functional analysis – Affine continuous surjective function between Choquet simplexes

As a generalization of the Banach Stone Theorem, A. J. Lazar proved that for any two simplexes $K_1$ and $K_2$, $A(K_1)$ and $A(K_2)$ are isometrically isomorphic if and only if $K_1$ and $K_2$ are affinely homeomorphic.

Now, for two simplexes $K_1$ and $K_2$, if we consider a linear isometry from $A(K_1)$ into $A(K_2)$(not necessarily onto) then does there exists an affine continuous surjection from $K_2$ to $K_1$? Is it possible to obtain such an affine continuous surjection from the above result by Lazar?

continuous integration – Clarifying the steps in a CI/CD, but namely if if unit testing should be done building a Docker image or before

I’m building at a Build and Deployment pipeline and looking for clarification on a couple points. In addition, I’m trying to implement Trunk Based Development with short-lived branches.

The process I have thus far:

  1. Local development is done on the main branch.

  2. Developer, before pushing to remote, rebases on remote main branch.

  3. Developer pushes to short-lived branch: git push origin main:short_lived_branch.

  4. Developer opens PR to merge short_lived_branch into main.

  5. When PR is submitted it triggers the PR pipeline that has the following stages:

    1. Builds the microservice.
    2. Unit tests the microservice.
    3. If passing, builds the Docker image with a test-latest tag and push to container registry.
    4. Integration testing with other microservices (still need to figure this out).
    5. Cross-browser testing (still need to figure this out).
  6. If the PR pipeline is successful, the PR is approved, commits are squashed, and merged to main.

  7. The merge to main triggers the Deployment pipeline, which has the following stages:

    1. Builds the microservice.
    2. Unit tests the microservice.
    3. If passing, builds the Docker image with a release-<version> tag and push to container registry.
    4. Integration testing with other microservices (still need to figure this out).
    5. Cross-browser testing (still need to figure this out).
    6. If passing, deploy the images to Kubernetes cluster.

I still have a ton of research to do on the integration and cross-browser testing, as it isn’t quite clear to me how to implement it.

That being said, my questions thus far really have to do with the process overall, unit testing and building the Docker image:

  1. Does this flow make sense or should it be changed in anyway?

  2. Regarding unit testing and building the Docker image, I’ve read some articles that suggest doing the unit testing during the building of the Docker image. Basically eliminating the first two stages in my PR and Deployment pipelines. Some reasons given:

    • You are testing the code and not the containerized code which is actually what will be run.
    • Even if unit testing passes, the image could be broke and it will be even longer before you find out.
    • Building on that, it increases the overall build and deployment time. From my experience, the first two stages in my pipelines for a specific service take about a minute and half. Then building and pushing the image takes another two and half minutes. Overall about four minutes. If the unit tests were incorporated into the Docker build, then it could possibly shave a minute or more off the first three stages in my pipeline.

    Would this be a bad practice to eliminate the code build and unit testing stages, and just moving unit testing into the Docker build stage?

Thanks for weighing in on this while I’m sorting it out.

Continuous daily exchange | Forum Promotion

real analysis – Can a sequence of absolutely continuous functions be rescaled to be equicontinuous?

Given a function $f: mathbb R to mathbb R$, we say $g: mathbb R to mathbb R$ is antopological rescaling of $f$ if $g = fh$ for some orientation preserving homeomorphism $h$ of $mathbb R$.

Given a sequence $f_n: mathbb R to mathbb R$ of absolutely continuous functions, do there exist topological rescalings $g_n$ of $f_n$ (where the choice of $h$ is allowed to depend on $n$) such that the restricted functions ${g_n}_{lvert(0, 1)}$ are equicontinuous?

dg.differential geometry – Sheaf-Like Reconstruction of a continuous function

Let $X$ and $Y$ be topological manifolds and let ${(phi_x,U_x)}_{x in X}$ and ${(psi_y,Y_y)}_{y in Y}$ be respective atlases of $X$ and $Y$; with each $phi_x:U_xrightarrow mathbb{R}^n,psi_y:V_yrightarrowmathbb{R}^m$ homomorphism onto their images and each $U_x,V_y$ open and non-empty.

Suppose that I’m given ${f_xin C(phi_x(U_x),mathbb{R}^m)}_{x in X}$. Can (when) I find a map $F:Xrightarrow Y$ (just a plane set-function) such that:
$$
psi_{F(x)}circ f_xcirc phi_x
,
$$

is a well-defined element of $C(X,Y)$?

Can this always be done if $X$ and $Y$ are topological manifolds? If not, what if we assume them to be $C^{infty}$?

Note: I added the tag ‘sheaf’ since I feel like that may be a reasonable (possible) way to approach this problem.

continuity – Is the Collatz function $xmapstodfrac{3x+1}{2^{nu_2(3x+1)}}$ continuous on $Bbb Z_2^times$?

Is the Collatz function $xmapstodfrac{3x+1}{2^{nu_2(3x+1)}}$ continuous on $Bbb Z_2^times$?

Let $Bbb Z_2^times$ be the 2-adic units.

Then e.g. $3mapstodfrac{10}2=5$

Attempt

Here’s what I know: The function above can be thought of as a map $xmapsto3x+2^{nu_2(x)}$ on all of $Bbb Z_2toBbb Z_2$ which descends to the quotient map $xmapsto{2^ixinBbb Z_2:iinBbb Z}$.

Since $xmapsto3x+2^{nu_2(x)}$ is continuous and the function $xmapstodfrac{3x+1}{2^{nu_2(3x+1)}}$ descends to the quotient, it must too be continuous on $Bbb Z_2/{sim}$

However, I think $Bbb Z_2/{sim}$ isn’t homeomorphic to $Bbb Z_2^times$. If it were, I could answer in the positive. There’s an obvious bijection between them but I think $Bbb Z_2/{sim}$ has the trivial topology because every element is arbitrarily close to $0$. So the trivial continuity in $Bbb Z_2/{sim}$ says nothing of continuity in $Bbb Z_2^times$.

real analysis – How to prove that a set of continuous functions is closed in (C([0,1]),d1).

I understand how to prove that for the set M = {f ∈ C((0,1)) : f(1) = 0} is closed in (C((0,1)),d(infinity)) using sequence of characterization but not sure how to prove it is not closed in d1.

There was a hint to draw a function fn and a function f (which is just f=1) and I am not sure how to a) know that this set F is not closed in the first place (so I know that I have to prove it is not closed) and b) how do I know what kind of graph to draw.

I also know I need to show three things:

  1. (fn)n ∈ M
  2. f ∉ M
  3. d1(fn,f) → 0

Thank you for all of the help!

fa.functional analysis – Is the bitranspose continuous for the $sigma$-strong topology?

Let $varphicolon Ato B$ be a bounded, linear map between C*-algebras. Is the bitranspose $varphi^{**}colon A^{**}to B^{**}$ continuous when the von Neumann algebras $A^{**}$ and $B^{**}$ are equipped with their $sigma$-strong topologies?

Motivation/Background: Note that $varphi^{**}$ is clearly continuous when $A^{**}$ and $B^{**}$ are equipped with their $sigma$-weak topologies, since these agree with the weak${}^*$-topology from the preduals, and $varphi^{**}$ is weak${}^*$-continuous (that is, $sigma(A^{**},A^*)-sigma(B^{**},B^*)$-continuous).

If $varphi$ if completely positive, then it follows that $varphi^{**}$ is a completely positive, normal map, and therefore is continuous for the $sigma$-strong topologies. Thus, the question is only interesting if $varphi$ is not completely positive.

On bounded sets of a von Neumann algebra, the $sigma$-strong topology agrees with the strong (operator) topology (SOT). If $Msubseteq B(H)$ is a von Neumann algebra, then a net $(a_j)_j$ in $M$ SOT-converges to $ain M$ if $|a_jxi-axi|to 0$ for every $xiin H$.

continuous integration – How should I think about infrastructure as code when using Kubernetes?

I’m trying to better think about our IaC configuration.

I work in a large enterprise, so our Kubernetes infra is managed by the Kubernetes team. We have our application deployments automatically deploy to this infra from our CI/CD pipelines after automated tests run, etc.

My question is – Where would something like Terraform come into play for me? I have a fair bit of deployment and test automation done for me completely via Gitlab and K8 YML specs.

Is there anything I should think about as far as TF goes to help benefit me even further? The one thing I see is maybe DyanmoDB and S3 buckets that we create. Does this automatically run as part of CI or just manually?

Thanks.