Top new questions this week:
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The Question is simple, yet I have encountered it multiple times in my mathematical life without finding an obvious answer, so I've decided to post it here.
Say $U, V$ are real vector spaces of ...
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$2^n$ players $P_1, \dots, P_{2^n}$, ordered in decreasing order of skill are placed uniformly at random at the leaves of a binary tree of depth $n$.
They play a knockout tournament according to the ...
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Let $f(s)$ be a Dirichlet series with algebraic coefficients, and suppose that:
it admits a meromorphic continuation to $\mathbb{C}$;
there exists $d \in \mathbb{N}$ such that $f(2n) \in
\...
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Denote $B_n$ as Bernoulli numbers, it is known that the following three identities hold (for $n\in \mathbb{N}$):
$$\tag{A}\label{504268_A}\frac{(2n)!}{(4n+1)!} \frac{-B_{6n+2}}{6n+2}= \sum_{k=0}^{2n} \...
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Let $C$ be a smooth curve of genus $g\ge 2$ over a number field $K$, and let $J$ be its Jacobian variety. Suppose we have an embedding $C\to \mathbb{P}^2$ so that $C$ is defined by a homogeneous ...
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Is the following (second-order) formula schema provable in ATR$_0$?
Let $\varphi$ be an arithmetical formula satisfying
For all $x, y\in \mathbb{R}$,
we have that $x=_\mathbb{R}y$ implies $\varphi(x)...
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Let $M$ be an orientable $n$-dimensional manifold with a smooth oriented atlas $A :=\{(U_\alpha,\Psi_\alpha)\}_{\alpha}$ so that every non-empty $U_\alpha\cap U_\beta$ is contractible.
Then we can ...
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Greatest hits from previous weeks:
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Reading mathematical articles, I sometimes see how mathematicians pull out amazing spectral sequences seemingly at will. While many are built using standard techniques like exact couples or filtered ...
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Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose $A$ is an abelian group such ...
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I am working on my zero knowledge proofs and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't ...
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How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
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I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
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The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
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I would like to ask if anyone could share any specific experiences of
discovering nonequivalent definitions in their field of mathematical research.
By that I mean discovering that in different ...
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Can you answer these questions?
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The following fact is well-known, and not hard to prove, but I do not know an explicit reference.
Let $R$ be the subring of complex numbers generated by all roots of unity. Then $R$ is free as an ...
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Set
$$
\lambda _{1}<\cdots<\lambda _{p}<0=\lambda _{p+1}<\cdots<\lambda _{n},
$$
consider the following system:
$$
\partial _{t}+\lambda _{i}\partial _{x}u_{i}=\sum _{j\neq i}\lambda _{...
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Let $p$ be a prime, $p\neq 2$. Let $Q$ be a cyclic $2$-group and $P$ an elementary abelian $p$-group of rank $n$. Suppose that $Q$ acts faithfully on $P$ (so $Q$ is isomorphic to a subgroup of $GL(n,...
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