Let $K/\mathbb Q_p$ be a discretely valued non-archemedean field, let $X$ be a smooth scheme over $\mathcal O_K$. To $X$ one can associate two rigid-analytic spaces over $K$:

1) the analytification $X_K^{\mathrm{an}} $ of the generic fiber,

2) the generic fiber $\mathfrak X^{\mathrm{rig}}$ of the corresponding formal scheme $\mathfrak X$.

One has a natural map $i_X:\mathfrak X^{\mathrm{rig}}\rightarrow X_K^{\mathrm{an}}$ which is an open immersion and an isomorphism in the case of proper $X$. For any $X$ one also has a comparison isomorphism between etale cohomology groups of $X_{\overline K}$ and $X_{K}^{\mathrm{an}}$ for torsion coefficients. So in the proper case one also has an isomorphism $H^i(({X_K})_{et},\mathbb Z/n\mathbb Z)\simeq H^i(\mathfrak X^{\mathrm{rig}}_{et},\mathbb Z/n\mathbb Z)$. Does one have this isomorphism in general or it holds only for $X$ proper? If not what could be a counterexample?

Let $Q_1, Q_2, R$ be quadratic froms over $\mathbb{Z}$ such that $Q_1 \oplus R \cong Q_2 \oplus R$ as quadratic forms. Is it necessary that $Q_1 \cong Q_2$?

I know that by Witt's theorem it is true for fields.

It is well known that given two nonconstant polynomials $f,g\in F[x]$ where $F$ is a field, there are unique polynomials $r_0,\dots ,r_n$ such that

$$f=r_n g^n +\dots+r_1 g +r_0,$$

where $\deg r_i <\deg g$ for all $i$. In other words, the radix expansion is possible in the ring $F[x]$. The proof is by induction on $\deg f$, similar to the existence of radix expansion for integers, and it uses the fact that in the division algorithm of polynomials the degree of the quotient is less than the degree of the dividend, when the degree of the divisor is bigger than zero.

But the above fact is not true in every Euclidean domain. For example in $\mathbb{Z}[i]$ we have

$$1=(1-i)(1+i)-1.$$

And the norm (degree) of the quotient $1-i$ is bigger than the norm of the dividend $1$. Hence the above proof does not work in $\mathbb{Z}[i]$.

**So my question is**, is there a Euclidean domain $R$ in which there are two nonzero nonunit elements $a,b$ such that there does **NOT** exist elements $r_0,\dots ,r_n \in R$ with $\deg r_i <\deg b$ so that

$$a=r_n b^n +\dots+r_1 b +r_0.$$

Note that I am not asking about the nonuniqueness of the coefficients of the expansion, since counterexamples to uniqueness are easy to build. Rather, my question is about the nonexistence of the expansion.

For any positive integer $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijections) $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For any $\pi\in S_n$ we let the *maximal displacement* be defined by $$\text{maxd}(\pi)= \max\big\{|k - \pi(k)|: k\in \{1,\ldots,n\}\big\}.$$
The expected value of the maximal displacements of all $\pi\in S_n$ is $$E^{\max}_n = \frac{1}{n!}\sum_{\pi\in S_n} \text{maxd}(\pi).$$

What is the value of $\lim_{n\to\infty} \frac{E^{\max}_n}{n}$?

(*Note.* The answer to this question seems to imply that $\lim_{n\to\infty} E^{\min}_n = 0$ if we define $E^{\min}_n$ in an analogous manner to $E^{\max}_n$ above, but I'm not sure this holds.)

For the Gamma function $\Gamma(x+1)$, we have beautiful approximations of the function in terms of elementary function, such as the Stirling approximation and its refinements, that give sharp estimates even for small values of the parameter, say $x\geq 1$.

Are there known approximations of the Polylogarithm $\mathrm{Li}_s(z)$ and the more general Lerch transcendent $\Phi(z,s,a)$ (as a function of $z$) in terms of elementary functions that give reasonably sharp estimates for all (or at least a wide range of) $z>0$? Ideally, I'm looking for sandwiching upper and lower bounds, but any estimator with good multiplicative error will be interesting.

Let $X$ be an infinite dimensional Banach space and $T:X\rightarrow X$ be a bounded linear operator. If $T$ is invertible and $\lVert T\rVert_e=\lVert T\rVert$, is it true that (or when is it true that) $\lVert T^{-1}\rVert=\lVert T^{-1}\rVert_e$? Here, $\lVert\cdot\rVert_e$ denotes the essential norm.

We all know 0.999... equals 1.0. Are there any other repeating decimals that can be represented as a non-repeating decimal?

The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix of a directed graph is number of vertex-disjoint cycle covers.

Given the importance of permanent as a $\#P$ complete problem that arises in various contexts such as lattices and polyhedra are there any other non-trivial interpretations of the permanent?

I would be interested in any interpretations to number theory (I know none and if there is one it would be very interesting) or algebraic geometry (however tenuous it may be) particularly since the latter is contemplated to be useful in studying the Permanent-Determinant problem.

I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:

$X$ has the same Betti numbers as $\mathbb{C}\mathbb{P}^{3}$ i.e. $b_{1}(X) = b_{3}(X) = 0$ and $b_{2}(X) = 1$ and all of its cohomology groups are torsion-free.

$\mathrm{Kod}(X) \geq 0$.

($\mathrm{Kod}(X)$ denotes the Kodaira dimension).

I have a problem that is generating a series ($d=2,4,\ldots,20,\ldots$) of pairs of $4 d$-degree palindromic (self-reciprocal) polynomials.

The first three members ($d=2,4,6$) of the first pair are: \begin{equation} \frac{32768}{3} \left(s^8-31 s^6-39 s^4-31 s^2+1\right) \end{equation} and \begin{equation} \frac{100663296}{5} \left(s^{16}-328 s^{14}+3223 s^{12}-3496 s^{10}+12505 s^8-3496 s^6+3223 s^4-328 s^2+1\right), \end{equation} and \begin{equation} \frac{85899345920}{21} \left(10 s^{24}-11775 s^{22}+677973 s^{20}-5688979 s^{18}+17249814 s^{16}-39697668 s^{14}+41914740 s^{12}-39697668 s^{10}+17249814 s^8-5688979 s^6+677973 s^4-11775 s^2+10\right). \end{equation} The first three members ($d=2,4,6$) of the second pair (now containing odd powers too of $s$) are \begin{equation} -\frac{4096}{315} \left(2843 s^8-15360 s^7-22 s^6-25600 s^5-6882 s^4-25600 s^3-22 s^2-15360 s+2843\right) \end{equation} and \begin{equation} -\frac{2097152 \left(3095503 s^{16}-36126720 s^{15}-441978818 s^{14}+1611939840 s^{13}+47805562 s^{12}-567361536 s^{11}-541456202 s^{10}+5031346176 s^9-2066788010 s^8+5031346176 s^7-541456202 s^6-567361536 s^5+47805562 s^4+1611939840 s^3-441978818 s^2-36126720 s+3095503\right)}{75075} \end{equation} and \begin{equation} -\frac{2147483648 \left(5660682116 s^{24}-106274488320 s^{23}-3585319303217 s^{22}+21392660889600 s^{21}+110882039596807 s^{20}-397070615445504 s^{19}-229641722669881 s^{18}+1033888042057728 s^{17}+383084268859914 s^{16}-2333132422905856 s^{15}+557684919386502 s^{14}+267997323722752 s^{13}-339034426148082 s^{12}+267997323722752 s^{11}+557684919386502 s^{10}-2333132422905856 s^9+383084268859914 s^8+1033888042057728 s^7-229641722669881 s^6-397070615445504 s^5+110882039596807 s^4+21392660889600 s^3-3585319303217 s^2-106274488320 s+5660682116\right)}{61108047}. \end{equation} I plan to input each of these series to the Mathematica commands FindSequenceFunction and/or FindGeneratingFunction, to discover their underlying rules--but am presently not too optimistic in these regards.

So, is it possible to exploit the palindromic property of these sequences in such a quest? (I, of course, can provide further members of these two sequence.) Are there any standard, recognized families of such polynomials that might be of potential interest?

These polynomials are constituents of an integrand I am seeking to evaluate over $s \in [0,\infty]$ for general integral $d>0$--and pertain to my previous question Extend a two-dimensional hypergeometric-related integration problem by allowing a parameter $d$ to be free. The variables $d$ and $s$ are the same in the two questions. (The members of the pairs are distinguished by the fact that those of the first pair are multiplied by $\log {s}$ in the integrand, and those of the second pair are not.)

In fact, if one chooses to first integrate over $s \in [0,\infty]$, rather than $t \in [0,1]$, one arrives at a "dual" set of pairs of palindromic polynomials. Then, we have the sequence ($d=1,2,3$) of degree $2 d+6$, \begin{equation} -512 t^8-8192 t^6-18432 t^4-8192 t^2-512 \end{equation} and \begin{equation} 16384 t^{14}+802816 t^{12}+7225344 t^{10}+20070400 t^8+20070400 t^6+7225344 t^4+802816 t^2+16384 \end{equation} and \begin{equation} -524288 t^{20}-52428800 t^{18}-1061683200 t^{16}-7549747200 t^{14}-23121100800 t^{12}-33294385152 t^{10}-23121100800 t^8-7549747200 t^6-1061683200 t^4-52428800 t^2-524288 \end{equation} and the companion (again $d=1,2,3$) sequence (corresponding to the terms not multiplied by $\log {t}$) \begin{equation} \frac{640}{3} \left(5 t^8+32 t^6-32 t^2-5\right) \end{equation} and \begin{equation} -\frac{4096}{35} \left(363 t^{14}+9947 t^{12}+48363 t^{10}+42875 t^8-42875 t^6-48363 t^4-9947 t^2-363\right) \end{equation} and \begin{equation} \frac{720896}{315} \left(671 t^{20}+41900 t^{18}+564975 t^{16}+2505600 t^{14}+3704400 t^{12}-3704400 t^8-2505600 t^6-564975 t^4-41900 t^2-671\right). \end{equation} (For the first "dual" set ($d=2,4,6$) the odd values ($d=1,3,5$) lead to intractable integrations.)

I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.

When I approached the speaker after the talk whether he could state in a general way what this concept says he was very resilient to state something that is universally true. He rather drifted off into examples in $l^1$ where he talked about non-compact symmetry groups and so on.

Although I appreciated this at the moment, I am a bit unsatisfied now, because I would like to see a very dense and general statement what concentration compactness is about.

To my surprise, also the internet seems to be full of rather vague explanations what this means on a general Banach space. I do not want to give references at this point, because I think many explanations are well written but do not answer my question:

Is there a comprehensive theorem stating the concept of concentration compactness in a most general way?

Put differently: What is the analogue of Banach-Alaoglu for concentration compactness?

As far as I know, it is still unknown whether there exists a (holomorphic) indecomposable vector bundle of rank $r$ on $\mathbb{P}^n_{\mathbb{C}}$ with $n\geq 6$ and $1<r< n-1$. What is the situation for hypersurfaces? Does anyone know a smooth hypersurface in $\mathbb{P}^{n+1}_{\mathbb{C}}$ ($n\geq 6$) carrying an indecomposable vector bundle of rank $r$ with $1<r< n-1$?

In the paper ''**A Nonstandard Model of Arithmetic Constructed by means of Forcing Method**'', Zhang Jinwen states the following in his abstract:

The first nonstandard model of arithmetic was given by Skolem. A. Robinson has introduced the concepts of standard, internal and external objects (sets, relations, functions, etc.) on the compactness theorem and concurrent relations, and has proved that if a set S is infinite, then S contains nonstandard internal objects. It is interesting to ask whether this is a common property of all non-standard modes of arithmetic. The author's answer to this question is in the negative. We have proved the theorem that there exists a nonstandard model of formal arithmetic in which there are infinitely many infinite internal subsets containing no nonstandard elements. This means that these infinite internal subsets are composed exclusively of finite natural numbers. In order to obtain this theorem we have made use of Cohen's forcing method.

The paper is in Chinese, and I could not understand it (I have a copy of it).

**Question.** I am wondering if someone can explain the main idea of his proof.

**Remark.** Based on Mathscinet, the paper is translated in English, but I could not find a copy of it.

The file can be seen here: A Nonstandard Model of Arithmetic Constructed by means of Forcing Method

I have one technical question on norm maps on Milnor K-theory.

When $K \subset L$ is a finite extension of fields, we know (by Bass-Tate and Kato) that there exists a norm map $N_{L/K} : K^M_n (L) \to K^M _n (K)$ on the Milnor K-groups for all $n \geq 0$. For instance, when $n=1$, this coincides with the "undergraduate algebra" level norm map $L^{\times} \to K^{\times}$.

My question pertains to the situation when this finite extension of fields is replaced by a finite ring extension. For instance when $A \subset B$ is a finite étale extension of semi-local rings with infinite residue fields, Kerz (Invent. Math, 09) constructed such norm maps from $K_n ^M (B)$ to $K_n ^M (A)$.

Here I wonder if there is a generalization of it in the following slightly more general circumstances:

(1) For instance, instead of the étale assumption, if $A \subset B$ is a finite extension of semi-local rings with infinite residue fields, where $B$ is a free $A$-module of finite rank, can we possibly still obtain a norm map $K_n ^M (B) \to K_n ^M (A)$?

(2) Or its special case: when $A \subset B$ is a finite extension of semi-local rings with infinite residue fields, such that the extension is "simple", i.e. $B= A[x]/(p(x))$ for a monic irreducible polynomial $p(x) \in A[x]$?

Of course, in the case of Kerz, it is known that his étale extension assumption implies that it is automatically simple, thus $B$ is a free $A$-module of finite rank. In general, (1) is more general than (2), and (2) is more general than the case considered by Kerz. Of course, (1) generalizes the situation of Bass-Tate and Kato for finite field extensions, too.

I suspect this might hold. I hope someone who knows well about this situation could kindly give some advice on this question.

Let $p$ be a prime and consider the ($p$-deprived) Hecke algebra $\mathbb{T}$ which the projective limit of the Hecke $\mathbb{Z}_p$-algebras $\mathbb{T}_k$ which act on modular forms of level $1$ with coefficients in $\mathbb{Z}_p$ and of weight at most $k$. (See $\S$2.1 of https://www.math.uchicago.edu/~emerton/pdffiles/padic.pdf)

Is the Krull dimension of $\mathbb{T}$ known when $p = 2$ or $3$? Is anything conjectured? The only references I can find deal with cases where the corresponding residual representation is absolutely irreducible, which excludes these cases.

Let $R$ be an integral domain. Consider the set $$S := \big\{a \in R\smallsetminus \{0\} : Ra+Rx \text{ is a principal ideal } \forall x \in R \big \}.$$ Is $S$ a saturated multiplicative closed subset of $R$? If in general $S$ is not saturated or multiplicative closed, what if we assume $R$ is a GCD domain? Is the claim true then ?

If we assume $R$ is local , let $x \in R$ , then $a \in S \implies \exists d \in R $ such that $Ra+Rx=Rd$ , where $d | a , d|x$ , then let $a'=a/d , x '=x/d$ , then $Ra'+Rx '=R$ , but $R$ is a local ring , hence one of $a' $ and $x'$ must be a unit i.e. either $a|x$ or $x|a$ . Thus in a local ring $R$ ,

$\big\{a \in R\smallsetminus \{0\} : Ra+Rx \text{ is a principal ideal } \forall x \in R \big \} $

$=\{a \in R \smallsetminus \{0\} : $ for every $x \in R$ , either $x|a$ or $a|x\}$ ; and I can show that in any integral

domain , $\{a \in R \smallsetminus \{0\} : $ for every $x \in R$ , either $x|a$ or $a|x\}$ is a saturated multiplicative closed set .

I have no idea what happens if the ring is not local .

Solve this functional equation:

$$\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$$

for $F(x)$ where $a > 0$ is a parameter. I know there is a trivial constant solution, $F(x) = 1$. Is there a non-constant solution?

I do not know if it has an analytical solution, and have no reason to expect that it does. It showed up in some calculations I was doing. But if there is a technique I can apply, it would be nice to know about it.

**Notations:** Let $L_\alpha$ stand for the Gödel constructible hierarchy ($L_0=\varnothing$ and $L_{\alpha+1} = \mathrm{def}(L_\alpha)$ is the set of definable subsets of $L_\alpha$ and $L_\delta = \bigcup_{\beta<\delta} L_\beta$ for limit $\delta$), and $J_\alpha$ for the Jensen hierarchy ($J_0=\varnothing$ and $J_{\alpha+1} = \mathrm{rud}(J_\alpha)$ is the rud-closure of $J_\alpha \cup \{J_\alpha\}$ and $J_\delta = \bigcup_{\beta<\delta} J_\beta$ for limit $\delta$). Let $M \mathrel{\preceq_1} N$ (if $M$ and $N$ are transitive sets mean “$(M,{\in})$ is a $\Sigma_1$-elementary submodel of $(N,{\in})$”.

Consider the following two relations between two ordinals $\sigma<\gamma$:

say that “$\sigma$ is $\gamma$-stable” when $L_\sigma \mathrel{\preceq_1} L_\gamma$,

say that “$\sigma$ is $\gamma$-J-stable” when $J_\sigma \mathrel{\preceq_1} J_\gamma$.

**Question:** Are the above relations equivalent? If not, is there still a way to define “$\sigma$ is $\gamma$-stable” in terms of the Jensen hierarchy and/or “$\sigma$ is $\gamma$-J-stable” in terms of the Gödel hierarchy?

Clearly, any one of the above relations implies that $L_\sigma = J_\sigma$ (because $\sigma$ is, at the very least, admissible $>\omega$), so the problem concerns the right-hand side, as $\gamma$ is not required to be admissible, or even a limit ordinal. Certainly $J_\sigma \mathrel{\preceq_1} J_\gamma$ implies $L_\sigma \mathrel{\preceq_1} L_\gamma$, so the question is whether the converse holds, or, if not, whether we can still find a way to express stability for one hierarchy in terms of the other.

I know that $\sigma$ is $(\sigma+1)$-stable iff it is $\Pi_m$-reflecting for each $m$, for example, but I don't see whether this also applies to being $(\sigma+1)$-J-stable (or, if not, what the corresponding criterion would be).

**Bonus question:** What if we replace $1$ by $n$ (i.e., $\Sigma_1$ by $\Sigma_n$) throughout?

Identifying antipodal points of an ellipsoid (with axes of different length) defines a Riemannian metric on the real projective plane $\mathbb RP^2$. Is there an **explicit** global isometric imbedding of this metric into Euclidean space $\mathbb R^N$? By explicit I mean using special functions, ellipsoidal harmonics, etc. The ambient dimension $N$ need not be 5 but perhaps not too large. Of course, the same question can be asked for any $\mathbb RP^n$ with an ellipsoidal metric.

*Reading several pappers to prepare my thesis I found the following problem:*

We considerer the following optimization problem $$ \left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[2pt] \text{s.t.} & \mathcal{A}x-b \in K \end{array} \right. \tag{1} $$

where $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ and $\mathcal{A}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ are linear functions non-zero, $b\in\mathbb{R}^{m}$, $\mathcal{C}$ is a convex cone in $\mathbb{R}^{n}$ and $K$ is a closed convex cone in $\mathbb{R}^{m}$.

**Question:** I need to find conditions over a set $U\subset\mathbb{R}^{n}$ such that if $\mathcal{C}$ is the convex cone generated by $U$, then problem $(1)$ is equivalent to the following problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in U} & f(x) \\[2pt]
\text{s.t.} & \mathcal{A}x-b \in K \end{array} \right. \tag{2}
?$$