Derivation of invariant Gibbs measures for nonlinear Schroedinger equations from many quantum states

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Derivation of invariant Gibbs measures for nonlinear Schroedinger equations from many quantum states
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Derivation of invariant Gibbs measures for nonlinear Schroedinger equations from many body quantum states We prove that Gibbs measures of nonlinear Schroedinger equations of Hartree-type arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. In dimensions d=2,3 these Gibbs measures are supported on singular distributions and Wick ordering of the interaction is necessary. Our proof is based on a perturbative expansion in the interaction, organised in a diagrammatic representation, and on Borel resummation of the resulting series
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as time and again and I think thank you very much for your invitation and the method would like to say something about the deterioration of environment it's measure for some university in it which in particular for Hartford types sequential starting from anybody .period on mechanics so so I'm going to say something to start about half deteriorating good to say something about many the quantum mechanics amenable right Title explaining the relationship resume duties in particular would like to present the Quran but we can rules woods so hot to here is based on their energy function which I wrote down here so we will consider this anti-French full-frontal Friday even as tools in his authority and readers in dimension 1 2 1 3 we values in Seaham is the kinetic part the energy external potential and this is interacting practice of the energy but want to concede so we will assume that the potential that external potentially effects is confined to a strategy of growth in Fiji and in a 2nd 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construction of this process construction of this measure here because it was going to be useful for what and to see you later sold the starting point to memory this exhibit over 1st constructed the free measures so that the invited measure associated with the free energy international human rights standards which can be written just as the expectation of the city operator AHL minus the past and present potential Prosser constant cut but even the state's fighting the move away from a fight it ain't so 2 this Friday the city manager it is useful to the yacht analyze the operator H Cecil we find that we we assume that the potential of these confining solar spectrum is proprietary so we can find that combined slammed on and off stage and that invectives you end of fate using curious black addition justly indicate the dominant projection on we will bright the and there and local forces it is confined even neutral was 2 upon spectrum but we need some more quantitative type of estimates so 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of each 1 of these Summer again is order 1 maybe 1 maybe something of matters is it's a here but this is proportional to the trace of 1 of which is the sum of it even the 2nd values of the Madonna age was Selena operator age so this is finite in 1 dimension because of the sentiment in the it is the in-state into 1 three-dimensional technical again you don't have enough became momentum to make sure that this is finite and 1 for and so it did because will not enough to win three-dimensional is even think so it means the Xanadu leaves on a space which is below and right the measure Museum Harvard tool is equal to 0 in you can avoid the Beechmont general computation and event the debates as normal as your number and HS of schools with respect to going Zamora a right is not going to be latest numbers will have an external protect just and manager was the competition and you find that this expedition here is a case of minus 1 process so we assumed for people to blame for the erect between sufficient amount to is financed so we see red that their expectation which minus 1 normal fight is fine in world for them and replaced so it means that there is a measure of New Zealand .period three-dimensional leaves somewhere between 2 and a minus game now so this is a different way to say to the national office to minus 1 is the full manager is 1 measure of a toll is 0 . 2 1 4 elementary one-dimensional image offered to his wife that went so it can be a little bit precise you make more of a say a two-year most about this external potential for example receivers didn't think about that is instead of taking external potential confine your pact because it just put your particles in a box and impose beyond wondered conditions wrestle you have this morning it is now the Anatolian plus a constant on the talk somebody dimension of talents and in this case we know exactly what American vectors and work with American functions of his operators so disciplined way so if you take away the 2 IP access and you active H on the data that these numbers you miss invites and be hazardous this decision condition coming from their duties at the condition of at around this this stuff OK so in this case incomplete exactly what is the expectation of H S Norman square with respect to the fleet measure New Zealander that is from Afghanistan fine you didn't mention the as satisfying response the ISO for example you see that include mentioned the measure newly released just below told him through dimension leave just below its 2 months 1 just be more precise statement with respect to the general statement follow from our assumptions on them as Elizabeth imagine I want to construct interacting party measures of how we do it the way they did it would like to know defines its the interacting in violent as big as an absolutely continuous measure with respect to amuse with density given by the 2 applicants and rate minus the interaction Odessa videos that it reminds it's a hard-to-find Johnny right does it reminds subsiding 0 times it reminds wnba to among the terms yearly absorbing the museum to what is left is Is this a midseason said density it man's knowledge of the death of as well you wanted to find the interacting countermeasures so how is it possible to it to do it that way I think for a 2nd fueling one-dimensional typical I have finally had to on almost all if you assume that the VW's in any field for example it is not needed to using W interaction W. user is almost surely fine right so it is fine you can do really define the French the measure on the form overflowed -dash measured by these former him by taking the New Zealand taking absolute continues measure with respect to with density even with functions and DM well-defined it's no problem if you're into wanted however we just discussed that they to 1 or more feisty preparing the carried so this interaction here imbecile partly because final missile part of this measure renews 0 is going to be the guy and I contend eradicating the expressing fears amongst press invaded the still 5 but is able to deal with it is difficult to define appropriate measures the dancers isn't incident occurred is almost surely what was applied so this is not going to toward this deficiency is not going to work if you are in the mansion .period and the solution of this problem was already during the 70 by the words of the Athens and so and it is still called the plays the introduction my week audit interruptions admitted to explain what is means so we fix a character of positive but officer it's efficient should think we're being that often the frequency but is actually it's in the energy spectrum was a British because with the Bennett is not just the square which means that we take received the fight we came as something of a little and the majority of asthma moderable but I would so we can define select escapist Infinity's is going be Oklahoma is that he was is going to die there's to compensate for his evidence so you define role came as the expectation of the square here and I was given released form here but that's available is a deterministic quantity and isn't going on is going to be verges on the it seemed that has not implemented the the president delivered as as escapist affiliated because just as some of the land grant and so what you want to use the phrase advantage of a divergence in the fight you over the burdens with whom the role and the nation concept to define kept up got off In the course of interaction but they can good wkt and but asserted that of they 5 5 faced by square with take 5 but on the bright X minus the several case point and the same thing .period why mines the corresponding rule of . 1 and should always efficient here now I'll let me repeat once again if you don't have the OK this is that this is going to have cables infinity and this has been delivered as it was in the what if you subtract the OK something exactly the right quantity in order for abuse that bounded escapist feed and then you can progress 2nd wkt finds Bush's sequence we should think of a W K as a function of the file is referred to as a function of the obligor right because in principle we defined our measure on the set of a possible sequences on so that the W K as a function of the is a cause she's sequence you know be followed the event as eliminating the need is independent of and we do not believe by W Bob Woodward's and maybe not the best condition but I just wanted to remind users the the order the and interact and using this new WOW can define their new the core the measure a heartily hacked in half to measure and noted the measure if new page novel and the corresponding expectations amended improvement measures environment with respect to the timing of the destinies of reconstruction at the heart of Lebanon nominees switch 2 men
aboard quantum mechanics and say something about this different you're here and at the end I will try to establish a relationship between the 2 parties what we need is a system of and particles in the quantum of evidence from anybody quantum level a system of and particles can be described by a wave function which is an element of these the space here the S in the bottom means that we ought to know that we're adjusting madly with respect to produce sold in the previous language remain a system of laws so we assume that we will live with phantom windows and a significant respect possible and then another and another .period but understanding we are giving the sitting with Fanchon which are normalized because it was fun to mechanics with Phantom probably think that petitioner take the absolute value squared of science is the probability that entity for funding the particles in the different regions so we always assume that we function is normal it would very interested in this system it is quantum system in the so-called mean field the regime and we cannot deny the mean season regime but taking him in his Dornan and the words an internal hearing it on the news set up a joint operator on the sea the space here and important American important think property discovering constant proportional to 1 of this is what characterized mean field division usually means that you have so many particles and particles each particles interact with almost all of the particles will we have an on N a collegians if you want but each 1 of these interaction is very weak it is aware of 1 of so that the total effect of these many weak interaction can be approximated by buying on iVillage midfield potential uses and this is why we think he's 1 of the but if you want to the preferred to fix indifferent to me you want to put 1 over and you have to make sure that so that the sleep after the Newtonian this operators here which Unisys a sum over and over and a party so it's a it's an object before end of this guy guy's some bedsores and number the father and squared but even right away and that of the same All so Lincoln did give rise to some as increased OK so this is a millstone and we have to look at so whatever properties of the system and let's start to see some relation with the hotter during time the pieces in the previous part of the top of the 1st a remark but you can make to see religion heartily relationship with France that is building the ground state announced a ban on the grounds that that the ousted factories Virgin vector rose 8 senators more possible item value and the gossip damages the corresponding then I took a lot of these type of pretentious a list of possible tension is quite easy to see people to check that the grounds that exhibits condensation meaning that even ground states that you can approximately granted by a state where all or particles having the same 1 particle state so you can approximate the allows states that don't buy a product of and copies of the same 1 particle with finally same 1 probably means that fight using it tool for the not the end of of the day the front of the fan of them invented something which is in direct space again eligibility for a 2nd that you have condensation that is correct in the ground state budget and onto the ground state energy just but taking the expectation of is doing in the stadium day so you have to take this operators here you have to take petition with respect to these estates here on this very simple competition to get exactly the front you get to hard-to-find Fountain Square with defend fight on if you want to go and take out there and and when 1 of his performance in this format he is not completely right but to still lives last from units and thousands behind the front of the the energy of fire OK so it means what is the ground state is a way to better the grouse that energy which of course he and his moralistic violence on installing and you divide by and use you just have to take the fight which minimizes the behind the front there is still so grass at energies giving embodied by the meaning of the heart function on the grounds that a rancid vector but respondents exhibits condensation in the meaning neither of these aren't so this is in some sense of 1st the relationship between Hartford is a 1st way in which high-altitude during his approximation of anybody want mechanic in these mean fewer as another example is fruit of than so that can sing many by the quantum mechanics the time evolution is governed by the surging equation whittled down here so many practical many bodies should so on the left you have attended the irritable with Frenchman 28 times and I and the variety of reaction of them is doing on the same wave front and side of the vessel easily question but the fact is that the logic can imagine it's not so easy to start beating you always a solution because you just suck you can just sort of it by applying the university group it reminds HT on the initial didn't always get a solution but took this to say something odd about as often as is very is and had had pursued the suspect in the sensitive improve following convergence to have the equation and this is the following statement so if you assume that the initial data is starting far and bottle with found agency meets approximate condensation and Naumann not assuming here that 5 million neither that can be any fight because it is if it Cianci the grounds have been doesn't move is also interesting but but point is that here if I can be any 1 particle with front and just have to assume that at times you look the many bodies taste approximately product of his fight you let it evolve over suspected this full of many believe economics amended planting you still have approximate condensation is still have it is the right approximately PCI a solution with the product of and copies of 1 particle with Fanchon fighting force of about 1 battle with what is not the same as the time equal to 0 you have to avoid the and evolution is described by the time-dependent categories which that so again we have a relationship between many but want to mechanics and the and the hearty and have you and and also here I II and III and I wrote some of the names of people who have been involved in the source of a fierce worker making the 70 by happen very Geneva available here and then there was the work of born and then many of the people and they will still have some work in this direction but maybe maybe this move on and and would let schools to which now to listen to the main question of the
stock right now we have we're definitely the relation between mini-budget when at 11 of grounds that energy and that 11 of the 9 extolled natural question that you can ask is is is there a way to to understand this environment manager you having Hertfordshire reducing violent measurement reconstructed in the 1st part of the of the talk as seen these measures as emerging from anybody quantum mechanics and what are the right order to what what what does correspond to the set police say invited measure at the level of many when me and for Mikhail one-dimensional system by quite precise answer to this question has been given by my mature Louisiana now amended ecologically who proved that he was about 2 years ago but that many body did state that in these mean feed the regime can be a park and approximate by very heartening bottom measurement that construction with part of the vessel that is the relationship with the measure if you look at the many body states the game so let me try to explain what are many body deep state that so many beleaguered states described Tourmalet even at positive temperature and in order to get this correspondence you have to take high enough at high temperatures as always see a uniform now in order to explain what I'm used to remotely state that'll tell you 1st what our enemy stating quantum quantum mechanics because this is something that didn't mention that and maybe not bodies from so when I told you that you can describe quantum system by way financial just correct but it is not the most generous this creature is positive standards result important the what what we need and describe your state by many states in state means that you don't know in which state you are you all you know that we are so that would be needed to be in several possible states guide and member states who are in is described by density matrix density matrix is the taste last non-negative operator on the where we worked with with the 1 so because of this essentially means that the density means except then always believed in as a combination of but projection the sun wake PGA but you can't interpret probably PGA of between 0 and 1 sun as for the day so if you're always given me somewhere here the interpretation is that your staying your system using the state side J. we've probability DPJ the it's a different thing when the mechanical forces 1 to make a gift you so if you have a set up sides and you can also take a new wave function but the combination of the 2 sides but this is not the same thing as taking the combination of the project there because if you take the Commissioner projection you dont have interferences between the difference with different but that "quotation mark the incoherent superposition of of states instead of a coherent vision anyway if you have your and a few of his interpretation is such a clear how you can compute the expectation of unobservable in such a state radicals because the issue of the exhibit a new idea state control where hot what is the expectation where we are in the state of seige April the DPJ so we better take the expectation of the sides and everything related evidence according to USA Waste's beads and if you think for a 2nd would find these some here is just betrays all the broader between 80 and the Soviet from here attests to how you take expectations with respect to mix that OK so why didn't abuse mixtapes because it clearly states that positive temperature have stating quantum mechanics and particularly from the temperature as 1 of the the living according to a state that is given by a normalizing constant times of exponential-of minors be depends at the time so if you have a misdemeanor channel we've Icahn-backed sides in the value 8 means that you have this format here it means I think the you are in this state PCI With probability to a man beat into what you need to know when you are positive that it will not only in exactly we stick to what you are doing all that you adding Boston many possible steadily some probability misses on the on information that you that to have but if you look at his farm political history is beta-testing fainting so the temperature fence 0 then the probability of going to collapse 1 of his Morris tenants vessel at the time of the attack and but lived legal status is just the ground state there and the other limit on the most obvious when the temperature goes to infinity where the schools to and so it means that all states have the simple will be OK would solve this other types of states which we gained in Leichter a single infinity pool to live a nonlinear needs different begin state associated with 5 security defined in the at beginning of the the guy and this is what we want to understand but but there are too if you think for a 2nd about the day with the Nunavut always important objective why cannot be true for these so the 1st thing and about is that if you fix me that positive so it means if you are that takes positive temperature then it is easy to see that the state of steel exhibits the same type of condensation as we have on the grounds became a remembered in the ground state or particles are described the same 1 particle would and it's not difficult to understand fix the Dow was up other defenders and his decision because if you have a system where it just advantages afforded but they not to take its additional this that energy by something 1 right which is what you see should take a fixed temperature to fix the Grandage's means you're not only the guns have you have a combination of states with energy which is a little bit above the ground much of the order what because if you have any farther than this weight he is still going to be very small in advancing OK but if you exactly grousing by order once it means that you can only move a finite number of particles out of the ground but they enjoyed excite the final number of practical
most of the time we have and particles in the system the back of a particle is the industry 1 but the day so because of his argument is guarantees they beat of thinks that the measure but it was yet the end of the Act unless there is going to get that the Phantom exactly the minimize around the front the guide about the practical all in the same want particles to that demented sit amid a different from these invited measure that we wanted to understand that so we have to do something that and the thing that you have to both with you have to take the temperature to grow old together we as defeated you don't want only the number of battles to win feeding but also the temperature and the right size turns out is given by the table 1 were told 1 of arrangements temperature is about to end look at this in a system that is the 2nd American and if you think for a 2nd back to live with a thing about the ground state the right immediately so that it is the relation between the number of particles many modern quantum mechanics and and they born Norma all the corresponding hottest data with 5 K so it means that if we consider system of a fixed number of particles and bottom exactly and particle that he didn't need then we will have a majority at a level where Red Bull Mondesi's seeks to we fixed as to not and this is not what we have what we had in the previous part of the talk so we have to switch to a different representation of despondency system where the number of particle is allowed to flock to along at to fixed minimum part wanted a combination of states with possibly a different number of particles the reason for this this whole year getting there the land use and related to this and this is all entities have it was the head of the initiative lonely and engine seen him as it's known on 1 of its own with variable available largest weight is always on the grounds that misses the stability of most energy but when you have a height of large of states so it will move up the managing of incentives refused because the temperature um a gasoline the physical languages means that you want to switch to a canonical ensemble cannot academies in number of particles fixed that the grand canonical ensemble were allowed a number of possible flock gay so you would think we take adages not only listed with different energy but also state of phenomenon by OK and grocery regard chronicled a setting the right way to do it in quantum mechanics is to switch to a fox priests at a presentation of the system so let me try to will fall said this performance said 1 position or canceled fast pace there was Wazzani folks space it means that instead of looking at the fixed and to space we take the dikes I'm all for possible bases or possible number of particles as it was something of a roll em up and we take semantic product of and copies of 2 right seeks to end the space here are the same as the space here that described the state said obviously stand we've exactly an article and now we're something over a possible and because we want to allow a number of of logical flock wooden went to fix OK so you have to Foxboro so so vectors in the Fox based sequences of with Fanchon side and for 4 different values of the day you have a different number of particles but there's other folks this is very useful to introduce creations in relations operators of these is the forum where it's not so important how we defined that interpretation is somewhat important so that a starter Fairfield mirrors the efforts mental Funchal on the streets 1 particle the 1 particle ruler with funds and acting with a style of music you are creating a new party believe that with front you would increase in the number of particle by 1 and if you have to use a which that joined the fray start you are decreasing the number of particle by 1 would invent a simple algebra and that's about to be useful creations and plate satisfy Canonica competition provision which hurtled down here so if you have a commentator between a and a start is just this kind of product in 2 between and and computer going to it is also useful to introduce the this emotion which called a effects and they start effects which create on an island about the Ogata .period Texan can easily disillusioned because because you have to view of new doubts Nevada indicating as a function of the West Bank couldn't quite make sense and the decision is so that these relations and in terms of these creations in relations that this emotion you have new very right Canonica condition relations based on human mind part of you can define what operators and 2 operators of England to need that a number of particles operator to another member but you know a system a number battle was not fix itself it's a viable readily accept it you can measure how many practical estate that has and the right way to measure apply Stability and if you think for a 2nd about that a start affects effects is the density of particles .period Texas given the total number of particles is given by integrating these things here over a possible bodies of OK so this is the number of patents and the other 1 is an with system and to find them Madonna why we have this past year we need to get a little exit visas settlement is additional these operators here now we suggest an additional fuel deferred you can check and that they are there the Amidon operator as defined in such a way that commuters a number of policy will not change the number of particles write sequel the understood and because needs to the number of creation is the same as the number of fun allegations of vessel the number of particles is not changing initiatives to make it to a fixed number of practical you get exactly the same operating as we had before but these things here which may look scary is just as some of the operator minus the class press external potential acting on a possible particles from 1 plant and in the same way misses the interaction is equivalent to these things OK so now we have an unbeaten operators so we can
construct the grand canonical agreed in state that so this is a states density matrix is a positive as non-negative tasteless operator was traceable to 1 and that is what 1 is guaranteed by this normalization constant fear and many say that it is proportional to its financial health -minus 1 over and then you have again the combination of energy and number of parts but it's a little things which episode revealed expanded the number of particles in sport and cap a constant Capus called the chemical potential right switched from canonical grand canonical remember but it has not fixed a so you have a new valuable census village onto form of this number of practical and that is that you can you have to fix the cap so that the expectation of a number of practical is the right word but don't fix the number of battered wouldn't fix expectation of an OK so this is a state that we are interested in indoor righty Tennessee that we divide by 1 of the men because of you we want to the temperature to be what went right to the beta is 1 of Iran's and that's why you have sex and 1 of where was adjusted reminds me that binds age plus number of so-called it since we divide the 1 of and it is useful to release gave re-creations annihilations operated by dividing it by 1 of us will of right because we remember that we have the HND Beagle back ones like that it's a sum of operatic and aquatic park but in front of aquatic but is already 1 of the men and I want to drive a living 1 of unending front so the 1st guy is coming in 1 of the main factor was analyzed 1 of Iran's quest for a solution to 1 of us will defending each day and they stopped all this and get absorbed into the OK so we define this new operators who use extra collided here and then the same state and the constant is the same as before and then this is the way to rewrite right now I have sort of and inside he's a staff and 85 days this measure but we want to look at the positive standards state which describes atomic leaving a temperature and look now what can we say about the state that well let's start by looking at the free practice so we don't of interactions with the W would lose the 0 right was likable with things that will as we were going for reacted your 1st look at the feet of the 3 measures off your 1st look at a few days by removing this interacting party OK so it could also fall under that displayed erratic expression meaning the stock but you can't they have analyzed by switching to luminaries is this is exactly the operator age that we had in the 1st part of that and amendment according Begum values landed today and begin director Susan right that in this way seasons when the standard because a study was a UGA is measuring the number of particles in the state Eugene and land as is at the theater director of red so if you some of the old J of number of battling the state use 8 times the added that the total energy released many particles a castle we use this condition kilometres off the state with the 3 parties in front of free normalization constant against what can we say about the state he arrest that the measure of the number of particles in the stateless settlement for example the number of particle in the state you are I right you either 1 but it was said you come in knew of and anybody we went from many the same particles and state he was would the sole reason expectation of this guy here expedition is that have to play the crazy always operated against these a density matrix he has put a matrix exceed and about an insulated as may factually twice the some of 700 right but or terms with Jay different than they simplify the cancer from Bolivia amended eliminated because their independent departed because there is no interaction in this in this setting so you can set everything and you are you and not just with this that the expression here and there and I would love to know that this operator he measured in the number of particle the 2nd values just ended and then there is the expectation that that is exactly these things tend to to compare with what we had in the heart of the case and have to consider member that we had 1 of the men died when it's not so these things because if you expand export entities while plus land Ireland and the something else so at this form and I which has a much smaller when and this looks exactly like 1 of them under so you start to see some relationships some similarity with the hard to measure what we had a Kansas went to the total number of practical many of the sum of all possible remotes right so far just measure the number of particles in the mood to you I know in my and something but I and II dividing by the end of the 1 to look at the risk of traditional divided by and is already moving a agent which carry the 1 response forget about this 1 and we need competition and you find the sound here right and then you know that since we in 1 dimension we assume with someone overland eyes finite so expanding this guy here recently from his new users submitted as something affordable into 1 three-dimensional built and where the sun is to find a way because even if land is very large you do get a cut right as soon as lenders must be given evidence is going to be very very large is going to make everything converge but it does not go over to right and state so what you get is a number which is fine at for every yet but it diverges languished in Ft the guys suddenly a similar items have accused the other differences but the heart of the hotter than other party were adjusting feet available normal adjusting the right until you have a natural cut off from movies and so you don't see that evidence for final knowledgeable evidences OK so these properties of the measure now if you want to pass to interactive measurement of inducing couple figures would remember haddock is a look at the interaction with the introduction right not the interaction is going to be expedition of international powers of interaction is going to be fine as long as in his final because we have this much dog but is look at what happens is endlessly in and then you get a divergence of gave that something which is fighting one-dimensional but is willing to be in the fundamental .period so also viewed is natural contributed to this report because the quality news that replacement interaction now we don't need to introduce into continues diskette off because this is not right that off again by so we just figuring that action you subtract from the store guys in the store and I say this as a quantity here which is just a few expectation of the problem of it so which is something we've whose whose Intergraph was 1 among movies that have beverages as anguished infinity .period again now with be using this be guarded interacted and finally called the canonical again canonical status which is what you would expect that and this is again estate on this on this floor space this many wonderful OK so now no land
to compare these state here we've got a measure of electrical started thinking about and that you should be there to be some relation between this guy and the final guy and then there are 3 and a half the measure the canal when it is just a comment and the skipper ,comma let's go so how have you compare this state this many body quantum state Weaver manager you look at the correlation function and look at the moment of the over all the higher measure wonderful for Travis it's all what you that whatever coalition financial they expectation with respect to this because of the many practical measure of brought looks of aid in the stock or observer of his many parts system can be beaten like that of course here we only look at a case where a number of creations is the same as the number of foundations operated just because altered its petition said it was soon and they want to compare these gun mind caves correlation function with their 1 constructed with the removal of important matters with that expectation with respect to the factory widened measure of products of this field firefighter OK OK and so conjecture is
as was defeated for the final K the salinity of for example measure is not here for example measuring in this latest apology should be able to see this should be the same dealing with an approximate anybody quantum Leap state that we we've made inviting hottest stated he would carry out the level of threat of expectation of the review of all these correlation front and this is exactly what was brought proven by the originally from one-dimensional case carried news of some results for Atlanta dimension but for a different type of potential sewn up mentioned here so what do you want to know or what we wanted to do was to put the same thing in dimensions to 1 the right where where you need to be confident in 1 dimension this simplification of course you don't need this because the technology look at unfortunately we cannot rule this conjecture here and there but we have to modify the need to be to be so many body functions state that and instead of looking at this stated it reminds for on and we have to look at is modified state with this parameter demonstrating just bigger events here when we take it with take out to waiter terms from here and we put it that this is a mistake here the 1 must await us longing multiplied speeches you not do not intend to Interlachen Cadiz 1 day soon but the greatest city in the east the submitter committed affordability and put everything together and you get exactly the same measure exactly the same state as we had in the previous liked the the fact that the state here is not the same as that was very deep state that we had before is a consequence of the fact that you cannot put the financial statements expansion of these not expansion of the possibly for non-payment of aggressiveness is state that we can look at fix up and for the state that we can prove what to expect selects the Cape article on a collision Funchal conversion to lift 1 correspondent will have the right to say other anglers fainting the distinction conversed with saved the maybe because that market that didn't make before so why
widely we expect this convergence to the point is that in terms of the they start that the court is not thing not commuting operators but the obvious 1 of the men Soviet almost commuting fees and as anguished fevered become committed fields and of course and of course then nothing gets the same decided to have at the after OK but that the 7 knockabout ,comma dividend but there in Fiji it doesn't matter what you write but whether readily agreed on it put everything together because in the middle colonialists and so you have only 1 no modification for that OK so maybe 1 means to from the broth was about approval proof is based on perturbation expansion so you have this being a blunt instrument but appointed state we're free interaction and and also you have got the measure which left the path and interrupts and then expand to both of them with respective introduction which means you can do it by the market has is escaped despite the absence of Boston
women the new Nunavut 1 expansions to expand everything in terms of free I'm Etonian and their interactions area in about right the settlement amount to much of a cylindrical construct this strange state wedding was modified state level subtract this guy and you will find that convert aside and you get that something like that exhibited deliver effect this modification is that the integrity of the invisible hand of CDs they do must start from 0 in what was about start from the Dutch and to 1 month OK so this is the expansion we look at and then we have to look at all of these terms here and abroad that each 1 of these stand here converted to the correspondent the film's pondering the expansion for the heart finished the day so well how do you do do it the way forget about this party stick with when you have here each 1 of these W this interaction is aquatic operators based on start AMD so you have a lot of a stock engage us into doing it and you have to take that expectation to complete his courage and also where can
also use this the during which testimony with respect with the manager the expectation of any probable fate in this stock is the summer over or possible pairing son of product couples of AT and this is the same is the same kind of rabbit you have forgotten measures and the alive is directly and convergence to the series of Augusta National for the last Russian measurement that in you day so you know what this expectation for that but that's part that the city you of these things here but that he had before that made convergence to to the classical little it will have to be too have to measure if you had it in the opposite .period the remember Duncan would because wearing wonderment cancel is ties to the fact that start but the difference is not so much as Ali was 1 of the Act because the sport was commuting problems so I think was defeated you can get to the the this guy here then compared each 1 of these guys we've 1 overreach that hiding behind the expansion and you contributor to expansions of it also is the need my collaborators on the skillful we've growing this docile so each 1 is that each 1 of you so legally lines there is a potential domino effects minus wide right there and then you can live with the alliance the described the pending review of 2 Tucson or possible pairings so you can bet these guys here these guys here this 1 in this land use land in this and so the only thing that you cannot do because of the economy is still bad 1 of these guys With this neighborhood the same next right and this is what you don't want to do because we should bear this guy with the same that Weaver a within Mexico if you go back for a 2nd to see it means that you have to think of expectation of the summit .period XX and I knew we needed but it'd take XX and you indicated that based on this operator here which is essentially a trace of 1 of which you know always seen feeding program management for the best of us what you want logical big only 1 operator the case of falling 1 operator we have to take at least all but this is what you have because of these structures of the past cancel using these items of interest was balanced and believe that the expectation of all of the spreading Hughes bounded up and then you can there you competitive improvement with the addition of each 1 of these spending give converges angles and feed it to the corresponding to amend expansion for Hatton and this 1 last thing that I want to see 1 last final obstacle is that is affecting for a 2nd What is the number of pending regular over you have your to where operators and you have to parents and the number of possible pairings of another of 2 went 4 2 OK so the monopoly of foreign because for each during the expansion of 1 more interactive induction squat 3 of and when creations implementations of so it is going to be proportional to plan for the dollar now you have something which adds to which is disintegration in the original simplex which gives a one-over and Victoria that's fine but state whether divided by a factor of behaving like an Victoria so this series is not converge gate so we can compare Eastern indices the wonders of the correspondent and the custody pending the for for the the measure but we don't have a convergence not succeeded to get to that it is not that we have and what say what saves as if this is the savior of Berenice summation itself which some additional information you have to prove some additional ICC all these financial review of coefficient in front of the coupling constant informed of interaction to me to be complexes and you have to prove analogies Indian some in some uploaded to the agent of the of the sponge and then and then the study of breast condition allows you to eat up this additional and Victoria over 200 expansion into get convergence that you that you want to cancel thanking them for Europe and few the question this is the 1st in lady the Hillary of no it's only because because of this that whatever technique is that media gives you convergence but before knowledge we used to kill people to get that you will need this CDs to be convergent amenable than incandescent vary but the material cannot just gets convergence pointwise convergence is not the the acceleration good both you can you see good I don't know I would have to think about it you see that the level of the ground state governor when love affair at situation I think that you have just sort because doing this at this temperature to be very large right so very much up in so I'm not sure about the Americans still have something like that too but I think not what that this and way the few new