Application

Application Technology

P v NP and Collaborative Mathematics

This is a nice article in the NYT about a recently proposed solution to the famously unsolved mathematical question “Does P = NP?”

http://www.nytimes.com/2010/08/17/science/17proof.html

Essentially this question is about how long it takes to solve certain kinds of problems: if a proposed solution to a problem can be checked in some reasonable amount of time, does that mean we can always solve the problem in a reasonable amount of time? [Warning: the definition of reasonable here may seem unreasonable.]

For example, it doesn’t require many operations to determine whether or not 7411 divides 748511; even by hand, you can work it out in a few steps. It requires significantly more operations, however, to find the prime factors of, say, 837751. Essentially, P v NP asks “are problems that can be checked by computers (maybe lots and lots of computers working in parallel) necessarily solvable by computers?” It is still an open question.

Another fascinating aspect of this particular open question is the role that the internet has played in bringing great mathematical minds together. Proposed solutions can be instantly accessed and vetted by those capable of evaluating the arguments. Such a community can work quickly and efficiently, not just to ascertain a proof’s validity, but to improve and refine it together.

By patrick honner, 14 years9 years ago

Application Challenge

Comprehensive Calorie Counting

There’s an interesting op-ed in the NYT about how mathematics is used to put food production and consumption into context:

http://www.nytimes.com/2010/08/20/opinion/20budiansky.html

For example, we can measure food transportation costs in calories (as a calorie is, indeed, a unit of energy), and so we can look at production and transportation costs of certain foods and compare them to the calorie content of the food itself. For example, it takes about 5000 calories to produce a 100-calorie head of lettuce.

The author’s intent is to poke holes in some of the common arguments used by proponents of “eating local”: Transportation costs for most foods are neglible compared to household storage and preparation costs, says the author, thus “eating local” is not an especially eco-friendly strategy. However, the author hilmself makes a number of weak and erroneous arguments, comparing “apples to rocks” in some cases.

The piece offers some interesting mathematical ideas and a good critical reading exercise.

By patrick honner, 14 years9 years ago

Application Geometry

Buckyballs Detected in Space

For the first time, scientists have verified the existence of “buckyballs” in space. Buckyballs are carbon molecules made up of 60 atoms arranged in a soccer-ball like structure

Notice the interlocking pentagons and hexagons. There are 60 vertices in this solid, so how many of each polygon?

Buckyballs are named after Buckminster Fuller, as they resemble the geodesic dome he made famous. Fuller was a creative, prolific man–a futurist–who was never short of whimsical ideas, like using blimps to drop bombs to make holes to plant tree-houses in.

Application History

How Do You Study Extinction? Commit Ecocide

I watched “Lord of the Ants” on PBS the other night, a documentary about biologist E.O. Wilson. Wilson possesses the characteristics of the great natural scientist: a never-ending fascination with the world, the persistence to keep asking questions and to keep looking for answers, and the discipline to focus on and master a specific domain. Wilson’s impact has been both deep and broad, and he’s even been at the center of a scientific-political-cultural controversy–another benchmark of greatness.

“Lord of the Ants” tells the story of his scientific life–past, present, and future–and it is viewable here. In Wilson’s story, a couple of cool math-y things caught my attention.

Wilson and Daniel Simberloff, a mathematician-turned-biologist, were interested in studying how ecosystems re-populate after extinction, so they fumigated a small island in the Florida Keys and watched what happened.

In particular, they wanted to know how re-population depends on the area of the region, and its distance from the “mainland”. Furthermore, they wanted to see if the same number of different species would return, if the same, or different, species would return, and if the relative populations of the various species would return to pre-extermination levels.

Later, Wilson goes on to describe an “Iron Law of Ecology”, namely that a 10-fold increase in habitat doubles the number of species that can be supported there. This quantitative analysis is obviously very useful for naturalists arguing in favor of preserving more and more natural habitat.

By patrick honner, 14 years9 years ago

Application Technology

Encrypting Secret Messages in Pictures

This is a very cool (and somewhat technical) description of a simple way to use Mathematica to hide a secret message in an image file.

http://blog.wolfram.com/2010/07/08/doing-spy-stuff-with-mathematica/

The basic idea is that you erase the last digit (in binary) of each pixel’s “color channel”, and then use that spot to store part of the secret message. Given an image, you can then recover the secret message by looking at that last digit in each channel.

This process does change the image somewhat, but not in a way that the normal human eye (with normal viewing equipment) would ever notice. In the post, the author subtracts the adjusted image from the original:

To most viewers, the difference looks like this:

Only under extreme contrast can you actually see the real difference in the two. Here, we see the invisible ink reappear!

All that’s left is to decode the colors and read your message. But be careful–sending and receiving secret messages might get you some unwanted attention.

By patrick honner, 14 years9 years ago

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