Ky Fan inequality
Hey there little one! Today, let's talk about ky fan inequality.
So imagine you have a big pile of sweet treats like cookies, candies, and chocolates. And you have two friends, let's call them A and B, who wants to divide the pile fairly.
Now, they can't just divide it in half because there's not always the same number of each treat. So, how can they divide it fairly?
This is where the ky fan inequality comes in. It's like a rule that tells A and B how they can divide the pile fairly based on how many of each type of treat there are.
The ky fan inequality states that if there are n treats with sizes x1, x2, ..., xn (where xi is the size of the i-th treat) and k friends who want to divide them fairly, each friend can take no more than the sum of the k-th powers of all the sizes divided by the sum of the k-th powers of the integers from 1 to k.
Woah, that's a lot of math-y words! Let me explain it more simply. Basically, if you have a bunch of different sweet treats and you want to divide them fairly between your friends, you can use the ky fan inequality as a guide to make sure everyone gets a fair share.
The rule says that each friend can take an amount of treats equivalent to the size of the treats raised to the power of k (the number of friends) divided by the sum of the k-th powers of the numbers from 1 to k.
So, let's say you have four types of treats with sizes 2, 3, 5, and 6. And you want to divide them equally between two friends.
Using the ky fan inequality, each friend can take no more than (2^2 + 3^2 + 5^2 + 6^2) / (1^2 + 2^2) treats.
This simplifies to (4 + 9 + 25 + 36) / (1 + 4) treats for each friend.
So, each friend can take no more than 74/5 treats each.
And that's how the ky fan inequality can help divide sweet treats fairly between friends!
So imagine you have a big pile of sweet treats like cookies, candies, and chocolates. And you have two friends, let's call them A and B, who wants to divide the pile fairly.
Now, they can't just divide it in half because there's not always the same number of each treat. So, how can they divide it fairly?
This is where the ky fan inequality comes in. It's like a rule that tells A and B how they can divide the pile fairly based on how many of each type of treat there are.
The ky fan inequality states that if there are n treats with sizes x1, x2, ..., xn (where xi is the size of the i-th treat) and k friends who want to divide them fairly, each friend can take no more than the sum of the k-th powers of all the sizes divided by the sum of the k-th powers of the integers from 1 to k.
Woah, that's a lot of math-y words! Let me explain it more simply. Basically, if you have a bunch of different sweet treats and you want to divide them fairly between your friends, you can use the ky fan inequality as a guide to make sure everyone gets a fair share.
The rule says that each friend can take an amount of treats equivalent to the size of the treats raised to the power of k (the number of friends) divided by the sum of the k-th powers of the numbers from 1 to k.
So, let's say you have four types of treats with sizes 2, 3, 5, and 6. And you want to divide them equally between two friends.
Using the ky fan inequality, each friend can take no more than (2^2 + 3^2 + 5^2 + 6^2) / (1^2 + 2^2) treats.
This simplifies to (4 + 9 + 25 + 36) / (1 + 4) treats for each friend.
So, each friend can take no more than 74/5 treats each.
And that's how the ky fan inequality can help divide sweet treats fairly between friends!