不知为何,突然想起测度论里的不可测度的维塔利集合

复制以我写的知乎文章

我在知乎上写的目前竟是一些有关于美国华人和ABC和犹太人的政治话题,自己快成了民族活动家了,其实对于学理工科的人而言,民族活动家比较贬义。民族活动家似的言论与活动,尤其在美国,其实是自然被有能力的人所藐视的,这原则很简单,它根本就是不“专业”的表现,甚至可以说是一种流氓耍诬赖的作为。在美国,中国人政治上都是特别老实的,从来不闹事儿,不抗议,就服从性的低调的埋头苦干。相反,我看到过一位根本不黑但有黑人血统的数学研究生,他的数学水平其实很差的,与其他人相比,可是他却公开的支持Black Lives Matter,然后学校媒体却非常支持他,以他宣传自己的diversity,公布的视频里还有他说I didn’t have to think about race。有意思的是他根本不黑,要他不说,其实都看不出来他是黑人。所预料,这些在学校没人敢说的,说了都怕给自己惹麻烦,其实好多人都为此感到不满,但不得不不了了之,最终政治赢者是谁就毫无疑问了。

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Implicit function theorem and its multivariate generalization

The implicit function theorem for a single output variable can be stated as follows:

Single equation implicit function theorem. Let F(\mathbf{x}, y) be a function of class C^1 on some neighborhood of a point (\mathbf{a}, b) \in \mathbb{R}^{n+1}. Suppose that F(\mathbf{a}, b) = 0 and \partial_y F(\mathbf{a}, b) \neq 0. Then there exist positive numbers r_0, r_1 such that the following conclusions are valid.

a. For each \mathbf{x} in the ball |\mathbf{x} - \mathbf{a}| < r_0 there is a unique y such that |y - b| < r_1 and F(\mathbf{x}, y) = 0. We denote this y by f(\mathbf{x}); in particular, f(\mathbf{a}) = b.

b. The function f thus defined for |\mathbf{x} - \mathbf{a}| < r_0 is of class C^1, and its partial derivatives are given by

\partial_j f(\mathbf{x}) = -\frac{\partial_j F(\mathbf{x}, f(\mathbf{x}))}{\partial_y F(\mathbf{x}, f(\mathbf{x}))}.

Proof. For part (a), assume without loss of generality positive \partial_y F(\mathbf{a}, b). By continuity of that partial derivative, we have that in some neighborhood of (\mathbf{a}, b) it is positive and thus for some r_1 > 0, r_0 > 0 there exists f such that |\mathbf{x} - \mathbf{a}| < r_0 implies that there exists a unique y (by intermediate value theorem along with positivity of \partial_y F) such that |y - b| < r_1 with F(\mathbf{x}, y) = 0, which defines some function y = f(\mathbf{x}). Continue reading “Implicit function theorem and its multivariate generalization”

A nice consequence of Baire category theorem

In a complete metric space X, we call a point x for which \{x\} is open an isolated point. If X is countable and there are no isolated points, we can take \displaystyle\cap_{x \in X} X \setminus x = \emptyset, with each of the X \setminus x open and dense, to violate the Baire category theorem. From that, we can arrive at the proposition that in a complete metric space, no isolated points implies that the space uncountable, and similarly, that countable implies there is an isolated point.

 

Arzela-Ascoli theorem and delta epsilons

I always like to think of understanding of the delta epsilon definition of limit as somewhat of an ideal dividing line on the cognitive hierarchy, between actually smart and pseudo smart. I still remember vividly struggling to grok that back in high school when I first saw it junior year, though summer after, it made sense, as for why it was reasonable to define it that way. That such was only established in the 19th century goes to show how unnatural such abstract precise definitions are for the human brain (more reason to use cognitive genomics to enhance it 😉 ). At that time, I would not have imagined easily that this limit definition could be generalized further, discarding the deltas and epsilons, which presumes and restricts to real numbers, as it already felt abstract enough. Don’t even get me started on topological spaces, nets, filters, and ultrafilters; my understanding of them is still cursory at best, but someday I will fully internalize them.

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