I'm interested in the behaviour of the following integral around $\delta=0$ which is similar to the Watson integral (the case where $\delta=0$ which diverges in n=1 and n=2 dimensions, but converges for $n\geq 3$)

$$I_n=\frac{1}{(2\pi)^n}\underbrace{\int_{-\pi}^{\pi}\ldots\int_{-\pi}^{\pi}}_{\text{n times}}\frac{d\theta_1\ldots d\theta_d}{1-n^{-1}\sum_{i=1}^{n}\left[\cos(\theta_i)+i\delta\sin(\theta_i)\right]}$$

In one dimension I expect $I_1\sim \delta^{-1}$, in two dimensions I expect $I_2\sim \log(\delta^{-1})$ (see for instance Lattice random walk under gravity) and above that I'm unsure but suspect $I_n\sim C_n + \alpha_n \delta^{n-2}$ where $C_n^{-1}$ is the $n$th Polya random walk coefficient and $\alpha_d$ a constant.

I note that the author of an answer here Lattice random walk under gravity notes such scaling for $n=2$ but doesn't mention a method.

I can't seem to rearrange it into a form where the canonical methods are helpful, but perhaps I'm missing something?

Can I use the normal rules to apply here?
I mean the rules about real number.

Can I apply them for matrices or not?

Show that

$$ A \begin{pmatrix} B & C \ \end{pmatrix} = \begin{pmatrix} AB & AC \\ \end{pmatrix} $$ for $A ∈ M_{m×n}$, $B ∈ M_{n×p}$, $C ∈ M_{n×q}$; $$ \begin{pmatrix} A \\ B \\ \end{pmatrix} C = \begin{pmatrix} AC \\ BC \\ \end{pmatrix} $$ for $A ∈ M_{p×n}$, $B ∈ M_{q×n}$, $C ∈ M_{n×m}$.

I cannot find the following equation $$u_t=u^2u_{xx}$$ anywhere in the literature. Could someone please point me in the direction of what's been published on this?

Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is nullhomotopic?

Let $X \in \mathbb{R}^{d}$ follows the standard Gaussian distribution $N(0, I_d)$. Let $Y = \max_{j\in[d] } X_j$. It is not hard to see that \begin{align} \mathbb{E}\left [ Y \cdot X\right] = \sum_{j=1}^n \mathbb{P}\left( j = \arg\max_{i \in [d]} X_i \right) \cdot e_j, \end{align} where $e_i$ is the standard basis in $\mathbb{R}^d$. Now I was wondering how to compute $$\mathbb{E} \left[ Y\cdot (X X^\top - I_d) \right ].$$ Is there a closed form solution?

Tit's Corvallis article introduces a map on special fibers of group schemes associated to the elements fixing sets pointwise in a building from $\bar{\mathcal{P}_\Omega}$ to $\bar{\mathcal{P}_{\Omega'}}$ where $\Omega' \subset \Omega$, but that is all he does with this map. In particular there is nothing about injectivity. Is it injective when we look at the maximal reductive quotients of the $\mathcal{P}$? By the discussion in $3.5$ it is enough to answer for facets.

I've looked in Bruhat-Tit's 5 articles and have not found anything about this, but it is very possible I am not looking in the right places for it.

Let $D$ be a bounded domain in $\mathbb{R}^{N}$ ($N\geq2$) and $E$ a closed subset of $D$ with empty interior. Suppose $f$ is a measurable function defined on $D$ and integrable on $D\setminus E$, i.e., $$\int_{D\setminus E}|f(x)|dx<\infty.$$ Can we say that at least either its positive part $f^{+}$ or its negative part $f^{-}$ are integrable on $D$?

A more general version of this question was asked: Combining Couplings of Random Variables, but I have additional constraints on my processes.

Given a fixed positive integer $n$, I have two random variables $$A(n)=2^{A_2}\cdots p^{A_{p_n}}, B(n)=2^{B_2}\cdots p^{B_{p_n}},$$ where $p_n$ is the largest prime number not exceeding $n$, $(A_p)_{p\le n}$ and $(B_p)_{p\le n}$ are two random processes, and the index $p$ ranges over prime numbers. **In addition**, $(A_p)_{p\le n}$ is a dependent process, $(B_p)_{p\le n}$ is an independent process, but the dependence of $(A_p)_{p\le n}$ isn't significant since $$(A_p)_{p\le n}\Rightarrow (B_n)_{p\le n}$$ in distribution as $n\to \infty$.

I am searching for a coupling of $A(n), B(n)$ such that $A\vert B$, and this holds if and only if we have the vector inequality $(A_2,A_3\ldots, A_{p_n})\le (B_2,B_3\ldots, B_{p_n})$.

Thus, I seek a probability space in which $A_p\le B_p$ for each $p\le n$.

For each $p$, I have a coupling of $A_p$ and $B_p$ such that $A_p\le B_p$. I.e., for each $p$ I have a joint distribution $(A_p',B_p')$ with marginals $A_p,B_p$ such that $\mathbb{P}(A_p=a, B_p=b)=0$ when $a > b$.

Is there any literature/technique on how to combine these couplings into a new probability space in which $A_p\le B_p$ for *all* $p\le n$? I know that this is not always possible.

My idea was to consider the joint distribution of the $n$ coupled random vectors $(A_p,B_p)$. Then the $p$th marginal is $(A_p, B_p)$ which only has positive probability when $A_p \le B_p$. However, I am not sure if this is the desired probability space since the marginals are not the $A_p$'s or $B_p$'s but are instead random vectors (and the $A_p$'s are dependent). I am unsure if this joint distribution gives the desired coupling of $A,B.$

Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin. Isoperimetric inequality says: $ Vol_{n-1} \partial A \geq Vol_{n-1} S^{n-1}_R $, where $S^{n-1}_R$ is the corresponding $n-1$ sphere.

I'm thinking of the following variation in terms of Wasserstein distance, $W_p$ (a better name is perhaps iso-Wasserstein distance inequality, if it were true). Let $B^n_r$ be another $n$ ball with radius $r$ centring at origin. Is it true that for all regions $A$ such that: $$ Vol_{n} A = Vol_{n} B^n_R $$ we have $$ W_p(\mathbb{P}(A), \mathbb{P}(B^n_r)) \geq W_p(\mathbb{P}(B^n_R), \mathbb{P}(B^n_r)) $$ where $\mathbb{P}$ denotes a uniform probability distribution on a region, i.e. $\mathbb{P}(A)$ has density $\frac{1}{Vol_{n} A}$.

Can you point out any references that deal with problems of the following type?

$$u_t(t,x) + (f(t,x,u(t,x)))_x = 0, \quad (t,x) \in (0,\infty)\times \Omega \\ f(t,x,u) \cdot \nu(\sigma) = g(u), \quad (t,x)\in (0,\infty)\times \partial \Omega^+\\ u(0,x) = u_0(x), \quad x \in \Omega, $$ where $\Omega \subset \mathbb{R}^2$ and $\partial \Omega^+$ is the subset of the boundary where $f(t,x,u) \cdot \nu(\sigma) >0$

Suppose $U=\{(p,q)\,|\, p,q$ are prime numbers$,\, q\le p\}$ & $(a_1,b_1),(a_2,b_2)\in\Bbb N\times\Bbb N$ that $a_1b_2\neq a_2b_1,$ & $L$ is a direct line contain the points $(a_1,b_1),(a_2,b_2)$ such that $0\le \frac{b_2-b_1}{a_2-a_1}\le 1$ now the question is what can state about cardinal of intersection of $L$ & $U$, although I guess nothing.

I thank you so much.

Ramanujan proved that $\pi(x)^2 < \frac{e x}{\ln x} \pi \left(\frac{x}{e} \right)$ for $x$ sufficiently large with $\pi(x)^2 - \frac{e x}{\ln x} \pi \left(\frac{x}{e} \right) = -\frac{x^2}{\ln^6 x} + O \left(\frac{x^2}{\ln^7 x} \right)$.

Is there a generalization for $a>0$ and Ramanujan inequality is special case when $a =e$? Does this estimate have a precision described by $\pi(x)^2 - \pi \left( \frac{x}{a} \right) f_a(x) = -\frac{x^2}{\ln^6 x} + O \left( \frac{x^2}{\ln^7 x} \right)$ or an even better bound?

If there is a paper addressing such a generalization, please cite the reference.

Let $E$ be an elliptic curve over a number field $F$. Assume that $E$ has a $F$-rational non-torsion point $Z$. For each prime $p$, let $\frac{1}{p}Z$ be the set of $X\in E(\bar{F})$ such that $pX=Z$. Is it possible to have

$F(\frac{1}{p}Z)=F(E[p])$

for infinitely many prime $p$?

I'm trying to understand the non-commutative Koszul complex, as can be found in Anick's nice paper "Non-Commutative Graded Algebras and Their Hilbert Series", J. of Algebra 78, (1982) and I'm stuck at two points, which are just where the paper "jumps" from the commutative case to the non-commutative one.

Let me set first some notation and assumptions. For the commutative situation, we have: $\mathbf{k}$ is a field, $R$ a connected commutative graded $\mathbf{k}$-algebra; that is,

$$ R = \mathbf{k} \oplus R_1 \oplus R_2 \oplus \dots $$

in which every piece $R_n$ is a finite-dimensional $\mathbf{k}$-vector space.

Let $\theta_1, \dots , \theta_r \in R$ be a *regular* sequence. That is, the ideal generated by $\theta_1, \dots , \theta_r$ is smaller than $R$ and each $\theta_n$ is *not* a zero divisor in $R/(\theta_1, \dots , \theta_{n-1})$ for all $n$. (Also the $\theta_n$ are homogeneous of positive non-zero degree.)

**First.** The main idea in Anick's paper seems to be replacing the notion of non zero divisors, with the help of the following characterization:

$$ \theta_1, \dots , \theta_r \quad \text{is a regular sequence}\quad \Longleftrightarrow \quad R \cong \mathbf{k}[\theta_1, \dots , \theta_r] \otimes \frac{R}{(\theta_1, \dots , \theta_r)} \ . $$

Here:

- $\mathbf{k}[\theta_1, \dots , \theta_r]$ is the polynomial algebra on $\theta_1, \dots, \theta_r$,
- the isomorphism is as graded $\mathbf{k}$-vector spaces,
- the tensor product is over $\mathbf{k}$.

My first question is about this isomorphism: Anick says it's "well known" and gives a reference: Stanley, "Hilbert Functions of Graded Algebras", Adv. in Math. 28 (1978). Ok, there the closest thing looking like this result is in page 63, where Stanley says: "This is essentially a well-known property..., though an explicit statement is difficult to find in the literature." And gives in turn references for a particular case.

So, ok, I'm trying to provide myself of some proof, with the help of what Anick does for the non-commutative case. You can easily produce a morphism of graded vector spaces

$$ \mathbf{k}[\theta_1, \dots , \theta_r] \otimes \frac{R}{(\theta_1, \dots , \theta_r)} \longrightarrow R $$

by sending each $\theta_i$ to itself in $R$. For the quotient part, you choose a $\mathbf{k}$-linear section of the projection $R \longrightarrow R /(\theta_1, \dots , \theta_r)$. Anick proves that you can pick no matter which section and the resulting morphism is an epimorphism. So far, so good.

Ok so, this is my first question: **how do you prove that this morphism is also a monomorphism?**

In another part of the paper, Anick uses an argument on dimensions: after all, we are dealing with finite dimensional vector spaces. So I'm trying to prove (first for the case $r=1$) that, degree-wise, what we have on both sides are vector spaces of the same dimension. It's a little bit messy (I hate counting!), but I think I got it. My only doubt is: is there a more simple, direct "well-known" proof?

**Second.** What's the problem of the notion of (two-sided) non zero divisor in the non-commutative case that forces Anick to replace it by that isomorphism? (Then he goes on talking about "strongly free" sets, which are those $\theta_1, \dots , \theta_r \in H$, such that you have an isomorphism

$$ H \cong \mathbf{k}\langle \theta_1, \dots , \theta_r \rangle \otimes \frac{H}{H(\theta_1, \dots , \theta_r)H} \ , $$

$H$ a connected non-commutative graded $\mathbf{k}$-algebra, $\mathbf{k}\langle \theta_1, \dots , \theta_r \rangle$ is the free associative algebra...)

Does the Hasse principle hold for quadratic forms over finitely generated fields (e.g. for the Henselisations/completions at height-$1$-primes or all places)?

Let $f: \mathbb{R}^d\rightarrow\mathbb{R}_{\geq 0}$ be a function which is convex and smooth (*i.e.*, in $C^{\infty}$). If $x^* \in \mathbb{R}^d$ is the (global) minimum of $f$, it is well known that gradient descent with a (sufficiently small) fixed step size $\eta$,
$$ x_{n+1} = x_{n} - \eta \nabla f(x_n), $$
converges to the global minimum $x_n \rightarrow x^*$.

**Question**: What is the asymptotic behavior of gradient descent when a finite minimizer $x^*$ does not exist (*i.e.*, when $\forall x \in \mathbb{R}^d : \nabla f(x)\neq 0$)?

My guess is that $x_n/||x_n||$ should converge to some finite solution, but I could not find a proof. Or perhaps there is some counter-example to this claim?

Thanks in advance!

Let $A(x)$ be a symmetric negative semi-definite matrix which depends continuously on the parameter $x\in\mathbb{R}^{d}$. We consider the differential equation $$\dot{x} = (I-xx^*)A(x)x$$ on the unit sphere. Is there anything known about the convergence behaviour of the trajectories for $t\to\infty$?

If the matrix is constant it converges to a stable equilibrium.

Let $BG$ denote the classifying space of a finite group $G$. For which group cohomology classes $c\in H^2(G;\mathbb{Z}/2)$ does there exist a real vector bundle $E$ over $BG$ such that $w_2(E)=c$?

In general when we talk about *controllability* we talk about proving the existence of control to remain the state to another state at desired time $T$, but in *stability* we prove that the solution tends to zero when $T$ tends to infinity.

My question is: Is there any relation between stability (exponential, polynomial, strong) and controllability (approximate or exact, null) of PDE?

Given the data of a triple $(G,h,k)$ where $G$ is a finite group, and $h,k\in G$ of the same order which together generate $G$, I'm interested in understanding the possible pairs $(i,\alpha)$, where $i : G\hookrightarrow \tilde{G}$ is an injection, and $\alpha\in\tilde{G}$ satisfying

$\tilde{G}$ is a finite group.

There is an $\alpha\in \tilde{G}$ such that $\tilde{G} = \langle G,\alpha\rangle$, and

$\alpha h\alpha^{-1} = k$.

There is a natural way to do this, which is to embed $G$ inside the symmetric group $S_G$ on $G$ via the left regular representation. I'll translate their proof into my situation: Let $H := \langle h\rangle$, $K := \langle k\rangle$. Now let $\sigma$ denote a pair of transversals $x_1,\ldots, x_n$ for $G/H$, and $y_1,\ldots,y_n$ for $G/K$, such that $y_1 = x_1 = 1$. Now define $$\alpha_\sigma : G\rightarrow G \qquad\text{by}\qquad \alpha_\sigma(h^jx_i) = k^j y_i$$ In particular, we have $\alpha_\sigma(h) = k$.

Then, if $\ell_t$ for $t\in G$ denotes the permutation given by left-multiplication by $t$, we wish to check that $\ell_k = \alpha_\sigma\circ\ell_h\circ\alpha_\sigma^{-1}$. To check this, for any $g\in G$, write $g = k^j y_i$, then note: $$\alpha(h^jx_i) = k^jy_i = g\quad\text{so}\quad h^jx_i = \alpha^{-1}(k^jy_i) = \alpha^{-1}(g)$$ Thus, $$(\alpha_\sigma\circ\ell_h\circ\alpha_\sigma^{-1})(g) = \alpha_\sigma(h\cdot h^jx_i)= k^{j+1}y_i = kg = \ell_k(g)$$

In particular, letting $L_G\subset S_G$ be the image of the left regular representation of $G$, then for any choice $\sigma$ as above, letting $G_\sigma := \langle L_G,\alpha_\sigma\rangle\subset S_G$, the pair $(G\subset G_\sigma,\alpha_\sigma)$ satisfies our desired properties.

Unfortunately, the link above doesn't provide any references for this proof.

**Some questions:**
What is the relation between such $G_\sigma$ and the infinite HNN extension $\langle G,\alpha | \alpha h\alpha^{-1} = k\rangle$?

Does this construction yield *all* pairs $(i,\alpha)$ satisfying (1),(2),(3)? (E.g., is every finite quotient of $\langle G,\alpha | \alpha h\alpha^{-1} = k\rangle$ isomorphic to some $G_\sigma$? If so, given a pair $(i,\alpha)$, how does one recover the transversals $\sigma$ and the permutation $\alpha_\sigma\in S_G$? If not, is there a more general technique for constructing the ones we're missing?

Does this construction satisfy some kind of universal property?

Does there exist a group $G$ for which any pair of generators of $G$ are conjugate (ie, the two generators are conjugate elements) in $G$?

Are any of these questions answered in the literature? (references would be appreciated).