The distance modulus of the Large Magellanic Cloud based on double-mode RR Lyrae stars

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Clump Distance to the Magellanic Clouds and Anomalous Colors in the Galactic Bulge

a r X i v :a s t r o -p h /9910174v 2 13 N o v 1999Clump Giant Distance to the Magellanic Clouds and AnomalousColors in the Galactic BulgePiotr PopowskiInstitute of Geophysics and Planetary Physics,L-413Lawrence Livermore National Laboratory,University of CaliforniaP.O.Box 808,Livermore,CA 94551,USA.E-mail:popowski@.ABSTRACTI demonstrate that the two unexpected results in the local Universe:1)anomalous intrinsic (V −I )0colors of the clump giants and RR Lyrae starsin the Galactic center,and 2)very short distances to the Magellanic Clouds(LMC,SMC)as inferred from clump giants,are connected with each other.The(V −I )0anomaly is partially resolved by using the photometry from the phase-IIof the Optical Gravitational Lensing Experiment (OGLE)rather than phase-I.The need for V-or I-magnitude-based change in the bulge (V −I )0is one optionto explain the remaining color discrepancy.Such change may originate in acoeﬃcient of selective extinction A V /E (V −I )smaller than typically assumed.Application of the (V −I )0correction (independent of its source)doubles theslope of the absolute magnitude –metallicity relation for clump giants,so thatM I (RC )=−0.23+0.19[Fe /H].Consequently,the estimates of the clumpdistances to the LMC and SMC are aﬀected.Udalski’s (1998c)distance modulusof µLMC =18.18±0.06increases to µLMC =18.27±0.07.The distance modulusto the SMC increases by 0.12to µSMC =18.77±0.08.I argue that a morecomprehensive assessment of the metallicity eﬀect on M I (RC )is needed.Subject Headings:distance scale —dust,extinction —Galaxy:center —Magellanic Clouds —stars:horizontal-branch 1.IntroductionMost of the extragalactic distance scale is tied to the LMC,and so the distance to theLMC (d LMC )inﬂuences the Hubble constant,H 0.For many years now there has been a division between the so called “short”and “long”distance scales to the LMC.Currently,the measured values of d LMC span a range of over 25%(see e.g.,Feast &Catchpole 1997;Stanek,Zaritsky,&Harris 1998).Paczy´n ski &Stanek (1998)pointed out that red clumpgiants should constitute an accurate distance indicator.Udalski et al.(1998a)and Stanek et al.(1998)applied the clump method and found a very short distance to the LMC (µLMC≈18.1).In response,Cole(1998)and Girardi et al.(1998)suggested that clump giants are not standard candles and that their absolute I magnitudes,M I(RC),depend on the metallicity and age of the population.Udalski(1998b,1998c)countered this criticism by showing that the metallicity dependence is at a low level of about0.1mag/dex,and that the M I(RC)is approximately constant for cluster ages between2and10Gyr.Stanek et al.(1999)and Udalski(1999)found a moderate slope of the M I(RC)–[Fe/H]relation of 0.15mag/dex.The only clump determination,which resulted in a truly long d LMC was a study of theﬁeld around supernova SN1987A by Romaniello et al.(1999).However,they assumed a bright M I(RC)from theoretical models and,additionally,the use of the vicinity of SN1987A may not be the most fortunate choice(Udalski1999).The value of M I(RC)in diﬀerent stellar systems is a major systematic uncertainty in the clump method.It is very hard to prove the standard character of a candle’s luminosity. However,it should be possible to check whether other stellar characteristics of a candle behave in a predictable fashion.Therefore,in§2I discuss the(V−I)0colors of the clump giants and RR Lyrae stars in the Galactic bulge.After making photometric corrections,I argue that the remaining color discrepancy between the Baade’s Window and local stars might have been caused by an overestimated coeﬃcient of selective ing corrected colors,in§3I derive a new M I(RC)–[Fe/H]relation for red clump stars and show its substantial impact on the distances to the Magellanic Clouds.I summarize the results in§4.2.Mystery of anomalous colors in the Galactic bulgePaczy´n ski(1998)tried to explain why the clump giants in the Baade’s Window have(V−I)0colors which are approximately0.2magnitudes redder than in the solar neighborhood(Paczy´n ski&Stanek1998).Paczy´n ski(1998)suggested super-solar metallicities of the Galactic bulge stars as a possible solution.However,there is a spectroscopic evidence(see Minniti et al.1995)that the average metallicity of the bulge is [Fe/H]∈(−0.3,0.0).Stutz,Popowski&Gould(1999)found a corresponding eﬀect for the Baade’s Window RR Lyrae stars,which have(V−I)0redder by about0.17than their local counterparts(Fig.1).A similar size of the color shift in RR Lyrae stars and clump giants suggests a common origin of this eﬀect.Does there exists any physical mechanism that could be responsible for such behavior?The bulge RR Lyrae stars and clump giants both burn Helium in theircores,but the similarities end here.RR Lyrae stars pulsate,clump giants do not.RR Lyrae stars are metal-poor,clump giants are metal-rich.RR Lyrae stars are likely to be a part of an axisymmetric stellar halo(e.g.,Minniti1996;Alcock et al.1998a),whereas clump giants form a bar(e.g.,Stanek et al.1994;Ng et al.1996).For RR Lyrae stars,Stutz et al.(1999)suggested that their very red(V−I)0might have resulted from an unusual abundance ofα-elements.Why should a clump population which emerged in a diﬀerent formation process share the same property?The solutions to the anomalous colors proposed by Paczy´n ski(1998)and Stutz et al.(1999)are not impossible but are rather unlikely.Alternatively,the eﬀect might be unrelated to the physics of those stars.The investigated bulge RR Lyrae stars and clump giants share two things in common.First,photometry of both types of stars comes from the OGLE,phase-I,project.Indeed,Paczy´n ski et al.(1999)showed that the OGLE-IV-magnitudes are0.021mag fainter and I-magnitudes0.035mag brighter than the better calibrated OGLE-II magnitudes.Therefore,the correct(V−I)colors should be0.056bluer. Additionally,the new(V−I)0from the more homogeneous Baade’s Window clump is bluer than Paczy´n ski’s&Stanek’s(1998)color even when reduced to OGLE-I calibration1.When the new OGLE-II photometry reported by Paczy´n ski et al.(1999)is used,the(V−I)0 anomaly shrinks and the remaining unexplained shift amounts to∼0.11both for the RR Lyrae stars and clump giants.Second,Paczy´n ski&Stanek(1998)and Stutz et al.(1999)use the same extinction map(Stanek1996)and the same coeﬃcient of conversion from visual extinction A V to color excess E(V−I).The absolute values of A V s are likely approximately correct(see equation1)because the zero point of the extinction map was determined from the(V−K) color and A V/E(V−K)is very close to1(Gould,Popowski,&Terndrup1998;Alcock et al.1998b).However,R V I=A V/E(V−I)is not as secure and has a pronounced eﬀect on the obtained color.Most of the current studies of the Galactic bulge use R V I=2.5.If a true R V I towards Baade’s Window equalsαinstead,then the adjusted Stanek’s(1996)V-band extinction, will be2:αA V,adjusted=1Udalski’s(1998b)data for the LMC,SMC,and Carina galaxy come from OGLE-II and therefore do not require any additional adjustment.2Equation(1)implicitly assumes that diﬀerential(V−I)colors from Stanek(1996)are correct.Whether it is the case is an open question.has been determined based on K-magnitudes3.The adjustment to the color,which follows from equation(1)is:∆(V−I)0=1αA V,adjusted=α−2.5A V,0-point−2.5∆(V−I)0.(3)Using∆(V−I)0≈−0.11as required to resolve the color conﬂict in Baade’s Window and A V,0-point=1.37(Gould et al.1998;Alcock et al.1998b),Iﬁndα≈2.1(Fig.2).This R V I=2.1is certainly low,but not unreasonably so.Szomoru&Guhathakurta(1999)ﬁnd that cirrus clouds in the Galaxy have extinctions consistent with A V/E(B−V)<∼2,which is more extreme than the change suggested here.If the extinction towards Baade’s Window is in part provided by the cirrus clouds,then the low R V I would be expected rather than surprising.The value and variation of R V I was thoroughly investigated by Wo´z niak&Stanek (1996).The essence of the Wo´z niak&Stanek(1996)method to determine diﬀerential extinction is an assumption that regions of the sky with a lower surface density of stars have higher extinction.Wo´z niak&Stanek(1996)used clump giants to convert a certain density of stars to an amount of visual extinction.To make a calibration procedure completely unbiased would require,among other things,that clump giants were selected without any assumption about R V I;that absolute V-magnitudes of clump giants,M V(RC),do not depend on their color[here(V−I)0];and that reddened and unreddened clump giants be drawn from the same parent population.None of these is true.A color-magnitude diagram(CMD)for dense Galacticﬁelds does not allow one to unambiguously distinguish clump giants from other stars.Diﬀerent parts of an intrinsically clean CMD overlap due to diﬀerential reddening and a range of stellar distances.Therefore,the selection of clump giants must involve some assumptions about R V I.Wo´z niak&Stanek(1996)adopt R V I=2.6.This procedure tends to bias the derived relation toward this predeﬁned slope. Wo´z niak&Stanek(1996)were fully aware of this eﬀect,and they performed a number ofsimulations,which are summarized in their Figure4.In brief,in the range2.1<R V I<3.1, the bias scales asδR V I∼0.4(2.6−R V I)and so may become very substantial for a low or high R V I.In particular,if the true R V I=2.1,Wo´z niak and Stanek(1996)wouldﬁnd R V I=2.3.Therefore,this eﬀect alone could account for half of the diﬀerence between the required and measured R V I.The intrinsic characteristics of the bulge clump stars are unknown,but I will assume they resemble the clump measured by Hipparcos(European Space Agency1997).Theﬁt to the local clump giants selected by Paczy´n ski&Stanek(1998)gives M V(RC)∝0.4(V−I)0. Therefore,the structure of the local clump itself acts similarly to extinction with R V I=0.4. In an ideal case,when the CMD locations of the entire clump populations in diﬀerentﬁelds are compared,the M V(RC)−(V−I)0dependence should not matter.However, when combined with the actual extinction and additionally inﬂuenced by the completeness function of a survey,this eﬀect may additionally bias the value of R V I.Because the smaller selective extinction coeﬃcient is not excluded by the current studies,one can assume R V I=2.1to match the(V−I)0colors of the bulge with the ones in the solar neighborhood.The color is a weak function of[Fe/H],so this procedure is justiﬁed because the[Fe/H]of the bulge and solar neighborhood are similar.This change in R V I will decrease the I-mag extinction,A I,by0.11mag.Therefore,the clump distance to the Galactic center would increase by the same amount.3.Recalibration of the clumpWhat is the bearing of the bulge results on the distance to the LMC?Let∆indicate the diﬀerence between the mean dereddened I-magnitude of clump giants and the derredened V-magnitude of RR Lyrae stars at the metallicity of RR Lyrae stars in the Galactic bulge. When monitored in several stellar systems with diﬀerent clump metallicities,the variable ∆,introduced by Udalski(1998b),allows one to calibrate the M I(RC)-[Fe/H]relation with respect to the baseline provided by RR Lyrae stars.The better photometry from Paczy´n ski et al.(1999)and a possible modiﬁcation of R V I inﬂuence the value of∆at the Galactic center(∆BW).It is important to note that one will face the same type of adjustment to∆BW whenever the anomalous colors in the Baade’s Window are resolved at the expense of the modiﬁcation of V-or I-magnitudes.That is,the modiﬁcation of R V I is not a necessary condition!It is simply one of the options.As a result of the change in∆BW, the M I(RC)-[Fe/H]relation for clump giants changes.Moreover,µLMC andµSMC will change as well because the M I(RC)–[Fe/H]relation is used to obtain the clump distances to the Magellanic Clouds.Here,I will modify Udalski’s(1998b)∆versus[Fe/H]plot and derive a new M I(RC) -[Fe/H]relation consistent with the new data and considerations from§2.I construct the Udalski(1998b)plot using his original points modiﬁed in the following way:—To match the change in(V−I)0,I modify∆BW by0.17mags(a combined change from photometry and some other,yet unrecognized,source,e.g.,selective extinction coeﬃcient).—I modify the[Fe/H]of the Baade’s Window clump giants,so that[Fe/H]=0.0(see e.g., Minniti et al.1995for a review on the bulge metallicity).The possible improvement to the above procedure would be a construction of Udalski’s (1998b)diagram based on clump giants in the LMC and SMC clusters,which would reduce the uncertainties associated with the reddening to theﬁeld stars.This more complex treatment is beyond the scope of this paper.I make a linearﬁt to the∆–[Fe/H]relation.I assume that a totalerror in dependent variable∆for the i-th point,σtotal,i,can be expressed asσ2total,i=σ2∆,i+ d∆Even though my approach in this paper is only qualitative,there are two important characteristics of this study:1)The calibration of M I(RC)–[Fe/H]relation,has been based on the homogeneous set of the OGLE-II photometry.Therefore,no corrections due to the use of diﬀerent telescopes, instruments and reduction procedures are required.Unfortunately,this makes the above calibration vulnerable to unrecognized systematic problems of the OGLE photometry.2)The M I(RC)value has been derived based on observational data and not simply picked from a family of possible theoretical models of stellar evolution.Romaniello et al.(1999)provide an independent source of clump photometry in the LMC, but due to the importance of photometric homogeneity I am not able to use their data in a way consistent with the rest of my analysis.With reference to point2),it is crucial to note that observationally calibrated M I(RC)is not subject to the modeling uncertainties which aﬀect the Romaniello et al.(1999)distance to the LMC.However,my calibration is only as good as the assumptions and data that enter the analysis.Reddening corrections to the original Udalski’s(1998b)diagram,which is partly based on theﬁeld stars in the LMC and SMC,may be needed.Therefore,a more comprehensive study of the metallicity eﬀect on M I(RC)is necessary.Udalski’s(1999)determination based on the local clump is an important step toward establishing a reliable M I(RC)–[Fe/H]relation.Andy Becker deserves my special thanks for many stimulating discussions about the extinction issues in the Galactic bulge.I am deeply grateful to Andy Gould for his very careful reading of the original version of this paper and a number of insightful remarks.I also would like to thank Kem Cook for his valuable comments and discussions.Work performed at the LLNL is supported by the DOE under contract W7405-ENG-48.REFERENCESAlcock,C.et al.1998a,ApJ,492,190Alcock,C.et al.1998b,ApJ,494,396Cole,A.A.1998,ApJ,500,L137European Space Agency.1997,The Hipparcos Catalogue(ESA SP-1200)(Paris:ESA) Feast,M.W.,&Catchpole,R.M.1997,MNRAS,286,L1Girardi,L.,Groenewegen,M.A.T.,Weiss,A.,&Salaris,M.1998,MNRAS,301,149 Gould,A.,Popowski,P.,&Terndrup,D.M.1998,ApJ,492,778Minniti,D.1996,ApJ,459,175Minniti,D.,Olszewski,E.W.,Liebert,J.,White,S.D.M.,Hill,J.M.,&Irwin,M.J.1995, MNRAS,277,1293Ng,Y.K.,Bertelli,G.,Chiosi,C.,&Bressan,A.1996,A&A,310,771Paczy´n ski,B.1998,Acta Astron.,48,405Paczy´n ski,B.,&Stanek,K.Z.1998,ApJ,494,L219Paczy´n ski,B.,Udalski,A.,Szyma´n ski,M.,Kubiak,M.,Pietrzy´n ski,G.,Soszy´n ski,I., Wo´z niak,P.,&˙Zebru´n,K.,1999,Acta Astron.,49,319Romaniello,M.,Salaris,M.,Cassisi,S.,&Panagia,N.1999,accepted to ApJ(astro-ph/9910082)Stanek,K.Z.1996,ApJ,460,L37.Stanek,K.Z.,&Garnavich,P.M.1998,ApJ,500,L141Stanek,K.Z.,Ka l u˙z ny,J.,Wysocka,A.,&Thompson,I.1999,submitted to AJ (astro-ph/9908041)Stanek,K.Z.,Mateo,M.,Udalski,A.,Szyma´n ski,M.,Ka l u˙z ny,J.,&Kubiak,M.1994, ApJ,429,L7Stanek,K.Z.,Zaritsky,D.,&Harris,J.1998,ApJ,500,L141Stutz,A.,Popowski,P.,&Gould,A.1999,ApJ,521,206Szomoru,A.,&Guhathakurta,P.1999,AJ,117,2226Udalski,A.1998b,Acta Astron.,48,113Udalski,A.1998c,Acta Astron.,48,383Udalski,A.1999,submitted to ApJ Letters(astro-ph/9910167)Udalski,A.,Szyma´n ski,M.,Kubiak,M.,Pietrzy´n ski,G.,Wo´z niak,P.,&˙Zebru´n,K.1998a, Acta Astron.,48,1Walker,A.R.1992,ApJ,390,L81Wo´z niak,P.,&Stanek,K.Z.1996,464,233Fig.1.—Left part presents(V−I)0colors of the Baade’s Window(open circles)and local (full squares)RR Lyrae stars as a function of(V−K)0.The(V−I)0shift between the two groups is of order of0.17mag.The standard extinction line(dashed)is parallel to the stellar lines and so will not cure the anomaly.The right panel shows a similar situation for clump giants.The vertical(V−I)0color axis should be treated as an abscissa.The two Gaussians are Paczy´n ski’s&Stanek’s(1998)ﬁts to the number densities of clump giants as a function of color:in the solar neighborhood(thick solid line)and in the Baade’s Window (thin dotted line).The(V−I)0diﬀerence between the maxima of the two clump groups is 0.21.Fig.2.—The expected adjustment to the(V−I)0as a function of the selective extinction coeﬃcient R V I.The full dot marks the standard extinction.The dotted lines indicate the new point which would explain the entire color anomaly seen in the Baade’s Window. Fig.3.—Udalski’s(1998b)diagram showing∆versus[Fe/H].In order of increasing[Fe/H], the data points correspond to the Carina,SMC,LMC,and Baade’s Window.The V-magnitude adjustment to RR Lyrae stars is limited to the OGLE-II-based correction in Baade’s Window.The possible changes to∆that move all the points on the diagram by the same amount can be ignored because they do not inﬂuence the calibration.0.51 1.52 2.50.20.40.60.81432100.811.21.41.61.82 2.2 2.4 2.6 2.8-0.2-0.1-2-1.5-1-0.50-1.4-1.2-1-0.8[Fe/H]。

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文档收集于互联网，已重新整理排版.word 版本可编辑,有帮助欢迎下载支持.1文档来源为:从网络收集整理.word 版本可编辑.Value of function ：函数值 Variable ：变数 Vector ：向量 Velocity ：速度Vertical asymptote ：垂直渐近线 Volume ：体积 X-axis ：x 轴x-coordinate ：x 坐标 x-intercept ：x 截距 Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点 T:Tangent function ：正切函数 Tangent line ：切线 Tangent plane ：切平面 Tangent vector ：切向量 Total differential ：全微分Trigonometric function ：三角函数 Trigonometric integrals ：三角积分 Trigonometric substitutions ：三角代换法 Tripe integrals ：三重积分 S:Saddle point ：鞍点 Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法 Second partial derivative ：二阶偏导数 Sector ：扇形 Sequence ：数列 Series ：级数 Set ：集合Shell method ：剥壳法 Sine function ：正弦函数 Singularity ：奇点Slant asymptote ：斜渐近线 Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线 Smooth surface ：平滑曲面 Solid of revolution ：旋转体 Space ：空间 Speed ：速率Spherical coordinates ：球面坐标 Squeeze Theorem ：夹挤定理 Step function ：阶梯函数 Strictly decreasing ：严格递减 Strictly increasing ：严格递增 Sum ：和 Surface ：曲面Surface integral ：面积分 Surface of revolution ：旋转曲面 Symmetry ：对称 R:Radius of convergence ：收敛半径 Range of a function ：函数的值域 Rate of change ：变化率 Rational function ：有理函数Rationalizing substitution ：有理代换法 Rational number ：有理数 Real number ：实数Rectangular coordinates ：直角坐标 Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数 Revolution , solid of ：旋转体 Revolution , surface of ：旋转曲面 Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何 Right-hand derivative ：右导数 Right-hand limit ：右极限 Root ：根 P 、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面 Paraboloid ：抛物面Parallelepiped ：平行六面体 Parallel lines ：并行线 Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程 Partial fractions ：部分分式 Partial integration ：部分积分 Partiton ：分割 Period ：周期Periodic function ：周期函数 Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律 Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理 Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题 Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理 Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则 Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线 Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法 Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数 Indefinite integral ：不定积分 Independent variable ：自变数 Indeterminate from ：不定型 Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度 Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值 Exponential Function ：指数函数 Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理 Factorial ：阶乘First Derivative Test ：一阶导数试验法 First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式 Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数 Horizontal asymptote ：水平渐近线 Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数 Decreasing sequence ：递减数列 Definite integral ：定积分Degree of a polynomial ：多项式之次数 Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数 Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数 Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分 C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标，一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成 Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数 Constant of integration ：积分常数 Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数 Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列 series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点 Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛 Absolute extreme values ：绝对极值 Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。

2004 AMC 10A Problems

2004 AMC 10A ProblemsProblem 1You and five friends need to raise 1500 dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?（A ）250 （B ）300 （C ）1500 （D ）7500 （E ）9000Problem 2For any three real numbers , , and , with c b ≠ , the operationis definedby:What is ? （A ）21- （B ）41- （C ）0 （D ）41 （E ）21 Problem 3Alicia earns 20 dollars per hour, of which 1.45% is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?（A ）0.0029 （B ）0.029 （C ）0.29 （D ）2.9 （E ）29Problem 4What is the value of if 21-=-x x ?（A ）21- （B ）21 （C ）1 （D ）23 （E ）2 Problem 5A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?（A ）211 （B ）141 （C ）212 （D ）71 （E ）72 Problem 6Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?（A ）22 （B ）23 （C ）24 （D ）25 （E ）26Problem 7A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack?（A ）96 （B ）98 （C ）100 （D ）101 （E ）134Problem 8A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places onetoken in the discard pile. The game ends when some player runs out of tokens. Players A,B and C start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?（A ）36 （B ）37 （C ）38 （D ）39 （E ）40Problem 9In the figure, ∠EAB and ∠ABC are right angles. AB=4, BC=6, AE=8, and AC and BE intersect at D . What is the difference between the areas of △ADE and △BDC ?（A ）2 （B ）4 （C ）5 （D ）8 （E ）9Problem 10Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?（A ）12829 （B ）12823 （C ）41 （D ）12835 （E ）21 Problem 11A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by 25% without altering the volume, by what percent must the height be decreased?（A ）10% （B ）25% （C ）36% （D ）50% （E ）60%Problem 12Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?（A ）24 （B ）256 （C ）768 （D ）40,320 （E ）120,960Problem 13At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?（A ）8 （B ）12 （C ）16 （D ）18 （E ）24Problem 14The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average would be 21 cents. How many dimes does she have in her purse?（A ）0 （B ）1 （C ）2 （D ）3 （E ）4Given that 24-≤≤-x and 42≤≤y , what is the largest possible value of xy x +?（A ）-1 （B ）21- （C ）0 （D ）21 （E ）1 Problem 16The 5*5 grid shown contains a collection of squares with sizes from 1*1 to 5*5. How many of these squares contain the black center square?（A ）12 （B ）15 （C ）17 （D ）19 （E ）20Problem 17Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?（A ）250 （B ）300 （C ）350 （D ）400 （E ）500Problem 18A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?（A ）1 （B ）4 （C ）36 （D ）49 （E ）81Problem 19A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?（A ）120 （B ）180 （C ）240 （D ）360 （E ）480Problem 20Points E and F are located on square ABCD so that △BEF is equilateral. What is the ratio of the area of △DEF to that of △ABE ?（A ）34 （B ）23 （C ）3 （D ）2 （E ）1+3Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is 8/13 of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: radiansis 180 degrees.)（A ）8π （B ）7π （C ）6π （D ）5π （E ）4π Problem 22Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicircle from C intersects side AD at E . What is the length of CE ?（A ）252+ （B ）5 （C ）6 （D ）25 （E ）5-5 Problem 23Circles A , B , and C are externally tangent to each other and internally tangent to circle D . Circles B and C are congruent. Circle A has radius and passes through the center of D .What is the radius of circle B ?（A ）32 （B ）23 （C ）87 （D ）98 （E ）331+ Problem 24 Let, be a sequence with the following properties. (i), and (ii) for any positive integer .What is the value of ?（A ）1 （B ）992 （C ）1002 （D ）49502 （E ）99992Problem 25Three mutually tangent spheres of radius 1 rest on a horizontal plane . A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?（A ）2303+ （B ）3693+ （C ）41233+ （D ）952 （E ）223+2004 AMC 10A SolutionsProblem 1There are 6 people to split the 1500 dollars among, so each person must raise1500/6=250 dollars. Problem 241311)3,1,1()213,132,321(-=+-=--⊗=---⊗ Problem 320 dollars is the same as 2000 cents, and 1.45% of 2000 is 0.0145*2000=29cent. .Problem 41-x is the distance between and ;2-x is the distance between and 2. Therefore,the given equation says is equidistant from and , so23221=+=x . Alternatively, we can solve by casework (a method which should work for any similar problem involving absolute values of real numbers ). If 1≤x , then 1-x =1-x and 2-x =2-x , so we must solve 1-x =2-x , which has no solutions. Similarly, if 2≥x , then 1-x =x-1 and 2-x =x-2, so we must solve x -1=x -2, which also has no solutions. Finally, if 21≤≤x , then 1-x =x-1 and 2-x =2-x , so we must solve x -1=2-x , which has the unique solution x =3/2 .Problem 5There are ways to choose three points out of the 9 there. There are 8 combinations of dots such that they lie in a straight line: three vertical, three horizontal, and the diagonals.Problem 6Solution 1Since Bertha has 6 daughters, she has 30-6=24 granddaughters, of which none have daughters. Of Bertha's daughters, 24/6=4 have daughters, so 6-4=2 do not have daughters. Therefore, of Bertha's daughters and granddaughters, 24+2=26 do not havedaughters. Solution 2Draw a tree diagram and see that the answer can be found in the sum of 6+6 granddaughters, 5+5 daughters, and 4 more daughters. Adding them together gives the answer of .Problem 7There are 5*8=40 oranges on the st 1 layer of the stack. The nd2 layer that is added on top of the first will be a layer of 4*7=28 oranges. When the third layer is added on top ofthe nd 2, it will be a layer of 3*6=18 oranges, etc. Therefore, thereare 5*8+4*7+3*6+2*5+1*4=40+28+18+10+4=100Problem 8Look at a set of 3 rounds, where the players have x x ,1+, and 1-x tokens. Each of the players will gain two tokens from the others and give away 3 tokens, so overall, eachplayer will lose token. Therefore, after 12 sets of 3 rounds, or 36 rounds, the players willhave 3, 2, and tokens, respectively. After 1 more round, player A will give away hislast 3 tokens and the game will end. 37Problem 9Solution 1Since AE ⊥AB and BC ⊥AB , AE//BC . By alternate interior angles and AA ∽, we find that △ADE ∽△CDB, with side length ratio 4/3. Their heights also have the same ratio, and since the two heights add up to 4, we have that 716744=∙=ADE h and 712743=∙=CDB h .Subtracting the areas, 4712621716821=∙∙-∙∙ Solution 2Let [X ] represent the area of figure X . Note that [△BEA]=[△ABD]+[△ADE] and [△BCA]=[△ABD]+[△BDC] . [△ADE] - [△BDC] = [△BEA] - [△BCA]=0.5*8*4-0.5*6*4=4 Problem 10There are 4 ways that the same number of heads will be obtained; 0,1,2 or 3 heads.The probability of both getting heads isThe probability of both getting head isThe probability of both getting heads isThe probability of both getting heads isTherefore, the probabiliy of flipping the same number of heads is: 12835128418121=+++ Problem 11When the diameter is increased by 25%, it is increased by 5/4, so the area of the base isincreased by .To keep the volume the same, the height must beof the original height, which is a36%Problem 12For each condiment, a customer may either order it or not. There are 8 condiments.Therefore, there are 25628= ways to order the condiments. There are also choicesfor the meat, making a total of 256*3=768 possible hamburgers. Problem 13If each man danced with 3 women, then there were a total of 3*12=36 pairs of a man and a woman. However, each woman only danced with 2 men, so there must have been36/2=18 women.Problem 14Solution 1Let the total value (in cents) of the coins Paula has originally be x , and the number of coins she has be n . Then and 21125=++n x . Substituting yields 80,4)1(212520==⇒+=+x n n n .It is easy to see that Paula has 3 quarters and 0 dimes.Solution 2If the new coin was worth 20 cents, adding it would not change the mean. The additional 5 cents raise the mean by 1, thus the new number of coins must be 5. Therefore there were 4 coins worth a total of 4*20=80 cents. As in the previous solution, we conclude that the only way to get 80 cents using 4 coins is 25+25+25+5.Problem 15 Rewritex y x )(+ as x y x y x x +=+1. We also know that x y <0 because and are ofopposite sign. Therefore, xy +1 is maximized when x y is minimized, which occurs whenis the largest and is the smallest. This occurs at (-4,2), so 21211)(=-=+x y x Problem 16Solution 1There are:1 of the 1*1 squares containing the black square,4 of the 2*2 squares containing the black square, 9 of the 3*3 squares containing the black square,4 of the 4*4 squares containing the black square,1 of the 5*5 squares containing the black square.Thus, the answer is 1+4+9+4+1=19 Solution 2We use complementary counting. There are only2*2 and 1*1 squares that do not contain the black square. Counting, there are 12 2*2, and 25-1=24 1*1 squares that do not contain the black square. That gives 12+24=36 squares that don't contain it. There are a total of 25+16+9+4+2+1=55 squares possible, therefore there are 55-36=19 squares thatProblem 17Solution 1Call the length of the race track . When they meet at the first meeting point, Brenda hasrun 100 meters, while Sally has run 1002-x meters. By the second meeting point, Sally has run 150 meters, while Brenda has run x -150 meters. Since they run at a constantspeed, we can set up a proportion : 1501001501002-=-x x . Cross-multiplying, we get thatx =350. Solution 2The total distance the girls run between the start and the first meeting is one half of the track length. The total distance they run between the two meetings is the track length. As the girls run at constant speeds, the interval between the meetings is twice as long as the interval between the start and the first meeting. Thus between the meetings Brenda will run 2*100=200 meters. Therefore the length of the track is 150+200=350 meters Problem 18Solution 1Let be the common difference. Then 9, 9+d +2=11+d , 9+2d +20=29+2d are the terms ofthe geometric progression. Since the middle term is the geometric mean of the other two terms, 0)10)(14(1404)292(9)11(22=-+=-+⇒+=+d d d d d d . The smallestpossible value occurs when d =-14 , and the third term is 2(-14)+29=1 . Solution 2Let be the common difference and be the common ratio. Then the arithmeticsequence is 9,9+d , and 9+2d . The geometric sequence (when expressed in terms of )has the terms 9, 11+d , and 29+2d , Thus, we get the following equations:119119-=⇒+=r d d r d r 22992+=Plugging in the first equation into the second, our equation becomes 07189221829922=--⇒-+=r r r r . By the quadratic formula, can either be31- or 37. If is 31-, the third term (of the geometric sequence) would be 1, and if is 37, the third term would be 49. Clearly the minimum possible value for the third term of the1 .Problem 19The cylinder can be "unwrapped" into a rectangle, and we see that the stripe is aparallelogram with base 3 and height 80. Problem 20SolutionSince triangle BEF is equilateral, EA=FC , and EAB and FCB are SAS congruent. Thus, triangle DEF is an isosceles right triangle. So we let x DE =. Thus 2x FB EB EF ===. If we go angle chasing, we find out that ∠AEB =75º, thus ∠ABE =15º. 42615sin -=︒=EB AE . Thus 4262-=x AE , or 2)13(-=x AE . Thus2)13(+=x AB , and , and . Thus the ratio of the areas is 2 Solution 2 (Non-trig)Without loss of generality let the side length of ABCD be 1. Let x DE = and x AE -=1. Then triangles ABE and CBF are clearly congruent by HL,So CF=AE and DE=DF . We find that 2x EF BE ==, and so, by the PythagoreanTheorem, we have 2221)1(x x =+- . This yields 222=+x x , so x x 222-=. Thus, the desired ratio of areas is 2122122=-=-xx xx Problem 21Let the area of the shaded region be S , the area of the unshaded region be U , and theacute angle that is formed by the two lines be . We can set up two equations between S and U : π9=+U S , U S 138=. Thus π91321=U , and 739π=U , and thus 724739138ππ=∙=S . Now we can make a formula for the area of the shaded region in termsof : 724)49(22)4(*2)(2*22ππππθπππθπππθ=-+--+ Thus 7733πθπθ=⇒= Problem 22Solution 1Let the point of tangency be F . By the Two TangentTheorem BC=FC=2 and AE=EF =x . Thus DE=2-x .The Pythagorean Theorem on △CDE yields21)2(2)2(22222=⇒+=+-⇒=+x x x CE CD DE Hence 5.2=+=x FC CE Solution 2Clearly, EA=EF=BG . Thus, the sides of right triangle CDE are in arithmetic progression. Thus it is similar to the triangle 3-4-5 and since DC =2, CE =5/2 . Problem 23Let be the center of circle for all and let be the tangent pointof B, C . Since the radius of D is the diameter of A , the radius of D is 2. Let the radius of B,C be and let . If we connect, we get an isosceles triangle withlengths 1+r, 2r . Then right trianglehas legs and hypotenuse 2-r . Solvingfor , we get r x r r x 44)2(222-=⇒--=. Also, right trianglehas legs r,1+x , and hypotenuse 1+r . Solving,98)1()441(222=⇒+=-++r r r r So the answer is Problem 24)1(12...22...)2(22)2(2)22()2(19899989899999999100f f f f f ∙∙==⨯∙=⨯=⨯=D ⇒===++++49502)100(9912 (9899222)Problem 25We draw the three spheres of radius 1:And then add the sphere of radius 2:2004 AMC10A11 The height from the center of the bottom sphere to the plane is, and from the center of the top sphere to the tip is2.We now need the vertical height of the centers. If we connect the centers, we get a triangular pyramid with an equilateral triangle base. The distance from the vertex of the equilateral triangle to its centroid can be found by 30—60—90 △s to be.By the Pythagorean Theorem , we have 3693)32(222=⇒=+h h . Adding the heights up, we get 21369++2004 AMC 10A Answer Key1. A2. B3. E4. D5. C6.E7.C8.B 8. B 10. D11.C 12. C 13.D 14.A 15. D 16.D 17. C 18.A 19.C 20.D21.B 22.D 23.D 24.D 25.B。

介观模拟相关翻译

How to predict diffusion of medium-sized molecules in polymer matrices. From atomistic to coarse grainsimulationsAbstract :The normal diffusion regime of many small and medium-sized molecules occurs on a time scale that is too long to be studied by atomistic simulations. Coarse-grained (CG) molecular simulations allow to investigate length and time scales that are orders of magnitude larger compared to classical molecular dynamics simulations, hence providing avaluable approach to span time and length scales where normal diffusion occurs. Here we develop a novel multi-scale method for the prediction of diffusivity in polymer matrices which combines classical and CG molecular simulations. We applied an atomistic-based method in order to parameterize the CG MARTINI force field, providing an extension for the study of diffusion behavior of penetrant molecules in polymer matrices. As a case study, we found the parameters for benzene (as medium sized penetrant molecule whose diffusivity cannot be determined through atomistic models) and Poly (vinyl alcohol) (PVA) as polymer matrix. We validated our extended MARTINI force field determining the self diffusion coefficient of benzene (2.27·10−9m2s−1) and the diffusion coefficient of benzene in PVA(0.263·10−12m2s−1). The obtained diffusion coefficients are in remarkable agreement with experimental data (2.20·10−9m2s−1 and 0.25·10−12m2s−1, respectively). We believe that this method can extend the application range of computational modeling, providing modeling tools to study the diffusion of larger molecules and complex polymeric materials.Keywords:Coarse grain . Diffusion .Molecular dynamics simulation . Multi scale models . Nanofiltration . Polymeric matricesIntroduction:Molecular dynamics (MD) simulations are a powerful tool in the material science field as they provide material’s structural and dynamics details that are difficult, cost- or time-consuming to be assessed with experimental techniques. In particular, MD simulations are a valid tool for the design of polymeric membranes with tuned permeability properties. Up to now, however, the design of barrier materials is based on trial and error experimental procedures, in which a large part of the effort is spent to synthesize and characterize materials and blends which finally turn out to be unsatisfactory.Atomistic simulation have been successfully applied in the past to obtain the diffusion coefficients of small molecules (like oxygen, carbon dioxide or water) inpolymeric membranes [1–7], polymeric blends [8, 9], biopolymers [10, 11] and organic-inorganic hybrid membranes [12]. However, despite the increasing computational power available to researchers and the improvements in the MD codes, atomistic simulations are still able to handle only systems with tens or hundreds of thousands of atoms and in the nanoseconds time scale. Several phenomena of interest at the material scale, however,cover time and space scales larger than those affordable with atomistic modeling. This is the case of normal diffusion regime. The diffusion coefficient D can be directly calculated from the motion of the particles extracted during a MD simulation, in particular from the mean square displacement (MSD) of the particles, using the Einstein equation [5]. This equation holds only in the case that the observation time (i.e., the simulation time) is large enough to allow the particles to show uncorrelated motion. This means that the MSD is linear with time, i.e., MSD∼t n where n=1. Conversely, if n<1 then the diffusion is in the anomalous diffusion regime. Therefore, in order to assess an accurate diffusion coefficient, MD simulations must reach the normal diffusive regime which, depending on the membrane and diffusive molecule, can be on time scales higher than few nanoseconds. This regime is therefore often difficult to reach for molecules larger than diatomic molecules, like water or benzene, hindering molecular simulations to assess an accurate diffusion coefficient [12, 13].Recently, the use of coarse grain (CG) modeling, in which a number of atoms are condensed into beads or interacting particles, has proven to be a suitable option to model large systems and long time scales, providing realistic results. The methods, assumptions and level of resolution greatly vary depending on the scope of the models and the properties of interests [14, 15]. CG models have been developed with particular focus on biomolecular systems, since biomolecules are often too large and their characteristic times too long to be treated with full atomistic simulations. However, there are no theoretical impediments to the application of CG methods to polymeric materials.In this view, of particular interest is the coarse grain force field developed by Marrink and co-workers and called MARTINI force field. This CG force field was originally developed to model the lipid bilayers forming the cellular membranes and then extended to proteins [16–18]. Unlike other CG models, which focus on accurate modeling of a particular state or a particular molecule, the philosophy of the MARTINI force field is to accurately parameterize the basic building blocks of the system (e.g., the single amino acids for proteins), thus allowing a broad spectrum of applications without the need of reparameterization.Relying on the same philosophy, in this work we present an atomistic-informed parameterization of the MARTINI force field for the modeling of penetrants diffusion in polymeric membranes. We used atomistic simulations to calculate the interaction free energy between the basic building blocks of the system, i.e., the penetrant molecule and the polymer monomers, and then we performed CG molecular dynamics simulations to assess the diffusion behavior. As a test case, we investigated the diffusion behavior of benzene in a matrix of Poly (vinyl alcohol), PVA. Benzene, an important industrial solvent and precursor in the production of drugs, plastics,synthetic rubber, and dyes, was chosen since it is a well known representative of medium-sized permeant molecules. On the other hand, the choice of PVA as the polymer matrix is based on the fact that this polymer is widely used in several fields and finds applications as membrane material due to its excellent chemical stability, film forming capacity, barrier properties and high hydrophilicity.Methods:Coarse grain mappingThe original MARTINI mapping scheme is based on the four-to-one rule, i.e., on average four heavy (nonhydrogen) atoms are grouped into a bead or interaction center. Here we used a similar mapping, where the benzene molecule is represented by one bead and the vinyl alcohol (VA) monomers are represented by a different bead (see Fig. 1). The mass of the two types of beads are calculated as the sum of the masses of the atoms grouped into the bead.Bonded and nonbonded interactionsThe chemically bonded beads interact through a bond interaction modeled as a harmonic potential Vbond(r):where r is the distance between two bonded beads, kbond is the force constant of the bond interaction and r0 is the equilibrium distance. In the system under investigation, only VA beads representing chemically bonded monomers along a PVA chain are subject to bond interactions.The beads i and j that are not chemically connected interact via nonbonded interactions, which are described by a Lennard-Jones 12-6 potential:where r is the distance between the two nonbonded beads, εij is the energy minimum depth (the strength of the interaction) and r0ij is the distance at the minimum of the potential.ParameterizationGiven the mapping scheme and the type of interaction, we needed to calculate one set of bonded parameters (for bonded VA–VA beads) and three sets of nonbonded parameters (VA–VA, benzene–benzene and VA–benzene). For each set we run a full atomistic simulation of the two groups involved in the interaction and we calculated the free energy of interaction between the two groups. The free energy calculationsare performed using the adaptative biasing force (ABF) framework [19] as implemented in the NAMD code [20, 21]. The atomistic MD simulations are carried out using the NAMD program and the all-atom CHARMM force field [22], for a simulation time of 40 ns (time step of 1 fs) at a temperature of 300 K. Nonbonded interactions are computed using a switching function between 20 and 22 Å. The free energy of interaction, computed via the ABF framework, is monitoredbetween the two groups of atoms as a function of their center-of-mass distance, in the range 2–20 Å using windows of 0.01 Å.Generation of the CG modelsWe generated three different molecular models: pure benzene, pure PVA and PVA with a small amount of benzene as penetrant molecule. All three systems were generated in the atomistic form using the Amorphous Cell construction tool of Materials Studio 4.4 (Accelrys, Inc.). The pure benzene system contained 500 benzene molecules in a cubic periodic box with initial density of 0.88 gcm−3. For the pure PVA system we considered six atactic PVA molecules (consisting of 200 repeat units) in a cubic periodic box with initial density of 1.25 gcm−3. Finally, the third system contained four PVA chains (of 200 repeat units) and 12 benzene molecules in a cubic periodic box with initial density of 1.25 gcm−3. The three atomistic systems are then converted into CG systems using the Coarse Grainer tool of Materials Studio according to the mapping scheme described above (see Fig. 2).CG simulationsCG molecular dynamics simulations are carried out using the Mesocite module and the MARTINI force field implemented in Materials Studio. Prior to the simulations, we modified the original force field including the parameters for bonded and nonbonded interactions between benzene and VA beads, as obtained from the ABF atomistic simulations. The CG systems are minimized for 1000 steps, then equilibrated for 1 ns (using a time step of 20 ps) at 300 K. Finally, production simulations are run for asimulated time of 10 ns (for pure benzene and pure PVA) and 200 ns (for benzene in PVA matrix). The diffusion coefficient Di of a single permeant molecule i is calculated by the Einstein relation, starting from the diffusion trajectory ~rietT which is determined during the production MD simulations:where represents the root mean square displacement (MSD) of the permeant molecule i averaged over all possible time origins and t represents the time. The computationally derived diffusion coefficient D for a given kind of permeant molecule is then obtained as average overthe diffusion coefficients for N permeant molecules:In this work the diffusion of all 500 benzene beads (for benzene self diffusion) and the 12 benzene beads (for benzene in PVA) was investigated during the CG simulations.Results:The parameters of the CG force field, i.e. the bonded and nonbonded interactions between the beads, are calculated through full atomistic simulations and by applying the ABF framework. The ABF calculations provide the free energy profile as a function of the distance between the group of atoms, as shown in Fig. 3. The free energy profile of two bonded VA monomers (Fig. 3a) is interpolated with a harmonic potential, giving the equilibrium distance and the force constant (see Table 1). On the other end, from the interaction free energy profiles between nonbonded VA monomers, benzene molecules and VA-benzene molecules (Fig. 3b–d) we obtained the three sets of r0 (energy minimum) and ε(energy minimum depth), used to feed the Lennard-Jones 12-6 potential of the CG force field (see Table 1).We used the CG approach to investigate three different systems: pure benzene, pure PVA and benzene molecules in a PVA matrix (see Fig. 2). The pure benzene system consisted of 500 benzene molecules, which were coarse grained into benzene beads. We measured the density of the system during the 1 ns equilibration obtaining a realistic value of 0.881±0.001 gcm−3 (where the experimental value is 0.876 gcm−3). After the equilibration we carried out a 10 ns simulation in which we monitored the MSD of the benzene molecules (see Fig. 4a) and, by applying Eq. 3, we determined the self diffusion coefficient of benzene, obtaining a value of 2.273±0.588·10−9m2s−1, very close to the experimental value of 2.203±0.004 ·10−9m2s−1 [23]. In the case of pure PVA the system consisted of a periodic box with six chains of 200 monomers each. The coarse grained system was equilibrated for 1 ns and the density of the CG PVA box was measured, giving a value of 1.305± 0.011 gcm−3, which lie in the experimental range (1.232– 1.329 gcm−3) [24]. Finally, the third system consisted of 12 benzene molecules diffusing in a PVA matrix. The density of the coarse grained system, measured at the end of 1 ns equilibration, was 1.293±0.003, similar to that of pure PVA. The MSD of the benzene beads was monitored during a 200 ns MD simulation (see Fig. 4b), and from the derivative of the curve, we calculated the diffusion coefficient of benzene in PVA. We obtained a diffusion coefficient of0.263±0.035·10−12m2s−1 which is in good agreement with the experimental value, that is 0.25·10−12m2s−1[12]. The 200 ns simulation of the 4 nm×4 nm×4 nm CG box (with 812 beads representative of ≈6000 atoms) took 36 hours on a single CPU. The results of the simulations as well as the experimental data are shown in Table 2.Table 1 Parameters of the CG force field, obtained fromfull atomistic free energy calculations.Within this approach bonded interactions (i.e., VA beads covalently connected) are modeled through a harmonic potential, while nonbonded interactions are approximated with a Lennard-Jones 12-6 potentialInteraction Bonded Nonbondedkbond (kcal/mol/Å2) r0 (Å) ε(kcal/mol) r0 (Å) VA–VA 40.36 2.76 1.306 4.115 VA–Benzene - - 0.957 5.51 Benzene–Benzene - - 0.9634 5.62 Table 2 Main results of the CG simulations. The conversion of the three systems under study fromatomistic to coarse grain reduces the interacting particles by a factor between 7 and 12, depending on the specific system. Despite the loss of atomistic details, the CG models feature realistic densities and are able to predict benzene diffusion coefficient very close to the experimentsSystems Totalatoms TotalbeadsFinal density(g/cm3)Experimentaldensity (g/cm3)PredictedDbenzene(m2/s)ExperimentalDbenzene(m2/s)500 benzene molecules 6000 500 0.881 0.876 2.273·10−9 2.203·10−9[23]6 PVA chains 8400 1200 1.305 1.23–1.32 [24] --4 PVA chains + 12 benzene 5744 812 1.293 -- 0.263·10−120.25·10−12[24]Discussion:In this work we present a multi-scale method for the parameterization of the MARTINI CG force field and its application for the calculation of the diffusivity of medium sized molecules in polymeric membranes. In the past, atomistic simulations have been already successfully applied for the calculation of diffusivity of small molecules in polymeric membranes. Nonetheless, the computational costs can hinder the ability of MD simulations to predict such a parameter. Indeed, the Einstein relation (Eq. 3) applied for the assessment of the diffusion coefficient can only be used when the simulation is in the regime of normal diffusion and this realm is reached when the slope of the function log[MSD(t)] = f [log(t)] equals 1. In the case of very small molecules, like oxygen or hydrogen, the normal diffusion regime can be reached within few nanoseconds [5] while, in the case of medium sized molecules like benzene [12] or even water [13], it cannot be reached within the limit of atomistic simulations. In this view, the use of CG simulations, where a number of atoms are condensed into beads or interacting particles, can be a useful approach to overcome the limitations of atomistic simulations.In order to test the feasibility of the CG approach, we investigated the self diffusion coefficient of benzene and its diffusivity in PVA matrix using an atomistic-informed CG force field (see Fig. 1). As shown in Table 2, the coarse graining reduces the number of interacting particles by a factor of ≈10. Furthermore,since the CG interactions are much smoother compared to atomistic interactions [16], it is possible to use a time step of 20–40 fs, much larger than that typical of classical MD (1–2 fs). Thus, the CG approach leads to a total speed-up factor of 200–400 with respect to atomistic simulations.In this work we did not rely on the standard MARTINI bead types but rather we defined two ad hoc bead types (one for benzene molecule and one for VA monomer) and we estimated the bonded and nonbonded parameters using atomistic free energy calculations.The predicted parameters are then used to feed the MARTINI force field. The extended force field is then used to perform CG molecular dynamics simulations of three different systems: pure benzene, pure PVA and benzene in PVA (see Fig. 2). The coarse grain systems underwent1 ns equilibration dynamics, during which they reached a stable density very close to the experimental values (Table 2), thus confirming that the outcome of the atomistic free energy calculations are reasonable. As a final validation of our approach, we ran longer CG simulations in order to assess the self diffusion coefficient of benzene and the diffusion coefficient of benzene in PVA. For the pure benzene, we ran a 10 ns simulations from which we estimated the diffusion coefficient. The calculated and the experimental values are shown in Table 2, and the comparison confirms that the CG model is able topredict the experimental value with good approximations. In the case of benzene diffusing in a PVA matrix, 10 ns of simulation were not enough to reach the normal diffusion regime, since the derivative of the function log[MSD(t)] =f [log(t)] was lower than 1. This result is in agreement with the observations of Pan et al. [12], which showed that MD simulations of a few nanoseconds are not long enough to reach the normal diffusion regime and provide a good estimation of benzene diffusivity in PVA. For this reason, we run a longer simulation of 200 ns, in order to reach the realm of normal diffusivity, as shown in Fig. 5. Indeed, the trajectory of this long simulation permitted us to obtain a diffusion coefficient very close to the experimental value (see Table 2). This result confirms the feasibility of the CG approach to reach the normal diffusion regime of medium sized molecules in polymer matrices and that our multi-scale approach is a valid method to treat this kind of problem.Fig. 5 Log(MSD) vs. log(t) plots (straight lines) and linear interpolation (dashed lines) for the self diffusion of benzene (panel a) and for the diffusion of benzene in PVA (panel b) obtained from CGsimulations. The plots show that in the case of pure benzene the normal diffusion regime is reached, as indicated by the slope very close to 1, already in the range from 10to 101ns, while in the case of benzene in PVA (panel b) a longer simulation time is required since the normal diffusion regime is reached in the range from 101.7 to 102.3ns (i.e., from 50 to 200 ns)The major limitation of atomistic computational techniques applied in the literature to solve diffusive problems is related to the restricted time scale and sample size which can be simulated, which are a few nanoseconds and a few nanometers, respectively. Thus, when the phenomena under investigation exceed these limits MD simulations fail to provide reliable values of the diffusivity. In order to overcome these limitations we developed a novel method, which consits of combining atomistic and coarse grain simulations in a multi-scale paradigm, where the parameters for the meso-scale model are derived from atomistic MD simulations. Similar techniques are increasingly applied for the study of biological problems, but to the best of our knowledge, have not been used for the investigation of diffusion problems. Here we showed that this technique can be successfully applied to investigate the diffusion of penetrant molecules in polymer matrices, reliably predicting experimental data. A similar method as used in this paper could be applied to study the diffusion of larger molecules, which require a longer time to reach the normal diffusion regime, or the study of complex polymeric materials, for which representative volumes are larger than a few cubic nanometersConclusions:In conclusion, the main focus of this work has been to develop and validate a novel multi-scale method for the prediction of diffusivity in polymer matrices. We demonstrated that atomistic-informed CG simulations can be a valid approach to treat problems where the computational limits of classical MD simulations are too restrictive while, at the same time, strictly atomistic details are not mandatory. Thus, the multi-scale approach presented in this work extends the application range of computational modeling and provide a useful tool to investigate phenomena at the micro-scale which determine macroscopic physical properties of polymeric materials.In this view, multi-scale paradigm here discussed can further help computation aided molecular modeling to reduce the extent of the experimental trial-and-error approach during the design and investigation of new materials, thus resulting in a more cost and time efficient process.Acknowledgments:This research was partially supported by the Italian Institute of Technology (IIT). The authors declare no conflict ofinterest of any sort.References:1. 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The Metric Nearness Problem

THE METRIC NEARNESS PROBLEM∗JUSTIN BRICKELL†,INDERJIT S.DHILLON†,SUVRIT SRA†,AND JOEL A.TROPP‡Abstract.Metric nearness refers to the problem of optimally restoring metric properties to distance measurements that happen to be non-metric due to measurement errors or otherwise.Met-ric data can be important in various settings,for example in clustering,classiﬁcation,metric-based indexing,query processing and graph theoretic approximation algorithms.This paper formulates and solves the metric nearness problem:Given a set of pairwise dissimilarities,ﬁnd a“nearest”set of distances that satisfy the properties of a metric—principally the triangle inequality.For solving this problem,the paper develops eﬃcient triangleﬁxing algorithms that are based on an iterative projection method.An intriguing aspect of the metric nearness problem is that a special case turns out to be equivalent to the All Pairs Shortest Paths problem.The paper exploits this equivalence and develops a new algorithm for the latter problem using a primal-dual method.Applications to graph clustering are provided as an illustration.We include experiments that demonstrate the com-putational superiority of triangle-ﬁxing over general-purpose convex programming software.Finally, we conclude by suggesting various useful extensions and generalizations to metric nearness.Key words.matrix nearness problems,metric,distance matrix,metric nearness,all pairs shortest paths,triangle inequalityAMS subject classiﬁcations.05C12,05C85,54E35,65Y20,90C06,90C081.Introduction.Most applications make some assumptions about the prop-erties that the input data should satisfy.Due to measurement errors,noise,or an inability to gather data completely,an application may receive data that does not conform to its requirements.For example,imagine taking measurements as a part of some experiment.The theory suggests that the quantities measured should represent distance values amongst points in a discrete metric space.However,measurements being what they are,one ends up with a set of numbers that do not represent ac-tual distance values,primarily because they fail to satisfy the triangle inequality.It might be beneﬁcial to somehow optimally massage the measurements to obtain a set of“nearest”distance values that obey the properties of a metric.It could also happen that experimental expenses and diﬃculties prevent one from making all the measurements.Before this incomplete set of measurements can be used in an application it might need to be tweaked,preferably minimally.As before, obtaining a“nearest”set of distance values(measurements)seems to be desirable.Both scenarios above lead to the metric nearness problem:Given a set of input distances,ﬁnd a“nearest”set of output distances that satisfy the properties of a metric.The notion of nearness is quantiﬁed by the function that measures distortion between the input and output distances.Matrix nearness problems[10]oﬀer a natural framework for pursuing the above-mentioned ideas.If there are n points,we may collect the measurements into an n×n symmetric matrix whose(j,k)entry represents the distance between points j and k.Then,we seek to approximate this matrix by another(say M)whose entries satisfy the triangle inequalities.That is,m ij≤m ik+m kj for every triple(i,k,j). Any such matrix will represent the distances among n points in some metric space.∗This research was supported by NSF grant CCF-0431257,NSF Career Award ACI-0093404,and NSF-ITR award IIS-0325116.A preliminary version of this work appeared at NIPS2004,Vancouver, Canada†Dept.of Computer Sciences.The University of Texas at Austin.Austin,TX,78712.‡Dept.of Mathematics.University of Michigan at Ann Arbor.Ann Arbor,MI,48109.12J.BRICKELL,I.S.DHILLON,S.SRA,AND J.A.TROPPWe calculate approximation error with a distortion measure that depends on how the corrected matrix should relate to the input matrix.For example,one might prefer to change a few entries signiﬁcantly or to change all the entries a little.This paper considers metric nearness problems that use vector norms for characterizing distortion.There is no analytic solution to the metric nearness problem.Fortunately this problem lends itself to a convex formulation,whereby developing algorithms for solv-ing it becomes much easier.However,despite the natural convexity of the formu-lations,the large number of triangle inequality constraints can make traditional ap-proaches or general purpose convex programming software much too slow.This paper provides solutions to the metric nearness problem that exploit its inherent structure for eﬃciency gains.The remainder of this paper is structured as follows.Section1.1highlights the principal contributions of this paper.Section2develops a convex formulation of the metric nearness problem.Following that,Section3provides eﬃcient triangle ﬁxing algorithms for solving the metric nearness problems described in Section2.An interesting connection of metric nearness with the All Pairs Shortest Paths(APSP) problem is studied in Section4.This connection leads to a curious new primal-dual algorithm for APSP(Algorithm4.3).Applications of metric nearness to clustering are discussed in Section5.Ex-periments highlighting running time studies and comparisons against the CPLEX software are given in Section6.1,whereas experiments illustrating the behavior of the primal-dual metric nearness algorithm are the subject of Section6.2.Section7.1discusses some variations to the metric nearness problem that may also be studied.Section7.2describes possible future work and extensions to this paper,while two open problems are mentioned in Section7.3.Finally,Section7.4 summarizes related work and concludes this paper.1.1.Contributions of this paper.In preliminary work[7],the authors pre-sented the basic ideas about convex formulations of metric nearness and triangleﬁxing algorithms.However,many of the details necessary for understanding and actually implementing the triangleﬁxing algorithms were missing.This paperﬁlls that gap by presenting a detailed derivation forℓ1(consequentlyℓ∞),andℓ2norm based metric nearness problems.Pseudocode for both theℓ1andℓ2problems is given along with the derivations.When one allows only decreasing changes to the input,then metric nearness becomes equivalent to the All Pairs Shortest Paths(APSP)problem[22].This paper studies this decrease-only version of metric nearness,and consequently obtains a new primal-dual algorithm for solving the APSP problem.This algorithm possesses some interesting characteristics related to its convergence behavior that are discussed in this paper.The paper discusses applications to the Max-Cut problem.We also developed eﬃcient C++code for metric nearness that outperforms CPLEX by factors of up to 30,and it may be requested from the authors.2.Problem Formulation.We begin our formulation with a few basic deﬁni-tions.We deﬁne a dissimilarity matrix to be a symmetric,nonnegative matrix with a zero diagonal.Such matrices are used to represent pairwise proximity data between objects of a certain type.A distance matrix is deﬁned to be a dissimilarity matrixMETRIC NEARNESS3 whose entries satisfy the triangle inequalities.Speciﬁcally,M is a distance matrix ifm ij≥0,m ii=0,m ij=m ji,and m ij≤m ik+m kj for distinct triples(i,k,j).We remark that symmetry,while part of the deﬁnition of a metric,is not crucial toour algorithms;asymmetry can be handled at the expense of doubling the runningtime and storage.The distance matrices studied in this paper are assumed to arise from measuringinter-point distances between n points in a pseudo-metric space(i.e.,two distinctpoints can lie at zero distance from each other).Consequently,distance matricescontain N= n2 parameters,and we denote the set of all n×n distance matrices as M N.We observe that the set M N is a closed,convex polyhedral cone.Assume that the input is a dissimilarity matrix D.Metric nearness seeks adistance matrix M that is closest to D,with respect to some measure of“closeness.”Formally,we seek a matrix M so thatM∈argminX∈M NX−D ,(2.1)where · is a norm.Though it is possible to use any norm in the metric nearness problem(2.1),we restrict our attention to the vectorℓp norms,wherein we treat the strict upper triangular part of our matrices as vectors.Theorem2.1(Attainment of minimum).The functional f(X)= X−D always attains its minimum on M N.Moreover,every local minimum is a global minimum.Proof.The latter claim follows immediately from the convexity of f.It remains to show that f(X)always attains its minimum on the cone M N.For convenience,we pass to the function g(Y)= Y .Notice that if g attains a minimum on M N−D, then f(X)attains a minimum on M N.The function g is a closed convex function, and it is homogeneous of degree one,so we can compute its recession function as(g0+)(Y)=limh→0(g(h Y)−g(0))/h=limh→0g(h Y)/h=g(Y).But g is non-negative,so its only directions of recession are directions in which it is constant.Since M N−D is a closed,polyhedral cone,we may apply[23,Theorem27.3] to conclude that g attains a minimum on this cone,whereby f attains its minimum on M N.2.1.Metric Nearness for theℓ2norm.We start with a formulation for the vectorℓ2norm based metric nearness problem.Given the input dissimilarity matrix D=[d ij](where d ij=d ji),we wish to obtain a distance matrix X that minimizes the squared error1 2i<j(x ij−d ij)2.Note that the sum above ranges over i<j,since the involved matrices are symmetric and have a zero diagonal.Let T n be the set of3 n3 triples,each of which corresponds to a triangle inequality that the entries of an n×n distance matrix must satisfy.Formally,T n={(i,j,k),(j,k,i),(k,i,j):1≤i<k<j≤n},(2.2)4J.BRICKELL,I.S.DHILLON,S.SRA,AND J.A.TROPPwhere the triple(i,k,j)corresponds to the triangle inequalityx ij≤x ik+x kj.With the introduction of an auxiliary matrix E=X−D that represents the changes to the original dissimilarities,theℓ2metric nearness problem can be rewritten as the following quadratic program:Minimizee ij 12i<je2ij,(2.3)subject to e ij−e ik−e kj≤d ik+d kj−d ij=v ikj∀(i,k,j)∈T n.(2.4) The triangle inequality constraints are encoded by(2.4).Since theℓ2norm is strictly convex,the solution to(2.3)is unique.The variable v ikj quantiﬁes the violation in the(i,k,j)triangle inequality.Note that non-negativity of x ij need not be enforced explicitly as it is implied by the triangle inequalities.2.2.Metric Nearness for theℓ1andℓ∞norms.When measuring approxi-mation error using theℓ1norm,we wish to minimizei<j|e ij|,(2.5)where e ij=x ij−d ij as in the previous section.However,to write the problem as a linear program,we need to introduce additional variables f ij=|e ij|.The resulting problem is the following linear program:Minimizee ij,f iji<j1·f ij+0·e ij ,(2.6)subject toe ij−e ik−e kj≤v ikj∀(i,k,j)∈T n,−e ij−f ij≤01≤i<j≤n,e ij−f ij≤01≤i<j≤n.(2.7)The fact that f ij=|e ij|is accomplished by the last two sets of inequalities in(2.7).Similarly,for theℓ∞nearness problem,we introduce a variableζ=max ij|e ij| that represents the vectorℓ∞norm of E.Theℓ∞nearness problem becomes: Minimizee ij,ζζ+ i<j0·e ij,(2.8)subject toe ij−e ik−e kj≤v ikj∀(i,k,j)∈T n,−e ij−ζ≤01≤i<j≤n,e ij−ζ≤01≤i<j≤n.(2.9)The last two sets of inequalities in(2.9)express the fact|e ij|≤ζfor all i and j.2.3.Metric nearness forℓp norms.Metric nearness may be easily formulated forℓp norms,where1<p<∞.The problem is the following convex program:Minimizee ij 1pi<j|e ij|p,subject to e ij−e ik−e kj≤v ikj∀(i,k,j)∈T n.METRIC NEARNESS5 Since theℓp norms are strictly convex for1<p<∞,the associated metric nearness problems have unique solutions.There is a basic intuition for choosing p when solving the nearness problems.Theℓ1norm error is computed as the absolute sum of changes to the input matrix,whileℓ∞reﬂects only the maximum absolute change.The otherℓp norms interpolate between these two extremes.Thus,a small value of p typically results in a solution that prefers a few large changes to the original data,while a large p typically results in a solution with many small changes.In practice,however,theℓ1,ℓ2,andℓ∞problems are computationally easier to solve than those using arbitraryℓp norms.Thus,we focus primarily on these three problems.3.Triangle Fixing Algorithms.The previous section formulated the metric nearness problem as a quadratic program for theℓ2norm,as a linear program forℓ1 andℓ∞norms,and as a convex program forℓp ing oﬀ-the-shelf software for these formulations might appear to be an attractive way to solve the corresponding problems.However,it turns out that the computational time and storage require-ments of such an approach can be prohibitive.An eﬃcient algorithm must exploit the inherent structure oﬀered by the triangle inequalities.In this section,we develop triangleﬁxing algorithms,which take advantage of this structure to eﬃciently solve the problem forℓp norms.These algorithms iterate through the triangle inequalities, optimally enforcing any inequality that is not satisﬁed.While enforcing the triangle inequalities,one needs to introduce appropriate correction terms to guide the iterative algorithm to the globally optimal solution.The details are provided below.3.1.Triangleﬁxing forℓ2metric nearness.Our approach for solving(2.3) is iterative,and is based on the technique described in[2].Collecting all the e ij values into vector e and the violation amounts v ijk into v,problem(2.3)may be rewritten asmin e 12e T esubject to Ae≤v,(3.1)where matrix A encodes the triangle inequalities(2.4),whereby,each row of A has one+1entry,and two−1entries.The Lagrangian of(3.1)isL(e,z)=12e T e+ z,Ae−v ,where z is the dual vector.A necessary condition for optimality of(3.1)is∂L∂e=0=⇒e=−A T z,z≥0.(3.2) Using(3.2)we see that the dual problem corresponding to(3.1)ismax z≥0g(z)=−12z T AA T z−z T v.(3.3)We solve(3.1)iteratively,wherein we initialize both e and z to zero as this choice satisﬁes(3.2).At each subsequent iteration we update the dual vector z one coordinate at a time,thereby resulting in a dual coordinate ascent procedure,while maintaining(3.2).Assume that the dual variable corresponding to inequality(i,k,j) is updated,i.e.,z′ikj=z ikj+θ.Then,the corresponding update to the primal is obtained via(3.2),i.e.,e′=e−θa ikj(using the fact that e′=−A T z′),where a ikj is6J.BRICKELL,I.S.DHILLON,S.SRA,AND J.A.TROPPAlgorithm3.1:Metric Nearness forℓ2norm.Metric Nearness L2(D,κ)Input:Dissimilarity matrix D,toleranceκOutput:M=argminX∈M N X−D 2.{Initialize the primal and the dual variables}e ij←0for1≤i<j≤n(z ijk,z jki,z kij)←0for1≤i<k<j≤nδ←1+κwhile(δ>κ){convergence test}foreach triangle inequality(i,k,j)v←d ik+d kj−d ij{Compute violation}θ∗←13(e ij−e ik−e kj−v)(⋆)θ←max{θ∗,−z ikj}{Stay within half-space of constraint}e ij←e ij−θ,e ik←e ik+θ,e kj←e kj+θ(⋆⋆)z ikj←z ikj+θ{Update dual variable} end foreachδ←sum of changes in the e ij valuesend whilereturn M=D+Ethe column vector containing the entries of the(i,j,k)row of A.Recall that a ikj has only three non-zero entries corresponding to the edges(i,j),(i,k)and(k,j).Thus, the update to e amounts to“ﬁxing”(enforcing)one triangle inequality at a time, hence the name of our procedure.The parameterθis computed by solvingmaxθg(z+θ1ikj),subject to z ikj+θ≥0,(3.4)where1ikj indicates the standard basis vector that is zero in all positions except the ikj entry,which equals ing(3.2)and(3.3),we may rewrite(3.4)asmax θg(z)−12a ikj 2θ2+(a T ikj e−v ikj)θ,subject toθ≥−z ikj.(3.5) Consider optimizing(3.5)in an unconstrained manner.It is easily seen thatθ∗=1a ikj 2(a T ikj e−v ikj)=13(a T ikj e−v ikj),(3.6)is the maximum.Ifθ∗≥−z ikj we are done,else the maximum of(3.5)will be achieved atθ=−z ikj.Thus,we obtainθ=max{θ∗,−z ikj}as the answer to(3.5). Algorithm3.1puts together all these ideas to give the complete iterative triangle ﬁxing procedure.Remarks.The procedure derived above ensures that at each iteration g(z′)≥g(z),i.e.,it is a dual coordinate ascent procedure.Following[2],it can be shown that in the limit,the Ae≤v constraints are satisﬁed.Since(3.2)is also maintained at each step,the KKT conditions,which are necessary and suﬃcient for this problem, are satisﬁed in the limit.Thus,the triangleﬁxing procedure converges to the optimal solution of(3.1).In fact,Algorithm3.1is an eﬃcient version of Bregman’s method for minimizing a convex function subject to linear inequality constraints[2].Our algorithm exploits the structure of the problem to obtain its eﬃciency.METRIC NEARNESS73.2.Triangleﬁxing forℓ1andℓ∞.Triangleﬁxing is somewhat less direct for theℓ1andℓ∞problems.The reason these norms pose an additional challenge is because they are not strictly convex;the convergence of the basic triangleﬁxing procedure depends on the strict convexity of the norm used.We illustrate only the ℓ1case;the development forℓ∞takes the same course.With the introduction of vector and matrix notation,theℓ1matrix nearness problem may be rewritten asmine,f0T e+1T f,subject to Ae≤v,−e−f≤0,e−f≤0.(3.7)The auxiliary variable f is interpreted as the element-wise absolute value of e.The violations to the triangle inequalities are again given by the vector v.To solve the linear program(3.7)without sacriﬁcing the advantages of triangle ﬁxing we replace it with an equivalent quadratic program.This replacement hinges upon a connection between linear and quadratic programs that may be motivated by the observation,argmin g g+ǫ−1c 2=argmin g(g T g+2ǫ−1g T c+ǫ−2c T c)≈argmin g g T c,ifǫis chosen to be suﬃciently small(so that the2ǫ−1g T c term dominates the objective function).The following theorem,which follows from a result of[17,Theorem2.1-a-i], makes the above connection concrete.Theorem3.1(ℓ1Metric Nearness).Let g=[e;f]and c=[0;1]be partitioned conformally.If(3.7)has a solution,then there exists anǫ0>0,such that for all ǫ≤ǫ0,argmin g∈G g+ǫ−1c 2=argming∈G⋆g 2,(3.8)where G is the feasible set for(3.7)and G⋆is the set of optimal solutions to(3.7). The minimizer of(3.8)is unique.From(3.7)one can see that the triangle inequality constraints involve only e and not f.This circumstance permits us to use triangleﬁxing once again.As before, we go through the constraints one by one.Theﬁrst3 n3 constraints are triangle constraints and are handled by triangleﬁxing.The remaining2 n2 absolute value constraints are very simple and thus are enforced easily.For theℓ2case,the dual variables(corresponding to each constraint)were repre-sented by the vector z.For(3.8),we let the dual variables be[z;λ;µ];vector z cor-responds to the triangle inequalities,while vectorsλandµcorrespond to−e−f≤0 and e−f≤0,respectively.Together,non-negative values of z,λandµcorrespond to the feasible set G alluded to by Theorem3.1.Our augmented triangle-ﬁxing procedure is as follows.First we initialize e,f,z,λandµso that theﬁrst order optimality conditions derived from(3.8)are initially true.Thereafter,we enforce constraints one by one to ensure the the dual functional corresponding to(3.8)is increasing andﬁrst order optimality conditions are main-tained.Written out as Algorithm3.2,this procedure becomes an eﬃcient adaptation of Bregman’s method,thereby,after a suﬃcient number of iterations,it converges to the globally optimal solution.8J.BRICKELL,I.S.DHILLON,S.SRA,AND J.A.TROPPAlgorithm3.2:Metric nearness forℓ1norm.Metric Nearness L1(D,ǫ,κ)Input:Dissimilarity matrix D;toleranceκ;ℓ1parameterǫOutput:M∈{argminX∈M N X−D 1}{Initialize primal&dual variables}e ij←0;f ij=−ǫ−1for1≤i<j≤n{Primal variables}(z ijk,z jki,z kij)←0for1≤i<k<j≤n{Dual variables–triangles}λij←πij←0for1≤i<j≤n{Dual variables–Other}δ←1+κwhile(δ>κ){convergence test}Do triangleﬁxing on the e ij as in Algorithm3.1{Enforce−e−f≤0and e−f≤0as follows}µ←12(e+f){Projection parameters}θ←min{µ,λ}{Update amount}λ←λ−θ{Update dual vector corr.to−e−f≤0}e←e−θ;f←f−θ{Update primal variables}ν←12(f−e)θ←min{ν,π}{Update amount}π←π−θ{Update dual vector corr.to e−f≤0}e←e+θ;f←f−θ{Update primal variables}{Update convergence test parameter}δ←sum of absolute changes in e ij.end.Remarks.Algorithm3.2depends on the parameterǫthat governs convergence to the true optimal solution.It is an open problem to obtain anǫthat guarantees convergence.However,upon experimentation with random dissimilarity matrices we found that settingǫ−1≈max ij d ij,worked well,i.e.,led to convergence,for Algo-rithm3.2.Furthermore,from Theorem3.1we know that there exists a range within whichǫcan lie,and in practice running Algorithm3.2a small number(2–3)of times (with early stopping to save time)helps to determine a suitable value forǫfor an arbitrary input matrix.3.3.Triangleﬁxing for otherℓp norms.We can go a step further and extend triangleﬁxing to solve the metric nearness problem for allℓp(1<p<∞)norms. The problem may be compactly stated asmin e 1pe p p subject to Ae≤v.(3.9)Recall that forℓ2metric nearness,at each iterative step we obtained e′from e by solving(3.2)after updating the dual variables z in a single coordinate.This update to e may be viewed as the result of an orthogonal projection of e onto the hyperplane deﬁned by a ikj,e′ =v ikj(ignoring inequalities for the moment).For the ℓp norm problem,we must instead perform a generalized projection,called a Bregman projection,which involves solving the following problemmin e′ϕ(e′)−ϕ(e)− ∇ϕ(e),e′−e such that a ikj,e′ =v ikj,(3.10)whereϕ(x)=1p x p p.We use(∇ϕ(x))i=sgn(x i)|x i|p−1to determine the projection(3.10)by solving∇ϕ(e′)=∇ϕ(e)+µa ikj so that a ikj,e′ =v ikj.(3.11)METRIC NEARNESS9 Since a ikj has only three nonzero entries,once again e needs to be updated in only three components.Therefore,in Algorithm3.1we may replace(⋆)by an appropriate numerical computation of the parameterµ,and replace(⋆⋆)by the computation of the new value of e as resulting from(3.11).As before,each iteration maintains the necessary condition∂L(e,z)/∂e=0while correcting the dual vector z,and the overall algorithm converges to the optimum of(3.9).4.Metric Nearness and APSP.The All Pairs Shortest Paths(APSP)prob-lem[3]is an important and well-studied problem in graph theory that still continues to interest researchers.For a given weighted graph G,APSP computes an associated matrix of distances M whose entry m ij gives the weight of a shortest path between vertices i and j.Optionally,shortest paths between all pairs of vertices corresponding to these distances are also obtained.On the surface,APSP appears to have no connection with the metric nearness problem.However,it turns out that APSP can be viewed as a special case of metric nearness.We develop this connection below.Note that in the previous sections we considered only symmetric matrices.However,in this section we consider asymmetric distance matrices,which are more natural for the APSP problem,as they correspond to directed graphs.4.1.The relation of Metric Nearness to APSP.Let the input be a weighted complete directed graph.We represent this graph by the(non-symmetric)matrix D, where d ij denotes the edge weight of edge(i,j).On D we perform a restricted version of metric nearness that permits only decreasing changes to the d ij values.Curiously this decrease only version of metric nearness is equivalent to APSP.Lemma4.1(Decrease only metric nearness is APSP).Let M A∈M N be the APSP solution for D.Then,M A is also the nearest“decrease only”metric solution. In fact,any metric solution M∈M N that is element-wise smaller than D,is also smaller than M A,i.e.,∀M∈M N,if M≤D then M≤M A.A proof of this lemma may be found in Appendix A.1.This connection between APSP and decrease only metric nearness(DOMN)suggests that the latter may be solved by using any oﬀ-the-shelf algorithm for APSP.More interestingly,one can turn the problem around and obtain a new method to solve APSP by solving the DOMN problem.In this section,we present a new algorithm for APSP based on solving a linear programming formulation of DOMN.APSP for dense graphs is commonly performed using the Floyd-Warshall algo-rithm,which has a complexity ofΘ(n3).Unlike the Floyd-Warshall algorithm that proceeds byﬁxing the triangles of the graph in a predetermined order,our DOMN algorithmﬁxes triangles in a data-dependent order.Empirically,our algorithm con-verges more quickly to the solution than Floyd-Warshall,despite having the same asymptotic worst-case behavior.4.2.The linear programming formulation of DOMN and its dual. Lemma4.1suggests that APSP solves the decrease only metric nearness problem regardless of the norm used to measure the error.We,however,focus on theℓ1 norm problem along with its linear programming formulation.The linear program is interesting both because it is a novel formulation for solving APSP,and also because its dual allows us to construct shortest paths,if desired.We apply the primal-dual technique for solving the resulting linear programs and obtain a new APSP algorithm as a consequence.10J.BRICKELL,I.S.DHILLON,S.SRA,AND J.A.TROPP4.2.1.Formulation.Let X represent a decrease-only distance matrix corre-sponding to the input matrix D.Then the entries of X must satisfy,x ij≤d ij for all(i,j),(4.1)x ij≤x ik+x kj for all(i,k,j).(4.2) Finding the matrix with the leastℓ1perturbation requires solving the problemminimizex ijij(d ij−x ij)subject to(4.1)and(4.2).Note that we are dealing directly with the values x ij rather than the error values e ij=d ij−x ij,as we did in sections2.1and2.2.Since the d ij areﬁxed we may replace this minimization by the equivalent problemmaximizex ijij x ijsubject to x ij≤d ij for all(i,j),x ij−x ik−x kj≤0for all(i,k,j).(4.3) The dual problem corresponding to(4.3)isminimizeπijijπij d ijsubject toπij+ k=i,j(γikj−γijk−γkij)=1for all(i,j),πij≥0for all(i,j),γikj≥0for all(i,k,j),(4.4)where the dual variablesπij andγikj correspond to the decrease-only constraints(4.1), and the triangle inequality constraints(4.2),respectively.It is illustrative to cast the linear program(4.4)as a networkﬂow problem,in which we must satisfy a demand for a single unit ofﬂow between every pair of vertices i and j.We can accomplish this either by sending theﬂow directly via the edge(i,j) (which corresponds to settingπij=1)or by bypassing the edge(i,j)and routing through some other vertex k(which corresponds to settingγikj=1);in the latter case,we increase the demand forﬂow between(i,k)and(k,j)by1.We note that while there is a unique optimal solution to the linear program(4.3), the linear program(4.4)has several optimal solutions,some of which involve non-integral assignments to theγikj variables.This non-uniqueness is not unexpected, because while there is only value that the shortest distance between two nodes in M can attain,whereas there can be several shortest paths that achieve this distance value(paths which may contain many intermediate nodes,each of which allows aγikj variable to assume a positive assignment).4.3.A primal-dual algorithm for DOMN/APSP.We apply the primal-dual method[19,16]to solve the linear programs for DOMN,and thereby obtain a new algorithm for APSP.Most treatments of the primal-dual method have a minimization of the primal problem and a maximization of the dual problem.Thus we will call (4.3)as the dual problem,and(4.4)as the primal problem.The primal-dual method begins with a feasible solution to the dual that is improved at each step by optimizing an associated restricted primal problem.In our case,weﬁnd it easier to optimize the associated restricted dual,whereby our method proceeds as follows:。

Cartan Calculus on the Quantum Space ${cal R}_q^{3}$

$Cartan Calculus on the Quantum Space ${cal R}_q^{3}$$

a r X i v :m a t h /0607383v 2 [m a t h .Q A ] 11 D e c 2006YTUMB 2006-02,July 2006CARTAN CALCULUS ON THE QUANTUMSPACE R 3qSalih C ¸elik 1,2,E.Mehmet ¨Ozkan 1and Erg¨u n Ya¸s ar 11Yildiz Technical University,Department of Mathematics,34210DAVUTPASA-Esenler,Istanbul,TURKEY.2E-mail:sacelik@.trABSTRACTTo give a Cartan calculus on the extended quantum 3d space,the noncommutativediﬀerential calculus on the extended quantum 3d space is extended by introducing inner derivations and Lie derivatives.1.INTRODUCTIONThe noncommutative diﬀerential geometry of quantum groups was introduced by Woronowicz[11,12].In this approach the diﬀerential calculus on the group is deduced from the properties of the group and it involves functions on the group, diﬀerentials,diﬀerential forms and derivatives.The other approach,initiated by Wess and Zumino[10],followed Manin’s emphasis[5]on the quantum spaces as the primary objects.Diﬀerential forms are deﬁned in terms of noncommuting coordinates,and the diﬀerential and algebraic properties of quantum groups acting on these spaces are obtained from the properties of the spaces.The diﬀerential calculus on the quantum3d space similarly involves functions on the3d space,diﬀerentials,diﬀerential forms and derivatives.The exterior derivative is a linear operator d acting on k-forms and producing(k+1)-forms, such that for scalar functions(0-forms)f and g we haved(1)=0,d(fg)=(d f)g+(−1)deg(f)f(d g)where deg(f)=0for even variables and deg(f)=1for odd variables,and for a k-formω1and any formω2∧ω2)=(dω1)∧ω2+(−1)kω1∧(dω2).d(ω1A fundamental property of the exterior derivative d isd∧d=:d2=0.There is a relationship of the exterior derivative with the Lie derivative and to describe this relation,we introduce a new operator:the inner derivation.Hence the diﬀerential calculus on the quantum3d space can be extended into a large calculus.We call this new calculus the Cartan calculus.The connection of the inner derivation denoted by i a and the Lie derivative denoted by L a is given by the Cartan formula:L a=i a◦d+d◦i a.This and other formulae are explaned in Ref.6-8.We now shall give a brief overview without much discussion.Let us begin with some information about the inner derivations.Generally,for a smooth vectorﬁeld X on a manifold the inner derivation,denoted by i X,is a linear operator which maps k-forms to(k−1)-forms.If we deﬁne the inner derivation i X on the set of all diﬀerential forms on a manifold,we know that i X is an antiderivation of degree−1:(α∧β)=(i Xα)∧β+(−1)kα∧(i Xβ)iXwhereαandβare both diﬀerential forms.The inner derivation i X acts on0-and 1-forms as follows:(f)=0,iX(d f)=X(f).iXWe know,from the classical diﬀerential geometry,that the Lie derivative L can be deﬁned as a linear map from the exterior algebra into itself which takes k-forms to k-forms.For a0-form,that is,an ordinary function f,the Lie derivative is just the contraction of the exterior derivative with the vectorﬁeld X:L X f=i X d f.For a general diﬀerential form,the Lie derivative is likewise a contraction,taking into account the variation in X:L Xα=i X dα+d(i Xα).The Lie derivative has the following properties.If F(M)is the algebra of functions deﬁned on the manifold M thenL X:F(M)−→F(M)is a derivation on the algebra F(M):L X(af+bg)=a(L X f)+b(L X g),L X(fg)=(L X f)g+f(L X g),where a and b real numbers.The Lie derivative is a derivation on F(M)×V(M)where V(M)is the set of vectorﬁelds on M:L X1(fX2)=(L X1f)X2+f(L X1X2).The Lie derivative also has an important property when acting on diﬀerential forms.Ifαandβare two diﬀerential forms on M thenL X(α∧β)=(L Xα)∧β+(−1)kα∧(L Xβ)whereαis a k-form.The extended calculus on the quantum plane was introduced in Ref.3using the approach of Ref.6.In this work we explicitly set up the Cartan calculus on the quantum3d space using approach of Ref1.2.REVIEW OF SOME STRUCTURES ON R3qIn this section we give some information on the Hopf algebra structures of the quantum3d space and its diﬀerential calculus[2]which we shall use in order to establish our notions.2.1The algebra of polynomials on the quantum3d spaceThe quantum three dimensional space is deﬁned as an associative algebra gener-ated by three noncommuting coordinates x,y and z with three quadratic relationsxy=qyx,yz=qzy,xz=qzx,where q is a non-zero complex number.This associative algebra over the complex number,C,is known as the algebra of polynomials over the quantum three di-mensional space and we shall denote it by R3q.In the limit q−→1,this algebra is commutative and can be considered as the algebra of polynomials C[x,y,z]overthe usual three dimensional space,where x,y and z are the three coordinate func-tions.We denote the unital extension of R3q by A,i.e.it is obtained by adding a unit element.2.2The Hopf algebra structure on AOne extends the algebra A by including inverse of x which obeysxx−1=1=x−1x.The deﬁnitions of a coproduct,a counit and a coinverse on the algebra A as follows [2]:(1)The C-algebra homomorphism(coproduct)∆A:A−→A⊗A is deﬁned by∆A(x)=x⊗x,∆A(y)=x⊗y+y⊗x,∆A(z)=z⊗1+1⊗z,which is coassociative:(∆A⊗id)◦∆A=(id⊗∆A)◦∆Awhere id denotes the identity map on A.(2)The C-algebra homomorphism(counit)ǫA:A−→C is given byǫA(x)=1,ǫA(y)=0,ǫA(z)=0.The counitǫA has the propertyµ◦(ǫA⊗id)◦∆A=µ′◦(id⊗ǫA)◦∆Awhereµ:C⊗A−→A andµ′:A⊗C−→A are the canonical isomorphisms, deﬁned byµ(k⊗u)=ku=µ′(u⊗k),∀u∈A,∀k∈C.(3)The C-algebra antihomomorphism(coinverse)S A:A−→A is deﬁned byS A(x)=x−1,S A(y)=−x−1yx−1,S A(z)=−z.The coinverse S satisﬁesm◦(S A⊗id)◦∆A=ǫA=m◦(id⊗S A)◦∆Awhere m stands for the algebra product A⊗A−→A.The coproduct,counit and coinverse which are speciﬁed above supply the algebra A with a Hopf algebra structure.2.3Diﬀerential algebraWeﬁrst note that the properties of the exterior diﬀerential d.The exterior diﬀer-ential d is an operator which gives the mapping from the generators of A to the diﬀerentials:d:u−→d u,u∈{x,y,z}.We demand that the exterior diﬀerential d has to satisfy two properties:the nilpotencyd2=0and the Leibniz ruled(fg)=(d f)g+(−1)deg(f)f(d g).A deformed diﬀerential calculus on the quantum3d space is as follows:the commutation relations with the coordinates of diﬀerentialsx d x=d x x,x d y=q d y x,x d z=q d z x,y d x=q−1d x y,y d y=d y y,y d z=q d z y,z d x=q−1d x z,z d y=q−1d y z,z d z=d z z.This algebra is denoted byΓ1.The commutation relations between the diﬀerentialsd x∧d x=0,d y∧d y=0,d z∧d z=0.d x∧d y=−q d y∧d x,d y∧d z=−q d z∧d y,d x∧d z=−q d z∧d x.This algebra is denoted byΓ2.A diﬀerential algebra on an associative algebra A is a graded associative algebra Γequipped with an operator d that has the above properties.Furthermore,the algebraΓhas to be generated byΓ0∪Γ1∪Γ2,whereΓ0is isomorphic to A.LetΓbe the quoitent algebra of the free associative algebra on the set{x,y,z,d x,d y,d z} modulo the ideal J that is generated by the relations of R3q,Γ1andΓ2.To proceed,one can obtain the relations of the coordinates with their partial derivatives using the expression+d y∂y+d z∂z)f.d f=(d x∂xConsequently one has∂x x=1+x∂x,∂x y=q−1y∂x,∂x z=q−1z∂x,∂y x=qx∂y,∂y y=1+y∂y,∂y z=q−1z∂y,∂z x=qx∂z,∂z y=qy∂z,∂z z=1+z∂z.Using the fact that d2=0,oneﬁnds∂x∂y=q∂y∂x,∂x∂z=q∂z∂x,∂y∂z=q∂z∂y.The relations between partial derivatives and diﬀerentials are found as∂x d x=d x∂x,∂x d y=q−1d y∂x,∂x d z=q−1d z∂x,∂y d x=q d x∂y,∂y d y=d y∂y,∂y d z=q−1d z∂y,∂z d x=q d x∂z,∂z d y=q d y∂z,∂z d z=d z∂z.We can deﬁne three one-forms using the generators of A.If we call themωx,ωy andωz then one can deﬁne them as follows:ωx=d x x−1,ωy=d y x−1−d x x−1yx−1,ωz=d z.We denote the algebra of forms generated by three elementsωx,ωy andωz by Ω.The generators of the algebraΩwith the generators of A satisfy the following rulesxωx=ωx x,xωy=qωy x,xωz=qωz x,yωx=ωx y,yωy=qωy y,yωz=qωz y,zωx=ωx z,zωy=ωy z,zωz=ωz z.The commutation rules of the generators ofΩareωx∧ωx=0,ωy∧ωy=0,ωz∧ωz=0,ωx∧ωy=−ωy∧ωx,ωy∧ωz=−ωz∧ωy,ωx∧ωz=−ωz∧ωx.The algebraΩis a graded Hopf algebra[2].2.4Lie algebraThe commutation relations of Cartan-Maurer forms allow us to construct the algebra of the generators.In order to obtain the quantum Lie algebra of the algebra generators weﬁrst write the Cartan-Maurer forms asx,d x=ωxy+ωy x,d y=ωxd z=ω.zThe diﬀerantial d can then the expressed in the formd f=(ωT x+ωy T y+ωz T z)f.xHere T x,T y and T z are the quantum Lie algebra generators.Considering an arbitrary function f of the coordinates of the quantum3d space and using that d2=0,weﬁnd the following commutation relations for the(undeformed)Lie algebra[2]:[T x,T y]=0,[T x,T z]=0,[T y,T z]=0.The commutation relations between the generators of algebra and the coordinates areT x x=x+x T x,T x y=y+y T x,T x z=z T x,T y x=qx T y,T y y=x+qy T y,T y z=z T y,T z x=qx T z,T z y=qy T z,T z z=1+z T z.The(quantum)Lie algebra generators can be expressed in terms of the generators of the quantum3d space and partial diﬀerentials:T x≡x∂x+y∂y,T y≡x∂y,T z≡∂z.The commutation relations of the Lie algebra generators T x,T y and T z with the diﬀerentials are followingT x d x=d x T x,T x d y=d y T x,T x d z=d z T x,T y d x=q d x T y,T y d y=q d y T y,T y d z=d z T y,T z d x=q d x T z,T z d y=q d yT z,T z d z=d z T z.The commutation rules of the Lie algebra generators with one-forms as followsT xωx=ωx T x−ωx,T xωy=ωy T x−ωy,T xωz=ωz T x,T yωx=ωx T y,T yωy=ωy T y−ωx,T yωz=ωz T y,T zωx=ωx T z,T zωy=ωy T z,T zωz=ωz T z.The Hopf algebra structure of the Lie algebra generators is given by∆(T x)=T x⊗1+1⊗T x,∆(T y)=T y⊗1+q T x⊗T y,∆(T z)=T z⊗1+q T x⊗T z,ǫ(T x)=0,ǫ(T y)=0,ǫ(T z)=0,S(T x)=−T x,S(T y)=−q−T x T y,S(T z)=−q−T x T z.2.5The dual of the Hopf algebra AIn this section,in order to obtain the dual of the Hopf algebra A deﬁned in section 2,we shall use the method of Refs.4and9.A pairing between two vector spaces U and A is a bilinear mapping<,>:U x A−→C,(u,a)→<u,a>.We say that the pairing is non-degenerate if<u,a>=0(∀a∈A)=⇒u=0and<u,a>=0(∀u∈U)=⇒a=0.Such a pairing can be extended to a pairing of U⊗U and A⊗A by<u⊗v,a⊗b>=<u,a><v,b>.Given bialgebras U and A and a non-degenerate pairing<,>:U x A−→C(u,a)→<u,a>∀u∈U∀a∈Awe say that the bilinear form realizes a duality between U and A,or that the bialgebras U and A are in duality,if we have<uv,a>=<u⊗v,∆A(a)>,<u,ab>=<∆U(u),a⊗b>,<1U,a>=ǫA(a),and<u,1A>=ǫU(u)for all u,v∈U and a,b∈A.If,in addition,U and A are Hopf algebras with coinverseκ,then they are said to be in duality if the underlying bialgebras are in duality and if,moreover,we have<S U(u),a>=<u,S A(a)>∀u∈U a∈A.It is enough to deﬁne the pairing between the generating elements of the two algebras.Pairing for any other elements of U and A follows from above relations and the bilinear form inherited by the tensor product.For example,foru′k⊗u′′k,∆U(u)=kwe have<u′k,a><u′′k,b><u,ab>=<∆U(u),a⊗b>=kAs a Hopf algebra A is generated by the elements x,y and z,and a basis is given by all monomials of the formf=x k y l z mwhere k,l,m∈Z+.Let us denote the dual algebra by U q and its generating elements by A and B.The pairing is deﬁned through the tangent vectors as follows<X,f>=kδl,0δm,0,<Y,f>=δl,1δm,0,<Z,f>=δl,0δm,1.We also have<1U,f>=ǫA(f)=δk,0.Using the deﬁning relations one gets<XY,f>=δl,1δm,0and<Y X,f>=δl,1δm,0where diﬀerentiation is from the right as this is most suitable for diﬀerentiation in this basis.Thus one obtains one of the commutation relations in the algebra U q dual to A as:XY=Y X.Similarly,one hasXZ=ZX,Y Z=ZY.The Hopf algebra structure of this algebra can be deduced by using the duality. The coproduct of the elements of the dual algebra is given by∆U(X)=X⊗1U+1U⊗X,∆U(Y)=Y⊗q−X+1U⊗Y,∆U(Z)=Z⊗q−X+1U⊗Z.The counity is given byǫU(X)=0,ǫU(Y)=0,ǫU(Z)=0.The coinverse is given asS U(X)=−X,S U(Y)=−Y q X,S U(Z)=−Zq X.We can now transform this algebra to the form obtained in section5by making the following identities:T x≡X,T y≡q X/2Y q X/2,T z≡q X/2Zq X/2which are consistent with the commutation relation and the Hopf structures.3.EXTENDED CALCULUS ON THE QUANTUM3D SPACEA Lie derivative is a derivation on the algebra of tensorﬁelds over a manifold. The Lie derivative should be deﬁned three ways:on scalar functions,vectorﬁelds and tensors.The Lie derivative can also be deﬁned on diﬀerential forms.In this case,it is closely related to the exterior derivative.The exterior derivative and the Lie derivative are set to cover the idea of a derivative in diﬀerent ways.These diﬀer-ences can be hasped together by introducing the idea of an antiderivation which is called an inner derivation.3.1Inner derivationsIn order to obtain the commutation rules of the coordinates with inner derivations, we shall use the approach of Ref. 1.Similarly other relations can also obtain. Consequently,we have the following commutation relations:•the commutation relations of the inner derivations with x,y and zx=x i x,i x y=q−1y i x,i x z=q−1z i x,ixix=q x i y,i y y=y i y,i y z=q−1z i y,yx=q x i z,i z y=q y i z,i z z=z i z.iz•the relations of the inner derivations with the partial derivatives∂x,∂y and ∂z∂x=∂x i x,i x∂y=q∂y i x,i x∂z=q∂z i x,ixi∂x=q−1∂x i y,i y∂y=∂y i y,i y∂z=q∂z i y,y∂x=q−1∂x i z,i z∂y=q−1∂y i z,i z∂z=∂z i z.iz•the commutation relations between the diﬀerentials and the inner derivations ∧d x=1−d x∧i x,i x∧d y=−q−1d y∧i x,ixi∧d x=−q d x∧i y,i y∧d y=1−d y∧i y,yi∧d x=−q d x∧i z,i z∧d y=−q d y∧i z,zi∧d z=−q−1d z∧i x,i y∧d z=−q−1d z∧i y,x∧d z=1−d z∧i z.iz3.2Lie derivationsIn this section weﬁnd the commutation rules of the Lie derivatives with functions,i.e.the elements of the algebra A,their diﬀerentials,etc.,using the approach of[1]as follows:•the relations between the Lie derivatives and the elements of AL x x=1+x L x,L x y=q−1y L x,L x z=q−1z L x,L y x=q x L y,L y y=1+y L y,L y z=q−1z L y,L z x=q x L z,L z y=q y L z,L z z=1+z L z.•The relations of the Lie derivatives with the diﬀerentialsL x d x=d x L x,L x d y=q−1d y L x,L x d z=q−1d z L x,L y d x=q d x L y,L y d y=d y L y,L y d z=q−1d z L y,L z d x=q d x L z,L z d y=q d y L z,L z d z=d z L z.Other commutation relations can be similarly obtained.To complete the descrip-tion of the above scheme,we get below the remaining commutation relations as follows:•the Lie derivatives and partial derivativesL x∂x=∂x L x,L x∂y=q∂y L x,L x∂z=q∂z L x,L y∂x=q−1∂x L y,L y∂y=∂y L y,L y∂z=q∂z L y,L z∂x=q−1∂x L z,L z∂y=q−1∂y L z,L z∂z=∂z L z.•the inner derivations∧i y=−q i y∧i x,ix∧i z=−q i z∧i x,ix∧i z=−q i z∧i y.iy•the Lie derivatives and the inner derivationsL x i x=i x L x,L x i y=q i y L x,L x i z=q i z L x,L y i x=q−1i x L y,L y i y=i y L y,L y i z=q i z L y,L z i x=q−1i x L z,L z i y=q−1i y L z,L z i z=i z L z.•the Lie derivativesL x L y=q L y L x,L x L z=q L z L x,L y L z=q L z L y.Note that the Lie derivatives can be written as follows:L x=x−1T x−x−1yx−1T y,L y=x−1T y,L z=T z.ACKNOWLEDGMENTThis work was supported in part by TBTAK the Turkish Scientiﬁc and Technical Research Council.REFERENCES1.Celik,Salih:J.Math.Phys.47(8):Art.No:0835012.Celik,Sultan A.and Yasar,E.:Czech.J.Phys.56(2006),229.3.Chryssomalakos,C.,Schupp P.and Zumino,B.:”Induced extended calculuson the quantum plane”,hep-th/9401141.4.Dobrev,V.K.:J.Math.Phys.33(1992),3419.5.Manin,Yu I.:”Quantum groups and noncommutative geometry”,(MontrealUniv.Preprint,1988).6.Schupp,P.,Watts,P.,Zumino,B.:Lett.Math.Phys.25(1992),139.7.Schupp,P.,Watts P.,Zumino,B.:”Cartan calculus on quantum Lie alge-bras”,hep-th/9312073.8.Schupp,P.:”Cartan calculus:Diﬀerential geometry for quantum groups”,hep-th/9408170.9.A.Sudbery,A.:Proc.Workshop on Quantum Groups,Argogne(1990)eds.T.Curtright,D.Fairlie and C.Zachos,pp.33-51.10.Wess,J.and Zumino,B.:Nucl.Phys.(Proc.Suppl.)18B(1990),302.11.Woronowicz,S.L.:Commun.Math.Phys.111(1987),613.12.Woronowicz,mun.Math.Phys.122(1989),125.。

人湿尺骨干在冲击压缩实验下的力学性能及规律

第27卷第3期2006年6月西安交通大学学报(医学版)J our nal of Xi πan J iaot ong U niversity (Medical Scie nces )V ol.27No.3J un.2006人湿尺骨干在冲击压缩实验下的力学性能及规律王玉梅1,张　堃1,韩少东2,祁　帜3(1.西安市红十字会医院骨伤科,陕西西安　710054;2.中国航天建筑设计研究院陕西分院,陕西西安　710014;3.长安大学理学院,陕西西安　710064)摘要:目的　探讨人湿尺骨密质骨在高应变率下的力学性能及其分布规律。

方法　用分离式Hopkinson 压杆(SHPB )技术对人湿尺骨密质骨进行了应变率在 ε=1.2×103/s 条件下的冲击压缩试验。

结果　尺骨密质骨在高应变率下的力学性能随其纵向位置变化而呈现出中间强两端弱的分布,最强的位置约在距近端1/3处,且近端强于远端。

与静态结果的比较表明,尺骨密质骨对应变率有较大的依赖性。

结论　在冲击压缩力作用下,尺骨的两端,尤其是远端容易骨折。

关键词:尺骨;冲击响应;应变率中图分类号:R681.7 文献标识码:A 文章编号:167128259(2006)0320240203Impact compressive experiment of human fresh ulnar compactWang Yumei 1,Zhang Kun 1,Han Shaodong 2,Qi Zhi 3(1.Depart ment of Ort hopedics ,Xi πan Red Cro ss Hospital ,Xi πan 710054;2.China Design Academy of Spacefilight Architect ure ,Shaanxi Branch ,Xi πan 710014;3.College of Science of Chang πan U niversity ,Xi πan 710064,China )ABSTRACT :Objective To investigate t he mechanical p rop erties a nd dist ributing characteristics of huma n f reshulnar comp act and its correlation t o st rain rate.Methods Imp act comp ressive experime nt f or human f resh ulnarcomp act bone at high st rain rateε=1.2×103/s by Split Hop kinson Pressure Bars (S HPB )were perf or med.Re sults　Themechanical p roperties of huma n f resh ulnar comp act bone varied along t he longitudinal direction ,t heultimate st ress ,breaking st rain a nd dynamic comp ressive modulus at middle section were st ronger t han t hose at bot h e nds ,t he st rongest at section occurs nearly at t he junction of p roximal and middle t hirds and t he p roperties at t he p roximal e nd were much higher t ha n t hose at t he distal e nd.The ultimate st ress at high st rain rate was largely depe nde nt on st rain rate.Conclusion U nder imp act comp ression ,t he ulnar comp act gets f ractured easily at bot he nds ,especially at distance section.KE Y WOR DS :ulna ;imp act p rop erty ;st rain rate收稿日期:2005205224 修回日期:2005210215作者简介:王玉梅(19662),女(汉族),学士,副主任医师. 骨的力学性能是固体生物力学研究的一个重要内容。

The distance to the Fornax Dwarf Spheroidal Galaxy

a r X i v :0707.0521v 1 [a s t r o -p h ] 4 J u l 2007Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 1February 2008(MN L A T E X style ﬁle v2.2)The distance to the Fornax dwarf spheroidal galaxy ⋆L.Rizzi 1,2,E.V.Held 3,I.Saviane 4,R.B.Tully 1,M.Gullieuszik 3,51Institute for Astronomy,University of Hawaii,2680Woodlawn Drive,HI 96822,USA2JointAstronomy Centre,660N.A’ohoku Place,University Park,Hilo,HI 96720,USA3Osservatorio Astronomico di Padova,INAF,Vicolo Osservatorio 5,I-35122Padova,Italy4European Southern Observatory,3107Alonso de Cordova,Vitacura,Casilla 19001,Santiago 19,Chile 5Dipartimento di Astronomia,Universit`a di Padova,vicolo dell’Osservatorio 2,I-35122Padova,ItalyReceived ...;accepted ...ABSTRACTA large multicolour,wide-ﬁeld photometric database of the Fornax dwarf spheroidalgalaxy has been analysed using three diﬀerent methods to provide revised distance estimates based on stellar populations in diﬀerent age intervals.The distance to Fornax was obtained from the Tip of the Red Giant Branch measured by a new method,and using the luminosity of Horizontal Branch stars and Red Clump stars corrected for stellar population eﬀects.Assuming a reddening E (B −V )=0.02,the following distance moduli were derived:(m −M )0=20.71±0.07based on the Tip of the Red Giant Branch,(m −M )0=20.72±0.06from the level of the Horizontal Branch,and (m −M )0=20.73±0.09using the Red Clump method.The weighted mean distance modulus to Fornax is (m −M )0=20.72±0.04.All these measurements agree within the errors,and are fully consistent with previous determinations and with the distance measurements obtained in a companion paper from near-infrared colour-magnitude diagrams.Key words:Galaxies:fundamental parameters –Galaxies:distances –Galaxies:in-dividual:Fornax dwarf spheroidal1INTRODUCTIONAlong with the disrupted Sagittarius dwarf,Fornax is one of the most massive satellites of the Milky Way.Fornax shows evidence of a complicated and extended star forma-tion history and several studies have contributed to clar-ify the variety of stellar populations that are present.This galaxy was one of the ﬁrst to provide convincing evidence of the presence of a conspicuous amount of intermediate-age stars,probed both by the presence of luminous carbon stars (Aaronson &Mould 1985,1980;Azzopardi 1999)and by the well populated red clump (Stetson et al.1998;Saviane et al.2000).Blue luminous stars populating the upper main se-quence indicate that Fornax has been forming stars up to very recent times (Buonanno et al.1999;Pont et al.2004;Saviane et al.2000;Stetson et al.1998).Finally,the detec-tion of a signiﬁcant number of RR Lyrae stars probes the presence of an old,metal-poor component (Bersier &Wood 2002;Greco et al.2005).Most recently,stellar population gradients and stellar metallicities have been studied over a wide area of Fornax by Battaglia et al.(2006),conﬁrm-ing the presence of at least three distinct stellar compo-⋆Based on data collected at the European Southern Observatory,La Silla,Chile,Proposal No.66.B-0615nents:a young population concentrated near the centre,an intermediate-age population,and an ancient population.The three components are found to be distinct from each other kinematically,chemically,and in their spatial distri-bution.The distance to Fornax marks a reference point in the distance ladder based on secondary distance indicators.To-gether with the Carina dwarf spheroidal and the Large and Small Magellanic Clouds,Fornax has been used as a bench-mark to investigate possible discrepancies between diﬀerent distance indicators,and to study their dependence on the properties of the underlying stellar populations (e.g.,see Bersier 2000;Pietrzy´n ski et al.2003).Fornax is particularly suitable for these kind of investigations because of the con-temporary presence of both old and intermediate-age/young stellar populations.The distance to Fornax has been determined using a number of diﬀerent indicators,and the estimates range from (m −M )0=20.59±0.22(Buonanno et al.1985)to (m −M )0=20.86(Pietrzy´n ski et al.2002).In particular,Saviane et al.(2000)found (m −M )0=20.70±0.12based on the Tip of the Red Giant Branch (TRGB)technique,and (m −M )0=20.76±0.04based on the luminosity of red Horizontal Branch (HB)stars.Bersier (2000)derived a slightly shorter distance modulus,(m −M )0=20.66,2Rizzi et al.Table 1.Fornax distance determinationsValueMethodE (B −V )Referencebased on TRGB and Red Clump (RC)methods.A largerdistance modulus,(m −M )0=20.86,was obtained by Pietrzy´n ski et al.(2003)using K -band imaging of RC stars.A recent analysis of a wide-area database of near-infrared data by Gullieuszik et al.(2006)resulted in (m −M )0=20.74±0.11from the mean magnitude of the Red Clump (RC),and (m −M )0=20.75±0.19based on a population-corrected measurement of the TRGB in the K band.Previ-ous measurements of the distance to Fornax are summarised in Table 1.The consistent results from the near-infrared photome-try of Gullieuszik et al.(2006)and the relatively large scat-ter of previous optical distance estimates motivated us to re-vise the distance to Fornax by exploiting a large multi-band BV I database of photometric observations of Fornax.Our data,covering 1/4square degree and containing ∼450.000stars,were obtained with the Wide Field Imager at the ESO/MPG 2.2-m telescope at La Silla,Chile.The range of distances found in the literature are based on diﬀerent data sets,each with its own ﬁlter set and photometric zero-point.Therefore,measuring the distance with diﬀerent methods on a single,homogeneous data set with high statistics may provide a standard case of the intrinsic uncertainties in dis-tance determinations in stellar systems with extended star formation histories.The paper is organised as follows:Sect.2presents the observations,the reduction techniques,and the resulting colour-magnitude diagram (CMD).The diﬀerent methods used to derive the distance to the Fornax dwarf spheroidal are then presented and the results discussed in Sect.3,4,and 5.Final remarks and discussion of the results are presented in Sect.6.2OBSER V ATIONS AND DATA REDUCTIONWide ﬁeld observations of Fornax were obtained with the WFI camera mounted at the Cassegrain focus of the ESO/MPG 2.2-m telescope at La Silla,Chile.The camera consists of eight 2k ×8k CCDs,closely mounted on a mo-saic pattern yielding a total ﬁeld of view of ∼32′×32′.The observations were carried out on October 21,2001,as a part of ESO proposal 66.B-0615.Both long and short exposures were obtained:3×900+1×120seconds in B ,2×900+1×120seconds in V ,and 2×900+1×120seconds in I .The absolute calibration was obtained by observing stars from the list ofFigure 1.(V −I,I )colour magnitude diagram of Fornax.Only stars with photometric error less than 0.2mag are shown.Landolt (1992)on each CCD.Pre-reduction was performed within the IRAF 1environment,using the MSCRED pack-age by Valdes (1998).The subsequent reduction steps made use of the WFPRED script package developed by two of us (L.R.and E.V.H.)at the Padova Observatory.This package eﬀectively deals with the problems of astrometric and pho-tometric calibration.Images were astrometrically calibrated using a polynomial solution computed on star ﬁelds taken from the list of Stone et al.(1999).Crowded ﬁeld stellar photometry was performed with the daophot ii /allstar package (Stetson 1987),with point-spread functions (PSFs)independently computed for each CCD and each ﬁlter.Aper-ture corrections were used to match the PSF photometry with the aperture photometry,using growth-curve analysis of bright isolated stars.Finally,the photometric catalogues were independently calibrated for each CCD using the ob-servations of standard stars.The RMS of the photometric calibration is ∼0.02mag.An extensive set of artiﬁcial star experiments was run to estimate the degree of completeness and the distribution of photometric pleteness levels drop to 10%at V ∼24.5where the photometric er-rors reach 0.25mag.The resulting (V −I ),I CMD is presented in Figs.1.The limiting magnitude is around V ∼24,corresponding to the position of the old main sequence turn-oﬀ.Note that the ﬁgure only shows stars with a photometric error less than 0.2,and is consequently limited to V ≈23.5.A number1The Image Reduction and Analysis Facility (IRAF)software is provided by the National Optical Astronomy Observatories (NOAO),which is operated by the Association of Universities for Research in Astronomy (AURA),Inc.,under contract to the National Science Foundation.The distance to the Fornax dwarf spheroidal galaxy3of known features are evident:the well populated luminous main sequence of stars as young as0.3Gyr,the wide and non-uniformly populated red giant branch(RGB),the con-spicuous red clump of intermediate-age stars in their helium burning phase,and the less populated but evident horizon-tal branch(see,e.g.,Stetson et al.1998;Saviane et al.2000; Pont et al.2004;Battaglia et al.2006).These features are used in Sect.5to derive informa-tion on the star formation history.This CMD oﬀers us the possibility to use several diﬀerent methods to estimate the distance to Fornax.The TRGB is bright,well deﬁned and well populated,with manageable AGB contamination.The RC is prominent and bright enough to be usefully adopted as a distance indicator.The HB is not quite as populated as the RC,but still contains enough stars to derive a precise average magnitude.3DISTANCE BASED ON THE TIP OF THE RED GIANT BRANCHIn low mass stars,the He ignition occurs under degener-ate conditions at almost the same luminosity,with very little dependence on metallicity or age.The observational evidence of this physics is a sharp cut-oﬀof the lumi-nosity function of the RGB,approximatively located at M I∼−4.Da Costa&Armandroﬀ(1990)and Lee et al. (1993)demonstrated the power of this distance indicator ap-plied to nearby galaxies.Lee et al.(1993)provided both an absolute calibration of the I-band luminosity of the TRGB (M TRGBI)and an objective method to estimate its position on a CMD,based on a digital Sobelﬁlter.This technique was reﬁned by Sakai et al.(1996),who replaced the binned luminosity function with an adaptively smoothed probabil-ity distribution.M´e ndez et al.(2002)introduced a new way to estimate the position of the TRGB,based on a maximum-likelihood approach that eﬀectively uses all the stars around the tip region.In a series of papers,Ferraro et al.(1999, 2000)and Bellazzini et al.(2001,2004)obtained a new ro-bust calibration of the magnitude of the tip,extended to higher metallicities(up to[Fe/H]=−0.2)and to infra-red pass-bands.A modiﬁed and optimised version of the maximum-likelihood approach has been recently presented by Makarov et al.(2006).The new method has the advantage that completeness,photometric errors,and biased error dis-tributions are fully taken into account.The application of this method to the CMD of Fornax is presented in Fig.2. The left panel of Fig.2shows the CMD and the limits of the CMD region selected for theﬁtting procedure.On the right half of theﬁgure the completeness function and the distri-bution of photometric errors derived from artiﬁcial star ex-periments are shown.A biased error distribution is evident at magnitudes fainter than I=24.The lower panel shows the result of theﬁt.The tip is detected at I TRGB=16.75±0.02,and the av-erage colour of stars at the tip is(V−I)=1.64±0.03.As a ﬁrst approximation,the colour of stars at the tip can be used to estimate the metallicity of the underlying stellar pop-ulation.By using the relation presented in Bellazzini et al. (2001)(their Figure1),this colour is converted into a metal-licity[Fe/H]=−1.50±0.04(the error is purely statisti-Figure 2.Detection of the RGB tip using the maximum-likelihood method of Makarov et al.(2006).The TRGB is indi-cated in the left panel,while the right panels show the complete-ness function(top panel),the distribution of photometric errors (middle panel),and the luminosity function near the RGB tip (bottom panel).The observed LF is indicated by the solid(black) histogram,while the thick(red)line shows the bestﬁtting model. cal and does not contain any systematic contribution).Us-ing this value and the TRGB calibration of Bellazzini et al. (2001)(their equation4),we obtain M TRGBI=−4.07.The corrected distance is then derived by adopting the redden-ing value E(B−V)=0.02from the infra-red dust maps of Schlegel et al.(1998)and the relation A I=1.94E(B−V). Using the relation(m−M)0=I TRGB−M TRGBI−1.94×E(B−V)the distance to Fornax from the RGB tip is then(m−M)0= 20.78±0.04.The main contribution to the error aﬀecting this deter-mination comes from the conversion of the mean colour at the level of the tip to metallicity.In the case of Fornax,this conversion is not actually necessary.Indeed,several deter-minations of the metallicity of Fornax exist in the litera-ture.Saviane et al.(2000)found that the average metallic-ity is[Fe/H]∼−1.0±0.15(on the scale of Zinn&West 1984),with tails extending to−2.0<[Fe/H]<−0.7. They also concluded that a model involving two popu-lations seems to provide a good description of the star content of this galaxy,with the older populations hav-ing[Fe/H]=−1.82.With a similar technique,applied to V−K colours Gullieuszik et al.(2006)found a mean age-corrected mean metallicity of[M/H]≃−0.9.Tolstoy et al. (2001)found [Fe/H] =−1.0,later conﬁrmed with high resolution spectroscopy(Tolstoy et al.2003).More recently, Pont et al.(2004)and Battaglia et al.(2006)derived the metallicity distribution of Fornax RGB stars from spec-troscopy in the CaII triplet region.Both studies agree on ﬁnding a metallicity distribution centred at[Fe/H]≃−0.9 (on the scale of Carretta&Gratton1997)with tails extend-ing to[Fe/H]∼−2.2and[Fe/H]∼−0.2.In summary,both photometric and spectroscopic determination agree in sug-gesting a metallicity[Fe/H]∼−ing this value,rather than the value derived from direct conversion of the averagecolour at the tip into metallicity,we derive M TRGBI=−4.00,4Rizzi etal.Figure 3.Measurement of the level of the horizontal branchin Fornax.The magnitude histogram of the stars in the outlined CMD region (most of which are RR Lyrae variable stars)is shown in the inset.Solid and dotted lines indicate the mean V magnitude of HB stars and the r.m.s.of the data,respectively.and the best estimate of the distance based on the TRGB method is then (m −M )0=20.71±0.07.The error we quote is quite large,and is derived by estimating the diﬀer-ence between this last distance measurement and the value derived by simple conversion between colour at the tip and metallicity,considering that the metallicity uncertainty is the dominant factor in the error estimate.4DISTANCE BASED ON HORIZONTAL BRANCH STARSThe average luminosity of RR Lyrae and HB stars is the most widely used Pop.II distance indicator.Our measure-ment of the HB level in Fornax is presented in Fig.3.Stars with 0.3<V −I <0.7and 20.8<V <21.8were selected,and their luminosity function computed.The average mag-nitude of these stars is V HB =21.38±0.04,where the error was inferred from the typical magnitude error at the corre-sponding level.Errors caused by the binning in magnitude,estimated by adding diﬀerent shifts to our choice of bins,were found to be negligible (<0.001mag).Assuming an absolute magnitude for HB stars (actually,RR Lyrae stars)requires knowledge of metallicity.A lot of work has been devoted to determine the metallicity depen-dence of the luminosity of RR Lyrae stars.In this paper,we adopt the recent calibration of Cacciari &Clementini (2003):M V (HB )=(0.23±0.04)([Fe /H]+1.5)+(0.62±0.03)We also consider the previous calibration from Carretta et al.(2000).In both cases,the metallicity is on the scale of Carretta &Gratton (1997).In the previous Section we mentioned a number of esti-mates of the metallicity of Fornax,based on the RGB stars.In this Section,however,the metallicity we are interested in is that of the old stellar population producing the HB and RR Lyrae stars.A search for variable stars by Bersier &Wood (2002)resulted in an average metallicity of the variables [Fe /H]=−1.64on the Butler-Blanco scale,which is known to be more metal-rich than the scale of Zinn &West (1984).This value is conﬁrmed by preliminary results of a study of RR Lyrae in the ﬁeld of Fornax by Greco et al.(2005),suggesting a metallicity [Fe /H]=−1.78on the scale of Carretta &Gratton (1997).This range of results clearly reﬂects the complex enrichment history of Fornax.For the absolute calibration of HB stars,we assume that HB stars belong to the oldest and most metal-poor stellar popula-tion in the galaxy.For this reason,and taking into ac-count the results of Saviane et al.(2000),Bersier &Wood (2002),and Greco et al.(2005),we adopted a metallicity of HB stars [Fe /H]≃−1.8on the scale of Zinn &West (1984),corresponding to [Fe /H]≃−1.6on the scale of Carretta &Gratton (1997).The resulting distance modulus was then derived as:(m −M )0=V HB −M HBV −3.315×E (B −V )Using the calibration of Cacciari &Clementini (2003)the absolute magnitude of HB stars is M HBV =0.59±0.04and the absorption-corrected distance is (m −M )0=20.72±0.06.The calibration of Carretta et al.(2000)would yield M HBV =0.52±0.04and a resulting distance modulus (m −M )0=20.79±0.04.It is well known that RR Lyrae stars spend most of the time near the faint limit of their light curve,so that a straight average of the magnitudes measured in a single epoch might be biased towards fainter values (resulting in an overestimate of the distance).To quantify this possible source of error,we can look at the average magnitude of RR Lyrae stars as measured by Bersier &Wood (2002)and Greco et al.(2005).In both cases,the reddening-corrected magnitude is V 0 ∼21.27,or V =21.34without reddening correction.This is 0.05mags brighter than the values we computed,and this is probably a reliable estimate of the errors related to averaging single-epoch observations rather than the complete light curve.5DISTANCE BASED ON THE RED CLUMP METHODCore helium-burning stars of intermediate age form a well deﬁned clump of stars at a magnitude slightly brighter than the HB and near the Hayashi line.This clump has been claimed to provide a very accurate standard candle (e.g.,see Udalski 2000a,and references therein).Once the mean I -band magnitude of the red clump,I RC ,is known,the ab-solute distance modulus can be derived using the relation:(m −M )0=I RC −M RC I −A I −∆M RCIwhere M RCI is the mean absolute magnitude measured for nearby red clump stars whose distances were measuredThe distance to the Fornax dwarf spheroidal galaxy5 Figure4.Measurement of the mean I magnitude of red clumpstars in Fornax.using independent techniques(such as trigonometric paral-laxes for the Hipparcos sample),A I is the interstellar ab-sorption in the direction of the object and∆M RCIis a pop-ulation correction term accounting for the diﬀerent mixtureof stellar ages in the local sample of stars and the Fornaxgalaxy.The correction term∆M RCIwas initially neglected(Paczynski&Stanek1998;Stanek et al.1998;Udalski et al.1998),but Cole(1998)and Girardi et al.(1998)pointed outthat it is non-negligible according to theoretical models ofclump stars.Empirical determinations of the dependenceof M RCIfrom stellar parameters by Udalski(1998a,b)givea linear relation between M RCIand[Fe/H]with little or nodependence from stellar ages.Udalski(2000b)suggested thefollowing relation for the RC luminosity:M RCI=(0.14±0.04)×([Fe/H]+0.5)−0.29±0.05Discussing the problem from a theoretical point of view,Girardi&Salaris(2001)found a clear and non linear de-pendence of M RCIboth from age and metallicity.They de-termined the mean value of the red clump stars by averag-ing the contribution from all the stars in the core heliumburning phase,and provided tables of M RCIas a functionof age for6diﬀerent metallicities ranging from Z=0.0004to Z=0.03.Following their precepts,we computed theexpected correction to the absolute magnitude of the redclump in Fornax,given its star-formation history and chem-ical enrichment law.The apparent magnitude of the RC in Fornax was mea-sured by selecting stars with0.8<(V−I)0<1.1and19.5<I0<21.0.After this selection the luminosity functionwas computed,a linear continuum subtracted,and Gaus-sianﬁtting used to measure the average magnitude of RCstars.According to our measurements,illustrated in Fig.4,the Fornax RC is found at I RC=20.29±0.03.The errorsare inferred from the typical magnitude error at the corre-sponding level.Errors due to the magnitude binning wereFigure5.The star formation history of Fornax dwarf spheroidaladopted in this paper.The star formation rate is normalised to anaverage value of317M⊙/Myr.Vertical error bars indicate the1σconﬁdence level returned by the best-ﬁt algorithm,while dashedlines show the1σuncertainty derived from an extensive set ofMonte Carlo simulations.estimated by adding a sequence of shifts to the zero-pointof the binning.Such errors were found to be less than0.001and were consequently neglected.We now need to determine the population correction tocorrect for the diﬀerence between the absolute luminosity ofthe RC in the Hipparcos sample and in Fornax.To derivethe population correction we follow these steps:(1)we per-form a full inversion of the CMD of Fornax,to derive itsstar formation history,(2)we measure the luminosity of RCstars on a simulated diagram that closely reproduces thestar formation history of Fornax(M RCI,F ornax),(3)we con-struct a simulated CMD based on literature studies of theage and metallicity distribution of RC stars of the Hippar-cos sample(M RCI,Hipparcos),(4)we compute the populationcorrection∆M RCI=M RCI,F ornax−M RCI,Hipparcos.To derive the star formation history of Fornax,we usedthe chemical enrichment law of Pont et al.(2004)and theCMD simulation technique presented in Rizzi et al.(2002,2003)to perform a full inversion of our Fornax CMD.Theresult is shown in Fig.5.Note that there is a very good agree-ment between the SFH derived here and the one presentedin Tolstoy et al.(2003).Using the results of this simulation,we applied the above described selection of RC stars to thesimulated diagram,and derived an absolute magnitude ofthe RC that fully takes into account the chemical evolu-tion history and the SFH of the galaxy.Gaussianﬁtting tothe background-subtracted luminosity function of simulatedstars in the RC region results in M RCI,F ornax=−0.42±0.03.The error was derived by Monte Carlo simulations,by re-peating500times the CMD inversion using a completelynew set of simple stellar populations.To derive the absolute magnitude of the RC in the6Rizzi et al.Table2.Distance to FornaxMethod Distance DistanceE(B-V)=0.02E(B-V)=0.05 Mean20.72±0.0720.65±0.07Weighted Mean20.72±0.0420.64±0.04The distance to the Fornax dwarf spheroidal galaxy7Cacciari C.,Clementini G.,2003,in Stellar Candles for the Extragalactic Distance Scale,Eds.D.Alloin&W.Gieren, Lectures Notes in Physics,Springer Verlag,635,105 Carretta E.,Gratton R.G.,1997,A&As,121,95 Carretta E.,Gratton R.G.,Clementini G.,Fusi Pecci F., 2000,ApJ,533,215Cole A.A.,1998,ApJL,500,L137Da Costa G.S.,ArmandroﬀT.E.,1990,AJ,100,162 Demers S.,Grondin L.,Kunkel W.E.,1990,PASP,102, 632Ferraro F.R.,Messineo M.,Fusi Pecci F.,de Palo M.A., Straniero O.,ChieﬃA.,Limongi M.,1999,AJ,118,1738 Ferraro F.R.,Montegriﬀo P.,Origlia L.,Fusi Pecci F., 2000,AJ,119,1282Girardi L.,Groenewegen M.A.T.,Weiss A.,Salaris M., 1998,MNRAS,301,149Girardi L.,Salaris M.,2001,MNRAS,323,109Greco C.,Clementini G.,Catelan M.,Held E.V.,Poretti E.,Gullieuszik M.,Maio M.,Rest A.,De Lee N.,Smith H.A.,Pritzl B.J.,2007,ApJ,submittedGreco C.,Clementini G.,Held E.V.,Poretti E.,Catelan M.,Dell’Arciprete L.,Gullieuszik M.,Maio M.,Rizzi L., Smith H.A.,Pritzl B.J.,Rest A.,De Lee N.,in”Re-solved Stellar Populations”,eds.D.Valls-Gabaud and M. Chavez,ASP Conf.Ser.(in press),ArXiv Astrophysics e-prints astro-ph/0507244Gullieuszik M.,Held E.V.,Rizzi L.,Saviane I.,Momany Y.,Ortolani S.,2007,A&A,accepted,ArXiv Astrophysics e-prints astro-ph/0703489Landolt A.U.,1992,AJ,104,340Lee M.G.,Freedman W.L.,Madore B.F.,1993,ApJ,417, 553Makarov D.,Makarova L.,Rizzi L.,Tully R.B.,Dolphin A.E.,Sakai S.,Shaya E.J.,2006,AJ,132,2729M´e ndez B.,Davis M.,Moustakas J.,Newman J.,Madore B.F.,Freedman W.L.,2002,AJ,124,213Paczynski B.,Stanek K.Z.,1998,ApJL,494,L219 Pietrzy´n ski G.,Gieren W.,Fouqu´e P.,Pont F.,2002,AJ, 123,789Pietrzy´n ski G.,Gieren W.,Udalski A.,2003,AJ,125,2494 Pont F.,Zinn R.,Gallart C.,Hardy E.,Winnick R.,2004, AJ,127,840Rizzi L.,Held E.V.,Bertelli G.,Nasi E.,Saviane I.,Val-lenari A.,2002,in Lejeune T.,Fernandes J.,eds,ASP Conf.Ser.274:Observed HR Diagrams and Stellar Evo-lution,p.490Rizzi L.,Held E.V.,Bertelli G.,Saviane I.,2003,ApJL, 589,L85Rocha-Pinto H.J.,Maciel W.J.,Scalo J.,Flynn C.,2000a, A&A,358,850Rocha-Pinto H.J.,Scalo J.,Maciel W.J.,Flynn C.,2000b, A&A,358,869Sakai S.,Madore B.F.,Freedman W.L.,1996,ApJ,461, 713Saviane I.,Held E.V.,Bertelli G.,2000,A&A,355,56 Schlegel D.J.,Finkbeiner D.P.,Davis M.,1998,ApJ,500, 525Stanek K.Z.,Zaritsky D.,Harris J.,1998,ApJL,500,L141 Stetson P.B.,1987,PASP,99,191Stetson P.B.,Hesser J.E.,Smecker-Hane T.A.,1998, PASP,110,533Stone R.C.,Pier J.R.,Monet D.G.,1999,AJ,118,2488Tolstoy E.,Irwin M.J.,Cole A.A.,Pasquini L.,Gilmozzi R.,Gallagher J.S.,2001,MNRAS,327,918Tolstoy E.,Venn K.A.,Shetrone M.,Primas F.,Hill V., Kaufer A.,Szeifert T.,2003,AJ,125,707Udalski A.,1998a,Acta Astronomica,48,383Udalski A.,1998b,Acta Astronomica,48,113Udalski A.,2000a,ApJL,531,L25Udalski A.,2000b,Acta Astronomica,50,279Udalski A.,Szymanski M.,Kubiak M.,Pietrzynski G., Wozniak P.,Zebrun K.,1998,Acta Astronomica,48,1 Valdes F.G.,1998,in Albrecht R.,Hook R.N.,Bushouse H.A.,eds,ASP Conf.Ser.145:Astronomical Data Anal-ysis Software and Systems VII The IRAF Mosaic Data Reduction Package.p.53Zinn R.,West M.J.,1984,ApJs,55,45。

Is Double Reionization Physically Plausible

Draft version February 2, 2008
arXiv:astro-ph/0409656v2 20 Jan 2005
ABSTRACT Recent observations of z ∼ 6 quasars and the cosmic microwave background imply a complex history to cosmic reionization. Such a history requires some form of feedback to extend reionization over a long time interval, but the nature of the feedback and how rapidly it operates remain highly uncertain. Here we focus on one aspect of this complexity: which physical processes can cause the global ionized fraction to evolve non-monotonically with cosmic time? We consider a range of mechanisms and conclude that double reionization is much less likely than a long, but still monotonic, ionization history. We ﬁrst examine how galactic winds affect the transition from metal-free to normal star formation. Because the transition is actually spatially inhomogeneous and temporally extended, this mechanism cannot be responsible for double reionization given plausible parameters for the winds. We next consider photoheating, which causes the cosmological Jeans mass to increase in ionized regions and hence suppresses galaxy formation there. In this case, double reionization requires that small halos form stars efﬁciently, that the suppression from photoheating is strong relative to current expectations, and that ionizing photons are preferentially produced outside of previously ionized regions. Finally, we consider H2 photodissociation, in which the buildup of a soft ultraviolet background suppresses star formation in small halos. This can in principle cause the ionized fraction to temporarily decrease, but only during the earliest stages of reionization. Finally, we brieﬂy consider the effects of some of these feedback mechanisms on the topology of reionization. Subject headings: cosmology: theory — galaxies: evolution — intergalactic medium

#x #

Abstract We consider uniqueness theorems in classical analysis having the form (+) ∀u ∈ U, v1 , v2 ∈ Vu G(u, v1 ) = 0 = G(u, v2 ) → v1 = v2 , where U, V are complete separable metric spaces, Vu is compact in V and G : U × V → IR is a constructive function. If (+) is proved by arithmetical means from analytical assumptions (++) ∀x ∈ X ∃y ∈ Yx ∀z ∈ Z F (x, y, z ) = 0 only (where X, Y, Z are complete separable metric spaces, Yx ⊂ Y is compact and F : X × Y × Z → IR constructive), then we can extract from the proof of (++) → (+) an eﬀective modulus of uniqueness, i.e. (+ + +) ∀u ∈ U, v1 , v2 ∈ Vu , k ∈ IN |G(u, v1 )|, |G(u, v2 )| ≤ 2−Φuk → dV (v1 , v2 ) ≤ 2−k . Such a modulus Φ can e.g. be used to give a ﬁnite algorithm which computes the (uniquely determined) zero of G(u, ·) on Vu with prescribed precision if it exists classically. The extraction of Φ uses a proof–theoretic combination of functional interpretation and pointwise majorization. If the proof of (++) → (+) uses only simple instances of induction, then Φ is a simple mathematical operation (on a convenient standard representation of X , e.g. on f together with a modulus of uniform continuity for X = C [0, 1]). Various uniqueness theorems in best approximation theory have the form (+) and are proved using only analytical tools of the form (++). We analyse the most common proof of uniqueness for the best Chebycheﬀ approximation of f ∈ C [0, 1] by polynomials of degree ≤ n given by de La Vall´ ee Poussin and obtain explicit moduli of uniqueness and uniform constants of strong unicity. In a subsequent paper two further proofs of this uniqueness will be analysed yielding better estimates (due to the fact that mainly (++)–lemmas are used) which allow us to improve results obtained prior by D. Bridges signiﬁcantly.

曼哈顿度量算法英文

曼哈顿度量算法英文English:The Manhattan distance algorithm, also known as the city block distance or L1 distance, is a metric used to measure the distance between two points in a grid or coordinate system. It calculates the distance by summing the absolute differences between the corresponding coordinates of the two points. In other words, it measures how many blocks or steps it would take to travel from one point to another if you can only move horizontally or vertically.To compute the Manhattan distance, you would start by identifying the coordinates of the two points, let's say (x1, y1) and (x2, y2), in a two-dimensional grid. Then, you would calculate the absolute difference between the x-coordinates (x2 - x1) and the absolute difference between the y-coordinates (y2 - y1). Finally, you would sum these two differences to obtain the Manhattan distance.The Manhattan distance is widely used in various fields, especially in computer science and mathematics. One major application is in thefield of image processing, where it is used to compare and classify images based on their similarity or dissimilarity. It is also frequently employed in computer vision tasks, such as object tracking or image segmentation, where measuring the spatial distance between pixels or image features is essential.In addition to its application in image processing, the Manhattan distance algorithm has other practical uses. For instance, it is commonly used in routing algorithms for finding the shortest path between two points in a city grid or road network. It can also be applied in cluster analysis, data mining, and machine learning algorithms as a distance or similarity measure to compare and group data points.In conclusion, the Manhattan distance algorithm provides a simple and intuitive way to measure the distance between two points in a grid or coordinate system. Its applications extend to various fields, including image processing, computer vision, routing algorithms, and data analysis. By summing the absolute differences between the corresponding coordinates of the points, the Manhattan distance caneffectively quantify spatial relationships and facilitate decision-making processes.中文翻译:曼哈顿度量算法，也称为城市街区距离或L1距离，是用于测量网格或坐标系中两点之间距离的一种度量标准。

tpo40三篇托福阅读TOEFL原文译文题目答案译文背景知识

tpo40三篇托福阅读TOEFL原文译文题目答案译文背景知识阅读-1 (2)原文 (2)译文 (5)题目 (8)答案 (17)背景知识 (17)阅读-2 (20)原文 (20)译文 (23)题目 (25)答案 (35)背景知识 (35)阅读-3 (38)原文 (38)译文 (41)题目 (44)答案 (53)背景知识 (54)阅读-1原文Ancient Athens①One of the most important changes in Greece during the period from 800 B.C. to 500 B.C. was the rise of the polis, or city-state, and each polis developed a system of government that was appropriate to its circumstances. The problems that were faced and solved in Athens were the sharing of political power between the established aristocracy and the emerging other classes, and the adjustment of aristocratic ways of life to the ways of life of the new polis. It was the harmonious blending of all of these elements that was to produce the classical culture of Athens.②Entering the polis age, Athens had the traditional institutions of other Greek protodemocratic states: an assembly of adult males, an aristocratic council, and annually elected officials. Within this traditional framework the Athenians, between 600 B.C. and 450 B.C., evolved what Greeks regarded as a fully fledged democratic constitution, though the right to vote was given to fewer groups of people than is seen in modern times.③The first steps toward change were taken by Solon in 594 B.C., when he broke the aristocracy's stranglehold on elected offices by establishing wealth rather than birth as the basis of office holding, abolishing the economic obligations of ordinary Athenians to the aristocracy, and allowing the assembly (of which all citizens were equal members) to overrule the decisions of local courts in certain cases. The strength of the Athenian aristocracy was further weakened during the rest of the century by the rise of a type of government known as a tyranny, which is a form of interim rule by a popular strongman (not rule by a ruthless dictator as the modern use of the term suggests to us). The Peisistratids, as the succession of tyrants were called (after the founder of the dynasty, Peisistratos), strengthened Athenian central administration at the expense of the aristocracy by appointing judges throughout the region, producing Athens’ first national coinage, and adding and embellishing festivals that tended to focus attention on Athens rather than on local villages of the surrounding region. By the end of the century, the time was ripe for more change: the tyrants were driven out, and in 508 B.C. a new reformer, Cleisthenes, gave final form to the developments reducing aristocratic control already under way.④Cleisthenes' principal contribution to the creation of democracy at Athens was to complete the long process of weakening family and clanstructures, especially among the aristocrats, and to set in their place locality-based corporations called demes, which became the point of entry for all civic and most religious life in Athens. Out of the demes were created 10 artificial tribes of roughly equal population. From the demes, by either election or selection, came 500 members of a new council, 6,000 jurors for the courts, 10 generals, and hundreds of commissioners. The assembly was sovereign in all matters but in practice delegated its power to subordinate bodies such as the council, which prepared the agenda for the meetings of the assembly, and courts, which took care of most judicial matters. Various committees acted as an executive branch, implementing policies of the assembly and supervising, for instance, the food and water supplies and public buildings. This wide-scale participation by the citizenry in the government distinguished the democratic form of the Athenian polis from other less liberal forms.⑤The effect of Cleisthenes’ reforms was to establish the superiority of the Athenian community as a whole over local institutions without destroying them. National politics rather than local or deme politics became the focal point. At the same time, entry into national politics began at the deme level and gave local loyalty a new focus: Athens itself. Over the next two centuries the implications of Cleisthenes’ reforms were fully exploited.⑥During the fifth century B.C. the council of 500 was extremely influential in shaping policy. In the next century, however, it was the mature assembly that took on decision-making responsibility. By any measure other than that of the aristocrats, who had been upstaged by the supposedly inferior "people", the Athenian democracy was a stunning success. Never before, or since, have so many people been involved in the serious business of self-governance. It was precisely this opportunity to participate in public life that provided a stimulus for the brilliant unfolding of classical Greek culture.译文古雅典①在公元前800年到公元前500年期间，希腊最重要的变化之一是城邦的崛起，并且每个城邦都发展了适合其情况的政府体系。

David Hilbert - Mathematical Problems

Mathematical ProblemsLecture delivered before the International Congress ofMathematicians at Paris in 1900By Professor David Hilbert1Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the "problem of the line of quickest descent," proposed by John Bernoulli. Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermat, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted, as is well known, that the diophantine equationx n + y n = z n(x, y and z integers) is unsolvable—except in certain self evident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors—a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very different region of research, I remind you of the problem of three bodies. The fruitful methods and the far-reaching principles which Poincaré has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.In order to throw light on the importance of certain problems, I may also refer to Weierstrass, who spokeof it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi's problem of inversion on which to work.Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics. But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. Thus arose the problem of prime numbers and the other problems of number theory, Galois's theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions; indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in this ever-recurring interplay between thought and experience.It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning. Indeed the requirement of rigor, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and, on the other hand, only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially when it comes from the world of outer experience, is like a young twig, which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem, the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor. Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigorous function-theoretical methods and the consistent introduction of transcendental devices. Further, the proof that the power series permits the application of the four elementary arithmetical operations as well as the term by term differentiation and integration, and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis, particularly of the theory of elimination and the theory of differential equations, and also of the existence proofs demanded in those theories. But the most striking example for my statement is the calculus of variations. The treatment of the first and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations. By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of the calculus of variations. For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum, the calculation of the second variation and in part, indeed, the wearisome reasoning connected with the first variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment. This opinion, occasionally advocated by eminent men, I consider entirely erroneous. Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation? Whocould dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content. Just as in adding two numbers, one must place the digits under each other in the right order, so that only the rules of calculation, i. e., the axioms of arithmetic, determine the correct use of the digits, so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen.2Some remarks upon the difficulties which mathematical problems may offer, and the means of surmounting them, may be in place here.If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems. The introduction of complex paths of integration by Cauchy and of the notion of the IDEALS in number theory by Kummer may serve as examples. This way for finding general methods is certainly the most practicable and the most certain; for he who seeks for methods without having a definite problem in mind seeks for the most part in vain.In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them bymeans of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and it seems to me that it is used almost always, though perhaps unconsciously.Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant C, or the existence of an infinite number of prime numbers of the form 2n + 1. However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility. I instance the problem of perpetual motion. After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;3 and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry. The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of thecontinuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky. I therefore first direct your attention to some problems belonging to these fields.1. Cantor's problem of the cardinal number of the continuumTwo systems, i. e, two assemblages of ordinary real numbers or points, are said to be (according to Cantor) equivalent or of equal cardinal number, if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other. The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving. This is the theorem:Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3,... or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line; as regards equivalence there are, therefore, only two assemblages of numbers, the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage; the proof of this theorem would, therefore, form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before b and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. But there are, as is easily seen infinitely many other ways in which the numbers of a system may be arranged.If we think of a definite arrangement of numbers and select from them a particular system of these numbers, a so-called partial system or assemblage, this partial system will also prove to be ordered. Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way, that not only in the assemblage itself but also in every partial assemblage there exists a first number. The system of integers 1, 2, 3, ... in their natural order is evidently a well ordered assemblage. On the other hand the system of all real numbers, i. e., the continuum in its natural order, is evidently not well ordered. For, if we think of the points of a segment of a straight line, with its initial point excluded, as our partial assemblage, it will have no first element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have a first element, i. e., whether the continuum cannot be consideredas a well ordered assemblage—a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out.2. The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms. The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them4 and in so doing replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows: that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.To show the significance of the problem from another point of view, I add the following observation: If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -l does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved. In the case before us, wherewe are concerned with the axioms of real numbers in arithmetic, the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum. Indeed, when the proof for the compatibility of the axioms shall be fully accomplished, the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless. The totality of real numbers, i. e., the continuum according to the point of view just indicated, is not the totality of all possible series in decimal fractions, or of all possible laws according to which the elements of a fundamental sequence may proceed. It is rather a system of things whose mutual relations are governed by the axioms set up and for which all propositions, and only those, are true which can be derived from the axioms by a finite number of logical processes. In my opinion, the concept of the continuum is strictly logically tenable in this sense only. It seems to me, indeed, that this corresponds best also to what experience and intuition tell us. The concept of the continuum or even that of the system of all functions exists, then, in exactly the same sense as the system of integral, rational numbers, for example, or as Cantor's higher classes of numbers and cardinal numbers. For I am convinced that the existence of the latter, just as that of the continuum, can be proved in the sense I have described; unlike the system of all cardinal numbers or of all Cantor s alephs, for which, as may be shown, a system of axioms, compatible in my sense, cannot be set up. Either of these systems is, therefore, according to my terminology, mathematically non-existent.From the field of the foundations of geometry I should like to mention the following problem:3. The equality of two volumes of two tetrahedra of equal bases and equal altitudesIn two letters to Gerling, Gauss5 expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.6 Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.74. Problem of the straight line as the shortest distance between two pointsAnother problem relating to the foundations of geometry is this: If from among the axioms necessary to。

火山爆发岩浆喷出英语作文

Volcanic eruptions are one of the most spectacular and powerful natural phenomena on Earth.When a volcano erupts,it is the result of intense geological activity deep within the planet.Here is a detailed description of what happens when a volcano erupts and the process of magma being expelled:1.Preeruption Seismic Activity:Before a volcanic eruption,there is often an increase in seismic activity.This is due to the movement of magma as it rises towards the Earths surface.Seismographs can detect these tremors,which can be an early warning sign of an impending eruption.2.Magma Formation:Magma is formed when the Earths mantle melts due to high temperatures and pressures.This molten rock material is less dense than the surrounding solid rock,causing it to rise towards the surface.3.Magma Chamber:The magma collects in a magma chamber,which is a reservoir beneath the volcano.Over time,the pressure in the chamber increases as more magma accumulates.4.Eruption Trigger:An eruption can be triggered by several factors,including the influx of more magma into the chamber,the release of dissolved gases,or the weakening of the overlying rock due to geological processes.5.Magma Ascent:As pressure in the magma chamber builds,the magma forces its way up through the volcanic conduit,which is a pathway from the chamber to the surface.6.Eruption:When the magma reaches the surface,it is called lava.The eruption can be explosive or effusive,depending on the viscosity of the magma and the amount of gas it contains.Explosive eruptions produce ash,pyroclastic flows,and other volcanic materials,while effusive eruptions result in the slow outpouring of lava.va Flows:Once the lava reaches the surface,it flows downhill,creating lava flows. These flows can travel great distances and can cause destruction in their path,depending on their speed and volume.8.Ash Clouds and Pyroclastic Flows:In explosive eruptions,the magma can fragment into small particles,creating ash clouds that can be carried by the wind for hundreds or even thousands of kilometers.Pyroclastic flows are fastmoving currents of hot gas and volcanic matter that can travel at high speeds and cause widespread devastation.9.Posteruption Activity:After an eruption,the volcano may continue to emit gases andsmall amounts of lava.The formation of new land can occur as lava cools and solidifies, and ash can contribute to the formation of fertile soils.10.Longterm Effects:Volcanic eruptions can have longlasting effects on the environment,climate,and human societies.They can cause shortterm climate changes due to the reflection of sunlight by ash particles in the atmosphere,and they can lead to the creation of new ecosystems as life returns to areas affected by eruptions.In conclusion,a volcanic eruption is a complex process that involves the movement, accumulation,and expulsion of magma from the Earths interior.The resulting lava flows, ash clouds,and pyroclastic flows can have significant impacts on the surrounding environment and human activities.Understanding these processes is crucial for monitoring volcanic activity and mitigating the risks associated with eruptions.。

Ferromagnetic ordering in graphs with arbitrary degree distribution

We present a detailed study of the phase diagram of the Ising model in random graphs with arbitrary degree distribution. By using the replica method we compute exactly the value of the critical temperature and the associated critical exponents as a function of the minimum and maximum degree, and the degree distribution characterizing the graph. As expected, there is a ferromagnetic transition provided k ≤ 2 k2 < ∞. However, if the fourth moment of the degree distribution is not ﬁnite then non-trivial scaling exponents are obtained. These results are analyzed for the particular case of power-law distributed random graphs.
I. INTRODUCTION
The increasing evidence that many physical, biological and social networks exhibit a high degree of wiring entanglement has led to the investigation of

庄子-秋水Autumn-Floods

Autumn Floods (英文版庄子-秋水全篇)In the time of autumn floods, a hundred streams poured into the river. It swelled in its turbid course, so that it was impossible to tell a cow from a horse on the opposite banks or on the islets. Then the Spirit of the River laughed for joy that all the beauty of the earth was gathered to himself. Down the stream he journeyed east, until he reached the North Sea. There, looking eastwards and seeing no limit to its wide expanse, his countenance began to change. And as he gazed over the ocean, he sighed and said to North-Sea Jo, "A vulgar proverb says that he who has heard a great many truths thinks no one equal to himself. And such a one am I. Formerly when I heard people detracting from the learning of Confucius or underrating the heroism of Po Yi, I did not believe it. But now that I have looked upon your inexhaustibility -- alas for me ! had I not reached your abode, I should have been for ever a laughing stock to those of great enlightenment!"To this North-Sea Jo (the Spirit of the Ocean) replied, "You cannot speak of ocean to a well-frog, which is limited by his abode. You cannot speak of ice to a summer insect, which is limited by his short life. You cannot speak of Tao to a pedagogue, who is limited in his knowledge. But now that you have emerged from your narrow sphere and have seen the great ocean, you know your own insignificance, and I can speak to you of great principles."There is no body of water beneath the canopy of heaven which is greater than the ocean. All streams pour into it without cease, yet it does not overflow. It is being continually drained off at the Tail-Gate yet it is never empty. Spring and autumn bring no change; floods and droughts are equally unknown. And thus it is immeasurably superior to mere rivers and streams. Yet I have never ventured to boast on this account. For I count myself, among the things that take shape from the universe and receive life from the yin and yang, but as a pebble or a small tree on a vast mountain. Only too conscious of my own insignificance, how can I presume to boast of my greatness?"Are not the Four Seas to the universe but like ant-holes in a marsh? Is not the Middle Kingdom to the surrounding ocean like a tare-seed in a granary? Of all the myriad created things, man is but one. And of all those who inhabit the Nine Continents, live on the fruit of the earth, and move about in cart and boat, an individual man is but one. Is not he, as compared with all creation, but as the tip of a hair upon a horse's body?"The succession of the Five Rulers, the contentions of the Three Kings, the concerns of the kind-hearted, the labors of the administrators, are but this and nothing more. Po Yi refused the throne for fame. Chungni (Confucius) discoursed to get a reputation for learning. This over-estimation of self on their part -- was it not very much like your own previous self-estimation in reference to water?""Very well," replied the Spirit of the River, "am I then to regard the universe as great and the tip of a hair as small?""Not at all," said the Spirit of the Ocean. "Dimensions are limitless; time is endless. Conditions are not constant; terms are not final. Thus, the wise man looks into space, and does not regard the small as too little, nor the great as too much; for he knows that there is no limit to dimensions. He looks back into the past, and does not grieve over what is far off, nor rejoice over what is near; for he knows that time is without end. He investigates fullness and decay, and therefore does not rejoice if he succeeds, nor lament if he fails; for he knows that conditions are not constant. He who clearly apprehends the scheme of existence does not rejoice over life, nor repine at death; for he knows that terms are not final."What man knows is not to be compared with what he does not know. The span of his existence is not to be compared with the span of his non-existence. To strive to exhaust the infinite by means of the infinitesimal necessarily lands him in confusion and unhappiness. How then should one be able to say that the tip of a hair is the ne plus ultra of smallness, or that the universe is the ne plus ultra of greatness?""Dialecticians of the day," replied the Spirit of the River, "all say that the infinitesimal has no form, and that the infinite is beyond all measurement. Is that true?""If we look at the great from the standpoint of the small," said the Spirit of the Ocean, "we cannot reach its limit; and if we look at the small from the standpoint of the great, it eludes our sight. The infinitesimal is a subdivision of the small; the colossal is an extension of the great. In this sense the two fall into different categories. This lies in the nature of circumstances. Now smallness and greatness presuppose form. That which is without form cannot be divided by numbers, and that which is above measurement cannot be measured. The greatness of anything may be a topic of discussion, and the smallness of anything may be mentally imagined. But that which can be neither a topic of discussion nor imagined mentally cannot be said to have greatness or smallness."Therefore, the truly great man does not injure others and does not credit himself with charity and mercy. He seeks not gain, but does not despise the servants who do. He struggles not for wealth, but does not lay great value on his modesty. He asks for help from no man, but is not proud of his self-reliance, neither does he despise the greedy. He acts differently from the vulgar crowd, but does not place high value on being different or eccentric; nor because he acts with the majority does he despise those that flatter a few. The ranks and emoluments of the world are to him no cause for joy; its punishments and shame no cause for disgrace. He knows that right and wrong cannot be distinguished, that great and small cannot be defined."I have heard say, 'The man of Tao has no (concern) reputation; the truly virtuous has no (concern for) possessions; the truly great man ignores self.' This is the height ofself-discipline.""But how then," asked the Spirit of the River, "arise the distinctions of high and low, ofgreat and small in the material and immaterial aspects of things?""From the point of view of Tao," replied the Spirit of the Ocean, "there are no such distinctions of high and low. From the point of view of individuals, each holds himself high and holds others low. From the vulgar point of view, high and low (honors and dishonor) are some thing conferred by others. "In regard to distinctions, if we say that a thing is great or small by its own standard of great or small, then there is nothing in all creation which is not great, nothing which is not small. To know that the universe is but as a tare-seed, and the tip of a hair is (as big as) a mountain, -- this is the expression of relativity."In regard to function, if we say that something exists or does not exist, by its own standard of existence or non- existence, then there is nothing which does not exist, nothing which does not perish from existence. If we know that east and west are convertible and yet necessary terms in relation to each other, then such (relative) functions may be determined."In regard to man's desires or interests, if we say that anything is good or bad because it is either good or bad according to our individual (subjective) standards, then there is nothing which is not good, nothing -- which is not bad. If we know that Yao and Chieh each regarded himself as good and the other as bad, then the (direction of) their interests becomes apparent."Of old Yao and Shun abdicated (in favor of worthy successors) and the rule was maintained, while Kuei (Prince of Yen) abdicated (in favor of Tsechih) and the latter failed. T'ang and Wu got the empire by fighting, while by fighting, Po Kung lost it. From this it may be seen that the value of abdicating or fighting, of acting like Yao or like Chieh, varies according to time, and may not be regarded as a constant principle. "A battering-ram can knock down a wall, but it cannot repair a breach. Different things are differently applied. Ch'ichi and Hualiu (famous horses) could travel 1,000 li in one day, but for catching rats they were not equal to a wild cat. Different animals possess different aptitudes. An owl can catch fleas at night, and see the tip of a hair, but if it comes out in the daytime it can open wide its eyes and yet fail to see a mountain. Different creatures are differently constituted."Thus, those who say that they would have right without its correlate, wrong; or good government without its correlate, misrule, do not apprehend the great principles of the universe, nor the nature of all creation. One might as well talk of the existence of Heaven without that of Earth, or of the negative principle without the positive, which is clearly impossible. Yet people keep on discussing it without stop; such people must be either fools or knaves."Rulers abdicated under different conditions, and the Three Dynasties succeeded each other under different conditions. Those who came at the wrong time and went against the tide are called usurpers. Those who came at the right time and fitted in with their age are called defenders of Right. Hold your peace, Uncle River. How can you know thedistinctions of high and low and of the houses of the great and small?'"In this case," replied the Spirit of the River, "what am I to do about declining and accepting, following and abandoning (courses of action)?""From the point of view of Tao," said the Spirit of the Ocean."How can we call this high and that low? For there is (the process of) reverse evolution (uniting opposites). To follow one absolute course would involve great departure from Tao. What is much? What is little? Be thankful for the gift. To follow a one-sided opinion is to diverge from Tao. Be exalted, as the ruler of a State whose administration is impartial. Be at ease, as the Deity of the Earth, whose dispensation is impartial. Be expansive, like the points of the compass, boundless without a limit. Embrace all creation, and none shall be more sheltered or helped than another. This is to be without bias. And all things being equal, how can one say which is long and which is short? Tao is without beginning, without end. The material things are born and die, and no credit is taken for their development. Emptiness and fullness alternate, and their relations are not fixed. Past years cannot be recalled; time cannot be arrested. The succession of growth and decay, of increase and diminution, goes in a cycle, each end becoming a new beginning. In this sense only may we discuss the ways of truth and the principles of the universe. The life of things passes by like a rushing, galloping horse, changing at every turn, at every hour. What should one do, or what should one not do? Let the (cycle of) changes go on by themselves!""If this is the case," said the Spirit of the River, "what is the value of Tao?""Those who understand Tao," answered the Spirit of the Ocean "must necessarily apprehend the eternal principles and those who apprehend the eternal principles must understand their application. Those who understand their application do not suffer material things to injure them. "The man of perfect virtue cannot be burnt by fire, nor drowned by water, nor hurt by the cold of winter or the heat of summer, nor torn by bird or beast. Not that he makes light of these; but that he discriminates between safety and danger, is happy under prosperous and adverse circumstances alike, and cautious in his choice of action, so that none can harm him."Therefore it has been said that Heaven (the natural) abides within man (the artificial) without. Virtue abides in the natural. Knowledge of the action of the natural and of the artificial has its basis in the natural its destination in virtue. Thus, whether moving forward or backwards whether yielding or asserting, there is always a reversion to the essential and to the ultimate.""What do you mean," enquired the Spirit of the River, "by the natural and the artificial?""Horses and oxen," answered the Spirit of the Ocean, "have four feet. That is the natural.Put a halter on a horse's head, a string through a bullock's nose. That is the artificial."Therefore it has been said, do not let the artificial obliterate the natural; do not let will obliterate destiny; do not let virtue be sacrificed to fame. Diligently observe these precepts without fail, and thus you will revert to the True."The walrus envies the centipede; the centipede envies the snake; the snake envies the wind; the wind envies the eye; and the eye envies the mind. The walrus said to the centipede, "I hop about on one leg but not very successfully. How do you manage all those legs you have?""I don't manage them," replied the centipede. "Have you never seen saliva? When it is ejected, the big drops are the size of pearls, the small ones like mist. At random they fall, in countless numbers. So, too, does my natural mechanism move, without my knowing how I do it."The centipede said to the snake, "With all my legs I do not move as fast as you with none. How is that?""One's natural mechanism," replied the snake, "is not a thing to be changed. What need have I for legs?"The snake said to the wind, "I wriggle about by moving my spine, as if I had legs. Now you seem to be without form, and yet you come blustering down from the North Sea to bluster away to the South Sea How do you do it?""'Tis true," replied the wind, "that I bluster as you say. But anyone who sticks his finger or his foot into me, excels me. On the other hand, I can tear away huge trees and destroy large buildings. This power is given only to me. Out of many minor defeats I win the big victory. And to win a big victory is given only to the Sages."When Confucius visited K'uang, the men of Sung surrounded him by several cordons. Yet he went on singing to his guitar without stop. "How is it, Master," enquired Tselu, "that you are so cheerful?""Come here," replied Confucius, "and I will tell you. For a long time I have not been willing to admit failure, but in vain. Fate is against me. For a long time I have been seeking success, but in vain. The hour has not come. In the days of Yao and Shun, no man throughout the empire was a failure, though this was not due to their cleverness. In the days of Chieh and Chou, no man throughout the empire was a success, though this was not due to their stupidity. The circumstances happened that way."To travel by water without fear of sea-serpents and dragons, -- this is the courage of the fisherman. To travel by land without fear of the wild buffaloes and tigers, -- this is thecourage of hunters. When bright blades cross, to look on death as on life, -- this is the courage of the warrior. To know that failure is fate and that success is opportunity, and to remain fearless in times of great danger, -- this is the courage of the Sage. Stop bustling, Yu! My destiny is controlled (by someone).Shortly afterwards, the captain of the troops came in and apologized, saying, "We thought you were Yang Hu; that was why we surrounded you. We find we have made a mistake." Whereupon he apologized and retired.Kungsun Lung said to Mou of Wei, "When young I studied the teachings of the elders. When I grew up, I understood the morals of charity and duty. I learned to level together similarities and differences, to confound arguments on "hardness" and "whiteness", to affirm what others deny, and justify what others dispute. I vanquished the wisdom of all the philosophers, and overcame the arguments of all people. I thought that I had indeed understood everything. But now that I have heard Chuangtse, I am lost in astonishment. I know not whether it is in arguing or in knowledge that I am not equal to him. I can no longer open my mouth. May I ask you to impart to me the secret?"Prince Mou leaned over the table and sighed. Then he looked up to heaven and laughed, saying, "Have you never heard of the frog in the shallow well? The frog said to the turtle of the Eastern Sea, 'What a great time I am having! I hop to the rail around the well, and retire to rest in the hollow of some broken bricks. Swimming, I float on my armpits, resting my jaws just above the water. Plunging into the mud, I bury my feet up to the foot-arch, and not one of the cockles, crabs or tadpoles I see around me are my match. Besides, to occupy such a pool all alone and possess a shallow well is to be as happy as anyone can be. Why do you not come and pay me a visit?'"Now before the turtle of the Eastern Sea had got its left leg down its right knee had already stuck fast, and it shrank back and begged to be excused. It then told the frog about the sea, saying, 'A thousand li would not measure its breadth, nor a thousand fathoms its depth. In the days of the Great Yu:, there were nine years of flood out of ten; but this did not add to its bulk. In the days of T'ang, there were seven years of drought out of eight; but this did not make its shores recede. Not to be affected by the passing of time, and not to be affected by increase or decrease of water, -- such is the great happiness of the Eastern Sea.' At this the frog of the shallow well was considerably astonished and felt very small, like one lost."For one whose knowledge does not yet appreciate the niceties of true and false to attempt to understand Chuangtse, is like a mosquito trying to carry a mountain, or an insect trying to swim a river. Of course he will fail. Moreover, one whose knowledge does not reach to the subtlest teachings, yet is satisfied with temporary success, -- is not he like the frog in the well?"Chuangtse is now climbing up from the realms below to reach high heaven. For him nonorth or south; lightly the four points are gone, engulfed in the unfathomable. For him no east or west - starting from the Mystic Unknown, he returns to the Great Unity. And yet you think you are going to find his truth by dogged inquiries and arguments! This is like looking at the sky through a tube, or pointing at the earth with an awl. Is not this being petty?"Have you never heard how a youth of Shouling went to study the walking gait at Hantan? Before he could learn the Hantan gait, he had forgotten his own way of walking, and crawled back home on all fours. If you do not go away now, you will forget what you have and lose your own professional knowledge." Kungsun Lung's jaw hung open, his tongue clave to his palate, and he slunk away.Chuangtse was fishing on the P'u River when the Prince of Ch'u sent two high officials to see him and said, "Our Prince desires to burden you with the administration of the Ch'u State." Chuangtse went on fishing without turning his head and said, "I have heard that in Ch'u there is a sacred tortoise which died when it was three thousand (years) old. The prince keeps this tortoise carefully enclosed in a chest in his ancestral temple. Now would this tortoise rather be dead and have its remains venerated, or would it rather be alive and wagging its tail in the mud?""It would rather be alive," replied the two officials, and wagging its tail in the mud.""Begone!" cried Chuangtse. "I too will wag my tail in the mud.Hueitse was Prime Minister in the Liang State, and Chuangtse was on his way to see him. Someone remarked, "Chuangtse has come. He wants to be minister in your place." Thereupon Hueitse was afraid, and searched all over the country for three days and three nights to find him.Then Chuangtse went to see him, and said, "In the south there is a bird. It is a kind of phoenix. Do you know it? When it starts from the South Sea to fly to the North Sea, it would not alight except on the wu-t'ung tree. It eats nothing but the fruit of the bamboo, drinks nothing but the purest spring water. An owl which had got the rotten carcass of a rat, looked up as the phoenix flew by, and screeched. Are you not screeching at me over your kingdom of Liang?"Chuangtse and Hueitse had strolled on to the bridge over the Hao, when the former observed, "See how the small fish are darting about! That is the happiness of the fish.""You not being a fish yourself," said Hueitse, "how can you know the happiness of the fish?""And you not being I," retorted Chuangtse, "how can you know that I do not know?""If I, not being you, cannot know what you know," urged Hueitse, "it follows that you, notbeing a fish, cannot know the happiness of the fish.""Let us go back to your original question," said Chuangtse. "You asked me how I knew the happiness of the fish. Your very question shows that you knew that I knew. I knew it (from my own feelings) on this bridge."[文档可能无法思考全面，请浏览后下载，另外祝您生活愉快，工作顺利，万事如意!]。

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a r X i v :a s t r o -p h /0011056v 1 2 N o v 2000A&A manuscript no.(will be inserted by hand later)ASTRONOMYANDASTROPHY SICS1.IntroductionThe determination of the distance moduli (DM )of the Magellanic Clouds plays an important role in establish-ing the cosmic distance scale.Mostly due to the analyses of the hipparcos data,there is a renewed eﬀort to pin down this crucial distance within an accuracy of better than ≈0.05mag.However,the question is far from being settled,and the LMC distance moduli obtained with var-ious methods may diﬀer by several tenth of magnitudes,stretching from the ‘short’(DM ≈18.2,e.g.,Stanek et al.2000)to the ‘long’(DM ≈18.6,e.g.,Feast 1999)distance scales.In the ﬁrst paper of this series (Kov´a cs &Walker 1999,hereafter KW99,see also Kov´a cs 2000b)we used2G.Kov´a cs:The distance modulus of the Large Magellanic Cloud based on double-mode RR Lyrae stars KW99simply leads to the distance modulus through thecomparison with the observed magnitudes.In the following we give a more detailed descriptionof the parameters entering in the calculation of the LMCdistance modulus.Together with the earlier discoveries of Alcock et al.(1997),the recent analysis of more than1300short-periodRR Lyrae stars led to a substantial extension of the knownRRd stars in the LMC(Alcock et al.2000).In the presentanalysis we include181RRd variables,which constituteall presently known RRd stars in the LMC.The machoinstrumental magnitudes have been transformed to thestandard Johnson V and Kron-Cousins R c colors accord-ing to the recipe of Alcock et al.(1999).The periods andthe average magnitudes are listed in Table1.1For an independent computation of the LMC distancemodulus we use globular cluster RRd data compiled byKW99.Relative distance moduli required by this methodare checked by the photometric data of Udalski(1998)andClementini et al.(2000b,hereafter C00b).The T eﬀ=f(color,log g,[M/H])relations are derivedfrom the stellar atmosphere models of Castelli et al.(1997,hereafter C97)with the zero point adjusted to the irfm re-sults of Blackwell&Lynas-Gray(1994,hereafter BLG94).The required color,log g and[M/H]data are obtainedfrom Clementini et al.(1995).It is important to remarkthat this approach assumes a uniform shift in log T eﬀbe-tween the irfm and theoretical scales,and that this shiftis applicable throughout the relevant parameter regime(i.e.,from dwarfs to giants).Additional diﬃculty mightoccur because of the inaccuracy of the[Fe/H]and log gvalues given by the irfm sources,or the neglect of thelog g dependence in some of those works(e.g.,Alonso etal.1996;1999,hereafter A96and A99,respectively).Fur-thermore,diﬀerent colors may yield diﬀerent shifts,pro-ducing inconsistency among the derived temperatures ata level of0.004in log T eﬀ.By considering various colorsand overlapping samples in the irfm publications,ourcurrent estimates for the zero point diﬀerences(in thesense of log T eﬀ(source)minus log T eﬀ(C97))are as fol-lows:−0.004(BLG94),−0.007(A96);−0.008(Blackwell&Lynas-Gray1998);−0.010(A99).Here we use the scaleof BLG94,because it is close to the one used in our pre-vious studies and in the Baade-Wesselink analyses(seeKW99).In Sect.3we will discuss the eﬀect of the T eﬀzero point on the distance determination.Theﬁnal for-mulae,adjusted to the BLG94scale are the followinglog T eﬀ=3.8804−0.3213(B−V)+0.0176log g+0.0066[M/H],(2)log T eﬀ=3.8928−0.4910(V−R c)+0.0116log g+0.0012[M/H].(3)G.Kov´a cs:The distance modulus of the Large Magellanic Cloud based on double-mode RR Lyrae stars3 Table2.Relative distance moduli(LMC minus IC4499)and reddenings of LMCﬁeld RR Lyrae starsField(ogle RRab):100 2.040.13RRd(macho):181 1.990.11RRd(C00b):10 2.030.094G.Kov´a cs:The distance modulus of the Large Magellanic Cloud based on double-mode RR Lyrae starsTable3.Derived distance moduli for the LMC.Columns5–8show the changes in the distance modulus,if E B−V, [M/H],log T0,P1/P0(observed)are changed by+0.03,+0.2,+0.005and+0.001,respectively.Column9shows the eﬀect of changing X from0.76to0.70M15−2.30.0718.520.040.050.070.050.01M68−2.00.0318.470.050.080.070.060.02IC4499−1.50.2218.500.060.150.080.050.04LMC−1.50.1118.520.040.140.080.060.04。