Lück claims in his preliminary book, that the left hand side of the Baum-Connes map is functorial in the group $G$. For the right hand side $K(A \rtimes G)$ this is clear for the full crossed product, as he himself points out.

How is this justified?

Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of diameter $<\varepsilon$ in $X$ such that $x\in C_1$, $y\in C_n$ and $C_i\cap C_{i+1}\ne\emptyset$ for all $i<n$. It is clear that $\sim_\varepsilon$ is an equivalence relation on $X$. The equivalence class $[x]_\varepsilon:=\{y\in X:y\sim_\varepsilon x\}$ will be called the *$\varepsilon$-connected component* of $x$.

**Problem.** What can be said about the $\varepsilon$-connected components of $X$? In particular, is each $\varepsilon$-connected component $[x]_\varepsilon$ dense in $X$? Is each $\varepsilon$-connected component $\sigma$-compact?

**Added in Edit.** It can be shown that each $\varepsilon$-connected component $[x]_\varepsilon$ is $\sigma$-compact. To prove this fact, fix a countable base $\mathcal B$ of the topology of $X$, which is closed under finite unions and let $\mathcal B_\varepsilon$ be the subfamily of $\mathcal B$ consisting of basic sets of diameger $<\varepsilon$. It can be shown that each connected subset of diameter $<\varepsilon$ is contained in the connected component of some set $\bar B$ with $B\in\mathcal B_\varepsilon$. Now fix an enumeration $\{B_n\}_{n\in\omega}$ if the family $\mathcal B_\varepsilon$. Given any point $x\in X$ let $C_0=\{x\}$ and for every $n\in\mathbb N$ let $C_n$ be the union of $C_{n-1}$ with all connected components of $\bar B_n$ that intersect $C_{n-1}$. It can be shown that the sets $C_n$ are compact and $[x]_\varepsilon=\bigcup_{n\in\omega}C_n$.

sui mother buy some share of A on day 0. On day 7 share price of A is $\$44.6$.If she sell all share of A and buy 2000 shares of B on day 7 she would receive $\$7400$. On day 12 Share price of A is $\$4.8$ and B is $\$0.5$ less than on day 7. If she sell her all shares of A and buy 5000 shares of B on day 12 she would have to pay $\$5800$. Find share of A and share price of B on day 12

Let $a_1,a_2,\dots$ be a sequence of positive numbers less than $1$, such that $$\sum_{n=1}^\infty a_i= \infty,$$ and $S^1 = \mathbb{R}/\mathbb{Z}$.

Suppose $I_1,I_2,\dots$ be random intervals with respective lengths $a_1,a_2, \dots$in $S^1$ such that the distribution of the centers of $I_n$ (for every $n$) are uniform and independent.

It can be shown that with probability $1$, $I = \cup_{n=1}^\infty I_n$ is a full measure subset of $S^1$. Is it true that "With probability $1$, $I = S^1$"? If this is not always true, does there exist a good characterization of the sequences $\{ a_n\}_{n=1}^{\infty}$ with this property?

**Edit**. A more precise question: "What happens in the special case $a_n = \frac1n$?"

Are there general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?

*Background:* By a theorem of Myers and Steenrod the isometry group of a Riemannian manifold is a lie group. The same is true for Semi-Riemannian manifolds where the idea of a possible proof (which also works for Riemannian manifolds) is to embed the isometry group into the bundle of orthonormal frames as a closed submanifold. In the case of Riemannian manifolds this immediately yields that the isometry group of a compact Riemannian manifold has to be compact. But this is in general false for Semi-Riemannian manifolds since the bundle of orthonormal frames is not necessarily compact in this case even if the manifold is. So I ask myself if there are general statements about the phenomenon of a closed Semi-Riemannian manifold with non-compact isometry group.

Edit: My goal is to understand a bit better what the intuition behind compactness / non-compactness of the isometry of a closed Semi-Riemannian manifold is. For example topological restrictions on the underlying space. An example of this kind would be the following theorem for Lorentzian manifolds:

**Theorem:** Let $(M,g)$ be a closed, simply connected Lorentz manifold and assume $M$ and $g$ are analytic. Then the isometry group of $M$ is compact.

Let $H=(V,E)$ be a hypergraph. We say that $C\subseteq E$ is a *cover* if $\bigcup C = E$. Let $H$ be a hypergraph with the following properties:

- $\bigcup E = V$,
- all members of $E$ are finite, and
- $d,e\in E$ with $d\subseteq e$ implies $d=e$.

**Question.** Does this imply that there is a minimal cover $C_0$ (that is, $C_0$ has the property that for all $c\in C_0$ the set $C_0\setminus \{c\}$ is no longer a cover)?

I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given a representative cocycle of $H^2(Q,N)$ corresponding to such an extension. I will use the example of $\mathbb Z_4$, which is a nontrivial central extension of $\mathbb Z_2$ by $\mathbb Z_2$. I have tried to explain my reasoning below.

So I start with the exact sequence

$0 \rightarrow \mathbb Z_2 \overset{i}{\rightarrow} \mathbb Z_4 \overset{p}{\rightarrow} \mathbb Z_2 \rightarrow 0$

and compute the $E_2$-page using $E_2^{p,q} = H^p(\mathbb Z_2,H^q(\mathbb Z_2,\mathbb Z))$. In the case of the trivial central extension, where the action of every group on its respective coefficient module is trivial, we get a page which vanishes for $q$ odd and is a checkerboard of $\mathbb Z_2$'s and 0's, except for $E_2^{0,0} = \mathbb Z$, when $q$ is even. This sequence stabilizes on the $E_2$-page giving the results expected by using, say, the Kunneth formula.

However, this cannot be the right $E_2$-page for $\mathbb Z_4$, because the higher cohomology groups would then be too large. Therefore we must have a nontrivial action of $\mathbb Z_2$ on the coefficient modules $H^q(\mathbb Z_2,\mathbb Z)$. This is where I am stuck. I have two questions:

1) What is the above action, and is there a systematic way to see it for large $q$?

2)Is there a systematic way to compute differentials in the $E_2$-page given the maps $i,p$ and a representative cocycle for this extension?

In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.

$H^2(M,\mathbb{Z})$ is the group of isomorphism classes of line bundles over $M$.

I guess they mean there is a natural isomorphism.

Can some one give a rough idea of what obvious second cohomology class we can think of given a line bundle over $M$ and what line bundle can we think of given an arbitrary second cohomology class.

Intuitive comments are also welcome.

I am familiar (not the proof details) with following result:

If $G$ is a group and $M$ is a $G$-module, then the $H^2(G, A)$ is in one-one correspondence with the set of equivalence classes of extensions $E$ of $M$ by $G$, in which the action of $G$ on $M$ induced by conjugation in $E$ is the same as the action defined by the $G$- module $M$.

I am expecting some intuitive explanation that looks similar to this.

I asked this question on Mathematics Stackexchange (link), but got no answer.

Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.

Recall that $A$ can be defined as the set of expressions of the form $\sum_ua_uu$, where $u$ runs over the set monomials in $x_1,x_2,\dots$, and each $a_u$ is in $K$, the addition and multiplication being the obvious ones.

Then $A$ is a local domain, its maximal ideal $\mathfrak m$ is defined by the condition $a_1=0$, and it seems natural to ask

Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?

I suspect that the answer is No, and that the series $\sum_{n\ge1}x_n^n$, which is clearly Cauchy, does *not* converge $\mathfrak m$-adically.

I would like to prove such a matrix as a positive definite one,

$$ (\omega^T\Sigma\omega) \Sigma - \Sigma\omega \omega^T\Sigma $$ where $\Sigma$ is a positive definite symetric covariance matrix while $\omega$ is weight column vector (without constraints of positive elements)

I would apply an arbitrary $x$ belonging to $R^n$ to the following formula, $$ x^T((\omega^T\Sigma\omega) \Sigma - \Sigma\omega \omega^T\Sigma)x > 0 $$

But how could I go further to prove such a inequality above?

Thanks,

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed.

Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$.

Does there exist an open neighbourhood $U$ of $p$, and a closed and co-closed $k$-form $\omega \in \Omega^k(U)$ satisfying $\omega_p=\alpha_p$?

This question is equivalent to the following question:

Do closed and co-closed frames for $\bigwedge^k(T^*M)$ always exist locally?

Indeed, if we can specify the value of a form in a point, we can take a basis for $\bigwedge^k(T_pM)^*$, and so obtain forms which form a frame at $p$. Since "being a frame" is an open condition, we have a local frame. On the other hand, suppose that local closed and co-closed frames exist. Then, by choosing a linear combination with constant coefficients of that frame, we can realize any given value at $p$.

*Comment:* In general we cannot expect such a frame to be induced from coordinates. Indeed, when we specialize to even dimension $d$, and $k=\frac{d}{2}$, then, for a generic metric $g$, there are no coordinate systems where even one wedge $\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_n}
$ is co-closed.

The question was motivated by this question of Anton Petrunin.

By a *metric continuum* we understand a connected compact metric space.

Let $p$ be a positive real number. A metric continuum $X$ is called *$\ell_p$-almost path-connected* if for any points $x,y\in X$ and any $\varepsilon>0$ here exists a family $\big((a_n,b_n)\big)_{n\in\omega}$ of pairwise disjoint open intervals in the unit segment $[0,1]$ and a continuous map $\gamma:[0,1]\setminus\bigcup_{n\in\omega}(a_n,b_n)\to X$ such that $\gamma(0)=x$, $\gamma(1)=y$ and $\sum_{n=0}^\infty d_X(\gamma(a_n),\gamma(b_n))^p<\varepsilon$.

It is easy to see that each almost $\ell_p$-connected metric continuum is $\ell_q$-almost connected for any $q\ge p$.

By my answer to the question of Anton Petrunin, each plane continuum is almost $\ell_1$-connected. By analogy it can be shown that each continuum in $\mathbb R^3$ is $\ell_2$-connected.

**Problem.** Is there a metric continuum which is not almost $\ell_1$-path connected? not almost $\ell_p$-connected for every $p<\infty$?

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of squares of $d$ dimensional Fourier harmonics up to degree $n$.

My question is if $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$ and can be represented as combination of spherical harmonics dimension $d$, then does there exist some spherical harmonic polynomials $g_1,\ldots, g_k$ of degree $n$ such that $p=g_1^2+\cdots g_k^2$ is a sum of squares?

i edit the question after Zach Teitler's comment.

The interval $[-\pi,\pi]^d$ means we concern the trigonometric polynomials positive on frequency domains.

The optimization problems about the polynomials positive on frequency domain $[-\pi,\pi]^d$ can be implemented via SDP approach(Gram matrix Rpresentation).

Given a positive polynomial represented as combination of spherical harmonics dimension $d$, Obviously, it is sum of squares of $d$ dimensional Fourier harmonics. Furthermore, it implies the symmetry relationship between $[-\pi,\pi] \times [0,\pi]$ and $[-\pi,\pi] \times [-\pi,0]$ on 2-sphere as an example. May be there is less information on sphere than cube?

So, is it the sum of squares of spherical harmonics?

Please feel free to provide any advices. Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $Rep_V(\mathfrak{g}) \subset Rep(\mathfrak{g})$ to be the smallest symmetric monoidal, idempotent complete, abelian subcategory with duals (so, closed under tensor products, retracts, direct sums and duals) which contains $V$.

**Question:** Does there always exist an irreducible $V$ for which $Rep_V(\mathfrak{g}) \cong Rep(\mathfrak{g})$? When it exists, is there
a unique minimal one (in terms of the order on the weights) such $V$ (up to dualizing)? If not is there a unique self-dual such representation? If it doesn't exist, what is the minimal dimensional $V$
(possibly reducible) which satisfies this condition? Is it unique in some sense?

I'm interested in the question for all $\mathfrak{g}$ of type $A,B,C$ and $D$ (the exceptionals are a luxury). I think the standard representation $V$ in the case of type $A$ generates the entire category in this sense so that the answer is positive for this case but i'm not sure about any of the other cases.

While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific.

I am writing a paper, in which we have a section called "Outline of Proof". (It's Section 2.)
The outline is fairly informal, and we omit some technical details, making approximations.
However, among these approximations, my co-author wants to state (and label) *important* definitions and results (lemmas, equations, etc). He then wants to, later in the paper when we are doing the corresponding part carefully and rigorously, refer back to these (say, "by equation (2.4)", or "by Lemma 2.2"). Moreover, he is very against redundancies, so does not like things being stated twice precisely (including in the outline) -- once precisely in the text and once approximately in the outline is fine.

To me, this seems insane. (Of course, I did not use such a phrase when speaking with him!) When I read a paper, I never carefully read the outline: I just read it, and try to get an overview (or 'outline') of the proof; if there are parts that I don't really understand, I don't get hung up on them, trusting that with the more rigorous explanation later I'll be able to make sense of what the authors are saying.

*However*, I'm a pretty junior author -- 2nd year of PhD -- while he is a first year postdoc. That doesn't mean that I haven't read a reasonable number of papers (and in fact my lack of experience and knowledge means that I am less able to understand poorly written papers); moreover, he has said that he feels writing papers well isn't his best attribute.

So my question is this:

(a) is it standard to read an outline of a proof carefully?

(b) is it standard (or at least not discouraged) to state precisely important, even key, results/definitions that will be referred back to in the main body of the paper when giving proofs?

Just as some extra comments... I'm not here to try to get people to tell me that I'm right and my co-author is wrong and/or being silly! I know that *sometimes* some people come to Stack Exchange for such comments (see, particular, Workplace/Interpersonal Skills SEs!). I should have perhaps made the following clear: *if everything my co-author does is standard in the field, and I'm in the wrong, I definitely want to know that and will accept it!* -- I'm here to learn :-) please have no qualms about hearing criticism! (assume that it's constructive, of course)

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\operatorname{ht}(\alpha^\vee)-1$, where $\ell$ denotes the length function and where $\operatorname{ht}$ denotes the obvious height function. Let $w_o$ be the longest element of $W$. Let $Q^\vee$ be the coroot lattice.

For a sequence of positive roots $\gamma_1,\ldots,\gamma_\ell$, define $$ \delta_i(\gamma_1,\ldots,\gamma_\ell)=\begin{cases} \hphantom{-}1\,,&\text{if }\gamma_i\in\tilde{R}^+\text{ and }\ell(s_{\gamma_1}\cdots s_{\gamma_i})=\ell(s_{\gamma_1}\cdots s_{\gamma_{i-1}})+\ell(s_{\gamma_i})\,,\\ \hphantom{-}0\,,&\text{if }\ell(s_{\gamma_1}\cdots s_{\gamma_i})=\ell(s_{\gamma_1}\cdots s_{\gamma_{i-1}})-1\,,\\ -1\,,&\text{otherwise}\,. \end{cases} $$

We say that an effective element $d\in Q^\vee$ satisfies Property $(\mathrm{A})$ if the following condition is satisfied: There exists a sequence of positive roots $\gamma_1,\ldots,\gamma_\ell$ and a sequence of simple roots $\beta_1,\ldots,\beta_\ell$ such that

- $w_o=s_{\gamma_\ell}\cdots s_{\gamma_1}$,
- $s_{\beta_1}\cdots s_{\beta_\ell}$ is a reduced word,
- $\beta_i$ is in the support of $\gamma_i$ for all $1\leq i\leq\ell$,
- $\delta_i(\gamma_1,\ldots,\gamma_\ell)\in\{0,1\}$ for all $1\leq i\leq\ell$,
- $\left(\sum_{j=1}^i\delta_j(\gamma_1,\ldots,\gamma_\ell)\gamma_j^\vee,\beta\right)\leq 2$ for all simple roots $\beta$ and for all $1\leq i\leq\ell$, and such that
- $d=\sum_{i=1}^\ell\delta_i(\gamma_1,\ldots,\gamma_\ell)\gamma_i^\vee$.

**QUESTION 1.** Can you compute the elements of the set
$$
\{d\in Q^\vee\text{ effective with Property $(\mathrm{A})$}\}
$$
which are *maximal* with respect to the natural partial order on $Q^\vee$?

**QUESTION 2** (maybe simpler)**.** Can you compute the number
$$
\mathscr{D}_R=\operatorname{max}\{\operatorname{ht}(d)\mid d\in Q^\vee\text{ effective with Property $(\mathrm{A})$}\}\,?
$$

**REMARKS.** If you can answer Question 2, it would be completely sufficient for me. I posed Question 1 only out of curiosity. If you can provide an implementation in *SageMath* or similar software which provides the number in Question 2 for type $\mathsf{A}_n$ and given $n$, I would be more than happy. I cannot do this for myself, unfortunately, because I am not experienced enough with computer algebra software.

I thank everyone for their help!

**EDIT.** The expected outcome for type $\mathsf{A}_n$ is as follows:
$$
\mathscr{D}_{R\text{ of type $\mathsf{A}_n$}}=\begin{cases}
\left[\tfrac{n}{2}\right]\left(\left[\tfrac{n}{2}\right]+1\right)\,,&\text{if $n$ is even}\,,\\
\left(\left[\tfrac{n}{2}\right]+1\right)^2\,,&\text{if $n$ is odd}\,,
\end{cases}
$$
where $[-]$ means the floor function. Can you confirm or disprove this expected outcome?

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product \begin{equation*} \prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}. \end{equation*} The product converges if and only if $\sum_{i=1}^{\infty}\log(\lambda_{i})$ converges.

Also, we consider the following zeta function given by a Dirichlet series \begin{equation*} \zeta(s)=\sum_{i=1}^{\infty}\lambda_{i}^{-s}. \end{equation*} We assume that the zeta function is absolutely convergent on some right-half plane, has an analytic continuation on $\mathrm{Re}(s)>-\epsilon$ for a positive number $\epsilon>0$, and is holomorphic at $s=0$. Then, for this case, we define a regularized product by \begin{equation*} \hat{\prod}_{i=1}^{\infty}\lambda_{i}:=\exp(-\zeta^{\prime}(0)). \end{equation*}

My question is that

If the ordinary infinite product converges, then does it coincide with the regularized product?, i.e. $$\prod_{i=1}^{\infty}\lambda_{i}=\hat{\prod}_{i=1}^{\infty}\lambda_{i}$$

The question is equivalent to the following

If $\sum_{i=1}^{\infty}\log(\lambda_{i})$ converges, then $$\sum_{i=1}^{\infty}\log(\lambda_{i})=-\zeta^{\prime}(0)?$$

I'm currently studying Iwasawa theory.

1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act. So I often face some facts on the group representation over local fields or p-adic integer ring. But I can't find any references yet.

Of course, there are articles on p-adic representation. But I want references that are not too deep. I want references using just easy-to-follow arguments of algebra and representation theory.

Can you suggest any references?

2) Currently, I'm reading the paper "On the Iwasawa Invariants of Totally Real Number Fields" written by Ralph Greenberg. There I cannot understand a line which I have underlined with red line.

I'm afraid that there are many counter-examples against the line.(For example we can take cyclic group of order prime to the order of the group of units.) Can you please explain the line to me?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.

Assume $d \ge 3$ and that $df=0$ at some point. Is it true that $f$ is constant?

A proof for the Euclidean case, can be found in "Geometric Function Theory and Non-linear Analysis", by Iwaniec and Martin. Their proof uses the fact both $M,N$ are Euclidean.

If I am not mistaken, a conformal map (in $d\ge3$) is determined by its 2-jet at a point, so it suffices to prove all the second derivatives at the point where $df$ vanishes are zero. (But maybe this wrong; it's possible that the "2-jet determination" only holds for conformal maps whose differentials are everywhere non-zero. I am not sure.)

I only know how to differentiate the conformal equation covariantly, which at a point where $df=0$ does not provide any information.

The ordinary circle packing problem in the variant with equal radii asks for the largest radius $r_{max}$ that allows placing $n$ non-overlapping circles with radius $r_{max}$ e.g. in the unit square, in the unit circle or, on the unit sphere (Tammes' problem).

Now, I would like to solve a somehow opposite problem:

**Question:**

given a number $n\in\mathbb{N}$, what is the smallest radius $r_{min}\in\mathbb{R}^+$ that permits a non-overlapping, rigid placement of $n$ circles with radius $r_{min}$ in the unit square, or in the unit circle or, on the unit sphere?

Under a rigid configuration I understand a configuration, where every open halfplane, resp. hemisphere defined by a hyperplane through a circle's center contains at least one contact point with another circle or, with the boundary of the containing region.

Are there already algorithms and/or theoretical results available for that problem?