Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\mathrm{sh}}$. Are $K_{1},K_{2}$ isomorphic?

Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \mathfrak a_{M,\mathbb C}^{\ast}$, we have the induced representation

$$I(\nu,\pi) = \operatorname{Ind}_P^G \pi q^{\langle \nu+\rho,H_M(-)\rangle}$$ of $G$. For $w$ in the Weyl group sending $\theta$ to $\theta' \subset \Delta$, and $P' = M'N'$ corresponding to $\theta'$, we have the intertwining operator $A = A(\nu,\sigma,w): I(\nu,\pi) \rightarrow I(w(\nu),w(\pi))$ defined by

$$A(f)(g) = \int\limits_{N_w} f(w^{-1}ng)dn$$

where $N_w$ is generated by the root subgroups of positive roots made negative by $w^{-1}$. The given integration takes place in the vector space $V$, and I am trying to understand:

What is the meaning of this vector valued integral?

Why does the integral converge (whatever that means, depending on the answer to my first question) for $\nu$ in a suitable cone?

I had asked a question about the meaning of the integral before, but I am sorry to say that after all this time I still do not understand what is going on. Paul Garrett provided an answer in which he suggested that we should not think of $V$ as having the discrete topology, but having a locally convex, quasi-complete topological vector space structure (coming as a colimit of its f.d. subspaces) in which one could make sense of the integral as a Pettis integral. That is, we should show that there exists a vector $v = A(f)(g)$ in $V$ with the property that for all $v^{\ast}$ in the algebraic dual of $V$,

$$\langle v^{\ast},v \rangle \rangle = \int\limits_N \langle v^{\ast}, f(w^{-1}ng)\rangle dn$$

He also suggested that taking a good maximal compact subgroup $K$ of $G$, so that we have $G = PK = P'K$, we could use the fact that elements of the induced representation are determined by their effect on $K$ to reduce to the case where the vector valued integrals are just finite sums. I still have not figured out how to do this, and wanted to ask math overflow again for help.

These intertwining operators are unfortunately still very much a mystery to me, and I have not seen any reference explain them rigorously.

Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d_C$ be the infimal distance function to $C$ and $P_C(x) = C \cap S[x; d_C(x)]$ be the metric projection. We say $C$ is *proximinal* if $P_C(x) \neq \emptyset$ for all $x \in X$.

I'm wondering if the intersection of two proximinal subsets of $X$ must be proximinal? To avoid trivial counterexamples, I need their intersection to be non-empty.

The answer is most likely "no", or at least, something that is difficult to prove in the affirmative if true. Asplund proved that, given any non-convex Chebyshev subset $C$ (meaning $|P_C(x)| = 1$ for all $x$), there exists a closed half-space $H$ such that $X \cap H$ is not proximinal. Therefore, were proximinal sets closed under intersection, this would prove the Chebyshev conjecture: that all Chebyshev subsets are convex.

I was wondering if someone had an explicit example of two proximinal sets whose intersection is not proximinal?

Let $V$ be an irreducible and smooth variety over a number field $k$ and let $f:V\rightarrow\mathbb{A}_k^n\backslash F$ be a surjective morphism with rationally connected generic fiber, where $F$ is a closed subset of codimension at least $2$. Let $\mathcal{A}$ be the variety of affine lines in $\mathbb{A}_k^n$. Can we find a non-empty open subset $\mathcal{U}\subseteq\mathcal{A}$ such that $f^{-1}(l)$ is rationally connected for every $l\in\mathcal{U}$?

This question came to my mind when I was reading D. Harari's paper "Flèches de spécialisations en cohomologie étale et applications arithmétiques", in which a question of similar kind was solved by Theorem of Bertini.

This is probably a very dumb question. Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ canonically isomorphic to the cocharacter lattice $L$ of $T$?

If $\operatorname{Spec}A$ is affine scheme, it seems to me that $p^*T=T\times \mathbb{G}_m$, so $$p_*p^*T(A)=p_*((T\times \mathbb{G}_m))(A)=T(A[t^{\pm 1}])\times \mathbb{G}_m(A[t^{\pm 1}]).$$ Say that $A=K$ is a field. Then $$p_*p^*T(K)=L\oplus\mathbb{Z},$$ but why is the image of $T(K)$ equal to $\mathbb{Z}$?

Where can I find a (readable and self-contained) proof of the following result?

Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\mathrm{div}(A(x)\nabla u) = \delta_0,$$ $$u|_{\partial \Omega} = 0,$$ where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$). Then on $B_{1/2}$, we have $$\frac{C_2}{|x|^{n-2}} \le u(x) \le \frac{C_1}{|x|^{n-2}}.$$

In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with (i.e. have the same parameter space as) the instantons in 2D for the theory in which the complex projective $n$-space $CP_n$, is replaced by the infinite-dimensional manifold ($\Omega G$ of loops on the structure group $G$."

Question (1): What is the precise restriction of $G$ in Atiyah's on Instantons in 2D and in 4D? Must $G$ be simple compact Lie groups? Or must $G$ be classical groups? Or how general the $G$ can be?

In Atiyah's work [Ref. 1], he also mentions that Donaldson's work [Ref. 2] gives the proof only for the classical groups but it seems likely that his result holds for all $G$.

Question (2): What is the precise restriction of $G$ in Donaldson's work here? Must $G$ be simple compact Lie groups? Or must $G$ be classical groups? Or how general the $G$ can be?

Question (3): The instanton study here in 2D for Atiyah's theory in which the complex projective $n$-space $CP_n$, is replaced by the infinite-dimensional manifold ($\Omega G$ of loops on the structure group $G$.) How is this story of $CP_n$ v.s. $\Omega G=\Omega SU(n)$ here related to the $G=SU(n)$-Yang Mills theory? Here $CP_n$ is finite $n$-dimensional complex manifold, while $\Omega G$ is said to be infinite-dimensional.

Refs:

Instantons In Two-dimensions And Four-dimensions, 1984 - 15 pages Commun.Math.Phys. 93 (1984) 437-451, M.F. Atiyah

Instantons and geometric invariant theory, Comm. Math. Phys. Volume 93, Number 4 (1984), 453-460. S. K. Donaldson

Note: Classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces

Where can I find a proof of the following result?

Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$ where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$) in $B(0,1)$ such that $u \le 1$ and $|\{ u \le 0\}| =a>0$. Then $$\sup_{B_{1/2}}u \le \mu(a) < 1.$$

I have found quite a few papers on asymptotic constancy of solutions of delay differential equations and integral differential equations (see e.g. this reference or this reference).

I am however most interested in methods for finding the constant to which these solutions converge. My question is for references which provide such a method or some type of review paper or book where I can read more about asymptotic constancy in general.

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-X$ is still clear. What is known about the action of $W$ on $L$? Is anything known about the orbit structure? References would be most welcome.

Let $A$ be a finite dimensional algebra with finite global dimension and with minimal injective coresolution $I_i$ of the regular module $A$.

The study of the projective dimensions of the $I_i$ is an important tool to test whether certain subcategories are extension closed or closed under submodules, see for example the article "Homolocial theory of noetherian rings" by Idun Reiten and https://www.sciencedirect.com/science/article/pii/0022404994900442 .

Questions:

Is there an easy example with $pd(I_i)=1$ for some $i>1$? (probably yes, but im too blind at the moment to construct an example. It necessarily has to have global dimension at least 3.)

Can we have $pd(I_i)=1$ for some $i>1$ in case $pd(I_0)=0$?

Can we have $pd(I_i)=1$ for some $i>1$ in case $A$ is a Nakayama algebra?

To my surprise my computer found no such example for a Nakayama algebra.

(of course this question has the danger that I oversee something obvious)

edit: The reason might be as follows when $pd(I_0)=0$:

We have $0 \rightarrow A \rightarrow I_0 \rightarrow \Omega^{-1}(A) \rightarrow 0$ and thus all indecomposable injective modules of projective dimension one appear in $\Omega^{-1}(A)$ and thus also in $I_1$. Now it is probably easy to see that they cant appear later again, but Im not sure why at the moment.

Consider $n \times n$ non-negative binary matrices ${\bf A}_i$ with $1\leq i \leq m$ over $\mathbb{R}$.

Assume that $1\leq k \leq n$ is selected as a fixed number.

Suppose that a subset of size $k$ of the set $\{1,2,\cdots , m\}$ is denoted with ${\bf I}_k=\{a_1,a_2,\cdots ,a_k\}$ and the related matrix to ${\bf I}_k$ is defined by ${\bf H}=(h_{i,j})=\prod_{t=1}^k\, A_{a_t}$.

*My Question:* With which conditions over ${\bf A}_i$'s we have the following property.

If for an ${\bf I}_k$ the matrix ${\bf H}$ be a positive matrix over $\mathbb{R}$ ($h_{i,j}>0$), then the value of $\sum_{i=1}^n \sum_{j=1}^n h_{i,j}$ be a fixed number (not depends on the value of ${\bf I}_k$)

Set : $$ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $$

I would like to know if there exists a projective variety $ H ( \mathbb{P}^n ) $ whose points are in a natural one-to-one correspondance with smooth projective varieties of $ \mathbb{P}^n $ ( i.e : with elements of $ \mathcal{H} ( \mathbb{P}^n ) $ ).

If it's the case, how to construct $ H ( \mathbb{P}^n ) $ ?

If there is a way to construct $ H ( \mathbb{P}^n ) $, is there any book treating in detail this subject of how to construct $ H ( \mathbb{P}^n ) $ and its properties ?

Thanks in advance for your help.

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally we provide an example that seems to contradict that claim. The question is

**Question**: Is the example given below a counterexample to the claim? And if not, why not?

Fix an Artinian monomial ideal $I$ of a polynomial ring $K[\mathbf{x}] = K[x_1, \dots, x_n]$. A monomial $\mathbf{x}^{\mathbf{b}}$ is a *socle monomial* of $I$ if $\mathbf{x}^{\mathbf{b}} \notin I$ and $x_i\mathbf{x}^{\mathbf{b}} \in I$ for all $i$. Let $\mathrm{MonSoc}(I)$ be the set of all socle monomials of $I$.

**Def**: $I$ is *reflection invariant* if there is a *canonical monomial* $\mathbf{x}^{\mathbf{K}}$ such that the map that sends a monomial $\mathbf{x}^{\mathbf{b}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}$ is an involution on $\mathrm{MonSoc}(I)$.

Following these definitions the authors note the following.

The Claim**Claim**: $I$ is reflection invariant with canonical monomial $\mathbf{x}^{\mathbf{K}}$ if and only if the monomial ideal generated by $\mathrm{MonSoc}(I)$ equals the Alexander dual $I^{[\mathbf{K} + \mathbf{e}]}$ where $\mathbf{e} = (1,1,\dots, 1)$.

Let $I = \langle a^4,~ab^2,~b^3,~a^3c,~abc,~c^3 \rangle \subset K[a,b,c]$ and let $\mathbf{K} = (3,2,2)$. Then

$$\mathrm{MonSoc}(I) = \left\{a^{3}b,~a^{2}c^{2},~b^{2}c^{2}\right\}.$$

By (the constructive proof of) Proposition 5.2 in this paper the ideal $J = \langle a^4, a^2b, b^3, ac, b^2c, c^3 \rangle$ is the unique Artinian ideal with

$$\mathrm{MonSoc}(J) = \left\{\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}} \mid \mathbf{x}^{\mathbf{b}} \in \mathrm{MonSoc}(I)\right\}.$$

Moreover, the same algorithm can be used to show that $I$ is the unique Artinian ideal with

$$\mathrm{MonSoc}(I) = \left\{\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{c}} \mid \mathbf{x}^{\mathbf{c}} \in \mathrm{MonSoc}(J)\right\}.$$

In particular, the map that sends a monomial $\mathbf{x}^{\mathbf{b}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}$ is an involution on $\mathrm{MonSoc}(I)$, so $I$ is reflection-invariant with canonical monomial $\mathbf{x}^{\mathbf{K}}$. We now get a contradiction to the claim above by computing the Alexander dual (in Macaulay2, for example) and noting that the minimal generators of $I^{[(4,3,3)]}$ are $\{a^4bc,~a^2b^3c,~ab^2c^3\} \neq \mathrm{MonSoc}(I)$.

Again, the question is

**Question**: Is the example just given a counterexample to the above claim? And if not, why not?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:

There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically just means a reflective subcategory of $C$.

There is also a more restrictive usage (as when talking about spectra), which requires $C$ to be monoidal, and means a reflective subcategory where the class of maps being localized at is of the form $\{X \mid E \otimes X = 0\}$ for some fixed $E \in C$.

In this question, I'm interested in the more restrictive usage (2).

In this sense, Bousfield localization makes sense in either an ordinary monoidal category or in a monoidal $\infty$-category (for that matter, the more general usage makes sense in either an ordinary category or in an $\infty$-category). But it's typically only discussed in an $\infty$-categorical setting (e.g. in model categories or triangulated categories).

**Question 0:** Is there a good reason why Bousfield localization for ordinary categories (in sense (2)) is rarely discussed?

I think the answer may be "yes" because the behavior of Bousfield localization may be quite different in the two settings, and it seems somehow "better" in the $\infty$-categorical setting. But I'm not sure how to articulate this.

Here are two examples of what I mean:

$E = \mathbb Z/p$:

When $C = Ab$ is the (ordinary) category of abelian groups and $E = \mathbb Z / p$, the Bousfield localization consists of the abelian groups which have no nonzero infinitely $p$-divisible elements.

But when $C = D(Ab)$ is the $\infty$-category of chain complexes of abelian groups (localized at the quasi-isomorphisms) and $E = \mathbb Z/p$, the Bousfield localization consists of chain complexes whose homology is $p$-complete.

$E = \mathbb Z_{(p)}$:

When $C = Ab$ and $E = \mathbb Z_{(p)}$, the Bousfield localization consists of abelian groups which are $\ell$-torsionfree for $\ell\neq p$.

When $C = D(Ab)$ and $E = \mathbb Z_{(p)}$, the Bousfield localization consists of chain complexes whose homology is a $\mathbb Z_{(p)}$-module.

By "different behavior", I mean, to a first approximation, that even though $D(Ab)$ is "the natural $\infty$-categorical counterpart to $Ab$", in these cases it's *not* the case that the restriction of the $E$-Bousfield localization in $D(Ab)$ to $Ab$ coincides with the $E$-Bousfield localization in $Ab$ itself.

Part of the problem is that I'm not exactly sure what qualifies as "being in the $\infty$-categorical setting". After all, an ordinary category is in particular an $\infty$-category. But maybe for concreteness, I'll ask a slightly less vague version of the question:

**Question 1:** If $T$ is a tensor triangulated category with a $t$-structure, and $E \in T^{heart}$, is there any reason to think about the Bousfield localization of $T^{heart}$ at $E$ rather than the Bousfield localization of $T$ at $E$?

Let $G$ be an $n \times n$ grid graph. Is there anything known about the asymptotic growth rate of the number of self avoiding paths from $(0,0)$ to $(n,b)$ (from the lower left corner to some arbitrary point on the right hand side)?

For example, is there anything known about the $b$ that maximizes this number? A limiting probability distribution on $[0,1]$ for the value of $b$ (if we pick a self avoiding walk from $(0,0)$ to $x = n$ uniformly at random)?

Some similar questions go unanswered:

Any approximation algorithms for self-avoiding walks?

How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?

Thank you!

I am studying theorem (1.11)* from this article: https://sci-hub.tw/https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-9.2.337

Theorem (1.11)*. Let $R$ be a Jacobson ring, then $S = R[x,\alpha]$ is a Jacobson ring.

In the proof, when $x \notin P$, the author takes the isomorphism $\dfrac{S}{(P\cap R)S} \cong \dfrac{R}{P \cap R} [x,\alpha]$ and says to **replace** $R$ by $R/(P \cap R)$ and **reduce to the case when** $S$ is a prime ring and $P$ is a prime ideal which satisfies $P \cap R = 0$.

My question is: I don't understand why we can only consider the case when $P \cap R = 0$.

Any help would be great.

Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder if this is statement is correct and whether it is written down in some book/article.

Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix.

There has been some study (e.g. see this paper) on the notion of diagonal and antidigonal $X$-ray sequences. Given a permutation matrix $\pi\in\mathfrak{S}_n$, starting from the bottom, construct the $k$-th diagonal sum $y_k$ of its entries for $k=1,2,\dots,2n-1$. Then, the sequence or word $y(\pi)=y_1y_2\cdots y_{2n-1}$ is called the diagonal $X$-ray of $\pi$.

For example, if $\pi=231\in\mathfrak{S}_3$ then $y(\pi)=10020$.

It is easy to check that the number $N(\mathfrak{S}_n)$ of distinct $X$-ray sequences of the set $\mathfrak{S}_n$ is less than $n!$ while the exact value is unknown.

**QUESTION.** Is there at least a $1^{st}$-order asymptotic estimate on the growth rate of $N(\mathfrak{S}_n)$?

Is there an analytical solution satisfying these 3 equations with non-constant z?

$$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$ $$\frac{dy}{dt}=z\cdot\sin(\omega t)$$ $$\frac{dz}{dt}=x\cdot\cos(\omega t) - y\cdot\sin(\omega t)$$

Pick a specific non-zero $\omega$ (e.g., 1 or $\pi$) if you must do so.