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## Detecting when a single group of blocks is broken into multiple groups

Let’s say you have an array of blocks (AABB), where each block is touching or intersecting any number of other blocks. Together they make up a larger object, and because they’re all touching, you can consider them to all be on the same “grid”. Each block can have any size or position, but is always axis-aligned.

Now let’s say you delete one of the blocks, somewhere in the middle, such that there is now a gap where blocks are no longer touching.

How would you go about detecting that you now have two clusters of blocks, two “grids”, instead of one?

My first thought is to loop through all blocks, perform AABB collision checks against all other blocks, mark the ones that touch, then recursively cycle through the touched blocks looking for others that touch, until you reach the end and can look for blocks that were not marked. But that sounds extremely inefficient.

This is in plain C, so I’m trying to keep memory use limited to the existing array of block structs.

## RDS Aurora Serverless “Parameter Groups”

I have an RDS Aurora Serverless MySQL cluster, and I am trying to change a MySQL setting (connect_timeout). Normally, you would use a Parameter Group to set the value on the DB instance. But, since this is serverless, the instances are all managed by AWS, so it seems I can only configure the cluster.

Is there a way to set the Parameter Group that is used by the instances that AWS creates?

## Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?

Given a morphism of Lie groups $$\theta:G\rightarrow H$$  and a principal $$G$$ bundle $$\pi:P\rightarrow M$$ there are (at least) two ways to assign a principal $$H$$ bundle.

1. See that the morphism of Lie groups $$\theta:G\rightarrow H$$ gives an action of $$G$$ on $$H$$ by $$g.h=\theta(g).h$$. Given an action of $$G$$ on manifold (Lie group in this case) $$H$$ there is an associated fibre bundle $$P\times_G H\rightarrow M$$ with fibre $$H$$. This gives a principal $$H$$ bundle.
2. For principal bundle $$\pi:P\rightarrow M$$, we can find an open cover $$\{U_\alpha\}$$ of $$M$$ and  (transition) maps $$g_\alpha g_\beta:U_{\alpha\beta}\rightarrow G$$ satifsying the cocycle condition $$g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$$ on $$U_\alpha\cap U_\beta\cap U_\gamma$$. Then the compositions $$\tau_{\alpha\beta}=\theta\circ g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G\rightarrow H$$ also satifies the cocycle condition $$\tau_{\alpha\beta}\tau_{\beta\gamma}=\tau_{\alpha\gamma}$$ on $$U_\alpha\cap U_\beta\cap U_\gamma$$. One can then produce a principal $$H$$ bundle over $$M$$ given this open cover $$\{U_\alpha\}$$ of $$M$$ and smooth maps $$\tau_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow H$$ satisfying the cocycle condition. This gives a principal $$H$$ bundle.

It is a good exercise (that I have not tried) to check that principal $$H$$ bundles obtained from above two methods are (naturally) isomorphic.

Given a Lie group $$G$$, let $$BG$$ denote the category of principal $$G$$ bundles. Objects are principal $$G$$ bundles and morphisms are $$G$$-equivariant morphisms.

Given a morphism of Lie groups $$\theta:G\rightarrow H$$, above construction gives a functor (at the level of objects) $$B\theta:BG\rightarrow BH$$. It is not difficult to see that, a $$G$$-equivarint map induce a $$H$$-equivariant map. This gives a functor.

I am trying to understand what can we say about $$\theta:G\rightarrow H$$ if we know that $$B\theta:BG\rightarrow BH$$ is an equivalence of categories? Does it have to be a diffeomorphism? Any comments are welcome.

## Generating rotating groups for a seminar

One of my teachers is planning a seminar for his English class and he asked me if there was a way to generate the groups for the days other than brute-force random generating. I really think there should be a way, but I can’t think of something.

Here is the situation:

• There are 18 kids and 22 days
• Each seminar has a group of 12 and a group of 6
• Goals:
• Maximize the variety of students in groups (i.e. that aren’t always in the same or nearly same group of people)
• Ensure all students are in both groups about the same amount of times

I was hoping to find a way of solving the problem instead of find just a solution. Basically I have two questions:

1. Does anyone know how this can be done?
2. Does anyone have any resources/similar problems/methods that I can look at that might help me solve this?

## Load balancing Availability groups with MSSQL Standard

So i have the current scenario, and it looks to be working fantastically, but i just want to get some input on the configuration. Is it smart? Are there any issues i am not thinking of?

We have MSSQL Standard, and as such, you can only have 1 DB per AG (we have 20 databases), and no Read-Only secondaries. Basically meaning, the primary server is doing ALL the lifting, with the secondary doing alot less. So essentially, you are paying for resources for Node 2, that are sitting at 10% workload, while node 1 is at 70/80% workload. Both nodes are fully licensed with regards to MSSQL Cores.

What i have done, to assist this, is slit the database primaries up. So about 50% of the databases are Primary on node1, while the other 50% are primary on node2.

The Results :

The applications all connect great to either node via their respective listener If a failover occurs, just the databases on the failing node are effected, and failover to the other node (We have tested this fairly well).

Each node, can now split the load, essentially load balancing. It is a manual process to set it up this way and when deploying new DB’s and groups they go to the lighter node. but a small price to pay for “more” hardware punch without much cost (Licenses which we already have and a bit of admin)

What are your guys thoughts on this?

## Topological groups containing the Sorgenfrey line

The Sorgenfrey line $$\mathbb S$$ is the real line endowed with the topology generated by the base consisting of all half-intervals $$[a,b)$$ for real numbers $$a.

The Sorgenfrey line is first-countable and non-metrizable and hence is not homeomorphic to a topological group.

On the other hand, the Sorgenfrey line $$\mathbb S$$ is homeomorphic to a subset of a topological group. For example, the free topological group $$F(\mathbb S)$$ over $$\mathbb S$$ contains a closed topological copy of $$\mathbb S$$. But $$F(\mathbb S)$$ also contains a topological copy of the square $$\mathbb S\times\mathbb S$$ and hence $$F(\mathbb S)$$ contains an uncountable discrete subspace. Is this situation typical?

Problem. Let $$G$$ be a topological group containing a topological copy of the Sorgenfrey line. Does $$G$$ necessarily contain a uncountable discrete subspace?

## Fixed points of the automorphisms of sporadic groups

Sporadic groups have very few outer automorphisms (in fact, $$|\mathrm{Out}(G)|\leqslant2$$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable outer automorphism) one can get a twisted group or the same group over a subfield, but what about sporadic groups? I consulted several sourced but could not find the answer (or, perhaps, I am just not fluent enough in the language of the Atlas).

## Working groups, 2 different approaches

I am building a working group module for a CRM. The goal of the module is allowing user to share anything under a specific domain.

## With working group module users can create:

• A sales team
• A projet team
• An organization (e.g people who run a specific campaign)
• A contract team (imagine that you must prepare a proposal for your client in a certain time and you create a temporary team for this task)
• A discussion team (e.g brainstorming)
• Or a department (this is the thing that ruins everything)

## Authorization of group:

• Can be public
• Can be closed
• Can be isolated

## Relations:

Group module must be adaptable for following modules:

• Document management system (Pelase think about authorized user).
• Other modules (leads, branches, email, contacts, etc.) which require authorized user for using them.

## I would like to choose one of approaches I indicated below.

1. Grouping module just like facebook

No pain for integrations of modules and their authorization. In this case each group has it’s own authorization system. But in this case I have to design a different module for departments of business. Thus, the question appears in my mind is why should I separate departments and other teams or organizations. They both seems like similar operations. (Sales(a department), Sales of “ABC product”… I want to allow users to share for “sub-departments” too)

2. Using CRM in a workspace

Creating departments or teams or organizations initially. Then choosing one of them which logged in user get involved. In this case all other modules (leads, sales, contacts etc.) and a complex authorization system must be integrated well with this workspace logic.

Which one seems good to you? And why?

## Different ways to split 8 people into at least two non-empty groups

In how many different ways can you split 8 people into at least two non-empty groups?

My attempt:
Since Bell number of 8 elements set is 4140 and we don’t need whole set in one partition, is the answer: 4140 – 1?