Stability of line solitons for the KPII equation
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Title 
Stability of line solitons for the KPII equation

Title of Series  
Part Number 
15

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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Subject Area  
Abstract 
The KPII equation was derived by Kadmotsev and Petviashvili to explain stability of line solitary waves of shallow water. In this talk, I will show stability of 1line solitons for perturbations in the class (1+x^2)^H^1(\mathbb ^2)

00:00
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01:52
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04:49
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Food energy
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Theorem
Conservation law
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Conservation law
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05:30
Dynamical system
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Resonator
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James Waddell Alexander II
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Summierbarkeit
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Resultant
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Spacetime
Functional (mathematics)
Eigenvalues and eigenvectors
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Line (geometry)
Student's ttest
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Momentum
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11:00
Eigenvalues and eigenvectors
State of matter
Line (geometry)
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Different (Kate Ryan album)
Boundary value problem
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James Waddell Alexander II
Exponentialabbildung
Addition
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12:27
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10 (number)
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Directed graph
17:18
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Infinity
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Körper <Algebra>
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Dimensional analysis
Line (geometry)
Cartesian coordinate system
Time domain
Numerical analysis
Degree (graph theory)
Arithmetic mean
Computer animation
Order (biology)
Universe (mathematics)
Theorem
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22:22
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24:39
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Resultant
Freezing
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25:43
Line (geometry)
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Line (geometry)
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27:04
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31:11
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32:47
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39:28
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00:03
as time and again and home and in the end and still like the sun colonizers for inviting me to last conference and give me a obligation to stay here for 2 months among them want to talk what typically question it is answered is the question this equation is very close through the KDB quit here we have exterior little kitty the question of other plants here which takes into the strong variation of some emotional race in the trust sponsor of I'm here sees Singh miser parameter which elected the southwest into system I suppose the promise it means surface tension is weak and the Cory KP question and that's what I'm going to cultivate and the singer Michael Myers 1 it is cause to be like wages and in that case suffered is presented is assumed to be strong and let's price of CB dysfunction mating then the subsea of X minus the 10th seed and the is that Syria was at the root of JTB wage and get the question is on the model of Big Spring transports of stability or instability of KDB Sowetan in that to the city and for that part of the equation was by couldn't walk and was different but history in 9 in 1970 and he they also did some former rises or stability of life sort of thing on the contrary boss KDB and keep questions are known to the integral system and this
01:54
is a political outsider borrowed from the website of professor of of its wits with the which is the picture of the ocean and here and here is that they're here many great waves which a uniform entranced was the directions motion and the they're closing and there has rated the questioning I used also used to describe that in the past it's alright so waste but today and Oracle wants the media oneline so it does he might object but that does not appear in any perception of sorts at 1st I would like
02:31
interviews conservation laws of Katie twoweek equations and in the in the conservation low I use this new which which is defined by reference to eat .period divided divided times you hot in the fluid Trostle and there's and it's this great flooring industry transformed this on is defined in this way and the laboratory as stated the question at the moment concern floral provide solutions and moreover it it hasn't Newtonian which is leading this fall answer is a pity equation I logged in the previous slide is for a great equilibrium to the fresher the body of this from modified brightest relies on the fact that the the DX and former Rays sees it is to see that this quantity is preserved and had it is also preserved and 4 kb inflation of Serie which Wednesday denied Eagles minus 1 of the 1st time has at the same time signed and as a result there upkeep is exercise I expect that there is a grand stage whispers provide on the wall and all about 4 kb take place in the 1st time has all but it was all this part time is those orders answering clearly entitles scaling 1 day but it is equivalent to the extent of it twice of extroverted so this time is for the same skating water with this 1 so this 2 times as priest bottom was this coming an end and 4 KB to wage and what to the town's half of this time for this leads in this could commit is not so useful 2 estimated time global pondered quantities for the other hand this was quite useful and also for and it is quite useful and the along lines and this quantity is almost breeds stopped and they can't prove it if he proves that the demonstration plant as which we were real and to his it has a good DJs
04:51
and from before I introduced more results on the equation I'm writing to summer famously stability all the KGB once items answering them somewhat not many but the X industry is also are quite famous on I'm the first one he's never lost a of in the energy costs and 3 the Anna is used the and use forced them
05:20
to conservation law and also the 121 for the KDB question unfortunately such announcement doesn't work because of the bottom edge of the 120 for the which so
05:33
unfortunate disbandment of the world the 2nd 1 is prove asymptotic studied of 1 1 1 1 sort of K B which he explains witness but what can 1 White said is the Lucent said will best make it quite aware space you jury in Nampula space because KDB is an Internet we don't we cannot be explained to include the destination in toothpaste but in you might have to face said it is up 1 function is awaits functionally pressed some weight the most we don't have some kind of Begiristain the thereby model and now there they proved the proved asymptotic studied the altered its origin age 1 and the the key in the paper was being as well as the others and in the spoke I don't use students and because we don't do anything to classification and but the video identity the uses of their identity by by then was quite useful In the argument of KB to ease the and for L 2 stability of clients once and it was full of fire Maryland they got in 2003 and what said it is their security by using some Muratov usually just submitted thought about and transformed and the comments sort solutions ways King solutions all modified the collisions and and by using another provisions they not since original modified solutions and actually gave us lot of ice isomorphism between these 2 because reasons so as a result it gives them openly like isomorphism being able to sense between the neighborhood of them and the and enable foot of 1 sort of so that so that this is the idea and this argument cannot be applied electric to the studied online so the I don't want to but at the argument the idea is to useful to investigate the leniency of the of prices so the next tries these small reserves for KB collisions and being instability will hit 4 kb twoweek which of course there is a former government but for the waterfront to the history that had somehow mathematical argument was done by bouts of forest much heed the advice is is by using the inverse scattering theory he finds some less than onetwentieth hiding most which is related to the dynamics always waiting lines origins and later returned uncontested facts appear once and for all that thing El to space and if you live Iraq is an email as a plate of KB delete was an online so it'll do not have any idea agenda is or continuous pistol other than a miser exits sold include new thrust and such a result would extend was approved by my heart nothing some 10 on the 2nd floor of the use computers at being it's viable and there a coincidence library as well the whole time but also there is a result of the lower and upper its who construct the longer all exhibitions around lines so by using invested salary and that is what I did with so that's to playoff is to use near transformation as use for to be decommissioned idea is almost as saying there is no 1 better and for this he this case you have estimates of amount available farewell because there's no resonance In their spectrum and actor in this case is near transformation gives local Moslem of mobilizing 1 prison in 2 sets and there is also serious on the White Witch that this is something that I would only do sums and the 1st 1 is down by the House in 1975 and what they did is to find an unstable Andean was foreign arrival later and Zaitsev and he proved to be doing any strategic later on there resents that investigate there are the stability of planner sort 420 insisting sustaining quite a systematic way and other times they proved also reminding that the stability I'm soulful KB to equation it is expected that lies at table on the other hand came line that is expected expected to be unstable being pool "quotation mark KB White Witch and widen the decision so different because it's so I sent it submitted to this difference sign Madonna and there we have
11:02
different silently Dominion and the 2 nasal anniversary for "quotation mark
11:10
PAP 1 The said it has a stable plans it was approved by 2 blunt and and for us for the will both miss the fossils out almost was thought to stand by guiding 90 89 and later by drew about using for it is the recent Florida restrictions or by himself he proved were poisonous of KB questioning dissident and later handed rate of Maureen sold go over to the schools and that I should get to like many other people but seafood is improved the rule and result in many boundaries in that was approved a senior at this time like this in different boundary conditions and what I used that is say that the world was dissolved around like Sodom and it was proved by Moroney and so once this school this and what they proved is that if you have some L towpath addition to this lines and people itself then the difference between solution online so it unstable state being able to look into space and this this quantity is continuous in Healdton along with respect to time and my
12:29
talk is also related to the more results on keeping creations of so I wanting to do some of them and the typical 1 a study 1801 King solution of these heat equation in our if any course once it has the king source which moves with a constant speed and the applying is an since this solution is at the cliches frustration by undercover the problem with the way this doesn't and is that there is sufficient evidence that also to the carnivals and inevitable so if you like the spectrum of that in your eyes operator for clinical 28 then suspects and used it's so there is an isolated on the international level and all the other spectrum rockets in stable half right around there's a spectral but if you think of all the transfer stability of this troubling way so it means that the it's not a uniform into we extend this erosion of onedimensional onedimensional King solution to inform readers the transverse objections and think and then what happened is that if you think of that trend whole industries because thanks to the content other directions it appears that the suspects from which it was his here so there's continues to trend and then there is not an isolated undermining more of course has enough there should be some stable but here either but there anyway for what means if you think of the transport of stability of King solution for this equation there is some putting a spectrum of and if you want this in this case that dynamics of solution which we used to make use of this and the and substitute see it on the into this equation then we get a solution of heat then we find that the gunman satisfies the heat equation and minus onedimensional sold the original program of being in main job on the motion of this 1 is described by in minus onedimensional heating stations and the need to getting this and
15:03
also the isolated Internet system which was obtained by kind what he did in a single business region and France mother is somebody onedimensional game against them this translation very this function is Fribourg balls are PD which is onedimensional smaller which locates hitting good image in this case because the original problem was threedimensional and that this 1 is constant and this satisfies the to the Wabeek which and because this is a solution to the way it it is expected to be combined with the government plans to deal at the infinity hence and now we need I want to return to their don't typically questions 2 unauthorized KB equation we make I make use of the funds as of this fall here sees this agree the outdated but it depends on not only on plenty but it also depends on trust because direction and also as phrase find here isn't depends not only on the but also on trial of ostentation so if you think the classics in view of I'm using this formula means that the center also the classics some theories here it's of 1 and C O P Y means a look around produced all right them sir and the fees the placed all right certain from as a vise and then it is because Excel depends all all depends also on drums of Oslo parameter why it is not necessary a straight fight and that is close to a straight right so if you think also why derivative of X this basic rate means the animal but there is going to sound not wanting to
17:22
use my of forests by saving you start really is alliance them plus some small part which is written as an X but it all went to function and the roof was not lost more In this sense stories from only a small but also as a minus half dead minus hostility but it won't be there located in to underscore not at alone is small and it is this assumption I'm very exists son was right in front of the X sites that you all TX why is it so is always close to this I was riding a bicycle all the time and the difference or local between these 2 is it is all the calls Elton on all the same I'm also if golf the difference between this was vision from the D 1 and there is no way originally wanted a life sentence it's difference is in a spin on being y it is awesome is also of this order for all the time and if you think all that all right so it's of 1 and the free square this quantity and until it forms of and the whole entire them and we also have this quantity sold at this IXC excelled vinyl the X is under a state in all of it accurately enough but not too long excelled why this is the sort of wars and this just says that except the is almost close to the truth of the Pansey c Astle basically except the means Speed was all of the motion of Liselotte so because if condition I'm also if single why did but with this 1 1 6 it's so why why all the nowhere except 1 private dances the university was being sold this means that so combining these 2 results we can say that In the EPD since this 1 ,comma bodies original speaker as the the and also we have some local compilations like this 1 if features that phase friendships here then the difference between these 2 ,comma bodies to sale a priest Lokeren in space so this just means other than the last 1 means that this is an excellent from fund this is the right solution and these are best so it's moving and then if you don't you think things in the moving need for this is thanks to land described it at the time and so in this local Frank here we have I think that it is I think that extending the rights and the the 1st assumption I made here is almost the same with this assumption that these same did allow all be there along the axis with line which is part of my answers is equal to 0 for all why this is not sunshine made in area and actually began address this subsumption by these assessments of soul is a part of it and the degrees for numerous space heating up but days what right
21:27
on the field instead of assuming some small Nestle this long weekend since more as of this moment and if you have small abysmal we have the same sort of result and this is a remarks which Springer reversal and skipped it said that the Basic carry my myself and says that the licensing Quanwan if you part of sort of it :colon voices Lokeren to the original wrong isolated as the closing maybe with some faced with doesn't depend on 1 but if he sees things here we limit myself ourselves to this domain soul this is owned the at risk for why this is for compost sector and increase seizing seeing the
22:24
entire I'm along like this things will be different this is another might result and under 2 standards without either in Iceland part of perturbation is expression localized and if perturbation is expression localized on also III Is it easier 1 in 100 the part of the Teresina want what we have is that the difference I beat ratings speed light certain and the region's speed or said if the angle of the class of the line so it is if you is a linear combination also ourselves in the solution of the brother of the great acts as the fossil this season result and he had this 1 means we might mean I mean this indeed the most and by dysfunction and answering this 1 is assessing the solution of this part of the solution so considered if the single also dynamics of HMOs waiting lines and its dynamics is described by ballot question that that's what I want to say and that Stratus Celeron is scary approved by using serial Kashima on policies by Aristide equations and this this kind of observation was not for them meet myself but a series of other a lot by physicists Pederson who shows that similar results can be obtained for business Systems and since assistant and liquidation us quite similar it has quite similar I singles out this type of result can be obtained for many this industry question means the country's if it's concede at a Tuesday described the case where the confidence in the southern station also Waterman is small and this is
24:32
a heuristic picture of what I brought in the previous Friday so he feared he
24:43
find integrate this formula in white ones
24:47
what the heavy security for the like the actress alarms at and around the center of right was around on the combat freeze in the composition of White was below the expectations it's on the because flat on Friday as people's the but I don't have any the series ship around here because this region is expanding in the and the and but what is the previous system says that this part can be eaten by I did or the sale solutions of 1 of the questions so as a result we have some doubts about given Jap here suggested a small difference of height here and here and he she estimates that to differences on was recognizing Oregon honoring rights and here then we should have some sort of means that the new too
25:44
so that's what we have here if you just singles I'm all right so all right so set of the said then zest should insist on a solution of the delete button who's different feet Freeman said send balls all lines were deficient self the guns goes up at Trieste industry and this happens the cost if the the spirits and was inquiries and I said this about a system but if a if he's single of someone if he drives musician or acquittal on X and the sea and the the disregarded the flattened about substrates a kind of diffusion time then it becomes a way the question it's fun and it's just been eaten then the solution can is candidate and by using them on the formula in this way and because of the stands appears from 3 ships like
26:53
here this year so this kind of result this ceremony is peculiar to the case Wang when the moderating question is that if the described to the 1 the wave which and now I would like to
27:07
explain their strategy of the People's fast I want to recall that that why 1 whilst thankful question and what they did is other studies have found the right sort the way In this said in this weight space and the feeling eyes the KGB greater once the solution we obtained this over suspect from all sitting around operator is however summer but I don't I still here so this is it this is Katie and went into the getting so anyway it has a spectrum of your written and because of the way that the country has been removed the stable of things and the spectrum is a sum this isolated in Beirut and the stable once this happens because that any oscillation tales of KDB solution moves slower than the means or the weight of the and also the story the way is smaller than the propagation speed is smaller than the main weight so what happened in the way this space is that the wave function growth explains Street and the what I suppose this is a friend yeah and Men's certain moves With will see the same speed with the scent outlines on most of the business and a want so if there is a small force way the propagation speed is much more and that means that when it is when the center all displayed function will this quantity is this brave is getting smaller and smaller In England yes thank you and because if we add some extra time something like this after its time realizes I saw there was more so if advocated acquittal on you tell thing we have this despite emissions under the computer X exceeded all this this this what is on the up then we have to look at city and stuff faster town is exactly the same as the as a matter hasn't yet it's so this also means that if there are some of the leading terrible KB to in the 3rd set of certain exists for their provisions be almost latent and thus was saving pale Sibusiso work than that of and which means that those waves are also explains the stable in wages space at really how if you of the of solution to the question we have explanations for the 18 waited space but as it has with the
30:40
same hit the and there should appear suddenly checked all of whom were transports deception and actuary for the Kaibito equation that appears some continuous spectrum all part of the state of the art world which caused them and if you truncate take this prospect from then we have the same measures to BT has frozen all Selena equation so that is a 1st step and the 2
31:13
walking explained that it with this space I am eating interpret what's on what I did anybody means in this space if the hunting you is a derivative of erectile function then dented inevitable view can be detained displayed by using this and sister company of the continent's function with complex support and since attacks in subsection of 20 functional full support on the it to the extent that to function is dense this expression is based the and also this integral is gives our abundant operator in dissuaded space we can extend this ponderous the victory through the latest thing so this is how we interpret the entered the liberty of being exploited workers and then
32:05
odysseys so be doing equations and will discuss this is because is not a single case history equations all of the players some continuous but from here and if you cut off these continuous more then we have to explain just agreed that there is a result here so there are continuous was like this 1 .period joint the knives inflation being our eyes open and if the cutoff continue most which 1 In his 1st then we have explained to the meeting and now I want to
32:50
the line was virtually equations the motion of migrating to describe most of moderating Minnesota so that readers use sons of selected this season was writing lines it on and this 1 is the part time and this season of a function and dysfunction is needed because if you substitute something like this in the kidney to equations what happens is that we have something like this 1 so she compute this quantity there appears something like this ,comma so this is a I didn't the deadline of functions and the disk function tends to the objects closely media but it doesn't it tends to constant exposed to minus infinity displays problematic the single or 2 if it if we need it remains at the kids and we cannot say that these the main senior L 2 4 was a kind so I need I use this adjustment to ensure we satisfy stand this function is a lot of freedom to choose but I'd anyway I needed to know that the difference of these 2 is equal to the difference the summer of the summer months of these funds the the said the soul with this assessment we can so that these days all the time and for all the time and 2 years glamorize today being spread awaited space I need to front some small small parts and what I want to do is this 1 is not being explanatory played a and if 1 doesn't belong to explain away his face so so I wanted from meaning something the so that this 1 belongs takes 21 and their candidate of that fund is a small solution of modified it's all small solution of the tributary question was initially that is exactly equal to the population which is added to the and if the court become boss said main B into a all B1 and B2 in this way the system must all be tilled the man is very small what is expected is up so if there is a crisis of them and devices very small so it's expropriation of beatings for so what is expected is that this 1 a lot far behind the rise of items and the free media have served as compliance than the rest of the pound belongs to exponentially weighted space and since the vise solution to the KB take place and we can use a result for small solutions and that for this for this beetle I use explanatory studied results by the phrase at that explains the study diocesan cynical and to use the expression study gives I need our 1st managed to me I need to cut off continue fighting was which belongs to these continuous spectrum so I need to link costs some assumptions like this 1 so this is this is appointing a joint continue joint called a joint finding was also Armenia operators so even the 2 leaders also not to 1 if he had presented the 2 is also wanted his money can buy the single best made at the 2 and to do this we choose parameters C and X from this it's this Altos space that suffered so as to whose freed from form has a compost support unnecessary I think he doesn't want to be very small and the flight data he doesn't have a very small what happened here is that I have seen this as something for he had don't don't the decision and this finding is very close to history that John rise Connell also a joint of leading eyes KDB which will over sorcery it can be a little bit after reading this is very close to these 2 funds price of Seat and saying did all of Tennessee and there's also the assassination the visit every time we that 3 by car thieves also wanted Dismore and also took to the and if the computer time derivative of this quantity and this equation into the resulting equation what the heaviest subsidies to I'm scenes since Elway is explicit written In this way we have to have a PD get PD also see and
38:26
and it's not so clear what we get back to you Diana arising out was racing was raising questions we can get some PT which is much better Of course I need to differentiate X into the mix of X is a single it's unwanted effects because otherwise it's saloon and maybe question and if it's not so easy to approximate but then if you think of this system these 2 functions we get about assistance with some mean that an and and I am writing this equation we have very good policy and uncertainties his mother the question has satisfied among people over this 1 sold this is enough country enough to approved that the remain Syria to no more than a minute and B is almost know what they need to do or is this something
39:30
that remained after the it is immense ego in summoned at to localize space so much this is what I need to do In the last of the time and in for the bond it's very easy because it's a solution of 1 is a small selection of activity which is submitted I just need to music the idea debating the people booked to fall still on behalf the title vests and so for the bumped up the hot disinformation and for the and to do the rest of the nicest I also knew that emissary normal the abandoning small but this can be done by using a series of I had that kind cost a series you approve scattering forced most urgent the fit equation is a critical space and then using the blue slowstarting government the detainee's rejects about abound of diesel and what I need to do is to estimate have investment for this 1 and will 2 instabilities I 1st truncate the 2 meanwhile frequencies might be an uprising to enforce that .period and the effect truncated the in it's little frequencies meanwhile then it's almost an equation of 1 the soul the test made this part by using the same respect is that they're saying is that the same way as being 1000 people so it's quite easy but the full high frequencies just the said a single direct this estimate we face some sort sort of mess up most of the day but it's so to get through the saddest thing we do need to use the identity so this is a video and into the fall the tool and there are some extra pounds coming from potentials and some others but since will be for this reason and since we already know from this type of Westerners in frequencies what we need to do is to obtain the same sort of dismissal dash his wife and 4 high frequencies seeing why the feed use used his authority transformed to to the redirect this past then the symbol becomes something like this and using the losses in gaudy we have some estimate that and if we wanted only high frequencies wanting what this means that despite these claims she was sold this part can be of salt into this Seoul for the fall in that way we can that Hi high frequencies in line also belongs to the funds of space and that this there is a sense that we can progress and stability of 1 sort of missing pieces missing I want to stop stop here the top 50 and the fastest way not only will the players that union leaders your initial Alonso regions would also ensure that distorted the facts in the investigation this is a list to what recent ascension to the distortion mutilation of women wear I think this 1 is simply small and also I haven't checked work to produce solo I might be wrong but I think this kind of assumptions this is very small more want curiously media his equation you have some pieces of the model something you discipline were word yes normally you ride the forest onslaught with the future of the year that was the last time you used to describe a little bit more widely here while you were out of the country in the world would be this you will be isolated little nest of life why would you find justification in the the 1st quarter and synchronization by a single there is expression deepest part of somebody I think this fashion this adventure is important so this is this commodity mixture of right away when reading lines with work so this power becomes far from the front as it was in the and this offer cannot move because of will cause because the other 1 is fun and that their used from there should be some kept a threeset here and there what my ousted pins that there is decision and what must it is just the use wave equation to to describe here but the parents and later says that existed if you derived from moderate includes the question more precisely says should be the fashion so this this shouldn't be something like jump but we there should be some Heatley greater would with to moderate here and that this recovery is expanding in of water I think this is of provision prepare them and that's what happened of on this and also from the spectrum the decimalization but I was written questions also want to give to the finals and the all this is not took feeling my but it cost me the problem so this is another issue that you wouldn't note History of those I'd animals history grows exponentially as exposed to mn minus infinity so if you single designee Morzine 2 space it never can never be a contest bold few new