The h-principle and a conjecture of Onsager in fluid dynamics

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The h-principle and a conjecture of Onsager in fluid dynamics
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I will explain some interesting connections between a well known conjecture of Lars Onsager in the theory of turbulence and a technique pioneered by Nash to produce counterintuitive solutions to (some) systems of PDEs.
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cool and I'm not and and thank you for the opportunity to speak at this wonderful place the 1st time for me and I'm thank you to the chairman who comes from 1 of the best universities in the work so I'm going to talk about the oil creations anti-Semitic and problems so you're ready soul in our blocks talk a lot about the only question so this is the compressible on questions and the question is whether the energy identity the energy conservation which is this identity here is very normal and the outsiders conjecture as the following so yes each you have the following conditions but but but but but but but but but but after that and all there are solutions which columns I had the energy conservation if the Commission is weaker OK and you see that's actually his original conjecture this constant here is the scene which is independent of X Y and the again so in a sense we could here we have the one-third anybody in space of the which is In addition it in time and space and here we have this guy one-third minus Etzioni and Phoenix OK and then you heard the the last fall all the results which were known so far about this conjecture so yes this is OK so Constantine a were actually the 1st approved in exactly the statement alongside the after-work Elaine can then being refinement and then 4 then all you have shares fell in 93 engineers Monday in 98 and uses the and then you have Laszlo and my says in 2008 and you have the infinity then again Larslan myself in 2011 and you have the insincere then slightly after 1 10th of a derivative of the Union lucidity this he said the enthusiasm 13 which 1 thing of anybody in anything the entities and by this time last and myself around the same time and then there is again to stun Larslan and myself and this time is clearly 1 3rd of the median but it's going to be in everyone's been signed anything In space and then there is the face time not and alliance and they he one-third of anybody which is and infinity in space and at an interesting time and to inspect cancel these days and the
results so far bonds Glad discussed that probably because that conjecture is that actually this Over here easier and the instead of being in anything at the end of that is the funny thing that actually 2 results in the police to 20 although we don't know how to do the work OK so as you see it so I had a lot of exponents and I want to give you a feeling about these exponents today but I want to start the day yet the origin of the methods that we applying to was to look at these of the ideas that we're using in order to prove his results and that goes back to the semantic embedding problem and OK so here you have a singer situation so you have the the semantic amending problem so this is and map which is a medical embedding many holding out randomly through not familiar with the problem is for instance what you can do with a piece of paper this is the fact that can I can bend it but I cannot tear it up or I cannot be extended because it's not the last so that's anti-Semitic amending ends up but for instance if you are is a medically amending flats manifold you actually know that if you order is American meddling in situ enough to be this has to be a gold surface ruled surface means for every point that must be a line which is passing through it and so you see the free since you cannot do anything better than sitting during in a sense if you want to would you many for the very small states flats Naish enzyme effects in the way I'm going to say that you come here subsequently Cleaver or could and money it me it should be sent so they don't in the 5th that at the time that you have so for every intermediate zeal and for any immersion the which is short which means that it satisfies this in the sense of operatic forms so nearby admission that the shots that exists it new which is going from energy into Australia in looking glum and in this case the committee's anti-Semitic amending in which it is uniformly close to you this and so obviously this company it's the rigidity theorem because I could take action for use as a piece of paper since its why decreasing unless thing tool the ball of the Concepcion and then by the feud nations quicker approximated it again by stating that still neighborhood and then pulled is already fairly my piece of paper in the 1 with the C 1 is honesty in the bowl of a distillates OK and it is obviously contradicting somehow of the Jedi so actually what happens is that if you're considering positively current surfaces so it is actually known when this case and is equal to the candidate mentioned the surface is equal to all energy has positive Goss curvature it was actually belonged in the face survival is off then the amending Saturday injured so you cannot fully unlike national EPA they are being seen 1 2 thirds class at and we had a 2nd tool for this together with surgical also in 2011 which is much shorter than what is offered and there's some connection with the proof of concept in a entity of the yes to on conjecture and singing loudly there are of them were was actually a claim by what is often 1963 so that's what he announced and so the next weeper phenomenon so next procurement can commitment for COC 1 of funding so if that for foreseen when I found out that makes you piece connecting Cheney's and that the exponent that he actually announced in the case of two-dimensional surface is enough to be eat so that I can give you the general exponent is 1 resentment but in fact proved a lot later so this was 2014 1 so 1 over 7 actually appeared in our paper together with the US and and and and and Lazlo and we can actually move out of the city of this magic and I can maybe give you an idea of why is assuming Jianli taken and what is that and more recently in the 2015 together agreed and it's a student of mine plows on my myself actually improved the exponent of 105 OK and up-to-date actually the best that you can do for two-dimensional surface is enough that there is a general term that you can prove for General Services in you know In some a nonplussed one-dimensional you could give space but then the exponents of going to the theater but many important soul but the meshed with the theorem can be proved for C 1 of 4 medics amenable opening this this this exporters over fear of that in the view of many folders for quality and that so if it is a from OK and I am the an ambitious goal of this talk would be to give you an idea of where these exponents come and by the way actually should intercede in these American meddling problem will move conjectures that the threshold which fall insiders conjecture is actually 1 over 3 should be 1 and that seems to be also what bodies believed the body's operation died so he can't answer to this question I not history seems to be would be OK so I'm going to actually focus on the exponents but on the construction site workers so the 1st thing that I want to show you is seoul actually could match proved that theorem over there and for the sake of the of of Of the Stoke actually let me assume that instead of going into any Ukrainian space you're going to a European space which is dimension more so it's a began his beauty :colon and bust soared as it is does not serve the purpose although some of them of all adult which women mean although problem over here which would be for two-dimensional services in the next section the improvement of but the competition's then there's sort of last year and more difficult to explain OK so I want to actually produce mine approximation as the sequence of successive approximations I want to blast from from you cure to you just 1 to you can propose to and OK and I probably want to do this well 1st of all I will be a short man along the way so I'm going to have this inequality all the time and all this actually that the Maddox notation these inequalities the fact that this Martic systematic mistakes over here it's less than G OK and I want to modify you killed when you were you to plus 1 in such a way that this thing over here so I want to to get these guys which substantially decreased so payments so is substantially smaller then this guy over here Bulgarian how am I going actually
to do this so this is the interesting
part so I'm going to take a which is my Medicare OK this is a positive definite of this is a positive definite I'm not thinks I'm going to the compulsory as son all coefficients that's a square and 1 meeting seized on which a positive Simitis OK so the fact that I can do do this fact can bulls this by having fixed Bechtel's over here and years of coefficients which that I index actually it is a simple exercising the OK now what they want to do is they want to modify my new Q way actually have to modified in a finite number of steps so what they want to do is I want to perturb you due to new group of swine by ending at the Commission and time that I edited to mention this but all that is going to look I I mean that the did that this product computer for the perturbed message is going to look like the previous 1 but I ending 1 1 of these guys will the sport and that this is a finite number of time and then I have something which is due to blast 1 which is going to have so the game is essentially to have something like this afterward will ever again committee would still and I actually doing this so this is not by what are called nowadays Nash files sold 40 cents for the 1st time that the mission UK you both X and you add the following precise formula so you have won over London and then you have the Aleksandr win to tell you in a 2nd with it is that you have 81 X "quotation mark Auld Lang the next book he won and then you're going to have 1 alumni and international banks in an 8 1 and then signed into law in the he won OK so now without B and B and then have to alter little vectors which are also ought to normal to my surface you so the use of total to end Indian and out 2 the dungeons To my image uh makes OK so
now what I want to show you his wife this speculation is going to work 10 days and widest activation is going to work is that when a computer D you queue last my perturbation so when they get is the following so it is that you OK and then you see that when the bidding but fits the "quotation mark again the land outside which cancels the land so I have something similar when here I think it's the other guy and I have some Arab terrorists unease in terms of all the 1 alumni because of anybody's heat the the other the other coefficients be sole wanted example of an area which is over fears for cents something which looks like this so did anybody then I'm dividing by none other than I had anyone and I have co-signed a case is that declared OK and now with Evans is that since these guys are actually told all to this guy over here when you compare would be all this guy transposed so what you're going to have is that you have known explored because they'll cancel each other so this is the ability of the weekend and you don't ideal indeed we do OK and then what you get is actually here he wanted so you 1 and here you would have 1 square and then you have the 1 study and and costs and inside square and is assigned to In 1 of ones 1 squared and signs with Bekele's lines going OK and you'll see that with Athens is that while I get mine he won Density 1 finds a monster it ended 1 of the all all the summons that they wanted to now it I do this a finite number of times and him but some OK so this is the basic education and you can actually put the excellent that and so one-sided trade you actually the state's essentially in in a couple of minutes you do you will debate or you can write mesh paper from 1954 it's all here this nothing deeper than its OK old Howley that actually from this construction too a C 1 of constructions all you get school and let me show you in this competition how old while we get an exponent of this meeting was over the audience on conjecture will appear In instead OK so let me so what is going to happen is actually the following so what you will prove his so this guide will actually become a sum of all CDs so that you can with convergence he won who came up with this guy most likely will go up and while most likely would go up you know it would be likely to be blocked because you have the rigidity theorem which is telling you you can't possibly prove that this is going to work in situ misinformation that because you see the point is that since had as well alarm about I shoot very high in might that conditions in the orchestra younger undervalued close to your initial met as as you want OK so 1st of all let me by doing like not actually being yesterday so let me call this guy which is going to be small medical actually that to wanted the 1 now if you look at if you look at our competition this guy over here is essentially a 1 right the size of a 1 and a 1 square is actually in that upstairs relating to the Medicare also what you actually get is that if you will make this OK so this right is essentially all sides then close OK now here when concluding ceremony but its rights so the most important part is when I feet again we did anybody with a 2nd but if my fossils relating to so I get OK so let me just put this guy this way OK know what I'm expecting is that this is converging 2 0 and is is actually going up so let us assume that the opposite astonishing and the convergence is of financial so let me assume that that that fuel is given by learned that he will do some power and phone Sony minus 2 this is going to be equal to London zeal To the minus 2 I'll Fazeel keep cancel the fault line that he was actually lumbers to the so this is the answer now if USC
wanted that politician estimated between these 2 guys were discoveries that In the found is necessary cold and that that you just won the 100 times Lambert who 1 To and and of course now if you look at what happens over here you have minus of fuzzy also this is going to win and this is going to convergence satisfies less than half of the soldiers of fuzzy here is exactly the kind of threshold that you can achieve we don't OK so um if that is what you can do all now you have to understand how you're going to kind of you know make at this convergence of the debt that was fastest possible given what you're going to choose for the London tube and when actually the point so the point is that I have to kick have to choose 1 that you lounge To make a certain analysts more so that's the end of the day have over right so if I made that have small then I have 1 so how small I have to make that kind of action so I have to make that there small compared to the news that that you plus 2 right so because that area is going to tell me how big is the new guy sold essentially what they have in the computation upstairs is that the cube lost 1 transposed signed the unit lost 1 -minus gene right so this is going to be as small as as as this big old 1 remember and I want to kind of quantified OK now if find at I have an example of 1 ever actually the story would be much more complicated because they have many other others but let's see what happens with that over OK so I have allowed announcing that he's the 1 land that he just 1 OK and I have a that all of vector that have chosen which is normal to my to my surface now the backstory is going to be as regular as my kind and space I chooses not right sold the delivery of the vector is going to be like the 2nd anybody you now the Senate the said on but he will Q was going up like an exponential so the ceremony with the rebuke you is the sum of these guys but when you signing a genetic thing what you see something which is comparable to the last guided FC OK so the 4th year I have been accused of one-time calamity OK but then you have the A 1 and you remember the 1 small is small as the Medicare or that you had a key OK and that is that the Cubans want the 1 OK now if you are system shot edition is consistent with this guy should better be actually as small as well as the of no smaller than the next step will be to want to achieve so your condition is the following is that their attitude to the 1 part of that that you lost 1 to the 100 among the queue London that you plus 1 minus 1 has to be less lethal than that but you blessed an obsolete I mean making it less suitable means to be up this even bigger so essentially you can put it equal that's the best you can do all for is the worst thing that could form to because it seems to me they on a war on Islam do all this is that this 1 again so the size of the previous average ensued the sides of the next activation so there is a mismatch of plus 1 it was like a couple of years before understanding this section you know we were always using accommodation of interest under the first one that really pointed out the notation which is consistent with the anyway so why don't know can actually answered your hands that's right and compute the log and computes what is minus 12 physiologist insert inside event sold out in a case over here I have alarmed as usual to the minors Q the last 1 that I have lumbers to the minors of fuzzy look you OK then I have on London's you go to the killed than I had planned you go to the minus Q minus 1 and this has to be equal to the London 0 to minus Alpha zeal the tool and Fazeel cute OK now I take the long which makes the land as it will disappear OK and now you notice that the the DQ will be tourism fueled the actually cancel rights of this cancels with that and I am sorry this minor suffers to minus of fuzzy took your mind store of fuzzier Q and here I have Q minus Q OK so
let me get upstairs where they get so I have minds of fuzzy on here I don't have anything here I don't have anything that I have minds 1 and on these other side they have minus fall as far as kill yeah as my minus 4 office OK so pulled the forces on the other side than the 1 on the other side for fuzzy or equal 1 in the office won 1 and of as you were work OK so this musical tradition we did in 2009 and from 2010 which is thought that the case this might explain actually another I level why you have you Cyprus conjecture so now why actually hear you were wife actually here you are degenerating while you are degenerating because I II III amazing sigh you only have to add 1 Mexican the perturbation actually have to have many right so I just made the competition with 1 ever but then I have to and the next and the next that road is going to have a farcical situation compared to before but I have to make faster and faster installations but the Medicare is not improving OK what actually happens if you may call the competition which I'm not going to give you is that essentially the facilitated the won over 1 last told a number of steps that you have to do this and stuff the was again so why if when he says that you have won over 7 over here is because the states of symmetric measures is three-dimensional and so if I want to write my significant optics as the sum of rank 1 must I mean that 1 matters is easily here of generator of the space of Semitic mothers is but I that history OK and I have 1 divided by 1 stadium that said and see that members you can actually think out for all possible image so in ahead can I actually improved 1 refined and that's because I can use differential geometry and by making the change all variables which is conformal at each step of the delegation I can actually buy analyze them and when I and II and the message I actually need only 2 but 1 never sees the light and Maddox in their own form so you need to steps and when I need to steps and applied in this formula I get the wonderful the thinking has evolved into a change in the religion of more than 2 billion the and here there's no don't no no goes to the senior because as I know yes that is actually going to kill me because I have to and many models unless you understand help blond them altogether which we have now with the story of your own on right and start is uniformly about right right so 40 cents in the case of n-dimensional services that if it is a topological Octavio we will be able to make this and stock equal to the dimension of this in place of Internet market which is 10 times and lost wonderment about king and this leads to the 1 over 7 foot 2 dimensional In the question you can use it to new highs and it was the 1st yet it is this is a leg you increase instructors from the last the less you put in there the veteran is winning OK so now it looks actually kind of funny right because you see that the world record without going to sort of different faces is the 1 refinery here the funny thing is that that 1 over 5 doesn't have anything to do with the other 1 5 but since some used use essentially 20 minutes when 30 minutes of my talk and we'll be able to tell you that 1 of 5 so but hopefully I will say I will be able to tell you these 2 guys OK so many new fake proof of the unsigned conjecture so many fighters set out to save iteration mechanisms for it over the use of the researchers health of the yeah that somehow this is only going to 1 over the so I you believe gone off and and this is not really changed all around you believe that there's a reason why the should be critical even in this case so here we go on like this idea chancellors so citing the word that some explanations for the 1 over 2 years from the GDP and it would take me when I would go extra time if I have to explain why 1 over to sexual interest in having 2 or 3 also although there is not a casual when it's not just a random number sold 2 or 3 1 over 2 and 1 over the they all have their own kind of internal reasons OK so how am I going to go on solutions of your integration so you knew you were not so it's any donation and the delegation is going to look like so that's already was shown by love so this knowledge that they might be significantly so all that is actually the connection of this to American bending problems are hold we come up with an idea like this so you can think about it in the following way so what is a shortness so short that is something in which you have the inequality instead of having the quality safe areas sequence of his American endings and you don't have any estimate of the OK what is going to happen is going to happen that you have a sequence of many faults which preserve the
length of curves but if you have a sequence of Kurds we just with the same length which is converting to another curve the current delineate less as land which is less of so you can interpret this short men this inequality BIU about the you as a relaxation of of of of the of these American banking problem so how can interpret this I can't you can interpret this by saying put this equal to 0 and assume I give you a sequence although of 1 of our solutions of your integrations which is converging weekly but most OK if it is converging on the weekly I mean if you only have abounded as to when you don't that your converging strongly you can take the limits of the contents of the kill OK and you will see that this will most likely drop below the the Q intensifies the need of the OK so it's a comeback seeking equality as much as that 1 over there is the complexity of so this is in some sense the relaxation of the problem started OK now what I want to do is they want a blazing of so I want to start from the queue a cynical deal and I want to generate Institute was 1 of the was 1 classical compressor OK so how am I willing to do that so 1st of all I know the following cell this is a problem which I can always sold I mean you give me a vector and they want a symmetric matrix whose divergence is equal to that effect then I always sold easily so as I can always sold the burdens of active field equal To what functions writes I sold a lot blush so that I can solve so I can still be more than 1 way you so I'm going to do identifying and went all by the emergence of the miners 1 some operator which is inverting the and if you look at this including fully airspace the it is overrated is going to have on their minds what right so if I want 40 cents vintages of mine is 1 of something like the 2 London cable access um OK and then I will have something like 1 over among of Kerry and square and Katie people the I lung cable attacks in in writing by bailout the right so that is the 1 over London which is coming so this is a good if if we help illegal border reminds 1 OK sold I can always Seaver these guys so I actually now want to good use In so this is going to be the 1 this is the destination of an ending OK so here that division among going to give a name and then I declared this to be the next guy because they sold the operator David mines so the whole point is that I want to choose value and they wanted to the perturbation for the pressure in such a way that this guy is is much smaller than before OK so as before when you see analogy the seasonal unreal which is the true blessed one-liners killed it's going to be estimated by the attitude 1 the 1 and since I will actually end the fossil lending terms the same 1 long will also have an estimate of the OK so how I actually am I going to construct is W OK so it is not it's going to look like this so the I'm going to make another that's so I'm going to say that is not new is going to look like a function being W of the tune I fuel so these are small variables and then I have fossil situation OK so fully as if you were Indians upstairs 81 which is a function which depends on the Medicare will be somehow the dependence in these vegetables and the old solution that I have so because signs of a textbook 1 would be his OK and something similar I do for the pressure so becuase one-liners speaking president of the something function the Cananea here the queue up here after just 1 angst in Cuba last month that this OK so what happened there and that the what are the set of conditions which in this context would made the Nasheed Nash teamwork so all I'm just going to write them down and they talk quite some time to understand why use of a set of conditions because it took us but sometimes to set up something which would resemble like Nash then I will show you how this set of conditions he is actually very neutral if you want to set up the invitations but said set of conditions the following so 1st of all OK so my function that now is going to be a
function of the 5 side and all and the 1st thing they want to solve is guy so this is the PD and I have a 2nd plant some since I want to Edelstein locations the function should actually be
pretty all the kinks inside OK and now I'm going to use the bracket for the averaging side of that so the adage of this should be seen on any eligible this minus it should actually be 1990 5 the traces back to life I didn't tell you but the tense over there I actually pretty so why these conditions under conditions which will ensure that they can run the but let us look at the reasons 1 1st thing so 1st of all my value is not necessarily the evidence and I want to actually edit the condition which is the regions In fact that mission that ending years this 1 which would not be the gorgeously and then I had something which is the OK know what they claim is that if I take this condition and that conditions OK the together so the data it is David and somebody was going to be very small and I have to correct it with something which is small so that is seasonal OK and why is that while because they can expand from season in Italy I excite advanced this would look like something like 1 advocate and then I will have coefficients CK which will depend on on on on In order OK and I have been to the eye land that you just 1 game .period Texas OK so the condition that the average about musical 2 0 it me that's easy is equal to the and the condition that the excited evidence of that is equal to 0 it does mean that when I am computing this guy I'm not heeding the fastest sedating exponential I'm heating would this guy so I did it so now if I want to invert the virgins from here and they want to find a new act of field which has this agents since it is fast oscillating gained 1 the tube 1 when I heard you and that will give me a collection that receipt which is small OK so that's actually the easy part of the of them I mean this is the thing which is easy to figure out so let us go and look actually at what happens to the most complicated part which is the part upstairs so from now on I will actually forget that these guys exist OK so all UCI not all that the PDQ or plastic Virgins Ecuador's you President Peter Beattie was equally classical Q so I can South after the operation for the fuel from the equation for the 2 plus 1 and what they have is the following so I against the not only will blast prediction that not really OK and I dont blast divergent all time you cancel the last stands of 1 1 has been killed so you see that somehow here on expanding the ball that over there so I've taken the Cumloden Dad's W but they have the value of so this is still missing the and then I have minus 1 and subplot in which he is the thing that stands between the divergent pharmaceutical OK now captains somehow let me look at so let me take this guy out this guy is going to be cold national and I'm going to discuss it in a moment and let me actually couple all of these guys to OK now we're unplugging my arms Arts and my guns arts events that the Commission has that for OK so when applied is overrated I am obliged to the fast variables and I'm applying to the slogan again and the condition that I have over here exactly as you that when applying it to the fast enables do better operator equal to OK so that's exactly the condition that I have over here so this will only be applied so this is only small Over 90 function being W and B B so that it would it means it means that I'm fit the role the Internet is not on the land that does 1 1 the bottom of the deck OK so now if I'm feeding this morning from hitting the small derivative what a century can always I can take my expressions over here albeit the expressions that appear an electronics plant Felicitas oxide OK and I can't bloodying folks side along the 2 plus 1 so I'm doing exactly the same trick that I've done over here and the reason why I was to gain the lumber to plus 1 was that the zeal coefficient of the 40 I was able to see OK so now let me make the computations so by this nomination slowly means I'm concluding admitting that he knew only on the on the entries of being BMW where I have to think you and are you OK so here I have the small BMW last for small BMW and I have divergent slower the company intends to W finds us OK what does it mean the writing has is the lighting means that if I explain everything so for reasons he fights spun everything here W errors a senior so I will have the CME lumber Q plus 1 acts which is not that much and I have declined antibodies which receding Michael efficiency the plant and when I'm actually expanding Felicitas W I know that since he was equal to 0 that's my condition that the adage about excitement was equal to the so I mean good business over here here I am also a good business because you see that's been W. enters nearly so again if the original job value and excitement was equal to 0 but here I'm not in the business and that is why not include businesses because they have a maximum term I haven't quadratic too right so if I had a quadratic terminology have to do is they have to expand the dominance of W as fully he oxides and then set out the condition that this guy is equal to 0 no here that is not Farstvedt medals so what actually the condition means is that is not a tense about who has to cancel this obstacle OK and that is the condition that we had here and actually it's a little bit fake because here I have a place for the Mavericks and your my Maddox is not the ball what actually will happen is that so since here a circle Cuomo is this means that sincere somehow that he's a divergent what they can do is I can actually add a constant which is independent of expertise depending on time and can make action this this so effective that the Commission is not really this 1 it's more something like this is subsequent blasts last summer call some function which I'm going to call yield to find you and so this is the real condition and it is kind of the incremental the energy fuel OK so as to when can do this convert my opening of labor just to the miners 1 Indian land OK so if you able to do that of course you would be staying with an Arab over here and this is the error which looks like a miniature so let me therefore
handle that from all of them so 1st of all you see from the sun's out that things I must think that the adage about a about who's equal to this obstacle so what actually happens is that OK sold there seems eager normal values so this was the scene on normal can goes one-liners the queue so this is although we said of the order them that you just wanted the 1 has so the compatible with this distinction we find exactly what we had for the National that this article Q actually has to be on the form that and so many good now if I had the sun's out so what is actually going to happen is that my natural which is computed on this guy OK so my national due to the following so I have to invert the operator labor just minus 1 so the data adjusted the minus 1 gives me the 1 that you personally and I have to compute the seasonal of this guy and this is your moment this guy the that actual plus 1 to the 1 house which is coming over here and I have to get into a video and if the becuase converging with the speed which is the subject of the one-half the gradient is just getting the resources and the remember what we had on this is exactly the same as all that we had so if this will work actually interviews conjecture this 1 so what is the problem so why can't we actually sold the inside conjecture by dismantled and that is that is essential in so it is possible and end and this was in some sense he'd been in in last all it is possible to make this dive small but not to make it exactly equal to 0 so if you're going to look for a function being that which is solving these guys over here you're not able to say that it exists will not winning the need exists but it seems very unlikely so what eventually doorways you can make this 1 small by adding a next up parameter and then by adding these extra parameter you can't fight to alter the mice with the other parameters land to plus 1 and would never never term which unfortunately brings you a away that will from from from this 1 over 3 addition and I mean if Eurojust coolant in being you waited as well over 10 if you are less crude brings you away but this 1 5 but OK so I guess it was just 1 beaches if I wanted to give you some ideas alone is 1 of 5 1 with national to lunch and this will the Youk a soul formally it's unlikely that you have a solution to this because in some sense of this guy had appears over here politically and his does not so it doesn't look it doesn't look kind of I mean you you could think OK so what happens the farcical Q is very small but these small then a farcical Q is very small but the smart you converging to a regime in which the passport term actually seems to win because it's it's way bigger than the other guys so you can make it in you can try to make it small because after all evidence and that is not exactly equal to 0 and which is this is what happens on our nominee directions but to solve it exactly the this is the the area even that 1 of course it's not necessarily so so it was like periodic solution actually would still be OK so the point is that they want to speaking land that you plus 1 instance they bounded and abolish the bounded the abounded with all the so if I had a solution which is that of being periodic it would be In a uniform rebounded with old the derivatives uniformly mounted ends a lot of the times when we have these epsilon we actually end up computing a a large number of the so all of the stock sold which is which is growing up as we get closer to the finish so was that 1 would be good enough well so it was like periodic solutions instead perfectly the phrase they use of own so here's the Edge principle this field as you better than actually epoxy made any relaxation of the all the problem with an actual solutions OK so this is 1 form of BHP 2 collections of each principal of also has a kind of power connecting the solution to the subsidy I don't know if this was rights missile some legacy so you you have a system of beginning this is like the point of view and it's the other system of PT is um and the system of BDM and for the fiscal system of is you have the following principle which holds if you have a solution of the relapse problem you can't approximated and we made any sum up to any exit with an actual so that is a form of each principle if you want OK so it's kind of telling you that although you have a system of these which should kind of give you some rigidity it's actually behaving more like an inequality than uh through right so if you have a solution to the PD inequality in nearby that is a true solution of the constraint of the .period OK so that is a form of each principal now OK now the Sabres really proven each principal box and that is the reason they provide the last long in them and decided any which who's also an exchangeable for this guy no 1 will be a feasible in this context in each visible in this context would be something like this so I told you I didn't tell you like I told you how widely the nation right inattentive from from from which point they stopped so here essentially internationally proteome is telling you start from a point in the relaxation and you can buy sell the paper by in that in all these evocations we're starting from a legal point starting from 0 0 0 and without the nation's using was using the relaxation soul in that the paper by EDS signed last will have a statement which characterize the point from which you can stuff OK and because essentially that all the points which satisfies certain you know what In the West the city will workers those who should relations hinders this is a sofa style 1st signing we actually this and that said well only reduction to Seoul and fees Act and blond have actually applied each tool the I'm old please install Arctic scullery creations which satisfy uh a certain condition on their own the on the singular integral crime and the condition of the condition which tells you that there is essentially no called partners local botanists students inside right so using it as a lactation problems I mean this relaxation insists this is working because you have some sort of lack of combat OK so Opel's video leader that the age principle is standing you there is no partners for your sequence of solutions otherwise wouldn't be able to approximate when something which is in the relaxation rights speculators events still other your you could sort of dream OK if I have a lack of compactness then again applied but of course the situation is very subtle because you see if I stopped OK I action black-on-black in here for the rigidity problems so flour if I have a sequence of solutions which were small and your solving being fully cents being anti-Semitic embedding problem for a positive surface than the dolls curvature is positive then you have complex and then you have an exercised and so actually your space of solutions as Compaq so the lack of but this is only when you will be lost a compression and on but so it's you in this case is all very interesting because some so what is the special lying in we don't have a good guess 404 where researchers uses this session is lying for the Saudis conjecture that we have the Osiris conjecture so we have the dual problems which is kicking you and 40 cents here is much less clear because that is not clearly included picture