gn.general topology – Does a compact ANR have a local equiconnecting function which connects distinct points by simple paths?

It is known that if $X$ is a (metric) ANR, then $X$ is locally equiconnected, that is, there is a neighborhood $V$ of the diagonal $Delta X subseteq X times X$ and a continuous function $$f colon V times (0,1) rightarrow X$$
such that

  1. For every $(x,y) in V$, the path $f(x,y,-) colon (0,1) rightarrow X$ starts at $x$ and ends at $y$.
  2. For every $x in X$, the path $f(x,x,-) colon (0,1) rightarrow X$ is the constant path at $x$.

(Side note: Local equiconnectivity is equivalent to the diagonal map $Delta colon X rightarrow X times X$ being a Hurewicz cofibration.)

Let us also assume that $X$ is compact. My question is: Can we choose the $U$ and $f$ such that when $x neq y$ in the 1st condition, the path connecting them is a simple path?

Remark: It follows from Lemma 2.1 of the paper “A remark on simple path fields in polyhedra of characteristic zero” by Fadell that the answer is yes when $X$ is a finite simplicial complex. I am interested in a (strict) generalization of this result.

mg.metric geometry – Gromov-Hausdorff limit of compact surfaces with same boundary of equal areas

Define $$ A = (0,0,0), B=bigg(frac{1}{n},0,0bigg), C=
bigg(frac{1}{n},frac{1}{n},0bigg) , D= bigg(
0,frac{1}{n},0bigg), E=
bigg(frac{1}{2n},frac{1}{2n}, frac{a}{n}bigg)
for some $a>0$.

Consider $Sigma=Delta ABE bigcupDelta BCEbigcupDelta
CDEbigcupDelta DAE $
. When $T_{ij}$ is a translation map on
$mathbb{R}^3$ by $T_{ij}(x,y,z) =( x+
, then define $$ M_n =bigcup_{0leq
i, jleq n-1} T_{ij}(Sigma)$$

When $d_n$ is intrinsic metric on $M_n$, then $d_n((0,0,0),(1,1,0))
> sqrt{2}$
are equal and areas of $M_n$ are equal. Here does the Gromov-Hausdorff limit of $
exists ?

reference request – How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $Lambda^3(mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe in his answer to the post A question on complex semisimple Lie groups and $(mathbb{C}^2, omega)$ a nice description of the complex $E_6$ as the group of symmetries of $V = Lambda^2_0 (mathbb{C}^8)^*$ endowed with a cubic form on that space. In that description, what is the real structure which gives the compact real form $E_6$? I conjecture it would be induced by the real structure $j wedge j$ on $V$, where $j$ is a quaternionic structure on $mathbb{C}^8$ such that $omega$ (see Prof. @RobertBryant’s answer in the above link) is real. It is just a guess. Is it correct please?

I do not really know how to realize the compact real forms of $F_4$, $E_7$ and $E_8$ as matrix groups. Your help is kindly appreciated. References are more than welcome (especially if they can be found online, and I hope “my” library has access to them!). If someone feels like writing a whole answer, then that would be great too. It is time for me to learn more about the exceptional Lie groups.

One last thing. I know that what I am looking for can be found in the relevant E. Cartan’s papers. However, while I would definitely learn a lot by going back to the source, yet I don’t have as much free time nowadays as I would like to (not to mention that reading Cartan is known to be difficult, and it is not the language barrier, in my case). So is there a simplified and modernized version of that part of Cartan’s work please, that would also discuss compact real forms?

data structures – Is there a known way to make an efficient, compact, and fully persistent stack or queue?

In the world of mutable/ephemeral data structures and imperative programming languages, one of the classic ways to implement a stack or queue is to use array doubling: use mutation to fill up or empty an array, doubling or halving to expand/contract. Such stacks/queues have several nice properties:

  1. They use at most twice as much memory as strictly necessary.
  2. They involve minimal indirection.
  3. They use cache efficiently.
  4. They have amortized $O(1)$ insertion and deletion operations.

In a purely functional system, this approach falls down quite flat, because “mutate the array to fill/empty” becomes very expensive: the array has to be copied each time. I was wondering if there might be some reasonable compromise approach, making something more compact than classical approaches (a la Okasaki) but still with constant amortized time operations. Stacks are simpler, so I started thinking about those. My first attempt (in Haskell notation) was

data Queue a
  = Shallow !(Array a)
  | Deep !(Array a) (Queue a)

with the rule that the array at depth $n$ must have either $0$ or $2^n$ elements. Unfortunately, this doesn’t look like it’s nearly good enough. It appears that insertions impose an $O(log n)$ amortized cost, since flipping from a 1 digit to a 0 digit gets more expensive the deeper it happens in the tree. My next attempt was the same, but using skew binary numbers instead of binary numbers. Same deal. Is there some trick I’m missing, or am I asking to have my cake and eat it too?

dg.differential geometry – Reductive Lie groups and existence of maximal compact subgroup

I am reading Knapp’s book “Lee groups beyond introduction” (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra $mathfrak g$ of $G$ is reductive and equipped with a involution and a nondegenerate symmetric bilinear form $B$ on $mathfrak g$ which is $theta$-invariant and $Ad$-invariant and the following hold.

  1. We have a decomposition $mathfrak g=mathfrak koplus mathfrak p$ with respect to the eigenspaces of $theta.$

  2. $G$ has a compact subgroup with Lie algebra $mathfrak k.$

  3. The map $(k,exp X)mapsto kexp X$ from $Ktimes expmathfrak p$ to $G$ is a diffeomorphism.

  4. The bilinear form $B_theta (X,Y):=-B(X,theta Y)$ is positive definite on $mathfrak g.$

  5. Every automorphism $Adg$ for $gin G$ is inner.

From this Knapp concludes that $K$ has to be a maximal compact subgroup. he argues as the following (Page 446). Let $K$ is properly contained in a compact subgroup $K_1$. Take $k_1in K_1setminus K.$ Then for some $kin K$ and $Xinmathfrak p$, we have $kexp X=k_1$ which implies that $exp Xin K_1.$ Since $(exp X)^n=exp (nX)$ is in $K_1$ the sequence $exp(nX)$ must have a convergent subsequence. I get easily to this point. Now Knapp says that this contradicts 3.! I do not understand this. Can someone help me out?

at.algebraic topology – Increasing connectivity of a compact set

In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $Ksubset X$ is compact, then there is a compact connected $Lsubset X$ such that $Ksubset L$. After revising that question (or rather Anton Petrunin’s answer) it occured to me that I don’t know how to make $L$ path connected. This motivated the following questions.

Let $X$ be as above and $pi_k(X)$ is trivial, for $k=0,…,n$ (resp $X$ is contractible). If $Ksubset X$ is compact can we find a compact $Lsubset X$ such that $Ksubset L$ and $pi_k(L)$ is trivial, for $k=0,…,n$ (resp $L$ is contractible)?

Regarding the “contractible version”, here is an idea that does not work: let $F:Xtimes(0,1) to X$ be a homotopy from the identity to a constant. It is tempting to try to show that $F(Ktimes (0,1))$ is the set that we are looking for. However, if we started with $K$ a singleton, the obtained set can be any Peano continuum, and so not necessarily contractible.

real analysis – Example of compact operator which has eigenvalues with 0 as limit point

I was reading Fredholm alternative which has following consequence

A compact linear mapping T of normed linear space into itself possesses a countable set of eigenvalues having no limit point except possibly 0. Each Nonzero eigenvalue has finite multiplicity

I wanted to find example of compact operator which has eigenvalues with limit 0

Any Help will be appreciated

Compact BaseForm Alternative that Extends to Bases > 36

I recently created a document listing the terms of OEIS sequence A256112 in BaseForm. Term #164 and #165 at the end of the document are the two solutions in base 35. As I continue to search for additional terms in the sequence I am about to hit the BaseForm limitation of bases needing to be integers less than 37.

A viable alternative is to list these numbers as comma-separated IntegerDigits. I would prefer something more compact, say a BaseForm that incorporates the Greek alphabet starting at base 37. I don’t know how to do that. Or, if it was possible to enclose each digit of an IntegerDigits list in a standard-sized square, that might look ok as an output. I’m asking for a good-looking, compact BaseForm alternative that extends the current limitation to bases beyond 36.