Why is No One Answering This Week's Problem?

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In summary, the problem for this week is a question or puzzle that requires scientific thinking and problem-solving skills to answer. There could be several reasons why no one is answering this week's problem, such as its difficulty, relevance, or lack of promotion. The possibility of a prize for solving the problem depends on the platform or organization. To improve your chances of solving the problem, you can do research, approach it scientifically, and use critical thinking skills. The submission process for the problem may vary, so it's best to follow the guidelines provided by the source.
  • #1
Euge
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Here's this week's problem!

________________

Problem. Let $\nu$ be the differential 2-form $dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n}$ on $\Bbb R^{2n}$. Show that $\nu^n = n!\, dx_1 \wedge dx_2 \wedge dx_3 \wedge \cdots \wedge dx_{2n}$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
No one answered this week's problem. You can find my solution below.
Let $\nu_j := dx_{2j-1} \wedge dx_{2j}$, for $j = 1,2,\cdots, n$. Then the nonzero contributions to $\nu^n$ are terms of the form $\nu_J$ for some ascending index $J$ of order $n$. For each ascending index $J$, there corresponds a unique permutation $\pi \in S_n$ such that $\nu_J = \nu_{\pi(1)} \wedge \cdots \wedge \nu_{\pi(n)}$. So $\nu^n$ is the sum of terms $\nu_{\pi(1)}\wedge \cdots \wedge \nu_{\pi(n)}$, as $\pi$ ranges over $S_n$. Since each $\nu_j$ is a two-from, $\nu_j \wedge \nu_k = (-1)^{2\cdot 2} \nu_k \wedge \nu_j = \nu_k \wedge \nu_j$ for all $k \neq j$. It follows that $\nu_{\pi(1)}\wedge \cdots \wedge \nu_{\pi(n)} = \nu_1 \wedge \cdots \wedge \nu_n$ for all $\pi \in S_n$. Thus

\(\displaystyle \nu^n = \sum_{\pi\in S_n} \nu_1 \wedge \cdots \wedge \nu_n = \#(S_n)\, \nu_1 \wedge \cdots \wedge \nu_n = n!\, dx_1 \wedge \cdots \wedge dx_{2n}.\)
 

FAQ: Why is No One Answering This Week's Problem?

What is the problem for this week?

The problem for this week is a question or puzzle that requires scientific thinking and problem-solving skills to answer.

Why is no one answering this week's problem?

There could be several reasons why no one is answering this week's problem. It could be a difficult or complex question that requires more time and effort to solve. It's also possible that the problem is not relevant or interesting to the audience. Another reason could be that the problem has not been shared widely enough to reach a larger number of people.

Is there a prize for answering this week's problem?

It depends on the specific platform or organization that shared the problem. Some may offer a prize or recognition for the first person to solve the problem, while others may not have any incentives.

What can I do to improve my chances of solving this week's problem?

To improve your chances of solving this week's problem, you can do some research and gather relevant information or resources related to the problem. It's also helpful to approach the problem with a scientific mindset and use critical thinking skills to analyze and solve the problem.

How can I submit my answer for this week's problem?

The submission process for this week's problem may vary depending on the platform or organization. Some may have a specific submission form or email address, while others may require you to post your answer in a designated forum or social media platform. It's best to check the guidelines or instructions provided by the source of the problem for the submission process.

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