October 26, 2011 — Samuel Chen, Technical Communication & Strategy
What do computer animation, oil exploration, and the FBI’s database of 30 million fingerprints have in common?
As of Version 8, wavelet analysis is an integral part of Mathematica.
Wavelets themselves are short-lived wave-like oscillations. Taking the Morlet wavelet, for example, we can see that unlike sines and cosines, this wave-like oscillation is localized in the sense that it does not stretch out to infinity.
January 26, 2011 — Jon McLoone, International Business & Strategic Development
Mathematica can make you feel like a computational superman. Armed with that attitude and some schoolboy knowledge of cryptography, I turned my attention to cipher breaking this week, only to discover buried kryptonite.
The weakness of ciphers (where you swap every occurrence of a particular letter in your message with the same different letter) is that they don’t change the patterns of letters. The simplest attack that exploits this fact is frequency analysis. The most common letter in English is “e”, and so it follows that the most common character in an encoded message (assuming the message is written in English) will correspond to “e”. And so on through the alphabet.
Mary Queen of Scots famously lost her head when Queen Elizabeth’s spymaster broke Mary’s cipher using frequency analysis. I figured that if sixteenth century spies could do it by hand, I should be able to automate it in Mathematica in about 10 minutes.
December 9, 2010 — Jon McLoone, International Business & Strategic Development
I just published a Mathematica package that provides an alternative, richer implementation of units and dimensional analysis than the built-in units package. You can get it here. Aside from being a really nice extension to Mathematica, it is also an interesting case study in adding a custom data “type” to Mathematica and extending the knowledge of the built-in functions to handle the new “type”.
First I have to explain the point by answering the question, “What’s wrong with the built-in units package?” Well, there is nothing actually wrong with it, it just doesn’t apply Mathematica‘s automation principles. It can convert between several hundred units and warn if a requested conversion is dimensionally inconsistent. But give it an input like…
and it does nothing with it until you specify that you want the result in a specific unit. The core reason is that it doesn’t teach the system, as a whole, anything about units, or even that the symbol “Meter” is any different than the symbol “x”. All of the knowledge about units and Meter in particular is contained in the Convert command.
November 30, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
Let’s flip a coin, over and over. Beforehand, the players each pick a sequence of flips. The sequence that occurs first wins. With HH vs. TH, HH will win if the first two flips are HH and will lose if any of those flips are tails. HH vs. TH has a 1/4 vs. 3/4 possibility of winning.
Phrased a different way: Suppose I offer a bet on a series of coin flips. One of these bets would be bad for you. Which one? The odds of the event occurring are given at the end. (The second bet is the bad bet to take.)
HTT appears before TTT. If it does, I give you $1. If not, you give me $4. (7/8)
HHT appears before TTT. If it does, I give you $1. If not, you give me $3. (7/10)
THH appears before HHT. If it does, I give you $1. If not, you give me $2. (3/4)
HTH appears before THH. If it does, I give you $1. If not, you give me $1. (1/2)
This is the strange world of Penney’s game. Here is a table of odds and facts for various 3-flip games. Calculating these odds can be both tedious and mathematically demanding—a natural job for Mathematica or Mathematica Home Edition.
October 1, 2010 — Jon McLoone, International Business & Strategic Development
Buried deep in the list of new technology in the Mathematica development pipeline was the item “integration of oscillatory functions (univariate, multivariate)—new algorithm”. I expect most people will overlook it, as I did, in favor of the new functions, new directions, big infrastructure, and the eye candy. Even worse, most people who will use it won’t even know—it will be selected automatically when needed, like many of Mathematica‘s algorithms. So I think it’s my duty to share my discovery that this algorithm is actually really cool.
Why is it so cool?
The first clue I had was when I read in the notes that this was the first time anyone had fully automated the algorithm into a very wide class of problems. Second, that it was a hybrid numeric-symbolic method (putting it beyond the reach of most numerical systems). And finally, that it was developed by the talented Wolfram Research developer Andrew Moylan.
September 27, 2010 — Jon McLoone, International Business & Strategic Development
Since I just heard that the video for Conrad Wolfram’s recent TED talk “Stop teaching calculating, start teaching math” will be coming out soon, I thought I would address the single biggest fear that I hear when I talk about using computers in math education.
The objection that using computers will “dumb down” education comes with the related ideas “students have to learn to do it by hand or how will they know they have got the right answer”, “they won’t understand what is happening unless they do it themselves”, and so on.
Well, let’s examine this by looking at a typical math question that I know I had to solve at some point in my education.
August 30, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
Back in 325 BC, Aristotle talked about which polyhedra can fill space, and noted that regular tetrahedra could fill space.
Around 1470 AD, Regiomontanus showed that Aristotle was wrong. He also found the spot where a statue on a pedestal appears the largest, as shown in the Demonstration “The Statue of Regiomontanus”.
In 1900, Hilbert tried the problem of tetrahedra packing and included it as a part of problem 18 in his list of unsolved problems. Hilbert is also famous for the Hilbert curve and “The Hilbert Hotel”.
July 13, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
Is it possible to have a pair of nonstandard dice with the same odds as regular dice?
Sure. You just need to know how to calculate the odds, and how to determine what different numbers could be on the faces to give the same odds. Let’s start with some tables.
The addition table is one of the first tables learned in school. Here is
one way to present an addition table in Mathematica.
June 30, 2010 — Wolfram Blog Team
Bruce Torrence, PhD, chair of the Department of Mathematics at Randolph-Macon College, says he’s engaging his students in mathematics more than ever before thanks to a single Mathematica command. That command is Manipulate.
Professor Torrence calls the ability to create instant dynamic interfaces a “real game changer” for helping students understand mathematics. He says, “Once you play with a Manipulate and interact with the sliders and buttons, you really develop your intuition as to how the underlying mechanisms are interacting and working.”
In this video, Professor Torrence shares an example of how he used Mathematica to turn a previously tedious lesson into a highly compelling, interactive classroom activity.
June 16, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
Somewhere, you’ve likely been forced to learn how fractions work, and how to calculate 2/7 + 2/5. To some extent, fractions have been falling out of favor in the world, losing out to decimals. The New York Stock Exchange gave up fractions on April 9, 2001.
Much of the time, a decimal is okay. Sometimes, though, especially in mathematics, exact values are desired. Instead of a value being 3.00000000…00727…, it is exactly 3. Or exactly 10/35 + 14/35 = 24/35. For fractions themselves, the Farey sequence is quite interesting—the reduced fractions between 0 and 1 where the denominator is less than or equal to a particular value, like 7. For example, the F7 Farey sequence is the the first row in the following block. The next row has the denominator. The third row is twice the reciprocal of the denominator squared. The fourth row is the denominator from the third row.