## pr.probability – Central limit theorem for chi-squared random field on \$mathbb R^p\$

Let $$X:x mapsto X(x)$$ be a centered stationary Gaussian process on the $$Omega:=mathbb R^p$$, such that $$X(x) overset{d}{=}X(x’)$$ for all $$x,x’ in Omega$$. Set $$sigma^2 := mbox{Var}(X(0)) = mathbb E(X(0)^2)$$. Let the random fields $$X_1,ldots,X_N$$ be iid copies of $$X$$, and define a random process $$Z_N$$ on $$Omega$$ by
$$Z_N(x) := frac{1}{sqrt{2Nsigma^2}}(sum_{i=1}^N X_i(x)^2-Nsigma^2),;forall x in Omega.$$

Question. Is there a central limit theorem (perhaps under further conditions on the base field $$X$$) for the limiting distribution of the random field $$Z_N$$ when $$N to infty$$ ?

## general topology – Is the empty set compact in \$mathbb R\$?

Here they say that the emptyset is compact. Nevertheless, in $$mathbb R$$, I know that compact sets are closed and bounded. So, indeed $$varnothing$$ is closed, but we have that $$inf_{mathbb R}varnothing =+infty quad text{and}quad sup_{mathbb R}varnothing =-infty ,$$
in partuclar, it doesn’t seem bounded. So, who is correct ?

## ag.algebraic geometry – Tate-Shafarevich groups of high-rank elliptic curves over \$mathbb Q\$

Assume the BSD conjecture. By checking various examples, it seems that the Tate-Shafarevich groups of elliptic curves over $$mathbb Q$$ satisfies the following propositions:

• If an elliptic curve E over ℚ has rank ≥ 2, then Ш(E)=1 or Ш(E)=4.
• If an elliptic curve E over ℚ has rank ≥ 3, then Ш(E)=1.

EDIT: The second proposition is false. The elliptic curve $$E:y^2 = x^3 + 1916840x$$ has rank 3 and Ш(E)=4, by the following SageMath computation:

``````A=EllipticCurve((0,0,0,1916840,0))
A.rank()                   #=3
A.sha().an_numerical()     #=4.0000000000
``````

Question: Are there references, heuristics, counterexamples, etc. to the first proposition above?

## general topology – Prove a homomorphism between \$(-1,1)\$ and \$mathbb R\$

I found a homomorphism in a book, between $$(-1,1)$$ and $$mathbb R$$, this is:

$$f(x)=frac{x}{1+|x|}$$

So i check the bijection.Firts, I have to check if $$f(x_{1})=f(x_{2})$$ then $$x_{1}=x_{2}$$ but i have this:

$$frac{x_{1}}{1+|x_{1}|}=frac{x_{2}}{1+|x_{2}|}$$

I don’t know how can i proceed with the absolute value and how can i see the continuity? Can you give some hint? Thank you.

## nt.number theory – What do the eigenvalues of a random element of \$mathbb Z_ell[Gamma]\$ look like?

Let $$Gamma = varprojlim Gamma_n$$ be a profinite group with $$Gamma_n$$ finite quotients. For concreteness, let us fix $$Gamma_n = operatorname{PGL}_2(mathbb Z/ell^n)$$ so $$Gamma = operatorname{PGL}_2(mathbb Z_ell)$$.

Let $$R = mathbb Z_ell((Gamma))$$ be the completed group algebra and consider it’s action on itself. If we pick a random element $$gamma in R$$, what does the distribution of its eigenvalues look like?

More precisely, call the image of $$gamma$$ in $$R_n = mathbb Z_ell(Gamma_n)$$ as $$gamma_n$$. Now $$gamma_n$$ acts linearly on the finite free $$mathbb Z_ell$$-module $$R_n$$ and let its set of eigenvalues be $$S_n$$. Can we say anything about the distribution of $$S_n$$ as $$nto infty$$?

For a simpler situation, we can consider $$Gamma = mathbb Z_ell$$ and then $$R cong Z_ell((t))$$ and if $$gamma = f(t)$$, then $$gamma_n$$ will have eigenvalues $$f(zeta_{ell^n}^i-1)$$ for $$i=1,dots, n$$ (unless I made a mistake in the computation).

I am not even sure what we can say about this set as $$nto infty$$.

## Measure in \$mathbb {C} ^p\$

If we have a non-constant holomorphic map $$f: C ^ p to X$$, where $$X$$ is a complex manifold. Let $$omega$$ be a metric on $$X$$, $$omega$$ is a $$(1,1)$$ – form positive definite.
$$f ^ * (omega ^ p)$$ Is it a measure over $$C ^ p$$?

## differential geometry – Is the set \${x in mathbb R^n : d(x, M) = c}\$ a smooth manifold for a small constant \$c\$ when \$M\$ is a smooth manifold embedded in \$mathbb R^n\$?

Not necessarily; the most important issue is that two points of $$M$$ which are “far apart inside $$M$$” might actually be very close together in $$mathbb{R}^n$$. For example, let $$M$$ be the image of the smooth embedding $$f : mathbb{R} to mathbb{R}^2$$ given by $$f(t) = (cos(2arctan(t)), sin(2arctan(t)))$$. For any $$c > 0$$, the set $${x in mathbb{R}^2 : d(x,M) = c}$$ contains two points which have no open neighborhood homeomorphic to any Euclidean space.

This issue is obliterated if $$M$$ is compact; then the result does hold.

## gr.group theory – Stabilizers in the action of \$mathrm{GL}(n, mathbb Z)\$ on \$mathbb Z^n\$

How can we calculate effectively the subgroups of $$G: = mathrm{GL}(n, mathbb Z)$$ which fix pointwise a given submodule $$S$$ of $$mathbb Z^n$$ in the action of $$G$$ on $$mathbb Z^n$$ by left multiplication?

Note: Suppose $$S$$ is freely generated by $$xi_1, xi_2, cdots, xi_s$$ then the clearly the problem is equivalent to describing the unimodular matrices fixing each of the tuples $$xi_1, cdots xi_s$$. If the $$xi_j$$s have a simple form, e.g., $$xi_j$$s are the standard basis vectors $$e_1, cdots, e_n$$ of $$mathbb Z^n$$ then the form of matrices fixing each $$xi_j (j = 1, cdots, s)$$ is easy to determine.
But in general the method to obtain the stabilizer is not clear.

## real analysis – Does the continuous function \$ H ^ 1 ( mathbb {R} ^ 2) \$ have to tend towards zero in infinity?

Here, $$H ^ 1 ( mathbb {R} ^ 2)$$ is the standard Sobolev space for $$L ^ 2 ( mathbb {R} ^ 2)$$ Functions whose weak derivation is one of them $$L ^ 2 ( mathbb {R} ^ 2).$$

My question in the title comes from the variational calculation. It is usually the case that a minimizer of a particular energy function is defined $$H ^ 1 ( mathbb {R} ^ 2)$$ is known as continuous (or even continuous) $$C ^ 2 ( mathbb {R} ^ 2))$$. I want to know the behavior of this minimizer in infinity.

If $$u in L ^ 2 ( mathbb {R} ^ 2),$$ then is known $$liminf_ {| x | to infty} u (x) = 0.$$ But it cannot say $$limsup_ {| x | to infty} u (x) = 0$$ because counterexamples exist.

If we accept $$u in H ^ {1+ epsilon} ( mathbb {R} ^ 2)$$ for some $$epsilon> 0,$$ then the inequality of the classic Morrey can imply a uniform older continuity of $$u.$$ So we can close $$limsup_ {| x | to infty} u (x) = 0$$ by proof by contradiction.

So my problem is the case $$epsilon = 0.$$ That is when
$$u in H ^ 1 ( mathbb {R} ^ 2) cap C ( mathbb {R} ^ 2)$$
(or $$u in H ^ 1 ( mathbb {R} ^ 2) cap C ^ 2 ( mathbb {R} ^ 2)),$$ is it true that
$$limsup_ {| x | to infty} u (x) = 0?$$

With evidence of contradiction, I think this should be true. Here is my non-rigorous evidence.

Let's not assume there is $$epsilon> 0$$ and $$x_n in mathbb {R} ^ 2$$ so that $$| x_n | to infty$$ and $$| u (x_n) | geq 2 epsilon.$$ Because of the continuity there is $$r_n> 0$$ so that $$| u (x) | geq epsilon$$ for all $$x in B (x_n, r_n).$$
Since $$u in L ^ 2, r_n to 0$$ how $$n to infty.$$
I don't think so rigorously
$$int_ {B (x_n, r_n)} | nabla u | ^ 2 gtrsim int_ {B (x_n, r_n)} ( frac { epsilon} {r_n}) ^ 2 = epsilon ^ 2$$
for big $$n$$ and
$$int _ { mathbb {R} ^ 2} | nabla u | ^ 2 geq sum_ {n , text {is large}} int_ {B (x_n, r_n)} | nabla u | ^ 2.$$
So they imply a contradiction $$int _ { mathbb {R} ^ 2} | nabla u | ^ 2 = infty$$

I am happy about every discussion.

## elliptic curves – why \$ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p \$

I have read an article that says: $$E_1 ( mathbb {Q} _p) approx mathbb {Z} _p$$

Where $$E$$ is an elliptic curve over $$mathbb {Q} _p$$ and $$E_1 ( mathbb {Q} _p) = {P in E ( mathbb {Q} _p): tilde {P} = tilde {O} }$$.

The author says the proof is in "Silverith Arithmetic of Elliptic Curves" on page 191, but it says here:

If $$E$$ is an elliptic curve over $$mathbb {Q} _p$$ and $$has {E}$$ is the formal group, then:

$$E_1 ( mathbb {Q} _p) approx hat {E} (p mathbb {Z} _p)$$

So I don't know a good reference for proving $$E_1 ( mathbb {Q} _p) approx mathbb {Z} _p$$.