Quasi  periodic standing wave solutions of gravitycapillary water waves
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Title 
Quasi  periodic standing wave solutions of gravitycapillary water waves

Title of Series  
Part Number 
13

Number of Parts 
21

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2016

Language 
English

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Abstract 
We prove the existence of Cantor families of small amplitude time quasiperiodic standing wave solutions (i.e. periodic and even in the space variable x ) of a 2dimensional ocean with infinite depth under the action of gravity and surface tension. In addition we prove that these solutions are linearly stable. Joint work with Riccardo Montalto

00:00
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03:19
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07:48
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12:46
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18:40
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21:38
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27:24
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36:19
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38:37
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44:13
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48:15
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52:05
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53:40
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54:25
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00:03
the the and and 2 and 3 and 2 and thanks to match result for the creation the more fruitful and also for me here at the least until 5 o'clock he said just looking the Navy's abuses the joint work with the government passed quarantine attitudes and make inroads in the office of the that the system is that the same was described the yesterday by Asian market so of that all equation for a Florida international fluent on the part monitor conditions the so there could have been the fluid that is the graph a wise equal and Aleksandr it is ruled by the order decorations and indicators of a mutation of velocity field that 1 this kind of thing that I of his field that has the gradient of the final which is also the potential it before of soontobe compressible and so if there's not 1 function and that the action so here we told that even if the that that could be and has also would found that the hearing things and then it's OK if there is a lack it the EPA is the fluid that moves and the action of vanity and then there which is due and then there are the surface tension the captain letting the effects of the free surface and so has this was discussed in the debate that always liquidation of the cancer was bound to visit only equation the pressure of the continuity of the pressure after the freeze and then there 1 also assumes that the last equation which is a key nomadic condition which stands at the particles which originally were in the on the surface it remains far from the surface so that no breaks off on the way in the way at the end of courses so they unknown so he too will know how the foods of all the answers but wanted to know it's a little city operations and this is as was a reminded that solicits any from the National than the canonical variables are the day to the function dialects and the victories of the boundary of the velocity and the fact that you fired 1 Mendoza the it so big shapes domain altered the main occupied by the fluid and then if 1 The nor was the value of the bonds of the velocity potential 1 that solves all kinds of money can become monarch function as the main occupied by the food with his mother a condition
03:20
and then we should follow the time this is the kind of critics information of which is this 1 so deep that an outside board so the effect is that on formulation of it and sigh this is the moral hazard this image but insisted that this is the and gradient of the unleashed on you which turns out to be expressed in wasn't yes the reminded initially Mulberry out of the 4 1st applied to see physically nobody site and then there is the 2nd equation which is on the contrary quite a long time and that they had been thorny unless it is what 1 should expect namely that done is the full energy system expressed the only through this ,comma convertibles which are at the site so it is the kinetic energy of the fruits that there when expressed the balance of the division on takes of his former mentor there is there a gravitational energy and then there is the energy due to the latter's forces OK so
04:29
effectively system has its own policies then but it has also other signatories 1 it is that it's not that simple assistant this serviceable system from the point of view of the thorn in meaning that simply means that the beneath done and is that even in the potential slack this is the same as the simple but quite the important has we'll have also seen estimates and with so that the Anatolian Agency is right left and right life there is not the action of the systems which is that this evil evolutionary as a square simply changes the sign of site and then there it is natural Luca to solutions which are reversible namely says that the other in time that are even more than that the union space in their in the 1st component which is the space but even abandoning and in that the loss of the potential impact is in all the functions said particularly at the bank is equal to 0 the velocity potential which is seen as a cost of the then there is another property is that the underwater ways like the film that listen valiant but besides Bissell functions which are even next so if the use of these substances left invited them and Though we shall confined to this case so despite these functions of forth I always and we should functions which are given access in this case said all also the vast potential fine is even next and hence said it's the velocity the extol born of the velocity the so the accent is all done in accidents lead managers accessible to the top and because of the bible the city it also mentions X is a by and so the situation is that the physically the fluid that has been in their velocity feeling that was going on and the Exco born and when Texas according to don't always vanishes so that it is like this that is not flexible image of food here the same had answer because of all the by being visited after X acquired by and then the the problem that we are expanding physically it means eventually that is not flocks of fluid out of the top of the wall suspects it is a quality 0 X is a "quotation mark bite and so we really we are studying the standing waves that so is there the evolution of a fluid that we could find between 2 worlds the result of walking in then it has with sadness and burial caves Lynn Mass is a prime means the gun and then in the case of the infinite Florida that is also another peculiarity many cases so we shall always confined to solutions which have also we shall so this is preserved for simplicity we put that the average of the physical because here as well as the average of the past
07:49
OK so we look for I was a bit of solutions there we want to say that it is a pity become a phenomenon that we should expect that in this case said to have most of this kind of solutions I remind what is it was a bit of a solution was a bit of that solution is if a solution defined for all times which has a specific former so that the is difficult position of calling the and frequencies and that means print frequencies so is the composition of all a function you which depends on the new angles so these are knocking ghosts and witches and belongs to the and so that function Luisa provide periodic few wanted by periodic fan combos that with the the building of most members so it should this might have also had their invitation with frequency omega 1 on an again so when and is equal to 1 that there is only 1 frequency his periodic functions with the frequency of the had to and so these are the uh interactions of more sober positions of more frequencies on day of secular frequencies with the listed on the 1 we so that if 1 has OK the frequency omega in order to be really quasiperiodic here has to be independent film to assumed that the Macintosh only has no leverage over relations so Omega not case different from 0 4 4 on integer came in that the on which of course not it's also shows of this kind of solutions global development the Fund for old times and from then on it the music the free space so that existed in a manifold ,comma times that guy and and also the 2nd phase facesaving measure of his basic this solution made mention bitumen starts here and then the solution will remain forever on this manifold and actually the fool applied to this initial did will be nothing but that it remains that it is nothing but suggested that operations officer for the leading all so the dynamics of restricted on the use of the dynamics of the notion that there is little manifold is just that this year this In the beginning
10:41
AFP wait Luca for solution solves and so of course it is a highly wanted to know what is the dynamics of the unionized equation the flats portion equal you we the a loss at the size equals cuisine winning eyes of the the order question and we get this system which in this constant coefficients since then where'd you observational but on the flat surface is just that the queen with the black models of the accident and then no 1 will care for it's actually so these are all its lenient who are all standing wave solutions which are even reversible for busily attic which saw even the next so we have expanding the nicest business of presenters of generics are unloading the origins based Michigan 1 the 1st of all and that the time user even the velocity potential the bands of all the accentuate are on so the linear equations actually has these the structure all the solutions of the region had the stature however this form here even with any the national forests here with infinitely many fragrances because he would exceed his Fallujah gratitude of 1 and as this kind of nonresidents conditions on the coauthor of course it would depend on the leaner frequencies on the values of the parameters gravity and so for Spanish according to to the values of June cakes that are different universes compassion and the question that there will be in the William boson is no actually whether the solutions of the question can be continued that was for the solutions from the 4 million integration for most of the fact so
12:47
the problem is that quiet their involving the and so effect we shall continue solutions so which are supported for based on fact in many mainly with Baker finitely many industry integers G 1 biology and and then we look at solutions which have bottom in 40 on these sites the other modes are authorized and then at a rephrasing but in this case a very billionyen equation has uh let the which are explicitly is 1 of the guys angles the 1 of the 2 feet and a which supported the frequencies at which are effectively the vector coming out of the question which I know it is in the wilderness also were known from the mechanics and the exceeds 100 years sentence even more that there are elements that were not less because the column of the 2nd war but the vector what is what 1 would be important to know whether the solutions which persists and that the Commission is to know whether the frequencies novel solutions satisfies strip the most prominent resonance conditions and particularly for each for the frequency of active satisfies the conditions for which Omega Gold and the is not the only difference resumed but from the quantitative point of view if this because a lower bound on display here which is a socalled indefinite and condition and then the point here is that the the building efficiencies that the call here are some of the new frequencies defense and he and King Kong and wrote that in the month J at the frequencies in indeed that 1 would want to ensure that the foremost the values for example of fixing Jake for most of areas of the city suspension but need that these vector of his and dimension vector actually for France this conditions so that along the corridor Omega capped by the vector in the midst encompasses a lot of points where this nation's so utterly of initial it's the initial them starting point in order to what hope for the call but this solutions would proceed schools in the nonlinear systems is satisfied so here we are in a situation which we have the meaning of season which is like an internal system and only narrative was 1 of the solutions is a scene has at best been so this kind of problem has been studied the considerably in the last the faith of the US under the name of overcame theory which has been quite developed the force equations like many people here and there but the fact that we expect that the stopping distance the total the year so there is motivated especially from waterways the usual for questioning because fact most of the questions of despite the end of the 1st
16:22
results of the thing the way the it's command and makeup don't go for and bipartisan concerning the fact that clearly equations so blasts a wasn't enough more than 2 and there have been infected with the developer filed a development visa a strategy with is what I would describe but by the fall the water OK so far with way said that out would original Nissan results of periodic solutions Starting from a series of remarkable works said the ball in the Balkans girlfriend just brought in installments from different cases so when the 1st step where the difference the water was indications of followers were spending waves finance or portion and the the of would find them who will be the 1st results proved and without capillary being the 1st results prove the existence of periodic solutions find the that's only under reaction of gravity and then various Poland in the case of more difficult case of the Napster users gives it is more difficult because it is an OK this indigenous can and then that's with the only drafts In the case that will the capillary in fact which is exactly this case there has been sold by violence and while proving their exactly for the system periodic solutions in time and then there are also some related that his answer all the traveling wave solutions which like Yukos and which in highest I think that the main problem with animation thinks a periodic his leadership from left what it is not as has altered the natural persons exactly continue there is on the violence among the blue weather was appeal solutions proposed periodic and we would like to to see that OK so for
18:42
concerning quasiperiodic solutions the previous works which trying to understand what is the fact that of the letters and then only denominated and the former vector of the warm weather but perturbation conveys many that this was done by quarks enforcement and company officials I have been here 1 day this is of course still listening migration because that is just 1 that imitating the information concerned with respect to kids and then informed the last such fall the playing Gordon equation there was a work by Bush again for periodic solutions of this is that some sensible mythical too because it is less dispensing equation and then there is the more difficult and then with the Kadeer's permitting approaches that will be given a generic example of quasiperiodic solutions for the Steelers playing God equations with singing preparations that emissions so we've got word of the of depends on 1 day but different than when they in space and this 1 has to assume stock shot which is measured section which is relevant that which is analogous immediately this is annoyed the sublimely funny structure of this equation but it is natural to expect DB structure the rest of us actually uses a structure which is compatible with the presence of the quasiperiodic of solutions to control the growth of stubble of norms and there is so also for present and equations effect and has a nation before the visitors answer will stay with around applicable about the concerned that given the blast and 1 other enemies daunting political division which has a city that is so it is with a so which is generated by no means dawn in where the density depends on a gaining on 1 space there and then you want when where and what writing down the vector field in the medical field 1 rates but it is now has the shape so it is advising at so now let's go back
20:52
to what the waves and wanted which is precisely the is out of prison .period solutions India is for this stuff so this article wishes and it's a noball almost spent so that was a bit of solution there the frequencies of that was a bit solutions is not fixed the priority it will change according to the nonlinear editing when and according to the amplitude so if the solutions with downstairs busier than on inherently weak the mine and we have on the frequency and so we look for it was a bit of solutions with frequency omega is that we that which I have to be found that the book and then to the
21:39
Internet is this 1 in his statement that is the usual resulted 1 would expect that namely that exist solution that was a bit of solutions of the system Bloomberg other than that in the following way as said that they take it any of fine subset of indices as such but they want G 2 barges and then I think was a periodic solutions said Out in space said is based mainly supported the on these other monarchs consignments objects there are mainly their amplitudes are mainly in the main part of the solution is by the leading equation In fact this once but the frequencies are shifted the fact that there would be if she staying the frequencies ,comma gadget that the frequencies omega GTE we've banned for billing at frequencies of oscillations when the altitude excitement was not so when acceptance year the frequencies With converge on this 1 the frequencies ,comma education that has a 74 has to satisfy streak stronger treatable nonresidents condition as a consequence of the solutions and not exist for all the values of the parameters but they exist before most of the hundreds on this planet and then we say like that to that effect so for most of our lives all the surface tension by for example considering the gravity G is equal to 1 of the solutions in fact that expected sort of quasiperiodic solutions the and the an extremely important the property for the process not only because it is interesting from an American point of view but also for the proof is that we know that the solutions and linearly state I want to can explain know what they mean and it is only stable in the meaning there of dynamic systems I would have liked so this is the typical the results of 1 can expect 1 can of course also rephrase that his answer saying there 1 can for most of values of the surface tension but I mean that seeks 150 equation for most of values of CSI there exists a solutions all that 6 now some comments
24:19
that it is natural when the 1st thing is that the fact that the bodies of restrictions is not technical candidate visitors distinction on the parameters is not thinking Yukos otherwise the 1 should expect completely free from this is the typical situation that 1 should encompass unit consistency in namely the outside the Morrisons phenomenon idea in the steroid and at the end I completed my friend that solutions and of solutions and the now there the least since then but there's been discovered before fact the even probably a form of proof lacks unless is lacking but it is an important animal or shorten the war that that only the Internet the commenter needed and the for example are also if 1 looks at the dynamics of the 5th order since then the approximate system get out there and that can be sources of instability is and calls can be and the dynamics and called the will of the proposed and this will lose the values of the past the choice of these values of the parameters of the G & cup of course of its control of boys and these effects is the charge for which 1 does not see this is that instead instability and similarly if he was there in fact has turning to political vision like that reform was the values of G & cup solutions within the shell zeal it exists for all if only for a long time if we don't know if solutions he's for troubled times and the sewers and censor the effect in the previous and 1 in fact is a selecting the initial conditions for which actually solution of Lloyd's forever altered residences and so it is survived by his ancestral past navy in between that can be decisions swearing on the contrary can be brought back from growers and would be this other kind of buybacks and so this is that picture very natural pixel at 1 would expect of the complicated dynamics between their standing in the and then the hyperbolic orbits horseshoes and the completed OK so I said that at the request apparently solutions a profound that in stable they are stable reading the sense of the dynamical systems were really the sense of that In the following sets what In this suitable set of coordinates the bed linen Isaac system that unfortunately things I
27:24
it so "quotation mark so in this sense of
27:29
an emphasis of something that always had an uneven insisted in the interview and then we will have a solution of time in this case it it is a possible solution and then will utilize the directive will now find that out to listen and solution lists the lives of pollution and then we have any questions which is of course a living with the band and the quasiperiodic but the question of the so this is an act you know up which depends on the not hitting it a of the question and the minister made clear is that the bulk of it was a barracks of an object like it can be very complicated and the whole point the main point of all the proof of is that we are able to tool 3rd goal to control computer that is a production with analyze so we are able to compute its this and that the expansion of all old its eigenvalues voice frock exported and this is a sense that Court of them so because it is denies a question this and I know I am able to control expected costs I'm able to control was holding a license to Kennedy and effects of the precise thinking it admitted that existence with the Bulls set of coordinates would coordinates which invigorated the inextricable set of coordinates the isn't very complicated the debate leading up to the guns this study is musicals and coefficient system namely OK the NY outside mansion coordinates even when the dollar rose from mention bones in the describing the tangential and the normal the dynamics of the Taurus is qualified fight if you want the mangoes but then let me see above the 2 agreed that you threedimensional about that visited the site In fact business meeting energy needs so that that now forgetting their action Angola dynamics is conjugated there were constant coefficient the absence of any kind the composing the mean infelicitous as also won that made so that this is a really are completely different denies in deciding once and whether they are simply out monarchists and we the language that I could injure we have in this case the real thing and for which in the next slide that which should also so innocent public expressions of dissent so because of the this so if this complicated the PGA denies the question of the system has been conjugated this is a sad for these constant revision pretty so this set of infinitely many the Monocacy Lagos the Cabinetlevel then of course and we new J which is very unlikely to then of course so for example 1 has estimated that the summit of normal often this system does not include 0 and I originally wanted could the weaknesses in the frock exponents of the some the focus points of the solutions so have cost final everything there is to know them with then that I completely controlled everything about the dynamics here and there what the infected demand and provide an asymptotic of the user food export this is the user ID and venues such as the following format would still can't this before the other 2 frequencies where footage is equal to 1 side executives 1 week ago because of the ones which have to be corrected the collected by a constant which is coastal land and when the solutions that Esposito Wisconsin's with Tempel 1 Scott act that these frequencies must converge to the frequencies of Western nations blasts is lower on the collections there will be another collections but the 1 thing that is a my cancelations due to the signatories of the equation that is not the the the the correction laserbeam next collections of all the onehalf again this is another reality small corrections other collections .period so what if I want to that would provide that at any order this symbolic expression of so the idea the food and 1 the use of the promoted by an artist bitterly fiction of course as it should be offered the lenient attitude with references and that is of course a fundamental information because it allows them to control and the preparation and the nearby dynamics the 2nd information :colon concerns at the change of valuable source said that the 1 can contribute to his linear B To this end constantly efficient then took to the system through a change of animals have to control and the point is that the discharge of vitamins actually maps always business expenses into himself or for any NES Howard logic why this is a relevant information because so it means that essentially is I wanted information of possible if said when I can achieve these conditions here which is simple and then there by the change of valuable I would have of course the informations about the summit that ended the illusion of the initial data in some of norms also from for this so these are the 2 fundamental information so that exist so no wonder it is expected that Inglewood that Spector analysis of the denies depressed so the denies the breadth of his conjugated through changes of light analysts which so was based command set for any subliminal into something deeper another system which is a device that and then we control in a constructive way and that precise with all of the asymptotic expansions again it's clear that this information so it enable to war and the program Canada's more divisive issue what this is a it is expansion White is a small device of the future paraphrasing it was quite as smooth as a problem we face it this was will look for 0 7 of the user the nonlinear function those of his own operator where 1 look affordable embedding sold 1 looks for an amending good part of the solutions but a blazing did begin with the Omega the feet of depicted the gaps and president of the sex and about the so 1 looks 1 is to look for the evils of the user opening when he was equal to 0 because we want a montage of solutions we have with faster from the flat portion with the 0 velocity potential and then for the EU's expansion only wanted to find that he does and the has a function of the Bandidos by 1 and then 1 it's interested in the eyes then the operator which is essentially what we did before and this operator is this 1 at a constant coefficients and for his basic and made is in this way but then
36:21
of course we see the initiative that we mentioned before namely the is the development of these medicines was once and here there is a need to John J is any integer Saugus numbers always reluctantly to the Zito when 1 cannonball was the foremost of vendors of frequencies and again by discussed the lower bound of on the site but so that 2 billion as the peripheries inedible but it's the best operator because of the presence of Faxon was there and so this is the usual to modernize because of that public resonances the investigators it and the continuation of these audits it is not based on compaction
37:10
and then a way to solve this problem is through nationalism implicit function the amendment which uses has his own knowledge is made that on a new fundamental the horror of looking at this year's profit function given the and proximate solution you when 1 look for a better approximation you enclosed land in such a way that this is something divided the intersection with the condensate and this is the it that the scheme but the fact that ended the advantage of this Kemakeza has it is well known this aquatic scheme for the distance between 2 successive approximations listed on the square although they have recently Canterbury was 1 these are so fanatic who this but expanded of convergence enabled to compensate that is more divisive difficulties but the difficulty is that in order to write for example the skin and we needed to know the denies the operator after a function which is not only is it up like that you different things and so we are in fact accounted for in this project problems to show that the building owners operators using vegetable for more survivors of the bombing and that it's the satisfies estimates that the namely that the the Napa Sullivan on there can fixable and it was meeting with the respect to the right .period you where we and
38:38
this is the end of the shift is better than the 1 bridetobe denies the operator for the what waves said he it has had bad the shape at is that what this is the 4th the way here due because the additional Predator B and the functions from the ingredients of the velocity potential after the free surfaces OK and so it is at the point is OK because it is exactly the operator here and that we would like to convert to prove that this is the foremost of of bonhomie doesn't approve estimates for last and so the proof is quite there is composed of main arguments that was not the to discuss was all alone but the real capabilities and nationals of proof that the accordingly has a here all had put the conjugation of amendment in order to to take away the vision with the parameters them that I will discuss a bit this past year namely the analysis of the denies that because of course of from this is an essential to the past OK about but only because the gender justice state 2 stinks of course has reset the body is all of this phenomenon has a view or not season on time really depends on a very complicated way on their campus at the time he doesn't affect practice which appear that I really complicated accountant and expand their soul really it's a very sensitive to know or not I hope to fulfill 25 nonresidents conditions on the front and but in the effective the 1st accord will governors out their work before weirdest that is stronger than on generous a condition that for example was not satisfied In a celestial mechanics for the for the commander for the solar system and that has come a lot of news and then factories for example was a motivation although we can indigenous to conditions in political life and that and that is that have the right figure also of the sources that we will not satisfy the weakening business conditions here I want to discuss in here which was a and use the surface tension but I them off easily from a physical point of view which is also equivalent we can fix cap but and the considerable has abandoned the wavelength in here I put the tube by about 2 by land that is the end the parameter which enters into the waves so I fix the equation this solutions with existed for more survivors of the wasteland of the over there with and in that sense the same or 1 Thinker also to for the index to the depths of the ocean except the OK in this case is that the ingredients which I have and what we used which are important solidified his we on the generous conditions and the fact that the visiting the frequencies are indeed the analytic functions of the surface tension and there at least suitable as in politics at Infinity group of Jakarta 5 3 house and then there is a suitable number Janessa conditions and promote the condition that they explain which is fact the school the Gaussian condition which I think it goes back to the wall was precisely going up there saying that a but also problems under Russian people and that it is just to say is that there and if 1 thinks of the fragrances 1 looks at the founded the state we don't want to end then that image is not contained in any heightened so it is a storefront conditions there's so that the curve Omega trapped isn't which is an date is not identical contained the 8th when I saw analytical for any vector in and not leave the function of city Scott products Omega account but is not identical so it is not containing hyperbole that is called the abortion condition ended the April in this case in fact it's a competition that 1 they get the frequency of actor and the 1 shows that it is not contained in and I think and 1 shows that these vectors are indeed more attainable by the Finnish issuance and indeed that limited act the from here comes out Monday among them them on the continent and then 1 check that there is this :colon montages conditions .period then
44:05
that's OK and this is what they say about the the body and then I started to make the about the pity but that has
44:15
said that concern the analysis the spectral analysis of these timedependent time and space dependent on which is the I think in this way and I repeat that the goal is to find their conjugation suitable change revivals fight which contribute to the sense when opening in a very simple British AlQaeda here I will be that putting the operator arrested at the border directions which I normally do they have tangential dynamics still OK and is that the quota is to compute the existing public expansion of and there always in the same spirit of the book market is abandoned in a conceptually different the steps the 1st step has the goal not reduce the sides all the good news is a constant revision but that sometimes but that the 1st step will be noted too worries about the acting thinks and to make it an atom square and so on but the 1st to reduce it to constant coefficients halfdozen without them which is in to EU relevant because when I hear that I was effects on highfrequency said for J 3 house is extremely high and that it is important that they should control at that time but it is a very old biggest contribution to the dynamics so the 1st step is to portable constant coefficients hazardous here the highest of landed 3 300 constant and then also will disappear the order liner again by the signatory of the city and the than the older than the X 1 half of which is always an unbounded contributions 1 hospital because of the fish when 1 of which is that all that is your people care about this a sufficiently while the effect that we have known to Union's situation at this point it is natural to stop a hairraising that bonds which are excellent and an end to all square at some 3 and saw no need to performances and not 1 4 times this is removing so a little steps 1st to break the news articles and coefficients because seemed was and then they have bid to to reduce the size of the to jeopardize the effect of these so we needed to have his presentation for that for the United States and that's OK this is the case it is news will be attended by conjugation changes of mind was banned abandoned changes of light if we have a system which is a them in this case quest began thinking time we conjugate with the change of mind which is in this case was a chaotic in time then it transforms into another is the form of the new operator is written here that is the conjugation of the space operator in jest Western attitudes Fernandez 1 8 feet and then there is a better which canceled the contribution of the gives this conclusion OK and then the goal activities is to find their 7 atbats formation aid which it in fact that we live in meaning that look for suitable any such a that abuse me does not depend on space on bank and that it is very that it is what that we would like to resolve was a of the use obtained by compositions of several transformations change the latter was over there to defend the nation this is the 1st
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year it is that I started from
48:18
assistant I make a change of animal and I think users this agenda but it with something that this is a very judges lot that is the rules on the whole changes part of time and the part of space and then I will chose a in order to no it won't be like that I would like to do that if I don't win 1 step but in all steps that meant that you came to the the idea says Sorry sorry I said I wanted I choose fee which is a valuable known also as I have the status of course I check nutrition that the system with the vector field a transforming consistent with the defendant is adjusted transformation nothing and now we have to choose and prepare with the with the naked with thinks here there local experts say that in this context that we appears they wouldn't neck in the following this is the unionized ability effectively this stresses that the analysis the same has another about it then it is really operator 18 there with these a valuable there were norm which needs the conjugating that system with just his Maddox amid the multiplication port functional but the system called the contributing another problem we this 1 which is a much nationalist semantic and has also by John here so they it's that eigenvalues are in this context a purely imaginative because of an audit by the by I so the agony message thanks to this with knows that invented so we can spend much of this is the 1st change of which OK has visa change of valuable 48 is a Howie we said that in the beginning there that the Belgian right property of the system are important here we try to preserve old ways that condition because it is important it is necessities obsessively that and finally that the recipient in order to really the existence of this kind of conditions also we want always to preserve his property that this transformation president of the city so that wouldn't normally reserved a Sunni actually so the net amount from President Barroso listened consistently that measures so that we could as but nevertheless there for rest of it is sufficient and that because I've always listened but the character of the transformation is the 1 could be that much and then again has another there are some changes of valuables to which I contributed the from office while the dollar was on the rebound position of time such a bad that we are able after this change of animals that do not have to constantly fission the behest of the facts the action buying explaining what 1 has obtained the Golan Heights tolerance and then OK 1
51:21
has lived still some jobs to do this effect is not an equation with this system in Asia "quotation mark and the then and also the next goal is to seem many of these up to smoothing over the last fall .period inundated the off they had another dams which acts on the Cole born of each to pollution to support the proposed already negative order so standardization activities moving the operators so that the new system will be constant coefficients still viable proficiency on the which I act on each they have another once but then and all the argument initial becomes extremely small will say
52:07
something very good and then the next goal we want to do uh we had Vernell with the affected of at the 1st daughter is still viable coefficients so in order to do the same we conjugated with the floor the produce the rights of like this 1 and the the floor to the pedia like is 1 since we'd aid which is a valuable the reason for the Boston between each incident which France is nice sat at this very much and then there the reconciliation that did the conjugated banker we wanted to to analyze it when it satisfies the duties and has an equation is 1 session and then is equation has a solution that can be sold into the casing symbols thanks to the fact that the gloomy data infect the gains 1 of the Act and that because of this 1 can solve this equation and find that there a contributor the data decreasing seemed lost the new denim in front of the accident Is that is modified and then 1 chooses the function aerial acts in order to put the user to cost as it turns out that 1 can put it OK and
53:28
the same significant the additional How much Primary One is finished OK and then as well that is of the Tibetan again but I want maybe I see the Sundance and I think and then
53:42
just 1 water so and after this conjugation because we wanted to because the sides at this point we are adding a singing out situation and then the transformation the
53:53
analog of the normal form as this 1 we want to now to find the transformations which in the committees that the preparation from sites excellent besides at square so what conjugate there with the floor Over a function W to be found the new operator can we analyze the gain as before but then we expand the really empowers of absent and then we look at the new Of the new excellent and then we want to choose W so that the Vista appears in doing
54:25
this the was sold it and the
54:28
1 that is able to delete that III for this function of you and everyone how to obtain his winter from the bitter mission 1 has to divide this through the finances of ligand some on the face of it in medicine and so appears so that it is a necessity to boys and this the analysis and proper approach progress but there is liberalization process that 1 has to impose verified these conditions which in the original has the 2nd of the monarch of Morrison's conditions if 1 is able to is 1 firing up there is that thank you and