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So first, I choose a volatility randomly, from some distribution of possible volatilities 2 from to, in this case, 0.2.
来决定的一个值，所以首先我先随机选择一个浮动值，从可能的浮动值中的分布进行选择,在这个例子中就是0。
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Lots of 6-year olds, 9-year olds, 12 year-olds, a couple teenagers, and I do have 2 adult students.
很多六岁、九岁、十二岁的学生，几个青少年，我也有两个成人学生。
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I first had a list of size 8, then 4, then 2, but then I had another problem of size 2.
首先是一个有8个元素的序列，接着变成了4个元素，接着2个元素，然后我就碰到了有2个元素的另一个问题。
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OK, that's not so bad. Moving a stack of size 2 if I want to go there, I need to put this one temporarily over here so I can move the bottom one before I move it over.
好，并不那么难，移动上面两个圆盘的话2,我需要把最小的盘子，临时先放到多余的柱子上来,这样才能在把它移过来之前,把最底下的盘子放到目标柱子上。
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In particular, if the other candidate were to choose position 1, I would get a higher share of the vote choosing 2 than I would have done if I had chosen 3.
如果对手选择立场1,我选立场2会比选立场3,得到更多的票数
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That makes sense if you think about this process of cell multiplication, that I have one cell it becomes 2 cells in a minute, it could become 4 cells in another minute, it could become 8 cells in another minute and that's all that this set of equations is representing.
如果你和细胞增殖的过程联系起来的话,这个是说的通的,我有一个细胞在一分钟里变成了两个细胞,那就可以再花一分钟变成四个细胞,再花一分钟就变成八个细胞,这就是这个方程所代表的意义
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This is a story that is predominantly in Genesis 2 and trickles into Genesis 3 and I'm going to look at it mostly in isolation from the first account.I'm going to be looking at it in light of an important parallel.
这个故事在《创世纪》2，3章中占据很重要的地位,我采用脱离对第一个故事的描述方法,将这两个故事，放在同等重要的位置上来解读。
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Now the answer seems to me, the only answer I can imagine Socrates and Plato are giving at this point is to say, look, I need a different 3 definition of invisible, not 2 but 3.
我认为答案是，我所能想到的唯一答案是,在这点上苏格拉底和柏拉图,给出的答案是，我需要一个,无形的不同定义，不是2而是。
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Then you're able to generalize that and say, if you multiply it by 2.6, I mean a vector 2.6 as long as A.
然后可以将其推广到更一般的情况,如果用2.6乘以它,所得矢量即为矢量 A 的2.6倍
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This young lady I worked graduated from high school with a 2.5 and she got into college.
向我咨询的小女生以2.5的平均分从高中毕业，并且顺利进入了大学。
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and she has really made a lot of progress. I've been teaching her for almost 2 years,
她真的进步很大。我教她将近两年了，
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So we have a total of 2, 4, 6, 8, 10 valence electrons, so I'll make sure I count to 10 as we fill up our molecular orbitals here.
我们一共有2，4，6，8，10个价电子，所以我一边填一边要确认,我数到10。
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A lot of people watch the price of their share everyday-- some people get neurotic about it-- and then suddenly it drops by $2 a share and they say, oh my God I'm worried.
很多人每天都会查看自己所持股股价,有些人因此神经过敏,突然间股价降低了2美元,他们会说,天啊,怎么办啊
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And at a distance of 2 r naught, I have a positive repulsive term.
在这2r圈的距离上，这里有一个不可忽略的排斥场。
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I'm gonna zoom in a little more and notice it's pretty much 1, 2, 3, 4, 5, 6, 7, 8, 9 lines of code and that's kind of rounding up because of the white space.
我现在放大一点，大家会看到，2，3，4，5，6，7，8，9行代码，多亏了空格,这些代码才能看起来如此美观。
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This is sorted, this is sorted, how do I now make a list of size 2?
这个是有序的，这个也是有序的，我怎样才能组成一个有2个元素的列表？
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And then I could also do a Gaussian one here, with the mean of and the standard deviation of volatility divided by 2.
然后我在这里再写一个高斯分布的函数，它的浮动值的平均值和,标准偏差值都除了2。
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So I want to figure out the best response of Firm 1 as a function of the price chosen by Firm 2.
我想找出公司1的最佳对策,并用公司2价格的函数表示
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And yet if a computer can only count up to 2 billion or maybe 4 billion, I mean, what do you then do?
如果一台计算机只能够支持20亿,或者40亿，那我们该怎么办？
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So if I start off with a list of length n, how many times can I divide it by 2, until I get to something no more than two left?
我能够除以多少次2呢？,直到我得到的长度不超过2么？,对数次，对吧？就像刚才那位同学说的那样？
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So here I have a new expression for Firm 1's profit and I could do the same for Firm 2, but I'm not going too because it's... I'm getting bored.
这就是公司1的利润表达式,接下来我们算公司2的,我就不算了,太没有意思了
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But in any event, Deuteronomy is not simply the concluding book of the Pentateuch, ; or the story that began in Genesis; it's also the first part of a much larger, longer literary work, as I mentioned last time, a work that runs from Deuteronomy through to the end of 2 Kings.
但无论如何，《申命记》不是摩西五经的终结篇，或者说是自《创世纪》开始的故事;,它也是一部更宏观长远的作品的开篇，我上节课提到过，这部作品从《申命记》一直到《列王记》结束。
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I want to just highlight what's in Chapter 2, because I told you we're not going to cover the details in Chapter 2 in the course, but I give it to you as a resource so that you you might have other books which describe this which you like, and you've read already and so but I'm going to assume that you understand this information to some extent.
我想强调第二章的内容,因为我告诉过你们,课堂上不会细讲第二章,但我会把它作为阅读材料给你们,你们可能已经有讲述相关内容的书,这些书你们喜欢看,并且已经看过了,我就当你们,在一定程度上理解了这些知识
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If I take the second derivative I get 2, but I want to get a and not 2.
如果我再求二阶导数就能得到2,但我想要a不想得到2
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Let's just... We'll get there, just to remind you, the way we read this is you give me a quantity of Firm 2, I find Firm 1's best response by going across to the pink line and dropping down.
我们当然也能算出来,提醒一下各位,这个图像的个意思是任意给出公司2产量,然后通过这条粉色的线,就可找出与之对应的公司1的最佳对策
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And we already know if I look at the probability of a ?, which is here, that the payoff I get from choosing, the expected payoff I get from choosing up against a is 2.5.
我们已经计算过了概率是?的情况,即在对手概率是的情况下,我选择上我从中获得的,预期收益是2.5
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It says, in either case in general, t of b-- and this is where I'm going to abuse notation a little bit but I can basically bound it by t, 12 steps plus t of b over 2.
我可以用一个,比12+t的数代表,这里有点不准确的地方,具体的步数依赖于奇数偶数，但是你们可以看到在两个case中。
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PROFESSOR 2: -- other, it could have said - who or zort or -- PROFESSOR: Yeah, sorry, that was part of the question, I could have a picked foobar could put anything in here.
教授2：另外一个对象？,这可以说是其他对象或者-,教授：对，很抱歉，这是问题的一部分，我可以写一个,可以放进任何东西的代用符。
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And I can give a name to that, so c p 1 and c p 2 are both going to point to that.
为这个实例分配了空间，现在它是空的,实际上也不完全是空的。
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