I am reading an article and I didn't find some of the definitions of the terms the author use in the article. If someone know I will appreciate ! What is the prime-to-$p$ part of a torsion subgroup on a module ? (I know what his a torsion subgroup but prime-to-$p$ part ? ) And what is a pro-$p$ quotient ? Thanks !

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-modules. The Auslander-Reiten quiver is a locally finite quiver whose vertices are indecomposable modules (up to isomorphism) and whose arrows are irreducible morphisms between indecomposable modules. It is given the structure of a translation quiver via the Auslander-Reiten translate $\tau$.

Two distinct vertices/modules of $\Gamma_A$ are said to belong to the same component if there exists a finite path/composition of irreducible morphisms between them. If $A$ is representation-finite (i.e. the category of finitely generated right $A$-modules contains finitely many non-isomorphic indecomposable objects), then $\Gamma_A$ consists of finitely many components, which each have finitely many vertices. However if $A$ is representation-infinite, then necessarily $\Gamma_A$ is infinite.

It is easy to construct an example of a representation-infinite algebra $A$ for which there exists a finite component of $\Gamma_A$. For example, consider the algebra $KQ/I$, where $KQ$ is the path algebra of the quiver $$ Q\colon \; 1 \begin{smallmatrix} \alpha \\ \rightarrow \\ \rightarrow \\ \beta \end{smallmatrix} 2 \begin{smallmatrix} \gamma \\ \rightarrow \\ \color{white} \gamma \end{smallmatrix} 3 \begin{smallmatrix} \delta \\ \rightarrow \\ \color{white} \delta \end{smallmatrix} 4$$ and $I$ is the ideal generated by the set $\{\beta\gamma, \gamma\delta\}$. This is representation-infinite because it contains the Kronecker quiver as a subquiver. But the relation $\gamma\delta$ 'cuts off' the right-hand side of the quiver, which means we have a finite Auslander-Reiten component $$\begin{matrix} S(4) & & & & S(3) \\ & \searrow & & \nearrow & \\ & & P(3)=I(4) & & \end{matrix}$$ (which is essentially $\mathrm{mod}\;K\mathbb{A}_2$) in $\Gamma_A$, where $S(v)$, $P(v)$ and $I(v)$ are the simple, indecomposable projective, and indecomposable injective modules corresponding to the vertex $v$ in $Q$ respectively. (One can also use the fact that this is a string algebra to easily compute $\Gamma_A$.)

**My question is this:** When is it the case that for a representation-infinite algebra $A$, every component of $\Gamma_A$ contains infinitely many vertices? Is this for example true if the algebra is self-injective? (I imagine the proof of the latter question is straightforward if true, but please point me to a reference if one exists.)

The lambda calculus is not upward confluent, counterexamples being known for a long time. Now, what about the interaction calculus? Specifically, I am looking for configurations $c_1$ and $c_2$ such that $\exists c: c_1 \rightarrow^* c\ \wedge\ c_2 \rightarrow^* c$, but $\nexists c': c' \rightarrow^* c_1\ \wedge\ c' \rightarrow^* c_2$.

Suppose $S$ is a surface, a Riemannian manifold in $\mathbb{R}^3$. Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics. If $T$ can be moved around arbitrarily on $S$ while remaining congruent (edge lengths the same, vertex angles the same), does this imply that $S$ has constant curvature?

I realize this is a naive question. If $S$ has constant curvature, then $T$ can be moved around without distortion. I would like to see reasoning for the reverse: If $T$ can be moved around while maintaining congruence, then $S$ must have constant curvature. What is not clear to me is how to formalize "moved around."

I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\Omega_r=\{x\in \mathbb{R}^2: d(x,\Omega)\leq r\},$$ $A$ denotes the area and $L$ the length. I tried to find a reference for this inequality, but I only found Steiner's formula which states that equality holds when $\Omega$ is convex. Can someone please give me a reference for that inequality?

I am trying to understand the behaviour of quotient stacks under the semidirect product.

Let $G\rtimes H$ be an algebraic group acting on a scheme $X$, over a field $k$. Assume that $G$ acts trivially on $X$. The canonical map $[X/G\rtimes H]\to [X/H]$ is a $G$-gerbe, and it admits a section $s:[X/H]\to[X/G\rtimes H]$ induced by the inclusion $H\subseteq G\rtimes H$. Is $s$ a $G$-torsor? By this I mean the following:

for any scheme $T$ and any map $T\to [X/G\rtimes H]$, the induced map $T\times_{[X/G\rtimes H]}[X/H]\to T$ is a $G$-torsor.

The map $s$ is clearly representable, so at least the question makes sense. Objects of $T\times_{[X/G\rtimes H]}[X/H]$ are $H$-torsors $P$ with an $H$-equivariant map to $X$ such that $P\times^H(G\rtimes H)$ is isomorphic to a fixed $G\rtimes H$-torsor (with compatibility of the maps to $X$), but I cannot define a $G$-action on this set.

Added: consider a map $S\to [X/G\rtimes H]$, corresponding to a torsor $P$. We have $S\rtimes_{[X/G\rtimes H]}X=P$, and therefore $S\times_{[X/G\rtimes H]}[X/H]=P/H$. This certainly becomes isomorphic (as a scheme) to $G$ after passing to a cover of $S$, but I don't see how one can define an action on it to make it a $G$-torsor.

Let $R$ be a commutative ring. Then there is a forgetful functor from the $\infty$-category of simplicial commutative $R$-algebras to the $\infty$-category of connective $E_{\infty}$-algebras over $R$. It's well-known that this functor admits left and right adjoint. Moreover, it's an equivalence if $R$ contains the field of rational numbers $\mathbb{Q}$. If I understand correctly, this means that SAG and DAG are (more or less) equivalent over such $R$.

On the other hand, if $R$ does not contain $\mathbb{Q}$ (say $R$ is a finite field), then the forgetful functor above is not necessarily an equivalence. This implies that SAG and DAG are very different in this situation. I wonder if there are any examples of problems where using SAG (resp. DAG) is more appropriate? Such examples would help to illustrate the difference between SAG and DAG in positive characteristic.

P.S.: related but not identical questions: $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras, Why do people say DG-algebras behave badly in positive characteristic?

In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $ \mathbb{Q} $ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article of Kozuma Morita (arXiv:1803.11074) claiming to prove that all elliptic curves with complex multiplication satisfy this conjecture, entailing a solution of the congruent number problem through Tunnell's theorem.

Assuming such a result holds, which lower bound to the proportion of rational elliptic curves satisfying BSD conjecture could be reached ?

[Sent here from Math.StackExchange by suggestion of an user.]

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological grupoids. Journal of Pure and Applied Algebra 130, 223-235, 1998) that topoi "with enough points" admit actually a representation as classifying topoi of topological groupoids.

Now my question is the following: take a well-known topos, as the étale topos for a scheme. This is the classifying topos of a localic groupoid, acrtually a topological groupoid since it is "coherent" and thus has enough points by Deligne's theorem; but which groupoid is this in concrete? Do you know if someone has ever investigated that? Thank you in advance.

P.S.: I have been suggested to look at the proof and try to reconstruct the particular case, which I am going to try.

I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\mid a\in {\mathbb Z}\}$.

A search in MathSciNet found a paper of Behr and Mennicke(A presentation of the groups PSL(2,p), *Can. J. Math.* 20, 1432-1438 (1968)) that gives a presentation for the special case of $n=k=2$; (add the diagonal matrix $(2,\frac{1}{2})$ as extra generator and describe its conjugation action on suitable generators of ${\rm SL}_n({\mathbb Z})$); but a reference search did not yield a further result.

Similarly mathOverflow carried the same question question for other rings, but not $R$.

(A follow-up question would be the same question for ${\rm Sp}$)

begin tl;dr: I just read this paper which gives the equations for the structure constants, braiding operators etc. for a generic quantum Lie algebra. I always found it very annoying that in the Kauffman abstract tensor formalism, you need caps and cups (read: creation and destruction operators, if you read it as Feynman diagrams), at least for undirected lines. Since in any graphical formalism (read: birdtracks - what I do since 30 years might be called quantum birdtracks), the "space" and "time" directions are equivalent. (OK, one could gauge in some left and right kink operators equal to cup and cap, but that's very artificial. But then, why should *I* be able to unify relativity and quantum theory :-)

Now these equations are even worse in that kind - when one writes the tensors graphically, even left-right symmetry is violated. Outside twistor theory, this is a no-no :-) Thus: Is there an equivalent formulation of quantum Lie algebras (as tensors) keeping as many symmetries as possible (as graphs)?

I am writing because I have a doubt about a cumulative distribution function having a pdf.

How can I derive the CDF?

Thanks in advice to everyone who help me.

Have a good day :)

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). There seems to have been earlier work (by Grégoire de Saint-Vincent in 1647 and Gabriel Cramer en 1750) that Gerono and Lissajous don't seem to have been aware of.

Historically which of the two 19th century Frenchmen has priority for the lemniscate? Was Lissajous aware of Gerono's work when he introduced his curves?

The question was posed a few days ago at hsm to little effect.

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to be $\sum\bar{a_g}g^{-1}$ and for $\beta=\sum b_gg\in\mathbb C[G]$, we have the (convolution) product $\alpha\beta:=\sum a_gb_hgh$. Let $\alpha=\alpha^*$ be an element in $\mathbb C[G]$, define $F(\alpha):=a_e-\sum_{g\neq e}|a_g|$. Consider the sequence $(A_n)$ defined by $$A_n:=F\left(\alpha^{2n}\right)\quad(n\in\mathbb N)$$ Does the sequence $(A_n)$ contain a nonnegative number?

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two topological spaces (say as nice as we want) spaces, I would like to understand how to construct the $E_2$ term $$E_2^{pq} = H^*(Y,\mathcal{H}^*),$$ where $\mathcal{H}^*$ is the pre-sheaf $U \mapsto H^*(f^{-1}(U))$. I know a bit about sheaf cohomology, but honestly I don't really understand how this pre-sheaf appears, or why/if we need to look at its sheafification.

I have several general questions regarding all this, some being even more general:

- Why are we always talking about the $E_2$-term and not the $E_1$-term of a spectral sequence?
- From where to I need to start in order to construct this $E_2$-term ?
- Suppose that there exists an open cover $\{ U_i \}$ of $Y$ such that $\mathcal{H}^*(U_i) = \mathcal{G}^*(U_i)$, where $\mathcal{G}^*$ is another pre-sheaf (for instance the constant one). Under what condition could we have: $$H^*(Y,\mathcal{H}^*) = H^*(Y, \mathcal{G}^*) ?$$

Does anyone have a nice reference to these concepts (important to note that I'm no specialist in algebraic geometry :))

Thanks a lot for your help !

Consider the expectation $E(G(v,X_v)|\mathcal{F}_t)$ for $t\leq v \leq T$ for a stochastic process $X_t$. We can impose one of the two following conditions :

- $E(G(v,X_v)|\mathcal{F}_t)$ has a uniform bound which is independent of $v$ for $t\leq v \leq T$
- $E(G(\tau,X_\tau)|\mathcal{F}_t)$ is bounded for all stopping times $\tau$

Are the two conditions above equivalent? If not then does one imply the other?

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is best possible? In other words, if $X \times Y =Z$ where $X$ is a manifold and $Z$ is a 3-manifold, then is $Y$ a manifold?

In Bing's example, $Z$ is not compact. Is there a compact example in dimension $4$?

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}|$$ Denote $z^*_n := \min_{S^{n-1}} f_n$.

Is it true that for sufficiently large $n$s (which depend on $S$), we have $z_n^* = 2(n-1)$?

*Remarks*:

$f_n(x)$ is sum of the absolute values of eigenvalues of a circulant matrix generated by $(0,x_1,x_2,\ldots,x_{n-1})$.

$f_n(x) = 2(n-1)$, when $x$ is all ones vector.

Above question stems from this question and may be related to the Littlewood problem, as a comment of Tao to the later linked question.

Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$

(The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Lviv Scottish Book).

Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\operatorname{T}$ has a Jordan canonical form, i.e., it is similar to the direct sum of a number of Jordan blocks. If a certain eigenvalue (i.e., root of a polynomial) is not inside $\mathbb{F}$ then in the corresponding Jordan block it is represented by its $\mathbb{F}$-matrix representation (that is, the Jordan block is a block matrix).

In infinite dimensions the existence of Jordan canonical form is not guaranteed. I consider only those operators for which it exists.

**Question:**

Let $V$ have arbitrary dimensions, and *assume* that $\operatorname{T}$ does have a Jordan canonical form, i.e., it is similar to the direct sum of (an arbitrary set of) Jordan blocks. Let now $W\subset V$ be an $\mathbb{F}$-vector subspace which is $\operatorname{T}$-invariant, $\operatorname{T}W\subset W$. Is it true that the restriction $\operatorname{T}|_W\in\mathrm{End}(W)$ also has a Jordan canonical form?

Thank you.