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2/3 Bounded gaps between primes
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Title  2/3 Bounded gaps between primes 
Title of Series  Summer school Analytic Number Theory 
Number of Parts  36 
Author 
Tao, Terence

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/17026 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
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00:03
after all and in the long and so began by finishing something that I wanted to do last lecture by that time which is to discuss the parity problem and sold last time we were talking about how to use egregious recent estimates on this available mostly in France and how to use this to prove themselves like Mike Vanderjagt OK so we we now have a quite long associated theoretically can isn't mostly based on the sole basis of fall and for converting estimates and crimes to to it and I think regressions out abiding interest so 1 the equities equities witnesses some crimes is that he consider sounds like this he takes it all in and between X and tunics and restricted ethnic aggression and many examples sample maybe I shifted version of the 4 mangled would function of all may be the prime function instead Walter also counters the true function once and for all or some of the more advanced the theory that stuff will and to consider factors the convolutions and on various functions often Bader of dismembered usually convolution I this so various functions here and needed the key thing they have in common is that they all depend basically on the prime factorization of an interstate I thought that result conditions but basically all the privatization and use some them in the summer of aggression is embracing connected to X and and have all these expressions the verdict is expected to be the main terms in this expression and something like maybe next time some some functions to the beaches of the wonders of plus narrative thank you for the a indeed at its meeting OK and by that could issue a quick decision on on on these functions are what what we mean is we have some some abound on the Skoda and maybe on average the case of Aziz's various types of all the of input of any computable these bans that you use using some of abounds maybe on average in some sense and so you have all these balance and the fact that many combinations of these grounds and you so you do the whole lot of superiors to choose lowflow very cleverly chosen the new combinations of all the sets of axioms at some point you insert the complete review fact that that we have a sequence of positive numbers summer's possible role in the basically if you do take many combinations and I knew you just keep using Monica Vitti than if you're lucky and you chose your system very there ultimately you can eventually get results such as that exists in critics until next such that in a savings of at least 2 and what he can do most but but the notice because the release of the and this is
03:53
about right so I mean or you can simply because all kinds of things but 1 of things you can do that this is for use again by by using the civil axioms and SNCLavalin accommodation them and positively to show that they you can find these 1 and in here says that that that to apply to shift of in a prime minister you bombing captured in practice because I sketched houses and worked last time and so 1 can hope that that maybe you could and you could use this to promote so but maybe you could you could prove that so so that we saw last time that you assume the best possible could assertion estimates into generalized Republican conjecture that he can get about adapting private 6 and so he can help may be contributing further so too is this is the optimal but they're just slightly less ambitious I just indigestion before can you can use it .period to at least show that you can find their prejudices .period and and unfortunately the answer is no still longtime nobody got exactly when Haiti's results and Solberg pointed out this disparity obstruction which prevents was only prevents us from the train and conjecture civility means that it also prevents proving about 4 I wanted to explained why I see basically set theory is a means to this end parameter is nothing interests of interesting happens to to end when you the winning combinations of all these always axioms and but you never miss a B and so on and so everything is linear and an arm and because menu in an hour you can you can replace the summation and work with all waited some of on and everything still works so that if you had a security guard units that talks of the equitization estimates and no negativity and produced gusts of both sides and was fault many the same argument must also hold with weights so some give anyone they could wait conscience if it goes away would have exactly the same Equity estimates as the UN with itself so that he can find a way for which you have the same maintenance and evidence basically the same upper bounds believes that was costly and they get the same upper bounds as a note with a weight and of course weighted songs songs among negative view was not a trivial negativity that the citizen is more negative than with his arms was not negative and so on inspect how superior items work and if you could prove a some estimate by than is also promising estimate will wait which means that particular value should also be able to prove not only can you find in an hour for which good examples of for but I will ask the budget but that in muscle Sabena support afterward OK so if you can prove a cap of fall by superior than Finnegan was approved with his way and and this unfortunately is too conjectural contradiction because I can cook up a waits for which you have what we believe we have all the same discretion estimates away to some as fast as fully and with some but for which there is no no number in in the support of the weight which you have this conclusion itself and say this is a indirect proof condition and you forget about that little bit of so so it's so a thing could prove a Coniston for if period it approved underweighted version but with a vicious false who believe false because we conclude the following an example weight so you take the full and wait I think I need 1 uh just 101 upswing of work now and however worry Herman said the problem OK look you the movie yeah I get on to the quickest way this specialists say I'm uh back here again In the mess ,comma right now you know that people didn't quite do things properly and come to the opposite the side of I tend to be in the back but instead this "quotation mark you had to put a clear sign that Israel I didn't set up properly when it appeared likely that it is system this is worse than it was was really important and I was looking point fortunately good this stupid obstruction went through that I have to do with it you know what I need to do something to pick and I think I did not serve a problem but I if you take away something like this this thing of exactly right now but if you take away like this and where land is that what is that we will function so that there is a minus 1 the number of all contract of and the as is basically Mobius function but there is still 1 0 up in the ones with the numbers and in particular squeezed 1 and so this this
10:07
weight what it does and solicit 3 it doesn't matter what is as you come this way so if you are industry what 3 it's only nonzero when any of us who have you had the opposite parity actors and and 2 1 in the 6th also have the opposite consistent was a size which 1 wins it works yeah yeah you want an investor Pfaelzer parity in pursuing the 6 also helps a parody which means that the numbers couples prime pursuing the 6 couples the price on the low end of the 6th these guys this despicable find that these kids :colon and some will be here and so the the way this is this week functions designed to set up a steady take any and all his supporters waved then you can't have any of them imposter was full of any 2 them the 16 0 204 residents of properly but but this weight we believed to have the same equities equitization properties as the unweighted 1 because he you expand others wait this is this is 1 of the miners land and and too 1 understand 1st to 6 and because members good as 1 plus them again and again 6 containers to muttering something similar is 1 of it's to see of his weight is a good 1 for the winning combination of nonconventional 2 factors all and we stick in there and we will so have example the 2nd oral and written and pursued so maybe these 2 guys come by worries that with this to you get to turn 1 of which involves privatization opposed to any other possibilities and we will function of come and we have a heuristic quality of the Mobius around this year's 2nd ministers that that that this function the London fortress so randomly it should basic tactics civil order cancelation with an expression that doesn't involve privatization and company said it was a little surprised opposition to the answer is not very precise you stick is not fear but we believe that any any expression which involves both the removal and and something of interest to involving the privatization was to enable the sport 6 abduction have lost as a consolation and so Buchenwald just get absorbed the evidence the I number that we be even believe again vicious little cancelation initially much better than the evidence that we have a play with so and so these extra terms should should just be be a BB noise in the area and so whatever actors who just missed we can hope to prove for a full of wit 1 should also be true is conjecture late for Phylis weight and then therefore any intercepted a it would also give the government fall well with his weight but squeezed is designed so that there were no fundamental gaps of enforceable way and so 45 butchered this decided that this is the term sobered expendable but I should have but this is 1 of those on the ground I can can have the idea but that's my fault OK market but I don't get the idea that you can make this argument that the hero but it does to you that there is no severe to would purely use theory 2 2 2 2 beat 6 and now you can do better if you don't use it just you use something else and particularly you don't just have a somewhat ended you break up until can do do something untrue insulation likes student dividing itself in the middle of that the so you have a whole lot of this is a list but so unfortunately is mothers will get you the trend and conjectured that that is something else is needed Arkansas as a parody of and confusion because of own on there was word about how we prove equity investment in price so somehow and phenomena shift from established to look at so we have some some wasn't Panama's Russians and there is a part of the main town which can rights project so we we we have discrepancy of Panama's enough progression minus the the expected to average and you want to boundaries of expression and so it is right that as humble and landed in and where it is it's just 1 of just a function of 10 between connected to and saying what the minus hit so there a steady awaited some obvious athletic function isn't depends on the privatization of and is very explicit and none of medical mission which just depends on on this ,comma and the size of exit has a again so when more general questions about how how can you control songs yet so that is extremely general questions and how can we control sums of this fall where 1 function is Manigat depends on the privatization and the other 1 is 1 of many defended its it just depends on of local factors of and local and again you know so well maybe we can expect a phase that we all the characteristics of it and how to control and to songs like this in general so I'm standard answers is to reach up 13 of vanity Kowalski of that will have head yeah because this is this serious hopes of this is closer to sons were priced at heart you might use some a nonAfrican to reprisals almost at the central question in and so we have a lot of techniques for a willful for doing these things and so they soulful fulfills the ways the crew trying 1 quality thing that was due to cancelation and abound and and this always give you some amount of participants supported but that at this is never healed but we want you to want to save like a big pile of the trivial this the benchmark is truly about it so we will regroup after rebounds here in this case using the site expands upon loading exiting 1 a gain in altitude college and generally public but it's so that there is a variety of techniques that that 1 can use and so forth sample and you have can begin decompose and and other tenders his foot and other terms into into good pieces of armor and very ,comma technique is used for insulation and so for expansion so the example of this particular function f and the institute's either by more Mauritania for analysis into and complicated for extension of the the on principal characters of purity and together and 3 D can all you can do the injury in Najaf for extension which defied mistaken that is something like is in cable complicated and this may be gives privatesector speculative doesn't think if correction to entities not opined the roughly speaking title depends on whether you can actually destiny and forced the culprit and it but is something for expansion that they can do it just like this 1 I think it here in the educators media posted Atlantic contact and so you can say is that the pieces and selected as the site this into lots of pieces maybe this formal only this and you can see the China quality and any substantive and like this In fact so you can you can take this this this expression Britain into characters publicly by the attestation separately usually
19:05
you can and you can't exploit cancelation any further prosecutors give it up with the final yesterday things that so and this unfortunate losers because the trial inflationary losses and so this this this step can be quite inefficient if from if you have a lot of terms and expansions if you have a duration which is not very close to being a character the new volatility here and you use a lot step but often you still have to do this because I'm the sort of game that get what you have done so character structure will be giving you get so this this and this is 1 thing you can do and because of that the health into pieces will see see Alex later on OK so there's that and if you at it is that these things in such a way that is not the kind of man and sword said month thinking in the control of a number theory OK so if if you're staring for example In order to someone this get have of character than modern with steps in and his ultranationalist in and the real function of population but fiduciary serious intent and and this this this this has some form of assistance the system ,comma expression in terms of social function and then you can use say the cost ,comma complex houses methods for nowadays penalties pretentious methods to start to controlling this expression in terms of what you know about the function of canceled so this is this is what you have to have a good character but unfortunate because our understanding of the denominator is not so much as good as we'd like this tends only work well when England New York and all the different in character a very small conductor likely and unconditionally or if you assume jury just when you can do better than that only assume students from conjecture this only what's the full of prickly characters of other small conducted on I would also to and if you have a lot of matches 1 function at the loss of consciousness and you call cable and average than the lots of inequality is your friend and had so if you think you could get a family of the songs and then for each single function f maybe all we can do is is is truly abound and the whole family at the start of his advice and these guys of behaves and instances of only to each other and and and very often get really good on the Swiss nearby but not the builder was a natural bound on this question which means that he beat rerun on average for most of his father and as long as you have seen some good you OK so I want to I have to say that you can do so you can do things like this and of course you have so that doesn't help you 1 of the many so all along we gaze at HUD dispersed about the 1 after that make on many objects so that nothing he can do what other tricks to have available right so another thing is is that if if he can if can decompose and this village think function this so it it depends on the privatization so typically or have some some factorization introduced a series that maybe you can become factorize a to come out and combination convolution of 2 other functions and wars and has combination of flotsam and find number of difficult solutions that that if you have found the factorization here which exploits the ethnic structure then this linear some now becomes a earlier some and when letters can so that you have a difficult loose attracted any instead of having a summit into a binding it some on and then and there's a lot more games and with the buying is something you can the so for instance if this is not like it is if you're planning to some against Kagame multiple character then then things are really nicely factor In Tetsuya product structure which is very nice so that's that's over and his funding that can happen at the opposite extreme if you don't have it you don't have had emoticons tragedy happens it added destruction if example this happens to be a bit inyourface example and that is the multiplicative but then and
24:16
office for cancelation just in this by the a colonel even if you don't know where much about these weights again as all of which is so yes we want which is just such a trivial but I think it will be important because Schwartz inequality so because he was in quality is is a way in which the crosscorrelation
24:39
cyclists becomes abated something else can be controlled by by by so coalitions that this will occur because was simply and this is great because of colleges often a lot easier to understand than across coalitions and so a Gattuso like this then we can do that but it and and often the so Indian in more diverse applications you double submitted proposals and the other the loss of ways emissions to endangered country Schwartz and knowing when it comes to the disarm and Kurdish is half the game if you want you really good and a good if estimates this is all part of what would cut but just you know a little about now if you have this some then on your around this then I would think that being out here cushy Schwartz can you can just the inspectors into cancer but balancing the standards as it is all in Chapter 13 of the vanished rusty but I think so they have a summit is that because you shorts OK and that the point is that greater here and these can ritual here but that you use them up OK that that was you know them joint some involving beaten down OK now knows that this
26:08
this is no negative this year's farmers and by Century about in Western monitors the cancelation was attributed to the shop and then here on cases of this you have to expand our history we typically do you you expand out 2 political crime in the the chemical general crime which typically it's good to see and hear and sycophants and tell them but the and so you have a question like this and then and it's common to Split up to this and in the cases so this the Diogo turned Ali was Prime Minister octagonal terms and use the treaty's differently in many applications actually years since the you is more there is a continuum is is just equal not equal but each instead of depending how many homes Hoke harmony Prime Factors of I could have in common all over how it affects the Bible myself from something that would not be speaking at a press box approximation you have a diadem and underground and I with the diameter and why I what religion then this and this and that no consolation it was it's not positive and soulful for this should be pretty much a force futurist in which it is it which which are very good but on the other hand that their dad was pretty small usually known if if Els being some of the range said hello to help the diagonal is only a portion 1 L of the whole some services as long as your your analysis is nontrivial alleviate this summer's something over some Montemayor range and the diamond contribution should still be better than the trivia about and the bigger the L the the small that I should will be the conversely you you have and the alternatives and he will you can do you them so we can do for the Kurdish forces of the political office often display you can just get lot of mileage actually of the 2nd trial inequality Baroness and trivial and then and then the next thing that some of this and the point here is that is that war is unknown as many countries like beta gamma gone and this is a function purely also very smooth functions and is as surprising as character or also millions Henry well and it's so you understand particularly expected some estimates get on the leaders Rangers center so close this section of the incomplete expeditious investments and then if you have good bounce in here that he that if an exhibit the cancelation and then he gets nontrivial contributed and begin of course you you lose something so you you giving up some consolation some potential deflation because of what it is to find a quality so it was very big then and then this may be an efficient so you got these these these 2 cases in this case the dynamic it is is better when I was big but was it was small and often accuses is the reverse effect the worst was big and and that was small and so is a whole lot all of picking your way with Schwarzer balances to grows in size buckets of blood on talk too much about so technical OK so this is it this is the basic strategy on 1 or more more and for this is the simplest model it often you have other promised averagealthough and so you you often don't just want sums I want to go for 1 of summary on average a sums which often leads to more conventional extension some estimates of and so to get the best results to but most cancelation you you often need to need 2 to 2 conflicting on the average and this requires often by highpowered expected some tools it a soso difficult thing that you might wanna care about is that we might have signed lease simpler conclusions and had been in what is now the average over the Amy parameters with some some weights or average of the you parameter can and hope to get some control on on on on more compounds and it was after that with and it it's it's harder to get Cecilia use Montreat techniques to do things like this and if you're still averaging over the new realities sort parameters and that the typical techniques you need art techniques from from the police have proved RIANovosti on on all the function fields and this is what for giant example and accusing and Norwegian polymath joint average of acute that's that's a whole different and then did this host defensive techniques used portable music forms it is for the people about dozen these things on the so that direction we don't use them In in in John's work and and followups and resistances to the Oval Office meeting is a deeper but that they have a weird restrictions in the face of this and only work when his face fixed we expect to receive the at that but for altercation need very will be large and and there is in constant exploit any sort of averaging queue up by the court on the Newfoundland and instead we have the means sport evident in these parameters which using more than cleanup methods I think so and so I guess that's that's 1 of the main the differences between July chunks onions and priests and previous sewing up before youngest the loss of work but you could buy from rebound fit and advantage
32:04
on getting acquisition estimates for things like that like like a piece of songs from for quite highmargin lied and that the only way they will usually powered if not mistaken usually buy it would hold more than half business in which it involved everything in exporting a cancelation queue up and as such there mostly restricted to 2 bounded modulate and another the new thing is that by using the lean instead of of what's going the media and stuff he can can get it can also control the roadway OK also the the overgenerous techniques and so the whole lot somehow ,comma put it together Faisal 1 of technique which have their own identities just to be nice to get the schedule is enchanting songs into the show given was to tease out solution its own set of problems so that these also the methods we have looking at the moment it's actually mean it noted United's these great what we have said that they they can't do everything in particularly with regard to exclude cancelations in almost anything of what these methods you put some cancelations usually have an opportunity to remember a little bit but not by by mother you want and it soaks old lady expanding often is a bit lost and country Schwartz offers little Aussie will have to remain at the Savannah losses on which is brew all 4 problems but we can do something OK so I said these are the basic methods and so some examples of public works a lot of the classic example of this is that the mommy of Nevada .period kept so the only bring about few the way it works is that the just as it did so huge that the discrepancy all example that once the pump having a mental function In some residents and known as this complexity but OK so you have to take to discrepancies worst once and for many similar orgy of after to some something and the claim is that about runs next templates to power should deal the jury about the power of them on as long as the is listening and so he can do much like shop this this this this and this is what version of the popular about look at so this is this is 1 of petitioners for the up to level 1 half of on this very general interface for minor abide by any convolution basically calls to functions with regional that's where you examples where the Elfers said mostly locally size times maybe some of the other guys attention and pretty much any of the interesting part of the conventional care about had this sort OK so it is really the generalization of wasted more harshly attacked and Arkansas highly precise like this so use have you will these increasing just the system which ones which would you pick and so is mileage approve this year to study the ingredients that go into it in an attempt to work out that works that if trickled in religion but the strategy is that the 1st decompose into multiple characters OK and so I think it's understandable that the characters he's trying monopoly and the grating ways the kind of quality here is that it shows a subpoena because of the age when you do the math expansion to be a comes and into this coefficient major captivated goes away so it's a very efficiently the soup and so on roughly speaking ,comma you end up with a dozen like this just so so you can you can because in character songs around and there's there's some normalization which I don't know maybe hospital get on the ballot this month is a tactical reduction here which is that these characters monopoly be Prince was a "quotation mark imperative that but it so you you give Peterson standard maneuvers to reduce the primitive tastes that the next signal that step as standard but then when you do that is this into effect of the citizen convolution of example migrant camps but others the movies can control block option for instance and then use this complicated the search show evidence of that that is going on OK this expression just lit up final up before you decided that this that some cases added that somewhere out there to live so he can chuckles about pigeonholing people always something instant elephants range the end end and be cement into an engine and then they the provisional by otherwise you don't need any country in sun I know that
38:02
this this some degenerate cases where health is very small but it is very small and in those cases on the is something that's on the bottoms of the general and and on gaining a cancelation there so is there's some cases in which the the alphabet is really really small arms right and notice of the fact you don't do this concept of a KSL actually have you you you don't do interviews the office of this expression otherwise you might Indian pollution snugly in the UK many function as conclusions drawn from itself and this is a useless accusations and it doesn't tell you giving a consolation you know and so you use Minor ,comma structure in this so that you and your gives the sums over there in them at the bigger than this not here so you you that expressions like this and if your characters has a very small conducted using his mother good number theory :colon was the will force there and you get very good bounce that signals interconnectors Big Arm & Sons of the ability of many many different different characters now because you'll be conducted and then what you do here is that you do capture Schwartz you know you can't control by by attrition existing similar touching the betas on and then would use that use lots of inequality which I still believe damages gone as well and yet but there but there's lots of an accord which allows you to control expressions like this sum the square some and the key points as long conductors a conductor lesson 6 1 half into characters will still be primitive afforded to each other on and next to exclude that the product was so less X and so security group you much less than and that the with you and will continue grow about Nelly aging muscles a cancelation energy crunch would have questions you eventually get the the people regard unit but because it relies very much on some of melody among goal of excellent Hopper's Horton completed "quotation mark so that's that's the sort good classical benchmark Secretary's refusal we have taken something onehalf I and all the more reason wasted can improve distribution basically do something completely different media they still use the same box of tools and has that but they don't even bother to characters they the limit when for expansion and catchy Schwartz the crash was still present we proved a lot of inequalities old country Schwartz and of but the fact that OK so now "quotation mark October next is how you and you decompose the sizes said yellow American unemployment would function and respecting the pieces sources said that this is a standard splitting which is obviously the Mobius function and launching this is the sister and OK In an analyst has said this is this is true it is that any function through using the direct order function of total and 1 it's 1 thing to all the and it is a complete the picture news was also have new movies articles and that's the the identity of smokers version follows the concentration 1 and yet so this splitting is not quite as the actual because it contains Diarra pieces of new can call the police on his goal is reaching 1 X and if if you if this is to close to 1 of us is to supposed 1 then you get a degenerate convolution with with no not enough cancelation but do anything so I just wanted to to have to do 2 years studying in which European which you have money indigenous summations it so you have you what you want a to of a minor function into pieces then may ,comma which the 2 pieces ,comma many pieces which someone money generated and and to this 2 ways in which you can gender and oneway Sierra convolutions well aware that the terms a counterbalance that both the lease terms as supporters on the large numbers I know this has broken into an into an incident and then it was 1 facility could expert Marianne is bigger than sex and power no I don't think this is a to 215 the cannon fund of but complementary to so you and saw 1 good turn that will come out dozens of convolution where but both of these terms are on top of mediumsized and then have a lot solution and Yukon calling it into many pieces the user's group campaigning to put 2 and 2 1 or both which are mediumsized firms and others so that these Michael so tied to some of the officers could be confusing they're not subject to such a cat and a lot of that I saw that Fontana comes up I and another to that that is good and so soaked having a degenerate imbalance somewhere 1 thing a small 1 of his big is bad and unless the debate is also smooth so that if if if you give a conclusion was supported on the very small and from logic and then ordinarily this is bad but if this function is very nice you visavis the smooth functioning the it's just a bit systems with functional program and this disaster so he could be the consumption 1 truncated incidents and slightly below is also boasts with function truncated some nice way he gave her a smooth function here and is also a very good time to have a new expansion because you do some of these if you if you're you know it's you kind of get the slid up into pieces OK and if distances some history smooth often to stick it into exponential some and and you can just computers things really precisely the proposal submission was that of on and on here so you control this system so well with those of the office's Ross the new parliament Yukon during much of the subsidized because the cancelation of a smooth then you can you have a good chance Zionism and often the Kentucky district of all knowledge something that values and can make a good bounce so the the sort of terms also pretty good so you you you want to end his music classical type 1 sums that I have for reasons which we conclude the wonders of the future Internet and the so you wanna stood up of 102 into pieces which even rough the balanced or announced this move and and these are the types of thing they want to do and so this we have and that we understand how do I not so Arkansas Arkansas on a possible wasted fugitives and she persuades score was an area that was his destiny with 4 for many applications In below the strongest is also promises of quick enough and so of the way this words is a depicted veterans during the summit between next and it was a typical values might be excellent for its very difficult and any consisted of among format would function into a big pieces for peace process and in this this is landowner in restricted to indignity and and administrative spending less and the energy to that this is really what to exit the assumptions of the same and that this is a 0 he also stood up Mobius wanted OK and more than the it take care of these 3 facts about land and and don't have new and you just play a little bit and you'll find out of the calculation that he has to the Michael functions into a version of and these 2 expressions and persistent get that day I said that is a certain amount of basic knowledge about the latest listen in the BBC pieces of order here it's not all that difficult to watch the whole that eventually but again you believe this the splitting the these terms are good and we want to be a good time .period hands because you need a small and this is as much a small couples move and assist small candles so fast that it's often a good but and this time it is the is a term of a good team or the other type it's it's it's logical large hinges on the yesterday with this 1 and 1 the students from the hands of process another term that that that that you can play with the would have to stop and so this is a very very useful splitting with the class replication of this to down special songs like this I think it's so this is 1 thing that can use it's not like the most efficient stating that is is out there on its own but and nowadays the use of the splitting somewhat really sharp exponent of things when 1st met mentioning employers don't use decomposition we don't use the morally used which is exciting Arkansas these latest said as he is to temporarily entries as a function and contribute have expansion of closing around Z equals 1 formerly the party's policies along gives use this expression of these little formal power series and so this this is valid for history legal questions for me is really big and so if you some of this this identity nuclear coefficients always given the final is say in decompose language functions into and incident certainly there will be during any of the series so this is the block function basically like 1 1 hand lock this this is 1 of the largest source of this this this is 1 man's daughter it's 1 except when did when it is 1 which is 0 so it's it's it's the large 1 and so is this is this is all this is this is reflected vise appointment comes down in some sense and then you trouble thinking it's like a 2nd device conscience of the opportunity of the and and so on and so forth and get the divisive function instead of cutting solutions to the people and technological is coming solutions to ABC was in OK so you have a standing on end in principle all these firms of a really good users of the service is smooth so if Yukiko into terms used his degenerate inches 1 1 goes to work wonders whether that small it's balanced and and so this is something is the Okrent decomposition and it morally reduces the study of equities version of Flamengo to the discretion of 1 and talent how to 3 and so forth but unfortunately this this is infinite and and because evaluated season is not immediately user directly so we never actually use Linux and any direct military action from users and others of concern here is the finest hour yacht is this rather than most X terminals will hold its terms the rather modest terms and and typically yeah we week we can only get behind rebounded in many instances it was 5 another there's no actual use of the Singapore so In the case of the the blue line all Of all the legislation to the floor he implicit in the alliance troops engaged that he is a chaotic that How can I get my handle the bulk of the actually because of this chapter in area I have this argument I should like to read it died of his users but I did before the war we do we don't use it but there is no new money from the provisions you lose and you know I used to live the waist most of them are now in the U.S. a full full obligations was whistled good litigator huge public will object is to game you a lot no the anything goes we have a problem the all so of that what please you can see absolutely much of the text that here they say about the future of lead severity of the almost rhymes ,comma some of this small and then he you would and from the college and the club and he's the only 1 of 4 currently has a friend who is in recent years to hear about it so that great yes the summary of the next year he was so pleased he was to arrive in Canada that has a good point here say it you pilots yet the philanthropists overseas investment project is an interview settles for factors in that then this this year than I agree this is because identities but had yet to start surrounds question has the inefficiency in in one's identity is that if possible somebody cheese no no matter what you choose with these unity but you have to at some point in time the study but you have to create terms which balance of and and an imbalance in the sense that the scale of this guy can be as low as 6 excellent the restricted flights situation so that the best convolution you that the vessel packed to something to hospitals where formidable onehalf and send out often and you knew you were moved to fall away from onehalf the becomes here because you to degenerate 1 just 1 couple ecstasy because doing so because he continued to generate a little bit less cancelation again on and so you want to minimize the distance onehalf and it doesn't matter how you structure was your your you always have to to do something in there that there's always a least 1 term of which is a disease 1 6 from falling from about and as it is in the best of the rest of the way candidates and that use heat from which only covers later than you can cut this down to 2 the onetenth of which is actually imported versions work and further at the cost of getting some extra terms and just type 1 and type 2 the total sum protect threesomes and the Chinese habitat not the same as what happened this the increasingly confusing and I'm not going to sit here as 1 of the kingdom Type 1 or Type 2 existing changed very shortly here and she get more descriptive name that wanted to disintegrating into information on all ready for an American hero the they go OK then you can get elected again on the attack within the linear songs it in the university this is a summary of the priority is what's it Linnaeus move this formula on the phone or more more accurately type linear forms a smooth coefficients pipe to his body from recollections from involves a smooth corporations and put in a bid have contacted me about the nature of the objects of the 101 users now and in in some sense actually it is all all but that it would governed by was issued on you and you can't do much better than than than that in the front of week as it turns out that OK so that it involves a student the case which can pick and it doesn't matter what cases so this was big enough to take was tend to be good enough applications ,comma so the way he proved this so the new drivers and sitting and Texas overseas for the movie's function into 2 pieces and so this this is it's a small piece would less than having to exclude the case and this process is the OK have to chase biggest is that these small factors and that this is the big factors on and there is a need to do this is that you observed that so the Flamengo pension is Mobius Campbell L which is obvious noticeable 1 can well consisted of function and he keep doing this and you eventually you can expand on you into a huge convolution involving Kato the Mobius skaters 1 working imagine Mendoza 1 and 1 of the lower so this is a big convolution and it's not in good form right now because it is a little hesitant at muffins and imbalances of all of but what you observed is that if if all the Museum of 4 big on the succession of his vanish identically on the of care about 2 X because it is because we knew the big support only 1 on on numbers beginning to explore explicate so the can of Times is usable on numbers because acts and probably more will not change that process vanishes so this is 0 this is what we want to know what you do is use rewrite this year minus the small guys you stickers in extended out to get 1 2 like this and all those terms and so after short calculation to get the Prodigy which is that the correct function actually splits into the combination of the small news will help in building a combination of a few more thank pencil 4th down to have a lot to and see Yellowstone's have been bitterly finalyear potatoes and the 1st Angelides L the 1st thank you could hang and only 2 ports on the the when we wouldn't have had to go get a research unit How can follow that that was not enough time to get up there but I get the and this term is soliciting the sort of advice I can tell some legal ones come over onetoone well water and sewer system identity doesn't extra terms but this small facilities initially and that they don't do you go with the notice of the become more of a nuisance at the end of bed time yeah but basically our leading use convolution means splitting up and down its it's like clinic but what about the motives and so you you know the plays the prosecution for all the victims of small and so you have a review you in really good shape of spilling into also good time so that it uses activities with KB cannot effectively given 5 I think of in particular what happens is that using this month against the land into the into the final pieces the various types of and so on Catholicism ,comma on skin and so on chairwoman of different that with so that the against the lending into 3 has pieces and 1 of the Linnaeus who sounds which included 0 you know the user have 1 on which of these sums will perform side by side smooth as a and a bigscale on support from skillet begin accepting so and even a small because you got a small rafting come over me so he's a really good terms tender and so it was the 1st year terms to be speaking this and then this standard by the sounds and the use of the optical beta cell Heltonville but supporters between the 2 bits and it's a figure that is within onetenth of scores acts as opposed to war that India which which is only 1 6 and and many others that have 3 terms which look like conclusions of the smooth functions on the status 41 and 1 and 2 and 3 and this something and then went to 3 at a reasonably big it and as it turns out the way said something between the 2003 this year involves something and this technical Commissioner thinkers have known in that between wanted to and the public's ability to feed its people OK so that they want to generate but that this somewhat small but it's clear that on and took it turns out to a certain routines :colon sincerity elegiac literate the that all the signs up indicted pieces from all the times you get will be on 1 of these 3 types and then there's this Fidesz was optimal on a joint even narrower than than than than onetenth that of get within 114 also of export introduced not just trying to conclusions but was attacked .period 5 sons version of confusion for her fight other we do understand that the 4th and 5th devise a functional and so on and this is the limit all kinds coveted technology so far so it turns out that using the Internet as you stood up at the moment into these 3 pieces and so to prove that we distribution reacted by finding many of them actually on its side .period bald and the system that unit and of splitting into armed but because you're going to get accused of doesn't matter using Wallstreet I'm and and so on In the last lecture artery how to prevent industry on 22 these individual songs industries China quality is so Connecticut systems on the but if B resistance to Washington but and
00:00
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03:47
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19:05
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24:38
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32:04
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