network profile
| bio | website | |
|---|---|---|
| location | Germany | |
| age | ||
| visits | member for | 2 years, 9 months |
| seen | May 4 at 7:50 | |
| stats | profile views | 30 |
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Apr 10 |
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Taylor's polynomial (existence?) proof He took a polynomial which we know exists (because it is explicitly given) and showed that it has the desired property. I mean, what else is there to prove? (I find the way the statement is written somewhat confusing, maybe that is your problem. Making $r$ a function of $(x-a)^n$ instead of $x$ is, in my eyes, rather pointless; I'm not even sure if it is strictly speaking correct. I'd go with $r(x)$ and $\lim_{x\to a}\frac{r(x)}{(x-a)^n}=0$.) |
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Apr 10 |
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Show that an element belongs to unique factorization domain. Too true, my memory was faulty. The order of proofs is exactly the other way round (and it seems that Gauss' Lemma is around 45 years older). |
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Apr 10 |
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Taylor's polynomial (existence?) proof He proved that $P_{n,a}$ has the required property. Which also proves that such a polynomial does exist – that is a non-trivial result. |
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Apr 9 |
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Show that an element belongs to unique factorization domain. Gauss' lemma, which follows from Eisenstein, implies that if a monic polynomial over a UFD $D$ has a zero over the quotient field of $D$, then that zero is an element of $D$, no? (I agree, a pointer to Gauss' lemma would have been more appropriate.) |
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Apr 9 |
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Show that an element belongs to unique factorization domain. Scrap that. I thought I had learned Eisenstein for UFDs, but cannot find a reference to back that up right now, so my memory is probably faulty. |
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Apr 9 |
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Show that an element belongs to unique factorization domain. If you're looking for a bigger hammer, the name Eisenstein might help. |
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Apr 9 |
answered | Taylor's polynomial (existence?) proof |
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Apr 9 |
awarded | Commentator |
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Apr 9 |
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Range of $ax+by$ where $\gcd(a,b)=1$ Is “$\mathbb{Z}$ is a principal ideal domain” acceptable as “some other method”? |
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Nov 19 |
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Analytic method for determining if a function is one-to-one Actually, what you should learn is that there is an inverse function for a one-to-one mapping. That does not necessarily imply that there is any way of writing it down as a formula. As a famous example, take $f = x\mapsto x \mathrm{e}^x$. As a map from $[0,\infty)$ to $[0,\infty)$, this is a bijective function, so it is invertible, but the inverse cannot be written with standard functions, which is why it's now called Lambert's W function. Or take a function like $x\mapsto x+sin(x)$, which is invertible but whose inverse probably didn't get a name yet … |
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Nov 19 |
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Re-arranging an equation with a square root If you want extra points for precision, note down “if $\Re a>0$” after your answer. For additional credit, extend that condition to the full set of $a$ for which $a=\sqrt{c}$ has an answer, it's very similar to the condition above … |
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Oct 20 |
awarded | Teacher |
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Oct 20 |
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What is an alternative formulation to a contour integral? Symbolically, differentiation is way easier than integration. Numerically, it's the other way round. |
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Oct 20 |
answered | θ = (length of arc)/(angle subtended by it). How? |
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Sep 9 |
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Is a field (ring) an algebra over itself? +1: Useful information, but still reminds the poster to try answering the question right from the definition. (Personally, I assume he did, but it would be useful to indicate that in the question, something like, “following the definition, I believe the answer is yes, can someone confirm that?”) |
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Sep 1 |
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Finding the Heavy Coin by weighing twice @Dan: Obviously not. E.g., any set of five coins will contain a heavy one. |
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Sep 1 |
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What is the value of $1^i$? @Carl: “Complex exponentiation is only ever well-defined relative to a choice of a branch of the logarithm.” – Absolutely true. So we should pick a branch (there is a common way of doing that, so if that choice does what we need, we should), then we can call the logarithm and $(a,b) \mapsto a^b$ a “function” again and continue with more interesting questions. |
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Sep 1 |
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What is the value of $1^i$? @LL: I do not agree with your last comment. I do not really see any reason to make $(-1+i)^{1+i}$ multivalued – it seems to me that useful computations are much easier when you pick one branch of $\ln$ and stick to it, making $a^b$ denote exactly one complex number for complex numbers $a$ and $b$ with $a \neq 0$ (or $b > 0$ for $a=0$). There's enough literature out there doing exactly that, try “Can Your Computer do Complex Analysis?” by Aslaksen or Knuth et al.'s papers on the Lambert W function. |
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Aug 20 |
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Why is $22/7$ a better approximation for $\pi$ than $3.14$? Since you are using regular continued fractions, why “essentially unique?” The a_n sequence is unique, is it not? |
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Aug 20 |
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Best intuitive metaphors for math concepts (of any level) I don't think the zooming in part is a metaphor for the completeness axiom. You can zoom in indefinitely on the rationals, too, without ever “seeing” a difference. (More formally: for any rational p < q, the rational interval [p, q] is isomorphic to the rational interval [0, 1], exactly as in the reals.) Perhaps I just don't find “doing something infinitely often” as intuitive as you might. |
