## Understanding Human History

I had the pleasure to read parts of Understanding Human History: An Analysis Including the Effects of Geography and Differential Evolution by Michael H. Hart. He has astrophysics PhD from Princeton, which implies that he is a serious intellectual, though it doesn’t seem like he was quite so brilliant that he could do good research in theoretical physics, though an unofficial source says he worked at NASA and was a physics professor at Trinity University who picked up a law degree along the way. I would estimate that intellectually, he is Steve Hsu level, perhaps a little below, though surely in the high verbal popularization aspect, he is more prolific, as evidenced by that book, among many others, such as one on the 100 most influential historical figures. He is active in white separatist causes (heh) and appears to have had ties with the infamous and now deceased Rushton.

## 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.

## On questioning authority

A couple years ago, my friend who won high honors at the Intel Science Talent Search told me that he was talking this guy who created some app that allows you to schedule a Uber ride for later, who was also at/near the top of the same science competition, who is extraordinarily versatile and prolific. I watched a little of a video of a TED talk he gave, wherein he explained what one can learn from ancient Hebraic texts. Overall, I wasn’t terribly terribly impressed by it, though it was quite eloquently delivered. Mostly because with those types of things, one is too free to interpret and thus, the lessons/messages given were overly generic so as to make them almost meaningless, one of which was how the Bible teaches the importance of questioning authority, with reference to the refusal to bow to the golden image of King Nebuchadnezzar by Shadrach, Meshach, and Abednego as an exemplary.

## Israel, China, and more

I figured that as interested in Jews and Jewish achievement (and shenanigans) as I am, I should at least learn something real about Israel, which I know little about at the detailed factual level. That part of the world has, predictably, always felt rather remote in my life, though it is in some sense the cradle of civilization. While on the bus with nothing to do, I was just last week, trolling some of my friends on Facebook with some Hebrew I copy pasted. Like, ברוך השם (Baruch HaShem), which literally means “blessed his name.” On that I’m pleased to say that I’m now sort of paying attention to the letters of the Hebrew alphabet when I visit English Wiki pages on Jewish matters with the English transliteration of Hebrew words alongside the Hebrew original. It’s kind of cute that it, like Arabic, reads right to left, a fact I had not known.

## Urysohn metrization theorem

The Urysohn metrization theorem gives conditions which guarantee that a topological space is metrizable. A topological space $(X, \mathcal{T})$ is metrizable is there is a metric that induces a topology that is equivalent to the topological space itself. These conditions are that the space is regular and second-countable. Regular means that any combination of closed subset and point not in it is separable, and second-countable means there is a countable basis.

## Jewish pro-Americanism

In America, people often bring up what they view as China’s suppression of free expression. I personally dislike strongly the usage of “free expression,” because it is meaninglessly vague. And there is no such thing as free expression in the strictest sense of it. Especially when you are in a job dealing with a boss who can fire you, which is why politics is generally supposed to be a no-no in the workplace, discussion wise. People avoid it out of prudent protection of their careers. One naturally feels at disease when what one wishes to express is such that is unwelcome or hostile in the environment of one’s residence. In such case, one feels that his or her right of free expression is being beaten down. This is very much the case in America right now, in many places.

## Riemann mapping theorem

I am going to make an effort to understand the proof of the Riemann mapping theorem, which states that there exists a conformal map from any simply connected region that is not the entire plane to the unit disk. I learned of its significance that its combination with the Poisson integral formula can be used to solve basically any Dirichlet problem where the region in question in simply connected.

## 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.