(via sakesi)

proofmathisbeautiful:

Waiting for a bus? Math may help

(Via CNN)

Georgia Tech student Alexandra Gaigelas takes a shuttle bus to get around the Atlanta campus. Many times, she waits too long for a bus.

“There’s nothing more frustrating than standing at a stop, waiting for 10 minutes, getting on the bus and seeing another bus directly behind you.”

And that second bus is largely empty. It’s called bus bunching, and it happens when buses are thrown off schedule because of traffic, weather or too many passengers at one stop.

And when those buses are off schedule, the drivers try to adjust. Student Sukirat Bakshi says he’s been victim of a bus “drive-by.”

“It happened to me where the driver just would not stop at a stop. They would just run off to catch up to the schedule.”

It turns out math can fix the problem. Georgia Tech professor John Bartholdi and University of Chicago professor Donald Eisenstein used complex algebra to develop a kind of anti-bus-bunching formula. They took what’s known as the Markov Chain through the wringer. It’s a math theory that shows predictable long-term behavior.

“The trick is to hold the bus for an adjustable amount of time at one stop,” Bartholdi said. “We simply control how long they wait at the end of the route, and then we tell them, ‘drive comfortable with the traffic to the other end. Don’t worry about where you are. Just flow with the traffic.’ “

Buses in the loop are all connected through GPS and a computer pad. It signals to the driver when it’s time to leave. Georgia Tech is testing the theory on its shuttle system.

“This tells me exactly when it’s time to go, and the communication between each other is done automatically, so it takes a lot of stress from us,” said Clarence July, who drives one of the gold and yellow Georgia Tech buses.

Drivers can ignore the schedule, and riders on campus can walk up to any stop and know that a bus will come within approximately six minutes. Bartholdi and Eisenstein say their math formula works for any shuttle system that runs in a loop in which buses are no more than about 12 to 15 minutes apart.

“Others have tried to control buses by asking drivers to try to adhere to a target schedule,” Bartholdi said. “What is new here is that the buses in effect coordinate themselves. No one needs to tell the drivers what to do; no one needs to worry about being off-schedule or how to recover a lost schedule.”

Georgia Tech plans to fully implement the no schedule bus system on campus this fall.

Here’s how Bartholdi explains the equations used to calculate the space between buses:

This equation is actually a bunch of equations: one for each bus. The first line describes how the headway (the space between buses) changes for the bus that is currently at the end of the route (the turnaround point). Alpha (in red) is a control parameter - a number, say, 0.5 - by which the bus manager chooses whether the bus should wait longer (and fix imbalances faster) or vice versa. The “v” is the average velocity of the buses.

The second line describes how the headways of the other buses change.

This collection of equations describes how the headways change from bus arrival t to the next bus arrival t+1. In other words, it predicts the future behavior of all the buses.

Don Eisenstein and I recognized that this set of equations has a very special algebraic structure: they describe a “Markov Chain,” which is a sequence of events for which the future can be predicted by knowing merely the current state (no history is needed). In our case, we only need to know the most recent headways to predict the next headways, and the headways after those, and so on.

The theory of Markov Chains allows us to conclude that, in the absence of disruptions, the headways will move inexorably and quickly toward a common value, which is given in the equation above. What this means in practice is that the buses will move away from each other, to space themselves more evenly. In other words, we will have created a force, a sort of “anti-gravity” that pushes the buses apart and so resists bunching.

proofmathisbeautiful:

Saddle up for maximum snack satisfaction (mathematically speaking)

Stephanie V.W. Lucianovic is a Bay Area writer and editor. Her first book Suffering Succotash: A Picky Eater’s Quest to Understand Why We Hate the Foods We Hate, a humorous non-fiction narrative and exposé on the lives of picky eaters, will be released by Perigee Books on July 3.

My husband is a calculus professor and one who brings food items into the classroom with surprising regularity. No, he doesn’t bring pies on Pi day - though he can recite the string up to a couple dozen digits - but he does bring Pringles. As a teaching aid.

This afternoon when I walked into his study, I nearly tripped over a plastic Safeway bag filled with six red cans of Pringles. “Is it Pringles Day already?” I asked, nudging the bag. Pringles Day is the day Dr. Mathra lectures on the classification of critical points in multivariable calculus, and he uses the saddle-shaped Pringles to illustrate his points.

After class, the students get to eat his illustrations. It’s their favorite day.

However, this Pringles Eve, Dr. Mathra is kicking himself because in addition to stocking up on Pringles, which were invented by Proctor & Gamble & heaven in the 1960s, he also got an oblong can of Lays Stax, the parvenu potato chip that’s only been around since 2003.

Personally, I’ve never been turned on by Lays Stax. Not only are they covered with the stink of being the unoriginal upstart that is so obviously trying to rip-off the adored-for-decades potato chip, but they’re not thin and delicate enough, they’re not oily enough, and they’re not addictive enough. However, none of the above is Dr. Mathra’s complaint with them.

“It’s ridiculous!” he fumed, “They set themselves up as a Pringles competitor, but it’s an entirely different curvature!”

The shape of the Lays Stax - known as a parabolic cylinder - is way less mathematically interesting than the hyperbolic paraboloid of a Pringles, which is also known as a saddle. In math, the Pringles saddle shape exemplifies how you can stand at the flat point of a surface and not be at the highest point of your surroundings or at the lowest point of your surroundings.

Basically, you could call the saddle “the taint” of critical points. T’aint the highest point, t’aint the lowest. “Um, sure. If you wanted to be crass about it,” Dr. Mathra mumbles.

The big three types of critical points in multivariable calculus are the bottom of a bowl (aka the local min), the top of a dome (the local max), or in the middle of a saddle (saddle point).

“The Lays Stax shape isn’t even as interesting as a bowl - it’s a wishy-washy bowl. I mean, you can make the Lays shape with a piece of paper,” Dr. Mathra explains. (In my twelve years of being married to him, I have frequently found that being able to make something with paper is met with derision.) See, you can’t replicate the Pringles saddle shape with a piece of paper without cutting the paper and actually adding more paper to it and that makes it more mathematically desirable.

Sensing he has my attention throughout all of this raving, Dr. Mathra continues, “They’ve got these Lays Stax right next to the Pringles as though they are equivalent. How can they do that? One is a positive semi-definite quadratic form and the other is an indefinite quadratic form - they’re not even the same definiteness!”

When I don’t react, he insists, “Oh, come on - that will KILL in class tomorrow!”

And why should you, the non-calculus student, care about the Pringles saddle form? The principal application of calculus is optimizing, or determining whether you are at a maximum. You use calculus whenever you want to optimize, well, anything. “If you are at a local max (the top of a dome), everywhere you go moves you down. If you’re at a saddle, there’s a way you can go that will take you up.” Knowing this is important when thinking about increasing filthy lucre, precious time, diminishing resources, or a supply of Pringles.

And that, my friends, is why Pringles will always, always beat Lays Stax.

Flavor is subjective. Math is irrefutable.

cacalita:

jessicavalenti:

This is how God will strike down Rick Santorum.

unknowablewoman:

I don’t know what this is from and I don’t care.

AHHHHHHHHHHAHAHAHAHHAHAHAHAHAHAHAHHAHAHAHAHAHAHAH.

This is a must have on my tumblr.

(via cacalita)