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7h
comment $\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials
In principal, you could write $f(t)$ as a linear combination of $f(t-2)$, $f(t-1-\frac{n-1}n)$, ..., $f(t-1-\frac1n)$, $f(t-1)$ (because finite-dimensional vector space). That would at least allow the integral $\int_1^2 f(t)^2\,dt$ to be written in terms of the values of $f$ between $-1$ and $1$....
8h
comment $\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials
I think they are both norms on the same finite-dimensional vector space, regardless of the fact that the integrals are over different intervals. So the OP's argument looks solid to me. ($C$ needs to depend on $n$, of course.)
8h
comment Largest prime factor of a Mersenne number with exactly two prime divisors
I recommend not getting too excited ... law of small numbers....
9h
comment Largest prime factor of a Mersenne number with exactly two prime divisors
I suspect such things will be very hard to prove; but the contrary statement, that $P(p)/\sqrt{M_p}$ is bounded, would be a massively surprising turn of events.
1d
comment Subfield Criteria - Proof or Counterexample
Jason, thank you for putting so much detail into your question! @MartinBrandenburg: interestingly, the closure of $\{x\}$ in $\Bbb F_2(x)$ is all of $\Bbb F_2(x)$: since $1,x\in X$ then all polynomials are in $X$, hence all reciprocals of polynomials. By one of Jason's lemmas, that means everything of the form $h(x)/g(x)^2$ is in $X$. In particular, $f(x)/g(x) = (f(x)g(x))/g(x)^2\in X$.
1d
answered Do primes modulo k form a normal sequence?
1d
comment Trouble Understanding Continuity Theorem
The domain $A$ having a hole isn't, in and of itself, an impediment to the function being continuous. To look at a counterexample, let $A$ be the circle in $\Bbb R^2$, and define $f\colon A\to\Bbb R$ to take a point to its angle (in the usual sense), so that $f(A) = [0,2\pi)$.
1d
comment Summing the sequence $a(n) = \sin(n x) \exp(-nt)$
Strants is addressing the first "challenge" in their answer. As for the second, of course such an example does not exist - a necessary condition for a series to converge is that the terms tend to $0$ in the limit. I never claimed that was possible. If that is what you meant by saying "$\sin \infty$ is not defined", then I for one don't find that terminology clear or accurate.
1d
comment Estimate for $n$th prime
I see no reason to think that each additional log in your integrals improves the accuracy of the estimate - unless the estimates are all underestimates, say, and slightly increasing the function gets closer just for size reasons (without approaching the true size), having nothing to do with the suitability of the complicated integrals to the task.
1d
comment Summing the sequence $a(n) = \sin(n x) \exp(-nt)$
I disagree. Yes, if you change the sum, then you can change it to something that unambiguously converges. But there are many choices one can make, and they lead to different answers. Also, "the expression $\sum \sin nx$ is not defined because $\sin \infty$ is not defined" doesn't make sense. No function on the real numbers is defined when one "plugs in $\infty$"; yet plenty of series converge.
2d
comment Maximum number of highways
It's certainly possible to have $\binom{19}2+1=172$ highways: include only one highway out of city #1 and all possible highways among cities #2-#20. Maybe try to prove that this is the most you can have?
2d
awarded  Nice Answer
2d
answered Thomae's function, doubt in continuous proof in the irrationals.
2d
answered Solutions of $a^{2} - 2b^{2} \equiv 0$ mod $p$
Nov
24
comment If $a \equiv b \bmod n$, then $\gcd(a, n)= \gcd(b,n)$
What have you tried?
Nov
24
comment Which integers are a sum of two relatively prime squares?
This is correct, and a proof can be found in lots of number theory textbooks - for example, Niven/Zuckerman/Montgomery.
Nov
24
comment When is it not safe to apply the approximation (1+a)^N = 1+Na (for a<<1)?
Yes, as you can check. Indeed, if $-1<x<0$ then you won't even need the $2^{N-2}$ term. (By the way, "$k$ is negative" is the kind of information that it's helpful to put in the question to begin with.)
Nov
23
comment How do I find vectors that are linear independent of another two vectors in $\Bbb R^5$
Note that "linearly dependent" applies to sets of vectors, not individual vectors. I would phrase the question as asking: find two vectors $u,v$ such that $\{x,y,u,v\}$ is linearly independent. And yes - randomly chosen vectors are extremely unlikely to be in the span of $\{x,y\}$, so just choose two vectors $u,v$ however you like and check whether $\{x,y,u,v\}$ is linearly independent (by putting them in a matrix and row-reducing to find its rank, for example - however you would normally do that).
Nov
23
comment Summing the sequence $a(n) = \sin(n x) \exp(-nt)$
"Mathematically allowed" depends on what you want to claim. If you want to claim that $\sum \sin nx$ converges to $\frac12\tan\frac x2$, that's not "allowed" because it's false. If you want to claim that the sum's value at $t$ has a limit as $t\to0^+$, and that limit equals $\frac12\tan\frac x2$, then that's "allowed" because it's true. If you have another statement in mind, then clearly stating it should allow you to determine whether it's true or false.
Nov
23
comment When is it not safe to apply the approximation (1+a)^N = 1+Na (for a<<1)?
Taylor's theorem, applied to the function $f(x)=(1+x)^N$ at $x=0$, tells us that $(1+x)^N = 1 + Nx + \frac12N(N-1)x^2 (1+\xi)^{N-2}$ for some $\xi\in[0,x]$. In particular, if $x\le1$, then $|(1+x)^N - (1 + Nx)| \le \frac12N(N-1)x^2 2^{N-2}$. (And you can clearly improve the constant at the end if you assume a smaller bound on $x$.)