I asked this question at MSE but I did not get any answer so I ask it here at MO:

Assume that $M$ is a smooth $n$ dimensional manifold with $n>1$.

Is there a lie algebra structure on $\chi^{\infty} (M)$, the space of all smooth vector fields on $M$, such that we have the following property:

For every vector field $X\in \chi^{\infty}(M)$ the operator $ad_X:\chi^{\infty}(M) \to \chi^{\infty}(M)$ with $ad_{X}(Y)=[X,Y]$ is an elliptic differential operator of positive order (non zero order) when we restrict it to non singular points of $X$.

A motivation for this question is the consideration of the adjoint operator by Loic Teyssier in the following talk. But I do not know if this talk imply that there is a relation between the number of limit cycles and the adjoint operator?Is there a pre print or published paper extracting from this talk?What is the role of the adjoint operator in investigation of the number of limit cycles?

https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt

My initial motivation for consideration of diff. operators associated with a vector field is the following note:

https://arxiv.org/abs/math/0408037

The note has a false part: " It is claimed that the codimension of the range of derivation operator is equal to the number of limit cycles" but the true version is that "This codimension is an upper bound for the number of limit cycles". The true part of the "Proof" of this note is that :"around a hyperbolic limit cycle, one can solve the PDE $X.f=g$ provided the integral of $g$ along the limit cycle be equal to zero. Another true part of the note is included in Remark 1, which actually contains a proof of the fact that the codimension of $D_X$ is an upper bound for the number of limit cycles" But after 13 years, I think that my dream of finite codimensionality is going to be collapsed. The reason is the following interesting comment by Lukas Geyer

I understand that my old dream is collapsed, because perhaps his argument, in the above MSE link, for two singularitis can be repeated for coexistence of one singularity and one limit cycle. Assume that a limit cycle $\gamma$ surrounds a singularity. Consider the space of all smooth (or analytic) functions vanishing at $\gamma$ and singularity. Then in this function space may be we can seperates all orbits in the interior of $\gamma$ by $\int_{-\infty}^{+\infty} g(\phi_{t})(x)dt$ hence we have infinite codimension. So I search for some other diff. operators associated with a vector field. This is a motivation to ask for some elliptic operator in the form of $ad_X$ for some other Lie structures on $\chi^{\infty}(\mathbb{R}^2)$

Let us assume the principal symbol of a nonlinear differential operator $E$ at a point $p$ is $$\sigma_{E}(p):\Gamma (T^*M)\times \Gamma (T^*M)\to \Gamma (T^*M)$$ which acts as follows: $$ \sigma_{E}(p) (a,\eta) = \sum_{k,l=1}^n \Big( a_k a_l\alpha_i\eta_j + a_k a_l \eta_i\alpha_j -a_i a_k\alpha_l\eta_j- a_i a_k\eta_l\alpha_j$$ $$-a_ja_k\alpha_l \eta_i-a_j a_k\eta_l\alpha_i +a_i a_j\alpha_k \eta_l +a_i a_j\eta_k\alpha_l \Big) \xi^j $$ $$- 2\rho \Big(a^t a_t \alpha_k \eta_l +a^t a_t \eta_k \alpha_l -a^ta^s \alpha_t \eta_s -a^ta^s \alpha_s \eta_t \Big)\alpha_i $$ In above the Einstein summation convention is used and $\rho$ is a real scalar.

The Problem is: I want to show this equation is not strictly parabolic.

I put $a=(1,0,\cdots ,0)$ and $\eta=(1,0,\cdots ,0)$. With this assumptions I only can show the first part is equal to zero.

Any suggestion is highly appreciated.

Consider the group $\mathfrak{S}_n$ of permutations on the letters $\{1,2,\dots,n\}$.

We say two permutations are *b-equivalent*, $\pi_1\,\pmb{\sim^b}\,\pi_2$, if one can be determined from the other by reversing a block of $b$ consecutive integers. For example, $617\pmb{5432}\,\pmb{\sim^4}\,617\pmb{2345}$.

**Question.** Is this true? The number $h_b(n)$ of $(b+1)$-equivalent classes is given by
$$h_b(n)=\sum_{j=0}^{\lfloor\frac{n}{b+1}\rfloor}(-1)^j(n-bj)!\binom{n-bj}j.$$

The special case $b=1$ recovers Theorem 2.2 of this paper.

Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar function, corresponds to a conformal immersion $f:M\rightarrow \mathbb{R}^3$ (See Friedrich 03). It follows that the eigenvalue problem $D \phi =\lambda_i \phi$, where $\lambda_i\in \mathbb{R}$, gives some special immersion of this surface $f_i:M\rightarrow \mathbb{R}^3$. My questions is, if $f_1$, the immersion corresponding to the first eigenvalue, has some special meaning? Actually I'm highly interested in the Willmore energy of this immersion. I guess the this immersion would have a small Willmore energy from the inequality in (Bär 98): $$\lambda_1^2\leq \frac{\int_M H^2}{\mathrm{area}(M)}$$ However it might be too optimistic to expect that this immersion is a round sphere. Anyone has some insights for this immersion? Thank you very much.

Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.

**What does the distribution of the determinants look like? Is it normal or skewed?**

**What proportion of such $n\times n$ determinants is singular?**

Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ consider the space $C_\#(X,\mathbb{C})$ of finitely supported complex functions on $X$. Then $C_\#(I,\mathbb{C})$ is algebraically isomorphic to $V$ by definition of a Hamel basis, $$ \forall v\in V,\quad v=\sum_{e\in I}\lambda_e^v e,\quad I\ni e\mapsto\lambda_e^v\in\mathbb{C}. $$

**Question:** Does there exist a topology on $C_\#(X,\mathbb{C})$, such that $C_\#(I,\mathbb{C})$ is isomorphic to $V$ as topological vector spaces?

In finite dimensions $I$ is finite and hence discrete, so that all reasonable topologies on $C_\#(I,\mathbb{C})=\mathbb{C}^I$ coincide and give a positive answer. The question therefore pertains to infinite dimensions.

I am mostly interested in separable Hausdorff $V$. Thank you.

I am trying to read Carayol's article on the construction of Galois representations associated to Hilbert modular forms (http://archive.numdam.org/article/CM_1986__59_2_151_0.pdf). The main geometric ingredient is a study of the reduction of (non-modular) Shimura curves associated to a quaternion algebra over a totally real number field $F$ over $Q$, where $F \neq Q$.

In that article the existence of a canonical model (in the sense of Deligne) is deduced from Deligne's general results about general Shimura varieties (https://publications.ias.edu/sites/default/files/34_VarietesdeShimura.pdf). If I remember correctly Deligne's proof is very general and works by showing the result for entire classes of groups $G$ at once.

I was wondering if there was an easier way to show existence in this particular case ?

Thanks!

I have two finite sets of events $\{x_1, ..., x_N\}$ and $\{y_1, ..., y_N\}$ that are sampled from the PDFs $f(x)$ and $g(x)$, respectively. I want to estimate the Legendre expansion of $r(x) = f(x)/g(x)$ using these events. What is the best method?

Let $F : \mathcal{C} \rightarrow \mathcal{D}$ be functor of Lawvere theories $\mathcal{C}, \mathcal{D}$ (i.e. cartesian categories where every object is isomorphic to some power of a chosen object) that preserves finite limits and the chosen object. Precomposition $- \circ F$ turns every $\mathcal{D}$-model $A : \mathcal{D} \rightarrow \underline{\text{Set}}$ into a $\mathcal{D}$-model $A \circ F : \mathcal{C} \rightarrow \underline{\text{Set}}$ in a functorial way. This functor preserves limits and directed colimits (because they are computed componentwise---use the fact that directed colimits commute with finite limits in $\underline{\text{Set}}$ for the latter). From Adamek and Rosicky's book on locally presentable categories, we know that

- Every Lawvere theory's category of set-theoretic models is locally presentable. (Thm. 1.46)
- A functor between locally presentable categories is right adjoint iff it preserves $\lambda$-directed colimits for some regular cardinal $\lambda$. (Thm. 1.66)

Therefore, the above-mentioned functor $- \circ F$ admits a left adjoint $G$. Now suppose that we have a model $A : \mathcal{C} \rightarrow \mathcal{D}$ and a way to decide whether two elements in its carrier are equal, and we can compute all functions induced by the structure as a model of $\mathcal{C}$. When can we construct $G$ in such a way that we can also decide equality in $G(A)$?

For example, this seems to be the case where $\mathcal{C}$ is the trivial theory whose models are sets, and $\mathcal{D}$ the theory of monoids, because we can easily decide whether two lists (elements of the free monoid) are equal if we can compare its elements.

The situation is less clear to me when $\mathcal{C}$ is the theory of monoids and $\mathcal{D}$ the theory of groups. This seems related to the world problem, but easier.

I'm also very much interested in the generalization to multisorted Lawvere/algebraic theories (i.e. with $\mathcal{C}, \mathcal{D}$ arbitrary cartesian categories), and to essentially algebraic theories (where $\mathcal{C}$ and $\mathcal{D}$ are finitely complete categories and $F$ preserves finite limits). For example, the free category over a graph has decidable equality, while I'm again unsure about the free groupoid over a category.

Answers to special cases are also welcome.

For any open set $D \subset \mathbb{C}^n$, the ring $\mathscr{C}_D$ of continuous, complex valued functions in $D$ has a natural topology, the topology of uniform convergence on compact subsets. That is, for any compact set $K \subset D$ and any $\epsilon > 0$, let $$U(K,\epsilon): = \{ f \in \mathscr{C}_D \ \vert \ \left| f(z) \right| < \epsilon \ \ \ \forall z \in K \}$$ form a basis of the topology.

Note that a Hausdorff topological vector space over $\mathbb{R}$ or $\mathbb{C}$ in which every neighbourhood of the zero element contains a convex neighbourhood of the zero element is said to be locally convex. Therefore, to show that the Frechet space $\mathscr{C}_D$ is locally convex, we need only show that the set $$U(K,\epsilon) : = \{ f \in \mathscr{C}_D \ \vert \ \left| f(z) \right| < \epsilon \ \ \ \forall z \in K \}$$ is convex. To this end, let $f,g \in U(K,\epsilon)$ and consider the chord formed between them. That is, $tf + (1-t)g$ for $0 \leq t \leq 1$. To see that the entire line segment is contained in $U(K,\epsilon)$, we observe that \begin{eqnarray*} \left| tf(z) + (1-t)g(z) \right| & \leq & t \left| f(z) \right| + (1-t) \left| g(z) \right| \\ & < & \epsilon t + (1-t) \epsilon = \epsilon. \end{eqnarray*} This shows that $\mathscr{C}_D$ is locally convex.

Can someone justify why $\mathscr{C}_D$ is not globally convex?

A problem led me to the following arithmetic counting
problem: Let $n$ be a fixed number, $k\in\{1,\ldots,n\}$ and $Q_{k}$
the set of integer tuples $(t,u,v,w)$ of "compositions" of $k$, meaning
$k=t+u+v+w$, *but where contrary to the standard definition*,

- we allow $t,u,v,w\geqslant0$ such that, e.g., $k=(k-1)+1+0+0$ and $k=(k-1)+0+0+1$ count as two different "compositions" and not as one, $k=(k-1)+1$
- we assume the constraints $n-k\geqslant u$ and $u\geqslant v$}.

(So the first bullet increases the number of possibilities compared to compositions of $k$, while the second one decreases again the number of possibilities.)

**Question 1**: What techniques would help me to obtain a good upper bound on the size of $Q_{k}$?

**Question 2**: Let
\begin{alignat*}{1}
f(k):=\sum_{(t,u,v,w)\in Q_{k}}\binom{k}{t}\cdot\binom{n-k}{u}u!\cdot\binom{u}{v}v!\,\cdot\,!w\cdot\left(\frac{1}{2}\right)^{t}\left(\frac{1}{2n-2}\right)^{k-t} & ,
\end{alignat*}

where, as usual, we denote with $!w$ the number of derangements of $w$-many objects. With what (general) techniques can I achieve some upper bounds, in closed-form, on $f$?

I am not well versed in combinatorics so the easier the potential technique, the better. I'd also be very grateful of the technique were citeable somewhere.

(**Subquestion 2.5**: There is a general technique to evaluate similar sums
of products involving binomials, by Egorychev, in his book "Integral
Representation and the Computation of Combinatorial Sum", translated
from Russian. It would probably take me weeks to really understand
his techniques (my complex analysis is rather rusty and he uses quite
a lot of it), so can you tell me, if his technique is likely to help
me establish an upper bound for $f$?)

I have a sequence of points $y_1,\ldots,y_n\in \mathbb{R}^3$ and want to approximately minimise $$ \sum_{i=1}^{n-1}|y_i-y_{i+1}|. $$ I have a budget of $\mathcal{O}(n\log(n)^p)$ for some $p\in\mathbb{N}$. What would be the best heuristic for this endeavour?

Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\in E)\{a,b\}\in e\}$, and for $S\subseteq A$ let $N_G(S) = \bigcup\{N_G(a):a\in S\}$.

A *matching* is a set $M$ of pairwise disjoint edges, and a *marriage* of $A$ is a matching such that $A\subseteq \bigcup M$. For finite bipartite graphs, Hall's marriage theorem says that there is a marriage if

(Hall's condition:) for all $S\subseteq A$ we have $|S| \leq |N_G(S)|$.

Let $A,B$ be infinite and disjoint and set $[A,B]_2 = \big\{\{a,b\}: a\in A, b\in B\big\}$. Consider the collection ${\cal E}$ of sets $E\subseteq [A,B]_2$ such that Hall's condition holds for $(A\cup B,E)$ but $(A\cup B,E)$ has no marriage.

**Question.** Does every element of the poset $({\cal E},\subseteq)$ lie below a maximal element?

Let $Y$ be a smooth projective curve defined over number field $K$. Let $P_1,\dots ,P_m$ be some $K$-points to which we will associate "multiplicities" $m_i\in\{ 2,3,\dots \}$ for $i=1,\dots ,m$. The other $K$ points (and even those not defined over $K$ ) are considered with "multiplicities" $1$. We want to find a smooth projective curve $X$ defined over some number field $M$, and a morphism $\phi :X\to Y$ such that $e_{T}=m_{\phi (T)}$ for all closed points $T$ of $X$. Here $e_T$ is the ramification at the point $T$. We assume the "characteristics" of $Y$ is negative, that is $2-2g-\sum _P(1-\frac{1}{m_P})<0$. Here $g$ is the genus.

In Serre's Topics in Galois theory, he solves the problem for the Riemann surfaces. He doesn't require that the first curve is projective. The universal cover is $\mathbb{H}$, the Poincaré half-plane, if the "characteristics" is negative.

The cover here is infinite. Of course, I need a finite. Therefore somehow I should quotient it by some group and get a smooth projective curve. Then I know how to use GAGA to finish the problem.

Any ideas how to make the missing step?

Given a group $G$ and a left $G$-set $X$, then we can make $X$ a right $G$-set defining the action as $xg:=g^{-1}x$ , or if you prefer we are considering the opposite group $G^{op}$ to make the left action a right action.

Now, talking about $G$-torsors, there is the notion of contracted product, i.e. a product of torsors. In the literature the definition of the product is:

*Given X a $(G,H)$-bitorsor and $Y$ a $(H,G)$-bitorsor then $X \times^H Y=\frac{X \times Y}{\Delta(H)}$ is a $G$-bitorsor*

Here $\Delta(H)$ is the diagonal action, or equivalently the equivalence relation is defined by $(xh,y) \sim (x,hy)$. One can show that this product is associative.

Now, my question is, here there is a definition based on bitorsors, but what if, given $X,Y$ left $G$-torsor, i take the quotient of the fibre product of $X,Y$ by the equivalence relation $(gx,y) \sim (x,gy)$? Why this is not a left $G$-torsor?

In the case where $G$ is abelian, we can note that every $G$-torsor is a $G$-bitorsor and the product will be a $G$-bitorsor as well. But in the non abelian case? I suppose that there could be a problem of associativity with this product, but I don't know if it is the case and why.

I'm quite new to stacks, so this might be very easy. In particular, if there is a canonical reference I can consult for these things, please feel free to point it out.

Let $f:X\to Y$ be a $G$-equivariant morphism of schemes, for $G$ an affine group scheme (please feel free to relax the assumptions if it is natural, but this is the case I care about). Under what conditions on $f$ is the map between quotient stacks $[X/G]\to [Y/G]$ an open/closed immersion? Does it suffice for $f$ to be open/closed? In case not, does that become true after adding some reasonable assumptions on $G$?

First, define a sequence $F_0,F_1,\dots$ of functions by $$F_0(x,z) = z,$$ $$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$ So, for example, $$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{xe^z},\dots$$ etc. Also, set $F_{-1}(x,z)=0$. Now, let $$G(x,z) = \sum_{k=0}^\infty \left(F_k(x,z) - F_{k-1}(x,x)\right).$$ That is, $$G(x,z) = z + \left(xe^z - x\right)+\left(xe^{xe^z} - xe^x\right) + \dots$$

What I would like to do is to get some information (it doesn't have to be amazingly strong information...) about the asymptotics of the coefficient of the $x^{n-j}z^j$ term in the power series for $G(x,z)$.

**Question:** Does anyone know whether I have any hope in extracting any information from this generating function? If so, any ideas about what I should do/try? Even a pointer to something in the literature which *might* help me would be great!

By the way, the function $G(x,z)$ is closely linked to the Lambert $W$ Function. In particular, (I think) it is not hard to see that $$G(x,x)=\sum_{n=1}^\infty \frac{n^{n-1} x^n}{n!}$$ and it is well known that this function is the solution to the functional equation $$G(x,x) = x\exp(G(x,x)).$$ The thing that makes this question tricky therefore seems to be the presence of the second variable, $z$.

**Remark:** By the way, the coefficient of $x^{n-j}z^j$ in
$$F_k(x,z)-F_{k-1}(x,x)$$
counts the number $n$-vertex trees rooted at vertex $1$ of height exactly $k$ such that there are exactly $j$ vertices at distance $k$ from the root. Therefore, the coefficient of $x^{n-j}z^j$ in $G(x,z)$ is the number of $n$-vertex trees (of any height) in which there are $j$ vertices at maximum distance from vertex $1$. If anyone knows anything about the number of such trees (independently of the generating function), then that would also be useful!

For any smooth continuous surface it is known that ** only** lines of curvature $ (\psi=0, \pm \pi/2, \pi...) $ have zero geodesic torsion, ( time being keeping aside great circles on a sphere) :

$$\tau_g= (\kappa_1- \kappa_2) \sin \psi \cos\psi $$

For a minimal surface we have $$\kappa_1<0,\,2 H =(\kappa_1+ \kappa_2)=0, \, \tau_g = H\, \sin 2 \psi =0 $$

So for a minimal surface ** all geodesics** have zero Torsion.

Is this statement true? If so, what are Differential Geometry textbook references? If not, what is conceptually incorrect in this context?

Thanks for your comments or answers.

Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\in E)\{a,b\}\in e\}$, and for $S\subseteq A$ let $N_G(S) = \bigcup\{N_G(a):a\in S\}$.

A *matching* is a set $M$ of pairwise disjoint edges, and a *marriage* of $A$ is a matching such that $A\subseteq \bigcup M$. For finite bipartite graphs, Hall's marriage theorem says that there is a marriage if

(Hall's condition:) for all $S\subseteq A$ we have $|S| \leq |N_G(S)|$.

**Question**. Assuming that we $|A|=|B| = \aleph_0$, is there $n\in\mathbb{N}$ such that:

if $|N_G(a)| \geq n$ for all $a\in A$ and Hall's condition holds for $G$, then $G$ has a marriage

?

**Background**: For $n=1$, the implication in the question fails: Let $A = \omega \times\{0\}$ and $B = \omega \times\{1\}$. Set $$E = \big\{\{(k,0), (k-1,1)\}: k\in\omega\setminus\{0\}\big\}\cup\big\{\{(0,0),(n,1)\}: n\in\omega\big\}.$$ Then it is easy to see that $G= (A\cup B,E)$ obeys Hall's condition, but $G$ has no marriage.

Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e.

- $V(x)=V(-x)$ for all $x\in\mathbb{R}$
- $\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$
- $V(x)\in \mathbb{R}_{2n}[x]$
- $V$ has two local minima $x_1,x_2$ such that $x_1=-x_2$.

Then the operator $H=-\frac{d^2}{dx^2}+V(x)$ has pure discrete spectrum $(\lambda_i)_{i\geq0}$ and associated to each eigenvalue there is an eigenfunction $\phi_i$.

**My questions are:**

I read that there are 2 eigenfunctions $\psi_1,\psi_2$ , associated to "each well of the potential" (localisation in one side or the other). What is the relation between those "more elementary eigenfunctions" $\psi_1,\psi_2$ and the ground state $\phi_0$. I understant that is a sort of linear combination, but, why we cannot say that it's $\psi_1$ the ground state instead of $\phi_0$?

With the above assumptions on the potential, the question asked here has a positive answer? Or can we at least make explicit the growth of a bound $M_k$ depending on $k$? i.e. $\lvert\lvert \phi_k \rvert \rvert_\infty \leq M_k $ and $M_k\sim k^{\rm something}$

Thanks!