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Hello and welcome. This is a
presentation of non linear functions.
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And I have developed this as a public
class at the Licium in December of 2014
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but had not recorded it back then.
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So, I'm recording it as a screen cast so
that I can upload it
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on to you tube and I can make it even
more available.
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This is the first time that I am doing
this converting my class
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into a screen cast so I would appreciate
your feedback.
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This presentation is on non linear
functions.
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Non linear functions are models,
mathematical models of
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physical phenomena that we
often encounter
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However, unlike linear functions they are
a little more harder to analyze and
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predict. The behavior is harder to
predict.
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In many circumstances when we try to model
physical phenomena we see that
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higher order terms have very very small
coefficients.
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So we just ignore them but those small
terms can actually affect
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the behavior of functions in certain
circumstances.
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So this presentation is going to be an
introduction to these types of
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functions and how they behave and what
you can do with them.
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So to start off with lets just do a quick
recap.
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A recap of basic ideas which I know many
of you may already be familiar with
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If you already know about all this you can
skip this and go to the next
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The basic ideas we are going to discuss
are functions.
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What are functions? Functions are
mathematical concepts which we use to
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describe how elements of one set change
into elements of another set.
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For example I can say F of x equal to
x squared.
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Which is a function that is squared and
simpled.
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Or you can say F of x equal to sine of x
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The trigonometric sine of x, expressed in
radians. Things like that.
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These are examples of functions,
simple functions
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These two are non-linear
but we can have linear ones as well.
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Say x + 5, or something like 2x
or something like that
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These are linear functions.
These last two are linear.
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The first two are non-linear because
they have higher order terms
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Higher order terms meaning terms with
a power of more than 1
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Now, usually when we talk functions we
talk about solving them
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So we say: y equal to x squared, and you
can plot it
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Or you can find the value of the function
for a given value of x, and something like that.
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But in this series we're going to be
mostly talking about iteration functions
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This is something which you might not
have done.
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So the basic idea is to take a function,
say this one which we have up here
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F of x equal to x squared
And apply this function repeatedly
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So if I were to take F(x) = x^2,
and I say F of 2 will be 4
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And then you apply F on this one,
so you say F of 4 equal to 4 squared, 16
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And then you say F of 16, it's the square
of 16, then you say square of that
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Then square of that, then square of that.
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This is called iterating. You apply a
function repeatedly on something.
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And this actually gives you certain kinds
of behaviour
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which we'll find interesting in the
upcoming segments.
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So let's say if I wanted to apply
F(x) on a function, so F(x) = x^2
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So if I apply this three times, it will be
F of F of F of x
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I want to apply the squaring function on a
number (say 2), three times
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You can write it like this. You will get
the third "iterate" of the number 2 under
the function F
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And as a matter of notation, we tend to
write it like this, F three times of 2.
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So this is F of F of F of 2
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So this is the notation which we're going
to use repeatedly in this series.
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So this is iterational functions.
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And what we're interested in is finding
behaviours of functions when we do this.
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Now let's move to the next phase.
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We have a notation and we have the basic
thing that we're going to do.
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What we're going to discuss right now is
what I call Orbits.
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The term might sound technical, but it's a
simple thing
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It's just the successive iterates of a
number under a function.
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So if I were to take a number, say in this
case let's just say a simpler function
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Let's just take F of x equal to 2x (double
the number). Then you can start with 1
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We'll start with Orbit of 1 under F.
The first thing would be 1,
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then you apply F once it would become 2.
You apply F to that, you get 4,
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you apply F to that you get 8, 16, 32, 64,
128, etc.
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These are the iterates of 1 under the
function F(x) = 2x.
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And this series is called the Orbit of 1.
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So the Orbit of 1 is successive iterates
of a number under a function.
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So we talk about what is the orbit of this
number, what is the orbit of that number,
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how does the orbit behave? And things like
that. So it's this orbit thing that we're
interested in.
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And we can actually make predictions about
orbits and where this thing will go
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if you start iterating and things like
that.
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Now given that we know what orbits are now
there is some interesting kinds of points
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We'll discuss many of them in the upcoming
segments
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but the first and simplest one is what's
called a "Fixed Point"
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It's something important, just in case.
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A Fixed Point is a point that is fixed.
This means that F(x) will be equal to x
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for that point. And this is something
which varies depending on functions.
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If I were to take F(x)=2x, this function
which I just talked about earlier
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Zero is a fixed point.
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Why? Because F(0) will be 0, and then you
apply that as many times as you want
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It stays there it doesn't move, it's fixed
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On the other hand, we just saw over here
that if you take 1, it's not fixed
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It just keeps on increasing
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If you take 0, it's what we call
a Fixed Point.
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And you can see how this behaves, you can
see how the functions behave now.
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Let's take a few simple examples and see
how this thing does.
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So consider this, we take an interpreter
so that we can do this easily
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So that I have square root, sine, cosine
and things like that.
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And take a number, so let's say 5. I'm
going to say square root of 5, we get this
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Square root of that, we get this.
And I keep doing this again and again
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So this is iterating manually
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And we can see that as I iterate, this
thing slowly becomes smaller and smaller.
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It tends towards 1, right? And 1 is under
square root of 1 a Fixed Point
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Because square root of 1 is 1
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Alright, consider the other one. Consider
sine of something expressed in radians
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So let's take sine of pi. Sine of pi is
this, it's infinitesimally small.
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Let's take sine of pi/2. This is 1.
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Sine of pi/1.9, just a little bit less.
If you iterate all of this...
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This is sine of some random number, just
do 2 or something
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Sine of 3, and then sine of this. So if
you apply sine repeatedly on a number
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you see that it slowly goes down lower and
lower and lower
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and ultimately it tends to 0.
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You can see this happening if you write a
small function
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Iterate, this is the function, this is the
start point, this the number of iterations
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And you can see S equal to start, and
say for i in range of iterations
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And you can say S equal to the function
applied on S
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And you just say Print. Simple function.
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So just try this, iterate sine starting
from 2, a hundrer times
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You can see that it starts off from 0.9
and keeps reducing
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Reduces, reduces, reduces it goes down
like this
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If I were to take this to 1000, it would
go extremely lower
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10,000 it's even lower.
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So it's slowly tending towards 0. And this
seems to hold for many of the points
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For example for 3, for example for pi.
It's going down to almost 0.
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Pi/2, anything. So anything you give,
it slowly tends towards 0
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And 0 of course is a Fixed Point.
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So 0 is a Fixed Point for this function.
0 is also a Fixed Point for sine of x.
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Now let's take another one. Let's try
cos of x which is looking more interesting
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Let's just take pi, and we'll take
cos of x
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Cos of x and cos of a function, and then
now we try this.
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You can see that it goes to 0.739085133215
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And this seems to happen for anything
usually: 3, 2, 1, anything.
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And this is a Fixed Point of cos of x.
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This seemingly random number is a Fixed
Point for cos of x.
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Because of my definition you know that
this is a point at which cos(x) equal to x
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But it's not straightforward what's so
particular about this number.
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So this is the concept of a Fixed Point.
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So that ends our first segment, which is
just an introduction to functions
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What is iteration, what is an Orbit, and
what is a Fixed Point.
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Now we'll discuss more types of points
in the next segment.