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2 answers
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I am trying to compute the limit $ \lim_{n \to \infty} \left(\frac{2n - 1}{3n + 1}\right)^n, $ and I would appreciate some guidance. My first attempt was to apply the theorem stating that if $a_n \to ...
Aldo's user avatar
  • 231
0 votes
0 answers
14 views

In triangle $\triangle ABC$, the points $L, M$ are the midpoints of $BC$ and $CA$, respectively, and $CF$ is the altitude from $C.$ The circle through $A$ and $M$ which $AL$ is tangent to at $A$ meets ...
John O'neil's user avatar
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-7 votes
0 answers
36 views

Suppose we were to beam a 1024x1024 bitmap into interstellar space (or send a probe with a golden plate) and we wanted to describe how advanced our current level of mathematics is. It's clear aliens ...
Kirill Osenkov's user avatar
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1 vote
0 answers
23 views

While contributing to OEIS, I noticed that entries display a “status” field such as “proposed”, followed by numbers like “+12 −2”. For example, an entry may show: “Status: proposed +12 −2” I ...
Igor Blokhin's user avatar
  • 11
1 vote
1 answer
26 views

Let $\phi: A\to B$ be a ring map, $(f, f^\#):X=\mathrm{Spec}\ B\to Y=\mathrm{Spec}\ A$ the induced map of affine schemes. Now, for each point $x\in X,y=f(x)\in Y$, there is the stalk map $\mathcal{O}_{...
Academic's user avatar
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1 vote
0 answers
69 views

We want to prove that $\Omega^k[\mathbb{P^n}]=0$. Let`s consider the Euler exact sequence $$0\to \Omega^1[\mathbb{P}^n]\to O_{\mathbb{P}^n}(-1)^{\oplus (n+1)}\to O[\mathbb{P}^n]\to 0$$ where $E=O_{\...
Redpoint's user avatar
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2 votes
0 answers
37 views

I have faced the following problem: Let $M$ be a nonzero Noetherian module over a Noetherian ring $R$. Prove that there exists such a sequence of submodules $0 = M_k ⊂ M_{k−1} ⊂ ... ⊂ M_0 = M$, where ...
Gleb's user avatar
  • 21
-5 votes
0 answers
40 views

I cannot understand the description underlined in red. Because we need to estimate the $L^2$ norm of the supremum w.r.t. $\lambda$ of $T_{\lambda}^+f$. But in the following equality (The inequality ...
Ultraman ACE's user avatar
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1 vote
0 answers
21 views

Let $X$ be a smooth projective curve over an algebraically closed field $k$. A vector bundle $\mathcal{E}$ on $X$ is semistable if $\mu(\mathcal{F})\leq \mu(\mathcal{E})$ for any subbundle $\mathcal{F}...
user393795's user avatar
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5 votes
1 answer
50 views

Informally, I would like to find an infinite product of rational numbers that evaluates to a nonzero rational number such that the multiplicity of each prime in the numerator is finite, while on the ...
Alex's user avatar
  • 51
1 vote
1 answer
81 views

I'm studying complex analysis via Conway's Functions of one Complex Variable. My teacher is following Stein's Complex Analysis instead so I'm learning things a bit out of order. I was given an ...
Arthur Farias Zaneti's user avatar
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0 answers
48 views

If $a,b,c \in \mathbb{R^+}$ and $a+b+c = 3$, prove that $\left(\sum_{cyc}\frac{1}{a^6+b^6+3c^3+4}\right)\leq\frac{3}{3+2\sqrt{ab}+2\sqrt{ac}+2\sqrt{bc}}$ I am not able to find a way to link it to an ...
Nineta's user avatar
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1 vote
0 answers
30 views

I'm reading Schur's theorem on primes in some arithmetic progression. https://projecteuclid.org/journalArticle/Download?urlId=10.7169%2Ffacm%2F1229442627 I do not understand a step in the proof of the ...
Jorge's user avatar
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0 votes
1 answer
92 views

While exploring the area of a regular $n$-gon inscribed in a unit circle ($r=1$), I arrived at the equation: $$\frac{n}{2}\sin\left(\frac{2\pi}{n}\right) = 1$$ or equivalently: $$n\sin\left(\frac{2\pi}...
Darko Lovrincevic's user avatar
  • 11
0 votes
0 answers
14 views

Let $$\mathcal{C}_n(T)=\#\left\{(a_1,\dotsc ,a_n)\in(\mathbb{Z}^+)^n:\underbrace{\gcd(a_i,a_j)=1}_{\forall i\neq j:}, a_1\dotsi a_n(a_1+\dotsc +a_n)\leq T\right\},$$ and also define $$P_n(T,q)=\frac{1}...
user1668001's user avatar
  • 176

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