Unitary Triangularization of a Nonsymmetric Matrix
证明开普勒猜想的原文(英文)并附译后汉语
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׈A×A),where P is the normalized Poincar´e line bundle on A׈A.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)
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.
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A
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characteristic arrangement of the zoOids and branches. Axial microzoöids 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
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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.