[00:00:05]
>> What they plan to do is to give a complete proof of most of this theorem and actually I want the proof is do we bury Quinn did that the exit so let me tell you what is their idea so the basic idea was that if you have a small devices this small device or sat very bad because they cause things to be unbounded I mean one of the things I'm bound as we saw in the things but if you don't try to do all of that or by your there are things but you try to do that I think I mean and you have a B.C.S. So this is one of the cases where when you have ambitious and then you try to eliminate all the ever at the same time these pay self and then you can get the even more sophisticated things so this is one of the cases where trying to do more actually beat up problems so it goes over the problems and then so we will post we will present is something that people call an approach to the earlier version of the theorem so all this is something in which we just assume that you have an approximate solution of the equation that satisfy some degenerate are sick on this and then you can find that resolution and moreover the difference between the 2 solution on the approximation is more OK your need to justify how do you get there proximity solution you could be doing some that the numerical method you could be doing so my proxy mission you could be waking up in the middle of the night and instead of the writing numbers in a piece of paper it doesn't much OK the only thing that you need to verify is that the every small and that you have some fun thing.

[00:01:43]
So in particular so this is one of the things that they want to brag a little bit about it. Is that this system doesn't need to be close to him think it also this works even when you try to do realistic things like this solar system or. Other things and then then on the university assumptions they don't depend on global assumptions from them up they depend only on properties of the approximate solution I mean many classical theory that the map is to be in on the Internet or the conditions of these lessons require that you just need proper of this of the approximate solution OK So let me also tell you that there is something that they will discuss a little bit with the math of the sufficient.

[00:02:32]
In the sense that you only need to deal with functions of before the point of view of numerics to leave only with functions from Mumbai able dealing with functions of the body able she is much worse than dealing with functions from one body of world in a medical and I least call it the curse of this mention only theory goes I mean if you want to discredit a scene agree with say for example right if you have functions of one body world then you can achieve one accuracy but if you want to have done the same accuracy you have to take they squared of the numbers of these 2 at the station I mean graphically speaking.

[00:03:04]
OK And then you will discard a scene in Terrace and then you will see that the one is there then you will get an order that any storage or the in Logan operations and one is there will be also whether I think the liquid version so if you think about did that mean or there in a story to mean if you take a 1000000.

[00:03:24]
Present your numbers a 1000000 prison numbers. Like. 8 megabytes that nowadays nobody cares about the megabytes. So you and if you have like a fact or 20 or 30 then this is perfectly the only been with our mothers computers so this criticizing and getting a 1000000 times is something that can be than even the most stupid laptop you can get all OK so let me tell you the few of them who will try to be a little bit I mean this is not the who are here and of the so I will assume you have all my being that goes from as I said from the ceiling there which is the circle grows the Rios and then you're still my say I said before so this is only a percent of being see them in flux and then eternally digging another main we tell you I'm not going to specify too much but these are mainly said extension of variables right so this is a silly endeavor so the ceiling there is all real but this ill in there is sits inside complexes complex numbers etc And then so are you assume that these map extends to a complex the main right of course it maps that we have seen to the right this is the analytic things and then I also need their your funding condition which I also explain what it was and they explain you a little bit clearly stickily way it's important then it's something that could have been guessed even from the study of thought about the expansions Lenstra C.D.'s OK so then this is the theorem so that's what I was saying so you give I K C though that belongs to least analytical spaces you remember the analytical space which functions that are and I live in a strip right on this case either as we discussed before maybe I should try to write the equation here for the people just in case that you forget the equation that we have trying to solve ease if compose with K. minus K. compose with the my god the Omega is the rotation.

[00:05:27]
Do you make of the feet the plaster made up. Again most one OK So this is so this is the assumptions that they want to make about none degeneracy one is of course the one you think the range of Casey go from the complex the main this is kind of well inside dentally deceit to the main because I will be correcting it except that and then you see it's very dangerous if you're getting very close to them at least this is the main There is one thing which is very mysterious that I am not going to discuss too much at the moment which is this quantity that they were calling us but this is our number and the only thing that I am saying is that this number over here is not 0 and this S. if you will see a little bit it will come out the length of the says is an explicit expression that you have day and by thinking that he wrote the case you know they can that he.

[00:06:19]
Would be playing them together except that I will give you the explicit expression so I could put it here just by copying need from where he is but it's not going to tell you anything so the only thing that I can tell you of the moment I think which is useful is that you have an expression that these extremely explicitly to not do but I get pression and then you can compute that the 1000 I'm sure of it OK and then the conclusion is that if this epsilon which is the ever is to fish only a small depending on those which I have not told you with the thirty's meal which is that your fun think on this young star which is that your fun thing conditions government which is this thing over here which is going this year and then the only assumption that they need on F. is that in class to leave out the eaves that abound in the in the Indian elite This would be the main.

[00:07:10]
OK so not this condition no assumptions in there and then the conclusion is that there exists a cave that solves exactly the equation and then I can buy the editor or I can bond somebody come by distance between the exact solution and the approximate solution seems her mine is that this is that out there that happens to be here is there no need to see the main I mean typically in many cases people take those that do we want half row which is then only disagree the main of the unison case and then you can not put it but I think it's.

[00:07:44]
To keep it there specially for an American things because then you can you can trade off this morning to some patients with ventilate the city that do it OK and then the thing that happens is that you get the dissolution and then you come down them again it's an explicit expression that depends on all these things and I put the would he have the explicit dependance on the your fun thing constant that mean they are merely deceit the laws that you make and this is useful for several other purposes that we will discuss lead to be later in particular you can get measure estimates because of this form and you can get finally differential things right so let me repeat again their main assumption is that you have on is more lever and then you have these 2 known the unit are 2nd these units and then you have that the average is more.

[00:08:33]
OK And then these have any explicit I mean this epsilon Istari is a fairly explicit expression to mean he's not completely explicit to write it in a piece of paper but these reasonably explicit program and they will come into things and we have that he thinks that there are so so let me read to so this is more or less for the proof so what they will do is the following so I will I will describe you the method you would do right OK And you would see that the method involves exactly 13 is Step 6 at their right and there are different I get expressions except there then they will give estimates for how much you have improved and then they will show you that you can eat that 8 there forever and this is what you do in the new Don't method already in any of these methods that you will do show that you can do one they step on the improved and then you take the things can be repeat the infinitely often and then that they think I'm but this and then at the end they will sort of makes one remarks about the fact that all these can be implement the unit computer and then it gives you 7 a very efficient that worth.

[00:09:39]
OK So let me just read to start of with their description of the method so I'm just going to be a little bit Coverly it and not take the mains and not take things but just tell you what are the money pure lation and why the way you do them the fact that they were going to require a little weed so this is the beginning that you start and they put their name so you give you some approximate things and the only thing that they have them here is I have given this name because he's going to keep on appearing all the time so so you have something and the thing that I was telling you is that you should be thinking of your son is more quantity under small function.

[00:10:14]
So his hand is more function. So this is what they give you so they give you something they give you the original equation that satisfies this. This is a small and I am going to tell you how to get something which is a smaller. OK So on a smaller mark and this is something that I am going to keep for future use to use it a little bit later which is that if you give me this then you can give me and you are giving me anything that I kind of thing by taking you know I could take a square or so both sides I could look at Lytham so I could think signs I could do anything and if you give me these then you're giving me what they were a do you see legitimate operations right OK most of these things have not going to be useful right but there is one which is useful which is what they have pain if I take that even at the right so the only thing that I am telling you is that if you give me this in the initial conditions you also have to give me these Ok I am using a prime for the lady but if this is the change route right if you think that what the welfare with respect to K. I think that anybody will feed that this is the way to Matrix and this is one backdoor.

[00:11:27]
OK And this is someone back to reevaluate this we thought My God And this is the prime So if you give me these then you have to give me the source right because I just of painting from the thing that you gave me by taking that he went to use on both sides Yes All right so I keep it there with here so I could I called these I 40 body ends and then these the because it's what they have taken by thinking that he what they will feel about Ian's right.

[00:11:56]
To say merely approximate the baby and then so the new good method of what then you can method suggests is that you make a correction which is very a small in such a way that you kill off the 1st thought of that expansions in Delta then you'll kill everything in the 1st of them so you change the F. compose with K. you change came to K. plus Delta the 1st thought of the change that you have pain is this OK.

[00:12:24]
The 1st part of the tame that you have Dana will hear because this is of course a linear operate these right OK so what would suggest you would need them would suggest you is that your study this equation over here. So the compose with K. would be played by note that mind of the composer with the R. make a sequel to minus E. Now this equation then is hard to study because this thread I'm over here is not constant provisions so this is kind of the the they fit a very distant over here that is not constant proficiency of course we have several lectures here about how to study these things with non-constant proficiency by fencing one but.

[00:13:11]
But. So but let me not try to do it over here so because that is I think that you can do it over here so let me just try to see on the the the main think that we are going to using is we are going to use the geometry to transform this equation into constant confusions equations that you study with before.

[00:13:34]
You explain you these things these things that we were calling cohomology equations so let me try to tell a little bit of. Their genetic identity so this is let me just read to interpret these things over here that I discuss before this lady about the of equation in turn so far.

[00:13:58]
A little bit dramatically so this is more or less the range of K.. So the range of K. is going to be something which is it's a set of course right and then came Prime is the tangent to the so called normal I.C. in such a way that these things over here right and so what does that even the eve of embody MC question tells me is that when you take that band then it gets marked into the something over here to buy them up right I mean if I think of vectors the way that they think of.

[00:14:33]
Thieves is like a pair of points that are infinitesimally close right so when you have these 2 points in the circle right you have a prime and then they map them into K. prime. With big what we've got right. Plus no matter which I mean but you ignore the evidence because the evidence we're going to be a small right OK Now let me tell do so now here we started with the league so this is a very good thing because if you think about it dramatically what this tells you is that there is one big dog in which this matrix transforms in the right way so to speak Right OK So this is I would think to have.

[00:15:17]
Because of the invariance equation but now what I am going to do is the following I am going to find the perpendicular to the Cape but I may have normalizing it so this is the perpendicular which is the same as the simpler form except that I have right but dramatically just the perpendicular so I am covering these back to the prime and I am drawing a normal and I am normalizing the normal normalizing the normal he's a little bit funny but so I'm normal icing the normal in such a way that it's actually orthogonal but it's.

[00:15:51]
Not my legs so that the area respond by they do things the sequel to one yes so I put it here and they put their I know their situation is the following so there's this like that girl has said they have one OK because they have said young kid is what I am using my assumption they made of them up we have one right they make 2 of these rectangle will go into another rectangle and the rectangle will have area one OK now you have 2 rectangles on both of them have area one write on their base of one is equal to the base of the other with can you say about the height where you can say about the heights is that the heights have equal yes.

[00:16:39]
Right this is just eliminate that he. Saw this but only pipe is spot on with this then you have that this. Has to be equal to be composed with the Omega except 2 editors that depend on their own the embody arms right OK so I was telling you the exciting if the there is one right OK And of course they embody and she's not exactly right because it's bounded by you ever thought of you prime OK so the whole argument has several sort of order that he but I'm OK we will do them in more the right but this is the key ideas so that you'll find the frame in which.

[00:17:21]
You find the frame. In which there are 2 vectors that behave in a recent I will wait right there is one back there which is embodied in which is the key but I'm right and then the perpendicular gets my ped into itself plus maybe something like us here.

[00:17:39]
OK And there's these famous that I was describing at the beginning as it stands for Shia. So this is sort of doing all of this a little bit I mean I was doing all this with pictures but let me do it. So you read this made the exam and I am just saying they're not they show no of love in which you put the will he have a K. prime and then you put the will he have these V. and remember that that he was precisely the perpendicular but not my life right so you have to make those and then you can concatenate them and then you get that way to Matrix.

[00:18:18]
Yes And so now the thing that you have Thing is these things over here vs compose with the play by AM is equal to. The plus Omega and then you're right multiply by one of these just tells you that this vector over here the same body and by that the body and and this other one the sort of factor one that you get in the lot where they are going on is precisely this genetic argument that comes from the upper side of Asian Yes OK And then the cess is something which I mean you can compute it and I will compute the minimum minute by their seeds many of them because I mean this is quite explicit and then you build the play by feet up let's all make out and then you get the factor there right OK And actually you can put all the formula there then when you compute the average of this this is precisely the number that we have in my hypotheses right.

[00:19:18]
It's a not direct expression so this is the. So and now this is the think I mean so the way that you think about this him is that the same gives you a frame near the total where. Those became very nicely. So the thing that I am going to do is take advantage of this frame and then write my editor or something the correction that I want to do I am going to express it into this good frame so if you have a good friend try to carve compute in it right OK so you have so the thing that you little is that you write Delta is equal to M W W is the expression of the correction in the good frame.

[00:20:01]
Yes So this is just the idea and then the new dimethyl simplifies very that I'm not the galley So let me tell you what these questions over here and and I say mention these things over here. And there's more lever but these 7 are just because of the body and sick way.

[00:20:19]
I mean there is one lever that they have to track but they have comes from the fact that this K. plan was not exactly in body and but it was there ever was bounded by E.-Prime and the rest is basically algebra and geometry etc so that all the times the cover never of that those bounded by you prime Yes I mean if the if was equal to seed all they need would be everything that they told you would be exactly true is not exactly true but I mean in the 1st step I was making I never sees it but I'm right OK and the rest is hard to break expressions than what they were so the whole thing takes E.-Prime ever right and I will do more explicit calculations late that except I haven't They want to explain what the story so now let me try to tell you what is the idea so the idea is no you put up with.

[00:21:08]
Equal M.W. So the equation becomes the equation that they want to solve becomes precisely this right so this is this is Delta and then this is Delta evaluate to their feet a plus My God this is equal to mine a C. This is the Newton equation. I have not done anything here except this thing in there yes but nothing says stuff to get a little bit interesting because now I realize that this factor here becomes the so that a factor there.

[00:21:41]
And then that he's decided that. OK And then that is the set of ave here. I know the things become very interesting because then you get these same feet up less so make us a common factor and this had a value here so let me tell you why you have thinking that this had a W is going to be a small so the thing that I am going to do and this is what it's called The question you don't method the question you don't methyl these lists ignore out of the.

[00:22:11]
OK I will prove that this works and I will give us 3 minutes and I will show you that this is actually not so above but clearly stickily the idea is that the SAT is basically the same order of magnitude those the ever right I mean OK it's time but.

[00:22:29]
Right so our it is a small more or less of the size of the editor and that we do which is the correction and I expect that the correction We'll also be of size of. Of the other so ignoring this term will produce adverse of have so it's OK because in the Newton method I am avoiding I am allowing for the square so this is not the new dimethyl it has an X. that there is not the need the method the class next to that there but this Newton method that has an extra 10 will be extremely easy to solve so I pay off because I a throw away some small thing but but then I have 10 something which is extremely easy to solve and I can show you and I mean it that they will be able to solve it using.

[00:23:18]
The questions all right so let me just try to do that so if you ignore the out of every new term then you're left with this equation over here right which is what amps one and remember that explicit having S. is an explicit thing the only and any sort of an explicit thing east and explicit things so the only thing that they have to do is to find the.

[00:23:42]
OK. So let me tell you more or less how you do it and this is exactly the same thing that was happening to us in there in their lives to insidious which is a very amazing thing which is that you have to solve the 2nd component of the 1st and then plug it into the 1st component right OK so you have to solve the 2nd component 1st OK and then it becomes this the 2nd component of these things over here right so I you know you do to show that they have an A.T.C. though I actually have evidence will not be C.E.O. but they will be quoted article you small right but the Newton method of there OK there.

[00:24:26]
Are Thank you yes so this is yes yes you're absolutely right thank you very much yes this is a mistake here so this would be futile Plus I mean I think you think you're so good that the audience is more careful than I am already so you solve this equation and then this equation becomes these 2 equations right so I wrote the will he have the 2nd component 1st and then the 1st component the 2nd and then you choose to do did in the 1st grade so the thing is you solve this equation and I will tell you how to solve it then you plug it into the 2nd one into the 2nd equation which is the 1st component and then you do will be able to solve it right.

[00:25:10]
Then you will see why you can do it so what are you to do I mean just to look at these things I owe you to show that these one over here this is more so than sorry you do you that this one has seen all of it OK and then you see the thing is the following The but let me postpone this because it's a little bit more tricky but the thing that you have pain when you solve this equation and here I put divide I mean that's very good thank you for noticing it right so this one that that means that you do this one that that means that when you do up to a night it is constant if you remember the theory that we have we had to pay a price that we had to sort of satisfy some conditions but then once we can solve it then we have our one family of solutions because we could have a constant to the to right I know here comes the beauty and this is where you need that is conditional here so you see you have the you do which is the No no but it's part and the average part right and then you can choose their absolute rule in such a way that the 2nd half.

[00:26:12]
Of it and you can do it provided of course that the in thing in all of us is not equal to 0 right so you change the habits here you change the habits here and then of course the advantage of the right hand side you can adjust it provided of course that the average of fish is not equal to seal.

[00:26:31]
So this is their recent why you need these I would it not be well to Seattle and so this is like a veteran of hypotheses but hypotheses just in the neighborhood of the. Yes. All right so this is the thing so the 1st equation that that means that we do have 2 a constant and then you choose the constant in such a way that the 2nd equation becomes sort of a ball right and then of course you can solve their 1st equation after a constant and I don't care about this other constant That's a good solution so this is something that I have to tell a little bit so I know you do sort of so why this is true and this is the reason that the claim of as more of it is true so if you think a little bit about it what the same cedar minus one which is the thing that we want to do well is the projection of the ever onto the 2nd component in the frame OK And the 2nd component does not have tickle right the 2nd component is this medical but if you look at the 2nd component in the article is precisely and then you integrate their belief in full would you have pain is that if it is between their last of the 2 things right so you get the difference between the areas between the 2 operations here let me yesterday to say these things over here.

[00:27:52]
OK so the 2nd component of a so if you take if you take a curve and then you look at M. governor and then you look at the difference between the you look at the. Perpendicular component and then you into going through is precisely the difference of there yes because you have this area that they was telling you about the no flex condition that turns out to be the C.E.O. So you see this is I was laying out my god did so and so on and I get to use them in the when I am painted into a corner right.

[00:28:26]
OK So this is the the thing so here. Is So this is what you have pain that this is the genetic argument that the thing over here is if you think they develop the kind of component would be played by their ways and then you hear in the great throughout the new of thing the difference between the areas.

[00:28:45]
Which. Are should take the 2nd component think you should put the word here yes thank you yes very cool perfect this is what they want to do the people help me to develop all these so let me try to tell you so I have tell you of all the operations that you need to do right and the rest is going to be well we can go either to try to do numerics or to try to do mathematics and do estimates so I will try to do both.

[00:29:14]
I have like maybe an hour or so but I just point out that the operations if you remember all the operations that I have done are. Thinking that he but they use them but they think things have around except that I may have algebra I corporations and so being called equations right so I will go through this a little bit more you need the again but they're all building elementary operations and these operations are all of them fast either in 40 of a space really in the realist space right so if you have a function and you have a discrete basis for your most then solving commodity equations is fast computing that event this is fast sifting is fast right.

[00:29:58]
On the other hand if you have the function discrete Thyssen agree with then multiplying it by a matrix is very fast except that except that of write so. All these operations are fast either in full of a space already in or in the this space. And of course this is fantastic because now are these of course thanks to their fast food your thirst for having.

[00:30:23]
And for you this space is extremely cheap operation right OK But but they have a of pain also I let me just point out that as you will see a house of 39 you don't methyl requesting it the method that reduces the average square that equally I never had to invent the Matrix or even a story that you see so you need so they store it is going to be you know you have to store all the socially or even those right like W M except that I but this is just a story back those not distorting mattresses that's a really big difference because if you discount they seen a 1000000 operations it's only point is storing a matrix is 4000000 is not at the alien Right OK So let me just try to do a little bit of mathematics solve that out of this the things that we need to do so I have told you more or less what we are going to be doing right.

[00:31:25]
I have told you what we are going to be doing and I have told you the recipe the recipes seem to use this matrix am multi-player cetera so let me try to obtain estimates right so and then there are 2 types of estimates that I will of think what he says to mates to show you that I have gained in when I step I have gained and I actually have gained a lot but the in some funny sense OK and then.

[00:31:53]
The the the thing that they will do in me of the afterwards is I will try to sort of I will so that these estimates can be repeat the orbit I know but I gain and then you keep on gaining right so this is. And then this will show you the do convert to a true solution OK.

[00:32:19]
All right so let me just read 2 so I think 6 that are so W one satisfies. Me or say all these things about here so I am missing one page or you're missing one pate. But you're missing one page I don't know what's happened to the page or so let me just tell you what the situation is so if you remember what I was doing maybe I will try to put it over here so this is W. 2.

[00:32:54]
Let me put the word here of the case when I have that are you doing the good thing so this is what W. do satisfies right so if you remember they. They estimate that they have for their cohomology equations right I can estimate that you do in other Maine grow minus the right and I can estimate that by the ever by this each will be played by the MIT thinks.

[00:33:21]
And so on. So what they have asked him is that they have is that W. dual is bound the in some domain. W. 2 is bounded in some of the main broke my nose. Is multiplied by my nose they'll die out. Multiplied by the size of the ever and then our course time which is the norm of a minus one so I mean I will assume inductively that this is about the uniformly and then you will proceed to do it with here and then that only one is again solving our corner of the equations and then I again of thing that this one is something which is to the mind of the here I loose some other factor that the minus Sigma about are you still cover never with this type of over here so this is something that I have to.

[00:34:17]
Mention that this is one method that there is a difference between for the experts in game theory that there are several questions of K. M. Some of them then you don't method requires to solve to come all of your questions and some of them if required so only one this is the case where you have to do to come ology equations but this tip.

[00:34:41]
So anyhow so the situation is the following and let me also put a way that if you use could see estimates and you loose a little bit of the main then you also have control on the least of the value Yes So you have control of value except there OK So this is what we are going to be claiming I mean it could be a little bit sad because in the early one in the lab you do you have a little bit of bit estimates in the value when you have what is 30 minutes but let me just claim that OK so I am taking the worst possible case for the both components and that's the story I'm now.

[00:35:20]
Now the interesting thing is the following so OK so these are the only estimates on this have true for everything that they're OK but if you remember when you're looking at these then you have to consider compose with K N F compose with Kate that's a problem because if you think Bill Perry a small and this if you think Bill W. small the belt that will be very big and will take you away from the main OK So that's the thing we have to do which I have to sort of if you have these situations that.

[00:35:55]
Terroristic that do roll minus 3 that are in the main Over here he says smaller than the distance to the domain of the finish and to the very end of the finish and then you can define the composition or fair for that right in general and so if composed with case that the distance it's at least that's from the range right so when you change when you change K. into K. plus Delta.

[00:36:20]
The range kind move a little bit yes but if they move if the motion of the range is much a smaller than the distance from the range to the got compliment of the definition then then you're perfectly fine OK. And of course this is the condition that you can define the things so let me just put this constant over here because this is a bound for the and if this is an upper bound for the Earth that if the upper bound for the bases smaller is more than these then you can define the composition OK Now let me try to do the estimates and then you see this is the story so let me just read to the estimate so this is the estimate for the correction OK.

[00:37:06]
This condition that allows me to do these things over here so this is the error. Right so this is the editor of the newest upright did serve. Minus K. plus they'll evaluate that the Obey got right so this is the new lever and then the thing that I am going to do is I am sure stacked the nodding and I am just acting and having the same things right so this is perfectly legitimate So this is the ever right but then rather than writing 3 times I am writing.

[00:37:38]
Like I haven't done writing 3 times have writing 7. But they're a little bit the stupid Right because how your mother answers that acting Yes but let me look at this column over here so when you look at this this reminds you of something writing that reminds you of course of Taylor for your employer OK So all these 3 things here right are kind of a small because of Taylor fear him so if I applied they look feel and this is what I have pain.

[00:38:08]
And this is where I use the fact that they kind of thing these things over here and this see our side was telling you before it in both they say the only thing that then tears is the 2nd that what they will fair for in the main yes.

[00:38:26]
Has to be 2nd that what they think you need has to be secondary with the wolf if yes every think so so these 3 terms and then these so there are times that they pull her over here and I will there is more there kind of a small because I chose very carefully developed.

[00:38:47]
You see this turns out over here I mean this is more always but this turns out is more because because of the way that idea of everything right if you do all this all this consolations right there in these terms if you had nothing nor the terror that they were telling you out of that would this would cancel exactly but they was not doing that I was kind of sad I was leaving out that that I'm out of W. that they was calling out of the right but if you look at this this had a value is bounded by they have a square because our is bounded by again by I could see estimates with a factor that the mine is one.

[00:39:29]
Basically made mine a C. class so this is where the of pain and this is have any wasteful estimate they don't care yes because it's to the main on their right yes so this is the things that he have the only time that sort of lives he said about you so if you put together these tariffs and this is why I was I mean actually this time we have seen this some is going to be a smaller than this one so I just modify the constant and that's it you see I mean this is the idea of estimates I mean this is not worth carrying out very carefully this was the main interest right.

[00:40:07]
OK. So in summary this is this condition that we need to have and then you can take any step and the error in this is more of the main is less I think well than the things over he have to the mind of right so the every square that I think but this is not in their regular new dimethyl because it's quite that thick in the sense that it is it is bound I mean this is a little bit of cheating in some sense in the sense that this norm is have any a strong Nordman this not me so we can ignore them.

[00:40:43]
And then you have this factor here that it's a little bit of a very complicated factor because I mean if you lose too little then stop right so you have to lose a little. But you can not last too little because then this too little will become a little bit of of the system and the amazing thing is that there is just filled room to try to give.

[00:41:06]
To try to give the to try to get through right I mean it will be very easy so if you lose too much the main then the system has become easier but then of course maybe you end up with something that doesn't have any domain which is something that he's an expert right and then but you're going to steal of taint results they buy out on the other hand if you lose too little then you don't get if you lose too little then you don't get so if you lose too little then you get the name to fear him and if you lose too much sorry if you lose too much then you end up with an empty theorem and if you.

[00:41:41]
Lose too little then there is nothing that you can prove but there is some margin of error so let me also tell a little bit of things that they want to sort of mention a little bit I mean this is a little bit of. Thing that I have to discuss.

[00:41:58]
Which is that there are these figures that out of these constants that I was trying to use and I was claiming that these things have bonded by the constants like the norm of fame or the norm of and minus one but. Then he went the row of decay except that So these things also change when you do this step but I claim that they don't change too much and you come down the steps and then these things will remain so for example when you look at these that event these days that event these notes don't change so much because if you have that event the even of the of them up which is the the original vector that they have so this is that he went he was capers that he went the overall mine of the sea not that which are you really told you that you come down OK saw this thing over here then anybody will so that if they're betting was like a regular game betting that has that event this then that anybody will of the correction these not the worse right and also the fact that you can also the you have an embarrass of the length then it is also true because he's 27 here and then you can do the corrections and the M. also doesn't change too much because it's a not to but I can express you and so on that the case.

[00:43:14]
And the saying with them by no one so none of these things change too much so all of these things if they just have to be good they haven't ever which is so they get affected by things that are of the size of the editor of the every engro would be played by those that do a Power OK and I claim that you can choose things in such a way that this is not too much.

[00:43:39]
OK So let me just read to tell you that so this step then you say is the editor of whether that together or more the law the fire that you have to breathe this that we will factors and then you have to pay the fact that the normal gets a little bit weaker but then let me tell you that you can eat that ate everything and of course there are many choices that you can chew so let me just try to put the will he have brought J.s they're the main claim at the State J. and then.

[00:44:08]
What they loose over here I am putting over here I am losing by placing exponentially fast I mean people who play games like I mean you could be big leasing is lowered you could be decreasing faster and that is quite a lot of room to maneuver right now but let me not complicate things too much and then I put the word here just use the main scene an exponential way.

[00:44:37]
So this is just the case and so this is the main that I am claiming at this step J. This is the editor of that I have to say that the state J. So let me just try to tell a little bit thinks that there are. Well that out of these constants that keep on changing from step to step so the thing that I mean this Constance would have like the size of fermion this a cell phone minus one so do all the estimates that correspond to the size of firm which have.

[00:45:09]
Twice the original ones and then I will sort of show that these this and change too much and I can so that there's a fine a number of conditions on the need to lever that you that everything is OK All right so let me tell you what that was so these are the finest in it so you see that this absolute right if has some power worse.

[00:45:33]
This is the thing that they have developed that is that is you know one but they have the pleasing exponentially faster right so then you have this factor which is subsequent game I know one and then you have this factor which is growing exponentially. OK With the steps All right so if you look at did this is see the 4 to the my nose Jay is right I know you apply the next step and then this is what you have deigned right.

[00:46:01]
I made a mistake of what he had because this is steps he learned to the this epsilon J. minus 2 should be inside this epsilon J. minus 2 should be sort of about that so that's another mistake OK so now if you put the told to go in there so you see there is this constant here and then you get the constant again the square so this is one plus 2 and then you get to do the 4 Sigma right and then you because J And then you Because j plus one over 2 and then this is Jay that it's without being inside and then you get to the 2 Yes OK Now you go with this again right then you should do these things over here and then you see that the thing that happens is that there should be something that leads to the 4th power so you get a far 4th power here.

[00:46:54]
And then you get j j minus one would be played by 2 and then you had a minus 2 would be played by 2 a square and then this one gets another power Q. OK so if you repeat it a lot of times then would you're going to get these these things so we'll hear that which is this constant will be getting all these 2 to the J. plus one this will be j j minus one would think by the way to minus 2 multiplied by 2 a square and then eventually do that one they violate the 2 to the j minus one and then you get to the 2 to the J.

[00:47:35]
OK really horrible because you do how all this exponents that are growing you how will this explain is that ever growing you how old this exporting is that are growing but that but now is we need to do like a little bit the fatherly good calculation because we have 7 things that are growing and 7 things that are vaguely sing and then we have to so that they would rise with the rather dice so what is this what where is the recent Well there may no salvation is these things over here which is that when you start shopping all these things this is something that you learned in kindergarten these are some of the you may think serious and this is a listen to to the days right.

[00:48:14]
When you write all these things are going to hear so this is J.J. minus one day to the to the day minus one you can take factor 2 to the J. and then you get this is this factor of 2 to the day and then you get something with the 2nd but it insidious and actually the sum is forceful then you can do the things.

[00:48:34]
OK so if you put all these things together so what you have 10 is that J. is less an equal than these but then Grace to the power of 2 to the J So if you take for example absolute 0 which is sufficiently small so that the 10 mean by emphases Cedar Point one then you get that this is Cedar Point one to the 2 to the J So which is very very fast this much faster than exponential right because it's exponential of exponential Yes.

[00:49:04]
Well when you have exponents has to explain and sell them so. Sorry. Deborah I J Remember that they were choosing this is where you I was putting I was thinking here then I J Which is equal Delta to the mind of J. Ok none of them be sort of novel apologizes things and that's why I was getting these these things you will hear of this is that that and then this is this fact or.

[00:49:31]
2 to the J.. So this is the thing over here these 2 the mind those 4 Sigma Yes So you see the point of these things is that when you get all these things the this is the factor over here and then if there is sufficiently is small right and then the conditions do taken a step that I was taking I mean these conditions that were bothering me where basically of the 4 epsilon J.

[00:50:04]
J 2 a power right but then to Jason exponential right so then to Jason exponential. To a power it's also an exponential and then this is an exponential of an exponential so this one beats everything right so if you assume that sufficiently a small then all these conditions that you need it to do they.

[00:50:29]
Get to be satisfied that all the steps right are more able they tend to see in these things that they was calling the quality quantities they're all bounded by epsilon de the power right and again this is going to be a convergence serious I mean 2nd Vatican City is which is many a smaller subset goes to 0 so that you can satisfy all these conditions to be less than twice the original quantities by imposing extra conditions so you're going to have like maybe 6 or 7 conditions and then Ciro right that it.

[00:51:08]
OK so one condition which is the most important one which is that these number is less than 0 and then several subsequent conditions that the you that the sums that you have to do to is to made the corrections. Of the quality quantities are going to be smaller than that Miss.

[00:51:29]
Is more than twice the things that are here so let me just try to do a little bit again more or less or build these things and then let me make a few remarks about the delegate of him. So what operations have been involved when I gain if you remember would we have been doing these constructing the might the exam and solving come all of the equations except that or so we have been doing compositions in but assess of matrix and then matrix times functions sifting solving come all your questions and then things that are being untruthful acting or multiplication by a scalar.

[00:52:05]
So there are these operations and I will go through them a little bit more carefully the remark that they are making is that they view this because they see nicely the points that they wrote over here in Blue Shield the systems that are in blue are fast if you do them in I agree the points and then the the steps that they rolled out of 5 to view them in full you have coffee since Right so if you have at the same time for your coffee.

[00:52:32]
If you have at the same time full of coffee since I'm getting AIDS which is now why they see more than computerless is not the extremely expensive because you have the fast food of transform which is very fast in theory and it's even faster in practice because. Many modern computers have a special hardware to do it or something.

[00:52:51]
Is one of the most up to myself going after so it's so these and Logan is not the theoretical estimate that. It's an understatement. OII done absolutely right they changed yes thank you so thank you so yes so these things over here and then of course I think the answer is through acting and multiplying by your scale of sufficed in any way you're like OK And then for sysadmin you can get.

[00:53:30]
So these are quite a sort of that in operations and out of it so let me just try to right there go to him it fits in one pace in you see there not that many steps right so the 1st step is to compute if SEATO except that then that he had I put a survey of these in reality based on blue is fast in space right computing the case you know then this is the normalization which is I mean I was putting all these things so these are the things that appear in the things and I'm here I put the extra things that intermediate the steps that you need to do in practice but they don't appear in the theory because in the fury just like in the expression but the 1st thing that you need to do is to compute the editor and the editor has one part of that is fast the info into your list based on another part of which is fasting for you this pays then you compute that what they've to compute the normalization by which is very fast if you do it.

[00:54:24]
In the early space then you form these made the exam which is I mean they this is of course forming a matrix which is fast anywhere but this is how multiplication. By number so it's fast and realist space right so that's why they put their thing in reality space then you need to compute their Even better of these metrics but these men think C C Not good either so you have to compute them best sort of 4 by 4 meant that he says enough of points right so this is fast in the early space because he's multiplying 4 matrices and actually they have made this issue of that that I mean on one so.

[00:55:02]
They didn't tell you but this is a metrics of that that I mean and one so computing the impetus is really just moving things around. OK so this could be fasting said then you need to sift it and then multiply the ever over by there a minus one then this is says which is taking these metrics and taking one of the elements and then this is solving the equations and then you put them equal to that of you so you see this is something that can be feet in in like one pate right so this is something that requests and all these are steps are something that if you have a sufficiently high level language then they can be implemented each of the steps can be implemented in a few lines or you need to write the function and so on so the whole thing for example I can discuss this a little bit in Octavia it takes like 200 lines to do it the whole things I mean is something that you can do in more ways like an afternoon or a couple of them tried and I mean many of them is would keep being accepted as you know the core of their good if it means maybe like 2 lines but then you need to keep being defined things this is global except that except that I set up another way of these things so this is a story so there are very famous I mention if you have a high level language that can handle vectors full of transforms so on and storage vectors so this is very easy.

[00:56:29]
To do list and if you discrete base a function even. In N. times they start it is sort of then on the operation count the sort of the end again and this is. So and then is by that the gully converge and so on you don't need to store anything so you see that even with the most the stupid laptop the you can have right then even like a 1000000 quid fish in 6 then the precision is not a big deal so I mean I was playing around with your laptop except there and so on and.

[00:57:04]
Even with $100000.00 or so they need merely seconds one step so. But the most important thing in my opinion is that this fear Him Actually there is I mean if you want to do our theorem. And I'll go with him which is. Just where it seemed then where everything works very well.

[00:57:26]
This is. Straightforward if you want to sort of do something which is much more challenging like going into an area where things are breaking down or where things are moving etc Then of course I mean the difference between a good algorithm and about algorithm is that they will go to or they will implementation is that it takes the lack of every.

[00:57:49]
Day sync and you're not discrete ice into my text at the exit there so you need to be doing diagnostics right and that's the difference between a professional quality program on a. Program that you just play around with right so playing around with and doing things in an area where things were well is extremely easy but then if you call thing of programs and the high quality programs is that they need to they know themselves when they have it working well and when they're not working well in that respect then I think that this is one of the bigger vantages of these these methods is that it's backed by a legal theorems that give you the quality measures the things that you need to monitor and you need the money thought of these norms and you need to monitor all of these things Blass of course money toting gold happens when you have to run case you're never on when you have a round off error 6 at etc but that least everything is in principle quite doable etc.

[00:58:50]
All right so this is more or less what they wanted to say about the theorem except that I was right and the last. Maybe. 15 minutes or so just give you I don't know if it had any questions or so or we put all the programs also in there when that.

[00:59:09]
So I can tell you a little bit of more or less I want to give an overview of the theory of you can take this as an a starting point but I mean the main idea it keeps on you can play that round and then do quite a lot of things and I put this to 30 L. is getting to grow to find the it's almost 200 pages now and I hope this summer I will have even a few more but they do things they follow and so let me try to tell you several extensions that you can do you see in the same circle of ideas more or less.

[00:59:41]
Which is the falling so the proof extends to have dimensions denotation is exactly the same except that rather than taking area preserving You need to take the matrix Ok saw this matrix who has this sadly the same form of the rotation but there is something which is called simplistic geometry which is something that Mr Saliers but in many of their actions at the same time so to speak.

[01:00:08]
And if you use if you're sure that the thing is that they might be simpler the gun that the map is. That it preserves this multi-dimensional area so to speak. It and there has exactly the same expression that they were doing before like cedar one minus one and it's even more general than that if you apply the same matter so this allows you to do what they should in multi dimensional and I'm going to favor mention so you can do it except there are so that's also true and then also this formulation that they call a posteriori in many fields of K. in theorem then you see that the joy of pain things for questing think I will systems.

[01:00:53]
This person that they gave you implies the version for the question think I will systems the only thing that you need to do is to take us an approximate solution of the 2 system the solution that you have trained for the quest figure I will see for the in thing level system so they in think about this solution that you have pain for then think I will system is an approximate solution for the question to get our system right or their belief system things etc and then well then on degeneracy conditions can be weak and there is something called Richmond's who fish in conditions that apply and this is something that these throw and then there there the here I am also there is something that they didn't want to discuss.

[01:01:39]
This hearing gives you local uniqueness of the thought I mean this equation that they were holding all the time never has unique solutions but not having unique solutions is actually a feature not a bug. The is the thing is the following if you change the origin of the parameterization then you've taken another solution and you and change that or right.

[01:02:00]
You can see that if you have that these things. If you have that these things are equal if you compose on the right by any of rotation then you just can make that a patient's jump right and then you have 3 in another new solution but it can be proved that this is the only known uniqueness the that exists in a neighborhood so that if you have 2 solutions that are close by then.

[01:02:26]
One is they sift of the other. But they have the same totals let me just add those who were here so that you can also do other things and this is something that I like saddo on. Underground. A little bit which is that if you want to the top of this is interesting I mean this is was something that was motivated by plasma fishes like feel more or so on or they will cost you that you can impose I mean if you have these signing a machine or something.

[01:03:01]
You have a lot of parameters that you cannot just write you cannot just they say prove your magnets or things like that and then you can doubt they thought I with extra properties that make them more robust etc And this is something that you can also do so you can express proper of these of the thought I that I decided I will.

[01:03:22]
And then you can take but I meant this and of them on there is also something which is theoretical which is actually quite useful and now you're going to give you a theorem which is that you have staying power. That out the medically. Dependent some but I meant this on the frequency right so this is something that So for example suppose that you have to there you frontin numb you frontin frequencies right and you have a solution for one well.

[01:03:53]
And then you tried to check whether the solution that you have 10 for one frequency is up it's an approximate solution for a nearby frequency so you have 2 frequencies that are close and then you have a solution for one of the frequencies so what they claim is that this solution that we have for Omega is an approximate solution for my god.

[01:04:15]
And the reason is very easy so you just algebra right and the reason he just told you were because you put the will he have this is the embody and situation for the make I feel the way but this is K. you'll make that this is because this is a solution for Omega and then you have think a Omega but in Omega and then make I think that and then of course these can be estimated by the event the in the play by your make up minus omega 3.

[01:04:45]
It's extremely easy so what do you have then these that this is just. C. way there bro. Over here I put over here the mind of Sigma then signal mines one to make a minus from a got into and then I apply in the post that really theorem and the uniqueness of the of thing that these 2 things are that the distance.

[01:05:07]
Bonded by this time so that the solutions depending Lipsitz way from the solutions depend in the lips is way from the frequencies. So dramatically they might bring that goes from Kate to. Being from the set of the you front the numbers with uniform Constance to imbedding and then you have pain that you can because it slips it's a nice close to your identity then you kind of pain measure estimates.

[01:05:36]
So you kind of thing the set of thought are you that the of thing it's a set of measure which is very close to fool if you do this in the case of in thing I will mapping sign your play these theorem then they know dramatically that this is the do go where are an area which is proportional which is basically you loose and they have Epsilon to the one quart that by where.

[01:06:01]
This is not optimal they have to multiply where these are constant tapes along through the one half. But this method doesn't give that. At least with this allays you have another 3 gigs that are so this is very good but they still tells you that they think of as positive measure and actually the measure cover the 70 thing and so on the proof can be adapted to find the difference several systems.

[01:06:29]
There is something which is. The thing is the following and I just going to give you the idea as I know that there are several people in the audience that experts like you young that you can. If you have difference have all systems then you can do either 2 things one is approximate the differential assistance where you analytic one sort of Superman the Newton or stay with him there's nothing in step so let me tell you one lemma which is an extremely useful lemma that was prove the originally by Julian motive but then that I have better proved by by the sender a little bit later so so this is a really useful limine do we allow analysis so that the function is out of place self and so this means our continues that what they've said and they are hold that it's see it's equivalent to saying that there exist a sequence of functions this is a function of 7 analytic but of the creasing domain and then you put the will he have this very small estimate so you put this system is that the difference between them is decreasing like exponentially fast on the exponent these basically there are minor sulfide which is that regularity and then they come but about over here and they have to model.

[01:07:53]
That you can put all the sick we will into these 5 your 5 mine or solve and then if you have. A problem like this you seen this lemma in real analysis so this is a limited real analysis that if I go teaching. Your analysis I could teach is in the provision of.

[01:08:10]
Very difficult actually. So if you have a finite lead the French have a problem is then you can get a sequence of finally to problems and then their idea is very simple because you think there is exactly Lucian of one as an approximate solution of the next right so you get a sequence of problems that that approximating these things and then everything matters perfectly well because when you get there's an output is that the solution is perfectly.

[01:08:39]
It is move so you see because the mission of one is the this is so this is the sequence of problems and the sequence of solutions will be satisfying more or less these but then will be played by a power of. Up I would have to show you have pain things which have less regularity and once you have this analytically sold if you use this.

[01:09:05]
Regularly dilemma that the. Income of money cannot then you can so that then a little resold in an upper studio before might imply all dramatically finally the financial results so that's the story and so on and then I get over things like we have thought Are you really think that I know that Jim at the gong thinks with similar things apply so for example many of us many of you have been hearing about formally simplistic systems or pressing black think and so on and I think that that's all that they want to say so I mean there are many things that need to be done that and then if there are people that they're in their debt to gigs at that week and try to do these things and what they were right out of course then a medical analysis is something that of course requires.

[01:09:58]
A lot of effort but these broader Seabury faster so I was telling you thanks to all these. Methods then Nowadays of course in one dimensional problems I think it's this thing that was can see that arson and practical problems like of painting estimates of the sizes of the of this then is becoming like you know it requires muscle and need to an important thing but nowadays in many this problem is people know they're great value of their breakdown with 4 figures pretty good so it's not like completely if you're at the Cholesky MIT You know it's something that can produce even for for yourself I.Q. or C. which is pretty groovy even for Apply people.

[01:10:44]
And of course if you ask me what happens in things like in 50 there mentions that it's. Still in the future but things are getting more and more applicable and in my opinion this thing allows you to take the Qayyum from a baby theoretical thing for crazy mathematicians to something which is actually quite the public of all and it leads you to even if he she and then go to him so OK Thank you.

[01:11:18]
Questions. They're difficult thing is not the number of the mentions the difficult thing is their number of thought the mention of the daughter so so actually for low dimensional thought are in infinite dimensions then the answer is yes. And it's even been done but infinite dimensional array that.

[01:11:50]
That's kind of I have read then something with the Kenyan exceed about it but the systems. And then there is of course be the hope next year having an expert at the P.D.. So he will be here next year so we will hope that we will have some more activity and we have last year also another one so.

[01:12:21]
So wait a couple years and maybe we will see but don't absolutely right that that is one of the places that we would like to see more it's absolutely correct. Yes. Our. Place simply think is something that this fire is. So. Well simpler for me simple can you see here so so so.

[01:12:56]
So let me put the word here rank of 4 May equal in equal the mention of a space let me put to work here let me yes let me just make it a little bit more clear. So me yes start again. So seem plenty is that if you have or may not of be solaced with the will here or make a simplistic or make a U. e equals C. No for every you implies.

[01:13:45]
This is the thing and then you have. The Omega equal to see it so this is the simplest thing so there are these 2 conditions for simplex think in their place implicated case. You think a way this so in the press implicated case take away is very out of course let me give you one example.

[01:14:12]
So one example of how pressing planting for them will be like you think is you know wired minus one C. No And then you put the would he have even more seat of columns so this will be a this will be an example of a place in Plec that things so fast so neutrally spaces solves why is this a satisfactory answer out of the people they are happy.

[01:14:51]
Yes it is yes yes. Yes that's what they were sterling by adding But I meant this for example and then with these servers mine conditions that are a little bit more sophisticated methods for example if you can have but I meant that if you have like a week month Easter something yes.

[01:15:33]
Yes yes yes yes. So you use it but I mean there is space and then you can use a finite dimensional implicit function Fearon to adjust that and you can use the frequencies Yes So we've wrote several papers about this one to store a year something which of course.

[01:15:56]
I want friends. Feel and they were very happy with the. 90 stories so this can be done but maybe it will take me a little bit longer to explain so yes. I mean they I wanted to give their full proof for but but they put the assumptions that can be but again no way but just to be able to give a full proof in like an hour and a half so I needed to do this but yes so you have absolutely yeah so any other questions No Well I think usually.