Math

What is the Carnival of Mathematics? Math bloggers submit articles they have written recently and each month a host writes a post linking to the submitted posts. The sister carnival, Math Teachers at Play, focuses on math education and on math up through high school level. For a more thorough description of the two carnivals and some FAQs, please see Mike Croucher’s article What is a Maths Carnival?

I’m taking a turn hosting this month. Tradition dictates that the host begin with some trivia about the number of the post. As this is the 62nd Carnival of Mathematics, here are a few facts about 62.

• 62 is the only number whose cube (238328) consists of 3 digits each occurring 2 times.
• The great rhombicosidodecahedron has 62 sides.
• Louis Pasteur developed the first rabies vaccination at age 62.

And now onto the posts.

Math and science teacher Cory Poole sends in a video that he created along with his partners and students. The video features a 64-foot Sierpenski triangle about of 12,000 tortilla chips. Read more about the story of the video. Also, here are the bloopers from making the video.

St. Swithun’s day is a sort of British analog of America’s Groundhog Day. If it rains on St. Swithun’s day, it is supposed to rain for the next 40 days. Is there some truth to the legend? See Jon McLoone’s article Mathematica Tests the St. Swithun’s Day Proverb posted at Wolfram Blog.

Edmund Harriss from Maxwell’s Demon presents Spirographs and the third dimension. Interesting visually and mathematically.

Rachel Thomas presents Beautiful symmetry provides glimpse into quantum world on News from the world of maths. This article reports on a low-termperature experiment that implies that the exceptional Lie group E8 is at work.

Did you know that sine and cosine are equal for all x? Heather (Xi) submitted a pseudo-proof in A=B implies that 1=1, therefore? by her colleague TwoPi at 360. (If there is ever a 360th Carnival of Mathematics, Heather should host it.)

Update: The 360 blog has agreed to host the 360th Carnival of Mathematics, tentatively scheduled for December 1, 2034. (Mike, I hope it’s OK that I scheduled this date without consulting you. )

Marc West presents The curse of the duck, a post about mathematics and cricket. His blog is Mr Science Show: Where Science Meets Pop Culture.

Dan M presents Appolonian Gaskets and Ford Circles on his blog mathrecreation. This post relates number theory and geometry as Ford circles are related to the Farey sequence.

Rick Regan presents Counting Binary and Hexadecimal Palindromes posted at Exploring Binary. Rick counts base 10 palindromes as a warm-up before diving into new territory with binary and hexidecimal numbers.

Gregory Astley wrote a guest post Maxima Tutorial – plotting direction fields for 1st order ODEs for Mike Croucher’s blog Walking Randomly. (Maxima is part of the SAGE mathematics system. Mike has written several posts about SAGE lately.)

Annarita Ruberto from Matem@ticaMente presents the article “How heavy the fish?” The original post was written in Italian, and here is Google’s translation of the page into English.

Matt McDonnell presents Mathematical Recreations: Tweetable Game Of Life, a guest blog for Loren Shure’s blog Loren on the Art of MATLAB.

For some mental arithmetic shortcuts and an explanation of why they work, see Sol Lederman’s post Trachtenberg speed multiplication: exploring why it works on his blog Wild About Math.

Politics impacts sphere of human activity, including math education. See Jason Dyer’s post Anatomy of a Political Math-Ed Reaction on The Number Warrior.

Peter Rowlett from Travels in a Mathematical World presents Substitution ciphers: Ancient – Renaissance, video included below.

You can keep up with Carnival of Mathematics news on Twitter by following @CarnivalOfMath. You may also be interested in daily math facts on Twitter from @ProbFact (probability), @AnalysisFact (real and complex analysis), and @AlgebraFact (algebra and number theory).

The most obvious way to compute the soft maximum can easily fail due to overflow or underflow.

The soft maximum of x and y is defined by

g(x, y) = log( exp(x) + exp(y) ).

The most obvious way to turn the definition above into C code would be

double SoftMaximum(double x, double y)
{
    return log( exp(x) + exp(y) );
}

This works for some values of x and y, but fails if x or y is large. For example, if we use this to compute the soft maximum of 1000 and 200, the result is numerical infinity. The value of exp(1000) is too big to represent in a floating point number, so it is computed as infinity. exp(200) is finite, but the sum of an infinity and a finite number is infinity. Then the log function applied to infinity returns infinity.

We have the opposite problem if we try to compute the soft maximum of -1000 and -1200. In this computation exp(-1000) and exp(-1200) both underflow to zero, and the log function returns negative infinity for the logarithm of zero.

Fortunately it’s not hard to fix the function SoftMaximum to avoid overflow and underflow. Look what happens when we shift both arguments by a constant.

log( exp(x – k) + exp(y – k) ) = log( exp(x) + exp(y) ) – k.

This says

log( exp(x) + exp(y) ) = log( exp(x -k) + exp(y-k) ) + k

If we pick k to be the maximum of x and y, then one of the calls to exp has argument 0 (and so it returns 1) and the other has a negative argument. This means the follow code cannot overflow.

double SoftMaximum(double x, double y)
{
	double maximum = max(x, y);
	double minimum = min(x, y);
	return maximum + log( 1.0 + exp(minimum - maximum) );
}

The call to exp(minimum - maximum) could possibly underflow to zero, but in that case the code returns maximum. And in that case the return value is very accurate: if maximum is much larger than minimum, then the soft maximum is essentially equal to maximum.

The equation for the soft maximum implemented above has a few advantages in addition to avoiding overflow. It makes it clear that the soft maximum is always greater than the maximum. Also, it shows that the difference between the hard maximum and the soft maximum is controlled by the spread of the arguments. The soft maximum is nearest the hard maximum when the two arguments are very different and furthest from the hard maximum when the two arguments are equal.

Thanks to Andrew Dalke for suggesting the topic of this post by his comment.

Related links:

Soft maximum
Anatomy of a floating point number
Avoiding overflow, underflow, and loss of precision

Here are ten things about numerical linear algebra that you may find surprising if you’re not familiar with the field.

1. Numerical linear algebra applies very advanced mathematics to solve problems that can be stated with high school mathematics.
2. Practical applications often require solving enormous systems of equations, millions or even billions of variables.
3. The heart of Google is an enormous linear algebra problem. PageRank is essentially an eigenvalue problem.
4. The efficiency of solving very large systems of equations has benefited at least as much from advances in algorithms as from Moore’s law.
5. Many practical problems — optimization, differential equations, signal processing, etc. — boil down to solving linear systems, even when the original problems are non-linear. Finite element software, for example, spends nearly all its time solving linear equations.
6. A system of a million equations can sometimes be solved on an ordinary PC in under a millisecond, depending on the structure of the equations.
7. Iterative methods, methods that in theory require an infinite number of steps to solve a problem, are often faster and more accurate than direct methods, methods that in theory produce an exact answer in a finite number of steps.
8. There are many theorems bounding the error in solutions produced on real computers. That is, the theorems don’t just bound the error from hypothetical calculations carried out in exact arithmetic but bound the error from arithmetic as carried out in floating point arithmetic on computer hardware.
9. It is hardly ever necessary to compute the inverse of a matrix.
10. There is remarkably mature software for numerical linear algebra. Brilliant people have worked on this software for many years.

Related posts:

Don’t invert that matrix
Searching for John Francis
Applying PageRank to biology
Matrix cookbook
What is the cosine of a matrix?

There is hardly ever a good reason to invert a matrix.

What do you do if you need to solve Ax = b where A is an n x n matrix? Isn’t the solution A^-1 b? Yes, theoretically. But that doesn’t mean you need to actually find A^-1. Solving the equation Ax = b is faster than finding A^-1. Books might write the problem as x = A^-1 b, but that doesn’t mean they expect you to calculate it that way.

What if you have to solve Ax = b for a lot of different b’s? Surely then it’s worthwhile to find A^-1. No. The first time you solve Ax = b, you factor A and save that factorization. Then when you solve for the next b, the answer comes much faster. (Factorization takes O(n³) operations. But once the matrix is factored, solving Ax = b takes only O(n²) operations. Suppose n = 1,000. This says that once you’ve solved Ax = b for one b, the equation can be solved again for a new b 1,000 times faster than the first one. Buy one get one free.)

What if, against advice, you’ve computed A^-1. Now you might as well use it, right? No, you’re still better off solving Ax = b than multiplying by A^-1, even if the computation of A^-1 came for free. Solving the system is more numerically accurate than the performing the matrix multiplication.

It is common in applications to solve Ax = b even though there’s not enough memory to store A^-1. For example, suppose n = 1,000,000 for the matrix A but A has a special sparse structure — say it’s banded — so that all but a few million entries of A are zero.  Then A can easily be stored in memory and Ax = b can be solved very quickly. But in general A^-1 would be dense. That is, nearly all of the 1,000,000,000,000 entries of the matrix would be non-zero.  Storing A requires megabytes of memory, but storing A^-1 would require terabytes of memory.

Related posts:

Floating point numbers are a leaky abstraction
Ten surprises from numerical linear algebra

James Burke had a television series Connections in which he would create a connection between two very different things. For example, in one episode he starts with the discovery of the touchstone for testing precious metals and tells a winding tale of how the touchstone led centuries later to the development of nuclear weapons.

I had a Connections-like moment when a calendar led to some physics, which then lead to Andrew Lloyd Webber’s musical Cats.

A few days ago I stumbled on Ron Doerfler’s graphical computing calendar and commented on the calendar here. When I discovered Ron Doerfler’s blog, I bookmarked his article on Oliver Heaviside to read later. (Heaviside was a pioneer in what was later called distribution theory, a way of justifying such mathematical mischief as differentiating non-differentiable functions.) As I was reading the article on Heaviside, I came to this line:

At one time the ionosphere was called the Heaviside layer …

Immediately the lyrics “Up, up, up to the Heaviside layer …” started going through my head. These words come from the song “The Journey to the Heaviside Layer” from Cats. I had never thought about “Heaviside” in that song as being related to Mr. Heaviside. I’ve never seen the lyrics in print, so I thought the words were “heavy side” and didn’t stop to think what they meant.

Andrew Lloyd Webber based Cats on Old Possum’s Book of Practical Cats by T. S. Eliot. The song “The Journey to the Heaviside Layer” in particular is based on the poem Old Deuteronomy from Eliot’s book. Webber used the Heaviside layer as a symbol for heaven, based on an allusion in one of T. S. Eliot’s letters. The symbolism is obvious in the musical, but I hadn’t thought about “Heaviside layer” as meaning “the heavens” (i.e. the upper atmosphere) as well as heaven in the theological sense.

An integer is either a perfect square or its square root is irrational. Said a different way, when you compute the square root of an integer, there are either no figures to the right of the decimal or there are an infinite number of figures to right of the decimal and they don’t repeat. There’s no middle ground. You can’t hope, for example, that the decimal expansion might stop or repeat after a hundred or so terms.

This theorem came up recently when I was talking to one of my kids about her math class, so I decided to look up the proof. It’s easier than I expected, not much harder than the familiar proof that the square root of 2 is irrational. Here goes.

Suppose a/b is a fraction in lowest terms, i.e. a and b are relatively prime, and a/b a solution to xⁿ = c where n > 0 is an integer and c is an integer. Then

(a/b)ⁿ = aⁿ / bⁿ = c

and so

aⁿ / b = c b^n-1.

Now the right side of the equation above is an integer, so the left side must be an integer as well. But b is relatively prime to a, and so b is relatively prime to aⁿ. The only way aⁿ / b could be an integer is for b to equal 1 or -1. And so a/b must be an integer.

This proof shows that what we said about square roots extends to cube roots and in fact to all integer roots. For example, the fifth root of an integer is either an integer or an irrational number.

Update: Note that there’s another proof in the comments, one that I believe is easier to follow.

How many gifts are there in the song Twelve Days of Christmas?

Day 1: 1 gift
Day 2: 1 + 2 = 3 gifts
Day 3: 1 + 2 + 3 = 6 gifts
…
Day 12: 1 + 2 + 3 + … + 12 = 78 gifts

The number of gifts on day n is the nth triangular number. The total number of gifts up to and including day n is the sum of the first n triangular numbers, known as the nth tetrahedral number. In the image below, the total number of balls is the fifth tetrahedral number. The number of balls in each layer are triangular numbers. (Image credit: Math is Fun.)

I’ll develop a formula for tetrahedral numbers and continuations of the pattern  such as the sum of tetrahedral numbers etc.

First, let T(n, 1) = n.

Next, let T(n, 2) be the nth triangular number. So T(n, 2) is the sum of the first n terms in the sequence T(i, 1).

Next, let T(n, 3) be the nth tetrahedral number. So T(n, 3) is the sum of the first n terms in the sequence T(i, 2).

In general, define T(n, k) to be the sum of the first n terms in the sequence T(i, k-1). You could think of T(n, k) as the nth k-dimensional triangular number. (A tetrahedron is a sort of 3-dimensional triangle. It’s a pyramid whose base is a triangle. T(n,4) would count balls arranged in a sort of 4-dimensional triangle, a simplex in 4 dimensions.)

Theorem: T(n, k) = n(n+1)(n+2) … (n+k-1)/k!

Corollary: There are T(12, 3) = 12*13*14/6 = 364 gifts in the Twelve Days of Christmas.

See these notes for a elementary proof by induction.

Update: Here’s more advanced proof that uses calculus of finite differences.  The more advanced proof requires more background, but it also gives a better idea of how someone might have discovered the formula.

Related posts:

Binomial coefficients
Splitting a convex set through its center
My favorite Christmas carol

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.

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

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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