Original Math Notes
All Seasons
Episode 301: Spree--Explore the Math
Original Math Notes
Episode 301: Spree
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 7:
POV -- CHARLIE, sandwiched between TWO BEDS OF NAILS -- a
SLAB OF CONCRETE on top of the upper bed.

LARRY, protective glasses on, has a sledgehammer over his
head, about to smash the concrete slab.

                 DON (CONT'D)
          Okay....
          
                 CHARLIE       
          This isn't what it looks like.
          
                 EDGERTON
          That's a relief... since it looks
          like your friend there is about to
          perforate you.
          
                 LARRY
          Actually, Professor Eppes is
          protected by the most impenetrable
          armor of all -- physics.
          
                 CHARLIE       
          My weight is distributed among the
          spikes at a very comfortable 1.7
          pounds each, with the kinetic
          energy of the hammer --
          
                 LARRY
          -- I believe, Charles, that the
          drama of demonstration will be more
          instructive.
          
                 CHARLIE       
          Right...
          
Larry brings down the sledgehammer...

                 DON
          Whoa! Larry --
          
... SHATTERING the concrete. Charlie is still for a beat.

                 DON (CONT'D)
          Charlie -- ?
          
Charlie grins -- enjoying the rush.

                 CHARLIE       
          Fine... I'm fine.
          
They remove the concrete, and help Charlie off the spikes.

Bed of Nails

A bed of nails can actually be comfortable, if there are enough nails. A person's weight is divided by the number of nails under them. Here, Mr. Slab weighs 200 pounds, so the default bed is actually extremely painful.
Scene 9:
ENTER CHARLIE VISION --
The map becoming a scattered, fragmented pastiche of images
that will explain themselves later on:
-- a BIPLANE flying through the air
-- a MOUSE scampering across a field
-- a CURVING GRAPH LINE

                 CHARLIE       
          Didn't you forget to mark one --
             (points to a spot in Wyoming)
          -- here?
          
Edgerton looks at the map, double checks the reports --

                 EDGERTON
          No... we don't have sightings
          anywhere in Wyoming.
          
Charlie is a little surprised --

                 CHARLIE       
          Oh... okay...
          
                 DON
          Why'd you think there was?
          
                 CHARLIE       
          Actually --
             (beat)
          -- I looked at the map and... just
          thought there should be a point
          there.
          
Larry looks at the map.

                 LARRY
          Did you see some sort of pattern,
          perhaps a Markovian path --

Finite State Discrete Time Markov Chains

Consider a system that is always in one of n states, numbered 1 through n. Every time a clock ticks, the system updates itself according to an n×n matrix of transition probabilities, the (i,j)th entry of which gives the probability that the system moves from state i to state j at any clock tick. A Markov chain is a system like this, in which the next state depends only on the current state and not on previous states.
Scene 17:
ALAN and LARRY playing chess.

                 LARRY
          Relocation is considered a highstress
          event, but in my experience
          it's been nothing less than
          exhilarating.
             (beat)
          When I sold my house... it was a
          Promethean shrug, the chains of
          earthly concerns falling from my
          shoulders.

Explaining Real Estate Price Bubbles

Bubbles form in investment real estate, leading to the questions: Where is the top? When will prices no longer rise? An answer to both of these questions is: when the money runs out. In real estate investment there are two sources of funds: the buyer's investment (down payment) and the lender's capital (the loan amount). The lender operates as a sort of governor, investing less as prices rise, thereby refusing to finance the speculative part of the bubble (that part unsupported by commensurate increases in net rent).
Scene 27:
     
                 CHARLIE       
          This is the Green River robbery.
          The missing point on a curve I
          didn't realize I was looking at.
             (beat)
          It's a complex variation of
          something called a pursuit curve...
          created when one point chases
          another across a graph. You see...
          
BACK TO AUDIENCE VISION --
of our WWI dogfight --

                 CHARLIE (CONT'D)
          -- if a fighter pilot flies
          straight at the point where he sees
          an enemy plane --
          
-- the biplane turns, but the the Red Baron flies straight --
seeing nothing but empty air --

                 CHARLIE (CONT'D)
          -- by the time he gets there, the
          plane will be gone.
          
-- turning and finding the biplane again -- this time
matching its maneuvers, banking when the biplane banks,
dipping when the biplane dips --

                 CHARLIE (CONT'D)
          Just to maintain his following
          distance, the chasing pilot has to
          constantly adjust his direction --
          matching the movements of his
          quarry.

-- then, when the biplane turns -- the Red Baron turns deeper
-- cutting into its path --

                 CHARLIE (CONT'D)
          If he wants to catch the plane, he
          has to anticipate its movement --
          aim his nose ahead of the lead
          plane --
          
-- and firing its machine gun, which hits the biplane
broadside; it goes down in a plume of smoke...

BACK TO SCENE
                 CHARLIE (CONT'D)
          (to Edgerton)
          You've been locked in a pursuit
          curve with the killers; constantly
          adjusting and readjusting your
          search pattern to anticipate their
          movement.

Pursuit Curves

Given a moving prey (black path) with unit speed and a time-dependent position (x(t), y(t)), a predator (colored path), moving with speed v, runs at all times in the direction of the current position of the prey. The resulting path is known as a pusuit curve.
 
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