general topology – Properties need to define Derivatives on Topological space

I just started learning topology and was curious about defining derivatives on general topological spaces.
Since we can define continuous functions on Topological spaces, my question is what additional properties one would need to define derivatives on Topological spaces.

I guessed one might only need converging sequences to define derivative, so space must have,

  1. metrizablity: to define some sort of distance between points so one can define converging sequences on space.
  2. Hausdorff property: so convergence would be unique.

but on the other hand, these properties are conserved under homeomorphism, where differentiability does not.
so, there should be some more properties (maybe other than topological properties) one would need to define derivatives, and I’m not sure what kind of property it would be that conserves differentiability.

I hope my question makes sense.
Thanks.

linux – How to collect selective dsks and unused space using ansible facts?

Here are my example codes, questions at bottom
vi hw.j2
System Total Memory : {{ ansible_memtotal_mb }}
Free Memory : {{ ansible_memfree_mb }}
System Total CPU : {{ ansible_processor_cores }}
System Virtual CPU : {{ ansible_processor_vcpus }}

Disks capacities
/dev/sda : {{ ansible_devices.sda.size }}
/dev/sdb : {{ ansible_devices.sdb.size }}
/dev/sdc : {{ ansible_devices.sdc.size }}

vi collecthw.yml

  • name: Collect information
    become: yes
    hosts: clients

    tasks:

    • name: use template from labs/hw.j2
      template:
      src: hw.j2
      dest: hw.{{ ansible_facts.hostname }}.txt
      owner: corona
      group: corona
      mode: ‘0600’

It works okay but ..

Now my questions are:

  1. How to loop across all /dev/sd* in managed nodes and collect their respective disk size?
  2. This is the disk size, how about unused disk space in the disk?

Thank you in advance for those that able to spend time and effort in getting the answer or perhaps other advice. Newbie here xD

bash – Is it possible to download a file while disk space is below a certain threshold?

Is it possible to download a file while disk space is below a certain threshold in a bash script? I’m asking because I’m downloading files from an online database, and would like to keep downloading until my storage passes a pre-determined threshold. Once that happens, I’d like to stop the download (i.e terminate the command), delete a few files, then restart the download (i.e rerun the command that downloads the file). Based on my understanding, you can’t “pause” a command in bash, and in my case it’s fine to just stop the download process and do it again.

Magento 2 space symbols in storefront error messages are replaced with plus characters

Does any one have the same question?
I saw this bug cannot be corrected by Magento.
Link: https://devdocs.magento.com/guides/v2.4/release-notes/release-notes-2-4-0-commerce.html

I wanna to ask is it possible to solve and how? any one sloved?

Storefront Demo

ag.algebraic geometry – Algebraic Space: Two equivalent constructions

According to Wikipedia
there are two common ways to define algebraic spaces:
they can be defined as either quotients of schemes by étale
equivalence relations,
or as sheaves on a big étale site that are locally isomorphic
to schemes.
Namely:

I) a la Knutson:

An algebraic space $X$ comprises a scheme $U$ and a closed subscheme
$R subset U times U$ satisfying the following two conditions:

  1. $R$ is an equivalence relation as a subset $U times U$;
  2. the two projections $P_i: R to U$ onto each factor are étale.

Knutson adds an extra condition that the diagonal map is quasi-compact.

II) as a sheaf:

An algebraic space $mathfrak {X}$ can be defined as a sheaf of sets
$$mathfrak {X}:(operatorname{Sch}/S)^{text{op}}_{text{ét}} to operatorname{Sets}$$
such that

  1. There is a surjective étale morphism $h_X to mathfrak {X}$;
  2. the diagonal morphism $Delta _{{mathfrak {X}}/S}:
    mathfrak {X} to mathfrak {X} times mathfrak {X}$

    is representable and quasicompact (thanks to David’s careful remark).

(Rmk: in II)1. we identified a scheme $X$ with its image $h_X$ wrt the Yoneda
embedding $X to operatorname{Hom}(X,{-})$.)

Two questions:

  1. About construction I). Wikipedia moreover says that if $R$
    is the trivial equivalence over each connected conponent of $U$
    (i.e. for all $x,y in U$ lying in same component then
    $xRy$ iff $x=y$) then the so defined algebraic space is a scheme
    in the usual sense. Why?

  2. Where I can find a proof/ reason that the constructions
    I) and II) are indeed equivalent?

dnd 5e – What is the interaction between Thunderwave and being required to target your own space?

The spell Thunderwave has the rather unusual target area of a 15-foot cube, the point of origin for which is the caster, which is generally understood to mean you can either position the cube anywhere adjacent to yourself, or (for whichever reason) include yourself in the area, but not if it’s the middle of it.

However, for homebrew reasons, the character trying to cast Thunderwave has the following limitation to their spellcasting:

(I)f the spell requires a somatic component, it must target you or a point in your space.

Thunderwave, with its components being V S, fits that description, and it does make wonder what would be the interaction of this spell and this rule – whether I can or can not place the spell that doesn’t result in targetting myself, specifically in a situation where I’m required to target “a point in my space”.

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dnd 5e – What happens when allies occupy the same space?

Unfortunately, Mike Mearls is not the proper authority on these matters – although he does try his best to help. Since he openly admitted he could not recall the rules on the matter and gave an educated guess, Jeremy Crawford recalled them for him.

As per the intended RAW in PHB pg. 191, under “Moving Around Other Creatures”:

“…you can’t willingly end your move in its space…”

Further explained by Jeremy Crawford in the tweet linked above:

“…you can’t willingly end your move —any part of it— in another creature’s space…”

So let’s first clear up some confusion as to the phrasing of this rule.

  • We know each creature gets a turn (1 turn) in the initiative each round; which starts being explained on PHB pg. 189.
  • We know each creature may take 1 action (Attack action, Cast a Spell action, Ready action) and possibly 1 bonus action (there are several) during that turn, with details also beginning on PHB pg. 189.
  • We know from PHB pg. 190 that you may also move an equal distance up to your speed, and can use as much or as little of your speed as you wish as well as breaking that movement in between various forms of travel (walking, climbing, jumping, etc) and that there is no ‘Move Action’.
  • We know from the next section on PHB pg. 190 that you can break up your movement in between attacks and other actions.

With what we know from the above clearly written rules, it then becomes clear that when Crawford says you can not end any part of your move (as detailed above) in a creature’s occupied space, he is referring to your Speed-based movement, and is not solely referring to ending your turn as a whole, which can include any of the above listed. Your only choice is finishing your movement in an unoccupied space near the creature in question, attack your target of choice, then move back through (if you have movement remaining) the occupied space making sure you (once again) do not end any part of your movement in that creature occupied space. Then you can end your turn. Keep in mind that each time you move through the space, it is difficult terrain, and costs you extra movement unless you have the ability to ignore difficult terrain.

With this in mind… the only relevant question you have remaining perplexes me, and I think there are only 2 outcomes.

  1. If Alice tries to move through an ally’s space, but she is stopped by a Ready-action grapple from a hostile creature near the ally’s space, then the Ready action you had in mind triggers (I am assuming?) the moment Alice is in range, interrupting her movement while she is still in her ally’s space and effectively reducing her speed to 0, as per the Grappled condition.

Her movement has just been forcefully (unwillingly) stopped in another creature’s space.

  1. Alice tries to somehow move through a hostile creature’s space but they Ready a grapple. In this case the Ready-action grapple would trigger when she got within attack range, because a grapple is always dependent on the Attack action, which you must forgo in place of a grapple. An enemy could never Ready a grapple to trigger when Alice steps into or onto their space.
  • There are no rules that apply any penalty for this outcome.
  • The space is difficult terrain anyway, so the addition of Alice to the space changes nothing.
  • We know a 5-foot ‘space’ is generally 5 feet of ‘effective creature threat’, and not a humanoid/creature 5 feet wide. So roughly 10 humanoids could stand grappled together in a 5-foot space, which is 25 square feet.
  • Stacking is another story entirely, and then you are dealing with distance/height in feet versus the creature’s speed (which becomes 0 and so couldn’t be tossed, thrown, or otherwise put on top of a grapple ‘pile’ higher than 1 foot tall).

algebraic geometry – Function Space for an Affine n-Space $ mathbb{A}^n $

the situation we study is as below:

we have an affine n-space $ mathbb{A}^n $ over a field $k$, and the Zariski topology on it (defined by taking the open subsets to be the complements of algebraic sets).

we are observing the case $ n=1 $. let $ A = k(x) $ be the function field over $ mathbb{A}^1 $. then, any $ f in A $ is of the form: $ f = c (x – r) = a_0 + a_1 x$.

but if we had $ n=2 $, what would we have as the function space? would it be $ A = k(x) $ with all $ f = c(x – r_1)(x – r_2) = a_0 + a_1 x + a_2 x^2$, or maybe $ A = k(x,y) $ with all $ f = c(x – r_1)(x – r_2) = a_0 + a_1 x^2 + a_2 x y + a_3 y^2 $ with coeficients agree with $ deg(f) leq 2 $?

What does the space $C^2([0,T]; H^2(Omega))$ mean?

Does the space $C^2((0,T); H^2(Omega))$ mean: $C^2$ in the time direction and $H^2$ in the space direction?

Thank you very much!