Neumann polynomial
Okay kiddo, have you ever played with Legos? Think of a Neumann polynomial as a big Lego structure made up of smaller Lego pieces.
Let's start by imagining a big square Lego block with a bunch of smaller Lego blocks stuck to it. Each of those smaller blocks has a number on it, like 1, 2, 3, 4, and so on. We'll call that big square block our "base."
Now, let's say we want to build another Lego structure on top of that base. But we don't want to just plop it on top – we want it to fit perfectly with the base. To do that, we need to build it out of smaller Lego blocks with numbers on them that match the numbers on the base.
If we do this right, we'll end up with a new Lego structure that looks like it's just a bigger version of the base, with some new parts added on top. And if we keep doing this over and over again, we can build a really tall pyramid out of Legos!
So, what does all of this have to do with Neumann polynomials? Well, just like we built our Lego pyramid out of blocks that matched the base, a Neumann polynomial is built using a base function and a bunch of other functions that "match" the base.
The base function is usually something simple and easy to work with, like x (which you might know from math class as the variable that stands for "some number"). Then we add on other functions that are built to match x in certain ways. They might have x^2 or x^3 or even higher powers of x in them.
By combining all of these functions together using some math tricks, we can build a new function that looks like a bigger version of our base function, with some new parts added on top. And just like with our Lego pyramid, we can keep doing this over and over again to build really complicated functions that do all sorts of cool things.
So, there you have it – Neumann polynomials are kind of like Lego pyramids, but with math instead of plastic bricks.
Let's start by imagining a big square Lego block with a bunch of smaller Lego blocks stuck to it. Each of those smaller blocks has a number on it, like 1, 2, 3, 4, and so on. We'll call that big square block our "base."
Now, let's say we want to build another Lego structure on top of that base. But we don't want to just plop it on top – we want it to fit perfectly with the base. To do that, we need to build it out of smaller Lego blocks with numbers on them that match the numbers on the base.
If we do this right, we'll end up with a new Lego structure that looks like it's just a bigger version of the base, with some new parts added on top. And if we keep doing this over and over again, we can build a really tall pyramid out of Legos!
So, what does all of this have to do with Neumann polynomials? Well, just like we built our Lego pyramid out of blocks that matched the base, a Neumann polynomial is built using a base function and a bunch of other functions that "match" the base.
The base function is usually something simple and easy to work with, like x (which you might know from math class as the variable that stands for "some number"). Then we add on other functions that are built to match x in certain ways. They might have x^2 or x^3 or even higher powers of x in them.
By combining all of these functions together using some math tricks, we can build a new function that looks like a bigger version of our base function, with some new parts added on top. And just like with our Lego pyramid, we can keep doing this over and over again to build really complicated functions that do all sorts of cool things.
So, there you have it – Neumann polynomials are kind of like Lego pyramids, but with math instead of plastic bricks.
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