gn.general topology – Statements related to Thurston’s work on the surface

If we have α and β be simple closed curves on a surface Σg. The intersection number i(α,β) is defined to be the minimal cardinality of α1 ∩ β1 as α1 and β1 ranges over all simple closed curves isotopic to α and β, respectively. We say α and β intersect minimally if i(α, β) = |α ∩ β|.

How to see thatα and β intersect minimally if there are no pairs of p, q ∈ α ∩ β such that the arc joining p to q along α followed by the arc from q back to p along β bounds a disk in Σg.

I think the converse is also true : “thatα and β intersect minimally only if there are no pairs of p, q ∈ α ∩ β such that the arc joining p to q along α followed by the arc from q back to p along β bounds a disk in Σg.”

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views – Format files related to node

I used Drupal a lot in my old days but now I’ve been like 10 years without using it and using it again.
What I want to do is to show the files that are related to a node of a specific content type in the format I want, that is all the files related to the node in a table at the bottom and with description and upload date. These files are in a unlimited file field.

The table format that the presentation of the content type lets me choose is not good for me, it only displays name and file size.
Is there a a way to specify what you want the table to show? I haven’t found any way.

Then two options have come to mind: make my own template and make a view block.

To make my own template I need to learn how these twig templates work now, and I would like to know where to find the default template for this table, so I can copy it and add the columns I want.

Fot the views approach I haven’t been able to get a successful relation that only shows the files attached to the node I am in.

I would appreciate help with any of these solutions, and in case it is possible to solve it in many ways, which one would be the best?

Big Sur update related [duplicate]

It is showing as compatible for this MacBook Pro but when I try to download the error message pops out that requested version of Mac OS is not available. then it checks for updates and shows the message as Your Mac is up to date.

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Please shrae good resource related to API Pentesting?

I have been eagerly learning about web security from past 2 months and move further to to API pentesting .But I am not getting any good resources when asking GOOGLE.IT would be great help if someone shares good resources about it.
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Are colored graphs and red-black trees related?

I’ve come across the concepts of colored graphs (register allocation) and red-black trees. They both seem to have this notion of "coloring", but I’ve never seen them being connected conceptually.

Do they have anything to do with each other?

mysql – How to design table to allow marking records as related to each other?

My web app’s main MySQL table has tens of thousands of rows. Pretend it has 2 columns: ID, Content.

I’m trying to design a new feature that allows rows to be marked as related to each other.

Obviously I could add a 3rd column: Related_IDs. It could either be a JSON array or could be a simple CSV string.

But that feels wrong. I’d need to update the Related_IDs field of each row of a group of N rows to be a comma-separated list of IDs of all of the rows in the group except itself. Every time any one of those records becomes unrelated, I’d need to iterate through the group and clean up that field for each.

What is a smarter approach?