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Q&A for professional mathematicians
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Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that $$\lVert(a_{ij})\rVert \le C \Bigl\lVert\s...
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Hottest Questions Today - MathOverflow
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Sum of modified $k$-fold divisor function
It is a well-known result (e.g. in Iwaniec-Kowalski's textbook) that\begin{equation*} \sum_{n\le x}\frac{d_k(n)^2}{n}\sim \frac{a(k)}{k^2\Gamma(k^2)}(\log x)^{k^2} \end{equation*}for an arithmetic factor $a(k)$, where $d_k(n)$ is the $k$-fold divisor...
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constructing a divisor of a projective bundle from a choice of sub-line bundle
Let $X$ be a smooth complex variety, and let $E$ be a locally free sheaf on $X$ of rank $r$. Let $\mathbb{P}(E)\simeq \mathcal{Proj}_X(\mathrm{Sym}(E^\vee))$ be the projective bundle of hyperplanes of $E$. If $L$ is a sub-line bundle of $E$, is there...
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Smallest $q$ such that $((2n+1)(2^m-1))\mid(2^q-1)$ with specific $m$
Let
$a(n)$ be the smallest positive integer $m$ such that $(2n+1)\mid(2^m-1)$.
$n$ be arbitrary positive integer. -
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Scaling laws and phase transitions in Tate-Shafarevich group distributions: empi...
In physics, dimensionless ratios (like Reynolds number) govern critical behavior; does arithmetic have analogous tools?
A Unified Theory connecting Elliptic Curves, Fluid Dynamics, and the Universal Constant 2/3
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