Beautiful Mind equation

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In summary, the equation introduced by Prof. Nash in the movie A Beautiful Mind is V={F:R3|X-->R3 so (Del x F)=0} and the question is whether it has a nontrivial solution and how to determine the dimension of V/W. The solution is trivial when X is empty, but when X is non-empty, the number of connected components of X may need to be considered.
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Loren Booda
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"Beautiful Mind" equation

In the movie A Beautiful Mind, Prof. Nash introduces his class at MIT to a problem (below) on the chalkboard and states "...for others among you it will take the term of your natural lives." What is this equation, is it nontrivial, and does it have a solution?

V={F:R3|X-->R3 so (Del x F)=0}

W={F=(Del g)}

dim(v/w)=?
 
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  • #2
Interesting; I thought about that too when I saw the movie. I couldn't figure out the solution yet. It's certainly trivial when the set X in the first line is empty, because then V=W and dim(V/W)=0.

And when X is non-empty? Maybe you have to deal with the number of connected components of X.

edit: the same again in clear latex:
[tex]V=\{ F:\mathbb{R}^3 \backslash X \rightarrow \mathbb{R}^3 \:so\: curl F=0\}[/tex]
[tex]W=\{F=\nabla g \}[/tex]
[tex]dim(V/W)=?[/tex]
 
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  • #3


The "Beautiful Mind" equation is a mathematical problem that was introduced by Prof. Nash in the movie A Beautiful Mind. It is a nontrivial equation that involves vector calculus and has a solution. The equation is written as V={F:R3|X-->R3 so (Del x F)=0}, which means the set of functions F in three-dimensional space that map from X to R3 such that the divergence of F is equal to zero. This equation is used to solve problems related to fluid mechanics and electromagnetism.

The second part of the equation, W={F=(Del g)}, introduces the concept of a gradient field and is related to the first part of the equation. The solution to this equation involves finding the dimension of the quotient space V/W, which is represented by dim(v/w)=?. This is a challenging problem that requires advanced mathematical skills to solve.

In the movie, Prof. Nash states that for some students, this equation may take the term of their natural lives to solve. This highlights the complexity and difficulty of the problem, emphasizing the brilliance and intelligence of Prof. Nash's mind.

In conclusion, the "Beautiful Mind" equation is a nontrivial mathematical problem that has a solution and is related to vector calculus and gradient fields. Its complexity and difficulty make it a perfect representation of Prof. Nash's genius mind.
 

1. What is the "Beautiful Mind" equation?

The "Beautiful Mind" equation, also known as the Drake equation, is a mathematical formula created by astronomer Frank Drake in 1961 to estimate the number of active, communicative extraterrestrial civilizations in our galaxy.

2. How does the "Beautiful Mind" equation work?

The equation takes into account factors such as the rate of star formation, the fraction of stars that have planets, the fraction of planets that could potentially support life, the fraction of those planets that actually develop life, the fraction of those life forms that develop intelligence, and the fraction of those intelligent life forms that develop technology capable of communication.

3. Is the "Beautiful Mind" equation accurate?

The accuracy of the equation is debated among scientists, as it is based on many unknown factors and assumptions. However, it serves as a tool for discussing the likelihood of extraterrestrial life and has sparked further research and discussions in the scientific community.

4. What is the significance of the "Beautiful Mind" equation?

The equation has sparked interest and discussion in the search for extraterrestrial life, and has also been used in the development of the SETI (Search for Extraterrestrial Intelligence) program. It also highlights the importance of considering multiple factors when contemplating the existence of alien civilizations.

5. Has the "Beautiful Mind" equation been updated or modified?

Since its creation, the equation has been updated and modified by various scientists to reflect new discoveries and advancements in the field. Some versions include additional factors such as the length of time a civilization is likely to survive and the likelihood of a civilization sending out signals that we are able to detect.

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