At first, I figured that an automorphism of a scheme $X$ would be a homeomorphism $f:|X| \to |X|$ of topological and an isomorphism of sheaves $f^{\#}: \mathcal{O}_X \to f_*(\mathcal{O}_X)$.

However, if $X= \mathbb{P}_k^n$, then the automorphisms of $X$ are actually defined as follows:

Let $\textbf{Aut}(\mathbb{P}^n)$ denote the functor taking a scheme $S=\operatorname{Spec}A$. where $A$ is a commutative algebra over an algebraically closed field $k$, to the group of automorphisms $\operatorname{Aut}_A(\mathbb{P}_A^n)$ of $\mathbb{P}_A^n$ over $A$.

I think the definition means that we need to look at the "automorphisms of all $A$-points of $\mathbb{P}_k^n$". I'm not sure if this last phrase is correct, I've just heard it being said.

However, how does this naturally follow from the "naive" interpretation of what an automorphism of a scheme $X$ would be? I can't figure out why or when the $A$-points of a scheme are of significance. For example, when Hartshorne discusses the automorphism group of $\mathbb{P}_k^n$ he determines it to be $PGL(n, k)$, i.e he really only considers the $k$-points of the group scheme $\textbf{PGL}(n)$.

Let $G(V, E)$ be a simple graph with $|V|=n$, and let $h$ be an integer in $[n]$.

We repeat $h$-many times the following operation in a sequential fashion, where the graph may change at each round. We denote by $G_t(V_t, E_t)$ the graph obtained just after round $t\in [h]$ (we have thereofore $G(V,E)=G_0(V_0,E_0)$):

At each round $t=1, 2, \ldots, h$, we select uniformly at random (with replacement) $h$-many vertices from $V_t$. Let $v$ be the first vertex selected. *If* at least one of the other selected vertices is adjacent to $v$ in $G_t$, then we remove from $G_t$ vertex $v$ and all vertices adjacent to $v$ (together with their incident edges) -- *otherwise* $G_t(V_t,E_t)=G_{t-1}(V_{t-1},E_{t-1})$.

**Question**: In expectation over the above random process, what is the maximum number of edges in $E_h$ over all possible $2^{n \choose 2}$-many input graphs $G(V, E)$ (when $n\to\infty$)?

(*I am especially interested in finding a tight* *upper bound**for $\max_{G(V,E)}\mathbb{E}\left[|E_h|\right]$*).

**Conjecture**: The maximum value for $\mathbb{E}\left[|E_h|\right]$ is equal to $\Theta\left(\frac{n^2}{h}\right)$.

(*In particular, I cannot find any graph $G(V,E)$ for which $\mathbb{E}\left[|E_h|\right]=\omega\left(\frac{n^2}{h}\right)$*).

I am studying some biology system and arrived at this simplified dynamical system:

\begin{align} x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\ x_2' &= a_5\frac{1+a_6x_3}{1+a_7x_3}x_1 - a_2x_2\\ x_3' &= a_8x_1 - a_9x_3 \end{align} Where as the context and notation dictate: all coefficients $a_i$ are strictly positive. I also showed all $x_i$ to be strictly positive and bounded.

I have obtained locally-asymptotically stability for the unique positive fixed point. I also tried various lyapunov function, but without much luck. I would truly appreciate any idea to approach this problem, perhaps a form of Lyapunov function, a direct method, etc that I should try.

Additionally, I have done extensive numerical simulations, which suggests the positive steady state is globally asymptotically stable.

Edit: for those interested in the original system - especially to back up the various claims that I made in my question. **Please note that the parameters do not match to the system in the question**. It's just a convenient way to write the parameters.

\begin{align} x_1' &= a_1x_2 -a_2x_1\\ x_2' &= -(f(y_2) + a_1) x_2 + a_3x_3 + a_2x_1\\ x_3' &= f(y_2)x_2 - a_3x_3\\ y_1' &= b_1x_2 - (b_2+b_3)y_1\\ y_2' &= b_2y_1 b_4y_2\\ f(y_2) &= \frac{c_1}{c_2} \frac{1 + c_2(y_2/c_3)^n}{1+(y_2/c_3)^n}. \end{align} The only conditions on the parameters are that they are positive and $c_2>1$ and $n=1$. Note that the first 3 equations are conservative, hence it can be reduced. And at any time, $x_1 + x_2 + x_3 = 1$. This system is part of a larger system, but is decoupled from the rest.

When I am studying some paper dealing with dispersive PDE(e.g. Wave, Schrödinger and Klein-Gordon equations), the potential $\frac {1}{|x|^2}$ which is called critical decay (or inverse square potential) is lots of handled.

They also consider Morrey-Campanato or Fefferman-Phong class to deal with that potential since it is contained those class.

The questions are here. Why is it called 'critical' and important or meaningful?

A compact HKT manifold is a hyperhermitian manifold $(M,I,J,K,g)$ such that either $\partial (\omega_J+i\omega_K)=0$ (if endomorphisms act on the left on the tangent space and $\partial$ is taken with respect to $I$) or, equivalently, the three Bismut connections for hermitian structures $(I,g)$, $(J,g)$, $(K,g)$ are all the same.

**My question** is related to the situation when the canonical bundle (with respect to $I$) is non trivial$^1$ (holomorphically). Can it happen that $\mathcal{K}_M$ has many section or it always has no sections at all, like for example for Hopf surfaces?

**The second question** is whether the answer is know for example for homogeneous examples of hypercomplex manifolds due to Joyce, $SU(3)$ for instance?

$1.$ From the work of M. Verbitsky it is know that a compact HKT manifold has a trivial canonical bundle iff the holonomy of the Obata connection is reduced to $Sl_n(\mathbb{H})$.

I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “internal perspective,” I mean a constant expression of worry from set theorists and logicians about the relevance of their work to the broader community / “real world”, with these worries sometimes leading to career-defining decisions on the direction of research.

For me, this situation is unwanted. I studied set theory because I thought it was interesting, not because I wanted to be a soldier in some kind of movement. Furthermore, I don’t see why an area needs defending when it produces a lot of deep theorems. That part is hard enough.

To what degree does there exist, in the various areas of mathematics, a widespread feeling of pressure to defend the relevance of the whole subject? Are there some areas in which it is enough to pursue the research that is considered interesting, useful, or important by experts in the field? Of course there will always be a demand to explain “broader impacts” to funding agencies, but I am talking about situations where the pressure comes from one’s own colleagues or even one’s own internalized sense of what is proper research.

I wanted to know how to go from baby Rudin to learn about Orlicz space

Given linear functions $f_1({\bf x}),\dots,f_n({\bf x})$ on ${\bf R}^m$, let $K = \{(a_1,\dots,a_n) \in {\bf R}^n:$ the $n$ halfspaces $\{{\bf x}: f_i({\bf x}) \leq a_i\} $ have nonempty intersection$\}$. If $K$ is compact, must it be a polytope?

(It's not hard to show that $K$ is convex. I imposed compactness to avoid questions about what I mean by a noncompact polytope.)

This question has probably arisen in linear programming, since it is natural to consider parametrized sets of linear programs, and to ask whether the set of parameter-values giving rise to feasible linear programs is itself characterized by linear inequalities.

Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the conditional probability given $B$, i.e. $\mu^B(A)=\mu(A\cap B)/\mu(B)$, and similarly for $\mu_n^B$.

Under what conditions (e.g. on $X$, $\mu$, and/or $B$) does it follow that $\mu_n^B\implies\mu^B$?

(a) Let $s>1$, $x>0$ be real. Then it is not hard to see that $$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$ basically because $x\mapsto 1/x^s$ is convex. (The naive bound would have $1/x^s$ instead of $1/2 x^2$.) By Euler-Maclaurin, this bound is tight, in the sense that the inequality would not be valid for large $x$ if $1/2$ were replaced by a smaller constant.

This bound looks as if it should be completely standard (in fact, known since the umpteenth century). Is there an easy reference? Also, what happens for real $0<s<1$? (Is the term $1/2 x^2$ still correct? It seems so to me.)

(b) Let $s = \sigma + i t$, $0<\sigma\leq 1$, $s\ne 1$. Let $x\geq |t|$ be real. After trying a little, my students and I showed that $$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O^*\left(\frac{c}{x^\sigma}\right)$$ with $c=5/6$, where $O^*(y)$ stands for a complex number whose norm is bounded by $y$. The bound is well-known with $c=1$. My question is: what is the optimal value of $c$? Again, this matter must be in some standard reference.

Is it possible to prove that the pair of consecutive primes $(479,487)$ is the only pair of consecutive primes $(p,q)$ such that $\text{rev}(p)=2q$? "rev" indicates the reverse digits. I checked all pairs up to $10^{11}$. I know that there are a lot of restrictions. How could be the chance of finding another pair of consecutive primes of this type below $10^{15}$?

What is an example of a manifold $M$ with $dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such that $D$ is a Lie algebra derivation on $\chi^{\infty}(M)$?

Does every manifold admit such an operator?

Is there a Riemannian manifold for which the Laplace operator $D=\Delta$, naturally defined on $\chi^{\infty}(M)\simeq \Omega^1(M)$, would be a derivation of $\chi^{\infty}(M)$?

Suppose we have a probability distribution $\pi : X \rightarrow [0,1]$ where $X$ is finite and let $Q : X \times X \rightarrow [0,1]$ be a Markov kernel that is reversible with respect to $\pi$. That is,

$$\pi(x) Q(x,y) = \pi(y) Q (y,x) $$

for all $x,y \in X$. Suppose we know the mixing time $t_x(Q, \varepsilon)$ of $Q$ to $\pi$ when started at $x$, defined as

$$ t_x(Q, \varepsilon) = \min \{ t \in \mathbb{N} : || Q^t(x, \cdot) - \pi ||_1 \leq \varepsilon \}. $$

Question: what can be said about the mixing time of the non-lazy version of this kernel?

That is, we can define $\tilde{Q}(x,y) = 0$ if $x=y$, and $\tilde{Q}(x,y)\propto Q(x,y)$ otherwise. Clearly $\tilde{Q}$ is still reversible with respect to $\pi$ and so has the same stationary distribution. So we can consider $t_x(\tilde{Q}, \varepsilon)$ and ask whether it is smaller (and by how much) than $t_x(Q, \varepsilon)$.

If anyone knew how to compare the two chains spectral gaps or log-Sobolev constants I would be particularly interested in that.

Motivation: I have a distribution $\pi$ and a Markov kernel $Q$ that mixes to $\pi$ quite slowly. However $Q $ is very often lazy, i.e. $Q(x,x)$ is close to $1$ for most $x$'s. I was hoping that there might be a way to show that the non-lazy version of my kernel mixes faster.

This question is about the variational principle in thermodynamical formalism.

Let $(X, T)$ be compact metric space. Consider $T:X\rightarrow X$ is a $C^{1+\epsilon}$ map. Let $\Phi:X\rightarrow \mathcal{R}$ be Holder continuous.

Broadly speaking the story is like that: in multifractal analysis we consider

$$E_{\Phi}(\alpha)=\{ x\in X, \lim_{n\rightarrow \infty}\frac{1}{n}S_{n}\Phi(x)=\alpha\},$$ where $S_{n}\Phi(x)$ is Birkhoff averages and $\alpha \in \mathcal{R}$. The map $\alpha \mapsto h_{top}(E_{\Phi}(\alpha))$, where $h_{top}$ denots the Bowen topological entropy, is call $\mathit{Lyapunov\hspace{0.2cm} spectrum}$.

We can easily find that $\alpha \mapsto h_{top}(E_{\Phi}(\alpha))$ is the Legendre transform of the pressure function $t\mapsto P(t\phi)$.

Suppose that $Log f, Log S :X\rightarrow \mathcal{R}$ are Holder continuous. Let $\chi_{f}(\mu) $ and $ \chi_{S}(\nu)$, where $\mu$ and $\nu$ are some equilibrium states for some potentials $tf$ and $qS$, are Lyapunov exponents. We consider following sets $$E_{log f}(\alpha)=\{ x\in X, \chi_{f}(\mu)=\alpha\},$$ and $$E_{log S}(\beta)=\{ x\in X, \chi_{S}(\nu)=\beta\}.$$ My question is as follows :

Assume that $\chi_{f}(\mu)>\chi_{S}(\nu)$. Can we show that $h_{top}(E_{log f}(\alpha))<h_{top}(E_{log S}(\beta))$ for any $\alpha, \beta \in \mathcal{R}?$

Is the answer to the above question known? If yes, could you give a reference? If no, under what condition we can prove it.

Thanks in advance.

Let us have two symplectic manifolds $(M, \,\omega)$ and $(N, \,\omega')$ and morphism between them: $$ \varphi \ :\ M \to N.$$ Then we geometrically quantize these systems:

we add a prequantum line bundle, $ L_M \to M$ and $L_N \to N$ correspondingly;

choose some real polarizations $P_M$ and $P_N$ such that $(\varphi_*P_M)(x) \neq P_N(x), \, \text{for }x \in \varphi(M)$;

let the quotient maps $\pi_{M}\ :\ M \to M/P_M$ and $\pi_N\ :\ N \to N/P_N$ be fibration.

Then the space of compactly supported sections of $L_M/P_M \otimes |T(M/P_M)|^{1/2} $ after completion realise Hilbert space $H_M$. The same procedure for $N$ gives $H_N$.

I don't understand how to build morphism between $H_M$ and $H_N$ from $\varphi$. Is there any natural way to do it?

I'm not sure whether it's a trivial question or not. I don't have deep understanding of geometric quantization, I've only read this review by Lerman. So if some comprehensible text about the question exists, I'll gladly accept a reference.

P.S. I'm reading this paper by Nekrasov which describes morphism between classical Calogero-Moser and Calogero-Sutherland models, and derives from it morphism between their quantum versions. Nekrasov briefly describes how he gets quantum morphism from classical one in the beginning of 4th section. But I don't understand it. So I'm interested whether there is a general way to do it. If it is possible only in this particular case, I'd like to get a more thoroughly explanation of Nekrasov's procedure.

Suppose we have a Banach space $X$ and have chosen a set $\Sigma$ consisting of some sequences whose members are in $X$. We can then say that $(x_n)_{n=1}^\infty\in X^\mathbb{N}$ is $\Sigma$-*convergent* to $x\in X$ if $(x_n-x)_{n=1}^\infty\in \Sigma$. We can say $C\subset X$ is $\Sigma$-*closed* if whenever $(x_n)_{n=1}^\infty\subset C$ is $\Sigma$-convergent to $x\in X$, then $x\in C$. We can say $U\subset X$ is $\Sigma$-*open* if its complement is $\Sigma$-closed.

Under some mild conditions on $\Sigma$, the collection $\tau$ of $\Sigma$-open sets is a topology on $X$ and $(x_n)_{n=1}^\infty$ is convergent to $x$ in the topology $\tau$ if and only if it is $\Sigma$-convergent to $x$.

Is there some reference for this process? It seems to me that this must be a well known, standard procedure for generating a topology.

There is a famous rule of Arnold which says that if the discovery of a mathematical object is attributed to a specific person, that person was not the first to discover it. The question is about a specific application of this rule. Was Drinfeld's upper half-plane considered before Drinfeld?

How do I show the following bounds on the mills ratio :

$\frac{1}{x}- \frac{1}{x^3} < \frac{1-\Phi(x)}{\phi(x)} < \frac{1}{x}- \frac{1}{x^3} +\frac{3}{x^5} \ \ \ \ \ \ \ $ for $ \ \ \ x>0$ where $\Phi()$ is the CDF of the Normal distribution , and $\phi()$ is the density function of the Normal distribution ?

Also , is there a similar bound when $x < 0$ ?

I am aware of the proof of the fact that the mills ratio is bounded below by $\frac{x}{1+x^2}$ and above by $\frac{1}{x}$ , but I am unable to prove this inequality .

Consider $H\mathbb{Z}[1/p]$ the Eilenberg-MacLane spectrum where we are inverting a prime $p$.

My question is:

Is it known the structure of the cohomology groups $H\mathbb{Z}[1/p]^{*}H\mathbb{Z}[1/p]$?

Let $M$ be a compact smooth Riemannian manifold. Then it admits a triangulation, i.e. a finite simplicial complex $K$ which is homeomorphic to $M$. Any such simplicial complex carries a natural metric - the path metric which concides with the usual metric on each simplex. Can we always choose $K$ so that with this path metric, it is bilipschitz equivalent to $M$ with the metric coming from its Riemannian structure?