Category Archives: Math

Innovation with Math

Invented by scientists at the nearby Los Alamos National Laboratory in the 1980s, complexity science is a gumbo of insights drawn from fields as diverse as biology, physics, and economics. At its core is the belief that any seemingly complex and utterly random system or phenomenon–from natural selection to the stock market–emerges from the simple behavior of thousands or millions of individuals. Using computer algorithms to stand in for those individual “agents,” scientists discovered they could build fantastically powerful and detailed models of these systems if only they could nail down the right set of rules.

When Brown arrived in town in the late 1990s, many of the scientists-in-residence at the Santa Fe Institute–the serene think tank dedicated to the contemplation of complexity–were rushing to commercialize their favorite research topics. The Prediction Co. was profitably gaming Wall Street by spotting and exploiting small pockets of predictability in capital flows. An outfit called Complexica was working on a simulator that could basically model the entire insurance industry, acting as a giant virtual brain to foresee the implications of any disaster. And the BiosGroup was perfecting agent-based models that today would fall under the heading of “artificial life.”

10 Lessons of an MIT Education

Very good, definitely worth reading – 10 Lessons of an MIT Education by Gian-Carlo Rota:

In science and engineering, you can fool very little of the time. Most of the sweeping generalizations one hears about MIT undergraduates are too outrageous to be taken seriously. The claim that MIT students are naive, however, has struck me as being true, at least in a statistical sense.

Last year, for example, one of our mathematics majors, who had accepted a lucrative offer of employment from a Wall Street firm, telephoned to complain that the politics in his office was “like a soap opera.” More than a few MIT graduates are shocked by their first contact with the professional world after graduation. There is a wide gap between the realities of business, medicine, law, or applied enginering, for example, and the universe of scientific objectivity and theoretical constructs that is MIT.

An education in engineering and science is an education in intellectual honesty. Students cannot avoid learning to acknowledge whether or not they have really learned. Once they have taken their first quiz, all MIT undergraduates know dearly they will pay if they fool themselves into believing they know more than is the case.

On campus, they have been accustomed to people being blunt to a fault about their own limitations-or skills-and those of others. Unfortunately, this intellectual honesty is sometimes interpreted as naivete.

Math and Nature

Can’t Knock It Down:

Give mathematicians such a toy, and they’re liable to turn it into a math problem.

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Next, the pair began to investigate whether all three-dimensional shapes have at least two stable and two unstable balance points. They tried to generalize their two-dimensional proof to higher dimensions, but it didn’t hold up. Therefore, it seemed possible that a self-righting three-dimensional object could exist. Such a shape would have only one stable and one unstable balance point.
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Once the pair had built their Once the pair had built their self-righting object, they noticed that it looked very much like a turtle. They figured that wasn’t an accident, since it would be useful for a turtle never to get stuck on its back., they noticed that it looked very much like a turtle. They figured that wasn’t an accident, since it would be useful for a turtle never to get stuck on its back.
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The mathematicians still face an unanswered question. The self-righting objects they’ve found have been smooth and curvy. They wonder if it’s possible to create a self-righting polyhedral object, which would have flat sides. They think it is probably possible, but they haven’t yet managed to find such an object. So, they are offering a prize to the first person to find one: $10,000, divided by the number of sides of the polyhedron.

Math’s Architect of Beauty

Math’s Architect of Beauty – How Terence Tao’s quest for elegance earned him a Fields Medal and a MacArthur Fellowship

The prime numbers do obey some simple patterns—for instance, all primes but 2 are odd numbers. The great achievement of the Green-Tao theorem is its use of subtle and novel methods of harmonic analysis (very much indebted to the work of 2002 Fields Medalist Timothy Gowers) to show that these simple patterns are essentially the only structure the primes possess. Beyond these basic structures, the primes look random—they are, in a sense, all noise and no music.

Related: Terence Tao – math related posts

248-dimension Math Puzzle

248-dimension maths puzzle solved:

What came out was a matrix of linked numbers, which together describe the structure of E8. It contains more than 60 times as much data as the human genome sequence.

Each of the 205,263,363,600 entries on the matrix is far more complicated than a straightforward number; some are complex equations. The team calculated that if all the numbers were written out in small type, they would cover an area the size of Manhattan.

In addition to facilitating further understanding of symmetry and related areas of mathematics, the team hopes its work will contribute to areas of physics, such as string theory, which involve structures possessing more than the conventional four dimensions of space and time.

Online Mathematics Textbooks

Online Mathematics Textbooks:

The writing of textbooks and making them freely available on the web is an idea whose time has arrived. Most college mathematics textbooks attempt to be all things to all people and, as a result, are much too big and expensive. This perhaps made some sense when these books were rather expensive to produce and distribute–but this time has passed.

A few years ago when I first posted a list of mathematics textbooks freely available on line, there existed only a handful of such books. Now there are many.

Including: Calculus by Gilbert Strang – Linear Algebra, Infinite Dimensions, and Maple by James Herod – Euclid’s Elements – Information Theory, Inference, and Learning Algorithms by David J. C. MacKay

Sudoku Science

Sudoku Science:

This places Sudoku in an infamously difficult class, called NP-complete, that includes problems of great practical importance, such as scheduling, network routing, and gene sequencing.

“The question of whether there exists an efficient algorithm for solving these problems is now on just about anyone’s list of the Top 10 unsolved problems in science and mathematics in the world,” says Richard Korf, a computer scientist at the University of California at Los Angeles. The challenge is known as P = NP, where, roughly speaking, P stands for tasks that can be solved efficiently, and NP stands for tasks whose solution can be verified efficiently.
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The route-finding algorithm that powers car navigation systems, for instance, was first demonstrated on the Sliding Tile puzzle, a child’s toy in which a player tries to move 15 tiles around a grid so that their surfaces form a picture. The same algorithm helps video game characters steer through virtual worlds. “This is an algorithm developed back in 1968 in abstract kinds of things,” says UCLA’s Korf, who himself has explored algorithms for the Rubik’s Cube. “It’s used all the time.”

Pixar Is Inventing New Math

Pixar Is Inventing New Math:

According to DeRose, Pixar is the first Hollywood studio equipped with it’s very own in-house scientific research facility. Mathematicians and computer scientists there are figuring out new mathematical ways to solve problems in animation.
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What they’re finding is that the interplay between academics and industry has been hugely successful. According to DeRose they now have more courage to explore scientific musings that would normally only have been possible in a university environment.

Seeing Patterns Where None Exists

I call data dredge studies the “Rorschach tests” of epidemiology, because researchers can pull out characteristics about people in almost unlimited combinations to find all sorts of correlations and conclude just about anything they set out to find. Just like the Rorschach test, seeing patterns where none exists, finding connections that are there but not as strongly as believed, and seeing what one expects to see, are common.

Page 8 of Statistics for Experiments by George Box, Willliam Hunter (my father) and Stu Hunter (no relation) shows a graph of the population (of people) versus the number of storks which shows a high correlation. “Although in this example few would be led to hypothesize that the increase in the number of storks caused the observed increase in population, investigators are sometimes guilty of this kind of mistake in other contexts.” And some might make it in this context 🙂

Declining Science and Maths Degrees in UK

Report: Core science and mathematics degree courses in the UK 1998-2007

In the decade to 2007, there has been a 10% reduction in the number of core, ie single honours, science and maths degree courses offered by UK higher education institutions.