In anticipation of this site being taken down this summer, I've transferred my blog to Wordpress

http://richbeveridge.wordpress.com/

Thanks Candice for all the work you've put in this year!!

I read a book this weekend called "What the Dormouse Said."  It tells the story of the birth of the computer industry in Silicon Valley during the 1960s and 1970s.  Most of the people involved were connected in some way with Stanford University because of all the computer research being done there.  Much of this work was government funded, with the bulk of the funding coming from the Pentagon. 

There was a naval air station in Sunnyvale throughout WW II that led to the need for research in advanced aeronautical technology and electronics.  A number of corporations set up offices in the area to help fill the need for advanced technological research.

Lockheed and Honeywell both set up research operations in Sunnyvale.

Shockley Semiconductor in Mountain View did extensive research in developing silicon chip technology and led to the founding of both Fairchild Semiconductor and Intel.

Ampex was based in San Carlos.

Hewlett Packard was founded in Palo Alto.

Xerox PARC (Palo Alto Research Center) was in Palo Alto.

Mathematics is essential to computer programming - Math is the the language that computers speak.

I'm getting ready to talk to MTH 060 about graphing in a few days and the connection between computer graphics and the Cartesian Plane occured to me.  All computer graphics are based on the pixels of the screen and what color they need to be to represent the object.

To communicate to the computer what to do with each pixel, the programmer must identify the pixel by its position on the screen.  This is usually done a little differently than the standard Cartesian Plane, but it is exactly the same idea.

In addition, any type of object that is projected on the screen can be moved by describing the movement to the computer as a series of mathematical formulas.  The more difficult movements are often established using motion capture technology in which real performers wear black bodysuits with white disks at key points of the body.  Those key points are the ones involved in making the movement appear natural.

There are a number of courses on Mathematics and Computer Graphics

At the University of Illinois Urbana-Champaign

At Georgia Tech

Here are links to several books on the subject

Mathematics for Computer Graphics by John Vince

Essential Mathematics for Games and Interactive Applications by James Van Verth and Lars Bishop

Mathematics for 3D Game Programming by Eric Lengyel

There is also a nice pdf on Mathematics and Computer Graphics here.

Wolfram Alpha is up and running!

Here is some discussion about the site.

And here.

I was also reading a New York Times article today about the web sites Course Hero and Cramster.

The future of education will, I believe, move in a direction that makes use of these tools, but still challenges students to learn and solve difficult multi-step problems.

I was saying to one of my students today that in difficult multi-step problems, knowing the answer is generally not all that helpful - it's making sense of the answer that is important.

Some skills never go out of style - for instance, just because we have cars, that doesn't mean we shouldn't learn to walk!

Even though simple calculators are ubiquitous, we should still develop number sense and learn to calculate and work with whole numbers, fractions, decimals and percentages by hand.

The same goes for elementary algebra - once hand-held computer algebra systems become common, I believe that we should still learn the basics of elementary algebra - for the same reason we should learn to compute without a calculator - critical thinking.

We can't think critically about something we don't understand.

I went to Lincoln City this past weekend for the ORMATYC Conference.  ORMATYC is the Oregon Mathematical Association of Two Year Colleges and is a part of the larger group AMATYC, the American Mathematical Association of Two Year Colleges.

Just about every year, we have a conference in Lincoln City - this is the fifth one I've attended.  It's a great experience to get together with other community college math instructors from Oregon to talk about math and math education.  It's a valuable forum to find out what other schools are doing in their math classes and how we compare with those other schools.

In attending this conference every year, I have learned more about both math and math education.  In some cases it was about how to present a particular topic in class, while in other cases it was something that provided additional background information about the topics I teach so that I can share these ideas with my students.

In addition to being exposed to new ideas about math and math education, another positive aspect of this conference is being part of a community.  After 5 years, I have gotten to know some of the other instructors and have a better idea about which talks to attend while I'm there.

This year, I saw Jim Ballard of OIT Klamath Falls give a talk on the mathematics of finance.  There was a lot of discussion about the current economic situation and he didn't really get a chance to talk about the Black-Scholes equations which are somewhat controversial, but have been used in mathematical finance for over 20 years.  I wrote about math and finance in an earlier post.

I also attended a session with Ron Wallace of Blue Mountain CC about deciding which topics to teach and which to leave out in the math curriculum.  He asked if anyone there had used the quadratic formula in their lives outside of teaching in the past five years.  I was the only one to raise my hand.

I did use the quadratic formula a few years ago when my Mom asked me about the cost of Medicare Part D programs.  The formula for pricing in Medicare Part D is quadratic in that it initially becomes more expensive the longer you wait to enroll, but the money you save by not enrolling right away can offset the higher premiums you end up paying.

On Friday afternoon I went to a presentation by Art Peck of Lane CC.  I had seen his talk last year about the connection between the Fibonacci Sequence and the Mandelbrot Set, which was excellent.  This year, he talked about applications of mathematics to environmental problems, including alternative energy.  There is a lot of mathematics involved in scientific research that is focused on the environment.  For particular examples, he mentioned a textbook and companion website that have been developed and have some great application questions.

This is an important time for alternative energy generation and research directed at the environment in general.  I wrote an earlier post about the Solar Tres project.  The development of electric car motors and batteries has reached a point that production of "all-electric" car models is happening now.  One of my calculus textbooks has a cover page addressed to the instructor saying "The first person to invent a car that runs on water may be sitting right in your classroom."

On Saturday morning, I saw a presentation by Geza Laszlo called "Rational Approximations of Roots of Polynomials."  This is a very interesting topic.  It has connections to some of the material we cover in MTH 111 about roots of polynomials, but it is more closely related to the ideas we discuss in MTH 116 about using Newton's method to approximate a square root.

Newton's method uses calculus, but the method itself was known to the Greeks, even though they did not have formal knowledge of the methods of calculus.  The idea of approximating irrational numbers with rational numbers is also of great importance in constructing a musical scale.  Attempting to approximate (log3)/(log2) with a rational number determines how an octave will be separated into notes and how accurate the scale will be.

Well, I've finally finished the second installment of Math and Music, and I scheduled a talk on this for Saturday May 16th.  There are power-point style slides from the first Math and Music here. 

The first Math and Music talk focused on the development of a musical scale.  This next talk will branch out a little and explore the "beats" that musicians often use in tuning and how these can be used to calculate the actual frequency of a note.

We'll also look at the Wave Equation - an important piece of mathematics that arises from attempts to understand the behavior of a vibrating string but which turns out to be useful in a wide variety of applications (including the invisibility equations!)

We'll also talk about harmony and harmonics, although I never did figure out exactly why woodwind and stringed instruments produce harmonic tones...

In MTH 116 (Pre-Calculus) we discuss some of the applications of a three dimensional parabaloid. The typical applications we cover usually include a satellite dish receiver, a parabolic microphone and a headlight.

The shape of the parabaloid is such that any type of wave or radiation, be it radio, sound or light is concentrated by the shape of the parabaloid to a point in the center called the focus.

Over spring break I saw a television program on a type of solar power generating system that uses the parabaloid shape to store solar radiation in order to generate a steady supply of electricity. A central tower is located at the focus of a field of mirrors arrayed into the shape of a parabaloid.

Molten salt in the central tower is heated to a high temperature (565 degrees Celsius in one example) by the concentrated solar energy reflected by the mirrors.  The salt is then moved to a storage tank and on to the electrical generator which uses the extreme heat to produce steam and consequently electricity.  Once the salt has cooled (but is still molten), it is pumped back to the tower to heat up once again.

The benefit of the molten salt system is that during the day, the solar radiation can be used directly to generate electricity, and at night, the heat stored in the molten salt is used.  The hot molten salt can be stored in an insulated tank for up to a week and still be used to generate electricity independently of the solar radiation.

As these technologies are developed more fully, they can become more efficient both in terms of cost and power output.

Here are some links to information about these projects

A 10 page pdf on the Solar Tres project in Spain

The Solar Project (Wikipedia)

A power point presentation on modern solar power

An article from the National Renewable Energy Lab

I don't know much about Wolfram Alpha except that its developer Stephen Wolfram is well known in the mathematical community as a ground breaking thinker.

His business is based on the Mathematica software package and he has written a book called A New Kind of Science about a way of approaching science that is closely related to fractal analysis and artificial intelligence.

Wolfram Alpha appears to be a Google like interface that will actually answer questions!

Coming in May 2009....we'll have to wait and see how well it works...

Article on Wolfram Alpha

Announcement from Wolfram Research website

Wolfram Alpha website

"Because the math is really complicated people assume it must be right." — Nigel Goldenfeld, whose company sells derivatives software.(from a NYTimes article March 9, 2009)

The cover article in this month's Wired magazine has to do with the applications of mathematics to finance and investing.  Specifically, it talks about what is called the Gaussian Coplua function, which supposedly allowed invesment banks, hedge funds, and other high-flying invesment professionals a quick, straight forward way to determine the risk in a pool of invesments.

The article starts out explaining that the mathematician who came up with the Gaussian Copula function was widely celebrated and that many people even thought that he might win a Nobel Prize.

"A year ago, it was hardly unthinkable that a math wizard like David X. Li might someday earn a Nobel Prize. After all, financial economists—even Wall Street quants—have received the Nobel in economics before..."

What the article fails to mention, and I think that this is a particularly egregious omission, is that the last mathematical/economics investment wizards to win the Nobel Prize were the team behind the Black-Scholes equation -

"Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society."

Great! So the mathematicians have finally figured out how to feed a computer equations and get it to spit out money in return!

Wrong.

From Wikipedia -

"Together with Myron Scholes, Merton was among the board of directors of Long-Term Capital Management (LTCM), a hedge fund that failed spectacularly in 1998 after losing US$4.6 billion in less than four months.  The Federal Reserve was so concerned about the potential impact of LTCM's failure on the financial system that it arranged for a group of 19 banks and other firms to provide sufficient liquidity for the banking system to survive."

Long-Term Capital Management was like a dress rehearsal for today's financial meltdown.  I think that it would have served the author well to at least mention that the last widely celebhrated financial risk assessment mathematicans were also spectacularly wrong.

Here's the problem - people want a short-cut.  Math has some great short-cuts.  Think of the quadratic formula in Algebra, or the power rule for differentiation in Calculus.  But, for some things, there is no short-cut.  There is only the grunt work of going in and examining and analyzing enough pieces of what you're working with so that you have a solid overall understanding of the big picture.

When David X. Li came out with his new short-cut for assessing invesment risk...

"The effect on the securitization market was electric. Armed with Li's formula, Wall Street's quants saw a new world of possibilities. And the first thing they did was start creating a huge number of brand-new triple-A securities. Using Li's copula approach meant that ratings agencies like Moody's—or anybody wanting to model the risk of a tranche—no longer needed to puzzle over the underlying securities. All they needed was that correlation number, and out would come a rating telling them how safe or risky the tranche was."

Normally, ratings agencies and/or investors have to do a lot of analysis to determine the quality of invesments.  This is what is known as DUE DILIGENCE.  This is a very important legal term that essentially means doing your homework and NOT taking short-cuts.

Two mathematicians who have their own ideas about applications of math to investing are Benoit Mendelbrot and Nassim Nicholas Taleb.  I wrote a blog entry about these two back in October.

Mandelbrot is the mathematical father of fractals and fractal analysis.  He published a book a few years ago about the application of the ideas of fractal analysis to the financial markets.

Nassim Nicholas Taleb's training and education were focused on the application of mathematics and statistics to business and finance.  He wrote the book The Black Swan, about how improbable occurrences  that we are unaware of can have a dramatic impact on what we expect the future to look like.

Both of these mathematicians point out the severe limitations of both the Black-Scholes model and the Gaussian Copula model.  The Wired article discussed Taleb's take on the role the Gaussian Copula function played in the recent financial meltdown.

"Nassim Nicholas Taleb, hedge fund manager and author of The Black Swan, is particularly harsh when it comes to the copula. 'People got very excited about the Gaussian copula because of its mathematical elegance, but the thing never worked,' he says. 'Co-association between securities is not measurable using correlation,' because past history can never prepare you for that one day when everything goes south. 'Anything that relies on correlation is charlatanism.' "

On the other hand, some people have used the Black-Scholes model successfully by understanding its limitations.

Paul Wilmott, who is quoted in the Wired article, has also been critical of attempts to quantify risk.  But, he has used the Black-Scholes model in the past to develop trading strategies.

"A couple of years after leaving academia I became a partner in a volatility arbitrage hedge fund, and this was the start of phase three. In this fund we had to price and risk manage many hundreds of options series in real time. As much as I would have liked to, we just weren’t able to use the ‘better’ models that I’d been working on in phase two. There just wasn’t the time. So we ended up streamlining the complex models, reducing them to their simplest and most practical form. And this meant using good ol’ constant volatility Black-Scholes, but with a few innovations since we were actively looking for arbitrage opportunities. From a pragmatic point of view I developed an approach that used Gaussian models for pricing but worst-case scenarios for risk management of tail risk. And guess what? It worked. Sometimes you really need to work with something that while not perfect is just good enough and is understandable enough that you don’t do more harm than good. And that’s Black-Scholes."

The problem is not necessarily the mathematics but that people often take mathematics (and science) as some kind of all-seeing oracle that will make decisions for them.  This blinds them to what is actually happening right before their eyes and they assume that they don't have to do the (intellectual) heavy lifting, because the equations and computers are doing that for them.

Remember the fundamental mantra of computer programming -Garbage In, Garbage Out

The Wired article says that

"Bankers should have noted that very small changes in their underlying assumptions could result in very large changes in the correlation number. They also should have noticed that the results they were seeing were much less volatile than they should have been—which implied that the risk was being moved elsewhere. Where had the risk gone?

They didn't know, or didn't ask. One reason was that the outputs came from "black box" computer models and were hard to subject to a commonsense smell test. Another was that the quants, who should have been more aware of the copula's weaknesses, weren't the ones making the big asset-allocation decisions. Their managers, who made the actual calls, lacked the math skills to understand what the models were doing or how they worked. They could, however, understand something as simple as a single correlation number. That was the problem."

And with the money rolling in, nobody wants to ask questions.

Georg Cantor was an important mathematician who lived during the late 19th century.  He devised a method for comparing different sizes of infinite quantities and demonstrated that there were differnet sizes of infinite sets.

The foundation of Cantor's ideas is the concept of a one-to-one correspondence.  This is a concept which is deeply rooted in the human psyche, as is shown by the fact that the word digit - meaning number - also means finger.  This one-to-one correspondence between enumerating objects by comparing them to the fingers and/or toes of the human body is the presumed origin of our base ten number system.

What Cantor did was to compare infinite sets by establishing a one-to-one correspondence bewteen the elements of the two sets.  If it was possible to establish a one-to-one correspondence, then Cantor concluded that the sets were the same size.  In 1873, Cantor published a paper showing that the rational numbers (which includes all fractions and whole numbers) is actually the same size as the counting numbers (1, 2, 3, ...and so on).  Any set that is the same size as the counting numbers is called "countable."

Another example of this is the set of even numbers.  The set of even numbers is the same size as the set of counting numbers.  It might seem as though there would be fewer even numbers, because the set of counting numbers includes all the odds, but the set of even numbers doesn't!

But, you can create a one-to-one correspondence between the coutning numbers and the even numbers.

1 (in the counting numbers) matches up with 2 (in the even numbers)

2 (in the counting numbers) matches up with 4 (in the even numbers)

3 matches up with 6

4 matches up with 8 and so on.  Any counting number N will match up with 2N in the set of even numbers.

For every counting number, there is a corresponding even number - therefore the sets are the same size.  Cantor called this size - aleph null, the smallest infinite size.  Cantor showed that the Real Numbers, represented by the number line and including all rational and irrational numbers, are NOT countable - in fact, there are more Real Numbers than Counting Numbers.  Cantor's famous Diagonal Proof is actually fairly simplistic.

Most mathematicians of the late 19th century hated Cantor's ideas, and he didn't receive much support intellectually while he was struggling with his work.  Cantor often became depressed toward the end of his life and spent time in and out of sanitariums, eventually dying in 1918.  By this time, however, many younger mathematicians looked up to Cantor and relied heavily on his work and ideas.

Today Cantor is regarded as one of the most important mathematicians to ever work.  His ideas regarding infinity are deeply philosophical and have had a profound impact on mathematics, science and philosophy.

There are two topics that I teach which allow me to discuss counting.  When we cover the order of operations (or what I like to call the hierarchy of computation, since exponentiation is not an operation), I begin by saying that the most basic form of mathemtics is counting.  By repeated counting, we arrive at a consideration of addition.  By repeated addition (despite some vigorous dissent) we arrive at multiplication and so on.

In teaching about the set of complex numbers, I begin by saying that the simplest form of mathematics is counting, so the most basic set of numbers is the set of counting numbers.  But a problem arises if we write an equation using the counting numbers - we can write an equation whose solution is not a counting number.

x+5=1

This then requires an extension of the Counting Numbers to the Integers.

When I discuss counting, I mention that almost all cultures count.  From there, different cultures develop different mathematics depending on their different needs.  Ethnomathematics is a very interesting field.  Researchers in ethnomathematics study not only how different cultures use math, but also how different professions use math.  Professor Tod Shockey at University of Maine is a specialist in ethnomathematics and wrote his dissertation on the use of math in the medical sciences.

I mentioned that ALMOST all cultures count.  There are some cultures that do not have words for numbers above 3 or 4.  Why not?  Probably because they don't need them.

One culture which seems not to count at all is the Piraha people of the Amazon basin.  Here is a link to a pretty good article on this.  Two researchers have argued about whether or not the Piraha count and whether or not they have words for colors.  One researcher who thought that the Piraha had words for numbers was contradicted by another who said that they just have words for smaller and bigger and that these were being mistaken for words for one and two.

What makes this particularly interesting is that this disagreement between the linguists also arose over whether or not the Piraha have words for colors.  In describing a color, they will say that it is the same color as something in their everyday life.  This seems to get at a fundamental property of the Piraha language - it does not have abstractions.  This is an insight into the way that this culture views the world, just as any language is a snapshot of the collective mind of the culture that uses it.