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Cycles that are suggested that you go through and aren't gone through in class also sometimes turn up on exams too, by the way.
顺带一提，那些我建议,你们去推导，而没有在课上,推导的循环有时也会。
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If you go inside, there's spiral staircase that goes all the way up.
如果你进去，会有一个旋转阶梯通向顶端。
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If these are my payoffs I should go ahead and choose Alpha because that way I end up getting 3.
如果我关心我成绩我应该直接选α,因为那样我能得到3单位效用
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That is to say I can now reroute that sodium plus chlorine reaction and go a different way, but I still end up with the same change in energy.
也就是说，我可以重新定义一种,钠与氯的反应途径,从另一种途径来实现反应，但是能量的变化过程相同。
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You can go all the way up to the origins of World War I to see.
这可以追溯到一战的起源
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So since it's a component of the angular momentum, that means that it's never going to be able to go higher than l is, so it makes sense that, for example, it could start at and then l go all the way up to l.
因为，它是，角动量的分量，这意味着，它不会，比l大，这是很容易理解的，比如说，它可以从零开始,一直到。
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So if earnings go up a lot, we'll raise our dividends, but only part of the way up.
所以若收益大幅提高,我们会增加分红,但增量有限
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They had to go from Italy, because much of Italy was controlled by Spain, through the Alps all the way up along the Rhine and finally get into the Dutch Republic.
他们不得不穿过意大利,因为当时许多意大利地区正由西班牙控制,他们从阿尔卑斯山一路,沿着莱茵河,最终深入荷兰共和国腹地
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And unlike n, l can start all the way down at 0, and it increases by integer value, so we go 1, 2, 3, and all the way up.
不像n，l可以从0开始取，然后每次增加一个整数，所以我们可以去1，2，3，一直下去。
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If stock price option -if stock options tend to be issued as announced by the company later, just before stock prices go up, there's no way that that can be right because nobody can know exactly when the stock price is going to go up.
如果股票期权价格...,如果就像公司事后宣称的一样,公司恰好是在股价上涨之前,发行的股票期权,这是不可能做到的,因为没有人可以确切知道,股票价格何时会上涨
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The way you would go about it, if you're a portfolio manager, is you have to come up with estimates of the inputs to these formulas-- that means the expected returns, the standard deviations, and the covariances.
你所要做的，如果你是一个资产经理，你要做的事情就是,对公式里面的一些参数进行估计-,那些参数包括预期收益，标准差，和协方差。
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