At oC, water boils spontaneously even though the reaction is endothermic. There must be another factor that helps determine whether a reaction is spontaneous.

Share via Email The Simpsons: This is not a figment of my deranged mind, which admittedly is obsessed with both The Simpsons and mathematics, but rather it is a concrete claim backed up in a series of remarkable episodes.

The first proper episode of the series in contained numerous mathematical references including a joke about calculuswhile the infamous "Treehouse of Horror VI" episode presents the most intense five minutes of mathematics ever broadcast to a mass audience. Moreover, The Simpsons has even offered viewers an obscure joke about Fermat's last theoremthe most notorious equation in the history of mathematics.

These examples are just the tip of the iceberg, because the show's writing team includes several mathematical heavyweights. Al Jean, who worked on the first series and is now executive producer, went to Harvard University to study mathematics at the age of just Others have similarly impressive degrees in maths, a few can even boast PhDs, and Jeff Westbrook resigned from a senior research post at Yale University to write scripts for Homer, Marge and the other residents of Springfield.

When they moved from academia to Fox Studios, these writers retained their passion for numbers and they have secretly planted mathematical references in dozens of episodes.

Until now, only extreme geeks have been aware that the writers have been smuggling mathematics into their scripts while the rest of the planet has been oblivious to the numerous nods to number theory and geometry.

The episode "Marge and Homer Turn a Couple Play", for example, contains a triple dose of secret mathematics. The storyline revolves around Marge and Homer's efforts to help baseball star Buck Mitchell and his wife Tabitha Vixx, who are experiencing marital difficulties.

The episode climaxes with Tabitha appearing on the Jumbo Vision screen at the Springfield stadium, where she publicly proclaims her love for Buck. More important, just before she appears on the screen, it displays a question that asks the baseball fans in the crowd to guess the attendance.

The screen displays three multiple choice options; 8, 8, and 8, These digits might seem arbitrary and innocuous, but in fact they represent a perfect number, a narcissistic number and a Mersenne prime.

The smallest perfect number is 6, because 1, 2 and 3 not only divide into 6, but they also add up to 6. The second perfect number is 28, because 1, 2, 4, 7 and 14 not only divide into 28, but they also add up to The third perfect number isand the fourth one is 8, which appears in this episode.

The fact that 8, can recreate itself from its own components hints that the number is in love with itself, hence the narcissistic label. Among the infinity of numbers, fewer than exhibit narcissism. More generally, Mersenne primes fit the pattern 2p —1, where p is any prime number.

Not surprisingly, several of the mathematical quips in The Simpsons relate to Homer and Marge's daughter, Lisa. She is proud to be a nerd and her grasp of everything from trigonometry to logarithms is recognised by Principal Skinner in the episode "Treehouse of Horror X" After a stack of bench seats falls on Lisa, he cries out: And so have the hopes of our mathletics team.

The entire storyline is rooted in statistics, but the most significant nerd reference appears and disappears in a blink of an eye.

Just before her first big game in charge, we see Lisa poring over piles of technical books. This extraordinary sight prompts a reporter to remark:Gravitational Entropy and Global Structure S.W.

Hawking∗ and C.J. Hunter† Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom by a universal formula. We illustrate with a number of examples including spaces with nut charge.

In these cases, the entropy is. Entropy and the Second Law of Thermodynamics.

The second law of thermodynamics describes the relationship between entropy and the spontaneity of natural processes. Second Law: In an isolated system, natural processes are spontaneous when they lead to an increase in disorder, or entropy.

Password entropy is the measurement of how unpredictable a password is. The mathematical formula is based off of how many different character sets are used as well as the length of the password.

The formula . In A Mathematical Theory of Communication, appendix 2, Shannon proves his Theorem 2, that this Boltzmann entropy is the only function which satisfy's the requirements for a function to measure the uncertainty in a message .

Based on that, a finite-time version of Pesin's entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived.

The Clausius-Clapeyron equation. Phase changes, such as the conversion of liquid water to steam, provide an important example of a system in which there is a large change in internal energy with volume at constant tranceformingnlp.come that the cylinder contains both water and steam in equilibrium with each other at pressure P, and the cylinder is held at constant temperature T, as shown in the.

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