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## representing an uncountable free group as a union of an increasing sequence of countable subgroupsLet $(G_\alpha)$ and $(K_\alpha)$ $(\alpha<\aleph_1)$ be strictly increasing chains of countable sets such that if $\alpha$ is a limit, then $G_\alpha=\bigcup_{\beta<\alpha}G_\beta$ and $K_\alpha=\bigcup_{\beta<\alpha}K_\beta$. Assume $\bigcup_{\alpha<\aleph_1}G_\alpha=\bigcup_{\alpha<\aleph_1}K_\alpha$. Does there exist a club $C$ such that $K_\gamma=G_\gamma$ for all $\gamma\in C$? In the paper ``The Abelianization of Almost Free Groups" (the end of the proof of Lemma 2.5 on page 1801), this assertion is made where the $G_\alpha$ and $K_\alpha$ are subgroups of a free group with some additional properties, but the author makes the assertion I want without any explanation, so I assume the reason must be simple and may not depend on the group theory. | |

## Infinite connected graphs isomorphic to their line graphFor any simple, undirected graph $G$, let $L(G)$ denote its line graph. $G=(\mathbb{Z}, E)$ with $E = \{\{k, k+1\}:k\in \mathbb{Z}\}$ has the property that $G\cong L(G)$. Is there a connected infinite graph $G$ such that every vertex has more than $2$ neighbors, and $G\cong L(G)$? | |

## Can $S_n$ be partitioned into subsets containing an involution and satisfying $∀σ≠τ, ∃j$ s.t. $σ(j)≠τ(j),σ^{−1}(j)=τ^{−1}(j)$?
Background
Let $\sigma, \tau \in S_n$. We will say that $\sigma$ and $\tau$ are We say $\sigma, \tau \in S_n$ are Does there exist a partition of $S_n$ into exclusive subsets such that each subset contains an involution? Does either Knuth equivalence or dual Knuth equivalence imply local orthogonality? Either would imply a positive answer to Q1. Has this notion of local orthogonality for elements of $S_n$ been defined anywhere before (presumably under some other name)?
I have verified Q1 by brute force for $n \leq 11$. MotivationThe notion of local orthogonality appears in a more general context as part of a necessary condition for multipartite quantum correlations. A positive answer to Q1 would imply that in the variant of the 100 prisoners problem (sometimes called the locker puzzle) in which $n$ prisoners may only open 2 drawers, the classical best solution cannot be improved upon using shared quantum entanglement between the prisoners. (Note that this is different from the quantum version of the game considered by Avis and Broadbent in which the prisoners are allowed to open a superposition of drawers.) | |

## Cancellation in this exponential sum?I would like to know whether it is possible to obtain cancellation in the sum $$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$ where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number. | |

## Pascal's theorem for spherical hexagonI draw a cyclic spherical hexagon and I check by geogebra that Pascal's theorem is true in this case.
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## Is there any charactrization for lifting clopen subsetsLet $Y$ be a subset of a topological space $X$, we say that clopen subset of $Y$ lift to $X$ whenever $L$ is a clopen subset of $Y$ then there exists a clopen subset $H$ of $X$ such that $H\cap Y=L$. Let $X$ be a compact and $T_0$-space and $Y$ be a closed of $X $, I am looking for equivalent conditions under which the clopen subsets of $Y $ lift to $X $. | |

## Von Neumann's theorem on realizing automorphisms of the measure algebraI'm looking for a proof, in English, of the following theorem due to von Neumann (which apparently originates in the paper Every automorphism of the Boolean algebra of (Lebesgue) measurable subsets of $[0,1]$ modulo null sets is realized by a Borel measurable bijection on $[0,1]$. Obviously, $[0,1]$ can be substituted with any standard finite measure space. I know that this is generalized by results of Mackey, Maharam, and others, but I'm really just interested in the original result and its proof. Alas, I cannot read German. | |

## Is $G$ non-solvable?Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that (1) there is a character $\chi\in\mathrm{Irr}(G)$ such that $2^5\cdot7|\chi(1)$; (2) there is a character $\theta\in\mathrm{Irr}(G)$ such that $5^2\cdot7|\theta(1)$; (3) there is a character $\xi\in\mathrm{Irr}(G)$ such that $3^3\cdot7|\xi(1)$; Question: Is $G$ non-solvable? | |

## Coloring circles in planeWe assume that all the circles in the plane are each colored with one of two colors: red or blue.
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## fractional Brownian Motion driven stochastic integralsWe consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem. Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.
- Itô formula
- Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )
- Regularity structures
- Malliavin Calculus (Skorohod integral)
- White Noise Analysis
- ...
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## Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$
I know that there is a finite tensor category (from minimal models) with the following relations that seem rather strange to me. In particular it seems now one cannot realize it as modules over a ring. Does anyone know an algebraic situation, where something similar occurs? (maybe in a derived category?). A $$0\to J\to \mathbb{1} \to Q \to 0$$ $$J\otimes Q=\{0\}$$ $$Q\otimes Q=Q$$ (for modules over a ring $R,\otimes_R$, this means $J$ is an ideal with $J^2=J$, thus $R$ often splits, see below). such that $J\otimes J$ is an extension and which acts somewhat like a Any hints what some of these situations are called in literature are also very welcome. Thanx very much for your help in advance! | |

## Example of a ring with non-finitely generated unit group?The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is
Is there any concrete counterexample? | |

## Latent Dirichlet allocation - math words digest ?Latent Dirichlet allocation - is quite a popular topic in data-mining. Wikepedia mentions thousands citations in few years.
Ideally I would be happy to see some theorem which would be important for some practical problem. | |

## Estimate for the binomial coefficients and bounds from below for the Beta functionLet $n\ge p\in \mathbb N$ and let $\binom{n}{p}$ be the binomial coefficient. I believe that $$ \binom{n}{p}\le 2^n\sqrt\frac{2}{π n}. $$ Question: is that true? Of course I would like it as a non-asymptotic result, valid for all integers $n,p$. A related estimate would be a bound from below for the Beta function with $$ B(x,y)\ge \frac{\sqrt{(y-1)(x-1)}}{2^{x+y-1}}\sqrt\frac{π(x+y-1)}{2},\quad x,y\ge 1. $$ | |

## Non-orientable 3-manifoldsI am reading "Non-orientable 3-manifolds of small complexity" by Amendola, Martinelli. In this work $\mathbb P^2$-irreducible complexity 6 manifolds are listed. There are 5 of them. I wonder about the following non-orientable manifolds. - Take $S^2\times I$ and glue its top sphere to its bottom sphere with the antipodal homeomorphism or with a reflection in plane homeomorphism. Let's denote it by $S^2\widetilde\times S^1$.
- $\mathbb P^2\times S^1$
I assume that those two manifolds are not $\mathbb P^2$-irreducible. I don't know how to embed $\mathbb P^2$ into the first one. The preposition 1.3 on page 5 of the abovementioned work says that a Stiefel-Whitney surface cannot be a sphere. It seems to me that it is sphere for the first manifold. Both of them have double cover $S^2\times S^1$, and the fundamental groups are $\mathbb Z$ and $\mathbb Z + \mathbb Z_2$. What are the fundamental groups of the 5 manifolds of complexity 6 in the above work? At the same time I have the following additional questions about non-orientable 3-manifolds.
The answer to my question | |

## Parity with feasibility Mixed LPIf $b\in\mathbb Z$ is an integer constant with $0\leq b\leq 2^t-1$ then is it possible to test for parity of $b$ in a feasibility Mixed Integer Linear Program with $2^{2^{O(t)}}$ real variables and constraints and $1$ integer variable $b$ where the variables and constraints depend only on $t$ and the same program works at all $b$ in $[0,2^t-1]$? Surely convexity is not issue since we can do this with both feasibility Integer Linear Program in $O(1)$ integer variables and constraints and with optimization Linear Program with $2^{O(t)}$ real variables and constraints but where $b$ is a constant which with strong duality gets different LP for each $b$. | |

## Cohen/Random reals over intermediate models in countable support iterationsAssume that the continuum is $\aleph_2$ in our ground model $V$. Suppose that $(P_n \, : \, n \in \omega)$ is a fully supported iteration of length $\omega$. Suppose further that the factors of the iteration are proper and add for every step $\omega_1$-many Cohen reals over the previous model. If we let $P_{\omega}$ be the inverse limit, then there will be new reals which were not added by one of the $P_n$'s. Will there be a real $r$ in $V^{P_{\omega}}$, such that $r$ is Cohen over every $V^{P_n}$? I am also highly interested in the dual situation, where we replace Cohen forcing with Random forcing in the above. Can we guarantee that there is a real $r$ which is Random over all $V^{P_n}$'s? If not, is it possible to feed in countably many proper factors to the iteration, such that we can exclude reals which are Random/Cohen over the $V^{P_n}$'s | |

## Generalized Smith Theorem for the torsion of cokernelsLet $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$. Let $I_k=(\det \operatorname{min}_k(A))$ be the ideal of $R$ generated by all determinants of minors of $A$ of size $k$. I wonder if \begin{equation} \operatorname{Tor}_1^R(K,Q/R) \simeq \prod_{k=1}^{\operatorname{rk} A} \frac{I_{k-1}}{I_{k}}. \end{equation} If $R$ is a PID, the above isomorphism holds by Smith Normal Form. Does it hold for other classes of rings? I'm particularily interested in the case of $R$ is an order in a quadratic imaginary extension of $\mathbb{Q}$. | |

## Map of stacks $\underline{M}\rightarrow B\mathcal{H}$ is given by principal $\mathcal{H}$ bundleI am reading Orbifolds as stacks. In page $22$ it says : Lemma : Let $M$ be a manifold, $\mathcal{H}$ be a Lie groupoid. Then any map of stacks $F:\underline{M}\rightarrow BH$ is naturally isomorphic to the functor $BP$ induced by a principal $\mathcal{H}$ bundle $P$ over $M$. Here $\underline{M}=B(M\rightrightarrows M)$, category of principal bundles and equivariant maps associated to groupoid $\{M\rightrightarrows M\}$. I am not really looking for proof with full details. Some rough sketch would also be sufficient for now. See that $(\underline{M})_0$ is the collection of principal $\{M\rightrightarrows M\}$ bundle. What we want is a principal $\mathcal{H}$ bundle $P$ over $M$. So, it is only natural to look for an element of $\underline{M}$ i.e., a principal $\{M\rightrightarrows M\}$ bundle say $Q$ and take its image under (object map of $F$) to get a principal $\mathcal{H}$ bundle $F(Q)$. We want principal bundle to be over $M$. As $F$ is such that $\pi_{\mathcal{H}}\circ F=\pi_M$ where $\pi_{\mathcal{H}}:B\mathcal{H}\rightarrow Man$ and $\pi_M:\{M\rightrightarrows M\}\rightarrow Man$ where both functors on object level sends a principal bundle to its base. Thus, to get a principal $\mathcal{H}$ bundle over $M$, it is only natural to look for a principal $\{M\rightrightarrows M\}$ bundle $Q$ over $M$. Then $F(Q)$ is a principal $\mathcal{H}$ bundle over $M$ which we might want to call as $P$ in the lemma. Now the In that paper, proof starts with saying I could not understand how $4.1$ assures what is said above. I tried to follow my nose and ended up some where close to Can some one suggest an outline of the proof of above lemma which is not along the lines of seeing objects of $\underline{M}$ as maps $Y\rightarrow M$. There are two notions of categories which has same notation : - Let $M$ be a manifold. We can associate it with Lie groupoid $\mathcal{G}:=\{M\rightrightarrows M\}$. Given a Lie groupoid $\mathcal{H}$, there is a stack associated with it namely $B\mathcal{H}$ whose objects are principal $\mathcal{H}$ bundles and morphisms are $\mathcal{H}$ equivariant maps. For $\mathcal{G}=\{M\rightrightarrows M\}$ we have associated category $B\mathcal{G}$ and we denote it by $\underline{M}$.
- Let $C$ be an object in a category $\mathcal{C}$. We have $$\underline{C}_0=\{f:C'\rightarrow C\text{ for } C'\in \mathcal{C}_0\} $$ Given $f:C'\rightarrow C$ and $g:C''\rightarrow C$, the morphism set $Mor_{\underline{C}}(f,g)$ is defined to be $$Mor_{\underline{C}}(f,g)=\{\alpha:C'\rightarrow C''\text{ such that }g\circ \alpha=f \}$$ In particular, for a smooth manifold $M$ seen as an object in category of Manifolds, we have $\underline{M}$
I am trying to see that these two notions are exactly the same(upto isomorphism/equivalence of categories). I have rough idea which has some gaps. Is there any quick way to see these two are one and the same? | |

## A projective $\mathbb{Z}\pi_1$-moduleSuppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces an isomorphism on fundamental groups, where $X$ is a finite wedge of circles. Is $\pi_2 (M_{h},X)$ a projective $\mathbb{Z}\pi_1 (X)$-module? Here $M_h =\frac{X\times I\sqcup Y}{(x,1)\sim h(x)}$ denotes the mapping cylinder of $h$. |