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.

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1
A special case of Eq. (1.1) has theu)ut − (h1 + h2u)ux − k0uxx + ϕiui = 0,
i=1
(1.3)
which, after the substitution of (1.2) and multiplication by F (x, t)3 becomes an equation of the third order of homogeneity with respect to the functions G(x, t) and F (x, t).
of a solution, there is considerable arbitrariness related to the “extra” function A(x, t), which must be excluded. Namely, this function must be added to the function H(x, t) according to the formula H(x, t)(new) = H(x, t)(old)+exp(ln(A)). Then the same notation H(x, t))(new) is preserved for the function H(x, t).
Invariant Properties of the Ansatz of the Hirota Method for Quasilinear Parabolic equations
K.A. Volosov Moscow Institute of Electronics and Mathematics
∂ν(·) = (·)ν, ν = x, t
(1.4)
We choose the solution (1.2) of Eq. (1.3) in the form
G(x, t) = exp(H(x, t))B(x, t) + Q(x, t), F (x, t) = 1 + A(x, t) exp(H(x, t)) .
1
We consider the quasilinear parabolic equation
(1 + b1u + b2u2)ut − (h1 + h2u + h3u2)ux − g0(ux)2
4
−(k0 + k1u)uxx + ϕiui = 0.
i=1
(1.1)
where bj(x, t), hj(x, t), kj−1(x, t), j = 1, 2, ϕi(x, t), i = 1, . . . , 4, g0(x, t) are smooth functions or constants.
B B2ϕ3 − 2k0(Ax)2 + B(−b1At + H2Ax)
+A B2(ϕ1 + ϕ3) + 2k0AxBx + B(−At + B1Bt + h1Ax − h2Bx + k0Axx)
+A2(Bϕ1 + Bt − h1Bx − K0Bxx) = 0.
(1.7)
2
The special case considered in [3], p. 51, Section 24, and in [2], p. 190, is the case of B = (±)A, where the equation has the form
Contsam@dol.ru
February 7, 2008
arXiv:math-ph/0103014v1 12 Mar 2001
Abstract
We propose a new method based on the invariant properties of the ansatz of the Hirota method whcih have been discovered recently. This method allows one to construct new solutions for a certain class the dissipative equations classiﬁed by degrees of homogeneity. This algorithm is similar to the method of “dressing” the solutions of integrable equations. A class of new solutions is constructed. It is proved that all known exact solutions of the FitzHygh–Nagumo–Semenov equation can be expressed in terms of solutions of the linear parabolic equation. This method is compared with the Miura transforms in the theory of Kortveg de Vris equations. This method allows on to create a package by using the methods of computer algebra.
We denote the obtained system of equations by (eq0), (eq1), (eq2), (eq3). Equation (eq0) is the same original equation (1.3) for the function Q(x, t). Hence Q(x, t) is a solution of this equation. Let us consider the equation obtained at a power of exp(3H(x, t)), namely, Eq. (eq3):
u(x, t)
=
G(x, F (x,
t) t)
.
(1.2)
After multiplication by F (x, t)4, for the functions G(x, t) and F (x, t) we receive a homogeneous equation of the fourth order of homogeneity with respect to all functions.
(1.5) (1.6)
All the other representations of the solution obtained by the Hirota method can be reduced to this form. There is only one possibility to extend this notation, namely, to add sums of series in powers of the exponentials exp(iH(x, t)), i = 2, . . .. Let us substitute (1.5) ﬁrst into (1.2) and then into (1.3) and equate the coeﬃcients of equal powers of exp(H(x, t)) (these powers are 0, 1, 2, 3) with zero.
− A2B + B3 − ABAt − 2BA2x + AB Axx + A2Bt + 2AAxBx − A2Bxx = 0. (1.8)
and is an identity. First, we present a lemma in which we prove that, in the conventional ansatz form
Special cases of this equation are widely known (see the extensive bibliography in [1, 2, 3, 11, 12, 14]).
If h1 = 0, ϕi = 0, i = 1, 3, then this is the Bu¨rgers equation in the wave theory [4]. If h1 = 0, h2 = 0, ϕ1 = −1, ϕ2 = −ϕ1, ϕ3 = 0, then this is the Fisher–Kolmogorov– Petrovskii–Piskynov (FKPP) equation (see the bibliography in [2, 3, 13]); If h1 = 0, h2 = 0, ϕ1 = −1, ϕ2 = 0, ϕ3 = −ϕ1, then this is the FitzHygh–Nagumo– Semenov (FhNS) equation in the wave theory (see bibliography in [1]–[8]), [19], p. 12), and the Allen–Cahn equation (which is close to the FhNS equation) in the theory of phase transitions [10] and in ecology [14, 15]. Equations in which the coeﬃcients of the Laplace operator depend on the function u appear in models of the theory of epidemic distribution [20]. Equations in which the coeﬃcients of the derivative ut depending on ux arise, for example, in models of processes of ocean water freshening [21]. The initial reasons on a theme of the given operation can be detected in operation [22]. The method proposed for constructing solutions of Eq. (1.1) ideologically goes back to the Hirota method. An essential distinguishing feature of the present paper is that a new invariance property of solutions of equations from a given class is discovered (cf. [16, 17]). The equations obtained from (1.1) after the substitution of (1.2) can be arranged in the order of homogeneity as embedded equations. From now on, we use the following notation for the derivatives: