Critical Wind Speed

Critical wind speed (~94 mph) for tree breakage

Data from storms suggest that the critical wind speed at which trees break is constant (≃42m/s), regardless of tree characteristics.

The research is summarized in this focus article.

Essentially, the authors performed experiments, and observed that wood breaks at a critical curvature radius, which depends on the diameter D and length L of the rod. Generally the relation between the diameter and length of a tree follows a \(D \sim L^{1.5}\) relation. Using a relatively simple mechanical model, and experimental data, and this relation between D and L, they found that the critical wind velocity increases very weakly with length, \(V \sim L^{1/8}\).

The speed does not depend on the elastic modulus of the wood, which is consistent with data showing that hardwoods (e.g., oaks) are just as wind susceptible as softwoods (e.g., pines).

Life Lessons From MCMC simulations

1. If your attempts are too successful, perhaps you are playing it too safe

2. Don’t fear rejection; in the long run, it is your friend!

3. In theory, it doesn’t matter where you start; if you are persistent, and persevere long enough, you will get to where you need to be.

4. Don't burn out before you burn in.

5. If you are lucky to begin from a good starting point, you are more likely to reach your goals.

6. Luck matters!

7. When the odds are good, the longer you stay in the game, the more it works for you. Ask Charles Darwin or Warren Buffett.

8. Hard problems become simpler, when you mix things up or adjust the thermostat.

9. Sometimes it feels like you’ve arrived. Before you celebrate, remember to check if you’ve arrived at the correct destination.

10. Knowing the answer is important; knowing how much to trust it is more important.

Janus Words

“Life after xyz is all downhill.”

Does that mean things get easier as in riding a bike downhill? Or does it mean things deteriorate and head south, perhaps like a stock chart?

Words like these that have somewhat opposite connotation, are amusing and frustrating in equal parts. Here are some interesting collections of such words.

I was at a conference last month, and in the polymer simulation talks I went to, I noticed that the word “explicit” could mean a calculation was easy or hard depending on the noun it modified.

In explicit solvent simulations of proteins or surfactants in water, for example, we explicitly resolve all the water molecules. Typically, the number of water molecules is very large, and this drives up the computational cost. This makes “explicit solvent” calculations more time consuming, than implicit solvent simulations, which model the effect of the solvent indirectly.

The reward for the extra effort is generally greater accuracy.

In the numerical solution of a differential equation (like Newton’s law), an explicit method, marches in time by using quantities that are known from the past. The canonical example is Euler’s famous method. In contrast, implicit methods, involve solving systems of equations at each time step, since the future value of a quantity is not expressed purely in terms of quantities known from the past. Implicit methods are typically more demanding to set up and execute than explicit methods of comparable accuracy.

The reward for the extra work is generally enhanced numerical stability.

Math Links

1. Marvin Minsky on "What makes mathematics hard to learn?"

There is a popular idea that, in order to understand something well, it is best to begin by getting things right—because then you'll never make any mistakes.  We tend to think of knowledge in positive terms—and of experts as people who know exactly what to do.  But one could argue that much of an expert’s competence stems from having learned to avoid the most common bugs.

2. A Poker-based primer on p-hacking

It is easy to get impressive results if you are selective about what you tell people. If you have two groups of people who are equivalent to one another, and you compare them on just one variable, then the chance that you will get a spurious 'significant' difference (p < .05)  is 1 in 20. But with eight variables, the chance of a false positive 'significant' difference on any one variable is 1-.95^8, i.e. 1 in 3.

3. A readable assessment of roundoff errors in floating point computation (pdf)

Error-analysis attracts few students and affords fewer career paths. Therefore almost all users and programmers of floating-point computations require help not so much to perform error-analyses (they won’t) as to determine whether roundoff is the cause of their distress, and where.

Gerschgorin's Circle Theorem

Let \(\mathbf{A}\) be an \(n \times n\) matrix.

Define the disks (i.e., circles) in the complex plane, for \(i = 1, 2,...,n\), by
\[\mathcal{D}_i = \left\lbrace z : |a_{ii} - z | \leq \sum_{j \neq i} |a_{ij}| \right\rbrace\] Then all the eigenvalues of \(\mathbf{A}\) lie in the union of the disks
\[\bigcup_{i=1}^n \mathcal{D}_i\] Moreover, if \(k\) disks are disjoint then there are exactly \(k\) eigenvalues lying in the union of these \(k\) disks.

For a diagonal matrix, these discs coincide with the spectrum.

Example

The python program attached below can be used to visualize the Gerschgorin discs \(\mathcal{D}_i\) and the actual eigenvalues on a complex plane.

A = np.matrix('5. 3. 2.;4., 6., 5.; -3., 1., 4.')
demoGerschgorin(A)

produces the plot:

The discs are centered around the diagonal elements (5,0), (6, 0), and (4, 0) with radius 5, 9, and 4 respectively. Since the matrix is real, the centers all line up on the real axis.

The eigenvalues of this matrix are approximately 0.92,  7.04+0.70j, and 7.04-0.70j, which are shown by the blue, green, and red dots respectively.

Python Program

Controlling Footnote Timing in Beamer Presentations

If you primarily use the \pause command to control the when objects appear in a Beamer presentation, you may have encountered some problems where footnotes appear before they are supposed.

A minimal working example:

\begin{itemize}

 \item Mary had a little lamb

 \pause

 \item Humpty Dumpty sat on a wall\footnote{be careful when you sit on walls}

 \pause

 \item ABCDEFG

\end{itemize}

The footnote associated with bullet-point 2 did not wait for the second bullet point to appear. It was present from the very first slide. The \only tag provides an easy way out.

\begin{itemize}

 \item Mary had a little lamb

 \pause

 \item Humpty Dumpty sat on a wall\only<2->\footnote{be careful when   you sit on walls}

 \pause

 \item ABCDEFG

\end{itemize}

Parenting Links

1. An enjoyable, and well-sourced trifecta on why parents matter; parenting - not so much! (Quillete.com: part1, part2, and part3)

Natural selection has wired into us a sense of attachment for our offspring. There is no need to graft on beliefs about “the power of parenting” in order to justify our instinct that being a good parent is important. [...] If you want happy children, and you desire a relationship with them that lasts beyond when they’re old enough to fly the nest, then be good to your kids. Just know that it probably will have little effect on the person they will grow into. 

2. Brian Caplan (WSJ), and on EconTalk

In research including hundreds of twins who were raised apart, identical twins turn out to be much more alike in intelligence and happiness than fraternal twins, but twins raised together are barely more alike than twins raised apart. In fact, pioneering research by University of Minnesota psychologist David Lykken found that twins raised apart were more alike in happiness than twins raised together. Maybe it's just a fluke, but it suggests that growing up together inspires people to differentiate themselves; if he's the happy one, I'll be the malcontent. 

[...] 

Critics often attack behavioral genetics with a reductio ad absurdum: "If it doesn't matter how you raise your kids, why not lock them in a closet?" The answer is that twin and adoption studies measure the effect of parenting styles that people frequently use. Locking kids in closets fortunately isn't one of them. It's also important to remember that most studies focus on kids' long-run outcomes. Parents often change their kids in the short-run, but as kids grow up, their parents' influence wears off.

3. I had picked up Judith Hariss' "The Nurture Assumption" somewhat randomly from a library shelf about 15 years ago. Not knowing too much psychology then (or now), I thought it was a great book. Here is a relatively recent interview with her.

Q: So if they can't influence the adults their children become, then what, if any, steps can parents take to help ensure their kids succeed? Or become "good" people?
A: I believe the most important function of parents is to give their children a happy home — not because it will make them more likely to succeed but because everyone has a right to a happy home life. Aside from that, there are other things parents can do, such as providing training in music or sports. Parents have some ability to decide where they will live and where their children will go to school. Some schools have an atmosphere that is more favorable to academic achievement.

Links

1. Bob the Engineer's heartbreaking story (NPR)

The night before the launch, Ebeling and four other engineers at NASA contractor Morton Thiokol had tried to stop the launch. Their managers and NASA overruled them. 

That night, he told his wife, Darlene, "It's going to blow up." 

When Challenger exploded 73 seconds after liftoff, Ebeling and his colleagues sat stunned in a conference room at Thiokol's headquarters outside Brigham City, Utah. They watched the spacecraft explode on a giant television screen and they knew exactly what had happened.

2. Don't be stupid! Keep your wisdom teeth, unless they are troubling you. (Fusion)

As of 2011, 10 million wisdom teeth get hacked out of the back of Americans’ mouths a year. 

For decades, the procedure was performed only when the teeth, also known as third molars, were causing real trouble, like in the case of appendectomies. After World War II, however, the ranks of dentists exploded, and with them recommendations that people get their third molars removed as a precaution. As dental care got more advanced—and the financial incentive to perform the procedures increased—wisdom teeth removal began to become as routine as getting braces.

3. A Noam Chomsky interview (SmashingInterviews) Nearly 90, but amazing faculties!

About half of the history of the country, there were two major problems that required guns. One was eliminating the indigenous population. They had to be eliminated or exterminated. They fought back which meant you needed guns. 

The other was that the United States was running the most hideous slave labor camps in human history in the South, which is a large part of the basis of their economy. It was not done just for the wealth of the plantation owners, the manufacturing system was based for a long time on textile production that was largely cotton based. The banks were developing credit for cotton. Cotton was the main commodity of the early part of the Industrial Revolution. Same in England. A large part of their economic wealth and power developed from the slave labor camps. Well, you know, running slave labor camps means you’ve got to be afraid of the slaves. Maybe they’ll erupt.

 4. A fascinating portrait of Bernie Sanders (Politico). Regardless of how you feel about his politics, his integrity is unimpeachable!

It’s always been that way with Sanders. The issues. The issues. Stick to the issues. The rich are too rich. Those with power have too much. The middle class is withering. Inequality is a crisis, and the system is rigged. With Sanders, what you see is what you get, insist the people who know him best — and that’s almost all you get. 

But if his positions are well known, the person, it turns out, is less known. Before Sanders was a U.S. senator, before he was a congressman, before he was mayor of Burlington — before he won one shocking election, then 13 more — he was a radical and an agitator in the ferment of 1960s and '70s Vermont, a tireless campaigner and champion of laborers who didn’t collect his first steady paycheck until he was an elected official pushing 40 years old.