1/4 Analytic aspects of CohenLenstra heuristics
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1/4 Analytic aspects of CohenLenstra heuristics

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Number of Parts 
36

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2014

Language 
English

Content Metadata
Subject Area 
00:00
Point (geometry)
System of linear equations
Musical ensemble
Observational study
Euler angles
1 (number)
Cohen's kappa
Student's ttest
Analytic set
Prime number
Number
Quadratic equation
Prime ideal
Goodness of fit
Manysorted logic
Natural number
Square number
Heuristic
Algebraic number field
Area
Addition
Theory of relativity
Moment (mathematics)
Theory
Staff (military)
Local Group
Field extension
Number
Summation
Computer animation
05:48
Area
Computer programming
Wage labour
Incidence algebra
Bilinear form
Field extension
Root
Lecture/Conference
Wellformed formula
Square number
Fiber bundle
Family
Subtraction
Resultant
Units of measurement
10:06
Principal ideal
Complex (psychology)
Group action
State of matter
INTEGRAL
Multiplication sign
1 (number)
Propositional formula
Parameter (computer programming)
Prime number
Mereology
Quantum state
Mathematics
Manysorted logic
Stress (mechanics)
Square number
Social class
Stability theory
Area
Process (computing)
Concentric
Thermal expansion
Price index
Perturbation theory
Sequence
Arithmetic mean
Order (biology)
Normal (geometry)
Right angle
Mathematician
Identical particles
Resultant
Directed graph
Point (geometry)
Computer programming
Divisor
Parity (mathematics)
Real number
Rule of inference
Number
Power (physics)
Prime ideal
Goodness of fit
Wellformed formula
Integer
Field (mathematics)
Subtraction
Units of measurement
Condition number
Alpha (investment)
Addition
Prime factor
Forcing (mathematics)
Exponentiation
Mathematical analysis
Dimensional analysis
Division (mathematics)
Incidence algebra
Mathematical model
Local Group
Voting
Thermal radiation
26:20
Frequency
Product (category theory)
Lecture/Conference
Hausdorff dimension
Forcing (mathematics)
Multiplication sign
Order (biology)
Mereology
Bilinear form
Resultant
Local Group
Number
34:27
Group action
Interior (topology)
Multiplication sign
Real number
Mass
Mereology
Food energy
Rule of inference
Grothendieck topology
Number
Expected value
Prime ideal
Explosion
Frequency
Mathematics
Goodness of fit
Manysorted logic
Positional notation
Meeting/Interview
Wellformed formula
Average
Square number
Integer
Field (mathematics)
Condition number
Area
Addition
Product (category theory)
Military base
Forcing (mathematics)
Exponentiation
Moment (mathematics)
3 (number)
Bilinear form
Functional (mathematics)
Automorphism
Local Group
Field extension
Lattice (order)
Commutator
Hausdorff dimension
Order (biology)
Table (information)
Coefficient
46:52
Musical ensemble
Confidence interval
Real number
Partition function (statistical mechanics)
Propositional formula
Student's ttest
Expected value
Lecture/Conference
Wellformed formula
Natural number
Oval
Square number
Selectivity (electronic)
Integer
Partition (number theory)
Physical system
Expression
Moment (mathematics)
Mathematical analysis
Infinity
Thermal expansion
Mortality rate
Bilinear form
Sequence
Functional (mathematics)
Field extension
Proof theory
Summation
Lattice (order)
Right angle
Directed graph
55:53
Group action
Equaliser (mathematics)
Multiplication sign
Mereology
Food energy
Theory
Vibration
Grothendieck topology
Number
Frequency
Wellformed formula
Square number
Flag
Field (mathematics)
Subtraction
Associative property
Partition (number theory)
Descriptive statistics
Position operator
Physical system
Area
Uniqueness quantification
Rate of convergence
Thermal expansion
Functional (mathematics)
19 (number)
Database normalization
Universe (mathematics)
Game theory
Family
00:02
around the top and the among the
00:10
a and warned that
00:14
some beautiful mind the nature will be these to share with you particularly recent don't always a pleasure I had to to studies this question about 0 2 number field put in this scene in question concerning ground when and uh when preparing easel which I was thinking of when I was a student I was to supervise all these really and even yet and if any given it was when you do not know what to do do closest number always say Would you do not know what to do clause group and it's fantastic really areas we know not everything is new and the to put up with the strange recommended intimate images we see Europe appeared to anew with a new start from golf's of course and so on and very exciting for the moment much more exciting to buyers then by prime numbers Suri solo stuff so the full sum by the facts so I stopped for finite extension of Q considering a interview and that we consider idea and so I consider ought to be the set of among the euro the idea of OK and then you say is that I N J of the clearance additional duties that exist on file and from Prime belonging to a subject when you multiply legalizing equally so it's McGreevey announced relation and what is very surprising is that if you reconsider the closet group and diseases you the world which is what it is he said with this irrational use of finite and abelian 1 the group falls at all the induced by Semitic edition the complications of nobles and duplication of ideas so that could succumb and what we get is a CAD what according seat and the sort you must not make confusion it's what recalled all the American people yes unity to key SOEs is syndicated by at least this recreation I would like to 3 . before automatic preacher interested buyers that 1st of 4 seats and abroad people from Asia Brighton on the surety and all we can say people from an entity analysts will be barred from competition our point of view good so there is a very interesting point of view and I shall be more concerned not only with the case of quadratic but the disease so you you impose Over Q it to and what is important is that the discriminants from them entirely screen and so I shall write it did so do you read the fundamental disagreement you start from the start square freely and you impose it to be so it's just square fridges ,comma to Austrian what and it is positive only at the end of the easy caught fundamental just for the fun of you obtained for the style and this is the humid applies so this is equal to the star of fall the staff finds him when discovery 1 what for instance discuss the pontiff to Wallstreet what so it's really you know exactly so what is the area of a case of good attitude we can enumerate fuses were extremism program so that means you you consider on the 2
05:50
of them the injuries urine which is a very easily and like OK Kelly evils he said of this is a serious and it is a on 0 in on the stimulus by square thanks to the people all square root of next season will depend on and it is the same type of the there's lots of continues Griffin on their knees and next and it's important you can this formula also tastes like material situation we use it you can't do it for the nicotine you could impose some ,comma unseeded did hold onto 1 waterfall quality ,comma onto Longwood area of people around the world that take you have exactly the same type of forming suitable because it's surprising for me just so what is it what is very few
07:08
you'd people must be very conscious of that there's this formula it has been accepted is very Herrera extending so my question is I can't and of X to Reza cardinality so of up to life in the Soviet Union meaning to this film is not subject to the streaming out of is that the capital excess so you're counting the extension of a D retains that you sheets on that instrument is an X and to question so for 2 weeks and of course you can't put a minus and very few results I check research that was a bit of so for Katie costfree this is 0 2 Davenport Ireland so you can see at the beginning of the 70's and Fourcade costs fall it's a is due to 2 to severed people alleviate Diaz India as Cohen In bundle AEK capos 5 very soon couple of toboggan and that's all so I insist about his programs for the people of labor to families by this type of as far as I know this type of generating reading that we do not know how to count so already there is a warning incidents so it I continue to Windgate these fuel spirit so I recall some form of artifacts such hearing of the danger you have an easy for me now which is With that woman got this on this press square divided by 2 Of course you have the normal eminent experts on the nite think the and so here we have to do with X and Y into and it already becomes a difference so of course we speak about that we are set of units it's completely different when the positive and he was
10:06
achieved here we you know that it's a reasonable W which is equal to the fall 6 which comes a number of units sold positive you of something like Pressel minus 1 multiplied by its the end belonging to that in this excitement is what we call a fundamental unique which is the increasing 30 questions which is a nightmare of we awkward at each Salazar and those of the Glasgow preacher arrive at see what they tilt serious and will of ordinary clothing group also fuel the you of this narrow crowded room and of course of which is you you consider you have a new radiation we send end and you ask that I fight for the 2 of finally with norm of other normal Alpha Prime was eating so this isn't narrow and sometimes a very hazard to guess what people have in mind was the speaker of the year the travel so as you can guess still the fault when Wendy's negative no program and what happened at the end of the month to that of the CIA and the ease of photography all the wisp index 1 of 2 grand villas and it's very difficult question to see when we are adequate it's soul in U.S. soldiers and is proposition so desire still draws a group coincided so so 1st possibility is that did the negative or the positive and the normal etc. and did in court minus 1 sober so so we may ask about what is reconnaissance herself or to see if you make confusion so my question is when he is phenomenal so is it's not not so easy as the treaty to say that ended at 5 under the this is all about is equivalent to the daily creation with parameter the style he is survived so you have to take a chill because when you think of uh the the integer you have been is not number 2 here which claimed many of the integration with the press of minus 4 and now the U.S. for this implies on the prime factor of the divided this stuff implies people was too all people around you 1 not an orderly 180 that is equal to convince his forces it's not to look at global solicited it's all over I arrived this seems the sort it's received the cardinality geology between 0 and Bates said but the results of the CND for choosing this year and costs you really feel very area the it is you process usually you we not visible group the cardinality as this is equal to 2 times to cut and gives you so what do we know most all we know is a customer customers so the is negative no confusion tho so we will usually refines innovation should be and this is W which is equal to 2 false each area 1 the square with the 2 5 and for positive the formula is only which recorded a the which ordinarily the trouser role and spirit of the anyone who knew her love of Eppes identity which is isn't a nightmare at the time yes another plan is a good thing which is interesting is that of course you are speaking a sinking of gulls girls was king of quality care for almost all of the can be interpreted or by the binary fall although said India is the action officers 2 of this good so I would like to know to give some conjecture so I would like to suggest what is it with the way is the nightmare because the nightmare easier because there so many former you have always formula when HOT ROD it suddenly on merit this is a case is very when can sit the exam well during and would not say cities saying that we are not approaching a difficult the real initiative state of Sing Sing Sing a chose the rule of logo at 70 no I write so moved some conjecture to MiGs state of fault conditions in which it was like to to to stress and they really concerned mainly it is the that so I suppose the positive the incident is the most famous ones either you want to golf that exists so question
18:14
indefinitely many police said Jack please as of people congruent to 1 more fall said Jack picture of the award 1 of the means that not mainly could reorder conducted she that we which a principal so of course when I say that means at the disposal tediously usually prime the 2nd 1 is strange I remember it was quoted to me by crew and as discussed above you that but that's what happens if we do not know these things so you can I would like to write the name of corn that exists in the area said John each of the different from what we do not believe that this is a strange because it says the strange we on this 1 is always privacy here we really made me someday is apparently there's a parity question the other so the people of course it is this 1 is was implicitly means of settling the brain of a lot of people are so I would like to so just had the same amount 3 people at a quarter Nordic good coincidences in the summer William Cohen were awarded to the 2 conjecture also is exists see once treated poorly to such that when you saw the class from the between you rule and banks it is as to some signaled the cake look squarely thinks so you see I was speaking to you of married and no they speak of you lost him so the commands the stability the existed the true the woman was behind you sighs fourthplaced but my book of the 1st set city so H of the story right did come around to 1 would fall his agreement on ICI's stock would be 1 of the aides hinted that the see 6 the sewage and analysis was refused so I I I write it in a very so he or the condemnations of several the mathematician Austria quoted so cruel has proved prove that its identity that most at 70 concentrate around the expansion of American of deep so I do not define what me concentrate and you can guess which is a nightmare for computers from the point of view duplicity to deal with an invertible with strategic and what is obscene and the the sees it is a difficult so in 5 years is conjecture on the weekend eyewitnesses like even under the czars essentially as soon as you have that you you you may seem that is just on formula problem will use it of ah yes loading silently is almost equivalent of the so that you see the U.S. consolation so unified that almost equivalent to None of these Wentzel 2 factors this orders is a number of classes which ordered the courses exponent is very around we all know that is going ended a shy spins out of Tulsa by prime numbers uranium that he has a sense areas the appeals of the physical His even part which is some that between bubble may go do something like a log of the film and the soldiers want so they're very challenging question and the the that I would say is that some of whom which is known as 0 what what I like about 1 in 2 so nursing precisely known about 1 and 2 but they have a restrained resident Research calving been about which say that there existed so it is your the feelings about conjecture there exists but for propulsion also people room to 1 model for all search ads Sri does not divide a of trophy years of so it's a very few reserves with it Schilsky infinitely and a political portion of prime and that it will ask me what the wood but the boat parity you but this sequence is very typical of it we can escape its you In the end you see you may ask the assault resisting statements in Humeston him of course and the it's so it was full around this year and the people know this result and it's always know when a question to prove that he should take the government to 1 would fall subject 3 doesn't divide H F minus key so easy so we have the feeling that the real quality field and much more complicated and the example of something we know about Riordan and it is not known militant vote complexity so is the ideas about would reduce the number of days a difference received resentment of hybrid of those a number of filament industry so now about this 1 this so I always have the visibility 1 ETA this year so it's arrival at the just wanted something that is due to my stand for the removal and after In improved by also so which voted for almost all the excitement the clues greater spending power 3 miners dead it's it's very with its ridiculous because when you compare this number with this 1 is like the type of my between 9 other hazardous onto the sun's rays of the sun nevertheless if someone who recent fall between the EPB accept eventual because this is a question as to which of course we also need to believe proves at which type of mathematics you need to deal with all Susan determined and committed cases innocent and arrived in his discretion OK there were only just so yes or no we
26:21
needed by the time we converging we Clinton tussle what about the indignities did negative message about his question we we know a much smaller and use of because of the ease of the order Paul please something like it's clear which of the so it's going pretty frantic last summer and this is the His is this 1 not fall most orders would fall or was he and so this is if you read this in the U.S. is in the details of the alleged that inflation is less British With the rest March the has hold this year but with the users along those results fully booked you have to to push to to get to free really looked at me the he has for almost already using the foremost words could end in richest comes to my mind crucial support of you know that the confusion in the prime of the phone and squid purse 1 many question here the head of the company this is not about playing the political and the military said the creation of a new group of use the kind of place the rest of the world and the region is this at the heart of the money or the names of the victims of Arabs in the form of the look at the history of those I think this is very special me that the alliance of the 2 to commit to a geostrategic I about this type of submissions it's what you have in my people people and result was the role the head of the I'm so now we we go to so always ability cheated their way do we know about the Jurong so 1st of 4 I'm as defined which is sold off can and billion group and which is Pierog overseas it is a dimension of the lead Of the development by options so I use also sometimes notation the Citigroup Of all the people knew and if you right means to 2 of the feuding so I suppose that you write it is as a product of the or the prime News to want to continue to all the psyche group in which each 1 a varying reasons the frequency of fucking isn't appearance is a summer off of opinion so what is will know and it is this surrender of gulls on ways in which says that the 2 off the see the is according to the number of crimes devices of the minus 1 so in some sense we know that any question about the to run is a question about McGovern further seemed to itself it and they know what about the literature what about 0 but is it to rank of synergy so here we have problems in literature because are the force proved correct 1 so that I know that the skin over the it is a fact of group of C D of the next 1 and 2 so it may be that flexible parts but it may affect the 2 Ronco may be effective C 8 may become effective because this is not sufficient to guess which is at Toronto so I recorded if you hours it is not so we have always had to travel to contained and it is used here and now as she knew to we finally approved grateful for also and so suppose that the positive and that's fine the peace accord 2 1 so as to the jazz group are not the same in to statement I provide the first one is a city of the disease according to the Citigroup multiplied by sihanouk these so that means that he is the 2 rock is lower by 1 and this is equivalent to that exists but please which divides such are the little room to 1 1 4 I just so in other words of the road see if they've got this is agreement lowered his exwife miners start please locally so remote but what is just curious users with the yes No so we have to guess so we know albeit touring of ordinary crimes no we arrive at Kremlin's fiery sticks with them do so at the FleetCenter it's published lecture notes that timidity this summer will never admit it shares in the lot of people of sold to boats is Uresti have been in arrives some of them are improving and extending the the situation so so is
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it well so it shall not telling the Aegean region depredation very superficial saying so the use a computer 1st of all just forward and the is accounts the proportion of the negative up the capacity of the computers at Getty said that the 3 parts also see the leader but the upon you seen I'm so is it true that the proportion is something like OK and all is to another group of cardinality 9th in year's they find that it is only 1 .period OK so uncle can you explain that to fester for you you but you must be careful in this approach recently computational because the convergence is very very slight we give you examples of so what is the ratio of these 2 number is 8 carries and a years correct and this is 9 years so you like kids and what he sees this is the commonality of automorphism the 3 the square divided by the company's future of automatic use of city ancestry these bases exponents is clear OK so this is what I find to explain you which is behind so there energy which is and was a brilliant intuition to affect to some groups some question whether you want to meet some gold note the cardinality eagerly the carbon entities other thing coming but is this way the college rest the cardinality so you see here is it is robust and much much more but amorphous mass then this 1 so it appears much of you're not in until the end of the so for me it is the sort to so some call manager want to do so you must be very careful which people are discarded entities please bear in mind this land Key lime please pay among plant is the end of it's a natural order a useful misconstrue is from this you To be natural really always find a formula with a tough sanctions so this function at the interior of politicians and so the first one is you take it that case for the use of the product this is the real function of Deboer minus 5 years goes out of cost is less than 1 and it electionyear he pored through the product additives 1 1 1 in the minors so this product is very critical we've been issued from and for his son's his notation is is constantly Jesus is people were area squared the a of the services when I write Corliss site conditions will not be surprised to see this dysfunction frequently but the boat so I wrote :colon inside Saudi teams that I liked 4 and a 3rd of the year's still the picnic so I thought that prime accused 3 so they were frightened by the Prime EP costs too because rules was awaiting them and so the 1st of 4 so we suppose that Deby used immediately so I use imitation number 5 because although conjecture is a probability so so I think you can guess what they mean that probability is that preserved the pier on all this CD is the call to arms yes the probability is a quarter the astonishing president of 190 billion managed care minus 2 from 2 of the 6 say average for I find integer without acknowledging the role is that of Orinda value or if you want expectations the expectation of this is crazy products not crazy the appearance of city minus case use some form this has petitioned the court to 1 OK so you when you see that you can remember always that remind me some accounting Oktoberfest it is this is the origin of the and for we accredited chief for field because accords them C 9 seats and thought he was a probability will be slightly modified between B T minus their characters swamped with it is almost invisible writing he a prodigy noting this once said of him and so he is also a great mistake of Quinlan soccer in zoology of paper a workable who we must make some comments which is known within the thing on the field the so when they have to support that is their conjecture at that time on computer astronaut and that found in this conjecture where approval on in 1 case so we are going to be there approval and in 1 case that means .period meetings time in Dublin talks table which is the socalled according already existing seedups counting to Chi's advertised for you they only do better is this big fresh here so they were so they will only proving that I can the city commuters communal forces since this is a case where this is your fault that she did not think we went on for a course to the use your for quarter justify which 1 it's OK it was only approved falls on it with with and as he doesn't much change everything but rather extension which confirmed intuition of questions so I would like to make some comments the first one is if you believe it OK
43:53
so you see this you must have in mind that this coefficient goes to 0 very quickly very quickly so for instance I doubt that someone a computer that is known but the Wii's really wrong the deal is equal to 10 . it means you was the 1st coefficient and the 2nd abound in the 1st option people work the ministry follow minus and miners underage is very very small so I see true the shortest of competition of ICH dismissed it it's not surprising and very difficult at end of the 1st of 4 winning here with my colleague cleanup would like you maybe you and I would not criticize of course but to and the seed and see 10 so we make some of the situation and the recalled good we I'd call N K the key to the cardinality also does a total number of the adult which is usually the orbital space season it subspecies of mention all of a by working yes kind of any dimension so it's sits in the computation is almost done so you have a the and the report was a song from at any cost 2 0 2 K and you over the issue of duty minus 1 PKK minus the is a minor 1 divided by he estimated mindful so it is a bit of function it is here that it actually function so sissy so it's only repaying we sat with their formula so since it is equivalent on 2 this fall or on farm floor and for encouraging the average values that museum what I want expectation also people were unfair see is according to the number of victims of the species this is who we are at the moment fall
46:54
someone for managing a mystery preferred computing moments and this is what it is Is this expression which has which is full of algebraic meaning and unaffordable and all of the 10 please equivalent for warrant for integer so it is a slightly modified and unfair press 1 killing miners and the form of peace for people and in something which I will explain you know will cool from 0 4 we screeners receive this victory speech that through year the but we see the scene of a costly look so no agreements on remark was pronounced because of advising that Quinonez ,comma not very precisely because it was the clear things on and off so it's not always fair proposition proposition you since it's been priced and said implied the 9 meetings it is so and I have this version of the sinks so as soon as you are able to compute awesome amount you know the problem he is also it's not surprising that it's not in it and started and end the controversy we can approve of course we do not know which is the rate of condoms altered his priority from poverty to assure that you cannot deduce the overrated he uses expectations so nevertheless I like disproved and because the uh I must admit that sometimes we need the use of paper brown concerning the 2 7 0 there is a certain type of solution and Will we meet UN and infuriated Linnell with so I worked 26 so I take pondered the nudity "quotation mark ended so were when using the the big community confidence so on and so on so I ecology our the probabilities of propulsion that there is it the arrival of CGT easy for 2 off so were with user will select countries fees at the Eden that uses that did so you you may discontinue analysis stand which is the 0 as you want president toward is to the the cost to ban is you will be which is 1 of scores of the sum of probability is equal to 1 of and after it's more the world proposed bill you want to speak to this disk where the 2 close PQ and so on and the world the protests and because of the peaceful so it's like I would say there and infinitive on their mortgage determined there is Broncos he doesn't like Avondale only screen of EU that we find those of the technique so I would use some 2 to prove that it doesn't have to check that it doesn't have the solution and in fact that the solution is unique so it so 1st of checked that's the always equal to that primary care ward which is to please close solution so I was very happy we have to open the 1st of 4 Albion right book is a partition function and so we re finds its way of course it's just a coroner for so you know he issues a partition and the and primes you know that this is 1 of the and it is exactly the function International 1 of the eggs and no real Jessica extension always in Abyan right which is the remark so suddenly say the same thing it was known to them so it is you know was only so disease I think the boardroom 1 business that the by wonderment this makes clear breasts the next long such a former now and in fact we knew we were required another 1 which is that we found it in this exuberant we can write his formula hours it crashing of 1 of Texas music or to the song X K square divided by it so we are in the saying the kingdom of politicians from Seoul to check this is not sufficiently strong trends we have to June arises in the quantity so we have to generalize it and reinforced were ready and it's the generalization is this 1 so so supposed the positive integer so we're really really is slightly easier as it is a sequence of Square here who is slightly modified this it will not
54:05
be all square but are all minus the budget to do something different which is always at the partition function Piaf N expansion so is the son from all across city to infinity expand their minds to it and then I write because and so you finish 1 minus base more it takes them the reason so it is not in alldigital would not end up in right so his formula you can use it to check that it satisfies its infinite that no question that system and legs a proof of that because you realize you are exhausted so we make a drawing of to explain the reasons which recorded Duffy Britain ,comma we found it in the book of contained so I take 19 29 and they decompose denied light a procedure and analysts for all principal players streamed through student protests 1 and they may forum drawing corresponding to 1 2 3 4 5 seeks Stephen 8 a play off got the bank said a pleasure
55:57
the quarter but it did so what is it so season usually the description of the Protestants of the legal may be in the 19th century nothing the so what is fearing dungarees always defect was what so you dig the largest Rotundo you can you can put inside is truly we their size sighs aah aah 1 so you it a draw uses following its fall 3 said tsunamistricken go is here 3 OK you can look at this is it there is 1 big difference between the not nearly enough final putting so when you I what uplands is that you are going to break to each partition time you when you read Associates 1st of food Adelphia redundant and that he should hear what happened here you out free OK so you notice that this at a number of years air there my mistake .period and then suppose that had any suppose that you knew of any integer and .period energy em What is it you could use this season drawing hit you on the partition of this number and we set some amount less than this number is 3 seats so you is you who you were anodized is growing lights out this year as a practitioner of the number and with someone less than a year it is here is less than 3 even and so you have a vibration and this year it is very sad in formula which is a key of the flags of the G R the solution in the article and the university so the university for me that is use restraint because you you are so I'd claims that it is is this system I'd called Base 1 so is the 1st bothersome 1 1 and this 1 based a minus as at most 1 solution the positive good and is a key to what is important is that ended it's OK please it's something like his best players could be all of these jails square would to squarely to concede that this is in the end what company that is a symbol what we call a soso members 2nd part of which is and at 1 PM and 4 goes to infinity but not too quickly and here we will use so it's the action of a career Jensen formula to create we to comprise the functions and is our system of that type which has a game in which have not a unique solution for instance so suppose here I replaced symposium I replace here but some seek a solution to the city you always see so he may have more than 1 solution many of the mall designed the unique positive solution so you so I shall take CJ equality is assigned the get when it signed off by the people would change ended then of course you rejected the excuse me so you will find the field to find solutions use of Taylor expansion unused the expansion of the functions Of signed by doubling the size of the area Kelly at about a K and you will find another solution he should consider Zedillo expansion of signed the bill when it signed off by brothers the EU's role the userdefined functions using Taylor expansion of these balls the function you will see that this is the case With the courses adjust the does value and the data crayfish and not the same and position and you see this function goes to infinity he got absolutely not the same site is something that takes control key it publicly so integral to infinity much much quicker services so it was a pleasure with finesse to make aginst informing workers that the broadens the ticket so some view we have the wisdom if you don't have to worry about the size of the no you do not know that the rate of convergence of quality With the stereo from unwanted can we are and the agencies that it doesn't theory of fomenting private Renault probabilities it's working if you have at rate of convergence if I say mall and that's a quality but the number is equal to pompom .period Brussels everywhere at having a 2nd letter to the thank them