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HA DANG: Today, I am
going to use Mathematica

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to visualize a nanoparticle
polymer network

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and to investigate how
the polymer length affects

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of probability that polymer
would cross the nanoparticles

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and how the polymer
length affects

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the nanoparticle polymer
network formation.

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Polymers and macromolecules
composed of hundred to thousand

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subunits.

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They can adopt many
different configuration,

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depending on the identity
of their subunits.

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Common everyday
examples of polymers

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are PTC, polystyrene,
Teflon, and so on.

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When particles and
polymers of are

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favorable interaction are
mixed, it is not always apparent

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whether they will form
a cross-linked network.

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If the polymer
configuration makes the end

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to end distance of
the polymer shorter

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than the distance
between two particles,

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no polymer-particle
network will form.

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This project will focus on
visualizing the interaction

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between particles and polymers
given polymer's length,

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or the number of subunits.

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In stage 1, we make polymer
by use the random walk method.

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We consider a particle
as many lattice sites.

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Polymer with start a
random lattice site

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This initiation part just
ensure that the polymer will

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truly random walk at
a certain distance

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away from the surface
of the particle.

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In the random walk
simulaton, the step size

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is the length of the
subunit of the polymer,

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which is about 10 times smaller
than the size of the particles.

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The step angle tell
you the direction

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that the poem will walk.

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The step tell you how
much the polymer will walk

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in the x and the y direction.

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The next position of the polymer
will be the current position

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of the polymer plus step.

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Each time the new polymer
position is computed,

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it is stored inside on
list which we call path.

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Once we have the
polymer path, we

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can visualize it inside the
particle matrix using Graphic.

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In stage 2, we want to
generate many polymers

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and visualize how they
distribute and interact

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with particles inside a
lattice of many particles.

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And in order to do so,
we combine all the steps

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in state 1 into one
function, polymerRandomwalk,

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which takes in
the polymer length

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and the lattice of particles
and returns a polymer path.

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We will generate 100 random
random walk polymers of 1,000

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subunit by using Table.

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We visualize the polymers and
particle matrix using Graphic.

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In stage 3, we want to calculate
the probability of crossover

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between nanoparticle
in polymers by counting

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how many polymers that wrap
around more than two particles.

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In a nutshell, we arrive a
function and crosslinkSuccess,

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which test whether each
subunit of the polymer

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would touch a neighboring
particle by using RegionMember.

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If there is a least a subunit
that successfully cross

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a particle, the polymer
connects two or more particles

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and the function return 1.

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If the polymer fails
to cross a particle,

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the function will return 0.

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So we run this function
over many polymers

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by using the PR function.

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And this is a
resource that we got.

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And we really want to know the
ratio of the cross-link polymer

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to a total number of polymers.

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And to do that, we
just do a simple math

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by dividing the
number of successfully

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cross-linked polymer to the
total number of polymers.

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Below is a function
that calculate

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the ratio of polymer that link
two or more particles together.

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Next, we calculate the ratio
of cross linking for polymer

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of various lengths--

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100, 200, 500, 800,
1,000 and 1,500.

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Since the operation
take a while,

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I'll show just the results.

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The most interesting thing that
we can obtain for this data

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is actually plotting
the ratio of cross-link

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versus the polymer length.

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What we find out is that
as the polymer length

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increases, the probability
to cross-link increases.

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This makes sense because
as a polymer gets longer,

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it can walk a larger distance
and touch a particle.

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What is even more
interesting is that the graph

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has a shape of a lock function.

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We go ahead and
use Fit to generate

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least square fit equation.

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And then we plot
it with the data.

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As you can see, the
square fit function

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give you a pretty
good fit of the data.

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What this fit tell
you is that first,

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there's a minimum
number of subunits

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that cross-link will occur.

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In our case, the
minimum polymer length

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will have to be between
100 and 200 subunits

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to be able to provide you
some type of cross-linking

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between polymer
and nanoparticles.

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Second, the graph plateau
at a certain polymer length.

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Any polymers that
are longer than those

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would not increase the
probability of cross-link.

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Although the analysis of
probability for polymer

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to cross-link it's
useful, it does not

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tell any information about
the network of polymers

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and particle as a whole.

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So in stage 4, I will
use cluster analysis

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to analyze network formation
given polymer length

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and particle distribution.

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In a nutshell, the function
singlePolymerCrossLinkTest will

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return all the particles'
lattice site which the input

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polymer can connect.

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Then we use a built-in function,
UndirectedEdge and Graph

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to visualize this connection.

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We want to do this simulation
for all the polymers

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and particle to visualize the
connectedness with the polymer

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nanoparticle network then use
a cluster analysis to analyze

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the network formed
by mixing polymers

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and nanoparticles together.

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In here, we write
a little function

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that basically run
the single polymer

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cross-link test over many
polymer of the same length.

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What I want to show you
is this GraphCommunityPlot

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and FindGraphCommunities.

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In this graph, you
see that there's

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many clusters which has a
very dense cross-linking

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between polymer
and nanoparticles.

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However, most of the clusters
do not connect it together.

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This suggests that a network
of polymer and nanoparticle

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may not be formed,
even though you

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have many cluster of
densely cross-linking

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between nanoparticles
and polymers.

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What I want to show
next is a comparison

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between a cluster network
of 100 subunit polymer

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and a cluster network of
1,000 subunit polymers.

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As you can see, as a
polymer length increase,

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there is a higher chance that
cluster can connect together.

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So definitely the
polymer length can

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affect the connectedness
of the system

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and give a higher
chance that polymer

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can connect nanoparticle
and form a cohesive network.

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In a realistic system of
polymers and nanoparticles

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which have favorable
interaction, one

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a polymer hits a
nanoparticle, it

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will stick to that nanoparticle.

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In the system that
we used above,

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polymers are allowed to walk
in such a way that does not

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take into account
the interaction

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between nanoparticles
and polymers.

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In state 5, we'll
attempt to take

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into account that interaction
by using this algorithm step.

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First, we make a lattice
of nanoparticle sites.

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One lattice is
randomly selected.

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And a random walker will spawn
from the site and can walk.

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If polymer walker
hits a charge site,

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including the original one, then
the polymer stick to that side.

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The modifiedPolymerRandomWalk
function

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is very similar to the random
walk polymer function above,

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except for its use RegionMember
to test whether or not

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a subunit touched a particle.

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If the subunit
touched a particle,

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the function will stop and
return the current path.

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Here I generate hundreds
polymer of 1,000 subunit

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by using this modified
polymer random walk function

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and using Graphic
to visualize it.

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As you can look
into this picture,

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the probability that polymers
will cross-link nanoparticles

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decreases because you restricted
it to stopping once the polymer

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hitting a nanoparticle.

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Although this approach takes
into account the interaction

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between nanoparticles
and polymers,

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it is still not a
realistic representation

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of a real system.

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In this approach, we
let a polymer subunit

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spawn on the particle surface
and grow into a polymer.

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In a real system, polymers are
thrown into a disperse solution

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of particles.

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The confirmation of polymers
in a dynamic process in which

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the entire polymer
twist, turns so

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that the two ends of a polymer
find nanoparticle surfaces.

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This project has not
taken into account

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the dynamic confirmation of
polymers in a dilute solution.

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For future research,
I would like

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to improve this project by
taking into account polymer

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confirmation when doing
polymer random walk

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and finding how
polymer concentration

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and nanoparticle concentration
affect the nanoparticle polymer

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network by doing
cluster analysis.

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I would like to thank Dr.
Keane for your guidance

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on this project.

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I would like to thank Professor
Carter for teaching 3016, which

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I found very helpful
for the other project

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I'm working on my
UROP right now.

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And I would like to tell
you all the TA's in 3.016

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say for helping me
during office hour.