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Jun
29
comment A good book on humankind’s understanding of primes?
Thank you for your recommendation. The book seems to be on freshman level, though, and only covers the Riemann hypothesis as you already stated. Anyway, thanks for bringing it up!
May
31
comment Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.
@Andrew You mean “$f(x) ∈ C$”. Other than that, the preimage of $C$ under $f$ is defined as $f^{-1}(C) = \{x ∈ X;~f(x) ∈ C\}$ and can be defined for any map $f$ – it doesn’t need to be surjective. (Maybe you thought of $f^{-1}(C)$ as the image of an inverse $f^{-1}$ of $f$ which would only exist if $f$ was already bijective. But that’s not how the symbol “$f^{-1}$” is used in this case.)
May
29
comment Why is the cartesian product so categorically robust?
I don’t know all of these categories, but – at least as far as I know – for most of them the corresponding forgetful functor has a left adjoint, thus is continuous. Morally speaking, I’d guess that so many of these forgetful functors have left adjoints because mathematicians (for historic reasons alone) tend to build stuff from sets and that already suggests a forgetful functor as well as an left adjoint for it.
May
28
comment How to not feel bad for doing math?
@AsafKaragila Why not? It’s a real concern to some and I guess it can be a real mood killer if you’re considering spending 3–5 years working hard on one particular problem/question. Even though I don’t pursue a career as a mathematician, I can relate to this concern. Why not take it seriously?
May
28
comment How to not feel bad for doing math?
@Did I added a comma and read the sentence as “It’s the only job I can think of, which doesn’t contribute to bettering the world somehow.”.
May
23
comment How to insert Gothic letters in Word?
That’s not a math question, though.
May
21
comment Proving that the ball is converx
Invoke the definition of convexity and the triangle inequality.
May
15
comment If a field extension contains a cyclotomic extension is it solvable?
Recall that a solvable group requires a subnormal series in which all factors are abelian. What can you really say about $\mathrm{Gal}(L/M)$? As an extreme case, take any non-solvable extension $L/K$ and choose $M = K$, which trivially is a cyclotomic extension.
May
9
comment Some very short clarification on quotient groups
@elDin0 No. Your problem is on set-theoretic level: It seems like you are not differentiating between an element of a set and an element of an element of a set. Take your favorite mathematical starting object $x$ (if you’re a minimalist, choose $x = ∅$, the empty set) and then consider $y = \{x\}$ and $z = \{y\}$. Then $z ≠ \{x\}$ because the only element of $z$ is $y = \{x\}$ and $x ≠ \{x\}$. Likewise, $\{A_3, (1~2)A_3\} ≠ \{ \text{elements of $A_3$}, \text{elements of $(1~2)A_3$}\}$.
May
9
comment Some very short clarification on quotient groups
Do you know the isomorphism theorem for groups? How is $A_3$ defined for you? Do you know how many elements $A_3$ and$S_3$ have?
May
8
comment Looking for Open Source Math Software with Poor Documentation
I upvoted for your charity intentions solely. Felt right.
May
7
comment Existence of a holomorphic function with the desired property
@learnmore Linear maps are holomorphic, what are the linear maps $ℂ → ℂ$? Which one of these restrict to $D → D$?
May
7
comment Existence of a holomorphic function with the desired property
@Blake $f(D) ⊂ (\overline D)° = D$.
May
7
comment Existence of a holomorphic function with the desired property
$\frac{3}{4}·\frac{1}{3} = \frac{1}{4}$ and $\lvert\frac{3}{4}\rvert < 1$. Otherwise, use the open mapping theorem and the Schwarz lemma.
May
7
comment Connected spaces where all subsets are either open or closed
@bof Ah, neat. Thanks. One-point spaces being the least interesting examples imaginable, though. But the link actually proves there are no other.
May
6
comment A path to more advanced math topics.
I already upvoted it, but I’d like to verbally support the recommendation made by Hasan as well: If you know about matrices, determinants, vectors, vector products and such and you are intrigued by algebra, chances are you will find linear algebra extremely enlightening. It’s probably the most natural thing to go to if you want to get introduced to more abstract and structural mathematics. It is also helpful, if not necessary, for studying multivariate calculus and abstract analysis – which would be the next step on the calculus path.
May
6
comment Definition of a prime
“Now, $6$ is not a unit in $ℤ$” is what that should read.
May
6
comment formal power series expansion for square root
@OneWingedPterodactyl And the index in the sum is $n$, not $i$.
May
6
comment formal power series expansion for square root
Hm, your series suffers from index sickness.
May
6
comment Why does an isomorphism need to be a homomorphism?
Also, I think this answer is missing the point of the question.