Never-before-seen images of the Lévy stable distribution

Analytic continuation to the complex plane of the Lévy stable distribution, as the Lévy index is varied from the Cauchy to the Gaussian limits.   We have used domain coloring to plot a complex function of complex numbers.

The $\alpha$-stable Lévy distribution has been known since the 1930s and been the subject of systematic and persistent study for many decades.  The most important special cases of the  Lévy stable distribution are the Gaussian distribution (also known as the normal distribution) and the Cauchy distribution (also known as the Lorentzian distribution).

My doctoral student Éric C. Rocha, as part of his thesis project, generated domain coloring plots of the analytic continuation to the complex plane of Lévy stable distribution.  Some of these images have been published in Physical Review E (see PDF file).

The animated GIF above shows the Lévy stable distribution as $\alpha$ varies from 1 to 2.   The $\alpha=1$ case is the Cauchy and the $\alpha=2$ case is the Gaussian. The complex conjugate poles on the imaginary axis are clearly visible for the Cauchy case. As the distribution is deformed towards the Gaussian, one can see a “palm leaf structure” form due to the presence of zeroes. Finally, the zeroes go off to infinity and the shape converges to the saddle structure of the Gaussian.

Enjoy!

I thank Wolfram for donating a licence for Mathematica.

Writing a paper in E-prime

Many top scientists communicate clearly, sometimes seemingly effortlessly. The papers by Einstein flow elegantly in clear and logical steps, almost as if choreographed, from one idea to the next. Some articles even have qualities more commonly seen in great works of art, for example, Dirac’s seminal book on quantum mechanics or Shannon’s paper introducing his celebrated entropy. What a pleasure to read! Most physicists similarly recognize Feynman as a master of clear communication.

Before I became a grad student, I had underestimated the importance of good and effective communication. My former PhD advisor, an excellent communicator, taught me the crucial role played by communication in scientific discourse and debate.

Let me explain this point in greater detail. As an illustrative example, imagine if Einstein had not written clearly. Then it may very well have taken much longer for his ideas to percolate and gain acceptance throughout the scientific community. Indeed, Boltzmann, in contrast to Einstein, wrote lengthy and admittedly difficult-to-read texts. Some of his critics perhaps  failed to grasp his seminal ideas. Disappointed and possibly depressed, he eventually committed suicide while still in his prime. Today, the top prize in the field  of statistical physics honors his name— the Boltzmann Medal. Nevertheless, it took many years and the efforts of other scientists (e.g. Gibbs) for the physics community to recognize the full extent of Boltzmann’s contributions.    Clear exposition can make a big difference.

In this blog post, I do not give tips or advice about how to write clearly. Good tips on how to write clearly abound.  Instead, I want to draw your attention to how this article does not contain a single instance of the verb “to be” or any of its conjugations or derived words, such as “being,” “was,” “is,” and so forth — excepting this sentence, obviously. The subset of the English language that remains after the removal of these words goes by the name E-prime, often written E’. In other words, E’ equals English minus all words derived from the above-mentioned verb.

Writing in E’ usually forces a person to think more carefully. Scientists need to communicate not only clearly, but with a slightly higher degree of precision than your typical non-scientist. I have found that fluency in E’ helps me to spot certain kinds of errors of reasoning. The key error of reasoning attenuated by the use of E’ relates to identification.   Too often, the referents of the grammatical subject and object become identified in standard English, where in fact no such identification exists in the real world.  E’ helps to reduce this improper identification, or at least to call attention to it.  The topic of E’, and of related subjects, such as  its ultimate historical origins in general semantics, the study of errors of reasoning, the nature of beliefs, cognitive biases, etc., would require too broad a digression for me to discuss here, so I recommend that interested readers research such topics on their own.

In my early 30s, soon after I obtained tenure in my first faculty position, I decided to write a full article entirely in E’.  What a wonderful and interesting exercise!  Of course, I did not find it easy to write in E’, but with few exceptions, the finished paper contained only E’ sentences.  Forcing myself to think and write in E’ helped me to give a better description of what we, as scientists, really did.  I would cautiously claim that writing in E’ benefited our paper, at least as far as concerns clarity and precision.  No longer do I publish papers in E’, but I learned a lot about how to write (and think) a little bit more clearly.

That paper, about an empirical approach to music, appeared in print in 2004 in the statistical physics journal  Physica A. It eventually ended up cited very well: 33 citations according to  Thomson Reuters’  Web of Science and 60 citations on Google Scholar, as of May 2016.  Most incredibly, it even briefly shot up to the top headline at Nature.com (click here to see)!  We had never expected this.

In that paper, my co-authors and I proposed a method for studying rhythmic complexity. The collaboration team included as first author Heather Jennings, a professor of music (and also my spouse). We took an empirical approach for comparing the rhythmic structures of Javanese Gamelan, Jazz, Hindustani music, Western European classical music, Brazilian popular music (MPB), techno (dance), New Age music, the northeastern Brazilian folk music known as Forró and last but not least: Rock’n Roll. Excepting a few sentences, the paper consists entirely of E’ sentences.

You can read the paper by clicking here for the PDF. A fun exercise: as you read the paper, (1) try to imagine how you would normally rephrase the E’ sentences in ordinary English; (2) try to spot the subtle difference in meaning between the English and E’ sentences.