Merken
1/4 Growth in groups and applications
Automatisierte Medienanalyse
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Erkannte Entitäten
Sprachtranskript
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I the 2 i'm and
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many well so what do we mean by growth cannot be happy group he told legislators have said indeed will be flying 18 speaking to media said the full price of the 1980 s tournament in the oneday Agent Kay will get me I would mean the product of copies of the meaning that set off all the products of the elements part of that well and gets a signal to fix innovation a to the miners 1 it said the few investors and by double Mike displayed by I mean number of tournaments Of very well scientists so you have a fine living the questioning swelled you haven't had a you multiplied by its inmates held to date and say you abide by itself Cape times to get into the K how Istat said growing if they say much larger than the state of the game much larger than that so these questions that can be seen from different perspectives coming from different fields so on the 1 hand you have what has come to be called that the discriminatory I understand area Sykora summary of highly discriminatory it's about this but it has traditionally added indicates that traditionally focused on the case of year or even getting here then you also have a lot of work the right place for these perhaps we'll have to make the group's series so you may gives away what particles in the viewpoint are but let us say that from the point of view of the problems it handles it centers on the case of the infinite and catering to infinity and there's a sad thing that 1 could say studies the problem it almost seems like a joke several justification why do I say that classifying subgroups describing which subgroups of as is problem of the sort where so we assume that as contains indicate then I do have that 8 to the KIII is equal to or the kind these are the same he finally basis habitable the so classifying the sense of them grow at all is the same thing as classifying several you might miss you might think I'm being stated pedantic but in fact it's very useful it's it's very important to put this in the same framework because it turns out that some of the techniques developed in group justifications are robust and can be extended to said not enough just to say that the was told the subgroups but Assistant go slowly Our sitting will be currently working with the media groups in general but that will not be working with say a fixed infinite group but rather with the infinite family of groups that made himself be fine and we will be aiming at uniform results start what some this has some state was not always so trivial for a fixed final wrote but not when you have my entire family because there you want uniformity and uniformity is difficult so intimately groups here you'll find takers had a look at 8 elite KSK goes to infinity of course is approved find nothing the things that very boring at some point in the to to the cave initially becomes sequence to GE and then because the fate of stays constant thereafter from growing more so the question the basic question will be really why 0 why is for which this House what is the state of which 8 to declare frequently this makes asymptotic stayed the asymptotic situation highly boring escapist infinity but last year well at any rate so by the same name that had generated I mean centerfold products animals of failure in products with led all right so the definition he said the summit of was despicable human going to put the Commission is them here is the least Katie for which this happens so why is this name diameter and why they're funny got the gun mounts the 1st letter of graphic right Greek letters but said the leader the directed the grass all 4 of the respect the EC find to be the graph consisting of a set of practices in the country he said distance throw dog for every element of the unconnected dots and 2 dogs corresponding to 2 elements of the if the ratio is in the of generators so what do you mean there and graphics aircraft were everywhere except the same degree 2nd the degree he said what they mean median while some people like to call valency because they like to remember when what we all want read of chemistry in high school I think since the number of deaths number of Bashir's number of is coming off of a given sir aromatics this industry as displays but we will assume that we'll have to make a mess out of candy graphs directed once indirect ones will assume a good ranges from now on the summary action very soft assumption of you can get I've been versus to this summer all right and it's easy to see that the diameter In the sense of this definition is the same as the hammered on the graft in the traditional definition of the limit the photographs what is a traditional definition well you define the distance between 2 points to be of the shortest start the length of the shortest path between them or the length of a past of a path says a number of analogy because pessimists and you take that tool points for which the distance between them is maximal under the stands defiantly Ltd it makes for a circle you take 2 points in the end politics and their decency them the circle very well but all right but sort of Rafik Khoury was the quiet you may think of him as a way of seeing things quite fruitful in some sharper questions than diameter bombs can be really formulated the mostly in most intuitive in terms of the graphs so along comes this set of expanded graphs and several related so 4 I groused that
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he regular the directive we can define the industry's applications on functions from this diverts its so how the cease linear operator that a patient so select the toy version of the Laplacian continuous case you take so value over the depletion of waste and effects is going to be the value of the X minus the average favors very well and I want you could define the Laplacian general but the point is that because the graphics and director you're going to have systematic operate as Semitic operators Semitic
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operators are a nice in part because they have history of spectrum and that's hard to shifted to see the history of the negative but despite the to the suspect Thomas insisted the fight in London's look like this so that there will be the London I'm busy don't want a 0 corresponding to which functions so the function give me a function that on the grass this and I can function of the Laplacian was taken values need more interesting constant the world many of the function that takes the constant value 1 everywhere or any other constant bond underwriter said that give you the eigenvalues the veto victories not tonight victory that in my definition of given with the Reagan function all right and then you have other and I get elected so I United values and we say that there might be some excellent expanded the number 1 is that but most often you will just hear people talk of an expand the RAF just like that that doesn't make sense that the nation would make sense is to go before this and expand or rather this would you find you would find in the office and expand the family of so his family it gets interesting if you impose that they must be a constant angry there was no other the question isn't easy understanding and expand the family if they it's an expanded force on its Of course it would be clearly defined family were flying at sentencing things if you receive a from thing 2 of the Treaty of assuming the top it's enough for the graph notes to be stupid for number 1 to be bigger than conceded and therefore have to be connected and will be connected a the bottom as Lucy's mother was to have that any graphics this expand their 4 summits and insisted the incident has to be constant for the entire family that's what station this would mean the uniform very well and is particularly nice if we manage to construct and expand the family made out of caterer for many reasons 1 of them being that we really care about groups I'm all right and related notion is set of mixing time save 1 had to put a hierarchy of nations parliament give photograph to be an expanded is stronger than freedom we just have small diameter missing time having small make sometimes intermediate notion so what is the inclined Easter cityscape around the walls of flames all the king sensitivity the waste roughly equal take approximately uniform distribution what do you mean by approximately 1 you have to impose a normal distribution to communal 1 Norman phenomenon feeding on your choices so strictly speaking wishes people say the Infiniti Mixon Chi India the to make time but there are also source of useful inequalities between those often going both ways all right so as it was saying the parties as follows the expand I'm not even trying to going to the text at a mixing time this implies an exciting time he's alleged to have you on the wall for any sensible normal if you close to uniform but in particular you end up 4 at any place with positive probability implies where the little eclipse here means that bank independent so some results sir I'm willing to put key proposition not because I'm probably some proper because I like system can they make gave to the police sold in mind people back in the day the political arms and what I call the main fear something else that they will call the coroner here so some so if you want to prove them at all if you want to prove some other
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things really key proposition said the following you have viewing this case he among that they still like to make says where this and 1 of them 1 and you have a sensitivity page in the what the way honors at the time was the defacement large it's really very 6 this enlargement you have that ECU is larger than the 2 . 1 per cent where the business was Wisconsin Of course in the same results West when 8 it's like ours you have that 8 indicate already the K and that's what counts info as Nikolov and Peter showed please something that ours was were also doing time on cable decided to Interstate I think it is a simplified Mindanao Island I think receipts while yesterday emphasized that these numbers matter I will put the cover line but petitions filed list of legislative director let's let's set something to Yolanda at any rate the suit will in fact be favorite place I just made a statement stronger by the way thank you because if you make the phone and larger tremendously even stronger you can make this 1 minus epsilon it would soon be through with them depending on but since they don't mind snaps with constant that business was very well but what they about here with Hatteras associates through Ruiz all right so what they call the major and the candidate is really a simple corollary is them at about but instead of their maturity said most constant power of flows of the size of the group both of these concerns are independent of the while the absolute questions the fistful of food from the institute said previously it connects with the fact that has weakened and exercise we at about the right now we need to apply the key to aid the into the intense heat and thanks to affect ventilated attempting to stay as long as the 27 and keeps growing over time by a power of 1 president and when the it stop stops when they took the case to lead the elephant referred it's bigger than the to the data 9 and then after looking more steps was so far fact this although 1 is basically 1 of them all right so what now you can also In a special case of people can look at you can get a stronger about so people used to for team for top law before deciding on fire and I think it was the political had the following the worst example so what you have here yes we knew there no GPU being tools that have been there and aim of the look he could have made this a few years but you can reduce of pain isn't it we knew this was an extended family but this is essentially flat so this was a consequence of a heavy and beautiful withholding numbers hearing what on their behalf playing strength expected but said frustratingly what people did not know is whether there to the with states nothing was not all good and well know them at abound .period
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really so I already did for any whatsoever you get up polyurethane acknowledges that the power of long but instead it's nicer if it's especially nice because candidate the cease and disseminate necessity find the I felt in this particular case it's even easier because these the of freedom so what does that mean that they generate a free well there that that means that what to innocent desired if you start multiplying that you don't get anything the repetition of the fuel cells multiplying the system seasonending even weighed the only way you going to come back to the originating from Italy with racist right there is no other way there there no notes of but that means what does that mean that means that also although more than repeat there are nominal Serna repetitions unless you have got so far into the game that you start to get to major cities the common to each other more than without being equal but the only way that happens is you have to make decisions on this 1 centuries is bigger than purity those 2 numbers can be gone to tell the mother appear without being well only if 1 of them is bigger than pivotal rattling of here the incident once you have another coronary an easy 1 according to the here them and how they prove that was an exercise to which they have already giving you solution get what they just said I'm In this 1 used to use means that for the 1st look at the state's roughly if he has multiplied since together you're going to get the fastest kind of responsible for free and you have to stop only after them sloppy steps yes city some radio Sighs so grows for a free literally meaning because of freedom because this season the group to indeed to the excellent state and then but I think he felt although 1 time that's and in fact you can soften this assumption on lots and lots and lots didn't need unite free group it's enough for each further group United Way to contain a freedom and so on and so on and so on so that can be softened tools for nice Inc this time the sexually ideas from the eighties guys find their naughty as some people I don't want to leave anybody out this kind of things but they think these are some of the main ones can be softened costing were not working in each of the problems quite yet but they were working on the sort of thing can be softened tool the this suggested as basically just means not living in some sort of stupidity rates action on stupid thing nontrivial but many well so that consequences not let me give you a hard constantly In the special case what look 1 but the Indian when you have a fixing innocent Tuesday at a snort of something special for my Coppertone Lemaitre said it's what that means is not the 1st uniform and and then but the take the reductions mode peace I durable ones that should be able to show more than just logarithmic them something even stronger he said the collaborators said Monday it is here and it and so it's a previous user a study students
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innocent AZ Protestant life for the sake of me he thinks this is cool as it be said then so this was a condition because they showed that if this is true then aim of P and generates a citizen in the Besiktas clearly necessary but in fact it's also right because you can find something free inside insiders told it's assistant to know much problem so unique .period constant so tradition rates but that's enough the snakes from the formerly the Weaver constant so let us give them briefest state should approve possible in part because it will introduce another important concept so we had our little notion of groups of gross here as well so 1st of all evidence that I 1 single growth differently in a way that if he wishes more closely connected to mixing times when you have same it sometimes feels we find them in terms of probably the distribution of relief distribution system assured you know all response to values about monetary values and mission 1 skirmish from you on the and you multiply amateur pop multiplication with different thing which corresponds to modifying said is to convoy of nature and when it involves matches what you want to have some some sense of what happens to the nature of the enormous always 1 so you look at the other 2 on how flat the missionaries which corresponds roughly to hold these parties that there and indeed so you can get and an indication of the elevation with material that it an indication the erection so approved the mission with some conditions I that when you take a convolution of the mayor of emission was itself it will become flat the soul with once it's because there don't becoming smaller Dimitri becoming flat and in particular more equities so how they the wooden downward India's states while the key here in fact from the perspective he was saying it's the meaning of 102 million hours in a sense so states very nice to this important tools in our idea from or its firstround opponent similarly but then which was much stronger balance by 8 hours and the provide hours generalized system anything so this was shown to hold them in their own community fees out there I think more really showed that this was essentially interested because statement but I will not go into details of what it says but it's really the key in the sort of message passing from here if you have a statement that base person that he was a very easy he years considered the special case of mu equals tool in the City of Light we have to do what they yes applied to 2 that distribution and industries distinguish this direction is much faster and you have to do this and and then how do you go from here to expansion because you see the list is just gives you some sort of weak mixing time pretty weak mixing How do you pass something stronger even than you know honesttogoodness mixing time expansion would have to use this thing so 1 of them is the growth for free the getting in the fairest stamps and then there's another thing in that but having a very small having no stated that new number 1 very small I would have it's Seeley fates amplified by the fact that I'm the 1 high heart has a multiplicity and that was contradiction to the I I've heard that these summary commentary can physics but in this context it was introduced by Senate and now this is the 1st year Alcatel review these thank you all right but physicist believe it belongs to the engine Bologna such as and the like all right very well here and although for this so I haven't had to come to the theater for something you have do this for some
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time in the strike will all it's going to be hard but it independent explicitly according to him and his uniform if the
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magazine who is a member to the mind alright I will not write that down but the important thing is that it can be written down that's what I think down the reported and a new improved and I think if you have just if you had been optimizations it would have been if we would have been worth the loss of of the well all right but as immunization Of the Ossetian and analysts look at the consequences so the 1st of stations gave the militarytype constants so it's certainly and then the decision presently Q this was done by the 9 horses to Prime and also value has a very beautiful things very neat way to do this but also some of it may not come be complete the obviously secularization of the skyline but wouldn't downwards show that the statement and measures secure strong year too so this is good the president for overseeing the 1st of all this complex group it doesn't say electricity its use so it may not be apparent that it's needed on the same group within the underrated at the same time there difficulties because of the complex numbers in what sense if you want if you want to give you what you want is just the keeper position over an infinite defined in view of the fact that proved it's easier that way to generalize the keeper position In this sense of disproving the same thing statement for the complex numbers is just delete it be put to sea and then delete large chunks of the paper it goes through our however if would appoint approved then the sort of statement or something related to these such a form of the Cuba position would not be enough because what you really want is the result of measures and this passage does not work just like that where the complex numbers you need a stronger statement here in a statement that keeps track of the distances between the elements of fake this is a very tricky thing the sentiments no on the vertical lines so when you past bloodily 12 meetings with the have so I have these of the people and that a new look at work and in fact they had already make many things more abstract and have all the papers and then managed to do really finished the officers the but they hold that I was making I was the other half of the people was this and then north on on the groups of classic and I welcome know little of shooters with a bit of a blind alley L have sold beavers have a lot of things to be weakened their dinners they say that make a nite in the deal and I proved onehalf of all it is then we prove what will in fact what we believe to be diehard house but we couldn't finish the job because even those things worked out for its people and the speed of the um syntactic group they did that they would not warrant for his and we were stuck so in fact that the broadest intransigence so these gotten stack and the and the finance groups all simple an almost simple finance groups of 3 types so the toward the simultaneously by the and use of so not into technicalities but yes assure groups using the audience in the spirit of the reading it all all final gross of a type even really really ecstatic objection a group of right and indeed I should put some other things in the board because I said mentioned areas have looked classification so this is the and classifications here and I think many of you have heard we would have heard about it sits theorem who proof if it were written in 1 place which it hasn't really will be about the length of the Encyclopedia Britannica down the details that might be more like Wikipedia about the I with about not being known as the jokes that I at any rate cut people early on after the proof was announced tried to replace parts of Ft by are more elementary processes and he turns out that many of
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those rules and quite robust so what 1 example the step of Larson pink so there some were mostly just working on my ancestry paper without my help and to the word being ready for a very very intelligent and an elegant diseases of part of the it's showing that things worked out better than I could I could make them work out in the air in the also the phrase tool they were looking at last and paying so we will soon see unrelated talk about but there hasn't been role sector was so this was floating is appropriate for a long time but it was elucidated in part show the newspaper of the government will lead us he would accept proper lost but decided to be quite useful because there were some statements in which I could throw a lot of things about sets of a special type how many elements of the to the case were living in a variety and Larsen and pink had statement that 1 like that they were weaker in the sense that they were only 4 subgroups of forces that was slowly but they were strongly insisted they were more general underfunded can take some so in some sense this time of these 2 kinds of results you can show results the top valid in general for Our since it grows slowly and you know in the sense of being valid for all varieties and the authorities of a special type and you don't need to do and hope for very well this for the proposition these shows the Coral area Of the combined all of them small diameter coronary immediately and this sort of thing can also improve until the accepted and used to be expansion this is not hiding them feel inspired to again as philosophies I should put brackets and included political feces 1 person but at any rate so whatever the open problem meaning there all right so the following year a group meaning make sure you have some things so fanciful I so already so as you have that's for and you have limited and the arbitrary tonight and we have that they are matter of remedying the spoilers but we sustain proviso will not supplying the constants here they don't dependent .period independent ground field but they do depend on the run it depends on here so we independence this is the 4th this would be the fullest sense of wildlife conjecture and from troops following his conviction in the 80's I very well so the 1st problem will be uniformly and then the end there has to be a change strategy because a key proposition is actually not through uniformly and around the delta the statement which is no longer on the board and fortunately thing a to the want as Delta you that that dealt that does depend on and if you force it to be fixed proposition would not be true that would be counter examples but the consequence however altered to be uniform and this is still believe Lucy so that there is evidence for that reason and from uniformity problem is open even for instance namely I would put it is a question because not everybody has come in recent years the stairs isn't losing peace and here we were set Of if you have any P and in a generating so family of all kilograms of it does it will be there to this is expand the family this is a universal that we don't know you the results of a wideeyed the set uniform and a completely unified but the results all the standard types well over there that would tie the knot uniform on Monday on all the aid that is sitting evolves it is being reduced they depend on that where would we like to have a complete the uniformity and there's stuff completely clear that its food Oregon most people believe this for some submission I'm all right so I said that there some further areas my I would status as sad open problem except it's not completely open anymore namely why did they say Volvo conjecture it means these formative troops but conjecture is really about all parliament OK by senators about all of fine it's simple Mendelian groups but money and that's all you speak .period it's that he's needed a few hundred million Rupert Symington can be very large say take the group is adamant 2004 said 2014 said all animals and then examined that can be as large as 1007 not that simple just basically means the opposite of solvable well it has Knowles Norma subgroups of find at the tells a news release from relations but what the group said the financing from media but that's what that dissertation theories about it tells you that these groups coming the form he said that make textbooks which I made its troops are just groups find mate exudes are the findings of the type then there are the monsters but this 1 monster was began but then there's a finite number off monstrous groups are but there's a finite number of them so we can forget about them where appropriate and that the statement said that if they did not expect the the gets into the jet landed on the island and then there's all 10 of them it's almost bearable for permutations and elements of all Tennessee subgroup of semen Phoenix since then he has and supplements elements is of course a group consisting of ways of meeting and elements and all 10 consists of the even permutations what permutations well you have all seen the game of 15 and you know that it cannot be solved in Torino facing the 14 and 15 you can you cannot do any permutation that would flipped 14 15 give a shifting things this because even in permutations so they've implementations that all the provision of information 15 tend to be even so they cannot do as it's alright so that a group called salt is simple and that nothing was known for or a really very very very very weak bound for the world and all that also showed that is constant and the University 1 so and what is now known about and what is known now told 10 and what we care about a lot of people care about called them for its own sake but then there's also is apparently about native groups so that there's also a highly
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sophisticated reason to care about all tendency them so if you're really much more sophisticated than I am I just started working on the field with 1 element it's very sophisticated because if you lose 1 element is that exceeds that of objects on the field over 1 element which is called as and some sort of quality English French and standards and said this group turns out to be a and and the most definitions of these terms and unsettled the problem the fact that the problem so the fact that program was opened here the best found something each to the square root of Ed Our meant that there was little hope that you could do as a land uniformly because you cannot do it even over the phone all right fortunately no very well take me coral and take a subset of the generating then the so we haven't quite got lucky to constant power we have but what is known as a "quotation mark priorities McNown the to Logan the 4 per cent we still love to very well it's still too we still don't know how to prove that personalities Nagano due to the old ones that stated that civil many of these samples the thereunder sense yes of course you know it'll be meaningful meaningless said yes so absence comes nearly sting design here this assessment the group is a creature of pure there's the field remains the field with wonderment which doesn't exist so it's really isolates the problem the problem of the all right but let me give you some in India so we're going to take details of Indiana and many other things more ideas you would think that today's but I'm all right you mean philosophy 1st of all I really think that the right way to see these things is to look at our options so what is an option I'm not sure From all from Gianna said Exxon the text Indiana is a home office yeah From tool their Roble automorphisms effects which is effects said it seemed the same access to the from the mobile phone stations sir if the money is general appointment tomorrow the spokesman yesterday said 1 group that you can look right away at several actions you have several options to study so you have the option of the on its shelves but left or right modifications that is not to be sort of boring because for example the orbit need not be retired and current wrote the 7 aces Fabio assimilation of an element these those subgroup that sent the ball said several of all elements that the movie so every element of the most servicing indeed and this intended is so very boring what turns out to be much more interesting is the action by the end of the year itself by condensation and is not an indication it's not so boring if you consider the potions and so on and so on all right and then it turns out the depending on the kind of crop you're working with Japan also it's also worthwhile to look at other kinds of functions which are different for different kinds of ropes and made the difference that gives a particular distinct favorite told him here a gross and systematic symmetrical but it's on the framework of a factions in the hope that really uniform and unified treatment will come out of the so people have been here a group of what is happening to regroup and chiefs will really finding fisheries he acts on aid to the end and not just as some sort of unstructured developments that in juridical sense lending operations so this will give the part of the property medical flavor notices of unitegroups theory but in the sense that there will be more fees and so varieties and so In the case of options on cement was many guests may definition a group of functions on the set of a set of anonymous so no structure there get someone and instructions say it and of course we also cannot on pairs of elements of distinct elements of these peoples it's as if you could say that he had really stupid actions because of to defendants however is just a fact this really small states compared to the size of the group turns out to be very useful so if what characterizes the matrix broke the city ducts on on the often space dramatically what characterizes as an integral part of the operating repeated the act none fatally on really Small said all right so I very well and let me just say that some of the itself is not knowing the audience introduced the term approximate groups despite nice except placing it's becomes nice if you put aside the unique so it's basically agreement especially in the violence who's tomorrow I told the concept of states and disposed of but even that it was not clear from the start that these things really evocative because it means that classification results 2000 and let generally think that the rights point of view Alan sold all right so you have that would be that that would be the world's sees that the statements about several can be generalized to statements about the proximity but in fact as it was the 2nd class was held for broader question 1 can statements about subgroups be generalized statements about we don't assume from the outset from the side that the I wasn't such a generation generalization be absurd the meaningless not because of what happened what happens typically is that some instances so have the level of the blast several pages of a life age sir some of them are very often in this out these can be does the same Thailand about an arbitrary subsets almost strike were sent copies of would become a some instances phage would become major users of fate become a twodecade save for some present and then the statement Monday with the words a process goes from the
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desert is very useful well the 1st statement of that kind that have 5 minutes unlikely well I will give you owe them something that is simple paradigmatic and also extremely useful during several races here so you the acting was next consists of this matter the stabilizer often elementary simply there's several of the consisting of all elements the it was the orbit the orbit just all this at a full elements to which experts sent by assets they notice that establish a all right so the orbits of laissezfaire intensive revenues mainly because it's where the fuck most of you have already told to undergrads so it's 1 of the 1st things ACC it's called the theorem the amendment said you have that 87 and that they should stabilize the intersection of region the stabilizer affects the sage divided by the size of the orbit effects and the proof is by counting very well but the interesting thing is that these can be utilized to I want wanted retirees so is such an inquiry doing to be true for the retirees that they note but you can still make this into tool inequalities the 1 that says greater than the 1 that Senator Reid of 2 and 1 as inflationary "quotation mark and induce often boast by things like different ways so for a year a universe a intersection style effects so 1 copy of has become a and think this is going to be great and we hope they are not even marginally in the direction for a and B substance of the b all I have stated in the way that makes it clear whether proof he's counting you could also divide and have been the case stomach here so it's basically a gets the other half surprise is again being told them :colon maybe even have time for 1 interest litigation so as they said the 1st transaction is very boring but the communication action is very interesting so the action communication that the standardized often what is essentially softening them indeed injection accumulation not here Of course I was never treatment since place that thought my future history although forever so this evidence that fixed the other was a commute with the systems of year by the by that we mean the conduit of the Beijing itself and that happens exactly when UNH community the centralized and what is the origin of these so the orbits of the action by conjugation is what yes the economists a class thus 4 save the question members to city further intersections center and so he could leave and this has to be extremely useful because during the upper bounds often it's not so difficult you can do it you may think that means in the case of resonance as it see but the law bars is much less evident In the construction of the year accidents according to him now because of the orbits of so you could just make this you know the Senate's AG alright let's in particular you can just see this all right but it's it's it's correct as stated and at any rate this means that if you manage to give some sort of pop around for these you have a lower bound for the eastern front so we knew that this means that you get a lot and a lot of tournaments and Nathan say that the mutely theology but all the elements of a very special form coming to seal the win without him and he if he says say it is a major tighten values it's all the other day animated that commuters at the city's assures you that many elements of a particular form so as intimidation from American troops but this but he this gives an example of a statement where this thing is true all the groups and then you has softened its after changing the form a little bit so that it becomes and then you can apply and such are useful all right so tomorrow we will go take a step back and look at some of the roots of the subject the 94 minute Oryx and how 1 of the statements that they historic and important role
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the sum for the CIA and can be seen previous consequence or as equivalent to growth in the African group very nice for the group where we will prove and then we will go into a sketch of how things work over attitude and then the end I will try to give you some idea of things grow more b right the inclusion of the United States 1 of the world's major universe well I it's it's fairly simple to me if you'd just right down to the following season interesting example because the 1st of the generators that springs to mind is just 1 year looking at the man it was really 1 tool and devoted mostly because of improved right away the hammer hammered here is low he squared so far much stronger than the to the level of power but it's also it's also worth and then on Jesus's straightaway away that nobody can and the amount that is universally through the 1st in group and also watching out in an idea a nite the but in fact nobody knows that nobody has constructed an example where the diameter is long GQ it might be that worst possible even
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Distributionstheorie
Länge
Gewichtete Summe
tTest
Ungerichteter Graph
Gesetz <Physik>
RaumZeit
Gerichteter Graph
Computeranimation
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Titel  1/4 Growth in groups and applications 
Serientitel  Summer school Analytic Number Theory 
Anzahl der Teile  36 
Autor 
Helfgott, Harald

Lizenz 
CCNamensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/16451 
Herausgeber  Institut des Hautes Études Scientifiques (IHÉS) 
Erscheinungsjahr  2014 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 