Challenge problem #5 [Olinguito]

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In summary, the conversation discusses the definitions of $A\circ B$ and $A^{[n]}$, as well as finding the sum of all elements in $\zeta_n^{[r]}$. It is concluded that $\zeta_n^{[r]}$ is equal to $\zeta_n^{[1]}$ for all $r$, and the sum of the elements in this set is equal to $0$.
  • #1
Olinguito
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If $A$ and $B$ are nonempty sets of complex numbers, define
$$A\circ B\ =\ \{z_1z_2:z_1\in A,\,z_2\in B\}.$$
Further define $A^{[1]}=A$ and recursively $A^{[n]}=A^{[n-1]}\circ A$ for $n>1$.

Let $\zeta_n=\{z\in\mathbb C:z^n=1\}$. Given a fixed integer $n\geqslant2$ and any positive integer $r$, find the sum of all the elements in $\zeta_n^{[r]}$.
 
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  • #2
[sp]
As $\zeta_n^{[1]}$ is closed under multiplication, $\zeta_n^{[r]}\subset\zeta_n^{[1]}$. On the other hand, as $1\in\zeta_n^{[1]}$, $\zeta_n^{[1]}\subset\zeta_n^{[r]}$.
The conclusion is that $\zeta_n^{[r]}=\zeta_n^{[1]}$ for all $r$. The sum of the elements of that set is the sum of the roots of $z^n-1$, which is $0$.
[/sp]
 
  • #3
Nice work. (Yes)

Alternatively, note that
$$A\circ B\ =\ \bigcup_{z\in A}\,\{z\}\circ B$$
and $\{\omega\}\circ\zeta_n=\zeta_n$ where $\omega$ is any $n$th root of unity.
 

Related to Challenge problem #5 [Olinguito]

1. What is an olinguito?

The olinguito (Bassaricyon neblina) is a small mammal in the raccoon family, native to the cloud forests of the Andes in South America. It was first discovered and described by scientists in 2013.

2. What makes the olinguito unique?

The olinguito is the first new carnivorous mammal species discovered in the Western Hemisphere in 35 years. It also has some unique physical characteristics, such as its woolly fur and long, sharp claws for climbing trees.

3. How was the olinguito discovered?

The olinguito was discovered by a team of scientists who were studying museum specimens of olingo, a similar species. They noticed that some specimens had different physical characteristics and DNA, leading them to realize that they were actually a new species.

4. What is the habitat and behavior of the olinguito?

The olinguito is found in the cloud forests of the Andes, at elevations of 5,000-9,000 feet. It is a solitary, nocturnal animal that primarily feeds on fruit, but also eats insects and small animals. It is an excellent climber and spends most of its time in the trees.

5. Is the olinguito endangered?

The olinguito is currently listed as "Least Concern" on the IUCN Red List of Threatened Species. However, its habitat is being threatened by deforestation and climate change, so conservation efforts are important to ensure its survival in the future.

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