q-Deforming Maps for Lie Group Covariant Heisenberg Algebras
千万别学数学：最折磨人的数学未解之谜
千万别学数学：最折磨人的数学未解之谜数学之美不但体现在漂亮的结论和精妙的证明上，那些尚未解决的数学问题也有让人神魂颠倒的魅力。
和 Goldbach 猜想、 Riemann 假设不同，有些悬而未解的问题趣味性很强，“数学性”非常弱，乍看上去并没有触及深刻的数学理论，似乎是一道可以被瞬间秒杀的数学趣题，让数学爱好者们“不找到一个巧解就不爽”；但令人称奇的是，它们的困难程度却不亚于那些著名的数学猜想，这或许比各个领域中艰深的数学难题更折磨人吧。
作为一本数学趣题集， Mathematical Puzzles 一书中竟把仍未解决的数学趣题单独列为一章，可见这些问题有多么令人着迷。
我从这一章里挑选了一些问题，在这里和大家分享一下。
这本书是 04 年出版的，书里提到的一些“最新进展”其实已经不是最新的了；不过我也没有仔细考察每个问题当前的进展，因此本文的信息并不保证是 100% 准确的，在此向读者们表示歉意。
这篇文章很长，大家不妨用自己喜欢的方式马克一下，一天读一点。
天使和恶魔天使和恶魔在一个无限大的棋盘上玩游戏。
每一次，恶魔可以挖掉棋盘上的任意一个格子，天使则可以在棋盘上飞行 1000 步之后落地；如果天使落在了一个被挖掉的格子上，天使就输了。
问题：恶魔能否困住天使（在天使周围挖一圈厚度 1000 的坑）？这是 Conway 大牛的又一个经典谜题。
经常阅读这个 Blog 的人会发现， Conway 大牛的出镜率极高。
不过这一次，Conway 真的是伤透了不少数学家的脑筋。
作为一个很“正常”的组合游戏，天使与恶魔的问题竟然一直没能得到解决。
目前已经有的结论是，如果天使每次只能移动一步，恶魔一定能获胜。
不过，天使只要能每次飞两步，似乎就已经很无敌了。
当然，魔鬼的优势也不小——它不用担心自己“走错”，每多挖一个坑对于它来说都是有利的。
话说回来， Conway 本人似乎仍然相信天使能赢——他悬赏了 1000 美元征求恶魔必胜的证明，但只悬赏了 100 美元征求天使必胜的证明。
物理学名词
1/4波片quarter-wave plateCG矢量耦合系数Clebsch-Gordan vector coupling coefficient; 简称“CG[矢耦]系数”。
X射线摄谱仪X-ray spectrographX射线衍射X-ray diffractionX射线衍射仪X-ray diffractometer[玻耳兹曼]H定理[Boltzmann] H-theorem[玻耳兹曼]H函数[Boltzmann] H-function[彻]体力body force[冲]击波shock wave[冲]击波前shock front[狄拉克]δ函数[Dirac] δ-function[第二类]拉格朗日方程Lagrange equation[电]极化强度[electric] polarization[反射]镜mirror[光]谱线spectral line[光]谱仪spectrometer[光]照度illuminance[光学]测角计[optical] goniometer[核]同质异能素[nuclear] isomer[化学]平衡常量[chemical] equilibrium constant[基]元电荷elementary charge[激光]散斑speckle[吉布斯]相律[Gibbs] phase rule[可]变形体deformable body[克劳修斯-]克拉珀龙方程[Clausius-] Clapeyron equation[量子]态[quantum] state[麦克斯韦-]玻耳兹曼分布[Maxwell-]Boltzmann distribution[麦克斯韦-]玻耳兹曼统计法[Maxwell-]Boltzmann statistics[普适]气体常量[universal] gas constant[气]泡室bubble chamber[热]对流[heat] convection[热力学]过程[thermodynamic] process[热力学]力[thermodynamic] force[热力学]流[thermodynamic] flux[热力学]循环[thermodynamic] cycle[事件]间隔interval of events[微观粒子]全同性原理identity principle [of microparticles][物]态参量state parameter, state property[相]互作用interaction[相]互作用绘景interaction picture[相]互作用能interaction energy[旋光]糖量计saccharimeter[指]北极north pole, N pole[指]南极south pole, S pole[主]光轴[principal] optical axis[转动]瞬心instantaneous centre [of rotation][转动]瞬轴instantaneous axis [of rotation]t 分布student's t distributiont 检验student's t testＫ俘获K-captureＳ矩阵S-matrixＷＫＢ近似WKB approximationＸ射线X-rayΓ空间Γ-spaceα粒子α-particleα射线α-rayα衰变α-decayβ射线β-rayβ衰变β-decayγ矩阵γ-matrixγ射线γ-rayγ衰变γ-decayλ相变λ-transitionμ空间μ-spaceχ 分布chi square distributionχ 检验chi square test阿贝不变量Abbe invariant阿贝成象原理Abbe principle of image formation阿贝折射计Abbe refractometer阿贝正弦条件Abbe sine condition阿伏伽德罗常量Avogadro constant阿伏伽德罗定律Avogadro law阿基米德原理Archimedes principle阿特伍德机Atwood machine艾里斑Airy disk爱因斯坦-斯莫卢霍夫斯基理论Einstein-Smoluchowski theory 爱因斯坦场方程Einstein field equation爱因斯坦等效原理Einstein equivalence principle爱因斯坦关系Einstein relation爱因斯坦求和约定Einstein summation convention爱因斯坦同步Einstein synchronization爱因斯坦系数Einstein coefficient安[培]匝数ampere-turns安培[分子电流]假说Ampere hypothesis安培定律Ampere law安培环路定理Ampere circuital theorem安培计ammeter安培力Ampere force安培天平Ampere balance昂萨格倒易关系Onsager reciprocal relation凹面光栅concave grating凹面镜concave mirror凹透镜concave lens奥温电桥Owen bridge巴比涅补偿器Babinet compensator巴耳末系Balmer series白光white light摆pendulum板极plate伴线satellite line半波片halfwave plate半波损失half-wave loss半波天线half-wave antenna半导体semiconductor半导体激光器semiconductor laser半衰期half life period半透[明]膜semi-transparent film半影penumbra半周期带half-period zone傍轴近似paraxial approximation傍轴区paraxial region傍轴条件paraxial condition薄膜干涉film interference薄膜光学film optics薄透镜thin lens保守力conservative force保守系conservative system饱和saturation饱和磁化强度saturation magnetization本底background本体瞬心迹polhode本影umbra本征函数eigenfunction本征频率eigenfrequency本征矢[量] eigenvector本征振荡eigen oscillation本征振动eigenvibration本征值eigenvalue本征值方程eigenvalue equation比长仪comparator比荷specific charge; 又称“荷质比(charge-mass ratio)”。
双语经验对大脑可塑性的影响
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b Centre of Cognitive Neuroscience, University Vita Salute San Raffaele, Milan, Italy
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c Division of Speech and Hearing Sciences, University of Hong Kong, Pok Fulam, Hong Kong
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monolingual peers. These findings suggest that L1 production in bimodal bilinguals involves an interaction 32
between L1 and L2, supporting the claim that learning a second language does, in fact, change the functional 33
Support vector machine_A tool for mapping mineral prospectivity
Support vector machine:A tool for mapping mineral prospectivityRenguang Zuo a,n,Emmanuel John M.Carranza ba State Key Laboratory of Geological Processes and Mineral Resources,China University of Geosciences,Wuhan430074;Beijing100083,Chinab Department of Earth Systems Analysis,Faculty of Geo-Information Science and Earth Observation(ITC),University of Twente,Enschede,The Netherlandsa r t i c l e i n f oArticle history:Received17May2010Received in revised form3September2010Accepted25September2010Keywords:Supervised learning algorithmsKernel functionsWeights-of-evidenceTurbidite-hosted AuMeguma Terraina b s t r a c tIn this contribution,we describe an application of support vector machine(SVM),a supervised learningalgorithm,to mineral prospectivity mapping.The free R package e1071is used to construct a SVM withsigmoid kernel function to map prospectivity for Au deposits in western Meguma Terrain of Nova Scotia(Canada).The SVM classification accuracies of‘deposit’are100%,and the SVM classification accuracies ofthe‘non-deposit’are greater than85%.The SVM classifications of mineral prospectivity have5–9%lowertotal errors,13–14%higher false-positive errors and25–30%lower false-negative errors compared tothose of the WofE prediction.The prospective target areas predicted by both SVM and WofE reflect,nonetheless,controls of Au deposit occurrence in the study area by NE–SW trending anticlines andcontact zones between Goldenville and Halifax Formations.The results of the study indicate theusefulness of SVM as a tool for predictive mapping of mineral prospectivity.&2010Elsevier Ltd.All rights reserved.1.IntroductionMapping of mineral prospectivity is crucial in mineral resourcesexploration and mining.It involves integration of information fromdiverse geoscience datasets including geological data(e.g.,geologicalmap),geochemical data(e.g.,stream sediment geochemical data),geophysical data(e.g.,magnetic data)and remote sensing data(e.g.,multispectral satellite data).These sorts of data can be visualized,processed and analyzed with the support of computer and GIStechniques.Geocomputational techniques for mapping mineral pro-spectivity include weights of evidence(WofE)(Bonham-Carter et al.,1989),fuzzy WofE(Cheng and Agterberg,1999),logistic regression(Agterberg and Bonham-Carter,1999),fuzzy logic(FL)(Ping et al.,1991),evidential belief functions(EBF)(An et al.,1992;Carranza andHale,2003;Carranza et al.,2005),neural networks(NN)(Singer andKouda,1996;Porwal et al.,2003,2004),a‘wildcat’method(Carranza,2008,2010;Carranza and Hale,2002)and a hybrid method(e.g.,Porwalet al.,2006;Zuo et al.,2009).These techniques have been developed toquantify indices of occurrence of mineral deposit occurrence byintegrating multiple evidence layers.Some geocomputational techni-ques can be performed using popular software packages,such asArcWofE(a free ArcView extension)(Kemp et al.,1999),ArcSDM9.3(afree ArcGIS9.3extension)(Sawatzky et al.,2009),MI-SDM2.50(aMapInfo extension)(Avantra Geosystems,2006),GeoDAS(developedbased on MapObjects,which is an Environmental Research InstituteDevelopment Kit)(Cheng,2000).Other geocomputational techniques(e.g.,FL and NN)can be performed by using R and Matlab.Geocomputational techniques for mineral prospectivity map-ping can be categorized generally into two types–knowledge-driven and data-driven–according to the type of inferencemechanism considered(Bonham-Carter1994;Pan and Harris2000;Carranza2008).Knowledge-driven techniques,such as thosethat apply FL and EBF,are based on expert knowledge andexperience about spatial associations between mineral prospec-tivity criteria and mineral deposits of the type sought.On the otherhand,data-driven techniques,such as WofE and NN,are based onthe quantification of spatial associations between mineral pro-spectivity criteria and known occurrences of mineral deposits ofthe type sought.Additional,the mixing of knowledge-driven anddata-driven methods also is used for mapping of mineral prospec-tivity(e.g.,Porwal et al.,2006;Zuo et al.,2009).Every geocomputa-tional technique has advantages and disadvantages,and one or theother may be more appropriate for a given geologic environmentand exploration scenario(Harris et al.,2001).For example,one ofthe advantages of WofE is its simplicity,and straightforwardinterpretation of the weights(Pan and Harris,2000),but thismodel ignores the effects of possible correlations amongst inputpredictor patterns,which generally leads to biased prospectivitymaps by assuming conditional independence(Porwal et al.,2010).Comparisons between WofE and NN,NN and LR,WofE,NN and LRfor mineral prospectivity mapping can be found in Singer andKouda(1999),Harris and Pan(1999)and Harris et al.(2003),respectively.Mapping of mineral prospectivity is a classification process,because its product(i.e.,index of mineral deposit occurrence)forevery location is classified as either prospective or non-prospectiveaccording to certain combinations of weighted mineral prospec-tivity criteria.There are two types of classification techniques.Contents lists available at ScienceDirectjournal homepage:/locate/cageoComputers&Geosciences0098-3004/$-see front matter&2010Elsevier Ltd.All rights reserved.doi:10.1016/j.cageo.2010.09.014n Corresponding author.E-mail addresses:zrguang@,zrguang1981@(R.Zuo).Computers&Geosciences](]]]])]]]–]]]One type is known as supervised classification,which classifies mineral prospectivity of every location based on a training set of locations of known deposits and non-deposits and a set of evidential data layers.The other type is known as unsupervised classification, which classifies mineral prospectivity of every location based solely on feature statistics of individual evidential data layers.A support vector machine(SVM)is a model of algorithms for supervised classification(Vapnik,1995).Certain types of SVMs have been developed and applied successfully to text categorization, handwriting recognition,gene-function prediction,remote sensing classification and other studies(e.g.,Joachims1998;Huang et al.,2002;Cristianini and Scholkopf,2002;Guo et al.,2005; Kavzoglu and Colkesen,2009).An SVM performs classification by constructing an n-dimensional hyperplane in feature space that optimally separates evidential data of a predictor variable into two categories.In the parlance of SVM literature,a predictor variable is called an attribute whereas a transformed attribute that is used to define the hyperplane is called a feature.The task of choosing the most suitable representation of the target variable(e.g.,mineral prospectivity)is known as feature selection.A set of features that describes one case(i.e.,a row of predictor values)is called a feature vector.The feature vectors near the hyperplane are the support feature vectors.The goal of SVM modeling is tofind the optimal hyperplane that separates clusters of feature vectors in such a way that feature vectors representing one category of the target variable (e.g.,prospective)are on one side of the plane and feature vectors representing the other category of the target variable(e.g.,non-prospective)are on the other size of the plane.A good separation is achieved by the hyperplane that has the largest distance to the neighboring data points of both categories,since in general the larger the margin the better the generalization error of the classifier.In this paper,SVM is demonstrated as an alternative tool for integrating multiple evidential variables to map mineral prospectivity.2.Support vector machine algorithmsSupport vector machines are supervised learning algorithms, which are considered as heuristic algorithms,based on statistical learning theory(Vapnik,1995).The classical task of a SVM is binary (two-class)classification.Suppose we have a training set composed of l feature vectors x i A R n,where i(¼1,2,y,n)is the number of feature vectors in training samples.The class in which each sample is identified to belong is labeled y i,which is equal to1for one class or is equal toÀ1for the other class(i.e.y i A{À1,1})(Huang et al., 2002).If the two classes are linearly separable,then there exists a family of linear separators,also called separating hyperplanes, which satisfy the following set of equations(KavzogluandFig.1.Support vectors and optimum hyperplane for the binary case of linearly separable data sets.Table1Experimental data.yer A Layer B Layer C Layer D Target yer A Layer B Layer C Layer D Target1111112100000 2111112200000 3111112300000 4111112401000 5111112510000 6111112600000 7111112711100 8111112800000 9111012900000 10111013000000 11101113111100 12111013200000 13111013300000 14111013400000 15011013510000 16101013600000 17011013700000 18010113811100 19010112900000 20101014010000R.Zuo,E.J.M.Carranza/Computers&Geosciences](]]]])]]]–]]]2Colkesen,2009)(Fig.1):wx iþb Zþ1for y i¼þ1wx iþb rÀ1for y i¼À1ð1Þwhich is equivalent toy iðwx iþbÞZ1,i¼1,2,...,nð2ÞThe separating hyperplane can then be formalized as a decision functionfðxÞ¼sgnðwxþbÞð3Þwhere,sgn is a sign function,which is defined as follows:sgnðxÞ¼1,if x400,if x¼0À1,if x o08><>:ð4ÞThe two parameters of the separating hyperplane decision func-tion,w and b,can be obtained by solving the following optimization function:Minimize tðwÞ¼12J w J2ð5Þsubject toy Iððwx iÞþbÞZ1,i¼1,...,lð6ÞThe solution to this optimization problem is the saddle point of the Lagrange functionLðw,b,aÞ¼1J w J2ÀX li¼1a iðy iððx i wÞþbÞÀ1Þð7Þ@ @b Lðw,b,aÞ¼0@@wLðw,b,aÞ¼0ð8Þwhere a i is a Lagrange multiplier.The Lagrange function is minimized with respect to w and b and is maximized with respect to a grange multipliers a i are determined by the following optimization function:MaximizeX li¼1a iÀ12X li,j¼1a i a j y i y jðx i x jÞð9Þsubject toa i Z0,i¼1,...,l,andX li¼1a i y i¼0ð10ÞThe separating rule,based on the optimal hyperplane,is the following decision function:fðxÞ¼sgnX li¼1y i a iðxx iÞþb!ð11ÞMore details about SVM algorithms can be found in Vapnik(1995) and Tax and Duin(1999).3.Experiments with kernel functionsFor spatial geocomputational analysis of mineral exploration targets,the decision function in Eq.(3)is a kernel function.The choice of a kernel function(K)and its parameters for an SVM are crucial for obtaining good results.The kernel function can be usedTable2Errors of SVM classification using linear kernel functions.l Number ofsupportvectors Testingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)0.2580.00.00.0180.00.00.0 1080.00.00.0 10080.00.00.0 100080.00.00.0Table3Errors of SVM classification using polynomial kernel functions when d¼3and r¼0. l Number ofsupportvectorsTestingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)0.25120.00.00.0160.00.00.01060.00.00.010060.00.00.0 100060.00.00.0Table4Errors of SVM classification using polynomial kernel functions when l¼0.25,r¼0.d Number ofsupportvectorsTestingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)11110.00.0 5.010290.00.00.0100230.045.022.5 1000200.090.045.0Table5Errors of SVM classification using polynomial kernel functions when l¼0.25and d¼3.r Number ofsupportvectorsTestingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)0120.00.00.01100.00.00.01080.00.00.010080.00.00.0 100080.00.00.0Table6Errors of SVM classification using radial kernel functions.l Number ofsupportvectorsTestingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)0.25140.00.00.01130.00.00.010130.00.00.0100130.00.00.0 1000130.00.00.0Table7Errors of SVM classification using sigmoid kernel functions when r¼0.l Number ofsupportvectorsTestingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)0.25400.00.00.01400.035.017.510400.0 6.0 3.0100400.0 6.0 3.0 1000400.0 6.0 3.0R.Zuo,E.J.M.Carranza/Computers&Geosciences](]]]])]]]–]]]3to construct a non-linear decision boundary and to avoid expensive calculation of dot products in high-dimensional feature space.The four popular kernel functions are as follows:Linear:Kðx i,x jÞ¼l x i x j Polynomial of degree d:Kðx i,x jÞ¼ðl x i x jþrÞd,l40Radial basis functionðRBFÞ:Kðx i,x jÞ¼exp fÀl99x iÀx j992g,l40 Sigmoid:Kðx i,x jÞ¼tanhðl x i x jþrÞ,l40ð12ÞThe parameters l,r and d are referred to as kernel parameters. The parameter l serves as an inner product coefficient in the polynomial function.In the case of the RBF kernel(Eq.(12)),l determines the RBF width.In the sigmoid kernel,l serves as an inner product coefficient in the hyperbolic tangent function.The parameter r is used for kernels of polynomial and sigmoid types. The parameter d is the degree of a polynomial function.We performed some experiments to explore the performance of the parameters used in a kernel function.The dataset used in the experiments(Table1),which are derived from the study area(see below),were compiled according to the requirementfor Fig.2.Simplified geological map in western Meguma Terrain of Nova Scotia,Canada(after,Chatterjee1983;Cheng,2008).Table8Errors of SVM classification using sigmoid kernel functions when l¼0.25.r Number ofSupportVectorsTestingerror(non-deposit)(%)Testingerror(deposit)(%)Total error(%)0400.00.00.01400.00.00.010400.00.00.0100400.00.00.01000400.00.00.0R.Zuo,E.J.M.Carranza/Computers&Geosciences](]]]])]]]–]]]4classification analysis.The e1071(Dimitriadou et al.,2010),a freeware R package,was used to construct a SVM.In e1071,the default values of l,r and d are1/(number of variables),0and3,respectively.From the study area,we used40geological feature vectors of four geoscience variables and a target variable for classification of mineral prospec-tivity(Table1).The target feature vector is either the‘non-deposit’class(or0)or the‘deposit’class(or1)representing whether mineral exploration target is absent or present,respectively.For‘deposit’locations,we used the20known Au deposits.For‘non-deposit’locations,we randomly selected them according to the following four criteria(Carranza et al.,2008):(i)non-deposit locations,in contrast to deposit locations,which tend to cluster and are thus non-random, must be random so that multivariate spatial data signatures are highly non-coherent;(ii)random non-deposit locations should be distal to any deposit location,because non-deposit locations proximal to deposit locations are likely to have similar multivariate spatial data signatures as the deposit locations and thus preclude achievement of desired results;(iii)distal and random non-deposit locations must have values for all the univariate geoscience spatial data;(iv)the number of distal and random non-deposit locations must be equaltoFig.3.Evidence layers used in mapping prospectivity for Au deposits(from Cheng,2008):(a)and(b)represent optimum proximity to anticline axes(2.5km)and contacts between Goldenville and Halifax formations(4km),respectively;(c)and(d)represent,respectively,background and anomaly maps obtained via S-Afiltering of thefirst principal component of As,Cu,Pb and Zn data.R.Zuo,E.J.M.Carranza/Computers&Geosciences](]]]])]]]–]]]5the number of deposit locations.We used point pattern analysis (Diggle,1983;2003;Boots and Getis,1988)to evaluate degrees of spatial randomness of sets of non-deposit locations and tofind distance from any deposit location and corresponding probability that one deposit location is situated next to another deposit location.In the study area,we found that the farthest distance between pairs of Au deposits is71km,indicating that within that distance from any deposit location in there is100%probability of another deposit location. However,few non-deposit locations can be selected beyond71km of the individual Au deposits in the study area.Instead,we selected random non-deposit locations beyond11km from any deposit location because within this distance from any deposit location there is90% probability of another deposit location.When using a linear kernel function and varying l from0.25to 1000,the number of support vectors and the testing errors for both ‘deposit’and‘non-deposit’do not vary(Table2).In this experiment the total error of classification is0.0%,indicating that the accuracy of classification is not sensitive to the choice of l.With a polynomial kernel function,we tested different values of l, d and r as follows.If d¼3,r¼0and l is increased from0.25to1000,the number of support vectors decreases from12to6,but the testing errors for‘deposit’and‘non-deposit’remain nil(Table3).If l¼0.25, r¼0and d is increased from1to1000,the number of support vectors firstly increases from11to29,then decreases from23to20,the testing error for‘non-deposit’decreases from10.0%to0.0%,whereas the testing error for‘deposit’increases from0.0%to90%(Table4). In this experiment,the total error of classification is minimum(0.0%) when d¼10(Table4).If l¼0.25,d¼3and r is increased from 0to1000,the number of support vectors decreases from12to8,but the testing errors for‘deposit’and‘non-deposit’remain nil(Table5).When using a radial kernel function and varying l from0.25to 1000,the number of support vectors decreases from14to13,but the testing errors of‘deposit’and‘non-deposit’remain nil(Table6).With a sigmoid kernel function,we experimented with different values of l and r as follows.If r¼0and l is increased from0.25to1000, the number of support vectors is40,the testing errors for‘non-deposit’do not change,but the testing error of‘deposit’increases from 0.0%to35.0%,then decreases to6.0%(Table7).In this experiment,the total error of classification is minimum at0.0%when l¼0.25 (Table7).If l¼0.25and r is increased from0to1000,the numbers of support vectors and the testing errors of‘deposit’and‘non-deposit’do not change and the total error remains nil(Table8).The results of the experiments demonstrate that,for the datasets in the study area,a linear kernel function,a polynomial kernel function with d¼3and r¼0,or l¼0.25,r¼0and d¼10,or l¼0.25and d¼3,a radial kernel function,and a sigmoid kernel function with r¼0and l¼0.25are optimal kernel functions.That is because the testing errors for‘deposit’and‘non-deposit’are0%in the SVM classifications(Tables2–8).Nevertheless,a sigmoid kernel with l¼0.25and r¼0,compared to all the other kernel functions,is the most optimal kernel function because it uses all the input support vectors for either‘deposit’or‘non-deposit’(Table1)and the training and testing errors for‘deposit’and‘non-deposit’are0% in the SVM classification(Tables7and8).4.Prospectivity mapping in the study areaThe study area is located in western Meguma Terrain of Nova Scotia,Canada.It measures about7780km2.The host rock of Au deposits in this area consists of Cambro-Ordovician low-middle grade metamorphosed sedimentary rocks and a suite of Devonian aluminous granitoid intrusions(Sangster,1990;Ryan and Ramsay, 1997).The metamorphosed sedimentary strata of the Meguma Group are the lower sand-dominatedflysch Goldenville Formation and the upper shalyflysch Halifax Formation occurring in the central part of the study area.The igneous rocks occur mostly in the northern part of the study area(Fig.2).In this area,20turbidite-hosted Au deposits and occurrences (Ryan and Ramsay,1997)are found in the Meguma Group, especially near the contact zones between Goldenville and Halifax Formations(Chatterjee,1983).The major Au mineralization-related geological features are the contact zones between Gold-enville and Halifax Formations,NE–SW trending anticline axes and NE–SW trending shear zones(Sangster,1990;Ryan and Ramsay, 1997).This dataset has been used to test many mineral prospec-tivity mapping algorithms(e.g.,Agterberg,1989;Cheng,2008). More details about the geological settings and datasets in this area can be found in Xu and Cheng(2001).We used four evidence layers(Fig.3)derived and used by Cheng (2008)for mapping prospectivity for Au deposits in the yers A and B represent optimum proximity to anticline axes(2.5km) and optimum proximity to contacts between Goldenville and Halifax Formations(4km),yers C and D represent variations in geochemical background and anomaly,respectively, as modeled by multifractalfilter mapping of thefirst principal component of As,Cu,Pb,and Zn data.Details of how the four evidence layers were obtained can be found in Cheng(2008).4.1.Training datasetThe application of SVM requires two subsets of training loca-tions:one training subset of‘deposit’locations representing presence of mineral deposits,and a training subset of‘non-deposit’locations representing absence of mineral deposits.The value of y i is1for‘deposits’andÀ1for‘non-deposits’.For‘deposit’locations, we used the20known Au deposits(the sixth column of Table1).For ‘non-deposit’locations(last column of Table1),we obtained two ‘non-deposit’datasets(Tables9and10)according to the above-described selection criteria(Carranza et al.,2008).We combined the‘deposits’dataset with each of the two‘non-deposit’datasets to obtain two training datasets.Each training dataset commonly contains20known Au deposits but contains different20randomly selected non-deposits(Fig.4).4.2.Application of SVMBy using the software e1071,separate SVMs both with sigmoid kernel with l¼0.25and r¼0were constructed using the twoTable9The value of each evidence layer occurring in‘non-deposit’dataset1.yer A Layer B Layer C Layer D100002000031110400005000061000700008000090100 100100 110000 120000 130000 140000 150000 160100 170000 180000 190100 200000R.Zuo,E.J.M.Carranza/Computers&Geosciences](]]]])]]]–]]] 6training datasets.With training dataset1,the classification accuracies for‘non-deposits’and‘deposits’are95%and100%, respectively;With training dataset2,the classification accuracies for‘non-deposits’and‘deposits’are85%and100%,respectively.The total classification accuracies using the two training datasets are97.5%and92.5%,respectively.The patterns of the predicted prospective target areas for Au deposits(Fig.5)are defined mainly by proximity to NE–SW trending anticlines and proximity to contact zones between Goldenville and Halifax Formations.This indicates that‘geology’is better than‘geochemistry’as evidence of prospectivity for Au deposits in this area.With training dataset1,the predicted prospective target areas occupy32.6%of the study area and contain100%of the known Au deposits(Fig.5a).With training dataset2,the predicted prospec-tive target areas occupy33.3%of the study area and contain95.0% of the known Au deposits(Fig.5b).In contrast,using the same datasets,the prospective target areas predicted via WofE occupy 19.3%of study area and contain70.0%of the known Au deposits (Cheng,2008).The error matrices for two SVM classifications show that the type1(false-positive)and type2(false-negative)errors based on training dataset1(Table11)and training dataset2(Table12)are 32.6%and0%,and33.3%and5%,respectively.The total errors for two SVM classifications are16.3%and19.15%based on training datasets1and2,respectively.In contrast,the type1and type2 errors for the WofE prediction are19.3%and30%(Table13), respectively,and the total error for the WofE prediction is24.65%.The results show that the total errors of the SVM classifications are5–9%lower than the total error of the WofE prediction.The 13–14%higher false-positive errors of the SVM classifications compared to that of the WofE prediction suggest that theSVMFig.4.The locations of‘deposit’and‘non-deposit’.Table10The value of each evidence layer occurring in‘non-deposit’dataset2.yer A Layer B Layer C Layer D110102000030000411105000060110710108000091000101110111000120010131000140000150000161000171000180010190010200000R.Zuo,E.J.M.Carranza/Computers&Geosciences](]]]])]]]–]]]7classifications result in larger prospective areas that may not contain undiscovered deposits.However,the 25–30%higher false-negative error of the WofE prediction compared to those of the SVM classifications suggest that the WofE analysis results in larger non-prospective areas that may contain undiscovered deposits.Certainly,in mineral exploration the intentions are notto miss undiscovered deposits (i.e.,avoid false-negative error)and to minimize exploration cost in areas that may not really contain undiscovered deposits (i.e.,keep false-positive error as low as possible).Thus,results suggest the superiority of the SVM classi-fications over the WofE prediction.5.ConclusionsNowadays,SVMs have become a popular geocomputational tool for spatial analysis.In this paper,we used an SVM algorithm to integrate multiple variables for mineral prospectivity mapping.The results obtained by two SVM applications demonstrate that prospective target areas for Au deposits are defined mainly by proximity to NE–SW trending anticlines and to contact zones between the Goldenville and Halifax Formations.In the study area,the SVM classifications of mineral prospectivity have 5–9%lower total errors,13–14%higher false-positive errors and 25–30%lower false-negative errors compared to those of the WofE prediction.These results indicate that SVM is a potentially useful tool for integrating multiple evidence layers in mineral prospectivity mapping.Table 11Error matrix for SVM classification using training dataset 1.Known All ‘deposits’All ‘non-deposits’TotalPrediction ‘Deposit’10032.6132.6‘Non-deposit’067.467.4Total100100200Type 1(false-positive)error ¼32.6.Type 2(false-negative)error ¼0.Total error ¼16.3.Note :Values in the matrix are percentages of ‘deposit’and ‘non-deposit’locations.Table 12Error matrix for SVM classification using training dataset 2.Known All ‘deposits’All ‘non-deposits’TotalPrediction ‘Deposits’9533.3128.3‘Non-deposits’566.771.4Total100100200Type 1(false-positive)error ¼33.3.Type 2(false-negative)error ¼5.Total error ¼19.15.Note :Values in the matrix are percentages of ‘deposit’and ‘non-deposit’locations.Table 13Error matrix for WofE prediction.Known All ‘deposits’All ‘non-deposits’TotalPrediction ‘Deposit’7019.389.3‘Non-deposit’3080.7110.7Total100100200Type 1(false-positive)error ¼19.3.Type 2(false-negative)error ¼30.Total error ¼24.65.Note :Values in the matrix are percentages of ‘deposit’and ‘non-deposit’locations.Fig.5.Prospective targets area for Au deposits delineated by SVM.(a)and (b)are obtained using training dataset 1and 2,respectively.R.Zuo,E.J.M.Carranza /Computers &Geosciences ](]]]])]]]–]]]8。
格点量子色动力学进展.pptx
The fDs puzzle?
Experimental results
Comparison
Independent determination of Vcs
More comparisons…
Is is a puzzle?
只有HPQCD的结果偏离最严重 Vcs需要单独确定 目前还不是很严重的问题…
理论方面的探讨
相关的唯象理论工作
Shallow bound state of two D mesons (S.L. Zhu et al, ) PRD77,034003 Tetra-quark resonance above threshold (X.-H Liu et al, PRD77, 094005) Threshold enhancement (J.L. Rosner, ) PRD76,114002
The Berlin wall & lattice QCD c o s t
m m
z
a
z 4, 7
Berlin wall crushing…
More physical…
Pion mass 700MeV below 300MeV where cPT starts to work…
Matrix elements
B_K 的计算 B ˆB KKBK K((8 0|/Q 3 ))L fSK s (L 232M )|(K K 2)0,2/Q 9L 1S L 2(s(43s)d()L)(sJd3)L
Renormalization dilemma
Continuum perturbation done in the MS-bar Naive Dimensional Regularization scheme (MS-bar NDR-scheme)
Numerical Linear Algebra
letters (and occasionally lower case letters) will denote scalars. RI will denote the set of real
tions to the algorithm, it can be made to work quite well. We understand these algorithmic
transformations most completely in the case of simple algorithms like Cholesky, on simple
LA
Numerical Linear Algebra
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1汉英力学名词(1993)
BZ反应||Belousov-Zhabotinski reaction, BZ reactionFPU问题||Fermi-Pasta-Ulam problem, FPU problemKBM方法||KBM method, Krylov-Bogoliubov-Mitropolskii method KS[动态]熵||Kolmogorov-Sinai entropy, KS entropyKdV 方程||KdV equationU形管||U-tubeWKB方法||WKB method, Wentzel-Kramers-Brillouin method[彻]体力||body force[单]元||element[第二类]拉格朗日方程||Lagrange equation [of the second kind] [叠栅]云纹||moiré fringe; 物理学称“叠栅条纹”。
[叠栅]云纹法||moiré method[抗]剪切角||angle of shear resistance[可]变形体||deformable body[钱]币状裂纹||penny-shape crack[映]象||image[圆]筒||cylinder[圆]柱壳||cylindrical shell[转]轴||shaft[转动]瞬心||instantaneous center [of rotation][转动]瞬轴||instantaneous axis [of rotation][状]态变量||state variable[状]态空间||state space[自]适应网格||[self-]adaptive meshＣ0连续问题||C0-continuous problemＣ1连续问题||C1-continuous problemＣＦＬ条件||Courant-Friedrichs-Lewy condition, CFL condition ＨＲＲ场||Hutchinson-Rice-Rosengren fieldＪ积分||J-integralＪ阻力曲线||J-resistance curveＫＡＭ定理||Kolgomorov-Arnol'd-Moser theorem, KAM theoremＫＡＭ环面||KAM torusｈ收敛||h-convergenceｐ收敛||p-convergenceπ定理||Buckingham theorem, pi theorem阿尔曼西应变||Almansis strain阿尔文波||Alfven wave阿基米德原理||Archimedes principle阿诺德舌[头]||Arnol'd tongue阿佩尔方程||Appel equation阿特伍德机||Atwood machine埃克曼边界层||Ekman boundary layer埃克曼流||Ekman flow埃克曼数||Ekman number埃克特数||Eckert number埃农吸引子||Henon attractor艾里应力函数||Airy stress function鞍点||saddle [point]鞍结分岔||saddle-node bifurcation安定[性]理论||shake-down theory安全寿命||safe life安全系数||safety factor安全裕度||safety margin暗条纹||dark fringe奥尔-索末菲方程||Orr-Sommerfeld equation奥辛流||Oseen flow奥伊洛特模型||Oldroyd model八面体剪应变||octohedral shear strain八面体剪应力||octohedral shear stress八面体剪应力理论||octohedral shear stress theory巴塞特力||Basset force白光散斑法||white-light speckle method摆||pendulum摆振||shimmy板||plate板块法||panel method板元||plate element半导体应变计||semiconductor strain gage半峰宽度||half-peak width半解析法||semi-analytical method半逆解法||semi-inverse method半频进动||half frequency precession半向同性张量||hemitropic tensor半隐格式||semi-implicit scheme薄壁杆||thin-walled bar薄壁梁||thin-walled beam薄壁筒||thin-walled cylinder薄膜比拟||membrane analogy薄翼理论||thin-airfoil theory保单调差分格式||monotonicity preserving difference scheme 保守力||conservative force保守系||conservative system爆发||blow up爆高||height of burst爆轰||detonation; 又称“爆震”。
结构化学_李炳瑞_习题
结构化学习题（选编）（兰州大学化学化工学院李炳瑞）习题类型包括：选择答案、填空、概念辨析、查错改正、填表、计算、利用结构化学原理分析问题；内容涵盖整个课程，即量子力学基础、原子结构、分子结构与化学键、晶体结构与点阵、X射线衍射、金属晶体与离子晶体结构、结构分析原理、结构数据采掘与QSAR等；难度包括容易、中等、较难、难4级；能力层次分为了解、理解、综合应用。
传统形式的习题，通常要求学生在课本所学知识范围内即可完成，而且答案是唯一的，即可以给出所谓“标准答案”。
根据21世纪化学演变的要求，我们希望再给学生一些新型的题目，体现开放性、自主性、答案的多样性，即：习题不仅与课本内容有关，而且还需要查阅少量文献才能完成；完成习题更多地需要学生主动思考，而不是完全跟随教师的思路；习题并不一定有唯一的“标准答案”，而可能具有多样性，每一种答案都可能是“参考答案”。
学生接触这类习题，有助于培养学习的主动性，同时认识到实际问题是复杂的，解决问题可能有多钟途径。
但是，这种题目在基础课中不宜多，只要有代表性即可。
以下各章的名称与《结构化学》多媒体版相同，但习题内容并不完全相同。
第一章量子力学基础1.1 选择题(1) 若用电子束与中子束分别作衍射实验，得到大小相同的环纹，则说明二者(A) 动量相同(B) 动能相同(C) 质量相同(2) 为了写出一个经典力学量对应的量子力学算符，若坐标算符取作坐标本身，动量算符应是(以一维运动为例)(A) mv (B)(C)(3) 若∫|ψ|2dτ=K，利用下列哪个常数乘ψ可以使之归一化：(A) K (B)K2 (C) 1/(4) 丁二烯等共轭分子中π电子的离域化可降低体系的能量，这与简单的一维势阱模型是一致的，因为一维势阱中粒子的能量(A) 反比于势阱长度平方(B) 正比于势阱长度(C) 正比于量子数(5) 对于厄米算符, 下面哪种说法是对的(A) 厄米算符中必然不包含虚数(B) 厄米算符的本征值必定是实数(C) 厄米算符的本征函数中必然不包含虚数(6) 对于算符Ĝ的非本征态Ψ(A) 不可能测量其本征值g.(B) 不可能测量其平均值<g>.(C) 本征值与平均值均可测量，且二者相等(7) 将几个非简并的本征函数进行线形组合，结果(A) 再不是原算符的本征函数(B) 仍是原算符的本征函数，且本征值不变(C) 仍是原算符的本征函数，但本征值改变1.2 辨析下列概念，注意它们是否有相互联系, 尤其要注意它们之间的区别：(1) 算符的线性与厄米性(2) 本征态与非本征态(3) 本征函数与本征值(4) 本征值与平均值(5) 几率密度与几率(6) 波函数的正交性与归一性(7) 简并态与非简并态1.3 原子光谱和分子光谱的谱线总是存在一定的线宽，而且不可能通过仪器技术的改进来使之无限地变窄. 这种现象是什么原因造成的？1.4 几率波的波长与动量成反比. 如何理解这一点?1.5 细菌的大小为微米量级, 而病毒的大小为纳米量级. 试通过计算粗略估计: 为了观察到病毒, 电子显微镜至少需要多高的加速电压.1.6 将一维无限深势阱中粒子的波函数任取几个, 验证它们都是相互正交的.1.7 厄米算符的非简并本征函数相互正交. 简并本征函数虽不一定正交,但可用数学处理使之正交. 例如，若ψ1与ψ2不正交，可以造出与ψ1正交的新函数ψ’2ψ’=ψ2+cψ12试推导c的表达式(这种方法称为Schmidt正交化方法).1.8 对于一维无限深势阱中粒子的基态, 计算坐标平均值和动量平均值,并解释它们的物理意义.1.9 一维无限深势阱中粒子波函数的节点数目随量子数增加而增加. 试解释: 为什么节点越多, 能量越高. 再想一想: 阱中只有一个粒子, 它是如何不穿越节点而出现在每个节点两侧的?1.10 下列哪些函数是d2/dx2的本征函数: (1) e x (2) e2x (3) 5sin x (4)sin x+cos x (5)x3. 求出本征函数的本征值.1.11 对于三维无限深正方形势阱中粒子, 若三个量子数平方和等于9, 简并度是多少?1.12 利用结构化学原理，分析并回答下列问题：纳米粒子属于介观粒子，有些性质与宏观和微观粒子都有所不同. 不过，借用无限深势阱中粒子模型，对纳米材料中的“量子尺寸效应”还是可以作一些定性解释.例如: 为什么半导体中的窄能隙（<3eV）在纳米颗粒中会变宽, 甚至连纳米Ag也会成为绝缘体?第二章原子结构2.1 选择题(1) 对s、p、d、f 原子轨道进行反演操作，可以看出它们的对称性分别是(A) u, g, u, g (B) g, u, g, u (C) g, g, g, g(2) H原子的电离能为13.6 eV, He+的电离能为(A) 13.6 eV (B) 54.4eV (C) 27.2 eV(3) 原子的轨道角动量绝对值为(A) l（l+1）2(B)(C) l(4) p2组态的原子光谱项为(A) 1D、3P、1S(B) 3D、1P、3S(C) 3D、3P、1D(5) Hund规则适用于下列哪种情况(A) 求出激发组态下的能量最低谱项(B) 求出基组态下的基谱项(C) 在基组态下为谱项的能量排序(6) 配位化合物中d→d跃迁一般都很弱，因为这种跃迁属于：(A) g←/→g(B)g←→u(C) u←/→u(7) Cl原子基态的光谱项为2P，其能量最低的光谱支项为(A) 2P3/2 (B) 2P1/2(C) 2P02.2 辨析下列概念，注意它们的相互联系和区别：(1) 复波函数与实波函数(2) 轨道与电子云(3) 轨道的位相与电荷的正负(4) 径向密度函数与径向分布函数(5)原子轨道的角度分布图与界面图(6)空间波函数、自旋波函数与自旋-轨道(7)自旋-轨道与Slater行列式(8)组态与状态2.3 请找出下列叙述中可能包含着的错误，并加以改正：原子轨道（AO）是原子中的单电子波函数，它描述了电子运动的确切轨迹. 原子轨道的正、负号分别代表正、负电荷. 原子轨道的绝对值平方就是化学中广为使用的“电子云”概念，即几率密度. 若将原子轨道乘以任意常数C，电子在每一点出现的可能性就增大到原来的C2倍.2.4(1) 计算节面对应的θ;(2) 计算极大值对应的θ;(3) 在yz平面上画出波函数角度分布图的剖面, 绕z轴旋转一周即成波函数角度分布图. 对照下列所示的轨道界面图, 从物理意义和图形特征来说明二者的相似与相异.2.5 氢原子基态的波函数为试计算1/r的平均值，进而计算势能平均值<V>, 验证下列关系:<V> = 2E= -2<T>此即量子力学维里定理,适用于库仑作用下达到平衡的粒子体系 (氢原子基态只有一个1s电子,其能量等于体系的能量) 的定态, 对单电子原子和多电子原子具有相同的形式.2.6 R. Mulliken用原子中电子的电离能与电子亲合能的平均值来定义元素电负性. 试从原子中电子最高占有轨道（HOMO）和最低空轨道（LUMO）的角度想一想，这种定义有什么道理？2.7 原子中电子的电离能与电子亲合能之差值的一半, 可以作为元素化学硬度的一种量度（硬度较大的原子，其极化率较低）. 根据这种定义，化学硬度较大的原子，其HOMO与LUMO之间的能隙应当较大还是较小？2.8 将2p+1与2p-1线性组合得到的2p x与2p y, 是否还有确定的能量和轨道角动量z分量？为什么？2.9 原子的轨道角动量为什么永远不会与外磁场方向z重合, 而是形成一定大小的夹角? 计算f轨道与z轴的所有可能的夹角. 为什么每种夹角对应于一个锥面, 而不是一个确定的方向?2.10 快速求出P原子的基谱项.2.11 Ni2+的电子组态为d8, 试用M L表方法写出它的所有谱项, 并确定基谱项.原子光谱表明, 除基谱项外, 其余谱项的能级顺序是1D<3P<1G<1S, 你是否能用Hund规则预料到这个结果?2.12 d n组态产生的谱项, 其宇称与电子数n无关, 而p n组态产生的谱项, 其宇称与电子数n有关. 为什么？2.13 试写出闭壳层原子Be的Slater行列式.2.14 Pauli原理适用于玻色子和费米子, 为什么说Pauli不相容原理只适用于费米子?第三章双原子分子结构与化学键理论3.1 选择题(1) 用线性变分法求出的分子基态能量比起基态真实能量，只可能(A) 更高或相等(B) 更低(C) 相等(2) N2、O2、F2的键长递增是因为(A) 核外电子数依次减少(B) 键级依次增大(C) 净成键电子数依次减少(3) 下列哪一条属于所谓的“成键三原则”之一：(A) 原子半径相似(B) 对称性匹配(C) 电负性相似(4) 下列哪种说法是正确的(A) 原子轨道只能以同号重叠组成分子轨道(B) 原子轨道以异号重叠组成非键分子轨道(C) 原子轨道可以按同号重叠或异号重叠，分别组成成键或反键轨道(5) 氧的O2+ , O2, O2- , O22-对应于下列哪种键级顺序(A) 2.5， 2.0, 1.5, 1.0(B) 1.0, 1.5， 2.0, 2.5(C) 2.5， 1.5, 1.0 2.0(6) 下列哪些分子或分子离子具有顺磁性(A) O2、NO (B) N2、F 2(C) O22+、NO+(7) B2和C2中的共价键分别是（A）π1+π1，π+π（B）π+π，π1+π1（C）σ+π,σ3.2 MO与VB理论在解释共价键的饱和性和方向性上都取得了很大的成功, 但两种理论各有特色. 试指出它们各自的要点 (若将两种理论各自作一些改进, 其结果会彼此接近).3.3 考察共价键的形成时, 为什么先考虑原子轨道形成分子轨道, 再填充电子形成分子轨道上的电子云, 而不直接用原子轨道上的电子云叠加来形成分子轨道上的电子云?3.4 “成键轨道的对称性总是g, 反键轨道的对称性总是u”. 这种说法对不对? 为什么?3.5 一般地说, π键要比σ键弱一些. 但在任何情况下都是如此吗? 请举实例来说明.3.6 N2作为配位体形成配合物时, 通常以2σg电子对去进行端基配位(即N ≡N→), 而不以1πu电子对去进行侧基配位。
Generalized WDVV equations for B_r and C_r pure N=2 Super-Yang-Mills theory
a r X i v :h e p -t h /0102190v 1 27 F eb 2001Generalized WDVV equations for B r and C r pure N=2Super-Yang-Mills theoryL.K.Hoevenaars,R.MartiniAbstractA proof that the prepotential for pure N=2Super-Yang-Mills theory associated with Lie algebrasB r andC r satisfies the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)system was given by Marshakov,Mironov and Morozov.Among other things,they use an associative algebra of holomorphic diffter Ito and Yang used a different approach to try to accomplish the same result,but they encountered objects of which it is unclear whether they form structure constants of an associative algebra.We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic differentials.1IntroductionIn 1994,Seiberg and Witten [1]solved the low energy behaviour of pure N=2Super-Yang-Mills theory by giving the solution of the prepotential F .The essential ingredients in their construction are a family of Riemann surfaces Σ,a meromorphic differential λSW on it and the definition of the prepotential in terms of period integrals of λSWa i =A iλSW ∂F∂a i ∂a j ∂a k .Moreover,it was shown that the full prepotential for simple Lie algebras of type A,B,C,D [8]andtype E [9]and F [10]satisfies this generalized WDVV system 1.The approach used by Ito and Yang in [9]differs from the other two,due to the type of associative algebra that is being used:they use the Landau-Ginzburg chiral ring while the others use an algebra of holomorphic differentials.For the A,D,E cases this difference in approach is negligible since the two different types of algebras are isomorphic.For the Lie algebras of B,C type this is not the case and this leads to some problems.The present article deals with these problems and shows that the proper algebra to use is the onesuggested in[8].A survey of these matters,as well as the results of the present paper can be found in the internal publication[11].This paper is outlined as follows:in thefirst section we will review Ito and Yang’s method for the A,D,E Lie algebras.In the second section their approach to B,C Lie algebras is discussed. Finally in section three we show that Ito and Yang’s construction naturally leads to the algebra of holomorphic differentials used in[8].2A review of the simply laced caseIn this section,we will describe the proof in[9]that the prepotential of4-dimensional pure N=2 SYM theory with Lie algebra of simply laced(ADE)type satisfies the generalized WDVV system. The Seiberg-Witten data[1],[12],[13]consists of:•a family of Riemann surfacesΣof genus g given byz+µz(2.2)and has the property that∂λSW∂a i is symmetric.This implies that F j can be thought of as agradient,which leads to the followingDefinition1The prepotential is a function F(a1,...,a r)such thatF j=∂FDefinition2Let f:C r→C,then the generalized WDVV system[4],[5]for f isf i K−1f j=f j K−1f i∀i,j∈{1,...,r}(2.5) where the f i are matrices with entries∂3f(a1,...,a r)(f i)jk=The rest of the proof deals with a discussion of the conditions1-3.It is well-known[14]that the right hand side of(2.1)equals the Landau-Ginzburg superpotential associated with the cor-∂W responding Lie ing this connection,we can define the primaryfieldsφi(u):=−∂x (2.10)Instead of using the u i as coordinates on the part of the moduli space we’re interested in,we want to use the a i .For the chiral ring this implies that in the new coordinates(−∂W∂a j)=∂u x∂a jC z xy (u )∂a k∂a k )mod(∂W∂x)(2.11)which again is an associative algebra,but with different structure constants C k ij (a )=C k ij(u ).This is the algebra we will use in the rest of the proof.For the relation(2.7)weturn to another aspect of Landau-Ginzburg theory:the Picard-Fuchs equations (see e.g [15]and references therein).These form a coupled set of first order partial differential equations which express how the integrals of holomorphic differentials over homology cycles of a Riemann surface in a family depend on the moduli.Definition 6Flat coordinates of the Landau-Ginzburg theory are a set of coordinates {t i }on mod-uli space such that∂2W∂x(2.12)where Q ij is given byφi (t )φj (t )=C kij (t )φk (t )+Q ij∂W∂t iΓ∂λsw∂t kΓ∂λsw∂a iΓ∂λsw∂a lΓ∂λsw∂t r(2.15)Taking Γ=B k we getF ijk =C lij (a )K kl(2.16)which is the intended relation (2.7).The only thing that is left to do,is to prove that K kl =∂a mIn conclusion,the most important ingredients in the proof are the chiral ring and the Picard-Fuchs equations.In the following sections we will show that in the case of B r ,C r Lie algebras,the Picard-Fuchs equations can still play an important role,but the chiral ring should be replaced by the algebra of holomorphic differentials considered by the authors of [8].These algebras are isomorphic to the chiral rings in the ADE cases,but not for Lie algebras B r ,C r .3Ito&Yang’s approach to B r and C rIn this section,we discuss the attempt made in[9]to generalizethe contentsof the previoussection to the Lie algebras B r,C r.We will discuss only B r since the situation for C r is completely analogous.The Riemann surfaces are given byz+µx(3.1)where W BC is the Landau-Ginzburg superpotential associated with the theory of type BC.From the superpotential we again construct the chiral ring inflat coordinates whereφi(t):=−∂W BC∂x (3.2)However,the fact that the right-hand side of(3.1)does not equal the superpotential is reflected by the Picard-Fuchs equations,which no longer relate the third order derivatives of F with the structure constants C k ij(a).Instead,they readF ijk=˜C l ij(a)K kl(3.3) where K kl=∂a m2r−1˜C knl(t).(3.4)The D l ij are defined byQ ij=xD l ijφl(3.5)and we switched from˜C k ij(a)to˜C k ij(t)in order to compare these with the structure constants C k ij(t). At this point,it is unknown2whether the˜C k ij(t)(and therefore the˜C k ij(a))are structure constants of an associative algebra.This issue will be resolved in the next section.4The identification of the structure constantsThe method of proof that is being used in[8]for the B r,C r case also involves an associative algebra. However,theirs is an algebra of holomorphic differentials which is isomorphic toφi(t)φj(t)=γk ij(t)φk(t)mod(x∂W BC2Except for rank3and4,for which explicit calculations of˜C kij(t)were made in[9]we will rewrite it in such a way that it becomes of the formφi(t)φj(t)=rk=1 C k ij(t)φk(t)+P ij[x∂x W BC−W BC](4.3)As afirst step,we use(3.4):φiφj= Ci·−→φ+D i·−→φx∂x W BC j= C i−D i·r n=12nt n2r−1 C n·−→φ+D i·−→φx∂x W BCj(4.4)The notation −→φstands for the vector with componentsφk and we used a matrix notation for thestructure constants.The proof becomes somewhat technical,so let usfirst give a general outline of it.The strategy will be to get rid of the second term of(4.4)by cancelling it with part of the third term,since we want an algebra in which thefirst term gives the structure constants.For this cancelling we’ll use equation(3.4)in combination with the following relation which expresses the fact that W BC is a graded functionx ∂W BC∂t n=2rW BC(4.5)Cancelling is possible at the expense of introducing yet another term which then has to be canceled etcetera.This recursive process does come to an end however,and by performing it we automatically calculate modulo x∂x W BC−W BC instead of x∂x W BC.We rewrite(4.4)by splitting up the third term and rewriting one part of it using(4.5):D i·−→φx∂x W BC j= −12r−1 D i·−→φx∂x W BC j= −D i2r−1·−→φx∂x W BC j(4.6) Now we use(4.2)to work out the productφkφn and the result is:φiφj= C i·−→φ−D i2r−1·r n=12nt n D n·−→φx∂x W BC j +2rD i2r−1·rn=12nt n −D n·r m=12mt m2r−1[x∂x W BC−W BC]j(4.8)Note that by cancelling the one term,we automatically calculate modulo x∂x W BC −W BC .The expression between brackets in the first line seems to spoil our achievement but it doesn’t:until now we rewrote−D i ·r n =12nt n 2r −1C m ·−→φ+D n ·−→φx∂x W BCj(4.10)This is a recursive process.If it stops at some point,then we get a multiplication structureφi φj =r k =1C k ij φk +P ij (x∂x W BC −W BC )(4.11)for some polynomial P ij and the theorem is proven.To see that the process indeed stops,we referto the lemma below.xby φk ,we have shown that D i is nilpotent sinceit is strictly upper triangular.Sincedeg (φk )=2r −2k(4.13)we find that indeed for j ≥k the degree of φk is bigger than the degree ofQ ij5Conclusions and outlookIn this letter we have shown that the unknown quantities ˜C k ijof[9]are none other than the structure constants of the algebra of holomorphic differentials introduced in [8].Therefore this is the algebra that should be used,and not the Landau-Ginzburg chiral ring.However,the connection with Landau-Ginzburg can still be very useful since the Picard-Fuchs equations may serve as an alternative to the residue formulas considered in [8].References[1]N.Seiberg and E.Witten,Nucl.Phys.B426,19(1994),hep-th/9407087.[2]E.Witten,Two-dimensional gravity and intersection theory on moduli space,in Surveysin differential geometry(Cambridge,MA,1990),pp.243–310,Lehigh Univ.,Bethlehem,PA, 1991.[3]R.Dijkgraaf,H.Verlinde,and E.Verlinde,Nucl.Phys.B352,59(1991).[4]G.Bonelli and M.Matone,Phys.Rev.Lett.77,4712(1996),hep-th/9605090.[5]A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B389,43(1996),hep-th/9607109.[6]R.Martini and P.K.H.Gragert,J.Nonlinear Math.Phys.6,1(1999).[7]A.P.Veselov,Phys.Lett.A261,297(1999),hep-th/9902142.[8]A.Marshakov,A.Mironov,and A.Morozov,Int.J.Mod.Phys.A15,1157(2000),hep-th/9701123.[9]K.Ito and S.-K.Yang,Phys.Lett.B433,56(1998),hep-th/9803126.[10]L.K.Hoevenaars,P.H.M.Kersten,and R.Martini,(2000),hep-th/0012133.[11]L.K.Hoevenaars and R.Martini,(2000),int.publ.1529,www.math.utwente.nl/publications.[12]A.Gorsky,I.Krichever,A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B355,466(1995),hep-th/9505035.[13]E.Martinec and N.Warner,Nucl.Phys.B459,97(1996),hep-th/9509161.[14]A.Klemm,W.Lerche,S.Yankielowicz,and S.Theisen,Phys.Lett.B344,169(1995),hep-th/9411048.[15]W.Lerche,D.J.Smit,and N.P.Warner,Nucl.Phys.B372,87(1992),hep-th/9108013.[16]K.Ito and S.-K.Yang,Phys.Lett.B415,45(1997),hep-th/9708017.。
第2章 夸克与轻子 (2)
第二章夸克与轻子Quarks and leptons2.1 粒子园The particle zoo学习目标Learning objectives：我们怎样发现新粒子？能否预言新粒子？什么是奇异粒子？大纲参考：3.1.1￣太空入侵者宇宙射线是由包括太阳在内的恒星发射而在宇宙空间传播的高能粒子。
如果宇宙射线粒子进入地球大气层，就会产生寿命短暂的新粒子和反粒子以及光子。
所以，就有“太空入侵者”这种戏称。
发现宇宙射线之初，大多数物理学家都认为这种射线不是来自太空，而是来自地球本身的放射性物质。
当时物理学家兼业余气球旅行者维克托·赫斯（Victor Hess）就发现，在5000m高空处宇宙射线的离子效应要比地面显著得多，从而证明这种理论无法成立。
经过进一步研究，表明大多数宇宙射线都是高速运动的质子或较小原子核。
这类粒子与大气中气体原子发生碰撞，产生粒子和反粒子簇射，数量之大在地面都能探测到。
通过云室和其他探测仪，人类发现了寿命短暂的新粒子与其反粒子。
μ介子（muon）或“重电子”（符号μ）。
这是一种带负电的粒子，静止质量是电子的200多倍。
π介子（pion）。
这可以是一种带正电的粒子（π＋）、带负电的粒子（π－）或中性不带电粒子（π0），静止质量大于μ介子但小于质子。
K介子（kaon）。
这可以是一种带正电的粒子（K＋）、带负电的粒子（K－）或中性不带电粒子（K0），静止质量大于π介子但小于质子。
科学探索How Science Works不同寻常的预言An unusual prediction在发现上述三种粒子之前，日本物理学家汤川秀树（Hideki Yukawa）就预言，核子间的强核力存在交换粒子。
他认为交换粒子的作用范围不超过10－15m，并推断其质量在电子与质子之间。
由于这种离子的质量介于电子与质子之间，所以汤川就将这种粒子称为“介子”（mesons）。
一年后，卡尔·安德森拍摄的云室照片显示一条异常轨迹可能就是这类粒子所产生。
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arXiv:q-alg/9710024v3 7 Jan 1998October1997LMU-TPW97-27q-DeformingMapsforLieGroupCovariantHeisenbergAlgebras1
GaetanoFioreSektionPhysikderLudwig-Maximilians-Universit¨atM¨unchenTheoretischePhysik—LehrstuhlProfessorWessTheresienstraße37,80333M¨unchenFederalRepublicofGermanye-mail:Gaetano.Fiore@physik.uni-muenchen.de
AbstractWebrieflysummarizeoursystematicconstructionprocedureofq-deformingmapsforLiegroupcovariantWeylorCliffordalgebras.1IntroductionAnydeformationofaWeylorCliffordalgebracanberealizedthroughachangeofgeneratorsintheundeformedalgebra[1,2].“q-Deformations”ofWeylorCliffordalgebrasthatwerecovariantundertheactionofasimpleLiealgebragarecharacterizedbytheirbeingcovariantundertheactionofthequantumgroupUhg,whereq=eh.Herewebrieflysummarizeoursystematicconstructionprocedure[6,7]ofallthepossiblecorrespondingchangesofgenerators,togetherwiththecorrespondingrealizationsoftheUhg-action.Thispavestheway[6]foraphysicalinterpretationofdeformedgeneratorsas“compositeoperators”,functionsoftheundeformedones.Forinstance,ifthelatteractascreatorsandannihilatorsonabosonicorfermionicFockspace,thentheformerwouldactascreatorsandannihilatorsofsomesortof“dressedstates”inthesamespace.Sincethereexists[7]abasisofg-invariantsthatdependontheundeformedgeneratorsinanon-polynomialway,butonthedeformedonesinapolynomialway,thesechangesofgeneratorsmightbeemployedtosimplifythedynamicsofsomeg-covariantquantumphysicalsystemsbasedonsomecomplicatedg-invariantHamiltonian.Letuslisttheessentialingredientsofourconstructionprocedure:
1.g,asimpleLiealgebra.2.ThecocommutativeHopfalgebraH≡(Ug,·,∆,ε,S)associatedtoUg;·,∆,ε,Sdenotetheproduct,coproduct,counit,antipode.
3.Thequantumgroup[4]Hh≡(Uhg,•,∆h,εh,Sh,R).4.Analgebraisomorphism[5]ϕh:Uhg→Ug[[h]],ϕh◦•=·◦(ϕh⊗ϕh).5.AcorrespondingDrinfel’dtwist[5]F≡F(1)⊗F(2)=1⊗2+O(h)∈Ug[[h]]⊗2:(ε⊗id)F=1=(id⊗ε)F,∆h(a)=(ϕ−1h⊗ϕ−1h)F∆[ϕh(a)]F−1.
6.γ′:=F(2)·SF(1)andγ:=SF−1(1)·F−1(2).7.Thegeneratorsa+i,aiofaordinaryWeylorCliffordalgebraA.8.Theaction⊲:Ug×A→A;Aisaleftmodulealgebraunder⊲.9.Therepresentationρofgtowhicha+i,aibelong:x⊲a+i=ρ(x)jia+jx⊲ai=ρ(Sx)ijaj.10.TheJordan-Schwingeralgebrahomomorphismσ:Ug∈A[[h]]:σ(x):=ρ(x)ija+iajifx∈gσ(yz)=σ(y)σ(z)
111.Thegenerators˜A+i,˜AiofadeformedWeylorCliffordalgebraAh.12.Theaction⊲h:Uhg×Ah→Ah;Ahisaleftmodulealgebraunder⊲h.13.Therepresentationρh=ρ◦ϕhofUhgtowhich˜A+i,˜Aibelong:X⊲h˜A+i=˜ρ(X)ji˜A+jX⊲h˜Ai=˜ρ(ShX)ij˜Aj.14.∗-structures∗,∗h,⋆,⋆hinH,Hh,A,Ah,ifany.2ConstructingthedeformedgeneratorsProposition1[6]Onecanrealizethequantumgroupaction⊲honA[[h]]bysettingforanyX∈Uhgandβ∈A[[h]](withX(¯1)⊗X(¯2):=∆h(X))
X⊲hβ:=σ[ϕh(X(¯1))]βσ[ϕh(ShX(¯2))].(1)
Proposition2[6,7]Foranyg-invariantsu,v∈A[[h]]theelementsofA[[h]]A+i:=uσ(F(1))a+iσ(SF(2)γ)u−1Ai:=vσ(γ′SF−1(2))aiσ(F−1(1))v−1.(2)
transformunder⊲has˜A+i,˜Ai.Asuitablechoiceofuv−1maymakeA+i,AjfulfilalsotheQCRofAh[7].Inparticularwehaveshownthe
Proposition3[7]Ifρisthedefiningrepresentationofg,A+i,AjfulfilthecorrespondingQCRprovided
uv−1=Γ(n+1)2(n+1+N2(n+1+NΓq2[12+l)]Γq2[12−l)]ifg=so(N),
(3)
whereΓ,Γq2areEuler’sgamma-functionanditsq-deformation,n:=a+iai,l:=Thealgebrahomomorphismfα:Ah→A[[h]]suchthatfα(˜A+i)=A+i,αandfα(˜Ai)=Ai,αiswhatisusuallycalleda“q-deformingmap”.ForacompactsectionofUgonecanchooseaunitaryF,F∗⊗∗=F−1.ThentheUg-covariant∗-structure(ai)⋆=a+iinAisalsoUhg-covariantinA[[h]]andhastheform(Ai,α)⋆=A+i,α,providedwechooseu=v−1andα⋆=α−1.Moreformally,underthisassumption⋆◦fα=fα◦⋆h,with⋆hdefinedby(˜Ai)⋆h=˜A+iIfHhisinsteadatriangulardeformationofUg,thepreviousconstruction
canbeequallyperformedandleadsessentiallytothesameresults[6],providedwechooseinthepreviousformulaeu≡v≡1.Thisfollowsfromthetrivialityofthecoassociator[5],thatcharacterizestriangulardeformationsHh.
AcknowledgmentsItisapleasuretothankJ.Wessforhisstimulus,supportandwarmhospitalityathisInstitute.ThisworkwassupportedthroughaTMRfellowshipgrantedbytheEuropeanCommission,Dir.Gen.XIIforScience,ResearchandDevelopment,underthecontractERBFMBICT960921.
ReferencesReferences[1]M.Pillin,Commun.Math.Phys.180(1996),23.[2]F.duCloux,Asterisque(Soc.Math.France)124-125(1985),129.[3]V.G.Drinfeld,DokladyANSSSR273(1983),531.[4]V.G.Drinfeld,ProceedingsoftheInternationalCongressofMathemati-cians,Berkeley1986,Vol.1,798.
[5]V.G.Drinfeld,LeningradMath.J.1(1990),1419.[6]G.Fiore,DeformingMapsforLieGroupCovariantCreation&Annihila-tionOperators,LMU-TPW96-20,q-alg/9610005.
[7]G.Fiore,Drinfel’dTwistandq-DeformingMapsforLieGroupCovariantHeisenbergAlgebras,LMU-TPW97-07,q-alg/9708017.