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Summer Vacation Research

Walter Hickey, BI, How to use Math to Crush Your Friends at ‘Risk’ Like You’ve Never Done Before, here.

Here are the five things to take away from this.

  • Holding the continent of Europe requires you to hold the fewest number of territories per bonus army. However, almost 60% of European territories lie on a border. The per-territory reward comes with risk.
  • That risk can be measured by the ratio of continental bonus armies per border territory. North America has the most favorable ratio.
  • Africa is easy to invade but difficult to hold. The amount you need to commit to its defense make it a particularly unsafe Australia.
  • When you know battle is inevitable, attack as soon as you have the same number of attacking armies as your opponent has defending armies. You hold the advantage.
  • The larger the battle, the larger the attacker advantage.

And of course, never attempt a land war in Asia. That’s just basic.

Steven G. Johnson, Photonic Crystals: Periodic Surprises in Elecromagnetism, 2003-8, here.

Has there been anything new in classical electromagnetism since Maxwell laid down the law in 1864? If so, can one learn it without wading through a vectorial mire of partial differential equations? Come and find out what solid-state physics has brought to 8.02 in the last 15 years: photonic crystals and the surprising new phenomena that arise when light propagates through a periodic medium. This crash course will introduce Bloch’s theorem for electromagnetism, photonic band gaps, the confinement of light in novel waveguides and cavities by synthetic optical “insulators,” startling sub-micron fabrication advances, exotic optical fibers, and will upend what you thought you knew about total internal reflection. We will focus less on gory differential equations than on high-level approaches such as linear algebra, variational theorems, conservation laws, and coupled-mode theory; the course should be accessible to anyone with a grasp of basic electromagnetism and who does not quake in fear at the word “eigenvalue.”

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