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Congratulations on surviving module one

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and welcome to module two
of modeling complex system.

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In this module, we're gonna tackle models

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with explicit structure,
structure and space

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which we did not have in module one

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and for a very interesting reason.

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So we're gonna try and
build the simplest possible

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spatially explicit models that we can.

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And those will be called
cellular automata.

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And because those are two very hard words

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to pronounce for me, we'll call them CA

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and the error that we
made, or it's not an error

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but this choice that we
made early on in module one

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that sort of sent us towards the path

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of dynamical system and away

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from spatially explicit model
was when we were building

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our mechanisms, our
putting parts of a system

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in boxes and and describing
mechanisms as arrows.

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We made the approximation
that these boxes and arrows

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were telling a complete story
about the state of the system.

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And so we could write down equations

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using a mean field approximation,

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where a given neighbor of an individual

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in a social network or a piece of land

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next to a fox or shark or a fish

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in the (indistinct)systems
was just randomly chosen

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based on the current state of the system.

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We were assuming that
things were well mixed,

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randomized and that we could
do a mean field approximation.

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That approximation got rid of space

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made so that our models
assume that things were either

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fully random or completely connected,

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either of which doesn't have

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any interesting spatial features.

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So what we wanna do is go back

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to the very basic elements
of our modeling recipe

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and try and avoid making that
mean filled approximation.

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We won't get explicit equations for now,

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but we'll get very
interesting models nonetheless

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and models that will allow
us to consider space.

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So we want to think about ingredients

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preparation of the system
and its presentation, right?

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So for spatially explicit
models, the goal here

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is no mean-field approximation,

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and then we start from scratch.

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So importantly, what this means is that

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before the updates rule for a
fish or susceptible individual

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or any piece of our system
was based on its own state

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and the state of the system

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and that mean field approximation

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was taken care of the rest.

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Here, we're gonna have to be sure

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to specify update rules that depend

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on the state of the parts

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and on the state of their neighbors.

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And that's how we're going to be able

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to keep track of space.

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All right, so we want to do
the simplest possible case

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of that where the
mechanisms are not as simple

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as S goes to I with some rate,

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but depend on what's around
the S individual, right?

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So what are the ingredients

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for the simplest possible case

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of a especially explicit model?

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Well, we're gonna make a a
discrete time approximation

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because that's often easier at first.

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We're gonna also make a
discrete space approximation

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and then we're gonna be
working in one dimension.

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So really discrete space in one dimension.

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We're working sort of on an
array, one dimensional array

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or a vector where you
have a bunch of cells

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next to each other that can
each take a given state, right?

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Something like this.

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So already it sort of makes sense

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that we're gonna be leaning more

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on computational model and
less so mathematical equations

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because that's a very
natural data structure

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to represent in a code, right?

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An array or a vector
or a list that contains

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all the possible spaces
that our model can occupy.

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And then the parts of our system will be

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the cells of this array
and they'll be able

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to take discrete states in
the simplest possible case,

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simply zero or one.

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So we could call that off
or on active or inactive.

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And then we'll have to
specify the neighborhood

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because our rule set will now depend

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on the state of explicit neighbors.

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In this case, it makes sense
that one array is connected

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to the two arrays next to it.

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So there's a symmetry, the left or right

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doesn't necessarily matter at first.

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You're connected to a
neighbor on both side.

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And then we need to
specify rules that govern

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the evolution of that system.

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Well, what are the rules?

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The rule set will have to
specify something like this.

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If my cell and I'm the center cell

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or index cell in this picture,

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if my cell is on and I
have two on neighbors

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maybe the rule set would say,
well, you turn off, right?

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And that's like changing the
state of the index cell only.

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So notice that in the output of this rule

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I don't specify what happens
on the left and right.

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The idea is if I'm on and
my two neighbors are on,

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I turn off, right?

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That could be overpopulation.

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I don't like it if there are
too many on people around me

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or maybe I'm modeling something
with a hipster effect.

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I wanna be different somehow.

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But one with two ones on
on each side turns off,

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goes to zero.

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And really there's only a finite numbers

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of rules that we need to specify.

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So I could be on my neighbor to one side

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could be on and the other
neighbor could be off

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in which case maybe I turn off.

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If my two neighbors are on
and I'm off, I stay off.

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So really how many rules
are there to specify

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where we have three cells that
can either be zero or one.

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So two options for this one,
two options for the middle one,

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two options for the last one.

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So that's two times, two times two options

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or eight rules that we need to specify.

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So here I'm giving one
example of such a rule set

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but really each of the boxes
at the end of every arrow

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could be either a zero or one.

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So essentially to specify

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these very simple
spatially explicit models,

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all I need is eight numbers
that are either zero or one.

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So two to the eight options
for possible models.

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Because that first rule is
either zero or one, two options.

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Two options for the second
rule, two options for the third.

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So that's two times
two times two times two

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times two times two times
two times two or 256 models.

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I'm stressing this here

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because it's not that big of a number.

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We never thought about the space
or the size of the ensemble

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of all possible models when talking

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about dynamical system in module one.

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But here, discrete state,
simple spatially explicit model

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there are only 256 that we can explore,

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which is already very, very different.

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The question is whether
those are interesting or not

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and that's a completely
different question.

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But that's something
we can explore, right?

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So that also means that we'll be ablet

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to represent these rule
set as binary numbers.

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And if you are not
familiar with binary number

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I invite you to watch
the quick intro video,

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the bonus video that's on it.

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If you need a refresher, watch it as well.

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It's just to understand
how binary numbers work

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to represent numbers.

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And so any sequence of zeros and one

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can be used to represent the number.

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And that means that we
will be able to refer

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to these spatially explicit models

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by the number corresponding
to their rule set.

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We can talk about model 121 or model 24

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and we'll see how to do
that in the next few videos.

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This one, for example, is rule 30

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and I'm gonna show you
what this model does

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at the end of the video and
we'll go into greater details

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in the computational part of the week.

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And so the key part here
is that these very simple

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or perhaps the simplest,
especially explicit models

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take only 256 flavors.

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And so we'll refer to
those as elementary CA

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or elementary cellular automata.

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They're cellular because
they use discrete space

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and cells to represent the system.

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They're automata because this
is a deterministic rule set.

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Give me the state of the system
and I can update all cells

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at once and that's gonna give
me a deterministic evolution.

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So they're like cellular robots in a way.

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And we call that these
elementary because we'll be able

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to add complexity to them either by making

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the deterministic rule
stochastic by growing space

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in two dimensions, by changing
the size of the neighborhood

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or by adding states
that the cells can take.

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And so in the next video,
I'll show you how to code

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some of these elementary
cellular automata.

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This is what comes out of the one

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that was just on the screen.

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It seems like trivial little
choices of zeros and one.

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Some, but not all still
have very complex behavior.

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And if anything, that's the main story

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behind cellular automata
is that very simple

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local rules can give rise to very complex,

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interesting global behavior
like the one on the screen now.

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So I invite you to join
me in the next video

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where we'll be exploring the
computational implementation

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of these models.

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So I'll see you in the next one.