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Loops and their multiplication groups
A thread in 15 parts
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A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
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- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples previously:
https://t.co/q5LjmxHEIF
https://t.co/UPHSMwQo75
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Rethinking Vector Addition
— Michael Kinyon (@ProfKinyon) December 1, 2020
or
How I Learned to Stop Worrying and Love Nonassociativity
A thread in 29 tweets
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Now back to the general case where Q is any loop.
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Loops can be studied via their multiplication and inner mapping groups. I will give one example of how this works.
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1. N is the kernel of a homomorphism,
2. N is invariant under Inn(Q),
3. N is a block of Mlt(Q).
Such an N is called a normal subloop of Q. Q is simple if it has no nontrivial normal subloops.
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Theorem (Albert 1941): A loop Q is simple if and only if Mlt(Q) is primitive.
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More from Maths
It is trying when mathematicians declare condescendingly that there is no point doing things because their models tell them so. Well maybe some of the assumptions don't hold up. How did that work out for the no additional risk from large events and no point in border controls...
During wave 1 cases fell very fast, faster than I think most people were expecting. Particularly in Scotland. Rt was probably ~0.5 until we started easing off.
This was despite a constant leak of cases coming out of hospitals and LTC facilities as we were rationing PPE and are policies were nowhere near ideal. There was insistence from infection control that droplet protections were sufficient. We have all learned a lot since then.
Not to mention we have learned to avoid the shit show of actively importing cases into care homes. We've learned not to repeat that. Other sectors have learned too.
We've learned a lot and there's no reason we can't control this new variant. But we will not manage if we don't try and act with clarity of purpose.
Oh for crying out loud. I don't know anyone who thinks we can get R below 0.9 with this new variant. It's 22 virus generations to even get from 50,000 cases to 5,000 at R=0.9 - that's 4 months. TTI is a complete fantasy right now: spend the money on the vaccine rollout. https://t.co/MyeBt8tC1w
— Oliver Johnson (@BristOliver) January 3, 2021
During wave 1 cases fell very fast, faster than I think most people were expecting. Particularly in Scotland. Rt was probably ~0.5 until we started easing off.
This was despite a constant leak of cases coming out of hospitals and LTC facilities as we were rationing PPE and are policies were nowhere near ideal. There was insistence from infection control that droplet protections were sufficient. We have all learned a lot since then.
Not to mention we have learned to avoid the shit show of actively importing cases into care homes. We've learned not to repeat that. Other sectors have learned too.
We've learned a lot and there's no reason we can't control this new variant. But we will not manage if we don't try and act with clarity of purpose.
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The entire discussion around Facebook’s disclosures of what happened in 2016 is very frustrating. No exec stopped any investigations, but there were a lot of heated discussions about what to publish and when.
In the spring and summer of 2016, as reported by the Times, activity we traced to GRU was reported to the FBI. This was the standard model of interaction companies used for nation-state attacks against likely US targeted.
In the Spring of 2017, after a deep dive into the Fake News phenomena, the security team wanted to publish an update that covered what we had learned. At this point, we didn’t have any advertising content or the big IRA cluster, but we did know about the GRU model.
This report when through dozens of edits as different equities were represented. I did not have any meetings with Sheryl on the paper, but I can’t speak to whether she was in the loop with my higher-ups.
In the end, the difficult question of attribution was settled by us pointing to the DNI report instead of saying Russia or GRU directly. In my pre-briefs with members of Congress, I made it clear that we believed this action was GRU.
The story doesn\u2019t say you were told not to... it says you did so without approval and they tried to obfuscate what you found. Is that true?
— Sarah Frier (@sarahfrier) November 15, 2018
In the spring and summer of 2016, as reported by the Times, activity we traced to GRU was reported to the FBI. This was the standard model of interaction companies used for nation-state attacks against likely US targeted.
In the Spring of 2017, after a deep dive into the Fake News phenomena, the security team wanted to publish an update that covered what we had learned. At this point, we didn’t have any advertising content or the big IRA cluster, but we did know about the GRU model.
This report when through dozens of edits as different equities were represented. I did not have any meetings with Sheryl on the paper, but I can’t speak to whether she was in the loop with my higher-ups.
In the end, the difficult question of attribution was settled by us pointing to the DNI report instead of saying Russia or GRU directly. In my pre-briefs with members of Congress, I made it clear that we believed this action was GRU.