For a Markov chain $X(k)$ on a separable, complete metric space, define an Feller operator $T(f(x))=E[f(X_1) \mid X_0=x] = \sum_{\theta \in \Theta} m(\theta) f(f_\theta(x))$ where the process is $X_n=f_{\theta}(X_{n-1})$ where $\theta$ is iid discrete rv from $\Theta$. The dual of the operator is $T^*\mu(\cdot)= \sum\limits_{k=1}^{m}m(\theta)\mu(f_{\theta}^{-1}(\cdot))$, where $\mu$ is a probability measure on $X$, Could anyone tell me how to show $T^*,(T^*)^2,\dots, (T^*)^n\dots$ is a tight sequence? $m$ is a probability measure on $\Theta$. All functions are continuous.

I was trying to read and this article but could not apply results, my bad. http://www.hairer.org/notes/Convergence.pdf

Let $V$ be the vertices of the cycle graph $C_{3n}$. Suppose there is a partition of $V$ into sets of $3$, i.e. $V=\cup_{k=1}^{n}{V_k}$ where $|V_k|=3$ for $k$ in $1..n$.

**QUESTION:** Is it possible to find an independent set of $V$ with exactly one vertex from each $V_k$?

By the Lovasz Local Lemma, it's possible if the $3$ is replaced with some larger number, say, $11$.

It is given that $ P (x) = ax ^ 4 + bx ^ 3 + cx ^ 2 + dx + e = a (x ^ 2 + p_1x + q_1) (x ^ 2 + p_2x + q_2) $ for some real $ a, b, c, d, e, p_1, q_1, p_2, q_2 $. It is required to prove that $ P (x) $ by replacing $ x = x (t) = \frac {\alpha t + \beta} {\gamma t + 1} $ can be reduced to the form $ \frac {(M_1 + N_1t ^ 2) (M_2 + N_2t ^ 2)} {(\gamma t + 1) ^ 4} $. I could prove it only for the case when $ P (x) $ does not have all real roots. If there is at least one complex root $ z $, that is, its pair is the conjugate root $ \bar {z} $ and then if $ x ^ 2 + px + q = (x-z) (x-z_1) $, where $ p $ and $ q $ are real numbers, then $ z_1 = \bar {z} $. This implies that $ p_1, q_1, p_2, q_2 $ are numbers uniquely determined by $ a, ..., e $. This greatly simplifies the situation, so by extracting the inequality $ p_1 ^ 2-4q_1 <0 $, one can prove the required. If we have 4 real roots $ x_1, x_2, x_3, x_4 $, then in the equality $ (x-x_i) (x-x_j) = M_1 + N_1t ^ 2 $ the numbers $ i $ and $ j $ can be any of the set $ \left \lbrace1,2,3,4 \right \rbrace $. How to be in this case, I do not know.

Phrased in the language of vector bundles, the Quillen-Suslin theorem states that vector bundles on $\mathbb C^n$ are algebraically trivial (for any algebraic vector bundle there exists an algebraic isomorphism to the trivial bundle).

For more general affine varieties, Grauert's Oka principle implies that the holomorphic and topological classification of vector bundles coincide. In particular, all algebraic vector bundles which are topologically trivial are also holomorphically trivial.

As far as I understand, it is not known whether topologically trivial algebraic bundles on affine varieties are *algebraically isomorphic* to the trivial bundle.

If this would be the case for an affine variety $X$, I would call this an analogue of the Quillen-Suslin theorem for $X$.

My question is whether such analogues of the Quillen-Suslin theorem have been proved for affine varieties other than $\mathbb C^n$.

Projection method is a traditional method to numerically handle problem of linear integral equation. The routine way is to do it in $ L^2 $ setting. For example:

Let $ A:L^2(a,b) \to L^2(a,b) $ be a compact injective operator. We introduce a sequence of basis subspace $ X_n := \{ \xi_k\}^n_{k=1} $, which is increasing and eventually dense in $ L^2(a,b) $, that is, \begin{equation*} X_n \subseteq X_{n+1}, \overline{\bigcup_{n \in \mathbb{N}} X_n} =L^2(a,b). \end{equation*} Then define a sequence of orthogonal projection operator $ \{ P_n \} $, which project $ L^2(a,b) $ onto $ X_n $. Now for $ y \in \mathcal{R}(A) $, $ A^\dagger_n y_n $ could be a natural approximate scheme to $ A^{-1} y $, where \begin{equation*} A_n := P_n A P_n : X_n \longrightarrow X_n \ \text{and} \ y_n := P_n y \in X_n \end{equation*} (Of course we could describe above system in a inner product form). The convergence result could be seen in [Theorem 13.6] of

R. Kress: Linear Integral Equations, Springer-Verlag, Berlin, 1989.

However, we notice that the convergence analysis permits the compact operator $ A $ to be defined in Banach space ($ L^p $), but we have not seen a numerical example with projection procedure in $ L^p $ setting. So we want to know if this convergence result in $ L^p $ setting is of practical value? or only of theoretical importance?

If the former, please indicate some references with numerical example which handle linear integral equation with projection method in $ L^p $ setting.

Thank you in advance!

If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$

An approximate solution of $\phi$ is obtained numerically by dividing the boundary into a finite number of segments ,N.

So we can write $$\phi(\vec x_j)=\sum_{i=1}^{N} [\phi(\vec x_i)\frac{\partial \ln(r_{ij})}{\partial n} -\ln(r_{ij})\frac{\partial \phi}{\partial n}(\vec x_i)]\Delta s_i $$ Where $\Delta s_j$ represents the boundary segment length and $r_{ij}$ is the distance between the $i^{th}$ and the $j^{th}$ segment

So It's easy to write $$\frac{\partial \ln(r_{ij})}{\partial n}=\frac{1}{r_{ij}}[-\frac{(x_i -x_j)}{r_{ij}}(\frac{\Delta y}{\Delta x})_j +\frac{(y_i -y_j)}{r_{ij}}(\frac{\Delta x}{\Delta y})_j ]\Delta s_j$$ And $$Z_{ij}=[\ln(r_{ij})]\Delta s_j$$

Prove that $$\lim_{j \to i} \frac{\partial \ln(r_{ij})}{\partial n}=[\frac{(-x_{ss} y_s + x_s y_{ss})_i}{2}]\Delta s_i$$

My try $$\lim_{j \to i} \frac{\partial \ln(r_{ij})}{\partial n}=\lim_{h\to 0}\frac{-(x_i -x_{i+h})(y'_{i+h})+(y_i - y_{i+h})(x'_{i+h})}{(x_i -x_{i+h})^2 +(y_i -y_{i+h})^2}$$

Using $$x_{i+h}=x_i +h x'_i +(h^2/2) x''_i$$ and $$x'_{i+h}=x'_i +hx''_i$$ Hence we get the required result

My question How to prove that $$\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$$

Thanks in advance .

Let $X$ be the affine building of $GL_n(\mathbb{Q}_{p})$. We call oreinetd chamber of $X$ every sequence $\overrightarrow{C}=(s_1,...,s_n)$ of vertices such that $C=\{s_1,...,s_n\}$ is a chamber of $X$. If $\overrightarrow{C}=(s_1,...,s_n)$ and $\overrightarrow{C'}=(s'_1,...,s'_n)$ are two oriened chambers of $X$, we say that $\overrightarrow{C}$ is directly linked to $\overrightarrow{C'}$ if : $$\forall~1\leqslant i\leqslant n-1,~s'_i=s_{i+1}.$$ We call directed gallery every sequence $(\overrightarrow{C_0},...,\overrightarrow{C_m})$ of oriented chambers such that for every $1\leqslant i\leqslant m-1$, $\overrightarrow{C_i}$ is directly linked to $\overrightarrow{C_{i+1}}$ and $\overrightarrow{C_i}\neq\overrightarrow{C_{i+1}}$. Obviously, if $(\overrightarrow{C_0},...,\overrightarrow{C_m})$ is a directed gallery of $X$, then the corresponding sequence of non-oriented chambers $(C_0,...,C_m)$ is a non-stuttering gallery of $X$.

I want to prove that the non-stuttering gallery $(C_0,...,C_m)$ corresponding to the directed gallery $(\overrightarrow{C_0},...,\overrightarrow{C_m})$ is necessarily a minimal gallery of $X$ and then it lies a same apartment of $X$. Thank you in advance for your help.

Permutations $\sigma$ in the symmetric groups $S_n$ can be characterized by their Cayley distance $C_\sigma$, being the minimal number of transpositions needed to convert $\{1,2,3,\ldots n\}$ into $\sigma$. The sign of the permutation is $(-1)^{C_\sigma}$.

For example, when $\sigma=\{2, 3, 4, 5, 1\}$, one has $C_\sigma=4$ and for $\sigma=\{1, 2, 3, 5, 4\}$ one has $C_\sigma=1$. Of the $5!$ permutations in $S_5$ there are, respectively, $1,10,35,50,24 $ with Cayley distance $C_\sigma=0,1,2,3,4$.

**Question:** What is the general formula that counts the number of permutations at a given Cayley distance?

This question was motivated by my attempt to check an integral formula in the unitary group.

Here the following is stated:

It's a basic fact in $p$-adic Hodge theory that any 2-dim. absolutely irreducible $G_{\mathbb Q_p}$-representation with distinct Hodge-Tate weights is uniquely determined by $a_p$.

Could somebody explain in maximum detail why is this true? Taking inspiration from the famous ELI5 communication style, explain like I am a generic first-year grad student.

When working on a research project, one tries to spend their time answering questions that have not yet been answered. There enters the terminology of "known" versus "unknown" results, which we generally take to mean whether a problem has already been solved. On the other hand, we know that mathematics is always a work in progress, including instances of "known" facts that have turned out to be wrong.

The proofs of some results are quite esoteric, requiring extreme specialization in the topic to be able to understand. It is feasible that a paper might be peer reviewed, accepted by the community, and its theorems entered into mathematical canon, only for everyone capable of following the arguments to then pass away leaving no apt descendants to maintain the knowledge. My question is whether those results are still considered "known." The deeper question is about the value of finding new and more accessible proofs for such results, such that they may be more widely known in the literal sense.

To make the question less subjective, let's focus on the etiquette of using this terminology. For a mathematician to publicly proclaim that something is "known," does it require them to have read and understood the proof, to know of someone who has read and understood the proof, and if the latter, must that person be alive? On the other hand, does "known" merely mean that a proof has been published in a peer-reviewed journal at some time in history, no matter how long ago?

If for any strong digraph $H$ we let $\lambda(H)$ to be the length of any shortest closed walk traveling over every arc in $H$ then what is the maximum value of $\lambda(D)$ for any strong digraph $D$ with $n$ arcs?

I.e. for any $n\in\mathbb{N}$ how well can we approximate $M_n=\max(\lambda(D):{\small D\text{ is strong and }|E(D)|=n})$?

I can prove $\frac{1}{4}n^2-17n^{3/2}\leq M_n\leq 2n^2$ so I'm curious if there exists $c\in\mathbb{R}$ for which $M_n\sim cn^2$.

there is a famous lemma which says: if $Y$ and $W$ are flat,projective schemes over $S$ and $s \in S$ be a geometric point and $Y_s$ and $W_s$ be fibers over $s$ and $f:Y_s \to W_s$ be a morphism then with some good conditions we have:

Dimension of every component of scheme $Hom_S(Y,W)$ at a point $f$ is at least:dim$H^0(Y_s,f^*T_{W_s})-$ dim$H^1(Y_s,f^*T_{W_s})+$ dim$S$.

Now suppose that $C$ is a nodal curve of genus zero and $\mu:C \to X$ is a stable map.Suppose that $\bar{C}$ be smoothing of $C$ over some base like $S$ and $\chi = X \times S$.

($X$ is convex,nonsingular variety)

My question is that how can we use above lemma to prove that $\mu$ lies in closure of locus of maps with irreducible domain?

Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment ($p>0, p \in \mathbb{R}$) of the integral $$ \int_0^T |X(t)|^2 dt? $$ (The integral is the pathwise integral.)

Why separate the complete Riemannian manifolds that admit and those that do not admit positive Green function? In summary, what is the motivation for studying parabolic and non-parabolic manifolds?

I have a constrained maximization problem (maximizing a functional), with number of constraints being uncountable infinite.

It looks something like this. I want to maximize the convex functional $C(f)$ over $f \in S$ with the constraint that look like $G(f,\phi) = 0 \forall \phi \in C^{\infty}(\Omega)$. Clearly these constraints are infinite in number (infact uncountably infinite).

I don't know how to use Lagrange multiplier technique in this context. Hence I request for a reference for a theory in this regard. I am not aware of any such concepts and no idea on what terminolgy I should search on google.

Edit : The example cited may not represent a typical case, neverthless I want a reference to the concerned generic theory.

Given a log scheme over $\mathbb{C}$ whose underlying scheme is locally of finite type, you can associate to it a ringed space called the Kato--Nakayama space. Is there a $p$-adic analogue of this construction (presumably something rigid-analytic)?

It is well known that there exists no non-trivial bounded solution of $-u''+u=0$ in $\mathbb R.$ Is this result even true, the problem $$ \bigg(-\frac{d^2}{dx^2}\bigg)^{s} u+u=0 $$ has no bounded solution in $\mathbb R$ where $s\in (0, 1).$ Here $$\bigg(-\frac{d^2}{dx^2}\bigg)^{s} u(x)=c\int_{\mathbb R}\frac{u(x)-u(y)}{|x-y|^{1+2s}}dy$$ where $c$ is positive constant depending on $s.$

I'm trying to read chapter 8 of the book on gradient flows by Ambrosio-Gigli-Savaré. In this context, I would like to better understand how the theory works for the following specific example. Take the family of probability measures on the real line $$\mu_t=t\delta_0+(1-t)\delta_1, \quad t\in(0,1),$$ where $\delta_x$ denotes the Dirac delta at $x$. It seems that this probability-valued curve is absolutely continuous with respect to the Wasserstein metric, but it does not seem to satisfy the conclusions of Theorem 8.3.1 in the book. In other words, there does not seem to be a vector field satisfying the continuity equation for this family of probabilities. As far as I can see the proof would already fail at equation (8.3.10), since it is possible to produce a test function with $\partial_t\phi\neq 0$ everywhere, yet $\partial_x\phi=0$ on the support of $\mu_t$, namely, $\{0,1\}$.

So my question is, what am I missing here?

Thanks a lot in advance.

Let $B_h$ be the set of rooted *perfect* binary trees having height $h$ (i.e. the binary trees with height $h$ in which all interior nodes have two children and all leaves have the same depth or same level). For any rooted tree $T$, we denote by $r(T)$ its root.

Let $F_{h,m}$ be the set of *all* possible forests that ** (i)** are formed by at most $m$ trees and

**Question**: How can we calculate (or bounding from above) the cardinality of $F_{h,m}$ asymptotically when $h\to\infty$ (as a function of $m$ and $h$)? Does the bound $|F_{h,m}|\le\sum_{i=0}^m \binom{2^h}{i}$ hold for all $h, m>1$?

I have submitted a paper to a journal on June 2017. The corresponding author is my coauthor. The first response of the editor was after a year, June 2018:

Your paper has been sent, consecutively, to four referees.

Of the preceding three, one declined to review it and two
never responded to the invitation or to multiple reminders.
The fourth one accepted; a report is expected in September [2018].

Since then, the editor disappeared. No notice at all, neither in the positive, nor in the negative. Obviously we tried to contact him again multiple times (approximately every two-three months) through the journal's platform. No response at all. We sent countless emails to the editor, either via the platform or through his personal email (my coauthor knows him in person). No response at all. We tried to contact the chief editor. No response at all.

In a few days, it will be exactly two years since the submission: this is an incredible amount of time, especially since we absolutely don't know what is the motivation for this delay.

What shall we do?

[I'm not posting this on academia, since this is a paper in Mathematics; but feel free to migrate the discussion elsewhere if you feel like so]

Unfortunately I am unable to comment as I lack sufficient reputation.

- Yes, "consecutively" means exactly that 3 people in a row refused to referee the paper; the fourth accepted and then disappeared since June 2018.
- The editor-in-chief should be aware of what's going on, more so because we wrote him an email two weeks ago or so. No answer.
- The journal is a pretty good and reputable one. At least until I spread this voice.