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And the solution to this equation looks like this where it is written in terms of a quantity called a wavefunction.
这个方程的解法是,看起来像是写成数学符号就是,波函数。
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so that's great that a lot of people have enough time to be in something that's not math or science related.
很多人投入了时间，来做与数学或科学无关的事情，这真的太好了。
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This is an abstract and powerful mathematical statement of the first law of thermodynamics.
这是热力学第一定律的一个抽象,而且具有很强数学性的表述。
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Again, this is a right-handed centric thing but if you're right-handed, language on the left, math and music on the right.
重申一遍,这是以右利手为基础的研究结论,如果你是右利手,那么语言功能便定位于左半球,而数学运算与音乐则定位于右半球
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It's 33, 32, it's only 32 times and you can check that math at home but here lies the power of algorithmic thinking, of computational thinking as they say.
你需要33，32，对，32次,你也可以在家里自己计算，但数学算法的力量是无穷的，这就是科学家们所谓的计算机思维。
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The concept of regression goes back to the mathematician Gauss, who talked about fitting a line through a scatter of points.
回归这个概念要追溯到数学家高斯,讨论的是从若干散点中切合出一条直线
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One solution for that is to get a copy of a textbook I wrote called Basic Training in Mathematics.
解决办法之一,就是买一本我编写的教材,书名叫《数学基础训练》
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I think math is the most obvious one that would be suitable for adapting to this format.
我觉得数学是最明显的,数学能很好地,适应这种方式。
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That's going to presumably be given--mathematically if we gave it a number, we'd slap a zero on that.
如果用数学解释,给它一个数字，给它一个零。
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At the end of the day, I wound up starting in Math and Physics at Harvard, finishing an Applied Math and Economics and doing business.
最后，我跑去哈佛学了数学和物理，念完了应用数学和经济，跑去经商了。
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And in particular, I'm going to cheat slightly I'm going to use a little bit of abusive mathematics, but I'm going to show you a trick to figure it out.
尤其要说明的是,我会有一点不属实在里面,我会滥用一点数学，但是,我会教会你们这个小把戏的。
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Plato, being a mathematician, as Aristotle was not, decided that the perfect polis would have 5,040 citizens.
而和亚里士多德不同,作为数学家,柏拉图认为完美的城邦应有五千零四十个公民
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So this is a little bit of math, a little formality, but is everyone okay with that?
这里只是稍微用了点数学符号,还有简单的公式,大家都听懂了吗
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And this is one of the times when you are actually going to be able to use some of that math they teach you.
这就是时候,用他们教你的,数学知识了。
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So you can see that we're starting to have a very complicated equation, and it turns out that it's mathematically impossible to even solve the exact Schrodinger equation as we move up to higher numbers of electrons.
所有你们可以看到我们得到了,一个非常复杂的方程，结果是它在数学上是,不可能解出确定的,薛定谔方程,当我们考虑更高的电子数目的时候。
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I don't really have the opportunities to study math and science, which, to some degree, I think is regrettable
我没什么机会继续学习数学和科学。我有时觉得有些很遗憾，
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I talked about life is not like a mathematic problem ? with you or in my other class ?
我说过生活不是个数学难题,是跟你们说过？，还是在我另外的课上？
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A famous mathematician once said "Every question must have a correct answer " for every question one answer."
一位著名的数学家曾说过,“每一个问题都必定有一个正确的答案,每一个人们所回答的问题“
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You know, if you are looking to go into business, learn math really well, learn economics really well.
例如，如果你将来想从商，你现在就要学好数学和经济学。
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Imagine you have a math test and this is the front of the math test.
假设要做一个数学测试,这是数学测试卷的正面。
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Always provides you with access to the math library because that's useful, too, so you don't always have to remember that.
总是能够让你能够使用数学程序库,那也是有用的，所以你不一定要把它记住。
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We have theories--mathematical theories-- that lead us to devise financial structures, which are complicated devices just like engines or nuclear reactors.
金融学中的理论,数学理论,能帮助我们构造金融工具,金融工具就像发动机,或者核反应堆一样,都是复杂的工具
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So the mathematical notions of infinity or logical notions like logical entailment are beyond a child of this age.
因此,对于无限这个数学概念,或者是,像逻辑蕴涵这样的逻辑概念,超出了该年龄阶段儿童的理解范围
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Without worrying about the economics of it a second, how would we find that just as a math exercise?
不需要考虑经济学的理论,用数学的方法怎么来求解啊
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So, all you will have the opportunity to solve differential equations in your math courses here. We won't do it in this chemistry course. In later chemistry courses, you'll also get to solve differential equations.
你们在数学课中有机会,遇到解微分方程,我们在这化学课里就不解了,在今后的化学课程里，你们也会遇到解微分方程的时候。
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And I think that's very important especially with electromagnetism, which is so numb toward if, so abstract and so much math flowing around.
我觉得这是非常重要的,尤其是对于电磁学来说，它是一门很难懂的，很抽象的学科,包含很多数学在里面。
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They read. And one of the people that was reading this literature was a high school teacher who was teaching mathematics in Basil, Switzerland.
其中一个读过这篇文章的人,是个教高中生的老师,在瑞士的罗勒教数学。
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I will use it in moderation and use it to the best effect possible rather than use it because it is there.
我会适量的运用数学以达到最佳的效果,而不是为了数学而用数学
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I'll show those again, but what I want to do mostly today is try to put a mathematical statement of the second law in place that corresponds to the verbal statements that we saw last time.
今天我还将跟你们讲这些，但今天的重点是,对上次口头表述内容中的热力学,第二定律,的数学推导,我们上次看到的。
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Yet even there, when I imagine an eternity of thinking about math--or for that matter, an eternity of thinking about philosophy, which I obviously like even more than math-- the prospect seems an unattractive one.
即使如此，当我想到让我永远,思考数学问题，或者说，让我永远思考哲学问题，显然和数学相比，我更钟情于它-,但这个提议也变得无趣。
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