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I really appreciate it when you take time to answer my questions. Thanks!


Jan
17
comment Representation and the central of a algebra
You can try to cook up some intertwining maps, for example, $z - c \cdot \mathrm{id}$ for suitable constant $c$. (Assuming that your field is algebraically closed)
Jan
17
comment branch of logarithm
@mrf, $1/z$ having an antiderivative in a domain would imply that the integral of $1/z$ along any closed curve in the domain to be zero. This has nothing to do with the simply-connectedness of the domain.
Jan
17
revised If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
added 44 characters in body
Jan
17
answered If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
Jan
17
comment Primes congruent to 1 mod 6
@Math2012pc, one possible answer is from the arithmetic of Eisenstein integers. There is probably an answer in the book "Primes of the form $x^2+ny^2$", but it would be better if you say a few words about your background, i.e. do you know what PID is? Do you know anything about splitting of primes? Quadratic reciprocity?
Jan
17
comment Is the number of alternating primes infinite?
@proximal, as Old John said, $a^2-b^2 = (a-b)(a+b)$ is all you need - not even modular arithmetic.
Jan
16
comment Is every contractible space a cone?
I think that this answer is wrong. You are probably thinking about math.stackexchange.com/questions/150751/… but a contractible space does have the homotopy type of a point.
Jan
15
comment What academic level would one need to be at to fully understand papers published on the twin prim conjecture?
I would say undergraduate is enough - this article is very well written and is self-contained. A plus is that there isn't much specialized vocabulary in this article. (when compared to algebraic geometry, for example)
Jan
14
comment Proof of floor of division
$t \leq b-1$ and $r < a$. What can you infer?
Jan
13
comment $f$ an endomorphism of $V$, $f^m=\operatorname{id}(V)$. Show $f$ is diagonalizable?
@JKH, yes, that would be sufficient to show that $x^m-1$ splits into distinct linear factors.
Jan
13
comment $f$ an endomorphism of $V$, $f^m=\operatorname{id}(V)$. Show $f$ is diagonalizable?
@JKH, if you are talking about complex numbers, then $x^m-1$ splits into distinct factors indeed.
Jan
13
comment $f$ an endomorphism of $V$, $f^m=\operatorname{id}(V)$. Show $f$ is diagonalizable?
Do you know that $f$ is diagonalizable if the minimal polynomial splits?
Jan
13
comment Real valuations on Dedekind domains
When you say valuations, do you mean discrete valuations?
Jan
13
comment Modular form weight 0
On the other hand, note that boundedness on upper half plane does not imply constant in general.
Jan
13
comment Modular form weight 0
A modular form should be regarded as a holomorphic function on the compact Riemann surface $\mathcal{H}^*/\Gamma$, which is $\mathcal{H}/\Gamma$ together with $\infty$. Open mapping theorem would then force it to be constant.
Jan
13
comment Positive Definite Matrix Question
You can assume that $M$ is diagonal. Can you solve it then?
Jan
13
comment Prove that: $\nu \ll \mu$ iff $|\nu| \ll \mu$
Hahn decomposition tells you that for measurable $E$, you can find $P \cup N = X$ and $P \cap N = \emptyset$ such that $v^+(E) = v(P \cap E)$, $v^-(E) = -v(N \cap E)$. Apply absolute continuity to $E \cap P$ and $E \cap N$.
Jan
13
comment Is my understanding of product sigma algebra (or topology) correct?
@Tim, at least for the case of topological spaces, (1) would give you a basis rather than a subbasis, which could be more useful in various situations.
Jan
13
comment The definition of normal covering in Hatcher book
That's right :) Now a normal covering, means that for any two points in the preimage of the same point, there's an automorphism of the covering (called deck transformation), which preserves the covering map and would send one point to another. In this case, it means that for the two preimages in the center of the covering/the two preimages on the left & right of the covering, there's a symmetry of the covering that sends one point to the other point. Is there any symmetry you can think of that does just that?
Jan
13
comment The definition of normal covering in Hatcher book
OK. In fact, what you said about lifts is slightly off-topic - in this context we are really talking about the preimage of the covering map, where $a$ is sent to the circle on the left of $S^1 \vee S^1$ and $b$ is sent to the circle on the right of $S^1 \vee S^1$. What is the preimage of a generic point of the left circle? What is the preimage of a generic point of the right circle?