Invariant Properties of the Ansatz of the Hirota Method for Quasilinear Parabolic equations

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Hawking and Page on the brane

a r X i v :h e p -t h /0412017v 3 2 S e p 2005UW/PT 04-24hep-th/0412017Hawking and Page on the braneA.Chamblin 1and A.Karch 21Department of Physics,University of Louisville,Louisville,KY 40292E-mail:chamblin@ 2Department of Physics,University of Washington,Seattle,WA 98195-1560E-mail:karch@ Abstract We show that the Hawking-Page phase transition of a CFT on AdS d −1weakly coupled to gravity has a dual bulk description in terms of a phase transition between a black string and a thermal gas on AdS d .At even lower temperatures the black string develops a Gregory Laﬂamme instability,which is dual to black hole evaporation in the boundary theory.11IntroductionGravitational theories on Anti de-Sitter space undergo an interesting phase transi-tion as the temperature is varied,the well known Hawking-Page phase transition[1]. At high temperatures a large black hole is formed,whereas at low temperatures a thermal gas in the AdS d space with curvature radius L is the preferred conﬁgura-tion.Aﬁrst order transition occurs between the two geometries at a temperature T HP=d−2(d−2)2−12The Hawking-Page Phase transition on the bound-ary2.1General SetupIn order to establish the phase transition,one calculates the diﬀerence of the on-shell gravitational action,I=−M d−2P l√L )ds2AdSd−1+dr2(2)one obtains as a dual a CFT on AdS d−1,or more precisely two copies of AdS d−1at r=±∞,which communicate with each other via their common boundary[2]as we will review in detail in the next subsection.Introducing a UV brane at r=r0serves as a UV cutoﬀin theﬁeld theory and in addition introduces dynamical gravity with an almost massless graviton,where the mass can be understood as being a CFT loop eﬀect[7].The curvature radius on the brane is l=cosh(r02.2Boundary Conditions:Interplay of two CFTsThe bulk metric only deﬁnes the conformal structure at the boundary.By picking a deﬁning function suggested by the slicing,in our case cosh(rone would expect that this does not inﬂuence the properties of the black hole:as longas the temperature in the two AdS d−1spaces is the same,it should not matter if theblack hole is in a thermal bath of gravitons that get reﬂected from the boundary or in a thermal bath of gravitons at the same temperature that freely move from one AdS d−1to the other.This expectation will be born out by computations in this paper. One interesting special case is r1→∞,in which case dynamical gravity switches oﬀin one of the two AdS d−1.Another one is r1=r0.In this case one can impose anorbifold projection r→−r.In the orbifolded theory the boundary is just one copyof AdS d−1and the transparent boundary conditions turn into the regular reﬂecting boundary conditions.Both those special cases are completely consistent with the analysis in this paper.Last but not least let us emphasize that even in the limit that both cutoﬀs are takento inﬁnity one does not simply recover the standard bulk Hawking Page physics.In the standard scenario one demands the metric to take the form S d−2×S1close to the boundary,corresponding to a CFT on the sphere atﬁnite temperature.Since we are interested in the scenario where gravity on the AdS d−1slices is dynamical,we allow for non-trivial geometry on those slices and onlyﬁx the boundary conditions on the S d−3×S1along which the two AdS d−1components of the boundary communicate.While in the cutoﬀto inﬁnity limit dynamical gravity decouples,one of course only obtains a smooth limit if one does not suddenly change the boundary conditions.In particular the black string solution we will introduce momentarily,which gives rise to two boundary AdS d−1Schwarzschild black holes,would be allowed by our boundary conditions and not by the standard bulk Hawking Page analysis.As is apparent from formula(4)the free energy of the black string is negative and grows exponentially in the cutoﬀr0, whereas the free energy of the large AdS d black hole is independent of the cutoﬀ.As soon as the boundary conditions allow for the black string,it completely dominates the thermal ensemble for large values of the cutoﬀand the physics is genuinely diﬀerent from the standard bulk Hawking Page analysis.2.3The high temperature phase–The black string solution The Hawking Page transition on the boundary occurs when the inverse boundary temperature isβbound=2πlL )ds2S−AdSd−1+dr2.(3)That is one simply replaces the AdS d−1with a S-AdS d−1metric on every slice,as5Figure1:The black string solution.depicted in Figure1.While forﬂat or dS slices this string suﬀers from a Gregory Laﬂamme instability,it has been argued in[5]based on thermodynamic arguments that for suﬃciently large horizon radius r H the S-AdS black string is actually stable. This has been conﬁrmed by a numerical analysis for d=5in[9].T HP corresponds to r H=L.In order to establish that the black string is the dominant conﬁguration for r H>L,one can use the euclidean path integral as in the original Hawking Page paper to establish that the diﬀerence in free energies associated with the metrics(2)and(3) is given by the diﬀerence in the value of on-shell actions(1)β∆F=I Black−String−I AdS d= r0−r1dr cosh d−3(r/L) I S−AdS d−1−I AdS d−1 .(4)But I S−AdS d−1−I AdS d−1is just the free energy diﬀerence that is governing the d−1 dimensional Hawking Page transition on the brane.It is negative for r H>L and positive for r H<L.To the extend that these are the only two geometries that correspond to a boundary CFT on AdS at a given temperature,this establishes that in the high temperature regime the black string is the dominant conﬁguration.We will address the issue of possible other geometries later.Note that the sign of this free energy diﬀerence is completely independent of r1. Both the black hole radius on the slice as well as the AdS d−1curvature radius of the slice grow with the same power of the warpfactor.The diﬀerence in free energies is either positive on all slices or negative on all slices.For generic values of r0and r1 the phase transition occurs when on every slice,including the two cutoﬀbranes,the black hole radius is equal to the AdS d−1curvature radius of the slice.In the special case of r0=r1with the orbifold projection imposed,corresponding to the standard reﬂecting boundary conditions in AdS d−1as we explained above,we can take the lower integration boundary to be r=0.This conﬁrms our expectation that the transparent boundary conditions do not alter the d−1dimensional Hawking Page physics at all.2.4The low temperature phase–Thermal AdSAt temperatures below T HP the dominant boundary metric should be thermal AdS d−1, that is AdS d−1with periodic euclidean time.This is what we get by simply replacing6the bulk AdS d with thermal AdS d.To compare temperatures,we have to examine the change of variables between(2)and a more conventional way of writing euclideanAdS d,e.g.ds2=cosh2(ρL)dΩ2d−2.(5)The change of variables has for example been presented in[2].The important point is that t=˜t and hence cosh(rL)=cosh(ρL )L a periodβbulk of thet coordinate implies an inverse temperatureβbound=cosh(r0d−3is what we obtain from a thermal bulk AdS d withβbulk=2πLd−2,is lower than the valuerelevant for the boundary Hawking page transition(7),and hence the corresponding temperature is higher.So by the time the boundary made the transition to thermal AdS d−1,in the bulk the S-AdS d black hole is already a subdominant conﬁguration–actually the temperature is even lower than T S,so the bulk black hole solution doesn’t even exist anymore.7S-AdS d could still be relevant for the high temperature regime.For the case r0,r1>> L we are discussing it is however clear that the black string will always dominate over the black hole.The diﬀerence in free energies between the string and the background AdS according to(4)grows exponentially with r0and r1,while the diﬀerence between the bulk S-AdS d and the background approaches aﬁxed value independent of r0and r1.The same argument also suppresses other geometries that would lead to induced metrics on the brane that diﬀer from thermal AdS and the S-AdS d−1black hole,like the bulk black hole which gives rise to an FRW like cosmology on the brane.For r0,r1>>L only the same two geometries as in the original Hawking-Page transition contribute on the brane,and the two bulk geometries we considered are the bulk extensions corresponding to those two boundary geometries.If we want to go beyond the r0,r1>>L regime,the bulk black hole certainly will start contributing.On the boundary side this maps to the strong correction due to matter eﬀects becoming important.3Quantum CorrectionsThe induced metric in the high temperature phase is exactly that of a lower dimen-sional AdS Schwarzschild black hole.In[6]however it was shown that a brane world black hole can not be a solution to vacuum Einstein equations on the brane.AdS/CFT predicts that classical gravity in the bulk is dual to the quantum CFT including pla-nar corrections on the boundary,and coupled to classical gravity if the UV brane is introduced.In particularM d−3P l,d−1=(M P l,d L)M d−3P l,d,where M P l,d L is a large number,e.g.N2/3in Maldacena’s d=5example.So for M P l,d<<M<<M P l,d−1bulk Hawking radiation is negligible,while on the brane Hawking radiation and its backreaction dominate∗From this point of view it is a puzzle why the black string gives a metric on the brane that is not corrected at all due to Hawking radiation.While this still remains a puzzle to us in general,in the r0,r1>>L regime we are analyzing,where the UV brane is close to the boundary this is not an issue.The4d Planck scale is still M2P l,4=M3P l,5L, whereas the4d curvature radius on the brane is l=cosh(r0∗The authors of[6]were mostly interested in d=5black holes with M>(M5L)3the large black hole is in thermal equilibrium with the heat bath generated by its own Hawking radiation.4The onset of the Gregory Laﬂamme instability Like the bulk,the boundary black hole also disappears at a spinodal at temperatureT bound S .While in the temperature range T bounds<T<T boundHPthe thermal AdS isthe dominant conﬁguration both in the bulk and on the boundary,the black hole solution on the boundary is still a valid local minimum of the free energy.But whenthe temperature is lowered below T boundS the black hole on the boundary has now anegative heat capacity and no longer even represents a local minimum.At the same time the Gubser-Mitra hypothesis[10],which states that classical stability of a black string should come hand in hand with local thermal stability,would now say that the black string in the bulk should develop a classical instability of the Gregory Laﬂamme type.While the numerical analysis of[9]is too crude to give a decisive answer to this question about the precise value at which the instability sets in,their result is consistent with this interpretation.On the boundary,the black hole will start evaporating,and will eventually disappear (as long as we keep the induced Planck scale large butﬁnite).At least the initial stage of this evaporation process is completely captured by the classical evolution of the unstable mode in the black string background.Once the black string becomes suﬃciently hot,d dimensional quantum eﬀects will become important and classical bulk gravity is no longer a good description,so theﬁnal stages of the evaporation process go beyond classical gravity.This is an explicit realization of the scenarioﬁrst envisioned in[8],where the classical dynamics associated with the black string in the bulk captures the physics of the black hole on the brane.Additional supporting evidence for our picture comes from the analysis in d=4,where the so called c-metric yields an exact solution describing the snapping string[5].Based on thermodynamic arguments theyﬁnd that for small enough mass(which for the negative speciﬁc heat black holes means large temperature) the black string becomes disfavored.Instead there exist two classically stable black hole conﬁgurations on the brane.These should be thought of as the endproducts of the classical evolution of the black string,any further evaporation process would be due to bulk quantum eﬀects.One can envision various scenarios for how precisely the black string evolves†.One still somewhat controversial aspect is whether the string snaps at all[11].If indeedtheﬁnal conﬁguration of the string is a stable non-uniform string,we would get a stable remnant as the endresult of the evaporation on the brane.In any case,the map to quantum black hole physics really shows how important it is to understand what happens at the Gregory Laﬂamme point.It is however safe to say that whatever the string does—that is precisely what the black hole is going to do on the brane! AcknowledgementsWe thank Roberto Emparan,Nemanja Kaloper and Toby Wiseman for very helpful discussions about black holes on the brane and for comments on the draft.The research of AK is supported in part by DOE contract#DE-FG03-96-ER40956and AK is also supported by NSF Grant SBE-0123552as an ADVANCE professor. References[1]S.W.Hawking and D.N.Page,“THERMODYNAMICS OF BLACK HOLESIN ANTI-DE SITTER SPACE,”Commun.Math.Phys.87(1983)577.[2]A.Karch and L.Randall,“Locally localized gravity,”JHEP05(2001)008,hep-th/0011156.[3]L.Randall and R.Sundrum,“An alternative to compactiﬁcation,”Phys.Rev.Lett.83(1999)4690–4693,hep-th/9906064.[4]R.Gregory and ﬂamme,“Black strings and p-branes are unstable,”Phys.Rev.Lett.70(1993)2837–2840,hep-th/9301052.[5]R.Emparan,G.T.Horowitz,and R.C.Myers,“Exact description of black holeson branes.II:Comparison with BTZ black holes and black strings,”JHEP01 (2000)021,hep-th/9912135.[6]R.Emparan,A.Fabbri,and N.Kaloper,“Quantum black holes as holograms inAdS braneworlds,”JHEP08(2002)043,hep-th/0206155.[7]M.Porrati,“Mass and gauge invariance.IV:Holography for the Karch-Randallmodel,”Phys.Rev.D65(2002)044015,hep-th/0109017.[8]A.Chamblin,S.W.Hawking,and H.S.Reall,“Brane-world black holes,”Phys.Rev.D61(2000)065007,hep-th/9909205.[9]T.Hirayama and G.Kang,“Stable black strings in anti-de Sitter space,”Phys.Rev.D64(2001)064010,hep-th/0104213.10[10]S.S.Gubser and I.Mitra,“The evolution of unstable black holes in anti-deSitter space,”JHEP08(2001)018,hep-th/0011127.[11]G.T.Horowitz and K.Maeda,“Fate of the black string instability,”Phys.Rev.Lett.87(2001)131301,hep-th/0105111.11。

Counting BPS Blackholes in Toroidal Type II String Theory

1. Introduction Supersymmetric indices have proven to be invaluable in the program of accounting for black-hole entropy using D-branes [1]. In particular, in those cases where the computation of BPS black holes can be related to counting functions in a conformal ﬁeld theory, the elliptic genus has been of particular use. Nevertheless, there are examples, notably toroidal compactiﬁcation of type II string, where the relevant elliptic genus vanishes, thus giving little indication about the D-brane BPS state degeneracies. Perhaps surprisingly, the degeneracies are therefore more subtle for compactiﬁcation on T 4 than for K 3. These degeneracies were ﬁrst seriously investigated in [2][3][4]. In this paper we study these degeneracies further in the case of the three charge system of [1] consisting of Q1 D1branes, Q5 , D5-branes and momentum N . Using a function closely related to the elliptic genus we derive E6,6 (Z Z) U -dual expressions for the case of primitive charges, i.e., charges such that gcd(Q1 , Q5 , N ) = 1. The formula is given in equations (6.2), (6.3) below and is easily derived from our central result, the counting formula for 1/8 BPS states given in equation (5.9) below, valid for gcd(Q1 , Q5 ) = 1. Our approach to the problem is to deﬁne an “index” in the same spirit as the “new supersymmetric index” of [5]. These authors investigated the traces in supersymmetric quantum mechanics deﬁned by: Eℓ = TrH (−1)F F ℓ e−βH , (1.1)

Gamma2

u ? d djN i = gA U (p) U (p) ; (1) where U (p) denotes the nucleon spinor. Assuming isospin symmetry, one nds (gA )p = ?(gA )n, and the value of (gA)p = 1:2573 0:0028, extracted from the neutron beta decay, has been quoted in the literature 1]. In nature, the isospin symmetryis broken by the current quark mass di erence as well as the electromagnetic interaction, and thus (gA )p 6= ?(gA )n . Previous studies of the nucleon isovector axial charge in the framework of external eld QCD sum-rule method have been made by various authors 2{4]. However, to our best knowledge, the isospin breaking e ects have been ignored in these studies. The goal of this Letter is to examine the di erence between (gA )p and (gA)n using the external eld QCD sum-rule approach, which has been used in studying various nucleon matrix elements of bilinear quark operators 2{13]. The isospin violation is re ected in mu 6= md and the isospin breaking in the vacuum condensates. Electromagnetic e ects will not be included. Invoking the Goldberger-Treiman relation, we also give an estimate of the isospin breaking in the pion-nucleon coupling constants, which is of interest in the investigation of charge symmetry breaking phenomena 14{17]. Let us start from the correlation function of the nucleon interpolating eld in the presence of a constant external isovector axial vector eld Z

现象学胡塞尔英文

现象学胡塞尔英文English:Phenomenology, developed by Edmund Husserl, is a philosophical method that focuses on the first-person experience of consciousness and the structures of experience. Husserl believed that knowledge begins with our direct experience of the world, and that by suspending judgment and focusing on the phenomena themselves, we can uncover the essential structures that underlie our perception and understanding of the world. Through the process of epoché, or bracketing, Husserl suggests that we can temporarily set aside our preconceived ideas and biases in order to grasp the pure experience as it presents itself to us. This method allows us to closely examine the different layers of experience, including sensory perceptions, emotions, and thoughts, in order to gain a deeper understanding of the fundamental structures of consciousness. Husserl's phenomenology has had a significant influence on various fields, including philosophy, psychology, sociology, and cognitive science, as it offers a unique approach to understanding the nature of reality and human experience.中文翻译:现象学，由埃德蒙德·胡塞尔发展，是一种专注于意识第一人称体验和经验结构的哲学方法。

计量经济学专业英汉词典

计量经济学专业英汉词典计量经济学专业英汉词典中文英文调整的R^2 （确定系数）adjusted R^2调整系数adjustment coefficient调整系数矩阵adjustment coefficient matrix赤池信息准则（AIC）Akaike’s information criterion (AIC) 阿尔蒙分布滞后模型Almen distributed lag model阿尔蒙滞后Almon lags备择假设alternative hypothesis方差分析analysis of variance辅助变量ancillary variable近似协方差矩阵approximate covariance matrix近似正态分布approximate normal distribution自回归模型AR model自回归过程AR process自回归条件异方差模型ARCH model自回归移动平均模型ARMA model假定assumption渐近χ2分布asymptotic χ2 distribution渐近协方差矩阵asymptotic covariance matrix渐近分布asymptotic distribution渐近有效性asymptotic efficiency渐近性质asymptotic properties渐近抽样特性asymptotic sampling properties渐近设定asymptotic specification渐近标准误差asymptotic standard error渐近检验asymptotic test渐近检验统计量asymptotic test statistic渐近逼近asymptotically approximation渐近有效估计式asymptotically efficient estimator渐近无偏估计式asymptotically unbiased estimatorADF检验，增项（增广）DF检验Augmented Dickey-Fuller test AEG检验，增项（增广）EG检验Augmented Engle-Granger test自相关方程误差autocorrelated equation error自相关autocorrelation自相关函数autocorrelation function自协方差autocovariance自协方差函数autocovariance function自回归autoregression自回归条件异方差autoregressive conditional heteroscedasticity自回归分布滞后模型autoregressive distributed lag (ADL) model自回归单整移动平均（ARIMA）autoregressive integrated moving average process 过程自回归（AR）摸型autoregressive model自回归移动平均（ARMA）过程autoregressive moving-average process自回归算子autoregressive operator辅助回归auxiliary regression平均值average行为方程behavioral equation贝拉－哈尔克（BJ）统计量Bera-Jarque statistic贝努利分布Bernoulli distribution最佳决策best decision最佳线性无偏估计式（BLUE）best linear unbiased estimator (BLUE)最佳线性无偏预测best linear unbiased prediction最佳无偏估计式best unbiased estimator偏倚bias偏倚向量bias vector有偏估计式biased estimator二元选择模型binary choice model二项分布binomial distribution二元正态随机变量bivariate normal random variable自举法，靴襻法bootstrap procedure博克斯－考克斯变换Box-Cox transformation博克斯－詹金斯方法Box-Jenkins approach布罗施－帕甘检验Breusch-Pagan test布朗运动Brownian motion典型相关canonical correlation因果性causality中心极限定理central limit theorem特征方程characteristic equation特征根characteristic root特征向量characteristic vector卡埃方分布chi-square distribution古典统计学classical statistics柯布－道格拉斯生产函数Cobb-Douglas production function 科克伦－奥克特方法Cochrane-Orcutt procedure“概率极限”概念concept of “plim”条件推断conditional inference条件概率conditional probability条件概率密度函数conditional probability density function 置信区间confidence interval一致性consistency一致估计式consistent estimator一致性检验consistent test消费函数consumption function同期相关contemporaneous correlation同期协方差矩阵contemporaneous covariance matrix同期扰动相关contemporaneous disturbance correlation同期独立随机回归自变量contemporaneous independent stochastic regressor 连续映射理论continuous mapping theorem 连续随机变量continuous random variable连续回归函数continuous regression function常规抽样理论conventional sampling theory依概率收敛converge in probability收敛convergence依分布收敛convergence in distribution相关correlation相关系数correlation coefficient相关矩阵correlation matrix相关图correlogram成本cost协方差covariance协方差矩阵covariance matrix协方差矩阵估计式covariance matrix estimator克拉美规则Cramér rule克拉美－拉奥不等式Cramér-Rao inequality克拉美－拉奥下界Cramér-Rao lower bound临界区域critical region临界值critical value截面数据cross-section data累积分布函数cumulative distribution function 数据data数据生成过程（dgp）date generation process数据标准化date normalization盲始模型dead-start model决策decision making决策规则decision rule决策规则选择decision rule choice决策理论decision theory演绎系统deductive system定义方程definitional equation解释程度degree of explanation自由度degree of freedom密度函数density function相依变量dependent variable设计矩阵design matrix检验方法detection methods方阵的行列式determinant of a square matrix确定系数，可决系数determination coefficient诊断校验diagnostic checking对角矩阵diagonal matrix对称矩阵的对角化diagonalization of a symmetric matrix 差分difference差分方程difference equation离散随机变量discrete random variable离散样本空间discrete sample space离散随机过程discrete stochastic process非均衡误差disequilibrium error不相交集disjoint set分布滞后distributed lag分布滞后模型distributed lag model分布distribution分布函数distribution function分布理论distribution theory扰动协方差矩阵disturbance covariance matrix扰动方差disturbance variance位移项drift虚拟变量dummy variable虚拟变量估计式dummy variable estimatorDW（德宾—沃森）统计量Durbin-Watson statisticDW（德宾—沃森）检验Durbin-Watson test动态模型dynamic model动态乘数dynamic multiplier动态回归dynamic regression动态联立方程dynamic simultaneous equation计量经济学，经济计量学econometrics经济变量economic variables经济学economics经济economy有效性efficiencyEG检验EG test特征值eigen value弹性elasticity椭圆ellipse空集empty set内生变量endogenous variableEG两步估计量Engel-Granger (EG) two-step estimate EG两步法Engel-Granger (EG) two-step method 方程误差equation error 方程识别equation identification均衡equilibrium均衡分析equilibrium analysis均衡条件equilibrium condition均衡乘子equilibrium multiplier均衡关系equilibrium relationship均衡状态equilibrium state遍历性ergodicity误差error误差分量error component误差修正机制error correction mechanism误差修正模型error correction model误差修正项error correction term误差平方和error sum of squares误差向量error vector估计量estimate估计estimation估计式estimator欧氏空间Euclidean space外生前定变量exogenous predetermined variable 外生变量exogenous variable期望算子expectation operator期望值expected value试验experiment被解释变量explained variable解释变量explaining variable解释explanation指数分布exponential distributionF分布 F distributionF统计量 F statisticF检验 F test因子分解准则factorization criterion反馈feedback最终形式final form有限分布滞后模型finite distribution lag model有限非奇异矩阵finite nonsingular matrix有限多项式滞后finite polynomial lag有限抽样特性finite sampling property有限方差finite variance一阶自回归模型first-order autoregressive model 一阶条件first-order condition一阶差分算子first-order difference operator 一阶泰勒级数first-order Taylor series拟合值fitted value固定回归自变量fixed regressor预测区间forecast interval预测区域forecast region预测方差forecast variance预测forecasting频数，频率frequency完全信息估计full information estimation完全信息极大似然法full information maximum likelihood method 函数形式function form函数空间function space泛函中心极限定理functional central limit theorem (FCLT)伽玛分布Gamma distribution伽玛函数Gamma function广义自回归条件异方差模型GARCH高斯白噪声Gaussian white noise高斯－马尔可夫定理Gauss-Markov theorem高斯－牛顿算法Gauss-Newton algorithm一般协方差矩阵general covariance matrix一般均衡general equilibrium一般线性假设general linear hypothesis一般线性统计模型general linear statistical model一般随机回归自变量模型general stochastic regressor model“一般到特殊”方法general to special method广义自回归算子generalized autoregressive operator广义最小二乘法generalized least squares广义最小二乘估计generalized least squares estimation 广义最小二乘估计式generalized least squares estimator 广义最小二乘方法generalized least squares procedure 广义最小二乘残差generalized least squares residual广义最小二乘规则generalized least squares rule几何滞后模型估计geometric lag model estimation总体极小值global minimum拟合优度goodness of fit格兰杰因果性Granger causality格兰杰因果性检验Granger causality test格兰杰非因果性Granger noncausality格兰杰定理Granger representation theorem增长率模型growth rate model豪斯曼设定检验Hausman specification test重（厚）尾heavy tail海赛矩阵Hessian matrix异方差误差heteroscedastic error异方差heteroscedasticity同一性homogeneity同方差误差homoscedastic error同方差homoscedasticity假设hypothesis假设检验hypothesis test同分布随机变量identically distributed random variable 识别identification识别规则identification rules单位矩阵identity matrix压缩矩阵，影响矩阵impact matrix影响乘数矩阵impact multiplier matrix非一致性inconsistency错误约束incorrect restriction独立同一分布independent and identical distribution (IID) 独立分布independent distribution独立事件independent event独立随机变量independent random variable独立随机回归自变量independent stochastic regressor独立变量independent variable间接最小二乘法indirect least squares不等式约束inequality restriction推断inference无限分布滞后infinite distributed lag无限累加算子infinite summation operator无限方差infinite variance有影响的观测值influential observation信息矩阵information matrix内积inner product新息过程innovation sequence投入产出关系input-output relationship工具变量instrumental variable工具变量估计instrumental variable estimation 单整integration截距intercept区间估计interval estimation区间预测interval forecast不变性invariance逆矩阵inverse matrix信息矩阵的逆inverse of information matrix可逆性invertibility可逆移动平均过程invertible moving-average process 投资investment迭代方法iterative procedure大折刀方法jackknife procedure雅可比变换Jacobian of the transformation联合置信区间joint confidence interval联合置信区域joint confidence region联合密度函数joint density function联合扰动向量joint disturbance vector联合假设检验joint hypothesis test联合区间估计joint interval estimation联合零（原）假设joint null hypothesis联合概率分布joint probability distribution联合被确定变量jointly determined variable恰好识别方程just identified equation核kernel凯恩斯消费函数Keynesian consumption function 凯恩斯模型Keynesian model克莱因－戈德伯格消费函数Klein-Goldberger consumption克莱因－鲁滨效用函数Klein-Rubin utility function柯依克变换Koyck transformation克罗内克尔积Kronecker product库恩－塔克条件Kuhn-Tucker condition峰度，峭度kurtosis滞后lag滞后长度lag length滞后算子lag operator滞后权数lag weight滞后变量lagged variable拉格朗日乘数Lagrange multiplier拉格朗日乘子检验Lagrange multiplier test拉普拉斯展开Laplace expansion大样本特性large sample properties全概率定律law of total probability前导模型leading indication model最小绝对离差least absolute deviation最小绝对误差估计式least absolute error estimator 最小平方偏倚least squares bias最小平方准则least squares criterion最小平方估计式least squares estimator最小平方法least squares procedure最小平方残差least squares residual最小平方规则least squares rule最小平方方差估计式least squares variance estimator左逆矩阵left-inverse matrix显著性水平level of significance杠杆率leverage似然函数likelihood function似然原理likelihood principle似然比原理likelihood ratio principle似然比统计量likelihood ratio statistic似然比检验likelihood ratio test线性代数linear algebra线性联系linear association线性相依linear dependency线性相依向量linear dependent vector线性等式约束linear equality restriction 线性方程linear equation线性方程系统linear equation system线性估计式linear estimator线性形式linear form线性参数linear in parameter线性无关向量linear independent vector线性不等式假设linear inequality hypothesis 线性不等式约束linear inequality restriction 线性损失函数linear loss function 线性算子linear operator线性概率模型linear probability model线性规划模型linear programming model线性约束linear restriction线性规则linear rule线性联立方程linear simultaneous equation 线性统计模型linear statistical model线性变换linear transformation线性无偏估计式linear unbiased estimator线性linearity局部极小值local minima罗基斯迪随机变量logistic random variable罗基特（Logit）模型logit model对数似然函数log-likelihood function对数线性函数log-linear function长期效应long-run effect损失loss损失函数loss function下三角矩阵lower triangular matrix矩（M）估计式M estimator移动平均模型MA model宏观经济学macroeconomics边缘分布marginal distribution边缘概率密度函数marginal probability density function 边际消费倾向marginal propensity to consume数理经济学mathematical economics数学期望mathematical expectation矩阵matrix矩阵分解matrix decomposition极大似然估计maximum likelihood estimation极大似然估计式maximum likelihood estimator极大似然法maximum likelihood method均值mean均方误差mean square error均方误差准则mean square error criterion均方误差矩阵mean square error matrix均值向量mean vector测量误差measurement error中位数median矩法method of moments极小极大准则minimax criterion使损失最小minimizing loss使风险最小minimizing risk最小绝对离差估计式minimum absolute deviation estimator 最小方差minimum variance最小方差无偏估计minimum variance unbiased estimation 错误设定misspecification混合估计mixed estimation众数mode模型model模型设定model specification模数module复数的模modulus of a complex number矩moment蒙特卡罗Monte Carlo蒙特卡罗数据Monte Carlo data蒙特卡罗试验Monte Carlo experiment蒙特卡罗模拟Monte Carlo simulation移动平均moving average移动平均（MA）模型moving average (MA) model移动平均过程moving average process移动平均表示法moving average representation移动平均季节过滤算子moving average seasonal filter多重共线性multicollinearity多项选择模型multinomial choice models多项分布multinomial distribution多元回归multiple regression多重解multiple solution多重时间序列分析multiple time-series analysis乘法multiplication乘子，乘数multiplier多元分布multivariate distribution多元函数multivariate function多元正态分布multivariate normal distribution多元正态随机变量multivariate normal random variable 多元随机变量multivariate random variable多元t 分布multivariate t distribution互斥集mutually exclusive set自然共轭先验概率密度函数natural conjugate prior probability density function半负定矩阵negative semidefinite matrix嵌套nest牛顿－拉夫森算法和方法Newton-Raphson algorithm and method非线性函数nonlinear function参数非线性nonlinear in the parameter非线性最小平方法nonlinear least squares非线性最小平方估计nonlinear least squares estimation非线性似然函数nonlinear likelihood function非线性极大似然估计nonlinear maximum likelihood estimation 非线性回归nonlinear regression非线性似不相关回归方程nonlinear seemingly unrelated regression equation非线性nonlinearity非负定矩阵nonnegative definite matrix非嵌套模型nonnested models非正态分布nonnormal distribution非正态误差nonnormal error非正定矩阵nonpositive definite matrix非纯量单位协方差矩阵nonscalar identity covariance matrix 非奇异矩阵nonsingular matrix非平稳nonstationary非平稳过程nonstationary process非随机变量nonstochatic variable正态分布normal distribution正态分布理论normal distribution theory正态误差的检验normal error testing正态线性统计模型normal linear statistical model正态概率密度函数的核normal probability density function 正态随机向量normal random vector正态变量normal variable正态向量normal vector标准化常数normalizing constant正态分布随机变量normally distribution random variable 多余参数nuisance parameter零（原）假设null hypothesis零矩阵null matrix空集，零集null set可观测随机变量observable random variable可观测随机向量observable random vector观测值样本observation sample观测上的等价模型observationally equivalent model阶order阶条件order condition普通最小二乘法ordinary least squares正交矩阵orthogonal matrix正交向量orthogonal vector正交orthogonality标准正交线性统计模型orthonormal linear statistical model 离群值outliers过度识别方程overidentified equation参数parameter参数估计parameter estimation参数方差parameter variance参数检验parametric test帕累托分布Pareto distribution局部调整分布滞后模型partial adjustment distributed lag model 偏（局部）调整模型partial adjustment model偏自相关partial autocorrelation偏自相关系数partial autocorrelation coefficient偏自相关函数partial autocorrelation function偏相关partial correlation偏相关图partial correlogram偏导数partial derivative局部均衡partial equilibrium分块逆规则partitioned inverse rule完全多重共线性perfect multicollinearity长期收入假设permanent income hypothesis分段线性回归piecewise linear regression分段回归函数piecewise regression function点估计量point estimate点估计point estimation点估计式point estimator点估计式性质point estimator properties多项式polynomial多项式滞后polynomial lag多项式矩阵polynomial matrix合并数据pooling data合并模型pooling model合并模型选择pooling model selection合并时间序列pooling time series合并时间序列数据pooling time series data总体population正定矩阵positive definite matrix正定对称矩阵positive definite symmetric matrix 半正定矩阵positive semidefinite matrix后验密度posterior density后验密度函数posterior density function后验分布posterior distribution后验信息posterior information后验均值posterior mean后验优势posterior odds后验优势比posterior odds ratio后验概率posterior probability后验概率密度函数posterior probability density function 后验概率区域posterior probability region假设过程postulation process功效函数power function检验功效power of a test前定变量predetermined variable预测误差prediction error随机分量的预测prediction of random components预测精度prediction precision主分量模型principal components model先验协方差矩阵prior covariance matrix先验分布prior distribution先验均值prior mean先验概率prior probability先验概率密度函数prior probability density function先验概率区域prior probability region概率probability概率密度probability density概率分布probability distribution离散随机变量的概率分布probability distribution for discrete random variable概率分布函数probability distribution function概率测度probability measure概率单位（probit）模型probit model积矩product moment积矩量矩阵product moment matrix积算子product operator生产函数production function生产过程production process比例响应模型proportional response model 伪样本数据pseudo sample data二次型quadratic form二次损失函数quadratic loss function二次矩阵quadratic matrix定量选择模型quantitative choice model 定量因素quantitative factors定量信息quantitative information随机系数模型random coefficient model随机分量预测random component prediction 随机误差random error随机试验random experiment随机变量random variable随机向量random vector随机向量分量random vector component随机游走random walk秩rank秩条件rank condition矩阵的秩rank of a matrix简化型reduced form简化型系数reduced form coefficient简化型扰动reduced form disturbance简化型方程reduced form equation简化型估计式reduced form estimator。

奈奎斯特定理的英文

奈奎斯特定理的英文The Nyquist Theorem, also known as the Nyquist Sampling Theorem, is a fundamental principle in the field of signal processing and telecommunications. It was first formulated by Harry Nyquist in 1928 and later expanded upon by Claude Shannon in his groundbreaking work on information theory.The theorem states that in order to perfectly reconstruct a continuous-time signal from its samples, the sampling rate must be at least twice the highest frequency present in the signal. This minimum rate is referred to as the Nyquist rate. If the sampling rate falls below this threshold, a phenomenon known as aliasing occurs, where the sampled signal becomes a distorted version of the original signal.The importance of the Nyquist Theorem lies in its application to digital signal processing, where analog signals must be converted into digital form for processing and storage. By adhering to the theorem's guidelines, engineers can ensure that the digital representation of an analog signal is accurate and free from aliasing.In practice, the Nyquist Theorem has implications for a wide range of technologies, from audio recording and broadcasting to medical imaging and seismology. It is a cornerstone concept that underpins the design of sampling systems and the development of anti-aliasing filters, which are used to prevent aliasing before the sampling process.The theorem also has a direct impact on the capacity of communication channels. In digital communication systems, understanding the relationship between the sampling rate and the frequency content of signals is crucial for maximizing the amount of information that can be transmitted without error.In summary, the Nyquist Theorem is a foundational principle that guides the process of sampling and reconstructing signals in digital systems. It ensures that high-quality digital representations of analog signals can be achieved, provided that the sampling rate is sufficiently high. This theorem has far-reaching applications and continues to be a key concept in the advancement of digital technology.。

Integrated density of states and Wegner estimates for random Schrodinger Operators

INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES
99
On the other hand, the electron could be moving in an amorphous medium, in which case there is no large group of symmetries of the Hamiltonian. However, from the physical point of view it is reasonable to assume that the local structure of the medium will be translation invariant on average. This means that we consider the potential which the electron experiences as a particular realisation of a random process and assume stationarity with respect to some group of translations. Moreover, physical intuition suggests to assume that the local properties of the medium in two regions far apart (on the microscopic scale) are approximately independent from each other. Therefore the stochastic process describing the potential should have a correlation function which decays to zero, or — more precisely — should be ergodic. There are interesting models which lie between the two extreme cases of latticeperiodic and amorphous media. They still have an underlying lattice structure which is, however, modiﬁed by disorder. Probably the best studied Hamiltonian with this properties is the alloy type model. We leave its precise deﬁnition for the next paragraph and introduce here a special case on the intuitive level. Consider ﬁrst the potential Vω (x) := uk (ω, x) Each k corresponds to a nucleus sitting on a lattice point. The function uk (ω, ·) describes the atomic or nuclear potential at the site k and depends on the random parameter ω which models the diﬀerent realisations of the conﬁguration of the nuclei. If there is only one type of atom present, which has a spherically symmetric potential, all the uk (ω, ·) are the same, and Vω is periodic. Now assume that there are two kinds a and b of atoms present, which have spherically symmetric atomic potentials of the same shape, but which diﬀer in their nuclear charge numbers. In this case the potential looks like Vω (x) :=

On non commutative sinh-Gordon Equation

On noncommutative sinh-Gordon equation
arXiv:hep-th/0605093v2 10 May 2006
U. Saleem1, M. Siddiq23 and M. Hassan4
Department of Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan.
1Байду номын сангаас
noncommutative version of Lax representation of the noncommutative sinh-Gordon equation. In Section 4, we expand the ﬁelds perturbatively and obtain zeroth and ﬁrst order sinh-Gordon equations, the associated linear system and a set of parametric B¨ acklund transformation (BT) of the sinh-Gordon equation. It has been shown that the compatibility condition of the associated linear system and the B¨ acklund transformation (BT) is the sinh-Gordon equation at the perturbative level. Section 5, contains our conclusions.
Abstract We give a noncommutative extension of sinh-Gordon equation. We generalize a linear system and Lax representation of the sinh-Gordon equation in noncommutative space. This generalization gives a noncommutative version of the sinh-Gordon equation with extra constraints, which can be expressed as global conserved currents.

Conley index theory

Research supported in part by NSF gran6
y Research supported in part by KBN, Grant 2 P03A 029 12.
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4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7
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'(0; x) = x
2
'(t; '(s; x)) = '(t + s; x) A set S X is an invariant set for the ow ' if '(R; S ) :=
t2R
'(t; S ) = S:
Much of the qualitative theory of di erential equations involves the study of the existence and structure of invariant sets. There are at least four di culties associated with the investigation of nonlinear systems: 1. Invariant sets are global objects, and therefore, their study involves obtaining global estimates of a nonlinear problem. 2. Invariant sets posses an in nite variety of di erent structures some of which can be extremely complicated (chaotic dynamics and fractal structures). 3. The structure of invariant sets can change dramatically with respect to perturbations (bifurcation theory, normal forms, catastrophe theory); 4. The bifurcation points need not be isolated (structurally stable systems are not dense). The fact that for a xed family of di erential equations all of these problems must be considered simultaneously makes the analysis of nonlinear systems difcult. Consider for instance the simplest example of an invariant set, that of a xed point or equilibrium, i.e. a point x 2 X such that '(R; x) = x. For many di erential equations proving the existence of xed points either analytically or numerically is a nontrivial task. Degree theory is by now a standard tool for this type of problem. The degree of a map can be thought of as a function from a set of continuous maps to the integers with three important properties: 1. It is de ned in terms of a region for which there are no xed points on the boundary. 2. If the degree of a map is nonzero, then the map possesses a xed point. 3. Degree is a continuous function. To successfully apply this theory requires a few essential steps. Direct computation of the degree of a map is often quite di cult. Therefore, the rst step is to choose a continuous family of maps going from the map of interest to a simpler map for which the xed points are explicitly known. One then chooses a region containing one or more equilibria of the simple map for which the resulting degree is nonzero. The third step typically involves analysis; one must show that each function in the continuous family of maps does not posses a xed point on the boundary of the region. Having done this one observes that since the degree is integer valued and continuous, it cannot change throughout the continuous 3

Solvable scalar and spin models with near-neighbors interactions

a rXiv:h ep-th/047274v22Se p24Solvable scalar and spin models with near-neighbors interactions A.Enciso,F.Finkel,A.Gonz´a lez-L´o pez ∗,M.A.Rodr´ıguez Depto.de F´ısica Te´o rica II,Universidad Complutense,28040Madrid,SpainCalogero–Sutherland (CS)models are one of the most extensively studied types of exactly solvable and integrable quantum Hamiltonians describing a system of N particles in one dimension with long-range two-body inter-actions [1,2,3,4,5].Apart from their intrinsic mathematical relevance [6,7,8],these models arise naturally in many diﬀerent ﬁelds,such as Yang–Mills the-ories [9,10],quantum Hall liquids [11],random matrix theory [12,13,14,15],propagation of solitons [16],fractional statistics and anyons [17,18,19,20,21],etc.Several generalizations of CS models to particles with internal degrees of freedom (“spin”)have been developed over the last decade using two main approaches,namely the supersymmetric formalism [22,23,24]and the Dunkl operator method [25,26,27,28,29,30].Spin CS models are intimately connected with integrable spin chains with long-range position-dependent interactions,like the celebrated Haldane–Shastry spin chain[31,32].Indeed,when the cou-pling constant of a spin CS model tends to inﬁnity the particles“freeze”in the classical equilibrium positions of the scalar part of the potential,thus giving rise to a spin chain of Haldane–Shastry(HS)type.This mechanism,usually called the“freezing trick”,wasﬁrst used by Polychronakos[33]to construct theﬁrst integrals of the original HS spin chain by taking the large coupling con-stant limit of the corresponding integrals of the Sutherland spin model.These ideas have also been successfully applied to construct integrable and exactly solvable spin chains of HS type from other spin CS models[34,35,36,37].In a recent paper[38],Jain and Khare proposed new solvable versions of the original Calogero and Sutherland scalar models featuring near-neighbors in-teractions.These new models are closely related to the so-called short-range Dyson models[39,40]in random matrix theory,in that the square of the ground-state wavefunction of the many-body system coincides with the joint probability distribution function for eigenvalues of the corresponding short-range Dyson model.Several generalizations of the Jain and Khare models ap-peared in a subsequent paper[41],including their extension to the BC N root system and to higher dimensions.The previous publications open a number of interesting questions,such as the existence of other solvable scalar models with near-neighbors interactions,or the construction of spin models with near-neighbors interactions and their corresponding spin chains of Haldane–Shastry type.These short-range spin chains with position-dependent interactions are of particular signiﬁcance,since they occupy an intermediate position between the well-known Heisenberg chain(short-range,position-independent interac-tions)and the usual HS-type spin chains(long-range,position-dependent in-teractions).Aﬁrst step in this direction is the recent work by Deguchi and Ghosh[42],in which several spin1/2models related to the scalar models of Jain and Khare were introduced and partially solved using the supersymmet-ric formalism.These authors also pointed out how to obtain the spin chains corresponding to these models by applying Polychronakos’s freezing trick.In this letter we present three new families of solvable scalar and spin N-body models with near-neighbors interactions.By contrast with Ref.[42],our approach is based on a modiﬁcation of the Dunkl operator formalism and provides a wide range of totally explicit solutions for all values of the spin. The potentials of these spin models are given byV1=2a2 i cot(x i−x i−1)cot(x i−x i+1)+2a i sin−2(x i−x i+1)(a−S i,i+1),(1a) V2=ω2r2+ i b(b−1)(x2i−x2i−1)(x2i−x2i+1)+4a i x2i+x2i+1+ i2a (x i−x i−1)(x i−x i+1)(z2i∂i−z2i+1∂i+1)z i−z i+1−2a i z i z i+1(z i∂i−z i+1∂i+1)z i−z i+1−a i z i+z i+1(∂i−∂i+1)z i−z i+1−2a i1ﬁrst-order operatorsJ−= i∂i,J0= i z i∂i.(3)IfΦ= s f s(z)|s ,where z=(z1,...,z N)and s=(s1,...,s N),is a state totally symmetric under permutations of both the spatial and spin coordinates of each particle,then K ijΦ=S ijΦ.Hence TǫΦ=T∗ǫΦ,ǫ=1,2,3,whereT∗ǫ=Tǫ|Ki,i+1→S i,i+1.The Hamiltonian Hǫ=− i∂2x i+Vǫof each of themodels(1)can be obtained by applying a suitable gauge transformation and change of variables to a linear combinationHǫ=c T∗ǫ+c−J−+c0J0+E0.(4) More precisely,we can writeHǫ=µ· Hǫ z i=ζ(x i)·µ−1,(5)where the constants c,c−,c0,E0,the gauge factorµ,and the functionζare given in each case by1)c=4,c−=0,c0=4(1−2a),E0=2Na2,µ= i sin a(x i−x i+1),ζ(x)=e±2i x,(6a) 2)c=−4,c−=−2(2b+1),c0=4ω,E0=Nω(4a+2b+1),µ=e−ω2r2 i(x i−x i+1)a,ζ(x)=x.(6c)The key idea in our approach to the solution of the models(1)is toﬁnd an increasing sequence ofﬁnite-dimensional linear spaces H0ǫ⊂ H1ǫ⊂···invariant under the gauge Hamiltonian Hǫ.Indeed,by Eq.(5)this implies that the corresponding Hamiltonian Hǫcan be diagonalized in each of its invariant subspaces H nǫ=µ H nǫ z i=ζ(x i),n=0,1,....The operators(2)and(3)preserve the space P n of polynomials in z of total degree at most n,for all non-negative integer values of n.In our recent work on spin CS models[29,30],the operators T cs analogous to Tǫalso leave P n invariant,and in addition commute with the total symmetrizer under particle permutationsΛ.This guarantees that the corresponding operators T∗cs preserve the space of completely symmetric spin functionsΛ(P n⊗S).In the present case,however,the operators(2)do not commute withΛ,and hence it is not clear a priori whether the operators T∗ǫleave invariant anyﬁnite-dimensional space of spin functions.In fact,since J−and J0obviously commute withΛ,the results of Jain and Khare[38]forthe scalar case indicate that the operators T∗1and T∗3possess at least“trivial”invariant spaces of the form Q⊗(ΛS),where Q is aﬁnite-dimensional subspace of the space of totally symmetric polynomials in z.As it turns out,each of the operators T∗ǫpreserves a nontrivial subspace T nǫ⊂Λ(P n⊗S)for all n, namelyT n1= f(τ1,τN−1,τN)Λ|s ,g(τN−1,τN)Λ(z1|s ),q(τ1,τN)Λ(z1···z N−1|s )|f11=f N−1,N−1=g N−1,N−1=q11=0 ,(7a) T n2= f(τ1,τ2,τN)Λ|s ,g(τ1,τN)Λ(z1|s )|f22=f NN=g NN=0 ,(7b) T n3= f(τ1,τ2,τ3)Λ|s ,g(τ1,τ2,τ3)Λ(z1|s ),h(τ1,τ2)Λ(z21|s ),h(τ1,τ2)Λ(z1z2|s′ )|f33=g33=0 .(7c) The proof of this statement,which is crucial for what follows,will appear in a separate publication.In Eq.(7),τk= i1<···<i k z i1···z i kis the k-th elementary symmetric polynomial,f,g,h,and q are polynomials of total degree in z less than or equal to n,n−1,n−2,and n−N+1, respectively,and(for instance)f k=∂f/∂τk.The spin states|s ∈S are arbitrary,while|s′ denotes a state such that the sum i|s′i,i+1 is totally symmetric,where|s′ij is deﬁned byΛ z1z2|s′ = i<j z i z j|s′ij .Note that the spaces(7)are also preserved by the operator J0.Similarly,the operator J−leaves the space T n3invariant,while it preserves T n1and T n2provided that f N−1=f N=g N−1=g N=q=0and f N=g N=0,respectively.From Eqs.(4)and(6),it easily follows that the gauge Hamiltonians Hǫpreserve the spaces H nǫgiven byH n1=T n1, H n2=T n2 f N=g N=0, H n3=T n3.(8)By diagonalizing the Hamiltonian Hǫin its corresponding invariant spaces H nǫ,n=0,1,...,one can in principle construct an inﬁnite sequence of exact eigenvalues and eigenfunctions,which shall be referred to as“algebraic”in what follows.We have found in this way all the algebraic eigenvalues of the spin models(1).We also present explicit expressions for the corresponding algebraic eigenfunctions,with the only exception of the spin eigenfunctions of the model(1c)not factorizing asµf(τ1,τ2,τ3)Λ|s .In particular,we obtain all the algebraic eigenfunctions of the scalar reductions of the spin potentials(1), thus considerably extending the results of Refs.[38]and[41].It should be noted,however,that the point spectrum could possibly include additional eigenvalues and eigenfunctions which are not algebraic.We shall now discuss in more detail each of the models(1).Case1.The Hamiltonian H1commutes with the total linear momentum P=−i k∂x k,so that it admits a basis of eigenfunctions with well-deﬁned total momentum.In fact,since in this caseτl N=exp ±2i l k x k ,multiplying an eigenfunction of H1with energy E and total momentum p byτl N simply “boosts”its energy and total momentum.We shall take advantage of this fact to“normalize”the algebraic eigenfunctions to zero total momentum.It easily follows from Eq.(7a)that when this normalization is performed one obtains only aﬁnite number of eigenfunctions of H1.In the scalar case,there are exactly four eigenfunctions with zero momentum,namelyψ0=µ,ψ1,2=µ i cos sin 2(x i−X) ,ψ3=µ NaN i x i is the center of mass coordinate.Their respective energies are E0(ground state),E1,2=E0+4(2a−1+1/N),and E3=E0+8(2a+1).These are essentially the solutions found in[43].In the spin case,to each scalar eigenfunction(9)there correspond 2M+N N factorized solutions of the formΨ(0)n=ψnΛ|s ,where|s ∈S is arbitrary. There are three additional families of algebraic spin eigenfunctions with zero total momentum,given byΨ(1)1,2=µ i cos sin 2(x i−X) |s i ,Ψ(1)3=µ ain fact shared by all the models(1).In the scalar case,for each n≥2there are two algebraic eigenfunctions with energy E n,namelyψ(0)n=µLα−1n(ωr2),n=0,1,...,ψ(1)n=µ ω2 N(α+1) i x4i−αr4 Lα+3n−2(ωr2)+n(n−1)αLα−1n(ωr2) ,n=2,3,...,whereα=N(2a+b+1/2)and Lλνis a generalized Laguerre polynomial of degreeν.Note,in particular,thatψ(0)0=µhas no nodes in the conﬁguration space0<x1<···<x N,and is thus the ground state wavefunction.In the spin case,for each energy level E n we haveﬁrst of all the factorized eigenfunctions of the formΨ(k)n=ψ(k)nΛ|s ,k=0,1.In addition,for each n≥1there is a family of genuine spin eigenfunctionsΨ(2)n=µ Lα+1n−1(ωr2)Λ x21|s −α2](t),l,m=0,1,...,(10)where t=2Nr2/τ21−1,α=N(a+1/2)−3/2,β≡β(m)=1−m−N(a+1/2),P(α,β)νis a Jacobi polynomial of degreeν,and[x]denotes the integer part ofx.The second family readsψ(1)lm=µτm−31L−βl(ωr2) P(α+3,β)[m−32N2ϕlm(t) ,l,m−3=0,1,...,(11) whereϕlm is a polynomial of degree[m/2]given byϕlm=m+2α+22(t)−P(α+3,β−1)mm−1P(α+2,β−1)m3P(α+3,β)mfor even m,while for odd m we haveϕlm=2P(α+2,β−1)m−12(t)+12(t)+m+2α+22(t)−m+2α+22(t).(13)Note that,as in the previous cases,ψ(0)0=µis the ground state.The above results show that for all N≥3the scalar reduction of the Hamiltonian H3possesses two diﬀerent families of eigenfunctions of the formµL−βl(ωr2)pν(x),with pνa homogeneous polynomial of degreeν≥3,cf.Eqs.(10)and(11)–(13). This was veriﬁed only up toν=6and N≥νin Ref.[41].In the spin case,the algebraic energies are of course the same as in the scalar case.As in the previous cases,each scalar algebraic eigenfunction gives riseto 2M+N N factorized spin eigenfunctions.The computation of the remaining algebraic spin eigenfunctions,which is considerably more involved than inCases1and2,is still in progress.Even without an explicit knowledge of theeigenfunctions,for each algebraic level E n one can easily compute the number of independent states of the form(7c)of degree n.If all the eigenfunctions of this model(and of its scalar reduction)were algebraic,the previous remark would imply that the degeneracy of the levels can be explicitly found.The method of Ref.[37]could then be used to compute the partition function of the associated spin chainH3= i(ξi−ξi+1)−2S i,i+1,where(ξ1,...,ξN)is an equilibrium of the scalar potentialU3=1(x i−x i−1)(x i−x i+1)+ i1References[1] F.Calogero,J.Math.Phys.12(1971)419.[2] B.Sutherland,Phys.Rev.A4(1971)2019.[3] B.Sutherland,Phys.Rev.A5(1972)1372.[4]M.A.Olshanetsky,A.M.Perelomov,Phys.Rep.94(1983)313.[5]J.F.van Diejen,L.Vinet,eds.,Calogero–Moser–Sutherland Models,CRMSeries in Mathematical Physics,Springer-Verlag,New York,2000.[6] A.Kasman,Commun.Math.Phys.172(1995)427.[7]pointe,L.Vinet,Commun.Math.Phys.178(1996)425.[8]T.H.Baker,P.J.Forrester,Commun.Math.Phys.188(1997)175.[9] A.Gorsky,N.Nekrasov,Nucl.Phys.B414(1994)213.[10]E.D’Hoker,D.H.Phong,Nucl.Phys.B513(1998)405.[11]H.Azuma,S.Iso,Phys.Lett.B331(1994)107.[12]B.Sutherland,J.Math.Phys.12(1971)246.[13]B.Sutherland,J.Math.Phys.12(1971)251.[14]B.D.Simons,P.A.Lee,B.L.Altshuler,Phys.Rev.Lett.72(1994)64.[15]N.Taniguchi,B.S.Shastry,B.L.Altshuler,Phys.Rev.Lett.75(1995)3724.[16]A.P.Polychronakos,Phys.Rev.Lett.74(1995)5153.[17]A.P.Polychronakos,Nucl.Phys.B324(1989)597.[18]Z.N.C.Ha,Phys.Rev.Lett.73(1994)1574.[19]Z.N.C.Ha,Nucl.Phys.B435(1995)604.[20]A.L.Carey,ngmann,Commun.Math.Phys.201(1999)1.[21]P.Shukla,Phys.Rev.Lett.87(2001)194102(4).[22]L.Brink,A.Turbiner,N.Wyllard,J.Math.Phys.39(1998)1285.[23]P.Desrosiers,pointe,P.Mathieu,Nucl.Phys.B606(2001)547.[24]P.K.Ghosh,Nucl.Phys.B595(2001)519.[25]C.F.Dunkl,Trans.Am.Math.Soc.311(1989)167.[26]A.P.Polychronakos,Phys.Rev.Lett.69(1992)703.[27]T.Yamamoto,Phys.Lett.A208(1995)293.[28]B.Basu-Mallick,Nucl.Phys.B482(1996)713.[29]F.Finkel,D.G´o mez-Ullate,A.Gonz´a lez-L´o pez,M.A.Rodr´ıguez,R.Zhdanov,Commun.Math.Phys.221(2001)477.[30]F.Finkel,D.G´o mez-Ullate,A.Gonz´a lez-L´o pez,M.A.Rodr´ıguez,R.Zhdanov,Nucl.Phys.B613(2001)472.[31]F.D.M.Haldane,Phys.Rev.Lett.60(1988)635.[32]B.S.Shastry,Phys.Rev.Lett.60(1988)639.[33]A.P.Polychronakos,Phys.Rev.Lett.70(1993)2329.[34]A.P.Polychronakos,Nucl.Phys.B419(1994)553.[35]D.Bernard,V.Pasquier,D.Serban,Europhys.Lett.30(1995)301.[36]T.Yamamoto,O.Tsuchiya,J.Phys.A29(1996)3977.[37]A.Enciso,F.Finkel,A.Gonz´a lez-L´o pez,M.A.Rodr´ıguez,hep-th/0406054.[38]S.R.Jain,A.Khare,Phys.Lett.A262(1999)35.[39]B.Gr´e maud,S.R.Jain,J.Phys.A31(1998)L637.[40]E.B.Bogomolny,U.Gerland,C.Schmit,Phys.Rev.E59(1999)R1315.[41]G.Auberson,S.R.Jain,A.Khare,J.Phys.A34(2001)695.[42]T.Deguchi,P.K.Ghosh,J.Phys.Soc.Jap.70(2001)3225.[43]M.Ezung,N.Gurappa,A.Khare,P.K.Panigrahi,cond-mat/0007005.。

Tzitz'eica transformation is a dressing action

4
ERXIAO WANG
and is essentially determined by them. Take the exterior diﬀerentiation of (2.2) to get
j
( dhij + hij ω3,3 − hik ωkj − hkj ωki ) ∧ ω j = 0,
´ TZITZEICA TRANSFORMATION IS A DRESSING ACTION
3
2. Indefinite affine sphere and its Lax representation Classical aﬃne diﬀerential geometry studies the properties of surfaces in R3 invariant under the (equi-)aﬃne transformations x → Ax + v , where A ∈ SL(3, R) and x, v ∈ R3 . There are three fundamental aﬃne invariants: the aﬃne (or Blaschke) metric, the Fubini-Pick cubic form, and the third fundamental form (or the aﬃne shape operator). These invariants satisfy certain compatibility equations and the Fundamental Theorem states that they then determine a surface uniquely up to aﬃne transformations. Let us ﬁrst review the deﬁnitions of these invariants (for more details see, e.g., [2], [13], [18]). The reader may also refer to [2], [3], [4] or [17] for an elementary description of aﬃne spheres. Let X : M ֒→ R3 be an immersed surface with non-degenerate second fundamental form. Let E = (e1 , e2 , e3 ) be a local SL(3, R)-frame on M such that e1 , e2 are tangent to M , and e3 is transversal to M . Let ω1 , ω2 denote the dual coframe of e1 , e2 , i.e., Let (ωAB ) denote the sl(3, R)-valued 1-form E −1 dE , i.e.,

Analyzing the CRF Java Memory Model

Analyzing the CRF Java Memory Model Yue Yang,Ganesh Gopalakrishnan,and Gary Lindstrom School of Computing,University of Utah{yyang|ganesh|gary}@AbstractThe current Java Memory Model[1]isﬂawed and has many unintended implications[2].It is a very chal-lenging task to design or understand a memory model at the programming language level.As multithreaded programming becomes increasingly popular in Java and hardware memory architectures become more aggres-sively parallel,it is of signiﬁcant importance to provide a framework for formally analyzing the Java Memory Model.The Murϕveriﬁcation system is applied to study the Commit/Reconcile/Fence(CRF)memory model[3], one of the proposed thread semantics to replace the present Java Memory Model.The CRF proposal is for-mally speciﬁed using the Murϕdescription language.A suite of test programs is designed to reveal interest-ing properties of the model.The results demonstrate the feasibility of applying model checking techniques to language level memory model speciﬁcations.Not only can it help the designers to debug their designs,it also provides a formal mechanism for Java programmers to understand the subtleties of the Java Memory Model. 1IntroductionJava is theﬁrst popular programming language that integrates thread support at the language level.As a type-safe and robust language,Java needs a solid memory model to specify how threads interact through shared memory.Unfortunately,the current Java Mem-ory Model as given in Chapter17of the Java Language Speciﬁcation[1],has manyﬂaws.The present model has been summarized by Pugh[2]as both being too weak and too strong.It is too weak because safety guarantees can be violated and no semantics are given forﬁnalﬁelds.On the other hand,the constraints of the current Java Memory Model are too strong in the sense they prohibit common compiler optimiza-tion techniques and introduce many unintended impli-cations.Toﬁx the problems,the Java Memory Model is cur-rently under review[4]and will be replaced in thefuture.Two replacement semantics for Java threadshave been proposed.Manson and Pugh[5]use a no-tation based on multimap to update and record his-tories of Java memory operations.Maessen,Arvind,and Shen[3]use the Commit/Reconcile/Fence(CRF)[6]framework to provide an operational semantics.Al-though notations are very diﬀerent,the two proposalshave similar design goals.They both present a generalguideline and a detailed speciﬁcation.However,nei-ther of them has applied formal analysis using modelchecking to ensure that the low level semantics actuallyachieve the projected guidelines.While formal veriﬁcation techniques have achievedsome success in verifying architecture level sharedmemory systems and cache coherence protocols,a lan-guage level memory model has not been formally ana-lyzed to date.In this paper,we present a formal frame-work for analyzing the CRF Java Memory Model.Wechoose the CRF notation because its precise rewriterules can be naturally transformed into more rigorousmathematical rules.Our method is based on the exe-cutable speciﬁcation idea developed by Dill,Park,andNowatzyk[7].The CRF model is formally speciﬁed inMurϕ[8],a description language as well as a veriﬁ-cation system.This forms the veriﬁcation engine.Asuite of test cases is executed with this engine to re-veal pivotal properties of the model.This mechanismenables one to exhaustively exercise a language levelmemory model on a particular program.Our Murϕveriﬁcation program for the CRF Java Memory Modelis available at[9].In the next section,we review the related work inmemory model veriﬁcations.It is followed by a dis-cussion of some of the problems of the current JavaMemory Model.The CRF model,upon which ourexecutable model is based,is described in Section4.Section5outlines how to formally specify the memorymodel and construct test programs.In Section6,ourpreliminary results are presented.We conclude by dis-cussing the limitations of our approach and proposingsome areas of future work that could help advance the software model checking techniques.2Related WorkA memory model describes how a memory sys-tem behaves on memory operations such as reads and writes.Much previous research has concentrated on the processor level memory models.One of the strongest memory models for multiprocessor memory systems is Sequential Consistency[10].Many weaker memory models[11]have been proposed to enable opti-mizations.One of them is Coherence[12].Informally, Coherence requires all memory operations at a given memory location to exhibit a total order that respects the program order of memory operations at that loca-tion in each processor.Another weak memory model, PRAM[13],requires the execution sequences contain-ing all instructions from a given processor and all write instructions from all other processors to exhibit a total order which also respects the program orders in each processor.With our veriﬁcation system,we have for-mally compared the CRF model with some of these conventional models.Several model checking tools,such as VeriSoft[14], Java PathFinder[15][16],JCAT[17],Bandera[18], and the SAL Model Checker[19],have been devel-oped for verifying multithreaded Java programs.These tools,however,are intended for real Java applica-tions and they do not speciﬁcally address Java Memory Model issues.The memory operations in these applica-tions are interpreted using intuitive behaviors instead of a strict memory model.Formal veriﬁcation techniques have also been ap-plied for verifying processor level memory design pro-tocols.The idea of executable speciﬁcation of abstract memory models is proposed in[7].Using this method-ology,one can specify processor memory models using Murϕ.Such a description provides a precise speciﬁca-tion of the machine architecture,both for implementors and programmers.The model can be executed with test programs to verify certain properties.Although this approach has been applied to some processor ar-chitectures such as the Relaxed Memory Order(RMO [20])[21],it has not been explored for software level memory models which have to deal with more com-plicated issues.In our work,we have followed similar techniques but applied them to the domain of Java Memory Model,with encouraging initial results.use Figure 1.Memory hierarchy in the currentJava Memory ModelInitially,p=nullThread1Thread2Synchronized(this){if(p!=null){p=new Point(1,2);a=p.x;}}can result in a=0Figure2.Premature release of object refer-ence3Problems with the Existing Java Memory ModelThe current Java Memory Model[1]uses a mem-ory hierarchy illustrated in Figure1.It deﬁnes pos-sible actions that may occur at each layer.Besidesthe actions shown in Figure1,two additional actions, lock and unlock,are cooperatively used by both the threads and the memory system to achieve synchro-nization.The speciﬁcation further outlines sets of rulesto impose constraints to the actions.There are several shortcomings of this speciﬁcation. The style of the constraints is to prescribe the program behaviors,which is very hard to be transformed to for-mal notations.In addition,the multiple layers used between threads and memory introduce unnecessary complexities.As a result,the model has some hiddenﬂaws and is very hard to understand.Under this model,an object reference is allowed to escape prematurely during the object construction if there exists a race condition.Takeﬁgure2as an ex-ample,when Thread2fetches the objectﬁeld without locking,it might obtain uninitialized data in the state-ment a=p.x.Consequently,ﬁnalﬁelds are not trulyﬁnal and immutable objects are not truly immutable since other threads might see their default values.Because of this loophole,some popular synchro-nization idioms are unsafe.For example,the double-checked locking [22]algorithm is often used for creating a Singleton object,which is only constructed once.As shown in Figure 3,this programming pattern only uses locking when the object is created to avoid unnecessary synchronization overhead.Unfortunately,because the assignment to helper may be visible to other threads be-fore the constructor completes,this algorithm is broken under the existing Java Memory Model.Another problem of the current model is that it pro-hibits important compiler optimization techniques such as fetch elimination.In [23],a proof is given to show that the Java Memory Model requires Coherence .Con-sider ﬁgure 4,the instructions in Thread 1eventually operate on the same variable due to aliasing.There-fore,the statement k =p.x can not be replaced by k =i by the compiler because Coherence requires a total or-der for operations on the same variable.Many existing JVM implementations violate this requirement.Compared with a processor level memory model,the Java Memory Model has many new challenges.First,it must maintain safety guarantees which can not be com-promised even under race conditions.Second,the Java Memory Model is mapped to many diﬀerent proces-sor architectures,some of which have very weak mem-ory models.For the Java Memory Model designers,it is a nontrivial task to identify the optimal require-ment that can be supported by all major architectures in a safe and eﬃcient manner.For the JVM imple-menters,the compliance with the Java Memory Model need to be ensured.Third,the Java Memory Model needs to address the semantics of many programming language level constructs,such as ﬁnal ﬁelds,volatile ﬁelds,constructors,and ﬁnalizers.Finally,run-time ef-fects such as aliasing can introduce more implications of the language level speciﬁcation.Because of these issues,a mechanism for formally reasoning about theclass foo {private static Helper helper =null;public static Helperif (helper ==if(helper helper }}return helper;}}Figure 3.Double-Checked Locking IdiomInitially,p.x =0Thread 1Thread 2i =p.x;p.x =1;q =p;j =q.x;k =p.x;Figure 4.Coherence prohibits fetch elimina-tionJava Memory Model would be highly valuable.4The CRF Java Memory ModelThe CRF framework is designed to describe instruc-tion reordering and data replication (caching)in a memory model.It captures the precise semantics of memory operations with or without race conditions.The CRF memory hierarchy has three layers,as shown in Figure 5.The elimination of the buﬀer layer in Fig-ure 1simpliﬁes the design.Each thread has a local semantic cache.Each cache entry maintains a tag with a state of either Clean,Dirty,Locked,or Frozen .We present the outline of the CRF Java Memory Model in this section.The full deﬁnition can be found in [3].4.1CRF InstructionsTo provide enough ﬁne granularity to specify Java memory activities,eight CRF instructions are used.These CRF instructions are summarized as follows:Figure 5.Memory hierarchy in the CRF modelOperation Regular Final VolatileStore a,v;Storel a,v;Storel a,v;Fence rw∗V R,a;Commit a;Commit a;Fence ww∗V R,a;Freeze a;Storel a,v;Commit a;v=Load a;Reconcile a;Reconcile a;Fence wr∗V,a;v=Loadl a;v=Loadl a;Reconcile a;Freeze a;v=Loadl a;Fence rr a,∗V R;Fence wr a,∗R;Table1.CRF translations for Java loads andstoresCRF Instruction DescriptionLoadl Load value from local cacheStorel Store value to local cacheCommit Ensure write back from cache to memory Reconcile Ensure update from memory to cache Fence Ensure ordering restrictionsLock Acquire a lockUnlock Release a lockFreeze Complete aﬁnalﬁeld operation4.2Translating Java into CRFJava memory operations are categorized as fourgroups depending on the type of variables they actupon:regular,volatile,ﬁnal,and monitor.These oper-ations are translated to theﬁne-grained CRF instruc-tions according to Table1and Table2.For example,the source level Store instruction for a regular variableis decomposed into a Storel(store-local)followed by a Commit.The fence operation takes two parameters as the pre-address and post-address and speciﬁes an ordering con-straint between them.It can also use a wildcard∗to denote all addresses.For example,Fence rw∗V R,a;re-quires that all previous Loadl instructions on all volatileand regularﬁelds must be completed before allowingStorel to a.The EndCon instruction indicates the completion ofa constructor.It is used to solve the reference escape problem,which will be further discussed in Section6.2.4.3CRF Reordering Rules and RewriteRulesIn the CRF model,the semantics of the Java Mem-ory Model is speciﬁed collectively from two aspects:the Operation Translation Enter l;Fence ww∗L,l;Lock l;Fence wr l,∗V R;Fence ww∗V R,l; Exit l;Fence ww∗V R,l;Fence rw∗V R,l;Unlock l; EndCon;Fence ww∗,∗V R;Table2.CRF translations for Java synchro-nization operationsreordering rules and rewrite rules.The former dictates how the CRF instructions can be reordered locally and the latter deﬁnes the eﬀects of memory activities.A reordering table[3,Figure6]is provided to spec-ify ordering rules between any CRF instructions.It is designed to allow maximum reorderingﬂexibility. For example,memory access instructions can be re-ordered if they access diﬀerent addresses or if they are both Loadl instructions.Memory rendezvous instruc-tions(Commit and Reconcile)can always be swapped. Memory fence instructions can always be swapped.All locking operations can be ordered freely unless Fence instructions are applied.As long as the reordering ta-ble is followed,any instructions of a given thread can be arbitrarily reordered.In addition to the ordering table,a set of rewrite rules[3,Figure4and Figure5]is speciﬁed.These rewrite rules include local rules,which deﬁne the ef-fects of executing the CRF instructions,and back-ground rules,which maintain the synchronization be-tween cache and shared memory.5Modeling the CRF Memory Model in MurϕTo better understand the CRF model,we have de-veloped a formal executable Java Memory Model in Murϕbased on the CRF speciﬁcation.Our veriﬁca-tion system consists of two parts.Theﬁrst part is the formal speciﬁcation of the CRF Java Memory Model using Murϕrules,which provides a“black box”that deﬁnes the thread semantics.The second part com-prises a suite of test cases.Each test program deﬁnes the corresponding Murϕinitial state and invariants.5.1Data StructuresThe global state of our system includes information about all threads and the shared memory.A variable can be declared as one of the following:REGULAR,VOLATILE,FINAL ,or LOCK ,which is referred to as the variable type in this paper.Each thread maintains its own local cache and an instruction buﬀer.There is a tag associated with each cache entry,which can be UN-MAPPED,CLEAN,DIRTY,FROZEN ,or LOCKED .The instruction buﬀer records all CRF instructions in pro-gram order.Each instruction uses a completion ﬂag to indicate whether it has been completed.It also main-tains a result ﬁeld to store the value obtained from a read operation.The scale of our program can be ad-justed for diﬀerent tests.For example,key parameters can be declared as follows:THREAD_NUM:2;--maximum number of threadsADDRESS_NUM:2;--maximum number of different addressesVALUE_NUM:2;--data rangeLABEL_NUM:6;--maximum number of instructionsLOCK_NUM:1;--maximum numberoflocks5.2Formal Speciﬁcation of the Java Mem-ory ModelOur Mur ϕprogram models all the interleaving be-haviors of a set of running threads.These threads can be executed concurrently on multiple processors.The nature of the processor architecture is abstracted out and thread semantics is only governed by the high level memory model.The CRF model uses rewrite rules to describe the ef-fect of completing any CRF instruction or background action.These rewrite rules are implemented using Mur ϕrules.For the local rules,an eligible instruc-tion from any thread can be nondeterministically cho-sen and executed.A background rule may also be ﬁred when its condition is satisﬁed.The Mur ϕveriﬁcation system automatically enumerates the state space and examines all the possible interleaving between these rules.In addition to the rewrite rules,the CRF ordering constraints are also speciﬁed.A Mur ϕfunction Or-dered(t:Thread;i,j:InstructionIndex)is implemented following the CRF reordering table to check if there ex-ists any ordering restriction between instruction i and j .It is used as one of the conditions for local rules to ensure an instruction can be executed.An instruction is only eligible to be selected when it can bypass all previous pending instructions.Although the process of transforming CRF to Mur ϕwas relatively straightforward,implementing the oper-ational semantics with strict mathematical rules forcedus to think harder about the CRF rules,which in turn helped us clarify many details.5.3Constructing Test ProgramsA test case consists of small concurrent programs comprised of two or more threads.The programs are abstracted to their memory operations.They accept source level instructions including Load,Store,Enter-Monitor,ExitMonitor ,and EndCon .The type of each variable is initialized in Mur ϕstartstate.A proce-dure AddInstruction is used to translate the high level instructions into the ﬁne-grained CRF instructions,which are later added to the instruction buﬀer of the corresponding thread.Our system can detect deadlocks and invariant vio-lations.To examine test results,two techniques can be applied.The ﬁrst one uses Mur ϕinvariants to specify that a particular scenario can never occur.If it does occur,a violation trace can be generated to help understand the cause.The second technique uses a special “thread completion”rule,which is trig-gered only when all threads are completed,to output all possible ﬁnal results.We use a conﬁguration ﬂag SHOW ALL RESULTS to select this output mode.Another conﬁguration ﬂag ALIASING is used to in-dicate if the variables might be aliased.This enables one to easily experiment with both scenarios.In the CRF reordering table,some ordering rules depend on if the two addresses are the same and every single reordering decision may be made dynamically.Since static analysis at the language level does not have dis-crimination power comparable to the architecture level where an address is always concrete,aliasing analysis could be hard and sometimes people have to make con-servative assumptions.As a veriﬁcation tool for high level reasoning,our system relies on the users to do the aliasing analysis themselves but enables them to know what to expect in either case.A third ﬂag DATA DEPENDENCE is used to spec-ify an additional ordering restriction during object dereference.When accessing an object ﬁeld,the ob-ject reference must be obtained before fetching the content of the ﬁeld.Since we model the reference and ﬁeld as separate global variables,we turn on the ﬂag DATA DEPENDENCE when simulating the sce-nario of object dereference.We add the following re-strictions to the original CRF reordering table when DATA DEPENDENCE is set:the CRF instruction Loadl a can not be bypassed by Reconcile b or Storel b if a and b has the reference/ﬁeld type of data depen-dency.Initially,A=0Thread1Thread2A=1;X=A;A=2;Y=A;Finally,can it result in X=2&Y=1?Figure6.Test program revealing violation ofCoherence6Analysis of the CRF ModelBy exercising our Java Memory Model engine with test programs,we are able to gain insights about the CRF memory model.To demonstrate the feasibility and usability of this approach,we present our prelimi-nary study of the ordering rules,the constructor prop-erties,and synchronization idioms.6.1Ordering RulesOne can better understand a memory model by com-paring it to other well known memory models.We have designed some test programs to reveal ordering prop-erties of the CRF Java Memory Model.6.1.1Comparison with CoherenceFigure6illustrates a test program that reveals viola-tion of Coherence.When we declare A as a regular variable and exhaustively execute this program with our memory model engine,a trace leading to the con-tradictingﬁnal result is discovered.The violation is due to the fact that a Loadl instruction can be freely re-ordered with another Loadl instruction,even if they use the same address.Therefore,this simple test program formally proves that the CRF speciﬁcation does not enforce Coherence.Since Coherence is strictly weaker than Sequential Consistency,it can be concluded im-mediately that the CRF Java Memory Model does not enforce Sequential Consistency either.6.1.2Comparison with PRAMFigure7is a program to test PRAM ordering proper-ties.Our veriﬁcation system reveals that the contra-dicting result exists for regular variables.A study of the violation trace indicates it is also because the Loadl instructions can be reordered.The result proves that PRAM is not obeyed by the CRF Java Memory Model.Initially,A=B=0Thread1Thread2A=1;X=B;B=1;Y=A;Finally,can it result in X=1&Y=0?Figure7.Test program revealing violation ofPRAMInitially,A=B=0Thread1Thread2X=A;Y=B;B=1;A=1;can it result in X=1&Y=1?Figure8.Test program for prescient store6.1.3Prescient StoreIn some circumstances,the CRF model can relax the read/write ordering such that a write may be per-formed before a read completes.Figure8is used to study such eﬀects.The results turn out to depend on the setting of ALIASING.It can be concluded that the CRF model only allows the prescient store optimiza-tion if A and B can never be aliased.6.1.4Write AtomicityFigure9veriﬁes write atomicity,which has passed suc-cessfully.Because the CRF model uses the shared memory as the rendezvous between threads and caches, it is impossible to have X=2and Y=1in Figure9. This example shows a main diﬀerence between the CRF model and some of the more relaxed memory models. For example,Pugh’s model[5]would allow X=2and Y=1.Initially,A=B=0Thread1Thread2A=1;A=2;X=A;Y=A;Finally,can it result in X=2&Y=1?Figure9.Test program verifying write atom-icityInitially,A=B=0Thread1Thread2Thread3A=1;X=A;Y=BB=1;Z=AFinally,can it result in X=1&Y=1?&Z=0?Figure10.Test program for CausalityInitially,A=B=0Thread1Thread2B=1;X=A;EndCon;Y=B;A=1;Invariant:it should never result in X=1&Y=0Figure11.Test program for EndCon6.1.5CausalityCausality requires thread-local orderings to be transi-tive,which might be expected by programmers intu-itively.Consider Figure10,if A and B are aliased at run time,the CRF model prohibits the reordering be-tween the two instructions in Thread2.If X=1and Y =1,Z must return1if Thread3respects the ordering of Thread2.But our veriﬁcation system reveals that even when ALIASING is set,the program can still re-sult in X=1,Y=1and Z=0.Therefore,the CRF Java Memory Model does not enforce causality.The above examples only demonstrated tests for reg-ular variables.By simply changing the type declaration of the variables,the same approach can be applied to analyze properties of diﬀerent variable types,such as ﬁnal and volatile.6.2Constructor PropertiesOne of the problems of the current Java Memory Model is the premature release of an object reference in the constructor.To prevent such an early release,the EndCon instruction is introduced in CRF to denote the completion of a constructor.Figure11shows the test program we designed to study the eﬀectiveness of this approach.We use A to represent the object reference and B to represent the objectﬁeld.Thread1simulates the essential activities in the constructor.The reference is assigned only after theﬁeld is initialized and EndCon is issued.Thread 2simulates a race situation by attempting to access the objectﬁeld without synchronization.The invariant ensures the premature release of the object reference can never occur.There is no violation found by this test.However, several subtle issues have been discovered:•If theﬂag DATA DEPENDENCE is not turned on,the test would fail.This implies the order-ing restriction due to data dependence between the reference andﬁeld is absolutely required.Al-though seemingly a reasonable assumption,this constraint is not enforced by all architectures.Un-der a very aggressive memory model,the content pointed by a reference might come out of a stale cache line.In eﬀect,this breaks the data de-pendence restriction.Therefore,we propose to strengthen the CRF reordering table by explicitly adding the constraints due to this kind of data dependency,as mentioned in Section5.3.•If we move the instruction A=1ahead of the End-Con instruction,the test would fail.Therefore, the EndCon operation must be inserted between initializing the objectﬁeld and assigning the ob-ject reference.In the original CRF speciﬁcation,it is not clearly stated when the EndCon instruction should be issued.•If the above items are taken care of,the test suc-ceeds regardless of the variable types.The same level of restriction is imposed to bothﬁnal and non-ﬁnalﬁelds.To make sure the requirement of EndCon is correctly implemented in an actual JVM implementation,additional memory barriers are required on some architectures with aggressive memory models,which will result in a performance penalty.Pugh’s proposal[5]is diﬀerent in this re-gard as it only impose safety constraints toﬁnal ﬁelds.6.3Synchronization IdiomsAs multithreaded programming becomes increas-ingly popular,many synchronization idioms have been established.However,an algorithm working under one memory model is not necessarily safe to use under an-other memory model.As previously illustrated,the Double-Checked Locking idiom is a good example.Our veriﬁcation system can be used for testing the validity of a synchronization idiom.The idiom needs to be abstracted to a test program.Violations can be detected by any contradicting results.To demonstrate this strategy,we abstract the Double-Checked Locking algorithm in Figure3to a test program shown in Fig-ure12.Again,A represents the object reference and B represents the objectﬁeld.We use Thread1to sim-ulate the thread that grabs the lock and creates the Helper class.After the thread enters the monitor,itInitially,A =B =0Thread 1Thread 2EnterMonitor;Y =A ;X =A ;Z =B ;B =1;EndCon ;A =1;ExitMonitor ;Invariant:it should never result in X =0&Y =1&Z =0Figure 12.Test for Double-Checked Locking id-iomﬁrst double-checks if the reference is still null .Then it completes the constructor before exiting the moni-tor.Thread 2simulates another thread that obtains a non-null helper by calling the get method.When it reads the object ﬁeld,it should never see uninitialized content.This test has passed successfully.Although this is not a complete proof of validity,it does help Java pro-grammers to gain some level of conﬁdence in the pro-gramming pattern.And we expect many broken idioms can be discovered this way.It should be pointed out that the EndCon instruction plays a crucial role to make the Double-Checked Locking algorithm work.In a closely related version,the ob-ject reference is obtained from a pre-allocated object instead of being created from scratch in the“double-checked”slot.To simulate this programming pattern,EndCon should be taken out from Thread 1in Figure 12.This would cause the failure of our invariant.6.4Performance StatisticsThe above test programs are executed on a Dell PC with a 366MHz Pentium processor and 128MB RAM running Windows 2000.The performance statistics are listed below:Test State Size (Bytes)Total StatesRules Fired Time (Sec)Coherence 14887524740.21PRAM 152294296760.57Prescient Store 152283491420.59(without aliasing)Prescient Store 152114834840.23(with aliasing)Write Atomicity 14893730370.22Causality 2283152610544712.06EndCon 188158446900.36Double-Checked500699224790 3.69Locking7ConclusionSpecifying the Java Memory Model with a Model Checking framework has several advantages over the original CRF speciﬁcation.First,the speciﬁcation is executable.It can beneﬁt the model designers to gain extra level of conﬁdence in their designs.It is also ap-pealing to programmers,JVM implementers,and com-piler writers because it can help them better under-stand the subtleties of the model.The beauty of this mechanism is that the memory model can be provided as a “black box”.The user of the veriﬁcation system is not necessarily required to understand all the details of the memory model.Second,the Model Checking technique enables one to exhaustively exercise simple programs.This can reveal some corner cases which would be very hard to ﬁnd through traditional simu-lation techniques.Finally,the speciﬁcation in Mur ϕis more rigorous,which further eliminates any ambi-guities.The process of formally specifying the model forces one to think harder about the speciﬁcation.Our experience of translating CRF to Mur ϕhas helped us clarify many details about the original design.On the other hand,it is inevitable that the Mur ϕprogram comes with some internal data structures and implementation details.Without a higher level descrip-tion,this speciﬁcation alone might be hard to follow.It would be a desirable feature to have a common de-sign interface for any executable memory model with all the implementation details encapsulated.Our approach also has some limitations.Based on the Model Checking techniques,it is exposed to the state explosion problem.We have to limit our test program to a small scale.A test with four variables and eight instructions per thread has run out of mem-ory on our 128MB PC.Even though our examples demonstrated how simple tests can help people under-stand complex memory models,state explosion prob-lem nonetheless limits our ability to verify bigger com-mercial Java code.Because our system is bounded,it is not designed for guarantee of correctness.Rather,it provides a debugging tool for understanding the un-derlying memorymodel.Some “advanced”features,such as thread creation,termination,and interruption,are not discussed in de-tail in the CRF speciﬁcation.Pugh has suggested that hidden volatile ﬁelds can be used to describe these kinds of thread communication semantics.This strat-egy can be integrated to our veriﬁer to model these scenarios.As an on-going process for studying the Java Mem-ory Model,we are continuing to develop more compre-hensive test programs to cover more interesting prop-erties.A well designed suite of test programs will be a。

Algebraic spin liquid as the mother of many competing orders

spinons is natural, they do not behave as free quasiparticles even at asymptotically low energy. Furthermore, the dynamic critical exponent z = 1. Alternatively, the longdistance, low-energy properties are controlled by an interacting, conformally invariant ﬁxed point. Such states were dubbed algebraic spin liquids (ASL) in Ref. 7. Here we show that, remarkably, several competing orders are uniﬁed within the sF state by an emergent SU (4) symmetry, and all have the same slowly-varying long-distance correlations. Two of the competing orders are simply the the N´ eel vector, and the order parameter for the columnar and box valence-bond solid (VBS) states – cartoon pictures of these orders are shown in Fig. 1. To be precise, con-
Algebraic spin liquid as the mother of many competing orders

advances in math的特征因子 -回复

advances in math的特征因子-回复Advances in Math: An Exploration of Key CharacteristicsIntroduction:Mathematics is a fundamental discipline that has shaped our understanding of the world for centuries. Over the years, numerous advancements in math have revolutionized various fields such as engineering, physics, computer science, and finance. In this article, we will explore the key characteristics of advances in math and discuss how they have contributed to our understanding of the world.1. Rigor:One of the key characteristics of advances in math is rigor. Mathematics is built on logical reasoning and precise definitions. Advancements in math involve developing rigorous proofs, theorems, and axioms that are consistent and reliable. This rigor ensures that mathematical concepts are solidly grounded and can be applied universally. For example, advances in calculus made by Isaac Newton and Gottfried Leibniz introduced rigorous methods for calculating rates of change and led to breakthroughs in physics and engineering.2. Abstraction:Abstraction is another crucial characteristic of advances in math. It involves removing unnecessary details and focusing on the essential properties of mathematical objects. By doing so, mathematicians create general theories that apply to a wide range of situations. For instance, the discovery of abstract algebra allowed mathematicians to study the properties of groups, rings, and fields, which have applications not only in mathematics but also in cryptography and coding theory.3. Generalization:Advances in math often involve generalizing existing theories to encompass broader situations. This characteristic allows mathematicians to build upon previous work and extend their understanding of mathematical concepts. Generalization is crucial in areas such as number theory, where discoveries made about prime numbers have been generalized to more complex structures, leading to breakthroughs in cryptography and data encryption.4. Computational Efficiency:As technology advanced, so did the need for computationallyefficient mathematical algorithms. Advances in math often focus on developing algorithms that can solve complex problems in less time. For example, the Fast Fourier Transform (FFT) revolutionized signal processing and significantly improved the efficiency of data compression and image processing techniques.5. Interdisciplinary Connections:Advances in math often transcend disciplinary boundaries, leading to fruitful connections with other fields. For example, the application of game theory in economics and social sciences has provided insights into strategic decision-making. Similarly, the use of mathematical models in epidemiology has helped understand the spread of diseases and develop effective prevention strategies.6. Visualization:Visualization plays a crucial role in math, allowing mathematicians to gain insights, test hypotheses, and communicate their findings effectively. Advances in math have often been accompanied by the development of visual tools such as graphs, charts, and diagrams to represent complex mathematical concepts. The use of visualizations in geometry, topology, and graph theory has revolutionized these fields and helped mathematicians makesignificant discoveries.Conclusion:Advances in math are characterized by rigor, abstraction, generalization, computational efficiency, interdisciplinary connections, and visualization. These key characteristics have played a crucial role in shaping our understanding of the world and have led to immense progress in various fields. As technology continues to advance, it is likely that further advancements in math will continue to drive scientific, technological, and societal progress.。

专八英语阅读

英语专业八级考试TEM-8阅读理解练习册（1）(英语专业2012级)UNIT 1Text AEvery minute of every day, what ecologist生态学家James Carlton calls a global ―conveyor belt‖, redistributes ocean organisms生物.It’s planetwide biological disruption生物的破坏that scientists have barely begun to understand.Dr. Carlton —an oceanographer at Williams College in Williamstown,Mass.—explains that, at any given moment, ―There are several thousand marine species traveling… in the ballast water of ships.‖ These creatures move from coastal waters where they fit into the local web of life to places where some of them could tear that web apart. This is the larger dimension of the infamous无耻的，邪恶的invasion of fish-destroying, pipe-clogging zebra mussels有斑马纹的贻贝.Such voracious贪婪的invaders at least make their presence known. What concerns Carlton and his fellow marine ecologists is the lack of knowledge about the hundreds of alien invaders that quietly enter coastal waters around the world every day. Many of them probably just die out. Some benignly亲切地，仁慈地—or even beneficially — join the local scene. But some will make trouble.In one sense, this is an old story. Organisms have ridden ships for centuries. They have clung to hulls and come along with cargo. What’s new is the scale and speed of the migrations made possible by the massive volume of ship-ballast water压载水— taken in to provide ship stability—continuously moving around the world…Ships load up with ballast water and its inhabitants in coastal waters of one port and dump the ballast in another port that may be thousands of kilometers away. A single load can run to hundreds of gallons. Some larger ships take on as much as 40 million gallons. The creatures that come along tend to be in their larva free-floating stage. When discharged排出in alien waters they can mature into crabs, jellyfish水母, slugs鼻涕虫，蛞蝓, and many other forms.Since the problem involves coastal species, simply banning ballast dumps in coastal waters would, in theory, solve it. Coastal organisms in ballast water that is flushed into midocean would not survive. Such a ban has worked for North American Inland Waterway. But it would be hard to enforce it worldwide. Heating ballast water or straining it should also halt the species spread. But before any such worldwide regulations were imposed, scientists would need a clearer view of what is going on.The continuous shuffling洗牌of marine organisms has changed the biology of the sea on a global scale. It can have devastating effects as in the case of the American comb jellyfish that recently invaded the Black Sea. It has destroyed that sea’s anchovy鳀鱼fishery by eating anchovy eggs. It may soon spread to western and northern European waters.The maritime nations that created the biological ―conveyor belt‖ should support a coordinated international effort to find out what is going on and what should be done about it. (456 words)1.According to Dr. Carlton, ocean organism‟s are_______.A.being moved to new environmentsB.destroying the planetC.succumbing to the zebra musselD.developing alien characteristics2.Oceanographers海洋学家are concerned because_________.A.their knowledge of this phenomenon is limitedB.they believe the oceans are dyingC.they fear an invasion from outer-spaceD.they have identified thousands of alien webs3.According to marine ecologists, transplanted marinespecies____________.A.may upset the ecosystems of coastal watersB.are all compatible with one anotherC.can only survive in their home watersD.sometimes disrupt shipping lanes4.The identified cause of the problem is_______.A.the rapidity with which larvae matureB. a common practice of the shipping industryC. a centuries old speciesD.the world wide movement of ocean currents5.The article suggests that a solution to the problem__________.A.is unlikely to be identifiedB.must precede further researchC.is hypothetically假设地，假想地easyD.will limit global shippingText BNew …Endangered‟ List Targets Many US RiversIt is hard to think of a major natural resource or pollution issue in North America today that does not affect rivers.Farm chemical runoff残渣, industrial waste, urban storm sewers, sewage treatment, mining, logging, grazing放牧,military bases, residential and business development, hydropower水力发电,loss of wetlands. The list goes on.Legislation like the Clean Water Act and Wild and Scenic Rivers Act have provided some protection, but threats continue.The Environmental Protection Agency (EPA) reported yesterday that an assessment of 642,000 miles of rivers and streams showed 34 percent in less than good condition. In a major study of the Clean Water Act, the Natural Resources Defense Council last fall reported that poison runoff impairs损害more than 125,000 miles of rivers.More recently, the NRDC and Izaak Walton League warned that pollution and loss of wetlands—made worse by last year’s flooding—is degrading恶化the Mississippi River ecosystem.On Tuesday, the conservation group保护组织American Rivers issued its annual list of 10 ―endangered‖ and 20 ―threatened‖ rivers in 32 states, the District of Colombia, and Canada.At the top of the list is the Clarks Fork of the Yellowstone River, whereCanadian mining firms plan to build a 74-acre英亩reservoir水库，蓄水池as part of a gold mine less than three miles from Yellowstone National Park. The reservoir would hold the runoff from the sulfuric acid 硫酸used to extract gold from crushed rock.―In the event this tailings pond failed, the impact to th e greater Yellowstone ecosystem would be cataclysmic大变动的，灾难性的and the damage irreversible不可逆转的.‖ Sen. Max Baucus of Montana, chairman of the Environment and Public Works Committee, wrote to Noranda Minerals Inc., an owner of the ― New World Mine‖.Last fall, an EPA official expressed concern about the mine and its potential impact, especially the plastic-lined storage reservoir. ― I am unaware of any studies evaluating how a tailings pond尾矿池，残渣池could be maintained to ensure its structural integrity forev er,‖ said Stephen Hoffman, chief of the EPA’s Mining Waste Section. ―It is my opinion that underwater disposal of tailings at New World may present a potentially significant threat to human health and the environment.‖The results of an environmental-impact statement, now being drafted by the Forest Service and Montana Department of State Lands, could determine the mine’s future…In its recent proposal to reauthorize the Clean Water Act, the Clinton administration noted ―dramatically improved water quality since 1972,‖ when the act was passed. But it also reported that 30 percent of riverscontinue to be degraded, mainly by silt泥沙and nutrients from farm and urban runoff, combined sewer overflows, and municipal sewage城市污水. Bottom sediments沉积物are contaminated污染in more than 1,000 waterways, the administration reported in releasing its proposal in January. Between 60 and 80 percent of riparian corridors (riverbank lands) have been degraded.As with endangered species and their habitats in forests and deserts, the complexity of ecosystems is seen in rivers and the effects of development----beyond the obvious threats of industrial pollution, municipal waste, and in-stream diversions改道to slake消除the thirst of new communities in dry regions like the Southwes t…While there are many political hurdles障碍ahead, reauthorization of the Clean Water Act this year holds promise for US rivers. Rep. Norm Mineta of California, who chairs the House Committee overseeing the bill, calls it ―probably the most important env ironmental legislation this Congress will enact.‖ (553 words)6.According to the passage, the Clean Water Act______.A.has been ineffectiveB.will definitely be renewedC.has never been evaluatedD.was enacted some 30 years ago7.“Endangered” rivers are _________.A.catalogued annuallyB.less polluted than ―threatened rivers‖C.caused by floodingD.adjacent to large cities8.The “cataclysmic” event referred to in paragraph eight would be__________.A. fortuitous偶然的，意外的B. adventitious外加的，偶然的C. catastrophicD. precarious不稳定的，危险的9. The owners of the New World Mine appear to be______.A. ecologically aware of the impact of miningB. determined to construct a safe tailings pondC. indifferent to the concerns voiced by the EPAD. willing to relocate operations10. The passage conveys the impression that_______.A. Canadians are disinterested in natural resourcesB. private and public environmental groups aboundC. river banks are erodingD. the majority of US rivers are in poor conditionText CA classic series of experiments to determine the effects ofoverpopulation on communities of rats was reported in February of 1962 in an article in Scientific American. The experiments were conducted by a psychologist, John B. Calhoun and his associates. In each of these experiments, an equal number of male and female adult rats were placed in an enclosure and given an adequate supply of food, water, and other necessities. The rat populations were allowed to increase. Calhoun knew from experience approximately how many rats could live in the enclosures without experiencing stress due to overcrowding. He allowed the population to increase to approximately twice this number. Then he stabilized the population by removing offspring that were not dependent on their mothers. He and his associates then carefully observed and recorded behavior in these overpopulated communities. At the end of their experiments, Calhoun and his associates were able to conclude that overcrowding causes a breakdown in the normal social relationships among rats, a kind of social disease. The rats in the experiments did not follow the same patterns of behavior as rats would in a community without overcrowding.The females in the rat population were the most seriously affected by the high population density: They showed deviant异常的maternal behavior; they did not behave as mother rats normally do. In fact, many of the pups幼兽，幼崽, as rat babies are called, died as a result of poor maternal care. For example, mothers sometimes abandoned their pups,and, without their mothers' care, the pups died. Under normal conditions, a mother rat would not leave her pups alone to die. However, the experiments verified that in overpopulated communities, mother rats do not behave normally. Their behavior may be considered pathologically 病理上，病理学地diseased.The dominant males in the rat population were the least affected by overpopulation. Each of these strong males claimed an area of the enclosure as his own. Therefore, these individuals did not experience the overcrowding in the same way as the other rats did. The fact that the dominant males had adequate space in which to live may explain why they were not as seriously affected by overpopulation as the other rats. However, dominant males did behave pathologically at times. Their antisocial behavior consisted of attacks on weaker male,female, and immature rats. This deviant behavior showed that even though the dominant males had enough living space, they too were affected by the general overcrowding in the enclosure.Non-dominant males in the experimental rat communities also exhibited deviant social behavior. Some withdrew completely; they moved very little and ate and drank at times when the other rats were sleeping in order to avoid contact with them. Other non-dominant males were hyperactive; they were much more active than is normal, chasing other rats and fighting each other. This segment of the rat population, likeall the other parts, was affected by the overpopulation.The behavior of the non-dominant males and of the other components of the rat population has parallels in human behavior. People in densely populated areas exhibit deviant behavior similar to that of the rats in Calhoun's experiments. In large urban areas such as New York City, London, Mexican City, and Cairo, there are abandoned children. There are cruel, powerful individuals, both men and women. There are also people who withdraw and people who become hyperactive. The quantity of other forms of social pathology such as murder, rape, and robbery also frequently occur in densely populated human communities. Is the principal cause of these disorders overpopulation? Calhoun’s experiments suggest that it might be. In any case, social scientists and city planners have been influenced by the results of this series of experiments.11. Paragraph l is organized according to__________.A. reasonsB. descriptionC. examplesD. definition12．Calhoun stabilized the rat population_________.A. when it was double the number that could live in the enclosure without stressB. by removing young ratsC. at a constant number of adult rats in the enclosureD. all of the above are correct13.W hich of the following inferences CANNOT be made from theinformation inPara. 1?A. Calhoun's experiment is still considered important today.B. Overpopulation causes pathological behavior in rat populations.C. Stress does not occur in rat communities unless there is overcrowding.D. Calhoun had experimented with rats before.14. Which of the following behavior didn‟t happen in this experiment?A. All the male rats exhibited pathological behavior.B. Mother rats abandoned their pups.C. Female rats showed deviant maternal behavior.D. Mother rats left their rat babies alone.15. The main idea of the paragraph three is that __________.A. dominant males had adequate living spaceB. dominant males were not as seriously affected by overcrowding as the otherratsC. dominant males attacked weaker ratsD. the strongest males are always able to adapt to bad conditionsText DThe first mention of slavery in the statutes法令，法规of the English colonies of North America does not occur until after 1660—some forty years after the importation of the first Black people. Lest we think that existed in fact before it did in law, Oscar and Mary Handlin assure us, that the status of B lack people down to the 1660’s was that of servants. A critique批判of the Handlins’ interpretation of why legal slavery did not appear until the 1660’s suggests that assumptions about the relation between slavery and racial prejudice should be reexamined, and that explanation for the different treatment of Black slaves in North and South America should be expanded.The Handlins explain the appearance of legal slavery by arguing that, during the 1660’s, the position of white servants was improving relative to that of black servants. Thus, the Handlins contend, Black and White servants, heretofore treated alike, each attained a different status. There are, however, important objections to this argument. First, the Handlins cannot adequately demonstrate that t he White servant’s position was improving, during and after the 1660’s; several acts of the Maryland and Virginia legislatures indicate otherwise. Another flaw in the Handlins’ interpretation is their assumption that prior to the establishment of legal slavery there was no discrimination against Black people. It is true that before the 1660’s Black people were rarely called slaves. But this shouldnot overshadow evidence from the 1630’s on that points to racial discrimination without using the term slavery. Such discrimination sometimes stopped short of lifetime servitude or inherited status—the two attributes of true slavery—yet in other cases it included both. The Handlins’ argument excludes the real possibility that Black people in the English colonies were never treated as the equals of White people.The possibility has important ramifications后果，影响.If from the outset Black people were discriminated against, then legal slavery should be viewed as a reflection and an extension of racial prejudice rather than, as many historians including the Handlins have argued, the cause of prejudice. In addition, the existence of discrimination before the advent of legal slavery offers a further explanation for the harsher treatment of Black slaves in North than in South America. Freyre and Tannenbaum have rightly argued that the lack of certain traditions in North America—such as a Roman conception of slavery and a Roman Catholic emphasis on equality— explains why the treatment of Black slaves was more severe there than in the Spanish and Portuguese colonies of South America. But this cannot be the whole explanation since it is merely negative, based only on a lack of something. A more compelling令人信服的explanation is that the early and sometimes extreme racial discrimination in the English colonies helped determine the particular nature of the slavery that followed. (462 words)16. Which of the following is the most logical inference to be drawn from the passage about the effects of “several acts of the Maryland and Virginia legislatures” (Para.2) passed during and after the 1660‟s?A. The acts negatively affected the pre-1660’s position of Black as wellas of White servants.B. The acts had the effect of impairing rather than improving theposition of White servants relative to what it had been before the 1660’s.C. The acts had a different effect on the position of white servants thandid many of the acts passed during this time by the legislatures of other colonies.D. The acts, at the very least, caused the position of White servants toremain no better than it had been before the 1660’s.17. With which of the following statements regarding the status ofBlack people in the English colonies of North America before the 1660‟s would the author be LEAST likely to agree?A. Although black people were not legally considered to be slaves,they were often called slaves.B. Although subject to some discrimination, black people had a higherlegal status than they did after the 1660’s.C. Although sometimes subject to lifetime servitude, black peoplewere not legally considered to be slaves.D. Although often not treated the same as White people, black people,like many white people, possessed the legal status of servants.18. According to the passage, the Handlins have argued which of thefollowing about the relationship between racial prejudice and the institution of legal slavery in the English colonies of North America?A. Racial prejudice and the institution of slavery arose simultaneously.B. Racial prejudice most often the form of the imposition of inheritedstatus, one of the attributes of slavery.C. The source of racial prejudice was the institution of slavery.D. Because of the influence of the Roman Catholic Church, racialprejudice sometimes did not result in slavery.19. The passage suggests that the existence of a Roman conception ofslavery in Spanish and Portuguese colonies had the effect of _________.A. extending rather than causing racial prejudice in these coloniesB. hastening the legalization of slavery in these colonies.C. mitigating some of the conditions of slavery for black people in these coloniesD. delaying the introduction of slavery into the English colonies20. The author considers the explanation put forward by Freyre andTannenbaum for the treatment accorded B lack slaves in the English colonies of North America to be _____________.A. ambitious but misguidedB. valid有根据的but limitedC. popular but suspectD. anachronistic过时的，时代错误的and controversialUNIT 2Text AThe sea lay like an unbroken mirror all around the pine-girt, lonely shores of Orr’s Island. Tall, kingly spruce s wore their regal王室的crowns of cones high in air, sparkling with diamonds of clear exuded gum流出的树胶; vast old hemlocks铁杉of primeval原始的growth stood darkling in their forest shadows, their branches hung with long hoary moss久远的青苔;while feathery larches羽毛般的落叶松,turned to brilliant gold by autumn frosts, lighted up the darker shadows of the evergreens. It was one of those hazy朦胧的, calm, dissolving days of Indian summer, when everything is so quiet that the fainest kiss of the wave on the beach can be heard, and white clouds seem to faint into the blue of the sky, and soft swathing一长条bands of violet vapor make all earth look dreamy, and give to the sharp, clear-cut outlines of the northern landscape all those mysteries of light and shade which impart such tenderness to Italian scenery.The funeral was over,--- the tread鞋底的花纹/ 踏of many feet, bearing the heavy burden of two broken lives, had been to the lonely graveyard, and had come back again,--- each footstep lighter and more unconstrained不受拘束的as each one went his way from the great old tragedy of Death to the common cheerful of Life.The solemn black clock stood swaying with its eternal ―tick-tock, tick-tock,‖ in the kitchen of the brown house on Orr’s Island. There was there that sense of a stillness that can be felt,---such as settles down on a dwelling住处when any of its inmates have passed through its doors for the last time, to go whence they shall not return. The best room was shut up and darkened, with only so much light as could fall through a little heart-shaped hole in the window-shutter,---for except on solemn visits, or prayer-meetings or weddings, or funerals, that room formed no part of the daily family scenery.The kitchen was clean and ample, hearth灶台, and oven on one side, and rows of old-fashioned splint-bottomed chairs against the wall. A table scoured to snowy whiteness, and a little work-stand whereon lay the Bible, the Missionary Herald, and the Weekly Christian Mirror, before named, formed the principal furniture. One feature, however, must not be forgotten, ---a great sea-chest水手用的储物箱,which had been the companion of Zephaniah through all the countries of the earth. Old, and battered破旧的，磨损的, and unsightly难看的it looked, yet report said that there was good store within which men for the most part respect more than anything else; and, indeed it proved often when a deed of grace was to be done--- when a woman was suddenly made a widow in a coast gale大风，狂风, or a fishing-smack小渔船was run down in the fogs off the banks, leaving in some neighboring cottage a family of orphans,---in all such cases, the opening of this sea-chest was an event of good omen 预兆to the bereaved丧亲者;for Zephaniah had a large heart and a large hand, and was apt有…的倾向to take it out full of silver dollars when once it went in. So the ark of the covenant约柜could not have been looked on with more reverence崇敬than the neighbours usually showed to Captain Pennel’s sea-chest.1. The author describes Orr‟s Island in a(n)______way.A.emotionally appealing, imaginativeB.rational, logically preciseC.factually detailed, objectiveD.vague, uncertain2.According to the passage, the “best room”_____.A.has its many windows boarded upB.has had the furniture removedC.is used only on formal and ceremonious occasionsD.is the busiest room in the house3.From the description of the kitchen we can infer that thehouse belongs to people who_____.A.never have guestsB.like modern appliancesC.are probably religiousD.dislike housework4.The passage implies that_______.A.few people attended the funeralB.fishing is a secure vocationC.the island is densely populatedD.the house belonged to the deceased5.From the description of Zephaniah we can see thathe_________.A.was physically a very big manB.preferred the lonely life of a sailorC.always stayed at homeD.was frugal and saved a lotText BBasic to any understanding of Canada in the 20 years after the Second World War is the country' s impressive population growth. For every three Canadians in 1945, there were over five in 1966. In September 1966 Canada's population passed the 20 million mark. Most of this surging growth came from natural increase. The depression of the 1930s and the war had held back marriages, and the catching-up process began after 1945. The baby boom continued through the decade of the 1950s, producing a population increase of nearly fifteen percent in the five years from 1951 to 1956. This rate of increase had been exceeded only once before in Canada's history, in the decade before 1911 when the prairies were being settled. Undoubtedly, the good economic conditions of the 1950s supported a growth in the population, but the expansion also derived from a trend toward earlier marriages and an increase in the average size of families; In 1957 the Canadian birth rate stood at 28 per thousand, one of the highest in the world. After the peak year of 1957, thebirth rate in Canada began to decline. It continued falling until in 1966 it stood at the lowest level in 25 years. Partly this decline reflected the low level of births during the depression and the war, but it was also caused by changes in Canadian society. Young people were staying at school longer, more women were working; young married couples were buying automobiles or houses before starting families; rising living standards were cutting down the size of families. It appeared that Canada was once more falling in step with the trend toward smaller families that had occurred all through theWestern world since the time of the Industrial Revolution. Although the growth in Canada’s population had slowed down by 1966 (the cent), another increase in the first half of the 1960s was only nine percent), another large population wave was coming over the horizon. It would be composed of the children of the children who were born during the period of the high birth rate prior to 1957.6. What does the passage mainly discuss?A. Educational changes in Canadian society.B. Canada during the Second World War.C. Population trends in postwar Canada.D. Standards of living in Canada.7. According to the passage, when did Canada's baby boom begin?A. In the decade after 1911.B. After 1945.C. During the depression of the 1930s.D. In 1966.8. The author suggests that in Canada during the 1950s____________.A. the urban population decreased rapidlyB. fewer people marriedC. economic conditions were poorD. the birth rate was very high9. When was the birth rate in Canada at its lowest postwar level?A. 1966.B. 1957.C. 1956.D. 1951.10. The author mentions all of the following as causes of declines inpopulation growth after 1957 EXCEPT_________________.A. people being better educatedB. people getting married earlierC. better standards of livingD. couples buying houses11.I t can be inferred from the passage that before the IndustrialRevolution_______________.A. families were largerB. population statistics were unreliableC. the population grew steadilyD. economic conditions were badText CI was just a boy when my father brought me to Harlem for the first time, almost 50 years ago. We stayed at the hotel Theresa, a grand brick structure at 125th Street and Seventh avenue. Once, in the hotel restaurant, my father pointed out Joe Louis. He even got Mr. Brown, the hotel manager, to introduce me to him, a bit punchy强力的but still champ焦急as fast as I was concerned.Much has changed since then. Business and real estate are booming. Some say a new renaissance is under way. Others decry责难what they see as outside forces running roughshod肆意践踏over the old Harlem. New York meant Harlem to me, and as a young man I visited it whenever I could. But many of my old haunts are gone. The Theresa shut down in 1966. National chains that once ignored Harlem now anticipate yuppie money and want pieces of this prime Manhattan real estate. So here I am on a hot August afternoon, sitting in a Starbucks that two years ago opened a block away from the Theresa, snatching抓取，攫取at memories between sips of high-priced coffee. I am about to open up a piece of the old Harlem---the New York Amsterdam News---when a tourist。

Non-associative gauge theory and higher spin interactions

a r X i v :h e p -t h /0412027v 2 10 F eb 2005MCTP-04-64QMUL-PH-04-08hep-th/041202702/12/04Non-associative gauge theory and higher spin interactions Paul de Medeiros 1and Sanjaye Ramgoolam 21Michigan Center for Theoretical Physics,Randall Laboratory,University of Michigan,Ann Arbor,MI 48109-1120,U.S.A.2Department of Physics,Queen Mary University of London,Mile End Road,London E14NS,U.K.pfdm@ ,s.ramgoolam@ Abstract We give a framework to describe gauge theory on a certain class of commutative but non-associative fuzzy spaces.Our description is in terms of an Abelian gauge connection valued inthe algebra of functions on the cotangent bundle of the fuzzy space.The structure of such a gauge theory has many formal similarities with that of Yang-Mills theory.The components of the gauge connection are functions on the fuzzy space which transform in higher spin representations of the Lorentz group.In component form,the gauge theory describes an interacting theory of higher spin ﬁelds,which remains non-trivial in the limit where the fuzzy space becomes associative.In this limit,the theory can be viewed as a projection of an ordinary non-commutative Yang-Mills theory.We describe the embedding of Maxwell theory in this extended framework which follows the standard unfolding procedure for higher spin gauge theories.1IntroductionWe formulate gauge theory on a certain class of commutative but non-associative algebras, developing the constructions initiated in[1].These algebras correspond to so called fuzzy spaces which reduce to ordinary spacetime manifolds in a particular associative limit.We ﬁnd that such gauge theories have a realisation in terms of interacting higher spinﬁeld theories.The non-associative algebra of interest A∗n(M)is a deformation of the algebra of func-tions A(M)on a D-dimensional(pseudo-)Riemannian manifold M.The∗denotes a non-associative product for functions on the fuzzy space whilst n∈Z+provides a quantitative measure of the non-associativity(in particular A∗∞=A).For simplicity,we take M=R D withﬂat metric.Most of our formulas will be independent of the signature of this metric, though we will take it to be Lorentzian in discussions of gauge-ﬁxing etc.Furthermore, although we focus on the deformation for R D,there is a conceptually straightforward gener-alisation for curved manifolds.For example,the deformation A∗n(S2k)has been used in the study of even-dimensional fuzzy spheres in[2].In section2we deﬁne the commutative,non-associative algebra A∗n(R D)which deforms A(R D),and give the derivations of this algebra.In this review,we recall that the associator(A∗B)∗C−A∗(B∗C)of three functions A,B and C on A∗n(R D)can be written as an operator F(A,B)acting on C or as an operator E(A,C)acting on B.These operators have expansions in terms of derivations of the algebra(given in Appendix B)and naturally appear when one attempts to construct covariant derivatives for the gauge theory.Weﬁnd that an inevitable consequence of this structure is that the connection and gauge parameter have to be generalised such that they too have derivative expansions(i.e.they can be understood as functions on the deformed cotangent bundle A∗n(T∗R D)).The inﬁnite number of component functions in these expansions transform as totally symmetric tensors under the Lorentz group.Consequently weﬁnd that this extended gauge theory on A∗n(T∗R D)is related to higher spin gauge theory on A∗n(R D).The local and global structure of this extended gaugetheory is analysed in section3.We observe that the extended gauge theory remains non-trivial even in the limit where the non-associativity parameter goes to zero.In section4we describe certain physical properties in this associative limit.In particular we construct a gauge-invariant action andﬁeld equations for the extended theory using techniques related to the phase space formulation of quantum mechanics initiated by Weyl[20]and Wigner[21].The inﬁnite number higher spin components of the extended gaugeﬁeld become just tensors on R D in the associative limit.We describe various aspects of the extended theory in component form in order to make the connection with higher spin gauge theory more explicit.From this perspective it will be clear that the extended theory(as we have presented it)does not realise all the possible symmetries of the corresponding higher spin theory on R D.We suggest that it could describe a partially broken phase of some fully gauge-invariant theory. We then compare the structure weﬁnd with that of the interacting theory of higher spin ﬁelds discovered by Vasiliev[14].A precise way to embed Maxwell theory in the extended theory is given.The method is identical to the unfolding procedure which has been used by Vasiliev in the context of higher spin gauge theories[16].It can also be understood simply via a change of basis in phase space under a particular symplectic transformation.In section5we describe how the extended theory in the associative limit described in section 4is related to a projection of an ordinary non-commutative Yang-Mills theory.We also describe connections to Matrix theory.We then discuss how one might generalise the results of section4to construct a gauge-invariant action for the non-associative theory.Section6 contains some concluding remarks.2The non-associative deformation A∗nWe begin by deﬁning the non-associative space of interest.Following[1],we consider the commutative,non-associative algebra A∗n(R D)which is a speciﬁc deformation of the commu-tative,associative algebra of functions A(R D)on R D(which is to be thought of as physical spacetime in D dimensions).Another space that will be important in forthcoming discus-sions is the algebra of diﬀerential operators acting on A∗n(R D).This algebra is isomorphic to the deformed algebra A∗n(T∗R D)of functions on the(ﬂat)cotangent bundle T∗R D.This correspondence will be helpful when we come to consider gauge theory on A∗n(R D).The space R D has coordinates xµandﬂat metric.The Euclidean signature metricδµνarises most directly in the Matrix theory considerations motivating[1]but the algebra can be continued to Lorentzian signature by replacing this with Lorentzian metricηµν.The algebraic discussion in this and the next section(and in the appendices)works equally well in either signature,but some additional subtleties related to gauge-ﬁxing discussed in section4are speciﬁc to the Lorentzian case.The deformed algebra A∗n(R D)is spanned by the inﬁnite set of elements{1,xµ,xµ1µ2,...}1,where each xµ1...µs transforms as a totally symmetric tensor of rank s under the Lorentz group.The commutative(but non-associative) product∗for all elements xµ1...µs is deﬁned in[1]and Appendix B(this appendix also deﬁnes a more general set of products with similar properties to∗).The explicit formula is rather complicated but the important point is that xµ1...µs∗xν1...νt equals xµ1...µsν1...νt up to theaddition of lower rank elements with coeﬃcients proportional to inverse powers of n(for example xµ∗xν=xν∗xµ=xµν+11In[1],the elements x were called z and the deformed algebra A∗(R D)was called B∗n(R D).nnot break Lorentz symmetry.One can deﬁne derivations∂µof A∗n(R D)via the rule∂µxµ1...µs=sδ(µ1µxµ2...µs),(1) where brackets denote symmetrisation of indices(with weight1)2.This deﬁnition implies that∂µsatisfy the Leibnitz rule when acting on∗-products of elements of A∗n(R D).This Leibnitz property also holds with respect to the more general commutative,non-associative products described in Appendix B.It is clear that composition of these derivations is a commutative and associative operation.In the associative n→∞limit,∂µjust act as the usual partial derivatives on R D.2.1FunctionsFunctions of the coordinates xµ1...µs are written A(x)∈A∗n(R D).Such functions form a commutative but non-associative algebra themselves with respect to the∗multiplication.A quantitative measure of this non-associativity is given by the associator[A,B,C]:=(A∗B)∗C−A∗(B∗C)(2) for three functions A,B and C.Since A∗n(R D)is commutative then the associator(2)has the antisymmetry[A,B,C]=−[C,B,A].The associator also satisﬁes the cyclic identity [A,B,C]+[B,C,A]+[C,A,B]≡0.An important fact noted in[1]is that such associators can be written as diﬀerential operators involving two functions acting on the third.In particular,one can deﬁne the two operators E(A,B)and F(A,B)via[A,B,C]=:E(A,C)B=:F(A,B)C.(3) The antisymmetry property of the associator implies E(A,B)=−E(B,A)and the cyclic identity implies F(A,B)−F(B,A)=E(A,B).These operators have the following derivativeexpansions(see[1]or Appendix B)E(A,B)=∞s=11s!(Fµ1...µs(A,B))(x)∗∂µ1...∂µs,(4)where the coeﬃcients Eµ1...µs(A,B)and Fµ1...µs(A,B)are both polynomial functions of the algebra transforming as totally symmetric tensors under the Lorentz group3.The properties quoted above follow for each of these coeﬃcients so that Eµ1...µs(A,B)=−Eµ1...µs(B,A)and Fµ1...µs(A,B)−Fµ1...µs(B,A)=Eµ1...µs(A,B).The reason there are no s=0terms in(4)is that the associators[A,1,C]and[A,B,1]are both identically zero.Thus since(4)are valid as operator equations on any function then including such zeroth order terms in(4)would imply their coeﬃcients are identically zero by simply acting on a constant function.Theﬁrst non-vanishing s=1coeﬃcients in(4)can be expressed rather neatly as associators,such that Eµ(A,B)=[A,xµ,B]and Fµ(A,B)=[A,B,xµ].In a similar manner,all subsequent s>1coeﬃcients in(4)can also be expressed in terms of(sums of)associators of A and B with coordinates xµ1...µs(though we do not give explicit expressions as they are unnecessary). An important point to keep in mind is that E(A,B)and F(A,B)vanish in the associative limit as expected.The algebra of the diﬀerential operators in(4)closes under composition and is non-associative (following non-associativity of A∗n(R D))but it is also non-commutative.Since E(A,B)and F(A,B)vanish in the associative limit the algebra of these operators becomes trivially com-mutative when n→∞.As will be seen in the next subsection,more general diﬀerential operators acting on A∗n(R D)also close under composition to form a non-commutative,non-associative algebra.However,this more general algebra remains non-commutative(but as-sociative)when n→∞.For example,the commutator subalgebra of diﬀerential operators acting on R D corresponding to sections of the tangent bundle T R D(i.e.vectorﬁelds over R D)is non-Abelian(even though R D is itself commutative).Indeed this is often how oneconsiders simple non-commutative geometries–as Hamiltonian phase spaces of ordinary commutative position spaces(see e.g.[19]).We will draw on this analogy when we come to construct a gauge theory on A∗n(R D).2.2Diﬀerential operatorsGeneral diﬀerential operators acting on A∗n(R D)are writtenˆA=∞ s=01the algebra of functions is commutative).The deﬁnitions(6)obey the identitiesˆE(ˆA,ˆB)≡−ˆE(ˆB,ˆA)andˆF(ˆA,ˆB)−ˆF(ˆB,ˆA)≡ˆE(ˆA,ˆB)+[ˆA,ˆB](where[ˆA,ˆB]:=ˆAˆB−ˆBˆA is just thecommutator of operators).These reduce to the identities found earlier in terms of functions whenˆA=A andˆB=B.In the associative limit,notice thatˆF(ˆA,ˆB)vanishes identically whilstˆE(ˆA,ˆB)reduces to the commutator[ˆB,ˆA].The explicit derivative expansion forˆE(ˆA,ˆB)is given in Appendix A for later reference (the corresponding expression forˆF(ˆA,ˆB)will not be needed).We should just conclude this review of the relevant algebras associated with A∗n(R D)by noting that,unlike(4),the operator expression forˆE(ˆA,ˆB)includes a non-vanishing zeroth order algebraic term.It is easy to see that this is so by considering C in(6)to be the constant function.In this case all derivative terms inˆE(ˆA,ˆB)on the left hand side vanish whilst the right hand side reduces to the non-vanishing functionˆBA−ˆAB(where A and B are the zeroth order parts ofˆA and ˆB respectively).Thus the zeroth order partˆE(ˆA,ˆB)=ˆBA−ˆAB,which vanishes when(0)ˆA=A andˆB=B as expected.3Non-associative gauge theoryWe begin this section by reviewing the subtleties raised in[1]associated with formulating an Abelian gauge theory on A∗n(R D).We show that a naive formulation is not possible on this non-associative space.Instead it is rather natural to consider an extension of such an Abelian gauge theory on the deformed algebra A∗n(T∗R D)of functions on the cotangent bundle.We describe the local and global gauge structure of this non-associative extended theory.We ﬁnd the structure to be similar to that of a Yang-Mills theory with inﬁnite-dimensional gauge group.We will return to the question of embedding an Abelian gauge theory on A∗n(R D)in this extended structure in later sections.3.1Abelian gauge theory on A∗n(R D)A necessary ingredient in the construction of any gauge theory is the concept of a gauge-covariant derivative.Consider aﬁeldΦwhich is a function of A∗n(R D)and deﬁne it to have the inﬁnitesimal gauge transformation lawδΦ=ǫ∗Φ,(7)whereǫis an arbitrary polynomial function of A∗n(R D).(One reason for the choice of(7)is that it is reminiscent of the inﬁnitesimal gauge transformation for aﬁeld in the fundamental representation of the gauge group in ordinary Yang-Mills theory.)An operator Dµthat is covariant with respect to(7)must therefore obeyδ(DµΦ)=ǫ∗(DµΦ).(8)Clearly the derivation∂µ(1)alone does not obey this covariance requirement sinceδ(∂µΦ)=ǫ∗(∂µΦ)+(∂µǫ)∗Φ.To compensate we must introduce a gauge connection Aµ,which we take to be a function on A∗n(R D)and which transforms such thatδ(Aµ∗Φ)=ǫ∗(Aµ∗Φ)−(∂µǫ)∗Φ.Clearly the existence of such an Aµwould imply thatDµΦ:=∂µΦ+Aµ∗Φ(9) indeed deﬁnes a covariant derivative on functions,satisfying(8).Using(7)then implies that we require Aµto transform such that(δAµ)∗Φ=−(∂µǫ)∗Φ+ǫ∗(Aµ∗Φ)−Aµ∗(ǫ∗Φ).(10) In ordinary gauge theory(10)would allow one to simply read oﬀthe necessary gauge transfor-mation for Aµbut here things are more complicated due to non-associativity.In particular, notice that the last two terms in(10)can be written as the associator[Aµ,Φ,ǫ]and therefore, using(3),we requireδAµ=−(∂µǫ)+E(Aµ,ǫ).(11) This requirement,however,leads to a contradiction since theﬁrst two terms in(11)are algebraic functions on A∗n(R D)whilst(4)tells us that the third term acts only as a diﬀerentialoperator on A ∗n (R D ).Therefore such an A µcan only exist when E (A µ,ǫ)=0,i.e.in theassociative limit where this would simply be an Abelian gauge theory on R D !As indicated in[1],themost conservative way to proceed is therefore to simply generalise thegauge connection A µfrom an algebraic function to a diﬀerential operator ˆAµwith derivative expansionˆA µ=∞ s =01s !ǫα1...αs (x )∗∂α1...∂αs .(13)As noted already,the algebra of such operators is both non-associative and non-commutative.Consequently we must take care when revising the arguments of this subsection in terms of these extended ﬁelds.This revised analysis is described,in the next subsection,within the framework of global gauge transformations for the extended theory.In concluding,it is important to stress that the generalisation we have made is a modiﬁcation of the original theory and therefore the extended theory need not trivially reduce to an Abelian gauge theory on R D in the associative limit.(Notice that the s >0terms in (12)and (13)do not vanish as n →∞.)Indeed we will ﬁnd it does not though we will give a precise way to embed the Abelian theory in its extension on R D .3.2Global structureConsider again aﬁeldΦwhich is a function of A∗n(R D)but now with inﬁnitesimal gauge transformation lawδΦ=ˆǫΦ,(14) whereˆǫis the extended diﬀerential operator(13).Formally this is similar to Yang-Mills theory where one then obtains the global gauge transformation by exponentiating the lo-cal(Lie algebra valued)gauge parameter to obtain a general Lie group element(or more precisely the fundamental representations of these quantities).The main diﬀerence here is that the algebra of local gauge transformations(14)is non-associative.Despite this,given a general diﬀerential operatorˆǫ,there still exists a well-deﬁned exponential exp(ˆǫ)[18].The construction essentially just follows the power series deﬁnition of the exponential map for matrix algebras but here one must choose an ordering for powers ofˆǫ(so as to avoid the potential ambiguities due to non-associativity).We follow[18]and deﬁne powers via a‘left action’rule so thatexp(ˆǫ)Φ:=Φ+ˆǫΦ+13!ˆǫ(ˆǫ(ˆǫΦ))+...,(15)for any functionΦ.It is then clear that the exponentiated operatorˆg:=exp(ˆǫ)is also a diﬀerential operator acting on the algebra(albeit a rather complicated function ofˆǫ)and we deﬁne the‘global’transformation ofΦto beΦ→ˆgΦ.(16)This transformation obviously reduces to(14)in some neighbourhood of the identity where ˆg=1+ˆǫ(the‘identity’here is the unit element of A∗n(R D)).The set of all transformations (16)does not quite form a group under left action composition since it fails to satisfy the associativity axiom(due to non-associativity of the algebra).However,all the other group axioms are satisﬁed4.The derivation∂µis not covariant with respect(16)since this transformation implies∂µΦ→[∂µ,ˆg]Φ+ˆg(∂µΦ).As noted at the end of the previous subsection,we therefore introduce a gauge connectionˆAµwhich must transform such thatˆAµΦ→−[∂µ,ˆg]Φ+ˆg(ˆAµΦ)in order thatˆDΦ:=∂µΦ+ˆAµΦ(17)µtransforms covariantly under(16).This necessary gauge transformation ofˆAµΦunder(16) can be realised provided the gauge transformation ofˆAµis deﬁned such thatˆAΦ→−[∂µ,ˆg](ˆg−1Φ′)+ˆg(ˆAµ(ˆg−1Φ′))(18)µunder the more general function transformationΦ→Φ′.This gives the desired gauge transformation whenΦ′=ˆgΦ.One can obtain the gauge transformation ofˆAµitself by using the operatorˆF(6)to rearrange the brackets in(18).In particular,notice that the right hand side of(18)can be written−[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) (ˆg−1Φ′)(19) = −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ˆg−1 Φ′−ˆF −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ,ˆg−1 Φ′. ThereforeˆAµmust have the following gauge transformationˆAµ→ −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ˆg−1−ˆF −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ,ˆg−1 .(20) Settingˆg=1+ˆǫin(20)leads to the inﬁnitesimal form of the gauge transformationδˆAµ=−[∂µ,ˆǫ]+ˆE(ˆAµ,ˆǫ).(21) Of course,at the inﬁnitesimal level,this transformation equivalently follows by the require-ment thatδ(ˆDµΦ)=ˆǫ(ˆDµΦ)under(14).Notice that(20)and(21)do not quite take the form one would expect by naively following the Yang-Mills analogy(that is they diﬀer from what one might expect by associator terms). This is a consequence of the non-associativity of the underlying algebra of functions.In the following section we willﬁnd that the expected Yang-Mills type structure follows exactly in the associative limit.In the discussion above we have only deﬁned covariant derivativesˆDµon functions and not on diﬀerential operators.Although not of the standard Yang-Mills form,(minus)the right hand side of(21)can still be taken as the deﬁnition for the action of the covariant derivative on operatorˆǫ,such thatˆDµ·ˆǫ:=[∂µ,ˆǫ]+ˆE(ˆǫ,ˆAµ).(22) This statement is partially justiﬁed by the fact thatˆDµthen satisﬁes the Leibnitz rule ˆD(ˆǫΦ)=(ˆDµ·ˆǫ)Φ+ˆǫ(ˆDµΦ)(for general operatorˆǫand functionΦ)5.µBased on the transformation law found above,we deﬁne theﬁeld strengthˆFµνasˆF:=ˆE(ˆDν,ˆDµ)=[∂µ,ˆAν]−[∂ν,ˆAµ]+ˆE(ˆAν,ˆAµ).(23)µνIt is clear from this deﬁnition thatˆFµνis indeed a diﬀerential operator which transforms as a two-form under the Lorentz group.In addition,since the gauge transformations above imply thatˆDΦ→ˆg(ˆDµ(ˆg−1Φ′)),(24)µunder(18),then it follows thatˆFµνΦ=ˆDµ(ˆDνΦ)−ˆDν(ˆDµΦ)transforms asˆFΦ→ˆg(ˆFµν(ˆg−1Φ′)),(25)µνand is therefore also gauge-covariant whenΦ′=ˆgΦ.The inﬁnitesimal form of the covariant gauge transformation ofˆFµνisδˆFµν=ˆE(ˆFµν,ˆǫ).(26)From the evidence above,it is clear that there are various subtleties related to the non-associative nature of the theory.Indeed the non-associativity complicates matters even further in the description of more physical aspects of the theory like Lagrangians,ﬁeld equations and the embedding of an Abelian gauge theory in this extended framework.Recall though that this extended theory should have a non-trivial structure,even in the associative limit.We therefore postpone further discussion of the non-associative extended theory to analyse its associative limit in more detail.4Gauge theory on T∗R D and higher spin gauge theory on R DWe begin this section by brieﬂy summarising the results of the previous subsection in the associative limit.We then describe how one can construct a gauge-invariant action and equations of motion for this theory.Writing the extended gaugeﬁeldˆAµin terms of com-weﬁnd that the extended theory describes an interacting theory ponent functions Aα1...αsµinvolving an inﬁnite number of higher spinﬁelds.When written in component form,it will be clear that the extended theory(as we have described it)does not realise all the possible symmetries of the corresponding higher spin gauge theory.We suggest that the extended theory could correspond to a partially broken phase of some fully gauge-invariant higher spin theory.A comparison of the structure weﬁnd with that of the interacting theory of higher spinﬁelds discovered by Vasiliev[14]is then given.We conclude the section by showing how an Abelian gauge theory can be embedded in this extended framework.The embedding is related to the unfolding procedure used by Vasiliev in the context of higher spin gauge theory[16].134.1The associative limitMany expressions found in the previous section retain their schematic form in the associative limit.For example,the gauge transformations for functions are just as in(14),(16)though Φis now simply a function on R D whilst operators likeˆǫin(13)now have the expansionˆǫ=∞ s=01also transforms covariantly.The inﬁnitesimal form of this covariant transformation beingδˆFµν=[ˆǫ,ˆFµν].(32)4.2Action andﬁeld equationsA simple equation of motion to consider for the extended theory in the associative limit is[ˆDµ,ˆFµν]=0.(33)This is theﬁeld equation one would expect from following the Yang-Mills type structure found for the extended theory in the previous subsection.The equation(33)is invariant under the gauge transformation(28).Moreover it is this equation(rather than,say,the also gauge-invariant equationˆDµˆFµν=0)which reduces to the correct Maxwell equation as we will see in section4.5.Following the Yang-Mills analogy further,a natural gauge-invariant action to consider is of the form−1taking the usual gauge-invariant trace(using the Cartan-Killing metric for the gauge group) followed by integrating over spacetime.However,we do not assume a priori that the map(35)can be factorised in thisway6.In the Yang-Mills case the symmetry property of Trsimply follows from the fact that the trace is symmetric.The symmetry of the trace is a rather general property ofﬁnite-dimensional representations–as one considers for Yang-Mills theories with compact gauge groups–since such representations can be expressed in terms ofﬁnite-dimensional square matrices(and for two such matrices X,Y,the trace of XY is just X i j Y j i=Y i j X j i).For the extended theory we are considering thoughﬁelds are valued in the algebra of diﬀerential operators on R D and the situation is very diﬀerent for the case of such inﬁnite-dimensional representations.For example,in quantum mechanics, if the Heisenberg algebra[ˆx,ˆp]=i had any representations ofﬁnite dimension n=0(and hence a symmetric trace)then it would imply the well-known contradiction0=in!The example above is quite pertinent since we will now show thatﬁelds in the extended theory we are considering are related to certain functions in the formulation of quantum mechanics based on the original work of Weyl[20]and Wigner[21]which was later developed by Groenewold[23]and Moyal[24](see[27]for a nice review).Within this framework,there exists a natural concept of the symmetric map Tr.In terms of the abstract canonically conjugate operatorsˆxµandˆpµ,a general operatorˆA of the form(27)is writtenˆA=A(ˆx,ˆp)=∞ s=0i s6As explained in[19],non-commutative gauge theories provide a counter example where such a factori-sation of Tr is not possible.16ordering prescription above7.Given this ordering rule,the Weyl homomorphism[20]says that every operator A(ˆx,ˆp)(37) is naturally associated with an ordinary c-number function˜A on the classical phase space R2D(spanned by coordinates(x,p)),such that1A(ˆx,ˆp)=(2π)2D dy dq dx dp˜A(x,p)yα1...yαs exp(i qµ(ˆxµ−xµ)−i yµpµ).(39) The trace Tr of the operator A(ˆx,ˆp)is deﬁned byTr(ˆA):= dx dp˜A(x,p).(40) This integral is only deﬁned for functions˜A with suitably rapid asymptotic decay properties. We will describe a particular Wigner basis for a class of such integrable functions in the next subsection.The inverse of the relation(38)can then be expressed in terms of this trace,such that˜A(x,p)=1−→∂∂pµ(−i)m ∂∂pµm˜A ∂∂xµm˜B .(43)m!Notice in particular that the m=0term in(43)is just the commutative classical product of functions˜A˜B.The m>0terms are not commutative but are invariant under the combined exchange˜A↔˜B and x↔p.Equation(43)implies that xµ⋆pν=xµpνandpν⋆xµ=xµpν−iδµν,thus conﬁrming that the⋆-product of functions preserves the structureof the Heisenberg algebra.It is also worth noting that partial derivatives(with respect tox or p)act as derivations on the algebra of classical phase space functions with⋆-product since they obey the Leibnitz rule when acting on(43).The deﬁnition(43)implies thatdx dp(˜A⋆˜B)(x,p)= dx dp˜A′(x,p)˜B′(x,p)= dx dp(˜B⋆˜A)(x,p),(44)where the primed phase space functions denote˜A′:=exp i∂xµ∂2∂∂pµ ˜B which are just multiplied with respect to the classical product in(44).Thus the trace(40)of the operator productˆAˆB is indeed symmetric,as required.The precise form of the gauge-invariant action(34)is therefore given by−14 dx dp˜F′µν(x,p)˜F′µν(x,p),(45) where the function˜F′µν:=exp i∂xµ∂4.2.1Wigner basis for integrable functionsWe will now brieﬂy describe a particular basis for a class of classical functions which have ﬁnite integrals over phase space(a more detailed review of this construction is given in[27]). This will show us how to restrict to the class of Weyl-dual operators for which the trace map Tr is well-deﬁned.Of course,this is necessary so that the gauge-invariant action(45)exists.Consider a complete orthonormal basis of eigenfunctions{ψa}for a given Hamiltonian H. To each such eigenfunctionψa(x)on R D,there is an associated Wigner function1f a(x,p)=proportional to exp −i ∂x µ∂4(2π)4D dxdp dydq dy ′dq ′exp −i 4(2π)2D dy dq exp (−i y µq µ)×Tr exp(−i q µˆx µ)ˆF αβexp(−i y µˆp µ) Tr exp(i q µˆx µ)ˆF αβexp(i y µˆp µ)。

The Standard Model and Fermion Masses

τ3 =
ψ L and isotopic matrices ψ R
I 0 0 I , τ1 = affecting the four upper and four lower components of 0 −I I 0
operator Ф1 . Now, equations (2) can be written as
Consider the density of Hamiltonian of a Dirac particle with mass m f interacting with an arbitrary boson field B µ ℋD = ψ † (α p + β m f + qα µ B µ )ψ = ψ † ( PL + PR ) (α p + β m f + qα µ B µ ) ( PL + PR )ψ =
GG † GG † † =ψ L p + qα µ B µ )ψ R + ψ L β m fψ R +ψ R β m fψ L (α p + qα µ B µ )ψ L +ψ R† (α
.
GG
GG
(1)
In (1) q − is the coupling constant; PL =
1− γ 5 1+ γ 5 , PR = − are the left and 2 2
∂ . Linear forms of SU (2) − invariant ∂t
equations for fermion fields relative to p0 can be obtained using the FoldyWouthuysen transformation [6] in the specially introduced isotopic space. Introduce the eight-component field operator Ф1 =

stolzen定理英文文献

stolzen定理英文文献The Stolz-Cesàro theorem, also known as the Stolz-Česáro theorem, is a fundamental result in real analysis that provides a powerful tool for evaluating limitsinvolving sequences. The theorem is named after Otto Stolz and Ernesto Cesàro, who independen tly discovered andproved the result in the late 19th century. The theorem is particularly useful in dealing with indeterminate forms, which arise frequently in calculus and other areas of mathematics. It provides a method for determining the limit of a sequence by examining the ratio of consecutive terms and their averages, allowing for the evaluation of limits that would otherwise be difficult or impossible to compute directly.The Stolz-Cesàro theorem states that if the sequence (an) is such that li m n→∞ (an+1 - an) = L and lim n→∞ (bn - bn-1) = M, where both L and M are real numbers or±∞, then the limit of the ratio of the two sequences can be expressed as lim n→∞ (an / bn) = L / M. In other words,the theorem provides a method for evaluating the limit of the ratio of two sequences by examining the limits of their differences. This result has wide-ranging applications in real analysis, particularly in the study of infinite series and improper integrals, where it can be used to determine convergence or divergence.One of the key insights of the Stolz-Cesàro theorem is its ability to handle indeterminate forms of the type 0/0 or ±∞/±∞, which frequently arise in the evaluation of limits. By examining the ratio of consecutive terms in the sequences (an) and (bn) and their averages, the theorem provides a systematic approach for resolving these indeterminate forms and determining the limit of the ratio. This makes it an invaluable tool for mathematicians and scientists working in fields where the evaluation of limits is crucial, such as in the analysis of algorithms, the study of differential equations, and the modeling of physical systems.The Stolz-Cesàro theorem has also found applicationsin the study of infinite series, where it can be used todetermine the convergence or divergence of a series by examining the limit of the ratio of consecutive terms. This is particularly useful in cases where the terms of the series do not have a simple closed form or where other methods of convergence testing are inconclusive. By applying the Stolz-Cesàro theorem, mathematicians can gain valuable insights into the behavior of infinite series and make important conclusions about their convergence properties.In conclusion, the Stolz-Cesàro theorem is a fundamental result in real analysis that provides a powerful tool for evaluating limits involving sequences.Its ability to handle indeterminate forms and its wide-ranging applications in real analysis and other areas of mathematics make it a valuable and important result. The theorem's systematic approach to resolving indeterminate forms and determining the limit of the ratio of sequences has made it an invaluable tool for mathematicians and scientists working in fields where the evaluation of limits is crucial. Its applications in the study of infiniteseries further demonstrate its importance and relevance in modern mathematics.。