uk – Is the BIVS still in effect during COVID times?

In the course of the Covid pandemic the Irish Short Stay Visa Waiver Programme has been suspended for an effectively indefinite amount of time. While pages on the SSVWP show a warning that it is on hold, BIVS explanations only state that

The British Irish Visa Scheme will build on Ireland’s Short-stay Visa Waiver Programme

but provide no further info on whether it is also suspended due to suspension of the SSVWP.

While I would assume that to be the case, is there any more definitive/legally binding source that can confirm or refute this assumption?

linear algebra – Determine the rank of $AB$, given matrices $A$ which is $m times n$ and $B$ which is $n times $p$ – proof assistence

Let $A$ be an $m times n$ matrix with rank $m$ and $B$ be an $n times p$ matrix with rank $n$. Determine the rank of $AB$. Justify your answer.

So to begin after doing some specialized examples I worked out that the rank of the matrix $AB$ will be $m$. Now I’m having difficulties proving it.

First off in the textbook I’m using (Linear Algebra, Friedberg, Insel, et al), the definition of the rank is the following:

If $A in M_{m times n}(mathbb{F})$, we define the rank of $A$, denoted $rank(A)$, to be the rank of the linear transformation $L_{A}:mathbb{F}^{n} to mathbb{F}^{m}$, where $L_{A}$ is the left multiplication transformation, $L_{A}(x) = Ax$ for $x in mathbb{F}^{n}$. And as a result $L_{B}: mathbb{F}^{m} to mathbb{F}^{p}$ is the left multiplication transformation, $L_{B}(x) = Bx$ for $x in mathbb{F}^{m}$

Since I deduced that the rank of $AB = m$, that means I have to show that $rank(AB) = rank (A)$. THis means:

$$rank(L_{AB}) = rank(L_{A}) \ Rightarrow dim(R(L_{AB})) = dim(R(L_{A})),\ text{where $R$ is the range of a linear transformation}$$

So it means I have to show that:

$$dim(R(L_{A}L_{B})) = m$$

So form here to me I deduced that somehow I have to show that $R(L_{A}L_{B}) = R(L_{A})$

And this is where I’m stuck. There was a proof I saw that used the idea of $L_{B}(mathbb{F^{p}}) = mathbb{F^{n}}$ but that seemed very much off because for that to be the case, it must be that $L_{B}$ is invertible, which it may not necessarily be. Perhaps my approach is wrong, but I feel I’m close to what I need to accomplish but can’t seem to see what last steps I need to take. Some assistance would be nice.

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general topology – Yes/No : Is the subspace $A times { b }$ of $A times B$ is homeomorphic to $A$?

Yes/No : Is the subspace $A times { b }$ of $A times B$ is homeomorphic to $A$?

My attempt : No

Here if we remove one point from $A times { b }$ then $(A times { b }) setminus{1}$ is connected since $mathbb{R}^n$ is connected if $n >1$

But if we remove one point from $A$ then $A setminus {1}$ is not connected

so $A times { b } notcong A$

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ux designer – Convey the ability to use an input many times

My site is a legal consultation platform. I am offering an option for visitors to clap answers they like, and they can clap up to 10 times.
Example of the page here: Mahkamati. Please go and try to clap the answer.

How can I convey to the users that they can clap up to 10 times in a user-friendly way?
Thank you.

dnd 5e – How many times do you roll damage for Magic Missile?

The “Damage Rolls” section of the rules (PHB p. 196, also in the online Basic Rules) says this:

If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell’s damage is rolled once for all creatures caught in the blast.

The Magic Missile spell (PHB p. 257, also in the online Basic Rules) has this effect:

You create three glowing darts of magical force. Each dart hits a creature of your choice that you can see within range. A dart deals 1d4 + 1 force damage to its target. The darts all strike simultaneously, and you can direct them to hit one creature or several.

  1. If I target three different creatures with the darts, the spell says quite clearly says that the darts strike at the same time and the Damage Rolls rule says that in that case (as multiple targets are being dealt damage at the same time) I roll only once for all of them. So my reading of that is that I roll my four-sided die just once, add 1, and apply that much damage to each target. Is that correct? It seems the straightforward reading, but then the next scenarios make me unsure.
  2. If I target just one creature with all three darts, then it seems that the “more than one target” clause of the Damage Rolls would no longer apply. So, does that mean that I effectively roll 3d4+3 (rolling my four-sided die three separate times, one for each dart), since there’s no longer anything saying to use the same roll for each dart?
  3. And how about if I target creature A with one dart, and creature B with two darts? Per the Damage Rolls section I should roll once for both of them, but that doesn’t make any sense they should be taking different amounts of damage, right?

I’m sure I’m just reading things too literally, but it all seems rather contradictory to me, and now I’m unsure how many dice should be rolled for Magic Missile in any scenario. Is there further direction somewhere I’m missing? (I currently only have the PHB, the Starter Box, and the online Basic Rules, so maybe there’s more guidance in the other books? I wouldn’t think so, though.)

List of Times and Dates a Specific App was Accessed on my Mac

I would like to know if it is possible to use the terminal in order to get info regarding when a specific app was accessed/closed (history of the app usage). I’m pretty sure that my mac stores this type of info (it uses it for screen time… but screen time only gives you the total time, not the detail regarding when the app was accessed…) but I’m not sure how to access it. Thanks!!