Why we have always the following fact for affine variety

If $M$ is an affine variety of dimension $n$ then there exists an $C^\infty$ exhaustion $\tau: M\to [0,r)$, $0<r\leq \infty$ such that

on $M^*=M\setminus \tau^{-1}\{0\}$ the function $u=\log \tau$ satisfies

1) $(\partial\bar\partial u)^n=0$

2)$\sqrt{-1}\partial\bar\partial u\geq 0$ and $(\sqrt{-1}\partial\bar\partial u)^{n-1}\neq 0$

outside of the ramification divisor

After all the discussion raised by this old question, I am wondering about a somewhat complementary one:

For any given rectangle, does there exist a finite set of pairwise *different* isosceles triangles which tile it?

It is easy to tile e.g. a $1\times a$ rectangle for $1<a<2$ by four isosceles triangles, but with two of them being equal. In the case that $a=\sqrt{\frac{5-\sqrt{5}}2}$, we are lucky and can split one of those into two smaller ones, obtaining a tiling into 5 different isosceles triangles (with all occurring angles being multiples of $\frac\pi{10}$). BTW, we can iterate that by splitting the blue triangle again etc., getting tilings of the same rectangle into $k$ different isosceles triangles for all $k\ge5$.

I am quite sure the answer to the initial question is no, and it may even be interesting to restrict it to the following:

For which other rectangles is such a tiling known to exist?

And possibly, it doesn't even make a difference if we allow an *infinite* set of pairwise different isosceles triangles!

Supposing that D is a bounded Lipschitz domain (and not smooth) in $\mathbb{R}^d$. From what I know, it is known that the trace operator is well-defined and continuous from $H^s(D)$ to $H^l(\partial D)$ when $l=s-1/2$ and $ 1/2<s<3/2$. My question is what happens when $s>3/2$, is the above result true? Also I am interested in versions concerning more general spaces like Besov or Tribel-Lizorkin. Any answer or reference will be appreciated.

Are there any useful characterisation of categories whose auto-equivalences are all naturally isomorphic to the identity? For example, I read in this thread, What are the auto-equivalences of the category of groups? , that the category of groups is one such category. Is there some nice general property that I can use to check whether or not a category admits of auto-equivalences that are not naturally isomorphic to the identity? If not, it'd be useful just to find some more interesting examples of categories with this property.

More specifically, what examples (if any) are there of auto-equivalences that send every object to an isomorphic object but are not naturally isomorphic to the identity? For instance, I know that any auto-equivalence of SETS must send every object to an isomorphic object, but is it true that any such equivalence is naturally isomorphic to the identity? If so, is this a general property of toposes or just a special case?

I am a masters student of mathematics. Me and my friends wish to organize a small seminar, with aim of understanding the poincare conjecture.

We do not wish to delve into the case of 3-manifolds, but only understand it for high dimension.

My question is: What resources (articles, books) should we use in order to understand the proof and the specific tools that are used in the proof?

(This question is a follow-up on an older one.)

A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest gets all of what is left.)

A guest is said to be **fortunate** if his share of pie is strictly greater than the average share (which is $1/N$ of the original pie). Let $f(N)$ denote the number of fortunate guests out of the total of $N$ guests. What is the value of $$\lim\sup_{N\to\infty}\frac{f(N)}{N}$$
?

Let $\emptyset \subsetneq S \subsetneq \{1,\cdots,n\}$ be a set with cardinality $s$, and $g\in\mathbb{R}^n$ be a vector such that $$\sum_{\{i,j\}\subseteq \{1,\cdots,n\}}{(g_i-g_j)^2} = s(n-s).$$

**Question.** Is this true?
$$\sum_{\{i,j\}\subseteq \partial S \\(g_i-g_j)^2\le1}{(g_i-g_j)^2} \ge \frac{s(n-s)}{n}$$
where $\partial S$ is the set of all $2$-subsets $\{i,j\}\subseteq \{1,\cdots,n\}$ that exactly one of $i$ or $j$ is in $S$.

Let $p$ be a prime and $f(x,k)= \sum_{p^{}\leq x} \frac{1}{p^k\log p}$ where $k\geq 1$ is an integer. Is there a known asymptotic expression for $f(x,k)$, even for $k=1$ ?

Motivation: If no such results are known, i'm intending to take this as my Bachelor's thesis problem.

In our scriptum we're talking about singularities. And there is the term "fat point" (for example of "tangent of fat point") . I cannot find any definition :-/ Has somebody an idea?

It is possible to approximate the action of a matrix exponential $exp(A)$ on a vector $v$ in the corresponding Krylov subspace, i.e. calculate $exp_{Kr}(A)v$ [Saad].

Some sources claim that it is possible to compute the action of $exp_{Kr}(A)V$, where $V$ is a matrix, preserving symplecticity [Lopez] (however, the algorithm they provide does not produce the results described).

Are there any works showing that the Krylov-type exponential $exp_{Kr}(A)v$ preserves structure (especially, symplecticity) as does $exp(A)v$, when $v$ is a vector? Or, maybe, there is a way to prove it which I do not see?

For example, if $A\in sp(2n)$, then $\Psi := exp(A) \in Sp(2n)$, and for any two vectors $v, w$ holds true $\omega(v,w) = \omega(\Psi v, \Psi w)$, where $\omega(v,w) = v^T J w$ is the symplectic form. Does it holds for Krylov subspace exp?

Stoll in

Stoll, W.: *The characterization of strictly parabolic manifolds.* **Ann. Scuola Norm. Sup. Pisa,**
VII, 87-154 (1980)

showed that if $M$ is a connected complex manifold of dimension $n$ with $C^\infty$ exhaustion $\tau: M\to [0,r)$, $0<r\leq \infty$ such that

1) $\sqrt{-1}\partial\bar\partial\tau>0$ on $M$

and on $M^*=M\setminus \tau^{-1}\{0\}$ the function $u=\log \tau$ satisfies

2) $(\partial\bar\partial u)^n=0$

3)$\sqrt{-1}\partial\bar\partial u\geq 0$ and $(\sqrt{-1}\partial\bar\partial u)^{n-1}\neq 0$

then $M$ with its Kaehler metric is biholomorphically isometric to the ball of radius $r$ in $\mathbb C^n$ with the Euclidean metric.

I am wondering about the singular case, i.e. if $M$ is a singular complex variety and $\tau$ is not $C^\infty$ then what can we say about $M$?

Let $G$ be a finite group of order $n$, and let $L := (\mathbb{Q}/\mathbb{Z})^d$ be a (not necessarily trivial) $G$-module (we assume that $d$ is finite).

By a direct limit argument, there must be a finite $G$-module $M$ (in $L$) such that $$H^2(G,M) = H^2(G,L).$$

How small can I take $M$ (in terms of $G$ or even $n$)?

In particular, since $H^2(G,L)$ is $n$-torsion, can I just take $M$ to be the $n$-torsion points of $L$?

The Fibonacci polynomials are defined recursively by $F_0(x)=0, F_1(x)=1$ and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$, for $n\geq2$.

While computing certain integrals, I observe the following (numerically) which prompted me to ask:

**Question.** For $n, k\in\mathbb{N}$, are these always integers?
$$\int_0^1F_n(k+nz)\,dz$$

To help clarify, here is a list of the first few polynomials: $$F_2(x)=x, \qquad F_3(x)=x^2+1, \qquad F_4(x)=x^3+2x.$$

I was reading a paper (the 2015 paper by A. Falkowsy and L. Slominski **Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation**, page 353, proof of the Lemma 3.1) for my master degree thesis.

The framework is the following: we fix $$ f:\Bbb R^d\to\Bbb R^d $$ continous and such that satisfies the linear growth condition $$ |f(x)|\le L(1+|x|),\;\;\;x\in\Bbb R^d $$ and if $1<p\le2$ take $$ g:\Bbb R^d\to\mathcal M_d(\Bbb R) $$ $\alpha$-Holder continous function, where $p-1<\alpha\le1$ i.e. $$ ||g(x)-g(y)||\le C_{\alpha}|x-y|^{\alpha},\;\;x,y\in\Bbb R^d $$ where $||A||:=\sup\{|Ax|\;:\;|x|=1\}$ is the usual matrix norm.

Fix then $a:\Bbb R_{\ge0}\to\Bbb R$ and $z:\Bbb R_{\ge0}\to\Bbb R^d$ right continous functions which admit finite left limit (RCFLF for short) such that \begin{align*} V_1(a)_{[0,T]}&:=\sup_{\pi[0,T]}\sum_{j=1}^n|a_{t_j}-a_{t_{j-1}}|<+\infty\\ V_p(z)_{[0,T]}&:=\sup_{\pi[0,T]}\left[\sum_{j=1}^n|z_{t_j}-z_{t_{j-1}}|^p\right]^{1/p}<+\infty\\ \end{align*} where obviously $\pi[0,T]$ denotes the generic subdivision of the closed interval $[0,T]$.

Fix then $l:\Bbb R_{\ge0}\to\Bbb R^d$ another RCFLF function and consider the following integral equation $$ x_t=x_0+\int_0^tf(x_{s-})\,da_s+\int_0^tg(x_s)\,dz_s+k_t $$ whose unknown are the RCLFL functions $x,k:\Bbb R_{\ge0}\to\Bbb R^d$, with the constraint given by $x_0\ge l_0$ (inequality taken componentwise); both integral are Riemann-Stieltjes ones.

Now the Lemma of the paper I refer to, says what follows: suppose there exists $b>0$ such that $$ \max\left[V_1(a)_{[0,T]},\;V_p(z)_{[0,T]},\;\sup_{t\le T}|l_t|\right]\le b $$ then there exists $\bar C$ depending ONLY on $d, p, \alpha, L, g(0), x_0, b$ such that, if $(x,k)$ is a solution of the integral equation above, then $$ \bar V_p(x)_{[0,T]}:=|x_0|+V_p(x)_{[0,T]} \le\bar C\;. $$

I will skip all the detail of the proof, going directly to the core of the problem (otherwise the post would be the longest ever!).

I am able to prove the following inequality: \begin{align*} \bar V_p(x)_{[0,t]} &\le (d+1)\left[|x_0|+LV_1(a)_{[0,t]}(1+V_p(x)_{[0,t]})+DV_p(z)_{[0,t]}(1+V_p(x)_{[0,t]})\right]+db\\ &= (d+1)\left[|x_0|+(LV_1(a)_{[0,t]}+DV_p(z)_{[0,t]})(1+V_p(x)_{[0,t]})\right]+db \end{align*} where $D$ is another absolute constant. Here $t\le T$.

Starting from here, we define $$ t_1:=\inf\left\{t>0\;:\;LV_1(a)_{[0,t]}>\frac1{4(d+1)}\;\;\mbox{or}\;\;DV_p(z)_{[0,t]}>\frac1{4(d+1)}\right\}\wedge T $$ from which immediately $$ \bar V_p(x)_{[0,t_1[}\le(d+1)|x_0|+\frac12(1+V_p(x)_{[0,t_1[})+db $$ and thus $$ \bar V_p(x)_{[0,t_1[}\le2(d+1)|x_0|+1+2db\;\;. $$ Next we accept that $$ |\Delta x_{t_1}|=|x_{t_1}-x_{t_1-}|\le(L(1+|x_{t_1-}|)+C_{\alpha}|x_{t_1}|^{\alpha}+||g(0)||+2)b. $$

Then the authors write there exists then $C_1,C_2>0$ depending only on $d, p, \alpha, L, g(0), b$, such that $$ \bar V_p(x)_{[0,t_1]}\le C_1+C_2|x_0|. $$ WHY?! I know that $\bar V_p(x)_{[0,t_1]}\le\bar V_p(x)_{[0,t_1[}+|\Delta x_{t_1}|$ but controlling $\Delta x_{t_1}$, terms depending on $x$ appear!!

Then they set $$ t_k:=\inf\left\{t>t_{k-1}\;:\;LV_1(a)_{[t_{k-1},t_k]}>\frac1{4(d+1)}\;\;\mbox{or}\;\;DV_p(z)_{[t_{k-1},t_k]}>\frac1{4(d+1)}\right\}\wedge T $$ thus for the same constants $$ V_p(x)_{[t_{k-1},t_k]}\le C_1+C_2|x_{t_{k-1}}|\le C_1+C_2\bar V_p(x)_{[0,t_{k-1}]} $$ Set now $m:=\sup\{k\;:\;t_k\le T\}$ and accept that $m$ is finite and absolute (in the sense it doesn't depend on $(x,k)$ but only on the constants already written).

How can I reach the conclusion from here? I am really confused: how can I get rid of the terms depending on $x$ in order to get my absolute estimate?

I am wondering if it is possible to obtain a closed-form formula for $$ f(\alpha) = \frac{1}{{\sqrt{2 \pi } \; \alpha }} \int^\infty_{-\infty} x^2 \cosh(x) \; e^{-\frac{\sinh ^2(x)}{2 \alpha ^2}} . $$ This integral came up when I tried to calculate the second moment of the random variable $$ \DeclareMathOperator\arsinh{arsinh} X = \arsinh(Z \cdot \alpha) $$ where $Z \sim \mathcal{N}(0, 1)$ is a normally distributed random variable and $\arsinh$ denotes the inverse hyperbolic sine.

*Motivation/Context:* The above came up as I was investigating whether $\arsinh$ could be a useful variance-stabilizing, non-saturating activation function for artificial neural networks. For more details, see this Reddit thread and the original research cited there.

I am encountering the following situation which is similar to the Abhyankar's higher dimensional conjecture on étale fundamental groups, but with much stronger assumptions:

Let $S$ be a finitely generated subring of $\mathbb{C}$, let $X$ be a smooth affine variety over $S$, and let $G$ be a finite group such that the following holds. For any large enough prime $p$ and a base change $S\to k$ to an algebraically closed field of characteristic $p$, the variety $X_{k}$ admits a Galois covering with the Galois group $G$. Does this imply that $X_{\mathbb{C}}$ also admits a Galois covering with the group $G?$

I believe the answer is "yes" if $X$ is a curve, or is a complement of divisors with normal crossings in a projective space (by a result of Abhyankar).

Any suggestions or references would be greatly appreciated.

Morel has defined the motivic Hopf map $\eta$ (in the motivic stable homotopy category $SH(k)$). I suspect that the following facts are valid for it and its topological "cousin"; please correct me if they are false and give me some (nice) references if they are true.

1) For the topological Hopf map we have $\eta^4=0$.

2) The action ot the topological $\eta$ on the values of oriented cohomology theories is zero.

3) If $k$ is the field of complex numbers then the "topological realization" of motivic $\eta$ is the topological Hopf morphism in $SH$ (also denoted by $\eta$?).

Most academic jobs involve some amount of teaching. Post-docs generally do not, but they are only short-term positions.

**Question:** in which countries can one obtain a research-only permanent position in mathematics? Please provide a link to a relevant website if possible.

Please mention only one country per answer, and since there is obviously no best answer, this is a community-wiki question.

Assume $\Omega$ is an open set in $\mathbb R^3$ such that the intersection of $\Omega$ with any horizontal plane is simply connected.

Can you prove that $\Omega$ is simply connected?

(Note that by the definition, simply connected set can not be empty.)

**Comments.**

- The proof given by Tom Goodwillie below is done with bare hands. I would prefer to find ready to use tool for answering this and similar questions.

In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-adic measures. In particular, he characterizes all $p$-adic measures on $\mathbb{Z}_p$, with values in $\mathbb{Z}_p$, which correspond to rational functions in $\mathbb{Z}_p[[T-1]]$ under the Iwasawa isomorphism. Then, on page 488, he makes the following observation:

*The measures $\mu_F$ we considered above correspond exactly to the rational functions in $\mathbb{Z}_p[[T-1]]$ (the "rationality of the zeta function"!).*

Assuming he is referring to the rationality of the zeta function in the Weil conjectures, my question is:

**What is the relation between $p$-adic measures and the rationality of the zeta function of an algebraic variety over a finite field?**

The exclamation mark in his claim disturbs me. Is this something obvious?

I have studied Dwork's proof (from his paper and from Koblitz book), but I'm not familiar with Deligne et al proof of the whole conjecture.

Thank you very much.