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A few recommendations: Fernando Q. Gouvêa's $p$-adic numbers: An Introduction. This text gives a good introduction to the $p$-adic number system and the properties of the space of $p$-adic numbers, vector spaces over $\mathbb{Q}_p$, and the metrically completed algebraic closure of $\mathbb{Q}_p$. There is also some discussion of $p$-adic analysis near ...


4

I do not believe there are any good such strategies, especially if you are in a Ph.D. program. I think your question is an important one because the comprehensive exam system at some places conveys some need to desperately learn lots of mathematics quickly, and as a result, many students feel this sort of pressure. The late Ken Appel told me once that nobody ...


3

3 blue 1 brown has an outstanding video on Fourier series: https://www.youtube.com/watch?v=r6sGWTCMz2k Thinking of a periodic complex valued map as a parametric curve in the complex plane, and thinking of the complex exponentials as "clocks" running at different speeds, the problem becomes: how do we draw anything using a bunch of revolving clocks ...


3

Another option (I have never attempted this) would be to claim the existence of a thing called "The Lebesgue integral", list some carefully chosen theorems about it as axioms (maybe just linearity, some inequalities, and dominated convergence), and treat it as a black box for your course. You could give an extremely high level sketch of how this ...


2

We can think of these transformations from two perspectives: as mappings of the plane or as graphs of pre/post composition of the function. For instance, the map $T: \mathbb{R}^2 \to \mathbb{R}^2$ given by $(a,b) \mapsto (a+2,b)$ is the translation of the plane two units to the right. Let $\phi: \mathbb{R} \to \mathbb{R}$ be the function $\phi(x) = x-2$. ...


2

The word covered is unproductive, as it is so teacher-oriented. Any amount of material can be covered in 12 hours, provided that you can speak and write fast enough, and when you reach your limit, you prepare slides full with formulas and flip them faster than anyone can even read. The material is covered, right? If you want a streamlined presentation, take ...


2

In my opinion whether this proof is correct or not depends on the actual wording of the question. This recursive argument shows that the statement is true for 100 numbers (since it is true for 2, 3, 4, so we eventually reach 100) It also shows that the statement is true for 1000 numbers (we eventually reach 1000). For any fixed natural number this is a ...


1

I'll give a less elegant answer: because the dilations and translations are easy to understand and implement. If we were interested in preserving shape under the transformations then surely rigid motions or orthogonal transformations would be a better scope for the discussion. Sadly, highschool students (in the USA) do not typically learn linear algebra, ...


1

Gunaydan. Here is a lit review of US math elementary textbooks evolution: https://www.jstor.org/stable/41103881?seq=1 Realistically your question is so broad that I think you need to learn more before you can even ask the right questions. But maybe this will get you thinking. For example, I would think about restricting your scope to two to three countries ...


1

The discussion seems to overlook this simple fact: the Lebesgue integral is not a generalization of and cannot serve as a substitute for the Riemann integral.


1

There's a really nice book by Svetlana Katok called "p-adic analysis compared to Real."


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