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this this class is talking about two  lectures both about protein folding  fixed angle chains things like that  there's a few questions about them one  is mostly about these open problems  equilateral equiangular obtuse 3d chains  fixed angle open problem is are they  locked so the question is about what  about any subset of those combinations  so this originally comes from an open  problem I think posed in 1999 one of the  first 3d linkage papers and it asked  whether equilateral universal joints can  lock and that's still open for for  universal joints these two constraints  don't make a lot of sense because who  cares if it's initially equiangular as  soon as you move it it will no longer be  equiangular and obtuseness I don't think  matters too much although it potentially  could so for that problem  I mean equilateral seems to be the core  for fixed angle though we know or we  conjecture that fixed angle equilateral  is not enough from this example it's  still not proved to be locked I don't  know if it's hard but it's probably  tedious so hasn't been done this is the  cross legs example all the edge lengths  are the same and if you don't allow the  touching part then all the angles are  also the same so this is everything  except the obtuse property and it's  probably locked so dropping obtuse is no  good the other things you could drop our  equilateral or equiangular this is if  you drop equiangular that I mean fixed  angle equilateral is not terribly  constraining because you can have you  can simulate a long bar by having a lot  of 180 degree angles so - this is obtuse  and it's equilateral but it's not  equiangular and it's locked for a  trivial reason the last so this is  dropping equi angular if I drop  equilateral  I can also just make a knitting needles  example I take a really long link and  then I take this thing is basically  string so I don't really care this  connection can be done with a lot of  obtuse angles and then have a really  long link so dropping any of the three  constraints makes the problem it makes  it easy to lock so for fixed angle  chains you need equilateral equiangular  and obtuse all these things together  potentially mean you're not locked but  we don't know that's the open problem of  course we don't necessarily need exactly  equally lateral or equal angular  hopefully within some small min to max  ratio would be enough but we don't know  next question is why did we model things  as why do we model the ribosome as a  cone that seems rather simple is this  realistic and probably when I taught the  class in 2010 I I didn't use any images  that I couldn't get permission for this  year I'm more lacks will ask for  forgiveness instead of permission but  this is a paper from science 2000 this  is what the ribosome actually looks like  there's many different figures of it but  this particularly liked this one because  it highlights a tunnel in the center of  the ribosome and so the idea is this  this is a machine for converting mRNA  into your proteins and the idea is that  protein comes through here there's a  little bump in the tunnel some people  conjecture this is where the amino acid  gets attached and then it feeds through  here and starts spitting out and there's  they say there's barely enough room here  for an alpha helix  so probably not too much folding happens  inside and the folding should just  happen over here and the observation is  if you have a reasonable size protein  it's only going to be about this big  then there's this big flat wall at the  exit so you have basically a half I have  a plane there and this half space is  more or less a big obstacle and so the  it's the alpha cone model where alpha is  180 I think I felt was the half angle so  it's not really a cone it's a cone  that's been opened up all the way to a  plane but that  one of the situations that is handled by  the theorems that we talked about so  that's why the cone model I mean we  generalized it cones just because it  works for general cones but the real one  is sort of flat cone that's where that  comes from  I don't yeah I think actually we proved  the theorem before we knew this but then  I looked it up and it was true and the  last question is about the electric  twenty one which some of you may not  know watch because it's optional but  there's this model called the HP model  for protein folding it's a model of  protein energy and it says you have a  chain of H&P nodes and basically the H  nodes are attracted to each other and  the P nodes don't care and the model is  that the H nodes are hydrophobic so they  want to be next to each other so they're  not next to water which is surrounding  the whole molecule and the P nodes let's  say don't care or they're hydrophilic  and it's known that if you have an HP  string and you want to find the optimal  folding into a 2d or a 3d structure  that's np-hard which is kind of weird  because somehow nature does it maybe  because it has found the easy instances  by evolution or maybe there's something  we're missing this model doesn't capture  reality we don't know but it's it  captures part of reality at least as we  observe it and unfortunately these  hardness proofs are a little too much  too complicated to cover here but I can  at least answer what are they reducing  from there's two proofs one is the first  one or I guess they're basically the  same time I think around 2001 first one  here is in 3d and these are by two MIT  professors and there's this big  construction but the reduction is from  bin packing so you have a bunch of fixed  size bins and you have a bunch of items  of varying sizes and you just want to  fit them all in and using the fewest  bins possible and the rough idea of the  construction this one this is how an  individual number is represented the  rough idea is to you have this big cube  you fill most of it which is sort of  stuff or you fill the sides with stuff  too  protect from the outside boundary the  insides are just used for connections  and then the front face here is this  stuff and you construct these bins and  you construct numbers which have to fit  inside the bins that's the rough idea  the details are complicated the 2d proof  by several people is from hamiltonicity  in maximum degree for graph so you have  a graph for every vertex as at most for  incident edges you want to find a  Hamiltonian cycle I think and this is  there's it's much harder to see this  picture I would say there's there's no  one diagram that kind of summarizes it  this is roughly the construction which  is quite complicated and I'll just leave  it at that so that's what they're  reduced from I think an interesting open  problem would be to find a simple or  cleaner proof of these results now that  it's known that they're hard it's  probably easier hardness proofs there's  also some open questions from lecture  like is it a px hard McKittrick symbol  to a 1 plus epsilon factor or 1 minus  epsilon factor for any epsilon or is  there some limit to approximability we  saw a nice approximation 4/3 whatever  3/4 whatever but can you do better still  open so that was the questions and then  well this is a question that you usually  ask in every lecture you didn't actually  ask it on this one press I copied and  pasted so there's some interesting  progress from this class two years ago  and I think in the open problem session  initially probably also a class project  related to this part so this is back in  lecture 20 so we had this proof just  kind of fun that flattening a fixed  angle chain deciding whether there was a  flat folded state is weakly np-hard  weakly meaning it depended a lot on what  these numbers look like and it was a  reduction from partitions you had to  split up the numbers into two equal  parts and if you did then this key would  fit in the slot and you're okay if you  didn't the key would collide with  something here so this this means the  problem is hard if your edge lengths are  vastly different they would have to  differ by a ratio of exponential in the  number of edges for this to really be  hard so a natural question is well what  if all the links are equal equilateral  chains turns out that is still hard and  this is a paper just published last year  with Sarah Eisenstaedt who took the  class then and there's a bunch of  results here so there's three all of  these are empty hardness results strong  np-hardness so it doesn't depend on the  numbers we've got flattening results  there's also a min flat span and Max  flat span once you have that flattening  is hard these are pretty easy to show  hard so I won't talk about them so much  but basically we consider different  ranges of angles in what you might allow  and in some cases we can get perfect  equilateral chains in some cases they  have to range between say 1 & 2 in  length and depending on the angles  I guess these natural question is obtuse  angles we don't quite know that it's the  best we have is like 60 degrees minus  epsilon and larger for this is nice for  orthogonal chains but it's not for  chains its for trees so still some open  questions here but lots of hardness  results I thought I'd show you roughly  what the hardness proofs look like  because they're kind of fun they all  follow this kind of structure this is a  gadget and it's a it's a fixed angle  chain here I'm going to show everything  with 90 degrees a lot easier to think  about and it's kind of like I don't know  a thinking plunger but something like  that you can kind of decide which parts  get pushed in and which parts get popped  out and this L shape can basically shift  left and right to three different places  if I did it right the  one this one this one and so that's a  useful construction as you might imagine  and the idea is you take that well and  you you add on these guys so I've got  another kind of plunge II thing like  this uh-huh  and yeah this guy can move left and  right to various extents well that's not  very intuitive so then this connects to  these little guys and these can just  flip up or down if it's a flat flat  embedding of a fixed angle chain and so  the idea is that you have three of these  pokey elements and they can attach they  bump into these different things and if  it's up then this can be up but when  they're down like this guy is currently  down but he could be pushed up then this  guy must be flipped down like that  that's the rough idea of how these parts  fit together and so then this is how you  you end up building some kind of 3sat  problem so you have some variables they  can either be true or false  and the idea is that the true guys are  on the top the bottom guys are on the  bottom and there's some complicated  interaction between them to make sure  that they can't both be true and false  basically you want these things to point  in whenever possible because then these  guys can go down otherwise you would  collide and if this guy's in the  corresponding guys in the bottom must be  down and that's enforced by this long  thing which is either completely down or  completely up so when it's completely  down these guys all have to be down  which is going to be a problem down  below if things aren't satisfied and  when it's up all of these guys would  have to be up some of them could be up  if you feel like it but really you've  probably put them all down whenever you  can because then you can take these guys  and stick them down so this is how you  set variables and then the other things  I showed where clauses essentially it  at least one of those pins had to be  pushed down and you don't know which one  and that's the hard part  it's a rough sketch of how this this  proof looks oh one thing is here  these dots for this to work these edges  can't be spinnable right so when I go  here I must immediately go back down I  can't flip it that's so these are rigid  edges that's not really the original  problem but you can simulate rigid edges  and this is where the angles get small  you can simulate rigid edges with the  very sharp zigzag because if you ever  flipped one of these it would collide  with the previous edge so for  sufficiently sharp angles you can force  parts to stay straight in this funny way  which is not great but it's one way to  force it we can also do it with trees or  various other techniques but that's how  the the proof looks so that is  flattening fixed angle chains strongly  np-hard we go next to one more topic for  fun this is the topic of flips yes I  think I'll switch to well I guess I have  a an image for this so this is a problem  posed by Paul air dish I think when he  was a student famous very famous  mathematician 1935 he asked this  question this is the whole question as  originally written but you have a  polygon and you take the convex hull the  polygon you take these regions which are  in the convex hull but outside the  polygon we call those pockets and you  imagine flipping those outside  reflecting through that supporting line  that tangent on the outside so then you  get this polygon here you get two  pockets maybe you flip both of them and  then you get a convex polygon now you  have no pockets and so you're done and  the question is as posed  prove that after a finite number of such  steps polygons will become convex so you  might call this a conjecture by arish  that it is finite  he'd never published a proof there's one  issue with the problem as stated you  can't actually flip both pockets at once  if you want to what collisions the  relation to linkages is we imagine these  flips could actually happen by a  rotation but you can also just imagine  them as reflecting instantaneously  there's this issue if you flip two  pockets at once you might have  collisions afterwards so the way this  problem has been interpreted by most  people is don't flip them all at once I  just flip them one at a time and then  you guarantee because this is a  supporting line a tangent line of the  convex hull when you flip one pocket out  it will go to the other side so it won't  can't intersect the rest of the polygon  so you would avoid collision in if you  do one flip at a time now this was  observed I guess a few years later this  finish and he also published the first  proof sorry before we get there another  a weird property and the why you might  worry about this being finite or  infinite is if you take this  quadrilateral and make this very narrow  this edge very small relative to the  horizontal edge then you can require  arbitrarily many flips to convexity so  even for N equals 4 you can require a  million flips so definitely there's some  dependence on the the ratio between the  longest length and the smallest length  though an open problem we still don't  know is whether you can bound the number  of flips in terms of N and that that  ratio is it pseudo polynomial who knows  so there have been actually many so it  turns out it's always finite and there  have been many proofs over the years of  this result that's kind of been  rediscovered many times so Nagy solved  originally in 1939 and he cited air dish  these two guys didn't cite anyone so  they may have come up with the problem  independently this paper these are two  Russian proofs then this paper cited  all they knew about everything they  cited all of them or so they said of  bruschetta niak I'm not sure I don't  think they necessarily knew about you  suppose then Vagner so independently  Kalu suppose the problem 1981 and  begginer solved it and then kind of  finally clean it up Greenbaum we saw  from uh none foldable knew about  everything and decided presumably this  is everything of course we might have  missed one but he knew all of the above  came up with his own proof and then  Godfrey Toussaint's father of  computational geometry one of them knew  about these and came up with yet another  simpler proof so the the story of this  is kind of fun when I taught this class  for the very first time 2003 so I  thought okay cool this is classic  theorem everyone should know it so I  thought I'd cover the latest proof that  presumably the best so I covered I got  free proof and after I'd so a little  unhappy with it but I finished writing  down the proof and then one of the  students asked raise their hands like is  that really right can you do that in  step two and the answer was no you can't  do that so it kind of basically skipped  a step and so I thought oh gee and  that's how I correspond with draw Roark  and Gottfried that weekend is like  something missing here maybe we should  go back to some of the other proofs cuz  we've got lots to choose from so we went  to the original which very short proof  it's only one page long maybe one and a  half pages and so oh this is fun you  know this is 1939 the way mathematics  used to be done and it's funny because  groom bomb proof was based on Nash's  proof and Toussaint's proof was based on  grim Bob's roof so in the end these  proofs were very similar in fact they  differed exactly in one the one step  which was kind of emitted here so  thought okay great here we have fill in  on how to do it and so next class you  know I went up and I presented it was  really happy isn't this cool you know  1939 mathematics you know it's really  awesome and it's like one line that's  like filled in the step the same student  raises  like I don't think that's true so now  the step was filled in but it was wrong  so it turns out actually most of these  proofs are wrong but not all of them  fortunately so the theorem is still true  so that was wrong this one was wrong  this one is wrong  this one skipped the step that was key  this one essentially also skipped the  step that was key so in the end there's  to correct proofs Rochette niak and Bing  and Catherine Catherine off which was  surprising I thought I'd show you one of  the errors in the in the nahji proof oh  there's one other interesting feature  here which is why wasn't this discovered  until our class the student was blaze  Gasol and so then we wrote a paper about  it and we have our own proof of course  now some people may have realized there  was an error in some of the proofs of  Bingen cows are not one of the correct  proofs but very nice one wrote here's  the original Russian sentence and the  English translation is the proof of this  theorem given by Nagy is incorrect  which leaves something to be desired and  Greenbaum who can read Russian we had to  get them translated mentioned this so he  noticed that point they remarked that  Nash's proof is invalid but there's no  basis for this claim so then it will  remain undiscovered until 2005 or  something so kind of funny good thing  there's so many proofs to choose for us  so this is Nash's original proof you can  see the example where two flips flipping  to Pakistan while taneous Lee causes a  crossing and yeah don't read the whole  thing every thing but there's one  sentence here which is if you take a  polygon and call it p0 and you flip a  pocket and you get p1 and you flip a  pocket and you get p2 and you take the  convex hull of pean you get c0 and you  take the convex hull of p1 and get C 1  and P 2 you get C 2 and then you  interleave these polygons so the polygon  is convex hull next polygon it's convex  hull because you're always flipping  out each of these polygons obviously  contains the foregoing ones and so that  seemed really nice and we thought how  this is so elegant this is they use this  to prove that the limit of the peas is  convex because at the limit of the peas  then would be the limit of the C's  because of this interleaving but it's  not true that these things contain the  previous ones because if you have two  pockets and you flip one of them it's  this one pocket versus multiple pockets  issue you flip one of them that will not  contain the convex hull of the original  because you haven't flipped them all I  mean after you flip if you flipped all  the pockets then you would contain the  convex all the original but that would  end up arguing that some flip sequences  work not all of them so this is annoying  and I think this is where I run out of  slides but we have some time so I can  give you a sketch of the proof of how  this is the Bing and Katherine off proof  or our version of it give you an idea of  how this works it's actually easy it's  an easy proof it's just easy also to get  it wrong or to skip one of the steps so  I will just do a proof by a picture I  think so suppose you have a polygon on  the plane and let's say you flip a  pocket so this would look colors  something like that a little hard to do  a reflection so this is the reflective  polygon and if we look at each of the  vertices over here these guys don't move  these guys moved over here reflecting  through that line okay  observation one if I take some point X  interior to the polygon then these  points that move get farther away from X  Y because if you look at this line here  let's say you look at a vertex and where  it goes  this line is the Voronoi diagram of  those two points right this is the  perpendicular bisector of this and this  that's the meaning of reflection so that  means everything to the left of the line  here is closer to this point than that  point everything to the right of the  line is closer to this point than that  point now X which is interior to the  polygon must be to the left of the line  because the whole polygon is the left of  the line so X distance from X to this  vertex increases okay and that's I mean  X will remain inside because as you flip  you only get bigger so if I take a point  X and I look at the distance to some  vertex it can only monotonically  increase it also can't get arbitrarily  large because the maximum it could  possibly be is like half the perimeter  of the polygon the perimeter of the  polygon is preserved so it can only  stretch so far okay so if you look at  this distance  it's monotonically increasing and it's  bounded because it could never get  bigger than the perimeter of the polygon  therefore it has a limit that distance  has a limit because monotone bounded  sequences always have a limit unique  limit so cool distance from X to some  vertex has a limit well I'm going to do  this for three different points X that  lie in some non degenerate triangle so  not all in a line and that means the  three distances from these points to  this vertex all converge to some limit  and therefore that point converges to a  limit namely the intersection of those  three circles  send it at those points and so this  proves that the polygon has a limit  because every vertex has a limiting  point cool now the kind of the tricky  part is to argue that limit is convex  and this is where everyone kind of had  an issue all right  the first thing we argue is that the the  angles converge to a limit this is kind  of a technicality because we know the  points converge to a limit so surely the  angle does the only issue is well if all  the points say converge to the same  point then the angle would not have a  limit but that's easy to argue can't  happen because these edge lengths are  preserved okay so I will just skip that  one I mean it follows from some of the  things we said so all right now we get  to the fun part  this polygon has a limit the angles have  limits so I want to look at the limiting  angles I want to in particular look at  the vertices that move is if this if you  flip an infinite number of times that  means some vertex must move an infinite  number of times because every time you  flip somebody moves these guys are  considered not moving even though  they're kind of involved in the flip so  if you look at the moved guys what is  their angle okay so before or some angle  here and afterwards the interior angle  is the reverse so if this was reflex  before it's convex now if there were a  convex angle over here like this it  would become reflex over here so you you  alternate between being less than 180  and greater than 180 every time you move  so if you have a vertex that is moving  infinitely many times it must it's angle  must alternate between less than 180  convex and greater than 180 a reflex  infinitely many times if that happens  and you have a unique limit angle your  limit angle must be 180 okay that's  interesting  so if  our vertex moves infinitely many times  then its limit angle equals 180  it must be flat very close to a  contradiction at this point so these  guys are gonna end up looking like this  well let's look at the other guys look  if you look at any so we have some limit  polygon in the limit polygon there are  some flat vertices but there must also  be some non flat vertices you can't just  go straight and hope to close a cycle so  there's some maybe some convex ones like  this there may be some reflex ones at  some point these vertices must stop  moving because everyone who moves  infinitely many times has a limit angle  flat so anybody who is we call it  pointed here though it's a little  different from pointed  pseudo-triangulations anyone who's not a  flat angle must stop moving after finite  time so let's go to that time when all  the when all of these guys have stopped  moving okay so now I'm imagine so your  limit polygon is something we don't know  whether it's convex or whatever could  have from any flat angles it's got to  have at least one of them let's move if  we assume there's something infinite  here this is the limit polygon okay now  at finite time we know that these guys  have stopped moving meaning we're done  with those guys so I've drawn the limit  here but also the finite thing which  must be on the inside right must be  something like this I don't know somehow  it's it's gonna flip and reach the  infinitely many times and reach this in  the limit  that looks weird so the way to argue  this in the clean way is if you look at  the convex hull of the limit let's say  convex hull the limit is this okay and  look at the convex hull of this finite  time when the squared vertices have  stopped moving convex hull will be well  it's got to be at least this because  these guys are already there the convex  hull is defined by the square points you  don't care about the flat vertices that  won't affect the convex hull so that  means the convex hull equals the limit  convex hull at this finite time now  you're about to do another flip which  means you're going to go outside that  convex hull in and contradiction exact  clear so maybe go through that part one  more time  so after finite time when all of the  non-flat forces have stopped moving we  have reached the final convex hull that  means you can't do anymore flips because  every flip makes a convex hull bigger so  you actually had to stop at that time it  was kind of a fun proof it the key is  really this part that if a vertex flips  infinitely many times then that limit  angle must be flat and so they really  don't participate in the convex hull and  this is kind of the big and Catharine  off key idea yeah  does anyone news  3d is a good question people have tried  to define flips for 3d and I guess I  think there's never really been a  successful definition did ya and exactly  how to flip it I mean you can define  pocket in the same way but then the  boundary won't be a single plane it'll  be some convex cap and so flipping you  can't really just reflect yeah that's  kind of annoying the other question was  so you got a secret these are the  sequence of simple operations that takes  you from convex to convex right does  anybody use that yeah it's so is this  useful for something shortest paths are  certainly not preserved perimeter edge  lengths are preserved of course we knew  how to do that with the Carpenters rule  theorem and just in staying in 2d so  it's not that exciting but the  operations are definitely a lot simpler  and well so I think the easy answer to  your question is there are many natural  simple moves and this is sort of the  first one that people consider but  actually there are a lot more and I'm  going to talk about those the main  application I know for this stuff is  that basically people wanted to generate  random closed walks in 3d typically and  so they wanted to find a small set of  operations they could just perform  randomly and hope that that was a  rapidly mixing Markov chain so  eventually you'd have a kind of random  thing I don't think there are any rapid  mixing results at least that I'm aware  of but in order to hope in order to hope  to get the space randomly you at least  have to be able to make anything so for  that the question is can you make can  you--can vex if I anything because if  you can connect to anything then at  least you can make anything more or less  so that's where these questions come  from we focus more in the universality  but I think what people care about is a  random generation business so in to that  end of course for random generation you  don't really care about crossings and so  another fun extension which was in I  think our paper for the first time  although there's a weaker version in the  Grim Bob  paper if you start with an on crossing  thing you can this proof still works you  need to add a little more to the  argument say either you decrease the  number of crossings which can only  happen a finite number of times or this  kind of stuff works so even for crossing  you can make crossing polygons - which  is kind of cool then so that's the end  of basic flips then we have something  called a flip turn just kind of fun and  in some ways better behaved and gets to  your question of what else is preserved  so let's say you have a pocket like this  so normally with a flip we would reflect  with the flip turn you reflect and then  also flip this way so it's the same as  rotating 180 degrees about this Center  in addition of preserving perimeter this  preserves the edge directions this  matches this this matches this this  matches this it does not preserve the  edge order however here we preserve the  edge order here we've reversed the order  of those at three edges so all this is  really doing is permuting the sequence  of edges the edge directions and edge  lengths are all the same so this means  you can make it most n factorial moves  here so immediately it's different from  this situation where you could have even  for N equals four you could have  arbitrarily many moves here it's at most  n factorial that was the original bound  it turns out every polygon convex flies  after order N squared flip turns and  it's a fun proof but I don't have time  to cover it here  that's flip turns then we go to  deflations  this is the inverse of a flip so suppose  you take a polygon and you do an  operation that if flipped would result  in the original polygon so exactly the  opposite of a flip this was conjectured  to also finish after finite time but in  fact it doesn't if you take any  quadrilateral satisfying Kawasaki  if you add up the opposite edge links  and their equal so 6 plus 3 is 9 4 plus  5 is 9 then there is at least a flat  limit possibly that's the Kawasaki thing  and in fact it will converge to that  flat limit and it will take infinite  time to get there  so that's kind of fun until it gets very  hard to draw the pictures this is really  the the main example of an infinitely  deflating polygon we have a paper called  deflating the Pentagon which got some  fun political views at some point it's  about a pentagon five sides and  essentially unless you have a flat  vertex in which case you are a  quadrilateral  there is no infinitely deflating  Pentagon which means you can deflate the  Pentagon always in finite time if it's  open for hexagons and higher whether  there is another example different from  the quadrilateral then there's the idea  of a pop this is an even simpler  operation than a flip you just take two  edges like here we're taking these two  edges and you just do a flip ignoring  crossings you just flip as if that were  a pocket lid so if this were a pocket  then you get this polygon now here you  can be forced to get crossings no matter  how you flip you might get a crossing  still we were wondering who cares about  crossings our pops enough to make  anything or two convex if I and the  answer is no there's this set of  polygons called alternating polygons  where the vertices alternate between the  x-axis and the y-axis and you can prove  that no matter what pop you do you are  still an alternating polygon and  alternating polygons you can also prove  are never convex so you're stuck this is  open for many years but finally solved a  few years ago and there's also pop turns  where you take two edges and you do  180-degree rotation like this and there  we can prove if you allow crossings you  can convince Phi any polygon I don't  have a figure of that because it's just  an algorithm we haven't actually run it  on a nice example  if you avoid crossings we can  characterize when it's possible when  it's impossible and those are pretty  much all the simple operations that at  least have been studied it's probably  more to think about but that's it and  that's the end of my part of the class  you

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this this class is talking about two  lectures both about protein folding  fixed angle chains things like that  there's a few questions about them one  is mostly about these open problems  equilateral equiangular obtuse 3d chains  fixed angle open problem is are they  locked so the question is about what  about any subset of those combinations  so this originally comes from an open  problem I think posed in 1999 one of the  first 3d linkage papers and it asked  whether equilateral universal joints can  lock and that's still open for for  universal joints these two constraints  don't make a lot of sense because who  cares if it's initially equiangular as  soon as you move it it will no longer be  equiangular and obtuseness I don't think  matters too much although it potentially  could so for that problem  I mean equilateral seems to be the core  for fixed angle though we know or we  conjecture that fixed angle equilateral  is not enough from this example it's  still not proved to be locked I don't  know if it's hard but it's probably  tedious so hasn't been done this is the  cross legs example all the edge lengths  are the same and if you don't allow the  touching part then all the angles are  also the same so this is everything  except the obtuse property and it's  probably locked so dropping obtuse is no  good the other things you could drop our  equilateral or equiangular this is if  you drop equiangular that I mean fixed  angle equilateral is not terribly  constraining because you can have you  can simulate a long bar by having a lot  of 180 degree angles so - this is obtuse  and it's equilateral but it's not  equiangular and it's locked for a  trivial reason the last so this is  dropping equi angular if I drop  equilateral  I can also just make a knitting needles  example I take a really long link and  then I take this thing is basically  string so I don't really care this  connection can be done with a lot of  obtuse angles and then have a really  long link so dropping any of the three  constraints makes the problem it makes  it easy to lock so for fixed angle  chains you need equilateral equiangular  and obtuse all these things together  potentially mean you're not locked but  we don't know that's the open problem of  course we don't necessarily need exactly  equally lateral or equal angular  hopefully within some small min to max  ratio would be enough but we don't know  next question is why did we model things  as why do we model the ribosome as a  cone that seems rather simple is this  realistic and probably when I taught the  class in 2010 I I didn't use any images  that I couldn't get permission for this  year I'm more lacks will ask for  forgiveness instead of permission but  this is a paper from science 2000 this  is what the ribosome actually looks like  there's many different figures of it but  this particularly liked this one because  it highlights a tunnel in the center of  the ribosome and so the idea is this  this is a machine for converting mRNA  into your proteins and the idea is that  protein comes through here there's a  little bump in the tunnel some people  conjecture this is where the amino acid  gets attached and then it feeds through  here and starts spitting out and there's  they say there's barely enough room here  for an alpha helix  so probably not too much folding happens  inside and the folding should just  happen over here and the observation is  if you have a reasonable size protein  it's only going to be about this big  then there's this big flat wall at the  exit so you have basically a half I have  a plane there and this half space is  more or less a big obstacle and so the  it's the alpha cone model where alpha is  180 I think I felt was the half angle so  it's not really a cone it's a cone  that's been opened up all the way to a  plane but that  one of the situations that is handled by  the theorems that we talked about so  that's why the cone model I mean we  generalized it cones just because it  works for general cones but the real one  is sort of flat cone that's where that  comes from  I don't yeah I think actually we proved  the theorem before we knew this but then  I looked it up and it was true and the  last question is about the electric  twenty one which some of you may not  know watch because it's optional but  there's this model called the HP model  for protein folding it's a model of  protein energy and it says you have a  chain of H&P nodes and basically the H  nodes are attracted to each other and  the P nodes don't care and the model is  that the H nodes are hydrophobic so they  want to be next to each other so they're  not next to water which is surrounding  the whole molecule and the P nodes let's  say don't care or they're hydrophilic  and it's known that if you have an HP  string and you want to find the optimal  folding into a 2d or a 3d structure  that's np-hard which is kind of weird  because somehow nature does it maybe  because it has found the easy instances  by evolution or maybe there's something  we're missing this model doesn't capture  reality we don't know but it's it  captures part of reality at least as we  observe it and unfortunately these  hardness proofs are a little too much  too complicated to cover here but I can  at least answer what are they reducing  from there's two proofs one is the first  one or I guess they're basically the  same time I think around 2001 first one  here is in 3d and these are by two MIT  professors and there's this big  construction but the reduction is from  bin packing so you have a bunch of fixed  size bins and you have a bunch of items  of varying sizes and you just want to  fit them all in and using the fewest  bins possible and the rough idea of the  construction this one this is how an  individual number is represented the  rough idea is to you have this big cube  you fill most of it which is sort of  stuff or you fill the sides with stuff  too  protect from the outside boundary the  insides are just used for connections  and then the front face here is this  stuff and you construct these bins and  you construct numbers which have to fit  inside the bins that's the rough idea  the details are complicated the 2d proof  by several people is from hamiltonicity  in maximum degree for graph so you have  a graph for every vertex as at most for  incident edges you want to find a  Hamiltonian cycle I think and this is  there's it's much harder to see this  picture I would say there's there's no  one diagram that kind of summarizes it  this is roughly the construction which  is quite complicated and I'll just leave  it at that so that's what they're  reduced from I think an interesting open  problem would be to find a simple or  cleaner proof of these results now that  it's known that they're hard it's  probably easier hardness proofs there's  also some open questions from lecture  like is it a px hard McKittrick symbol  to a 1 plus epsilon factor or 1 minus  epsilon factor for any epsilon or is  there some limit to approximability we  saw a nice approximation 4/3 whatever  3/4 whatever but can you do better still  open so that was the questions and then  well this is a question that you usually  ask in every lecture you didn't actually  ask it on this one press I copied and  pasted so there's some interesting  progress from this class two years ago  and I think in the open problem session  initially probably also a class project  related to this part so this is back in  lecture 20 so we had this proof just  kind of fun that flattening a fixed  angle chain deciding whether there was a  flat folded state is weakly np-hard  weakly meaning it depended a lot on what  these numbers look like and it was a  reduction from partitions you had to  split up the numbers into two equal  parts and if you did then this key would  fit in the slot and you're okay if you  didn't the key would collide with  something here so this this means the  problem is hard if your edge lengths are  vastly different they would have to  differ by a ratio of exponential in the  number of edges for this to really be  hard so a natural question is well what  if all the links are equal equilateral  chains turns out that is still hard and  this is a paper just published last year  with Sarah Eisenstaedt who took the  class then and there's a bunch of  results here so there's three all of  these are empty hardness results strong  np-hardness so it doesn't depend on the  numbers we've got flattening results  there's also a min flat span and Max  flat span once you have that flattening  is hard these are pretty easy to show  hard so I won't talk about them so much  but basically we consider different  ranges of angles in what you might allow  and in some cases we can get perfect  equilateral chains in some cases they  have to range between say 1 & 2 in  length and depending on the angles  I guess these natural question is obtuse  angles we don't quite know that it's the  best we have is like 60 degrees minus  epsilon and larger for this is nice for  orthogonal chains but it's not for  chains its for trees so still some open  questions here but lots of hardness  results I thought I'd show you roughly  what the hardness proofs look like  because they're kind of fun they all  follow this kind of structure this is a  gadget and it's a it's a fixed angle  chain here I'm going to show everything  with 90 degrees a lot easier to think  about and it's kind of like I don't know  a thinking plunger but something like  that you can kind of decide which parts  get pushed in and which parts get popped  out and this L shape can basically shift  left and right to three different places  if I did it right the  one this one this one and so that's a  useful construction as you might imagine  and the idea is you take that well and  you you add on these guys so I've got  another kind of plunge II thing like  this uh-huh  and yeah this guy can move left and  right to various extents well that's not  very intuitive so then this connects to  these little guys and these can just  flip up or down if it's a flat flat  embedding of a fixed angle chain and so  the idea is that you have three of these  pokey elements and they can attach they  bump into these different things and if  it's up then this can be up but when  they're down like this guy is currently  down but he could be pushed up then this  guy must be flipped down like that  that's the rough idea of how these parts  fit together and so then this is how you  you end up building some kind of 3sat  problem so you have some variables they  can either be true or false  and the idea is that the true guys are  on the top the bottom guys are on the  bottom and there's some complicated  interaction between them to make sure  that they can't both be true and false  basically you want these things to point  in whenever possible because then these  guys can go down otherwise you would  collide and if this guy's in the  corresponding guys in the bottom must be  down and that's enforced by this long  thing which is either completely down or  completely up so when it's completely  down these guys all have to be down  which is going to be a problem down  below if things aren't satisfied and  when it's up all of these guys would  have to be up some of them could be up  if you feel like it but really you've  probably put them all down whenever you  can because then you can take these guys  and stick them down so this is how you  set variables and then the other things  I showed where clauses essentially it  at least one of those pins had to be  pushed down and you don't know which one  and that's the hard part  it's a rough sketch of how this this  proof looks oh one thing is here  these dots for this to work these edges  can't be spinnable right so when I go  here I must immediately go back down I  can't flip it that's so these are rigid  edges that's not really the original  problem but you can simulate rigid edges  and this is where the angles get small  you can simulate rigid edges with the  very sharp zigzag because if you ever  flipped one of these it would collide  with the previous edge so for  sufficiently sharp angles you can force  parts to stay straight in this funny way  which is not great but it's one way to  force it we can also do it with trees or  various other techniques but that's how  the the proof looks so that is  flattening fixed angle chains strongly  np-hard we go next to one more topic for  fun this is the topic of flips yes I  think I'll switch to well I guess I have  a an image for this so this is a problem  posed by Paul air dish I think when he  was a student famous very famous  mathematician 1935 he asked this  question this is the whole question as  originally written but you have a  polygon and you take the convex hull the  polygon you take these regions which are  in the convex hull but outside the  polygon we call those pockets and you  imagine flipping those outside  reflecting through that supporting line  that tangent on the outside so then you  get this polygon here you get two  pockets maybe you flip both of them and  then you get a convex polygon now you  have no pockets and so you're done and  the question is as posed  prove that after a finite number of such  steps polygons will become convex so you  might call this a conjecture by arish  that it is finite  he'd never published a proof there's one  issue with the problem as stated you  can't actually flip both pockets at once  if you want to what collisions the  relation to linkages is we imagine these  flips could actually happen by a  rotation but you can also just imagine  them as reflecting instantaneously  there's this issue if you flip two  pockets at once you might have  collisions afterwards so the way this  problem has been interpreted by most  people is don't flip them all at once I  just flip them one at a time and then  you guarantee because this is a  supporting line a tangent line of the  convex hull when you flip one pocket out  it will go to the other side so it won't  can't intersect the rest of the polygon  so you would avoid collision in if you  do one flip at a time now this was  observed I guess a few years later this  finish and he also published the first  proof sorry before we get there another  a weird property and the why you might  worry about this being finite or  infinite is if you take this  quadrilateral and make this very narrow  this edge very small relative to the  horizontal edge then you can require  arbitrarily many flips to convexity so  even for N equals 4 you can require a  million flips so definitely there's some  dependence on the the ratio between the  longest length and the smallest length  though an open problem we still don't  know is whether you can bound the number  of flips in terms of N and that that  ratio is it pseudo polynomial who knows  so there have been actually many so it  turns out it's always finite and there  have been many proofs over the years of  this result that's kind of been  rediscovered many times so Nagy solved  originally in 1939 and he cited air dish  these two guys didn't cite anyone so  they may have come up with the problem  independently this paper these are two  Russian proofs then this paper cited  all they knew about everything they  cited all of them or so they said of  bruschetta niak I'm not sure I don't  think they necessarily knew about you  suppose then Vagner so independently  Kalu suppose the problem 1981 and  begginer solved it and then kind of  finally clean it up Greenbaum we saw  from uh none foldable knew about  everything and decided presumably this  is everything of course we might have  missed one but he knew all of the above  came up with his own proof and then  Godfrey Toussaint's father of  computational geometry one of them knew  about these and came up with yet another  simpler proof so the the story of this  is kind of fun when I taught this class  for the very first time 2003 so I  thought okay cool this is classic  theorem everyone should know it so I  thought I'd cover the latest proof that  presumably the best so I covered I got  free proof and after I'd so a little  unhappy with it but I finished writing  down the proof and then one of the  students asked raise their hands like is  that really right can you do that in  step two and the answer was no you can't  do that so it kind of basically skipped  a step and so I thought oh gee and  that's how I correspond with draw Roark  and Gottfried that weekend is like  something missing here maybe we should  go back to some of the other proofs cuz  we've got lots to choose from so we went  to the original which very short proof  it's only one page long maybe one and a  half pages and so oh this is fun you  know this is 1939 the way mathematics  used to be done and it's funny because  groom bomb proof was based on Nash's  proof and Toussaint's proof was based on  grim Bob's roof so in the end these  proofs were very similar in fact they  differed exactly in one the one step  which was kind of emitted here so  thought okay great here we have fill in  on how to do it and so next class you  know I went up and I presented it was  really happy isn't this cool you know  1939 mathematics you know it's really  awesome and it's like one line that's  like filled in the step the same student  raises  like I don't think that's true so now  the step was filled in but it was wrong  so it turns out actually most of these  proofs are wrong but not all of them  fortunately so the theorem is still true  so that was wrong this one was wrong  this one is wrong  this one skipped the step that was key  this one essentially also skipped the  step that was key so in the end there's  to correct proofs Rochette niak and Bing  and Catherine Catherine off which was  surprising I thought I'd show you one of  the errors in the in the nahji proof oh  there's one other interesting feature  here which is why wasn't this discovered  until our class the student was blaze  Gasol and so then we wrote a paper about  it and we have our own proof of course  now some people may have realized there  was an error in some of the proofs of  Bingen cows are not one of the correct  proofs but very nice one wrote here's  the original Russian sentence and the  English translation is the proof of this  theorem given by Nagy is incorrect  which leaves something to be desired and  Greenbaum who can read Russian we had to  get them translated mentioned this so he  noticed that point they remarked that  Nash's proof is invalid but there's no  basis for this claim so then it will  remain undiscovered until 2005 or  something so kind of funny good thing  there's so many proofs to choose for us  so this is Nash's original proof you can  see the example where two flips flipping  to Pakistan while taneous Lee causes a  crossing and yeah don't read the whole  thing every thing but there's one  sentence here which is if you take a  polygon and call it p0 and you flip a  pocket and you get p1 and you flip a  pocket and you get p2 and you take the  convex hull of pean you get c0 and you  take the convex hull of p1 and get C 1  and P 2 you get C 2 and then you  interleave these polygons so the polygon  is convex hull next polygon it's convex  hull because you're always flipping  out each of these polygons obviously  contains the foregoing ones and so that  seemed really nice and we thought how  this is so elegant this is they use this  to prove that the limit of the peas is  convex because at the limit of the peas  then would be the limit of the C's  because of this interleaving but it's  not true that these things contain the  previous ones because if you have two  pockets and you flip one of them it's  this one pocket versus multiple pockets  issue you flip one of them that will not  contain the convex hull of the original  because you haven't flipped them all I  mean after you flip if you flipped all  the pockets then you would contain the  convex all the original but that would  end up arguing that some flip sequences  work not all of them so this is annoying  and I think this is where I run out of  slides but we have some time so I can  give you a sketch of the proof of how  this is the Bing and Katherine off proof  or our version of it give you an idea of  how this works it's actually easy it's  an easy proof it's just easy also to get  it wrong or to skip one of the steps so  I will just do a proof by a picture I  think so suppose you have a polygon on  the plane and let's say you flip a  pocket so this would look colors  something like that a little hard to do  a reflection so this is the reflective  polygon and if we look at each of the  vertices over here these guys don't move  these guys moved over here reflecting  through that line okay  observation one if I take some point X  interior to the polygon then these  points that move get farther away from X  Y because if you look at this line here  let's say you look at a vertex and where  it goes  this line is the Voronoi diagram of  those two points right this is the  perpendicular bisector of this and this  that's the meaning of reflection so that  means everything to the left of the line  here is closer to this point than that  point everything to the right of the  line is closer to this point than that  point now X which is interior to the  polygon must be to the left of the line  because the whole polygon is the left of  the line so X distance from X to this  vertex increases okay and that's I mean  X will remain inside because as you flip  you only get bigger so if I take a point  X and I look at the distance to some  vertex it can only monotonically  increase it also can't get arbitrarily  large because the maximum it could  possibly be is like half the perimeter  of the polygon the perimeter of the  polygon is preserved so it can only  stretch so far okay so if you look at  this distance  it's monotonically increasing and it's  bounded because it could never get  bigger than the perimeter of the polygon  therefore it has a limit that distance  has a limit because monotone bounded  sequences always have a limit unique  limit so cool distance from X to some  vertex has a limit well I'm going to do  this for three different points X that  lie in some non degenerate triangle so  not all in a line and that means the  three distances from these points to  this vertex all converge to some limit  and therefore that point converges to a  limit namely the intersection of those  three circles  send it at those points and so this  proves that the polygon has a limit  because every vertex has a limiting  point cool now the kind of the tricky  part is to argue that limit is convex  and this is where everyone kind of had  an issue all right  the first thing we argue is that the the  angles converge to a limit this is kind  of a technicality because we know the  points converge to a limit so surely the  angle does the only issue is well if all  the points say converge to the same  point then the angle would not have a  limit but that's easy to argue can't  happen because these edge lengths are  preserved okay so I will just skip that  one I mean it follows from some of the  things we said so all right now we get  to the fun part  this polygon has a limit the angles have  limits so I want to look at the limiting  angles I want to in particular look at  the vertices that move is if this if you  flip an infinite number of times that  means some vertex must move an infinite  number of times because every time you  flip somebody moves these guys are  considered not moving even though  they're kind of involved in the flip so  if you look at the moved guys what is  their angle okay so before or some angle  here and afterwards the interior angle  is the reverse so if this was reflex  before it's convex now if there were a  convex angle over here like this it  would become reflex over here so you you  alternate between being less than 180  and greater than 180 every time you move  so if you have a vertex that is moving  infinitely many times it must it's angle  must alternate between less than 180  convex and greater than 180 a reflex  infinitely many times if that happens  and you have a unique limit angle your  limit angle must be 180 okay that's  interesting  so if  our vertex moves infinitely many times  then its limit angle equals 180  it must be flat very close to a  contradiction at this point so these  guys are gonna end up looking like this  well let's look at the other guys look  if you look at any so we have some limit  polygon in the limit polygon there are  some flat vertices but there must also  be some non flat vertices you can't just  go straight and hope to close a cycle so  there's some maybe some convex ones like  this there may be some reflex ones at  some point these vertices must stop  moving because everyone who moves  infinitely many times has a limit angle  flat so anybody who is we call it  pointed here though it's a little  different from pointed  pseudo-triangulations anyone who's not a  flat angle must stop moving after finite  time so let's go to that time when all  the when all of these guys have stopped  moving okay so now I'm imagine so your  limit polygon is something we don't know  whether it's convex or whatever could  have from any flat angles it's got to  have at least one of them let's move if  we assume there's something infinite  here this is the limit polygon okay now  at finite time we know that these guys  have stopped moving meaning we're done  with those guys so I've drawn the limit  here but also the finite thing which  must be on the inside right must be  something like this I don't know somehow  it's it's gonna flip and reach the  infinitely many times and reach this in  the limit  that looks weird so the way to argue  this in the clean way is if you look at  the convex hull of the limit let's say  convex hull the limit is this okay and  look at the convex hull of this finite  time when the squared vertices have  stopped moving convex hull will be well  it's got to be at least this because  these guys are already there the convex  hull is defined by the square points you  don't care about the flat vertices that  won't affect the convex hull so that  means the convex hull equals the limit  convex hull at this finite time now  you're about to do another flip which  means you're going to go outside that  convex hull in and contradiction exact  clear so maybe go through that part one  more time  so after finite time when all of the  non-flat forces have stopped moving we  have reached the final convex hull that  means you can't do anymore flips because  every flip makes a convex hull bigger so  you actually had to stop at that time it  was kind of a fun proof it the key is  really this part that if a vertex flips  infinitely many times then that limit  angle must be flat and so they really  don't participate in the convex hull and  this is kind of the big and Catharine  off key idea yeah  does anyone news  3d is a good question people have tried  to define flips for 3d and I guess I  think there's never really been a  successful definition did ya and exactly  how to flip it I mean you can define  pocket in the same way but then the  boundary won't be a single plane it'll  be some convex cap and so flipping you  can't really just reflect yeah that's  kind of annoying the other question was  so you got a secret these are the  sequence of simple operations that takes  you from convex to convex right does  anybody use that yeah it's so is this  useful for something shortest paths are  certainly not preserved perimeter edge  lengths are preserved of course we knew  how to do that with the Carpenters rule  theorem and just in staying in 2d so  it's not that exciting but the  operations are definitely a lot simpler  and well so I think the easy answer to  your question is there are many natural  simple moves and this is sort of the  first one that people consider but  actually there are a lot more and I'm  going to talk about those the main  application I know for this stuff is  that basically people wanted to generate  random closed walks in 3d typically and  so they wanted to find a small set of  operations they could just perform  randomly and hope that that was a  rapidly mixing Markov chain so  eventually you'd have a kind of random  thing I don't think there are any rapid  mixing results at least that I'm aware  of but in order to hope in order to hope  to get the space randomly you at least  have to be able to make anything so for  that the question is can you make can  you--can vex if I anything because if  you can connect to anything then at  least you can make anything more or less  so that's where these questions come  from we focus more in the universality  but I think what people care about is a  random generation business so in to that  end of course for random generation you  don't really care about crossings and so  another fun extension which was in I  think our paper for the first time  although there's a weaker version in the  Grim Bob  paper if you start with an on crossing  thing you can this proof still works you  need to add a little more to the  argument say either you decrease the  number of crossings which can only  happen a finite number of times or this  kind of stuff works so even for crossing  you can make crossing polygons - which  is kind of cool then so that's the end  of basic flips then we have something  called a flip turn just kind of fun and  in some ways better behaved and gets to  your question of what else is preserved  so let's say you have a pocket like this  so normally with a flip we would reflect  with the flip turn you reflect and then  also flip this way so it's the same as  rotating 180 degrees about this Center  in addition of preserving perimeter this  preserves the edge directions this  matches this this matches this this  matches this it does not preserve the  edge order however here we preserve the  edge order here we've reversed the order  of those at three edges so all this is  really doing is permuting the sequence  of edges the edge directions and edge  lengths are all the same so this means  you can make it most n factorial moves  here so immediately it's different from  this situation where you could have even  for N equals four you could have  arbitrarily many moves here it's at most  n factorial that was the original bound  it turns out every polygon convex flies  after order N squared flip turns and  it's a fun proof but I don't have time  to cover it here  that's flip turns then we go to  deflations  this is the inverse of a flip so suppose  you take a polygon and you do an  operation that if flipped would result  in the original polygon so exactly the  opposite of a flip this was conjectured  to also finish after finite time but in  fact it doesn't if you take any  quadrilateral satisfying Kawasaki  if you add up the opposite edge links  and their equal so 6 plus 3 is 9 4 plus  5 is 9 then there is at least a flat  limit possibly that's the Kawasaki thing  and in fact it will converge to that  flat limit and it will take infinite  time to get there  so that's kind of fun until it gets very  hard to draw the pictures this is really  the the main example of an infinitely  deflating polygon we have a paper called  deflating the Pentagon which got some  fun political views at some point it's  about a pentagon five sides and  essentially unless you have a flat  vertex in which case you are a  quadrilateral  there is no infinitely deflating  Pentagon which means you can deflate the  Pentagon always in finite time if it's  open for hexagons and higher whether  there is another example different from  the quadrilateral then there's the idea  of a pop this is an even simpler  operation than a flip you just take two  edges like here we're taking these two  edges and you just do a flip ignoring  crossings you just flip as if that were  a pocket lid so if this were a pocket  then you get this polygon now here you  can be forced to get crossings no matter  how you flip you might get a crossing  still we were wondering who cares about  crossings our pops enough to make  anything or two convex if I and the  answer is no there's this set of  polygons called alternating polygons  where the vertices alternate between the  x-axis and the y-axis and you can prove  that no matter what pop you do you are  still an alternating polygon and  alternating polygons you can also prove  are never convex so you're stuck this is  open for many years but finally solved a  few years ago and there's also pop turns  where you take two edges and you do  180-degree rotation like this and there  we can prove if you allow crossings you  can convince Phi any polygon I don't  have a figure of that because it's just  an algorithm we haven't actually run it  on a nice example  if you avoid crossings we can  characterize when it's possible when  it's impossible and those are pretty  much all the simple operations that at  least have been studied it's probably  more to think about but that's it and  that's the end of my part of the class  you
 

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250
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252
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254
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255
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256
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257
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258
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259
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260
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261
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262
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263
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264
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265
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266
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267
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268
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269
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270
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271
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272
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275
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276
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277
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279
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280
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281
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282
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283
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284
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285
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286
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287
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288
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289
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290
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291
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292
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293
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294
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295
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296
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297
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298
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299
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300
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301
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302
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303
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304
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305
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306
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307
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308
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309
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310
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311
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312
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313
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314
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315
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316
00:14:00,760 --> 00:14:02,770

 

317
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318
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319
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320
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321
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322
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323
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324
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325
00:14:26,350 --> 00:14:33,070

 

326
00:14:33,070 --> 00:14:39,070

 

327
00:14:39,070 --> 00:14:41,890

 

328
00:14:41,890 --> 00:14:45,220

 

329
00:14:45,220 --> 00:14:47,260

 

330
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331
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332
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333
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334
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335
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336
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337
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338
00:15:06,340 --> 00:15:08,020

 

339
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340
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341
00:15:13,630 --> 00:15:15,100

 

342
00:15:15,100 --> 00:15:18,790

 

343
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344
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345
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346
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347
00:15:28,300 --> 00:15:32,800

 

348
00:15:32,800 --> 00:15:34,750

 

349
00:15:34,750 --> 00:15:37,060

 

350
00:15:37,060 --> 00:15:40,330

 

351
00:15:40,330 --> 00:15:42,190

 

352
00:15:42,190 --> 00:15:44,140

 

353
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354
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355
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356
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357
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358
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359
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360
00:16:01,300 --> 00:16:04,150

 

361
00:16:04,150 --> 00:16:06,040

 

362
00:16:06,040 --> 00:16:08,200

 

363
00:16:08,200 --> 00:16:10,360

 

364
00:16:10,360 --> 00:16:12,460

 

365
00:16:12,460 --> 00:16:14,290

 

366
00:16:14,290 --> 00:16:16,150

 

367
00:16:16,150 --> 00:16:18,310

 

368
00:16:18,310 --> 00:16:19,900

 

369
00:16:19,900 --> 00:16:23,470

 

370
00:16:23,470 --> 00:16:26,260

 

371
00:16:26,260 --> 00:16:31,980

 

372
00:16:31,980 --> 00:16:38,530

 

373
00:16:38,530 --> 00:16:43,240

 

374
00:16:43,240 --> 00:16:46,270

 

375
00:16:46,270 --> 00:16:47,860

 

376
00:16:47,860 --> 00:16:49,510

 

377
00:16:49,510 --> 00:16:52,810

 

378
00:16:52,810 --> 00:16:54,910

 

379
00:16:54,910 --> 00:16:57,520

 

380
00:16:57,520 --> 00:17:01,090

 

381
00:17:01,090 --> 00:17:03,370

 

382
00:17:03,370 --> 00:17:06,460

 

383
00:17:06,460 --> 00:17:08,860

 

384
00:17:08,860 --> 00:17:10,960

 

385
00:17:10,960 --> 00:17:12,940

 

386
00:17:12,940 --> 00:17:14,650

 

387
00:17:14,650 --> 00:17:16,840

 

388
00:17:16,840 --> 00:17:20,550

 

389
00:17:20,550 --> 00:17:25,960

 

390
00:17:25,960 --> 00:17:28,600

 

391
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392
00:17:30,280 --> 00:17:31,540

 

393
00:17:31,540 --> 00:17:35,980

 

394
00:17:35,980 --> 00:17:39,730

 

395
00:17:39,730 --> 00:17:42,220

 

396
00:17:42,220 --> 00:17:43,420

 

397
00:17:43,420 --> 00:17:47,290

 

398
00:17:47,290 --> 00:17:50,690

 

399
00:17:50,690 --> 00:17:53,480

 

400
00:17:53,480 --> 00:17:55,460

 

401
00:17:55,460 --> 00:17:57,110

 

402
00:17:57,110 --> 00:17:58,400

 

403
00:17:58,400 --> 00:18:02,150

 

404
00:18:02,150 --> 00:18:04,370

 

405
00:18:04,370 --> 00:18:07,700

 

406
00:18:07,700 --> 00:18:09,950

 

407
00:18:09,950 --> 00:18:13,970

 

408
00:18:13,970 --> 00:18:16,250

 

409
00:18:16,250 --> 00:18:17,420

 

410
00:18:17,420 --> 00:18:20,390

 

411
00:18:20,390 --> 00:18:23,630

 

412
00:18:23,630 --> 00:18:26,450

 

413
00:18:26,450 --> 00:18:29,420

 

414
00:18:29,420 --> 00:18:31,100

 

415
00:18:31,100 --> 00:18:34,850

 

416
00:18:34,850 --> 00:18:37,010

 

417
00:18:37,010 --> 00:18:41,860

 

418
00:18:41,860 --> 00:18:44,450

 

419
00:18:44,450 --> 00:18:47,330

 

420
00:18:47,330 --> 00:18:49,670

 

421
00:18:49,670 --> 00:18:52,490

 

422
00:18:52,490 --> 00:18:56,480

 

423
00:18:56,480 --> 00:18:58,520

 

424
00:18:58,520 --> 00:18:59,600

 

425
00:18:59,600 --> 00:19:01,910

 

426
00:19:01,910 --> 00:19:04,010

 

427
00:19:04,010 --> 00:19:06,440

 

428
00:19:06,440 --> 00:19:08,150

 

429
00:19:08,150 --> 00:19:11,330

 

430
00:19:11,330 --> 00:19:13,160

 

431
00:19:13,160 --> 00:19:14,860

 

432
00:19:14,860 --> 00:19:17,150

 

433
00:19:17,150 --> 00:19:18,920

 

434
00:19:18,920 --> 00:19:21,260

 

435
00:19:21,260 --> 00:19:25,010

 

436
00:19:25,010 --> 00:19:26,990

 

437
00:19:26,990 --> 00:19:29,660

 

438
00:19:29,660 --> 00:19:32,060

 

439
00:19:32,060 --> 00:19:35,110

 

440
00:19:35,110 --> 00:19:37,430

 

441
00:19:37,430 --> 00:19:38,960

 

442
00:19:38,960 --> 00:19:40,610

 

443
00:19:40,610 --> 00:19:42,230

 

444
00:19:42,230 --> 00:19:45,020

 

445
00:19:45,020 --> 00:19:47,270

 

446
00:19:47,270 --> 00:19:49,820

 

447
00:19:49,820 --> 00:19:53,330

 

448
00:19:53,330 --> 00:19:55,130

 

449
00:19:55,130 --> 00:19:56,630

 

450
00:19:56,630 --> 00:19:58,880

 

451
00:19:58,880 --> 00:20:00,770

 

452
00:20:00,770 --> 00:20:02,990

 

453
00:20:02,990 --> 00:20:03,000

 

454
00:20:03,000 --> 00:20:03,700


455
00:20:03,700 --> 00:20:07,930

 

456
00:20:07,930 --> 00:20:10,200

 

457
00:20:10,200 --> 00:20:12,279

 

458
00:20:12,279 --> 00:20:15,399

 

459
00:20:15,399 --> 00:20:17,409

 

460
00:20:17,409 --> 00:20:20,230

 

461
00:20:20,230 --> 00:20:21,700

 

462
00:20:21,700 --> 00:20:23,289

 

463
00:20:23,289 --> 00:20:25,930

 

464
00:20:25,930 --> 00:20:28,269

 

465
00:20:28,269 --> 00:20:31,600

 

466
00:20:31,600 --> 00:20:35,230

 

467
00:20:35,230 --> 00:20:39,909

 

468
00:20:39,909 --> 00:20:42,990

 

469
00:20:42,990 --> 00:20:45,399

 

470
00:20:45,399 --> 00:20:47,230

 

471
00:20:47,230 --> 00:20:49,750

 

472
00:20:49,750 --> 00:20:52,690

 

473
00:20:52,690 --> 00:20:55,620

 

474
00:20:55,620 --> 00:20:59,350

 

475
00:20:59,350 --> 00:21:01,600

 

476
00:21:01,600 --> 00:21:03,220

 

477
00:21:03,220 --> 00:21:06,190

 

478
00:21:06,190 --> 00:21:09,220

 

479
00:21:09,220 --> 00:21:10,539

 

480
00:21:10,539 --> 00:21:13,180

 

481
00:21:13,180 --> 00:21:17,169

 

482
00:21:17,169 --> 00:21:20,260

 

483
00:21:20,260 --> 00:21:23,919

 

484
00:21:23,919 --> 00:21:25,659

 

485
00:21:25,659 --> 00:21:27,460

 

486
00:21:27,460 --> 00:21:30,159

 

487
00:21:30,159 --> 00:21:34,299

 

488
00:21:34,299 --> 00:21:39,940

 

489
00:21:39,940 --> 00:21:41,409

 

490
00:21:41,409 --> 00:21:44,200

 

491
00:21:44,200 --> 00:21:46,690

 

492
00:21:46,690 --> 00:21:48,220

 

493
00:21:48,220 --> 00:21:52,269

 

494
00:21:52,269 --> 00:21:53,950

 

495
00:21:53,950 --> 00:21:56,440

 

496
00:21:56,440 --> 00:21:59,799

 

497
00:21:59,799 --> 00:22:01,360

 

498
00:22:01,360 --> 00:22:03,340

 

499
00:22:03,340 --> 00:22:06,190

 

500
00:22:06,190 --> 00:22:08,799

 

501
00:22:08,799 --> 00:22:10,899

 

502
00:22:10,899 --> 00:22:14,230

 

503
00:22:14,230 --> 00:22:15,909

 

504
00:22:15,909 --> 00:22:17,650

 

505
00:22:17,650 --> 00:22:21,250

 

506
00:22:21,250 --> 00:22:25,240

 

507
00:22:25,240 --> 00:22:26,830

 

508
00:22:26,830 --> 00:22:28,630

 

509
00:22:28,630 --> 00:22:31,450

 

510
00:22:31,450 --> 00:22:33,160

 

511
00:22:33,160 --> 00:22:34,390

 

512
00:22:34,390 --> 00:22:37,000

 

513
00:22:37,000 --> 00:22:40,000

 

514
00:22:40,000 --> 00:22:41,890

 

515
00:22:41,890 --> 00:22:43,900

 

516
00:22:43,900 --> 00:22:45,700

 

517
00:22:45,700 --> 00:22:47,890

 

518
00:22:47,890 --> 00:22:49,570

 

519
00:22:49,570 --> 00:22:50,800

 

520
00:22:50,800 --> 00:22:52,570

 

521
00:22:52,570 --> 00:22:54,310

 

522
00:22:54,310 --> 00:22:56,650

 

523
00:22:56,650 --> 00:22:58,510

 

524
00:22:58,510 --> 00:23:02,710

 

525
00:23:02,710 --> 00:23:04,660

 

526
00:23:04,660 --> 00:23:11,280

 

527
00:23:11,280 --> 00:23:14,670

 

528
00:23:14,670 --> 00:23:17,620

 

529
00:23:17,620 --> 00:23:22,930

 

530
00:23:22,930 --> 00:23:25,960

 

531
00:23:25,960 --> 00:23:27,760

 

532
00:23:27,760 --> 00:23:30,730

 

533
00:23:30,730 --> 00:23:33,460

 

534
00:23:33,460 --> 00:23:40,540

 

535
00:23:40,540 --> 00:23:43,930

 

536
00:23:43,930 --> 00:23:57,670

 

537
00:23:57,670 --> 00:24:00,410

 

538
00:24:00,410 --> 00:24:03,740

 

539
00:24:03,740 --> 00:24:05,860

 

540
00:24:05,860 --> 00:24:11,170

 

541
00:24:11,170 --> 00:24:15,800

 

542
00:24:15,800 --> 00:24:17,480

 

543
00:24:17,480 --> 00:24:23,050

 

544
00:24:23,050 --> 00:24:26,870

 

545
00:24:26,870 --> 00:24:30,550

 

546
00:24:30,550 --> 00:24:37,150

 

547
00:24:37,150 --> 00:24:39,620

 

548
00:24:39,620 --> 00:24:39,940

 

549
00:24:39,940 --> 00:24:42,770

 

550
00:24:42,770 --> 00:24:44,030

 

551
00:24:44,030 --> 00:24:45,710

 

552
00:24:45,710 --> 00:24:47,870

 

553
00:24:47,870 --> 00:24:49,370

 

554
00:24:49,370 --> 00:24:52,400

 

555
00:24:52,400 --> 00:24:53,600

 

556
00:24:53,600 --> 00:24:55,370

 

557
00:24:55,370 --> 00:24:57,650

 

558
00:24:57,650 --> 00:24:59,300

 

559
00:24:59,300 --> 00:25:00,980

 

560
00:25:00,980 --> 00:25:04,310

 

561
00:25:04,310 --> 00:25:09,950

 

562
00:25:09,950 --> 00:25:12,080

 

563
00:25:12,080 --> 00:25:14,630

 

564
00:25:14,630 --> 00:25:16,220

 

565
00:25:16,220 --> 00:25:17,810

 

566
00:25:17,810 --> 00:25:19,670

 

567
00:25:19,670 --> 00:25:22,910

 

568
00:25:22,910 --> 00:25:25,490

 

569
00:25:25,490 --> 00:25:27,470

 

570
00:25:27,470 --> 00:25:28,730

 

571
00:25:28,730 --> 00:25:32,840

 

572
00:25:32,840 --> 00:25:33,470

 

573
00:25:33,470 --> 00:25:35,480

 

574
00:25:35,480 --> 00:25:36,890

 

575
00:25:36,890 --> 00:25:38,090

 

576
00:25:38,090 --> 00:25:40,310

 

577
00:25:40,310 --> 00:25:42,590

 

578
00:25:42,590 --> 00:25:45,890

 

579
00:25:45,890 --> 00:25:48,680

 

580
00:25:48,680 --> 00:25:51,680

 

581
00:25:51,680 --> 00:25:54,470

 

582
00:25:54,470 --> 00:25:56,480

 

583
00:25:56,480 --> 00:25:58,850

 

584
00:25:58,850 --> 00:26:01,070

 

585
00:26:01,070 --> 00:26:03,500

 

586
00:26:03,500 --> 00:26:06,320

 

587
00:26:06,320 --> 00:26:07,970

 

588
00:26:07,970 --> 00:26:09,320

 

589
00:26:09,320 --> 00:26:11,600

 

590
00:26:11,600 --> 00:26:13,220

 

591
00:26:13,220 --> 00:26:14,779

 

592
00:26:14,779 --> 00:26:19,310

 

593
00:26:19,310 --> 00:26:20,899

 

594
00:26:20,899 --> 00:26:23,629

 

595
00:26:23,629 --> 00:26:26,989

 

596
00:26:26,989 --> 00:26:31,249

 

597
00:26:31,249 --> 00:26:33,229

 

598
00:26:33,229 --> 00:26:35,539

 

599
00:26:35,539 --> 00:26:37,129

 

600
00:26:37,129 --> 00:26:42,830

 

601
00:26:42,830 --> 00:26:44,090

 

602
00:26:44,090 --> 00:26:45,349

 

603
00:26:45,349 --> 00:26:47,149

 

604
00:26:47,149 --> 00:26:48,499

 

605
00:26:48,499 --> 00:26:50,899

 

606
00:26:50,899 --> 00:26:54,139

 

607
00:26:54,139 --> 00:26:59,659

 

608
00:26:59,659 --> 00:27:00,379

 

609
00:27:00,379 --> 00:27:02,899

 

610
00:27:02,899 --> 00:27:06,470

 

611
00:27:06,470 --> 00:27:08,210

 

612
00:27:08,210 --> 00:27:11,060

 

613
00:27:11,060 --> 00:27:13,820

 

614
00:27:13,820 --> 00:27:16,099

 

615
00:27:16,099 --> 00:27:17,960

 

616
00:27:17,960 --> 00:27:19,940

 

617
00:27:19,940 --> 00:27:21,049

 

618
00:27:21,049 --> 00:27:23,149

 

619
00:27:23,149 --> 00:27:25,580

 

620
00:27:25,580 --> 00:27:30,470

 

621
00:27:30,470 --> 00:27:33,979

 

622
00:27:33,979 --> 00:27:37,009

 

623
00:27:37,009 --> 00:27:39,680

 

624
00:27:39,680 --> 00:27:43,070

 

625
00:27:43,070 --> 00:27:48,769

 

626
00:27:48,769 --> 00:27:50,960

 

627
00:27:50,960 --> 00:27:54,729

 

628
00:27:54,729 --> 00:27:58,759

 

629
00:27:58,759 --> 00:28:01,820

 

630
00:28:01,820 --> 00:28:04,970

 

631
00:28:04,970 --> 00:28:07,070

 

632
00:28:07,070 --> 00:28:09,619

 

633
00:28:09,619 --> 00:28:11,840

 

634
00:28:11,840 --> 00:28:15,710

 

635
00:28:15,710 --> 00:28:15,720

 

636
00:28:15,720 --> 00:28:16,340


637
00:28:16,340 --> 00:28:19,770

 

638
00:28:19,770 --> 00:28:25,230

 

639
00:28:25,230 --> 00:28:33,340

 

640
00:28:33,340 --> 00:28:42,940

 

641
00:28:42,940 --> 00:28:49,150

 

642
00:28:49,150 --> 00:28:52,020

 

643
00:28:52,020 --> 00:28:54,700

 

644
00:28:54,700 --> 00:28:57,760

 

645
00:28:57,760 --> 00:28:59,620

 

646
00:28:59,620 --> 00:29:01,480

 

647
00:29:01,480 --> 00:29:03,490

 

648
00:29:03,490 --> 00:29:05,830

 

649
00:29:05,830 --> 00:29:07,510

 

650
00:29:07,510 --> 00:29:11,280

 

651
00:29:11,280 --> 00:29:14,830

 

652
00:29:14,830 --> 00:29:17,620

 

653
00:29:17,620 --> 00:29:19,840

 

654
00:29:19,840 --> 00:29:23,560

 

655
00:29:23,560 --> 00:29:24,549

 

656
00:29:24,549 --> 00:29:25,390

 

657
00:29:25,390 --> 00:29:27,460

 

658
00:29:27,460 --> 00:29:30,430

 

659
00:29:30,430 --> 00:29:33,070

 

660
00:29:33,070 --> 00:29:35,230

 

661
00:29:35,230 --> 00:29:40,240

 

662
00:29:40,240 --> 00:29:41,919

 

663
00:29:41,919 --> 00:29:44,200

 

664
00:29:44,200 --> 00:29:46,120

 

665
00:29:46,120 --> 00:29:47,890

 

666
00:29:47,890 --> 00:29:49,390

 

667
00:29:49,390 --> 00:29:53,260

 

668
00:29:53,260 --> 00:29:55,600

 

669
00:29:55,600 --> 00:29:58,620

 

670
00:29:58,620 --> 00:30:04,480

 

671
00:30:04,480 --> 00:30:07,090

 

672
00:30:07,090 --> 00:30:09,669

 

673
00:30:09,669 --> 00:30:14,710

 

674
00:30:14,710 --> 00:30:17,200

 

675
00:30:17,200 --> 00:30:20,049

 

676
00:30:20,049 --> 00:30:22,519

 

677
00:30:22,519 --> 00:30:28,979

 

678
00:30:28,979 --> 00:30:31,379

 

679
00:30:31,379 --> 00:30:36,479

 

680
00:30:36,479 --> 00:30:42,509

 

681
00:30:42,509 --> 00:30:44,279

 

682
00:30:44,279 --> 00:30:46,589

 

683
00:30:46,589 --> 00:30:49,439

 

684
00:30:49,439 --> 00:30:51,419

 

685
00:30:51,419 --> 00:30:53,009

 

686
00:30:53,009 --> 00:30:55,829

 

687
00:30:55,829 --> 00:30:57,239

 

688
00:30:57,239 --> 00:30:59,129

 

689
00:30:59,129 --> 00:31:01,439

 

690
00:31:01,439 --> 00:31:05,069

 

691
00:31:05,069 --> 00:31:07,349

 

692
00:31:07,349 --> 00:31:08,549

 

693
00:31:08,549 --> 00:31:12,359

 

694
00:31:12,359 --> 00:31:14,879

 

695
00:31:14,879 --> 00:31:15,419

 

696
00:31:15,419 --> 00:31:18,229

 

697
00:31:18,229 --> 00:31:20,789

 

698
00:31:20,789 --> 00:31:23,519

 

699
00:31:23,519 --> 00:31:24,899

 

700
00:31:24,899 --> 00:31:27,899

 

701
00:31:27,899 --> 00:31:30,689

 

702
00:31:30,689 --> 00:31:32,249

 

703
00:31:32,249 --> 00:31:34,229

 

704
00:31:34,229 --> 00:31:36,509

 

705
00:31:36,509 --> 00:31:38,009

 

706
00:31:38,009 --> 00:31:39,689

 

707
00:31:39,689 --> 00:31:41,249

 

708
00:31:41,249 --> 00:31:44,720

 

709
00:31:44,720 --> 00:31:46,890

 

710
00:31:46,890 --> 00:31:50,410

 

711
00:31:50,410 --> 00:31:52,900

 

712
00:31:52,900 --> 00:31:54,580

 

713
00:31:54,580 --> 00:31:59,230

 

714
00:31:59,230 --> 00:32:00,910

 

715
00:32:00,910 --> 00:32:02,920

 

716
00:32:02,920 --> 00:32:05,140

 

717
00:32:05,140 --> 00:32:09,130

 

718
00:32:09,130 --> 00:32:11,530

 

719
00:32:11,530 --> 00:32:14,320

 

720
00:32:14,320 --> 00:32:16,290

 

721
00:32:16,290 --> 00:32:18,730

 

722
00:32:18,730 --> 00:32:21,070

 

723
00:32:21,070 --> 00:32:32,470

 

724
00:32:32,470 --> 00:32:34,750

 

725
00:32:34,750 --> 00:32:36,760

 

726
00:32:36,760 --> 00:32:38,530

 

727
00:32:38,530 --> 00:32:39,670

 

728
00:32:39,670 --> 00:32:41,500

 

729
00:32:41,500 --> 00:32:43,120

 

730
00:32:43,120 --> 00:32:44,560

 

731
00:32:44,560 --> 00:32:47,800

 

732
00:32:47,800 --> 00:32:50,860

 

733
00:32:50,860 --> 00:32:53,590

 

734
00:32:53,590 --> 00:32:54,910

 

735
00:32:54,910 --> 00:32:56,590

 

736
00:32:56,590 --> 00:32:59,550

 

737
00:32:59,550 --> 00:33:01,750

 

738
00:33:01,750 --> 00:33:03,850

 

739
00:33:03,850 --> 00:33:07,690

 

740
00:33:07,690 --> 00:33:09,610

 

741
00:33:09,610 --> 00:33:11,020

 

742
00:33:11,020 --> 00:33:12,580

 

743
00:33:12,580 --> 00:33:14,230

 

744
00:33:14,230 --> 00:33:15,790

 

745
00:33:15,790 --> 00:33:18,490

 

746
00:33:18,490 --> 00:33:19,960

 

747
00:33:19,960 --> 00:33:23,740

 

748
00:33:23,740 --> 00:33:25,990

 

749
00:33:25,990 --> 00:33:28,090

 

750
00:33:28,090 --> 00:33:30,940

 

751
00:33:30,940 --> 00:33:32,800

 

752
00:33:32,800 --> 00:33:33,940

 

753
00:33:33,940 --> 00:33:36,220

 

754
00:33:36,220 --> 00:33:38,530

 

755
00:33:38,530 --> 00:33:40,930

 

756
00:33:40,930 --> 00:33:42,700

 

757
00:33:42,700 --> 00:33:47,170

 

758
00:33:47,170 --> 00:33:48,700

 

759
00:33:48,700 --> 00:33:50,980

 

760
00:33:50,980 --> 00:33:53,080

 

761
00:33:53,080 --> 00:33:54,760

 

762
00:33:54,760 --> 00:33:57,280

 

763
00:33:57,280 --> 00:33:57,760

 

764
00:33:57,760 --> 00:34:00,010

 

765
00:34:00,010 --> 00:34:03,340

 

766
00:34:03,340 --> 00:34:05,260

 

767
00:34:05,260 --> 00:34:08,649

 

768
00:34:08,649 --> 00:34:10,060

 

769
00:34:10,060 --> 00:34:12,490

 

770
00:34:12,490 --> 00:34:15,700

 

771
00:34:15,700 --> 00:34:17,860

 

772
00:34:17,860 --> 00:34:22,930

 

773
00:34:22,930 --> 00:34:27,010

 

774
00:34:27,010 --> 00:34:31,690

 

775
00:34:31,690 --> 00:34:33,639

 

776
00:34:33,639 --> 00:34:34,930

 

777
00:34:34,930 --> 00:34:38,639

 

778
00:34:38,639 --> 00:34:42,040

 

779
00:34:42,040 --> 00:34:44,980

 

780
00:34:44,980 --> 00:34:48,419

 

781
00:34:48,419 --> 00:34:53,940

 

782
00:34:53,940 --> 00:34:57,250

 

783
00:34:57,250 --> 00:35:00,430

 

784
00:35:00,430 --> 00:35:02,560

 

785
00:35:02,560 --> 00:35:03,880

 

786
00:35:03,880 --> 00:35:05,740

 

787
00:35:05,740 --> 00:35:07,960

 

788
00:35:07,960 --> 00:35:10,450

 

789
00:35:10,450 --> 00:35:12,670

 

790
00:35:12,670 --> 00:35:15,640

 

791
00:35:15,640 --> 00:35:17,620

 

792
00:35:17,620 --> 00:35:20,560

 

793
00:35:20,560 --> 00:35:23,380

 

794
00:35:23,380 --> 00:35:25,210

 

795
00:35:25,210 --> 00:35:26,230

 

796
00:35:26,230 --> 00:35:27,850

 

797
00:35:27,850 --> 00:35:29,920

 

798
00:35:29,920 --> 00:35:32,800

 

799
00:35:32,800 --> 00:35:36,720

 

800
00:35:36,720 --> 00:35:38,950

 

801
00:35:38,950 --> 00:35:42,300

 

802
00:35:42,300 --> 00:35:48,540

 

803
00:35:48,540 --> 00:35:48,550

 

804
00:35:48,550 --> 00:35:49,750


805
00:35:49,750 --> 00:35:52,990

 

806
00:35:52,990 --> 00:35:55,330

 

807
00:35:55,330 --> 00:35:57,550

 

808
00:35:57,550 --> 00:35:59,830

 

809
00:35:59,830 --> 00:36:02,620

 

810
00:36:02,620 --> 00:36:05,590

 

811
00:36:05,590 --> 00:36:07,650

 

812
00:36:07,650 --> 00:36:11,090

 

813
00:36:11,090 --> 00:36:13,070

 

814
00:36:13,070 --> 00:36:16,550

 

815
00:36:16,550 --> 00:36:20,510

 

816
00:36:20,510 --> 00:36:23,500

 

817
00:36:23,500 --> 00:36:26,990

 

818
00:36:26,990 --> 00:36:28,370

 

819
00:36:28,370 --> 00:36:29,500

 

820
00:36:29,500 --> 00:36:31,820

 

821
00:36:31,820 --> 00:36:34,310

 

822
00:36:34,310 --> 00:36:36,170

 

823
00:36:36,170 --> 00:36:40,210

 

824
00:36:40,210 --> 00:36:44,170

 

825
00:36:44,170 --> 00:36:50,420

 

826
00:36:50,420 --> 00:36:53,660

 

827
00:36:53,660 --> 00:36:55,580

 

828
00:36:55,580 --> 00:36:56,780

 

829
00:36:56,780 --> 00:36:56,790

 

830
00:36:56,790 --> 00:36:57,410


831
00:36:57,410 --> 00:36:59,300

 

832
00:36:59,300 --> 00:37:00,980

 

833
00:37:00,980 --> 00:37:05,450

 

834
00:37:05,450 --> 00:37:07,070

 

835
00:37:07,070 --> 00:37:08,960

 

836
00:37:08,960 --> 00:37:11,840

 

837
00:37:11,840 --> 00:37:13,910

 

838
00:37:13,910 --> 00:37:16,370

 

839
00:37:16,370 --> 00:37:19,610

 

840
00:37:19,610 --> 00:37:24,260

 

841
00:37:24,260 --> 00:37:25,820

 

842
00:37:25,820 --> 00:37:27,800

 

843
00:37:27,800 --> 00:37:29,840

 

844
00:37:29,840 --> 00:37:31,370

 

845
00:37:31,370 --> 00:37:32,630

 

846
00:37:32,630 --> 00:37:34,970

 

847
00:37:34,970 --> 00:37:37,580

 

848
00:37:37,580 --> 00:37:40,160

 

849
00:37:40,160 --> 00:37:42,260

 

850
00:37:42,260 --> 00:37:43,910

 

851
00:37:43,910 --> 00:37:45,380

 

852
00:37:45,380 --> 00:37:47,990

 

853
00:37:47,990 --> 00:37:50,930

 

854
00:37:50,930 --> 00:37:53,200

 

855
00:37:53,200 --> 00:37:55,220

 

856
00:37:55,220 --> 00:37:58,130

 

857
00:37:58,130 --> 00:38:00,350

 

858
00:38:00,350 --> 00:38:04,570

 

859
00:38:04,570 --> 00:38:08,120

 

860
00:38:08,120 --> 00:38:11,930

 

861
00:38:11,930 --> 00:38:14,000

 

862
00:38:14,000 --> 00:38:16,160

 

863
00:38:16,160 --> 00:38:17,630

 

864
00:38:17,630 --> 00:38:19,130

 

865
00:38:19,130 --> 00:38:20,950

 

866
00:38:20,950 --> 00:38:23,499

 

867
00:38:23,499 --> 00:38:24,819

 

868
00:38:24,819 --> 00:38:26,890

 

869
00:38:26,890 --> 00:38:28,839

 

870
00:38:28,839 --> 00:38:30,700

 

871
00:38:30,700 --> 00:38:33,759

 

872
00:38:33,759 --> 00:38:44,720

 

873
00:38:44,720 --> 00:38:44,730

 

874
00:38:44,730 --> 00:38:46,790