Sunday, October 17, 2010

EQUATIONS are now a monthly staple in Wired

 A great teaching tool is now appearing every month in the media!  Starting back in May, mathematics writer Julie Rehmeyer started writing a new column in Wired Magazine.  An entire page features one equation along with a fabulous graphic (much better than the silly one I made to the right) to help bring it to life.  There are even sliders on the variables to help readers understand the range of possible scenarios modeled by the equation.

May: Carbon Emissions
June: Phantom Traffic Jams
July: 3-D Rendering
August: Power of Waves
September: Roller Coasters

Monday, October 4, 2010

It hasn't been that long since my last post

    It's been about 7 months, maybe 200 days or so, since my last post.  And while you may not have been counting the 4800 hours, you probably aware of the 5 million barrels or 210 million gallons of oil that were being dumped into the gulf.  This recent SEED slide show highlights the our deficiency as humans when it comes to comprehending these sorts of massive numbers.  Perhaps this is part of the reason we cannot stop buying drinks in disposable containers.  The images in the slide show come from Chris Jordan's 2009 book entitled Running The Numbers .  Of course, the idea of using technology and imagery to help us wrap our minds around the gargantuan nature of our world is nothing new.  The short video Powers of Ten from the 1960's highlights the wonders of the universe by expanding our field of view by one power of ten every 10 seconds.  In the video, we see a square that is 10^8 meters on a side framing the earth and a square that is 10^-8 meters per side framing a coil of DNA.

   Recently, while teaching my Math 124 Calculus Course, I came up with a little related rates program to help the students wrap their minds around the spreading of the oil in the gulf coast.  According to a NY times article from this summer, anywhere from 12 to 25 thousand barrels of oil per day were being dumped.  Wikipedia's site on oil slicks asserts that an oil slick is no thicker than about .002 millimeters.
There are 42 gallons of oil in a barrel.  So we will go with the round and reasonable number of about 1 million gallons per day.  A quick conversion gives us

(1 millions gallons/day) (3.79 liters/gallon) (10^6 mm^3/liter) = 3.79 (10^12) mm^3/day

Now we make a few fairly broad assumptions:

  1. The oil spreads in a circular manner and always has uniform thickness. 
  2. The oil is being spilled at a constant rate.
Most of us have experienced that as liquid spills out onto a surface, it spreads quickly at first and then slows down over time even if we do pour at a constant rate.  But how exactly is the rate of change of the radius of the spill related to the current radius of the spill?

We now answer that question with a little calculus.
Suppose V(t) denotes the volume of oil spilled at a time t where t is measured in days.  According to the assumptions we made,
V(t)=pi* r(t)^2 h where r(t) is the radius of the slick in millimeters at time t (in days) and h is the thickness of the slick.  So, using the chain rule, we have V'(t)= 2pi*h*r(t)*r'(t).
Because we know that the rate of change of volume of the oil spilled is constant at 3.79 (10^12) mm^3/day, we can determine the rate of change of the radius with respect to time in terms of the radius of the slick at a particular time.  So we see that r'(t)*r(t) is approximately 302(10^12)mm^2/day.  If we choose to measure r(t) in kilometers, then r'(t)*r(t) is approximately 302 km^2/day.
In other words, the rate of change of the radius of the slick is inversely proportional to the current radius.
So, when the slick is 1 km in radius, it is spreading at a rate of 302 kilometers per day.
But when the radius is 302 km, the radius is changing at a rate of only 1 kilometer per day.
By the way, 302 km is about 188 miles.  But the rate of spread is slowing, so how long would it take for the oil to reach shore if the spill occurred 100 miles off shore?  If the rate of the spill really does stay constant (i.e. no successful clean up) then we see can approximate r(t) as 25 t^(1/2).   When is 100=25t^(1/2)?  After about 16 days. So a major oil spill, even if it occurred 100 miles out into the ocean could reach the shore in a few weeks!
   I can only hope that the spread of information is as successful and uniform.  And I also make the observation that I waited about the same number of days to post as the number of kilometers that the oil slick would have grown in even one day.

P.S. How accurate is this little estimate?  Certainly we made many simplifications and overlooked all clean up efforts.  Looking at an interesting app from the NY times, I see that it did indeed take about three weeks for the spill to be noticed on the shore of Mississippi about 100 miles away from the source.

Tuesday, March 16, 2010

The Last Year of Grad School

  Nobody tells you about the emotional part of leaving grad school.  The time I've spent here has been longer than the time I've spent anywhere outside of the town I grew up in.  I've developed relationships here that will (I hope) last me a lifetime.  I have become a wife here, an adopted "auntie", and a teacher to many.  As I bike into my apartment complex, I wave left and right at familiar faces.  I see children who were babies when I started grad school here.  I stop to talk with former students who are themselves moving on to grad school, new jobs, teach for america, or travel abroad.  Getting ready to go isn't just about writing my thesis -- it's about saying goodbye to the people and places I've bonded with.
   Granted, I'm happy right now to be finishing the umpteenth grade and to be starting a new chapter in my life.  But I'm just warning you -- steel yourself against the fact that you will be processing a lot more than just your mathematics as you write up that thesis.   You'll be thinking about what  to keep and what to toss, you'll be spending time with friends, reconnecting with those you've lost touch with, visiting with those who are just near enough to drive to see and just far enough to not bump into.  You'll be wondering if anyone else will carry on the traditions you've started.  You'll see your surroundings with different eyes.  Try it now...It might be an interesting exercise.

Monday, March 8, 2010

Manifold Fashion

   This week Dai Fujiwara, a fashion designer, presented a collection based on Topology, specifically Fujiwara discussed his collection with Professor Bill Thurston, who is famous for his Geometrization Conjecture .  Here is a blog entry with pictures that discusses the role of geometry in this years fashion.  Another designer is also mentioned as incorporating geometric ideas in the NY Times.  Also see the YouTube interview of Thurston and Fujiwara .  

Wednesday, March 3, 2010

Census and Sensibility

    With the 2010 Census about to take place in the middle of this month during what is Spring Break for many college students, we might start to think about fairness.  As people receive so much junk mail, it's important to convey the importance of filling out the ten questions (which is the fewest number to ever appear on a census).  According to the Census2010 website $400 BILLION is distributed according to the numbers recorded in the census.  Just recently, a house committee decided to table a proposal for a bipartisan commission to help decide the redistricting that will occur when the census is over.  So how will redistricting be decided fairly by those people who have themselves benefitted from possibly unfairly drawn districts?
    Here we see the 12th congressional district that was approved in 1994, and which was controversial for being racially "gerry-mandered" .  Now, we have to ask ourselves if there is anything wrong with a district having an "bizarre" shape, whether this is somehow giving an unfair advantage.  Perhaps requiring that districts be "sensibly shaped" in some way is a reasonable way to ensure fairness.  Perhaps not.
Let's also take into consideration that "There's nothing Maryland can do about its bizarre shape," as economist Christopher Chambers said at the AAAS symposium on Fairness and Mathematics.  Professor Chambers is one of the authors of an academic article entitled "A Measure of Bizarreness"  that will soon be published in the Quarterly Journal of Political Science.  So however we may choose to answer the question of "How Bizarre is that shape?" it must take into consideration that the shape may already live in a somewhat bizarre larger one.  The way to measure this according the the authors is to use a path-based measure of convexity.  Given a district, its bizarreness is determined by the probability that it contains the shortest path within the state that joins two randomly selected points in the district.  The higher the probability, the lower the bizarreness.
Here's a January article from Slate magazine concerning this matter.

Monday, February 22, 2010

What's Massive about Media?

    Media is the plural of medium, and is a substrate through which information flows.  During the AAAS Mass Media Luncheon, speaker Dr. Jeffrey Kirsch , the Executive Director of the Fleet Science Center , asked us for our reactions to the phrase coined by Marshall McLuhan in the 60's: "The medium is the message."  Just the day before in the Counter-terrorism Symposium, Keith Devlin spoke about the difficulty of quantifying "information" and its reliability.  These discussions led me to think about information like light -- a pure and difficult to measure substance whose appearance is determined by the substance through which it passes.  Rather than thinking of information as being stuck in some sort of box to which an elite few hold the key, I recognize that information is difficult to confine.  Just as we created the light bulb, neon signage, fiber optics, and lasers to channel light which is typically free to move about, we design means by which we channel information so that it can be viewed, touched, heard, smelled, and sensed as we see fit.
    We control some of the media through which information passes: our own mental framework, a blog, a podcast, an imax movie, a newspaper article, our social interactions.  But we don't have nearly as much control over the information itself, which is floating everywhere around us in more or less dense and disorganized clouds.  So people who choose to be involved in media are attempting to corral information, this unruly light-like substance, so as to harness some of it's power and help others use it as a tool to brighten their lives.
     With this metaphor in mind, I see that there is power in heterogeneity. With a variety of media, what remains to be seen is which media will excel at fulfilling which roles.  While some media will replace others (like compact fluorescents are increasingly replacing incandescent bulbs), many will coexist or work in concert.
   Still there is the concern that people will be blinded by so much information, as if we are in a world covered in a thick blanket of pure white snow.  Well, I think that's what sunglasses are for!  In other words, people will have to squint for a while before they realize which media they need in order to function, and many members of the public are probably in that squinting phase right now.  So when they realize that they need a way to filter out the intensity in order to focus in on some of the details and survive, they will be scrambling to find good journalists.

Saturday, February 20, 2010

Frenemies and Functiononmeters - Countering Terrorism

   According to Dr. Gordon Woo, a catastrophe risk consultant who has also studied natural disasters, "It is the social networks between terrorists which ultimately are their undoing." Approximately 1 out of every 20 of a terrorists friends is either someone being watched by a security agency or an informant.   He concludes that terrorist cells with six or more members have about an 80% chance of being caught.  So one method of decreasing terrorist activities may be to model their extended social networks and the internal mechanisms of their cells.  Colonel Steve Horton's research focuses on the question of whether you can use raw data to infer social structure.  As a starting point he looks at the records over the last 30 years of bridge game scores to determine relationships between players.  
    On a different note, Dr. Paul Tannenbaum attempts to link the Research and Development world to the Battle Field by creating "functionometers" for the devices used by soldiers.  His hope is that by partially ordering the various functions of a device by their importance to a given mission, he can create a tool which quickly diagnoses the situation for the user.  In other words, the tool would be like a gas gauge of functionality ranging from "There's no way you can complete this mission with this device" to "Go for it!".
   Considering that several of the speakers for this Symposia were sick, a lot of information was presented.  I would loved to have heard more about how mathematics, as one speaker put it, "helps win over the hearts and minds" of insurgents' communities.  There are also ways to think of the mental attitude that condones terrorism as an "infection" that can be modeled in the same ways as disease.  For information on this and other research done in this area, check out Consortium for Mathematical Methods in Counterterrorism which was founded by Professor Jonathan Farley, the organizer of the symposium.

Higher School Mathematics

   So, there's the high school mathematics we all know and then there's the mathematics that the students pictured here are doing -- let's call it higher school mathematics!  These students were participating in the AAAS high school poster session, and had the opportunity to attend a conference right along side experts in many areas of science.
    Justin Lozano focused on solving the rubik's cube and variations using a computer programming approach.  Yesterday afternoon, he was among several students whose posters were on display in the exhibit hall.  When we talked about how he might use group theory to think about the rubik's cube, he commented "It's interesting that you think of the problem from a different point of view."
   Aishwarya Vardhana's poster features her work using linear programming methods to create a program that would show a company how to optimize their use of green energies such as wind and solar energy.    And, last but not least...

Varuna Rao worked on facial recognition.  She created a database with photographs for reference, and then wrote a computer program that assigns a number to a photo based on certain measurements obtained from the photo and compares different photos by comparing their numbers.  She says that now that she's learned about mathlab, she could see redoing her project using that tool, and that in the future, she would like to use more than one number to compare photos to improve her current recognition rate of 45%.
    Good luck to all of these students!

Gangs and Statistical Mechanics?

   The Los Angeles Police Department may have a new ally who, while not as all-knowing as Charlie Epps from Numb3rs, will help reduce crime by predicting where and when it might occur.   This morning, Professor Andrea Bertozzi  spoke about models that she and her team of post-docs at UCLA are developing to model gang violence.  Using statistical mechanics, bifurcation theory, and partial differential equations, she aims to predict where and when hot-spots of gang-related activity will emerge.  These tools have a long history in the physical and biological sciences, and are similar to those tools used to study the behavior of swarms of insects, which is the subject of some of her past research.
Dr. Bertozzi's newest paper on modeling gang activity will soon be published in the Proceedings of National Academy of Sciences.
   In order to model crime occurrence, Dr. Bertozzi consults the LAPD as well as Anthropologist Jeff Brantingham , whose research shows that criminals tend to commit their crimes near their own homes in areas with which they are familiar.  The model is designed with this in mind, and consists of a grid with a "house" situated at each vertex, freely moving "burglars", and an "attractiveness" function that depends on both space and time.  Different factors determine the attractiveness of a house to a burglar -- these include how close the house is to the burglar's home, whether the house was recently burgled, whether any of its neighbors have been victimized recently, and constants like the presence of graffiti or the type of housing prevalent in the area.  Two behaviors emerge from the simulations:  one in which a police presence would simply displace the criminal activity and one in which a police presence would actually eliminate the problem.  The model compares favorably to data collected over the period of a year in a particular LA neighborhood, and Dr. Bertozzi sees this as a first step in applying mathematical modeling to other social science issues.
    Where are the gangs?  Gang-related violence modeling is done by post-doc Alethea Barbaro , who studies gang networks, patterns in tagging, and how rivalries arise and dissipate.  In response to a question concerning how this work might inform decisions concerning the balance between hiring more policemen and "cleaning up the streets" to reduce the "attractiveness", Dr. Bertozzi responded "Sounds like a good research proposal!  These questions are excellent and hopefully will employ people like us for years to come."

Friday, February 19, 2010

World of Mathcraft?!

  Imagine your avatar immersed in a complex world in which mathematics is the greatest weapon, in which you can "turn the world off", sit down to build a new virtual tool with other players, and pick up again when you are ready to test the workability of your new tool.  Mathematician Keith Devlin thinks that such a game is entirely feasible, and could be a new way to attract students to mathematics.  Unfortunately, he also thinks that to develop such a game might be an expensive undertaking -- on the order of $100 million.  "But if we think of this as a national initiative then compared to Apollo it's not so expensive."
    Why should this be a national initiative?  The slide above was an illustration that 98% of students' responses to mathematical questions are correct when the questions are posed in a context relevant to the students' everyday lives while the students correctly answered only 37% of the SAME questions in a paper and pencil exam.  So one argument for creating math-based games for learning is that they have the potential to replicate the physical experience of using mathematics in everyday life much more effectively than typical textbook problems. Linguist James Paul Gee looks at textbooks as analogous to the dry manuals that accompany games, pointing out that "If you play the game you cannot fail to understand the manual."  Furthermore, he distinguishes between quality games and "skill and drill", saying "If we keep doing skill and drill, the only class your kids will care about is Mandarin."  However, he acknowledges that many of the people playing technical games are upper class, and that for a game to be successful as an educational tool, there must be an accompanying social network to which players can rely for mentorship and camaraderie.  Some of the games mentioned that you can learn about and try on-line were:

  • Portal -- a game that builds ones physical intuition
  • Fold-it  (Zoran Popovic )-- a game that helps researchers better understand protein folding
  • The products of Dream Box , a company whose CEO, Lou Gray, spoke in the session and whose products have been influenced by the research of math educators like Cathy Fosnot
But how can students really appreciate the mathematics behind the game itself?  Professor Frank Wattenberg has his students build simple games or models themselves in which they can see the relevance of exponential decay and differential equations up close and personal.  Similarily, Professor Brianno Coller teaches students the basics of control theory by having them learn to "steer" a virtual car -- you can see the simulations he uses on his website.  
     Commenting on the importance of understanding the nuts and bolts of mathematical models (including their limitations), Dr. Wattenberg commented that to update the old saying "There are three kinds of lies: lies, damn Lies, and statistics", many people would replace the word statistics with "modeling".  Perhaps when viewing games as a component of education, it is key to recognize that games themselves are models-- useful models that can enhance our understanding of reality.

Sea Ice, No See Ice

   We're on thin ice around the arctic, and the most well-respected mathematical models have underestimated the loss of sea ice shown by current measurements.   In particular, the photo above shows the loss of "multi-year" ice from March 2004 to March 2008.  Many people may wonder about whether we have reached a "tipping point" or point at which the loss in sea ice has passed the point of no return.  As Donald Perovich, a researcher with a US Army Laboratory, said during his presentation: "It's really a question of whether we will fall off the edge of the stage or fall down the stairs, bumping along the way"
    What's the difference between sea ice and the ice from your fridge?  The tiny channels of brine which carry algae and make modeling sea ice considerably different from modeling a liquid.  In addition to being different on the microscopic scale, sea ice covers vast expanses of water and melts because of effects from underneath (the ocean currents), above (the sun, snow, and atmosphere), and within (topography, permeability).  The "albedo" that I talked about in the last post is a ratio of reflected to incident sunlight, and the average albedo of the sea ice changes as the mosaic of ponds and ice evolves through the seasons.  First-year ice has a lower albedo and therefore absorbs more heat and melts quite differently from multi-year ice.   This creates a feedback loop that causes more ice to melt and lowers the albedo more....   Many of the presentations focused on finding better ways to model the albedo by better understanding pond-formation.
   But the view from the picture above is only one slice of the story.  I'll say more about ice volume and flesh out more of the picture about Sea Ice later tonight!

Thursday, February 18, 2010

In Search of Mathematics at the AAAS Annual Meeting: A PREVIEW

Watching the ocean views, graffiti, and tract housing of So Cal pass by my window, I am headed to The American Association for the Advancement of Science Annual Meeting in San Diego. There I will join over 10,000 scientists, journalists, and members of the public as we attend symposia ranging in topic from Astrobiology to Zoology. My mission is to attend as many mathematical Symposia as possible at this years meeting, whose theme is "Bridging Science and Society". Here is a list of the symposia I plan on attending. Each is about three hours long with several speakers from all over the United States and beyond.
Tune in tomorrow to learn about the "ice-albedo" feedback loop, and how we can model the melting of our polar ice caps. And yes, "albedo" and "albino" have the same root --meaning "white". The albedo of an object is a measure of how reflective it is. And I hopethat there will be some reflection by readers on my writing this week. Feel free to comment!