A subway topologist

One of my favorite books when I was growing up was the Mathematics volume in the LIFE Science Library. I didn’t own the book, but my uncle did, and I’d browse through the book whenever I visited him. I was too young at the time to understand much of what I was reading.

One of the pages that stuck in my mind was a photo of Samuel Eilenberg. His name meant nothing to me at the time, but the caption titled “A subway topologist” caught my imagination.

… Polish-born Professor Samuel Eilenberg sprawls contemplatively in his Greenwich Village apartment in New York City. “Sometimes I like to think lying down,” he says, “but mostly I like to think riding on the subway.” Mainly he thinks about algebraic topology — a field so abstruse that even among mathematicians few understand it. …

I loved the image of Eilenberg staring intensely at the ceiling or riding around on a subway thinking about math. Since then I’ve often thought about math while moving around, though usually not on a subway. I’ve only lived for a few months in an area with a subway system.

The idea that a field of math would be unknown to many mathematicians sounded odd. I had no idea at the time that mathematicians specialized.

Algebraic topology doesn’t seem so abstruse now. It’s a routine graduate course and you might get an introduction to it in an undergraduate course. The book was published in 1963, and I suppose algebraic topology would have been more esoteric at the time.

 

Read More

Bringing bash and PowerShell a little closer together

I recently ran across PSReadLine, a project that makes the PowerShell console act more like a bash shell. I’ve just started using it, but it seems promising. I’m switching between Linux and Windows frequently these days and it’s nice to have a little more in common between the two.

I’d rather write a PowerShell script than a bash script, but I’d rather use the bash console interactively. The PowerShell console is essentially the old cmd.exe console. (I haven’t kept up with PowerShell in a while, so maybe there have been some improvements, but it’s my impression that the scripting language has moved forward and the console has not.) PSReadLine adds some bash-like console conveniences such as Emacs-like editing at the command prompt.

Update: Thanks to Will for pointing out Clink in the comments. Clink sounds like it may be even better than PSReadLine.

PowerShell logo

Read More

Making change

How many ways can you make change for a dollar? This post points to two approaches to the problem, one computational and one analytic.

SICP gives a Scheme program to solve the problem:

(define (count-change amount) (cc amount 5))

(define (cc amount kinds-of-coins)
    (cond ((= amount 0) 1)
    ((or (< amount 0) (= kinds-of-coins 0)) 0)
    (else (+ (cc amount
                 (- kinds-of-coins 1))
             (cc (- amount
                    (first-denomination
                     kinds-of-coins))
                     kinds-of-coins)))))

(define (first-denomination kinds-of-coins)
    (cond ((= kinds-of-coins 1) 1)
          ((= kinds-of-coins 2) 5)
          ((= kinds-of-coins 3) 10)
          ((= kinds-of-coins 4) 25)
          ((= kinds-of-coins 5) 50)))

Concrete Mathematics explains that the number of ways to make change for an amount of n cents is the coefficient of z^n in the power series for the following:

\frac{1}{(1 - z)(1 - z^5)(1 - z^{10})(1 - z^{25})(1 - z^{50})}

Later on the book gives a more explicit but complicated formula for the coefficients.

Both show that there are 292 ways to make change for a dollar.

Read More

A puzzle puzzle

Jigsaw puzzles that say they have 1,000 pieces have approximately 1,000 pieces, but probably not exactly 1,000. Jigsaw puzzle pieces are typically arranged in a grid, so the number of pieces along a side has to be a divisor of the total number of pieces. This means there aren’t very many ways to make a puzzle with exactly 1,000 pieces, and most have awkward aspect ratios.

Since jigsaw pieces are irregularly shaped, it may be surprising to learn that the pieces are actually arranged in a regular grid. At least they usually are. There are exceptions such as circular puzzles or puzzles that throw in a couple small pieces that throw off the grid regularity.

How many aspect ratios can you have with a rectangular grid of 1,000 points? Which ratio comes closest to the golden ratio? More generally, answer the same questions with 10^n points for positive integer n.

More puzzles:

A knight’s random walk
Peculiar property of 3909511
Roman numeral problem
A perspective problem

Read More

Ellipsoid surface area

How much difference does the earth’s equatorial bulge make in its surface area?

To first approximation, the earth is a sphere. The next step in sophistication is to model the earth as an ellipsoid.

The surface area of an ellipsoid with semi-axes abc is

A = 2\pi \left( c^2 + \frac{ab}{\sin\phi} \left( E(\phi, k) \sin^2\phi + F(\phi, k) \cos^2 \phi\right)\right)

where

\cos

and

m = k^2 = \frac{a^2(b^2 - c^2)}{b^2(a^2 - c^2)}

The functions E and F are incomplete elliptic integrals

 F(\phi, k) = \int_0^\phi \frac{d\theta}{\sqrt{1 - k^2 \sin^2\theta}}

and

E(\phi, k) = \int_0^\phi \sqrt{1 - k^2 \sin^2\theta}\,d\theta

implemented in SciPy as ellipeinc and ellipkinc. Note that the SciPy functions take m as their second argument rather its square root k.

For the earth, a = b and so m = 1.

The following Python code computes the ratio of earth’s surface area as an ellipsoid to its area as a sphere.

from scipy import pi, sin, cos, arccos
from scipy.special import ellipkinc, ellipeinc

# values in meters based on GRS 80
# http://en.wikipedia.org/wiki/GRS_80
equatorial_radius = 6378137
polar_radius = 6356752.314140347

a = b = equatorial_radius
c = polar_radius

phi = arccos(c/a)
# in general, m = (a**2 * (b**2 - c**2)) / (b**2 * (a**2 - c**2))
m = 1 

temp = ellipeinc(phi, m)*sin(phi)**2 + ellipkinc(phi, m)*cos(phi)**2
ellipsoid_area = 2*pi*(c**2 + a*b*temp/sin(phi))

# sphere with radius equal to average of polar and equatorial
r = 0.5*(a+c)
sphere_area = 4*pi*r**2

print(ellipsoid_area/sphere_area)

This shows that the ellipsoid model leads to 0.112% more surface area relative to a sphere.

Source: See equation 19.33.2 here.

Update: It was suggested in the comments that it would be better to compare the ellipsoid area to that of a sphere of the same volume. So instead of using the average of the polar and equatorial radii, one would take the geometric mean of the polar radius and two copies of the equatorial radius. Using that radius, the ellipsoid has 0.0002% more area than the sphere.

Read More

Pi and The Raven

Michael Keith rewrote Edgar Allen Poe’s poem The Raven to turn it into a mnemonic for pi. Keith’s version follows the original quite well considering his severe constraints. The full poem has 18 stanzas. Here I include only the first and last. The full version can be found here.

***

Poe, E.
Near a Raven

Midnights so dreary, tired and weary,
Silently pondering volumes extolling all by-now obsolete lore,
During my rather long nap — the weirdest tap!
An ominous vibrating sound disturbing my chamber’s antedoor.
“This,” I whispered quietly, “I ignore.”

So he sitteth, observing always, perching ominously on these doorways.
Squatting on the stony bust so untroubled, O therefore.
Suffering stark raven’s conversings, I am so condemned, subserving,
To a nightmare cursed, containing miseries galore.
Thus henceforth, I’ll rise (from a darkness, a grave) — nevermore!

***

The number of letters in most words encodes a digit of pi. Words with 10 letters encode a zero. Words with more than 10 letters encode two consecutive digits of pi. The poem encodes the first 740 digits of pi.

Read More

Classical programming

The classical education model is based on the trivium of grammar, logic, and rhetoric. See, for example, Dorothy Sayers’ essay The Lost Tools of Learning.

The grammar stage of the trivium could be literal language grammar, but it also applies more generally to absorbing the basics of any subject and often involves rote learning.

The logic stage is more analytic, examining the relationships between the pieces gathered in the grammar stage. Students learn to construct sound arguments.

The rhetoric stage is focused on eloquent and persuasive expression. It is more outwardly focused than the previous stages, more considerate of others. Students learn to create arguments that are not only logically correct, but also memorable, enjoyable, and effective.

It would be interesting to see a classical approach to teaching programming. Programmers often don’t get past the logic stage, writing code that works (as far as they can tell). The rhetoric stage would train programmers to look for solutions that are not just probably correct, but so clear that they are persuasively correct. The goal would be to write code that is testable, maintainable, and even occasionally eloquent.

 

Parthenon replica in Nashville, TN.

Read More

Iterative linear solvers as metaphor

Gaussian elimination is systematic way to solve systems of linear equations in a finite number of steps. Iterative methods for solving linear systems require an infinite number of steps in theory, but may find solutions faster in practice.

Gaussian elimination tells you nothing about the final solution until it’s almost done. The first phase, factorization, takes O(n^3) steps, where n is the number of unknowns. This is followed by the back-substitution phase which takes O(n^2) steps. The factorization phase tells you nothing about the solution. The back-substitution phase starts filling in the components of the solution one at a time. In application n is often so large that the time required for back-substitution is negligible compared to factorization.

Iterative methods start by taking a guess at the final solution. In some contexts, this guess may be fairly good. For example, when solving differential equations, the solution from one time step gives a good initial guess at the solution for the next time step. Similarly, in sequential Bayesian analysis the posterior distribution mode doesn’t move much as each observation arrives. Iterative methods can take advantage of a good starting guess while methods like Gaussian elimination cannot.

Iterative methods take an initial guess and refine it to a better approximation to the solution. This sequence of approximations converges to the exact solution. In theory, Gaussian elimination produces an exact answer in a finite number of steps, but iterative methods never produce an exact solution after any finite number of steps. But in actual computation with finite precision arithmetic, no method, iterative or not, ever produces an exact answer. The question is not which method is exact but which method produces an acceptably accurate answer first. Often the iterative method wins.

Successful projects often work like iterative numerical methods. They start with an approximation solution and iteratively refine it. All along the way they provide a useful approximation to the final product. Even if, in theory, there is a more direct approach to a final product, the iterative approach may work better in practice.

Algorithms iterate toward a solution because that approach may reach a sufficiently accurate result sooner. That may apply to people, but more important for people is the psychological benefit of having something to show for yourself along the way. Also, iterative methods, whether for linear systems or human projects, are robust to changes in requirements because they are able to take advantage of progress made toward a slightly different goal.

Related post: Ten surprises from numerical linear algebra

Read More

Multiple zeta

The Riemann zeta function, introduced by Leonard Euler, is defined by

\zeta(k) = \sum n^{-k}

where the sum is over all positive integers n.

Euler also introduced a multivariate generalization of the zeta function

\zeta(k_1, \ldots, k_r) = \sum n_1^{-k_1}\cdots n_r^{-k_r}

where the sum is over all decreasing k-tuples of positive integers. This generalized zeta function satisfies the following beautiful identity:

 \zeta(a)\,\zeta(b) = \zeta(a, b) + \zeta(b, a) + \zeta(a+b)

The multivariate zeta function and identities such as the one above are important in number theory and are the subject of open conjectures.

Source

Read More

Benchmarking C++, Python, R, etc.

The other day Travis Oliphant pointed out an interesting paper: A Comparison of Programming Languages in Economics. The paper benchmarks several programming languages on a computational problem in economics.

All the usual disclaimers about benchmarks apply, your mileage may vary, etc. See the paper for details.

Here I give my summary of their summary of their results. The authors ran separate benchmarks on Mac and Windows. The results were qualitatively the same, so I just report the Windows results here.

Times in the table below are relative to the fastest C++ run.

Language Time
C++ 1.00
Java 2.10
Julia 2.70
CPython 155.31
Python with Numba 1.57
R 505.09
R using compiler package 243.38

 

The most striking result is that the authors were able to run their Python code 100x faster, achieving performance comparable to C++, by using Numba.

Read More

Weak static type systems

Comment by Simon Peyton Jones in an interview:

People often dislike static type systems because they’ve only met weak ones. A weak or not very expressive type system gets in your way all the time. It prevents you from writing functions you want to write that you know are fine. … The solution is not to abandon the type system but to make the type system more expressive.

In particular, he mentions Haskell’s polymorphic types and type inference as ways to make strong static typing convenient to use.

Read More

Closed-world expertise

Venkat Rao has an interesting take on the ideas of deliberate practice, flow, and the 10,000 hour rule. In The Deliberate Practice of Disruption he points out that these ideas of expertise are always presented in closed worlds.

The real problem is that research on expertise focuses on fields where “expertise” is a well-posed and objectively codified notion. This means mature fields that are closed and bounded, and can be easily observed, modeled and studied under laboratory conditions. So it is not surprising that the work … is based on fields like “medicine, music, chess and sports” … all sharply circumscribed and regulated domains.

Rao’s essay may be a bit too harsh on closed-world domains and a bit too romantic about open-world domains, but the distinction between the domains is important. You don’t become a successful entrepreneur, for example, the same way you become a successful violinist. Closed worlds place a much higher emphasis on error-elimination, at least initially, than do open worlds. In an open world, the concept of an error may not even make sense. Where there is no law there is no sin.

Consulting is a more open world than academia. As Rao notes, academia can close off an otherwise open world through “bureaucratic productivity measures like publications and citations.” Clients are happy if you solve their problems. They could not care less whether your solutions are publishable and all that implies. Original and thoroughly footnooted work that doesn’t solve their problems is not appreciated.

Clients are not going to give you an oral exam to see whether you’ve mastered some canon. And they don’t care if you cross academic boundary lines to use something “outside your field.” They do care about credentials sometimes, but in a pragmatic way: they may need someone with the right credentials to review something. In that case, your credentials are part of the solution.

Related posts:

How to know it all

Evaluate people at their best or at their worst?

Read More

Every exercise in the book

When I did an independent study course with Ted Odell, he told me to get a copy of De Vito’s Functional Analysis and work every exercise. I don’t recall whether I actually worked every problem, though I believe I at least did most of them. I heard of someone who learned algebraic geometry by working every problem in Hartshorne.

Doing all the exercises in a book isn’t a bad way to learn something, though it depends on the book, what you’re trying to accomplish, and on the quality and quantity of the exercises.

Have you ever gone through a book working every exercise? If so, what book? How was your experience?

 

 

Read More