Some Landau damping results for the HMF model and its discrete time approximation
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Some Landau damping results for the HMF model and its discrete time approximation
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Some Landau damping results for the HMF model and its discrete time approximation

Title of Series  
Part Number 
12

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 consider solutions of the VlasovHMF model starting in a small Sobolev neighborhood of a spatially homogeneous stationary state satisfying a linear stability criterion and prove a scattering result (Landau damping). We then consider time discretizations of these solutions based on splitting methods between the linear and nonlinear part of the equation and we prove that the numerical solutions converge weakly to a modified state which is close to the continuous one. We also prove that our numerical scheme is uniformly convergent, with a convergence rate of order one for Lie splittings, and two for Strang splittings. We will also consider the case of nonhomegeous states for which actionangle variables can be used

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Point (geometry)
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08:52
Logical constant
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Gradient
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23:42
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29:40
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40:13
Area
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43:13
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50:25
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as and if you thank you thank you for please and thank you and think I would like to thank the organisers of this invitation and then so I would like Zulu division of the of this work comes from I am an American and and I am particularly interested in them quantitatively every often America's schemes 12 and I would explain these modern days but uh if I wanted to go to to find some of the results concerning the longtime leader of numerous schemes applied to aggressively equations known in of the question so the idea and ending this is driven by the London Dumping effect also the very 1st thing I did was to open the will and new paper on and and and and that you would have been to to supplement earlier the parameters for the discrete parameters defined and stands for you don't want to do it of course so I tried to find some sincere example aware of when you can study the longtime Universal's along the dumpling units in a single word and my and the and this is a reservation of mothers reading from France's is the simplest because of weather and extend and 2nd motivation is I really wanted to play with 110 structure of the press of equation and pointed to some moment and and in end yeah it is completely healed and could be the technique it has nothing to do with any assistance but I read give kind of interpretation of that on the dumping and doubled in term of from OK so this is that the introduction so I would stop is something extremely database just as essential to introduce patients so so when you have a dynamical systems can be interested either in the computing categories uh or computing the statistical solution it's and from the new legal point of view is it is very very different because you have to solve it would mean that you don't have .period wanted and that and that's stood to work toward enemies assistance you can also associated transport equations which means that you if you want to look at that the statistic of this edition so that it would never own observable on the dynamical system then you end up with a new relinquished ended up in the heightening case said that they will you considering the and the rest of the talk stood in the transport equation can be retained his way so here this is Michael for some market extend the euro also X is the space of valuable invisibility July to each other dynamical systems which is Hamilton and you can write dissuasive J. here a syntactic too matrix and because survivors screen is essentially essentially is only really equation and the transport equation of essential Grenoble will focus on more to assign and you can make it this way so and in classical mechanics typically you have on you will have such 120 year became acute care could up and potential bomb and then there was the transport aggression is written this thank thinking he recognizes that the plan spots in the transport equations that will appear in the reserve equation OK so the satisfied the question for you if if you have a fixed uh fixed onion uh that's nice she should consider if you want to make some humiliations and very large systems when the when ended becomes extremely large essentially you you cannot you cannot use the for for the corresponding to the it's just but you you cannot solve the root looking at the transport equation in dimension more than 10 inches and then there and so then you have to rely on and on and on other models I also agree you make some some kind of modernization and you end previous Yoshio models will tentatively rest the incident and that up but in many many cases so so we will give you some examples then you have for this nonlinear transport equation so it just means that you will have a lowdimensional transport equation here means that you X and geologist we'll be of dimension of 1 or 2 but the Mets the Hamilton is no longer constant in terms of its missiles 110 edge here depends on of it when the functioning sensitivities anointment money upon spot equation that we consider all the obvious 1 thing you can do is wait and Ed Skyler uses the and energy and usually if you did this but that this holiday here is the of overuse of functions so you will have its from from the classical mechanics he inherited a sort of and so the benefits of his other 2002 migrations 22 more that's into simple truth so issue in in in in twodimensional should get order a question like this then facts of issuing posted to Bureau of their divergent free you are 100 and this is the magic of dimension to it so that if you just take about a city of about of you you can be right to question this way so that other Nevada cities transport might buy that institution and then the issue of the future the brightest and use of abusive are you in the prison which the fact that that's the height you is 110 yen and 110 in is just enough that a process of forming a sort of migration can be it in this way and in a very convulsed with and because this is the structure joint presented before a new in the sense that this is officially chief of the enough on this is which is quite radical and the and the fact that that there is something very strange indeed in all this in all of these things because if you have so if you have a 118 I have attended the funeral In fact it you will so issue we should be fine this operator so you take things a person like it's a passport operator but then if you look at the commutator also mentioned in the commutator of 2 guys like this and if you look but the the operator associated then it is the same as the comic commentator of the 2 agreed so in fact all these equations have at all the next batch of judges and have found the bodies of 4 probably just saying this is you just cannot do anything with this expert but that essentially this this way of writing the equation just is the same as the she writes it's using this formula in terms of the operator or if it is just saying that you have an expressed through its new coaching thing of all you have experienced it but it's not but said that the OK so so now all these equations to through all this equation written this way have I I think that it just say mortgage it does not really state more than these other than the fact that in order to preserve the volume and sense and there was a and got of the flow and and in fact that you have the preservation of all the to In fact so so this is the the highmaintenance hot the morning of transport equation considered so you of this defines here you wish for some market between 2 to between 2 functional and you recognize you but this is the right hand side of the morning of
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transport equation and so another way to write this equation is just to say that for all fall functional the evolution of the functional is just given by the person market proves that each where inches Hamilton conceded and so it is just an and operator so this is a very very very structure that you have 4 4 for me is the wonderful pulled the rest of equation is of course energies preserved and if you look at all the observable that are made of Jia festering the deal then in fact the frigid is in his G prime injured pride of 5th commuters as so the Justice Kansas so you have an infinite number of Annadiane which from the numerical point of view is of course complicated so in Patagonia hobbled so in the present case and another case of the functional H so so so that the edge of a freeze on DVD square even to press a potential the event of of the honor F a depends on F this through a convolution here in in X in XI so you have a governor here the and the rise of West migration so that these just a related to interests the levels of operator parliament and and the energy that can be the energy functional is the sum of 2 terms so this is the first one also the fashion that is just the square and this is the 2nd 1 the rigidity is at this 1 an and it is quite logical and that fall from the point point of view of Hamilton and then exist it is there you can see you can see the this credited backed as a perturbation of indicated the connectors OK so I was elected as tool to to say something about the time discretization of of these equations are they said the flow of such integration is essentially an exponential and because if you if you remember you're here so did you equal to other edge of genes and that and if you means that a new hope of course to gives meaning to this this exponential and usually you have to 20 the review the sense of what percentage of expansion but if you if you want to do numerics in some sense it is the same as solving imaging that now why user is a matrix it is the same as a sort and in any major addition efficient integration certainly group for example because of a has a structure for for example stressing that and you have tons of methods as is the very first one for example to will to freeze here this is going here and to solve the exponential on this and at the level of the means that if you have ever end of XV which is supposed to be an approximation of the exact form of solution at the time and Powell then freeze the new soldiers equation starting in F and and then you end up with offenders and this is typical of the kind of method that reduce and so on so formerly essentially you you can write this if is interests the problem here is how do you compute there what's the weather which is he exponential hustle how the conduits this definition of this equation we should expect under control and the solution you know what you give Rangers you just use characteristics as in the 1st in and you have the solution the problem is how to compute and and and this is a disease that these are essentially those 3 ways to to do this either you completely discretized so edges given you want to have a solution of this equation in short time so identities because commuting time using finer differences become involved but then you essentially you you get you have the troubles because this is an unbounded operator to introduce critized you really have a very severe restriction restriction the in the Times that you can use but economic Dudley essentially drops Patagonia and unit and them evolve through the floor and something you try to get some information to the problems the dropping practical because you have statistical and you have seen in Midland which are so the ladies now you fixes agreed and you you use characteristics to talk to 2 moves point as agreed and the new project and agreed the main problem here is that so if if White Jr the the point of agreed that and she lets if you will let's this .period evolved through a flow then you do not independently so you have to make some interpretations and when you I couldn't operation you create some our numerical it when you lose computers instructor and so typically you will get tout as a P which is essentially the the and the precision of your skiing and a space that all of which are conveyed this way which is due to the interpretation because each step you making an interpretation on all the dikes but you can play with these despair In fact use extremely high S who assume that on some sense very close to 0 a true story and just say OK so looking flat and committees and focus on now I will talk about the visit was stored in the reservoir Jennifer accretion so here as ability to mention so the reader significant reference may cause and 52 the universe and uh said we were reading with regret was that we were reading some the judge or those of those events of equations because of this all references and I think it is between you and the skin often opportunities and possibly remember and you had a reference on the paper may and then is reference to be a and never see it again in the discussion and this guy is announced last time knowing in Baton Basil E. I'm not sure if he's alive and if he wants to build you would think he's probably if he leaves his career more than 100 and he wrote tons of books but everything in so there is a big a "quotation mark where and when you have a approval of existence of the community's future what it once so if you have a lot of other names were mentioned to drop out from a winter of also could have been thinking of course of course of course is 1 of the interesting but was areas what it but that just the opposite and isolated pattern was to see if someone would have to mention think have a case Silbert said so as those of the Hindustan says tell them I just want to mention is that I will work in such spaces so from the father spun off the Nigerians and within you have to control also moments this community support and the support of the function of the any facts so here to begin this is the kind of results that you have a essentially season it was 1st written by manipulative so if you start in summer dash would have known and then you look really cheesy Union Minnesota phenomena work force of the law index and an abounded in this industry for assignment OK so essentially if you future the show's most enough for you have local existence of in highs with you and this if you
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want to do numerical energies this this is enough and because you if you want to to have a congressional approval of approved then each enough so here I just right now so what what all the best schemes for the rest of Western equations I think I think that there are no based on is the competition between the the field of Sicilian and nephew and it looked as in many examples in fact you can solve them explicitly up to space discretization so all this 1 it's clear and it's just a characteristic exact solution and for this 1 taking the cue because the integration you have if you look at some the money now opting for would his guns if you just sort and he it's the same should this equation without the rest of the year In fact it can prove that the size constant because it is just the way it does not depend here if you should change you to Eupora something that depends on excellent as business change we'll have 2 the exact exact it's exhibitions and then you can make some highorder speaking so you combine these 2 and use bigger common over the phone line you can derive reply on them and so it essentially you increase the the time the time it all prosecutors decrease the attainable and ends in a few users McAuliffe the negligent and then you can use height although on a operations so that any of the techniques to the there can it can be a century of accompanied blotch on the C17 that is used OK and the focus so typically so so these 2 sides just to 2 to finish should to talk about them the other than American so I think France incense in some sense there is a there is a 0 and that that produces many people with the various that you have been that is a rookie year cheats look and would pose on the way with in in this summer's space the chess and if you use a middle the middle of Order of classical although he fondled the men you have and that's the have at times the Linus and representation in which this is accompanied by constant that and finite time helping the end of course something if not engaged viscous 100 loss of success as prime as promptly as this trying so this is this is a typical convergence prove that because convergence result for a kidney you know you are you fix the finite Televisa is is there is a discretization parameter of consultation comes in town so you fix a finite time you lose I'd see during the estimated new control but there and in each you compare with the classical American gave this is a room in classical mechanics of course it chest and it's been fine could be compelled but so it's interesting only if she wants to go to to compute exactly a project on which was which was done indictees from which want to send satellites to the meeting of the warmest you have to to be extremely precise Humphrey and or and on but of course this constant here is exponential expansion of expression of genes and in some cases and in the classical case is just exponentialof of key but if you can have an expanded at an extremely large constant here at this and that if you want to come to the project area he that the Florentine means that you will have to take a italics is essentially expansion of ministers so you just can't so if you know if you want to increase the time I think we need to which is a much more than a good question you know that you use Atlantic but you want to have some of the globalization drove the fuel that is president and this is the history of this question in fact so it it's started especially in Estonian moleculardynamics because moleculardynamics you have other tons of of of of possible trajectories if make use of moving everywhere so the numerical scheme cannot catch them absolutely but what you want to know is that global property of simmering fury so what you want to know who he is more than a soso what is more important for the physicist is to do it have a preservation of the energy have a preservation of the momentum nor has the beauty of the of the products of solutions and so it is in the in the in the 90's said it was the assistant integrators in the case of the motion of classical mechanics were more and more users and it was only in the 90's the 2 that we understood why you have a much better behavior for syntactic integrators and the proof of this meant that the important thing here is that if you use a synthetic and in fact it is it is a very old I think it
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goes back more than once in 68 this it would say that if you can imagine that you have an act like this but she is close to identity endit and synthetic and eclectic then essentially Phi Phi Tau Fido coincides with the exact through to up at the door where fighting that here it is the solution of the other at the time so if you so so it if you if you mentioned that you have a discreet but the practical difficulty with mother said is that OK you can interpolate this trajectories using the 120 and 110 inflow independent at the time because you can also you can always do this issue arose among some time dependence you but in a minute and the time it is much more difficult and you have to pay the by an exponentially sport and so must just wrote in the 68 paper Kate works and then it was only given a job duties in the nineties who proved that who made the world bring speed estimates with the exponential remained winning the and then it was also approved by the high beach in India in any of the 90 close that if you have a synthetic integrator then you have a G U of a modified energy which is close to the exact form and his energies prison and once you have a piece of energy then you can play then you can use an educated if you want to work for some stability during the year .period of you can do OK until the woman in finding dimensions uh and this was extended to 0 2 0 2 2 standing up to the on waves and so on the value of essentially by but the end and ideal for this city so in this case you also have been is essentially this is an extension of the but you have to work so you need to see if conditions you have a restriction between them the time and the space and the spectacle and in all of this you have you have some numerical you have numerical resonance that appeal and the numerical resonance event understand understand them very easily he imagined imagine that you have something like this compressor may be pressed so close on Monday nights and long days it is a huge metrics if you want to understand the behavior of these the problem is to to to look at the person residents of this distance just in the enough cases you will engage until he or she do some leveraging the essentially you don't have to control all this sort of this sort of crisis and they have to be different from but in fact that the object if you do some new )right parenthesis this is not the but disease wiped out and level of to fill this exponential of outer times from but you fixed up to essentially this is this is the map Of course here in this case this is an exact filled with imagine that the letter you have prevented the sleek here many approximate played "quotation mark but so in fact that you not only have this residences but all the residents that from here and it and now we are on the stock we have much more personal residence it's that it's not this but more like doubt multiplied by the number of about 1 per person that he has to be no way From time to time which we are special numerical reasoning was that that's worsens the situation and so now about the next goal was to extend this to to the standing of transport equations to the very 1st question is what is the analog of forget submission to leave came to the abridging which is an antelope for holders of which if you if you have no incentive to some money now the Narnia plenary questions you Kansas City In an energy to to provocative the stability so you need to know what do what do we need for the questions so now and I'm going to show you some typical behaviorists of fomenting forms but equations so the very first one is so I was just consulate polygon equations and then we start is a good move all through school through I think it's a swath through it is known to be in stable still so you you start with the new stops was your similar
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caution skin perfusion inconsistency OK so the new workers accuse last year have creation of smaller structures but that killed by the numerical numerical dissipation and then you end up with something in this something is very particular because he looks bad Swiss this mission and this is an important reflective on on on and then it is extremely stable because and you see you can continue the table to time this time around talent so it is extremely stable signals that the public that they exist because now we have we believe we are computing rebreather on and that's the the districts because of the new and because of the new visited the station because now it in some sense inserted in the station WBZ prior conditions resulted in proved that no you have to have a solution of the and I want disease after you have it seems that you have many guiding many solution to the migration Endo and then there are stable he was another funny example the which 1 is this is this 1 and so when the beginning I fault for some personal reasons I agreed motions on an almost flat on was she wrote and then she should led this involve you have the very same which intolerance for some reason that I cannot explain it as a solution replies and you have some more .period solutions more to see that it is not it is not any more in the line X always originated which is more like a turning solution but you have to you and so if you look if you look now at 2 2 but this is the only thing both for both of them and there and these are the products of the fully account for of the Of the fully transform of the of
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the solutions so you you look at the modem and the evolution of the mode restricted time sewing registers absolute value of more than in blue and green these are the real and real and imaginary part so you see that you have a lot of children Ontario and then it is completely stable and computer and if you make Africans synergies so using a sumptuous private worshipping at a computer value and students this is completely buried and issued you should increase the size of the assembled to warn most Fremont which you can observe is that the barrier here is at the top that is more and more on the way so this is difficult for a forum at a typical phenomena for the week conventions which means that this is speed of convergence depends only on the size of the on the level of so uh so sort maker conjecture on for most of this is a good thing of contractors have to consume most of the solutions in some sense of how we weekly dumped to 2 Oscars debate exclusions and in fact so if we go so this is the 1st question is do we have product solutions in fact is very easy to cause panic solution to all of this was surprising I wouldn't eat it was not existing but it is distributed in Lufkin but you compare with the swimming I question whether you have to use the opinion on whether to approve the and traditions in just across the floor just take a shit floor he was a fact but this that that explain the monthly figures should flow "quotation mark was a flat box this is the all own and then you complete complete my opponent among the condition and you rule uh environments especially state of the 2002 was compatible often have to and then it travels at a speed that depends in front of because he is is the you out in the region such that such that you have do this and this means that the velocity here is constant and insincerity the new constructed the higher in of any dimension in the 2 Of course the question to traditional as on that is much more complicated the book is and how it happens for the for the for the rest of for the release of case so for the rest of case then in this case we can but actually proved which threeyear expects and I mean With this is probably 1 of the only example where we were when we can prove some such kind of results to weaken dumping to Watson and but this is essentially due to the fact that if you look at this issue at these operator here the the is completely rebel and you have and you have some fun don't switch which is surrounded dumping it was pulled by essentially Maine did in the 1st time that it's it's it's ETA is stable in issue by knocking out that emissions of farther for the rest of and so I just would like to to show you what you can mean OK so this could just just explained his slider so if you look at the free flow so this is the solution of this equation then the solution is an explicit he should come to the isn't the edges other solution then you have if you go specialty with 1 time spent with Stefan TCU have creation of illustrations but if you look carefully and then you just have it is um the so I'm waiting for you and for you to take a case could fully patient is just tiny way that moved moves at the speed cake and the number companies is just saying that if you look at these objects because there is a move in the f notice move then you have these indicate so you can just reasonable so when he goes through 2019 but at a speed here that depend on cases essentially with clear which we itself for the orderly question but in a much more complicated than that the sitting volume because you were convergence to walk affiliations where he accomplished after Wells's fashion but still it's a and and going to show you some in summer what it means look so here I started as a who is a solution that is essentially here this long scientists full X merger by indeed and then I lets the the it's the the free flow involved so his extending the is here and here Seoul you have creation of of smaller structures dedicated to interpolate numerical problems here that you can see but usually just to prove boulders that crime and the number dunking is just saying that to fixer here the modem is able to it's just the scale if the shoe fits the mold that at some point so you see nothing because the pockets just disappear him of the weak convergence just mean that you fix 11 presented this 1 here then the number of new which agreed to unseal the sky just because the packet fly away OK so now so this is for the 5th rule and now it happens in the kitchen if Kazan Schamus no less of us as of so if you don't on the 1990 should turn on the nights so you have a similar phenomenon but you have creation of moving in X office for the Southern was an existentially you have this disease the mood Preston minus forcefully creates present minus 8 and then was and that was you always is not dumped so when you wish to do that depends on the on context you are not traveling for you so so this is just this is just an illustration of the resort that that you weekly the 2 ,comma weekly Converse too well something that is essentially depends on you and me so you have this morning on exchanges that's but in the end it is more a sling officials and sewage it's reminds me a lot the fact that the fact that you should children competing to cover system in In the classical mechanics and then essentially here you generically remains for complete there is an underused a soso Resistencia Nacional Madeira that it would consider is just that the case where you have only 1 modem in there in the morning on its authorities here you have only 1 wants all of this is just Christian and not and not to the reminding us of the press operator and that and in this case I and just show you will I'm sure the sort of thing that observed In this case this season going the same way and this is exactly the same picture as a source of that was the case so sweet dream really retains the same the same thing but it
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is much more of a simple it is a much more simple model OK so but the the thing so these are all cases this is just a slight consequently concerning the the the existing resort on on the time you had your own Of the reserves and migration the orchids so essentially just explain this right through essentially means that if you don't own the 1990 issue were around some stable states that depends on India then you behave like a fruitful deficiencies which then there is the whether there is an extension of and in the case of Stoyanovich was equipped for the summit fulltime exclusion of which culminated in the coup terrible chaos I just explained because I would like to to 20 things to say and I just
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would like to explain prodigy so the IDE so you have this you have this reservation integration and the you look at this at this function so you just take the poor but if it exceeds soldier in world areas of Federation of recessionary phase depends on unity this is a stationary states in Britain backed by the maze of the fuel to essentially here what you do is you this is just in terms of Soviet if you consider that energy is the son of edge the total energy prisoners some of tension true let's say that pursuant this 1 is just going to on the square and and and this 1 is the of a compared to they said the idea is just at the end of of act an act of act which is the 2nd operator ends at the end and so you write down the equation for these views and you see that it is she forgets that if you imagine that the days will just change the potential buyer timedependent potential and depends on the on the time this week so he recognize the exact role of the of the moon so essentially what it means news that the especially those that you know I I have a statement is OK and then you then you want to have some assume understand ends
43:14
and the final result is catching reason isn't Saskatchewan urging in the sense that went egos twinkling this that depends on Texans OK so that the originality of the vessel modern Achim estimate that is that we can work with someone I figured I'd would I mean that in trying to seize taste buds in the morning then and then and then in the middle and mass media usually a lot of energy criminality in the JFK's because you have a very small scale and you can use the word so this is essentially the idea and then you do not have an exponential decay but manager decay that depends on the on the possible if we unite but this is a circuit and the firm so it means that in fact this guy has the president and I would like to have so this result implies a roses the dumping in the sense that should go back to the original viable so it should take the lead and if you go back to the original valuable In issue looking for you then you have the same gave us for the but the people which means that is this is a solution of the of the rise of equation then essentially what what what's what this campaign is about say is that a fullfledged it will percentage including college team is the same as before more and if you start with some if not here a Modified Modified which is essentially another way of writing down the escaping visit for the for the analysts case for example this gave very well defined and this is just the mention of IG that doesn't mean that In the in the other cases in the event of case this day is better seen in 4 years and that the new extends so that eminent explained idea in many the days because I would like to essentially the idea is this 1 and if you look at the question this equation if you forget it does so it is it is only now but then the value our transport equation and this that it depends on who is depends on GE only through this coefficient GK which is afraid of some of the G U K is an X and this season UK of cases and the argument is just effective GK is a stable to this most but they may have adhered decay of the D hat so I have a decaying so here's Montmartre 131 of up to 2 years In front of this guy so I can that makes into convention when economics into darkened I have solidarity in its stock but it it was tough talking this is centuries old and then you have to To Cogan almost will to expand this bit of arguments and which is essentially the same as it was useful .period but on of the fewer stages but Wilkinson
46:28
knows I wouldn't explain this in terms of number phone because I have been searching for a number from for 2 years and you just to discover that affects what the still not saying he's OK you can see the red of question as cancelation of anything other and then then it tried to solve some of merger could equation that was independent try to reduce the flow for example to change these by an education does not presage said Epson's Grant who would you want to win in Game and facts the best choice this choice is to use this transformation here firefight edge they will minus M and 5 it's still for the president to essentially died at the time em at La notched the last time and I think that if you if you rightly should point to to to find the best change of valuable you end up with this with this transformation but at lunchtime and not completely laughter but now and the effect of the is contained in this in this formula so if you if you look if you city if he imagined that this is the transformation from like exactly like normal funds you preserve the hunting structure and so you look it's the the the shape of the energy after the transformation to a century Yukon through these but these guys preserve this energy an edge to the wall it's preserved by this flow so essential the new energies this 1 the judge you just change PDIP composers of and people poses a free flow in the eyes of Gates it a very good object because you you essentially have the GK of K M and because it is a time this is a guy that we have in this in the right hand side and then so you see that if any is larger than these days very small so the effect of the fuel is to be so we can to weaken the Monday nite and when you go in and if you now try to make some the duration of the then you choose mr for example a wealth of history for this new noted but this is a the minus 1 over 7 makes that this guy is squarely dividend interest with identity this you can imagine he should go for a time that is if they want to squelch the One of Us "quotation mark of and then you see only to be the 1990 but not too much so for use to close to identity because it's a small but you trusted larger so so you can essentially you can go up to any equal absence a minus7 and and then you can directs the process because here here this is evolution after time after time after that I am and here in this case and amenities and then you can go and do exactly like him iterations book but each was the was the 1st pools but the fortunately we find we found the nonIndians and is not music Derek you don't need to prove to the whole is nationally but I like this mission inside like to present because it it might be helpful in some cases to use his face into a 2 2 hope to have something like a Biskoff girlfriend sophomore Florida long enough time but you cannot exactly makes a convergence maybe there is a case where you it's more flexible OK and then I hope that
50:27
essentially you know what when when when we obtained these results so we we obtained there the reason for the Americas schemes but I would like to end users was something funny I think from
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this is a few stocks for this is still a chairman from uh let's see you start is a tool to to large again to logic in In the focusing Gazans doctors into larger engine that so is essentially like for all you know you stop Christensen said something stable and you end up with something banking hearing from police you use your status as the this a kind of picture that ended in this picture is essentially and this this 1 which is when in the market so this is a very special ponder In fact of course that he it's it that it's about the simulations you you have to to prove too much of numerical because and it turns out that there exist for a chairman of a stationary states that are under the form of the square and but so that no such care that unstable In the reopen of cents and so on and there is no hope now to to it's to do the same servitude to pull the same sort of result around this stationary state why because they include everybody's entitled to usually gets if you look at this system then you can put it 2 To connection underlined but and this is a this is a gravity of the problem and to do this is you in all are essentially at his locker he always does so I made the order to the organizing abilities maybe this is because of this and future that's only God I think should get a it as a function of the body into something like this and so of course they hear is and is it is something that is this is coming out of the so essentially in the good valuables the free flow is a and the doctor said he would again which should very much like like like the TV that we had the virtues of legalities noted now but if we have no flat spots here we can expect from us and UK but the only you when your close to home and Kasey but when you look 1st to the harmonica Sidoti you cannot really a Internet don't so it means that you have a case for these guys you have dumping that blocked 2 1 erupted you have a long cannot have its financial or whatever science Wenatchee 2 because at this point it's OK because I'm the guys very early moves so you complete his pledge you you don't but close to the Indian interested Dominicus literally on every step but it will use where a lot I think that also questions you will on June the agent of the question whether unilaterally the international involvement in the the reason is the financial year of yes but yes and no the emissions of the but what is sure is that what is sure is that we have it if we have an end end theologians and instead of having an instant here we use for the United States so here you use of something that depends on after that you can try to optimize bits I don't want to play his game more from a more interesting is if you if you are between gym if an order 2 issues if you will become to depend individual under maybe not many may be at much maybe with compatible very nice but and then there is a world the between essentially we are we are looking at 1 of our case that lets you have a abusing United's on migration because of the kind depends on the context and is a singular in extended so it could be a nice going and only the the authenticity of the the deceased interest in places like this the Commission do you think that mn as