# Tag Info

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The null factor law is as follows: $$ab = 0 \Rightarrow (a = 0) \vee (b = 0).$$ This law applies for real numbers, as well as polynomials which is where the law is most commonly envoked. I have seen far too many instances of the following incorrect generalisation: $$ab = c \Rightarrow (a = c)\vee(b = c).$$

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Recently, a friend of mine pointed out the following to me: The open unit disc $D\subset\mathbb{C}$ is not biholomorphic to all of $\mathbb{C}$. Indeed they are diffeomorphic, but we can easily see that they are not biholomorphic since if there was a biholomorphism $\phi:\mathbb{C}\rightarrow D$, then consider the function $f:D\rightarrow\mathbb{C}$ given ...

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If you want to understand well topological vector spaces (TVS), and learn all the basic theorems of functional analysis in their most abstract version, which is usually in the context of TVS and not of normed spaces, I am afraid there is no speedy way. (No royal way to geometry, as the ancient greeks said.) I would recommend the book of W. Rudin, Functional ...

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I would recommend the following book: Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev

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As a high school student, it appears that there is some well defined natural order of mathematics, algebra, geometry, algebra II, ect... This is undoubtedly false. There is quite a bit of interesting and enjoyable math which is not covered in your average high school curriculum, but does not require any basis in either calculus, linear algebra or frankly ...

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I would begin going over the new material, and review on a need basis, as I go over new topics.

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Why do professors go through proof after proof with no rhyme or reason? One theory is that this is an "easy" way to give a lecture (to be negative about it, a "lazy" way.) This may be true in some cases. But on the other hand, much of the instructor's education might have been this way, and maybe they even think the experience is valuable. So, they ...

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One could argue that it is up to you as the student to learn. The best way to learn mathematics is by doing mathematics. By doing exercises, good exercises, the context and power of the material should come to life. A lot of the proofs and exercises in a ring theory course are just careful application of the definitions and theorems. Good luck.

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I don't think there is a definite answer to that question. It depends on a lot of factors, most important of which is the way you learn best. I'm currently doing my physics PhD and I have classmates who learn the underlying math best by learning it through the physics. That's fine. Before starting my physics PhD, I had only one math class at the ...

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Another way to look at this is the following: recall that $$\liminf_{n \to \infty} \frac{c_{n+1}}{c_n} \leq \liminf_{n \to \infty} \sqrt[n]{c_n} \leq \limsup_{n \to \infty} \sqrt[n]{c_n} \leq \limsup_{n \to \infty} \frac{c_{n+1}}{c_n},$$ where $c_n > 0.$ Then you obtain the desired result for the constant sequence $c_n \equiv p.$

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The log trick is always helpful, when your base is non-negative: $p^{1/n}=e^{(1/n)lnp}$. Now, $lnp$ is fixed, and $(1/n) \rightarrow 0$ , so you can find an $M$ , so that for $n>M$ , $\frac {1}{n}<\frac{1}{\epsilon(lnp)}$ .... EDIT: Another useful trick is that of sequential continuity, which works on the Reals: the idea is : given a sequence $x_n ... 1 Here is another avenue. Suppose$p \ge 1$. Then by the Geometric Series Theorem, $$\root{n}\of{p} - 1 = {p- 1\over \sum_{k=0}^{n-1} p^{k/n}}\le {p-1\over n}.$$ Hence$\root{n}\of {p} \to 1$as$n\to\infty$. If$0 < p < 1$, you can apply this result to$1/p\$.

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I guess it depends on your preferences. I once read into the first pages of Mac Lane and t.b.h. I didn't like it at all - the definitions and theorems aren't numbered, which I consider a sin for any math book. They don't even really stand out typographically, except for being written in italics. And there's way too much babbling around for my taste. Don't ...

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The aperiodical (http://aperiodical.com/) is basically a maths news site. They mention most major mathematical news stories (recent big stories are the abc conjecture, the odd Goldbach conjecture and the ongoing prime gaps work). The website also has a lot of fun mathsy articles which are worth checking out in addition to news. I hope this is the sort of ...

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