Blowup for supercritical equivariant wave maps

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Blowup for supercritical equivariant wave maps
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We consider equivariant wave maps from R^ into S^d for d\geq 3. Using mixed analytic and numerical tools we describe the dynamics of generic blowup and the threshold for blowup. We hope that our plausibility arguments will stimulate rigorous studies of this problem. This is joint work with Pawel Biernat and Maciej Maliborski
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at the time and we want to help thanks to 2 In or 2 and thank you thank you for inviting me to come and speak here I am going to talk about singularity formation for thank violently joint work was told former students probably not who's now oppose involvement much-admired Boesky who support spoke in understands so so here is the wave might be questioned for functioned erectile function fine dishes so-called exterior formulation in the domain of the way of the East Minkowski spacetime in Class 1 dimensions these the number of special dimensions and the target is three-dimensional which is think about this embedded in the close one-dimensional nuclear inspectors and this was not satisfied this seminar the semolina with inclination was granted and no midnight so this is a very nice geometry with equation which is interesting for us I mean past physicists working just because its shares many features we signed the nuclear nations which are of course more complicated and it helps to understand these features in particular so-called critical behavior twice the liquidations I have had something to say more about this I'm going to soon that's the way it music reviving on which eased distances over them uh and then there is away floppy question reduces to this simple scalar the quake equation In 1 plus 1 dimensions when valuable I Grand which spoils so we would like to understand a global dynamics for initial for this the question this was initially Naco Finance a song and that of the basic question is whether the solutions can develop singularities In finite or infinite uh so told answer this question about singularity formation we need to know to say at 1st that the recent
entry which is concerned which is written all over them it's manifestly positive and you see from this expression that for the energy to be finalized a deal with the dissolution you last go to a multiple of boasted 0 and definitive solution and we take conventions of this function 0 0 or so therefore the value of K which is a multiple 5 Infiniti is Cesar degree a topological degree of these solutions and this is preserved in the evolution which is sometimes referred to as topological stability so this means that the kosher problem breaks into infinity nannies topological sector was measured by the Labor the greatest integer case the 2nd important feature of this equation which is obvious
from the from the question is this :colon environment so if you are very scared of time and are by a positive constant longer than I good solutions and if you look how an edgy skaters will use when I do carrying you see that the energy scales as homogeneous function the minus tool so the unit which means that the mention tool in which energy does not see the scaling is critical and he mentions 3 and supra critic no most of the default or what will have very heavy the past 15 years or so uh has been concentrated on the critical cases starting with some numerical simulations then breaks were resolved by the comptroller who shouldered singularity formation must have a formal club wanting more shrinking wasn't politically to 0 and then there are some works which which refined Bloomberg information about where 1st of all that for singled out his door fall and derived surveyed the speed of singularly formation English In contrast the supercritical dimensions are much less understood so there are some results in 3 dimensions which it's from physical point of view was interesting maybe should stated in physics with optical signals so so these results are mainly about the existence of some similar solutions which are explicit examples of singularities and stability but as far as I know there's almost nothing has been all about higher dimensions so to set the stage for the glow of analysis and blow
I need to tell you 1st the boats of similar solutions because you will see the state really govern singularity formation for this quick so said similar solutions by definition are invariant on this case so they are functions of 1 variable which we call why which witches are over a cup of tea minus-10 cup of tea is just for convenience and disallowed by it but translation invariant in town so if you will apply the sun's us into the wave might be questioned you get in all the this form written here now Saudi has the singularity of their origin it has a singularity Hawaii ,comma 1 which corresponds to show much recruited to the past light called off the 1 of the singular .period 2 0 0 In some this is 1 of the key to his you want this is a past like corn which corresponds to what was 1 this equation also has seen variety of infinity so no but find that by fine at the speed of propagation what really matters is what happens inside the past by home and on the past light :colon so so also was so analyzes can be restricted to interval from 0 to 1 close intervals from 0 to 1 and if we have a small solution of this equation on the scene all then this is an explicit example singularity formation in finding because for such a solution is a gradient of solution at the center blows up as 1 over the capital T minus the As Time Goes to Capitol uh maker remounted the event's organizers is restricted to the interval from 0 to 1 it is essential at least if you want the solutions to participate in the dynamics of the dissolution remains small cell sites the local all the way to otherwise they do not play a role in dynamics and there are examples of such weapons and similar solutions which are also which I find inside but our singular outside like call and for some subcritical the way questions but not here no look at salute like to find if there are some small solutions for this equation question there and 1 way to approach this use these cities by kind of shooting guard argument so it means we we start solution which is which is good small the origin said solutions for 1 part of of the family which is parameter by the gradient of some of the center which I do know say 1 can easily so that the solutions this local solutions at origin extend all the way told told the opening of a revival from 0 to 1 but that dot the general notes malls that's that's and the like ,comma but there is 1 that before I discuss the new construction of solutions well it's always worse to see this explicit solution which correspond to a particular vial of the parameters of sees you all so this is this an explicit some similar solutions this was 1st found in 3 dimensions well 1st proved to exist biracial power and then found in explicit form but Wilkins program and and and the last year it was found that will be generalized to higher dimensions yeah we believe that this solution is only so similar solutions for dimensions 7 and a high so I come back to this uh uh uh and this should be contrasted with the harmonic marble floor for which you dimensions 7 hired tunnels have some solution some 1
slight how the solution of 0 also actually filed so this was inspired by may have been Cox talking bonds 2 years ago when he consider the journalist played a In even who empowers with Swiss slightly supercritical powers people four-plus epsilon S and similar here we can sink about the mentioned as a continuous parameter and introduce Epsilon a positive parameter option which is the minus tool and change viable and with which age valuables and rewrite our all the then we obtained this all the even in new viable after and X and on the left hand side this is exactly the static equation for the from in marked in 2 dimensions and on the right hand side you have a perturbation so this is so that the idea was to stop the perturbation analysis and construct this solution protected from the units along with the 1st step be turned out that the right hand side is 0 so therefore basis an exact solution so so you can say about this as an educated guess please but so
how about other solutions To understand other solutions 1 has to understand and electricity properties of solutions that equal 1 and at this very much depends on the dimension actually in these dimensions but it's it's it's different so if you take I'm sorry if you
take this equation a and won't apply it by "quotation mark and one-liners why squirt and differentiate and keep differentiating you would get
a a hearing all the questions and I will just 1st tool he over them In which have to be satisfied for the solution to be and from the 1st equation you see the dimensions 3 distinguished because when the history then after all of that 1 must be the multiple plot should which must be part of that now that so this is this what is written so that the dimensions 3 solution which is small at a market 1 is Palmer tries just by gradient of all but 1 In the mention 5 . 2 possibilities I there in the 2nd equation Isaac boast themselves 0 what will the 1st time resolve this is when you mention fiber the 2nd time could be 0 or but but all or and this is the city's but this is this discussed the the end of the month for compliance with or we have to solve this system of equations and we get this behave an extreme this explicit solution which I showed you before has this exactly this expansion of In even dimensions Donal restrictions like that in the didn't even dimensions the solution apparent tries by the value at 1 so once we understand this and electricity properties so we have 1 parameter family and origin 1 parameter family at the light :colon we have too much stuff in this most men so and here's the CEO of getting into dimension from 3 to 6 there is an infant sequence of these shooting parameters centered to the solutions the a wonderful interval from 0 to 1 and this is a pretty standard shooting guard went in he mentioned in only mentions being even mentioned it slightly more complicated but still criticisms no there is another matter of of constructing some similar solutions or more generally is solving following solutions for all these which is a variation on that and this is that this is functional the 4 which serve similar solutions are critical .period and the but you see he In order for this to be be finite you have to normalized it so you have to subtract the value of the solutions at once but if you don't know the value of priority this is a problem so for this reason the solutions have not been found before because the weather that there is a variation of proof of existence of 1 of the solutions of this infinite family you mentioned 5 by biracial Tower President Abbas's Fatah faction doesn't there but in this case if 1 use just pie hops which by the way is convexity regions of the it's a quick so so this solutions probably well it will be hard to find them by this valuation techniques
now once we have said similar solutions the next step is to honor life the spectral stability so the spectrum of small perturbations uh because this is essential in understanding the role they play in dynamics the solid standards to introduce so-called slow times of the blowup happens when St slow time is infinite and change variables of these S and why similarity firewalls and in this similarity violence away from the equation takes this fall and so on so similar solutions are just solutions which do not depend on time so the recessionary solutions of this for which the right-hand side his 0 so so the standard procedure is William arising from this that these this Simpson of solutions the convention is such that uh eigenvalues law under which a positive or unstable so we leave Will arise around then we get a quadratic I can value problem but rather because it involves London London School and did Qantas Asian conditions for you want solutions of this Leonora all the which are small from the internal close interval from 0 to 1 this is a condition for quantification of like about and this is the conjecture formal the spectrum and I will then give a for that in a moment so 1st of all the is the so-called gage malt which is always present in similarity viable variables which is due to the fact we just we don't know this time capital the so by shifting this time but by time translation would generate a which is explicit and this has identified 1 so these use these eigenvalues positive this is not a real instability so it's that's why it's called gage mall the conjecture is that there aren't exactly an unstable eigenvalues so positive eigenvalues biggest strictly bigger 1 and the Infiniti many negative eigenvalues which correspond to stable directions so what is the evidence for this conjecture so 1st let
me discuss the solutions this explicit solution of 0 4 which this conjecture applies as well so when this case we have a luxury that we know the solution in close fall so we can change variables I like this and then this equation we we had
we had here becomes
a so-called .period equation which is a generalization of hyper geometry equation which it has for Single-A singularities and 0 1 the minus-1 1 and infinity but this is not very helpful to the cause of the so-called connection problem for .period equation USA unsolved so In contrast to hype Joletta integration I mean it's not helpful in terms of approving this conjecture it is very helpful actually intent of computing eigenvalues because for example Maple Knoll .period equations can compute the rights scandal to salute
suggests is that the North American presentation of solutions to this is that that's what the hell is this is the reason why the organization plans to cut the connection
problem means that the connection problem you visited me if I have a singular uh bounded by problem and I take a solution which is good at 1 and the solution would be a superposition of wouldn't but solution other and connection problem is well for -dash geometry equation the coefficients in front of all and are explicit not in this so there are explicit in themselves of some hyper geometry functions but still look at some important but it but there is a way around it so and actually we can't take this solution which is good at at the origin of this that and take a power service expunged this is found the forks analyzes that we take these good solution in terms of power series around this with victory hauling .period functions but we don't know when this this power series is guaranteed to have a radius of convergence because this is the nearest singularity but in general it is not the most but what and the way to rule gets most most of these policies there isn't 1 is to look at us and politics of the coefficients In the index functions and this coefficient satisfied as three-term recurrence relations which can be solved and these are asymptotic solutions of this recurrence relation and this is the best solution for which say it is this coefficients you won his Ponzi role this was missiles with 1 but and this is a good solution soon so that privatization conditions actually did this coefficient is 0 then this can be used to compute eigenvalues with great precision and confirming this conjecture racial due that and quite recently the cost and bonding and luggage threefold rigorously that this it does not have positive fruits a but as I said this stick speak very much depends on explicit form of solution of 0 so we don't know other solutions in closed form no actually sell so what we do
fall 4 other solutions 1st of all we can rewrite this problem in a certified joint there so this is the 1st by by a change of variables we can bring to this quadratic eigenvalues problem into a standard selfish moving problem With operator a N which is cellphone joint in this field but sp To achieve this is very much related to the low class and on the hyperbolic space L now the point however release Z that eigenvalues of this problem which I call mule and eigenvalues landed of my original problem but the subtle differences exist so because in 1 case I what I want solutions which belongs to the functions they say next in that case I want solutions and and if you analyze this you can see that these eigenvalues Ituri coincide but only if the eigenvalues land is bigger than the minus 1 of 2 and in this case so sold by analyzing this problem we can write and learn something about the eigenvalues of our problem but only in this range and it's not difficult to see that if we take the gage malt which I mentioned before and fast forward to this function outside the good function like that which is an exact solution of the Swiss eigenvalues the minus-2 now From the shooting construction of the Senate similar solutions we actually have a complete control of the number of oscillations of the substantive solutions therefore we know the number of zeros of dysfunction so we can applies to most selection To and from this it follows so this is different in diamonds and 3 and 4 and 5 and 6 dimensions 3 and 4 are exactly dysfunction has exactly and 0 so there exactly and eigenvalues the deal and so on In this case around my neck like an hour so they give us the exact number of eigenvalues for land bigger than the minus 2 but this leaves open the interval from 0 to deny there is a gap but actually maybe I should mention so that to was appropriate by the way bye bye Colo a rougher than chapter in which say for a very similar problem was Hawaii heat flow problem whether the Soria should flow for semolina a wave equation the facility that will heat flow fulfilled for Powell lonely narratives shows that there are no GOP eigenvalues and it's very likely to disprove can be extended to the City Council no now so there is this guy up so in this sense I don't have approved its way :colon conjecture however numerical calculations show that the infected on the like environments and his GOP at at actually there is 1 the most surprising thing is that there that in dimension 6 the the there is exactly 1 eigenvalues of our problem which is not an arrogant vital decent drawing from so
so he hears a table computed new numerically this numbers where physicist so we want to have quantitative results of this stumbles will play a role you know moment so you see that this different dimensions there is this gage Oregon value 1 always and all eigenvalues on negative this is a list of diagonal value which will play a role and this useful for solution we swung announced 1 instability which would play a role in dynamic because this the call the focal dimension 1 then and it has 1 positive I can value the gage more than all other Oregon vise naked and this is the eigenvalues dimension which is somehow surprising that it's not it doesn't it is Copenhagen value of the corresponding 7 for trying
but so so we have said Souness solutions and and which are specter stable that have no longer mauled so this gives rise to conjecture that those are natural candidates for instructors and the conjecture is a victory this explicit solution of 0 is so universal the In a sense that if I take a generic generic in a sense well you vaccinations so we take some initial that and if they are large enough that would blow up and they will always blow up along along the solution of zeroing in this sense of their approach locally near the center of this but this is a non-contributory results misstated it is universal for all nations even in the case of data which are close to this and 0 actually there is approved for fleeing on loneliness stability of this blog but due to the money a industry dimensions management this is very likely that he's technique can be extended to higher dimensions so there is numerical evidence for this behavior which I'm going to show you now that so and and this totally with more because it says that the dynamical solution approaches they serve similar provide exactly with the radiator dictated by deletes them more which is here soon but what about the coefficient of course depends on the initial data and the title In addition the so this
is this this a new glaring and for that so we see what we see here so this has snapshots at this initial telephoned by Blake curve this isn't similarity variables and adult at the red line is the 7th similar profile and you see as at this time it goes on the solution approaches this and similar profile and that you see this approach as it is for use in the dishes that this is alike called 1 click approaches it's also also light this is a more quantitative evidence for the same result so so what what we show here is the gradient of the dynamic solution minus the gradient of the tractor or and in in a logarithmic scale and this shows that this it is decreasing in time with this when lumber minus 1 its invited which was computed independently by the specter that if and this shows this is a snapshot of solution at some later time and we compared the solution With the service similar solutions and the difference is the profiles all of these lower known so this is the evidence and this is just from hearing for the mentions but the same is true in dimensions 5 and 6 so this means that the solutions this behavior like that and of course we could say that more without encouraging so so this
was about what we could generic goal was stable now but we know that for this equation sufficiently small solutions the remaining globally regularly time so there is the question What is a borderline between a solutions which go up and solutions which don't at and there is a very a the the very straightforward the strategy to I'm threshold namely you take it there 1 time with the family of initial data which interpolated between small and large and then you try using by section of the parameter along this family could be a ghost hunters and amplitude Yukon fine-tuned for the critical value and and look of the what is evolution for solution which is has hazardous the newly critical the company to then so in this case that we have so this is a dynamic consumption and the candidate for the critical solution which somehow sits at the border line is obviously the 7th solution which was exactly 1 unstable which is the 1st solution family which is not known in closed for sold so we have this critical solutions have similar solution it has so exactly 1 unstable direction we think environment the 1 and all other modes of decaying so but by white by fine-tuning we can make this coefficient as smaller switch and if we if we if this coefficient is is very small them for long time we don't see this instability is a stunned 7 . and 1 from this follows of various scaling laws for example if you look at solutions which don't blow up but there but they're there and they are a marginally subcritical so they have a slightly less than the critical value the gratitude of the solutions and center can grow to over large can can become very large and and how large it is it will escape justice so for this this isn't so this gives rise to conjecture that the sensational similar solution with 1 unstable is a critical solutions who scored mention 1 stable wonderful separates blocks from destruction so here is a
schematic picture all of this behavior so so so so this here Hewitt had this call mentioned once they wonderful to reach we imagine as small Cyprus another infant dimensional phase space and here is the sense similar solution which has exactly 1 up on stage direction which is transverse to these and all other directions are stable and here is a careful finished data which intersects these wonderful and the data so "quotation mark war corresponding the 4 point of intersection will flow to the east and the data which would slightly will approach it well and then will would go either to envelop all known regularity surprisingly so I mentioned that the it helps to understand critical became vise any questions because it feared this picture which hatcheries where alone were well known in physics office transitions because there is nothing goes this picture exactly explains universality of all of the 2nd phase transitions like internal magnets so this was observed twice and equations and in this case the question of the the on the local bloggers is the call for relations so this is a numerical simulations illustrating the behavior I guess this is in 6 dimensions so these these adult line is is this set of similar solutions Of course the mansion once and this is a dynamic of solutions of blueline and which is fine tune through the threshold with this decision and that trade is that this is a pair of initial data on both sides of all of staff so they have altogether because a close but here you see that the unstable mold has grown to finite size of this separated and 1 would blow up another 1 with respect September 1 with blocked in which 1 would expect and this is again and again this illustration of this phenomenon now we look at the gradient of the dynamical solution at the center of the gradient which should approach the granting and all of this so similar contract or last this summer .period was the unstable was very small amplitude like 10 to minus 26 class stable more and again this solution will go up and it would so this was in
dimensions from 3 2 6 even dimensions 7 and higher I said before there are no sense similar solutions to accept this explicit solution and feral so there is no natural candidate for this according mention 1 critical solutions and the reason actually this set similar solutions ceased to exist in dimensions 7 use the spectral all the Quaker so there is no way that the wave of equation has a trickle constant solution pilot 2 which is not to the equator the solution is singular and this origin there but if you look at the spectral Paul this solution then I can functions are oscillating below dimensions 7 but above dimensions 7 the number of eigenvalues around solution is final and this final it's exactly responsible for that that you rules this and similar solutions but at the same time you gain the solution in a sense that this solution has finally could dimensions and actually it has called the mention 1 if will neglect is debatable so and ended the numerology here is a cheap you have the same phenomenon for what all different kinds of equations so hot money mopped floors and and and this suggests everything is about winning this case whether this is the square minus a B-plus is positive from negative and these changes sign somewhere between mentioned 6 cents that now so what we can do so he I want to describe a completely fall 1 construction of the threshold at at which the which is actually do Will William this for for money might flow To further on Velasquez unpublished paper many many years ago so so we take in and out of solution so that out of solution is just this singular solutions plans perturbations such as that week June await single unstable directions and the inner solution when this solution is singular the sentence so it's not good so near the center we attached victory static solutions which is really scared by Our priority on function of the solution is not good at infinity because it has infinite energy and the whole point is that there are some topics of these out the solution nearer the center and there are some politics of this static solution nearing the infinity much to have the same take much so they can be much this is called Munchausen's politics this is what of Alaska's this and by this fall 1 analysis we predict that the rate of blowup which is given by the and this is title up in the sense that this function of the goes to 0 faster than so this is accomplished formal argument uh and that should this argument doesn't work in dimension said because in dimensions 7 these eigenvalues allow that to happen to this 0 so maybe just another it's become connection could corrections this we don't but actually this probably is not so much worth pursuing because now there is a new approach to to blow up which was developed by by biographical the key for and the lessons and apply to other questions in particular to the supercritical wing creation by and it's very likely this approach which which does not use and matching a can be adapted to the screen uh but still I believe was the result was the same results OK so let me
finish with by mentioning some more open problems so said the fault of the critical dimension is used is well understood but there is 1 exception namely we don't know what is the threshold of in 2 dimensions there is the following sentence sold it is known is the when this was this result by through over there it's eateries blowout in critical dimension it must have fallen off a harmonic not shrinking cousin Celtic and it's 0 so this is a profile of harmonic mopping up for this function of spittle from abroad must go to Air Force stable up well and this this the function of as forms solution tied to bloggers was 1st arrived formally by Ovchinnikov and signal and then rigorously by will next game there however as far as I know it is not known what is this function of the so it is not known what is despicable the tradition as far as I know so fulfill for these problems so what it is not known even if this is this is this a power role doesn't have a logarithmic reduction so but it certainly is faster on the T minus the square so this is clearly seen numerics but resulted analytic prediction it's impossible to detect some corrections the 2nd problem which I want to mention use the continuation beyond blocked so whenever you have a solution which from which former singularity there in finite time there is a question can you continue beyond and in this case where you can always because this equations are the violence on the time reflection so when you have a better course of similar solution you can attach it to a forwards similar solutions trivially and then you would get weak solution which is single just a single point and this was such a phenomenon of in that solutions immediately recovers most Nozawa known mainly for parabolic equations and for a corresponding 1 cannot appreciate and so and that you we have numerical evidence that this happens and it and that is probably an interesting pattern of block types with the sequence of blocked the solution blows up and it goes up again and again another problem I wanted to mention use when we changed so did the way if we change the way also the wave I consider tool twiddle Maine which is bicycle parked all bonded to effectively bonded so we lose the the blowup does not really see the geometry of the demand because the blowup USA completely local phenomenon there however was a blot Wilkerson does depend on geometry and that we have some preliminary results for wave months with the domain the so-called onto the seat of space so I don't have time to go usually what what would be the space is we will only have so this is so you can take care of ACT UP and units here or even a nuclear and space and restricts when there were opportunities so this should 0 and you take it around metric conditions and then say might take that the line of work but which goes the equator and Parliament's rising by latitude Ted which is changes from 0 to apply half of the poll then the metric units inquiry but at to where agencies around metric on this fear this would be an alias metric units In Tokyo 1 dimension modified change 4 dimensions students various metrics In the end the following mentions which we have here well as you can see from the lecturing the constant precisely hyperbolic space so that the stones from and a point here to the boundaries infinite the boundary is just an idea bundle the hyperbolic space this is like at this moment for hyperbolic space however not geodesic struggling finite time so this is prominent .period here to the bone this space which is of great interest in streaks fury that is effectively the compact more effectively so but from the point of view of so if you take a sensible way and consider the same equation as uh we find that's evidence that arbitrarily small initial data blow up and the time of the role of the scales with the size of initial data swung over "quotation mark so this the conjecture said there is no factual although he did so with the yesterday with itself and the outside nationalizations sections of
all this year's actually yeah during this project the use of the century the city's moderated by trying to understand the stability of this that space as a solution of ice and questions that to conjecture is descended arbitrarily small generic emission data I need to do well in the case of ice and equations to block off elections that actually there is uh corresponding problem is for the critical cases will from Radius 3 2 as tools for which it is known that there is a threshold for blow for the same reason there isn't fashionable but here so so to have blocked critical dimension you must concentrate energy which is at least energy of the harmonic marks which means if you think the way flops in this case in the critical cases we cannot have low of 4 the Arbitrator small initial data well but it does not excuse told Interfax that solutions remains most of the radius of final at this city's shrinks the 0 infinite style and this is contract he indicated that In the critical cases of radius of unknown at this city shrinks to 0 he knew the time boastful this model and for the full license questions and finally I'd like to mention my favorite model which is a supercritical I spend way system this is a rather complicated system was very each fragment phenomenology air this system has been mentioned as parameters and that should that's why wait not so susceptible to modify spent namely for I stayed there is a coupling constant which is you can't constant kind of genes and mobs reserve coupling constant work called because "quotation mark and because of this they have the same scaling properties this coupling constants have the same dimensions 1 is the interests of the wasn't so therefore this is dimension and for this system there is so when they show everything is numeric except for existence of substantive solutions there is a numerical simulations indicated that if this parameter is small enough in particular was when this is 0 we are there's no gravity when this parameter is small enough we have soon Abdullah up very similar to what they described but these blow up appears when the parameters notching up and this is an example of gravitational decimalization so this is for the Kostic censorship to start working the parameter must be large enough this is not a counterexample too costly solution because the wave mobs have summarized this already result grow so gravity is not expected to help but the treaty does not give the color becomes legislature and what's even more interesting insufficient for coupling constant windows so similar solutions what we get in so-called discreetly said similar solutions which are only a single recently would have been studied in buttons for full systems result grass can take the people of the United
States was Japan in the by the locks
on issues was yeah I didn't see that in talking to sink about when you know all the results are on a heavily heavily rely on numerical simulations and single of the simulating single formation this numerically nontrivial so and so so we we don't know how to do it in the without these assumptions were so that's why all our results are restricted to recruit source many of those who
have resisted even then as saying that 108 so but the company president of the Nations that I don't think you have information that got the setting of solutions to the end of the game before for the Giants to do what
we well I I would bet they don't exist but I have no proof so it's wonderful decisively so no I was from we'll fall for this season all the year after all right so for these all this is an absolutely over handing numerical and and there are no substantive solutions such as the USA
Track and fall on his face
tells us that the proof of existence breaks down there for
the rest of the way this has been a lot of
places that went for the pollen early 90 the model Slattery too because you can't played with dimension and the power of the millenium ride so so here I have just 1 parameter which is dimension but of course I could I could generalize this tool hire crew violence and then I would have the 2nd parameter and then I would have a similar picture it was b and the thigh