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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.

Suppose you have a projective manifold $M$, a very ample bundle $\scr L$ and a transverse holomorphic section $s \in H^0(\scr L)$. Then the zero set of $s$ is a complex submanifold $S_M$.

Can we have a embedding of the the projective manifold $M$ in some projective space such that image of $S_M$ will not be contained in a hyperplane? You can assume $M$ has complex dimension $2$.

If R(E/Q_\infty) is the fine Selmer group and Y(E/Q_\infty) is its dual, then we know that Y(E/Q_\infty) is a finitely generated \Lambda-module and by a theorem of Kato, it is also torsion. My understanding is that it should thus make sense to want to attach a p-adic L function.

Can we say what this p-adic L function is? Or does it just follow (trivially) from the Iwasawa main conjecture?

Let $M$ be a matroid with ground set $E$. Deletion and contraction in matroids commute with each other and with themselves, i.e. for all $e,f \in E$ one has

$(M/e)\setminus f = (M\setminus f)/e$, $\hspace{0.1cm}$ $(M\setminus e)\setminus f = (M \setminus f) \setminus e$ $\hspace{0.1cm}$ and $\hspace{0.1cm}$ $(M/e)/f = (M/f)/e$.

Are there any matroids, aside from uniform matroids, which have the following property:

For all $e,f\in E$

$(M/e)\setminus f = (M/f) \setminus e$ ?

Problem statement

Let $0 \in K \subseteq \mathbb R^n$ be a non-empty closed and convex set with non-empty interior and piecewise smooth boundary $\partial K$. Fix $\lambda > 1$ and define $\lambda K := \{ \lambda x : x \in K\}$. For $x \in \mathbb R^n$ denote by $P_C(x) := {\arg\min}_{w \in C} \|w - x\|_2$ the nearest-point projection onto the convex set $C$ (as measured in the standard $\ell^2$ norm).

For any $x \in \mathbb R^n \setminus (\lambda K)$, prove that $\|P_K(x)\|_2 \leq \|P_{\lambda K} (x)\|_2$.

ProgressFor notational ease, let $x_1 := P_K(x)$ and $x_\lambda = P_{\lambda K}(x)$. Define the normal cone of $K$ at $x_1$ by $$N_K(x_1) = \{ y \in \mathbb R^n : \langle y, w-x_1\rangle \leq 0 \text{ for all } w \in K\}$$ If $x_\lambda - x_1 \in N_K(x_1)$ then the result follows by examining $\frac{\mathrm d}{\mathrm d t}\|t x_\lambda + (1-t) x_1\|_2^2$ at $t = 0$. Indeed, this shows that the norm of $tx_\lambda + (1-t) x_1$ is increasing on $t \in [0, 1]$, hence $\|x_\lambda\|_2 \geq \|x_1\|_2$.

ExamplesIt is easy to show that the result holds if $K$ is the $\ell^1$ or $\ell^2$ ball, or if $K$ is an origin-centered ellipse. Finally, using that derivative trick again, I can show the result holds as long as $\langle x_1, x_\lambda - x_1\rangle \geq 0$, but have been unable to verify that this identity always holds. However, I have been unable to craft a toy set $K$ where it seems like $\langle x_1, x_\lambda - x_1\rangle < 0$ without it also seeming necessary that $x_1$ was not chosen optimally to be the $\arg\min$.

I am wondering if any convex geometers/probabilists have looked at the following question:

Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each point $x_i\in\{x_1,\ldots, x_n\}$, draw $N$ points uniformly on the sphere $S^{d-1}$ with radius $r>1$ centered at $x_i$, denote as $x_{i, 1}, \ldots, x_{i, N}$. Let $C$ be the convex hull of $x_{1,1}, \ldots, x_{1,N},\ldots, x_{n,1},\ldots,x_{n,N}$. What is the probability that $\forall i\in\{1,\ldots, n\}$, $B(x_i, 1)\in C$?

So in other words, for every original point $x_i$, we draw a unit ball around it, how likely that $C$ contains all these unit balls?

I found Probability that a convex shape contains the unit ball was asking a similar question. According to the comments, if $N$ is exponential in $d$, then my question holds with probability $1$, because for each $i$, the convex hull of $x_{i, 1},\ldots, x_{i, N}$ already contains a unit ball. My question is different that we have $n$ points, and I imagine the neighboring vertices help each other to enlarge the convex hull. So perhaps a tighter bound exists?

Has this problem been studied before? What are the assumptions that people put on the distribution of $x_i,\ldots, x_n$? Thanks for any comments/answers!

I need to derive the following expression: $$ y = C^{T}ACx $$ w.r.t the matrix $C$

where $C$ is a rotation matrix, $x$ is a vector in $R^{3}$ and $A$ is a selection matrix in $R^{3\times3}$

$$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} $$

I've tried to directly apply the derivative of multiplication but i'm not sure if it applies and I end up with dimensionality errors. I'm quite stuck, and am checking the result against a numerical approximation of the jacobian.

Thanks for all the help

I am trying to understand the proof of weak Bezout theorem presented in Donu's algebraic geometry book. I wanted to ask where in the proof is he using the fact that we are working in projective space ? Is it when he assumes that translating the line at infinity if necessary ? Essentially the proof would work as well in complex plane if we have the point infinity in it as well right ? It would be nice if someone could also present couple of more intuition for this proof.

given $X_1,\cdots,X_n\overset{iid}\sim F$, where $F$ is a truncated normal, I wonder if there's something known about the distribution and specifically about the sub-Gaussianity of $X_i-\overline{X}$ and $\sum_{j=1}^k X_{i_j}-\overline{X}$? Thanks.