I apologize in advance if this question has an obvious answer.

Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$ is the pull back of the canonical symplectic structure of the cotangent bundle via the obvious diffeomorphism between $TM$ and $T^* M$ which is defined by the inner product $g$.

The standard structure of $T\mathbb{R}^n=\mathbb{R}^n \times \mathbb{R}^n$ is denoted by $\omega$.

For every Riemannian manifold $(M,g)$, is there an isometric embedding $j$ of $M$ in some $\mathbb{R}^n$ such that $j^*(\omega)=\omega_g?$

This is an analog of an older question:

What characterizations of relative information are known?

With the modification that I’m interested in the case when the distribution is over something that’s not a finite set. For example, for compactly supported distributions over an interval equipped with some measure. The definition of the KL divergence in this case is found as the third equation in the defintions section in the relevant wikipedia entry.

**I would like to know whether there’s an axiomatic characterization of this, generalizing the characterizations in the discrete case.**

My limited intuition (as a non-information-theorist) is that this could be tricky, for I’m reminded that there’s a nice characterization of ordinary entropy of discrete distributions due to Fadeev, which lacks an obvious generalization to the differential/continuous entropy. There’s a relevant discussion of this issue in another older post.

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point.

Let $p: \mathbb{R}^n \to \mathbb{R}$ be a polynomial function. We define the Hamiltonian $H$ on $\mathbb{R}^n \times \mathbb{R}^n$ as follows:

$$H(x,y)= \sum_{\alpha}\frac{1}{\alpha!} D^{\alpha} p(x) y^{\alpha}$$ where $\alpha$ varies over all multi integer indices and $x,y \in \mathbb{R}^n.$

What can be said about the dynamics of the corresponding Hamiltonian vector field?

**Is there an example of such a Hamiltonian vector field which possess a non trivial periodic orbit?**

We observe that the function $\sum x_i + \sum y_i$ is a first integral. Are there some other first integrals, independent of $\sum x_i + \sum y_i$? In particular is this Hamiltonian completely integrable?

Now we try to extend this questions on an arbitrary Riemannian manifold. So we consider a kind of Taylor series and we cut the series at its second term. So our question would be the following:

Let $(M,g)$ be a Riemannian manifold and $f:M \to \mathbb{R}$ be an smooth map. We define the following Hamiltonian on the tangent bundle $TM$ with its obvious symplectic structure

$$H(x,v)=f(x)+df(x).v +\frac{1}{2} Hess(f)(x)(v,v)$$ where $Hess(f)$ is the $2-$ linear form on $T_x M$ with the formula $$Hess(f)(x)(v,w)=g(\nabla_v \nabla f, w)$$ where $\nabla$ is the LC connection associated with the Riemannian metric. Is there an example of such a Hamiltonian with a non trivial periodic orbit?

There was a similar thread on the neighbour forum StackExchange on sufficient conditions for a topological space to be *completely* regular $T_{3^1/_2}$.

Please, let me know any known **condition(s) that a topological space is regular $T_3$**. Any known approach or a standard strategy for showing that a topological space is regular would be very welcome and helpful answer too.

Thank you in advance!

Take two identical cuboids in two copies of $\mathbb R^n$. Cut the cuboids by same hyperplane in both spaces. Keep the convex polyhedra $P_\leq$ that arises by intersection of cuboid and half space given by $\leq$ of the hyperplane in one space and the convex polyhedra $P_\geq$ that arises by intersection of cuboid and half space given by $\geq$ of the hyperplane in othetr space.

The tensor product of the polyhedra is in $\mathbb R^{n^2}$ and is not convex. However if we want to cover the tensor product with convex semialgebraic inequalities of $\leq$ type and equalities each of degree at most $poly(n)$ and coefficient size $2^{O(n)}$ and linear inequalites of general type so that the fraction of volume of the covering not containing the tensor product is at most $\epsilon$ then how many convex semialgebraic inequalities does one need?

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $x \in M$ there exists $p \in P$ such that $\|x - p\|_2\leq \epsilon$. What is known about lower bounds on $\mathcal{N}(\epsilon)$?

If $M$ were $d$-dimensional then there is a simple volume argument that shows that
$\mathcal{N}(\epsilon) \geq \frac{\operatorname{vol}_{d}(M)}{\operatorname{vol}_{d}(B^d_{\epsilon})}$ in terms of the $d$-dimensional volume of $M$ and a $d$-dimensional ball $B^d_{\epsilon}$ of radius $\epsilon$. However I'm interested in the case where $M$ is $k$-dimensional for $k < d$. In particular I'd like a bound in terms of the $k$-dimensional volume $\operatorname{vol}_{k}(M)$ *and* the $k$-dimensional ball $B^{k}_{\epsilon}$. However due to the curvature of $\mathcal{M}$ it doesn't like the same bound applies with $B^{k}_{\epsilon}$, but using $B^{d}_{\epsilon}$ may lead to a less tight result.

I'd also be interested in any known lower bounds, even ones that do not use volume arguments.

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

- $K_v/\mathbf{Q}_v$ is unramified for almost all $v$
- the degree $(K_v : \mathbf{Q}_v)$ is uniformly bounded
- add more as needed

It’s necessary to add more conditions. Do there exist conditions such that the answer to this question is “yes”?

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?

Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence:

$0\longrightarrow H^0(C,T_C)\longrightarrow Def_R(f)\longrightarrow Def(f) \longrightarrow 0$

Where $Def_R(f)$ is the first order deformation of $(C,{p_i},f:C\longrightarrow X)$ with $C$ held rigid.

1)what does it mean(($C$ held rigid)? Does it mean we consider $C$ fixed?

2)Why we have this short exact sequence?

(If f is closed immersion and ignore marking and my comment on 1 is correct we have $0\longrightarrow T_C \longrightarrow f^*T_X \longrightarrow N_f\longrightarrow 0$ so we will get the short exact sequence because $P^1$ is rigid.But in general i cant reach to this short exact sequence )

Let $V_1,V_2,\cdots,V_n$ be finite sets satisfying $$|V_i\cap V_j|\leqslant\frac{1}{2}\max\{|V_i|,|V_j|\},\ \forall1\leqslant i<j\leqslant n.$$ We say $C=(C_1,C_2,\cdots,C_n)$ a cycle sequence on $V_1,V_2,\cdots,V_n$ if $C_i$ is a cycle on $V_i$, $1\leqslant i\leqslant n$; and let $$t(C)=\max_{e\in K_V}|\{1\leqslant i\leqslant n:e\in E(C_i)\}|$$ where $K_V$ is the complete graph on $V\triangleq V_1\cup V_2\cup\cdots\cup V_n$.

Furthermore, let $t(V_1,V_2,\cdots,V_n)$ be the minimum value of $t(C)$ as $C$ ranges over all cycle sequences on $V_1,V_2,\cdots,V_n$.

**My question**

Does there exist a upper bound for $t(V_1,V_2,\cdots,V_n)$ as $V_1,V_2,\cdots,V_n$ ranges over all such finite sets ?

Separation principle $\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$ claims that any two disjoint pointsets of boldface class $\mathbf\Sigma^1_n$ are separated by a $\mathbf\Delta^1_n$ set. Separation principle $\mathbf{Sep}(\mathbf\Pi^1_n,\mathbf\Delta^1_n)$ is understood similarly. It is known classically that $\mathbf{Sep}(\mathbf\Sigma^1_1,\mathbf\Delta^1_1)$ and $\mathbf{Sep}(\mathbf\Pi^1_2,\mathbf\Delta^1_2)$ hold, while $\mathbf{Sep}(\mathbf\Pi^1_1,\mathbf\Delta^1_1)$ and $\mathbf{Sep}(\mathbf\Sigma^1_2,\mathbf\Delta^1_2)$ fail. The axiom $V=L$ (anyway, the existence of a good $\mathbf\Delta^1_2$ wellordering of the reals) implies that $\mathbf{Sep}(\mathbf\Pi^1_n,\mathbf\Delta^1_n)$ holds, while $\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$ fails for all $n\ge3$. The axiom of projective determinacy $\mathbf{PD}$ implies some other behaviour. But very little seems to be known about $\mathbf{Sep}$ in various generic models, including such best known ones like adding $\kappa$-many Cohen reals, $\kappa\ge\omega_1$. Basically, afaik this is 1) Harrington's unpublished handwritings of 1974-75, and my own APAL, 2016, 167, 3, 262–283. I wonder can someone share some other results in this direction.

*Kanovei, Vladimir; Lyubetsky, Vassily*, **Counterexamples to countable-section $\varPi_2^1$ uniformization and $\varPi_3^1$ separation**, Ann. Pure Appl. Logic 167, No. 3, 262-283 (2016) ZBL06529281 MR3437647

**Part 1: a single finite place.** Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

**Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, here)** Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer.
Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?

Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is infinite?

Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.

Assume that the action $\Gamma$ on $\mathbb{S}^\infty$ has no dense orbits. Is there a universal constant $\varepsilon >0$ such that there are two orbits of $\Gamma$ on distance at least $\varepsilon$ from each other?

In other words, is there a constant $\varepsilon>0$ such that $$\mathrm{diam}\, (\mathbb{S}^\infty/\Gamma) >0 \quad\Longrightarrow\quad \mathrm{diam}\, (\mathbb{S}^\infty/\Gamma) > \varepsilon\ ?$$

**Comments.**

- If true then it would be generalization of the main result in The curvature of orbit spaces by Claudio Gorodski and Alexander Lytchak.

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear combination of at most $k$ delta functions $\delta_{b_i}$, $b_i\in \mathbb{Q}$. Can we compute $(F\ast g)(n)$ for $l$ consecutive integers $n=n_0+1,\dotsc,n_0+l$ in time roughly linear to $\max(k,l)$ (say, $O(\max(k,l) \log \max(k,l))$)?

In the case that I am truly interested in, all of $a_1,\dotsc,a_k,b_1,\dotsc,b_k$ can be written as fractions with denominator $Q$, where $Q$ is much larger than $k$ and $l$ but not enormous ($\log Q$ is small). In this case, it seems plausible to me that one could adapt a Fourier-transform-based method. Still, I'd be surprised if the question hasn't been considered before, so I'd appreciate references (or any indication of how to solve the problem more cleanly or efficiently).

Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable: $$A,B\in \mathcal A, A\neq B \Rightarrow A\cap B \in \mathcal I. $$

Let us observe that maximal ideals have this property as only the empty family or the singletons meet this requirement.

Are there further examples of ideals with this property?

(As observed by Leonetti every such ideal must be non-meagre when regarded as a subset of the Cantor set.)

Let $C(X)$:space of continuous functions on a compact space.Topology $C(X)$ is generated by sup-norm($||T||=sup_{v}\frac{||T(v)||}{||v||}$).

Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper semi continuous.

We say that $f$ is upper semi continuous at point $x_{0}$ if $liminf_{x\rightarrow x_{0}}f(x)\geq f(x_{0}).$

The function $f$ is called upper semi-continuous if it is upper semi-continuous at every point of its domain. A function is upper semi-continuous if and only if $\{T \in C(X): f(T) < \alpha\}$ is an open set for every $\alpha \in \mathbb{R}$.

suppose for every $T\in C(X)$ set of $f(T)$ and set $(f+g)(T)$ are closed interval(connected set).Can we say set of $g(T)$ is closed interval(connected set),as well?

If Not,under which condition we have it.

In importance sampling, one proposes to compute an integral $I:=\mathbb E_{x \sim P}[h(x)]$ by rewritting it as $$ I=\mathbb E_{x \sim Q}\left[w(x)h(x)\right],\text{ with }w(x):=\frac{p(x)}{q(x)}, $$ for some appropriately chosen sampling distribution $Q$ with density $q$. Note that $w(x)$ is itself a random variable.

The general procedure is as follows:

- Sample $x_1,\ldots,x_N$ iid from $Q$.
- Compute $\hat{I}^{(N)}_q := \dfrac{\sum_{i=1}^nw_ih(x_i)}{\sum_{i=1}^N w_i}$, where $w_i = w(x_i) := p(x_i)/q(x_i)$.

It's known that the key condition (in addition to some finite-moment conditions) for the $\hat{I}^{(N)}_q$'s converge $Q$-a.s. to $I$ is that $$\operatorname{supp}(p) \cap \operatorname{supp}(h) \subseteq \operatorname{supp}(q), $$ where $\operatorname{supp}(q)$ is the support of $q$. In fact, under the above condition one has (see Theorem 2.2 of this material)

$$ \begin{split} \mathbb E_{x \sim q}[\hat{I}^{(N)}_q] &= I + \frac{\mu \operatorname{Var}_{x \sim q}[w_q(x)] - \operatorname{Cov}_{x \sim q}[w_q(x), w_{q}(x)h(x)]}{N} + \mathcal O\left(\frac{1}{N^2}\right),\text{ and }\\ \operatorname{Var}_{x \sim q}[\hat{I}^{(N)}_q] &= \frac{\operatorname{Var}_{x \sim q}[w_{q}(x)h(x)] - 2\mu\operatorname{Cov}_{x \sim q}[w_q(x), w_{q}(x)h(x)] + \mu^2 \operatorname{Var}_{x \sim q}[w_q(x)]}{N}\\ &\quad\quad\quad + \mathcal O\left(\frac{1}{N^2}\right) \end{split} $$

Thus asymptotically, the estimates $\hat{I}^{(N)}_q$ are unbiased and have no variance.

On notes that, if (*other things being equal*) $\operatorname{supp}(q)$ is very large compared to the support of the integral $I$, namely $\operatorname{supp}(p) \cap \operatorname{supp}(h)$, then the estimates $\hat{I}^{(N)}_q$ will not be very "efficient" as most points sampled for the computation, will not yield any information, and $w_i$'s will have high variance, etc. In an attempt to understand more formally how $\operatorname{supp}(q)$ affects the performance of the estimates $\hat{I}_n$, for $0 \le \epsilon < 1$ and $C \supseteq \operatorname{supp}(q)$, define the ** contaminated** density
$$
q_\epsilon = (1-\epsilon) q + \epsilon u_C
$$
where $u_C$ is the uniform distribution on $C$.

- How does $\operatorname{Var}_{x \sim q}(w_{q}(x)$ compare to $\operatorname{Var}_{x \sim q_\epsilon}(w_{q_\epsilon}(x))$ ? I suspect the former is larger than the latter, but don't have rough guess of the orders of magnitude.
- How does $\operatorname{Var}_{x \sim q_\epsilon}(w_{q_\epsilon}(x)h(x))$ compare to $\operatorname{Var}_{x \sim q}(w_{q}(x)h(x))$ ? (same remarks as above).
- How does $\operatorname{Cov}_{x \sim q_\epsilon}(w_{q_\epsilon}, w_{q_\epsilon}(x)h(x))$ compare to $\operatorname{Cov}_{x \sim q}(w_q(x), w_{q}(x)h(x))$ ? (same remarks as above).

Dealing with relations in a set theoretic context, i.e. as just sets of ordered pairs what would one call a function $f:\text{fld}(R)\to\text{fld}(L)$ for any relations $R$ and $L$ in each of these three scenarios:

$$\forall a,b\in \text{fld}(R)\left[aRb\implies f(a)Lf(b)\right]$$ $$\forall a,b\in \text{fld}(R)\left[f(a)Lf(b)\implies aRb\right]$$ $$\forall a,b\in \text{fld}(R)\left[aRb\iff f(a)Lf(b)\right]$$

From a graph theoretic context (i.e. associating every relation with a unique loop digraph containing no isolated vertices) case one is a homomorphism, while if we assume that $f$ is bijective then case three is an isomorphism. Though this still says nothing about case two and case three when the function $f$ is not injective and surjective onto $\text{fld}(L)$. So what would be the proper terminology in each of these three cases? The page on nlab https://ncatlab.org/nlab/show/relation even seems some what ambiguous. Also they deal with relations as correspondences, that is a binary relation for them is not just a set of ordered pairs but a triplet $R=(H,A,B)$ where we have $H\subseteq A\times B$

For a cardinal $\kappa$, let $V_\lambda\prec_{cp:\kappa}V_\mu$ whenever there is an elementary embedding from $V_\lambda$ to $V_\mu$ with critical point $\kappa$ (unless $\lambda\leq\kappa$ in which case $V_\lambda\prec_{cp:\kappa}V_\mu$ iff $V_\lambda\prec V_\mu$.) Then, let $Ext(\kappa)$ be the structure $(\{V_\theta|V_\kappa\prec_{cp:\kappa}V_\theta\}; \prec_{cp;\kappa})$.

If $Ext(\kappa)$ has a chain of length $\alpha$ beginning on $V_\kappa$, then it is clear that $\kappa$ is $\beta$-extendible for every $\beta<\alpha$. Therefore, if $Ext(\kappa)$ has chains of arbitrarily long length beginning on $V_\kappa$, then $\kappa$ is extendible.

This is analogous to unfoldable cardinals, except unfoldables have one extra predicate on the ranks inside the structure, and instead of $\prec_{cp;\kappa}$ use $\prec_{eee}$ (the end-elementary substructure relation).

Let $\kappa$ be **$\beta$-protractible** whenever $Ext(\kappa)$ has chains beginning on $V_\kappa$ of length $\beta$. $\kappa$ is **protractible** whenever $\kappa$ is $\beta$-protractible for every $\beta$, and **long protractible** whenever $Ext(\kappa)$ has chains of length ORD. Here are some facts:

- Every cardinal is 1-protractible (using $\{V_\kappa\}$ as the chain)
- A cardinal $\kappa$ is 2-protractible iff it is 0-extendible
- Let $\kappa$ be $\alpha$-extendible with target $\theta$ iff $V_{\kappa+\alpha}\prec_{cp;\kappa}V_\theta$. Then for $\alpha>2$, $\kappa$ is $\alpha$-protractible iff there is an $\alpha$-sequence $0=\theta_0,\theta_1...$ such that: $$\forall \gamma<\beta<\alpha(\kappa\text{ is }\theta_\gamma\text{-extendible with target } \theta_\beta)$$
- Every long protractible cardinal is protractible, and every protractible cardinal is extendible
- An interesting exercise proves every extendible cardinal is protractible; to see why, use transfinite induction along with the fact that any embedding $j:V_{\theta_\beta}\prec V_{\theta_{\beta+1}}$ could be used to generate the embedding $j\upharpoonright V_{\kappa+\theta_\gamma}:V_{\kappa+\theta_\gamma}\prec V_{\theta_{\beta+1}}$ for any $\gamma<\beta$. Obviously such an embedding preserves the critical point.

So I'm left with just long protractibility. **What is the consistency strength of the existence of a long protractable cardinal? Is it equivalent to the existence of an extendible cardinal?**

Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the space of geodesics on $\widetilde\Sigma$. It is well-known that $\mu$ can only have atoms at closed geodesics. We can also look at 1-dimensional subsets, namely *pencils* $P(a)$, the set of geodesics with one endpoint at $a \in \partial\widetilde\Sigma$. If $a$ is not one of the limit points of a closed geodesic, then $\mu(P(a)) = 0$; see, e.g., Martelli, "An Introduction to Geometric Topology", Proposition 8.2.8.

**Question.** What is an example of a geodesic current without atoms so that $\mu(P(a)) \ne 0$ (where $a$ is necessarily the endpoint of a closed geodesic)?