Fixed points, kernels, and generators

[bananas]

http://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is a point that is mapped to itself by the function. That is to say, x is a fixed point of the function f if and only if f(x) = x.

If f(x)=0, then x is also a kernel:

http://en.wikipedia.org/wiki/Kernel_%28mathematics%29

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:

* The kernel of a mapping is the set of elements that map to the zero element (such as zero or zero vector), as in kernel of a linear operator and kernel of a matrix. In this context, kernel is often called nullspace.
* More generally, the kernel in algebra is the set of elements that map to the neutral element. Here, the mapping is assumed to be a homomorphism, that is, it preserves algebraic operations, and, in particular, maps neutral element to neutral element. The kernel is then the set of all elements that the mapping cannot distinguish from the neutral element.
* The kernel in category theory is a generalization of this concept to morphisms rather than mappings between sets.
* In set theory, the kernel of a function is the set of all pairs of elements that the function cannot distinguish, that is, they map to the same value. This is a generalization of the kernel concept above to the case when there is no neutral element.
* In set theory, the difference kernel or binary equalizer is the set of all elements where the values of two functions coincide.

Kernel may also mean a function of two variables, which is used to define a mapping:

* In integral calculus, the kernel (also called integral kernel or kernel function) is a function of two variables that defines an integral transform, such as the function k in

(T f)(x) = \int_X k(x, x') f(x') \, dx'.

* In partial differential equations, when the solution of the equation for the right-hand side f can be written as Tf above, the kernel becomes the Green's function. The heat kernel is the Green's function of the heat equation.
* In the case when the integral kernel depends only on the difference between its arguments, it becomes a convolution kernel, as in

(T f)(x) = \int_X \phi(x - x') f(x') \, dx'.

* In probability theory and statistics, stochastic kernel is the transition function of a stochastic process. In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function.
* Kernel trick is a technique to write a nonlinear operator as a linear one in a space of higher dimension.
* In operator theory, a positive definite kernel is a generalization of a positive matrix.
* The kernel in a reproducing kernel Hilbert space.

Generators are extremely important, they are where f(x)<>0 instead of f(x)=0 like kernels, without them there wouldn't be much of anything interesting, but there wasn't a wikipedia entry for generators.

http://new.music.yahoo.com/fleetwood-mac/tracks/over-my-head--574099

You can take me to paradise
And then again you can be cold as ice
I'm over my head but it sure feels nice

You can take me anytime you like
I'll be around if you think you might
Love me baby and hold me tight

Your mood is like a circus wheel
You're changing all the time
Sometimes I can't help but feel
That I'm wasting all of my time

Over my head

Your mood is like a circus wheel
You're changing all the time
Sometimes I can't help but feel
That I'm wasting all of my time

Think I'm looking on the dark side
But everyday you hurt my pride
I'm over my head but it sure feels nice

I'm over my head but it sure feels nice
I'm over my head but it sure feels nice

[byronius]

1. Ha! Trick question! But the Fleetwood Mac gave you away.

IIf((f(x) = x), x = x, x = 'Blue Oyster Cult')

And I am a Kernel, by the way. Synchrolicious!

[BadgerKid]

2. The Eagles were covert mathematicians

"Take it to the limit one more time."

[zeemike]

3. f(x)=0 is the loneliest number you can ever do.

Obviously.