Look out the window or walk down the street to nearly any river or stream in Minnesota right now and you are likely to observe two things about the river:

1. it is getting deeper (or “rising” in relation to the banks); and
2. it appears to be moving faster.

You can, of course, confirm these observations by investigating reports from gauging stations along these rivers, maintained by the U.S. Geological Survey. (See data for the gauging station serving downtown St. Paul.) But what is really happening?

It may be high and fast...: ...but (as of today) the Mississippi at St. Paul is still in a bankfull state.
Courtesy Liza Pryor

Until a river flows over its banks, it is considered to be in a “bankfull” state. In this state, the water flowing through the river is confined to a relatively fixed channel area. Simply put, floods occur because more water is being introduced into this channel from upstream, due to snowmelt, heavy rains, or a dam breach. As this added volume of water moves through a fixed area, it both increases in velocity and in depth until it overflows the banks, at which point some, but not necessarily a lot, of the volume and velocity moving through the channel are reduced.

Scientists call the rate of flow through a channel “discharge." Discharge is defined as the volume of water passing through a given cross-section of the river channel within a specified period of time.A simple equation for determining discharge is

Q = D x W x V

where Q = discharge, D = channel depth, W = channel width and V = velocity.

Looking at this equation, it is easy to see that if discharge becomes greater and channel width is fixed, then an increase in both volume and depth (or height relative to the banks) is likely to be the cause. Discharge can be measured in cubic feet per second or cubic meters per second, for example.

But is the river flowing at the same rate at the surface as it does along its banks and beds? Understanding this requires investigating some more detailed equations, as the banks and bed introduce friction, which affects the rate of flow.

To learn more about rivers and how they flow, you may want to check out the works of Luna Leopold, and M. Gordon Wolman. In particular:

• Leopold, Luna B. (2006, reprint). A View of the River. Harvard University Press; and
• Leopold, Luna B.; Wolman, M. Gordon; and Miller, John P. (1995). Fluvial Processes in Geomorphology. Dover Publications, both classics for understanding how rivers work.

Also, check out our full feature on the 2010 Mississippi River flooding.

Tags : water, flooding, flood, discharge, thaw, St. Paul, river, natural disasters, natural disaster, mississippi river, melt, flow, floods

read more... | add a comment | share

Are sticky floor mats to blame?: Not likely.
Courtesy Ian Hampton
There's been a lot of news lately about "unintended acceleration" -- cars suddenly gaining high speed and drivers unable to stop them. Some observers question whether the problem lies with the car or with the driver. But whatever the cause, unintended acceleration is a deadly danger to the driving public.

Or is it?

Popular Mechanics crunched the numbers. They found unintended acceleration causes 3.2 deaths per year. This compares to:

• 1,550 deaths caused by drivers falling asleep at the wheel
• 6,000 deaths from distracted driving -- texting, cell phone use, etc.
• 7,400 traffic deaths attributable to bad weather
• about 11,800 deaths every year due to drunken driving

If you find your car accelerating, slam on the brakes, throw it into neutral, and steer to the side of the road. But don't waste time worrying about it. Instead, you should spend your effort avoiding bad weather, distractions, and above all not driving under the influence.

Tags : math, cars, Toyota, unintended acceleration

read more... | add a comment | share

Does this make you safer?: A glock 19 handgun.
Courtesy crossprocessedsoul

Tomorrow, the United States Supreme Court will hear arguments in the case McDonald v. Chicago. Chicago has some of the strictest gun laws in the country; McDonald (and others) argue that this violates their Second Amendment right to keep and bear arms.

In addition to the legal arguments, voices on both sides of the issue also talk about safety. Advocates for stricter gun control claim that reducing the number of guns on the streets will reduce the number of gun-related deaths. The Brady Campaign to Prevent Gun Violence recently released their annual survey of state gun laws, giving their highest marks to those states with the strictest regulations.

Others argue the opposite. They claim that gun-control laws only affect the law-abiding citizens who obey them. Criminals still have weapons, but the public is defenseless, leading to more deaths than if the public were armed.

Various people have tried to resolve this issue over the years, with little success. When the Brady list came out recently, blogger Jay Tea noted that some states with strict gun laws (such as California) actually had higher rates of gun death, while some states with looser laws (such as Utah) had much lower rates. (The "rates" are gun homicides per 100,000 people, and not total deaths. This allows us to compare large states and small states fairly.)

However, Mr. Tea failed to note that the reverse is also true -- that there are also states with strict laws that have low rates of gun violence, and states with loose laws that have high rates.

So, which is it: do gun controls make you safer, or put you in more danger?

To address this issue, we pulled out our old friend from math class, the coefficient of correlation. We last used this in an attempt to see if there's a connection between warm winters and warm summers (there is). This formula looks at two sets of numbers and determines how closely connected they are. Do they both move up and down together (a positive correlation)? If one moves up, does the other move down (a negative correlation)? Or, is there zero connection between them? So, we crunched the numbers, using the Brady Scorecard and the gun homicide statistics, and we found...

Nothing.

We came up with a coefficient of 0.00187. This tells us there is absolutely no connection between the Brady scores and the gun death rate: a state with strict laws is just as likely to have a high rate as a low one. The same goes for a state with loose laws.

The highest possible coefficient is 1.0. That indicates a direct one-to-one connection. In a complex system with many variables, such as human behavior, you want a score of at least 0.5 to say there is a strong connection, and a score of 0.3 to say there's even a weak connection. This score, however, was almost a perfect 0.

So, what does this all mean? Simply that neither side can use this as an argument. Gun-control advocates cannot use it to argue that regulations save lives; gun-control opponents cannot use it to argue the opposite, that regulations are dangerous.

Now, this all hinges on the Brady scoring system. It is possible that other ways of quantifying "strict" and "loose" laws could produce different results. And none of this has any bearing on the legal and Constitutional arguments being made. All we can say is, that in this case, the math is unambiguously neutral.

Tags : math, correlation, coefficient of correlation, gun, law, Supreme Court, Second Amendment, guns

read more... | add a comment | share

When we decide to go from point A to point B, we have a plan. If driving, we would decide which route to take, which intersections to avoid, and we will estimate how much time it will take. As we do drive from A to B, we will, intuitively and deliberately, make changes – we might change our route because of a traffic jam, or speed up to try and cover time lost. Thus, as we encounter new information and unexpected situations, we, intuitively, make changes to our plans, for the better.

Scientists and engineers have been trying for years to bring this capability – of automatically changing plans – to machines. Research in robotics has focused on equipping machines with sensors and radars to detect changes and respond to that change. For example, Toyota Motor corporation equips its Lexus LS460 with a radar which detects vehicles and obstacles on the road ahead. When the radar detects the possibility of a collision, the vehicle retracts the seatbelts, warns the driver, and applies brakes to reduce collision speed. Such systems are dubbed “sense-response” systems, and today are even available in automated vacuum cleaners such as the Roomba. However, taking the sense-response system a step further, such that it is applicable to entire supply chain and production systems has proven difficult.

A key breakthrough in developing automated “sense-response” systems, that would enable developing automated production decision support, sales and marketing recommendations, and several other automated systems has recently been achieved. Dr. Nazrul Shaikh, a post doctoral researcher at University of Pennsylvania has worked on the characterization of the planning and re-planning problem and have developed solutions for several classes of sense-response problems. The research, which has survived the extensive peer review from academicians, in now all set to make an impact to homeland security and corporate America. It will change the way the automated planning systems are used and implemented in industry.

Dr. Shaikh’s quest to characterize the sense-response problems began in 2002 when they were devised an approach to develop an intelligent sense response system for supply chain event management for IBM. This was in the wake of 9/11 when Corporate America realized that they need to be able to handle exceptions to plans better. The research, called Exception Analytics looked at the deviation between the planned and the actual values of several system variables and attempted to infer the “action” required to compensate for the deviation. Applications have been developed for supply chain event management, homeland security, quality control and marketing planning. Keen interest has been displayed by several companies, especially in marketing where the consumer and competitor dynamics influence the demand drastically. The basic concepts have been published in several journals after clearing rigorous peer review.

Tags : Applied Operations Research

read more... | add a comment | share

GOCE Satellite: The Gravity field and steady-state Ocean Circulation Explorer
Courtesy ESACan it be true? Yes, for a mere $5,544 dollars round-trip airfare to Greenland! In March 2009, the European Space Agency launched the Gravity field and steady-state Ocean Circulation Explorer (GOCE) into orbit around our planet, which is now transmitting detailed data about the Earth’s gravity. The GOCE satellite uses a gradiometer to map tiny variations in the Earth’s gravity caused by the planet’s rotation, mountains, ocean trenches, and interior density. New maps illustrating gravity gradients on the Earth are being produced from the information beamed back from GOCE. Preliminary data suggests that there is a negative shift in gravity in the northeastern region of Greenland where the Earth’s tug is a little less, which means you might weigh a fraction of a pound lighter there (a very small fraction, so it may not be worth the plane fare)!

In America, NASA and Stanford University are also working on the gravity issue. Gravity Probe B (GP-B) is a satellite orbiting 642 km (400 miles) above the Earth and uses four gyroscopes and a telescope to measure two physical effects of Einstein’s Theory of General Relativity on the Earth: the Geodetic Effect, which is the amount the earth warps its spacetime, and the Frame-Dragging Effect, the amount of spacetime the earth drags with it as it rotates. (Spacetime is the combination of the three dimensions of space with the one dimension of time into a mathematical model.)

Quick overview time. The Theory of General Relativity is simply defined as: matter telling spacetime how to curve, and curved spacetime telling matter how to move. Imagine that the Earth (matter) is a bowling ball and spacetime is a trampoline. If you place the bowling ball in the center of the trampoline it stretches the trampoline down. Matter (the bowling ball) curves or distorts the spacetime (trampoline). Now toss a smaller ball, like a marble, onto the trampoline. Naturally, it will roll towards the bowling ball, but the bowling ball isn’t ‘attracting’ the marble, the path or movement of the marble towards the center is affected by the deformed shape of the trampoline. The spacetime (trampoline) is telling the matter (marble) how to move. This is different than Newton’s theory of gravity, which implies that the earth is attracting or pulling objects towards it in a straight line. Of course, this is just a simplified explanation; the real physics can be more complicated because of other factors like acceleration.

Albert Einstein
Courtesy noneSo what is the point of all this high-tech gravity testing? First of all, our current understanding of the structure of the universe and the motion of matter is based on Albert Einstein’s Theory of General Relativity; elaborate concepts and mathematical equations conceived by a genius long before we had the technology to directly test them for accuracy. The Theory of General Relativity is the cornerstone of modern physics, used to describe the universe and everything in it, and yet it is the least tested of Einstein’s amazing theories. Testing the Frame-Dragging Effect is particularly exciting for physicists because they can use the data about the Earth’s influence on spacetime to measure the properties of black holes and quasars.

Second, the data from the GOCE satellite will help accurately measure the real acceleration due to gravity on the earth, which can vary from 9.78 to 9.83 meters per second squared around the planet. This will help scientists analyze ocean circulation and sea level changes, which are influenced by our climate and climate change. The information that the GOCE beams back will also assist researchers studying geological processes such as earthquakes and volcanoes.

So, as I gobble down another mouthful of leftover turkey and mashed potatoes, I can feel confident that my holiday weight gain and the structure of the universe are of grave importance to the physicists of the world!

Tags : climate change, gravity, math, nasa, ocean, physics, relativity, satellite, science, space, spacetime, time

read more... | add a comment | share

Philadelphia Phillies' second-baseman Eric Brunlett made an amazing and unassisted game-ending triple play against the New York Mets. The extremely rare play came in the ninth inning with the Mets trailing by two runs but threatening with runners on first and second base and no outs. Both base runners were stealing on a 2-2 pitch when the Mets batter hit a line drive right to Brunlett, who caught the ball for the first out, stepped on second base for the second out, then tagged out the runner from first. It's only the 15th unassisted triple play in major league history, and only the second to end a game. The poor Mets. The odds of this happening must be astronomical, but I'll let someone else figure that out.

Tags : sports, odds, mathematical odds, baseball science, baseball, amazing achievement

read more... | add a comment | share

We've all heard statistics about how boys are better than girls when it comes to math. Especially the kinds of advanced math it takes to find solutions to complex problems, to win important prizes and to invent world-changing technologies. According to some people, you can blame this gender gap on basic biology. Female brains are smaller than male brains, which means males are just naturally smarter, and really, what else do you need to know?

It's easy to believe assumptions and stereotypes about girls and math when you look around classrooms where advanced technical subjects are taught. Fewer women fill the seats, and in the top math and science positions, men outnumber women by a dramatic margin. It seems like no matter how many women prove that female brains can be every bit as good at math and science, we still hear that women are just not cut out for crunching numbers.

In other words, if you're a girl and you like math, you should probably quit now, because you will never be as good as the boys. And if you're a girl and you just don't understand math, it's okay, you won't need math anyway. Like Barbie says, Math Is Hard! Let's go shopping!

Or not? Could it be that fewer women excel in math and science fields because they have fewer opportunities? Or because everyone tells them they will do poorly, so they never really try? Is biology to blame for the math-science gender gap, or is culture the culprit?

A new study published in the Proceedings of the National Academy of Sciences says girls are not born to be bad at math. Instead, the authors say, the gender gap stems from cultural inequalities that put girls and women at an unfair disadvantage. Fewer educational and professional opportunities, negative stereotypes, and classroom or workplace dynamics all hinder the potential of girls and women to excel in math.

The authors of this study came to their conclusion by comparing data on gender inequality and math scores from around the world. They found that in countries where men and women had more equal opportunities, women did much better at math. If you're wondering where the United States stands, in 2007 the US was number 32 on the World Economic Forum's Global Gender Gap Report. Not the greatest place to be, but in the US it appears as though the math-science gender gap is narrowing. According to a recent article in the New York Times, there are now more opportunities than ever for women in science and math. If only they paid as well!

Tags : gender and science, gender and math

read more... | add a comment | share

Hey, it's Walk/Bike to Work Day tomorrow. And in honor of that, here's a little math problem to keep you occupied, courtesy of Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin by Lawrence Weinstein and John Adam.

"What are the relative costs of fuel (per kilometer or per mile) of New York City bicycle rickshaws (human-pedaled taxis) and of automobiles?"

Bicycle rickshaw: What does it take to fuel this?
Courtesy ARoberts

You might need a few hints.

1. How far does a bicycle rickshaw travel in one day?
2. What does it cost to fuel (i.e. feed) the bicyclist?
3. How far does a car travel on a gallon of gasoline?

What'd you get? Post your answer as a comment. Once a few folks post answers, I'll post the one from the book, as well as the "work."

A research group led by Dirk Brockmann at Northwestern University has created a computer model that predicts the spread of the 2009 H1N1 influenza virus in the US. (It uses a complex set of mathematical equations to describe the movement of people and virus.)

How can you track and predict the movement of something so small?: Follow the money, of course! (This is a colorized negative stained transmission electron micrograph (TEM) showing some of the ultrastructural morphology of the A/CA/4/09 swine flu virus. Got that? Good.
Courtesy CDC/C.S. Goldsmith and A. Balish

(Brockmann was a guest on Minnesota Public Radio's Midmorning show today, and you can listen to it online.)

The good news is that, based on what we know now, and assuming that no one takes any preventive measures, we could expect to see some 1,700 cases of swine flu in the next four weeks. Because of the preventive measures being taken wherever a suspected case of H1N1 flu has popped up, we should actually see fewer cases. (You can see Brockmann's models here.) That's lousy if you're one of the folks who picks up the virus, but not a devastating number of cases. Of course, this is a rapidly developing, fluid situation, and things may change. Still, tools like Brockmann's model help to ensure that emergency supplies and other resources get to the places likely to need them most before they're needed.

Professor's Computer Simulations Show Worst-Case Swine Flu Scenario from Northwestern News on Vimeo.

Don't have faith in computer models? Well, a second research group at Indiana University is using another model, with different equations, and getting very similar results. That's a pretty good indication that the predictions are reliable.

You might remember Brockmann from a 2006 study that used data from WheresGeorge.com, a site that allows users to enter the serial numbers from their dollar bills in order to see where they go, to predict the probability of a given bill remaining within a 10km radius over time. That gave him a very good picture of human mobility, reflecting daily commuting traffic, intermediate traffic, and long-distance air travel, all of which help to model how a disease could spread.

Tags : swine flu, spread, simulation, scenario, public health, Northwestern University, modeling, model, Mexican flu, influenza, infectious diseases, infectious disease, health, H1N1, flu, epidemiology, disease, cases, 2009

read more... | add a comment | share

The hounds of spring are on winter’s traces: It appears that winter and summer temperatures are yoked together. (This photo is for you, Thor. I'm a cat person myself. Title explained here.)
Courtesy sbpoet

Last week, Candace asked whether there’s a connection between winter temperatures and summer temperatures. She noted that the winter of 2007-08 was pretty cold by recent standards, and the following summer was cool as well. Is there something going on here?

Liza searched the Web but couldn’t find anything definitive. I (after pooh-poohing the idea that this has been a warm winter – if you want to pay my heating bill, you’re welcome to it!) decided to crunch the numbers.

First, I went to this site. It records monthly average temperatures going back to December 1978.

Actually, it records temperature anomalies – whether the observed temperature in a given month is higher or lower than the average. They use the 20 years of 1979-1998 as their baseline. A reading of 1.00 means the temperature was 1 degree Centigrade (1.8 °F) warmer than expected. (One degree may not sound like much, until you realize it means 1 degree of every minute of every hour of every day. It quickly adds up to a lot of heat.) A reading of -1.00 means it was 1 degree cooler.

Using temperature anomalies is good for this exercise, as it removes the effects of global warming. Because global temperatures rose during the period under study, a “warm” winter in the late ‘70s might be considered only “average” today. In fact, that’s exactly what happened last winter. It was almost perfectly average by historical standards, but because recent winters have been so much warmer, it felt cold to us.

Getting back to Candace’s question: does a warm winter predict a warm summer? To answer this question, we have to calculate my all-time favorite statistical formula, the coefficient of correlation!

It sounds like a mouthful, but it’s a pretty easy concept to grasp. The coefficient of correlation measures how tightly two sets of numbers go together. For example, if you surveyed 100 people, and asked each one what year they were born, and how old they were, you would find that every single person born in 1990 was the exact same age. The first number (year of birth) and the second number (age) are linked together 100%.

OTOH, if you asked those people for the last digit in their telephone number, you would find no relationship whatsoever. A person born in 1990 is just as likely to have a phone number ending in 9 as ending in any other number, and the same goes for people born in every year.

Calculating the coefficient of correlation (or “coco,” as I affectionately call her), requires wading through a truly horrific battery of equations all to arrive at a number between 0 and 1. A coefficient of 1 means the two sets of numbers are perfectly synched together; a coefficient of 0 means there is no connection whatsoever.

So, I went back to the temperature data. First, I defined “winter” the same way the weather bureau does: December, January and February, the three coldest months of the year. (None of that solstice-equinox nonsense here!) I defined “summer” as the three warmest months: June, July and August, again following weather bureau standards. Using the Northern Hemisphere Land figures (sorry, they didn’t have anything Minnesota-specific), I came up with an average anomaly for every winter and every summer. I crammed the numbers into the formula, turned the crank, and came up with a coefficient of…

(drum roll, please)

0.71

OK, now what does that mean?

Well, in general, a score below 0.30 is considered inconclusive. It’s too close to zero—the “relationship” could just be random. A score between 0.30 and 0.50 is generally considered moderate—there’s a connection there, but it’s somewhat weak. A score over 0.50 is generally considered strong—there’s definitely something important going on there.

(This is especially true in highly complex systems, like weather, where a lot of different factors can affect your results. In a very simple system, you’d probably want a result much closer to 1.)

It all boils down to this: we can be more than 99% certain that, yes, there is a connection between a warm winter and a warm summer, or a cold winter and a cool summer. How much of a connection? For that, we need another figure, the coefficient of determination.

This one is much easier. Just square the coefficient of correlation. 0.71 squared yields 0.5041. That means 50% of the variability in summer temperatures is determined by the winter temperatures.

And “variability” is the key. Like I said, weather is an extremely complex system. Lots of things can affect the temperature for a day, a week, even a season. The fact that this winter is warmer than last winter does not guarantee that this coming summer will be warmer than last summer. (For example, the winter of 2003-04 was one of the warmest on record, but the following summer was one of the coolest in the study period.)

What this number does mean is, that of all the factors that will affect next summer’s temperatures, half of them seem to be connected to winter temperatures. And this winter was warmer than last winter.

Just for fun, I also ran the calculations the other way, to see if a warm summer predicts a warm winter. The coefficient of correlation was 0.54, and the coefficient of determination was 0.29. So, again, there is a connection, but it seems to b a good deal weaker.

A word of caution: one thing statisticians like to say is “correlation is not causation.” Partly because it’s fun to say, but mostly because it’s true. Just because two things are correlated does not mean one causes the other. We have not proven that warm winters cause warm summers. It could be that winter temps and summer temps are both boosted by some other factor – El Nino, perhaps. All we can say is that there is some sort of connection going on, and that it probably wouldn’t hurt to lay in some tanning cream now.

Tags : winter, weather, summer, statistics, math

read more... | add a comment | share