[00:00:05]
>> OK thank you very much 71 of the 4 coming here saw what I am planning to go to the loo or do lectures and try to get their complete proof of from Best himself a theory. On so. All these so K.M.A. stands for Comodo of Nolan motive who were the 1st ones that developed all these have a human more or less between the late fifty's and early sixty's.

[00:00:37]
And then since then it has become an important tool in in mathematics and it goes beyond what they will be covering is there but duration of question here the core of it so I will define what that requests a penny of the good bits but the ration of pretty only quarter beats is something that we've even even there got a scene and there are other courses because basically it's a fixed point and then detained of a fixed point the only god which have more complicated but they're more interested interesting because they cover quite a lot of faces space but that's also what makes it kind of there so we will discuss what all these means so they way that they have prepared these East lectures in one lecture they will cover their fundamentals and their background I mean the theory you will see it requires geometry number theory and analysis.

[00:01:27]
And so I prepare one lecture which is the background no the lecture which is the actual proof so if you think a little bit of a if you if you like there James one more piece so this Fed lecture will be like when danger one goes on besides Q. and then the U.V. molding idea to set the next set that out right and of course it doesn't make sense I mean you have our lease of sewer and then you have a blog that explodes accept it or but then when you make the plot then they will come in the right places in the plot late that right so this would be the 2nd part which will be to morrow very So do they I'm going to tell you all sort of mathematics that would more or less known in the 19th century but then bringing them together was the genius of God Morgado final and most of it OK so there's this story so maybe there are references I wrote some set of notes which had to lead them to be so they would publish the proceedings I would $969.00 that to me stayed 69 so they were published by the A.M.A.'s and then that he said that they did very soon in my home page.

[00:02:36]
And I had to tell him that all of this is basically I've referred you do there so these are compendium of the lectures but they tell you go to the you know lectures go to the original papers except that are in paper there are many other views. OK so I said I mean so we're going to try to go east today East goal but of the analyses the number theory and the German theme of getting bowled we do all this proof and then late there I have to tell you that and I will also try to quote a little bit of the motivation.

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In the medical implementations except there are so we can try to sort of discuss all of these how little of it all right so let me tell you a little bit I mean so the definition is the thing that I have seen that they told the 1st definition which is a little bit subtle is what this whole question of the function so you know really functions sort of basically things that have 7 frequencies.

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So there are things that have set of frequencies and then so basically are things that can be expressed as a 40 of C.D.'s right so this is the way that they write the 40 of C.D.'s But the interesting thing is that these things would he have has had he saw that out of the frequencies and then you're expending in functions that they have similar frequencies you are combining then we think the yes and then you do the I mean if he if he was equal to one good B. does the normal 40 have C.D.'s right so would be the norm one for your C.D.'s and then you're multiplying by T. but then I am taking the and this is what makes it the least that we've heard of it as we will see that in a minute.

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OK And now the thing that they want to point out is that you can vital this more concisely and this is what gives you barley is the geometry that if you write this function K. your feet that you do right feet are for the side of human thought here you can buy this S.K.U. for make off the web get your feet that is precisely this function over here so you can think of this as a function on that those case a function and it thought it was because 3 days a function in the D.V. mention of dollars and then values into L. right and then when you look at them a got the ease up you got round in the totals right the totals you think of the total square with this I say then to 5 and then this function is like in the all day be the 0 games when you hit the side then you come from the other side right OK so in the or eliminate that to be the games that some of us played when we were kids.

[00:05:15]
So you had all these things and then this is defunct and so these have of course the Premier League functions and you see the question but you know the functions of functions I mean they're much more interesting than purely functions purely functions just go with our circle this one's kind of feel.

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The mention out of Toto's but kind of then slowly play itself is this is sometimes called is the least as you figure sort of something like that right so you see that they have up around them to be gross I mean when you look at the makeup the then the combinations of the angles can be in any place if they are make us have irrationality and relate.

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One is welcome bench in these and you can always assume that this will make other gays not equal to see you know for any I mean ice would have said that gays different from Seattle because if they where then you could just express one function in terms of the one frequency in terms of the others and then you could be using less.

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OK Is all this clear debris will the new hold these more or less all these have course functions. So this is the picture and I so this is the big tour that. This is there so to speak there but only question of the dollars and then you have a motion looking around and then in the realist space this gets the form by the function K. and the function K. gives you a real daughters which is more complicated and that this they seem to say though the function so people talk indistinctly to embody and thought I what about functions.

[00:06:47]
This is basically the same so in body and thought I thought what they should know motion is the same a 2nd video of the function OK you could be having embody and thought I either have more complicated than they nomics but people use the may mean body and thought I mostly do to mean all these things OK So let me also discuss some of these could time because that is I mean people sometimes when they normally go systems talk about differential equations sort of these clepe time as you will see that's much more efficient to talk about this because time solve in this good time is the same only that.

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These clues by the world rather than being a continuous function and then you're going to write this up but in fifty's but you have a sequence of points that evolving and this sequence of points indexed by N. we in these these 3 time. They can also be expressed as a 40 of C.D.'s but there again is the same if I mean this is just the previous lie there go back to the previous is like is just that.

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You can. The previous is like the only thing that you lose that you change and on do change when and then you have been exactly the same thing but the most geometries a little bit different because I mean you're making jumps and then the nice years like cutting things except that I mean to jump is not that continues lines or you have to change all my documents around but you will see that this is but the 2 things are equal and in many ways and I will describe a little bit better so this is the picture of though goes into this you have again there but effect autos then you can buy they didn't do this base and then these lines that were kind of distribute there then jumping around a jumping only slightly more complicated ways these things make a bombs and make up things except that are.

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It's a little bit more ugly than the perfect Dodos they have raped it's a little bit more complicated the right has bombs and so on it's less shiny and. Regular But the course leads to the bomp 6 at that are out of would make it possible to reuse the case of 6 and yes saw this video of the functions and then so the main question though we have this is the there thing.

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This is the main question that we type of we have this is suppose that you have a system that has one of these things and I will give you some examples of systems that have these things floating around. You part of the system can you find them all that one in the prototype system right so this is the goal of part of the relation theory and we will see we will do even a little bit with them so that if you have these things.

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When you change the system a little bit I mean and this happens in physics all the time because you don't have the exact mall that I mean typically you have approximate mall then and then you get the more refined products you may see and so you keep on having extra terms like either north or like D.V.D. order or some extent I faked except there and then when you have this system faked will just a question of the function persist OK And so this is the question that we are going to be these cussing and let me give you for the financial equation so I will just try to tell you the main thing that we have this is one important thing that.

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Maybe before that if you have the financial equation that meets the exclusion if. If the case satisfies this equation over here right so. This term over here is there that event he was gay when you're moving with the frequency Omega in the straight and then this is the differential equation that we have Having right so if you look at the picture of the.

[00:10:52]
Way the thing that you want is that these lead the lateral here corresponds to THE LEAD the lateral through the differential equation so through the embedding K. OK So there's the. This situation and then for these good dynamical systems. You know the solution is he says K. you know the gives your question pretty unique solution a frequency Omega if and only if the case satisfies these equation a bit here.

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OK so from the analytic point of view what we are going to try to lure and this is what. I mean and I least sort and when it collapsed at least they like functional equations right then and you these could face them or your play fixed point theorem sort your play.

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And I leave the techniques then this situation that they want to describe is that suppose that you have a set then they have the meets one of these solutions if you change the rules for me to another solution of the same form Right OK So we have I mean all this is so little read I think.

[00:12:03]
A little bit more precise formulation of the problem right they have explained you what that question of the solutions and then I have reduced the problem of existence to of course you put your exclusions to this solution of certain functional equations for a. I mean this is progress in the sense that we have formulate the brawling kind of precisely right and you can see that this is going to be like OK so let me give you some examples and this is one of the main motivations they saw this is.

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An example which I think it's one of the 1st brawling seen in mathematics really that it's the standing it's the of right it's a little bit of a very simple problem I mean people I mean from from ancient times people were describing people to realize that there were these things called planets and the plan is have a certain frequency so the planets cover certain frequency.

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If you ignore the interaction someone there but 92 of them are updates with a certain frequency of I mean people who are measuring these frequencies even from. Thousands of years ago and then when you think about the system consisting of their employment it's OK with only 8 planets right.

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So you have something that is what they doing with N $88.00 frequency so each blood if has a face which is one of the Court of the units and then each of the faces is rotating with 8 OK And then the question that you see I mean but this is of course an approximation because don't we know that the planets accept the forces on each other except that are slowly that I reach other Gracie except that Argon you're still fine.

[00:13:53]
Solution C. in the blind it was you take into account the interactions between them I mean of course these would have a hard problem in their. In their 17th century except that I because people said wearily it is how those sort of theological implications I mean of this all of system didn't exist for over there.

[00:14:13]
Then all sorts of theological implications were coming into play etc So people got very edgy they to them about this problem. Right and actually we will see that density 16 be complicated I mean I will not try to describe them over here but they are serious terrorist event that the they are not so complete.

[00:14:34]
They're a little bit surprising and I will not be covering them for the moment but I'll just point out that in the case of 2 planets Addinall proved the 16340000000 of my sis and then even in planets then you're stealing need the assumptions of as more masses it's how much more recently sold and that obviously developments going in there I anticipate the you cannot keep the same frequencies you have to adjust the frequencies and then there is some mysterious thing that happens in the solar system in the planets in the forces which is one of the big headaches which is that when you look at the planet the only one frequency but its will have to.

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And this is one of the big sources of HIV AIDS etc And that's why these this and apply so anyhow so so let me try to tell a little bit of an explanation so in this lecture I will be discussing and in the next lecture I will be discussing more they're discrete time case.

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There are 7 other reasons why nice of course that I will give you at the end there not a human saying I mean once you have the results for mumps for these great but I'm then you kind of thing that are sold for flows and this is the vice that people have seen before I mean clearly to lead to be more complicated but this is an idea that was due to bunk earlier that some day I mean that you cut the flow and then you're going to see the return mop I mean many they now he says no the stick but the result for mops imply that he sold for floats the other ways a little bit more complicated he talks with the robot and now that is an important thing is specially for that going to come things which is that once you're a starving discreet dynamical systems sound with their mops then you're using less dimensional objects so from the pedagogical point of view you seem less dimensional objects it's nice of thing because they need to allows you to make big into their mentions and pictures in 3 dimensions and you will see that my pictures into their mentions have a really pretty about their picture Cynthia mentions will be horrible so it's a little bit all these things and it's also computationally much more efficient so this is something that we have been discussing with several people if you take a surface of section and then you leave the complexity of representing an object good old very fast with that I mentioned grows exponentially and exponentially with heavy gets peninsular So if you cut there I mentioned by one then you can get things that are much more efficient and then also I will point out that the mathematics had a little bit easier because you see the.

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I can make their Owings by comparing points rather than comparing 2 points and the victor field so it's a little bit more complicated so that we will discuss mainly then these great time but I say I'm pointing to you. The results for maps in play that are sold for floats and actually they computations have a very good so let me try to give you the motivation and I will try to give you several examples that you are the result cannot be true unless you both have it are some since right so what they want to do is to give you several examples The motivate the assumptions that they will be making right so I want to motivate the example so let me try to tell you this is sick kind of the easiest example of.

[00:18:07]
So this is very deep because of the things that you find in celestial mechanics in making mechanics you have one body of will which is one goal essay was telling you to face but then there is another body of will which moralist tells you how fast you go or something like that and this variable is not the face it's a real number right so all young going to be gone see that NG are they numb ecosystem.

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One body of will which is how mangled one body all that it's a real number right. There number of the dynamical system is like these right so X. goes to express why and why goes to Y. OK so if you think a little bit about did you can imagine that this is a system described in the ceiling their way so a circle which is the angle and then an interval which is the ceiling then right and then the height in this is in there is they why and then this is preserved so in their the scene think it aids in 2 circles like that so let me lay and then do what they in each of the circle Jude rotate and then you rotate at the speed that is proportional to the velocity OK so.

[00:19:22]
All this is very similar to what physicists call in thinking I will systems there you have a concept of quantity which is that Y. and then the needs of the level surfaces of the constant then you get that rotation and that rotation maybe fasted or a slower depending on the value of the concept of quantity which is the why OK So this is an in thing and I will system.

[00:19:45]
This is what they put there would he have still the orbits of the thing and I will systems had a very simple do take X. plus in Y. after you eat that 8 right the San Ysidro. Descend do you do have to take it all more than one because it's a circle right so their faces have these things.

[00:20:05]
So let me just try to make an is more so this is this is then that I am going to be considering and of course if you pick your favorite He number why right then you get anybody on circle that has these frequencies right OK So let me now try to make up at the base Ian OK my part of the ration is going to be that they just multiply you by one man of steps he learns So this is already a small part of the ration right so this is already a small part of the ration but if you look at there is that eights right if you look at they did it's well this part of the wood here is very easy to take this part of the wood here this one minute eat to the end why Right and this one over here have their feeling I leave it blank because but because it is not so important but you see the thing that happens is that this is the action of that we have that you have a lot of embody and things that are what they believe right then in collapses and then all of then go do well to see it right so we used to have a lot of put to raise your right so we are sorry we used to have lots of people in the quarter beats with all sorts of frequencies right but then we make an is more but the ration and then almost all of them they out.

[00:21:22]
Yes So there is no hope of getting persistence of these things unless you put some makes the assumption so let me just try to say that one just 2 examples like this then what we will do is you can see there are things and they preserve some area right that mean the thing that happens is that this has some squeezing down of William and then everything goes down to the audience so let me ask you that these things so we need some assumption that excludes example one right that their faces place volume constructs on these that the police have said yeah and this is an assumption that happens to be correct in many of in celestial mechanics I mean for the people that know these then this is sometimes called the theory right so we will see another legal theory but this is the you will theorem that there is you that face is based volume is conserved in celestial mechanics and in many time in Ternium systems so these assumption is correct yes.

[00:22:29]
OK microphone nice OK now but no you. Are but there is somebody saying that they cannot hear so. Some some somebody from Jet Propulsion Lab is saying that they can. OK. OK So this is an assumption. Fortunately this assumption is very natural in celestial mechanics so we had a really happy to make it but but you see unless you make these hypotheses then that are made easy counterexamples right so let me give you another example do.

[00:23:16]
And we teach you make up at the base and then the thing that you do is the doing please they why by epsilon brain so you think epsilon 10 to the minus 10 and then you get to these things over here. That these things are will he have the keep on increasing SOL with all due out of its escaped or what is infinity re this thing over here.

[00:23:37]
Well I pointed out that's an exercise that they put over here of these blanks right so if you want to get this had an exercise fill in the blanks except that or so this is an exercise in summing Jamaica exist I mean things that's not there yet all right so we can exclude the likes of for example do but imposing another condition which is a little bit more complicated to explain but they tried to explain it now over here so this is something that people call.

[00:24:08]
Seed only in flux or condition or people call it also see. The entire They mentions OK So let me tell you for example what is it what does this thing over here mean so you take a circle over here ever right and then play them up OK So this is in the cylinder so so you take this circle so you take the circle over here then go play them up and then you get something the goes up and goes down so some of the points of the level of the circle and some points go below the circle right so you take the area.

[00:24:47]
So you take that Legion below the circle right and then you count the area of the things that go up and you come their way of the things that go their own and then just like these 2 A.V.'s OK And then I require that this area that these different see SEATO So the same area of them goes up in the same city of the sound OK so you can leverage in that this is like.

[00:25:12]
Like this is like a pipe and then you're flowing it just flowing what they are and so on and then you see how much and what that is how do you date there than doing all sort of crazy things right and then you're going to measure how much has to go in.

[00:25:26]
Might go. Out right and then you require that they have seed of flux Yes So this is what people call this you know mean flux and again I am pointing out that this is. A natural conditioning in celestial mechanics. Well you're going to argue well but I mean what happens if you take another circle if you take another set of coral if you are presenting it then this condition doesn't depend on the circle that you're taking because I mean if you were thinking of the circle that goes up a little bit of both then you see that there is something that used to go up on go down but if you take the difference between the 2 then if has to be equal to see you know if they're processed.

[00:26:16]
I mean if their volume is preserve and you have no flux then it doesn't matter where you're measuring their they want that right if you want to measure how much what they would have you use do they can this really pull right so you can measure of the flux seen one part of the pipe or you can measure of the flux in another part of the pipe and they should give you the same.

[00:26:39]
So that's why it doesn't depend on these same people. So let me give you another example which is kind of I think that this is a very interesting example and that's kind of them what the weighting example rights are one example that satisfies the 2 conditions of the 2 cases so there is this formula so that if you take the interesting thing is that if you take why why goes into why plus epsilon be prime affects and I put the word here to be prime or fix and then you can see that this.

[00:27:13]
Is satisfy so to magically the scene I mean flux property right because and B.'s happy view of the function so if you compute the flux or with any line it's going to be the same thing at all of the prime and then taking all of the prime the sequel to see though right.

[00:27:30]
Because it's a preview of the function OK so why get this place into the long blessed by primal fakes and then the 2nd pointer will he ever write it just takes eggs to eat then why that you used to have here so would you have is that the Fed she learned by these 2 things one is sort of shifting up a little bit and then this is just the standard to east that it's the in thinkable mob we've discussed before right so this is how well main example of of in thing I will mop and the 1st thing is that before applying the unthinkable mop then we apply some we shift some things up and some things down right so there's the so there's them up saw this is called this done that I'm up on there is one theorem that actually that's very important which is I mean this is an example that I can tell you this is B. recently sold saw there is a theorem that was proved by our own figured out look a 2017 they tell you that if you fix Omega which is their golden mean so this is Have any of us in a number.

[00:28:41]
So for epsilon equal to 9716 then you have pain. 49716 then you have 10 anybody on Circle right. There is nobody on Circle 4 epsilon bigger than 9718 I mean I should have given credit here so this is due to it being June Bryce in 1980 something so this is one of the things I will have to fix so this is Vice and so on and there is also something which is very I mean you see there is this a smart there so that if you chase like 4 figure a little bit then the resoled maybe maybe falls right so this is one of the conditions that you need and.

[00:29:25]
So geometry play several. And then we will see that very subtle number theoretic properties play a role and then the size of the participation is play several Right OK And so sometimes you'll hear lean in people tell you all but nobody knows the constants Well this used to be true maybe that but if you're in now it gives you even 2 for FICO score correct so we can guarantee that resells with 4 figures and the thresholds the boundaries of existence of non existence are known with 4 figures he in several cases right so and this is so so the this is the means that the theorem is nonconstructive then not so good so let me give you also on all the.

[00:30:16]
OK that's more or less the so what I am trying to pull is explain you what are the type of assumptions that we are going to need and we are going to need your metric assumptions that that area preservation and tsunami influx and we have are going to require something about sizes of things and then also before explained you the next type of assumptions I want to explain you calculations that astronomers were making in the 19th century early even before in the 17th or something you know which is something that is called instead serious OK So remember that what equation was so this was what a great question so what people the 1st one of the 1st things that people try to Little is to try to solve we didn't do.

[00:31:03]
Power serious way into formal power C.B.S. had much power serious So this is one of their evilly S. that thems and he was actually quite deficient in celestial mechanics and people were very happy so you can find examples of this type of technique called Levy in Newton's papers and I've even looked before because of course analysis is that there with new terms with the I mean it's certainly Newton was using things like that.

[00:31:29]
And doing impose these purely conditions that they were sir putting a bit here so you pose that when you increase the things by one then you got to the same point but the angle core of the unit increases by one and they will tickle call of the net increases by SEATO So these throws you do metrically that these closer circle that wraps around the angle but the process subtle Yes So that's where you have these one so I thought of the seed Well it's very easy to see what you get I thought of their C.E.O. of day in that case you know is equal to feed are NOT make it right so this is again I'm thinking this map that they was thinking.

[00:32:08]
Starting with the IF of the in Example 3 OK so I don't have see though you have tamed the case into a sequel to fit an Omega and when you look at the then I'm not going to do the calculation but you see you do look at what happens what is there to fish into for them well the only place where Cayenne is going to appear is the wild in the expansion here of the only with the time that is going to appear is going to we appear in you know we're here and then the rest is going to be a whole complicated expansion but the important thing is that only up to K. and minus one are going to appear so OK yen is going to be our only here right and then on the other side of these Medici right than the other side this very city goes Cayenne when you look at the dividend then you'd think a yen is equal to K. of plus or make.

[00:33:01]
Us. Saw this is what happens if you try to do these things and then you see my goodness these have complicate there the equations but they're complicated equations but the good thing is that they're all linear and they're all kind of the same type of equations so that's a good thing so this is what people were doing and so on in the case that we looked at the year for case you know it's a constant right it's just these up and triangular constant we thought of course he seems one right this is the thing the happen and so these equations that we have to enough to order then it just amounts to 2 equations right so it amounts to 2 equations the 1st component these I mean actually this is one of the cases that it's much better to solve the 2nd equation 1st and then the 1st equation 2nd just to confuse people so the 2nd equation which is the one that appears here every This is what you do when you have this equation drug where it's equal to and then you have equations like these and all these equations cov this a mistake children and this is what I'm going to study next so the next equations there are equations that you have the things of our rate of up to mangle my nose the things are violated that obligation and this is something that these prescribed OK So this is the equations that you have taken and if you have a theory about how to solve these equations then you can compute all these solutions and have them by means that leads to.

[00:34:34]
So these are called for reasons that I'm not going to justify So this article cohomology equations by cohomology Well OK so I'm not responsible for that so so this is the type of things that we have going to study OK these equations out of what was going to motivate that were lost assumptions because we're going to require assumptions seem very lengthy and we're going to require some seams in number theory but you can see the origin of these equations already.

[00:35:03]
In these type of things that appeared here all right so when you study equations like these one equations that involve translation so it is so I will call these equations cohomology equations. C 8 OK so this equation symbol translation so questions at him will translations is very natural to use fully of C.D.'s So it's very natural to use for C.B.S. like.

[00:35:31]
So if you think for your city is fully of city is so you take 40 of City is how you write it is use these not they should do they do by a thread that right then it do the 2 by and then I put the food in confusions with a hot Yes So this is the same in K. So when you think for your coffee since the foot of confusion is it OK on the foot of ISN'T of the translation is just multiplying by these factor here.

[00:36:00]
So the fully or serious of these you just plug it in and then you use the magic properties of the experience of in the sequence of these so when you study equations like this is the same as a starting that when they give you these sick ways you know here they give you the alpha case and then you have to find their back ace.

[00:36:20]
Right OK giving you the alpha is the same as giving you the alpha case and finding the is the same must find in the attack case and then the relation between the that they have here is the same as this relation into the Alpha case on the at that case.

[00:36:38]
So let's look at this a little bit. OK so if you study this OK there is 2 cases so you have to study the case when case equal to 0 and the case when case is not equal to 0 right so when to see though then the thing is very easy because I mean this equation this thing or what he had to be comes out in things he's become C.E.O. right there but emphasis becomes legal so you see that if other fussy though is not equal to see though there is no way that you can get a solution right and if it is equal to 0 if the Lucic want to see you know then you can get.

[00:37:19]
Any you know that you like because it's going to be multiplied by C. No and they mean much as you go right so you have I mean this is something which is very similar to what people in linear algebra call the freckle Multan at the right that if you have a constraint except that you do do do do satisfy one condition in there you need one condition to be in the capital but then you have many solutions right.

[00:37:48]
I mean if you have one of these fine it leave the Fed made this is then you need one condition but then you have a family of solutions except that are so this is exactly the same so in some ways these equations are as your going to see they behave like fighting a dimensional matrix equations that know that but in some ways they were some bold finally dimensional matrix equations when k's not equal to seal then you see that you can divide them then this is provided that these number is not 0 you can divide OK.

[00:38:21]
When I make I.C. last you know you're going to always divide that you see that's the point because the this this this thing here to becomes one when the exponent becomes an integer any form A is an irrational number it never becomes an entire year when you multiply by an integer but.

[00:38:43]
On the other hand then you see there is have any interesting problem which is that one my nose into the 2 by your maker it becomes I mean if you have any of us who know the number and then you multiply it by Into Yes right you can get it to be very close to an integer.

[00:39:00]
OK if this is really close to 0 and if the exponent these are very close to any of the IF they care you'll make eyes very close to an integer then these numbers he's going to be very close to see No So when you have to be waiting by a number which is very close to see no then things come blow up.

[00:39:18]
You see so that is going to be a problem that you're going to have which is that if they can YOU MAY GOD can get baby close to rescue not of can get surgery if they can get very close doing that yes then you have a very you do have a small denominator so this is called a small denominators and this is more denominators cannot be.

[00:39:40]
Cut know the avoid this in the least the serious and then you will see that this is the things you would hear so we are going to so let me just read to tell you a little bit about what these when we can follow the solutions of these commodity equations right.

[00:39:56]
Requires that the right hand side has to live in it and in this case when he says solution then you get the. Do get most allusions by adding constants right so the solutions have to be fine up to a constant and then you get the small devices where no make I stick with the seal OK So there are several questions that you can ask I mean you see I mean in order to be able to solve these equations right you just see the problem is that these equations that we were doing over here right so they depend on these expressions but this expressions depends on the previously computed ones right OK so they depend on the previously computed once and then I mean how can I sort of but maybe I get to paint myself in a corner that I keep on computing up to that 50 and then when I get bored of the 51 then I get that the average is not 0.

[00:40:55]
Saw this is a question I mean of course there are people that have a study the electrodynamics or something this is what people call the normal I say reality to a lot of this right. But that's very similar so there are several questions can one make sense of all the time seen but the race in theory because you see you have these functions and if you have VERY be small denominate those right the then maybe just start with have any reason I will function and then when you have Deniz not even of these 3 reassuring or something like that right then so even if you can solve it then you need that you solve them in terms of reasonable functions you don't want that I thought of that 57 then you have things something which is these 3 reassuring or even more complicated that you cannot multiply and do things like that and of course the golden question is do the serious converts OK so all.

[00:41:48]
This questions whether it's studied by Billy systematically and by others before so this is one of my favorite books. So they said the magic out of your mind that they don't know if I will be able to explain it here but we can put it that there is that if you have any episode of Being and see them in flags the average of all the times can be done it is going to be equal to see no provided that they are functions.

[00:42:20]
So if the so this idiocy is defined to a lot of there's in I mean it's a little bit more complicated than this but but if the serious is the fine and give you solutions to one or there then that I found safe as you know of it and this is if you think a little bit about it it's.

[00:42:39]
It's a very nice geometric argument because it does you that the case don't change the C.E.O. of its proper mean flick this it'll mean flacks is going to be the average of the 2nd going born into of the case and if this is C.E.O. of it it is through to a lot of those they need this through always will.

[00:42:59]
They come by 10 sexually let me just try to study this. This problem of punk and actually I have to put it this is a mistake. They are either very inconclusive so I left the sentence I mean this comes from point A COPY THE But I mean the story that they didn't put it in the right.

[00:43:21]
But. If you want to the people that read French he said example of our knowledge being of God of the in the other bed because he asks in double negative good cup in that did this that's in combat except that I said that out and then at the end says well I don't know.

[00:43:41]
It's extremely God of the sentence so you see the people who are hesitating Bob On the other hand says well there is these we had a case the good happened that maybe when the frequency solution of get equation with him to get a coffee since and so on this is we have but this is really we have so I recommend that you read the point that it's up there when he says he they think I mean this is a very interesting book because it's obvious that it was not it was written just without correcting anything because when you read what they were to Chapter 5 says well by the way in chapter 3 I forgot to put the factor 2 so please add these things into up there so it's it's a will which is kind of written from the heart so it's a lot of fun to read but sometimes people that do these things.

[00:44:31]
So I mean one of the things that they recommend have done it in some places and I can do it when we discuss the I leave but the things is very easy to write these expressions to compute the least the C.D.'s for their standard of my picks at that and then you can do it and several people here have done it right like them other people and then this is a story so in order to do to make sense of all this we are going to need rights which are going to be what last 2 assumptions we have already told you that Germany place are all right and sites play several right and now I have to tell you that analyses play some role and number theory play several And this is one of the things that made people extremely never boast about told this theorem is so we need to do 2 things One is to sort of quantify this small device of some of this small devices right and this is going to require a number theory and then define its basis in which to do estimate so if you're an analyst and you study a functional equation the 1st thing that you Louise list very finite myspace or something like that right so this is the type of things that I am going to try to do so let me start with the number theory because the number theory cell it'll be more fun.

[00:45:49]
So I am going to try to tell you one approach one theorem one definition so you're saying that I'm putting it in when they mention only where you say that the number economic guy is that you're finding if you can find cost dance in such a way that the distance of these number to the integers is bigger than.

[00:46:09]
K. through the N. So what this means is the following Ok so you know that if you have an irrational number when you move to show you can multiply and get it close to an integer but the thing that we are going to say is that if you want to get really close to an integer is then you have to pay a price that you have to take then multiple Barry large Let me put it in another way every number can be approximated by Russia and US.

[00:46:39]
Every number can be approximated by rationals but if this out of the you front the numbers are hard to approximate by rationales in the sense that if you want to prove she made that by a rational brain accurately you have to pay the prize the denominator of the the denominator of the Russian of has to be very large so that's what it does you over here that they were so putting over here so or make a minor send over to K. has to be bigger than these so the only way that you can make all the small is that you have to take a very large so if you want to get any with approximation to omega the only way that you can do it this way taking things we've very large to nominate those OK So this has.

[00:47:25]
7th Century technology when people were making the race right so you have what they think around and the only way that you can approximate the frequencies you put the wheels with his pockets right so if you want to get something that's what they do very irrational number and you want to proceed made it very accurately you have to make wheels with lots of his pockets.

[00:47:48]
OK and wheels with lots of fish broken its breakdown all the time and. So. And they are expensive so the people that were making mechanical devices and and wanted to rotate the device quite a lot of efficient methods of finding things in such a way to get a good approximations with a small device with a small denominators except that are but they will not describe all these things over here right so this is there you're fronting numbers so let me just tried to do a little bit that are quite a lot of your funding numbers so this is one of the so this is one definition but I mean does it make sense well let me yes the spend a little bit of time describing the there are lots of value fund the numbers so let me just denote by these there you frontin numbers of type new one pal is as I described before is the numbers in which.

[00:48:44]
We put over here so this is an divided by power he's bigger equal than this number for every N.. So all of this is that they're finishing OK Now let me try to discourage rather than their Your fund the number so this is the definition of the your fund the numbers and story that they wrote these these would be and then.

[00:49:09]
So it's much easier to study what that of the non-value frontin number so you see that your fund the numbers are correct that are used by the numbers that satisfy infinitely many inequalities. OK So what is the complement of these the complement of the things that satisfy infinitely many inequalities well is the union of all the things that that failed one of them right I mean this is something that we do in the introduction to prove courses but it's always tricky so then known by your fund the numbers out of the numbers that fail one of these inequalities have a sort of put this to big but it doesn't have to OK.

[00:49:48]
But if you think about it this place is where the thing fails it's an incredible right and it's an incredible Whose we if is OK to the mind of style. OK so all the things about here so that's their main observation is that this is an interval and if you look at what is the measure this is the measure of that these things.

[00:50:10]
And then on that you find the numbers is going to be the union of all these things but they have an estimate on the measure but they have an estimate on the measure so the measure of the number you're frontin you see no one is the sum of all these things so what he have right well so you're some 1st and then and then you some 1st think A and then you some late that in K. right so when you're summing then then you have.

[00:50:36]
The numerators can go from one to K. is right so that's from sort of go from 0 to K. minus one but so you put the factor again here and then you have all these things where they are right and then they interesting thing is that if this sum is bigger than I put the Sam is bigger than one then this is half and the number multiplied by a new right it's a find a number that want to play by a new disease.

[00:51:06]
But you see the thing about here is that there exist a new one power right so the you fund the numbers of type A new one if you fix the hour which is bigger than one then they have a measure which the number you're finding numbers have a measure which is less than a constant new so when you're allowed to be any any possibility then you get the basic Elise that they don't they haven't the numbers have seen or measure right so there are lots of the IF and the number so there is some other Leo There is another theorem of Louis and so you really was not only mechanician but he was also a number theorist.

[00:51:43]
OK and actually gives you quite a very. Efficient way of computing that you frontin numbers and actually taste up with guess the point that I was making. So they'd tell you the following so this loop will feel them and don't get confused with the you will feel them about mechanic's right so this is Lou really in theory so that is you that if a number satisfy sample you know be an equation with integer coefficients and he said not that I asked him a number then it is your phantom number OK So this is the theorem actually this is the theorem actually tells you for example that the golden mean which satisfies other you frontin equation X. is had the face an equation exists plus X. minus one is equal to 0 this is going to be other you frontin number right actually the golden mean these are the most they are fun to number in many senses of it thing to have so let me give you the proof because the proof is very simple right so all you have Sigma this is what the ME is so this is what we assume that B. of seek my sequel to see you know and when you're seeing that people I'm of sigma is not equal to 0 for simplicity this is not really me there but then there was a ration is the following you put over here so when you take P. or N. over K. and then you will be played by Kate 2 they need these the degree of the polynomial.

[00:53:11]
OK So this is not 0 but you take a polynomial with integer coefficients do able weighted in N. Owhere K. and then you multiply by a K. to the degree of the polynomial What did you obtain its I mean to write and it's an integer which is not 0.

[00:53:33]
So when you get them into your Which is not 0 you have that this has to be bigger equal than one enough salute I see the put absolute value right yes. So this is the key so if you have an entire year and they need to get is not equal to see you know the new fast to be bigger than one OK and then the rest is very simple because when you look at the people I minus be over in caves right so this is the gate to the right this is being recalled then what this is 0 OK but the mission I put it is because you realize that this is going to be people I'm of Sigma this is going to be roughly the the what the and then would be by Basic my mind of saying cable to play by K 2 the so what they have pain is that this is bigger than one and then this gives me the future fantastic I mean very simple argument thank you leave you when you said to prove that several famous numbers are not the.

[00:54:33]
Are. A polynomial equation so for example you this was used to show that by us in solving the polynomial equation so you can not do the right tool of the circle Well another interesting remark is that if you have these that you frontin numbers this is something that gets used so I am going to make it the word here which is that they are fun to numbers have satisfied by in an equality and that this in an inequality can can be said to rate that but it cannot be said to rate the very often and the reason is the following so suppose that these were in these various more supported as this one is really a small that the make you start a minus in a study is very small so so something that becomes very close to said too late during the the by you said to rating the you frontin bounds rights or something that is many a small so what they claim is that all the numbers needed by out of 25 from side to late in the round.

[00:55:35]
You see so the when you have a one of these is Mall devices this small devices repel each other so that's very good because he was telling you this mall device source kind of if you want to prove convergence and study convert it into this small device or solve your enemies and you see that once you get one of your body by the enemies then then the other ones are going to be soft OK they're really bad guys repel each other and this is a very useful thing so if these one is very small their separation is that you can't write those These So this is really truly a small when you Kate when you sort of more they fight them a little bit when you modify them a little bit then basically these things cancel out and the only thing that you're left out these there is more pressure to brace and so when you put all these things over here then this is the one that is going to be the main one thing the Katie so you cannot be very small because it satisfies the conditions of the you frontin ass right there you frontin nest tells you that you have to you can bond these by Kate that to the minus a power so if you have something that is really very small then nearby you're going to start from 0 again and then things are going to be repeating themselves so just to be big or not to various more OK So this is what they want to say about number theory so this is one hypothesis in number theory that allows us to control the small devices that appear in the part of the racial theory when you try to look then I've put the ratio of theories so the next thing that we have to say is this class or this basis this is a little bit more of a choice.

[00:57:21]
People are doing this even find a difference here or less basis or even the knowledge because places where they like to settle it is basis well because they give you their strongest resold. OK analytical spaces give you the strongest results this is what happens and actually it's also the simplest leaseholds this is the simplest response to proof.

[00:57:45]
I mean those of you some of you here have been working with find of the 1st level functions and you have to count that even these formulas and so on and and you need to write formulas on the paper on the side of them so in the analytic case then this is not right but so this gives you an idea this gives you a bit of results in some way.

[00:58:09]
That he's had technique that was developed by you to get more said I know there are people that if you haven't a lead to good results in celestial mechanics you can get find it leave the financial results but they would maybe not discuss So there are quite a good resource to study this I mean I could say well I only have like an hour and a half so that's what I am good to go so you know but I am pointing out that this this assumption is kind of a little bit negotiable a little bit anyhow so let me just add to this guy this.

[00:58:43]
And I interviews this notation with a row OK So these are places of functions which are pretty good because 1000000001 there are not. That right I mean technically I just assume that they are in a little band minimality means differential in the complex sense even an open set which has side with.

[00:59:10]
Class and we throw from the from the axis right OK I am assuming that they extend us a continuous function to the to the boundary right to the things over here so you can. You have the realign and you have this is the circle right when you identify the 2 sides and then there is a complex analytic in and the small band.

[00:59:35]
They are living in a small one and technically you assume that they are not live taking on this moron but let me put that they extend to the boundary in the wrong that either you extend. Continuously I mean there have been a little functions that could have been the one that you know something way that I don't want that yes it doesn't matter too much except that I exit there and then once you have this is space of functions I mean you know that you cannot them you can multiply them you can even multiply them right.

[01:00:07]
In this case over here is so I can so the thing that they do is I define the norm. Define the norm. As the supremum in this in this set over here so unified the norm in the supremum and then there is 74 of the people for the mathematicians then that is something that makes mathematicians many happy is that once you put these not then this becomes have an attic space OK so the only thing that you have to prove basically is that the limits of our functions that are converging unique so this norm is uniform in combat against and then difficult thing is to prove that the limit of finally think functions in the uniform norm is social and political that prove by You've seen the Go T. take it I'll accept that as of this is.

[01:01:01]
Done in any complex analysis course. Yes if you have not seen it except that it's a little big technical but this is something that makes analysts very happy because. One of his bases is the things that allow us to take limits and to study things. I think that that's kind of the last thing really and that they are going to this class and then so let me tell you several facts that are important and that they are going to use so there are these things.

[01:01:35]
And so these are things that are called could see violence and of course there are infinitely many things that are called. So this is the same thing which is the thing so that I have 2 of them that are going to useful for us one is that. Let me just try to say these things you would hear that if you have control of the function in the space you can get control on that anybody but me slightly a smaller space so this is one of the things that are happening I mean when your study of real analyses they tell you where live you know the size of the function you know nothing about the in complex analysis this is not true if you know something about the function then you know things of that he what these but the nearness Laidley a small of a space in them is lately a smaller of the main surgery and then this is something that is going to be extremely useful for us when we study Kemal A-G. equations.

[01:02:30]
Which is that if you know information about the size of the functions then you know information about the. About the fully of coffee since right. OK. You see the interesting thing is that so this is the supremum of the function and then this is more or less their proof of their knowledge the city the main So the functions if this bounce out of exponentially as small in their knowledge the city the main if you know the supremum.

[01:03:03]
And you know the functions so they're exponentially a small exponent depends on this K. right so the fuller coffee is in so if you have just the control and only small analogously the main then you know that the food of proficiency is going to decrease belly very fast with K. exponents fairly small they're going to be exponentially as bowling so if you have control on the function then the full year of reasons are going to be decreased exponentially faster and this is what is going to allow us to understand so you see when we were looking at the this is the game that we have to play when we were looking at the solutions of the Camorra equations like we putting over here.

[01:03:47]
Let me see if there is going to be any funny interplay I just anticipate the ideas if there if there. Are or I think I wrote them. I know every line is that they made any big mistake this would be the other way around right yeah so back a sequel to Alpha Katie by the by you all these 3 OK now you see the interesting thing is that if this thing over here is a little bit analytic right then things are going to legally sex punishingly fast but if the things out of your fun thing they only can grow.

[01:04:28]
Like a power play so the things are going to make sense right so this is the big key thing that ties them up to where that right so we have going to get estimates on the speed of the Nikkei right. Because under the K. of the coefficients because of the analysts in the then the number theory is going to give us how fast the denominator scum grow.

[01:04:55]
And that the things are going to allow us to make these things so let me try to sort of get the proofs and this is just for fun and I think this is more or less with the last thing that I saw this is a formula that those of you who have taken complex analysis know right so 5 prime This is one of these formulas it's the intake of all of these things over to see minus a square and then you can do it through any path like this right so you can do these through any path and then so if you are in a distance from minus that and then your C. is there so you see you can go all the way up to at this time 0 and then you integrate over this path this part of the over here and this other part of the will of the other cancel because the I mean we have a wrapping around in the when we do see if by one right so you're up around and then when you integrate the weight of the path then you have pain that.

[01:05:55]
If I plan C. is less an equal than one or 2 by 5 of C. over or over these and then you integrate the worth of this loop over there and when you do the same thing at all over the loop then you have DE in that 5 pm is less than equal than this.

[01:06:14]
That is. Think. So well you. Are well but I don't want to use but I want to work on a thought of I could use ball social right. And the supremum on the wall that OK so that. That's even better. For some grease on these gives me the this gives me that the minus 2 and I don't know why.

[01:07:02]
Yes. OK Very good. For the other one. This is exactly the SO for the other one let me just read to tell you what these the the proof is this formula for the full of professions that you know the interviewed all of these things and then the thing that I am going to use is that the function this function over here.

[01:07:33]
It's it's analytic. So the function is so I can see if the path of integration. So you can see if the path of integration so I can take the path of integration and deliver them being a path of integration over the over in the over the over the line I can make it a little bit up or a little bit down right so this is what I am thinking Sigma right so I can integrate the world of the over the Sigma right the.

[01:08:07]
Second they form the path and do the integration between Cedar plus a Sigma to one plus a sigma of the path and then over here of course the thing the company is that they K.X. gets a factor Sigma and then you see so this is some identity. For every Sigma but this identity is 70 years for me in different cases so if the if it's hard but they just for me to take Sigma.

[01:08:39]
If case positive disadvantages for me to take Sigma which is negative and then I get them is small factor here right. And if they use negative weight they Sigma positive. And then once you take this with the right sign then you can make them as large as more or less role play.

[01:09:00]
Yes So you see the ones you get the. Ones you get the few K. you take absolute values this is bounded by the supremum right by the norm right and then this exponential this is true for every seat by that you take but it's 710 years to take to take Sigma depending on the side of K. and I can take Sigma to be as close as throw or minus row us I want yes OK And then if you do this then you have.

[01:09:35]
The Guts you vouch for the exponent. OK So then I come to the last the last think about I am going to try to discuss right and this is. So this is the last thing I am going to discuss because this is kind of the last think that they are going to use and this dice up their analysis on the number theory OK I mean so that can make sense of.

[01:10:06]
Of. Of the participation expansions OK So there are some shims are the following so us. That. Just started with our right hand side of the the Kemal of your questions that belongs to an analytical space and you're seeing that they don't think it only sequel to see you know right we discussed already that this is a solution right yes and then we can find the unique at that satisfies the model of the equation and that satisfies the average it's equal to C.D.O. right I mean I put the 2 conditions because I say so if you don't put the 2nd condition then you can either constant right and of training estimates of something that you can be are in constant hopeless gameplay OK So are you have these equation and then.

[01:11:05]
This equation sold and then the conclusion is that for every Delta bigger than Seattle then you have this estimate so you have 10 estimates of the solution in and it's likely a small of the main and this goes like if you were losing. About one of which is plus one of the so the estimates blow up.

[01:11:30]
The estimates blow up as if you don't loose any of the main right but they blow up in a kind of a controlled way and they will walk more or less as the these good blow up. So to speak don't plus one that he with these yes so that's the moral of the story so you kind of tame estimates of the equation not in the same domain because it will be impossible right but you can get this timid scene only slightly a small of the main.

[01:11:59]
You have to pay a price but the price is kind of controlled Yes So that's the moral of the story. A footnote is that I mean this experiment is not optimal then many beautiful argument by the 1st man that solves that you can improve the exponent I am not going to do it but it uses the fact that this fact that they were telling you that.

[01:12:25]
This is more devices don't happen very often right and if you take it into account then you can improve the exponent so I leave it like that they certainly have a command that do read the original papers. And the proof is actually very simple so here we put the formulas correctly right so you have it back a try it of PETA is equal to there at the heart multiply by you if we have seen is right.

[01:12:55]
There at the heart. We've discovered the sequel to the Alpha heart divided by small denominators right. So this is the solution. I have to prove the estimates starting with I mean these formulae have not shown that it can but just but they will do estimates that the you that it actually can but it is right so this is the formal solution but if I estimate the general that I'm of the serious and I prove that it can but it is then this is the True for that to solution right yes OK so let me just add to the estimates and then you will see OK it's a little bit of a painful thing but it's not too painful OK like 5 lines yes OK So let's start to just roll out what is leaps on but you see this is going to be just you're seeing play and go inequalities things like that right so that's not a big deal so you have it like a pleasant that you seek well do these things over here right and then the thing that they put over here is the norm grow my nose they'll tell the exponential so they think that they put over here have sought out for K L 5 K. this is the amount that they had to port to their small denominators right so the exponential can only get very close to one if the your make are gets very close to an integer right but it can not go very close to an integer because this is so this is how close can it go to and in the year and this is how bound on the exponential way.

[01:14:30]
And then this thing over here we just brought my nose the norm of dysfunctional here because I mean this is an exponential so how big can you go if you allow the imaginary part to be rolled minus those that So this is how big it can go right.

[01:14:48]
OK So this is the story so now you have to look at these things. And now here is what I am going to use coach evolves right so this is just the more or less I use over here just this I mean the only the thing that they use here is this mall the devices things right and the rest is easy calculations of what is the normal for the explain themselves in the next thing I am going to usually something deep which is the estimate so the coach he estimates tell me that this thing over here every gate to the Tao is just bound the sorry Alpha K. is bounded by this exponential.

[01:15:25]
And then the rest I just copy Yes well I just copy but I just sort of factor out these exponents elves and now the interesting thing and this is the magic thing is that this exponential kind says this whole exponents have right so this exponential curve comes from the coach of our own kind says this whole exponential that I used to have here right yes and then I just take things out of the out of the some and then I have 10 that this is into the minor scale will deploy into the mind of scary they'll time want to play by K. to the.

[01:16:06]
Yes. Well we're in good shape I mean this this actually combat exists right this actually combated this because for every positive that the because one is son negative exponents Hill and the other Lisa power and then you can get the things to do combine it right. Well so let me just try to tell you a little bit about the think but I want to find out how it gone by just so the thing that I want to say is that of course if I do take the so how do I get what is the rate of commercial gents.

[01:16:44]
Well this is one is more think because I mean this is like completing the factors so you have over here and you say well if I have a K. they'll die there then fish would be very easy because then you can make a change of valuable so to speak in the in thing lives right so this could be like anything about so their salvation is the following Ok so let's put a little you have in there so live with the weather here out of the K. Delta so it was over here the now you think I made a mistake in here and I didn't put the K. minus that they have sorry so you put the weather here that that too that K. on I put if they have so it's perfect so you put the word here that that K. and then you pull does that do the minus came minus one and then you put the health.

[01:17:28]
Right yes saw this does that to the minus K. minus one is what gets out and then the only thing that I am going to say I mean this requires a little bit of work but this is something that you can give to the analysis one undergrad students and they think it's an easy exam that this think I'm but just to the remaining thing at all right so this is going to become the remaining thing that all of the function in think about from cedar to infinity into the minus 6 X. to the right yes.

[01:18:03]
I mean he's not completely obvious to do it but this is an exercise in analyses one except there so that when then time goes to when delta goes to small all these think can be approximated by the scene take it out right. Then and then this is the bounce that they want.

[01:18:21]
This goes like alpha male to the minus one to the mind of stop plus one and then multiply by a constant OK. Anyhow so this is the so this is the bounce for the so this is the bounce and then so we have already all the elements so this is more or less the main tricks that we are going to be using right remind you that I have motivated for you several things I mean that we are going to need some geometry assumptions like any oppressive regimes right tsunami in flux right we are going to need.

[01:19:04]
Some estimations some sizes right. And we are going to need some that you're finding estimates right and then we are going to be doing guesstimate scene and I'll leave the case basis right OK And then I will be able to show you the whole thing compared to something for that and so on and so I am going to just leave it here right OK and there's going to leave it here so the next time and unless there are some questions accepted out or something that I now when we go for the full feel him then we will just need to sort of try to get all the I mean so these are I want to get it straight and then I have to play them in the right time 6 at that and then pull them together.

[01:19:52]
This is going to take us I know that I would have 0 right. But the thing that they will try to have pain is not only a proof of a few of them but also a very efficient medical algorithm. Then America logarithm on the implementation I will discuss maybe a 3rd lecture.

[01:20:14]
OK So I think this is all that they want to say today. OK. Any questions or comments and you will be so good too fast will. OK So these are the main tricks and once you have these 3 eggs then the rest is going to be. Playing them in the right of the.

[01:20:50]
Yes there is something that I would call the blue note on this illness right. Yes So there are these brutal conditions and in many cases they don't over here we have. A very talented glow this do then and now is supposed to look in Intel Now when you check and see who is giving lectures about how to do the C. in the case but then see how the.

[01:21:18]
I mean when beyond there were no conditions fantastic that needs but required I just think but I'm at this if you do the things. That's right yes now there were no conditions suffered through in any number of frequencies in any number of frequencies Yes that's correct that was there were no condition worse there is a little bit of confusion except that I mean because there are 2 The Phoenicians of brutal conditions one using continued fraction approximations and the other of not using continue fraction approximations the one you're seeing going to new fraction about commissions is not equivalent to the other one in entire They mentions but both of them work.

[01:22:10]
There of the Mallee in the sense that if you make up at the invasion then you can destroy them but one of their discoveries of the late that years is that if they if they if they if they put the race in class but I mean there's then you can adjust the parameters and they still get that even beyond that so.

[01:22:31]
That's something that do you honestly even understand very well so far but I am hoping that my friends explain it to me a little bit better so. Fin find one who is not here but. Writing papers on this with him we need to start so you could talk to infant.

[01:22:57]
Questions comments and. Yeah well I'm going to prove. Maybe not because it takes me a little bit of time to explain but what I am going to prove is the most astute theorem. In the various you know if I posted earlier version except maybe i think i don't think I have it but.

[01:23:23]
But yeah so OK so I can tell you more or less what they are going to prove the remember that this is the condition that. Let me tell you what is the story and actually this is OK so what I am going to blow visa following suppose that you have a cave when you plug it in here has any small lever or right then that he said K. that satisfies the equation exactly I mean you will need a few extra assumptions of non degeneracy right but the theorem tells you that if you have something that when you plug it in here satisfies this equation with very good accuracy and Omega is there your frontin and some non degeneracy conditions are met then there is the true solution we are by.

[01:24:09]
With the same frequency yes that he'll make is fixed so this is the theory and that we are going to prove that if you have an approximate solution then you can find though the corrections to get it into a true solution and there are going to be several feet both bases of regularity etc etc But that's the story and the proof will be very soon are very explicit about the converging algorithm which can be implemented in like about 200 lines of Matlab.

[01:24:42]
And I will demonstrate it a little bit so. You went to the toilet Yeah yeah and they and they were young together they leave happily ever after the exhibit. Guys get caught they said that all right. OK Thank you.