In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the solution to this problem would be optimal transport with cost-function given by the *commute time distance* of the chain. I'm opening this new question to try and explore the OT connection better.

Consider an ergodic Markov chain with state space $S$. Let $\pi_0$ and $\pi_1$ be two distributions on $S$. Let $\Gamma(\pi_0,\pi_1)$ be the coupling polytope of joint distributions on $S \times S$ with marginals $\pi_0$ and $\pi_1$ respectively. For example, if the state space $S$ is finite, then this is just the set of all nonnegative matrices with row sum $\pi_0$ and column sum $\pi_1$. Since the *commute time distance* (i.e the average time it takes to move from one state to another and back) defines a distance on $S$, it follows that the mapping

$$\Delta(\pi_0, \pi_1) := \operatorname{minimize}_{\gamma \in \Gamma(\pi_0,\pi_1)}\mathbb E_{(x,y) \sim \gamma}[\operatorname{CommuteTimeDistance}(x,y)],$$ defines a distance between distributions on $S$.

QuestionIs there an efficient way to compute such a distance ?

Poorman's solutionIn finite dimensions, it should be possible to use Sinkhorn iterations to solve an entropy regularized version of the above problem (this appears to be the SOTA method for solving large scale OT problems). I was wondering whether we can do better. That is, maybe a more "analytic" way to go about the problem.

Euclidean embeddingThis paper shows that if the state space $S$ is finite, say has $n$ elements, then it can be embedded into a euclidean space of dimension $\le n - 1$ via the mean commute time as follows. Let $L = U\Lambda U^T$ be the eigen-decomposition of Laplacian of the chain (note that $L$ is positive semi-definite). Finally, for $i \in S= \{1,2,\ldots,n\}$, let $e_i$ be the $i$th unit vector in $\mathbb R^n$ and set $\psi_i := U{\Lambda^\dagger}^{1/2}e_i$. Then

$$\operatorname{CommuteTimeDistance}(x,y) = \|\psi_x - \psi_y\|_2^2. $$

Deterministically finding primes in $[N,2N]$ in $N^{o(1)}$ time is a big deal. Probabilistically we can locate them in $O(\log N)$ time using $PNT$.

- How difficult is to find an algorithm to beat the $O(\log N)$ bound by reducing finding primes to finding integers of size $o(N\log N)$ in intervals of cardinality $o(N\log N)$ with one big prime factor in $[N,2N]$ and at most $o(\log\log N)$ small factors in $[0,o(\frac{\log N}{\log\log N})]$ with probability $\frac1{\omega(\log N)}$?

It may be that such an algorithm is amenable to derandomization than an algorithm based on $PNT$ alone and so much more valuable than trivial randomized algorithm.

The probability a given integer in $[0,n]$ is a square is $\frac1{\sqrt n}$. What is the probability that if you take two integers uniformly then their product is square?

I know the main term is $\frac1n$. I am also looking for correction terms. The difficulty is an average integer has $\omega(\log\log n)$ factors which can go as high as $\frac{\log n}{\log\log n}$ and as low as $2$. Then we are looking at the probability that the prime factors with odd parity in both integers come in pairs. Is there a way to organize the computations and get one precise quantity?

In general what is the probability that product of $m$ integers is a $k$ power where each integer is in $[a,b]$?

Could anyone define the meaning of infinite -dimensional evolution equation in the simplest way, for instance in the context of the Navier-Stokes equation?

I understand the concept of what an evolution equation is, but what does the infinite-dimensional refer to specifically?

Let $\sigma$ be the spectrum of an unbounded operator $T$. It is well known that $\sigma$ is a closed set of $\Bbb{C}$.

Mu question is: if $\Bbb{R}\subset \sigma$ and $i\Bbb{N}\subset \sigma$ with $i^2=-1$ and $\sigma\subset\Bbb{R}+i\Bbb{N} $.

Can we say that $\Bbb{R}+i\Bbb{N}\subset \sigma$.

**The Coxeter transformation of a generalized Cartan matrix**:

In the paper The spectral radius of the Coxeter transformations for a generalized Cartan matrix, Claus Ringel defines a Coxeter transformation as follows, in several steps:

A *generalized Cartan matrix* of size $n$ is a matrix $A \in M^{n \times n}(\mathbb{Z})$ such that for all $i \neq j$ the following properties are satisfied:

- $A_{ii} = 2$
- $A_{ij} \leq 0$
- $A_{ij} \neq 0 \Leftrightarrow A_{ji} \neq 0$

He then goes on to define the reflection $R_i: \mathbb{R}^n \to \mathbb{R}^n$ as the linear map (depending on $A$) which is given on the canonical basis by $e(j) \mapsto e(j) - A_{ji}e(i)$.

Now if $\pi: \{1, \dots, n\} \to \{1, \dots, n\}$ is any permutation, he calls the product $C(A, \pi) := R_{\pi(n)} \cdots R_{\pi(1)}$ a *Coxeter transformation* for $A$.

**The Coxeter transformation of a finite dimensional algebra**:

Let $k$ be a field and $A$ be a finite-dimensional $k$-algebra. Let $P(1), \dots, P(n)$ and $I(1), \dots, I(n)$ be the indecomposable projective $A$-modules up to isomorphism and the indecomposable injective $A$-modules up to isomorphism, respectively (in such a way that the orders correspond, i.e. $P(i) = Ae_i$, $I(i) = D(e_iA)$ with the same idempotent $e_i$). Assume that the dimension vectors $\underline{\dim}(P(1)), \dots, \underline{\dim}(P(n))$ form a basis of $\mathbb{R}^n$. Then we can define the *Coxeter transformation* for $A$ (with respect to the ordering of the projective modules) by $\Phi_{A}: \mathbb{R}^n \to \mathbb{R}^n, \ \underline{\dim}(P(i)) \mapsto -\underline{\dim}(I(i))$,

**Question**:

What precisely is the relation between these two notions of Coxeter transformations? How do properties of a generalized Cartan matrix and properties of a finite-dimensional algebra translate via this relation?

Let $\mathcal A$ be a conformal net, and let $\mathcal J$ be the set of all proper open sub-intervals of $S^1$.

We say that $\mathcal A$ satisfies **von Neumann density**, if for any representation $\pi$ of $\mathcal A$ and any $I\in\mathcal J$, $\pi(\mathcal A(I)\vee\mathcal A(I'))=\bigvee_{J\in\mathcal J}\pi(\mathcal A(J))$. ($I'$ is the open complement of $I$.) The importance of von Neumann density is that any representation of $\mathcal A$ is determined by the corresponding $\mathcal A(I)$-$\mathcal A(I')^{\mathrm{opp}}$ bimodule. So Connes fusions of representations of $\mathcal A$ cannot be well defined without the requirement of von Neumann density.

We say that $\mathcal A$ is **strongly additive**, if for any $I\in\mathcal J$, and any pair of disjoint open intervals $I_1,I_2$ obtained by removing a point on $I$, we have $\mathcal A(I)=\mathcal A(I_1)\vee\mathcal A(I_2)$.

It is easy to see that strong additivity implies von Neumann density. Question: are these two conditions equivalent?

Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or *primitive sequence*.

For example it is easy to see that $\lfloor \frac{n}{2} \rfloor+1, \lfloor \frac{n}{2} \rfloor+2, \ldots ,n-1, n$ is an $M$- sequence with terms not greater than $n$. Actually we cannot hope from an $M$- sequence to contain more than half of the numbers $\le n$ as Erdos has proved.

My question is:

Are there other examples of non trivial $M$-sequences (prime numbers, squares of primes, etc) in the literature which we can form ** explicitly**?

The theta series for a lattice $\Lambda$ is defined by $$\displaystyle \Theta_\Lambda(q) = \sum_{x \in \Lambda} q^{x \cdot x}.$$ Setting $q=e^{-\pi\tau}$ yields the (maybe more usual) related theta series $$\displaystyle \theta_\Lambda(\tau) = \sum_{x \in \Lambda} e^{-\tau\pi\|x\|^2}.$$ By Poisson formula one gets the functional equation fulfilled by $\theta$: $\theta_\Lambda(\tau) = \frac{1}{\tau^{n/2}\textrm{det}(\Lambda)}\theta_{\Lambda^\lor}\left(\frac{1}{\tau}\right)$. It allows to derive some decay information on $\theta_\Lambda$ such as: $\theta_\Lambda(\tau)\leq \tau^{-n/2}\theta_\Lambda(1)$.

If we want to generalise the definition of the theta series to sublattices (instead of only vectors) we could think of something like: $$\displaystyle \theta^{(r)}_\Lambda(\tau) = \sum_{\begin{aligned}\ell~ \textrm{su}&\textrm{blattice of}~\Lambda\\ &\textrm{rk}(\ell) = r\end{aligned}} e^{-\tau\pi\cdot\textrm{det}(\ell)^2}, $$ for a fixed integer $1\leq r\leq \textrm{rk}(\Lambda)$. One could also restrict the latter sum to only primitive sublattices.

However since the set of sub-lattices of fixed rank is not a lattice itself (it is only embedded in the r-exterior power lattice constructed from $\Lambda$), harmonic analysis such as Poisson formula seems compromised.

Is it possible to derive a functional equation similar to the classical one for regular $\theta$? Or maybe in a direct manner get some decay bounds ?

Note: in the case of primitive sublattice one can relate the $\theta^{(r)}_\Lambda$ series with $\theta^{\textrm{rk}(\Lambda)-r}_{\Lambda^\lor}$ by relating a rank $r$ primitive sub-lattice $\ell\subset\Lambda$ with a sub-lattice of corank $r$ in $\Lambda^\lor$ of determinant $\det(\Lambda)\cdot\det(\ell)$

Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the poset $\mathbb{Z}$. There's a canonical functor:

$$Gr:Fun(\mathbb{Z},\mathcal{C}) \to \underset{n \in \mathbb{Z}}{\coprod}{\mathcal{C}} \subset Fun(\mathbb{Z},\mathcal{C})$$

$$X \mapsto \underset{n \in \mathbb{Z}}{\coprod}{Cofib(X(n-1) \to X(n))}$$

Define the category $Fil(\mathcal{C})$ of filtered objects of $\mathcal{C}$ as the localization of $Fun(\mathbb{Z},\mathcal{C})$ w.r.t. morphisms $f: X \to Y$ which induce equivalences on graded pieces $Gr(f):Gr(X) \to Gr(Y)$.

(I've taken this definition from the following paper. I have no claims to originality for this or for anything else in this question for that matter).

There's an obvious functor on $Fil(\mathcal{C})$ which shifts sequences. For notational convenience we will denote it $T: Fil(\mathcal{C}) \to Fil(\mathcal{C})$ with $T(X)(n):=X(n-1)$ (this convention will be useful later).

There's also the suspension functor coming from $\mathcal{C}$ which acts objectswise. We'll denote it as usual by $\Sigma$. Notice that $T$ and $\Sigma$ commute with each other.

The functor $Gr$ above sits in the fiber sequence of functors:

$$T \to Id \to Gr$$

Or equivalently:

$$Id \to Gr \to \Sigma T$$

Precomposing this sequence of functors with the tail $\Sigma T$ we get

$$Id \to Gr \to \Sigma T \to Gr \circ \Sigma T \to \Sigma^2 T^2$$

By repeating this process we get an infinite sequence of functors:

$$Id \to Gr \to \Sigma T \to Gr \circ \Sigma T \to \Sigma^2 T^2 \to Gr \circ \Sigma^2 T^2 \to \Sigma^3 T^3 \to \dots$$

One can just as easily extend this sequence in the other direction. Notice that inside this sequence sits the sequence:

$$\dots \to Gr \to Gr \circ \Sigma T \to Gr \circ \Sigma^2 T^2 \to Gr \circ \Sigma^3 T^3 \to \dots$$

Which is a "complex" in the sense that the composition of any two consecutive natural transformations is null-homotopic.

If $\mathcal{C}$ is provided with a t-structure then one should be able to make the sequence above the $E_1$ page of a spectral sequence with values in $\mathcal{C}^{\heartsuit}$.

In summary we have managed to make the construction of the $E_1$ page of a filtered object completely functorial. Here's the question:

**How far can we take this?** Meaning: Is there a functorial construction which gives some "object" naturally coming from $Fil(\mathcal{C})$ (can be a functor, a sequence of functors etc.) and independent from any t-structure on $\mathcal{C}$ s.t. after choosing a t-structure we get a spectral sequence with values in $\mathcal{C}^{\heartsuit}$ (with all the pages and all the differentials) for every object in $Fil(\mathcal{C})$?

Since $\zeta(s)$ is not so easy to handle, so the better choice is to study a modified function rather than $\zeta(s)$ itself, as an analogy, it is more convenient to deal with $\Gamma(s)\Gamma(1-s)$, which leads to an elementary function $\frac{\pi}{\sin\pi s}$ (acturally this fact enlights me to ask the following question).

By the integral representation of $\zeta(s)$ for $\Re(s)>1$: $$\zeta(s)=\frac{1}{\Gamma(s)} \int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$ $$\zeta(s)\zeta(1-s)=\frac{\sin \pi s}{\pi} \int_0^{\infty}\int_0^{\infty} \frac{x^{s-1}y^{-s}}{(e^x-1)\cdot (e^y-1)}dxdy$$ or $$\zeta(s)\zeta(1-s)=\frac{\sin \pi s}{2\pi} \int_0^{\infty}\int_0^{\infty} \frac{x^{-s}y^{-s}(x^{2s-1}+y^{2s-1})}{(e^x-1)\cdot (e^y-1)}dxdy$$ By this integral expression and its analytic continuation or its Euler's product form $$\zeta(s)\zeta(1-s)=\prod_p(1-p^{-s}-p^{s-1}+p^{-1})^{-1}$$ we may get a formula on function $\zeta(s)\zeta(1-s)$.

So the **question** is: Is it possible that function $\zeta(s)\zeta(1-s)$ can be simplified to an easily handled function(may be a combination of elementary functions, and some well-studied functions like hyperbolic functions, Bessel functions and so on), just like $\Gamma(s)\Gamma(1-s)$?

**THE FRAMEWORK:** let us consider a real topological vector space $V$.

We denote with $\mathscr C_k(V)$ the set of all continous functions $f:[0,T]^k\to V$ such that $f_{t_1\cdots t_k}=0$ whenever $t_i=t_{i+1}$ for some $0\le i\le k-1$.

We define the operator $\delta_k:\mathscr C_k(V)\to\mathscr C_{k+1}(V)$ as follows: $$ (\delta_kf)_{t_1\cdots t_{k+1}}:=\sum_{j=1}^{k+1}(-1)^jf_{t_1\cdots \widehat t_{j}\cdots t_{k+1}} $$ where the hat $\widehat{\cdot}$ means that argument is omitted.

So if $f\in\mathscr C_1(V)$ then $(\delta_1f)_{ts}=f_t-f_s$ and if $g\in\mathscr C_2(V)$ then $(\delta_2g)_{tus}=-g_{us}+g_{ts}-g_{tu}$.

Next we define the following norms: if $g\in\mathscr C_2(V)$ then, for $\mu>1$ we set $$ \|g\|_{\mu}:=\sup_{0\le s<t\le T}\frac{|g_{st}|}{|t-s|^{\mu}} $$ while for $h\in\mathscr C_3(V)$ we first set $$ \|h\|_{\mu,\rho}:=\sup_{0\le s<t\le T}\frac{|h_{tus}|}{|t-u|^{\rho}|u-s|^{\mu-\rho}} $$ (here $0<\rho<\mu$) and then define the norm: $$ \|h\|_{\mu}:=\inf\left\{\sum_j\|h_j\|_{\rho_j,\mu-\rho_j}:h=\sum_jh_j,\; h_j\in\mathscr C_3,0<\rho_j<\mu\right\}\;. $$

Finally let us denote for $k=2,3$ $$ \mathscr C_k^{\mu}:=\{h\in\mathscr C_k:\|h\|_{\mu}<+\infty\}. $$ and accept that $$ \ker\delta_k=\operatorname{Im}\delta_{k-1} $$ (this holds for every $k$).

**THE PROBLEM:** let us take $h\in\mathscr C_3^{\mu}$ such that $\delta_3h=0$. Then it is not difficult to prove that there exists $B\in\mathscr C_2$ such that $\delta_2B=h$. Now fix $0\le s<t\le T$ and consider on $[s,t]$ a sequence of partitions $\{\pi_n\}_n$ whose mash tends to zero.

To fix ideas we write $$ \pi_n=\{s=r_0^n<r_1^n<\cdots<r_{k_n}^n<r_{k_n+1}^n=t\} $$

Define then $$ M_{ts}^{\pi_n}:=B_{ts}-\sum_{l=0}^{k_n}B_{r_{l+1}^nr_l^n}\;. $$ Now accept this last one converges (up to passing to a subsequence): how can we show that the limit does not depend on the particular sequence of partitions chosen?

In the next minutes I'll write my attempt, however in the meanwhile I post my problem, so if someone has some good hint, I will thank you!

PS: this is taken from a proof contained in the 2010 paper by M.Gubinelli and S. Tindel ROUGH EVOLUTION EQUATIONS at the end of page 9 (they say to see another paper, but I didn't found nothing on it!)

**MY ATTEMPT:** let us take another sequence of partitions whose mash tends to zero, say
$$
\sigma_n=\{s=u_0^n<u_1^n<\cdots<u_{h_n}^n<u_{h_n+1}^n=t\}
$$

and prove that $$ |M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}|\to0\;\;\;n\to+\infty. $$

Now \begin{align*} M_{ts}^{\pi_n}-M_{ts}^{\sigma_n} &=\sum_{l=0}^{h_n}B_{u_{l+1}^nr_l^n} -\sum_{l=0}^{k_n}B_{r_{l+1}^nr_l^n}\\ &=\sum_{l=0}^{k_n} \left(B_{t_{4l+4}^nt_{4l+2}^n}-B_{t_{4l+3}^nt_{4l+2}^n}- B_{t_{4l+4}^nt_{4l+3}^n}\right)- \left(B_{t_{4l+3}^nt_{4l+3}^n}-B_{t_{4l+2}^nt_{4l+1}^n}- B_{t_{4l+3}^nt_{4l+2}^n}\right)\\ &=\sum_{l=0}^{k_n} (\delta_2B)_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}- (\delta_2B)_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}\\ &=\sum_{l=0}^{k_n} h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}- h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}\\ \end{align*} supposing wlog that $h_n\le k_n$ and setting, for $l\le h_n$ \begin{align*} t_{4l+1}^n&:=r_l^n\\ t_{4l+2}^n&:=u_l^n\\ t_{4l+3}^n&:=r_{l+1}^n\\ t_{4l+4}^n&:=u_{l+1}^n\\ \end{align*} while, for $l>h_n$ the definition of the odd terms stay the same, while $t_{4l+2}^n=t_{4l+4}^n:=t$.

Then we can write \begin{align*} |M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}| &\le\sum_{l=0}^{k_n} |h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}|+ |h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}|\\ &=\sum_{l=0}^{k_n} \left|\sum_jh_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}^j\right|+ \left|\sum_ih_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}^i\right|\\ &\le\sum_{l=0}^{k_n}\left[ \sum_j\left|h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}^j\right|+ \sum_i\left|h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}^i\right|\right]\\ &=\sum_{l=0}^{k_n}\left[ \sum_j\left|\frac{h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}^j}{|t_{4l+3}^n-t_{4l+2}^n|^{\rho_j}|t_{4l+4}^n-t_{4l+3}^n|^{\mu-\rho_j}}\right| |t_{4l+3}^n-t_{4l+2}^n|^{\rho_j}|t_{4l+4}^n-t_{4l+3}^n|^{\mu-\rho_j}\right]\\ +& \left[ \sum_i\left|\frac{h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}^i}{|t_{4l+2}^n-t_{4l+1}^n|^{\rho_i}|t_{4l+2}^n-t_{4l+2}^n|^{\mu-\rho_i}}\right| |t_{4l+2}^n-t_{4l+1}^n|^{\rho_i}|t_{4l+3}^n-t_{4l+2}^n|^{\mu-\rho_i}\right]\\ \end{align*} and since this is true for every $\{h^j\}_j,\{h^i\}_i\subset\mathscr C_3(V)$ such that $\sum_jh^j=\sum_ih^i=h$ and $0<\rho_j,\rho_i<\mu$, passing to the $\inf$ on these parameters, we get \begin{align*} |M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}| \le \|h\|_{\mu}\left[\sum_{l=0}^{k_n}\left(\max\{|t_{4l+3}^n-t_{4l+2}^n|,|t_{4l+4}^n-t_{4l+3}^n|\}\right)^{\mu}\\ +\left(\max\{|t_{4l+2}^n-t_{4l+1}^n|,|t_{4l+3}^n-t_{4l+2}^n|\}\right)^{\mu}\right]\\ \le\|h\|_{\mu}\left[\sum_{l=0}^{k_n}\left(|t_{4l+3}^n-t_{4l+2}^n|+|t_{4l+4}^n-t_{4l+3}^n|\right)^{\mu}\\ +\left(|t_{4l+2}^n-t_{4l+1}^n|+|t_{4l+3}^n-t_{4l+2}^n|\right)^{\mu}\right] \end{align*}

From this, I tried as follows: call $m_0=1$ and take $\pi_{m_0}$; then since the mash tends to zero, there exists $m_1>m_0$ such that $m_0$ elements of $\sigma_{m_1}$, call them $\widetilde{\sigma_{m_1}}:=\{u_{l_s}^{m_1}\}_{s=1}^{m_0}$ are closer as we want to the elements of $\pi_{m_0}$, say $|r_s^{m_0}-u_{j_s}^{m_1}|<\varepsilon_1$.

We have to work on $\sigma_{m_1}$: there exists $m_2>m_1$ such that considering suitable elements of $\pi_{m_2}$, say $\widetilde{\pi_{m_2}}:=\{r_{j_s}^{m_2}\}_{s=1}^{m_1}$, we can approximate the elements of $\sigma_{m_1}$, say $|u_s^{m_i}-r_{j_s}^{m_2}|<\varepsilon_2$, and so on.

Then I did the obvious: $$ M_{ts}^{\pi_{m_{2\beta}}}-M_{ts}^{\sigma_{m_{2\beta+1}}} =\left(M_{ts}^{\pi_{m_{2\beta}}}-M_{ts}^{\widetilde{\sigma_{m_{2\beta+1}}}}\right) +\left(M_{ts}^{\widetilde{\sigma_{m_{2\beta+1}}}}-M_{ts}^{\widetilde{\pi_{m_{2(\beta+1)}}}}\right) +\left(M_{ts}^{\widetilde{\pi_{m_{2(\beta+1)}}}}-M_{ts}^{\sigma_{m_{2(\beta+1)-1}}}\right)\;\;; $$ now the first and the last summand go to zero as $\beta\to+\infty$, but the second term doesn't a priori, since it is an approximation of the LHS.

I thought that maybe there is a way to modify the $\widetilde{\pi}$ and $\widetilde{\sigma}$ and/or exploiting the absolute continuity of $M$ in oder to compare the LHS and the second term of RHS, and maybe writing something like $$ \left|M_{ts}^{\widetilde{\sigma_{m_{2\beta+1}}}}-M_{ts}^{\widetilde{\pi_{m_{2(\beta+1)}}}}\right| =a_{\beta}|M_{ts}^{\pi_{m_{2\beta}}}-M_{ts}^{\sigma_{m_{2\beta+1}}}| $$ for some $0<a_{\beta}<1$ such that $\limsup_{\beta\to+\infty}a_{\beta}<1$, thing that allow me to conclude, but for the moment I'm stuck.

**Some definitions and Notations**

- Group Action see this link.
- $G$-invariant partition $B_i$ means $\forall \sigma \in G, B_i^{\sigma} = B_i$.
- Minimal block system see page no - 30

Let $V$ be a $k$-partite set, in the sense that $V = V_1 \sqcup V_2 \sqcup \cdots \sqcup V_k $. Let $ G \le \mathrm{Sym}(V_1) \times \cdots \times Sym(V_k)$ be a subgroup of the direct product. Now consider the group action $\pi : G \times V \to V$ of $G$ on the set $V$; assume each $G_i$ is transitive on each $V_i$:

Then each $V_i$ has a unique minimal block system $B_i$ on which $G$ acts as either as a Symmetric group or as an alternating group.

Consider the group action of $G$ on the $k$-partite set $(B_1,B_2, \cdots,B_k)$. We say $V_i$ and $V_j$ are **block-linked** if the $G$-action on $B_i$ and $B_j$ are linked.

I can add more information if needed. I am not able to understand the term " block-linked ".

For more detail see the paper of **Babai and Codenotti** with title " **Isomorphism of hypergraphs of low rank in moderately exponential time**" see page - 6

Maybe it is not a math question.

I got my math PhD degree more than 10 years ago, and my major is algebra. I have published some papers in some journals such as Adv. Math.. I really love math, but I am still thinking I haven't got a good and complete math education, and do not have a good picture of math. Someone may argue that I can teach math to myself. The problem is that I am a teather in a university now; I have to teach a lot of classes and have other duties. Now I have an idea which sounds crazy: I want to do another math PhD. But the problem is I am almost 40 years old. So I am wondering if this is possible. Does someone have experience of such a second PhD?

In a quantum-information-theoretic context, I've encountered the problem of integrating over $r \in [0,1]$, the function

\begin{equation} r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2};r^2\right) \left(1-\varepsilon ^2 r^2\right)^{d/2}, \end{equation}

where the parameters $d \geq 1$ and $0<\varepsilon<1$. Mathematica does not appear to directly succeed with this problem. I have tried using certain Pfaff transformations in this regard.

For fixed even values of $d$, the integration yields a polynomial of degree $ d$ in $\varepsilon$, while for odd values of $d$, the integration, quite differently, yields results involving inverse hyperbolic and polylogarithmic functions of $\varepsilon$.

The question pertains to the problem of constructing the functions $\tilde{\chi}_d (\varepsilon)$, raised in https://arxiv.org/abs/1610.01410 and further studied in https://arxiv.org/abs/1701.01973, concerning certain conjectures as to the (rational) values of generalized two-qubit Hilbert-Schmidt separability probabilities''. ($\varepsilon$ is thesingular value ratio'' and $d$ is a ``Dyson-index-like'' parameter, associated with random matrix theory.)

If we set $\varepsilon=0$, then the requested integration yields \begin{equation} \frac{\, _3F_2\left(-\frac{d}{2},\frac{d}{2},d;1+\frac{2}{d},d+1;1\right)}{2 d}, \end{equation} while if we set $\varepsilon=1$, then we obtain \begin{equation} \frac{\Gamma \left(\frac{d}{2}+1\right) \Gamma (d) \, _3F_2\left(-\frac{d}{2},\frac{d}{2},d;1+\frac{2}{d},\frac{3 d}{2}+1;1\right)}{2 \Gamma \left(\frac{3 d}{2}+1\right)}. \end{equation} So, we want a function $f(\varepsilon,d)$ that, in some sense, interpolates between these two endpoints, yielding those polynomials (as indicated by Carlo Beenakker) that result from the integration for even $d$, such as \begin{equation} f(\varepsilon,4)=\frac{13 \varepsilon ^4}{672}-\frac{31 \varepsilon ^2}{630}+\frac{1}{30}. \end{equation}

I need to solve the following system of quasi-linear partial differential equations, where the unknowns $f(x,t), g(x,t)$ are smooth functions on $\mathbb R^2$,

$$ \dfrac{\partial}{\partial t}f=\dfrac{\partial}{\partial x}((f^2+g)f),$$ $$ \dfrac{\partial}{\partial t} g=\dfrac{\partial}{\partial x}((f^2+g)g),$$

I think I am supposed to use the method of characteristics to find a characteristic surface, but I don't know how to do this.

(I asked a more or less equivalent question on Math Stackexchange here, and got no answers)

Consider the infinite 3-regular tree. Pick a vertex $C$, the "center". For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced subgraph on the set of vertices given by this ball. The finite tree $T_L$ rooted at $C$ has $L+1$ layers or generations. It has $3\times 2^L-2$ vertices and they all have degree $3$ except the $3\times 2^{L-1}$ leaves in the top layer. My question is:

Is it possible to add $3\times 2^{L-1}$ edges between the leaves in oder to get a graph $G_L$ with spanning tree $T_L$ such that $G_L$ is vertex-transitive?

For $L=1$ and $L=2$, I can do that which gives me the **tetrahedron** and the **Petersen graph** respectively.
Does this kind of problem arise in geometric group theory and the study of locally finite graphs?

Also, when looking at OEIS A032355, the relevant numbers of vertex-transitive graphs seems abnormally low. Even more surprising, they immediately precede a big surge (look at the values for $n=2,5,11,23,47,95$ in that list). Is there a known explanation for this phenomenon?

**Edit in view of the awesome answers by Aaron and Gordon:**
I still have to digest the theory around Moore graphs and bound. My motivation for this question comes from statistical mechanics on the widest possible class of lattices. In particular it has to do with the question of defining **"periodic boundary conditions"** for given finite subsets.
In this regard, an equally useful (for me) weakening of my question is as follows:

Is there a constant $c\in (0,1)$ independent of $L$ such that one can build $G_L$ as above to make the size of the orbit of the center $C$ no smaller than $c\times(3\times 2^{L}-2)$?

In other words I want to make this orbit of macroscopic size. The orbit is of course wih respect to the action of the automorphism group of $G_L$. A quick remark is that if one adds no edges, i.e., one takes $T_L$ itself, this orbit is the singleton $\{C\}$ and the automorphims group is a wreath product $\mathbb{Z}_2\wr\cdots\wr\mathbb{Z}_2\wr\mathbb{Z}_3$ with $\mathbb{Z}_2$ appearing $L-1$ times.

Let $\Omega$ be a Lipschitz domain in $\mathbb{R}^2$, and let $\mathscr{T}:H^2(\Omega) \to L^2(\partial \Omega)$ be the trace operator defined in the usual way. Is there a characterization of its image? It is clearly a subspace of $H^1(\partial\Omega)$; is it necessarily a closed subspace?

Let $A_h \in \mathbb{R}^{n \times n}$ be the matrix corresponding to a finite element discretization of some nonselfadjoint, bounded and indefinite bilinear form corresponding to a second order operator satisfying a Garding inequality.

Let $A_H \in \mathbb{R}^{m \times m}$ be a matrix corresponding to the same finite element discretization on a coarser grid (grids are nested) and let $R_H$ be the $L_2$-projection from $h$ to $H$.

What $L_2$ bounds can I get for the numerical range of $R_H^T A_H^{-1} R_H A_h$ as a function of $h$ and $H$?

I am using a nonoverlapping symmetric interior penalty discontinuous Galerkin discretization, if it is relevant.