东北师大附中理科学霸英语笔记_2014高考状元笔记

合集下载

高考状元笔记整理打印版_高中英语：语法学习记忆口诀

高中英语：语法学习记忆口诀高中阶段的语法知识在现行的教材中分布得较分散、零碎，学生学起来颇感吃力，难以记牢。

一般说来，学习语法知识的途径很多，但教学中常用的主要有如下二种：一是通过教师的讲解，对所学语法规则的概念、结构，用法有个确切的了解；二是通过大量的练习，在英语实践中正确、熟练地掌握语法规则的用法。

仔细惦量这两种方法，其实都强调了同一个极其重要的东西，即是―记忆‖。

记忆的方式、方法很多，诸如分类记忆法，直观形象记忆法、奇持联想记忆法，特征记忆法等等。

这里，笔者主要从词法与句法两大块对巧记英语语法作些阐述。

一、词法（一）巧记名词变复数的规则：单数变为复数式，一般词尾加-s；下列句词词尾后，要加-s先加-e。

发音[f] 、[ t ]、[s]和[z]，或是某些辅音加-o时。

有些名词变复数，词尾变化要注意。

y前字母是辅音，一律变y为-ies.遇到f和fe，有时需要变-ves.少数名词不规则，特殊情况要强记。

说明：1．名词变复数形式，一般在词尾加-s.eg: bag-bags banana-bananas bird-birds pen-pens, ....2．词尾发音为[f , t s, z]的名词（即以字母sh, ch, s, x结尾者）在变复数时，要在词尾加-es，eg: watch-watches, box-boxes, bus-buses,etc)3．若词尾字母o的前面是辅音字母，变为复数时，有些加-es，eg: hero-heroes, Negro-Negroes, patato-patatoes, tomato-tomatoes.这四个词可组成一句话来记忆：Heroes and Negroes eat potatoes and tomatoes（英雄和黑人吃马铃薯和西红柿。

简为二人吃二菜。

）但有些以o结尾的名词则加-s，eg: photo – photoes, piano –pianos等。

2014高考秘宝：历年高考状元独门笔记英语

2014高考秘宝：历年高考状元独门笔记英语高考英语2014-08 2129高考状元答高三学子问诚然，进入高三阶段，同学们来自各方面的压力逐渐增多。

或许你对高考认识不足，或许你在心理上有恐惧和恐慌，或许你在复习规划上有迷茫，或许你在学习上有不足从而造成自卑和信心不足，或许你对家长和老师的过于频繁的说教有反感，或许各方面的压力更造成你的心理紧张从而不能做到有效地复习，或许你存在着偏科，比如说英语可能就不是你的强项。

那么，该如何缓解这些压力呢?在此，将高考状元们的一些学习经验与大家一同分享，希望会对大家有所帮助。

学生甲：与高一、高二阶段的学习相比，高三阶段的学习策略有何不同呢?何润夏(2014年福建文科状元)：如果说高中生活是一场马拉松长跑的话，那么，高一便是兴奋的起点，高二算是难熬的中段，高三则是最后的冲刺。

高一、高二是打基础的阶段，基础越厚，高三才能冲得越高。

凭我自己的经历，在高中阶段，每天学习时间的投入(包括上课时间、课外学习时间)如下：高一、高二≥2014小时，高三≥12 小时。

高中生应当树立时间至上的观念，学会挤时间，抢时间。

在这方面，制定一份作息时间表和一份短时期的计划有较大意义。

如果说时间保证是表面上的刻苦，那么，好的方法便是本质上的动力。

其一，学习时间内心平气和，“静而后能安，安而后能虑，虑而后能得”，在翻开学习资料的同时，排除一切杂念，专心致志地思考。

其二，学会正确安排时间，高一高二应当文理相间，先课内后课外，先作业后练习。

高三我采取集中复习法，每天两门，每周完成两轮大循环，自我感觉效果不错。

三点一线总是很单调的，尤其对于高三学生，每日面对高强度的练习及记忆总不免会有几分害怕。

这时候，“坚持” 尤为重要，我高中时习惯写日记，高三更是每天必记，现在翻看，确有触目惊心之感。

常有的内容是发泄自己的坏心情，向笔记本诉苦，然后又鼓励自己前途是光明的。

出现最多的句子便是“锲而不舍，金石可镂”，“坚持就是胜利!”大概就是凭着日记的鼓舞和心里那一份对大学的渴望，我每天都始终如一地坚持做应该做的事，直至高考。

英语状元笔记必修五-6.docx

哈三中状元班笔记匚HRMPIDN CLH55 NOTES DF ND.3 MIDDLE 5匚HDQL SCM 仏f 触%勺$妙3勺1胁＜勺 Model 6 Animals in Danger . 1 endanger （\ft ） cause danger to; put...in danger △直 _____________<：处涨配 fi .啲 ________ _____ _（吸烟危害傩康）.■ *爼Q 林必：_严说-込一- a eo^A 伴必—学.6^4 __________ I （大熊猫是*而临绝紳危险的动物）. M2 spccies （c ）（pl ）J 自缄、对..* __ft 拟血么够沁亡_学乞扭:__..（ _种濒危物种）* 址一泅 _________________ .一（一种）an 也lope （羚杀） |3.rarvf»ilj ）noloften happening or seen; unusual.细盘犖）由aclv,_也幽 __:』趾必加 _ ■ *…一 GUr&iit _（稀客）* ______ 0,—”尿:血堺冷 ___________ （稀有M 溺蝶）*Only rarely ^_血_7 ________________ （我极少）eat in restaurantj 劭働 I Reserve （y.）.鈕竝耐迪 _____________________ n_屜淫」屈牆j_；cscrvalion ：_解尊― served 妁）__心壇酝，血g 肉「 *I ）oyou have io _廊£”池也也化说3皿 __________________________ （预先订票吗）？ | *I'd like lo reserve a table 和亡 two （一张两人的餐桌）. . * These scats are reserved … _______ the elderly an<l disabled. " *Wc always keep some money________ 仙 _______ reserve. | ____ a 一沁 ________ ©ildlifb 一 _怂业^ * a reserved girl 曲同龙扱 *1 am sorry but this seat is resci-ved _ ■5. cxthct adj...点业1 命 ________________ ；_毎曲3叙（*3 ___________________ |• * —上创诚（已灭绝的物伸）t 丄血就址!注一喊沁來两座死火山）-6. strURKle （ven ）珠* _____________ ；—亠半 ____________________________________ 1 * Ihc shopkeeper struggled _ 由[th. _______ the tliicf. x ）r |乜辺血哪h 一 L<ujut* llie two leaders are struggling power. 丸必 I "Tie is struggling__衣）.債&坷 ___ （bring ） - __ a family onavety low income. J7 protect （vt ）_^> _______* lroopil^c been Sent to protect 屈魅vorkcis _____ aja/ns t ：^一_a«ack...* You n 翹Qhrm cloihes to pr （xc<l留?d 严M the cold.山責"'p 砂泌蚀、.广- _一 ” - 丁 ⑴两如n &1皿kwn （此吗.幣匕牝咄*——严七一必嗨血也亠（放下科把刀）*_冲—述_血如_ r»严t 此以—'沖_ogsa 一.皿.血；\ J ___________________________________________________ .（镇压叛乱）Mt 9 althougi ） surpri:-d: 〃如呼k /u S … 愴询并淋诵：駅从平宦 * Md0eV_（movc ） lc ）tears, the bo ；; stood still a （ t]ie door.____ （鸟类淚护区）超 ctos 。

高考状元学习秘籍

1、福建北大保送生吴盛祥：重点抄本子上随时看吴盛祥同学就以全国化学竞赛金奖的成绩保送上北京大学。

他向记者透露了自己的学习秘诀：上课认真听，做作业要挑自己的弱项先做，不能先做自己喜欢的题目，否则光做得爽，但却无法提升自己的弱项。

在化学学习方面，他总是把知识要点进行归纳总结，找出一定的规律来背，他说，这样就容易记住。

同时，他喜欢把课本中的重点抄在本子上，随身携带，有空就看看。

仅高三期间，他就抄了40多个手抄本。

【理科状元】陈思恒，男，裸分691，各科分数为：语文118分，数学137分，英语144分，理综292分；曾楚元，女，裸分691，各科分数为：语文125分，数学135分，英语147分，理综284分。

均毕业于厦门外国语学校。

陈的经验：多练习，做题保持手感。

曾的经验：遇到难题与老师妈妈沟通；淡定、冷静善于与老师交流。

吴灈杭，女，裸分691，各科分数为：语文129分，数学138分，英语142分，理综282分。

毕业于泉州市第五中学。

刘泰然，男，裸分691，各科分数为：语文124分、数学139分、英语140分、理综288分。

毕业于福州一中。

经验：每做一道题目都要真正弄懂；学习英语要在生活中使用它。

【文科状元】张翔雁，女，裸分667，毕业于泉州市第五中学。

2、吉林【理科状元】耿天毅，男，裸分706分，各科成绩：语文129分，数学149分，理综285分，英语143分。

毕业于吉林油田高中。

喜欢的格言是：别想一下造出大海，必须先由小河开始。

学习方法上，一是，做题时，吸取经验，保证做过的错题不会再错。

二是，英语成绩一开始不是很好，后来，天天多做卷练习，靠日常积累成绩逐渐提高。

三是，没有准备错题本，他觉得平时保证不粗心大意，基本上就能保证数学分在144分以上。

【文科状元】刘伊恬，女，裸分664。

毕业于东北师大附中。

经验：在学习上是一个稳扎稳打的孩子，很有思想，又乖巧可爱，是老师眼里是完美优秀的好学生。

平时学习状态特别好，认真、扎实，喜欢积极主动地找老师问问题。

m SOLUTIONS GOOD, m 1 SOLUTIONS BETTER

Applied Mathematical Sciences,Vol.x,200x,no.xx,xxx-xxxm SOLUTIONS GOOD,m−1SOLUTIONS BETTERLuc Longpr´e,Vladik KreinovichDepartment of Computer ScienceUniversity of Texas at El PasoEl Paso,TX79968,USAlongpre@,vladik@William GasarchDepartment of Computer ScienceUniversity of MarylandCollege Park,MD20742,USAG.William Walster22116Dean Ct.Cupertino,CA95014-2723,USAwalster1@AbstractOne of the main objectives of theoretical research in computational complexity and feasibility is to explain experimentally observed diﬀer-ence in complexity.Empirical evidence shows that the more solutions a system of equations has,the more diﬃcult it is to solve it.Similarly,the more global maxima a continuous function has,the more diﬃcult it is to locate them.Until now,these empirical facts have been only partially formalized:namely,it has been shown that problems with two or more solutions are more diﬃcult to solve than problems with exactly one so-lution.In this paper,we extend this result and show that for every m,problems with exactly m solutions are more diﬃcult to solve than problems with m−1solutions.Rephrasing Orwell’s“Four legs good, two legs better”,we can describe this result as“m solutions good,m−1 solutions better”.Mathematics Subject Classiﬁcation:68Q17,68Q15,90C60,65G202L.Longpr´e,V.Kreinovich,W.Gasarch,G.W.Walster Keywords:computability,optimization,solving systems of equations, computational complexity1IntroductionIn many real-life situations,we want toﬁnd the best decision,the best con-trol strategy,etc.The corresponding problems are naturally formalized as optimization problems:we have a function f(x1,...,x n)of several variables, and we want toﬁnd the values(x1,...,x n)for which this function attains the largest(or the smallest)possible value.Many numerical algorithms have been proposed for solving optimization problems.Unfortunately,many of these algorithms often end up in a local maximum instead of the desired global one.•In some practical situations,e.g.,in decision making,the use of local maximum simply degrades the quality of the decision but is not,by itself,disastrous.•However,in some other practical situations,missing a global maximum or minimum may be disastrous.Let us give two example:•In chemical engineering,global minima of the energy function often de-scribe the stable states of the system.If we miss such a global minimum, the chemical reactor may go into an unexpected state,with possible se-rious consequences.•In bioinformatics,the actual shape of a protein corresponds to the global minimum of the energy function.If weﬁnd a local minimum instead,we end up with a wrong protein geometry.As a result,if we use this wrong geometry as a computer simulation for testing recommendations on the medical use of chemicals,we may end up with medical recommendations which harm a patient instead of curing him.For such applications,it is desirable to use rigorous,automatically veriﬁed methods of global optimization,i.e.,methods which never discard an actual global maximum;for a survey of such methods,see,e.g.,[12].These methods usually start with a large“box”on which a function is deﬁned(and on which global maxima can be located),and produce a list of small-size boxes with the property that every global maximum is guaranteed to be contained in one of these boxes.Most of such guaranteed methods use(a version of)interval computations. The main idea of interval computations is as follows:To solve a given numericalm Solutions Good,m−1Solutions Better3 problem(e.g.,an optimization problem),numerical methods typically generate better and better estimates for diﬀerent quantities related to the problem–such as the actual global maximum of the function,the value of its partial derivatives at diﬀerent points,etc.•In traditional numerical techniques,for each approximated numerical quantity,an approximation is a real number,with no guarantees on the approximation accuracy.•In interval methods,at any given moment of time,for each approximated quantity x,we compute not only the approximate value x,but also the upper bound∆on the possible approximation error,i.e.,a number∆for which we are guaranteed that|x− x|≤∆.In other words,at any step, we have not only an approximate value x of the approximate quantity, we also have an interval x=[ x−∆, x+∆]which is guaranteed to contain the(unknown)actual value of x.As we have mentioned,rigorous methods of global optimization start with a large box as a location of the unknown global maxima and gradually replace it will a smallﬁnite collection of small boxes.The decrease in a box size is usually achieved by dividing one of the boxes into several sub-boxes and eliminating some of these sub-boxes.When can we eliminate a sub-box B?At every stage of the optimization algorithm,we have already computed several values of the optimized function f(x1,...,x n),so we know that the global maximum of the function f cannot be smaller than the largest M of these already computed values.Thus,if we can guarantee that the maximum of the function f on a box B is smaller than M,we can thus exclude this box from the list of possible locations of a global maximum.To get such a guarantee,we canﬁnd an enclosure for the range of the function on a subbox,e.g.,by using methods of interval arithmetic[12], when we parse the computation of the function f and replace each arithmetic operation by the corresponding operation with intervals.In some real-life problems,we are not yet ready for optimization,e.g., because the problem has so many constraints that evenﬁnding some values x=(x1,...,x n)of the parameters x i which satisfy all these constraints is an extremely diﬃcult task.For such problems,we arrive at the problem of satisfying given constraints,e.g.,solving a given system of equations.In many such problems,it is important not to miss a solution.The complexity of locating global maxima of f is empirically known to depend on the number of these global maxima:the fewer global maxima,the easier the problem[8,9,10,11,12,28].It is desirable to formalize and explain this empirical fact.Previously,this result have been formalized only partially:namely,it was shown that global optimization is easier when we have exactly one global max-4L.Longpr´e,V.Kreinovich,W.Gasarch,G.W.Walster imum than when have several.In this paper,we extend this result and show that optimization with m global maxima is,in some formal sense,more dif-ﬁcult than optimization with m−1global maxima(and the same is true for solving systems of equations).2Formalization of the Problem and Previously Known ResultsIn order to formulate this result,we must recall some basic deﬁnitions of computable(“constructive”)real numbers and computable functions from real numbers to real numbers(see,e.g.,[1,3,4,5,24]):Deﬁnition2.1A real number x is called computable if there exists an algo-rithm(program)that transforms an arbitrary integer k into a rational number x k that is2−k−close to x.It is said that this algorithm computes the real number x.When we say that a computable real number is given,we mean that we are given an algorithm that computes this real number.Deﬁnition2.2A function f(x1,...,x n)from real numbers to real numbers is called computable if there exist algorithms U f andϕ,where:•U f is a rational-to-rational algorithm which provides,for given rational numbers r1,...,r n and an integer k,a rational number U f(r1,...,r n,k) which is2−k-close to the real number f(r1,...,r n),and|U f(r1,...,r n,k)−f(r1,...,r n)|≤2−k,and •ϕis an integer-to-integer algorithm which gives,for every positive integerk,an integerϕ(k)for which|x1−x1|≤2−ϕ(k),...,|x n−xn|≤2−ϕ(k)implies that|f(x1,...,x n)−f(x1,...,xn)|≤2−k.When we say that a computable function is given,we mean that we are given the corresponding algorithms U f andϕ.Theorem2.1[13,14,15,18,20]There exists an algorithm U such that:•U is applicable to an arbitrary computable function f(x1,...,x n)that attains its maximum on a computable box B=[a1,b1]×...×[a n,b n]at exactly one point x=(x1,...,x n),•for every such function f,the algorithm U computes the global maximum point x.m Solutions Good,m−1Solutions Better5 Theorem2.2[17,18,19,20,21,22,23,24]No algorithm U is possible such that:•U is applicable to an arbitrary computable function f(x1,...,x n)that attains its maximum on a computable box B=[a1,b1]×...×[a n,b n]at exactly two points,and•for every such function f,the algorithm U computes one of the corre-sponding global maximum points x.These results partially explain the above intuition because they show that the problem of locating global maxima is easier if we have a single global maximum and more diﬃcult if we have several global maxima.These results, however,do not completely explain this intuition because they do not explain why,say,a problem with three global maxima is more complex than a problem with two global maxima.Similar results hold for roots(solutions)of a system of equations:Deﬁnition2.3By a computable system of equations we mean a system f1(x1,...,x n)=0,...,f k(x1,...,x n)=0,where each of the functions f i is a computable function on a computable box B=[a1,b1]×...×[a n,b n].Theorem2.3[13,14,15,18,20]There exists an algorithm U such that:•U is applicable to an arbitrary computable system of equations which has exactly one solution,and•for every such system of equations,the algorithm U computes its solution.Theorem2.4[17,18,19,20,21,22,23,24]No algorithm U is possible such that:•U is applicable to an arbitrary computable system of equations which has exactly two solutions,and•for every such system of equations,the algorithm U computes one of its solutions.Comment.It is known that the general problem of solving a system of polyno-mial equations with rational coeﬃcients is NP-hard[24](it is even NP-hard for quadratic equations).The problem ofﬁnding the unique solution to a system of equations(or the unique point where the maximum is attained)is as compli-cated as the problem ofﬁnding the unique satisfying vector for a given propo-sitional formula[24].The latter problem(it is usually denoted by USAT,from u nique sat isaﬁability)is known to be“almost”NP-hard in the sense that every6L.Longpr´e,V.Kreinovich,W.Gasarch,G.W.Walsterother problem from the class NP can be probabilistically reduced to USAT;so if we were able to solve all the instances of USAT in polynomial time,we would have a probabilistic polynomial-time algorithm that solves almost all instances of all problems from the class NP.(Exact deﬁnition are somewhat complicated so,due to the lack of space,we refer the interested reader to[7]and[27].Note that in[2],arguments are given that this problem may not be NP-hard.)3New ResultsThe following modiﬁcation of the above results provides the desired complete explanation:For two global maxima,we cannot pinpoint one of them,but,as we will show,we can compute the next best thing:namely,we can compute two locations with a guarantee that one of them contains a global maximum. For functions with three global maxima,we cannotﬁnd neither one nor two such locations,but we canﬁnd three locations one of which contains the global maximum,etc.Thus,we have formalized the intuitive idea that the fewer global maxima,the easier the global optimization problem.Deﬁnition3.1We say that a box B=[a1,b1]×...×[a n,b n]is of size≤εif each of its sides is of size≤ε,i.e.,if b i−a i≤εfor all i=1,...,n.Theorem3.1Let m>1be an integer.Then:•There exists an algorithm U such that:•U is applicable to an arbitrary computable function f(x1,...,x n)on a computable box B=[a1,b1]×...×[a n,b n]that attains itsmaximum on B at exactly m points,and•for every computable real numberε>0,the algorithm U returnsm boxes of sizeεwith a guarantee that each global maximum iscontained in one of these boxes.•No algorithm U is possible such that:•U is applicable to an arbitrary computable function f(x1,...,x n)on a computable box B=[a1,b1]×...×[a n,b n]that attains itsmaximum on B at exactly m points,and•for every computable real numberε>0,the algorithm U returnsm−1boxes of sizeεwith a guarantee that one of these boxes containsa global maximum.Theorem3.2Let m>1be an integer.Then:m Solutions Good,m−1Solutions Better7•There exists an algorithm U such that:•U is applicable to an arbitrary computable system of equations whichhas exactly m solutions,and•for every computable real numberε>0,the algorithm U returns mboxes of sizeεwith a guarantee that each solution is contained inone of these boxes.•No algorithm U is possible such that:•U is applicable to an arbitrary computable system of equations whichhas exactly m solutions,and•for every computable real numberε>0,the algorithm U returnsm−1boxes of sizeεwith a guarantee that one of these boxes containsa solution.The problem becomes even more complicated if we do not know the actual number n(f,B)of points where the maximum of a function f on the box B is attained–only the upper bound m on this number.In this case,n(f,B) can take any value from the set{1,...,m}.It turns out that not only we cannot predict n(f,B)for a given f,we cannot even limit the m-element list of possible cardinalities to a smaller sublist:Theorem3.3Let m>1be an integer.Then,no algorithm is possible which,given a computable function f on a computable box B which attains its maximum at n(f,B)≤m points,returns a set S(f,B)⊂{1,...,m}with #S(f,B)<m elements which contains the value n(f,B).A similar result holds for solutions:Theorem3.4Let m>1be an integer.Then,no algorithm is possible which,given a computable system of equations s which has n(s)≤m solutions, returns a set S(s)⊂{0,1,...,m}with#S(s)<m+1elements which contains the value n(s).4Proofs4.1Proof of Theorem3.11.It is known that there exists an algorithm which,given a computable function on a computable box,and a givenδ>0returns a rational number M which isδ-close to max f[1,3,4,5,24].Let us reproduce the main idea of this proof.8L.Longpr´e ,V.Kreinovich,W.Gasarch,G.W.Walster1.1.First,we prove that there exists an integer m for which the 2−m -approximation δm to δexceeds 3·2−m .Indeed,since δ>0,we have δ>2−k for some k .Therefore,for the2−(k +2)-approximation δk +2to δ,we get |δk +2−δ|≤2−(k +2)henceδk +2≥δ−2−(k +2)>2−k −2−(k +2)=3·2−(k +2).So,the existence is proven for m =k +2.This m can be algorithmically computed as follows:we sequentially try m =0,1,2,...and check whether δm >3·2−m ;when we get the desired inequality,we stop.1.2.Let us now show that for the integer m computed according to Part 1.1of this proof,we have δ>2·2−m .Indeed,since δm >3·2−m and |δ−δm |≤2−m ,we can conclude thatδ≥δm −2−m >3·2−m −2−m =2·2−m .So,if we can ﬁnd a rational number M which is 2·2−m -close to max f ,this rational number will thus be also δ-close to max f .1.3.Let us now use this m to compute the desired δ-approximation to max f .1.3.1.By using the second algorithm ϕin the deﬁnition of a computable function,we can ﬁnd a value ϕ(m )such that if |x i −x i |≤ϕ(m )for all i =1,...,n ,then|f (x 1,...,x n )−f (x 1,...,x n )|≤2−m .For each dimension [a i ,b i ]of the box B ,we can then take ﬁnitely many valuesr (1)i ,r (2)i =r (1)i +ϕ(m ),r (3)i =r (2)i +ϕ(m ),...,r (N i )i =r (N i −1)i +ϕ(m )(separated by ϕ(m ))which cover the corresponding interval.Then,each valuex i ∈[a i ,b i ]will be diﬀerent by one of these values r (k i )i by ≤ϕ(m ).1.3.bining the values corresponding to diﬀerent dimensions,we get a ﬁnite list of rational-valued vectors r (k 1)1,...,r (k n )nwith the property that every vector (x 1,...,x n )∈B is ϕ(m )-close to one of these vectors.Due to the deﬁnition of ϕ(m ),this means that each value f (x 1,...,x n )is 2−m -close to one of the values f r (k 1)1,...,r (k n )n.Therefore,the desired max f is 2−m -close to the maximum of all the values fr (k 1)1,...,r (k n )n .By using the algorithm U f ,we can compute each of these values with the accuracy 2−m .Thus,the maximum M of thus computed rational val-ues U f r (k 1)1,...,r (k n )n,m .is 2−m -close to the maximum of all the values fr (k 1)1,...,r (k n )n,and hence,2·2−m -close to max f .Thus,M is indeed δ-close to max f .The ﬁrst part is proven.m Solutions Good,m −1Solutions Better 92.Let us now prove that the existence of the desired algorithm U .2.1.First,we prove that for every function f with exactly m global maxima x (1),...,x (m )in whichf x (1) =...=f x (N ) =M def=max B f (x ),there exists a real number δ>0such that for every x ∈B ,if f (x )is δ-close to the maximum M ,then x is (ε/2)-close to one of the global maxima x (i )(1≤i ≤m ).Indeed,if such a δdoes not exist,then,for every δ=2−k ,there exists a pointp (k )for which f p (k ) ≥M −2−k and ρ p (k ),x (i ) >ε/2for all i =1,...,m (where ρdenotes the distance).Since the box B is closed and bounded,it is a compact,so the sequence p (k )has a convergent subsequence p (k i )→p ∈B .In the limit k i →∞,we get:•f (p )≥M =max f (x ),hence p is the global maximum point,and •at the same time,ρ p,x(i ) ≥ε/2>0,so p is diﬀerent from all knownm global maxima.The contradiction shows that such a δ>0must exist.2.2.From 2.1,we can conclude that is more than (ε/2)away from one of the local maxima,then f (x )<M −δ.2.3.Now,we can present the desired algorithm.To get the desired boxes,for every N =1,2,...,we do the following:•We divide the original box into N ×N ×...×N subboxes B (N )1,B (N )2,....•We compute the maximum M (N )1,M (N )2,...of f on each of these subboxes with an accuracy 2−N .The maximum M (N )of these values is the 2−n -approximation to the actual (unknown)maximum M of the function f .•We dismiss all the subboxes B (N )i for which M (N )i <M (N )−2·2−N ,because for these subbboxes,the actual maximum is guaranteed to be snaller than M and hence,these subboxes cannot contain any global maxima.•Finally,we check whether all these subboxes can be subdivided into m groups of size ε.If they can be thus grouped,we get the desired m subboxes.If not,we increase N by one,and repeat the same procedure.10L.Longpr´e,V.Kreinovich,W.Gasarch,G.W.Walster The fact that this algorithm stops for some N follows from the Part2.1ofthis proof:if N becomes so large that2−N<δ/2and2−N<ε/4,then for every box B(N)iwhich is more than(ε/2)away from one of the global maxima,its actual maximum is<M−δ.Hence,its2−N-approximate value M(N)i is<M(N)−2·2−N,so this box will be dismissed.As a result,all remaining boxes will indeed beε-close to the actual global maxima.The existence is proven.3.Let us now prove the impossibility of an algorithm which would return m−1boxes such that one of these boxes contains a global maximum of a given function f.We will prove this impossibility by reduction to a contradiction.Let us assume that such an algorithm exists.3.1.First,let us prove that we will then have an algorithm I which,given an arbitrary everywhere deﬁned computable sequence a:I N→I N from natural numbers to natural numbers,returns an integer I(a)∈{1,...,m}with the following property:if∃n(a(n)=0),then I(a)is diﬀerent from theﬁrst non-zero value of a(n).3.1.1.First,for every such sequence a,we construct a computable functionf a:[0,2m−1]→[0,1].This function f a(x)is constructed as the sum of the constructively converging functionsf a(x)=f(0)a (x)+f(1)a(x)+ (1)for appropriately deﬁned functions f(i)a (x).In the following,we describe analgorithm which,given i and x,computes the value f(i)a (x)of i-th componentfunction.The values of i-th function is from the interval[0,2−i]and therefore, it is easy to prove that their sum is indeed a computable function in the sense of the above deﬁnitions(for exact proofs,see,e.g.,[1,3,4,5]).For constructing the component functions f(i)a ,we will use an auxiliary“trapezoid”function t:[0,1]→[0,0.5]which is deﬁned as follows:•t(x)=0for x≤0;•t(x)=x for0≤x≤0.25;•t(x)=0.25for0.25≤x≤0.75;•t(x)=1−x for0.75≤5≤1;•t(x)=0for1≤x.This trapezoid is of height0.25=0.5·(1−2−1)with an upper horizontal part of length0.5=2−1.As long as we havea(0)=...=a(i)=0,(2)m Solutions Good,m−1Solutions Better11we take,as f(i)a(x),a functionF i(x)def=t i(x)+t i(x−2)+...+t i(x−2·(m−1)),(3)wheret i(x)def=2−i·t2i·(x−0.5)+0.5.In particular:•For i=0,we get t0(x)=t(x),hencef(0)a(x)=t(x)+t(x−2)+...+t(x−2·(m−1)),i.e.,f(0)a (x)is the sum of m identical trapezoids concentrated on theintervals[0,1],[2,3],...,[2·(m−1),2·(m−1)+1=2m−1].•For i=1,adding f(1)a (x)to f(0)a(x)means that we add a small trapezoidon top of each of the previous m trapezoids.The parameters of both trapezoids are selected in such a way that their sides perfectly align,so for each of m intervals[0,1],...,[2m−2,2m−1],the graph of the sumf(0) a (x)+f(1)a(x)is also a trapezoid,of height0.5·(1−2−2)with an upperhorizontal part of length2−2.•Similarly,for every i>0,if a(0)=...=a(i)=0,then on each of mintervals,the sum f(0)a (x)+...+f(i)a(x)is also a trapezoid,of height0.5·(1−2−(i+1))with an upper horizontal part of length2−(i+1). When i→∞,the height tends to0.5,and the width of the upper horizontal part tends to0.We have deﬁned f(i)a (x)for the case when the condition(2)is satisﬁed.Let j be theﬁrst value for which this condition is not satisﬁed.Then,as we have mentioned,on each of m intervals[0,1],,...,[2m−2,2m−1],the sumf(0) a (x)+...+f(j−1)a(x)is a trapezoid,of height0.5·(1−2−j)with an upperhorizontal part of length2−j:•For theﬁrst interval[0,1],the horizontal part is centered around the midpoint0.5,i.e.,it corresponds to the values[0.5−0.5·2−j,0.5+0.5·2−j].•For k-th interval[2k−2,2k−1],the horizontal part is similarly centered around the midpoint2k−1.5,i.e.,it corresponds to the values[(2k−1.5)−0.5·2−j,(2k−1.5)+0.5·2−j].For the further construction,we compute k0=min(a(j),m);thus deﬁned k0is an integer whose possible values range from1to m.We will halt the construction of our“pyramids”on all intervals except the interval#k0,and on this particular interval,we will build m small identical“pyramids”on top12L.Longpr´e,V.Kreinovich,W.Gasarch,G.W.Walsterof the horizontal part[x−,x+]of the corresponding trapezoid,where x−def= (2k0−1.5)−0.5·2−(j−1)and x+def=(2k0−1.5)+0.5·2−(j−1).Speciﬁcally,for every i≥j,we deﬁnef(i) a (x)=x+−x−2m−1·F i−j2m−1x+−x−·(x−x−),where F i(x)is deﬁned by the formula(3).3.1.2.Let us now show that,for every computable everywhere deﬁned sequence a(n),the corresponding computable function f a(x)has exactly m global max-ima,and describe the location of these global maxima.When a(0)=...=a(i)=...=0for all i,the graph of the limit function (1)is the sum of m triangular functions each of which is located on the the corresponding interval[0,1],...,[2m−2,2m−1],and each of which attains the maximum0.5at the midpoint of the corresponding interval.Thus,for a(n)≡0,the function f a(x)has exactly m global maxima:0.5,2.5,..., 2(m−1)+0.5,each of which is located within the corresponding interval[0,1], ...,[2m−2,2m−1].When a(n)≡0,the function f a(x)depends on theﬁrst non-zero value a(j): Namely,on j and on k0=min(a(j),m).Similarly to the case a(n)≡0,thegraph of the sum f(i)a (x)+f(i+1)a(x)+...is a sum of m triangular functions.Hence,for a(n)≡0,the sum(1)attains maximum at exactly m points all of which are located within the k0−th interval[2k0−2,2k0−1].3.1.3.Let us now describe the desired I(a).The hypothetic algorithm U returns m−1boxes,one of which is guaranteed to contain a global maximum.Whenεis small enough(e.g.,≤0.5),each box can only contain points from one of the m intervals[0,1],...,[2m−2,2m−1]. Thus,when a(n)≡0,the m−1boxes resulting from applying U to f a intersect with at most m−1of these intervals.Thus,at least one of these intervals [2i−2,2i−1]does not intersect with any of the m−1boxes.As I(a),we then take the ordinal number of one of these non-intersecting intervals.3.1.4.To complete the proof,let us show that if a(n)≡0,then I(a)is diﬀerent from theﬁrst non-zero value a(j)of the sequence a(n).Indeed,if a(n)≡0,then one of the m−1intervals must contain a global maximum.Since all global maxima are located in the interval[2k0−2,2k0−1] (where k0=min(a(j),m)),one of the intervals from U(f1)must intersect with this interval[2k0−2,2k0−1].Since I(a)is the ordinal number of an interval which does not intersect,and the interval#k0does intersect,we conclude that I(a)=k0.By deﬁnition,I(a)≤m.So:•if a(j)>m,then I(a)=a(j);m Solutions Good,m−1Solutions Better13•if a(j)≤m,then k0=a(j)hence also I(a)=a(j).In both cases,I(a)=a(j).The statement is proven.4.To complete the proof of the theorem,let us prove that the algorithm I(a) described in Part3of this proof is impossible.We will prove this impossibility in a way which is similar to the standard proof of the undecidability of the halting problem.Let f n by a(partial)recursive function#n(or Turing machine#n,etc.),and let d be an integer.For each t,we can deﬁne a(t)as follows:•a(t)=0if the computation of f on d has not stopped by time t(i.e.,in t steps),and•a(t)=f(d)is the computation of f on d stopped by time t.By applying the algorithm I to this sequence a,we get a value I(a)with the property that if f is applicable to d(!f n(d)),then I(a)=a(j)=f(d).Let us denote this value I(a)by v(n,d).Thus,we have an algorithm,which,given two integers n and d,always returns a value v(n,d)such that if!f n(d)then v(n,d)=f n(d).In particular,the diagonal function v(n,n)is also computable and everywhere deﬁned.Thus,it has a number in the ordering of all recursive functions,i.e.,there exists a number c for which v(n,n)=f c(n)for all n. Then,we get the desired contradiction:•On one hand,from v(n,n)=f c(n),for n=c,we get v(c,c)=f c(c).•On the other hand,here!f c(c),hence from the above property of v,we conclude that v(c,c)=f c(c).The theorem is proven.4.2Proof of Theorem3.2Theorem2follows from Theorem3.1if we take into consideration that the problems of solving a system of equation and of locating global maxima can be naturally(and computably)reduced to each other in such a way that the solutions to the system of equations become global maxima and vice versa (and thus,the number of solutions becomes the number of global maxima and vice versa):•If we know how to solve systems of equations,then the problem of locat-ing global maxima of a function f(x1,...,x n)can be reformulated as a problem ofﬁnding all solutions to an equation f1(x1,...,x n)=0,wheref1(x1,...,x n)def=max f−f(x1,...,x n).14L.Longpr´e,V.Kreinovich,W.Gasarch,G.W.Walster •Vice versa,if we know how to locate global maxima,then the problem of solving a system of equations f1(x1,...,x n)=0,...,f k(x1,...,x n)= 0can be reformulated as a problem ofﬁnding all global maxima of a functionf(x1,...,x n)def=−(|f1(x1,...,x n)|+...+|f k(x1,...,x n)|).4.3Proof of Theorem3.3This proof uses the following result about bounded queries([25],see also[6]):Let#An (p1,...,p n)denote the number of elements in the intersection A∩{p1,...,p n},and let F∈EN(m)mean that there is an algorithm that will, given x,enumerate≤m possibilities for F(z)one of which is correct.It isknow that if A is not recursive,then#An ∈EN(n).In other words,we cannotalways eliminate one possibility for#An .Assume that for some m,there is an algorithm which returns the set S(f,B) with#S(f,B)<m.Let us then show that for the halting set A,we get#Am−1∈E(m−1)–in contradiction with the above result.To show this,we will follow the construction from[26],in which we construct,for every program p,a non-negative computable real number x such that x>0if and only if p halts.(This construction follows the spirit of Turing–who introduced the notion of a Turing machine in an attempt to formalize the notion of a computable real number.)Speciﬁcally,for every program p,we deﬁne x as follows:x k=2−k if p did not halt by time k,and x k=2−t if p halted at some moment t≤k.For every m−1Turing machines(or programs)p1,...,p m−1,we can then deﬁne the corresponding real numbers a1,...,a m−1and a computable function f(x)def=t(x)+(1−a1)·t(x−1)+...+(1−a m−1)·t(x−(m−1)), where t(x)=x for x∈[0,0.5],t(x)=1−x for x∈[0.5,1],and t(x)=0for all other x.This function consists of m triangular-shaped pieces of heights1(for x=0.5),1−a1(for x=1.5),...,1−a m−1(for x=m−0.5).Since a i≥0,the maximum of this function is1,and the number of points where this maximum is attained is equal to the number of values a1,...,a m−1which are equal to0. Since a i=0if and only if the program p i halts,any limitation of the numberof points where f attains maximum would thus lead to a limitation on#Am−1.The theorem is proven.4.4Proof of Theorem3.4This proof is similar to the proof of Theorem3.3,with the only diﬀerence that instead of the function f(x),we consider the computable system consisting of。

YW01.2014高考吉林理科状元语文笔记 115面胶装

第 1／115页
第 2／115页
第 3／115页
第 4／115页
第 5／115页
第 6／115页
第 7／115页
第 8／115页
第 9／115页
第 10／115页
第 11／115页
第 12／115页
第 13／115页
第 14／115页
第 15／115页
第 16／115页
第 17／115页
第 18／115页
第 19／115页
第 20／115页
第 21／115页
第 22／115页
第 23／115页
第 24／115页
第 25／115页
第 26／115页
第 27／115页
第 28／115页
第 29／115页
第 30／115页
第 31／115页
第 47／1 50／115页
第 51／115页
第 52／115页
第 53／115页
第 54／115页
第 55／115页
第 56／115页
第 57／115页
第 58／115页
第 59／115页
第 60／115页
第 61／115页
第 62／115页
第 63／115页
第 64／115页
第 65／115页
第 66／115页
第 67／115页
第 68／115页
第 69／115页
第 70／115页
第 71／115页
第 72／115页
第 73／115页
第 74／115页
第 75／115页
第 76／115页
第 77／115页
第 78／115页
第 79／115页

On many black hole space-times

a rXiv:g r-qc/2113v 129Oct22On “many black hole”vacuum space-times Piotr T.Chru´s ciel ∗Albert Einstein Institute †Golm,Germany Rafe Mazzeo ‡Department of Mathematics Stanford University Stanford,CA 94305USA Abstract We analyze the horizon structure of families of space times obtained by evolving initial data sets containing apparent horizons with several connected components.We show that under certain smallness conditions the outermost apparent horizons will also have several connected com-ponents.We further show that,again under a smallness condition,the maximal globally hyperbolic development of the many black hole initial data constructed in [9],or of hyperboloidal data of [19],will have an event horizon,the intersection of which with the initial data hypersurface is not connected.This justiﬁes the “many black hole”character of those space-times.1IntroductionThere is an ongoing eﬀort to construct “many black hole”solutions of the vac-uum Einstein equations numerically (see e.g.[2,25,26]and references therein).In practice this means that one numerically evolves initial data which contain trapped surfaces for as long as the computer allows.The question then arises whether the resulting space-time does indeed contain more than one black hole,or for that matter,any.Several issues arise here:a)The notion of a black hole is usually tied to the existence of a conformal completion of the space-time (but see [8]for alternative proposals).It is far from clear that the vacuum solutions,which are in principle asso-ciated with their numerical counterparts discussed in [2,25,26],possess suﬃciently controlled conformal completions,if any.b)Even assuming the issues in point a)do not occur,consider an initial dataset(S,g,K)which contains several trapped,or marginally trapped,sur-faces.If yet another trapped or marginally trapped surface S0encloses all the previous ones,then the geometry enclosed by S0is hidden from ex-ternal observers by the null hypersurface˙J+(S0).Numerical calculations of tentative radiation patterns inside J+(S0)have absolutely no relevance to the data detected at I+.Thus if one is willing to associate black hole regions to apparent horizons,then only the outermost apparent horizons are relevant.In this context a condition for a multi-black-hole space-time would be that the outermost apparent horizon has more than one component.c)In any case,the event horizon itself might have nothing to do with theapparent horizons,even the outermost ones.Under appropriate hypothe-ses,the existence of an apparent horizon implies the existence of a black hole region,but this can be much larger than the region enclosed by the outermost apparent horizon.In particular one might imagine a situation in which the outermost apparent horizon has several components,all of which are enclosed by a connected event horizon,so that the space-time contains only a single black hole region.The object of this paper is to point out that the issues raised above can be analysed in a reasonably satisfactory way for the“many Schwarzschild”initial data constructed in[9],or for the data obtained by the gluing constructions of [19,21],or for families of initial data sharing certain qualitative properties,as made precise below,similar to those of[9,19].Theﬁrst main result here is that for a rather general class of“small-data”families of black-hole space-times, the outermost apparent horizon A will have several connected components. We prove this both on the usual asymptoticallyﬂat initial data hypersurfaces and on hyperboloidal ones.For the initial data of[9]the relevant smallness condition holds when the mass parameters m i,i=1,...,I,of the individual Schwarzschild black holes are small enough as compared to the distance pa-rameters r i.For the initial data of[19]the smallness condition holds when the gluing necks are suﬃciently small.One of the features of the initial data sets of[9]is that these metrics are exactly Schwarzschild outside a compact set,and this guarantees that for any one of these,the associated maximal globally hyperbolic development(M,g) necessarily has a I+which is complete to the past.1As already indicated,the existence of I+is the usual starting point for a discussion of black hole regions. The second main result here is the proof that,for certain conﬁgurations and again for mass parameters small enough,the intersection:=˙J−(I+)∩S(1.1)E+Sof the future event horizon˙J−(I+)with the initial data hypersurface S has at least I components.(Indeed,we show that A has exactly I components andbelieve that this should also be true for E+S ;a proof of such a claim about E+Swould require complete control of the global structure of the resulting space-time,which is well beyond the range of techniques available nowadays.)2“Many black hole”initial dataThere are several constructions of families of initial data containing apparent horizons,see[1,3,4,14]and references therein.In this section we brieﬂy de-scribe three such families of“many black hole initial data”.Before doing this, it is useful to recall how apparent horizons are detected using initial data(com-pare[4]):let,thus,(S,g,K)be an initial data set,and let S⊂S be a compact embedded two-dimensional two-sided submanifold in S.If n i is the ﬁeld of outer normals to S and H is the outer mean extrinsic curvature2of S within S then,in a convenient normalisation,the divergenceθ+of future directed null geodesics normal to S is given byθ+=H+K ij(g ij−n i n j).(2.1) In the time-symmetric caseθ+reduces thus to H,and S is trapped if and only if H<0,marginally trapped if and only if H=0.In the hyperboloidal case with K ij=g ij we obtainθ+=H+2.2.1Brill-Lindquist initial dataProbably the simplest examples are the time-symmetric initial data of Brill and Lindquist.Here the space-metric at time t=0takes the formg=ψ4(dx2+dy2+dz2),(2.2)withψ=1+I i=1m i2We use the deﬁnition that gives H=2/r for round spheres of radius r in three dimensional Euclidean space.2.2The“many Schwarzschild”initial data of[9]There is a well-known special case of(2.2),which is the space-part of the Schwarzschild metric centred at x0with mass m:g= 1+mB(0,4R1),R1≥0B(0,R2),R1=0,and radii r i,i=1,...,2N,so that the closed ballsB( x i,r i) .(2.4) We assume that the x i and r i are chosen so thatΩis invariant with respect to the reﬂection x→− x.Now consider a collection of nonnegative mass parameters, arranged into a vector asM=(m,m0,m1,...,m2N),where0<2m i<r i,i≥1,and in addition with2m0<R1if R1>0but m0=0if R1=0.We assume that the mass parameters associated to the points x i and− x i are the same.The remaining entry m is explained below.Given this data,it follows from the work of[13](as pointed out in[9], compare[10])that there exists aδ>0such that if2Ni=0|m i|≤δ,(2.5) then there exists a numberm=2Ni=0m i+O(δ2)and a C∞metricˆgM which is a solution of the time-symmetric vacuum con-straint equationR(ˆgM)=0, such that:1.On the punctured balls B( x i,2r i)\{ x i},i≥1,ˆg M is the Schwarzschildmetric,centred at x i,with mass m i;2.On R3\3This will hold if the gluing regions are made small enough.It follows from(2.1)that in this setting trapped or marginally trapped surfaces are characterised by the conditionθ+=H+2≤0.(2.7) Fixing a polar coordinate r on the standard three-dimensional hyperboloid,the constant curvature−1metric takes the form1g=1+r−2.Now suppose we glue together two hyperboloidal initial data sets.From the point of view of far away observers sitting on the other side of the ensuing neck, the inner pointing normal for a geodesic sphere on one half of this conﬁgura-tion is actually pointing towards them,thus outer-pointing as far as they are concerned;hence the quantity−H+2=−2/(r2+of Section 2.2,this amounts to removing from the manifold that part which lies on the other side of the connecting necks.)We suppose furthermore that around each boundary component there is an annular ‘neck region’A i ,i =1,...,I ,equipped with a diﬀeomorphism Φi :S 2×[−1,1]→A i ,such that Φi (S 2×{−1})=(∂S )∩A i .ThusS =E (η)∪A 1∪...∪A Iis a union of manifolds with boundary,intersecting only along the submanifolds (∂E (η))∩A i =Φi (S 2×{+1})so that the A i are mutually disjoint.We call Φi (S 2×{−1})and Φi (S 2×{+1})the outer and inner boundaries of A i .The end E (η)is diﬀeomorphic to an exterior region in R 3,and we ﬁx a family of diﬀeomorphismsΨη:R 3\∪i B ( x i ,ρi (η))−→E (η),and assume that the radii of these balls ρi (η)tend to 0as ηց0.(It is only a matter of convention that we think of the annular regions as ﬁxed,whereas E (η)is identiﬁed with an η-dependent region.However,the metric g ηvaries nontrivially on each of these regions.)Our hypotheses on the metrics g ηare as follows:a)[Metric convergence on the distinguished end:]If K is any compact subset of R 3\∪i { x i }i =1,···,I ,then for some α∈(0,1)lim η→0 Ψ∗η(g η)−δ C 2,α(K )=0;here δis the Euclidean metric on R 3.b)[Mean outer convex necks and small minimising cycles:]For ηin a suﬃciently small interval (0,η0),both the inner and outer boundaries Φ−1i (S 2×{±1})of A i are mean outer convex with respect to g η;further-more,there exists a smoothly embedded sphere S i which represents thefundamental class σi ∈H 2(A i ,Z )and with area |S i |→0as η→0.Each of the three constructions outlined in Section 2.2produce familiesof metrics satisfying these hypotheses.For example,for the construction in Section 2.2,if M 0:=(m 0,m 1,...,m 2N )is a (2N +1)-tuple of nonnegative numbers and M(η)=(m (η),η M 0)is the associated mass-parameter vector from that construction,then g η:=ˆg M (η)satisﬁes both these hypotheses.Similarly,the initial data of Section 2.3satisfy the hypotheses here if ηis a suﬃciently small parameter controlling the outer radii of the I necks across which the gluing is performed.We begin with a geometric result which holds under slightly more generalhypotheses:Lemma 3.1Let g be a Riemannian metric on A =S 2×[−1,1]such that the two boundaries S 2×{±1}are mean outer convex.Fix a generator σA for H 2(A,Z ).Then any surface Σwhich is absolutely area minimising in this homology class is smoothly embedded,lies in the interior of A ,and consists of a single component of multiplicity one.Proof:The existence of a homological area-minimiserΣin the class of integral currents in a manifold with mean outer convex boundaries,and the regularity ofits support,is a standard result in geometric measure theory,cf.[30,Theorems37.2and37.7].(These arguments work equally well for domains with meanouter convex boundaries,cf.[29],and by the maximum principle,the supportof the resulting minimiser is disjoint from∂A.)In particular,the support ofΣis aﬁnite union of smooth,oriented,connected surfacesΣ1,...,ΣJ,where eachΣj appears with some non-vanishing integer multiplicity k j.Thus on the levelof homologyk1[Σ1]+...+k J[ΣJ]=σA,whereas|Σ|=|k1||Σ1|+...+|k J||ΣJ|.(3.1) We claim that the support ofΣhas only one component,and this occurswith multiplicity1.To prove this,noteﬁrst that any componentΣj dividesS2×[−1,1]into precisely two components.This may be seen by‘capping oﬀ’the boundary S2×{−1}of A by adding a3-ball;the interior of the resulting manifold A∪B3is diﬀeomorphic to R3.By the Jordan separation theorem,anysmooth,oriented,connected surfaceΣj embedded in A,hence in R3,dividesthis space into an‘inside’and an‘outside’.For example,a point p lies in theinner component if(all)generic pathsγconnecting p to the outer boundaryS2×{1}intersectΣj an odd number of times.In any case,this decomposition shows that in homology,[Σj]=±σA or else[Σj]=0for each j.If anyΣj is null-homologous,then we can obviously discard it,since it adds a positive amount to the area ofΣwithout contributing to the homology class;possibly changing orientations,we can therefore assume that each[Σj]=σA.Finally,amongst theΣj select one,Σ′,with smallest area.Then from(3.1),|Σ′|≤|Σ|,and equality holds only ifΣ′is the only component,and occurs with multiplicity1.ThusΣ′is the connected homological area-minimiser,as required.Now let us return to the more general ing this lemma,werepresent the generatorσj=[Φj(S2×{+1})]of H2(A j,Z)by a homologically area minimising surfaceΣj;according to hypothesis b),σj is also represented by the sphere S j.BothΣj and S j are smoothly embedded,connected surfaces of multiplicity one.(Since gηhas nonnegative scalar curvature,it is known [5,29]thatΣj–or indeed any stable minimal surface–must be either a sphere S2,or possibly a torus T2if gηisﬂat in a neighborhood ofΣj.)By assumption, |S j|→0,and hence|Σj|→0as well.It is proved in[18,Lemma4.1]that with the hypotheses above,for every0<η≤η0there exists a unique outermost minimal surface Sη,which is aunion of embedded stable minimal spheres of class C k+1,αif gηis of class C k,α. Furthermore,if we denote by S′the exterior of Sηin S(i.e.the unbounded component of S\Sη),then Sηis absolutely area minimising in its homology class in S′and moreover,S′is simply connected.Theorem 3.2There exists η1∈(0,η0]such that if η∈(0,η1],then S η⊂∪I i =1A i and the intersection of S ηwith each annular region A i is nonempty.Hence S ηhas at least I connected components.If we assume that there do not exist any stable minimal homologically trivial surfaces in any of the regions (A i ,g η)when ηis small enough,then S η∩A i contains exactly one component,and hence S ηhas precisely I components.Proof:Let S (0,R )denote a large sphere in R 3which contains all of the points x i ,and let Ωdenote the part of S interior to this sphere.Coherently orienting the fundamental classes σj (H 2(A j ,Z )),we have that [S (0,R )]=σ:=σ1+...+σI ,where we regard σj ∈H 2(A j ,Z )֒→H 2(S ,Z ),as induced by the inclusions A j ֒→S .From [18,Lemma 4.1],we know that S ′is diﬀeomorphic to the complement of a ﬁnite number of spheres in R 3,and hence S ηmust be homologous to S (0,R )as well,i.e.[S η]=σ.For each ηwe choose area-minimising representatives Σj (η)of σj in A j ,as in the preceding Lemma.By hypothesis a),S (0,R )is mean outer convex for g ηif ηis small enough,since it is strictly convex for the limiting Euclidean metric δ.Thus we have|Σ1|+...+|ΣI |≤|S η|≤|S (0,R )|.The ﬁrst inequality holds because ∪Σi is absolutely area minimising in its ho-mology class in S ,while the second inequality follows from the fact that S ηis absolutely minimising in its homology class in S ′.We claim that for ηsuf-ﬁciently small,S ηlies in the union A 1∪...∪A I .Granting this claim for the moment,let us prove that S ηhas at least I components.Choose for each j a smooth embedded curve γj which connects the inner boundary Φj (S 2×{−1})of A j to S (0,R ),does not intersect any of the other annular regions A i ,i =j ,and which represents the Poincar´e dual of σj in H 1(Ω,∂Ω).Then the homological intersection number of γj with [S η]equals[γj ],σ = [γj ],σ1+...+σI =1.On the other hand,if γj is in general position,then this intersection number is also computed by counting the signed geometric intersections of this curve and this surface.Therefore this geometric intersection is nontrivial,which shows that S η∩A j =∅for each j ,and hence S ηhas at least I components.To prove the claim,suppose there exists a sequence ηℓ→0such that S (ℓ):=S ηℓcontains a point q ℓ∈Ω\∪A i with q ℓ→ q ∈R 3\{ x 1,..., x I }.The interior curvature estimate for embedded stable minimal surfaces proved by Schoen [28]states that there is a uniform upper bound for norm squared of the second fundamental of S (ℓ)with respect to g ηnear q ℓ.More precisely,for any p ∈S (ℓ)with ρ( p )=min i {| p −ρi (η)|}≥δ>0for ℓsuﬃciently large,there exists a constant C >0,independent of ℓ,such that |II S (ℓ)(p )|2≤C .By standard calculus,this implies that the portion of S (ℓ)in a ball of radius ρ( p /2)around p may be written as a graph with uniformly bounded gradient over a disk of radius ρ( p )/4in T p S (ℓ).In particular,the area of S (ℓ)is uniformly bounded below by a positive constant.Applying these bounds to a ﬁnite covering of Ω\∪i B ( x i ,ρ)for any ρ>0,andthen taking a diagonal subsequence for some sequence ρj →0,we may extract asubsequence S (ℓ′)which converges to a nontrivial smoothly embedded minimal surface S (∞)in R 3\{ x 1,..., x I }.Since all of the S (ℓ′)are unions of spheres,and the number of components is uniformly bounded,the limiting surface must have ﬁnite genus.In addition,S (∞)is compact and has bounded area.We may now apply a well-known removable singularities theorem for minimal surfaces,see [6,Prop.1]for a proof,which shows that S (∞)is a nontrivial compact embedded minimal surface in R 3.Since no such surfaces exist,we have reached a contradiction.We have now proved the ﬁrst assertion,and hence that S ηhas at least one connected component in each A i .For the remaining assertion,write S i (η)=S η∩A i ,and suppose that thissurface has more than one component for some i ,i.e.S i (η)=∪J j =1S ij (η),where J >1and the S ij (η)are smooth embedded surfaces.By the same argument as in Lemma 3.1,each S ij (η)separates A i into two components.If A i contains no null-homologous stable minimal surfaces,then each component of A i \S ij (η)must contain exactly one of the two boundaries Φi (S 2×{±1}).However,the components S ij (η)are disjoint,and so if there are at least two,then any one must be contained in either the interior or exterior region of another;since their union is an outermost surface this is impossible.We conclude that S i (η)is connected.This completes the proof.In the case of data of Section 2.2the hypotheses of the second part ofTheorem 3.2are veriﬁed:Corollary 3.3Let I ∈N , M 0∈R I ,and consider initial data of Section 2.2,with M (η)=(m (η),η M 0)and g η:=ˆg M (η).If ηis small enough,than theoutermost apparent horizon is precisely the union of the Schwarzschild horizons | x − x i |=m i /2.Proof:Let A i be small annular regions around the x i ’s,chosen so that the metric is exactly Schwarzschild there,then by Theorem 3.2we have S η⊂∪i A i for ηsmall enough.The result follows now from the following fact:Lemma 3.4The only compact embedded minimal surface in a Riemannian Schwarzschild metric (2.3)is the sphere | x − x 0|=m/2.Proof:The Riemannian Schwarzschild metric is foliated by spheres of constant mean curvature.These are outer mean convex with respect to the normal pointing away from the neck.We may now apply the maximum principle.If S is any compact embedded (or even immersed)minimal surface,then there is some outermost such sphere which makes ‘ﬁrst contact’with S ,which is a contradiction.The only alternative is that S coincides with one of these spheres,and since it is minimal,it must be the central one.We may also argue using Lorentzian methods.In fact,standard causalitytheory shows that a compact embedded minimal surface within a time sym-metric Cauchy surface cannot be seen from I +,and so we may obtain the conclusion by inspecting the well known conformal diagram for the Kruskal-Szekeres extension of the Schwarzschild space-time.Using[18,Lemma4.1]one last time,each component of S i is a sphere,and it is plausible that these must agree with the homologically area-minimisingsurfacesΣi⊂A i,whose topology is a priori either that of a sphere or a torus. In each of the examples in the last section,the annular regions A i are smallperturbations of rescalings of the Riemannian Schwarzschild metric,and so onemay construct a foliation by constant mean curvature spheres using the implicitfunction theorem;from this it follows just as before that there is a unique stableminimal surface representingσi,so that S′i=Σi for all i.However,it is notclear that this is true in more general cases.There is an analogue of Theorem3.2concerning trapped surfaces for asymp-totically hyperboloidal initial data sets.Suppose that S has the same topologyas before,but that the metrics gηare asymptotically hyperboloidal.Metrics of this sort,with many necks,can be constructed as in Section2.3.We suppose that the diﬀeomorphismΨ−1ηidentiﬁes E(η)with the complement of aﬁnite number of balls in H3(or indeed any asymptotically hyperboloidal manifold with constant negative scalar curvature);we also replace the hypotheses a)and b)by:a’)[Metric convergence on the distinguished end:]If K is any compactsubset of H3\∪i{ x i}i=1,···,I,then for someα∈(0,1)lim η→0 Ψ∗η(gη)−g H C2,α(K)=0;here g H is the standard hyperbolic metric on H3.b’)[Neck boundaries with controlled mean curvature:]Forηin a suﬃciently small interval(0,η0),the outer boundariesΦi(S2×{−1})have mean curvature h<−2(with respect to the inward-pointing unit normal).We shall be using the maximum principle in the following form.Let S1and S2be two oriented,connected,embedded surfaces with constant mean curvature H1and H2,respectively.Suppose that these surfaces are tangent at a point p and their normals are equal at this point,and that in some small neighborhood S1lies on the‘interior’of S2(with respect to the normal).Then necessarily H1≥H2,and if H1=H2,these surfaces must coincide.As a slightly weaker statement,if H1and H2are now possibly variable and if H1>H2everywhere, then this one-sided tangency cannot occur.As an immediate application,let Σbe any compact oriented surface in H3which contains all of the points x i in its interior,and which has mean curvature everywhere greater than−2with respect to its outward normal.(For example,we could letΣ=S(0,R),a large sphere.)This mean curvature remains greater than−2when computed with respect to the metric gηwhenηis small enough.Hence Sηcannot be internally tangent to this sphere,and this shows that in particular Sηis contained in a ﬁxed neighborhood of the convex hull of the x i.Proposition3.5Under hypotheses a’)and b’),there is at least one(smooth, embedded,oriented)surface Sηwhich is homologous to S(0,R)⊂H3(for suf-ﬁciently large R)and which has mean curvature−2with respect to the normal pointing into the unbounded component of S\Sη,i.e.is marginally trapped.Proof:Since S is a manifold with boundary,the volume form dV gis exact,ηhence equals dΛfor some(non-unique)2-formΛ.Now deﬁne the functionalL(S)=A(S)+ SΛ,Note that changingΛalters L by a constant in each homology class,but this is irrelevant for our purposes.This functional was studied,for example,in[32], and it follows from(2.14)in that paper that if S is a smooth stationary point of L,then the mean curvature of S is equal to−2.Henceforth,let S(0,R)denote any large geodesic sphere in H3which encloses all of the points x i,and which we identify with a surface in S usingΨη.We may apply the usual geometric measure theory arguments,as follows,to conclude the existence of a smooth minimiser in the homology class of S(0,R).First, it is clear that L(S(0,R))increases without bound as R→∞.Next,when looking for a minimiser S,we may as well assume that S lies in the bounded component U of S\S(0,R),for if this were not the case,we could replace S by a homologous surface S′on which L assumes a smaller value.For example, if V is the bounded component of S\S,then∂(4This follows from convexity:if one lets S1be the portion of S outside the sphere,andΠthe projection from the exterior onto the surface of the sphere,thenΠ(S1)has less area than S1,because the Jacobian ofΠis everywhere less than1.So the sphere contribution to L is reduced;clearly the volume contribution is reduced as well.this special feature more strongly and show directly that S′is smooth;however, this is not so important for our purposes.)We may use the same removable sin-gularities theorem as before,or rather its proof,to show that each of the S′j aresmooth at the points x i.However,each S′j is compact and has constant meancurvature−2.But one could thenﬁnd a horosphere tangent to S′j,for exampleby bringing it in from inﬁnity(in any direction)until it reaches a point ofﬁrstcontact,and this would contradict the maximum principle.Hence S′j could notexist.(An alternative nonexistence proof is to note that if such S j’s existed,then Minkowski space-time would contain non-empty black hole regions.)We have now reduced to the case where Sη⊂∪A i.The same intersection theory argument as in the proof of Theorem3.2shows that each of the inter-sections Sη∩A i is nonempty,and so Sηmust have at least I components.Note that each A i contains an area-minimising surfaceΣi which is homologous tothe outer boundary,and the maximum principle implies that Sηis contained inthe region betweenΦi(S2×{−1}andΣi.One can impose various geometric conditions on the metric gηon the A iwhich would ensure that Sηhas exactly I components.A rather stringent one,which however is satisﬁed for the asymptotically hyperboloidal initial data setsof[19]forδsmall enough,is:c’)The diﬀeomorphismsΦi can be chosen,now possibly depending onη,so that each sphereΦi(S2×{t})has constant mean curvature H i(t),and that each H i is a monotone function on[−1,1]with values in some interval [−h(η),h(η)],where h(η)>2.To see that the initial data sets of§2.3have CMC foliations on each neck region,one can argue as follows.The quantitative estimates for the metric gηon these neck regions from[19,§8]show that if we scale(A i,gη)to have aﬁxed neck size(e.g.to have injectivity radius always equal to1),then this annulus is C2quasi-isometric,with constant tending quickly to1asη→0, with the neck region for the Riemannian Schwarzschild space(scaled to have the same normalisation).This latter space has a global CMC foliation,and by the implicit function theorem we can produce such a CMC foliation in any ﬁxed neighborhood of the neck.The outermost leaves of this foliation will have mean curvature±h,say,and when rescaled down to the original size,these leaves now have mean curvature±h(η),where h(η)→∞.We use this CMC foliation as follows.Consider the component S i,η= Sη∩A i.Chooseτ′andτ′′so that S i,η⊂S2×[τ′,τ′′],and such that this is the narrowest band with this property.Then S i,ηis tangent to both boundaries, and its outward unit normal at these points lies in the same direction as∂t. Denoting by H′and H′′the constant mean curvatures of those two boundaries, then the maximum principle gives that H′≥−2≥H′′.Butτ′≤τ′′and so H′=H′′andﬁnally S i,ηmust coincide with a leaf of the foliation,and hence is connected.4Sections of event horizons have at least I compo-nentsIn this section we analyze the global structure of the maximal globally hyper-bolic developments of families of initial data sharing certain overall propertieswith those of Section2.2,when the mass parameters are suﬃciently small.Thisquestion is rather diﬀerent from the one raised in the previous section,becausethe existence of apparent horizons involves only the geometry of the initial data, which is fairly well controlled.On the other hand,the notion of the event hori-zon involves the global structure of the resulting space-time,about which onlyvery scant information is available.Before proceeding further,the followingshould be said:because gravity is attractive,and because the Schwarzschildregions of the initial data of Section2.2are initially at rest with respect to eachother,one expects that those regions will“start moving towards each other”,leading either to the formation of naked singularities,or to a single black hole.In particular the resulting event horizon,if occurring,is expected to be a con-nected hypersurface in space-time.Nevertheless,the properties of the maximalglobally hyperbolic developments(M,g)of the data which we present belowlead us to conjecture that there exists no slicing of M by Cauchy surfaces Sτwhich are asymptoticallyﬂat in all their asymptotic regions and in which allthe intersections E+∩Sτare connected.This seems to be the proper way of making precise the many-black hole character of certain families of black holespace-times.While we do not prove such a conjecture,it follows from what issaid below that for some conﬁgurations there exist natural slicings of M whichdo have this property.Recall that the black hole event horizon E+is usually deﬁned asE+:=˙J−(I+;( M, g)).(4.1) Here the causal past J−is taken with respect to the conformally rescaled space-time metric g on the completed space-time with boundary M:=M∪I+. Thus,the starting point of any black hole considerations is the existence of a conformal completion at future null inﬁnity I+.In this context one usually assumes that I+satisﬁes various completeness conditions[16,17,31](compare the discussion in[8,11]).As already mentioned,for the metrics of Section2.2 past-completeness of I+is guaranteed by the fact that the initial data are ex-actly Schwarzschild outside of a compact set.However,the current understand-ing of the global properties of solutions of the Cauchy problem for the Einstein equations is insuﬃcient to guarantee any future completeness properties of the resulting I+.Nevertheless,we shall see that for some of those metrics the conformal boundary I+can be chosen suﬃciently large to the future so that deﬁned by(1.1)will have more than one component.(This feature willE+Spersist upon enlarging I+,and will therefore also hold for a maximal one.) Before passing to a proof of this fact let us point out that the existence time of the solution,deﬁned as the lowest upper bound on the existence time of all geodesics normal to S,goes to zero as the mass parameters go to zero.In or-der to see that,letΓbe a maximally extended future directed timelike geodesic。

YY01.2014高考河北文科状元英语笔记 171面胶装

第 62／171页
第 63／171页
第 64／171页
第 65／171页
第 66／171页
第 67／171页
第 68／171页
第 69／171页
第 70／171页
第 71／171页
第 72／171页
第 73／171页
第 74／171页
第 75／171页
第 76／171页
第 32／171页
第 33／171页
第 34／171页
第 35／171页
第 36／171页
第 37／171页
第 38／171页
第 39／171页
第 40／171页
第 41／171页
第 42／171页
第 43／171页
第 44／171页
第 45／171页
第 46／171页
第 163／171页
第 164／171页
第 165／171页
第 166／171页
第 167／1Байду номын сангаас1页
第 168／171页
第 169／171页
第 170／171页
第 171／171页
第 1／171页
第 2／171页
第 3／171页
第 4／171页
第 5／171页
第 6／171页
第 7／171页
第 8／171页
第 9／171页
第 10／171页
第 11／171页
第 12／171页
第 13／171页
第 14／171页
第 15／171页
第 16／171页
第 92／171页
第 93／171页
第 94／171页
第 95／171页

What Is Information Discovery About H. A. Proper

What Is Information Discovery About?H.A.ProperID Research,Groningenweg6,2803PV Gouda,The Netherlands.E-mail:E.Proper@P.D.Bruza*Distributed Systems Technology Centre,Building78,Staff House Road,University of Queensland, Brisbane4072,Australia.E-mail:bruza@.auThe Internet has led to an increase in the quantity and diversity of information available for searching.Further-more,users are bombarded by a constant barrage of electronic messages in the form of e-mail,faxes,etc. This has led to a plethora of search engines,“intelligent”agents,etc.,that aim to help users in their quest for relevant information,or shield them against irrelevant information.All these systems aim to identify the poten-tially relevant information in among a large pool of avail-able information.No unifying underlying theory for infor-mation discovery systems exists as yet.The aim of this article is to provide a logic-based framework for infor-mation discovery,and relate this to the traditionalﬁeld of information retrieval.Furthermore,the often ignored user receives special emphasis.In information discov-ery,a good understanding of a user’s(sometimes hid-den)needs and beliefs is essential.We will develop a logic-based approach to express the mechanics of in-formation discovery,while the pragmatics are based on an analysis of the underlying informational semantics of information carriers and information needs of users.If you knowwhat you are looking forwhy are you lookingand if you do not knowwhat you are looking forhow can youﬁnd it?—Old Russian proverb1.IntroductionWith the increased use of the Internet(the net)comes an increase in quantity and diversity of information carriers offered on the net.Most visible is the increased use of the World Wide rmation carriers accessible through the net include web pages,newsgroups,mailing-list ar-chives,networked databases,applications,business ser-vices,as well as indexing services.For users of the net, these carriers are at their disposal for doing business, searching for other information,educational purposes,or relaxation.The net can therefore be seen as a large market-place where information demand meets information supply. Since the net literally spans the world,the number of accessible information carriers is astronomical.This makes life rather difﬁcult for the average user who shops around to discover information carriers that fulﬁll his or her given information need.Existing Internet search tools return many information ers are still required to manually wade through large result sets in search of relevant infor-mation carriers.On top of this,most users are bombarded by a(mostly unsolicited)stream of messages in the form of e-mail, notiﬁcations of new WWW pages,news-feeds,faxes,and phone messages.This constant,and still increasing,bom-bardment of information has led to a feeling of information ers need mechanisms to shield themselves from irrelevant information.On the eve of what is sometimes called the information age,already two serious long-term problems can be identi-ﬁed:Discovering the relevant information in a huge ocean of information,and simultaneously shielding ourselves from irrelevant information coming at us.No unifying un-derlying theory for information discovery systems exists as yet.The aim of this article is to provide a framework of understanding for information discovery,and relate this to the traditionalﬁeld of information retrieval.rmation DiscoveryThe problem of discovering information carriers on the net is related to the classicalﬁeld of information retrieval*To whom all correspondence should be addressed.This work was performed while P.D.Bruza was employed at Queens-land University of Technology,Brisbane,Australia.Received June10,1997;revised August19,1998;accepted September10,1998.©1999John Wiley&Sons,Inc.JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE.50(9):737–750,1999CCC0002-8231/99/090737-14(Rijsbergen,1979).However,there are a number of clear differences as well.The information retrieval ﬁeld has traditionally focused on searching relevant documents in ﬁxed document collections;usually textual -ers are presumed to have a very clear understanding of their information need.Although it is acknowledged in,e.g.,the Cranﬁeld tests (Cleverdon,1991)that users have difﬁculty in expressing this need in a formal language,the fact that searching for information is more of an interactive process of learning and discovery is not taken into account.This latter limitation of the information retrieval ﬁeld is most apparent in the way systems are evaluated.The effective-ness of an information retrieval system is measured in terms of precision and recall 1for a ﬁxed set of queries on a standardized document collection.Information retrieval can clearly been distinguished from information discovery.For example,information discovery is performed in an open networked environment.As a consequence,the document collection is not ﬁxed.More-over,the documents,or rather information carriers,are not necessarily textual but may be of a heterogeneous or aggre-gated nature.Aggregation makes the problem of discover-ing the right information carriers to fulﬁll a user’s needs even harder.We agree with Lynch (1995)that information discovery isa complex collection of activities that can range from sim-ply locating a well-speciﬁed digital object on the networkthrough lengthy iterative research activities which involve the identiﬁcation of a set of potentially relevant networked information resources,the organization and ranking re-sources in this candidate set,and the repeated expansion or restriction of this set based on characteristics of the identi-ﬁed resources and exploration of speciﬁc resources.There has been much recent work on web-based infor-mation discovery,for example,Chen,Houston,Sewell,and Schatz (1998)and Desai (1997)are recent expositions in this rmation discovery is sometimes equated with the term resource discovery.The latter term is prevelant in digital library circles.We will adhere to the term informa-tion discovery in this article.This brings us to the information discovery paradigm.Figure 1portrays the essential aspects of the information discovery problem.On one side (the right hand side),there are information carriers as provided by the collections of information carriers that are at our disposal.These informa-tion carriers,which may be aggregated,are characterized in some way to facilitate their discovery.Note that even though we shall use the term information carrier,the car-riers actually only carry data.The data carried does not become information until a user interprets the data.Never-theless we will adhere to the term information carrier.Facing the information carriers is the user with an infor-mation need.The user expresses this need in terms of an information request;a query.The query will usually only be a crude description of the actual carrier(s)needed to fulﬁll the given information need.Therefore,it is also useful to allow further reﬁnements of this need as the search pro-ceeds.This reﬁnement process is usually referred to as relevance feedback.1Recall is the ratio of relevant retrieved objects to retrieved objects,whereas precision is the ratio of relevant retrieved objects to retrieved objects.These effectiveness criteria are generally applied in a controlled experimentalenvironment.rmation discovery paradigm.738JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE—July 1999The need for information can be caused by a number of reasons.The focus in this article is when the information need arises from a gap in the user’s knowledge.For exam-ple,the user needs to know something in order to complete a task.Relevant information is discovered and then ab-sorbed by the user toﬁll the knowledge gap.A knowledge gap may also arise out of idle curiosity.For example,some users of the Internet begin surﬁng the Internet with no speciﬁc goal and then encounter some topic that engages their curiosity in the sense they want to learn more about it.The knowledge gap can range from being fairly speciﬁc such as learning the latest price of19micron wool,to the very broad,such as learning about the theory of relativity.A speciﬁc need can usually be satisﬁed by a small collection of facts,while a broad need usually requires a wider variety of facts.Observe that during the search process users may learn more and more about their knowledge gap,and may thus discover aspects of this gap they were initially not aware of.This means that the actual information need of a user may evolve as they are exposed to new information.Given a query,a selection of information carriers that are considered relevant can be made.This selection mechanism can be compared to an automatic brokering service,match-ing demand to supply.Initially,only a limited number of the selected carriers can be shown to the user to obtain rele-vance feedback from the user to further reﬁne the query.The information discovery problem boils down toﬁnd-ing the right information carriers that willﬁll the user’s given knowledge gap.Three issues play a central role in the information discovery problem:1.Formulation of information requests;2.characterization of information carriers;3.selection of information carriers.The formulation of information requests involves two important issues.First of all,it requires some formal lan-guage in which to express the query.Secondly,a precise formulation of the true information need is required.Ob-taining such a formulation has proven to be a non-trivial task(Cleverdon,1991).Good characterization of information carriers is impera-tive for effective information discovery,as poor character-izations inevitably lead to the retrieval of irrelevant infor-mation,or the missing of relevant information.An impor-tant question is,of course,which properties to include in a characterization.A useful property to include seems to be what an information carrier is about.In addition,properties like authorship,price,medium,etc.,may be included.In the literature,standard attribute sets to characterize information carriers can be found in the context of metadata standard-ization efforts(Berners-Lee,1994;Sollins&Masinter, 1994;Weibel,Grodby,Miller,&Danierl,1995).The selection of relevant information carriers for a given query q is a well-understood problem.Forﬁnding unstruc-tured information carriers,theﬁeld of information retrieval has developed a number of retrieval models.However,this ﬁeld is still very much at the stage of simply returning information carriers which the user must then peruse in order to glean the information thatﬁlls their knowledge gap.A next step would be to support knowledge discovery.In knowledge discovery,one would try to derive the exact fragments of knowledge the user is after from the relevant information.So users would not have to read entire docu-ments,but the system would give an exact and concise answer to the user.This would require the system to some-times interpret the information found and autonomously infer new information.rmation RoutingBesides actively searching for information,users and organizations are confronted with a constant stream of elec-tronic messages.These messages range from simple notiﬁ-cations,via e-mail messages and notiﬁcations of new WWW pages,to voice mail.For this,a more passive form of information discovery is required.Incoming messages need to beﬁltered in order to partition the potentially relevant messages from the irrelevant ones.Conceptually,messages can be seen as a pointer to a freshly created information carrier(the actual body of the message).This view concurs with the view that modern software for messaging seems to take with,for example,a universal inbox for all incoming messages be it e-mail,fax, or voice messages.These messages need to be routed to the appropriate message-box(es)of the right person(s),and should then be prioritized within the message-boxes.This means that informationﬁltering,discovering relevant mes-sages in the incoming stream,involves two activities:Rout-ing and ranking.In Figure2,it is illustrated how we view this process of routing and ranking.Each incoming message passes through a layer of routing modules that select the appropri-ate message-box(es),which could be from a multitude of users.Each message-box has an associated ranking module that ranks the messages currently in the message-box using user-speciﬁed criteria.In the remainder,we shall use the term information discovery for the process of actively searching information, as discussed in subsection1.1,and the term information ﬁltering for passively discovering relevant information in an incoming message stream.The theory that will be devel-oped in this article is focused on a reasoning mechanism for relevance of information carriers.This theory will then be applicable to the selection process of information discovery, as well as the routing and ranking of messages for informa-tionﬁltering.1.3.Structure of the ArticleIn section2,the philosophical preliminaries(way of thinking)are discussed,and special attention is paid to the user in the discovery process.A generic reasoning mecha-nism for the relevance of information carriers is provided inJOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE—July1999739section3.A discussion on how user and information need speciﬁc requirements can be introduced in the discovery process is put forward in section4.As we will argue, aboutness plays an important role in determining the rele-vance of an information carrier.In section4.3,we therefore take a closer look at this property before concluding the article.2.Towards a Theory for Information DiscoveryIn developing a theory for information discovery,one mustﬁrst resolve two fundamental questions.What is an information carrier,and what is the information carried by it.The latter question of course raises the issue of what is information?This section aims to provide our view on these questions.2.1.What Is an Information Carrier?Thus far,the term information carrier has been used without actually providing a deﬁnition.In the context of the net,an information carrier can be deﬁned as:Any entity that is accessible on the net,and which canprovide information to other entities connected to the net.A deﬁnition that truly supports the open character of the net.Examples of information carriers included are:●Web pages(including free text,sound,images,and videofragments)●Free text databases●Traditional(relational,object-oriented,...)databases.Both the databases as such,as well as their instances●People’s e-mail addresses●Information about the location of non-electronic informa-tion carriers●Aggregations/groupings of information carriersA very special class of information carriers are aggre-gated information carriers.An obvious example of an ag-gregation is a database.A database in itself is an informa-tion carrier.However,it can also be seen as a collection of information carriers since each of its instances in itself carries information.Besides database-based aggregations, one can imagine creating general collections of information carriers that are strictly based on some thematic common-ality,or some common rmation carriers can obviously be present in multiple aggregations.manticsWhat information exactly is has been studied intensively before.Different authors have provided alternativetheoriesrmation routing.740JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE—July1999of information(Barwise,1989;Landman,1986;Losee, 1990,1997;Shannon,1948).The goals of this article do not include a deﬁnition of what information exactly is.We take a very modest approach to information theory.It is only assumed that information can be conceptualized as consist-ing of information particles called infons as suggested by Barwise(1989),and applied to theﬁeld of information retrieval by Huibers(1996);Huibers and Bruza(1996); Huibers,Lalmas,and Rijsbergen(1996);Rijsbergen and Lalmas(1996).The set of all such infons is denoted byᏵᏲ. Some information particles will contain more information than others,therefore it is reasonable to presume the exis-tence of an information containment relationՠwith,as intuition:If fՠg then f contains at least the same information as g.Two special elements inᏵᏲare presumed to exist.One element that represents the most information:ׅi,and one that contains the least information:Ќi.The subscript i is used to denote that these elements belong to the infon space.Infons express information about objects.Therefore,we also introduce a set of objectsᏵᏻ.In Barwise(1989)a concrete notation for infons is proposed.For example,con-sider the infon iϭ͗͗R,o1,...,o A n;1͘͘.This denotes that the objects o1...o n stand in relation R to each other. (In predicate logic,this would be denoted as R(o1,..., o n)).In this article,we are only interested in the objects that play a role in a given infon.To this end,a function Involve:ᏵᏲ3ဧ(Ᏽᏻ)is presumed to exist.By way of illustration Involve(i)ϭ{o1,...,o n}.The conceptualization of information discussed above can be captured formally by what will be termed infon space:Ᏽ᏿Supply U ͑c ͒*where for any x ʦᏵᏲand X ʕᏵᏲ,we use the following deﬁnition of infomantic closure of X :x*ഫx ʦX x *.The intuition behind infomantic closure is the following.Given some information X (represented by a set of infons),the closure is all information contained in these infons.The infomantic closure captures implicit information.In this way,we can model,for example,that in the infomantic closure of “salmon”we have the information “ﬁsh.”The information need N will be satisﬁed by a set of infons which relieves the need.The infon set is referred to as the demand of the need,denoted Demand U (N ).The subscript U reinforces the intuition that the demand is user dependent.A notion of relevance between an information need N and a carrier c can be modeled as a supply and demand of infons:c Relevant To UN$c ʦᏵᏯ͉c Relevant To U N ͖.This paints an idealistic situation.In practice,these De-mand U and Supply U functions will not be available as concrete and well-deﬁned functions.The above discussion on the nature of information dis-covery allows us to highlight a key difference between information ﬁltering and information discovery.In the case of information ﬁltering,the information needs (the informa-tion interests)involved have a more static and persistent rmation need in the context of information dis-covery tends to have a more temporary and ad hoc nature.With regards to information ﬁltering,the view can be taken that a user (or work-group)may have a number of different interests they would like to be kept informedabout.These information interests can be expressed as a set of (dormant)information needs N 1,...,N n .A set of incoming messages X can then simply be viewed as a subset of all (at that moment)known information carriers ᏵᏯ.If X is a set of such incoming messages,then for each interest N i ,the relevant messages are given by:Filter U ͑N i ,X͒N ͘.IfcNd .The second requirement states that c is preferred over d as it under-supplies the information need less than d does:͓PO2]Supply U ͑d ͒*ʚSupply U ͑c ͒*ʕDemand U ͑N ͒*fca means of gleaning a preferential ordering on situations (sets of infons).Bruza and Linder(in press)and Wonder-gem(1996)propose using a query by navigation mechanism to distill positive and negative user preferences for infor-mation.A correspondence theorem shows how these pref-erences identify a preferential ordering on the underlying set of documents.In short,preferential orderings are an emerging semantic framework for information retrieval ter in this article,we will show how they can be used to underpin a theory of information discovery.2.5.Preferential Ordering in Information FilteringIn the case of informationﬁltering,preferential ordering can be used to provide an ordering on the contents of message-boxes;i.e.,the ranking modules from Figure2. Each message-box has an associated information interest N i.This information interest will be used by the routing modules to do the actual routing,but it can also be used to provide a ranking on the messages within a message-box.2.6.SummaryThis section has presented a formal framework for de-ﬁning some essential concepts in information discovery. The user’s information need has been conceptualized as a supply and demand situation involving information parti-cles.More speciﬁcally,the information need is a demand for information rmation carriers supply infor-mation particles.Relevance is deﬁned as supply meeting demand.Additionally,it is proposed that preferential order-ings on the information carriers is a consequence of the fact that some information carriers meet the demand of the information need better than others.3.Logical Foundations of Information DiscoveryInformation discovery has its roots in theﬁeld of infor-mation retrieval.Over the last30years,a number of infor-mation retrieval models have been developed.These have mostly been numeric models conceived solely for driving the information retrieval process.Such models have ad-vanced theﬁeld of information retrieval from a practical point of view,but have not proven to be instructive in answering the more fundamental questions about informa-tion retrieval itself.This has led some researchers to turn to logic as a means toﬁnd the answers to these questions.In recent years the logic-based approach to information retrieval has clearly come to the fore as a framework for investigating such questions(Bruza,1993;Crestani&Lal-mas,1996;Huibers,1996;Lalmas,1996;Lalmas&Rijs-bergen,1992;Nie,1990;Rijsbergen,1993;Rijsbergen& Lalmas,1996).Recent surveys of the area have been pre-pared by Lalmas(1998),Lalmas and Bruza(1998),and Sebastiani(1998).These investigations appeal as they place information retrieval in a neutral framework(independent of any given retrieval model)and allow it to be described at a level of semantic detail hitherto not possible.Revealing insights have thus been gained,and as a by-product,an underlying theory for information retrieval is beginning to take shape.For the above reasons,as well as clarity of exposition,we propose a logic-based approach to information discovery. This logic will be based on the preferential ordering intro-duced in the previous section.3.1.Carrier LogicWhen judging whether a given information carrier is more preferred than another carrier,a userﬁrst needs to determine the relevance of the carriers involved.When humans judge the relevance or irrelevance of information carriers,they tend to do so in terms of proper-ties they observe the carriers to have.These properties are collectively referred to as metadata,and each of the indi-vidual properties as a metadata attribute(Weibel et al., 1995).Metadata attributes may range from fairly simple such as:Authorship,medium,pricing,quality,and location, to extremely complicated such as:The information provided (infomantics).No explicit choice on the set of metadata attributes for information carriers will be made in this article;a more general approach is adopted.The following signature format is used as a basis for for the syntax of the carrier logic:͸which requires the information carrier to be written by an author with aﬁrst name starting with an E.One relevance criterion does deserve explicit attention.This is the aboutness of information carriers,i.e.,a repre-sentation of the infomantics.Aboutness of information car-riers is at the very heart of information discovery.The underlying hypothesis is that if an information carrier is about the request from the user,then there is a high likeli-hood that the information carrier is indeed relevant to this need.For any information carrier,it is relevant to discuss its infomantics in terms of what it is about.An aboutness speciﬁc metadata signature is deﬁned as follows:͗᏷ᐃ; ;About͘where᏷ᐃis a set of keywords and Q is used to combine simple keywords into composed keywords.For example:tigerprawnoverᏵᏯϫᏯᏸ.It is not the aim ofthis article to go into detail on the deﬁnition of(and thus implicitly onthe way it is“implemented”)than limiting ourselves to one particular approach.To summarize,for a given signature¥of metadata,we have the following carrier logic:ރތ͘.3.2.Carrier ReasonerWith a carrier logic,we have a logic with which we can reason about the relevance of information carriers.How-ever,the logic is not complete without a set of formulae,a theory,which deﬁnes the semantics of the different meta-data attributes and operations.For example,for the meta-data attribute Price,and operationsϩandϪ,we would expect to hold:͑aϩb͒Ϫbϭ͑aϪb͒ϩb.Formally,a carrier reasoning system(carrier reasonerfor short)can now be deﬁned as a tuple:ރޒ⌽.3.3.Measuring the Quality of the Carrier ReasonerInspired by the recall and precision measures found intheﬁeld of information retrieval,quality measures for car-rier reasoners can be formulated.The satisfactionrelationship␾iff user U observes c to support␾in the context of information needN.In other words:Would the userﬁnd carrierc to be relevantfor query␾when trying to satisfy information needN?When␾only deals with simple metadata attributes likeprices,authors,etc.,the user-based semantics will generallybe clear and most likely be an exact match to the semanticsof the carrier reasoner.However,in the case of aboutness,these semantics become less obvious due to the subjectivityof aboutness.Therefore,this is not yet a satisfactory deﬁ-nition of the user’s view.We can go even one step further.When formulating their information need,users will expressthis in terms of some formulae taken fromᏯᏸ.Theseformulae are referred to as clues as they provide the carrierreasoner with clues on the information need of the user.3.4.Formulating Information NeedsIn the remainder of this section,we will look at howrealistic the assumption that users can formulate clear cluesabout their information need really is.We will also highlighthow a system can help users with the task of providing theseclues.The clues which a user U is able to give us about theirinformation need can be captured by a predicateHasClue UʕᏵᏺϫᏯᏸ,whereᏵᏺis the set of possible informationneeds andᏯᏸis the carrier logic language.744JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE—July1999Suppose a student is writing a report on river-pollution in The Netherlands.However,the student is not familiar with pollution at all.The student does know that Green-peace has,on numerous occasions,shown their concern about the pollution,and therefore assumes this is the case.To ﬁnd more information,the student turns to an infor-mation discovery system to learn more about pollution.All this student knows about the needed information at this stage is that it must deal about pollution of rivers in The Netherlands.So if the student’s information need related to the task of writing the report is N ,then we have:HasClue U (N ,pollution of rivers in The Netherlands).To the information discovery system,this is the ﬁrst clue about the student’s real information need.With the above predicate,and the Relevant To U predi-cate as introduced in section 2.3,the following more exact deﬁnitionofc Relevant To U N ∧N HasClue U ␾.In other words,the user would say that carrier c supports ␾,iff carrier c is about an information need N with as clue ␾.In Figure 3this deﬁnition is put in context.What should not be forgotten is that while a user is searching for infor-mation,they may already be learning more information that is relevant to their knowledge gap,possibly leading to a change in the actual information need.Using the above user-based semantics,two quality mea-surements for a carrier reasoner can be asserted.A carrier reasoner is called precise if:ifc␾and exhaustive if:ifcrelation depends on individual users and theirspeciﬁc information needs.This would imply that the qual-ity of a carrier reasoner needs to be evaluated for each individual user and information need.In practice,this is obviously very hard.An often used pragmatic way to cir-cumvent this is to assume a deﬁnitionofthatcan be used to evaluate a carrierreasoner.FIG.3.A user’s view on relevance of information carriers.JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE—July 1999745。

【高中英语】高考状元笔记：英语学习经验与大家一同分享

【高中英语】高考状元笔记：英语学习经验与大家一同分享答学子问诚然，进入高三阶段，同学们来自各方面的压力逐渐增多。

或许你对高考认识不足，或许你在上有恐惧和恐慌，或许你在规划上有迷茫，或许你在上有不足从而造成自卑和信心不足，或许你对和的过于频繁的说教有反感，或许各方面的压力更造成你的紧张从而不能做到有效地，或许你存在着偏科，比如说可能就不是你的强项。

那么，该如何缓解这些压力呢?在此，将们的一些经验与大家一同分享，希望会对大家有所帮助。

甲：与、阶段的学习相比，高三阶段的学习策略有何不同呢?何润夏(2021年福建文科状元)：如果说生活是一场马拉松长跑的话，那么，高一便是兴奋的起点，高二算是难熬的中段，高三则是最后的冲刺。

高一、高二是打基础的阶段，基础越厚，高三才能冲得越高。

凭我自己的经历，在阶段，每天学习时间的投入(包括上课时间、课外学习时间)如下：高一、高二≥10小时，高三≥12 小时。

生应当树立时间至上的观念，学会挤时间，抢时间。

在这方面，制定一份作息时间表和一份短时期的计划有较大意义。

如果说时间保证是表面上的刻苦，那么，好的便是本质上的动力。

其一，学习时间内心平气和，“静而后能安，安而后能虑，虑而后能得”，在翻开学习的同时，排除一切杂念，专心致志地思考。

其二，学会正确安排时间，高一高二应当文理相间，先课内后课外，先作业后练习。

高三我采取集中复习法，每天两门，每周完成两轮大循环，自我感觉效果不错。

三点一线总是很单调的，尤其对于高三学生，每日面对高强度的练习及总不免会有几分害怕。

这时候，“坚持” 尤为重要，我高中时习惯写日记，高三更是每天必记，现在翻看，确有触目惊心之感。

常有的内容是发泄自己的坏心情，向笔记本诉苦，然后又鼓励自己前途是光明的。

出现最多的句子便是“锲而不舍，金石可镂”，“坚持就是胜利!”大概就是凭着日记的鼓舞和心里那一份对的渴望，我每天都始终如一地坚持做应该做的事，直至高考。

学生乙：自从进入高三以来，我一直都在努力，但努力却没有得到回报。

高考状元经典英语纠错笔记(doc 11页)

高考状元经典英语纠错笔记(doc 11页)更多企业学院：《中小企业管理全能版》183套讲座+89700份资料《总经理、高层管理》49套讲座+16388份资料《中层管理学院》46套讲座+6020份资料《国学智慧、易经》46套讲座《人力资源学院》56套讲座+27123份资料《各阶段员工培训学院》77套讲座+ 324份资料《员工管理企业学院》67套讲座+ 8720份资料《工厂生产管理学院》52套讲座+ 13920份资料《财务管理学院》53套讲座+ 17945份资料《销售经理学院》56套讲座+ 14350份资料《销售人员培训学院》72套讲座+ 4879份资料【典例】—Where does your sister work, Jack?—She works in a .A. shop of clothB. cloth’s shopC. shop with clothesD. clothes shop错因分析有些考生会因为对名词作定语的用法运用不当而错选B。

其实，clothes"服装"只有复数形式，而单数形式"布店"应用shop for cloth或cloth shop，因此，根据语境可知，正确答案选D。

名词作定语时一般用其单数形式，然而，名词parents, clothes, sports 等，作定语时必须要使用其复数形式。

另外，man, woman作定语时，如果中心词是单数，则用其单数形式；如果中心词是复数，则用其复数形式。

易错点4 对不可数名词的应用判断失误【典例】—I find it very difficult to read novel you lent me last week.—Yes. It’s necessary to have good knowledge of history.A. the；不填B. a；不填C. the; aD. a; a错因分析考生可能以为第一空是特指对方上周所借给"我"的小说，故应填定冠词；第二空后面是抽象名词，为不可数名词，不填冠词，于是错选A。

looking for the brightside.x

Looking on the Bright Side:Children’s Knowledge About the Beneﬁts ofPositive Versus Negative ThinkingChristi BamfordJacksonville UniversityKristin Hansen LagattutaUniversity of California,DavisFive-to 10-year-olds (N =90)listened to 6illustrated scenarios featuring 2characters that jointly experience the same positive event (and feel good),negative event (and feel bad),or ambiguous event (and feel okay).Afterward,one character thinks a positive thought and the other thinks a negative thought.Children pre-dicted and explained each character’s emotions.Results showed signiﬁcant development between 5and 10years in children’s understanding that thinking positively improves emotions and thinking negatively makes one feel worse,with earliest knowledge demonstrated when reasoning about ambiguous and positive events.Individual differences in child and parental optimism and hope predicted children’s knowledge about thought–emotion connections on some measures,including their beliefs about the emotional beneﬁts of think-ing positively in negative situations.Mary Engelbreit,renowned children’s book illustra-tor and artist,once wrote,‘‘If you don’t like some-thing,change it.If you can’t change it,change the way you think about it’’(Engelbreit &Regan,2006).Most adults at least tacitly agree that thoughts are powerful and that it is far preferable to think positively than to think negatively.Self-help books,inspirational quotes,and motivational speakers also emphasize the importance of men-tally reframing negative situations into positive ones (Carnegie,1936;Peale,1952).The beneﬁts of positive thinking on emotions and overall subjec-tive well-being are so well documented by both psychological research (e.g.,Marshall &Brown,2006;Scheier &Carver,1992;Snyder,Cheavens,&Michael,2005)and anecdotal evidence that know-ing we should think positively goes almost without saying,though the practice of actually thinking positively is much more difﬁcult.Even though the importance of positive thinking seems obvious to many adults,it is not necessarily evident to young children.Knowledge about the mind and its relation to emotions develops signiﬁ-cantly between the ages of 5and 10(Band &Weisz,1988;Flavell,Green,&Flavell,1995;Lagattuta,2007,2008;Lagattuta &Wellman,2001;Pons,Harris,&de Rosnay,2004),so it might be espe-cially difﬁcult for young children to understand that thinking positively can make a person feel good,especially when the situation itself is nega-tive.The present study adds to this growing litera-ture by investigating developmental changes during this age period in children’s knowledge of the effects of thinking positively versus negatively on people’s emotions.In the present study,children predicted the emotions of two story characters who experience positive (e.g.,playing at the beach),neg-ative (e.g.,getting hurt),or ambiguous events (e.g.,meeting a new teacher).For each trial,one charac-ter thinks a positive thought and the other character thinks a negative thought.If children understand the emotional effects of thinking positively versus negatively,then they should predict more intensely positive emotions for the character thinking a posi-tive thought and more intensely negative emotions for the character thinking a negative thought,even though both characters face the same event.This research was submitted in partial fulﬁllment of the requirements for a doctoral degree at University of California,Davis (CB).Portions of this research were presented at the bien-nial meeting for the Society for Research in Child Development,2009.Support for the preparation of this article for KHL was pro-vided by a grant from the National Science Foundation (0723375).We thank our research assistants Kristine Aphugh,Millie Copara,Yasmin Hashemzadeh,Siera Levinson,and Malay Phoong for their assistance with this project,and Theresa Wong and Jonah Cox for drawing the illustrations for the stories.We especially thank Dr.Liat Sayfan for her advice during all phases of the project,as well as Drs.Simona Ghetti and Robert Emmons for commenting on an early draft of the manuscript.Correspondence concerning this article should be addressed to Christi Bamford,Department of Psychology,Jacksonville Univer-sity,2800University Blvd.N.,Jacksonville,FL 32211.Electronic mail may be sent to cbamfor@.Child Development,March ⁄April 2012,Volume 83,Number 2,Pages 667–682Ó2011The AuthorsChild Development Ó2011Society for Research in Child Development,Inc.All rights reserved.0009-3920/2012/8302-0023DOI:10.1111/j.1467-8624.2011.01706.xPrevious research indicates that children develop an increased understanding of connections between the mind and emotions during the preschool and primary school years.Three-to4-year-olds can pre-dict and explain emotions that are typical of situa-tions,such as feeling happy at a birthday party(see Harris,1989).Preschoolers also show knowledge about how desires mediate between situations and emotions:Getting what you want elicits positive emotions and not getting what you want makes you feel bad(Wellman&Banerjee,1991).By the ages of5and6,children demonstrate greater awareness of connections between thinking and feeling.They frequently explain negative emotions such as worry,fear,and sadness as caused by inter-nal cognitive states such as beliefs(Pons et al., 2004;Sayfan&Lagattuta,2008)as well by thinking about events from the past or the future(Lagattuta &Wellman,2001;Lagattuta,2007).Five-to6-year-olds also understand the reverse relationship:that feeling sad or mad about a previous event(e.g.,los-ing a favorite toy)can impair how well a person can currently concentrate or think(Amsterlaw,Lag-attuta,&Meltzoff,2009).Interestingly,children exhibit earlier knowledge about how negative thoughts can make a person feel bad in a positive situation than about how thinking positively can make a person feel good in a negative situation(Lagattuta&Wellman,2001). Similarly,young children demonstrate earlier knowledge about the potential impairing effects of negative emotions on thinking than the facilitative effects of positive emotions(Amsterlaw et al., 2009).Linguistically,young children also use a greater number of terms to describe negative versus positive emotions and they talk more frequently about causes of negative versus positive emotions (Lagattuta&Wellman,2002).More generally,nega-tive emotions and negative events are highly salient to young children(Vaish,Grossman,&Woodward, 2008),arguably making it difﬁcult for them to com-prehend that a person could feel good in a clearly negative situation.It is typically not until age7years or later that children frequently suggest using mental strategies (e.g.,distraction,mental reframing)to cope with negative situations(Altshuler&Ruble,1989;Bam-ford&Lagattuta,2010;Sayfan&Lagattuta,2009). At these older ages,children also demonstrate greater understanding that people can interpret the same situation in different ways(Carpendale& Chandler,1996)and have different thoughts about the same events(Flavell et al.,1995,2000).Older children can also better introspect on their own thought processes,and they show greater apprecia-tion that intrusive negative thoughts can be difﬁcult to control compared to younger children(Flavell et al.,1995,2000).Despite this strong research base on children’s reasoning about mind–emotion connections,there has not been systematic investigation of children’s beliefs about how here-and-now mental reframing of negative,positive,or ambiguous events can change a person’s emotions.Based on previous ﬁndings,we expect that whereas5-and6-year-olds will exhibit consistent knowledge about the emo-tional disadvantage of pessimistic thinking(i.e., thinking negative thoughts can make a person feel worse),it will not be until the ages of7–8years that children will demonstrate understanding that thinking positively in clearly negative situations can lead to improved emotions.Potentially,chil-dren may reveal strongest knowledge about thought–emotion connections in ambiguous situa-tions.That is,because ambiguous situations do not elicit unequivocal emotions,the salience of charac-ters’thoughts may be augmented.Age-related changes in children’s understanding of thought–emotion connections are expected to appear both in their predictions and in their explanations.That is, with increasing age,we expect children to explain emotions less often in relation to the objective situa-tion and more often in relation to the content of the person’s ongoing thoughts(see also Lagattuta& Wellman,2001;Sayfan&Lagattuta,2008,2009).Although children’s knowledge about the emo-tional beneﬁts of positive thinking is likely con-nected to their developmental level,individuals can vary in their overall optimistic or pessimistic approach to life.Adults higher in optimism and hope have better health,greater longevity,higher quality of life,less depression,more motivation, and more effective coping skills(Carver,Scheier,& Segerstrom,2010;Snyder,1994,2000)compared to adults lower in optimism and hope.Similarly,opti-mistic and hopeful older children(age8to adoles-cence)exhibit more effective coping styles,have greater self-esteem,do better in school,and have lower rates of depression(e.g.,Ey et al.,2005; Fischer&Leitenberg,1986;Nolen-Hoeksema,Girgus, &Seligman,1992;Snyder,2006;Snyder,McDer-mott,Cook,&Rapoff,1997;Seligman,Reivich, Jaycox,&Gillham,1995)compared to those lower in optimism and hope.In the present study,optimism refers to the general expectancy that things will turn out for the best(Carver&Scheier,2002),and hope refers to an individual’s beliefs about his or her capacity toﬁnd pathways to reach goals as well668Bamford and Lagattutaas his or her sense of agency to accomplish these goals(Snyder,2000).Although optimism and hope are typically correlated(at least in adults),they can predict different outcomes,and,as a result, researchers typically do not combine or aggregate these two measures(Bailey,Eng,Frisch,&Snyder, 2007;Bryant&Cvengros,2004).Given the limited previous research on unique correlates of optimism and hope in young children,we did not have spe-ciﬁc a priori hypotheses about whether each mea-sure would differentially predict performance on the experimental tasks.We assessed children’s level of optimism and hope(child report and parent report)to investigate whether these dispositions predict individual dif-ferences in children’s reasoning about the effects of mental reframing on emotions.In particular,per-haps children who more typically expect positive things to happen(optimism)or who believe that they can readilyﬁgure out solutions to problems or setbacks(hope)will more often predict improved emotions for characters thinking positively in nega-tive situations.Children higher in optimism and hope may also more frequently explain differences in the way two people feel as caused by the focus of their thoughts,particularly in negative situations. Children’s reasoning about negative events is cen-tral to our hypotheses regarding individual differ-ences because these are the situations where optimistic reframing enables positive emotion regu-lation.Parental levels of optimism or hope may also predict children’s knowledge about the emotional outcomes of positive thinking,due to parental modeling or direct teaching of positive reframing (Bandura,1977;Gillham,Reivich,Jaycox,&Selig-man,1995)or even to potential inherited tendencies that bias them toward or positive(or negative) interpretations of events(e.g.,temperament or per-sonality traits;Schulman,Keith,&Seligman,1993). We expect such individual differences to be strong-est for interpreting negative events because these are the ones in which it is most difﬁcult to use posi-tive thinking,due to the salience of the negative situation and its impact on emotions.Thus,chil-dren with parents higher in hope and optimism may demonstrate greater understanding of the beneﬁts of thinking positively,particularly for neg-ative events where thinking positively can promote coping.In summary,the current study investigated5-to 10-year-olds’knowledge that mentally reframing events in negative or positive ways can inﬂuence emotions:Two people can experience different emotions due to differences in the way they think about the same situation.These ages were targeted because previous studies have identiﬁed signiﬁcant developmental changes in understanding of mind and emotions during this age span.We hypothe-sized that between the ages of5and10children would more consistently predict emotions in line with the thought and not the situation,judge greater differences between the emotions of charac-ters thinking different thoughts in the same situa-tion,and more often explain emotions in relation to characters’internal mental states,including their thoughts.Reasoning about the impact of thoughts on emotions was expected to develop earlier for ambiguous and positive events(by the ages of5–6years)and later for negative events(age7years and older).Finally,we hypothesized that individ-ual differences in child and parental optimism and hope would predict children’s emotion predictions and explanations,particularly their reasoning about the emotional beneﬁts of thinking optimistically in negative situations.MethodParticipantsParticipants included90children between the ages of5and10years who lived in or near a California university town.They were divided into three groups:5-and6-year-olds(N=30; range=5.03–6.89years;M=5.94years),7-and8-year-olds(N=30;range=7.16–8.92;M=7.99),and 9-and10-year-olds(N=30;range=9.04–10.89; M=9.65)based on previous research on children’s understanding of mind–emotion connections.Each age group consisted of15males and15females. For ease of discussion,these three groups will be referred to as6-year-olds,8-year-olds,and10-year-olds based on the rounded mean age for each group.Participants were recruited through lists of fami-lies that had participated in previous research stud-ies,local farmer’s markets,word of mouth,and researchﬂyers and advertisements.Children were 80%Caucasian,9%mixed ethnicity,and11%from other groups(African American,Latino,Asian, Native American,and Middle Eastern).Parents reported the highest degree for the child’s father and mother:45.5%of fathers and37.6%of mothers had graduate degrees,41.1%of fathers and42.2% of mothers had college degrees,and11.4%of fathers and18.2%of mothers had a high school diploma(2.2%did not report).Children’s Knowledge About Positive and Negative Thinking669Materials and ProceduresMental Reframing Reasoning TaskChildren were introduced to a7-point pictorial Likert emotion scale.The pictures on the scale ran-ged from a very sad face(0)to a neutral face(3)to a very happy face(6),and the experimenter labeled them as very bad,medium bad,a little bad,ok(not good or bad),a little good,medium good,and very good.The experimenter veriﬁed that the child could accu-rately identify the emotions on the scale by saying each emotion label(e.g.,‘‘very bad’’)in random order and asking the child to point to its picture referent.Training continued until the child cor-rectly identiﬁed all points on the scale.Children were then presented with a series of six stories with two illustrated cards(4.5in.·5.5in. per story.Each scenario involved two protagonists of the same gender who experienced the same event.Two trials depicted a negative event(two girls break their arms,a cat knocks over two boys’glasses of milk right next to their homework)that resulted in both characters feeling‘‘medium bad’’(though neither of them chooses literally to‘‘cry over the spilt milk’’).Two trials depicted a positive event(two girls go on a vacation to the beach,two boys get the pet puppy they wanted)that caused both characters to feel‘‘medium good.’’Two trials featured an ambiguous event(two girls meet a substitute teacher, two boys learn their mother is having a baby)which lead both characters to feel‘‘ok,not good or bad.’’We explicitly told participants how both characters felt after each event to standardize the baseline emo-tion across all scenarios—making it equally intense (i.e.,‘‘medium’’)across all positive and negative scenarios and clearly neutral for the ambiguous sce-narios.Figure1illustrates one of the negative events.See online supporting information Appen-dix S1for the speciﬁc wording for each scenario.After experiencing the same baseline emotion, the two characters had divergent thoughts.One character had a positive thought,one that framed the event in a positive light(e.g.,‘‘Now we’ll get new casts and all our friends will get to write their names and draw pictures on them!’’),and the other character had a negative thought,one that framed the event in a negative light.(e.g.,‘‘Now we’ll have to wear itchy casts and it will be hard to play with our friends’’)Each character’s thought was described verbally as the experimenter placed a blank thought bubble(see Flavell,Green,&Flavell, 1993)above each character’s head.After the presen-tation of each thought,the participant was asked ‘‘How does[character]feel right now?’’and‘‘Why does[character]feel that way?’’The presentation of the thoughts(negative vs.positiveﬁrst)was coun-terbalanced across stories and across participants. Participants predicted each character’s emotions using the7-point pictorial Likert scale.If the child just pointed to the face on the scale,the experi-menter provided its label to verify that it was the emotion the child intended to choose.Finally,after predicting and explaining each character’s emotion,children were asked the direct comparison question,‘‘Why does character X feel bet-ter than(or the same as)character Y right now?’’The exact format of this question was determined by the child’s emotion predictions for each charac-ter.We included this comparison question because recent research has shown that asking children to explain between-person emotion differences signiﬁ-cantly increases the frequency at which children explain emotions in relation to internal mental states such as thoughts,beliefs,and desiresAmy and Erica are riding a big two-person bike in front of their house.Amy thinks“We broke ourarms!Nowwe’ll have towear itchycasts and itwill be hard toplay with ourfriends.”Suddenly, their bike falls overand they both break their arms!Amy and Erica both feel badright now [pointing to mediumbad on scale], and they bothbroke their arms.Erica thinks“We broke ourarms!Now we’llget cool castsand our friendscan write theirnames and drawpictures onthem.”Figure1.Reframing story example:Negative event.670Bamford and Lagattutacompared to when children are asked to explain each character’s emotions separately(see Sayfan& Lagattuta,2008).Coding of explanations.Children’s emotion expla-nations for each character were coded dichoto-mously(0,1)for the presence or absence of two central categories of explanations:(a)situation expla-nations described the situation as the cause for the emotion(e.g.,‘‘Because she has a cast’’;‘‘The cat spilled milk everywhere’’),and(b)internal mental state explanations made explicit reference to an inter-nal mental state(e.g.,thought,desire,preference) as the reason for the character’s emotion(e.g.,‘‘He likes cleaning’’;‘‘She knows that casts aren’t that bad’’).Internal mental state explanations were additionally coded as thinking explanations if the child made explicit reference to thinking or to the content of thoughts as the cause for the emotion (e.g.,‘‘Because she thinks she will have an itchy cast’’;‘‘He thinks it’s going to take a long time to clean up the milk’’),including making direct refer-ence to the beneﬁts of positive thinking or the costs of negative thinking(e.g.,‘‘Because she’s thinking of the good side of things’’).These categories were chosen because we were interested in developmen-tal changes in children’s understanding of internal mental states(especially thoughts)versus situations as determining emotions.A child’s explanation for any particular story trial could be coded into multi-ple categories.For example,‘‘Because the cat spilled the milk,and he did not want the cat to do that’’would be coded as both a situation and as an internal mental state explanation.Explanation scores were summed across the two trials of each Event Valence·Thought Valence story combina-tion,yielding a minimum score of0and a maxi-mum score of2.Children’s responses to the direct comparison question were also coded for the presence or absence(1,0)of the same explanation categories (situation,internal mental state,thinking).Scores were summed across the two trials of each story event valence(negative event,positive event, ambiguous event),so that scores ranged from 0to2.Three coders(Coders A,B,and C)jointly coded children’s explanations for10%of the trials.Chil-dren’s explanations for the remaining90%of trials were independently coded by two coders.Reliabil-ity between Coder A and Coder B across categories ranged from j=.79to j=.97(p s<.001).Reliabil-ity between Coders A and C ranged from j=.78to j=.96(p s<.001).Reliability between Coders B and C ranged from j=.80to j=.94(p s<.001).All disagreements were resolved by group discus-sion prior to analysis.Individual Difference MeasuresWe included six measures as potential predictors of individual differences in reasoning about thought–emotion connections for the mental refra-ming reasoning task.Two were child reports of their own optimism and hope(designated by CR), two were parent reports of the child’s optimism and hope(designated by PR),and two were parent self-reports of their own optimism and hope(desig-nated by PSR).Child self-report measures.Children completed modiﬁed versions of the Youth Life Orientation Test(CR-YLOT;Ey et al.,2005)and the Young Children’s Hope Scale(CR-YCHS;Berkich,1996). Because neither scale has been validated for chil-dren under age8due to problems administering paper-and-pencil questionnaires to young children, we modiﬁed the scales in several ways.First,we read the items to children individually.Second,we changed some of the complex wording to make the items more comprehensible to young children. Appendix S2(available as online supporting infor-mation)displays the original and revised wordings for the CR-YLOT,and Appendix S3(available as online supporting information)displays the origi-nal and modiﬁed items for the CR-YCHS.We also changed the response scales.The origi-nal YLOT had a4-point Likert scale including the items true for me,sort of true for me,sort of not true for me,and not true for me,while the YCHS had a 3-point scale with the choices never,sometimes,and always.We replaced these scales with a single 4-point pictorial Likert scale that depicted a series of four boxes that ranged from empty to full,and we asked the participants to point to whether the statement on each measure was like them none of the time(empty box),a little of the time(box was one-third full),a lot of the time(box was two-thirds full),or all of the time(box was completelyﬁlled). Children were trained to use the scale by asking about their preferences and daily habits(e.g.,‘‘I eat pizza for dinner’’;‘‘I wear clothes to school’’;‘‘I like to eat worms’’).Training continued until the child used all4points on the scale and could accurately identify each scale point.Parent-report child measures.The two modiﬁed CR measures were reworded to third person(‘‘My child…’’)for parents to report on their children’s optimism to the best of their knowledge(PR-YLOT and PR-YCHS).Response items were given in aChildren’s Knowledge About Positive and Negative Thinking6714-point(nonpictorial)Likert scale ranging from none of the time(1)to all of the time(4).Parent self-report measures.Parents also self-reported their own levels of optimism and hope by completing the Life Orientation Test Revised(PSR-LOT-R;Scheier,Carver,&Bridges,1994),which is the adult version of the CR-YLOT,and the Hope Scale(PSR-DHS;Snyder et al.,1991),which is the adult version of the CR-YCHS.General ProcedureTesting took place at in a quiet room at a univer-sity laboratory.Children responded verbally to items for the CR-YLOT and CR-YCHS prior to com-pleting the mental reframing reasoning task.This order was chosen so that children’s predictions and explanations regarding the effects of positive versus negative thinking would not affect their willingness to admit that they sometimes have negative thoughts on the self-report measures.Children also participated in a separate experimental task that was part of a different study.The two experimental tasks were counterbalanced across participants for order of presentation.Parents completed the ques-tionnaires in a separate room.Children received a toy and a$10gift card for their participation.The session lasted approximately30min.ResultsAll children successfully completed the training of the response scales for the self-report measures of optimism and hope(pictorial Likert scale)and for the mental reframing reasoning task(emotion scale).Therefore,we do not exclude data from any children.The parentﬁlling out the parent-reported questionnaires was the mother the majority of the time(84%of participants).Data are missing for parent-reported measures for3participants due to parents electing not toﬁll out these surveys.For all analyses,we set the signiﬁcance level(alpha)at.05, we used Tukey’s honestly signiﬁcant difference post hoc comparisons to follow-up signiﬁcant main effects,and we used simple effects tests to follow-up signiﬁcant interactions.Mental Reframing Reasoning Task Preliminary AnalysesTo conﬁrm that children treated the two trials of each Event Valence·Thought Valence combination similarly,we ran paired t tests(two-tailed)for chil-dren’s emotion predictions.The only signiﬁcant dif-ference was for emotions following negative thoughts in the positive events.Children predicted that a person thinking negatively in the‘‘beach’’scenario would feel worse(M=2.06,SD=1.26) than a person thinking negatively after getting a new puppy(M=2.69,SD=1.18),t(89)=)4.10, p<.001.Nevertheless,emotion predictions after thinking negative thoughts in these two positive scenarios were still positively correlated,r(88)=.29, p<.01.Thus,we collapsed data for the two trials of each Event Valence·Thought Valence combina-tion.Children’s similar responding across the two trials of each Event Valence·Thought combination further verify that telling children the characters’baseline emotions prior to thinking(i.e.,positive events=‘‘feel medium happy,’’negative events=‘‘feel medium bad,’’ambiguous events=‘‘feel okay,not good or bad’’)was successful in equaliz-ing the two scenarios of each type.Emotion PredictionsEmotion predictions for each character were scored from0(very bad)to6(very good)for each trial.We analyzed these predictions in two ways. First,we looked at their mean emotion ratings across the two trials of each story Event Valence·Thought Valence combination(e.g.,thinking posi-tive thoughts in negative situations).Next,we examined the mean emotion difference between the emotion ratings for characters thinking positive thoughts versus negative thoughts within events of the same valence(across the two trials of each event valence).Whereas,the former provides an absolute measure of how positively or negatively children judge that characters will feel after think-ing each kind of thought in each kind of situation, the latter provides a relative measure of children’s judgments about the degree to which a person’s emotional reaction to a positive,negative,or ambig-uous situation can be modiﬁed by reframing the event positively versus negatively.Preliminary analyses showed no signiﬁcant main effects or interactions for gender,so this factor was removed.Mean emotion ratings.A3(age group:6-,8-,and 10-year-olds)·3(event valence:positive,negative, ambiguous)·2(thought valence:optimistic,pessi-mistic)repeated measures analysis of variance (ANOVA)for children’s emotion predictions resulted in a main effect for event valence,F(2,174) =139.45,p<.001,g p2=.62;for thought valence, F(1,87)=1165.39,p<.001,g p2=.93;and for age, F(2,87)=5.02,p<.01,g p2=.10,qualiﬁed by an672Bamford and Lagattuta。

extended

David J. Salmond QinetiQ Cody Technology Park Farnborough, Hants GU14 0LX United Kingdom djsalmond@
relative to the line-of-sight (LOS) between the observer and the target. The incorporation of the target (elliptical) shape parameters into the state vector, however, poses some diﬃculties for the resulting nonlinear ﬁltering problem, as reported in [11]. In this study we examine in more depth this nonlinear ﬁltering problem using the Cram´ er-Rao lower bound (CRLB) [9, Ch.4] as an analysis tool. Then an Extended Kalman ﬁlter (EKF) and an Unscented Kalman ﬁlter (UKF) are developed for this application. The error performance of these two nonlinear ﬁlters is compared to the theoretical CRLB and their consistency is analysed by normalised estimation error squared (NEES) [2]. It is important to emphasise that the purpose of using target extent measurements in a tracking system is not so much to improve the track accuracy, but more importantly to estimate a very useful target feature: the major and minor axes of its (elliptical) shape. These target shape estimates are extremely useful both for track maintenance [11] (e.g. in dense target/false alarm environments) and for target identiﬁcation. The remaining part of the paper is organised as follows. Section 2 presents a mathematical formulation of the nonlinear ﬁltering problem for tracking an ellipsoidal target using measurements of its position and extent. Section 3 describes the derivation of the theoretical CRLB for this nonlinear ﬁltering problem. Section 4 is devoted to an analysis of CRLB as a function of measurement accuracy, target-observer geometry and the inﬂuence of prior knowledge of shape parameters. Section 5 presents an analysis of EKF and UKF error performance. Section 6 summarises the main results of this study.

A note on Cartan matrices for symmetric groups

J rn B. Olsson Christine Bessenrodt
Abstract
1 Introduction
In general, by a theorem of R. Brauer ( 4], Theorem IV.3.11), the elementary divisors of the p-Cartan matrix of a nite group G, p a prime, are known to be the orders of the p-defect groups of the p-regular conjugacy classes of G. In case of the symmetric group Sn explicit calculations were made in 7]. These included formulae for the multiplicity of any given power of p as an elementary divisor in the p-Cartan matrix of Sn and in the Cartan matrix of any p-block of Sn. From this information it should in principle be possible to compute the determinants of these Cartan matrices, but the calculations for speci c blocks appear to be rather complicated. In this paper we show that these determinants may be computed by very simple formulae using generating functions, see Theorems 3.3 and 3.4. A P special r^le is played by the generating function T (q) = n t(n)qn , where o t(n) is de ned as the number of divisors of the integer n.

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。