Take some self map on the unit disk , . If , has a removable singularity at . On , , and with the maximum principle on , we derive everywhere. In particular, if anywhere, constancy by the maximum principle tells us that , where . with the removable singularity removed has , so again, by the maximum principle, means is a constant of modulus . Moreover, if is not an automorphism, we cannot have anywhere, so in that case, .
I took a graduate course in complex analysis a while ago as an undergraduate. However, I did not actually understand it well at all, to which is a testament that much of the knowledge vanished very quickly. It pleases me though now following some intellectual maturation, after relearning certain theorems, they seem to stick more permanently, with the main ideas behind the proof more easily understandably clear than mind-disorienting, the latter of which was experienced by me too much in my early days. Shall I say it that before I must have been on drugs of something, because the way about which I approached certain things was frankly quite weird, and in retrospect, I was in many ways an animal-like creature trapped within the confines of an addled consciousness oblivious and uninhibited. Almost certainly never again will I experience anything like that. Now, I can only mentally rationalize the conscious experience of a mentally inferior creature but such cannot be experienced for real. It is almost like how an evangelical cannot imagine what it is like not to believe in God, and even goes as far as to contempt the pagan. Exaltation, exhilaration was concomitant with the leap of consciousness till it not long after established its normalcy.
Long time ago when I was a clueless kid about the finish 10th grade of high school, I first learned about Euler’s determination of . The technique he used was of course factorization of via its infinitely many roots to
The Cayley-Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation. That is, with the identity matrix, the characteristic polynomial of
The implicit function theorem for a single output variable can be stated as follows:
Single equation implicit function theorem. Let be a function of class on some neighborhood of a point . Suppose that and . Then there exist positive numbers such that the following conclusions are valid.
a. For each in the ball there is a unique such that and . We denote this by ; in particular, .
b. The function thus defined for is of class , and its partial derivatives are given by
Proof. For part (a), assume without loss of generality positive . By continuity of that partial derivative, we have that in some neighborhood of it is positive and thus for some there exists such that implies that there exists a unique (by intermediate value theorem along with positivity of ) such that with , which defines some function . Continue reading “Implicit function theorem and its multivariate generalization”
In a complete metric space , we call a point for which is open an isolated point. If is countable and there are no isolated points, we can take , with each of the 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.
The Urysohn metrization theorem gives conditions which guarantee that a topological space is metrizable. A topological space 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.
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.
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.
I’ve been reading some algebraic topology lately. It is horrendously abstract, at least for me at my current stage. Nonetheless, I’ve managed to make a little progress. On that, I’ll say that the path lifting lemma, a beautiful fundamental result in the field, makes more sense to me now at the formal level, where as perceived by me right now, the difficulty lies largely in the formalisms.
Path lifting lemma: Let be a covering projection and be a path such that for some and , Continue reading “Path lifting lemma and fundamental group of circle”