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Collatz variant: odd(3n+1), odd(3n+1), odd(n+1)
Notation: odd($3n+1$) is the standard Collatz operator: for odd $n$ build ($3n+1$) and then divide by $2$ until an odd number results. Analogously odd($n+1$) is defined.
The Collatz conjecture says ...
- reputation score 77
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Examples of dense pointless sublocales of $\mathbb{R}^m$, or other ways to build intuition about them
(Experts can immediately jump to the ‘❦’ sign below, or even directly to the question, skipping the following 3 paragraphs, which are included for completeness of MathOverflow, for the benefit of ...
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Can a submanifold with boundary always be extended?
Let $M$ be a $m$-dimensional submanifold of $\mathbb{R}^n$ with boundary. Is it always possible to find a $m$-dimensional $\tilde M$ submanifold of $\mathbb{R}^n$ without boundary such that $M\...
- reputation score 363
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Does a Hausdorff fixed-point property on Jaccard metric spaces imply Frankl's union-closed sets conjecture? [closed]
I have developed a proof that resolves and generalizes Frankl's union-closed sets conjecture by shifting the problem from a purely combinatorial framework to a topological one. Specifically, I ...
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Why totally real cubic field, $3$ completely splits with $A'_0\ne 0$ is very rare?
Recently, I had the opportunity to find and read the paper on $\mathbb Z_p$-extensions written by Taya: On $p$-adic zeta functions and $\mathbb Z_p$-extensions of certain totally real number fields. ...
- reputation score 383
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$f(s) = \prod_{p>3} (1 + \frac{2}{p^s - 2}) \implies f(1 + t i) \neq 0$ for real $t>0$?
Let
$$f(s) = \prod_{p>3} (1 + \frac{2}{p^s - 2})$$
And I conjecture
$$f(1 + t i) \neq 0$$
for real $t>0$
how to show this ?
Does the log trick for the regular Riemann zeta function and the PNT ...
- reputation score 753
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Group homomorphism $f:(\mathbb{R},+)\to(\text{Sym}(\mathbb{N}),\circ)$
Let $\newcommand\G{(\text{Sym}(\mathbb{N}),\circ)}\G$ be the group of all bijections $\varphi:\mathbb{N}\to\mathbb{N}$ with composition as group operation.
Is there a group homomorphism $f:(\mathbb{R},...
- reputation score 58225
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Pushouts of small presheaves
Let $\mathcal{C}$ be a locally small category. I do not assume that any limits or colimits exist in $\mathcal{C}$. Let $F$ be a small presheaf on $\mathcal{C}$, and let $E$ be a subsheaf such that the ...
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What is an unchanging derivatives in calculus [closed]
It was given to me by our lecturer, we are doing calculus and he gave us the topic unchanging derivatives to tackle.
He also said we should present one real item that can be used to represent ...
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Reference request: W. Floyd's paper on Heegaard splittings
I am having trouble tracking down reference [2] in Masur's paper "Measured foliations and handlebodies" (Ergod. Th. Dyn. Syst. 6, 1986):
[2] W. Floyd. Hyperbolic manifolds, 3 manifold ...
- reputation score 692
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Log-Sobolev type estimate in one dimension? $L^p$ integrability of $\log(|u|)\mathrm Du$
I am able to prove the following:
Theorem 1: Let $u\in C^1([0,1])$ be real analytic, witb $u(0)=0$, $u>0$ in $(0,1]$ . Then, $$\log(|u|)\mathrm Du\in L^p((0,1)) \\ \text{for all}~1\leq p<\infty$...
- reputation score 603
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Is the category of small categories and exponentiable functors presentable?
Recall Conduché's theorem a functor $E \xrightarrow F B$ is exponentiable if and only if, for every composable pair $x \xrightarrow f y \xrightarrow g z$ in $B$, the commutative square
$\require{AMScd}...
- reputation score 68530
Score of 3
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Finding all permutations "without rising or falling 2-sequences"
The number of permutations on $\{1,2,...,n\}$ without any adjacent numbers being consecutive is counted in Abramsons and Mosers work. According to them the number for $n = 7$ should be $462$ (p.9). ...
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Pro-algebraic completion of a product of groups
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, if you prefer you may choose $\mathbb{K}=\mathbb{C}$.
For clarity, I recall the following definitions.
Definition 1. A ...
- reputation score 1028
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Why does spacetime curvature not become zero due to space expansion and eliminate gravity? [closed]
My question is: if space is expanding, then why does the curve not become zero? Meaning, if we take a flat space and place a Sun in it, and then pull space from all sides, the curve will become zero. ...
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