Original Math Notes
All Seasons
Season 5
Episode 515: Guilt Trip
Solutions of a chaos-describing partial differential equation
Scene 14:
Charlie reaches into the glass, pulls out a fist full of gel- 
like spheres completely hidden in the water.
          
                    CHARLIE (cont'd)
          Hydrophilic spheres.  They start off
          as hard crystals, but drop them in
          water and they expand 300 times
          their size. Since the sphere
          crystals are mostly water --
          
He drops them back in the cup, they disappear again.

                    CHARLIE (cont'd)
          They become invisible when
          submerged due to an identical index
          of refraction with the liquid.

Refraction from a Submerged Light Source »

A light beam is produced by a source submerged in an aqueous solution of refractive index n. This Demonstration shows how the light source appears for different beam directions, divergences, and indices of refraction. The same effect causes a stick obliquely submerged in water to appear bent.
Scene 17:
CLOSE ON A LAPTOP SCREEN -- an ANIMATED MODEL of 12 numbered 
balls; it resembles the POWERBALL LOTTERY. 

Charlie works "jury" math on a board.  ALAN enters -- 

                    ALAN
            (re: the model)
          Hope you'll share the jackpot.
          
                    CHARLIE
          I'm not predicting the lottery.  And
          for the last time -- I can't.
          
                    ALAN
            (kidding)
          Can't... or won't?
          
                    CHARLIE 
          It's a Spatial Dynamic Jury
          Model... a statistical model of
          jury behavior.
          
                    ALAN
          Was it developed by twelve angry
          mathematicians?

A Spatial Dynamic Jury Model »

In this spatial and dynamic model of jury behavior, jurors initially vote to convict or acquit the defendant based on a random variable drawn from a distribution affected by: (1) the defendant's actual guilt or innocence, and (2) the location of each juror in a two-dimensional conceptual space that represents the juror's ability to accurately perceive the situation as well as his or her predilection towards conviction. On subsequent ballots (the second stage), the jurors revise their votes based on the weighted votes of the other jurors. The weights for each juror are determined by a distance measure between that juror and the other jurors and a "stubbornness factor" that gives a special weight to the juror's own prior vote. The balloting continues until a fixed point is reached or until ten ballots have been taken, at which point the jury is assumed to have reached exhaustion.
Scene 33:
                    CHARLIE
          Which, by definition, presupposes
          statistical analysis.  It comes
          down to group dynamics.  Kinda like -
          - herding cattle.
          
ENTER AUDIENCE VISION:
Charlie walks across a western landscape. 

                    CHARLIE (cont'd)
          In the Old West, you'd have maybe
          10 cowboys to drive thousands of
          cattle.
          
BEYOND Charlie, a massive herd of CATTLE on the move. 

                    CHARLIE (cont'd)
          Instead of trying to control an
          entire herd, they'd pick a lead
          steer.  They'd well, steer him...

3D Boid Model »

The Boid model is a famous multi-agent model that was invented by Craig Reynolds. Each Boid is an agent following three simple rules: (1) Align with other neighbors; (2) Try to be close to neighbors; (3) Avoid collision with other neighbors. These rules can make the flock of Boids reproduce the behaviors of real bird flocks or fish schools. Push the Play button to trigger the simulation, and adjust the number of Boids, the size of the world and other parameters by using the sliders, then enjoy yourself.
Scene 40:
                    CHARLIE
          I ran an asymptotic analysis of the
          program's Mersenne Twister
          algorithm -- the system used to
          pick jurors out of the county pool.
            (explains)
          Broke it down to see if it's
          working properly.
          
                    DON
          It wasn't?
          
                    CHARLIE
          No, when Hooper hacked the
          software, he planted Cerf into the
          system.

Mersenne Twister and Friends »

The Mathematica function SeedRandom comes with a variety of different methods. There are two basic types, often identifiable by sight. The pseudorandom methods (such as the Mersenne Twister) seem random, while the quasirandom methods (such as Sobol) seem to have a pattern, with less clustering.
 
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