Merken
4/4 Lfunctions
Automatisierte Medienanalyse
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Erkannte Entitäten
Sprachtranskript
00:03
Hey In
00:07
1 in an and slow continue from last time out we we were talking about the of bonds fall moments of functions and so everything that as saying the stock is sort of a joint with vaccine so I told you last month that the principle here is complementary to to what I described earlier for a long long way if you if you have an accident reports some woman you get more bang for the current global tire moments he said the complimented principle that if you have an asymptotic for some moment plus up to the right more I'm going to illustrate this by talking about but 1 political family let's think of a family of 5 .period Atticus open come something more important for the moment they say could be given by West Flanders have effects for some cubicle on the left or the integers and and we I want to do which are fundamental discriminants for which could occur the sign functionally question is why Clinton delegates will still function as trivial and then we won't understand say moments also have the twisted by Heidi or again if you think of the analogy with that would sell and theorem the skating state niches they might also want to understand the distribution was flawed Telegraph the Pacific standup so this is an example of a family where we can compute where little so I usually you can compute 1 or 2 moments in this case 1 can actually compute exactly 1 moment of snow no but this is no longer with us some degree of flexibility that you can put them on the shoulder ship alone will effective on the family longer the ship milk and still evaluate the false smaller so that's the 1 thing that kind of situation that you so you but again this fall high all low moments like the 1st you would get the right way but that really comedian extensions no set the stage for a separate probe of the industry for a moment I tried to indicate that we can the following upper bound fourfold 1 the up most of the ride the conviction asymptotic which in this case escape entertainment and secondly so so I indicated that in this this or that the sale of a flurry of this state each of which is the analog of summits ceremony which would say that if you look at long have the twisted bikeriding met the conductor of cheating and slaves approximately with me about miners have friends longer and variance logo on it so I'm thinking of convinced being of sizable tracts at the end of the talks and mentioned that this is a very powerful conducted because it implies Gold was connected with the sign of the functionally question is positive then 100 per cent of the time that the rancour 0 so this is a refinement of conjecture life and that was so weak this is all about he said that the book I think you did come In the 1st half we are but we we are in the family was signs as possible and actions of the water you agree with that ,comma personal interest among Israel's for a few of the family or friends of the New York Kennedy right what I and specialize in the family here for this political family of political the sign from the being 1 of the convicted of employers this conductor of both of them 100 per cent of the time that comes and I don't know the history of the way you think that somebody may not be you so you proving proving hinted at it quite hard because he was let's safely assume generates we don't amount to prove that 100 per cent of the time the summons or what we
07:13
can look what we can prove it is 1 all 4 of the skating 2nd lecture namely that we can prove that there is an upper bound so far from the state from the time of the year that I grew up with the goal of from the days of the year and the year and the way it was maybe maybe to make the point that the room where the family is worrisome because it just depends on the progression of the that the license if convicted of the crime let's kind of India's fixed to be some major should also specify that these something things but let's assume that coprime to and and also on the reasonable simple thing to write down what the science functional question and it just went on from there changes in the face of the earth is you have to so so this is what we can prove that if you look at the number of after that the life With that have not signed the function equation 1 and let's look at those for which the log of real functions so we want to so we want to take into account that the meeting is expected to be the negative number so let's add Liselotte supplied the mean which is the same as adding half log log divided by the variance come with the quantities now we don't know so you might say this assumed for example that the value not to this but what the group was part of the public but this is the most large number of the flying lying which is so what we would like very much would be formed equality here which we prove that we can prove a onesided bound to accept conjecture no now for the family of kinetic you can get some kind of interpretation of the all the chopper which led to like the buttons that let you really don't need to assume all of of which will connect you on the assuming ranks where it's almost appearing but not quite yeah but the only thing is that almost all of the group to be finite but on the other hand we don't know exactly what powers so where small crimes divide the group so you can't really control the aura of you because you you don't know how many calls to divide the right it's you know I tried to look at the literature some clear statement of this but it doesn't really think that's all I'm going to write that this sequence 2 so that's what would be convicted but which would combine for that this should be equal to the sale valued at times various about and so the square of the of pouring group what Q and then there's some real period for of the and then there's some time ago effective from we think mostly so so we know how to understand that this is some bonded object which is not wearing a hard to understand how this behaves as a function of the the real roughly speaking like 1 of the this just goes to show you the Times and sales and then the last thing to understand is this some time ago effective sorry 6 the case is the target of factors it as a profitable various crimes of some function the people of the 2 local calculations of most of these factors and you essentially have to worry only about the primes upbeat that divides the maybe also away with the kind of delight that the conductor and but that's effects of crimes would have have to worry about the safety divides the the least number keeping you can be described 1 class the number of solutions ,comma oneself effects where the all that appears from West no so now he conceded that that the conductor of keeping in snake on Wall normality of the should be translated to a conjecture on the long arm of the of as you this family of so well Everything is easy except for this summer ,comma dollar factor on the total factor is not so hard because it only depends on in a nice way on the prime
14:21
subdivide divide the so if you carry that out the connection that that maximum age is the following so you have to understand how the summit factors became so that future to points of there some feel whose degree acuity the 1 2 3 or 6 so it's a Gallo fields and the of thought of skilled you saves his some Group G the GAO group of subgroup due to a light on that I think of it as a of so it could be the the work be generated by 2 sides along the cycle could be all history and then we can define the following parameters and the policy of G 2 be 1 plus the number of fixed points of view as an element mystery so this could be a number which is 0 1 2 exactly as set this could be a number which 1 2 4 4 that supposed to be like the like these people I want to to find 2 parameters of Muirfield so this is going to be the main author of the takeover which groups of those In the slogan about the connector that I want to make this minus 1 the miners have indicating state conjectures and then it'll be modified by the expected value of this Loxley of cheese so that's a factor coming from these Gambardella factors which appear in the denominator the variance is going to be defined as when the Keating state variance with whatever 1 for the longer on the Net there should be a variance coming from the summit of factors which is just that so this is the kind of thing that you would expect if you put on the trip to think of some of that argument and the case's pall was that this kind of object logarithm of its calcium with the means and this would be the main and that OK and then if you put everything together than the conjecture would be that then as the worries in this family From on the same Lord of shall divided by square of the Real .period sizable real but this is approximately With Maine Muirfield times and let's what conjecture this is kind of thought related to somewhat by To that someone somewhere related conjectures that he may fall moments of shelf ED but he didn't quite culminated in this and in this sense of being global now and that it would suspend that show what we can prove sought again we can prove onehalf of this conjecture namely you can you can get an upper bound for the frequency of large value they bigger the for possible values the written down on paper right but you know this is a kind of a unified way of thinking what what this fall when you but was clearly if he has complete tutorial which the figure out what this says is just half minus locked for form you on them 1 square right on then for those cases you have to kind of look at what they're the small I mean several don't always fall for 4 cases a year don't see any obvious In this is 1 way of describing like I think I also realize that community is just and Silvestri guy I don't know that this summer silly ways in which you can adults of jails with 2 which is the same group right which is maybe more natural if you think of what proportion .period but there's nothing terribly under 1 thing terribly illuminating I asked Prime Congress was that well he for illuminating expression I think of miners newfie times well below decks Over the variance so signal the start of of wildlife next but the proportional time for which this is bigger than me is up most of what you would expect the number of discriminants of 2 acts the unique so the easy part of these conjectures about the somebody else industries I would development I'm only
21:31
it's something that Washington the answers right right In to noted that your Oakmont calcium serial way but there are looking for you to say that right that I don't know if you have the feeling that this is he said this is natural this is a natural way to normalize having gone in this case that you could do other things like you could you know that in some sense a chemical factories an artifact of the fact that we're looking at almost all them just the same kind of good and right and math and we proves and lots of appearance for primary screening also then we can provide optimum result in that case all this kind of new evil disappear unnoticed because replaced by 1 by 1 but that was kind of nice to write something which focus he has reassumed ranks Soroka which I think all a mild assumption because it might be removed any day OK so let me tell you something about the proofs of the use of these 2 runs and tried to connected back also with the with the conditional films offered approved for moments when we can safely assume during chicken proved the right of abode for all moments of thanks to my earlier work and Harper's we support and these techniques also give you a a way of thinking about that they're actually quite closely related to the 2 Hopis work but we we were working on this we have we had these 2 results independently and some realize that we're close to each other no so let me start maybe with the operation of the center because approved profits with extremely simple and in fact the same techniques will also give much proof of some the source of the normality of the ability to have the site eh and the proof is also kind of going backwards in history because of the problems in this function 100 years but people started thinking about these problems by modifying using on the work of Boris Johnson and others were really on the last time progressed people start replacing the on its way like some books animal modification and so on and now we're going to go backwards and the placed a fisheries by on products again so is so much easier to deal with incidents and with all the products are really much more flexible and thinking about many of these are these kinds of problems and it also led to the Simpson in a way that I'll explain so so I think it's insane discussion of some books and I made the case that you can fight and I don't want them to functions and pretend that is where the 5th and computer moments of that stupid someone so that's what we did so to define what you've got to be some of of the idea of we goes to some more .period so it's up to you something like that said next to the 1 on Dec square this is a very small power factor now I claim that that this object I can understand everything about on average so if I if I averaged over all would be up to some .period decks with deep in my heart for fundamentals commands I can compute asked many moments of the of the as I like I can compute the case moment of and I can evaluate you can evaluate them so AFP is signed by how my function them to cope which I'm going to write as a on the other other end of the year except said so we understand that in such a way that more too From so we can evaluate this for a wide range of values of of K so for all chaos to log logic squared essentially let's let's put forward 10 here if you want to make the case that it's a very small difficult along that you're trying to average when you expand this out you get terms would only go excellent 110 you can expect you can do all of the in life of the of such a shot no 1 will know the answer when we tried evaluate this but if you it out so when the average over the summer real characters you get the call that we get a contribution only when you're averaging over over spreads square value cutie stop a tidy L is saw small a square the maintenance so if case and all the number than when you expand this out you get all the number of crimes that appeared to appear here so you get complete cancelation so the old moments all vanished and what happens with even moments when expand this out to an even power you need to have a product of P 1 to P K Casey that's not going to be a square and the best way in which you can have K Prince multiplying obvious question is that if half the prime match but the other half of the prize a number of ways in which it is basically the same as the if that is so so these moments will will work out and they will match the moments of Augusta the name of the girls if you just take the 1st moment of being but the variants will be you need these crimes to match with some other crimes so you get us some or will it be up to this point Exelon lot logic square square overlooking and I think and this is just as important well log long so you can see what is good for these problems because it's blog exude square but that's exactly the same I was very little but so they might take shot from temptation of these formerly them that cation has
29:21
Galston distribution means and variance logic but in mean 0 because I did not include hear the Peace Square which would be watered contribute miners have to count and which the Commission I claimed that 1 of the following the following must happen you know this polymer this you the is bigger than the minus epsilon times square of so after all this is really what I would like because I would like to say that the of function really be like this on the appeal once I supply built this meeting which is from the peace Cueto OK so that's really what I would expect and then I'll make a slightly technical condition of the expertise neutral explained or maybe that this happens small this was just a nuisance it's not very important to think about this what happened is that 1 it still happens that but if you take half cut it with bike idea times loyalty to the house that from the time exports minus of the service the assumption is that this quantity is based and also assuming that this 1 is not too so this quantity must still be bigger than the exponentialof epsilon times square of this could still be pretty vague so 1 of these 3 cases must happen and if they can estimate the probe probity with which these cases happened then I'm happy well the false case we know already because I know that few of the is sort of normal With the right from the right means that the probity with which this happens is exactly dominated by the Council this action equal to the Council it's approximately that of so I don't realize that you in with the other people think of me as well now this on the this is also where you need to handle but if you if you just use the fact that you know this normalized by squares of obligates the calcium although bigger than squared off on so therefore this is still on the table the calcium so this happens happens rarely so now the last thing that you have to do is to wonder how often can it happen that this but this is not to under these gets lost OK now no 1 would understand that would be to deliver the 1st moment France's experts might have been less forgettable dialog acts like and if I can evaluate the kind of distance or to be smaller than I know that it's not going to be very often that this product times along its lobbied to have it gets lost In the bonding this so evaluating this would be enough but this is not so easy to evaluate because this is what I mean by taking up and another product of the product of what not but that usually series wanted them all a product this is likely on the part of the Prince of some point of some fun look at what is this the business that's right that's right yeah that's exactly why we what we deal with all the props to tumult of the facilities were so that's trick but so far oil products have had this advantage that I will I'm able to kind of Inwood the exponential office which I would not be able to do with the offices of the Board we can get we can get back to their facilities from the other product and you know something can you effects then In the next actually will approximated by right
35:39
so if you ride on the Taylor series of reading the is going to be a good approximation the Act's provided Texas not too big compared to where you approximate let's say excess less than I am aware of the square hello the Union of but now we exactly in this case where are we going to assume if 1 to fail it agrees appeal the small it's it's a little bit large it's not too small in a negative direction so their fall I can take where was beautifully .period there so therefore I let out a litany of maybe let's say I defined by faking deal he opted Exelon would tend Mullennex Quella forgot but regained some of what all of what you see can now replays so using this I can replace export minus 3 of the by the summer over Jake goes from 0 to local also minus 1 collegiate wanted factorial of In the case of I'm interested in where 1 and 2 failed this is a good approximation exponential might be if you make this an even number of what you have here is also going to be always positive so that be useful so instead creations of of evaluating this moment which I don't know how evaluate I can replace essentially by edifices approximation and then compute the moment and that's enough and the key fact that you want but you want to know is that since I chose this PDB source more even taking this to the part to low that is only going to make me something only of size extra 105 5 logic which is where he shot so I can evaluate everything so so that's the idea of approved and to prove the other central appear on for shot you just say it's a it's a very similar argument to this you combine this with the modish cats argument to prove that the time factors behave nicely and you also have to worry that the factors don't interfere with the old function but that's easy enough to warrant well but so now I will try to describe quickly the argument for hours for a moment the the the hearing will lose something but you don't what but in the next argument goes for you why you lose something OK so I'm committed to working with them so that accept that the fact that people that we stop you at some small of facts usually means that reform more associated with I that's true you would some parts of it but that's OK because this could be a twomonth is much but than powerful but you but next want to say that we can get the truck bound for moments of functions where you can often be powerful we say that we tell you how that argument works and then explain where the argument comes from I want to explain to you where the argument comes from 1 of the things concede to the argument that it comes from problems with we can talk about that but so let's think of the Brands which just purebreds supposed to become world prime acts and you want to buy Prime do some OK so you want to count the number of crimes up to act sibling by by the time the sum was underplayed by my inclusion exclusion and then you would want to understand something about the delights and the the whites a product of a kind of busy feet right but the symbol for atrocities and brooms idea was that you could get out of here by restricting yourself Due to the whole which you stop stepped in and exclusion from the worlds of all the world and this is a powerful man and what is important here how you choose we colorful and you have to go to another tool balancing forces when you carry this argument 1 at each stage of the so when you when you have your age you know which is size bigger 1 in this case so the total and that you have b on the order safe the number of India's up to ax off with which have most took a prime factors all the prime factors going up disease so up most of the time took a taking a very crude bound for the and then there's going to be a Maine which is going to look like the side the divide the the utmost tax on the new house and mules deal with the and will this condition may be spot and there's still a condition that only get these most right the question you know is going to
41:56
approximate for United which we would like we want is going to approximate the product of a Prince of disease 1 month which it should be a good case for this so you
42:15
want to choose came large enough the system the case small enough so that the on the to bring right for that that's the game and principles no and the urinal ideal bronze and if this is not a historical ,comma justice so in the case of someone he was so and so let's pretend let's think of the product to the more this like this like thinking the experiment minus B the 1 when and if I think golf Toms here with only is some someday the device obscene minus 1 what was the last thing this is like taking the Taylor expansion of this undertaking the G8 vision so what you see this approximate if the 1st Cape Town the Taylor approximation to the series will approximate that it would question is when will the 1st to a thumbs that history's approximate this exponential well OK has to be larger than the song which is just so in other words we get we get a good approximation here so this is good yeah it's so so no OK now we can see what Bruins argument would be the case like so this has to be so if this has to be small because has to be like the chosen to be something like excellent can gunmen to cater the true log longer new so you have assumed that works now the only problem with this is that you will not be able to set up some more effects which would which is what we would like it or not but we have to stop the much OK but now but now you can actually occurring this appearance of the season for Ucon for freedom consumer objects the oneofakind now that's the next step that I'm going to do that and will use this so 1st that the 2 using the effects 1 always tend to gobble what and the say status of this indeed part of the solidity even so I take this and then I put in my next set of crimes which are going to be those that divide so the divides the implies that he is not going to lie so you'll see what I want to do here can lower decks and subjects the waterway can log log log x and now you see that I can get away with this by only going up to twice we all log log but so I think I can do this with more losses because once again that the numbers involved are going to be outside Mexico wanted him the excellent and many going up it's the 1 I I want to do this carefully if I want don't want export of more than access to the evidence would be completely under control and why will truncating up to this triple lobby a good approximation because it is something because of the prime in this range is going to be extremely small it's only going to be like it's for the 1 0 loss to Bologna Israel's log and no you can't keep reiterating that kind of thing the but at each stage you get some number of prime time but again I might want to do this way because I wanted to know this is what I'm going to transport for the moment OK so undermined by keep integrating them serve well so in other words you kind of to maybe you start with some L 1 which might be something like 100 log log it too soon and then you take L 2 which is going to be Lobelville 1 for or 100 times that of the military which is 100 times longer fell to and that the state use the prime the that are going to lie between Exelon work with state 1001 square and here they going lie between 6 along with 1001 square and excellent on what can help to square the OK and so in this case you'd take numbers were placed 1 type of most Western backed by so that's that's version of of Prince kind of pure and it will not have to go to work we will gain back any power supply wanted to loosen the usual Branson no so now you can take this argument and use it to proves the bonds on moment because I can basically be place so I can take when I want to use a lot of pride I will use them on product that in some range from here and some range from their indifferent as to the power Protestant different pieces so that we right of I injuries and that like PGA of me to be the sum of all the crimes that life in the in northeastern doubled Exelon was no 10 L J M 1 1 square it found what I would like to say is that I want to be able to so I wanted to get together ball so I want to I'm going to write down a kind of thought .period ways for some kids moment I want to write down the bond which is good and wall which is great involving the formation of the 1st moment of time meal plus some of difficulties it's that's going to be there that's going to be the aim and I don't quite know how to do that because it's a difficult introduced a series here I would like to put the universal Woodruff literary flair and added a multitude a competent most what I can do it With oil products so I could take hell have To the right and if I had won the products I could try to express abounds this needs to to be worked out running out of time that so I want to be able to write something here with exports came minus 1 times the soil .period factors the 1 of the best feature of the plus long and then plus the exponential off the Cape Coral small apartments this is true but I can't get anything at this stage so it's OK it's going to be available of 1 right now less someone and the only 1 that I can only about at the 1st moment in the family right the suitcase anomalous someone's services some negative exponent of you like and would increase the moment
51:56
Pollack and I don't some interpolation inequality but I can't do it with the 6 areas I do that all the products I can ride on a bond of the site but I cannot do you know the position cases the case is being only 1 right but it was a wholesome and various ways more so so so now I have to worry about this of is the quality some of world numbers responded by something which responds to the operate so it's in the pudding and 1 of the 2nd it's India to inequality Almaden on the inequality but I can write on 20 the so what you want to know how you want to see the idea is that I will replace each of these exponential but the exponential Part 1 of these days reminds 1 market doesn't matter what the jury of the I will now want to replace goods by the Taylor series the going upstairs the L world factorial with the Taylor series going up to these number right so energy as here all the it were even as well so if you take them to be with you can you can replace 1 of these exponentially by this as an operable most of the time they always most OK so this is so if you like the idea of the prince transported to the hospital so what you want to do that then you will be left with history and don't be shocked as indispensable argument and therefore you can evaluate everything that you want on Monday and if you arrange this carefully there also in the matter and you get exactly I only a few minutes of that meeting so maybe I will tell you the rudimentary analysis that goes into making inequalities of this type stopped the software he opted to make that precise solemnly tell you the last few minutes how you can get bounced fault moments if you're willing to assume that he might pots and you can get tight bound for all Wall moments you generate the key ingredient here is that if you look at at the law office and have a good idea for example there was a lot might have seemed to work with the function with what happens is that this is actually To be interpreted as negative infinity the only so the similarities which come from also fell functions in the halflight are helpful to you when it comes to getting up about and they don't so that 1 can say that might convince you that you can prove a general upper bound fall the taking the values up to some .period X you then you would have term from the From the prime square which will give you something of that time Texas is a free can take acts to play Of course if you choose to be more and come which is going to be on the size of the conduct of the complex divided by the effect is like a power of the kind that constantly but small you take access the bigger your losses so this is time the proved In my work ,comma on 1 moment and and it's very flexible because you have complete freedom of of how you can choose so so no you said now you want to find an analog of this Prince of argument about which is stopped for them to the 1 that you can just take you can take extra be some small effects of like Catholics alone OK then you can compute so if you play around with the stood by With choosing axes little elects small enough you can compute as much as you want about this and then you can prove you can make precise a uniform motion of the film would type and so in this way can prove that the frequency but the number of the for which shall the held have twisted by Katie is bigger than some number well timeslot excellent efforts said that this is bigger than Italy into the B you prove that this the behaves like the 2 the minus we squared or what to well below decks this is true used by the but this is true for explosives by for example the state of matter 1st explained but it's not going to be true if you just want some crude but this time with the 1 before 1 the only true uniformly all be going up it's time for a lot longer than that to which I established an Yuriko calculation that on the connection between experiment and moments you only need to know something like this for me being Cape Times of actual case the moment that you find it and the law blog went infinitely subtle beat every case this would would give you the right up above and apart from the to the at the idea of Harper and getting rid of the epsilon is that you don't have to go to the center and use that to get moment you can stay at the moment all the time you know by themselves From the start and that argument is very close related to this side of the argument that I sketched here the idea is that the idea is that you take out a series a sequence of of fact may be the first one will be like 5 x In the 101 of the square the next and will also square so next and you consider the oil product up to these heights and you checked at any point of time if the this if the sum of all the public is small enough if it small you try to use a product of that and if you can't stop the 1st thing that you have done the job do you underestimate the front there this not the end people who will perform .period think theft what is the main well I gave you a so there was 3 things that I mentioned what was that about that you get tight former moment for moments up to 1 of the 1st moment in this family and the 2nd 1 was that you get the legal onehalf of the killings state connections to get out bounds at Keating's state several so in furniture that you that you can prove that the that the frequency of large values is nominated by the expected frequency which is out of the calcium on the 3rd 1 was the argument for the analog for sure which we conjecture is going to be log normal the
1:01:40
southern of variants of the family and once again we can prove an upper
1:01:44
bound for the large welts of there are tied up along for the large and then the other is overturned savory much was that owned G H without the right upper bound now for all values of k 1 of my work of copper refining margins for the 1st time in fact that the fact that this year so the the flexibility of the 380 kind of thing goes the exponential series approximation right I mean you can you can get in the way in which you know is the size of a pact with but that the number of prime factors you want to kind of keep well I mean you can play along with other portions of the conspiratorial so I'm sure you can khamenei strength but yeah but there is some of what the right answer is you know it's like it is kind of S's strangely backwards argument that's going back it in terms of you answer so this will be because any civil could write down right but it's but it actually gets the right result what it for you I'm just because that and the reason for that so that in the case of prime it's interesting that you we cannot we cannot handle the 1st moment we don't know Explorer but on the other hand you can still do it because you can combine it with the somewhat so so you can get proper balance of the right over magnitude for the 1st moment in the family of crimes even with amplifier because you could put in the social work so for free so all the the rest of the argument will go through and you get the right of the loans for more than 10 minutes you also get the a sample of the on the way to the final precisely the development refused to so you can you can do a primary there if you look at it if you look at the altruistic yeah the 1 other ingredient that's used is the possibility of the about I'm not emphasizes the full and floral prime that's still OK you by "quotation mark gave positive evaluation that and and everything else as I look at this question mentioned in my last lecture that too but that the industry's problem would be to get more for the small moments of my moments life and 1 now we have the opportunity become law but if you want all the more moments than 1 than you would get unconditionally possible portion of the money which this it's very worried that you have in mind of responsibly consumer it is the real extent of transcripts of all he has the only practicable composed of 3 or 4 it is expected that this is the way to the a White where did you grow some Walk of Montreal kind of related things would you the proportion of old approach was
00:00
Distributionstheorie
Lineares Funktional
Erweiterung
Momentenproblem
Wasserdampftafel
Gruppenoperation
Familie <Mathematik>
Strömungsrichtung
Computeranimation
Vorzeichen <Mathematik>
Sortierte Logik
Rechter Winkel
Vorlesung/Konferenz
Analogieschluss
Varianz
Aggregatzustand
07:13
Folge <Mathematik>
Gewicht <Mathematik>
Punkt
Momentenproblem
Extrempunkt
Klasse <Mathematik>
Gruppenoperation
Gruppenkeim
Familie <Mathematik>
Zahlenbereich
Bilinearform
Element <Mathematik>
Funktionalgleichung
Untergruppe
Arithmetischer Ausdruck
Negative Zahl
Logarithmus
Rangstatistik
Arithmetische Folge
Reelle Zahl
Vorlesung/Konferenz
Figurierte Zahl
Varianz
Leistung <Physik>
Einfach zusammenhängender Raum
Parametersystem
Bruchrechnung
Lineares Funktional
Erweiterung
Mathematik
Primideal
Rechnen
Frequenz
Teilbarkeit
Arithmetisches Mittel
Objekt <Kategorie>
Quadratzahl
Verbandstheorie
Differenzkern
Rechter Winkel
Dreiecksfreier Graph
Mereologie
Körper <Physik>
Normalvektor
Aggregatzustand
21:28
Resultante
Punkt
Momentenproblem
Zahlenbereich
Abgeschlossene Menge
Term
Mathematische Logik
Gerichteter Graph
Spannweite <Stochastik>
Rangstatistik
Reelle Zahl
Mittelwert
Unordnung
Vorlesung/Konferenz
Leistung <Physik>
Fundamentalsatz der Algebra
Lineares Funktional
Nichtlinearer Operator
GrothendieckTopologie
Matching <Graphentheorie>
Mathematik
Güte der Anpassung
Biprodukt
Fokalpunkt
Frequenz
Objekt <Kategorie>
Quadratzahl
Dezimalsystem
Rechter Winkel
Beweistheorie
Normalvektor
29:21
Distributionstheorie
Gewichtete Summe
Punkt
Momentenproblem
Gruppenoperation
Zahlenbereich
Mathematische Logik
Gerichteter Graph
Division
Richtung
Faktorzerlegung
Abstand
Inklusion <Mathematik>
Varianz
Lineares Funktional
Parametersystem
Approximation
Exponent
Tabelle
Gebäude <Mathematik>
Güte der Anpassung
Reihe
Disjunktion <Logik>
Primideal
Biprodukt
Teilbarkeit
Summengleichung
Quadratzahl
Forcing
Verbandstheorie
Rechter Winkel
Konditionszahl
Mereologie
Ordnung <Mathematik>
Normalvektor
Grenzwertberechnung
41:55
Stereometrie
Subtraktion
Einfügungsdämpfung
Gewichtete Summe
Momentenproblem
Gruppenoperation
Familie <Mathematik>
Zahlenbereich
Spannweite <Stochastik>
Spieltheorie
Vorlesung/Konferenz
Grundraum
Leistung <Physik>
Parametersystem
Approximation
Exponent
Güte der Anpassung
Reihe
Physikalisches System
Primideal
Biprodukt
Teilbarkeit
Objekt <Kategorie>
Quadratzahl
Menge
Rechter Winkel
Mereologie
Wärmeausdehnung
LieGruppe
Aggregatzustand
51:56
Folge <Mathematik>
Einfügungsdämpfung
Gewichtete Summe
Prozess <Physik>
Punkt
Momentenproblem
Ortsoperator
Gruppenoperation
Besprechung/Interview
Familie <Mathematik>
Zahlenbereich
Term
Gesetz <Physik>
Gebundener Zustand
Ungleichung
Schwebung
Analogieschluss
Analysis
Leistung <Physik>
Einfach zusammenhängender Raum
Lineares Funktional
Parametersystem
GrothendieckTopologie
Güte der Anpassung
Reihe
Ähnlichkeitsgeometrie
Rechnen
Biprodukt
Frequenz
Unendlichkeit
Energiedichte
Quadratzahl
Verbandstheorie
Interpolation
Flächeninhalt
Rechter Winkel
Mereologie
Faktor <Algebra>
Explosion <Stochastik>
Wärmeleitfähigkeit
Grenzwertberechnung
Aggregatzustand
1:01:43
Randverteilung
Resultante
Parametersystem
Erweiterung
Approximation
Exponent
Momentenproblem
Relativitätstheorie
Reihe
Familie <Mathematik>
Zahlenbereich
Primideal
Term
Summengleichung
Faktorzerlegung
Reelle Zahl
Rechter Winkel
Stichprobenumfang
Vorlesung/Konferenz
Größenordnung
Metadaten
Formale Metadaten
Titel  4/4 Lfunctions 
Serientitel  Summer school Analytic Number Theory 
Anzahl der Teile  36 
Autor 
Soundararajan, Kannan

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  10.5446/16991 
Herausgeber  Institut des Hautes Études Scientifiques (IHÉS) 
Erscheinungsjahr  2014 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 