applied-math

3

The Heap Sort Algorithm

The Heap Sort Algorithm has a best and worst case time complexity of O(nlog(n)). This means that given a list of numbers n it will take Heap Sort only n*log(n) steps to sort the numbers. If the numbers are already in a heap, then Heap Sort is even more efficient. The sudocode for a Head Sort is as follows:

  • for the Wikipedia article with this sudocode [click here]
  • for an implementation of Heap Sort in C++ [click here]

Credit: Wolfram Alpha / Wikipedia

One thing I want to get across when I’m talking with people is that I view a mathematics library the same way an archaeologist views a prime digging site. There are all these wonderful treasures that are buried there and hidden from the rest of the world. If you pick up a typical book on sheaf theory, for example, it’s unreadable. But it’s full of stuff that is very, very important to solving really difficult problems. And I have this vision of digging through the obscure text and finding these gems and exporting them over to the engineering college and other domains where these tools can find utility.
—  Robert Ghrist, professor of mathematics and engineering at UPenn

what i really want is an university au

dEIDARA IS THE MOST OBNOXIOUS ART STUDENT EVER like he thinks anyone who isnt taking ceramic/crafts is beneath him

ino wants to be a nature artist and likes more traditional media like oils and pastels

(sasori is studying engineering/robotics and calls it “art” which makes deidara really mAD)

vimeo

Awesome. To be seen in full screen on high definition. But to be accurate, I have to say that all the things featured in this video actually are Applied Mathematics, a crucial branch of Pure Mathematics, but not “just” Mathematics.

By Yann Pineill & Nicolas Lefaucheux

Seen in Gizmodo

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A Universal Power Law Governing Pedestrian Interactions

"To overcome the challenges that I mentioned already and robustly quantify the interaction law of pedestrians we employed a novel approach rooted in condensed matter physics. We initially measured the probability that any pair of pedestrians in the data has of maintaining a certain separation distance. We basically hypothesized that similar to charged particles, the interaction between pedestrians is distance-dependent. However, we found that the probability plots were very different for different walking speeds; when two pedestrians approach each other very fast they tend to maintain a larger separation distance than when they move slowly, as opposed to particles. As such, we started looking into different variables that can describe the interactions between pedestrians and we found that the time-to-collision is a sufficient descriptor. The probability plots were the same for different speeds as well as different orientations at which pedestrians approach each other. In addition, the time-to-collision measure naturally accounts for pedestrians coming relatively close to one another when moving in roughly the same direction–e.g. a pair of friends walking line-abreast. Eventually, by analyzing all the data, we inferred a simple energy law for the interactions between pairs of pedestrians."
- Dr. Ioannis Karamouzas

There’s an accessible (read: not math heavy) article from The Speaker about this research and the paper itself (which is not too complicated either) can be found here (pdf)

Youtube videos of the above gifs (1, 2, 3)

I alone cannot change the world, but I can cast a stone across the waters to create many ripples.
—  Mother Teresa

A ripple as a philosophical idea is something that has been discussed by many of the great minds from history, but how many of these people would know anything about how to measure ripples, or predict how quickly they will spread? Prepare to be amazed, as two mathematicians explore and explain: ‘Radial Spreading of a Surfactant on a Thin Liquid Film’.

For more maths, read these free articles chosen by the Editor of Applied Mathematics Research eXpress.

Is history just “one damned thing after another”? If not, if there are general principles shaping the broad patterns of history, we should be able to capture them with mathematical models. But would you believe that a mathematical model can predict, with startling accuracy, where and when major civilizations arose? (x)

University of Connecticut professor Dr. Peter Turchin has been leading the way on cliodynamics, a relatively new science at the intersection of marcosociology, economic history, and mathematical modeling. The idea is to use mathematical ideas to better understand how societies have risen or fallen throughout history.

Those working in cliodynamics have found consistent patterns of social instability. They often see that societies follow cycles resulting from both demographic trends (population growth, falling wages, age distribution) and violence trends. These trends form feedback loops that can be understood using differential equations in the same way predator-prey systems are understood in biology. 

While Turchin claims most of the mathematics is not too advanced, this is a great example of how mathematical thought can be applied to uncover patterns in other disciplines. A full description of cliodynamics and its research can be found here.