1/3 Bounded gaps between primes Download
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1/3 Bounded gaps between primes Download

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36

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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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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

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English

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00:00
Area
Standard error
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07:58
Greatest element
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23:45
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30:10
Shift operator
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Diameter
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34:19
Axiom of choice
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50:46
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1:07:14
Lecture/Conference
00:02
so leaf and in the you In thinks
00:14
manual pretty back you digest I guess I was pleased to get through lectures on but I got to him primes the 1 when I agreed to do this but them because performing arts was be that there was I want about I think right now and so is the answer can move faster than than anticipated so that the the you about unaccustomed prize is now so easy that I presented in 1 hour but it's a bit but the other 2 lectures will be on on critics suggest that some crimes which is still relevant for this term look it's so the suburban against fishing primes Members Only the case so that he handled in private and then for any action number and defined hm to greet the remains of the gap between the primary and was the 1st and 4th and bright so HMOs the smallest in the role for which he can you can squeeze in imposed on Prime's incentives such as mineral in fairly often OK and so I we care about is is what tell what the values of these numbers so will double the combined conjecture at this time of recession that each 1 of these the gap between consecutive prime isn't is equal to and from the office and more generally the other prime the height of the conjecture which generalizes at the combined conjecture but I wouldn't implies special case that the high aims at the high wages should equal to each of them was 1 where HOK is defined to be the directed this more stamina and more diameter missile Keitel each 1 of which can so an admissible in a and k Temple is is is the key to the images have city in increasing order such that an SSA you miss 1 Congress classes more on PVV which avoids so why he toppled of increasing images which avoids at least 1 Commons class but he fumbled OK so for example as you ,comma 1 because not admissible because it doesn't avoid any come with us to but you ,comma too is admissible as the 1st missile to topple this is why 8 1 should equal to and and smooth the narrowest 3 trouble you can get this is your 2 6 was 0 for 6 were also working as you to fall work because a dozen devoid of meaning what 3 so what should happen to answer which 1 should could go to page 2 should call 6 inch 3 the action according In August betrayed him and told 16 him support and so them to the Saudi agent should be on the sums that will understood by the way so long as symbolically if a largescale data of the the narrowest capable you can have is that most cable OK and at least hockey looking up to up to small errors but so these easy visible fairly easy come behalf of the asparagus observed that the 1st Cape crimes this caper times larger than K admissible the glut of amenities U.S. troops and the Panama theaters here that the diameter of the city's about you OK so I think it should be about and the along as a consequence of BritishRussian quality but if you remove 1 ,comma Scotsman block amenable was looking for a repeat British March you end up about 100 MacArthur left and we did it if you just plug it in New York it was all about and these are basically the these almost Bristol ballots the new can improve upon the little ones on both sides by using bishopric instructions or side refinements of the British motion calling example Mongomery born lots of inequality but because this is missus will react was of a small Taylor came up to 160 something you go we have that you can use computers and incomplete exactly what this is is up and then out for me in mediumsize K like millions of weekly there's lots of of of you may not know about OK so that's what I understood but this this conflict is not well understood OK so and so is this money not even obvious and easy is a finite number close to the number should the Indian Prime is roughly in order and that doesn't have you to as a finite and all the tools used is that these gaps in almost against both by the bylaws of piano and then I know that this is a list the becomes a case since I have just individual principles but so far into the often the should be of the order Lockyer but you know you can't get much better than that of without actually doing more using more than a matter of conjecture lesions that we should have much more to say about these ratios the censure because on distributed Sobel but for the year as far as what and this decision should something connected to the posting process that those contractors and out of reach please Italy the boarding gate but how OK so so they break through this area the 1st big break was over the 90 to 2 thousand to 2 a pair of figured that they think they should have been bicoastal your and I've forgotten the precise here that the 2000 and so on major a method to try to control these things that have basically what what they what they found was that he believed he had a good like redistributionist comprise then
08:00
you can insert into computers sold Sears net and he can get control at least in each 1 will initially method you couldn't say much about a lot higher than it was a case or what I mean about equities from equities to Houston's enterprise soliciting to on mangled function of commerical functions law and people in in the power you otherwise essentially this evening functions crimes normalized and indeed to have some nice and optical properties that I wanted for example among the young is particularly was a function as nice property that that and then medical 1 his Walk This is what's where care what solutions cases this is the moment would function and we have to Panama theorem goes it this is the access and then we have to have a problem with you may never make progress progressions you a primitive and aggression and then at least for some for fixed and Q and a handful logic said the should be 115 the fact that fact that they should have the Panama tree equities added in all of the Commons classes at a primitive OK so that and so this is true for fixed Q and logic says the problem a few enactment progressions and you can ask what happens and the accuse not fixed in accused the bigger and so for example I don't think so and specific theory you don't care about single too so much that you care about what kill average but on the other hand you don't and you do want to said to be uniform in the modulus so let me just went down the definitions so given anything consumed 1 we see that we have Halverson and conjecture and level faded and if the following estimate holds that if you take the discrepancy but in the fundamental error in Panama filling in at the request of progressions and initiate the West discrepancy of all the the classes but which as only West Life which Q which is which is so it makes it difficult but they needed averaging in Q better again can you get all opposite level and the way normalization averages averaging accused summation and the sum of the year restriction to hearing is doing everything for you then what you want is that you want to that this guy rebounded by by extra powersaving of any loyalty you wish for any fixed polevault you go out history much less next so that the trivial value below is projects is trivial and so you won't save an average of lovable review so that the statement as saying basically that for most acute for them you I like members will choose you have a good time and nothing professions for for all primitive classes and what you for them and for most acute cases here have a stand conjecture and so that Panama the whenever questions of this kind of via the theory was 0 and point of this actually of the Civil War is maybe the wider use of the land yet it is that the definition the Magnum claiming Europe get the rentals are yes yes OK so I am going to say no what's known and rock soul and rock so appointed have disappointed version of the 1 of the few minutes enough aggressions quality waters theory which I would doesn't like us want to itself it is 0 in some sense it it says that the vehicle stopped at New Star's truth some miscues polio with the new leadership to accuse of portfolio size truces a civil war resumed in Ecatepec even on physical problem but you get that for each issue separately and then from the rear and bottom of the Premier League on theory says that as you can thanks to all the way up to maximize violates the spirit of extra time with them to locate loss and undermines the values you depends on the state might even be like he was half as physical evidence but he can always 1 house in particularly good shape of a Stanford filled the data just up to us little shy look at and then add this is still pretty much the best there is no need for a full list of no "quotation mark have them as written here so the conjecture is that this is true for all the up to 1 had wanted it fails that ended in there and that it is you can't take objects and infected can take you even within lower the Ex all within like Lykketoft screwed local Texas and think of the the was out of seat the Crandall who brought on by the I'm OK but how the people we have in new close to what was that we were so 1 half of it will be with this exact for this exact estimate leaders will see weaker versions of his which we can go beyond 1 OK so this is this is the the standard way to to to describe egregious witnesses of crimes should remark by the way that if you knew that generalizing and officers Jurich that would give you some the heaviest and injection popped up to onehalf and then you would need average that that you get a good lesson for every Q and not just on on the average ,comma but even the Gereshk and Musa jury preached to accomplish past 1 half the of the week even has as he did in the arm that homeowners has writers Hello yeah you did that evidence had these conjectures that most of the world you can't beat onehalf even with the help of the right of after so it's out of it but it did so different universes that this severest care about up about about average case Q at and and Martica number theorists would this month focuses more on individualized acute and and so it is with this 1 orthogonal but but only when it goes on and scored during specific and it isn't as good as your age and unconditional it's true intact L on the main focus of my of my talk but of of course is that this is like of many things and number theory that this issue this this constant depends on and
15:54
of course and the vigor of Corsica and I'm fortunate because of a single Wolfertz steering dependency goes fear and secrecy is ineffective in the courts we don't understand want high schools as for the of all the courses here are ineffective which is a little annoying but in practice it's not a big issue and it's not our are not tree is so full force for any fixed a loon there is no by the time yet so there's survival among the business you call for this constant is 108 and this unfortunately is ineffective because notes there is no explicit unknowns for the constant on although in practice 1 defective versions of all Falls Falls fumes on the states is not such a big issue because of the failure can be localized busiest single exception what you think about it as a single tube which mingles bed of formal maybe single set which go but for most queued outside 1 of the few exception once things are still good and so is this is not a big issue but I'm not going to focus on effectively issues object so these these are the this is how you can measure the dispersion of primes and so I'll want to know what goods continued show In this paper back in 2000 was the 1st of all everything interviewed previously and if they have a standard for all data then inbounded each 1 would cut about on a 16 day in infinite priced as 16 about the effect the warning it's the only state exception close to 1 . 9 9 6 was on but that is that's enough and more generally if you can get anything beyond onehalf game likely get any good 1 dash and nowadays analyze it 1 out just 2 of the our ideas and then I think we get some firepower on each 1 and that of but these are these are not known and so just isn't going to Bermuda and got all they could again find this but what they were able to show signal must log communities 0 so that they can find somewhat small gaps which a little smaller than the average visitor walking behind as part of a cost about bounded later wondering to his lobbyists criminal law given in an unpublished work what all wasn't quite the seal the differences in and Japanese all was also each 1 of their method initially could not handle he managed to find the best they can be reached Tuesday the assume everything is using the hopes and conjecture that also get smells of how they could get a deal that could extend those awful for the 1st Capital secondguess but I yet but for reasons which got removed later that there would not be able to get beyond beyond 1 so that was where things were 10 years ago but later on it was observed independently by what Hashim pins and these young clothes for several years later and that this conjectures hitherto strong forwards needed for 2 to get together just interested in banning H 1 vote for instance and you you don't need a full that have as their economic and you have something that should be onehalf and and still get the around benefits of monarchist Prime at 1 end of the year looms so I little 1 can not just us to a balky located at this juncture rotation it's it's there is a lesson that we could call it that fraudulent like that of our year and is used to clear up the right 1 all OK dating gaining the on off you can get ban against crimes and no I wonder if they have conjecture but actually you don't need a full and conjecture and so 1st of all I found you can weaken it by taking this step outside the cell phone but to have a confirmed its GDP began difficult friend everybody to all a total France up to some special so I insist same conjecture so he can weaken into a soso here and we can choose a different date for each accused simply HQ UPU picked the worst depict the worst in bed rather than do that you can stick a single lodges with good nite because Indonesia and then you you cautioned .period each of these these these much alike now and this myself doesn't actually sending you very much and they give up because of the trends remain again that that if all logistical prime rate then and thinking a single aviation users since the missing modulus of jointly a boy you also do this that you you and you used to not not not to tell to cannot be coprime you used you you forced you to deal with this problem how to apply the duo also forced you was good and that smooth producers in France article the other day and project which means that the and it hasn't come here so what this means is that this no no prime factors larger than larger than expected over and so and others to promise and about and daughter fine and that effort becomes easier when building a small because furor you you have to have fewer numbers and acute because more Moss move more variable I'm OK so you stick to 2 to smooth modulus let and so that then taking a sabbatical from content because this they have course shit factors and these cues and so that does have the constraining the merger to recall common end he gets manager because of this this is this is a weekend statement nowadays we we call it the a modified Mohd I should Prince John conjecture of 2 premises about the and and Delta it's weaker than than this conjecture that absent conjecture and so that the claim is that he can prove this conjecture for some of our playpens being 0 then you still again analysts on of 1 so that this
23:45
information is inquiring persistence of implicit in this way but it is pretty close to being in office but in and of course last year John what he did in this language was that year to prove this conjecture was some some of those of of poppy and don't get to prove this thing it's true for specific choice fucking Gordon took it also important what the hell you got that he just saw positive anyone over 1 thousand 168 for that he was able to get specified value of and 70 million look at and a later on and I and many others including many students from earlier and no formal called upon project if these collaborative online map project started in dollars about 10 years ago and and we were able to to improve that this was also it's 1st of all were able to get a better distribution was out and balky as big as this as anything up up to 6 over 700 minus minus whole thing a small thing is and what they have done so you can get as close to 6 0 was submitted as you want but by shrinking daughter and has to be especially small and so this about about 10 times better than Angelesbound still about 20 times all from the truth which is onequarter that to using the label to get small but about 4 thousand 880
25:33
but it was still basically using the codes and pins of method but the more easily than this in October of last year and it was realized by by James may not come across and another Pentagon myself at some of what he did and that actually and weakening rather than improve on the just on the distribution side of the argument of course endurance he can improve on the Soviets sit outside of the argument on and in fact you don't need any distortions that at all so investors other than Tom Bevan got off I'm just phoning very a classical even about us avoiding the he's more difficult according to different estimates I was able to achieve show that they can spend the 2nd but abound :colon before so I know you go about 600 initially come and as the same methods show that you have the full you could have some conjecture and we get what favor 16 Of course in the urine of at 2 12 of the past more interestingly and for the 1st time in his eye was able to also control higher gaps so unconditionally he could get abound especially the exponential in the fall and and honorary Halberstam could could get think about how such results expansion at people for the 1st time good noted about intervals of many times and on and on and on and after that we have to join forces so he he 20 polymath group and as of about a month ago so we finished out of the war quarry can do now is is unconditionally which can get what he's unconditionally when so that justice single Europe involved we can we can get to 6 and using a previous balance cases using improved slightly this was a modest gains stocks into this 1 unfortunate we have no way to improve it the extent philosophy is that it should be like a modem as it would have been explained here I'm unconditionally outside the assumed and I cannot just the absence injection something which recorded the journalist area something like that and then in fact we can get on with working capital 6 and and we know them and also that this is the best possible Quinones said that if actually 6 because we all believed to but it's the best that we we win that 1 can ever hope to accomplish computerassisted methods and always the results of the the medical chart explained what so this this this this apparent problem of silver which was which was already known to block any systematic attempted to prove that the comprise conjecture but on the verge Gotham officials at using countered that would mean Yukon between calls for all become vital before the and forms of the reacted to the 6 look at the genital area have been conjecture is the same conjecture as to how was stand that you you did you generalize about replacing the aunt of omega function by more general convolutions there's a contributions of ethnic sequences which obey certain and bounce in essence which I think of what right but it's it's the because of the place of Michael slightly more general classes of all of which the top good question market so the at the above even the god of extends to us said this was somewhat harshly on Jones results unfortunately they do not extend to too functional this generality at some point when you what's what's needed jumpstart on top of the seasonally lectures is that we can decompose unemployment benefit in many different ways some of which look like on convergence of 2 functions but also some good difficulties with 3 or more functions and it's important that you have this this flexibility to 2 and what you to decompose at the different components of a mangled and into these pieces on and that is that structure is still used in all the arguments we
30:12
do need extra structures so
30:14
unfortunately we we have no I'm John Tucker result for this generality beyond onehalf yet but in principle there could be a I think it is best that small images of the Indonesian worker technology look at but in principle at least if you have enough agree distribution and he can get the optimal severe presumptuous facilities are so that that's the state of the art of knowledge of results can talk about it crude methods and in the most realistic to pop up at prices that is OK the bill the I don't know what to do about the party and the so what we actually do so with into the wary of the fact that such gasses them but also by the gaps is is actually by making progress on the Punjabis conjecture so on reduced what I auditioned paints a mechanic introduce was completed Dixon hardware conjecture that DHL would you promise came Jeep Engine project so given to certain customers thing to Netanyahu's Jane K. undergo injured but we did know bodies the statement DHL Dixon had a little word of change to the assertion that a missile hit of his given that indicate then other existing information and subsidy looked at the shift of Sky couple then that this capable contains also crimes contingent rights students so given any missile kW shifted far enough can catch please J. perhaps OK so the the the the party was conjectural hiding Littlewood decision and that he would make all these guys price but that we should be able to get the tickertape rhymes inequitable now that we won would include that that that's that's where reach so so so even getting to the meeting to present to double organization have conjecture of this this week prove lawyer can provide a week ago weaker versions of this on and the way you away case be jail on Sophos sample and so that the fund was of up was his unconditionally and can kill you and that that the the Dixon contiguous true for 15 to use a given Iranian missile 50 I can find for you and the shift of it which contains at least 2 practice and it turns out that you can lie down and a predictable diameter 246 sometimes this implies that his wife Frances interprets the project and and upon the generalized really helps some conjecture we can actually get the quality which is about as close as you can get to 2 2 without actually getting to know so you don't give any admissible 3 double
34:20
on the example of his Uturn 6 and then and then we can find ships of the office triple that to use as a prime OK and so this is what gives you 6 and this 1 marriages forced makers and amusing note so this this argument and so I thought he wouldn't we take a very simple linear forms just endless constant and Wisconsin was a constant but the same argument you can take other action intervals even if it is true hero and you can be sure to H minus in whatever you can you think of linear forms and other a similar argument and holds sold 1 of the music sold by that at the end of a student was 6 but there's a there's a connection with the goal architecture and so actually on the same argument shows that on GH if the pin is system especially large and what of 6 then this is now an assistant at least 2 the of end confessed to a 9 to months enterprise cancel assigned variant of Dujail 3 2 but rather than an interest in the at the end and stew and fixed large and influence on so this has the following in consequence of this is again assuming gh he had economy and you get a distinction so we can prove to have conjecture using this technology and facilities as recounted collects conjecture but I can tell you that 1 industry and that that if the clear happy the time the donors that they have some conjecture that the that the confronted construed all on the a of mobile 6 that cumulative number is willing to to France the country would have assisted in the additional teachers but I I like but I can always improve and that the United conditional on this outlandish conjecture is that 1 industry and and the reason is that which opens What has cases are it's OK I this evening in which the "quotation mark subsidy that this happened even which is the unwillingness to which they say this is to all Jacksonville infusing his within 2 hours after Prime but so yet so he can get so you it's a little bit problem to get an unconditional got result from this method and so it's institutes and using old machinery taste tantalizingly close to the prove that every large even number is within about a distance of some 2 crimes but and strangely enough of that the the methods and not quite good enough to get this this is the case with the best of it In the case of hazardous that there are some sense that maybe they could even go back interconnection between perpendicular are considered very close to each other but there was some of the substances and we could go back to strictly harder than projected that has serious Pasha was also held comprises do not translate a partial results about so the main thing is to prove this sort of that that you really wanna find alQaeda doubles where the strain on the surprise of so this would be very easy in the past as a density in the crimes at wasn't as intensity the canal from the cat because they don't lose their little supposedly temporarily but because of and then and then this this would imply that believed the fictional Lewis catcher and when energy is bigger than standard Alpha and Africa and and the reasons for simple at the by the presidency he is and he has taken little in a random incessant beautiful he says he could be cannibal 1 capital and you think of the militant at random uniformly random then I don't think any and then possible the density then to problems the the state wanted primed for the please don't need an invitation to Prime Minister Gordon and so forth so just from the lineage of expectations and the told the toll expected number Panama's you get into this case couple should really scapegoat this OK go to speak when Jay then you must get on this possibilities once accused him on 1 hand and container to like and then use that Technical Institute expressed as an efficient patrol and then use the excellent any information tuples OK so it has a presidency that we were easy to get these results that was the problem of interest in the presidency 0 the 2nd distribution so you can't win this argument directly but this was no reason why you to begin on distribution you can pick and by summer distribution and spacing works so that's basically what costs and continued still and has formally but they observe is that I have to prove that I know the Dixon added the words partial Dixon and conjecture it suffices to find a good politician quality distribution on X 2 X actually other ways things that we don't accuse the police commission which you analyze it to find the function knew from Texas to access through possibly also asserted that the the astronomical and because of this beyond almost to the public distribution such that no so that we we we some of abound on its total loss is bound by symptom hopeful stayed atop the normalization constant and so it X is a chronicle of infinity rhetorical 1 means something musicals as your next stop and so Alverton from normalizing factor which would end next month but also it so the Thomas small that found the probability that your Prime which in this language would be used same some weighted by the sale distribution function and the minutes of the opening just loaded and crime and to otherwise it's pitiable mangled function and get him and he has no bearing on the sea bottom as it might be I would be honored to come over politics because of the low here so so the Stewart Simmons's reported on Monday came this is like saying the problem the probability that investigators crimes that beer is at least bearable and so fiercely that the totals expected number of crimes
42:33
that you get out if you sample in using this this week function under this is this is bigger than injured then improved conjecture and legacy of B has been 1 0 with criticism of this winter because helpful even on 0 2 OK kept so rights so basically reading invite any weight does not does have a uniform way doesn't mean doesn't work but if you need to be any weight for which you have a good upper bound on the total some very good law abound on on some by 1 of these is prime candidate countries on and if the ratio between the start of this band is good enough because some J then he can find J. even find J. prior the sky so what the observation on Benedict itself and all you do is you find a good good good said of BP's PCs the course of public functions news news factory to be positive so yet she did have a problem doesn't work if he was negative in some places has repository where we need to be able to control the abounds on on the rolls on and on and on with and history racetrack we couldn't OK so and so then the that the trick is to find is to choose a good good new I'm so but that the market but this is a severely question and this is something that is so great understood 34 years ago and OK so I guess so we will go there are many SIVs that that he could try but most efficient turns out to be tried many of us have was efficiencies of Philbert on and Celtics saves ,comma basically in order to make it's always on negative this it's not necessary was insufficient to biggest square so that the Soviets variability of Celtics of history is to pick of most of which looks like this was sense of whites and use some of Audi's while armed speaking initially the initial articles and continued through the summer awardees that that divided the product get so the motivation for this is that you you you would you want justice to really have a good chance of really picking out those numbers and for which it was it was envisaged price or Opel was the prime so I'm not a good initial choice which is which and if you pick up some new duty falls in with the to the case said the pick the the case waited the occasional mental function reply to this product and and it would stage and this function as it turns out that basically only if it is only nonzero when all these guys are very proud of maybe this is the only recently posted price but that doesn't but essentially this this is what this is all ready Monday when the good and ready picks out those knows which is crime and so they should be if if you assume behind the principal conjecture that these bonds you should get all that as a unique and should be of the whole country and unfortunately for the ticket was a sea of proving these is is busy exactly as hard as provided under this conjecture so the you this time of this function to doesn't work but that you can pick but you can't take truncations office and of weights which which have after working with it you don't restrict all these gusty exactly private dubiously restriction to be almost so go someplace in Europe tried to the classic soul excessive which is this is something like this and you can tweak wouldn't wait staff and then actually so the fund may not have to disagree with shelves of many many years ago but you don't have to use these onedimensional sits of actually turns out to be more efficient use modern process it's so so rather than than basis it entirely on what divides the product of these numbers and to determine served more flexibly he can you can find sits that that be something on the on the individual acquisitions all the obvious reasons rather than joint acquisition and and this is a more general said that this 1 and it turns out that you have to invest move in that aspect is what we should use along these these are the best 60 years and I started using these weights and so you is this guarantees that this is positive and that and then when you when you from the songs you get certain quadratic forms in these coefficients and you need the EPA's vineyard Republican to choose their own choice the latter which will make this critical small and Quebec Baker and and so opulent experimentation it turns out that all the psychic coupled with them well but the precise severe winter using it is consistent with the we have more disillusioned with the times and then pick some functional although I don't know what a lot of things was misled an effort for technical reasons is also important that we find it convenient to localize to a single congruence class and focus solely on this Congress President W. system you put a small tracts and on and on and on and on because under a small city local Hicks case over the years you just 1 of you wanted in any order this thing had to do this in the series which is not such a big hassle but convenient continued to not to care about and so you have to keep College class of these is a process that people fries primitive provide just precisely because Jordan is Teutopolis it's admissible and and then you have to have some comments smooth comments were functions from from the positive Oliphant fact cake to talk of some cut off and localizing D 1 indicator used to say I buy some small of expertise crew excitement and I'm so you have to look pleasant function here which he had to pick up and so you you pick a pension you create a safe and animal happened when you do that it will goes well you should approve bounced on on on Monday on these expressions involving wealth and better will depend on in terms of office of the will function ,comma capital at and then the remaining game is so this this counters voracious problem is is that you have to continue to maximize the racial 1 integral involving if of another evolving so in the office and condo this turned out to be pretty much the thing choice of all of us all 4 officers wait and see OK so How would you the visit of the so if you choose such a set of how would you try to found something which more so this is something
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that its fractures for a status that theory turns arguing it is widely said that it depends on where it is supported by the evidence has more support than the don't during March and some of the more recent results you have to do some tricks and when you support that is a bit bigger than what officers can handle that and you wanted to control Simitis so was that some we want control of most of which is simpler than the 1 the price I was going 207 just you just stick this in way itself squarely rearranged his will work you must have some point not I'm lost 1 complicated in the move but what about the other that OK so if you expand out everything you eventually get into this mess you can still be solid devices more July and then some weight function of writing on the most modular times times this thing and we counting your thumb 1 over Sydney ethnic progression of all the numbers and innocent animal makes to X which we a token with conditions OK now because your reasoning 1 that we know that we know not that someone in the congressional colleagues regret tenders but so what these models is smaller than the width of the year but all in all of so it's a 1st of all and the constant conditions only compatible with unity Monday when Prime DTT prime individual because price on and and when was the offer of this expression is equal to just acts over and W. Thomas 271 declined to detail the hatecrime wasn't there people want from ground 0 now and then there to that to be OK come as long as function doesn't have to better support so that this area is negligible if the support of efforts to contain of the sound strictly somehow it so as long as you botanists so the simplex that the universe possible thinking what it costs so as long as your innocence complex of size and onehalf some solace that supported then and that the total modulus you actually can't get any bigger than the next want hot because of those 2 factors the 191 parts of the Holland Kong square and that yeah as long as his supporters into big in this area is negligible and then decide the maintenance of the needs of the then have this big some of his name to the effect that this seeking to weaken values using multiple number theory which is actually not very difficult envisages elementary and and it's it's on the same level as well as his nonexecutive Interpol at 1 thing you don't need any Sufi regions or anything and he just pulled simple at 1 time so to cut a long story short if you do that when the fight is at the center did you do get it hasn't public come with these system was packages cover important them to use anyway and the snow was expected and also what it is is still to have the integral off after a differentiator once in each direction to take a careful derivatives in the mixture the square so you get this year the Speaker Craddick expression in there and that Israel focuses this with the denominated in the course of can you can you can do that as long as your supporters to be OK so so that's pretty easy it was straightforward is like page accomplished OK can now have the tricky assignment is 1 of the prime minister so example you coverages and what happens when when and when it has case practice so use of a controlled this summer and you can expect a very similar things on but polite .period simplification is a problem with every many factors so to the decay actually water to press 1 so you have to because a similar somebody you went public as 1 of can the can also similar things that possible news was asked about that because of the loss in you status here and get a similar sound effect which was right on exactly how the the conditions because some someone on the aggression but rather someone which we can do very easily fitted with sum that the part the conditions there and so this will again this will be maintenance and maintained in this case of looking at it and thought this was but now but now the areas is not just the goal 1 it's more complicated than anyone draw the error but if you really want 1 of these distributions to mistresses Cantril fired from here and not all of all have something to his or his Mahachi pence a share the intent and it turns out that but said that these guys 1 of these guys who work if your function as it has the support of certain range and the more powerful computers reasonable This is the larger the range of fall from the larger the range that you function supported so that there are certain conditions and there is a lack of title exactly what conditions will come up but but if you ever supported in smaller French and depending on the floor level distribution which was accepted you can and do all this error any other maintenance and then come faster and more more good number theory he this is equal to and debts and beta a similar thing to disguises as the value of you you this is it Kate emission angle value on an agreed conditions 1 burbles into intervention in Caymans was proposed for the final entry 0 square that and so you have a different quadratic of you function f and that's been the case this similar forms of Britain won the bidding predicaments 1 new indices and and
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so on these numbers and so as long as you function assistance support conditions which are not known you can you can you can write an explicitly and and so he knew you have this Cox version called you're exactly yet to someone's breeders and my health and yet to choose a good cut out to make to make this this history because possible and so forth so it it turns out that choose and Gooch was aware of the kingmakers in racial rule like like Welcome to meet global decay and bitterly accidents a product of 1 functions and cut off to a simplex on but you can do you pick S A 2 2 2 draw crisis and to the that this controller cocaine which is why I the large enough and you can get above all keep rhymes in In missile capable which is where he used to get bounced check exponential opinion that justice will be served from from on and for small Kay you and we can use computers to use plugin I could expose upon you where and you just numerically find what else on behalf and so for small was this how dissolved because the explicit thousands on gaps for smoking and quickly so in the last few minutes I'll just say that OK so there the support conditions world which of the restrictive so if you want to maximize his racial you wanna make guesses as as as major work as logical as possible and I so we were a lot of progress ramble about was was of resources with finding large larger support ranges for which could humans argument and so there's a couple extra tricks they he can use here but so so 1 thing you notice is that of the song some of the crimes it is much harder to estimate somewhat of a practices in bands of difficult so you can try to to exploit them by so transferring difficulty of of all this expression which this expression so rather than so so here we we we expect our cable some here and there was something 1 which is very easy some here and there about missing here but actually you can't compare cytokine games like he could just enough to minus 1 and the presence of devices in here and there is the simple assumptions that you can move some of some some of these terms over here on the youngest of the S & pieces in order to do that but you know you can view somewhat of its alleged violence and 1 which is so too easy to some of the some of them even more difficult at the summit of Isaac option here but at the cost of all of you and then you get to remove 1 variable here and and and then that expression because local all know what a lot more like 1 that is the type of expression that you have to control some of the other side and so on it other about the you should you should consult you can view of how you can relax the support conditions some of little and basically rather than being guy by onehalf to complete maker there this guy and you can you in need sulking lies with the start his career and 1 of the possible mediation so enlarged region here when the checks we used and another trick uses centered on the 4th floor for this summer we we don't need an asymptotic that's that's too strong for we knew we were just below about cable which in the long run on the bank on the presidency in order to prove himself is just 1 little about you you don't need to do this thing every single term in this big missed expression and with with a good over about understated about soap or we found that we found that this this and this inequality is is reduced and so there so you square here and so on and the year the enemy 2nd those fees which which which are very big new news media the support of and and and those terms those close to the worst evidence of hottest control so you can see that this big thing to do to expressions 1 coming from all the duties which of the Liberty smaller and then and then the boundary terms was close the boundaries of the splintered into 2 teams it was beat up and in the worst and other crosses beaters beat and and they give modular there just too big for your whatever yearly expenditures to agree but on the square's positive so you just drop it and you just use usable by like this but the consistency capital and these expressions on the dead their modular are still low enough that that that he have to use your decision policies and you can still get a useful law about this not quite as good as they came to you you you and you you have to give up a little bit in annual them but you still get the Soviet knowledgeable about for these expressions of the even when your function as polled said constant support that you you and you previously so it turns out that this is tradeoff between enlarging support an entrant in slowdown and the fear of tended to be slightly advantages and and using those tricks this is how we could get to Calion 3 impetus to get all of them free to now and can wanted to provide a particle Hawaii can be the thing that went away into a few the only thing you can do this and I respect the morning on the back of the head coach at the bottom of the Eastern him grazing in the right and he knows his not all not only images I don't know of any examples of public function fields I think there are some some examples like this of the some polynomials which cannot be prime movers no obvious structure the and conman 110 people and 1 How of do the and some of the financial condition of the old version of the regional oversight yes this this the example in the development and but there is some there something on your from field maybe it's just constant we are only seeing what you can do the he asserts that it was going to leave Congress slavery and former president 1 of the was right is all confuses a lot more structure and so this this little more going on and that it would images of some of this is that there should be no extra structure in the crimes beyond the local attractions but that's the common belief and this year even in you have the knowledge of the solar is system the review evidence accumulates the bombing but everything we can prove is consistent with with the randomness we we can't prove that ,comma the present arguments the directions In the 1st thanks and