove part(b).Let M→B be an S1-bundle over a compact bounded surface B.Choose disjoint arcs in B whose union decomposes B into a single disc.Let A be a union of disjoint vertical annuli in M that are the preimages of the collection of arcs under the bundle projection.Without loss of generality we can assume that the surface S was isotoped in M so that the number of components of S∩A is minimal.Then there are no simple closed curves in S∩A that are homotopically trivial in either S or A,because such curves can be used to reduce the number of components in S∩A(here we are using the fact that every bounded S1-bundle is irreducible). Similarly arcs in S∩A that are∂-parallel in either A or S do not occur because both surfaces are∂-incompressible and∂M is incompressible in M(otherwise M would be a solid torus and S would have to be a disc,which is not a one-sided surface).It follows now that S∩A consists only of vertical circles or horizontal arcs,i.e.spanning arcs in the annular components of A.Let M1be the solid torus M−int(N(A))and let S1be the surface M1∩S.There can be no trivial simple closed curves in(∂M)∩M1coming from∂S1,because S has no disc components.So we can conclude that∂S1consists either of horizontal or vertical circles in the torus∂M1(a horizontal simple closed curve is the one that intersects eachﬁbre in ∂M1transversely).The surface S1has to be incompressible in M1.Every compression disc D for S1in M1yields a disc D′in ing irreducibility of M we could isotope S(rel∂D)so that D′becomes D.If D′were not contained in S1,this move would reduce the number of pieces in S∩A.Claim.If∂S1consists of horizontal simple closed curves,then S1is a disjoint union of meridional discs in M1.It is enough to show that under the hypothesis of the claim the surface S1has to be∂-incompressible in M1.This is because the only connected incompressible∂-incompressible surface in a solid torus is its meridian disc.Assume to the contrary that S1is∂-compressible.Let D be a∂-compression disc for S1in M1.We will modify D,in a thin collar of∂M1,so that the arc D∩∂M1lies in an annulus from ∂M∩∂M1.This isotoped disc therefore lives in M and is a∂-compression disc for the surface S.Like before this leads to a contradiction,because we can use the isotoped disc to construct an isotopy in M that will reduce the number of components in S∩A.So to prove the claim we need to isotope the disc D.Letαbe the embedded arc D∩∂M1, running between two points from∂S1.Notice that∂M1is an alternating union of annuli coming7from two families:one is M 1∩∂M and the other one is M 1∩N (A ).We will now isotope αinto the interior of M 1∩∂M .Assume ﬁrst that αis contained in the interior of an annulus from M 1∩N (A ).Then,since ∂S 1is horizontal in ∂M 1,the segments of ∂S 1in this annulus have to be spanning arcs.αcan either run between two distinct spanning arcs,or it can run around the annulus to hit the single spanning arc from two diﬀerent sides.In either situation we can push αinto the interior of an adjacent annulus from M 1∩∂M .This isotopy can clearly be extended to the collar of ∂M 1,thus producing the desired ∂-compression disc.If,on the other hand,αis not contained in the interior of an annulus from ∂M 1,then we must somewhere have the situation as described by ﬁgure 2.of the boundary S M 1α1boundary ofannulus in disc EFigure 2:An annulus in ∂M 1containing the disc E with the following property:the intersection E ∩αis a subarc of αcontaining precisely one point from ∂α.We can construct an isotopy of α,and hence of D ,using the disc E ,that will reduce the number of points in the intersection α∩∂(M 1∩∂M ).By repeating this move we arrive at a contradiction and the claim follows.It follows directly from the claim that S 1with the horizontal boundary can be isotoped (rel ∂S 1)so that it is horizontal in the ﬁbration of M 1.This isotopy induces an isotopy of S in M that makes it horizontal.If ∂S 1is vertical,S 1can only consist of annuli bounded by ﬁbres of M 1.We can therefore isotope S 1(rel ∂S 1)so that it becomes vertical.This concludes the proof.23THE MAIN THEOREMLet’s start by deﬁning precisely the class of Seifert ﬁbred 3-manifolds that we shall consider.The manifold M has to be compact and orientable.It has to ﬁbre into circles over a possibly non-orientable compact surface B .The case when M contains no singular ﬁbres will also be considered.Since we are only interested in general Haken 3-manifolds,i.e.the ones that are irreducible and that contain an injective surface diﬀerent from a 2-sphere,we need to exclude some Seifert ﬁbred spaces.The ones to go ﬁrst are S 1×S 2and the connected sum R P 3#R P 3since they are not irreducible (the latter manifold ﬁbres as an S 1-bundle over a projective plane).Another obvious family that we have to disqualify are the lens spaces including the 3-sphere,because they contain no injective surfaces.Those are all Seifert ﬁbred spaces over a 2-sphere with at most 2singular ﬁbres.Among the spaces that contain no vertical essential tori are also manifolds that ﬁbre over a projective plane with at most one singular ﬁbre.Since the Klein bottle over the orientation reversing curve in R P 2is not injective,they contain no injective vertical surfaces.There are no horizontal surfaces either.The existence of such a surface would imply that the only singular ﬁbre is of index 0。

Novel modulators of amyloid-β precursor protein processing

MINI-REVIEW Novel modulators of amyloid-b precursor protein processingBor Luen Tang*and Yih Cherng Liou*Department of Biochemistry,Yong Loo Lin School of Medicine and Department of BiologicalSciences,National University of Singapore,SingaporeAbstractProteolytic processing of the amyloid precursor protein (APP)is modulated by the action of enzymes a-,b-and c-secretases,with the latter two mediating the amyloidogenic production of amyloid-b(A b).Cellular modulators of APP processing are well known from studies of genetic mutations (such as those found in APP and presenilins)or polymor-phisms(such as the apolipoprotein E4e-allele)that predis-poses an individual to early or late-onset Alzheimer’s disease. In recent years,several classes of molecule with modulating functions in APP processing and A b secretion have emerged. These include the neuronal Munc-18interacting proteins (Mints)/X11s,members of the reticulon family(RTN-3and RTN-4/Nogo-B),the Nogo-66receptor(NgR),the peptidyl-prolyl isomerase Pin1and the Rho family GTPases and their effectors.Mints and NgR bind to APP directly,while RTN3and Nogo-B interact with the b-secretase BACE1.Phosphorylated APP is a Pin1substrate,which binds to its phosphor-Thr668-Pro motif.These interactions by and large resulted in a reduction of A b generation both in vitro and in vivo.Inhibition of Rho and Rho-kinase(ROCK)activity may underlie the ability of non-steroidal anti-inﬂammatory drugs and statins to reduce A b production,a feat which could also be achieved by Rac1inhibition.Detailed understanding of the underlying mechanisms of action of these novel modulators of APP processing,as well as insights into the molecular neurological basis of how A b impairs leaning and memory,will open up multiple avenues for the therapeutic intervention of Alzheimer’s disease.Keywords:Alzheimer’s disease,amyloid-b,Munc-18inter-acting proteins,Nogo,Pin1,Rho.J.Neurochem.(2007)100,314–323.Alzheimer’s disease(AD)is the most common course of dementia in the elderly worldwide,and is not surprisingly the neurodegenerative disease that has attracted the most research focus in the past rgely idiopathic,AD has two distinct pathological features–extracellular amyloid plaques and intracellular neuroﬁbrillary tangles.According to the amyloid hypothesis(Hardy and Selkoe2002), extracellular plaques consist of aggregated b-amyloid(A b) peptide generated from proteolytic cleavages of the amyloid precursor protein(APP)and are the etiological agents of AD pathology.It is now known that either intracellular or extracellular soluble oligomeric forms of A b could initiate synaptic malfunctions and the onset of AD symptoms (Selkoe2002;Wirths et al.2004;Cuello2005).A b generation from APP occurs via a two-step proteolytic process involving b-and c-secretases(Haass2004;Wilquet and De Strooper2004).The b-site APP cleaving enzyme (BACE1),a member of the pepsin family of aspartyl proteases,ﬁrst cleaves the ectodomain of APP to generate a membrane bound C-terminal fragment.A subsequent cleavage by the c-secretase activity further generates peptides,mainly of40or42amino acids in length,termed A b40and A b42.Both types of peptide could be found in amyloid plaques,but the latter is apparently more directly neurotoxic(Zhang et al.2002;Zou et al.2003)and has a greater propensity to aggregate.APP could also be proteo-lytically processed in other ways,notably by a-secretases (Allinson et al.2003).a-secretase cleavage is non-amyloid-Received June21,2006;revised manuscript received August11,2006; accepted August17,2006.Address correspondence and reprint requests to Bor Luen Tang, Department of Biochemistry,Yong Loo Lin School of Medicine National University of Singapore,8Medical Drive,Singapore117597.E-mail:bchtbl@.sgAbbreviations used:A b,b-amyloid;AD,Alzheimer’s disease;APP, amyloid precursor protein;BACE1,b-site APP cleaving enzyme;CR, calorie restriction;Mint,Munc-18interacting protein;NGR,Nogo-66 receptor;NSAIDs,non-steroidal anti-inﬂammatory drugs;PS,preseni-lins;PTB,phosphotyrosine binding;ROCK,Rho-kinase;RTN,reticu-lon;sAPP,soluble APP;siRNA,small-interfering RNA.Journal of Neurochemistry,2007,100,314–323doi:10.1111/j.1471-4159.2006.04215.x 314Journal CompilationÓ2006International Society for Neurochemistry,J.Neurochem.(2007)100,314–323Ó2006The Authorsogenic,occurs at a site that will preclude subsequent BACE cleavage,and generates a soluble sAPP a fragment that is actually neuroprotective(Mattson1994;Meziane et al.1998; Tang2005).The c-secretase activity responsible for the generation of A b from a BACE-cleaved APP is found in a complex which mediates a process of regulated intramembrane proteolysis (RIP;Weihofen and Martoglio2003)of several proteins, notably APP and Notch(Selkoe and Kopan2003).The complex consists of at least four proteins–presenilins(PS)-1 or-2at the catalytic core,nicastrin,anterior pharynx-defective phenotype1(APH1)and PS-enhancer2(PEN-2) (De Strooper2003;Edbauer et al.2003).c-Secretase cleavage of b-secretase-cleaved APP also releases an APP intracellular domain(AICD)fragment,with possible tran-scriptional functions(Cao and Sudhof2001,2004;V on Rotz et al.2004).Any factor that might inﬂuence A b production would very likely impinge on some aspects of APP processing and A b clearance.Known genetic predispositions to AD include mutations in APP itself and the presenilins,all of which increase A b generation or the A b42/A b40ratio(reviewed extensively by Tanzi and Bertram2005).Genetic links between AD and BACE1mutations have also been sugges-ted(Bertram and Tanzi2004).(Meta analysis performed for case-controlled BACE1’s genetic association with AD can be found at /res/com/gen/alzgene/ meta.asp?geneID=51).The e4allele of apolipoprotein E,a risk factor for late onset AD,inﬂuences A b etiology in the ageing brain,perhaps by enhancing A b oligomerization or lowering A b clearance(Huang et al.2004;Tanzi and Bertram2005).Interestingly,several diverse groups of molecules which appear unrelated to AD pathology have recently been shown to inﬂuence A b levels in vitro and in vivo.These include the neuronal Munc18-interacting protein (Mint)/X11protein,reticulon family members,the Nogo-66 receptor,the Rho family GTPases and their effectors,as well as the peptidyl-prolyl isomerase Pin1.Other than the Mints, none of the other proteins was previously known to interact with APP itself.We review and discuss below recent insights of how these proteins may inﬂuence APP processing and A b levels in the brain,as well as how theseﬁndings may open new therapeutic windows for AD.The APP C-terminal domain interacting proteins and APP processingThe C-terminal cytoplamic domain of APP,which could be present in the form of different proteolytic fragments released by secretases cleavages,is known collectively as the APP intracellular domain.This domain has several known inter-acting proteins(Koo2002;Kawasumi et al.2004;Russo et al.2005).Selective tyrosine phosphorylation of APP’s C-terminus may regulate its interaction with several cytosolic phosphotyrosine binding(PTB)domain or Src homology2(SH2)domain-containing proteins that are involved in cell signalling.These include the Fe65family of proteins,the Mints/X11s,mammalian Disabled,as well as the c-jun N-terminal protein kinase-interacting protein(JIP).Some of these may also bind APP in a tyrosine phosphorylation-independent manner.Fe65is an adaptor protein with a WW domain and two PTB domains.The Fe65-APP complex is known to trans-locate and signals to the nucleus transcriptionally in a manner analogous to how Notch functions(Cao and Sudhof2001, 2004;V on Rotz et al.2004).The target genes of this complex include BACE1and APP.However,tyrosine phosphorylation of Fe65itself by active c-Abl could also stimulate APP/Fe65-mediated gene transcription(Perkinton et al.2004),and there is evidence that Fe65may regulate transcription fairly independently of its interaction with APP (Yang et al.2006).c-Secretase cleavage of APP also does not appear to be required for transcriptional activation(Haass and Yankner2005).A possible role for FE65in AD pathogenesis was illustrated by increased Fe65immunoreactivity in the hippocampus of AD brain,which correlated with the severity of the disease,and which co-localized with tau proteins in neuroﬁbrillary tangles(Delatour et al.2001).Studies in vitro had shown that Fe65over-expression in cultured cells increases A b production(Sabo et al.1999;Tanahashi and Tabira2002;Chang et al.2003),although the opposite(i.e.a reduction of A b secretion)has also been observed(Ando et al.2001;Hoe et al.2006).In a recently reported in vivo study,transgenic mice expressing human Fe65driven by the neuronal Thy1promoter were crossed with transgenic APP mutant mice carrying both the Swedish and London muta-tions(APP751SL;Santiard-Baron et al.2005).Interestingly, APP751SL/Fe65double transgenics showed a lower A b accumulation in the cerebral cortex and a lower level of APP C-terminal fragments compared with the APP751SL trans-genics.However,a reduced A b secretion was also observed in primary neuronal cultures of Fe65knockout-APP trans-genic(Tg2576)hybrids(Wang et al.2004).In another recent report,a signiﬁcant decrease in brain A b42levels was observed for Fe65/Fe65L1double-knockout mice compared with wild type amongst males(but not females),while there was no signiﬁcant change in A b40levels(Guenette et al. 2006).These rather contradictory results illustrate that the role of Fe65in APP processing is likely to be complicated. Fe65may modulate APP trafﬁcking,while it is known that it also inﬂuences APP processing by a-secretases(Guenette et al.1999).Another major class of ACID interacting proteins,the Mints/X11s,consists of mammalian homologues of the Caenorhabditis elegans lin-10protein.These proteins con-tain PDZ domains in addition to the PTB domain,and neuronal isoforms amongst these have been implicated in polarized trafﬁcking in neurons(Setou et al.2000)andNovel modulators of APP processing315Ó2006The AuthorsJournal CompilationÓ2006International Society for Neurochemistry,J.Neurochem.(2007)100,314–323synaptic vesicle exocytosis(Butz et al.1998).Over-expres-sion of X11in co-transfected non-neuronal cells changes the subcellular distribution of APP.Its immunoreactivity is found to be associated with AD plaques,but not neuroﬁbrillary tangles(McLoughlin et al.1999).Over-expression of the neuronal isoforms of X11(X11a and X11b)had been consistently demonstrated to attenuate A b production in vitro (Borg et al.1998;Sastre et al.1998;Tomita et al.1999). More recent analyses in vivo using double transgenics of X11a(Lee et al.2003)or X11b(Lee et al.2004)and the APP Swedish mutant(APPswe)indicate that over-expression of X11s indeed decreased both A b levels and amyloid deposition.Interestingly,another recent analysis using small-interfering RNA(siRNA)-mediated silencing of X11a and X11b also demonstrated a lowering of A b levels in vitro(Xie et al.2005).These observations suggest that the effects seen with over-expression of X11s are likely to be a result of the proteins functioning in a dominant-negative fashion.The exact mechanism by which X11s reduce A b produc-tion is currently unclear.It is conceivable that the known function of X11in mediating neuronal trafﬁcking may be applicable to both APP and BACE1,with its over-expression inﬂuencing their subcellular dynamics,transport itinerary, membrane co-localizations and consequently proteolytic processing.One possible way in which X11s might affect APP processing directly is through a previously demonstra-ted interaction with presenilin-1(Lau et al.2000).X11a over-expression has in fact been shown to rather speciﬁcally inhibit c-secretase and but not b-secretase cleavage of APP (King et al.2004).This speciﬁcity is extended by the X11 siRNA silencing data,alluded to above(Xie et al.2005), which indicated that the A b-lowering effect of X11a silencing,but not X11b silencing,occurs via attenuation of c-secretase cleavage.It is conceivable that the interaction of X11s with APP modulates its exposure to c-secretase activity and,perhaps in rather complex ways,inﬂuence its processing.The reticulons/Nogo and the Nogo-66receptor–possible roles in APP processingThe reticulons(RTNs)are a family of four membrane proteins with various spliced isoforms found in higher vertebrates.RTNs share a high degree of homology in their respective transmembrane domains-bearing C-termini.Sev-eral reticulon-like genes are also found throughout the eukaryote kingdom(Oertle et al.2003a).The endogenous functions of most reticulon family members are unknown, although both RTN1(Van de Velde et al.1994)and RTN3 (Moreira et al.1999)are interestingly brain-enriched.RTN4, or Nogo,has been under intensive investigation in recent years for its pathological role in adult myelin-associated inhibition of neurite growth and axonal regeneration after injury(Schwab2004).RTN4/Nogo has three major splice isoforms(A,B and C).The longest of which,Nogo-A,is CNS enriched and appear to inhibit neuronal regeneration via several domains(Oertle et al.2003b),acting through at least one known neuronal receptor,the Nogo-66receptor(McGee and Strittmatter2003).Nogo-B,which is not CNS-enriched, has been found to be highly expressed in cultured endothel-ial/smooth muscle cells as well as blood vessels,and was implicated in vascular homeostasis(Acevedo et al.2004). Interestingly,Nogo-A is also found to be expressed in some CNS neurons(Huber et al.2002;Liu et al.2002;Wang et al.2002).A role for RTN4/Nogo in AD or related dementia has been speculated based on its possible contributions to pathologic and compensatory plasticity in the CNS(Strittmatter2002; Teng and Tang2005).He et al.(2004)ﬁrst showed evidence for possible roles of RTN family members in modulating APP processing.In a protein–protein interaction screen for BACE1interacting partners,the authors identiﬁed RTN3and RTN4B/Nogo-B in BACE1immunoprecipitate.All four RTNs could in fact be co-immunoprecipitated with BACE1 when these are simultaneously over-expressed in cells. Intriguingly,all four RTNs when over-expressed in APP-swe-expressing cultured cells signiﬁcantly reduced A b secretion.Over-expression of RTN3decreases measurable BACE1activity and APP processing,whereas knockdown of RTN3increases APP processing and A b secretion.RTN3 does not bind APP directly,but its interaction with BACE1 appears to either sequester it away from APP or blocks BACE1-APP enzyme–substrate interaction.For their analysis,He et al.(2004)had focused mostly on RTN3because RTN3is found in neurons(as are BACE1and APP),and it appears to co-fractionate with BACE1in a sucrose gradient analysis of cellular membrane fractions.The cellular and physiological relevance of this RTN3–BACE1 interaction,however,deserves further scrutiny.Like all members of the reticulon family,RTN3has a dilysine motif at its C-terminus,and is largely endoplasmic reticulum-localized.Over-expression of RTN3apparently blocks transport at the early secretory pathway(Wakana et al. 2005),manifested by the dispersion of Golgi markers and a block in Golgi-endoplasmic reticulum retrograde transport. The cellular trafﬁcking itinerary of APP and BACE1are similar.Both traverse the secretory pathway and tend to cluster within plasma membrane lipid rafts(Ehehalt et al. 2003),and could be internalized into endosomal compart-ments.The plasma membrane and endosome are presumably the major sites where APP cleavages by BACE1take place. BACE1is synthesized as a pre-pro-enzyme and is activated by a furin-like protease.It is not particularly clear how early along the exocytic pathway this activation occurs,but furin proteases are typically localized at the trans-Golgi and trans-Golgi network.In this way,both the reductions in measur-able BACE1activity and A b secretion observed in RTN3 over-expressing cells could be because of a rather non-speciﬁc block of both APP and BACE1as they traverse the316 B.L.Tang and Y.C.LiouJournal CompilationÓ2006International Society for Neurochemistry,J.Neurochem.(2007)100,314–323Ó2006The Authorsearly secretory pathway.That knockdown of RTN3in cultured cells increases A b production and appears more physiologically relevant and speciﬁc,but it is unclear if RTN3knockdown also results in a general increase in APP surface transport and secretion.It is also not known if the interaction between RTN3and BACE1has a regulatory function on the enzymatic activation of the latter by furin proteases,and whether RTN3’s modulation of BACE1 activity is substrate speciﬁc(i.e.will it also attenuate the processing of substrates other than APP?).In any case,the results of He et al.(2004)are still of considerable interest because it suggests that the activity of BACE1on APP processing could be modulated by its interaction with another membrane protein,or family of proteins.Regardless of the physiological signiﬁcance of this interaction,it could poten-tially be exploited therapeutically to modulate A b secretion. Another recent report by Park et al.(2006)has demon-strated an interaction between two other proteins related to RTN3and BACE1above,namely Nogo-66receptor(NgR) and APP.The authors observed that both Nogo and NgR are mislocalized in post-mortem AD samples.APP appears to interact physically with NgR,and over-expression of NgR in N2a neuroblastoma cells reduces A b production.Import-antly,when an AD mouse double transgenic for the APPswe mutation and presenilin-1[APPswe/PSEN-1(D E9)]was crossed with NgR-null mice,the NgR-null background mice signiﬁcantly increase A b levels and amyloid plaque deposition compared with wild type.Infusion of an NgR dominant-negative reagent[NgR(310)ecto-Fc]into APPswe/ PSEN-1(D E9)mice also reduced amyloid plaque deposition.A potential caveat of the results,but also a point of interest,is that NgR appears to bind many parts of the APP, including the BACE1cleavage product sAPP b and various lengths of the A b peptide.Furthermore,NgR appears to affect APP processing by both b-and a-secretases.An NgR antagonist reduces sAPP a production in culture cells as well as in mice,and levels of sAPP a are increased in the double transgenic APPswe/PSEN-1(D E9)mutant mice with an NgR-null background.These observations,while suggesting a certain degree of non-speciﬁcity with regard to APP–NgR interaction,are again interesting and useful from a therapeu-tic standpoint.The Rho family GTPases and their roles in APP processingA number of pharmacological agents,including non-steroidal anti-inﬂammatory drugs(NSAIDs;Hoozemans et al.2003; Gasparini et al.2004;Szekely et al.2004),HMG-CoA reductase inhibitors(or statins;Parvathy et al.2004;Pedrini et al.2005)andﬂavonoid compounds(Levites et al.2003; Colciaghi et al.2004)have been shown to be potentially beneﬁcial in either preventing the onset,or delaying the progression,of AD.These agents could reduce A b produc-tion and amyloid load in the brain via two routes.Theﬁrst involves direct inhibition of c-secretase activity(Eriksen et al.2003;Weggen et al.2003).The second appears to be an emerging,but less well-understood,mechanism involving the channelling of APP towards non-amyloidogenic process-ing by a-secretase.NSAIDs had been shown to stimulate the secretion of the non-amyloidogenic sAPP a(Avramovich et al.2002).Inter-estingly,at least some of the actions of NSAIDs appear to involve Rho and its effector Rho-associated kinase(ROCK). Zhou et al.(2005)showed that NSAIDs can lower A b42 levels by inhibiting the activity of Rho through the pertur-bation of its isoprenylation.Accordingly,geranylgeranyl pyrophosphate treatment of SH-SY5Y cells expressing the APPswe mutant increased A b42production,and this was blocked by NSAIDs.A dominant-negative form of Rho (T19N),but not Cdc42or Rac1,decreased A b42,as did Rho inactivation by C3transferase,or inhibition of ROCK by Y-27632.There appears to be a clear correlation between the ability of individual NSAIDs to negatively regulate Rho activity and their A b42-lowering capacity.In relatedﬁndings,Pedrini et al.(2005)demonstrated, using APPswe-expressing N2a mouse neuroblastoma cells, that ROCK modulates shedding of sAPP a induced by statins.sAPP a shedding is promoted by a farnesyl transferase inhibitor,FTI-1,synergistically with statin treatment,but is reduced by arachidonic acid,which activates ROCK.A constitutively active ROCK mutant diminished sAPP a shedding from both untreated and statin-treated cells,whereas a kinase-dead ROCK mutant on its own activated sAPP a shedding,probably by functioning in a dominant-negative manner.Both NSAIDs and statins may therefore enhance sAPP a production through the attenu-ation of a signalling pathway that is modulated by Rho and ROCK.Another interesting recent result connects ROCK to amyloid deposition.Calorie restriction(CR)is known to prolong the lifespan of multiple model organisms through the activation of the Sir2/SIRT1histone deacetylase(Bordone and Guarente2005).CR in mice has also been associated with a reduction in amyloid pathology in the brain(Patel et al.2005;Wang et al.2005).An interesting study by Pasinetti and colleagues now revealed that CR-induced SIRT1expression promotes a-secretase activity,sAPP a generation,and attenuates A b generation by neurons from Tg2576mice and CHO cells expressing APPswe in culture (Qin et al.2006).This effect of SIRT1appears to be dependent on ROCK1.Accordingly,ROCK1levels are markedly reduced in the brain of CR mice,while viral-mediated expression of dominant-negative SIRT1increases ROCK1levels in Tg2576mice neurons.In contrast,a constitutively active ROCK1attenuated SIRT1-induced sAPP a generation.Constitutive expression of SIRT1in transgenic mice conversely reduced ROCK1levels,with a concomitant increase in a-secretase activity.Novel modulators of APP processing317Ó2006The AuthorsJournal CompilationÓ2006International Society for Neurochemistry,J.Neurochem.(2007)100,314–323There is some evidence that APP processing by a-secretase is also inﬂuenced by signalling from Rac,another member of the Rho family(Maillet et al2003;Robert et al. 2005).Gianni et al.(2003)had shown that over-expression of dominant-negative Rac(RacN17)mutant inhibited,while the constitutively active mutant(RacQL)enhanced,c-secretase-mediated APP processing.De´sire´et al.(2005) recently reported that a novel small molecule inhibitor of Rac,EHT1864,blocked both A b40and A b42production in SH-SY5Y cells.The drug does not seem to affect either a-secretase or BACE1,and appears to target c-secretase processing of APP via a non-competitive mechanism. Importantly,it does not affect Notch processing and was able to signiﬁcantly reduce cortical A b levels after intraperi-toneal introduction into guinea pigs.The role of Rho and Rac in modulating APP processing is still far from being clear mechanistically.It is not known if NSAIDs and statins could elevate expression and/or activity of a-secretase,or they may in some manner facilitate subcellular co-localization and contact of a-secretase with APP(or attenuate that between BACE1and APP),thereby enhancing non-amyloidogenic processing.Neither APP nor a-secretases are known as direct targets of ROCK’s kinase activity.How Rac activity promotes APP processing by c-secretase is also unclear.In any case,the phamacological action of EHT1864further highlights a role for Rac in A b production,If without systemic toxicity,it may turn out to be a promising pharmacological agent for lowering brain A b load in the aged or genetically predisposed individuals,either alone or in conjunction with NSAIDs and statins.Pin1regulation of APP processingPin1is a member of the parvulin subfamily of peptidyl-prolyl cis-trans isomerases(Go¨thel and Marahiel1999).It is a highly conserved enzyme that isomerizes phosphorylated Ser/Thr-Pro bonds in a myriad of proteins(ranging from cell cycle regulators to signalling proteins),thereby inducing post-phosphorylation conformational changes.Such changes represent novel modes of signalling that are pivotal in the regulation of many cellular functions,and have important implications in cellular pathologies ranging from cancer to neurodegenerative diseases(reviewed extensively in Ryo et al.2003;Lu2004;Wulf et al.2005).The relationship of Pin1regulatory dysfunction and AD isﬁrst illustrated by Pin1’s action on tau(Lu et al.1999).Pin1binds to tau phosphorylated speciﬁcally at the Thr231-Pro site,and was shown to co-purify with paired helicalﬁlaments.This essentially results in a depletion of soluble Pin1in the AD brain,and sequestration of Pin1into neuroﬁbrillary tangles. As Pin1-depletion is known to cause mitotic arrest and apoptosis(Lu et al.1996),this loss of functional Pin1even contributes to the apoptotic death of post-mitotic neurons. Pin1could in fact restore the ability of phosphorylated tau to bind microtubules and promote microtubule assembly in vitro.Conformational changes on phosphorylated tau induced by Pin1may also promote dephosphorylation by the phosphatase PP2A(Lu et al.1999;Galas et al.2006).Pin1-knockout mice showed age-dependent neurodegeneration associated with tau hyperphosphorylation,neuroﬁbrillary tangle-like formation and motor-behavioural deﬁcits charac-teristic of tauopathy and neuronal loss,as seen in AD(Liou et al.2003).Indeed,immunohistochemical analyses showed that Pin1co-localizes with phosphorylated tau in brain sections of human AD and other tauopathies(Ramakrishnan et al.2003).Human genetic evidence for Pin1’s role in AD progression includes associations of reduced Pin1expression resulting from Pin1promoter polymorphisms with increased risk of late-onset AD(Segat et al.2006).The action of Pin1on the processing of phosphorylated APP and its effect on A b production has,however,only recently come to light.Akiyama et al.(2005)examined the production of A b in both wild-type and Pin1-null mice,and found that both soluble and insoluble levels of A b40and A b42were signiﬁcantly reduced in the Pin1-knockout mice compared with wild type.This suggests that Pin1promotes A b production in vivo.To conﬁrm that phosphorylated Thr668-Pro of APP is responsible for Pin1binding,the authors over-expressed C99,the direct precursor of A b in cultured cells.The results obtained suggested that Pin1not only interacts with phosphorylated Thr668-Pro,but also modulates the amyloidogenic process of APP by increasing A b production.A detailed study on the role of Pin1in regulating A b production has appeared more recently (Pastorino et al.2006).However,the results are contradict-ory to that of the above.Pastorino and co-workers found that Pin1binds to a phosphorylated Thr668-Pro motif of APP both in vitro and in vivo.The authors showed by NMR spectroscopy that Pin1binding to this phosphorylated site drastically accelerates its isomerization and regulates con-formational changes of the APP intracellular domain.In CHO cells expressing wild-type APP,there appears to be substantially detectable Pin1-APP co-localizations at the plasma membrane and clathrin-coated vesicles.This co-localization strengthens the notion that Pin1could inﬂuence APP processing.In fact,over-expression of Pin1in these cells reduced A b secretion.Furthermore,a cell line derived from Pin1-null mice secretes about threefold less sAPP a compared with wild-type cells,but about sevenfold more A b. The authors performed further examinations on their Pin1-null mice pertaining to A b levels at different ages.While Pin1knockout did not signiﬁcantly change soluble and insoluble A b levels at2–6months of age(when neuropathy is not yet detectable),levels of insoluble A b42were increased by32%in15-month-old Pin1-null mice compared with wild type.Crossing both the Pin1-wild type and Pin1-null mice with the APP Swe transgenic line Tg2576revealed that insoluble A b42is markedly elevated(by46%)in the Pin1-null background compared with Pin1-wild type.This A b42is318 B.L.Tang and Y.C.LiouJournal CompilationÓ2006International Society for Neurochemistry,J.Neurochem.(2007)100,314–323Ó2006The Authors。

Defect of a unitary matrix

Keywords: unitary matrices, bistochastic matrices, critical point, Fourier matrices, complex Hadamard matrices. MSC-class: 58K05, 51F25, 15A51, 65T50, 05B20, 15A90
From the unitarity condition, U U ∗ = ½, it follows that the resulting matrix B is bistochastic (also called sl doubly stochastic), since the sum of elements in 1
December 17, 2007
Abstract We analyze properties of a map f sending a unitary matrix U of size N into a doubly stochastic matrix B = f (U ) deﬁned by Bi,j = |Ui,j |2 . For any U we deﬁne its defect, determined by the dimension of the image Df (TU U ) of the space TU U tangent to the manifold of unitary matrices U at U under the tangent map Df corresponding to f . The defect, equal to zero for a generic unitary matrix, gives an upper bound for the dimension of a smooth orbit (a manifold) of inequivalent unitary matrices mapped into the same doubly stochastic matrix. We demonstrate several properties of the defect and prove an explicit formula for the defect of the Fourier matrix FN of size N . In this way we obtain an upper bound for the dimension of a smooth orbit of inequivalent unitary complex Hadamard matrices stemming from FN . It is equal to zero iﬀ N is prime and coincides with the dimension of the known orbits if N is a power of a prime. Two constructions of these orbits are presented at the end of this work.

The composition of the Earth

1. Introduction An accurate and precise model of the chemical and isotopic composition of the Earth can yield much information regarding its accretion processes, and global-scale differentiation processes, including: core segregation, possible mineral fractionation in a primordial magma ocean and crust-mantle differentiation. With such a model we can also constrain compositional estimates for present-day reservoirs in the Silicate Earth and thus provide insights into their evolution. (The terms Silicate Earth and Primitive Mantle are synonymous.) There are three main approaches which have been used to model the composition of the Earth: ( 1) using the seismic profile of the core and mantle and their interpretation; (2) comparing the compositional systematics of primitive meteorites and the solar photosphere to constrain the solar nebula composition and from this estimate the composition of the inner rocky planets; and ( 3 ) using chemical and petrological models of peridotite-basalt melting relations (i.e. the pyrolite model). The seismic velocity structure of the Earth, in combination with mineral physics data for phases at the appropriate pressures and temperatures, provide important information about the average density and from this the bulk composition of the crust, mantle and core. These data yield basic insights into the gross compositional characteristics of these regions, but cannot be used to constrain the minor- and trace-element composition of the Earth. Compositional models based on primitive meteorites relates elemental abundances in the bulk Earth to those observed in chondritic meteorites in general, but particularly the CI carbonaceous chondrites, the most primitive of the chondritic meteorites. These meteorites are free of chondrules, possess the highest abundances of the moderately-volatile and volatile elements rela-

不饱和脂肪酸与炎症性肠病因果关系的孟德尔随机化分析　

不饱和脂肪酸与炎症性肠病因果关系的孟德尔随机化分析*李　健1　高建淑1,2　赵可可1,2　高鸿亮1,2#新疆医科大学第一附属医院消化病二科1（830054）　新疆医科大学研究生学院2背景：炎症性肠病（IBD ）是一种慢性复发性胃肠道炎症性疾病，包括溃疡性结肠炎（UC ）和克罗恩病（CD ）。

目前尚不清楚不饱和脂肪酸与IBD 之间是否存在因果关系。

目的：采用两样本孟德尔随机化分析探究不饱和脂肪酸与IBD 之间的因果关系。

方法：不饱和脂肪酸和IBD 的全基因组关联研究（GWAS ）数据均来源于网络公开数据库。

采用逆方差加权分析法进行两样本孟德尔随机化分析，使用加权中位数法和MR⁃Egger 回归分析验证因果效应，以OR 及其95% CI 评价不饱和脂肪酸与IBD 风险的因果关系。

结果：ω⁃6脂肪酸与CD 无直接因果关系，与UC 有直接因果关系，逆方差加权分析结果显示ω⁃6脂肪酸基因水平每增加一个标准差，UC 风险增加16%（OR =1.16，95% CI : 1.00~1.36，P =0.04）。

而ω⁃3脂肪酸、单不饱和脂肪酸与IBD 之间均未发现因果关系。

结论：ω⁃6脂肪酸可能仅与UC 存在因果关系，ω⁃3脂肪酸、单不饱和脂肪酸与IBD 之间均未发现因果关系。

关键词　脂肪酸类，不饱和；　脂肪酸类，ω⁃6；　炎症性肠病；　结肠炎, 溃疡性；　Crohn 病；　孟德尔随机化分析Causal Association Between Unsaturated Fatty Acids and Inflammatory Bowel Disease: A Mendelian Random ⁃ization Analysis LI Jian 1, GAO Jianshu 1,2, ZHAO Keke 1,2, GAO Hongliang 1,2. 1The Second Department of Gastroenterology, the First Affiliated Hospital of Xinjiang Medical University, Urumqi (830054); 2Graduate School of Xinjiang Medical University, UrumqiCorrespondence to:GAOHongliang,Email:*************************.cnBackground: Inflammatory bowel disease (IBD) is a chronic recurrent inflammatory disease of gastrointestinal tract including ulcerative colitis (UC) and Crohn's disease (CD). It is unclear whether there is a causal association between unsaturated fatty acids and IBD. Aims: A two ⁃sample Mendelian randomization analysis was used to explore the causal association between unsaturated fatty acids and IBD. Methods: The data of the genome⁃wide association study (GWAS) of unsaturated fatty acids and IBD were obtained from web ⁃based public databases. Two ⁃sample Mendelian randomization analysis was performed by using inverse⁃variance weighted analysis, and weight median estimator and MR⁃Egger regression were conducted to validate the association of the causal effect. The causality of unsaturated fatty acids on the risk of IBDwas evaluated by OR and 95% CI . Results: No direct causal association was found between ω⁃6 fatty acids and CD, and a direct causal association was found with UC. Inverse⁃variance weighted analysis showed a 16% increase in the risk of UC for each standard deviation increase in ω⁃6 fatty acid gene levels (OR =1.16, 95% CI : 1.00⁃1.36, P =0.04). However, no causal association was found between ω⁃3 fatty acids, monounsaturated fatty acids and IBD. Conclusions: ω⁃6 fatty acids may be onlycausally associated with UC, and no causal association is found between ω⁃3 fatty acids, monounsaturated fatty acids and IBD.Key words Fatty Acids, Unsaturated;　Fatty Acids, Omega⁃6;　Inflammatory Bowel Disease;　Colitis, Ulcerative;　Crohn Disease;　Mendelian Randomization AnalysisDOI ： 10.3969/j.issn.1008⁃7125.2023.01.003*基金项目：新疆维吾尔自治区自然科学基金杰出青年科学基金项目（2022D01E25）炎症性肠病（inflammatory bowel disease, IBD ）是一种免疫介导的胃肠道慢性炎症性疾病，包括溃疡性结肠炎（ulcerative colitis, UC ）和克罗恩病（Crohn's disease, CD ），临床特征以腹痛和腹泻为主。

Unitarily invariant norms related to factors

Don Hadwin
Keywords: semi-ﬁnite factors, unitarily invariant norms, s-numbers, Ky Fan norms. MSC2000: 46L10; 46L51
1
Introduction
F.J. Murray and J.von Neumann [15, 16, 17, 26, 27] introduced and studied certain algebras of Hilbert space operators. Those algebras are now called “Von Neumann algebras.” They are strong-operator closed self-adjoint subalgebras of all bounded linear transformations on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator. Every von Neumann algebra is a direct sum (or “direct integral”) of factors. Thus factors are the building blocks for general von Neumann algebras. Murray and von Neumann [15] classiﬁed factors into type In , I∞ , II1 , II∞ , III factors. Type In and I∞ factors are full matrix algebras: Mn (C) and B(H). Type In and II1 factors are called ﬁnite factors. There is a unique faithful normal tracial state on a ﬁnite factor. 1

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* Received June, 1958 339
340
ALSTON
S.
HOUSEHOLDER
I t is as easy to discuss the general complex case, and the method is based upon the almost self-evident LEMMA. For a n y vector a ~ O, and a n y u n i t vector v, a u n i t vector u exists such
The proof exhibits the computation. Let a = ]]a]] > O. This represents one of the necessary square roots. I t is required t h a t
a 2 ( u * a ) u = av.
(1)
Let
= 2u*a. (2)
Then
~u = a ~v, (3)
2
2a(c~
v'a),
(4)
which accounts for the other square root. Clearly
c~ v*a >= O,
since a is the Euclidean length of a, and v*a the projection of a upon the unit vector v. Hence a and g are both real and can be taken non-negative. If a = v'a, then the lemma is verified with u = 0; otherwise take g > 0 defined b y (4), and one verifies easily t h a t u defined b y (3) is effective. T h e single reciprocation • . --1 necessary is in g N o w let a be the first column of A and take v = e~, the first column of the identity. Application of the l e m m a provides a unitary matrix U1 = I - 2ulu1* such t h a t the first column of U1A is null except in the first element. The result is equivalent to the application of n -- 1 plane rotations, with one slight difference, that det ( I -- 2uu*) = --1, whence this transformation reverses the orientation of the configuration• One continues after suppressing the first row and first column of the transformed matrix. After n -- 1 steps, at most, the matrix A is triangularized:
that (I 2 u u * ) a = II a [[ v,
where [[ a [[ represents the E u c l i d e a n n o r m : [[ a ]]2 = a*a. The c o m p u t a t i o n requires two square roots and a single reciprocation•
/ \
Inversion of R requires ( n +3 2 ) multiplications; and if R -1 is multiplied by the U~ in reverse order, then (n -- 1)(4n 2 + 7n + 12)/6 multiplications are required. Altogether, for the formation of A -~, the number of multiplications is (3n 3 --F 4n 2 + 5n - 10)/2 -+- 3n3/2, as compared with approximately 5n3/2 multiplications required if one were to form (A*A)-'A*. K y Fun has pointed out that half the square roots and all the reciprocations are evaded in the triangularization proper if one forms t~I-
R = DU,
where U has a unit diagonal. The purpose of the present note is to point out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order n, at most 2(n - 1) square roots are required, instead of n ( n - 1)/2. For n > 4, this is a saving of (n - 4)(n - 1)/4 square roots.
Unitary Triangularization of a N o n s y m m e t r i c Matrix*
ALSTON S. HOUSEHOLDER
Oak Ridge National Laboratory, Oak Ridge, Tennessee
A method for the inversion of a nonsymmetric matrix, due to J. W. Givens, has been in use at Oak Ridge National Laboratory and has proved to be highly stable numerically but to require a rather large number of arithmetic operations, including a total of n ( n -- 1)/2 square roots. Strictly, the method achieves the triangularization of the matrix, after which any standard method m a y be employed for inverting the triangle. The triangular form is brought about by means of a sequence of n ( n -- 1)/2 plane rotations, whose product is an orthogonal matrix. Each rotation requires the extraction of a square root. The advantage in using the method lies in the fact that an orthogonal matrix is perfectly conditioned. Hence the condition of the matrix cannot deteriorate through successive transformations. In fact, if ore defines the condition number of a matrix A to be [1] ~(A) = [] A ][ [I A-I[[, where the norm is the spectral norm, then for any orthogonal matrix W, 7(W) = 1 and the condition of any matrix is preserved under multiplication by an orthogonal matrix:
U • U n - l U n - 2 . . " U1,
UA = R.
Evidently det A = ( - 1 ) ~-1 det R.
UNITARY
TRIANGULARIZATION
341
An interesting byproduct of the reduction is a simple proof of the theorem of Hadamard, that I d e t A I ~ H]]a~]l, 1 where a, is the ,th column of A. This results from the fact that the Euclidean norm is preserved under multiplication b y a unitary matrix, and that each diagonal element of R is the norm of the projection of that column upon a certain subspace. I t is evident that when the reduction is applied to a matrix A of n columns and N > n rows, there results again an upper triangle of non-null elements, and a trapezoid of zeros, the triangle representing the factorization of the normal matrix A*A required for a least squares solution. Returning to the case of a square matrix, if the vectors u, are stored as generated and applied to the successive columns of the matrix, there are required the n - 1 reciprocations, 2(n - 1) square roots, and (n -- 1)(2n 2 + 5n + 9)/3 ~ 2n3/3 multiplications for the formation of R. In this count it is assumed that one forms o~, t~-1, u, and then each u'at and uu*a~, i > 1.

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