CHARLIE (V.O.) Entropy. A measure of randomness, a parameter of disorder... energy broken down in irretrievable heat. What might appear to be chaos... even decay... is really a system's way of smoothing out differences -- its search for equilibrium. Uncorrelated parts interact... find their connections in an evolving system...... so, from one perspective, entropy is a clock... charting the irreversible.

The entropy of a list Q measures its amount of disorder. The initial condition is a finite list of random bits. The
entropy can be used to study the amount of information in the evolution of a cellular automaton; it is lower in
ordered systems and higher in chaotic (disordered) systems.

Charlie is reading a newspaper as Amita and LARRY come in, bringing coffee -- AMITA Thought you snuck out early to work on that lecture... CHARLIE I made the mistake of buying a paper with my coffee. Remember that idea I had for a finding in Complex Polynomial encryption? Withers' group just patented it. AMITA It wasn't Withers... it was his collective. In fact, a civil engineer suggested attacking it through Riemann's Hypothesis.

Four points in the complex plane can be the roots of a complex polynomial of degree four. Solid lines indicate where the real part is zero and dashed lines indicate where the imaginary part is zero. These lines intersect at the chosen roots. The successive derivatives of the complex polynomial behave similarly.

DAVID (thinking) 127 rolls of floss -- which is a restricted item in prison for exactly this reason -- CHARLIE We might be able to apply a Simplex Algorithm -- the amount of time it would have taken to build the ladder, based on difficulty of access -- LARRY -- they had limited time to work on it, when guards and other prisoners weren't watching them -- AMITA -- they would have had to find a way to hide the empty containers, throw them out...

This Demonstration illustrates the graphical solution to several linear programming problems, all of which have the same set of constraints; you can vary the objective function. When two corner points are optimal, so are all the points on the line segment connecting them. The region shaded in blue is the feasible region and the colored lines correspond to the constraints. The black line represents the chosen objective function set to the slider value.

Larry and Amita give him a "no sale" look -- CHARLIE (cont'd) Okay... I was seven years old, and I asked my dad to help me figure out a good estimate for the remainder term in a Taylor expansion of the hyperbolic cosine. I remember our eyes meeting, and this... tacit understanding that we'd crossed the Rubicon. (beat) When you're seven, your father needs to be seven feet tall and infallible. So I separated math from my Dad... irrational as it may be, I'd prefer to keep it that way. As Larry and Amita exchange a look...

Choose the maximum degree of the Taylor polynomial to use to approximate a function. You can choose from a variety of functions and manipulate the expansion point. To see the error in the approximation, select the "error" radio button and use the slider that appears under the graph.

AMITA (exasperated) Yet again, male communication tests the limits of Shannon's source coding theorem.

Using the second law of thermodynamics, it is possible to use random variables to calculate the information entropy (or
Shannon entropy) of a message, which is a measure of the amount of information in the message. The probabilities that A
and B occur in the message are P(A) and P(B).

CHARLIE Sure... where's Maxwell's Demon when you need him. ALAN Maxwell's Demon -- CHARLIE You know -- the man who stands alone at the door. In two adjoining rooms, the temperature and pressure are the same... a state of perfect equilibrium. Every time the Demon opens the door, he admits only those molecules he chooses... heating one room and cooling the other... in violation of the Second Law of Thermodynamics.

Thermal energy is the energy of an object due to random motions of its atoms and molecules. The hotter the object, the
greater its thermal energy. Thermal energy is an extensive variable, proportional to the size of the object. The
individual molecules can have different kinetic energies, but a hot object has a higher average value. In a gas or a
liquid, molecules can move freely in all directions; in a solid, molecules execute small vibrations in all directions
about fixed positions.