Bit of a maths puzzle.

Ugh! This comes out nicely on my blog but looks like crap in Google reader, presumably because the javascript can't run.
Define a function:
\[ f(x) = \frac{2x}{1 - x^2} \] In case that maths layout thing isn't working, that's f(x) = 2x/(1-x^2)
Now, \[ f(0) = 0 \] so that repeats. Also \[ f(\sqrt 3) = -\sqrt 3 \] and \[ f(\sqrt -3) = \sqrt 3 \] so that repeats too.
The question is, what values repeat if you keep applying f to the result? I thought I knew the answer but now I think maybe it's more tricky.

The medical miracle of checklists.

From this New Yorker article

Michigan’s infection rates fell so low that its average I.C.U. outperformed ninety per cent of I.C.U.s nationwide. In the Keystone Initiative’s first eighteen months, the hospitals saved an estimated hundred and seventy-five million dollars in costs and more than fifteen hundred lives. The successes have been sustained for almost four years—all because of a stupid little checklist.
Pronovost and his colleagues monitored what happened for a year afterward. The results were so dramatic that they weren’t sure whether to believe them: the ten-day line-infection rate went from eleven per cent to zero. So they followed patients for fifteen more months. Only two line infections occurred during the entire period. They calculated that, in this one hospital, the checklist had prevented forty-three infections and eight deaths, and saved two million dollars in costs.

I work as part of a team that helps keep several giant websites up and running. An error by any of us can result in millions of users seeing error pages. That's not to say that every error is that dangerous, there are plenty of backups, fail-safes and checks but there are still opportunities for massive accidental destruction. Checklists and documentation are essential for us, although we prefer to produce software that removes the need for manual following of steps where possible. We aim to leave humans dealing with the big picture. We also want to free up our brains to make the best decisions we can during a crisis. When the site is melting, the last thing we want to be doing is to try to follow a procedure we haven't thought about for six months. With more to think about, we are more likely to make poor decisions and more likely to get the procedure wrong.

Most medical care is a long way from being automatable and in lots of cases human judgement is critical but it seems also that lots of common procedures are well understood and have well known procedures for minimsing risk. Errors in these are more far more common than you'd hope and can result in pain, disability and even death. For those interested in the bottom line (which is everyone in Ireland these days), all of the above lead to increased expense, usually for the tax payer. In the case of a 5 step procedure for inserting a sterile line the article says that "In more than a third of patients, they skipped at least one [step]." and when they introduced mandatory checklists the infection rate went from 11% to 0%, saving lives, huge amounts of money and freeing up staff and beds to look after new patients. Before the checklist, these staff would instead have been treating those who became even sicker because of an infection.

It's also important to note that when freed from thinking about the minor procedures, staff can apply better judgement about the big picture. They can also attempt more complex procedures knowing that all their effort won't be wasted just because 1 of the 10 sub-procedures is going to be fluffed and end up doing more harm than good. When you discuss a procedure with a doctor they explain it all and tell you the odds of success, failure, injury and death and you give informed consent or decide that it's not worth the risk. Those odds are based on previous cases and they are not just the odds that the specific cut or splice that the surgeon wants to make will work. They include complications that arose from all the other procedures involved, anaesthetics, blood transfusions, keeping you sterile, not leaving equipment inside you, stitching you back together etc. If you could guarantee that all of those other procedures would go perfectly, that could change the overall odds quite a bit. Enough that procedures that were previously high risk, high reward would now be much lower risk and well worth opting for.

Every now and then, someone in work quips, "it's only a website" but really we take it very seriously and do everything we can to improve the outcome, learn from previous mistakes and remove any element of chance. Most people would be horrified to hear a doctor say "it's only a patient". With checklists we have a simple technique that is uncontroversial in my line of work, that has a big impact on success or failure and that has now been proven to have the same impact in medicine. I'm horrified to find out that it has been adopted widely and that its introduction can meet significant resistance.

I guess I shouldn't be too surprised though, anaesthesiology has had a lot of engineering know-how applied to it as well as standardised user interfaces on their machinery with huge improvement in outcomes but that was also met with great resistance (I don't have a link for that, I read about it in a book once, so it must be true.)

The original article is from 2007, here's another from 2010 on the spread of this checklist since then. Here's another report on a similar checklist for general surgery.

I would love to know if checklists are being adopted in Ireland. It seems like a no-brainer. I will ask the next medical professional I meet.