BC MATHS
Saved by @CodyyyGardner

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.
We also learned from wave 1 not to abandon contact tracing and to test people as early as possible to prevent transmission chains. I.e. TTI. That still holds true. Let us not repeat the mistakes.
In fact, in March we got Rt down to that level with effectively no targeted support from TTI because we were only testing on admission (way too late). And we had so little testing capacity.
In addition, in March there was approximately zero prior immunity. Now there is a non-zero amount and that helps eat into some of the increase in transmissibility.
We have light at the end of the tunnel with vaccines and can target vaccines at those who cannot isolate (essential workers) with more scope to do this the less urgently we need to vaccinate the vulnerable. This helps psychologically and practically.
Practically, for example, having vaccinated workers in LTC and vaccinated people support the elderly is a game changer for isolation.
Rapid testing is another tool we now have that could help alongside TTI and vaccines. We have lots more at our disposal to work with.
Chasing down every case is still going to become *more* not less important. It is perfectly possible that if vaccine evading strains don't arise in the UK we will import them at some time. Will need TTI then. Goes without saying that this will also not be the last pandemic.
I suspect one reason why the TTI is constantly underestimated is that it is in practice just naturally much better at finding cases in high transmission events than no/low transmission cases. And maybe this is hard to model.
For very many human reasons that are just intuitively apparent to anyone who has trained in taking a history from a patient: you have a shared agenda to get to the most important information about the most important events. And you build a shared understanding of what those are.
Perhaps the cases missed are much much more likely to be in the 80% or so of people who don't infect someone else.
For all the obsession with overdispersion of SARS2 and HIT, I actually think this is where it makes the most difference. I think its why many mathematicians seem baffled or underenthused by the East Asian response... sure it works in practice, but it makes no sense in theory.
In addition, just because something doesn't solve the problem on it's own doesn't mean it doesn't help solve the problem.
I don't mean to single out @BristOliver for this rant, so sorry for that. I find your takes on the whole very interesting so hope you don't block me. This just really bugged me on the back of a recent chat with a friend who is a public health trainee.
She has been working very hard on TTI all year and feels drained, and, I think, underappreciated. And I feel strongly that we have not championed this essential work enough. So much so that, I'm told, it is now increasingly common that when contacting people they get vile abuse.
A bit less of this: "there's no point in doing that thing that has been the backbone of every competent outbreak response for over 150 years and that WHO insists is key and that all Covid-controlling countries have invested in" would be nice. (Paraphrasing no-one in particular.)
Even if only out of respect for all the people working their asses off to get it done in incredibly difficult circumstances.

More from Maths

Artem Chernikov
Artem Chernikov
@archernikov
Saharon Shelah and number 4, a thread.

Shelah is an incredibly prolific mathematician working mostly in mathematical logic who has (co-)authored around 1500 papers so far. His other superpower is the ability to discover number 4 where it has absolutely no reason to be. 1/32


Example 1. "There are just 4 second order quantifiers".
In first-order logic, one is allowed to form statements about structures, such as graphs, groups, fields,
etc., with only quantification over elements allowed. 2/32

So you can say "exists x, ..." or "for all x, ..." with x ranging over the elements of the structure M, but you can't say "for all subsets of M, ..." or "for all binary relations on M, ...", etc. Allowing such
quantifiers puts us in the context of _second order logic_. 3/32

It was well-known in logic and computer science that one can express more complicated properties by quantifying over binary relations on M rather than just over the unary ones. But can we say even more quantifying over ternary relations? 4/32

And how does this compare to quantifying over functions, bijections, or any other infinitely many possible types of relations on M one can think of? In his 1973 paper Shelah proved that there are exactly 4 (four) possible types of second order quantifiers. 5/32
MATHS
Riuji
Riuji
@weakconvergence
When I was starting out as a grad student, I remember being frustrated with analysis problems where it didn't use DIRECTLY the things we spent so much time proving in class.

As an example, let's look at a standard real analysis problem:

(1/23)


The problem looks like it can be done using the dominated convergence theorem.

This is an important theorem and is used in a lot of places outside a grad real analysis class. A student would go through a lot of buildup leading to this theorem.

(2/23)

Now, let's see whether the conditions for DCT are satisfied.

First, for fixed 00 as n->infinity.

So we have a.e. convergence of the integrand in [0,1].

To the student, this builds a stronger case to use DCT to solve the problem.

(3/23)

Finding an integrable dominating function is where the problem starts.

It would seem NATURAL (at least for me) to bound it with:

p.s. it feels "natural" to me because it is reminiscent of squeeze theorem problems, math students encounter in calculus.

(4/23)


However, 1/x^n is not integrable in (0,1) and hence cannot be (further) dominated by an integrable function.

So, what I would do is since I know it converges a.e. to 1, maybe it converges uniformly to 1?

This leads me into a BAD detour.

(5/23)
MATHS
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T. Benjamin
T. Benjamin
@1FreeInhabitant
Common Core math was the pinnacle of their evil genius, and low-point of our complacency... It disrupted the minds of an entire generation on how even the most basic things work.

Readin', Writin', & Rithmatic

How far did they get? Let's see.

1895 8th Grade Final Exam, Kansas:
MATHS
Gro-Tsen
Gro-Tsen
@gro_tsen
OK, I may be guilty of a DoS attack attempt on mathematicians' brains here, so lest anyone waste too much precious brain time decoding this deliberately cryptic statement, let me do it for you. •1/15


First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15

Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15

Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15

… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
MATHS
Jonathan Beardsley
Jonathan Beardsley
@JBeardsleyMath
In light of my tweet thread about the category of finite sets and commutative monoids (https://t.co/jnY0wZZbxq), I thought I might try to say what the analogue is for braided monoidal things (although much of this is still somewhat hypothetical).


It's also just kind of a cool combinatorial structure! I've been talking to @CreeepyJoe about this lately, as well as @grassmannian.

The first thing you have to know is that, in a braided monoidal category you can still have commutative monoids. Since a braided monoidal category C has a "twist" map for every object β(x):x⊗x→x⊗x, if x is a monoid you can ask for the following diagram to commute:


Remember that being symmetric monoidal just means that if you take the twist map above and do it twice, you get the identity map, but braided monoidal doesn't mean that. But it's okay! You can still define commutative monoids here.

But so anyway, we can talk about commutative monoids in braided monoidal categories.
MATHS

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