Analytic extensions of frequencies of integrable PDEs and applications

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Analytic extensions of frequencies of integrable PDEs and applications
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In form of a case study for the mKdV and the KdV2 equation we discuss a novel approach of representing frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include properties of the actions to frequencies map as well as wellposedness results in spaces of low regularity.
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data to and the and 2 2 2 2 OK thank you so much so as the speakers said before me I would like to to thank the organisers for putting together wonderful conference so the goal of my talk today uses to discuss an oval approach of how To extend frequencies of integral to spaces of functions of lower regularity or spaces most distributions but before I state the main results of on this topic we have obtained I would like to give an an application To the about pose knows of and KGB to entity modified KTV equations and let me mention right away that all of what I talk about to today's joint work with his young life the let me begin with this the KGB to equation so we consider the KDD to equation on the circles period 1 because integration is as an evolution equation dispose of evolution equation of the order it comes up in the long wave approximation of water wave equations of course the coefficients chosen Europe written down here are very specific nature and in the approximation many more of these type of equations ,comma but this specific choice of coefficients makes that this equation is actually an integral PD 1st it's it's Anatolia PD and the specific choice of equations makes that it's actually the kite KGB hierarchy therefore it can be written in the Hambletonian ways so did you did you receive equal to the X to gradient of the Hambletonian each tool that's the KGB to Hambletonian it is given by this expression so involved so that torn or move the 2nd derivative of USO H 2 about defined on the Sobel a space-age tool and has some knowledge about the execution the possible structure introduced by God so I recall this is the 2nd Hambletonian indicating the hierarchy so the zero-sum Tony using voters age 0 is yet to norm and the first one is the famous KGB Hambletonian show the belt Posada's this result which comes from this analytic extension I was talking to at the beginning for the KGB equation and to reads as follows it's about post on 2 despite the fact that is 5th ordering and then with Tony and of 2nd order so it's a different audience and tools so for the convenience restricted the same practically off someone of spaces that this average heroes of this causing you possible wreckage the same results would be true on any of such book and then also have to introduce submanifolds and analysts 0 diesel these are the functions you in some space space-age library averages 0 so that their true enormous constant and because of the added to along is the zeros Hambletonian the KGB hierarchy it is invariant on the KGB to flow so this manifold is invariant under the KGB to so the reason about wasn't as result for the KGB to equation reads as follows the KDB to its globally in time to see 0 well pose the H S 0 for any sp Eureka 2 0 it means that the solution may have continuously 2 from more regular data that it is known that the solutions exist in extends continuously to a map of the following types 0 initial are in age as Iraq goes to solution the continuous function the time interval minus for all opportunity into HS 0 and this map is continued however this matter is nowhere locally uniformly continues on H S 0 means that you give me any open or not uh nonempty set you want Hsu role than a very restricted as to said you it's not uniformly continues In addition if I take sp Eureko to 1 have indeed restricted positive otherwise consists of just over 1 potential the 0 entries uninteresting then this map actually is uniformly continues on this many about Yukon contrary if S is between 0 and 1 hand then again in the solution that s is nowhere local uniformly continues even on this amendment old even if I fix them out to on it is being nowhere local uniformly continues look and then the 2nd result group of results concerns EPO was innocent to describe these results have to introduce you normalize equation which I referred to as KGB to shop so it is again the Hambletonian equation and I subtract this term here this is a corresponds to the Hambletonian aged who shot but I subtract 10 times age school age 0 square so again this is an integral PDA because this is and zeros animal tone indicated the arrogant and therefore the difference of course is again intervals so then the resultant give poses goes as follows the modified the re-normalised skating with equation is globally in time to see 0 well post on each cellar space H as 0 for any s strictly because the minus 1 so for all distributional spaces agents 0 KTV to shop is still the outpost again it means that the solution that best shot continuously extend from Agence Europe To the space of continuous function the time interim honesty the waitresses as indicated correlating it's a that KGB to move must be imposed the and 2 you want to blow up a face factor so is presented for this series and there is a solution of KDB to shop in 40 years serious let's write it down like this so this would be he ends coefficient all the solution at time t of KDD to far with this initial they don't then if these coefficients compared to the corresponding Fourier coefficients of the solution of the case of the original KGB to equation with the same initials they tell me as follows the ends Fortier coefficient is obtained From the ends with coefficient of the KGB to shop equation times this face factor and of course spillway to this faced effective blows up because age to the tune on getting OK so the results improve when early results on and on on Delphos on this article and that is not being so much resulted in the patriotic case but more recently a lot of results on on the line particularly vocal Quan Quan and can and feel that they looked at solutions on the energy space contingent aged focus so much about the
KGB to no corresponding results for the D focusing and KGB equation yells some results on the focusing on but I don't want to talk about right now so the focusing and KGB equation everybody knows and the surprise given by this equation so weak that differs from the KGB equation by the fact that instead of you I have you use square again it's Tony impeding vis-a-vis the syntactic gradient where DX again is apostle the structure from Guanare and this is gradient then KGB Hambletonian is given by this expression it's a TXU square was due the force and Katie you would have you to this reunion was so here we have an imposing this result of the MKT equation below where too this goes into the stated again I have to be normalized the KGB equation call it and KDD shop so it is OK and KGB shop is obtained from and KGB by replacing you square that Bayou square that miners to warm again this equation is Hambletonian H. opposites Hambletonian it's H minus 6 times H 0 square and again age 0 if there's 2 enormous conserved by the end KGB flown and therefore it would be easy to compare the 2 floats and in order to stay dowry Posada's it also have also to introduce the formula spaces since have a PPD between 1 and Infiniti so this is a space of periodic distributions was 48 coefficients of sequence of 48 coefficients orange peel so Of course due to possible also identity for peak was to what effort was just the usual led to space but for peace stick to them that this is a big spaces be expensive distribution and it gets bigger and bigger as goes to infinity in particular for P bigger or than 2 In this contains elements which are not even measures OK so the result for the MKT goes as follows and KDD shop on S & L 48 pvs P between 2 1 0 at the following properties it's locally in times 0 well post meaning that for any initial date W P that exists in neighborhood and at time KB so that the solution that sharp continuously extending this way as before In the case these equal to 0 so we can have a strict oversight a strong result namely that uh the equation is not only locally in time with a sincere about post but actually globally in times of postal and he's so this will be true for any key in so again as indicating the 2 equation we have an immediate correlating saying that and KGB equation is in post below and 2 Due to blow up 2 of the face factor so again at 5 48 take the food the presentation of the solution of and KGB shop and ride it in this way initial data via fax then and I can compare the Fourier coefficients of Arcadia and KDD sharply the bonds of and KGB in the following fashion that hands-free coefficient of the corresponding and KGB solution is the 1 of the key and Katie shop equation times this phase factor so again if I'm below lowered to August Facebook affected gets instant so of course there
have been many of them come the and cagily of 1st few comments so we expect and KGB shop conclusion to be also globally in time 0 0 then post everywhere unaffected but there is 1 thing you among don't at the moment we don't know to prove the moment and I also mentioned right away because I might not have time and again during this election to explain anything of this that the key ingredient in the approve the and to to view the end KGB Hambletonian as that corresponding Hambletonian and even the analysts hierarchy so it's the forests Hambletonian so analysts is the 3rd Hambletonian in the analysts hierarchy and and KGB the fall so this is the the result of love formal homeless for the yes because there are so many just absolutely the actually to extend the frequencies to the 2 this is recalling and KGB system because it's actually of course you have to be in a neighborhood of the real because they cannot be explained by review Whittaker following because there's probably of hyperbolic structures as of course there's the many many Burkin and KGB on on the circle essentially it is the outpost for HSS bigger equal to 0 so the novels of the works so maybe I don't want to spend too much time on this but I just mentioned maybe you and even morning detail 1 resolved by the on imposing itself and KGB Belo a and to also be proved that and KGB imposed in H S for S strictly bigger than 0 in the following sentence the solution that is not continuous as an MEP from seasonal infinity reason Norman H. S calling from the Sublett spaces as strictly smaller than 0 into a city 0 minus TT periodic distribution so we can fuel corollary to was sharpening his lyrics results saying that imposes news Soviet is really that is due to the fact that you have a blowup of France focus
so give me and let me give you a go short outline of what I wanted to win these talks so the method to be really really proved this results are in the so-called nonlinear fully transformer pinnacle medical normal form transformation so it makes these on new coordinates which make that the equations when expressed in these coordinates are actually media and the dynamics of described to buy frequencies on so the crucial part in improving this about posing as the opposing as result results is that by extending these frequencies in the corresponding spaces off the functions of low regularity on distributions so is applications of these analysis of these extensions of these frequencies may have already mentioned are about Posada's of people estate debt but also that actually was the module originally motivation of why we started to do this is it applies to Hambletonian perturbations of such so it isn't as important implications on the properties of the action to frequency and also it allows to an entity could extend the Hambletonian sent to discuss its convexity properties too important for me ,comma type results for perturbations so as to keep it the exposition this simple politely explained the method actually for that to prove for the KGB frequencies but here in this case actually improve known results at the end of last time over cause many briefing to forgive focusing and Cagiva equation to see how well and how it connects with the analyst OK so just thought that had to refuel quickly this this known media freedoms for more these normal coordinates now for KDD actually it is these coordinates are good for the whole hierarchy for any Hambletonian into possible algebra it it generated by the KGB equation so that is set up is the scale of solid spaces Hsu C stands for complex valued at periodic functions Green Claude sp Greco 1 so some that negative assets are included here so as I said we have the possum bracket which was introduced by Gordon so these 2 functions that thingy on this basis is sufficiently regular and 2 gradient this have puzzled Reckitt defined in this fashion and noted that the average is a constant new meaning that for any functional if I put the averages of the functional cheese that antiquities Iraq so this causing nearly it's practically sending the sequel only restricted the 0 Leeds 0 averaged 0 and again this year refers to the complex the amendment to the state of result normal couldn't cells have to introduce sequence spaces so these R & PP sequences complex valued to these days for and they have weight so this is the way it went to the ESP and 0 stands for that and there for the equivalent of the sequence the element of the sequence in the middle is actually is you have to do with the 0 OK and then IV also introduces the real sup space of these consisting of all complex sequences which have the property like 448 sequences that is united stand a complex conjugate CIA and then to each of such a sequence to be associated sequence of actions for were and bigger equal to 1 their obtained by multiplying the and this the miners and for all and to 1 so just as an exercise because of the use later we will have review considers sequences in and peace bases was S-Pulse have then the corresponding actions has doubled weights of visas to as most and Katz heft because of the square and it as a special case if we have the real stop spaces but then this is a complex conjugate the fist so I a speaker or equal to 0 and then the sequence of actions is actually the positive quadrant of this and Pete sequences OK and then the interoperability of the KGB hierarchy can be formulated as follows so as our base space me take H minus 1 C officers of the real space of functions the space whose average zeroing in on sequels minus 1 retained and there exists a complex neighborhood of suspects in this complex space is complex obelisk space and a that from this neighborhood into the corresponding sequence based associating to each potential corresponding coordinates complex because goodness referred to is complex because of so that the following properties men be restricted smacked to the cellar space agency 0 for NES their equal to 1 that actually the real analytic from morphism onto the sequence base missing Espace one-half to 2nd fees canonical meaning that apostle brackets of the cordon the functions are the standards 1 of Christie at have complex cornered so we don't have 1 but I instead all the brackets managed and value the most important properties that the Hambletonian syndicating the arcade or real analytic functions of the actions owned so it means that in NHK TV so if this is the Hambletonian is defining H so the corresponding sequence of actions is in weighted and 1 space weights so if my potential is positive and the actions on the positive quadrant at 3 1 the correspondingly if I have the KGB to equation then I get this kind of real thickness and lastly actually fee of 0 seen on the differential the at curaco 0 is essentially the 48 for opposites little weight which calls from the possible says is that this man is usually referred to the government and the coordinates is complex because "quotation mark the principle property of the it's this story means that the delivery arises the PD using the that indicated the hierarchy so then we come to the frequency is the topic of my talk so the frequencies of communicated to equations are given by so we take the Hambletonian and the ads frequent is obtained by the partial derivative of the Hambletonian with respect to the and the actions for any and the Greek 2 1 and then the equation as I said linear rises in these coordinates and they read as follows so the and scored at the time derivative of the and scored in his mind's eye this comes from the possum bracket only gone KGB CNG on a grand scale it not guarding variants of the flow because the actions are invariant enhancing frequencies and hence these equations can be easily integrated the same for KB too so keen ingredients in the process here wants him to are these the extensions of the of the frequencies to work spaces of functions of low regularity and distributions and in addition hasn't optics of the frequencies as angles to infinity so the set of the chore to achieve this extension is as follows a new writers so as I said I'd do this only for the KGB frequency just to save a little bit of time so that the right to at all only KTV and as this is the frequency of the area equations of the linear part of the KGB equation blasts Let's say remained beach recalled all-male KTV star and of course it's enough to extend this ,comma only the KGB stock and in order to obtain asymptotic swore on a gun actually want to extend not each frequency by itself but 1 extended as a match so we look at this as a sequence so we have to determine from beach space to reach space this uh free it's so usually called the frequency map on the action to frequency and changeable systems and in the Division series plays an important role focus here is the
main result so for the KGB KGB star frequencies so the only other KGB star analytically extends as follows so we can view the frequencies easily insolvent spaces all on the corresponding spaces affection so I rolled the 2 would at the same place because this might be a little more difficult to die safer than just the supplements but so is the extent to H minus 1 0 so on the level of actions this means voting the positive quadrant of anyone sequences these weight minus 1 and it goes not quite to the same space for S equals minus 1 but it it would be the largest based mainly at an air or a space this rate minus 1 for any all or some of to than what but the fastest speed between in the opening chewable minus 1 and minus 1 hand then the goal from the spaces and to West was 1 long into the same space and if excess speed or equal to minus 1 half and then again we have somewhat because space on this side of this Cesare do because this has asymptotic expansions in 1 over and it appears that some polling and you cannot improve OK then they have asymptotic say so essentially the and frequency are only going and KTV star is minus 6 I ends and this is the end section variable blasts and editor which can be estimated in this way and this at a time depends on the debate on the ass so in Armenia there are more precise estimates but just to keep it simple like Ira stated in this way so investors between minus 1 and minus 1 street this is a form of the time they get and devices bigger equal to minus street it actually doesn't improve again due to topics which you cannot change in 1 way focus surgeon but we have seen that if he's sp minus 1 and minus 1 have then omigod KGB starters map from the space to itself and therefore can look at its differential and we can prove that different was actually fractal it's actually minus seeks identity lost Compaq and then I mentioned also because for 1 application I meet this that they have a special case actually can also extend the frequencies to fully Lebec spaces but that just take 1 example this would be fully Lebec S equals minus 1 have been P equals 4 which leads to free and actions space of and to blast so also in this case all-male relocating stock can expect extends real analytic map and the differential is threatened OK so this is the essential the result on the frequencies and I will discuss a little bit of the proving the 2nd of of my talk but 1st I want to
give applications so of course I want application they have already discussed but I would like to go back a little bit so fast that they can use these frequencies to analyze the PDE the piece considered the integral piece considered umbilical cord in it's so it allows to analyze the dynamics of the angles in a very precise way so we decompose any of such frequencies which come up in integral piece in the following fashion AT and was speedy and was only legal and start a N is supporting duty and Wisconsin coefficient the yen is appointing on an endless flow invariant coefficients and all and stock is of the form leading port was renamed the just to give an example in the for the KGB frequencies that agents would be the frequencies of the any equation if I consider it on not only on the 0 these but on the whole space actually I get to be Yemen which is 12 times the average of cucumbers and and then only stories minus 6 and plus the remainder and similarly for the KGB to equation this would be the frequency of the linear iced equation 0 BN would be this kind of ,comma expression and again only against would be here and I am plus made so the the and the only glands are possible causes for the PDE for not being uniformly 0 well post or not being local uniformly C 0 well the growth of these coefficients might create faced the coherence mandatorily to this the criticize notions of offal well and of course it will lead to impose innocent case of talking then of course the urban can analyze this year using these frequencies the PDG original solace faces so if Estes the flow the Sobel a space agency regional by St Peter corresponding flow met in the Duke of coordinates and then the floor mat honesty can be obtained from the 1 from the vehicles made by conjugation buyers through the through this difficult due to the fact that suspected so how can this be used that illustrate this just for illustrative purposes for the KGB to equation on Hsu's sp equal to 1 so 1 can prove an that only got to start is actually a continuous met from the space of actions into an infinity which is uniformly continues unbounded subsets and so are the corresponding dip of maps and it's in the and there had because it would be added his constant on the lead this Dedeke was constant and the KGB to uniforms Mississippi uniformly seizure outpost of this sort manifolds and unity for analyzing the frequencies in in this way 1 gets this type of office of results OK then just maybe as as Markos that let me mention from this analysis it suggests to think of a possible role of this great critical Sobolev exponent SET in analysis of self-imposed this so if I look at the 48 hour if I could write down the solution for the space then I can write it in this way so and of treason amplitude exponentially community the face factor expansion I to play acts so a possible scenario for imposing chance is well and the possible role of the overly critical Sobolev exponent is if S is smaller than 0 there from what we have is that it was an astute too the amplitudes blowup of the altitudes effectively have already imposed this before a might be due to the fringes so that's essentially true In the end KGB equation throwing our results are
expected OK so 2nd application as I mentioned is that the convexity of the KGB Hambletonian so to explain you so this is a cagey via the Anatolia inferior reliance by subtracting the term which gets very large effect going the functions of was spaces of distributions and then because KGB KDB Hambletonian is defined on H 1 therefore the corresponding saw spaces of actions is a positive quadrant of and 1 sequences whose Wade Street is real and it takes so there was a conjecture by saying that a Each stock ATV analytic and extends into strictly concave on to pass so they have to classes a space affections I was mentioning in 0 3 per cent the music later for applications United's now in earlier resulted big Biafran cooks proved that this Hambletonian sexual content so they proved this for finite get potentials and then continuity show that it's concave once you know that you can extended continuously and then lost the election in the approved good collaborators that should stock ATV actually extends to L 2 blasts and it's strictly concave media I equals 0 so that the origin and using our theorems Serino the property for actually the conjecture cities almost in the following sentence so this really re-normalised KGB Hambletonian is strictly concave on opened and subsets all of to blast uh reduce the origin being an element of this opened said it means the following that if I ever election is in his open said then the Hessian all of the KGB the star I mean this is the same actually a session of the original long because I just deducted in the near term the Hessian of the penalized pages the Hambletonian is strictly concave in the sense that I can bounded from below by miners the and to normal J for any changes to where the constancy can be chosen local uniform on Monday yes absolutely focus on our dimension connected to properties relevant for stability results of country OK so maybe there it's
also application to the objection to the continent in using them to be able to explain a little bit ,comma this time with losers contrary to use Dujarric area flats that so it's opened and so we don't know what I mean said thinks it's everywhere but I don't see any reason why there should not be some 0 In between progressive failure yeah that's it's it's always concave but strictly concave on these open dense substance so for local from perturbations in animal question enough of course but and the fact is the proof that at the differential of the frequency that it is fragile and that allows us to work together to together recently it is you of the old convincing OK so an outline of the industry so that I recall the industry was the analytic extension of the frequencies so I need 1st to introduce what are called the folk exponents sold on the realigned the corresponding notions would be probably the Transmission Co so this have been already introduced in the Senate think about intellectually 75 so procuring WSL WSP is open neighborhood of H minus 1 0 we look at the shredding operators of this would be a distribution you can do spectral serious stand for this and that the spectrum use discrete died in values ,comma payout the arrived so it is real so this is the lowest end their lives Q real than they are on the realignment and they satisfied they can be awarded In this way so the pair's never of coincide with another 1 but that the 2 were elements in the pair Michael inside then defined the gaps so this would be his injury here and the gap plants which would be the size of the lands of the Middle of the intervals and then we have hit another gap G 0 of which is to the left of housing which is infinite book and then I have to introduce to discrimination some of the things so of course if Q is recommended commuters given by the trace of the flock can matrix and 1 can show of sufficiently regular then I actually tested product this product expansion involving the height values of the world shouldn't operator and in case you're no distribution you use directly you define the annulment of the using this product expansion annually approximate mentioned will be all over New England the analysis of and the and the estimates of all the quantities involved and then introduce a canonical mood of square outlines for this usually gives rise to the hyper-elliptic surface associated with the spectacle problem and at the reason this is well-defined on C minus the gaps and from pure equals 0 just to give an example it's a Science Times Square Root of longer than the flocking exponents and so they say all the expert so we have the idea flocking multipliers would be the eigenvalues of the flocking matrix and if exponential form you can't focus exponent and 1 of them can be written in this way so many queue again even if focuses doesn't make sense because Delta makes sense you can this you can define the way well-defined also doesn't even if there a singularity of this is end OK so that these are a list of important properties of the flock in exponent so as of London's analytical way from the gaps which are open so if the 2 eigenvalues Yukon signed actually is located the of along extends analytically into the stubble height and and we have specific values that the periodic guiding values at London's 0 by definition 0 because we start integrating and London's 0 blast and here 1 can show due to to the fact that the pair potentials patriotic I get your mind's eye and pie and then it be introduced a version of this low-key exponent by normalizing indifferently Efron of longer so that at long the equals Blas Minor along the plus-minus and these actions 0 then when can prove that Geffen changes signed across again the year of different signs but there are limits as we approach the 2 sides of gap and hence the square of efforts actually analytically extends to neighborhood of Jr we have estimates he can indeed have an explicit form of the gradient and the Athens a related to the actions by these confident it will go would be in contravention of the guests book and then most importantly there is innocent TalkTalk expansion of half of longer so actually review was slated on only a full of long desire right down the expansion of rights for this and of course we have little problem if you do this for distribution it's enough to do it for real fine get potentials finer get potential means that all only fine idea many of these eigenvalues simple and the rest are all close and it means that the potentially seeing things OK then 1 can show that there is an expansionist and goes to infinity Of these forms so it's actually a metamorphic function has a poll at infinity you wanted to similarly but it's sold see that at the point of the residue b KGB Hambletonian is involved similarly for them KGB to equation reviews instead of after the 4 after the 6 and this makes up the KGB a Hambletonian appears again at the place of the residue later on if you do contrary integration it's a way of speaking of Hamilton OK now called an integral over 4 rounds simple and values because we know outside the city's interviews in it's recent even power so we know it's analytic in particular its analytic here so because I find entered many gaps have opened I know that outside the days this is analytic and therefore I can use contrary integration to pick up the residue here and on the right hand side of it have Facundo defamation just integral about condos on the open conveyor pairs of eigenvalues so as is a set of cases that the eigenvalues long decay miners on the Cape loss of simple meaning the complicates and from from 0 OK so this gives us the full corresponding formula for the KGB to equations and then to put the idea is to develop this expressions further in order to get a formal offer for the KGB to equation 11 explains a little bit became more date heating up here is the crucial idea so from Hambletonian formalism of you know that then I express the equation action dangles then the answers frequencies nothing else than the derivative in time with respect to the yen's angle and this is equal to the parcel bracket of the HKT the Hambletonian the time so of course in order to have the angles defined I need to know that the end skeptic has not just cannot compete not collapsed so this goes only for simple like that OK so this allows us to get a formal offer for the on because of the kg before the end skating frequency indicates that and corresponds to an open deck so it is equal to 0 this expression I substitute dour expression we got from the residue calculus for educating with his expression then to buy the property and find fine know-how to compute the gradient I certitude and apostle bracket and I get this now use a very important identity from our book saying that the current rate of the Apostle Bracken of Delta
focus on the merits said so this is at CNN is this has this product expansion to keep these are actually related to Wilmore the differential on surface so these are very well known quantities and play anyway a very important role in the community of of the sky but they're not OK so here I sit mind K acted out year was the zeros are completely determined by these equations so this is a normalization of the Office different and they can be easily shown in case a potential is really have to be in the game for and differently for different OK so we and we get this expression you know the expect and Ethel really as F K B minus acre by Q and uh get former loss for getting this formal offer and kg of Acadia for the ants KDB frequency so for this to express this introduced at the amps moment off of Of the FKI house and this an animal the novel uses of importance symmetry properties of these moments and that is actually why we can succeed in extending to such low regularity so 1st of all for all all to these moments are identically 0 and this has determined to implement them because I'm running out of time so also is gone OK In this the sequel to 0 then these moments of that for all and then 0 0 for the specific and finally the zeros moment is nothing else than the integral of this the House Moffatt differentials over the Gatski could market which is announcing his house and you know that the times to pass so using all
this week get the following fall mom for any real fine that potential of average 0 we have this form of all medical and KGB is to and Q so actually this comes from expanding from the 6 I explained this expression get this tournament is an intriguing precisely when and correspondingly the only God knows only that the frequency of the KGB To Hambletonian the Indians as an express form along the form only paid to the 5th visit being a part of this Paul beach the new me already cannot opposition to discussed frequencies gloss on we're going to start the only constant can be expressed of the moments of K of the moments 2 1 4 OK so the strategy for approving seriously really is the useful last uh of theater and 6 dramatically extend on KDD and only going to buy and analyzing the details these moments by showing that it is serious about ensuring that they individually the entity could extend to which minus want 1st of all 2nd by showing convergence of this series by expanded by by the rising as and topics of these and for this use of Minnesota quite involved in new estimates for product expansions of terms which involved it is the moment focus so maybe I'm doing this and then just maybe even Vermont lost little things so coming back to the analysts system and the KGB equations I II and so I said I the very and if I have time I would briefly say something how this can be In and KDD comes up in general assistance sorry called dental assistant is a system of Hambletonian PG use this for the last minors nature suggests companies valued at even by his expression it's the Hambletonian PDD Hambletonian is given censure by the analysts Hambletonian but with these 2 components and apostle bracket is the usual and with respect to minor side because in the complex version so if I restrictive system To real invariant such spaces I get actually focusing analysts in the focusing analysts equation the virus trick to feed blasts equals the miners complex conjugate I get the folk give focusing on less and if I restricted to fee plus equals minus the minus complex conjugate than I get focusing on and this and that the system has the analysts hierarchy it starts his age 0 H 1 this is the moment and then aged Louisiana lessons as I said age for it's related to the and KGB equation recorded the and KGB system so this would be the 1st equation of 1 of the 2 4 additional tooling and then this uh Hambletonian system has also invariant real space namely indicates that in addition to a feat last being the Kompas quantitatively 9 as if they are equal so you is actually real then we obtain the focusing and taking the equations and the feet past the miners is real but today good differ by a sign that the obtained the focusing and continue equation so
then and the idea is To look at the Hambletonian age for the actually expand to call normal form theory 240 Olympic spaces are fed he this piece negative is between toward infinity to spaces of distributions and then extended to the frequencies Of the head and KGB assistance to these 40 Olympic spaces and then of course we have to make a little analysis because it's a sop space they are looking at for and KDB how the umbilical cord and its restricted in this case of the real substances lately I mean that case you it features yes should hold it can these frequencies to the wider space for him back here for the biggest most or all small museums is probably easiest because it through containing a so that it is essentially limited them additional yes Yukon yes there was to bit of arrest of interest but can be done for instance you can do everything on the pianist yes absolutely not after has proved to be the firing the president Of those recent interest in the army nobody because this was career of audience that almost 3 solutions in and that it involves the use of force and In other words to the whole course and all of sudden it wasn't a strict rules during the passage of reflection was 1 of of the members time evolution seems to be released this sounds like this is machinery that would work for them question is that also I think this is really just a normal work the Primera I must admit I did not look into it but I think the machinery should be extendable to these conditions absolute so what more thinking off by looking at the action to frequency infinite dimensions so that it can also show that this but this is the pedagogic said that the matter is actually local from so that means you potentially you can do OK and the agrees infinitely a deplorable things because you can control the friendly infinitely many frequencies through the action can go back and forth "quotation mark where will the union continued