Math

A while back I wrote a post on three views of the negative binomial distribution. This post adds a fourth view.

One of the shortcomings of the Poisson distribution is that its variance exactly equals its mean. It is common in practice for the variance of count data to be larger than the mean, so it’s natural to look for a distribution like the Poisson but with larger variance. We start with a Poisson random variable X with mean λ, but then we make λ itself random and suppose that λ comes from a gamma(α, β) distribution. Then the marginal distribution on X is a negative binomial distribution with parameters r = α and p = 1/(β + 1).

The previous post said that the negative binomial is useful because it has more variance than the Poisson. The derivation above explains why the negative binomial should have more variance than the Poisson.

For details, see updated notes on the negative binomial distribution.

Related links:

Three views of the negative binomial distribution
Student-t as a mixture of normals
Diagram of distribution relationships
General binomial coefficients

You can express a Student-t distribution as a continuous mixture of normal distributions. Some properties of the t distribution are easier to prove in this form. Here are notes with details.

I ran across this tidbit reading Bayesian Data Analysis by Gelman et al.

Related post: Beer, Wine, and Statistics (origin of the Student-t distribution)

How can we extend the idea of derivative so that more functions are differentiable? Why would we want to do so? How can we make sense of a delta “function” that isn’t really a function? We’ll answer these questions in this post.

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Here are some notes on upper and lower bounds on the probability P(Z > t) for a standard normal random random variable Z. I wrote up these notes to settle a issue that came up in a probability class I’m teaching. It’s surprising that there are simple functions that provide efficient bounds on the normal distribution function.

The opening chord of the Beatles song “A Hard Day’s Night” has been something of a mystery. Guitarists have tried to reproduce the chord with limited success. Turns out there’s a good reason why they haven’t figured it out: the chord cannot be played on a guitar alone.

Jason Brown has digitally analyzed the chord using Fourier analysis and determined that there must have been a piano in the recording studio playing along with the guitars. Brown has determined what notes each member of the Beatles were playing.

I heard Jason Brown’s story on the Mathematical Moments podcast. In addition to the chord discussed above, Brown talks about other things he has discovered about the Beatles and about the relationship between music and math in general. Unfortunately, Mathematical Moments does not make it easy to link to individual episodes. Here is a link to a PDF file of show notes with the audio embedded. The file is slow to download, and your PDF viewer may not support it. Here’s a link directly to just the MP3 audio file.

The Mathematical Moments podcast also does not make it obvious that you can subscribe to the podcast; they only provide links to individual episodes with fat PDF files. However, you can subscribe by using the URL http://www.ams.org/rss/mathmoments.rss.

The Mathematics Genealogy Project keeps track mathematics PhD students and advisors. I was surprised to find that such information has been preserved for hundreds of years. I was able to trace my mathematical lineage back to Marin Mersenne (1588–1648) of Mersenne prime fame.

I did my PhD under Ralph Showalter, who studied under Tsuan Wu Ting, and so on back to Siméon Poisson (1781–1840).

Then things start to become more complicated. Poisson had two advisors: Joseph Louis Lagrange and Pierre-Simon Laplace. Lagrange also had two advisors: Leonhard Euler and Johann Bernoulli. Etc. One line goes back to Mersenne. Another line goes back to Demetrios Kydones (1324–1397).

Update: Thanks to Frederik Hermans for creating a graph by crawling the Mathematics Genealogy Project site and using Graphviz. It’s too big to view as an ordinary image; the graph gets very bushy in the 16th century. Here’s a PDF version that lets you zoom in and out to see the whole thing.

I was surprised to see Erasmus on the graph. I didn’t run across him when I was just clicking around the web site.

Related post:

Six degrees of Paul Erdős
Honeybee genealogy

Mark Ronan wrote a popular book Symmetry and the Monster about the story behind the classification of finite simple groups, or as Ronan calls such groups, the atoms of symmetry. All finite groups can be built up from simple groups somewhat like the way composite numbers are built up from prime numbers. Groups describe symmetries, and so the fundamental building blocks of groups are reasonably called atoms of symmetry. Ronan gave a lecture summarizing his book. Audio, video, and a transcript of his talk are all available here.

The classification of finite simple groups can be seen as a theorem whose proof is spread out over hundreds of articles and thousands of pages. A precise statement of the classification theorem is available here. One major piece of the puzzle is the theorem of Feit and Thompson. An entire journal issue was devoted to the 255-page proof of this one result. There is a project to simplify the proof, eliminating some of the redundancy between papers, etc. But it appears that the revised proof will still contain hundreds of pages of highly technical reasoning.

I enjoyed reading Ronan’s book a couple weeks ago. The biographical sketches in the book are the best part. The book begins with the study of symmetry via group theory, starting with the work of Évariste Galois and Sophus Lie. Someone with very little background in math could read most of the book. However, toward the end of the book when Ronan gets to the classification theorem and the role of “the monster,” he goes into more detail and there I believe he loses his audience. He goes into more detail than a non-mathematician would want to read, but not enough detail for a mathematician to understand exactly what he’s talking about.

I recommend starting with Mark Ronan’s lecture. If you want to go further, read the book, but feel free to skim over details toward the end.

(I haven’t seen a definitive count of the number of journal pages that comprise the classification proof. Ronan quotes one source who says the number of pages is “at least 3,000.” Other sources say “tens of thousands of pages.” Maybe it is unclear which papers should be included.)

How many basic trigonometric functions are there? I will present the arguments for 1, 3, 6, and at least 12.

The calculator answer: 3

A typical calculator has three trig functions if it has any: sine, cosine, and tangent. The other three that you may see — cosecant, secant, and cotangent — are the reciprocals of sine, cosine, and tangent respectively. Calculator designers expect you to push the cosine key followed by the reciprocal key if you want a secant, for example.

The calculus textbook answer: 6

The most popular answer to the number of basic trig functions may be six. Unlike calculator designers, calculus textbook authors find the cosecant, secant, and cotangent functions sufficiently useful to justify their inclusion as first-class trig functions.

The historical answer: At least 12

There are at least six more trigonometric functions that at one time were considered worth naming. These are versine, haversine, coversine, hacoversine, exsecant, and excosecant. All of these can be expressed simply in terms of more familiar trig functions. For example, versine(θ) = 2 sin²(θ/2) = 1 – cos(θ) and exsecant(θ) = sec(θ) – 1.

Why so many functions? One of the primary applications of trigonometry historically was navigation, and certain commonly used navigational formulas are stated most simply in terms of these archaic function names. For example, the law of haversines. Modern readers might ask why not just simplify everything down to sines and cosines. But when you’re calculating by hand using tables, every named function takes appreciable effort to evaluate. If a table simply combines two common operations into one function, it may be worthwhile.

These function names have a simple pattern. The “ha-” prefix means “half,” just as in “ha’penny.” The “ex-” prefix means “subtract 1.” The “co-” prefix means what it always means. (More on that below.) The “ver-” prefix means 1 minus the co-function.

Pointless exercise: How many distinct functions could you come up with using every combination of prefixes? The order of prefixes might matter in some cases but not in others.

The minimalist answer: 1

The opposite of the historical answer would be the minimalist answer. We don’t need secants, cosecants, and cotangents because they’re just reciprocals of sines, cosines, and tangents. And we don’t even need tangent because tan(θ) = sin(θ)/cos(θ). So we’re down to sine and cosine, but then we don’t really need cosine because cos(θ) = sin(π/2 – θ).

Not many people remember that the “co” in cosine means “complement.” The cosine of an angle θ is the sine of the complementary angle π/2 – θ. The same relationship holds for secant and cosecant, tangent and cotangent, and even versine and coversine.

By the way, understanding this complementary relationship makes calculus rules easier to remember. Let foo(θ) be a function whose derivative is bar(θ). Then the chain rule says that the derivative of foo(π/2 – θ) is -bar(π/2 – θ). In other words, if the derivative of foo is bar, the derivative of cofoo is negative cobar. Substitute your favorite trig function for “foo.” Note also that the “co-” function of a “co-” function is the original function. For example, co-cosine is sine.

The consultant answer: It depends

The number of trig functions you want to name depends on your application. From a theoretical view point, there’s only one trig function: all trig functions are simple variations on sine. But from a practical view point, it’s worthwhile to create names like tan(θ) for the function sin(θ)/sin(π/2 – θ). And if you’re a navigator crossing an ocean with books of trig tables and no calculator, it’s worthwhile working with haversines etc.

Related posts:

Mercator projection
Why care about spherical trig?
Three trigonometry topics
What is the cosine of a matrix?
Connecting trig and hyperbolic functions without complex numbers

In my earlier post on the Mercator projection, I derived the function h(φ) that maps latitude on the Earth to vertical height on a map. The inverse of this function turns out to hold a few surprises.

The height y corresponding to a positive latitude φ is given by

h(φ) = log( sec(φ) + tan(φ) ).

The inverse function, h^-1(y) = φ gives the latitude as a function of height. This function is called the “Gudermannian” after Christoph Gudermann and is abbreviated gd(y). Gudermann was the student of one famous mathematician, Karl Friedrich Gauss, and the teacher of another famous mathematician, Karl Weierstrass.

The Gudermannian function gd(y) can be reduced to familiar functions:

gd(y) = arctan( sinh(y) ) = 2 arctan( e^y ) – π/2.

That doesn’t look very promising. But here’s the interesting part: the function gd forms a bridge between hyperbolic trig functions and ordinary trig functions.

sin( gd(x) ) = tanh(x)
tan( gd(x) ) = sinh(x)
cos( gd(x) ) = sech(x)
sec( gd(x) ) = cosh(x)
csc( gd(x) ) = coth(x)
cot( gd(x) ) = csch(x)

By definition, gd(x) is an angle θ whose tangent is sinh(x).

In the figure, tan(θ) = sinh(x). Since cosh²(x) – sinh²(x) = 1, the hypotenuse of the triangle is cosh(x). The identities above follow directly from the figure. For example, sin(θ) = sinh(x) / cosh(x) = tanh(x).

Finally, it is easy to show that gd is the inverse of the Mercator scale function h:

h( gd(x) ) = log( sec( gd(x) ) + tan( gd(x) ) ) = log( cosh(x) + sinh(x) ) = log( e^x ) = x.

Related links:

Mercator projection
Gudermannian on MathWorld

A natural approach to mapping the Earth is to imagine a cylinder wrapped around the equator. Points on the Earth are mapped to points on the cylinder. Then split the cylinder so that it lies flat. There are several ways to do this, all known as cylindrical projections.

One way to make a cylindrical projection is to draw a lines from the center of the Earth through each point on the surface. Each point on the surface is then mapped to the place where the line intersects the cylinder. Another approach would be to make horizontal projections, mapping each point on Earth to the closest point on the cylinder. The Mercator projection is yet another approach.

With any cylindrical projection parallels, lines of constant latitude, become horizontal lines on the map. Meridians, lines of constant longitude, become vertical lines on the map. Cylindrical projections differ in how the horizontal lines are spaced. Different projections are useful for different purposes. Mercator projection is designed so that lines of constant bearing on the Earth correspond to straight lines on the map. For example, the course of a ship sailing northeast is a straight line on the map. (Any cylindrical projection will represent a due north or due east course as a straight line, but only the Mercator projection represents intermediate bearings as straight lines.) Clearly a navigator would find Mercator’s map indispensable.

Latitude lines become increasingly far apart as you move toward the north or south pole on maps drawn with the Mercator projection. This is because the distances between latitude lines has to change to keep bearing lines straight. Mathematical details follow.

Think of two meridians running around the earth. The distance between these two meridians along a due east line depends on the latitude. The distance is greatest at the equator and becomes zero at the poles. In fact, the distance is proportional to cos(φ) where φ is the latitude. Since meridians correspond to straight lines on a map, east-west distances on the Earth are stretched by a factor of 1/cos(φ) = sec(φ) on the map.

Suppose you have a map that shows the real time position of a ship sailing east at some constant rate. The corresponding rate of change on the map is proportional to sec(φ). In order for lines of constant bearing to be straight on the map, the rate of change should also be proportional to sec(φ) as the ship sails north. That says the spacing between latitude lines has to change according to h(φ) where h’(φ) = sec(φ). This means that h(φ) is the integral of sec(φ) which equals log |sec(φ) + tan(φ)|. The function h(φ) becomes unbounded as φ approaches ± 90°. This explains why the north and south poles are infinitely far away on a Mercator projection map and why the area of northern countries is exaggerated.

(Update: The inverse of the function h(φ) has some surprising properties. See Inverse Mercator projection.)

The modern explanation of Mercator’s projection uses logarithms and calculus, but Mercator came up with his projection in 1569 before logarithms or calculus had been discovered.

The Mercator projection is now politically incorrect. Although the projection has no political agenda — its design was dictated by navigational requirements — some people have gotten bent out of shape over the way it exaggerates the area of northern countries.

For more details of the Mercator projection, see Portraits of the Earth.

Related posts:

What is the shape of the Earth?
Finding distances using longitude and latitude
Spherical trigonometry

Suppose you’re waiting for a friend and you have nothing to do. After a few minutes of boredom you pick up a pencil and some scrap paper. You start listing the prime numbers.

2, 3, 5, 7, 11, 13, 17,19, 23, 29, 31, …

Next you write down the forward differences, subtracting each number in the sequence from the one that follows it.

1, 2, 2, 4, 2, 4, 2, 4, …

Your friend is running late and so you repeat the process starting with the sequence you just created.

1, 0, 2, -2, 2, 2, 2, 2, 4, …

Hmm. That time you got a negative number in the list. You’re just doodling and you don’t want to think too hard, so you decide you’ll ignore signs and just write down the absolute values of the differences. So you erase the negative sign and take differences again.

1, 2, 0, 0, 0, 0, 0, 2, …

Your friend is quite late, so you keep doing this some more. After a while you notice that every new sequence has started with 1. Will every sequence start with 1? That’s Gilbreath’s conjecture, named after Norman Gilbreath who asked the question in 1958. I ran across the conjecture in The Math Book by Clifford Pickover. Gilbreath wasn’t the first to notice this pattern. François Proth noticed it in 1878 and published an incorrect proof of the conjecture.

Gilbreath’s conjecture has been verified for the first several billion sequences, but nobody has proved that every sequence will start with 1. Paul Erdős speculated that Gilbreath’s conjecture is true but it would be 200 years before anyone could prove it. I find Erdős’s conjecture more interesting than Gilbreath’s conjecture.

Here’s what I imagine that Erdős had in mind. While the process Gilbreath created is very simple, it is also a strange thing to study. It’s not the kind of thing that people have proven theorems about. No one knows how to approach the problem. There are far more complicated problems in the mainstream of mathematics that will probably be resolved sooner because they are related to previously solved problems and researchers have some idea where to start working on them.

Other posts on number theory:

Constructive proof of the Chinese remainder theorem
How to solve linear congruences
How many numbers are squares mod m
Connecting probability and number theory

Other posts on proofs:

Why proof by examples doesn’t work
Errors in math papers not a big deal?
Jenga mathematics
In praise of tedious proofs
Proofs of false statements

Last spring I wrote a post on spherical trigonometry, the study of triangles drawn on a sphere (e.g. the surface of the Earth).

Mel Hagen left a comment on that post a few days ago saying

I am revisiting Spherical Trig after 30 years by going back over some of my books that I have collected over the years. …

I asked Mel via email why he was revisiting the subject. He wrote an interesting reply that I am including below with his permission.

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A woman nearly became violent in a math class I taught several years ago.  I was going over homework problems and she wanted to know whether a certain problem was a “permutation” or a “combination.”  She knew how to solve two kinds of problems and was irritated when I told her that her homework problem didn’t fall into either of her two categories.

She insisted that I tell her which of the two techniques would solve the problem and nearly lost control when I repeated that neither would work. The rest of the students and I were shocked. A little nervous laughter broke the tension and we resumed going over the homework.

The angry student had implicitly come to believe that if a counting problem contains two numbers, n and k, there are only two possible answers: P(n, k) and C(n, k). Here P(n, k) = n!/(n-k)! and C(n, k) = P(n, k)/k!. She was not alone. Students commonly believe this, and for good reason: most homework problems can be solved this way. For example, a club with 12 members can elect five distinct officers in P(12, 5) ways and they can select a committee of five members in C(12, 5) ways. It’s easy for an instructor or textbook author to think of dozens of homework problems in these patterns and unintentionally imply that these are these are the only possibilities. However, the following problem contains the numbers 12 and 5 but the solution is neither P(12, 5) nor C(12, 5).

Suppose you have a class of 12 students. Each student will receive one of five letter grades: A, B, C, D, or F. At the end of the course, you tally up how many students received each grade. How many different ways could the tally turn out?  For example, one possibility would be all A’s.  Another would be three A’s, four B’s, four C’s, no D’s, and one F.

The grade tally problem is representative of a class of problems that come up fairly often in application but that lie just outside what students typically learn. The general solution is written up in these notes on counting selections with replacement. The notes include the famous “stars and bars” explanation by William Feller.

By the way,  there are 1,820 possible grade tallies for 12 students and five grades.

Related links:

Binomial coefficients
Four uncommon but handy notations
How to compute binomial coefficients
Richard Stanley’s 12-fold way

A few weeks ago, Sterling Publishing sent me a copy of Cliff Pickover’s new book The Math Book. I enjoyed reading the book (see my review) and set up the interview that follows.

JC: The Math Book is your first book that I’ve read. Is it typical of your writing? How would you summarize the topics you’ve written about?

CP: My past 40 books cover many different topics. A number of these books concern the beauty of mathematics. Others cover topics at the borderlands of science, roaming far and wide on topics ranging from creativity, art, mathematics, and human intelligence, to higher dimensions, religion, strange realities, time travel, alien life, and science fiction. You can see a listing of my other books here. This should give your readers a flavor for the kinds of topics on which I enjoy writing.

Of course, The Math Book is serious mathematics, but I hope I’ve introduced an element of art and playfulness as well — the topics flow from fractals, to Rubik’s cube robots, to the infinite monkey theorem! For me, mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos.

With respect to my other books, some of which may be more at the fringes of science, I’d point out that “fringe” research is crucial — not just for its educational value but because significant discoveries can come from such study. At first glance, some topics in science or sociology in my other works may appear to be curiosities, with little practical application or purpose. However, I have found these experiments useful and educational, as have the many students, educators, artists, and scientists who have written to me. In fact, science is filled with hundreds of great discoveries that have emerged through chance happenings and serendipity, for example: Velcro, Teflon, X-rays, penicillin, nylon, safety glass, sugar substitutes, dynamite, and polyethylene plastics.

Several of my past books explore a variety of topics to test your curiosity and powers of lateral thinking. Robert Pirsig wrote in Zen and the Art of Motorcycle Maintenance, “It’s the sides of the mountain which sustain life, not the top. Here’s where things grow.” This also applies to the joy that writers experience when letting their minds drift and when wondering about humanity’s place in the universe.

Beltrami’s pseudosphere by Paul Nylander, included in The Math Book

JC: You’ve written a lot of books, especially for someone who has a full-time job in addition to writing. How do you manage your time?

CP: When people ask me how I manage my time, I reply: “Some people play golf on the weekends. Instead, I prefer to write.” Of course, my prolific writing pales in comparison to American novelist, lawyer, and workaholic Erle Stanley Gardner (1889-1970), who once worked on seven novels simultaneously and dictated 66,000 words a week! Gardner would never start to dictate until he had worked out the entire plot of his novel. He actually hired six secretaries to handle his dictation, which he found more efficient than typing. His best-known works focus on the lawyer-detective Perry Mason.

I don’t know how writers like Isaac Asimov were so prolific before the age of the computer. I would have a very difficult time writing books, and doing all the necessary text rearrangements and editing, without a word processor. According to the New York Public Library Desk Reference (4th ed.), Isaac Asimov wrote over 400 books and is the only author with a book included in every major Dewey-decimal category. I sit in awe of Asimov, but a few people have exceeded his book output. Lauran Paine (b. 1916) has published over 900 books under more than 90 pen names. Paine spent his youth working as a cowboy, and today at least 500 of his books are Westerns.

JC: How do you write? Do you have a set schedule and place for writing? Anything unusual about your environment or equipment?

CP: French writer Marcel Proust composed his books in a haphazard fashion. He did not start at the beginning and finish at the end. He did not write linearly. Instead, ideas came to him in flashes as he went about his daily routine. Most of my own books are composed in the same way. As ideas come to me during the day or in the realm between sleep and wakefulness, I jot them down and continue to fill in details in the book. For me, writing is exactly like painting, adding a spot of color here, a detail there, a twig on this tree, a bit of foam on that ocean wave… No painter starts at the top of the painting and finishes at the bottom.

My approach to filling in detail, like a painter dabbing paint, is fine in the age of word processors, but it was amazing that Proust used the same approach so well. He would dictate to his stenographers who would type an initial manuscript. Then, he would crowd the margins with additional details and establish links between scenes and characters. He would paste in new pages and have the new work typed again and again. Edmund White notes in his biography of Proust, “If any writer would have benefited from a word processor, it would have been Proust, whose entire method consisted of adding details here and there and of working on all parts of his book at once.” As for my books, there’s nothing special about the tools I use and nothing special about my environment. These days, I use Microsoft Word.

JC: Are you writing a book now?

CP: I am finishing a book in the style of The Math Book — one page of text facing one page of illustration. Entries are in chronological order. Let’s wait to see how well The Math Book sells. If it sells a sufficient number of copies, perhaps I can convince a publisher to consider this newer work that covers a particular array of topics in science, art, history, and popular culture.

JC: Would you be interested in writing a computer science analog of The Math Book?

CP: I very much enjoyed creating The Math Book with my publisher, Sterling, and the $19 price offered by Amazon.com is amazing for a 528-page all color hardcover. I would welcome doing another book of this kind if we feel that such a book has not been done before and that it is marketable.

JC: Who are some of your favorite authors, either for content or style?

CP: My favorite tales of parallel worlds are those of Robert Heinlein. For example, in his science-fiction novel The Number of the Beast there is a parallel world that appears identical to ours in every respect except that the letter “J” does not appear in the English language. Luckily, the protagonists in the book have built a device that lets them perform controlled explorations of parallel worlds from the safety of their high-tech car. This is my favorite novel, and the only one that I’ve read over five times — although I could never finish it the first few times. It’s a novel that many readers dislike, can’t finish, or understand. The final section is nearly incomprehensible. But for me, it provides a sense of mystic transport as the brainy characters enter parallel worlds, fleeing from danger.

JC: There is a scene in the movie Good Will Hunting where Robin Williams’ character, Sean, asks Matt Damon’s character, Will, what he likes to read. Will’s response is “Hey, whatever blows your hair back.” What blows your hair back? Any books, blogs, podcasts, etc. that you turn to for inspiration?

CP: These days, I’m enjoying CDs and DVDs from The Teaching Company – on subjects ranging from the history of mathematics, to the history of the world, to an introduction to Judaism. Some of their classes on the history of mathematics are awesome mind-bogglers.

My most popular blog, Reality Carnival, highlights the kinds of topics and stories that interest me.

JC: Your writing indicates you have broad interests. Have you struggled to find where you want to be along the continuum between Renaissance man and specialist?

CP: I prefer to be a generalist. In fact, if I had to manage a foundation that gives money to scientists, I would also consider high-quality “generalists” as recipients. Experts have become very specialized, and science popularizes are often frowned upon by their more “serious” colleagues. Sometimes, specialists develop blind spots after years of intense focus on a single topic. Thus, I would devote a portion of my money to training “generalists” who traverse several fields and then bring together ideas in ways that specialists may be unable to do. They will also look for overlaps between different domains of research and try to solve shared problems with a single approach. As our rate of technological progress skyrockets in the 21st century, these Facilitators will study the multidisciplinary implications of this acceleration and work on technologies or new ways of seeing that help humanity assimilate advances that outstrip our comprehension and the restrictions of our intuition.

Other interview posts:

Dan Bricklin, co-creator of VisiCalc, technologist, author
Carl Franklin, musician, software developer, podcaster