Global regularity and scattering for energy critical geometric wave equations
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Global regularity and scattering for energy critical geometric wave equations
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Title 
Global regularity and scattering for energy critical geometric wave equations

Title of Series  
Part Number 
16

Number of Parts 
21

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License 
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 
I will present recent results, and report on some new progress, concerning global regularity and scattering for geometric wave equations in the energy critical case. The emphasis will be on the MaxwellKleinGordon equation and the YangMills equation in (4+1)dimensions. This talk is based on joint work with D. Tataru.

00:00
Axiom of choice
Complex (psychology)
Group action
Presentation of a group
Multiplication sign
Direction (geometry)
Sheaf (mathematics)
Water vapor
Mereology
Mathematics
Manysorted logic
Different (Kate Ryan album)
Moving average
Physical law
Circle
Körper <Algebra>
Descriptive statistics
Social class
Physical system
Rotation
Geometry
Maxima and minima
Food energy
Connected space
Category of being
Curvature
Linearization
Equation
Nichtlineares Gleichungssystem
Spacetime
Surjective function
Geometry
Point (geometry)
Functional (mathematics)
Statistics
Divisor
Real number
Maxima and minima
Mass
Hand fan
Wave
Frequency
Centralizer and normalizer
Regular graph
Term (mathematics)
Natural number
Operator (mathematics)
Nichtlineares Gleichungssystem
Condition number
Graph (mathematics)
Prisoner's dilemma
Variance
Division (mathematics)
Basis <Mathematik>
Line (geometry)
Vector potential
Computer animation
Bellman equation
Fiber bundle
08:00
Axiom of choice
12 (number)
Group action
Functional (mathematics)
Existence
INTEGRAL
Connectivity (graph theory)
Multiplication sign
Real number
Sheaf (mathematics)
Mass
Infinity
Regular graph
Food energy
Lattice (order)
Dimensional analysis
Theory
Frequency
Hexagon
Duality (mathematics)
Mathematics
Manysorted logic
Lecture/Conference
Operator (mathematics)
Square number
Energy level
Modulform
Series (mathematics)
Theory of everything
Physical system
Condition number
Boolean algebra
Addition
Variance
Algebraic structure
Staff (military)
Complete metric space
Connected space
Degree (graph theory)
Equation
Object (grammar)
Resultant
Spacetime
Directed graph
19:11
Logical constant
Proof theory
Frequency
Finite element method
Computer animation
Lecture/Conference
Multiplication sign
3 (number)
21:19
Group action
State of matter
Multiplication sign
Direction (geometry)
1 (number)
Kontraktion <Mathematik>
Water vapor
Parameter (computer programming)
Icosahedron
Mereology
Food energy
Perspective (visual)
Grothendieck topology
Explosion
Strategy game
Manysorted logic
Different (Kate Ryan album)
Square number
Moving average
Cloning
Physical law
Position operator
Concentric
Block (periodic table)
Ext functor
Staff (military)
Variable (mathematics)
Sequence
Measurement
Oscillation
Proof theory
Pi
Arrow of time
Normal (geometry)
Absolute value
Arithmetic progression
Boiling point
Resultant
Spacetime
Point (geometry)
Trail
Existence
Divisor
Maxima and minima
Translation (relic)
Similarity (geometry)
Mass
Event horizon
Hand fan
Theory
Force
Finite element method
Hexagon
Population density
Lecture/Conference
Wellformed formula
Operator (mathematics)
Modulform
Analytic continuation
Stationary state
Dependent and independent variables
Focus (optics)
Scaling (geometry)
Graph (mathematics)
Prisoner's dilemma
Forcing (mathematics)
Physical law
Algebraic structure
Line (geometry)
Mortality rate
Radius
Vector field
Combinatory logic
Universe (mathematics)
Musical ensemble
Object (grammar)
Maß <Mathematik>
38:46
Point (geometry)
Statistical hypothesis testing
Finitismus
Flock (web browser)
Group action
Multiplication sign
Direction (geometry)
Connectivity (graph theory)
Regular graph
Food energy
Rule of inference
Area
Frequency
Goodness of fit
Population density
Manysorted logic
Lecture/Conference
Wellformed formula
Term (mathematics)
Boundary value problem
Cuboid
Cloning
Physical law
Körper <Algebra>
Summierbarkeit
Physical system
Focus (optics)
Scaling (geometry)
Concentric
Prisoner's dilemma
Weight
Model theory
Physical law
Algebraic structure
Basis <Mathematik>
Mortality rate
Multilateration
Limit (category theory)
Sequence
Right angle
Moving average
Figurate number
Musical ensemble
Game theory
Nearring
Resultant
Spacetime
Maß <Mathematik>
49:45
Point (geometry)
Group action
Multiplication sign
1 (number)
Maxima and minima
Translation (relic)
Insertion loss
Power (physics)
Manysorted logic
Lecture/Conference
Wellformed formula
Term (mathematics)
Modulform
Körper <Algebra>
Nichtlineares Gleichungssystem
Analytic continuation
Flux
Coalition
Sampling (statistics)
Thermal expansion
Complete metric space
Category of being
Computer animation
Large eddy simulation
Linearization
Right angle
Absolute value
Family
55:03
Lecture/Conference
00:06
In some of the the introduction and also
00:17
solid start by thanking the organizers for the invitation to adjust my 1st center and also along for this onto talk that this 1 conference so long as the suddenly the subject of my talk with the uncle worry Larry and on scaring properties of onto money away reasons which are all of which you cannot think of realization of the mass of origin so we'll start from there so long and so on that so prison is a system the near questions which concern the real value nonperformance on space time which is supposed to be the a dramatic potential on and on on in my talk MySpace time always been closely spaced peoplesone 1 written cost space and although this is defined what school on the community field which is nothing would be long curl of a city and his part of the gradient of and the matzo present reads as follows steely please find evidence all the when you feel is equal to the on charged current practices as the intelligence term for the last on the 1st loan you present there will subject all 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the cruise boards viable body like kind but then once you always hear evidence this you realize that all actually this man's this phenomenon of occasion residents the difference on the revision of these equations right so which you think is all in fact massacring increase the context you'll hear all can be thought of as a section of a complex line on look at where all you which can be identified with the company's value function by choosing a frame to frame by frame its mean a choice the basis of the fire at each point in this case would just be the choice of which factor should be deemed this 1 in the context play right and then on this occasion variances nothing from the rotation of the frame .period all and this all the steered it is nothing but a clue where narrative on sections of comic final the reason why I this along with animals equation is a generalization of the muscle Pridgen on in the case when the victor own bundle has 80 on gage group which is more general than just on 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due to the mountains activities of the each of and animals aggression reads on Is the animal also
08:01
massive presence with on 0 source on which system with the wrong wearing curator all secure in spacetime all this the evidence of that is equal to 0 whereby is on that song you view them as they section all the cancer all of all take the initiative so that this only when the medicine is on side at but at the bar In this case the almost that there is the same Father's on the which is given as well so all we should take a new and transform it by the following on functions or take any function from the spent time with its value and now the owner of the group of larger Utah and if you make a change in this then the all decrees and means covered the an image interpretation is of course that on these are the same objects but just the frames are located by this year so all will be concerned about large data theory for long on July the problem for these equations course assigned .period is the young conserved quantities on disagreed and so on the Baltic regions have on energy in the case of all language creators he committed that kind of the ultimate solution on which consists of a fee the on has the form facial injuries were slice constantly all basically please all square called the songs squares of the each component of this also I just chose it so that only the simplest 1 example of a donor billion Hong Group also that honors the Jews and list all nontrivial example of Of all the presence of vehicle Corsican from integration for either the group's went for completeness just accept it never baseball the reserve energy for the Mexican Britain on basically just the ultralong of S and the corridors of the and the because on energy for animals in Britain is basically just beyond Baltimore Colts so I write in this case on its toes all of this technology these deservedly studied and on both systems also have only oversold all so the the All in the faced here on and always in the space on you have on stemming endurance in the although both systems are to the systems variance in the following statements on the backs of Mexican engaged with long massacre was 0 dimensions given where you map a defeat into protests In other words on both in the skill all like congress of length it was just on this natural because they all this at the level of connection provisions In the young animals case on it's also they also still light on on his dimension of enormously and all of this you can see how the conserve energy of behave on the existing brands of system and realize that comparable equations the honesty of the on Football it is a critical dimension is all when these occur before so Indian space dimensions support for so on this with a case on there will be on constituted on the front and the rest of it also has to say on that in fact there are real wood vision will be to try to understand the young knows aggression of on but then all and that's a language region there has been traditionally sort of on agency of shares a lot of and also with the animals which the soul of the viewpoint will take the matter .period 1 Britain would be a bottle problems and about a problem for the animals operation and in that sense because the
13:36
animals accretion on his necessary there's a lesson there was agreement of the old master meter in the car and to be 0 so if more properly speaking when she called this the Mac's offspinner fuel depression but on virtually the literature of even the masses cases all to stick to the much that's a lot of Greece the on this year and now I'm not lying about anything here already sold In the only so dimensions for our own awful images of critical dimensions and in those cases on their the results about ordinary all of these aggressions from early on in the 80's and 90's but the work of 4 of the Moncrief Moncrief and on Clinton market on a somewhat but on the understanding all the critical dimension on and on at the regularity has been obtained even the maturity of all very recently so in the case of Mexican border regions on this theory due to lot .period during its power plant in 2012 that on small global existence hold on editors regularity in the for it so long saying these would come along the Mexican border region with 0 last fall on top sport on the some additional days conditionality thing I'll get back the some so but on this goal would bolster his staring holds this the only original data is sufficiently small look at and all I under on this mission so actually sold this condition is important to have on your actually have you know formidable resistance so this is a R and this has to do with the on various that explained to already here sold by the present various just write down the degrees and then you realize that it's the undetermined all if you have too few credence to have a on what the local system so you have to along the way you fix it is you impose more condition that you want your own but you want your own connection to satisfy and all of this is 1 way of doing it and this is called the employment conditions and the choice here is crucial because on it's in the following days where all the fine although structure diversified when Gration is apparent the sincerity of observation that goes back to on time market on in the 90's on the sale of responding there 4 all the Britain In this was proven when on last year this administration seems awful quick so with this small result hands on they are gold of this to all this the on large data problem for these acquitted but it's a good question on sold OK so long so it'll be a whole series of me not to get into too much detail but staring on here would just mean that on the interstate on the solution to the young and to the problem of with converged symbolically to racing clear all along you abrasion of but that the new system will be actually not exactly the on the way credit but all in all there yet only not statement with this all OK sold the question for a large data on the large data Christian for the language written on the subject of a result that long all group with funding from clergy their staff and all last year and we can prove we can show them along the result is that the
19:13
Will regularly and scattering on whatever that means holds for all arbitrarily on final so for no Poland for than a all for all marches the plan so all 1 goal the stock would be trying to give you a onward you all the proof of this there again Schumacher that's all on basically at the same time and independently there is a wanted proof all of this there by on fears and rumors all foolproof who follows the 3rd tenant narrow on constant tightness and resume your approach to romp through the began and more precisely the tape they follow the on the finance the development of the scheme due to long periods all of which were done on earlier in the week matches all that is you see on the the approach that already is on different
21:19
and on the 1 hand on it as you see it all is a bit more direct and on the other hand on the 2 proof on their views on relies more on the menu structure of the predicament structural automatically when prison and has them all the but along OK so this is the on the picture for Mexican were and they might ask what would be the on what would be the young responding on results for the animals and costs in the young young music abrasions on you cannot hold for on such a unconditional was result because there are applying independent solutions to the animals aggression on which don't always refineries and therefore doesn't scare began so all brand prisons and the thing was on instant answers the are sojourn to be in the thick of fourdimensional on and always have been studied on east in Georgia on community and because and moreover on it it is also known that on and constantly on the true can have unserviceable fall for finding and lots of their own explicit constructions all find it a lot no on due to rough and nasty following this case so in this case the proper on problem the large data results would not be inconsistent results rather it all on the question untill which energy on all those who oppose instance variables so this will be the problem for all animals at all and the on and the problem this falls in the volatile and or candles were opposed and have scattering for an Uzi all "quotation mark to ensemble "quotation mark threshold and DVD on the presidential threshold always attracted by on To conjecture would be that holds the 1st the on grounds that the young listeners a cultural oscillation Timex lost timedependent solutions throughout this also the whole Michael Colby is also in the ball on my talk altered explained the on the proof strategy of approved all of this theory all in served 2 purposes on the 1 hand Faustino although be aerial inspection of the proof of the there when other hand on the deeper revision is I want on this be this get this would be on a on a book about 2 or possible proof all the on this problem all along for the animals approach and then if tank permits all along report on the island recent progress on this problem which is staying on by collaboration with the questions what all when up that means distract beyond strategy Alster approved all of the global closeness and scary there for idiomatic language version and this is a all this builds upon the earlier strategy on due to Polish Germans and Farrow which was long carried out in the case of the without which and all this so for the purpose for appropriateness all focus on just the taste of global forces working on trade along unclear how local prove so the young whole argument is a person who traditionally been so will soon that will put spells or in other words find Walker and they will try to drive in a competition and as is well known all owned by the Our on combination all the small data results and so on finance operation beginning the state that if you have a fine then we must happen by energy concentration aired on the street finally many plants OK so that's what I mean is the fight so long there was some kind on before invading where you're social response so yeah finance and law he couldn't find a point on the phone final sliced 1st that if you look at the ongoing influence of a point to the past so we like ,comma and move from this point on a new measure the that's called a crosssectional of because the intersection of the along pleasantly slice and this Comiskey the energy on measures on this as the long does not go 0 is you as he approaches the young find time and position on because is it consistent with the case when on this point is a part of the regular of the site on which cases by continuity this would of course "quotation mark 0 then the result is a much more and this only goes in line with action the picture that this crisis on its name which is that because they deplore this constant happens on in In particular form so for proved is that you can extract the mobile on Armstrong this all planet of what with a particular pulls up again so I mean this is a fight so the same as that find the sequence of times and on space Hornets and long scales on the end of it .period sold Gershon mine is the they sank on its which approach this thing on like all piano which is key plus please extend remain in the long enough in the cold and on their sequence also was scales Jessica was 0 faster than although the radius of the young ,comma so With little that if you bought this picture and made the song skill and I want to be unit gets which means that we use scaling through anything about the following me spilled on altered so we re still everything by London center at here in existence then on the sequence converge something to another solution to their alter the were prison on the Internet and all strong local it's 1 sensed told clients the word precisely on on the
30:04
Pulitzer unit pinable interval going on where this object so long as the mastermind when Britain and once more is a stationary solutions to the Mexican border region and in fact I could even give you all know in which direction is stationary and so on by passing to some salsa you can assume it also generally better on these points on also told to some like curved booking which of the gentle like this cannot incinerator alternately his wife and all these the solution is not to be staged respect that wide all the directions within all Wall assurances because phone nothing can travel nothing with massive trouble at the speed of light because of financial planners say yes but all this is important actually on the low pressure ,comma Sylvia all and buys the seriousness of the following the contraction of white with foam formed by the 0 and whether it will feed suspected library the since July and then I always say the following OK well on we can always Lawrence transformed too wide to make it a duty rate and then the claims this claim the 2nd place this is monitoring the complaint is that all there does not exist any timedependent where sealed stationary Montreal final energy solution to that problem it is a reasonable on you apply the Bermuda proof to apply the always has prevented then you realize that the use of all they all the following a brilliant I think we're all in it looks former like the massive Iowa aggressive but this is a question on far for ages realize you can multiply this suppression might sign he reports disclosed that greater fight the the difficulty of parts In this tells you that the decrees in couple In there for a long and must be trivial so you can take on B B 0 and silences and of course on what the reason why do this on User parts is because you have financial all so it all boils down to to Europe on showing this bubble extraction on track actually I say the sport of this proposed Seoul and both unwilling to burn Korans to on on the school attraction is a on very beautiful once a for a lot for the on supplying water Britain which allowed 21 explained but he was solicited it is possible for this is on something like this August but I need to 1st offered some patients again so on many for convenience on translates me all the the need for communist translate the the similarity to the dollar origin priceless became a the media flipped the pine direction so that I could get on something that pulls all onto the home to the past going I just want to work with some of the of this have space then we also soon on for simplicity that I I've call for everything that happens outside on this all this organism so that essentially the energy outside is on very small for canceled so that only only concentrate on what happens here at this can be done on by some gluing on of initial excision arguing with and all in order to state principle of the neon that some going so of then introduced the vector field of musical acts which is the on normalized steering Butterfield good so this is a on scaling back fail but then multiplied by a factor so that on its new calls the norm is always a compromise the Princess Royal on PolishGerman told Peaceford dynasty .period expert in only a few plot this vector field all here uses his intellect espresso you always fall .period in scaling direction but then along you'll see some degeneration as you approach the on the other the funny like because their longdistance secure the newly introduced all variables you which is according Mike mentioned King Midas events in the which is on plus a survivor was so undertakers chosen that this fund rate like Coke is the few Costa 0 blocks look at it with this I'm ready to for unity on the key point is a formula for long this is actually looking so this perspective so that on page 2 time slices of pork as you want and future OK so long I want to sit formalizing make so morally morally the following 1 from room holds so on the society weighted on 1 negative on energy .period EP such that on its integral on this place on the bonds from below the animal all the the same intensity from on this slice from the others placed closer to the similarity all In fact the difference is given by a spacetime role on or off certain 20 consider chocolate chip which is also on the negative also America which goes to 0 as long as you approach the attack again and on here this density of the space Tyrol Is there also on 1 overall times the interior clones I hope that that's the on drafting of this expert fuel to the metal spirit 1st of all this so similar the are of the square In this case they survived .period please is some waited as economy where all but true but to combat result sort of it's all this wait times the following are inversely pp on the square the universe where finds on you the same quality hardened burst clear part of the squared plus staff please wait times
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on the and unit of all plus party and for the war the and actually but there is also a contribution all on some terms which involved on a which is on the negative side along a lot of what this thing OK so on a few remarks before the result of science of regular soldiers out so that was the 1st man in space so force on puzzled how gave its studiously seeing won't listen for a lot a reversible prior reversible on prison rape greater rate and cost me all the demands it lies in this territory which on is actually the contribution of a fox In fact the reasons why have this decade so that you have this sort of on all this summer months when a model based on air Is that all your query have detailed flocks and the your fault systems that just comes from the fact that the floods density on its books and CDs nonnegative sold unifying also take all you yet smaller than its music are sold there is a saying that this terror is viewed through looks at the use of fostercare which is my future on this aggression on and 2nd the news saying that highlight a list yes yes yes OK so I should also say that tightening of adopted on the time I should say that on this formula is actually rule only narrowly chose because on the way I would the wait on the sea on actually goes to infinity that about the boundary focus on you need some legalization procedure but also ministers with the on figure by the the message of the 2 of you hope be that on that suspected thing that work the big but on British soul what you get out of it on 1st of all is the following that's on you have effect .period independent music improve on that this is on a roll these intervals the following is funded from the local both by Hong Song supplies and find the energy focus of this something that's all that's always found all in the CEO and the on its approach to security and what this tells you is that this space time in a row at all is finite all the road and this all is this signifies Feisal look there is a on the way here which causes 0 on conveniently on his approach the great so this signifies that the the fact that this new rule is finite tells you that there's some already decayed .period on this integrin as it was 0 right otherwise on you on this cannot be so this is actually the on the team crucial ingredient for the on portable extraction as operated by now so what this tells you is that morally the during this sort of thing like you know on durative on on in so many words sense as a 1st appeared right so you would make use of that you say that no 1 can extracts they all the solution you know on to limits falsehoods fall sequences solutions and say that you know that was the stationary bike that's on due to the decay all stood at all which which comes from this right but then you have dividing into on cases 1 they prevent such also realizes that there is this enters the on as you all of this but the field is equal to the boundary of the call but so this is only effective arm in a fall in a fixed your time home not in the hole ,comma so you have to cases 1 when there is on when their sustained on fixed Tenneco's gentle on the field of footsteps all song luncheon aluminum energy remains on in the panic this should be all parties 1 and on your that on 2 of the cases went on this doesn't happen by time acquires meaning a clone with opening which is to be smaller than than than the snow in on the insiders lid from the you can also so great so I only concerning on energy on and and this on the music so that's what this is going to be with his friends say that this is the case with the song energy on remaining in a fixed on time a copy of it on and this is the case when on the exits all on fixed like panic and on the news as saying that's all so what happens is that in the case of 1 the decay all of such a on Pennebaker field of disaffected again and this allows you to go along find the desired sequence of Long Beach time scales which it gives you the whole structure you see all this explains why you have this particular on direction of stationary freight because all this is that it was so imagine that you found the sequences right on annual imagine that's all you can do it in such a way that on the same incumbent on the case looking on in these on boxes and then you realize that on that experts on along with politically because this wider Phil might be the on the direction the next time I factor in the on the direction of the school support at home and like me that and but then so game on the basis of the political use of it in the sense of right but then on youngsters right right but then the pointers on fact it so you can you can actually extract such a sequence by king who had 1 of the tools we wages stresses across if you if this holds for any fixed of anyone you know any fixed on chemical so the case that they cannot really cannot use that mountains a formula is exactly when you cannot find such a call so in other words the energy exits every time I call on his it's for public toward the goal of the season In fact this is precisely the on the the on the case where you have to think about this is precisely the case when everything sort of his on traveling at the speed of light right and this is on the case where your prairie with things that on there cannot be any Monday near bubbles right because if if there were any bubbles then you know because a finite energy assumptions you cannot in
47:31
of trouble it's through life in fact but you can do this you cannot look at on this day this statement that all this waited on court appointee is always it and realize that on the tests were very good weight on certain components of the the pretzel in particular to the leader of the taste endangered of the text In this tells you that in fact on there cannot be any on malleable a more positive than say the following facts all is 1 way to quantify this revised so you 1 later wall fines a concentration employment contrition would be through long look at and on your past this law integrated ensemble Over the years on London Gabaldon awaited so that on its admissions quality so you wanna take something like think on this and say that this is very small what is aware until I didn't think 800 has also I think of this and on this the music that I told you on base because to the statement that such appointees uniformly for all on London and all ex post 0 as you approach this is the time to get and of course was happening is that no outside the because of anything that's happening inside there's no is chemical and on and read into this fall and there is no Money evolves after smallness of there have been United's which did you on this and the key half come all the way from Korea interim loan that he proved precisely enters at this point so all the result is that if this is small enough so that we can continue
49:46
sojourn on past 0 so is a contradiction in this fall's so this is a contradiction in this all means the summary of the probe I think I rented since I can't think about him was also stopped he and his family and so that would lead to the loss of his formula in which itself was not among the artists had no idea that desert is the view of this the only solution is to use the cases using it going of Yunnan was because they said the yard so it's a really it's a very weak decay of your statement that it was read on tells you is that on the song this snow along this funding is founded right so obvious tells you that sort of form the the interim actually decays and some already stands at the center of the field and coalition soldiers although so what you can do it all because you know that all of this is so I am saying that finds the Burmese official but this is all responded and this means that you can take care of you can take it to the to 0 right all it's usually don't and so on they will tell you that this is terrible on spot this is the 1st of his his yeah yeah but it's only a in it's on the left side right I think that this is uniformly bounded for any future that's something that you can prove all the improved so you can take just the 2 of us yes yes yes the completion this is the 1st of these is that the Americans who all have had and so on and so and so of course but also and we also have to use fall sort of on by the nearest minute stand and so on so all by the complicated because of all the way to the same thing with now that's a very good point so in fact because his guests news Of the releases of Speaker of the National so Soviet he's more of a yet so all died rare exactly so to see on the field equation if you stand has a on a term which is 8 times during coffee and I'll you rights of this is a complete fidelity for on 17 years for the apartment linearity because in the end there which they say is that of pneumonia he became circuits to say that on you have all you have very good also diagonal decay in you decompose into onto the key suspect in the point is that this on translates to a smallness of some vessel more yet and that's it but they share it you realize that because of the presence of jury and way Britain gains you wonder if on there's no expansion decay between the low frequency partners and then the height of his powers so this is actually the difficulty the ones yeah that is positioned to actually I think the problem that we had to do was this what we have now as soon as the group that has been used approved have the government whose you say that you get the you know that the people who the right right but then read right but then that's when all you you construct a minimal all kinds of rare so right I mean I guess the point is this is more a more direct approach so we can also mean that we have a minimal kind of resuming the canvas any kind sample that so far this year more than this the use of continuation of course instead we have said we have said do from the outside outside actually been because of property of all and false details so it is very easy this you hold on to the market in the language the continue the minutes a
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beer