In the preface to his book Strength of Materials, J. P. Den Hartog says
After the alphabet and the tables of multiplication, nothing has proved quite so useful in my professional life as these six little expressions.
The six expressions he refers to are nicknamed the vergeet-me-nietjes in Dutch, which translates to forget-me-nots in English. They are also known as Dr. Myosotis’s equations because myosotis is the genus for forget-me-nots. The equations give the angular and linear deflections of a cantilever beam.
Imagine a beam anchored at one end and free on the other, subject to one of the kinds of load: a bending moment M at the opposite end, a point force P a the opposite end, or a force w distributed over the length of the beam. The equations below give the rotation (angular deflection) and displacement (linear deflection) of the free end of the beam.
Here E is the modulus of elasticity, L is the length of the beam, and I is the area moment of inertia.
From The Book of Strange New Things:
… I said that if science could come up with something like the Jump it could surely solve a problem like that. Severin seized hold of that word, “science.” Science, he said, is not some mysterious larger-than-life force, it’s just the name we give to bright ideas that individual guys have when they’re lying in bed at night, and that if the fuel thing bothered me so much, there was nothing stopping me from having a bright idea to solve it …
This is a thumbnail version of a large, high-resolution image by Ulysse Carion. Thanks to Aleksey Shipilëv (@shipilev) for pointing it out.
It’s hard to see in the thumbnail, but the map gives the change in velocity needed at each branch point. You can find the full 2239 x 2725 pixel image here or click on the thumbnail above.
My interest in the Anil Potti scandal started when my former colleagues could not reproduce the analysis in one of Potti’s papers. (Actually, they did reproduce the analysis, at great effort, in the sense of forensically determining the erroneous steps that were carried out.) Two years ago, the story was on 60 Minutes. The straw that broke the camel’s back was not bad science but résumé padding.
It looks like the story is a matter of fraud rather than sloppiness. This is unfortunate because sloppiness is much more pervasive than fraud, and this could have made a great case study of bad analysis. However, one could look at it as a case study in how good analysis (by the folks at MD Anderson) can uncover fraud.
Now there’s a new development in the Potti saga. The latest issue of The Cancer Letter contains letters by whistle-blower Bradford Perez who warned officials at Duke about problems with Potti’s research.
From Zero to One:
Eroom’s law — that’s Moore’s law backward — observes that the number of new drugs approved per billion dollars spent on R&D has halved every nine years since 1950.
Update: Here’s an article from Nature that gives more details. The trend is pretty flat on a log scale, i.e. exponentially declining efficiency.
Seth Juarez quipped in an interview that when people say they’ve got something “down to a science,” they probably don’t mean what they’re saying. Science is making guesses and testing to see whether they’re right.
Related post: Take chances, make mistakes, and get messy
Here’s a totally impractical but fun back-of-the-envelope calculation from Bob Martin.
Suppose you have a space ship that could accelerate at 1 g for as long as you like. Inside the ship you would feel the same gravity as on earth. You could travel wherever you like by accelerating at 1 g for the first half of the flight then reversing acceleration for the second half of the flight. This approach could take you to Mars in three days.
If you could accelerate at 1 g for a year you could reach the speed of light, and travel half a light year. So you could reverse your acceleration and reach a destination a light year away in two years. But this ignores relativity. Once you’re traveling at near the speed of light, time practically stops for you, so you could keep going as far as you like without taking any more time from your perspective. So you could travel anywhere in the universe in two years!
Of course there are a few problems. We have no way to sustain such acceleration. Or to build a ship that could sustain an impact with a spec of dust when traveling at relativistic speed. And the calculation ignores relativity until it throws it in at the end. Still, it’s fun to think about.
Update: Dan Piponi gives a calculation on G+ that addresses the last of the problems I mentioned above, sticking relativity on to the end of a classical calculation. He does a proper relativistic calculation from the beginning.
If you take the radius of the observable universe to be 45 billion light years, then I think you need about 12.5 g to get anywhere in it in 2 years. (Both those quantities as measured in the frame of reference of the traveler.)
If you travel at constant acceleration a for time t then the distance covered is c^2/a (cosh(a t/c) – 1) (Note that gives the usual a t^2/2 for small t.)
I’ve started a new Twitter account @UnitFact for tweets about units of measurement, constants, dimensional analysis, etc.
Cancer research is sometimes criticized for being timid. Drug companies run enormous trials looking for small improvements. Critics say they should run smaller trials and more of them.
Which side is correct depends on what’s out there waiting to be discovered, which of course we don’t know. We can only guess. Timid research is rational if you believe there are only marginal improvements that are likely to be discovered.
Sample size increases quickly as the size of the effect you’re trying to find decreases. To establish small differences in effect, you need very large trials.
If you think there are only small improvements on the status quo available to explore, you’ll explore each of the possibilities very carefully. On the other hand, if you think there’s a miracle drug in the pipeline waiting to be discovered, you’ll be willing to risk falsely rejecting small improvements along the way in order to get to the big improvement.
Suppose there are 500 drugs waiting to be tested. All of these are only 10% effective except for one that is 100% effective. You could quickly find the winner by giving each candidate to one patient. For every drug whose patient responded, repeat the process until only one drug is left. One strike and you’re out. You’re likely to find the winner in three rounds, treating fewer than 600 patients. But if all the drugs are 10% effective except one that’s 11% effective, you’d need hundreds of trials with thousands of patients each.
The best research strategy depends on what you believe is out there to be found. People who know nothing about cancer often believe we could find a cure soon if we just spend a little more money on research. Experts are more sanguine, except when they’re asking for money.
John Ioannidis stirred up a healthy debate when he published Why Most Published Research Findings Are False. Unfortunately, most of the discussion has been over whether the word “most” is correct, i.e. whether the proportion of false results is more or less than 50 percent. At least there is more awareness that some published results are false and that it would be good to have some estimate of the proportion.
However, a more fundamental point has been lost. At the core of Ioannidis’ paper is the assertion that the proportion of true hypotheses under investigation matters. In terms of Bayes’ theorem, the posterior probability of a result being correct depends on the prior probability of the result being correct. This prior probability is vitally important, and it varies from field to field.
In a field where it is hard to come up with good hypotheses to investigate, most researchers will be testing false hypotheses, and most of their positive results will be coincidences. In another field where people have a good idea what ought to be true before doing an experiment, most researchers will be testing true hypotheses and most positive results will be correct.
For example, it’s very difficult to come up with a better cancer treatment. Drugs that kill cancer in a petri dish or in animal models usually don’t work in humans. One reason is that these drugs may cause too much collateral damage to healthy tissue. Another reason is that treating human tumors is more complex than treating artificially induced tumors in lab animals. Of all cancer treatments that appear to be an improvement in early trials, very few end up receiving regulatory approval and changing clinical practice.
A greater proportion of physics hypotheses are correct because physics has powerful theories to guide the selection of experiments. Experimental physics often succeeds because it has good support from theoretical physics. Cancer research is more empirical because there is little reliable predictive theory. This means that a published result in physics is more likely to be true than a published result in oncology.
Whether “most” published results are false depends on context. The proportion of false results varies across fields. It is high in some areas and low in others.
“Progress in science depends on new techniques, new discoveries, and new ideas, probably in that order.” — Sidney Brenner
I’m not sure whether I agree with Brenner’s quote, but I find it interesting. You could argue that techniques are most important because they have the most leverage. A new technique may lead to many new discoveries and new ideas.
This tweet from Luis Pedro Coelho says so much in 140 characters:
“Oh, the intellectual freedom of academia” he thought while filling out a time sheet which checks that he does not work on non-grant science.
From The Philosophical Breakfast Club
When Coleridge, the most famous poet of the day, wrote his tract on scientific method in 1817 it was not considered an oddity; by 1833, the time of the third meeting of the British Association for the Advancement of Science, it was already remarkable, and in the years that followed it was almost inconceivable.
Related post: How the term “scientist” came to be
Interesting perspective on information theory:
To me, the subject of “information theory” is badly named. That discipline is devoted to finding ideal compression schemes for messages to be sent quickly and accurately across a noisy channel. It deliberately does not pay any attention to what the messages mean. To my mind this should be called compression theory or redundancy theory. Information is inherently meaningful—that is its purpose—any theory that is unconcerned with the meaning is not really studying information per se. The people who decide on speed limits for roads and highways may care about human health, but a study limited to deciding ideal speed limits should not be called “human health theory”.
Despite what was said above, Information theory has been extremely important in a diverse array of fields, including computer science but also in neuroscience and physics. I’m not trying to denigrate the field; I am only frustrated with its name.
From David Spivak, footnotes 13 and 14 here.
I’ve enjoyed reading The New York Times Book of Physics and Astronomy, ISBN 1402793200, a collection of 129 articles written between 1888 and 2012. Its been much more interesting than its mathematical predecessor. I’m not objective — I have more to learn from a book on physics and astronomy than a book on math — but I think other readers might also find this new book more interesting.
I was surprised by the articles on the bombing of Hiroshima and Nagasaki. New York Times reporter William Lawrence was allowed to go on the mission over Nagasaki. He was not on the plane that dropped the bomb, but was in one of the other B-29 Superfortresses that were part of the mission. Lawrence’s story was published September 9, 1945, exactly one month later. Lawrence was also allowed to tour the ruins of Hiroshima. His article on the experience was published September 5, 1945. I was surprised how candid these articles were and how quickly they were published. Apparently military secrecy evaporated rapidly once WWII was over.
Another thing that surprised me was that some stories were newsworthy more recently than I would have thought. I suppose I underestimated how long it took to work out the consequences of a major discovery. I think we’re also biased to think that whatever we learned as children must have been known for generations, even though the dust may have only settled shortly before we were born.