» Math

# PJM Lifestyle

Friday, January 24th, 2014 - by Charlie Martin

Let’s take a road trip. We’re going to visit all the capitals of all the 48 contiguous states, starting from Denver.

Now, since we’re taking vacation days to do this, we don’t want to visit any capital more than once, and we want to do this in the least time, or in the shortest distance traveled, which is pretty much the same thing.

Starting from Denver, if we only want to visit one other state capital, planning the trip is easy. Denver to Cheyenne. Boom. Two capitals is easy — you can either go Denver to Cheyenne to Topeka, or Denver to Topeka to Cheyenne. Add in Santa Fe, well, there are several routes. Ignoring, for the sake of keeping my readers awake, some details, basically as we expand the number of cities, we have to explore every possible ordering of the cities. So if we stay in Denver, visiting exactly one capital, we have exactly one route. Visit one other city, we have two choices of trip plan — Denver Cheyenne Denver, or Cheyenne Denver Cheyenne. Visit two other cities — so three cities total — and there are six choices. (Remember this includes trips where someone wants to start and end in Santa Fe or, Gods forbid, Topeka.) So, no other cities, you have 1 choice. One other city, you still only have 2 choices. Two other cities, you have six route choices.

Three other cities? Well, you can use all the routes for two other cities, and go to the new city from each of them. So you multiply the number of routes you’ve already got by the total number of cities. In other words, if we have n cities, we want n × n-1×n-2 … 1.

Many of you already recognize this as n! — “n factorial” — which is an important idea in a lot of different areas of math.

## 13 Weeks: Diets and Black Swans

Saturday, January 11th, 2014 - by Charlie Martin

Nassim Nicholas Taleb’s book The Black Swan introduced an old term and then, annoyingly, redefined it. For Sir Karl Popper, the black swan was an observation about logical quantification: if you assert “all swans are white” then the observation of a single black swan falsifies the assertion.

Taleb’s observation is different, although related: he’s observing that really unexpected events are unexpected: we have a model of the world that says “The US mainland is secure from attack” that seems perfectly plausible on 10 September 2001; we believe “Islamist terrorism is on the run” and then a bomb blows up in Boston.

(There’s a more sophisticated way to deal with all of these called Bayesian inference. We’ll leave the details for a science column, but in a few words, a Bayesian starts with an assumed a priori estimate of the probability of an event. After observation, they have a new a postieriori estimate that incorporates new experience.)

But there’s yet a third way to think about these that shows us how mathematics and probability can show us surprising things.

(Yes, this is a diet and exercise column, just a little further down.)

## The End of Poverty in America

Thursday, December 5th, 2013 - by Charlie Martin

President Obama, on Wednesday, made a big speech about “economic inequality” and vowed to spend his last three years in office working to increase the federal minimum wage, as well as a lot of other things.

Just as an aside, every time I hear talk about increasing the minimum wage — there’s a strike on today at some fast food places to raise their wage to \$15 an hour as well — I have a conversation something like this.

“I think increasing the minimum wage is a wonderful idea. In fact, let’s raise it to \$100 an hour.”

“Oh, you’re being silly.”

“No, imagine. Raise minimum wage to \$100 an hour. That way, everyone will be making \$200,000 a year. We’ll all be rich!

“Racist.”

Okay, I’ll grant that it usually takes two or three more exchanges before someone calls me racist, or a tea-bagger, or even an economic royalist if they’re of a classical turn of mind. The one thing I’ve never had anyone do is explain to me why if a \$15 an hour minimum wage is a good idea, a \$100 an hour minimum wage is a bad idea.

I suspect it’s because they realize that if they do, the jig is up: if they raise the minimum wage that high, companies won’t be able to pay the wage, and either there will be massive unemployment or massive inflation, as companies try to make up the difference. Mostly unemployment and shutdowns, because the money supply can’t grow that fast without a Weimar meltdown. But the trade-off is basically a linear function — raising the minimum wage by a lesser amount just means fewer people lose their jobs or go out of business. In the case of fast food workers, what would happen is that hamburger-making machines would become cheaper than burger-flippers. (In fact, that break-even is already past, the burger-flippers just don’t know it yet.)

In any case, though, this seems to be a solution in search of a problem, because there is no poverty in America, and I can prove it. According to a Cato Institute study published last year, the combined expenditures for federal and state governments directed to means-tested public assistance — “welfare” — is approximately \$1 trillion (yes, with a “T”) a year.

There are approximately 48 million people in the U.S. with incomes at the poverty level or below.

The application of advanced mathematics — long division, and I did it in my head thank you very much — tells us that’s about \$21,000 per person per year. Obviously, that’s \$84,000 for a family of four.

That’s got a problem, though. According to the 2013 Federal Poverty Guidelines, the poverty level for a family of four is \$23,950. The total of \$84,000 is roughly 380 percent of the federal poverty guidelines.

Obviously, there’s no poverty left in America.

Unless, of course, that money isn’t actually being spent on the poor people at all. I wonder where it goes?

More:

## 3 Reasons Why Dating is Especially Hard in Washington, D.C.

Tuesday, November 5th, 2013 - by Becky Graebner

At a recent convening of the “female minds” during a birthday party celebration, I was reminded of the challenges posed by the D.C. dating scene. A fellow friend at this birthday dinner was regaling the group with her predicament: she had to leave the birthday party early for a date.

Normally, this topic is the launching pad for well-wishes, compliments, and giggles. In this case, the poor girl was dreading her impending date. Subsequent conversations with the male in question after agreeing to the date had made her a little wary.  He was cocky and pushy–which made her question if he was interested in anything more than a quick hook-up.  However, she didn’t want to back out of the date 40 minutes before they were supposed to meet up.

We tried to psyche her up. It’s great to meet new people! A night on the town will be fun!

No go. She was all frowns and pessimism as she slid off her stool and collected her coat and purse.

“Why is dating in D.C. so hard?” she asked as she turned for the door.

We all knew from personal experience what she meant, but none of us had an answer…

Washington D.C. is always a nominee for those lists with titles like “worst city for singles” or “worst city for dating.” It’s not surprising, really. Washington, D.C. is not a normal city. Although the representatives of the nation live and work here, The Capital is in a fantasy land of its own, shielded from the real-world by a thick bubble. It makes sense that this removal from reality in the workplace would also translate to the playground. I do know good people who have met, dated, and married people that they met while living in D.C. However, these people seem to be either part of the lucky minority or are D.C.-dating-warriors who persevered after several harrowing attempts.

Here are three reasons why dating in D.C. is particularly difficult:

## The Math-iest Math Joke

Wednesday, October 30th, 2013 - by Charlie Martin

Q: What does the “B” in Benoit B. Mandelbrot stand for?
A: Benoit B. Mandelbrot.

## Infinity: Big and Bigger

Thursday, October 17th, 2013 - by Charlie Martin

On the Internet, you can never go wrong by quoting the The Hitchhiker’s Guide to the Galaxy.

Space is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.

Now, it’s kind of a cheat, because i’m not going to talk about that kind of space, I’m going to talk about spaces in a mathematical sense. But I’m offering something in exchange, because I’m going to talk about spaces that are much bigger than mere physical space.

The point of this is really to talk about (echo effect) infinity. And beyond.

Mathematically, space is much simpler than the thing in which your coffee cup is located just out of reach and that keeps your cat from being exactly where you’re sitting, no matter how much he tries. In mathematics, a space is simply a set of some sort with some kind of additional structure. (A set is just some collection of things with no duplicates, like {1, 2, 3, 4, 5}. By convention, we put sets into braces like that example.)

So far, that’s not a space — we haven’t said anything further about it than there is a bag full of things. But — since I’ve chosen a set we conveniently already know a lot about — we know that the set is ordered because we agree that 5 is bigger than 4. And we have a space.

Okay, it’s a pretty boring space, but it’s a space.

There are some other rules we think we know, like addition — 1+2=3. But in our little space, we immediately run into trouble, because 3+4 equals what? Oh, 7, but 7 isn’t in the set. To take care of 3+4, we need to expand the set to be at least {1,2,3,4,5,6,7} and then we’re immediately going to have the problem of 4+5, or for that matter, 7+1.

Now, with nothing more than the idea of addition (we talked about ordering, but we can define an order in terms of addition) we’ve run into our first experience with infinity. There is a set N that we can define like this:

• 0 is part of N
• For anything that is part of N, which we’ll call n, n+1 is also in N.

We call N the natural numbers.

Now, N is pretty big. After all, no matter what n we pick, there’s always something bigger. This is what we call infinite. And all is well, until we think about subtraction: we know 3-1=2, and we know 2-1=1, and we know 1-1=0, but 0-1 isn’t in our set. So we define a new set called the integers which has new elements -1, -2, -3, and so on. We can throw in multiplication now, and all is good, but when we put in division we’re in trouble again: 2÷3 and 1÷2 aren’t in there. So we define another set called the rational numbers, Q.

Now, we’ve pretty much defined all the numbers anyone had any use for until the Greeks and Egyptians screwed it all up by trying to measure fields and distances.

## Clown Car Web Design

Thursday, October 10th, 2013 - by Charlie Martin

A Programming Sutra

This is the way I heard it. (That’s the way all sutras start.) Long long ago — about 1997 and I’m not naming names to protect the innocent and because I figure the statute of limitations is up for the guilty, and the company I’m going to talk about has been through bankruptcy and several acquisitions so it’s not the same company anyway — a major toy retailer ToysForKids (TFK) with stores in malls all over America heard about this nifty new thing called “the web.” As I heard the story, two programmers in IT had the idea that TFK should be selling toys on the internet. They got permission to do a sort of side project, semi-bootleg, to build a demonstration e-commerce web site, ToysForKids.com. (By the way, that domain name is now owned by a domain-squatter in Hong Kong called “iGenesis Limited”, but then ToysForKids never existed anyway.)

They built the web site on a desktop server using a scripting language called `tcl`, and demonstrated it. It looked so good they got permission to take it live, and they happily started making dozens of sales a day with it. It really was a lovely site, too, won lots of awards.

The CIO was so pleased that he arranged a demo for the CEO. The CEO was so pleased that he arranged a big advertising buy for Thanksgiving Day during the football game — as I recall, \$50 million — so that everyone would know about the new ToysForKids.com.

Everyone did. And everyone’s mom, wife, and girlfriend that had a computer went and tried to start their Christmas Shopping sometime in the first quarter.

Now, remember this is 15 years ago. The desktop server they were using wouldn’t make a good iPad now, and the Internet connection, while good for the time, had less capacity than Comcast promises me.

And everyone who was bored with football and had computer access was trying to use it. The site pretty much melted down; it wasn’t long before the programmers had found different jobs, the CIO wanted to spend more time with his family, and the CEO, um, retired.

## Cocktail Napkin Website Planning and Obamacare

Thursday, October 3rd, 2013 - by Charlie Martin

Just for variety, today’s science column is about something I actually have some professional qualifications to write about.

No, that really hasn’t ever slowed me down, I just wanted to note it.

In fact, my master’s thesis, “A Software Performance Engineering Environment,” was about tools to allow software engineers to develop performance models of software alongside the software. I spent some years in IBM and Sun’s consulting practices, usually dealing in one way or another with web-based businesses. I had a very popular talk, “Capacity Planning on a Cocktail Napkin,” which I later wrote as an article for SmartBear Software.

There really is only one explanation for the meltdown of the Obamacare exchanges since 1 October, and that’s utter incompetence.

Now, lemme ‘splain.

Back in the old days, at the Very Beginning Of The Web, all a web server could do was deliver a static piece of text. It was a brilliant hack by Tim Berners-Lee, who realized that he could build a little editor for a simple markup language and add one little change — a special tag that could address another page of text in the same markup language. To make it work, he needed a program that could return those files and a simple way the editor could ask for the files it wanted. The markup language was a subset of a commonly available commercial standard, SGML, called the Hyper Text Markup Language, HTML, the server program using a very simple text-based protocol called the Hyper Text Transfer Protocol, HTTP, and the rest, as they say was ….

Yes, class, that’s right, “history.”

From this simple hack the whole World Wide Web was made.

Now, with the same foresight that led me to buy Borland stock over Microsoft when they both went public, I thought at the time that it was a mildly amusing notion, but I didn’t see much future in it. I was head-down in category theory working on my dissertation.

## The Greatest Genius No One Has Heard Of

Friday, September 13th, 2013 - by Charlie Martin

In the 1930s, “computer” was a job description: someone, usually a woman of mathematical bent, with an adding machine and a big sheet of columnar paper who performed a rigorous routine of hand calculations, using paper and pencil, slide rules and tables of logarithms. Stone knives and bearskins weren’t involved, but to modern eyes they might as well have been.

Large research organizations and the Department of War had a few special purpose mechanical computers intended to integrate differential equations. Vannevar Bush (who deserves his own article someday) brought a young grad student to MIT to work on the differential analyzer, a relatively advanced version of these. This video shows a version of the differential analyzer being applied to a problem for which it was utterly unsuited in Earth vs. the Flying Saucers:

This young man, a recent graduate of the University of Michigan, was named Claude Shannon, Jr. Shannon, while working on the differential analyzer, had the insight that these same computations could be done using combinations of a few simple circuits that performed basic logical operations on true and false values. He described how this could be done, and invented the whole concept of digital circuits, which derive from from Shannon’s thesis on what he called switching theory.

His Master’s thesis.

## Flipping Coins and Cancer Cases

Thursday, July 18th, 2013 - by Charlie Martin

Since science is all about making things predictable, it is sort of a surprise that many of the advances in science in the last hundred years have been made using mathematics about things which are inherently and intrinsically unpredictable: the mathematics of probability, and its applied-math stepchild, statistics.

The usual example of something that’s inherently unpredictable is flipping a fair coin. Take a quarter from your purse, flip it, and it comes up either heads or tails.

Now, because I know my readers, I can tell someone is getting set to write me a comment about how there’s no such thing as a perfectly fair coin, or that it can also land on an edge, or explaining how they learned to flip a coin so it made exactly one turn and so they could always predict how it would land, so let me just say: this is mathematics, it’s a perfectly fair coin, and we’re going to be catching it in the air so it never lands on edge. So just stop.

As I said, when you flip this perfectly fair coin, it either comes up heads or tails. The next time you flip it, it also comes up either heads or tails, and which comes up doesn’t depend on the previous flip at all. Technically, we’d say it “has no memory”, it’s memory-less. Random things with this memory-less property are going to be important, so remember the word.

The gambler’s fallacy is imagining that something like a fair coin actually has memory — in other words, if you’ve had a run of heads, you’re “due for” tails to come up. The truth is that every time you flip a coin, what comes up is independent of all the previous flips. What makes you think you’re “due for” a tails is that over many coin flips, the likelihood of getting a run of many heads or tails gets smaller, and it gets smaller quickly.

## Why Does Classical Music Make You Smarter?

Tuesday, April 23rd, 2013 - by David P. Goldman

Thirty-six million Chinese kids now study classical piano, not counting string and woodwind players. Chinese parents pay for music lessons not because they expect their offspring to earn a living at the keyboard, but because they believe it will make them smarter at their studies. Are they right? And if so, why?

The intertwined histories of music and mathematics offer a clue. The same faculty of the mind we evoke playfully in music, we put to work analytically in higher mathematics. By higher mathematics, I mean calculus and beyond. Only a tenth of American high school students study calculus, and a considerably smaller fraction really learn the subject. There is quite a difference between learning the rules of Euclidean geometry and the solution of algebraic equations: the notion that the terms of a convergent infinite series sum up to a finite number requires a different kind of thinking than elementary mathematics. The same kind of thinking applies to playing classical music. Don’t look for a mathematical formula to make sense of music: what higher mathematics and classical music have in common is not an algorithm, but a similar demand on the mind. Don’t expect the brain scientists to show just how the neurons flicker any time soon. The best music evokes paradoxes still at the frontiers of mathematics.

In an essay for First Things titled “The Divine Music of Mathematics,” just released from behind the pay wall, I show that the first intimation of higher-order numbers in mathematics in Western thought comes from St. Augustine’s 5th-century treatise on music. Our ability to perceive complex and altered rhythms in poetry and music, the Church father argued, requires “numbers of the intellect” which stand above the ordinary numbers of perception. A red thread connects Augustine’s concept with the discovery of irrational numbers in the 15th century and the invention of calculus in the 17th century. The common thread is the mind’s engagement with the paradox of the infinite. The mathematical issues raised by Augustine and debated through the Renaissance and the 17th-century scientific revolution remain unsolved in some key respects.

## 13 Weeks: Getting Animated

Saturday, March 30th, 2013 - by Charlie Martin

Week 8 of my second 13 week season: low carb diet and more exercise, tracking my weight, blood glucose, and body fat. You can follow me at my 13 Weeks Facebook page for daily updates, and you can join Fitocracy (free!) and follow my daily exercise, and maybe even start tracking your own.

A few days ago, PJ Lifestyle ran an excerpt from Leonard Mosely’s book Disney’s World, in which Walt Disney, in a letter to his partner Ub Iwerks, expressed his frustration with the his first sound cartoon, the now-iconic Steamboat Willie.

He’s pretty depressed. he doesn’t like Hollywood, he doesn’t like being away from home, and he’s losing confidence in the still-unfinished film.  You can see why, when he was having trouble selling the idea, and animation is a frustrating process anyway.  This was in the days of the most primitive hand-drawn animation, where every frame of the film had to be hand drawn on clear acetate, with tiny changes from frame to frame. Twenty-four times for each second of film.  In this 7 minute 23 second film, that’s something like 10,600 frames.  He was tired, and he was bored, and he had trouble seeing any progress.

Why did this strike me, he asked rhetorically?  Well, it reminds me of my ongoing glucose/bodyfat/weight project.  Here I am, eight weeks into my second season, 147 days since I first started tracking this, and it’s a little frustrating and hard. I’ve been less diligent about the exercise, and I do find myself missing things I used to eat.  Like chocolate. And pasta. And bread.  And while I have lost some weight, it’s slow and the day to day variations make it hard to see.  It’s like Disney must have felt — another 24 frames, another day’s work, and what did he have? Another lousy second of film.  That no one wanted to distribute.  He was past the initial excitement and into the slog.

Right now, this project feels much the same.  I’m actually losing weight, and I can see changes — more muscle coming back to my arms, and to put it bluntly, my boobs are smaller.  I’ve lost six inches around my waist, and I can feel that every time I put on a pair of pants that were in the back of the closet because I hadn’t been able to wear them.  But at the same time, the progress is a little slow and hard to see, and it’s a little hard to explain why it should matter to anyone — especially me.

But then I got thinking, and a little Excel-fu got me this.  Here’s my actual weight, charted over the last sixty days, with a trend line.  This is very much like the other charts I’ve been posting.

My actual weight for the last 60 days.

Trend line is down. This is good. It’s not down very fast, and the added muscle certainly explains that — but also notice that individuual weights vary pretty wildly around that trend line. So here’s another chart.

## VIDEO: October is ‘The Month of the Flexagon’

Monday, October 1st, 2012 - by PJ Lifestyle Science

Hat tip I09

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