3/3 Bounded gaps between primes
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Title 
3/3 Bounded gaps between primes

Title of Series  
Number of Parts 
36

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License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

Content Metadata
Subject Area 
00:00
Point (geometry)
Group action
State of matter
Euler angles
Multiplication sign
Water vapor
Parameter (computer programming)
Weight
Order of magnitude
Number
Glatte Funktion
Latent heat
Goodness of fit
Lecture/Conference
Energy level
Absolute value
Newton's law of universal gravitation
Injektivität
Area
Chemical equation
Projective plane
Sampling (statistics)
Algebraic structure
Division (mathematics)
Cartesian coordinate system
Limit (category theory)
Functional (mathematics)
Sequence
Measurement
Convolution
Category of being
Computer animation
Normal (geometry)
Object (grammar)
Figurate number
Discrepancy theory
Arithmetic progression
Resultant
Local ring
Chinese remainder theorem
09:09
Group action
Musical ensemble
Multiplication sign
Range (statistics)
1 (number)
Sheaf (mathematics)
Insertion loss
Parameter (computer programming)
Mereology
Grothendieck topology
Glatte Funktion
Maxima and minima
Plane (geometry)
Positional notation
Manysorted logic
Stress (mechanics)
Square number
Moving average
Physical system
Stability theory
Area
Product (category theory)
Constraint (mathematics)
Curve
Oscillation
Functional (mathematics)
Substitute good
Proof theory
Arithmetic mean
Right angle
Arithmetic progression
Resultant
Point (geometry)
Free group
Observational study
Divisor
Canonical ensemble
Event horizon
Number
Power (physics)
Prime ideal
Frequency
Thermodynamisches System
Lecture/Conference
Helmholtz decomposition
Term (mathematics)
Queue (abstract data type)
Lie group
Subtraction
Field (mathematics)
Units of measurement
Condition number
Addition
Scaling (geometry)
Chemical equation
Projective plane
Mathematical model
Convolution
Estimator
Summation
Voting
Ultimatum game
Combinatory logic
Object (grammar)
Game theory
25:57
Group action
Multiplication sign
Insertion loss
Complete metric space
Weight
Grothendieck topology
Maxima and minima
Mathematics
Roundness (object)
Manysorted logic
Velocity
Forest
Square number
Position operator
Amenable group
Social class
Process (computing)
Constraint (mathematics)
Block (periodic table)
Moment (mathematics)
Basis (linear algebra)
Thermal expansion
Variable (mathematics)
Functional (mathematics)
Flow separation
Arithmetic mean
Order (biology)
Right angle
Cycle (graph theory)
Resultant
Point (geometry)
Observational study
Translation (relic)
Dichotomy
Prime ideal
Root
Lecture/Conference
Term (mathematics)
Natural number
Liquid
Subtraction
Rational function
Condition number
Alpha (investment)
Addition
Focus (optics)
Shift operator
Scaling (geometry)
Chemical equation
Expression
Division (mathematics)
Incidence algebra
Ring (mathematics)
Mathematical model
Local Group
Estimator
Summation
Combinatory logic
Game theory
Coefficient
Local ring
Gradient descent
42:26
Group action
State of matter
Euler angles
Multiplication sign
Range (statistics)
Sheaf (mathematics)
Kontraktion <Mathematik>
Parameter (computer programming)
Grothendieck topology
Manysorted logic
Process (computing)
Product (category theory)
Thermal expansion
Perturbation theory
Order (biology)
Right angle
Point (geometry)
Divisor
Rule of inference
Event horizon
Goodness of fit
Lecture/Conference
Term (mathematics)
Queue (abstract data type)
Associative property
Subtraction
Rational function
Units of measurement
Condition number
Statistical hypothesis testing
Series (mathematics)
Shift operator
Forcing (mathematics)
Physical law
Expression
Exponentiation
Content (media)
Division (mathematics)
Incidence algebra
Mathematical model
Cartesian coordinate system
Convolution
Estimator
Field extension
Summation
Voting
Game theory
Coefficient
Discrepancy theory
55:56
Product (category theory)
Vapor barrier
Process (computing)
Conservation of energy
Variety (linguistics)
Multiplication sign
Expression
Exponentiation
Translation (relic)
Insertion loss
Mereology
Weight
Rule of inference
Functional (mathematics)
Local Group
Theory
Estimator
Goodness of fit
Lecture/Conference
Natural number
Average
Figurate number
Fiber bundle
Coefficient
1:01:53
Mathematics
Lecture/Conference
Multiplication sign
Mereology
Local Group
00:03
so if a 10 we can you can do
00:15
and today I'm going to talk about how we proved that could history dismissal crimes but so do you were aware that the main conjecture in there area with an injection which is significantly greater than the 1 fixed you take the woman would function manipulations it's a region to some codecs pleasures discrepancy was discrepancies in the modulus and some will warm water up to his level figure the conjecture is that you can round which is Exelon acts like any Powell called that you please get and so is the treehouse and projection if we have this conjecture that we could do a lot of things especially we had a certain technical generalization of like being prepared dependents or 6 . 6 for instance of the all currently we we we have a bomb even about euro which cruises this step up to onehalf of and we don't actually have any progress even after all of what young on the have of for anything bigger than what effect the unique was 1 of just little and that people want to what we do have is on the following with our young that there exists some small parameters which he calls about the indirectly discourse involved in his paper he takes and that please spent equal in less than but 1 what 1 thousand or so and this small consists of Bobby and daughter such that you have a weaker version of this and so have the superstore side and accused this can get a little bit about onehalf to no small love onehalf we should and like that of smooth or instead of travel which means that he will present the divide is due this affects the Delta so you only takes with numbers swiftly you do the same thing new measure distributors not to I had no idea this of all 8 of the Copa prompted to everything associated with the physical crimes less clinics don't so if you leave you wishing to gain input "quotation mark images of something that doesn't doesn't matter and so you can take that and this discrepancy and this guy in of a of OK so this is this is what I'm good good at and then later the budget improved to hold correctly especially Quebec worked and has been that is the thing that they have yeah I headed down to move the idea of receded because above all a year history was the specific label the proposed that if you have against and with a bit of luck with the military Jenkins again and so on yeah so I'm evil you know fully application to ban against version crimes specific and unique comes from a southern Corsican with causes the company Chinese remainder theorem but that he can obstructed at reconstruct away that hypothesis cyclical but I think it is the Clean Water State the good news is this is not the conductor using a special properties of a inmates would take the obdurate facility the key point is canny unbounded 8 and for fixed a a previous results like this and buy fruit 3 and on very fit and Vaneaton not exactly like this and the didn't quite have the absolute value which is very useful for us and instead eating the assigned some weighted by someone some from Nice weight but I guess that is essential to them not to is near because Kurdish was a few more times here today but but as the empathy the pudding is that the state can grow effects can be buried there so we don't assume founded and yet we have about uniform and vessel the novelty in in in 1 in ensure that cancel previous techniques which relied mostly on windfall methods they needed if a fixed OK so that's the objective and so on the 1st step is to see this is what I said before the 1st using the founder and To get and what this does is it this the would function into a budget pieces have received only long pieces together and each piece and I was 1 of 3 types as it turns out it was you couldn't do so this time 0 . com this would jungle type 1 and type 2 there's a distinction which he is not showing up yet but in fact would mention at all it's also important that the attitude festival was the same in time talking so sometimes you some pieces of the bone 1 4 men would function look like convolutions of 2 weights and this has supported on the losing end to end and you to and and actually what is also important that the and in fact is you don't actually make it into 2 regular but narrower than those that can actually know what has reported and market so this this is just another fear of which we has something that has some some balance ,comma local inside the office on mobile devices on the market so it's it's it's basically bounded up to Bulgaria's and this guy Smith himself so there's something should be done coming from some smooth functions supported only assistance with function with that reviews abounded and so it is with pungent managed to end the sample images the use of discretized smooth functions as a nice from sequence that is your friend this is roughly the same size bounded it they can also play in a way that that is start was not in the extraction and we we don't try to exploit any structure says the disqualification market so far but had wanted to use a like this win is a win him in his concluding text in his previous tragedy we give you the limited days because threefifths look at that time would have 2 sons similar now moving to functions to against what attended to him attitude it's weapons and so we're looking to the and actually on side don't confuse myself local you know what the other Justice Scalia and I know I was the 1st time an I'm OK so I want 1 of these guys in our office they obeyed and magnitude balance but no smoothness and and again involved quite a lot next and the word of normalizes attends a small guidance the big guy so this is so small as is white and the between it's the threefifths of institutions OK so you to rough functions that distills close to onehalf within life onetenth of its onehalf of Soviet
09:11
Yukon 2 Afghans guys together easily influenced by the sons and that is distinctly have when it had to be set it in terms of a technical reasons that when an enema bulb very X and the argument for type 1 doesn't work in a few days later tonight but that's the way stability minor point and then finally the type 3 sons look like equitable convolution so this guy's rough and supported some scale and to em 1 is that the smooth supporter said that skills in 1 and 2 and 3 but then again that the mobile objects but was it won't contribute to them and these are the very big guesses on with right so and these are the result woman mediumsize action once before in 2003 ,comma connected to next 1 but their products a big so I can run into contact with at least in the fetus and so did the the precise and and also important but like this is a mediumsized and a good example to keep in mind using just drop this 1 should think of just such a combination of all these guys supporters curious once that's as a good moral case and this is also a component of give the signal the demise of the newly commissioned Haitians attempt to adopt we have give it you ,comma wonders of 3 times economic sense you could healthy and then you get pieces that look like this and yet so that have the sends a similar to the devise a function which has been studied quite intensively and so in effect on them before Young's work they will were existing results on the dispersion of 3 things 1st by Friedman undervalued and then some he's Brown and then from 3 costume shop In and around infected with her for the young had a different side of a wave of of dealing with that somebody actually turns of that the recent work of vehemently Commerce Commission is quite good results for this for this result to the extent like that that we have some quite well in this is no longer the worst someplace that would have to things that actually least OK so this is what the ground decomposition gives you I should remarked that if Europe large the range that had 1 nite to some something instead of between have been between to good percent previous you enlarge its cost to life in a 1 could twothirds produces 164 1 rather than 110 of the need only have weakened at times anymore and then action to seize more than to get the the decomposition the need you don't need this more difficult a lot of people he said and because of that there there's something games you can play you know if you can if you don't like 1 easily if you 1 4 proof of the 2nd Isringhausen rigor was study of these have opted out of parameters you could just proved that when an actress and and not the happiest and this has been a Holy year reality these are the numbers of we're we optimize them although the judges site different numbers here that there was a fear of yeah it evolved over time because of a sense that it has this is different from as technology improving campuses will be improved the Barnsley but this won't have 1 have to another business and we have that did we really move the the the Opel officials between the 2 likable this is this is what it is ,comma In Susan onto nowadays this the have this was a proven ultimately the main tool that comes in here all part of the it's histories working officers I want to write you a final fields which grew allows you to to control high which Aleksandra sums and can we don't know of any waited and think we have any way to do that sums without using them the input of D'Alema now and the time when the countries and you can use to meet you want to get the best balance but actually under the terms of the deal you don't have to the economic environment .period have seized 2 sons using the more and more control of the easier removal system curves and you don't which gives you control 1 additional sums for stress that enough for that happened to you can stick in the more advanced lean estimates and 2 2 with the expense of little more want and which means that you when you serve you must around the thresholds and those that have the best estimates using war that ended badly you can if you want proof some equidistant estimates without ever having to to to use to these but the thing exploiting it was the best of the best results in new OK so as highly approved so so now we have to do is eat approved of equities necessary to the story so pieces again so the other US a very easy and began at times you'll see that thing is that is that this this and this and the discovery has so smooth a wider to widen a threefifths vote is widens Cuba's secure members can be physically X 1 half of the little bit was basically X 1 half of that that you will need only during the period covered the just goahead screwed up engages the spirits summoned a double sound and basically sigh supported on 1 but it seems that some skills reversing cues Leskanic 1 office a little bit so that found anything restricted to some of my progression want you which is which is much smaller in scale than the scale which smooth function is living organisms adjustable was aided by almost any technique will work here today said something in you can you use opossums summation all quoted Schroeder whatever would also look for something like in both of his work and you get great advances and so you can ask you know and this is this is just a it so other times you use something of a difficult ones and the government have won some and finished board psychosis and start over the actual details and bookkeeping a really messy and some are not likely show you actual computations with any precision of his history the flavor of the things you have to go wide while wildfires importance of this it's just raised it was important that you have square for instance or the social shortly but so too has said the Type 1 sums years the type of thing that you have a summary we care about is the sum of all hues just a little bit bigger onehalf of course 1 of plus extra free risk article smooth collectively house moved so an even better value than usual as the summit was aimed at you all the pollution in off to rough pieces unless he was sick of compromise that this guy is rough but to say that the scarcity U.S. industry stocks to the extreme case of hope but of all the time .period sums Aminus commune which I'm not not not explicitly and you control the source of light so 100 so the 1st thing is that we wanted and we subsidize annoying so I would lead them to do we need to pay a price which is a public official just this just inside was a minus 1 depending on size and we don't
18:12
control much about about the signed agreements the to customers 1 some eventually it would be willing we do it at some point as you can about quality like Schwartz better but we don't do that just yet because we need to do some other investors get so we unpack that and then again in this of course you can use them to double some canonical status ultimatum them come on OK and then this this this and this cute also without using the adjustment itself and smooth easily although the prime means at that your you're the part of many many people factors and I'm so sends 1 you accuse Prime then there's no factor to remind you that you smooth the what's most effective June 1 and you say it puts something exit smooth so far fact that no practice because if they don't then it's easy to greedy I wouldn't bet on it if you take any intermediate number say from off and on since exclude there's so many factors of cuter that than must get these 1 1 factor of Q between enabled our not elected or for any of you because it is refined the factors of Q 1 time for the past you 1 time and and because it has always sort of assistance which he must fall into this it's been getting intermediaries and the speaker cell so so basically is the 5th and forgetting the sector don't have basically you can factorize Q where you want to make use of this this this is both actual and and project so you can speak to this the new conscripts huge into 2 factors which are being square feet musical price and you can basically placed in review what I ended so given the precedent that we wait 1 2nd so it turns out that exercise onehalf plus a little bit and you back into pieces commitment to Chris Myers little bit services can be content with too little bits to to cancel out here so we can affect any queue duties to pieces are actually on the case because the notation of this book it was this was called Q Prime Minister and Justice is better can allow cannot confuse myself if I don't follow my motivation so because the Eugen what you listen to models of lies about excellent hands and so much as a substitute to fears about minds a little bit and the plane was minus is that it's just a little bit solicited the skill asking his entire needed loading scale and and that this turns out to be the important OK so because that you and you factor as this market and so we do that 2 2 To see useless some activity should do that before you take activities to decide that the kid gets base units splitup assignment and then an and instant into aside so if you do all that well the substance of this sighs it's certainly looks kind of scary Acasuso many some summations over that actually having losses emissions as your friend and because he had many opportunities to 2 2 2 2 2 2 council and shortens OK and those of others from which looking right at so we now have to my July 1 and we have to move on from this year's candidate constraints that you are at it and so do you have you had this subject so I can so the whole game is to rearrange and in such a way that you can actually do something useful like cushy Schwartz was moving so the story of reform agenda that I imagine he must pursue Cholla area for many years section is not obvious always have power to experience these things in such a way that you would like to get somewhere ,comma but I work in a war the previous works of community events that prompted some things on so the wearer gonna sure this is the 1st of all there are some in the UK for damaging an all that actually can even get happiness emissions without Virginia sometimes significantly ourselves so this fiscal year the US mission and then I want to rearrange the some of the Amazon from but there is acute and as a constraint the clothing of them was not estates and had them and then we have lost the surveillance of political officials and the military so you some of now I know why why is this the right way to rearrange things holidays of size which affects minds but if we can count how many children this summer so this sounds and will end use here is about to execute this and exclude onetime costs of it but we're not something of them did and we have to we have to portion of because we Congress condition if we only seminal those you for which an end user could aim what you up and you think you can endure random the that the chance that this Congress conditions has either side 1 you so reasonable terms you should be this time this divided you Kuala which is due to you on the size and on his side and so on because you chose not to solicit try this and other you have a nontrivial some here so there's this is a tiny bit of gained by this a tiny bit of oscillation going on this is that this is a slap interviews some of that in the event of a tiny bit of summation and they can set up what what is called Linux of dispersion of that which is basically at this particularly by country Schwartz can because of because no 1 another even Ontario some here he can't he can apply Schwartz when he invited Faustino you look trivial some union in introducing me to say this to you the flycatcher
25:57
Schwartz Society of Uganda's only understand and then the year and so the whole place they get you give a 1 Rough Guides Alpha was always incident standing between it eliminated actually done amenities and strictly speaking you don't completely eliminated Europe's income that and we are just out of high that it would this summer should be the focus of a smooth start again so you know you'd get because listing understand and you just can't understand this sort of so you can expand as a double some and ,comma Cunanan mean and crime ,comma Q security to all dangerous conditions and you have some sometime between a million and a said last time when you have a square that the 2 terms is a diagonal term when these guys the same and offtarget all of said and hold a hole where having a slight nontrivial and some here is is that I don't I was a little bit smaller than about and we went into the game and I have just a few blocks from your back so that I don't turn negligible because we have an interest in it and so that we can we can dispose the songs and we can do to insignificant we can do with them of the case in this case are not not the same so if you do that what you end up looking at him module terms the and OK so OK so how we should get History of an M which is that the size and primary residence size connected to a certain size and then in his with a more Q 1 and and crime and people too the more to opt and any custom coefficients focus was on the brink of a candidate and Prime CCube asking looks kind of bad here because like we don't control and anything about about the corporations but it is a marvelous there still is some consolation in the summertime and that this is the sort of Midas you get this circular translation goes on and on about not so apparent that look at so that so little time perturbed that I was 5 by 2 stories we can captains and little bit of grant of any imminent famine the same 1 off between the Indian prime minister what you know you can ensure that the reciprocal crap and forays on every day of the soso income actually have to differ by mobile are but ended in various sites in the amid 250 and also I think it was minus 1 so that the best the engines and reversal my and finding a vigorous emphasis will shift Thailand and analysts more and so does the simplicity of I'm just ignore the shift in sentiment as the good and Prendergast and that is a very practical so I to just simplicity I will just look at the dichotomy that that's that's ready the typical political terms so that a little bit OK so I would have taken that decorative and these these guys at different because there was still no because we assume that in and 1 in and prompt you to a different security to a different the action with a bit of work you can you can insure that only a chunky to different electrical products and Milanese signal that this is a technical step that corporate you have to go into detail you go back to that position Mitchell accused of small school Pentecostal Wallace will protect isn't all that and then I'm OK so how can you have a trash like this but I still haven't we rearrange again so you know this expression is kind of messy but the amount is 1 of the ways in which things are really good which is can because nothing has never been so and so we can rearrange itself can market so this assistance some here the 1st as a constraint and has to be of the order of day over and what Q so this is going to constrain the answer should be racist yet a combined about Chinese Minister of Indian descent like that and yet he always believed more complicated but that was and then what now how big it is this modulus here so hours of size expertise minus and these guys of Southeast onetenth just a little bit so put together and as much as 6 of the 5th plus a little bit he would and so this is pretty good at the time spacing here is only just a little bit wider than the and then there was the officer so it's it's it's this is almost unique that is easy to compute that this was with smaller this that process Nacional whatever would would would work really well and we get a very good balance on the sound basis a little bigger answers in soap and which which rejected the killer about what this visit this summer's Iraqi this this this and this and 1 of the John McKay to much and so you know In this in in sickly group here you your something really big of a really great is also full size and is so dense that formulate methods begin to become effective so what you should do is you actually they should because of this ,comma function that the cut off to to the scale and break up into 4 remote and and indices separately to cast the values you lose a based on the ratio of all of us in this but that's something that this a portable lost so what you should do they should you should do for expansion will cost estimation was saying things and so if you break this up by afford modes but what is happening is that he starts doing it from the In case of the Georgian this thing then you have a standard looking at some of these guys have changed so Slovenia's scary coefficients here attended out but this is it it because basically an inexpensive faced some magnets so as you were working basically Group Q 1 Q Q and the human tutor of anything this tube is a constant character the nature some forum ordered it isn't well major of size about the racial things the changes because the small man accused of age anyone we really efficient
34:12
natural on export Abidjan alleged the that's not to talk to age so this this and this image its major nonzero because there was this poses some maintain that we kept hiding from you and that she had at this point that the major haven't exactly equal to 0 mode and Catholic construction this and 1 0 8 years and so you don't know you expression begins with more investment yourself we could expect to somebody you also have you what you divide and so on and constrained equal this this this Congress class so he spent over 4 expansion you you you get something like this 1 no the new model by the for a coefficient of Scott function of aging some of which but but only a REM cycle yeah yeah yeah this is the end of this year division in in this ring this is this is not done yet again the extreme also use you used to this location elections but in an hour OK so you had this summer so yeah these rough corporations such as so you have to deal with them and so on right so you you have to cut Schwartz a few times to to to get rid of the last few things don't sound so what you do 1st is that you plug in summer this said Fazio OK so how you pull this out you because she Schwartz to give it a ticket to get rid of of all the debate over time and you do that you stop staring at the was looking to buy complicated moment but you have to this is my faith and and that there was a candidate communities of an optimistic result this committee context so maybe it is that things up enough this is no time will stagger beginning more unknown coefficients no less than you doing this so did you catch Schwartz and if you get an expression of books like this but pockets of more terms but finally there is the ones variable for which and which is not getting any optical precision that languages and OK so you rearrange that finally it was almost a billion US summations floating around the catfish was eventually wins with small OK was given some consideration of listings Regeneron I'm a little you know you because you're using things elsewhere in when we lost forest expansions of what it is not always clear that the beginnings of weight loss so this is a delicate interplay I mean that as you when velocities and delivery small about towns that this will OK so when you rearrange basically it all comes down to understanding and some expression this so the 2 can understand the severity of these guys and get a good round and he can be the driven by a by a lot of the new shape now know this combined by by some and 2 next to the conservative look at is so what's sold sold sold in this incompetence and pricing trends remain here and in the worst cases because local pride and listen the terms of the new can combine all these terms into into a single question like this 1 so this becomes an incomplete list of studies from 1 of his equipment some of it in general is not quite as simple those as well as the big 0 earlier men and actually it's actually this In general to assign more complicated something that was so just so it's so here so the continue staring at is a song like this ended the sizes modulus OK so you do the math forgiveness of some sort was instituted minds a little bit these guys are all the more intense plus a little bit and then so did this adds up to about Of course it's possible I would have to write because of the wiry when can we get like 1 onehundredth of 1 here followed orders and so forth again so the year she had 77 said that the short some the the little songs about the square root of all of .period but now we would be in addition to there actually slightly smarter ways to cut Schwartz ,comma in such a way that when we do all this you actually have a sum which is a little bigger than Scriba here the way was listless great but I'm biggest money maker of investors because grip and wheezing good because yes all right so I felt that our problem you face so this is a special case of a more general question of of any control X an incomplete expressions sums of some rational functioning as well as the modulus which is on nice speaks the modulus I'm always some some some ranging in ages ago within think you so it's very incomplete unchecked so on the meaning of this disease is your friend and contains some death is not completely trivial and the removal was gives you grounds of spirit you Mexico's Askew not very practical such a case the possible loss of life that essentially screwed cancelation of liquidity things have not yet incomplete some of them understand the ways to convening of the summit to complete some the played about method which is basically just a take cut off any further expanded and sold the conferees extends kind combinations of the complete sounds and a bunch of these guys that that conditions will be inefficiently this inefficiency which is based on the ratio between the scale the scale and so you could lead to the completion and you you buy about what you get is an exciting and queuing constitutional subscribe but which unfortunately ourselves list this type of this have a play that I got uptight about this the nontrivial only for you and took only him of opportunity that is that Texas that's standard screwed to limit to call even got it so you wanted your shorter sounds so in
42:27
general the hard and his associates with you so you can vote the singlesession characters if you want to shrink below which you and you can do it give never Burgess which is which is not to particularly bloody it's it's it's it's really tricky for characters and you don't apply for Christmas some still vote again you can in principle that that it is it is quite tricky and so on but strangely enough Steve for the year but the conventional wisdom is is that there is a prime sums us including CompuServe sums because we wished because we haven't we office upon something and we will try to to do presence 1st but actually there are some cases in which having every complicit will smooth modulus is very hopeful and silhouette rather than use the biggest members more efficient to use and different methods the agreement Ringoes most Peter Brown a little more physically cut relationship between that and it's a song that you medical witness so that it's not a very good wages to control this mediumlength exponential sums it up again a short extension sound like this and then we can do something ingenuity averaging so and we haven't had suspended but just what the imaging method which is that the images and some 10 additional having promised to release of these are much smaller in size and take a summer shifted around by some 18 in some premises that gravity on and then Deakin interchange with business saying the conditions something catchy Schwartz and at the end of the day what you end up with In this movie was originally going on here and so that you can catch Schwartz from 1 of these these sums into it was emotional this times another Kurdish was to that be used in the rule of what is now the kidnappers Apollo Milton is ball in trick and this was very well because only if it the parliament agreed differences of a law degree and so that this is a good estimates in our case it is a rational function and intervention rational function you things get worse the agreed it gets bigger and smaller so ordinarily this isn't this is in the wind except that you get to pick will you ship and so the and the idea of a Cuban call that method that is that if accuse very smooth between factored into into pieces and I need an assistant for you can pick where the fact hot and the country can choose your shifts to models 1 of 1 of these 2 terms to said that the state a model more posts of heart so you only should promote was and the reason you do that is then this this and this difference because divisible by off and so rather than being over this big modulus this reduces to yield of a smaller modulus of some other polynomial which depends on various parameters but I like it but you can replace 1 expression sung by another unexpected summed over small modulus on but where the the land gets of the summer changed and so if you get a short so he can touch wasn't into about them into a relatively larger some political modulus than we thought before and in particular you can get past this McCue special deployed in the Gulf so if you apply this method ,comma turns out Sosa deployed another method of getting onto resourceful when the event is bigger than cute and you apply the chemical method once and they can't get the enlarge the range which an interview about and given orders on all voters from the shortest to 1 that I try to fight again as he did this and you can get your 4th and 5th in support of law in practice on because Schwartz and 305 is method the game you get past and actually and it turns out that there is no more when actually we will we went only medical once the section we will experimented with the moral corporate victim was exponents than than than we thought it would have been you can actually improve on electric utilities you can with range which means you can get exponents in fact there is also is the sector could accuse man and complicate the process and the psychology that it will be process which is which is cultural and and any dividing both in get will oldschool appears in letters but I'm a bit of that didn't for this application you think it will deploy spiritually and so so so this this this will be enough at least if you prefer your original coffin made a pretty close to 2 to skirt affects you wanna get this large a range of 10 points of the next 1 next because then have give users twice 3 from but but basically using this this method you can get 1 2 rebounds before the sun which will eventually feed back to give you enough cancelation to contribute have 1 estimate as police have template some OK so the last few minutes of talk what have sons which actually I'm not lacking was there if you are willing to take the wings on the black box of the actually simpler than this this a lot this was cut Schwarzenegger still lots of Kurdish forces and for effective if that is not quite intricate OK so have to legislate at the moral case Prius take a triple convolutions with these kinds of psychological comfort care and this year this might something and then at the value market which poses same sticking incidents and coefficient a console we wish to understand an expression of the discrepancy like this would have 3 sons OK so you expand the Celts and accuse the size onehalf as little OK so expands out there know what you get is some and 1 and 2 and 3 AllStars about onethird Q I just figure it out and then it's something this year because of content and this Congress must petition England into 2 OK so the the
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modulus is much bigger than that each 1 of these ranges of ranges are living in some sort of reason small evolves you accusing you like but so what what he can do and this is expensive and if you can try expand these guys into you do for expansion about using that this is not a particularly efficient because the the with reasonable 6 1 3rd is quite a lot narrower than the total modulus here so you know there's something here and so you you need a payback with with quite a bit later on and this is where you need deletion because means the the only guy continue lossless cancelation but it but it is expands out in the 1st series start looking at expression this so that this summer each 1 and 2 at 3 just size about the racial these guys licensing of home and the news from 2 and 1 and 2 and 3 in a cyclical thing come with conditions and if someone might explain some look at this so as to accomplish it so this is what you get off for expansion of its it's inefficient to be bombers trivially you would not recover drove around here there's a copy of the of the pricing of the inefficiency here is that so is the ratio between them the slip with reasonable and what was here 3 times to aside 1 6 3 times so you you you lose about 1 half of it it because for expansion became so if you've lost the whole spirit and possible but more than that but OK we keep going you know expression on these pages you can change Robles to another units ring you just died begins by the ages Aliyev to covet his image has a contract to technical you really went beyond cable as a signal that but if they discovered accuse you can skills out and they just come in here get on and this has has a nice feature think this was 4 6 with a heat from is that most of those lost that in the in the women's the following are some of the more understandable and it will OK I didn't realize that I intend right at it so it is not yet sold it to the next week is powered by an old trick which is that this whole some depends on only 20 should receive only for the product so you can't you get this durable and seeking concatenate 3 small sums into 1 big seller argued to keep it small prices to pay this essentially buys a collection that it's a truncated was appointed but because of some of the city here but but it's still win but so you end up with an expression that this now this is as secure expression somewhat Fairfield County has a name that has got a hyper Christman so that the 3rd have the Christmas song also a pH attitude which is normalized is actually but it is expected to have so that the behind the customers some kind of acute Kansas some of than 1 and 2 in the he ECU 12 look this expression he says special summed over about square terms here because this this QQ terms you want ,comma salacious queues good terms but I'm so if you expect screw cancelation then they should shouldn't give size some of which we about sites at the services have expression that you know you have and so this is what think that that that don't lean into the deal with unsold views even a publicist for the the status of this year will you find such a secondround coopted include workers because of all the factors of you but added that this is as the bounded understood that you can get screw cancelations because it going to choose which of the 2 things correctly should exactly balance the right it's a little better because there was I was a 6 softening in acute onehalf again accused the test subject right again at you which is why onehalf half the occasion so this so much consolation injustice that years of this dispute the truth about this because you can't use of size as as excellent
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half so you've almost recovered all the loss they have before now this doesn't end the story because it didn't live a short and and so we will will almost back to where we started from but you just need is a little more cancelations than we have here and anything you would do something trivial and so basically for usages have established home so by the way the neutrality health war but it doesn't work because doing doesn't figure enough to cope ,comma lost and we just don't want to call for an adult just just wouldn't know the reach of all these methods I'm OK so and lasting more than an hour so you so you do a little again use for this crudeness of of curious that you to pieces was assigned excellent products 1 The songs from artists away the you can exploit you want the art historians say the here some and so if you do this you end up with some expressions which looks like this and so this is everything looking at many catchy Schwartz to give it this this this this annoying thing so the Schwartz OK yet the then you get a bunch of coefficient critical scary um but the Nunavut what is considered to have the case because the product of all these 2 cushion weights and then I think the 2nd donates her so we should do they should put this guy here with something in and you catch a short 1 more time or not actually leads to the yeah it's politically nature of their so what you have here this is like an incomplete exponential some but rather than having an exponential sounds a single piece of crystal and so on With the official function of this fight is more complicated expressions which is still bounded by this of generalized exponential faces and the cost of score trace weights on and it turns out that if you understand theory was enough you can you can you can get cancelation musicologists Well I guess maybe when you talk about this in week or 2 so it turns out that you can use to being 1 more time and get analysts and drew cancelation quite a quite good documentary conservation here and and the new process through you what you will get back quite a good detective the estimates its it's 0 so strongly that there is no longer the the dominant barrier to improving for the group Institute these estimates and because union we would Schwartz New Year's much summed up as as the other and also explain really strong translation screw a cancelation themselves elected by the government is to 6 games of of that it's not sold so strong but this publications and I will of course Marshall and to use but he together and you achieved in rule estimates on on the could estimates on the part of the history of the In the days that of you have a your yes we did for John instead .period that it some more efficient Schwartz so he she Justice Party it's important that we did his official such Schwartz and and then go do we used everything to get the best possible that so the whole thing to do on the floor and it has but I don't think that we will not move in and out right now that's when the biggest gains here but we will also spoil what vision and and engendered there's a lot premises averaging just threw away from which would also threw away here for simplicity but that Sable below the averages on street any doesn't the the modulus of 30 just affects whatever's in inside the parentheses is 76 .period abide by the selectors some methods what is difficult is this is tragic because solution when you very modulus and and with Greek unusual moving methods for a variety of reasons as soon as you know would do that if we don't do that this book the University of Minnesota Mining and and that
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this is a small part of the group laughter I want you can this it was the 1st of the year in which you In this case the changes in the world of this is the an man on a the new international we just want to be in the 1st half 1 of the most and that's the end of this year the 1st to you used to have we reached the In the riots this is a list of your bones which in this day and perhaps this is just a move the problem is that most of the day and the chance to move to the beach locations in and around the time of just the long ,comma so I will talk about this in the next room and French initial costs on this phenomenon and there is no immediate loser but most of all this the security of all this