real analysis – Compact Sobolev embedding with boundary conditions

Let $X$ be some metric measure space on which Sobolev spaces can be defined in a reasonable way. In many cases, $H^1(X)$ is compactly embedded in $L^2(X)$ (e.g., if $X=Omega$ is a bounded open set of $mathbb R^d$), and in that case, $H^1_0(X)$ is of course compactly embedded in $L^2(X)$, too. In many cases, on the other hand, $H^1_0(X)$ is not compactly embedded in $L^2(X)$ (e.g., $X=Omega=mathbb R^d_+$), let alone $H^1(X)$.

My question is now, whether structures $X$ are known such that the embedding of $H^1_0(X)$ in $L^2(X)$ is compact but that of $H^1(X)$ is not.

Spectral theory of compact operators on normed spaces

Given an infinite-dimensional normed space $X$, is it true that every (linear) compact operator on $X$, $T in {cal B}_infty(X)$, have at least one eigenvalue $lambda ne 0$? If yes, how many? Does the number of eigenvalues have to match $dim X$ (in this case $aleph_0$ I suppose)?

How to prove the quotient group is compact?

Assume $G$ is an abelian topological group. Let $Bsubset A$ be its two subgroups, $A$ closed and $B$ discrete. $C$ is a compact subset in $G$. Suppose $Asubset BC$, can we prove $A/B$ is also compact?

I want to show $A/B$ can be identified with a closed subset of $C$, but I can not give a strict proof. Could you help me prove it or give a counterexample?

ag.algebraic geometry – Is there an example of a compact complex (Kähler) manifold of general type with trivial topological euler charcteristic?

Topological Euler characteristic is given by the top Chern class of the tangent bundle up to sign.

Comments: Examples don’t exist in dimension 2 due to Bogomolov–Miyaoka–Yau inequality. Is that true in higher dimensions? Examples are known if one assumes a weaker condition that the holomorphic Euler characteristic is zero.

When is Cech cohomology with compact support isomorphic to Singular cohomology with compact support?

This is a specific question regarding the understanding of a section of a paper. Having never posted here before I’m not sure that this is the right forum for this sort of question, but I hope someone can help.

Reading the paper by Wan https://core.ac.uk/download/pdf/82686957.pdf, I got stuck trying to understand why he requires Cech cohomology for Corollary 5.1. Here, he is considering a submanifold, possibly with boundary, of a smooth manifold without boundary. And for this submanifold he calculates Cech cohomology with compact support, when throughout the previous parts of the paper Singular cohomology had been used.
The submanifolds, called stable manifolds here, seem fairly nicely behaved, and my intuition was that Singular cohomology should coincide with Cech cohomology here.

Later, this is repeated in Lemma 6.2 as well.

Why is he forced to use Cech cohomology here? He refers to a theorem about relative cohomology, which requires Cech cohomology, but again, only if for the space in question the two cohomology theories differ.

Could it happen that for the stable manifolds Cech and Singular cohomology differ, with or without compact support?

network – In Bitcoin Core, are compact blocks pre-filled with more than just the coinbase?

In the “How are expected missing transactions chosen to immediately forward?” section of https://bitcoincore.org/en/2016/06/07/compact-blocks-faq/ it states that “To reduce the number of things that need to be reviewed in the initial implementation, only the coinbase transaction will be pre-emptively sent.”

I found https://github.com/bitcoin/bitcoin/blob/master/src/blockencodings.cpp#L23 which still lists pre-filling more than the coinbase as a TODO. I was wondering if that was in fact correct or whether it was an erroneous TODO. If not, is anyone working on that functionality or is there a reason it was never implemented?

real infinitesimal weights in Representations Of Compact Lie Groups.

I’m currently reading Representations Of Compact Lie Groups by T. Bröcker and T. Tom Dieck. In the section to Representations and Lie Algebras (p.112) they introduce the notion of (infinitesimal) weights of a (complex) $T$-module $V$, where $T$ is a maximal torus and $V$ a euclidian vector space. They define a weight of $V$ as follows:

A homomorphism $varthetacolon Tto U(1)$ is called a weight of $V$
if the corresponding weight space $$V(vartheta) = {vin Vmid xv =
vartheta(x)cdot v text{for all} Xin mathrm{L}T}$$
is nonzero.

In Notation (9.7) they state ($L_X$ refers to the Lie-derivative at $X$):

An $mathbb{R}$-linear form $alphacolon mathrm{L}Tto mathbb{R}$
is a real (infinitesimal) weight of the $T$-module $V$ if $2pi i
alpha$
is an infinitesimal weight of $V$. The weight space of $alpha$ is then $$V(alpha) = {vin Vmid mathrm{L}_Xv = 2pi i alpha(X)cdot v
text{for all} Xin mathrm{L}T}.$$

For clarification, i provide the definition of an infinitesimal weight as in Definition 9.3 on p.112:

An $mathbb{R}$-linear form $Thetacolon mathrm{L}T to
operatorname{LU}(1) = imathbb{R} subset mathbb{C}$
is called an
infinitesimal weight of the $T$-module $V$ if the corresponding weight
space $$V(Theta) = {vin Vmid mathrm{L}_Xv = Theta(X)cdot v
text{for all} Xin mathrm{L}T}$$
is nonzero.

My question: What is meant by

if $2pi ialpha$ is an infinitesimal weight of $V$

?

According to the definitions, an infinitesimal weight of $V$ is an $mathbb{R}$-linear form $mathrm{L}T tooperatorname{LU}(1)$. So what is meant by saying that $2pi ialpha$ is an infinitesimal weight of $V$?

Thanks for any help!

differential geometry – Why the Euler characteristics of a compact connected lagrangian submanifold of $mathbb{R}^4$ is zero?

Let’s consider space $mathbb{R}^4$ with the standard symplectic structure. There is a fact that if $Lsubseteq mathbb{R}^4$ is an embedded compact connected lagrangian submanifold, then it is a torus, in other words, $chi(L)=0$. I saw this fact in many books and articles, but all proofs I could find are either not very detailed or too difficult for me. Could you write this proof using the simplest methods.
I will give some information about my knowledge and problems that I have reading proofs of this fact.
I know about Weinstein’s neighborhood theorem. I understood that I have to use that some neighborhood of $L$ is symplectomorphic to some neighborhood of zero-section in cotangent bundle $T^*L$. Also, many proofs use isomorphism $TL cong T^* Lcong N L$. I understand that these isomorphisms exist, but in proofs are used such phrases as “canonical” isomorphism and “orientation preserving/reversing” isomorphism, which I understand not confidently and do not understand if it is important or not. Then, I know about Euler characteristics as a number of zeros (with signed or modulo two) of a generic vector field on a manifold. Also, I am a bit familiar with the intersection index of two submanifolds, so as I understand Euler characteristics of submanifold is the intersection index of the zero-section in its tangent bundle with itself, but I am not sure about a sign in this formula. In some proofs are used homology classes, but I do not know this theory enough to use it.
In general, it seems to me after reading all proofs I found that I have all instruments to prove this fact, but some steps I do not see or understand.

functional analysis – How to prove that support of $u_{r}$ is compact?

Let’s ${u_r}:{mathbb{R}^k}to{mathbb{R}}$, $varphi in C_{c}(mathbb{R}^n)$, $Jf$ ( jacobian of $f$, with $f: mathbb{R}^{k}to mathbb{R}^{n}$ Lipschitz ), and $chi_E$ (characteristic function on $E subset mathbb{R}^{k}$ where $E$ borel bounded subset),
begin{equation*}
u_r(w) := chi_E(z+rw)varphiBigl(frac{f(z+ rw)-f(z)}{r}Bigr) J f(z+rw).
end{equation*}

Statement: there are $ r_0> 0 $ and $ R> 0 $ such that $ supp(u_{r}) subset mathbb{B}(0,R) $ for $ r in (0, r_{0}) $.

My attempt:
In fact, since $ f $ is derivable into $z$ and $ Jf(z)> 0 $, there are $ s_{0}, lambda > 0 $ such that
begin{equation}
|{f(z’)-f(z)}| geq lambda ||{z’-z}||
end{equation}

for every $ z ‘in mathbb{B}(z,s_{0}) $. On the other hand, if $ rho> 0 $ is such that $ supp (varphi) subset mathbb{B}(0,rho) $, then
begin{equation*}
|{f(z+rw) -f(z)}|leq rrho
end{equation*}

for all $win supp(u_{r})$. Hence, if $ w in supp(u_{r}) $ with $ r < s_{0} / rho $, you have $ z + rw in mathbb{B}(z,s_{0}) $, so $ r rho geq |{f (z + rw) -f (z)}| geq lambda ||{z + rw-z}|| = lambda r ||{w}|| $, then $ ||{w}|| leq rho / lambda $, which proves statement with $ r_{0}: = s_{0} / rho $ and $ R: = rho / lambda $.

I’m not sure what $ z + rw in mathbb{B}(z,s_{0}) $ , I really appreciate it if someone could give me an idea how to improve this argument.

Another idea that I had, was to prove that the $supp(u_{0})$ is compact, which I did, in order to arrive at that the $supp(u_{r})$ is compact, which would be another way to conclude that statement in another way. But unfortunately I could not find that relationship between the $supp(u_{0})$ and the $supp(u_{r}) $I really appreciate the attention given.

axiom of choice – Does ZF + BPI alone prove the equivalence between “Baire theorem for compact Hausdorff spaces” and “Rasiowa-Sikorski Lemma for Forcing Posets”?

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $mathbb{P}$ (i.e. $mathbb{P}$ is a reflexive transitive relation) and for any countable family of dense subsets of $mathbb{P}$ there is a generic filter which intersects all dense subsets of the countable family. It is well-known that this statement is equivalent to the Baire Category Theorem for Complete Metric Spaces – and thus it is also equivalent to the Principle of Dependent Choices.

A masters student of mine has found in the literature the following statement: “Rasiowa-Sikorski Lemma is equivalent to the Baire Category Theorem for Compact Hausdorff Spaces, modulo the Boolean Prime Ideal Theorem”. We understood this as the assertion that the theory ZF + BPI alone is able to prove the equivalence between the Baire Category Theorem for Compact Hausdorff Spaces and the Rasiowa-Sikorski Lemma.

Well, I asked my student to verify such claim, and at first glance I suggested him to follow the results 3.1 to 3.4 of Chapter II of Kunen’s book, where there are proofs for some equivalences of Martin’s Axiom at $kappa$, MA($kappa$): the idea was to discard the hypothesis “c.c.c.” and adapt the reasoning, arguing for $kappa = omega$. It turns out that it was not a good suggestion, because in 3.1 a kind of Downward-Lowenheim-Skolem argument is done, to show that it is equivalent to work with a restricted form of the forcing axiom, considering only partial orders of bounded cardinality. However, such argument seems to require the Axiom of Choice, or some part of it other than BPI.

Does any of you know if it is indeed possible to prove the equivalence between “Baire Category Theorem for Compact Hausdorff Spaces” and “Rasiowa-Sikorski Lemma for forcing posets” from ZF + BPI alone ? Any suggestions or references would be appreciated.