Meet the symbots.

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The frenetic little bubble animated above is a symbot, Occupy Math’s own name for a type of super-simple robot. These robots exist only in the computer; we don’t actually make physical versions. There is an interesting book about this type of robot and its more complex cousins: Vehicles: Experiments in Synthetic Psychology by Valentino Braitenberg. These robots have sensors and wheels; the input to the sensors controls how fast the wheels turn. The interesting thing is the number of different behaviors that you can get out of even really simple robots. The robot above can sense the flashing light and is trying to approach it. It lacks the ability to slow down, so it’s learned to run the light over repeatedly. Think Labrador puppy.

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The factorial: not an excited number.

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The factorial of a number is what you get when you multiply that number and those smaller than it (down to one) together. That means that five factorial is 5x4x3x2x1=120. The mathematical notation for factorial is to use an exclamation point: 5!=120. Occupy Math was teaching a course that used factorials to count things and one of the sharper students kept getting problems wrong. Occupy Math wrote “5” on the board and asked “what number is that?” The student replied “five”. Occupy Math added an exclamation point to get “5!” and again asked the student what the number was. The student replied “FIVE!” This was a third-year university student — hence this educational post. This week’s Occupy Math looks at what factorials do (e.g.: they count things). Factorials also provide an example of something that grows faster than exponentially.

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Why Top Journals Are More Likely to be Wrong

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Occupy Math is going to look at a simple piece of math that is ignored or, worse, abused by researchers in many fields. It amounts to an example of ignorance of statistics that leads to publishing results that are bogus and so impossible to replicate. This problem is called the replication crisis because many important results seem to disappear when other researchers try and reproduce them. Occasionally this is the result of actual fraud — but more often ignorance of simple facts about statistics can let you publish a paper whose results cannot be replicated (because its results are actually wrong) without even noticing you’re doing it. There is also a separate problem — it is very difficult to completely describe an experiment, which means that the people trying to reproduce your results may not be doing quite the same experiment. That last is a big problem, but not what Occupy Math is looking at today.

The core message of today’s post is that peer-reviewed results in a top journal are sometimes wrong because we don’t teach statistics properly.

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The Curse of Dimensionality

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Occupy Math often tries to find click-baity titles for his posts. This week is not an exception, but it is unusual in that a phrase that Occupy Math heard more than ten times at the IEEE 2017 Congress on Evolutionary Computation in Donostia-San Sebastian Spain last week. In other words, the curse of dimensionality is a real thing. This week’s post looks at the very strange behavior of normal-seeming objects when we create higher-dimensional versions of them. This strangeness often corresponds to problems getting much harder as we increase the dimension, hence the use of the word “curse”.

Big issue – your intuition is shaped by two-dimensional and three-dimensional objects and is just wrong in higher dimensions.

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Supreme Court: Not Possible! Math: Possible!

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This week Occupy Math takes a trip to the land of Freedonia which is beset by a vile dragon that menaces its democracy. A small African nation with a diverse population and the magnificent port city of Great Haven, Freedonia is a constitutional democracy modeled on the American experiment, an active, participatory, free democracy and, until recently, with a vibrant and open economy. Founded in the early 20th century by a largely peaceful revolution organized by tribal leaders — advised by the famous explorer and polymath, Captain Spaulding — Freedonia has been synonymous with hope for generations. Recently, however, the economy has been experiencing problems with corruption. Nepotistic awards of government contracts to incompetent nephews and corrupt back-room deals have taken the economy away from the hard-working farmers, shopkeepers, and craftsmen who have been the backbone of Freedonia society. UN monitors certify each of the biannual elections as free and fair but, somehow, in spite of public outrage, the Lucarian party ekes out a bare majority and restores the corrupt Prime Minister, Joseph Cagliostro to power. What dark force is subverting democracy in Freedonia? Let’s ask no lesser authority that the Governator himself!

Gerrymandering is a subtle way of subverting democracy — and the vorpal sword that can slay it is edged with mathematics.

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