The key assumptions made
in deriving the error bounds were:
bounded loss function
finite collection of candidate functions
The bounds are valid for every
${P}_{XY}$ and are called
distribution-free.
Deriving bounds for countably infinite spaces
In this lecture we will generalize the previous results in a
powerful way by developing bounds applicable to possibly infinitecollections of candidates. To start let us suppose that
$\mathcal{F}$ is a
countable, possibly infinite, collection of candidate functions.Assign a positive number c(
$f$ ) to each
$f\in \mathcal{F}$ , such that
However, it may be difficult to design a probability distribution
over an infinite class of candidates. The coding perspectiveprovides a very practical means to this end.
Assume that we have assigned a uniquely decodable binary code to
each
$f\in \mathcal{F}$ , and let c(
$f$ ) denote the codelength for
$f$ . That
is, the code for
$f$ is c(
$f$ ) bits long. A very useful class of
uniquely decodable codes are called prefix codes.
Prefix Code
A code is called a prefix codeif no codeword is a
prefix of any other codeword.
The kraft inequality
For any binary prefix code, the codeword lengths
${c}_{1}$ ,
${c}_{2}$ , ...
satisfy
$$\sum _{i=1}^{\infty}{2}^{-{c}_{i}}\le 1.$$
Conversely, given any
${c}_{1}$ ,
${c}_{2}$ , ... satisfying the inequality
above we can construct a prefix code with these codeword lengths.We will prove this result a bit later, but now let's see how this
is useful in our learning problem.
Assume that we have assigned a binary prefix codeword to each
$f\in \mathcal{F}$ , and let c(
$f$ ) denote the bit-length of the codeword for
$f$ . Set
$\delta \left(f\right)={2}^{-c\left(f\right)}\delta $ . Then
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?