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      9天前 北大袁萌最新黑洞M87*的发现证实广义相对论的正确性

黑洞M87*的发现证实广义相对论的正确性
认真阅读、读懂广义相对论的人,恐怕不多。在撰写广义相对论时,爱因斯坦非常谨慎,但是,字里行间露出“理论自信”,值得我们学习。
早在一百多年前,爱因斯坦论证了光纤在引力场中会发生偏转。在人类文明发展史上,这是破天荒第一次。由此,预见到宇宙黑洞存在的必然性。
本文附件完整地记录了爱因斯坦逻辑推理的缜密思想,值得认证阅读。
    广义相对论开启了二十世纪人类社会大发展的大门。
袁萌  陈启清  4月14日
附件:广义相对论第22节如下:
A Few Inferences from the General Principle of Relativity
The considerations of Section 20 show that the general principle of relativity puts us in a position to derive properties of the gravitational field in a purely theoretical manner. Let us suppose, for instance, that we know the space-time " course " for any natural process whatsoever, as regards the manner in which it takes place in the Galileian domain relative to a Galileian body of reference K. By means of purely theoretical operations (i.e. simply by calculation) we are then able to find how this known natural process appears, as seen from a reference-body K1 which is accelerated relatively to K. But since a gravitational field exists with respect to this new body of reference K1, our consideration also teaches us how the gravitational field influences the process studied.
For example, we learn that a body which is in a state of uniform rectilinear motion with respect to K (in accordance with the law of Galilei) is executing an accelerated and in general curvilinear motion with respect to the accelerated reference-body K1 (chest). This acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to K. It is known that a gravitational field influences the movement of bodies in this way, so that our consideration supplies us with nothing essentially new.
However, we obtain a new result of fundamental importance when we carry out the analogous consideration for a ray of light. With respect to the Galileian reference-body K, such a ray of light is transmitted rectilinearly with the velocity c. It can easily be shown that the path of the same ray of light is no longer a straight line when we consider it with reference to the accelerated chest (reference-body K1). From this we conclude, that, in general, rays of light are propagated curvilinearly in gravitational fields. In two respects this result is of great importance.
Part II: The General Theory of Relativity
Albert Einstein 66
In the first place, it can be compared with the reality. Although a detailed examination of the question shows that the curvature of light rays required by the general theory of relativity is only exceedingly small for the gravitational fields at our disposal in practice, its estimated magnitude for light rays passing the sun at grazing incidence is nevertheless 1.7 seconds of arc. This ought to manifest itself in the following way. As seen from the earth, certain fixed stars appear to be in the neighbourhood of the sun, and are thus capable of observation during a total eclipse of the sun. At such times, these stars ought to appear to be displaced outwards from the sun by an amount indicated above, as compared with their apparent position in the sky when the sun is situated at another part of the heavens. The examination of the correctness or otherwise of this deduction is a problem of the greatest importance, the early solution of which is to be expected of astronomers.1)
In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlinlited domain of validity ; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light).
Since it has often been contended by opponents of the theory of relativity that the special theory of relativity is overthrown by the general theory of relativity, it is perhaps advisable to make the facts of the case clearer by means of an appropriate comparison. Before the development of electrodynamics the laws of electrostatics were looked upon as the laws of electricity. At the present time we know that electric fields can be derived correctly from electrostatic considerations only for the case, which is never strictly realised, in which the electrical masses are quite at rest relatively to each other, and to the co-ordinate system. Should we be justified in saying that for this reason electrostatics is overthrown by the fieldequations of Maxwell in electrodynamics ? Not in the least. Electrostatics is contained in electrodynamics as a limiting case ; the laws of the latter lead directly to those of the former
Part II: The General Theory of Relativity
Albert Einstein 67
for the case in which the fields are invariable with regard to time. No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.
In the example of the transmission of light just dealt with, we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes, the Iaws of which are already known when a gravitational field is absent. But the most attractive problem, to the solution of which the general theory of relativity supplies the key, concerns the investigation of the laws satisfied by the gravitational field itself. Let us consider this for a moment.
We are acquainted with space-time domains which behave (approximately) in a " Galileian " fashion under suitable choice of reference-body, i.e. domains in which gravitational fields are absent. If we now refer such a domain to a reference-body K1 possessing any kind of motion, then relative to K1 there exists a gravitational field which is variable with respect to space and time.2) The character of this field will of course depend on the motion chosen for K1. According to the general theory of relativity, the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. Even though by no means all gravitationial fields can be produced in this way, yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. This hope has been realised in the most beautiful manner. But between the clear vision of this goal and its actual realisation it was necessary to surmount a serious difficulty, and as this lies deep at the root of things, I dare not withhold it from the reader. We require to extend our ideas of the space-time continuum still farther.
 
 
Footnotes
Part II: The General Theory of Relativity
Albert Einstein 68
1) By means of the star photographs of two expeditions equipped by a Joint Committee of the Royal and Royal Astronomical Societies, the existence of the deflection of light demanded by theory was first confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix III.)
2) This follows from a generalisation of the discussion in Section 20
Part II: The General Theory of Relativity
Albert Einstein 69
Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
 
Hitherto I have purposely refrained from speaking about the physical interpretation of space- and time-data in the case of the general theory of relativity. As a consequence, I am guilty of a certain slovenliness of treatment, which, as we know from the special theory of relativity, ––––is far from being unimportant and pardonable. It is now high time that we remedy this defect; but I would mention at the outset, that this matter lays no small claims on the patience and on the power of abstraction of the reader.
We start off again from quite special cases, which we have frequently used before. Let us consider a space time domain in which no gravitational field exists relative to a referencebody K whose state of motion has been suitably chosen. K is then a Galileian referencebody as regards the domain considered, and the results of the special theory of relativity hold relative to K. Let us supposse the same domain referred to a second body of reference K1, which is rotating uniformly with respect to K. In order to fix our ideas, we shall imagine K1 to be in the form of a plane circular disc, which rotates uniformly in its own plane about its centre. An observer who is sitting eccentrically on the disc K1 is sensible of a force which acts outwards in a radial direction, and which would be interpreted as an effect of inertia (centrifugal force) by an observer who was at rest with respect to the original reference-body K. But the observer on the disc may regard his disc as a reference-body which is " at rest " ; on the basis of the general principle of relativity he is justified in doing this. The force acting on himself, and in fact on all other bodies which are at rest relative to the disc, he regards as the effect of a gravitational field. Nevertheless, the space-distribution of this gravitational field is of a kind that would not be possible on Newton's theory of gravitation.1) But since the observer believes in the general theory of relativity, this does not
Part II: The General Theory of Relativity
Albert Einstein 70
disturb him; he is quite in the right when he believes that a general law of gravitation can be formulated- a law which not only explains the motion of the stars correctly, but also the field of force experienced by himself.
The observer performs experiments on his circular disc with clocks and measuring-rods. In doing so, it is his intention to arrive at exact definitions for the signification of time- and space-data with reference to the circular disc K1, these definitions being based on his observations. What will be his experience in this enterprise ?
To start with, he places one of two identically constructed clocks at the centre of the circular disc, and the other on the edge of the disc, so that they are at rest relative to it. We now ask ourselves whether both clocks go at the same rate from the standpoint of the nonrotating Galileian reference-body K. As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation. According to a result obtained in Section 12, it follows that the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. as observed from K. It is obvious that the same effect would be noted by an observer whom we will imagine sitting alongside his clock at the centre of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest). For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with respect to the body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case, but I do not wish to go any farther into this question.
Moreover, at this stage the definition of the space co-ordinates also presents insurmountable difficulties. If the observer applies his standard measuring-rod (a rod which is short as compared with the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileian system, the length of this rod will be less than I, since, according to Section 12, moving bodies suffer a shortening in the direction of the motion. On the other hand, the measaring-rod will not experience a shortening in length, as judged from K, if it is applied to the disc in the direction of the radius. If, then, the observer first measures the circumference of the disc with his measuring-rod and then the diameter of the disc, on
Part II: The General Theory of Relativity
Albert Einstein 71
dividing the one by the other, he will not obtain as quotient the familiar number ! = 3.14 . . ., but a larger number,2) whereas of course, for a disc which is at rest with respect to K, this operation would yield ! exactly. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length I to the rod in all positions and in every orientation. Hence the idea of a straight line also loses its meaning. We are therefore not in a position to define exactly the co-ordinates x, y, z relative to the disc by means of the method used in discussing the special theory, and as long as the co- ordinates and times of events have not been defined, we cannot assign an exact meaning to the natural laws in which these occur.
Thus all our previous conclusions based on general relativity would appear to be called in question. In reality we must make a subtle detour in order to be able to apply the postulate of general relativity exactly. I shall prepare the reader for this in the following paragraphs.
 
 
Footnotes
1) The field disappears at the centre of the disc and increases proportionally to the distance from the centre as we proceed outwards.
2) Throughout this consideration we have to use the Galileian (non-rotating) system K as reference-body, since we may only assume the validity of the results of the special theory of relativity relative to K (relative to K1 a gravitational field prevails).
Part II: The General Theory of Relativity
Albert Einstein 72
Albert Einstein: Relativity Part II: The General Theory of Relativity



 

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10天前 北大袁萌爱恩斯坦的天才何在?

爱恩斯坦的天才何在?
 1916年,爱恩斯坦在其“广义相对论”首次提出惯性质量与引力质量等价,发问:是否存在又“有限”与“有界”的宇宙?提出宇宙的时空结构问题,…..这分显露了爱恩斯坦的天才,因为,在人类发展历史上没有人这趟提问过。
  请见广义相对论章节的中译文如下:
  Part II:
        The General Theory of Relativity
        18. Special and General Principle of Relativity
   相对论的狭义与广义原理
        19. The Gravitational Field
        引力场
20. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity
       作为广义相对论假设推理的惯性与引力质量的等价性
   21. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?
      22. A Few Inferences from the General Principle of Relativity     几个相对论一般原理的推论
23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
时钟与参考体测量杆的行为
 24. Euclidean and non-Euclidean Continuum
    欧几里得与非欧几里得连续统
        25. Gaussian Co-ordinates
        高斯坐标系
26. The Space-Time Continuum of the Speical Theory of Relativity Considered as a Euclidean Continuum
  时空连续统作为n-维欧几里德连续统      
 27. The Space-Time Continuum of the General Theory of Realtivity is Not a Euclidean Continuum
m
       广义相对论的时空连续统可能不是欧几里德连续统
28. Exact Formulation of the General Principle of Relativity
        相对论一般原理的准确陈述
29. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity
        基于广义相对论原理的引力问题解
        Part III: Considerations on the Universe as a Whole
关于作为一个整体的宇宙 
30Difficulties of Newton's Theory
     关于牛顿理论的宇宙学的困难
31?The Possibility of a "▊inite"   and yet "Unbounded" Universe
关于是否存在又“有限”与“有界”的宇宙?
32. The Structure of S一ace According to the General Theory of Relativity
      基于广义相对论的空间结构
袁萌  陈启清  4月13日
附件:广义相对论全文
Part II:
The General Theory of Relativity
Albert Einstein 54
Part II The General Theory of Relativity
Special and General Principle of Relativity
The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. Let as once more analyse its meaning carefully.
It was at all times clear that, from the point of view of the idea it conveys to us, every motion must be considered only as a relative motion. Returning to the illustration we have frequently used of the embankment and the railway carriage, we can express the fact of the motion here taking place in the following two forms, both of which are equally justifiable :
The carriage is in motion relative to the embankment, (b) The embankment is in motion relative to t


 

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11天前 北大袁萌宇宙中黑洞产生的原因

宇宙中黑洞产生的原因

    根据广义相对论(请见其英译文电子版第105页)的结论:光纤被宇宙引力场中必定发生偏转。掌握这是黑洞产生的根本原因。   

近日,黑洞照片的公布,证明了广义相对论的正确性。

注:广义相对论英译文电子版,请见无穷小微积分网站。

袁萌 陈启清 412

附件:原文105

(b) Deflection of Light by a Gravitational FieldIn Section 22 it has been already mentioned that according to the general theory of relativity, a ray of light will experience a curvature of its path when passing through a gravitational field, this curvature being similar to that experienced by the path of a body which is projected through a gravitational field. As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the

Appendix: Relativity: The Special and General Theory

Albert Einstein 109



 

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12天前 北大袁萌拍摄黑洞相片,意味着什么?

拍摄黑洞相片,意味着什么?

   拍摄到 M87黑洞的相片是当代“大科技”的重大成果。黑洞里面有什么物理过程?为什么选定M87黑洞?请看附件吴庆文教授的解答。

    注:等吴教授说完了,我们再来分析。

袁萌   陈启清  412

附件:

北京时间10日晚9时许,包括中国在内,全球多地天文学家同步公布首张黑洞真容。图片来自欧洲南方天文台(ESO)官网

这是人类诞生以来,第一次见到黑洞的照片。

黑洞呈圆环形,中心灰暗,外围被一圈橙黄色的光晕包裹,下方比上方更明亮。有人说它像甜甜圈,有人说它像猫的眼睛,还有人说它像铸造中的魔戒。

而在一年前,吴庆文见到它时,第一反应是:爱因斯坦的预言真的灵验了。

吴庆文是华中科技大学物理学院的一名教授。2016年,他加入了“事件界面望远镜”项目,和来自全球的200多名科学家一起,计划为黑洞拍一张照片。

从理论分析、实际观测到数据处理,他们分成数十个小组,花了三年多的时间,把黑洞的样子展现在了世人眼前,也终于证实了黑洞的存在。

“在2016年发现引力波之后,人们寻找到了爱因斯坦广义相对论最后一块缺失的拼图”,吴庆文说。这意味着人类对黑洞的研究、对宇宙的探索迈入了全新的阶段。

411日,在黑洞照片发布的第二天,我们与这位在照片拍摄、冲洗过程中,承担理论分析工作的教授聊了聊。

我们要给黑洞拍照片

剥洋葱:你为什么会对黑洞感兴趣?

吴庆文:我研究生在中科院上海天文台,期间一直做的课题就是黑洞,还有吸积盘。黑洞本身是大家关注的,它因贪婪闻名于世,是一个时空漩涡,由弯曲的空间和弯曲的时间构成。黑洞有一个视界,其内部时空高度扭曲,所有物质掉入它的范围,都会消失得无影无踪,奔向奇点,连光都不例外。我觉得这很好玩,就一直坚持下来了。

剥洋葱:你是什么时候加入“事件视界望远镜”项目的?这个项目的目标是什么?

吴庆文:早在2015年,我们就开始陆陆续续讨论,怎么样才能够看到黑洞,需要用多大的望远镜来观测等等问题。大概是2016年左右我正式加入,参与项目的有来自全球的200多名科学家,中国有十几个人。

这个项目提出来就是为了观测黑洞、给黑洞拍照。目前,有很多的间接证据已经证明了宇宙里有非常多黑洞,2016年,人类观测到了引力波,知道了黑洞“听上去是什么样子”,但我们更想知道,它“看上去是什么样子”。

剥洋葱:从理论上来说,黑洞是看不见的,为什么还能拍到黑洞的照片?

吴庆文:虽然黑洞本身并不发光,但它具有强大的引力,可以将周围的物质吸引过来,形成绕其转动的吸积盘。吸积盘可以将吸积物质的引力能变成辐射,从而可以被我们看到。

剥洋葱:“事件视界望远镜”项目有200多名成员参与,你们是怎么分工、协作的?

吴庆文:从数据处理、观测到后边理论分析,其实有各种各样的小组,分成了很多个科学工作组。我们中国大概有十几个人参与,大家做的事也不同。

 

最早开始做的时候,我们200多人聚在台湾开过一次会,后面陆陆续续也在不同的地点开过。平时主要是通过电话、视频会议,或者邮件,来和其他人沟通进展。

剥洋葱:你主要负责哪些工作?

吴庆文:我是做理论分析。拍摄之前,我研究的是黑洞里边会发生什么样的物理过程;拍摄之后,根据传回来的图像,我开始分析黑洞边上那些看到的光环大概是哪里来的?为什么是那样?做这种理论的计算。

后期我们花了很长的时间,一直在算,一直在提高、比对:我们算的东西是不是正确的?跟看到的东西是不是完全一样的?

剥洋葱:在理论分析的过程中,有没有遇到过什么困难?

吴庆文:因为黑洞里的时空是弯曲的,是完全扭曲的,我们看到的这个黑洞,看上去它是那样,但和它真实的样子并不相同,那些光子是在弯曲的时空里走的,我们要把弯曲时空里边的一些东西算出来,这是有一点难度的。

剥洋葱:参与这次拍摄对你来说有什么意义?

吴庆文:我感觉就是好玩,它本来就是人类比较好奇的东西,看到了就有趣了。

被选中的黑洞

剥洋葱:既然科学家们很早之前就预测出了黑洞的样子,为什么直到2017年才成功拍到黑洞的照片?

吴庆文:我们很早以前就可以算出来在什么条件下,才能用望远镜看到黑洞。2000年左右,就有科学家预测,经过技术的发展,十几年后我们可以看到黑洞。

2017年以前,一直都在尝试,不过早期的望远镜分辨率比较低,只能看到黑洞外边那些东西。这些年我们一直就在尝试慢慢地提高分辨率,到前几年,才能说我们已经完全达到分辨率了。

剥洋葱:是因为我们和黑洞的距离太远,所以才要提高分辨率吗?

吴庆文:分辨率简单来说就像我们的眼睛一样,近视了可能就看得不太清楚,戴上眼镜可能就看得很清楚,所以望远镜越大,就能把黑洞看得越清楚,能够把不同的黑洞分辨开。

打个比方,一台汽车有两个车灯,如果你离车很远的话,你只能看到一个亮点,只有它靠近你、你能看得清楚的时候,才会发现是两个车灯。

剥洋葱:每个星系中心都有很多个黑洞,为什么最终选定的是银河系中心黑洞和M87黑洞?

吴庆文:因为我们望远镜给出的是一个分辨率,我刚才提到,如果两个车灯离你非常远的话,你是分辨不清的,所以要找到合适的黑洞。很早之前,我们就算了黑洞在天上的投影,近邻的宇宙里所有黑洞在我们看上去有多大都算出来了。我们发现银河系中心黑洞和M87黑洞,在天上的投影面积是所有的黑洞里边最大的,所以选了它们。

剥洋葱:确定这两个黑洞的位置难吗?

吴庆文:不难,其实这是一个非常重要的工作,就我们银河系中心的黑洞来说,最早十几年前,有的恒星绕着中间的某一个地方转了一圈,利用牛顿的理论“开普勒三定律”就可以把中间的黑洞质量算出来,测得很准。

当然技术难度挑战非常的大,因为我们大气里的云一直在抖动,所以望远镜的分辨率再高,也会被云给扭曲掉,就让人看不清。后来发展出了一种技术:打一束激光上去,可以快速算出来云怎么动,再让后端的望远镜跟着再做改正。这样,云的抖动就去掉了,去掉之后就可以让黑洞附近的恒星的位置定得非常准。

剥洋葱:黑洞的照片具体是怎么拍的?

吴庆文:经过测算,我们需要用相当于地球大小的超级虚拟望远镜(相当于一台上万公里巨型望远镜)来拍摄。这些望远镜分布在全球不同的地方,因为地球是转的,所以总是有的地方会转过去,因此范围分布越广越好,当然要选择一些高海拔的地方,因为第一云稍微少一点,第二温度也低一点。

协调全世界近十台望远镜去观测它,是最难的一步。要不同的国家把所有的望远镜在同一时间都对准这一个天体,成本是非常高的。像美国的有一个镇,它的设备总投资是150亿美元、将近1000亿人民币,每一夜的价格都非常昂贵。而且什么时候看黑洞、能否看到黑洞都是不确定的。我们必须说服他们,随时准备把设备给我们,去看一个从来没人见过的物体。

最终,我们用了八台望远镜来拍摄,其分辨率达到了20微角秒,比哈勃望远镜高近2000倍,可以分辨出38万公里外月球上的一个乒乓球大小。

协调好这些望远镜后,需要一个时间窗口。就像发射火箭一样,哪时候发射最好,这一年里边可能只有十天。这十天中,还有几天可能天气不好,找这种机会比较难。2017年的45日到14日,基本上八九天是不错的,我们在这段时间内进行了拍摄。

剥洋葱:你们拍摄了两张黑洞的照片,为什么这次只公布了M87黑洞的照片?

吴庆文:我们用了五六天去拍摄这次公布的M87黑洞,剩下两天是拍我们银河系中心黑洞,时间分配上,M87的拍摄时间远远多于银河系中心黑洞。

另外,M87黑洞是正面拍摄,银河系中心黑洞侧面拍摄,看上去不如正面的直观,而且其中还有一些不确定的地方,将来确定了,银河系中心黑洞的照片可能也会发布。

冲洗黑洞照片花了近两年

剥洋葱:黑洞拍摄好之后,冲洗花了近两年的时间,为什么需要这么长时间?

吴庆文:观测完了之后,得到的数据量是非常大的,每一晚的数据量达2PB1PB=1000TB=1000000GB),和欧洲大型对撞机一年产生的数据差不多。这些数据量怎么去处理出来,是非常困难的。我们把观测到的数据用三种完全独立的流程、建立多个独立小组进行处理,来保证结果的准确性。每一个工作组,可能大概都得花半年到一年的时间去处理,得到一个初步的结果,然后再进行下一步的细化,最后把这个结果发给某一个负责人,然后他们去对各个组的结果进行对比。

剥洋葱:昨天照片公布是你第一次看到黑洞的样子吗?

吴庆文:其实基本上一年前我就看到了。因为我是自己做这个的,所以说结果一出来我就知道了。

但那时候只是我们中国团队做出来的初步结果,还不太确定最终的图像是什么样的。我们分成了三个大的组,每个组里又分成了几个完全独立的小组。只有不同的组做出来的结果是一样的时候,才能说这个图像是可信的。

昨天公布的这张照片,跟我们初步看到的结果基本上是接近的。

剥洋葱:第一次看到黑洞的照片的时候,你的心情怎么样?

吴庆文:当然非常激动了。我们中国团队的十几个人当时讨论得热火朝天,我们在想,为什么它会呈现这样的结构?里面告诉了我们哪些信息?

这些年让我激动的,除了看到黑洞照片,就是2016年观测到了引力波,我们也是先在理论上预言了那样的引力波,结果就看到了。当你想象中的东西第一次出现在眼前时,真的会非常高兴,如同美梦成真。

剥洋葱:这次看到的黑洞图片,和科学家们预言的也一样吗?

吴庆文:对。这个黑洞圆环大小约为40个微角秒,与广义相对论预言几乎完全一致。亮环呈不对称结构,左下角比右上角亮10倍以上,也与广义相对论预言一致,是由多普勒效应导致的,其中朝向我们运动的等离子体辐射会变亮,而远离我们的辐射会变暗。

所以,今天我们看到的黑洞的样子,和从爱因斯坦的广义相对论计算、推断出的黑洞的样子几乎完全一样,因此真正的黑洞的图片其实大家是非常熟悉的。

剥洋葱:在一些科幻电影中,也有黑洞的图像,据说《星际穿越》里的黑洞,是最接近黑洞真实面貌的想象,但看上去似乎和今天发布的黑洞照片有些区别。

吴庆文:当年《星际穿越》里面那张图片是标准算出来的黑洞图像,导演找了2017年诺贝尔物理学奖获得者索恩,也就是引力波的发现者,他们一起召集了一个计算机团队,利用几千台计算机,去模拟我们看到会是什么样子。那张图是从侧面看过去的,而且吸积盘,就是黑洞边上发光的盘,是一个很薄的盘,比较好计算。

我们这次看到的黑洞是一个厚盘,黑洞周围的物质比较少,而且是从正面看过去的图像,所以从外观看上去,二者之间有点区别。

剥洋葱:黑洞照片出来之后,很多网友觉得它还比较模糊,主要原因是什么?

吴庆文:分辨率还是不够高,如果再高几倍的话,那么基本上就可以看得非常清楚了。未来的几年可能会发布更清楚一点的图片,但是也不会多清楚。大概这次是一毫米的,下次可能会观测零点几毫米的,就是亚毫米波的一个图。这个也是很难的一个事情,我估计这还是需要好几年的时间。

剥洋葱:这次黑洞照片的成功拍摄,对整个物理学界来说有什么意义?

吴庆文:它是爱因斯坦广义相对论最后一块缺失的拼图,他预言的这个东西,这次就让我们真正的看到了,这个是比较有趣的地方。

我相信会推动整个黑洞相关研究,因为黑洞是整个天文学和物理学最前沿的一个课题。中国将来可能会有几个大的卫星项目,2017年已经发射的第一颗X射线卫星,马上2021年会发射一颗爱因斯坦探针卫星,还有2025年会发射慧眼第二代卫星,这几个卫星会为黑洞研究提供非常大的帮助。

2014年,罗俊院士提出来了“天琴”计划,“天琴”计划将来是更好地研究黑洞的一个观测设备,宇宙里面非常多的黑洞,通过它,小的也能看到,大的也能看到。包括宇宙爆发之初的时候,大概136亿年或者135亿年的地方的那些黑洞,将来的空间设备都能够看得到。

新京报记者 周小琪 编辑 陈晓舒 校对 李世辉

 



 

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13天前 北大袁萌为准备全国“六卓越一拔尖”大会,快忙疯了

为准备全国“六卓越一拔尖”大会,快忙疯了

    刚才,我们电话询问教育部有关微积分教改问题,得到的回答是:当前,他们为准备429日召开的全国“六卓越一拔尖”大会,快忙疯了。我们的问题要等教育部“一流专业课程《双万计划》”5-6月发布之后再说。

    我们耐心等待!

袁萌  陈启清  411



 

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13天前 北大袁萌向《双万》计划呼吁,现行微积分教育必须改革

向《双万》计划呼吁,现行微积分教育必须改革

众所周知,建设一流专业点必须强化基础理论教育,而微积分课程是必修基础理论课程。

    今天,我们将向国家教育部电话反映这一心愿。

  关于《双万》计划,教育部指定联系人及电话:教育部高等教育司,朱蓓蓓、徐健,010-66097823;教育部高等教育教学评估中心,郭栋、南方,010-8221339082213395

袁萌   陈启清  411

附件:

教育部办公厅发布《关于实施一流本科专业建设“双万计划”的通知》。

教育部办公厅关于实施一流本科专业建设“双万计划”的通知

教高厅函〔201918

各省、自治区、直辖市教育厅(教委),新疆生产建设兵团教育局,有关部门(单位)教育司(局),部属各高等学校、部省合建各高等学校:

  为深入落实全国教育大会和《加快推进教育现代化实施方案(20182022年)》精神,贯彻落实新时代全国高校本科教育工作会议和《教育部关于加快建设高水平本科教育 全面提高人才培养能力的意见》、“六卓越一拔尖”计划2.0系列文件要求,推动新工科、新医科、新农科、新文科建设,做强一流本科、建设一流专业、培养一流人才,全面振兴本科教育,提高高校人才培养能力,实现高等教育内涵式发展,经研究,教育部决定全面实施“六卓越一拔尖”计划2.0,启动一流本科专业建设“双万计划”,现将有关事项通知如下。

  一、主要任务

  20192021年,建设10000个左右国家级一流本科专业点和10000个左右省级一流本科专业点。

  二、建设原则

  面向各类高校。在不同类型的普通本科高校建设一流本科专业,鼓励分类发展、特色发展。

  面向全部专业。覆盖全部92个本科专业类,分年度开展一流本科专业点建设。

  突出示范领跑。建设新工科、新医科、新农科、新文科示范性本科专业,引领带动高校优化专业结构、促进专业建设质量提升,推动形成高水平人才培养体系。

  分“赛道”建设。中央部门所属高校、地方高校名额分列,向地方高校倾斜;鼓励支持高校在服务国家和区域经济社会发展中建设一流本科专业。

  “两步走”实施。报送的专业第一步被确定为国家级一流本科专业建设点;教育部组织开展专业认证,通过后再确定为国家级一流本科专业。

  三、建设方式

  1.国家级一流本科专业建设工作分三年完成。每年3月启动,经高校网上报送、教育主管部门或高校提交汇总材料、高等学校教学指导委员会提出推荐意见等,确定建设点名单,当年10月公布结果。

  2.省级一流本科专业建设方案由各省级教育行政部门制订,按照建设总量不超过本行政区域内本科专业布点总数的20%,分三年统筹规划,报教育部备案后与国家级一流专业建设同步组织实施。每年9月底前,各省级教育行政部门将本年度省级一流本科专业建设点名单报教育部,当年10月与国家级一流本科专业建设点名单一并公布。

  3.入选省级一流本科专业建设点的专业,如同时入选国家级一流本科专业建设点,按照国家级一流本科专业建设点公布。空出的省级一流本科专业建设点名额可延至下一年度使用。

  4.根据20192020年一流本科专业点建设情况,2021年将对各专业类国家级一流本科专业的建设数量和建设进度进行统筹。

  四、报送条件

  (一)报送高校需具备的条件

  1.全面落实“以本为本、四个回归”。坚持立德树人,切实巩固人才培养中心地位和本科教学基础地位,把思想政治教育贯穿人才培养全过程,着力深化教育教学改革,全面提升人才培养质量。

  2.积极推进新工科、新医科、新农科、新文科建设。紧扣国家发展需求,主动适应新一轮科技革命和产业变革,着力深化专业综合改革,优化专业结构,积极发展新兴专业,改造提升传统专业,打造特色优势专业。

  3.不断完善协同育人和实践教学机制。积极集聚优质教育资源,优化人才培养机制,着力推进与政府部门、企事业单位合作办学、合作育人、合作就业、合作发展,强化实践教学,不断提升人才培养的目标达成度和社会满意度。

  4.努力培育以人才培养为中心的质量文化。坚持学生中心、产出导向、持续改进的基本理念,建立健全自查自纠的质量保障机制并持续有效实施,将对质量的追求内化为全校师生的共同价值追求和行为自觉。

  (二)报送专业需具备的条件

  1.专业定位明确。服务面向清晰,适应国家和区域经济社会发展需要,符合学校发展定位和办学方向。

  2.专业管理规范。切实落实本科专业国家标准要求,人才培养方案科学合理,教育教学管理规范有序。近三年未出现重大安全责任事故。

  3.改革成效突出。持续深化教育教学改革,教育理念先进,教学内容更新及时,方法手段不断创新,以新理念、新形态、新方法引领带动新工科、新医科、新农科、新文科建设。

  4.师资力量雄厚。不断加强师资队伍和基层教学组织建设,教育教学研究活动广泛开展,专业教学团队结构合理、整体素质水平高。

  5.培养质量一流。坚持以学生为中心,促进学生全面发展,有效激发学生学习兴趣和潜能,增强创新精神、实践能力和社会责任感,毕业生行业认可度高、社会整体评价好。

  五、报送办法

  国家级一流本科专业建设点以学校为单位组织报送。教育部直属高校直接报教育部,其他中央部门所属高校经主管部门同意后报教育部;地方高校由省级教育行政部门统一报教育部。各地各高校报送专业点数(比例)分年度下达。

  六、组织保障

  (一)构建三级实施体系。教育部等14个“六卓越一拔尖”计划2.0负责部委(单位)统筹一流本科专业建设“双万计划”组织实施工作,指导各地、各高校落实有关文件要求,加强一流本科专业建设,推动构建国家、地方、高校三级实施体系。

  (二)完善经费保障。中央部门所属高校应当统筹利用中央高校教育教学改革专项等中央高校预算拨款和其他各类资源,各地应当统筹地方财政高等教育资金和中央支持地方高校改革发展资金,支持一流本科专业建设。

  (三)建立动态调整机制。教育部和省级教育行政部门加强对计划实施过程跟踪,针对一流本科专业建设中存在的问题,提出改进意见建议,对于建设质量不达标、出现严重质量问题的专业建设点予以撤销。

  七、关于2019年国家级一流本科专业建设点报送工作

  1.报送数量。中央部门所属高校、部省合建高校2019年度报送的专业点数不超过本校本科专业布点数25%;各省级教育行政部门2019年度报送专业点数量不超过本地所属地方高校本科专业布点总数的15%

  2.在线登录账号和密码。高校使用“高等教育质量监测国家数据平台”的登录账号及密码。各省级教育行政部门、中央有关部门(单位)教育司(局)须明确工作联系人,于2019415日前将姓名、单位、座机、手机、电子邮件、传真号码报至教育部高等教育司文科处,获取报送系统登录账号及密码。

  3.在线报送时间和网址。在线报送时间为2019420日—630日,请登录“国家级一流本科专业建设报送系统”(网址:http://udb.heec.edu.cn),按照系统提示填报。

  4.在线审核和提交。各省级教育行政部门、中央有关部门(单位)教育司(局)须在2019630日前,登录报送系统,严格按照限额,完成所属高校报送信息的在线审核和提交工作。

  5.纸质材料报送。高校在线报送完成后,请导出《国家级一流本科专业建设点信息汇总表》,加盖本校公章。教育部直属高校、部省合建高校材料直接报教育部;中央部门所属高校材料加盖主管部门公章后报教育部;地方高校材料由省级教育行政部门加盖公章后统一报送教育部。请于201971日前(以邮戳时间为准),将材料寄北京市西城区西单大木仓胡同35号教育部高等教育司文科处,邮编:100816

  联系人及电话:教育部高等教育司,朱蓓蓓、徐健,010-66097823;教育部高等教育教学评估中心,郭栋、南方,010-8221339082213395

  附件:1.国家级一流本科专业分专业类建设规划

     2.国家级一流本科专业建设点信息采集表

教育部办公厅

201942

来源:教育部

(责任编辑:刘潇翰)

 

 



 

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13天前 北大袁萌校园漫步,巧遇卡通人小无穷

校园漫步,巧遇卡通人小无穷

    411日上午,雨过天晴,袁萌漫步在校园中。突然,身后传来一个声音:“老爷爷。”袁萌赶忙回头一看,发现呼唤我的竟然是一个卡通人。

我问它:“你是谁?”

它回答道:“我认识您啊!昨天,您在文章中不是说我在校团中游四方吗?”“我是小无穷啊!”

我很吃惊地说:“你是无穷小吧?”“怎么变成了小无穷呢?”

卡通人回答:“我在校园中散发无穷小微积分,害怕被别人误解。我就叫小无穷了。”

    后来,卡通人还向我夸耀自己是鲁宾逊的小孙孙。

……袁萌从梦中醒来。

袁萌  陈启清  411



 

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14天前 北大袁萌无穷小带来什微积分么大礼包?

无穷小带来什微积分么大礼包?

    当今,超实无穷小在我国普通高校的校园中四处游荡,散发微积分“大礼包”

,帮助男、女同学学习微积分。

   微积分大礼包里面有40余个微积分重要定理、600多张精心绘制的示意图片、800多道练习题以及完备的微积分名词索引(微积分大百科全书)。

    坦率地说,学微积,用手机,是无穷小的大功劳。

袁萌  陈启清 410

附件:无穷小带来的大礼包,

电子版微积分教材的名词索引,一千余条。

Index

Absolute convergence, 521 trapezoidal, 226 surface, 825 Absolute value, 12 within E, 284 surface of revolution, 328 of a complex number, 876 Arc, 323, 365 under a curve, 188 Absolute value function, 12 length, 324, 366, 420 Argand diagram, 876 Acceleration, 94 arccos, 384 Argument, complex number, vector, 565, 623 Arccosecant, 384 876 Addition formulas for sine derivative, 387 Associative Law, 905 and cosine, 372 Arccosine, 384 inner product, 597 Addition Property, 190, 303 derivative, 387 scalar multiples, 570 variables integral, 392 vector product, failure, 602 two, 712, 720 arccot, 384 vector sum, 568 three, 760 Arccotangent,………

 



 

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15天前 北大袁萌8+8+8,无穷小游四方

8+8+8,无穷小游四方

第一个“8”:第一个“8”:2018810日,向全国普通高校拖放“无穷小微积分”电子版教材正式启动;

第二个“8”:时至今日,2019410日,经过8个月的轮番投放,一万套电子版微积分投放成功;

第三个“8”:百度一下“无穷小微积分”关键词,百度为您找到相关结果约76,100(近8万个搜索结果)。

由此可见,无穷小游四方已经成为现实。

袁萌  陈启清   410

 




 

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17天前 北大袁萌超越教学大纲,畅谈基础研究

超越教学大纲,畅谈基础研究

    去年8UI月,正式启动向全国普通高校轮番投放电子版微积分教材行动。

    该电子版微积分教材第十三章向量微积分,其中,第6节的10张示意图谈的全是斯托克斯定理,不超出现行微积分教学大纲(2017年版本),投放行动是合法的。 

超越教学大纲,畅谈基础研究,谈什么?

根据大数据搜索,我们发现:英国大数学家霍布金斯有几位世界知名大弟子,其中斯托克斯与麦克斯韦名列前茅。

    事实是,麦克斯韦根据

斯托克斯定理推导出“麦克斯韦方程组”,最终实验证明了无线电磁波的存在、

   进入本世纪,无线移动互联网无处不在,……共享单车、快递小哥,….

超越教学大纲,畅谈基础础研究,………到此为止!

袁萌  陈启清  47

 



 

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18天前 北大袁萌格罗滕迪克是什么人?

格罗滕迪克是什么人?

    格罗滕迪克(Alexander Grothendieck,1928-2014)是世界知名的法国数学学派的传承、发展人、现代代数几何学的奠基人。

南京大学数学系教同佟文庭是代数几何方向的专家、博导。他的去世是我国数学发展事业的重大损失。

注:袁萌与佟文庭是南京大学数学天文系57级代数专门化小班同学。

袁萌  陈启清  43

附件: 格罗滕迪克简介

Alexander Grothendieck

Born

28 March 1928

Berlin, Prussia, Weimar Republic

Died

13 November 2014 (aged 86)

Saint-Lizier, France

Nationality

none

French[1][2][3]

Alma mater

University of Montpellier

University of Nancy

Known for

Renewing algebraic geometry and synthesis between it and number theory and topology

Awards

1966  Fields Medal

1988  Crafoord Prize (declined)

Scientific career

Fields

Mathematics

Institutions

Institut des hautes études

scientifiques (IHÉS)

University of Montpellier[4]

University of São Paulo[5]

Thesis

Produits tensoriels topologiques et espaces nucleaires (1953)

Doctoral advisors

Laurent Schwartz

Jean Dieudonné

Doctoral students

Pierre Berthelot

Pierre Deligne

Michel Demazure

Pierre Gabriel

Jean Giraud

Luc Illusie

Michel Raynaud

Hoàng Xuân Sính

Jean-Louis Verdier………



 

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18天前 北大袁萌微分几何与斯托克定理

微分几何与斯托克定理

今年33日,希尔伯特《几何基础》电子版上传

“无穷小微积分”网站,使其向下“对接”微积分获得成功。

   微积分向上 “对接”什么?答案是:微分几何。那么,什么是微分几何呢?答案是:斯托克定理也。   

   百度一下“无穷小微积分”网站,下载“elem…”电子版微积分教科书,查找第十三章第六节第一张示意图Figure 13.6.1 ,立即可见斯托克定理的精确阐述。

   请见:附件1与附件2

袁萌   陈启清  45

附件1p.833  Figure 13.6.1

Stokes' Theorem relates a surface integral over S to a line integral over the boundary of S.

附件2

斯托克斯定理(英文:Stokes' theorem)是微分几何中关于微分形式的积分的定理,因为维数跟空间的不同而有不同的表现形式,它的一般形式包含了向量分析的几个定理,以斯托克斯爵士命名。

1 ³ 上的斯托克斯公式

2 流形上的斯托克斯公式

 

 

 



 

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19天前 北大袁萌清明时节,缅怀老友张锦文研究员

清明时节,缅怀老友张锦文研究员

张锦文研究员是我国老一辈倡导无穷小微积分的先行者,至今,离开我们已经进三十个年头了。

秉承张锦文研究员的遗愿,袁萌继续为无穷小微积分下放中学而战斗!

袁萌   45



 

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20天前 北大袁萌K-理论是什么?

K-理论是什么?

    2007年,K-理论导引“上网”,读者只要懂得拓扑空间基本概念,看懂K-理论导引,一般不会有困难。

    K-理论导引,请见附件,

袁萌  陈启清   43

附件:K-理论导引(英文原文PDF电子版)

An Inroduction to K-theory

Eric M. Friedlander

Department of Mathematics, Northwestern University, Evanston, USA

Lectures given at the School on Algebraic K-theory and its Applications Trieste, 14 - 25 May 2007

LNS0823001

eric@math.northwestern.edu

 

Contents

0 Introduction 5

1 K0(−), K1(−), and K2(−)    7

1.1 Algebraic K0 of rings    7

1.2 Topological K0  9

1.3 Quasi-projective Varieti . 10

1.4 Algebraic vector bundles . . . . .. . 12

1.5 Examples of Algebraic Vector Bundles. 13

1.6 Picard Group Pic(X) . . . . . . . 14

1.7 K0 of Quasi-projective Varieties . . . 15

1.8 K1 of rings . . . . . .. . . 16

1.9 K2 of rings . . . . . . . . . . . . . 17

2 Classifying spaces and higher K-theory 19

2.1 Recollections of homotopy theory . . . . . . . . 19

2.2 BG . . . . . .  . . . . . 20

 

2.3 Quillen’s plus construction .  . . 22

 

2.4 Abelian and exact categories . . .. . 23

2.5 The S−1S construction . .  . 24

2.6 Simplicial sets and the Nerve of a Category . . . . 26

2.7 Quillen’s Q-constructio . . . 28

3 Topological K-theory 29

3.1 The Classifying space BU ×Z .  . 29

3.2 Bott periodicity . . . 32

3.3 Spectra and Generalized Cohomology Theories . . . . . . . . 33

3.4 Skeleta and Postnikov towers .  . . 36

3.5 The Atiyah-Hirzebruch Spectral sequence . . . . 37

3.6 K-theory Operations . . . .. 39

3.7 Applications . . . . . . . . . . . . . . . . . 41

4 Algebraic K-theory and Algebraic Geometry 42 4.1 Schemes . . . . . . . . 42

4.2 Algebraic cycles . . . . .  . . . 44

4.3 Chow Groups . . . . . . . . 46

4.4 Smooth Varieties . . . . . . . .  49

4.5 Chern classes and Chern character . . .. . . . 51

4.6 Riemann-Roch . . . . . . . . . . .  . . . . . . . 53

5 Some Dicult Problems 55

5.1 K(Z) . . . . . . . . . 55

5.2 Bass Finiteness Conjecture . . . . . 57

5.3 Milnor K-theory . . .  . 58

5.4 Negative K-groups . . . . . . . . . 59

5.5 Algebraic versus topological vector bundles . . . .  . 60

5.6 K-theory with nite coecients . .. 60

5.7 Etale K-t. . . . 62

5.8 Integral conjectures . . . . . 63

5.9 K-theory and Quadratic Forms . . . . . . . . . 65

6 Beilinson’s vision partially fullled 65 6.1 Motivation . . . . . . 65

6.2 Statement of conjectur. . 66

6.3 Status of Conjectures . . . 67

6.4 The Meaning of the Conjectures . . . 69

6.5 Etale cohomology . . . .  . . . . 71

6.Voevodsky’s sites . . . . . . . . . . . . . . . . .. . . . 74

References 75

An Introduction to K-theory 5

(全文请见“无穷小微积分”网站)



 

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21天前 北大袁萌佟文庭教授在天之灵,安息吧!

佟文庭教授在天之灵,安息吧!

今晨惊闻南京大学数学天文系小班老同学佟文庭教授去世,心中十分悲痛。

佟文庭教授是国内知名K-理论专家(代数拓扑)、博导.

    佟文庭教授

天之灵,安息吧!数学自有后来人!

袁萌  42                                                                                              



 

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22天前 北大袁萌什么是连续函数?

什么是连续函数?                

   在微积分中,什么是连续函数?这是一个根本问题,马虎不得。

   科普中国,完全避开拓扑结构,只谈自变量与因变量的微小变化,……让人哭笑不得。这就是中国的可科普水平?

     问题的实质是:给定函数f : XYXY的拓分分别是T1T2,问题变为

   f : (X,T1)(Y,T2)

   其中(X,T1)(Y,T2)是两个不同的拓扑空间。连续函应该在拓扑空间层面上来定义。请见附

   袁萌  陈启清  41

   附件:(摘自拓扑导论)

   Continuous Functions and Subspaces.

   Let (X,T1) and (Y,T2) be topological spaces. A function f : X Y is said to be continuous provided that f−1(U) T1 whenever U T2. To summarize this situation we will sometimes write

   f : (XT1)(Y.T2) : X Y is a continuous function.

  

   f : X Y

                                                     

   在微积分中,什么是连续函数?这是一个根本问题,马虎不得。

   科普中国,完全避开拓扑结构,只谈自变量与因变量的微小变化,……让人哭笑不得。这就是中国的可科普水平?

     问题的实质是:给定函数f : XYXY的拓分分别是T1T2,问题变为

   f : (X,T1)(Y,T2)

   其中(X,T1)(Y,T2)是两个不同的拓扑空间。连续函应该在拓扑空间层面上来定义。请见附

   袁萌  陈启清  41

   附件:(摘自拓扑导论)

   Continuous Functions and Subspaces.

   Let (X,T1) and (Y,T2) be topological spaces. A function f : X Y is said to be continuous provided that f−1(U) T1 whenever U T2. To summarize this situation we will sometimes write

   f : (XT1)(Y.T2) : X Y is a continuous function.

  

   f : X Y



 

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23天前 北大袁萌缅怀恩师莫绍揆先生

缅怀恩师莫绍揆先生

记得,1957年,袁萌在南京大学数学天文系一年级学习数学。当时,南京大学数学天文系教授莫绍揆先(1917-2011)担任我们的线性代数课程的任课老师。

每逢莫老师讲课时,袁萌必定“抢座”,坐在教室的最前排中间位置、讲台的正对面,认真听讲,记笔记。课间休息时,总是向莫老师问这问那。时间长了,我们师生之间产生了亲密“师生情”。莫老师讲课时的音容笑貌,至今“清晰不忘”。

毕业之后,每逢回母校,袁萌必定前去看望莫老师,其间,谈论数理逻辑问题。

坦率地讲,正是莫绍揆先生引领袁萌走上了数理逻辑的人生道路。莫老师,你非在天之灵,安息吧!

袁萌  陈启清 41

附件:莫绍揆奖学金设立

日前,一项由个人出资捐赠的奖学金设立仪式在南京大学鼓楼校区举行。南京大学发展委员会副主任张晓东女士代表南京大学教育发展基金会,与来自美国的史念东教授签署了捐赠协议。这则以南京大学数学系已故教授莫绍揆的名字命名奖学金的消息,并未在校内引起轰动,但其背后却隐藏着一段与“感恩”有关的动人故事。

奖学金捐赠人史念东并非腰缠万贯的富豪,也不是南京大学毕业的校友。数十年前,他只是南京市的一名只有高中学历的普通工人。然而今天的他,是美国宾州斯特劳兹堡州立大学(East stroudsburs university of Pennsylvania)数学系的教授,从事数理逻辑方面的研究工作,曾多次到南京大学数学系和北京师范大学等校讲学,并参与《中国数学大辞典》的编写工作。

一封邮件传递感恩之情

今年5月,南京大学教育发展基金会收到史念东教授发来的一封邮件。邮件中提到,他年轻时因求学心切,曾登门求教于我校数学系已故教授莫绍揆,得到莫老师的无私帮助和指导。怀着对莫老师的感恩之情,史教授希望在南大设立以莫绍揆教授名字命名的奖学金。

69日,南京大学发展委员会邀请史教授来到南大,正式设立奖学金,并向他颁发了捐赠证书。仪式结束后,记者赶在史教授回国之前进行采访,了解到事情的始末。坐在沙发上的史念东教授头发花白,神态安然,微笑着谈起与莫老师的忘年之交,语气中充满了怀念与敬重。“感恩”,是他此行的最大心愿。

原来,早在1958年,家住南京的史念东就带着高考失利的遗憾,成了工厂里的一名普通工人,但是他心里对知识的渴望却从未动摇。文革前,工厂开办了职工子弟学校,有着高中学历的史念东被安排为数学老师。“晚上有课的时候,白天自己备课,不用上班。”史念东娓娓道来,“那时候就有了很多时间自学。但数学毕竟有难度,自学起来遇到很多疑难问题”。他说,当时在报纸上看到一篇有关南京大学数学系某位教授的报导,于是试着写信过去提问。“那时谁也不认识,有问题也找不到人问。”史念东笑着回忆,“我心想,师者,传道授业解惑,于是在信封写上‘南京大学数学系X X教授收’,就寄了出去。”令史念东没想到的是,这位教授不但很快就给他回信答疑,更是在此后长达十多年的时光中,在学习上和生活上,持续地给予他热情而无私的帮助。而这位恩师,就是我国数理逻辑教育和研究的开拓者之一,南京大学数学系的莫绍揆教授。

学有所成不忘回报师恩

记者了解到,莫绍揆教授不仅是一位优秀的数学家,而且也是杰出的教育家,1950年留学回国后,在南京大学任教,创建数理逻辑专业,于201110月辞世。他在教育领域辛勤耕耘50多年,为国家培养出了许多优秀人才,还很关心中学生的科普工作,曾引导不少像史念东这样需要帮助的年轻人,走上了研究数学的道路。

史念东叹道:“当年就是因为莫老师,我才知道了数理逻辑这个方向,后来考取了数理逻辑的研究生,又出国读博并留在国外任教。”提起莫绍揆教授,史念东的感念之情溢于言表:“莫老师培养了很多很多出色的人才,现在在数学界和计算机科学界,许多颇有建树的学者都曾是莫老师的学生。”

去年史念东回国时,得知莫绍揆教授去世的消息后感慨万千,决定以自己的方式纪念并回报莫老师。“我去找他时,他会立刻把手上的工作停下来帮助我。”沉浸在回忆中的史念东不禁动容,“其实他从来没有帮助我的义务,他是不求任何回报地帮助我,无论是学习上还是生活上。”史念东教授心怀感恩,在实现由工人到数学家的完美转身后,他拿出个人的工资收入,来到莫老师生前工作过的南京大学,踏上了自己的感恩之旅。

史念东说,以捐赠“莫绍揆奖学金”的方式表达自己的感恩之心,一是为了鼓励数理逻辑方向的博士生们潜心治学,为国家培养更多优秀的基础学科人才,同时也为了纪念莫绍揆老师,希望将他诚朴雄伟、诲人不倦的精神发扬并传承下去。采访中,记者还了解到,史念东教授曾详细询问过南京大学奖学金的管理方式。他笑着说:“奖学金代表着对莫老师的感恩,一定要确保真正用在人才培养上。现在我可以放心了,莫绍揆奖学金会一直存在。”

据了解,“莫绍揆奖学金”每学年评审一次,每学年奖励2名,每人的奖励金额为500美元额度的人民币。奖励对象为南京大学数学系全日制非定向博士研究生,数理逻辑专业品学兼优的博士研究生优先。(校报 李凌霄)

 



 

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24天前 北大袁萌学微积,用拓扑,越用拓扑越明白,不做糊涂人

学微积,用拓扑,越用拓扑越明白,不做糊涂人

            大家知道,现代微积分建立在欧几里德拓扑之上。因此,学微积,用拓扑,是当然之理。

   因此,学微积,用拓扑,而且,越用拓扑越明白,不做糊涂人。

    对于拓扑学必须把有“敬畏之心”。拓扑学不是小儿科,请见本文附件(全文在“无穷小微积分”网站)。

袁萌  陈启清  3300

附件:

Introduction to Topology Winter 2007(必须使用200冬季发布的这个电子版!)

Contents

1 Topology    9

1.1 Metric Spaces . 9

1.2 Open Sets (in a metric space) . . . . . . 10

1.3 Closed Sets (in a metric space) . . . . . 11

1.4 Topological Spaces . . . . . . 11

1.5 Closed Sets (Revisited) . .  12

1.6 Continuity   13

1.7 Introduction to Topology.  14

1.8 Homeomorphism Examples . .. 16

1.9 Theorems On Homeomorphism . . 18

1.10 Homeomorphisms Between Letters of Alphabet . . .. 19

1.10.1 Topological Invariants . . . 19

1.10.2 Vertices . . . .  19

1.10.3 Holes . . . . . .. 20

1.11 Classication of Letters . .  . 21

1.11.1 The curious case of the “Q”  22

1.12 Topological Invariants . . .. . 23

1.12.1 Hausdor Property . . . 23

1.12.2 Compactness Property    24

1.12.3 Connectedness and Path Connectedness Properties . . . 25

2 Making New Spaces From Old 27

2.1 Cartesian Products of Space  27

2.2 The Product Topology .  28

2.3 Properties of Product Spaces . . 29

3

2.4 Identication Spaces . . . .. 30

2.5 Group Actions and Quotient Spaces  34

3 First Topological Invariants 37

3.1 Introduction  . 37

3.2 Compactness . . . 37

3.2.1 Preliminary Ideas . . . . . .. . . 37

3.2.2 The Notion of Compactness . . .. 40

3.3 Some Theorems on Compactnes   . 43

3.4 Hausdor Spaces . . . .47 3.5 T1 Spaces . .. .. 49

3.6 Compactication . .. . 50

3.6.1 Motivation . . . 50

3.6.2 One-Point Compactication . .  50

3.6.3 Theorems . . 51

3.6.4 Example  55

3.7 Connectedness . . 57

3.7.1 Introduction . 57

3.7.2 Connectedness . . .  . . 58

3.7.3 Path-Connectedness . . 61

4 Surfaces 63

4.1 Surfaces . . . . . . . . . 63

4.2 The Projective Plane . . . .  . 63

4.2.1 RP2 as lines in R3 or a sphere with antipodal points identied. . . . . . . 63 4.2.2 The Projective Plane as a Quotient Space of the Sphere . . . .  65

4.2.3 The Projective Plane as an identication space of a disc . . . . . .  66

4.2.4 Non-Orientability of the Projective Plane . . . . .. . 69 4.3 Polygons  69

4.3.1 Bigons . .  71

4.3.2 Rectangles . . . . 72

4.3.3 Working with and simplifying polygons . . . 74

4.4 Orientability . 76

4.4.1 Denition  . . 76

4

4.4.2 Applications To Common Surfaces . .   77

4.4.3 Conclusion . . . . . . 80

4.5 Euler Characteristicn. .. .80

4.5.1 Requirements .  . 80

4.5.2 Computatio. . .  81

4.5.3 Usefulness . . . . 83

4.5.4 Use in identication polygons . . . . . . 83

4.6 Connected Sums . . 85

4.6.1 Denition . .  . 85

4.6.2 Well-denedness . . 85

4.6.3 Examples . . .. . 87

4.6.4 RP2#T= RP2#RP2#RP2 . .88

4.6.5 Associativity .  . 90

4.6.6 Eect on Euler Characteristic . . . . . . 90

4.7 Classication Theorem . . .  92

4.7.1 Equivalent denitions . . . .  . 92

4.7.2 Proof . . . . . 93

5 Homotopy and the Fundamental Group 97 5.1 Homotopy of functions . . . . 97

5.2 The Fundamental Group .  . 100

5.2.1 Free Groups . .  . 100

5.2.2 Graphic Representation of Free Group . .. . 101

5.2.3 Presentation Of A Group . . . . . 103

5.2.4 The Fundamental Group .. . 103

5.3 Homotopy Equivalence between Spaces . . . . . 105 5.3.1 Homeomorphism vs. Homotopy Equivalence . 105

5.3.2 Equivalence Relation . . .  . . 106

5.3.3 On the usefulness of Homotopy Equivalence   106

5.3.4 Simple-Connectedness and Contractible spaces . . . . 107

5.4 Retractions . . . . 108

5.4.1 Examples of Retractions . . . . . 108

5

5.5 Computing the Fundamental Groups of Surfaces: The Seifert-Van Kampen Theorem . . .  110

5.5.1 Examples: . . . 112

5.6 Covering Spaces . . . 113

5.6.1 Lifting . . 117

 



 

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26天前 北大袁萌开集名称,由何而来?集合


开集名称,由何而来?集合

首先开机民本工程,我们考虑实数集合R一个集合::

T = { IR }

其中:字母“I”代表没有端的实数区间。

容易验证,拓扑T包含三个元素,而且确实满足拓扑的4项基本要求(公理),因而集合TR上的一个简易拓扑,。

所以,I,作为拓扑T的元素,称为实数集合R中的“开区间”。

    所以,开集(Open set)名称由此而来。

百度一下“无穷小微积分”,进入该站,下载“T

opology”,打开文件,查找第10页,即可见到本文附件文字。

概念错误,越学越糊涂,

袁萌  陈启清  329考虑日

附件:拓扑概念的正规定义

5. Denition of Topology and Basic Terminology.

Let’s start by repeating the denition of a topology on a set X:

Definition 5.1.

Let X be a set and let T be a family of subsets of T which satises the following four axioms:

(T1) The empty set is an element of T.

(T2) The set X is an element of T.

(T3) If Uj T for each j J thenSjJ Uj is in T.

(T4) If U1 and U2 are in T then U1 U2 is in T.

then we say that T is a topology on the set X.

Here the axiom (T3) is referred to by saying T is closed under arbitrary unions and axiom (T4) is referred by saying T is closed under pairwise intersection.

(请注意!)

A set X together with a topology T on that set is called a topological space, more formally we will refer to this topological space as (X,T). If (X,T) is a topological space then the elements of T (再次请注意!!)are called open sets in X. Theorem 5.2

(T is closed under nite intersections.). Let T be a topology on a set X, and let U1,U2,...,Un be open sets in this topology where n is a positive integer. Then the intersection U1 U2 ∩···∩Un is an open set.

Proof. We prove this statement by induction on n. If n = 1 then the collection of open sets has only one set U1 and the intersection is just U1 which is an open set. This shows that the statement is true when n = 1. Now suppose the statement is true for a positive integer n (this is the induction hypothesis). To complete the proof by induction we need to show that the statement is true for n+1. So suppose that U1,U2,...,Un+1 are open sets. Then we can write U1 U2 ∩···∩Un+1 = V Un+1 where V = U1 U2 ∩···∩Un. Then V is an open set by the induction hypothesis, and V Un+1 is open by axiom (T4). This shows that the statement is true for n+1 (that is, U1 U2 ∩···∩Un+1 is open) and completes the proof by induction.

10

If x is an element of X and U is an open set which contains x then we say that U is a neighborhood of x.

Theorem 5.3.

A subset V X is an open set if and only if every element x V has a neighborhood that is contained in V .

Proof. () Suppose that V is an open set in X. Then for every element x V , V is a neighborhood of x and V V . () Suppose that V is a subset of X and each element x V has a neighborhood Ux such that Ux V . Then V =SxV Ux and since each Ux is open then the union is open by axiom (T3). This shows that V is an open set and completes the proof. [In the second part of the previous proof, carefully explain the equality V =SxV Ux?] If T1 and T2 are two topologies on a set X and T2 T1, then we say that T1 is ner than T2, or that T2 is coarser than T1. If T2 ( T1 then T1 is strictly ner than T2. If neither of the topologiesT1 andT2 are ner than the other then   

we say that the two topologies are incomparable. We end this section by describing a number of examples of topological spaces.

Example 5.4.

For any set X let Tdiscrete = P(X) = {A | A X}.

Clearly axioms (T1) and (T2) hold since and X are subsets of X. The union and the intersection of any collection of subsets of X is again a subset of X and this shows that axioms (T3) and (T4) hold as well. Therefore Tdiscrete forms a topology on X and this is called the discrete topology on X. So in the discrete topology, every subset of X is an open set. Therefore Tdiscrete is the largest possible topology on X, which is to say that the discrete topology is ner than every topology on X.

Example 5.5.

For any set X let

Ttrivial = {,X}. It is easily veried that Ttrivial is a topology on X and this is called the trivial topology on X. This topology contains only two open sets (assuming that X is a nonempty set). Therefore Ttrivial is the smallest possible topology on X, which is to say that every topology on X is ner than the trivial topology.

Example 5.6.

Just a reminder that for each positive integer n, Teuclid is a topology on Rn called the Euclidean topology. In particular, taking n = 1 gives the Euclidean topology Teuclid on the real line R, referred to briey as the Euclidean line. Here Teuclid = {U R| for each x U there is > 0 such that (x−,x + ) U}. With this denition it is not hard to show that every open interval in R is an open set in the Euclidean topology. [Be sure you can write out an explanation for this.] But note carefully that there are open sets in the Euclidean topology that are not open intervals. For example, the union of two or more disjoint intervals (such as (−1,1) (2,π)) will be an open set which is not an interval.

Example 5.7.

Consider the collection of subsets T` of the real line R given by T` = {U R| for each x U there is  > 0 so that [x,x + ) U}.

11

This set forms a topology T` on R called the lower limit topology on R. It is not hard to show that

(1) each nite half-open interval of the form [a,b) where a < b is an open set in the lower limit topology but not an open set in the Euclidean topology, while

(2) every open set in the Euclidean topology on R is open in the lower limit topology. Thus the lower limit topology on R is strictly ner than the Euclidean topology on R (that is Teuclid (T`).

Example 5.8.

Let X be a set and let x0 be an element of X. Dene T to be the collection of subsets of X consisting of X itself and all subsets of X which do not contain x0. Thus T = {X}{U X | x0 / U} . This forms a topology on X which is called the excluded point topology.

Example 5.9.

Let X be any set and dene Tcofinite = {}{U X | X−U is a nite set } = {}{X−F | F is a nite subset of X}. Then it is not hard to show thatTcofinite forms a topology on X, called the conite topology on X. (We will explain this using closed sets in the next section.) Note that if the set X itself is nite then every subset of X will be nite, and so the conite topology on a nite set X is the same as the discrete topology on X.

 

 

 



 

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26天前 北大袁萌拓扑概念的正规定义

拓扑概念的正规定义

经过多方查证与对比研究,可以确认科普中国关于拓扑概念的定义的不正确的。

百度一下“无穷小微积分”,进入该站,下载“Topology”,打开文件,查找第10页,即可见到本文附件文字。

概念错误,越学越糊涂,

 

 

 

 

袁萌  陈启清  328

附件:拓扑概念的正规定义

5. Denition of Topology and Basic Terminology.

Let’s start by repeating the denition of a topology on a set X:

Definition 5.1.

Let X be a set and let T be a family of subsets of T which satises the following four axioms:

(T1) The empty set is an element of T.

(T2) The set X is an element of T.

(T3) If Uj T for each j J thenSjJ Uj is in T.

(T4) If U1 and U2 are in T then U1 U2 is in T.

then we say that T is a topology on the set X.

Here the axiom (T3) is referred to by saying T is closed under arbitrary unions and axiom (T4) is referred by saying T is closed under pairwise intersection.

(请注意!)

A set X together with a topology T on that set is called a topological space, more formally we will refer to this topological space as (X,T). If (X,T) is a topological space then the elements of T (再次请注意!!)are called open sets in X. Theorem 5.2

(T is closed under nite intersections.). Let T be a topology on a set X, and let U1,U2,...,Un be open sets in this topology where n is a positive integer. Then the intersection U1 U2 ∩···∩Un is an open set.

Proof. We prove this statement by induction on n. If n = 1 then the collection of open sets has only one set U1 and the intersection is just U1 which is an open set. This shows that the statement is true when n = 1. Now suppose the statement is true for a positive integer n (this is the induction hypothesis). To complete the proof by induction we need to show that the statement is true for n+1. So suppose that U1,U2,...,Un+1 are open sets. Then we can write U1 U2 ∩···∩Un+1 = V Un+1 where V = U1 U2 ∩···∩Un. Then V is an open set by the induction hypothesis, and V Un+1 is open by axiom (T4). This shows that the statement is true for n+1 (that is, U1 U2 ∩···∩Un+1 is open) and completes the proof by induction.

10

If x is an element of X and U is an open set which contains x then we say that U is a neighborhood of x.

Theorem 5.3.

A subset V X is an open set if and only if every element x V has a neighborhood that is contained in V .

Proof. () Suppose that V is an open set in X. Then for every element x V , V is a neighborhood of x and V V . () Suppose that V is a subset of X and each element x V has a neighborhood Ux such that Ux V . Then V =SxV Ux and since each Ux is open then the union is open by axiom (T3). This shows that V is an open set and completes the proof. [In the second part of the previous proof, carefully explain the equality V =SxV Ux?] If T1 and T2 are two topologies on a set X and T2 T1, then we say that T1 is ner than T2, or that T2 is coarser than T1. If T2 ( T1 then T1 is strictly ner than T2. If neither of the topologiesT1 andT2 are ner than the other then   

we say that the two topologies are incomparable. We end this section by describing a number of examples of topological spaces.

Example 5.4.

For any set X let Tdiscrete = P(X) = {A | A X}.

Clearly axioms (T1) and (T2) hold since and X are subsets of X. The union and the intersection of any collection of subsets of X is again a subset of X and this shows that axioms (T3) and (T4) hold as well. Therefore Tdiscrete forms a topology on X and this is called the discrete topology on X. So in the discrete topology, every subset of X is an open set. Therefore Tdiscrete is the largest possible topology on X, which is to say that the discrete topology is ner than every topology on X.

Example 5.5.

For any set X let

Ttrivial = {,X}. It is easily veried that Ttrivial is a topology on X and this is called the trivial topology on X. This topology contains only two open sets (assuming that X is a nonempty set). Therefore Ttrivial is the smallest possible topology on X, which is to say that every topology on X is ner than the trivial topology.

Example 5.6.

Just a reminder that for each positive integer n, Teuclid is a topology on Rn called the Euclidean topology. In particular, taking n = 1 gives the Euclidean topology Teuclid on the real line R, referred to briey as the Euclidean line. Here Teuclid = {U R| for each x U there is > 0 such that (x−,x + ) U}. With this denition it is not hard to show that every open interval in R is an open set in the Euclidean topology. [Be sure you can write out an explanation for this.] But note carefully that there are open sets in the Euclidean topology that are not open intervals. For example, the union of two or more disjoint intervals (such as (−1,1) (2,π)) will be an open set which is not an interval.

Example 5.7.

Consider the collection of subsets T` of the real line R given by T` = {U R| for each x U there is  > 0 so that [x,x + ) U}.

11

This set forms a topology T` on R called the lower limit topology on R. It is not hard to show that

(1) each nite half-open interval of the form [a,b) where a < b is an open set in the lower limit topology but not an open set in the Euclidean topology, while

(2) every open set in the Euclidean topology on R is open in the lower limit topology. Thus the lower limit topology on R is strictly ner than the Euclidean topology on R (that is Teuclid (T`).

Example 5.8.

Let X be a set and let x0 be an element of X. Dene T to be the collection of subsets of X consisting of X itself and all subsets of X which do not contain x0. Thus T = {X}{U X | x0 / U} . This forms a topology on X which is called the excluded point topology.

Example 5.9.

Let X be any set and dene Tcofinite = {}{U X | X−U is a nite set } = {}{X−F | F is a nite subset of X}. Then it is not hard to show thatTcofinite forms a topology on X, called the conite topology on X. (We will explain this using closed sets in the next section.) Note that if the set X itself is nite then every subset of X will be nite, and so the conite topology on a nite set X is the same as the discrete topology on X.

 

 



 

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