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 $.