Lattice Theta series
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Formal Metadata
Title 
Lattice Theta series

Title of Series  
Part Number 
2

Number of Parts 
6

Author 

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

Release Date 
2016

Language 
English

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00:00
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Series (mathematics)
Decision theory
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Bilinear form
Thetafunktion
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Subtraction
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02:32
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04:51
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08:28
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09:39
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14:22
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15:27
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19:53
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22:40
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24:49
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27:54
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29:15
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30:19
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31:27
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33:45
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35:58
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42:37
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44:56
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47:44
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50:33
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00:03
the top and in the end of the day is devoted to accept whatever to is show you and how all this kind of tumor lies the decision that can be very fruitful for analyzing the many communications systems and also designing the some once I as we learn as we used to so 1st a topic that we will consider is the use of these the the functions In fact that we use the terms the the series of the definitions depending on what we're doing with and OK I wish you 1 difference but we can consider these truer objects has been the same preference so 1st 1st of all of these can be used to analyze performance of communication systems encoding is shaping secrecy here that means for example and that means a physical security and then it can be very interesting another to design something coding scheme or shaping Mitzi shipping scheme as well it appeared and then 2nd secondbottom so we note that these 2 matches of this because we are working on this area as 1 way so it's something which is a little bit under secretly said right now but I will show you How could the functions can be also used I was away forms In order to transmit information over nonlinear channels as for example the fiber optic communication channels can be look at the start of the and is part sodium Melissa spot is more related to call right Tokyo let's start with a very simple function which is the Jacoby the function and then that we see how we can construct a on the family and the warm familial of the the functions appear interested we can OK
02:33
so what we have here this function feed them depending on 2 variables as and and OK it is exactly this series of and then I will do something sorry sometimes we use the style violence In this rule which is this 1 year and sometimes we use the Q horrible instead of the top of our board and the relationship between the 2 of these 2 barbers is given by this exponential by Powell and Cuiaba's jobs the development in for a series of these functions and many other viable is In the which in this case is a scholar but the more generally as we received can be effective OK and does so this is what can call the elected by 1 and the other 1 is the mother of our and in fact what we see that in many cases of the fetuses that would consider can constructive From just this functions so just 1 function so if we are able to move the maker of the table of this function which can be a sufficiently good then too we can derive all of them the definitions of the interest for us a signal plume of course not all of them all the definitions of interest as a function of this single through the function Jacoby so the 1st
04:12
conviction that we can make is that through some of Golestan measures which would be very important for example and physically EU security but also in some of the topics related to mainly I much use a communication using and so this is the same as some of Goshen measures so it's just the sum of Goshen which are in fact a shifted by a game which is an integer so does that this exponential and then we get this relation between the sum of Gulshan measures and the job could be in the form of soap here is an example into
04:52
dimensions because here it was in 1 dimension that means that if inks instead of being discussed is a vector twodimensional Victor we have this book and this is the sum of Goss emissions when the Seamus where you cost . 3 and then you can see would a straight course .period succeed tends to be something almost flat and of course if the violence uh increase is more than a year it then said to To listen to something uniform OK there
05:25
flexible 2 the feeder series of lies and let's start with something very simple for so what is it that is in general so let's consider what a and ndimensional letters would consider full rent that means that this is not Our sector that our but it said that the tying the whole space not just that leaving the injustice of stories became so it's just a discreet and it gives a group of and yeah and it can be defined in fact by basis which means that itself and the linearly independent vectors he wanted to Union and we can define from this basis to generate a matrix so just that the contents of gene would be that these victims rights and that this can be defined as a GE plants and the integer value vectors a onedimensional and OK so note this 2 the feeder series the Landau and we have many many times because of interest the time the characterize amenities and 1 of them is is fetuses so we will distinguish 1 by 1 of the 2 viable so the 1 viable fetus series of land is just in fact the generator functions of the European norms all of the bonds in the lies slander OK so as you see I will use of power here for the viable but to do so we have to see the same relations between Cairo and Q which is streak was exponentially bite OK and then the 2 Lionel feeder series so we added now not defective added that the detective stories the viable this year and we haven't this definition of the 2 viable fetuses of London which is just that so we just added this exponential to iPod that's been the product of between as easy and takes and is a inside the latest blunder OK so end of up fact now and what will follow will only consider the 1 viable fetuses at the end of when we consider the transmission of ammonia natural and we needed to the it's OK but we are reminded to OK so In fact we concede defeat a series of I mean the onedimensional goes 1 viable fetuses of the it is just that the Jacoby feed functions evaluated at points he was 0 OK and so it is also called the the the 3 of the top OK because I you you see
08:29
you have just a few takers equals 0 some disturbances this disappears and you just have OK cute bulky tumor gave the reasons he can so it is the
08:43
the series of so it is so the 1st ingredients that we need to construct a or B of the thesis is that we need afterward the 2nd ingredient is the feeder series of zebras onehalf so what is the best were not just add onehalf to or in the jury is and you will have and not let this matter shifted like this and what translated letters and then finds that the Japanese investors function we can express the 1 dimension of the decisions of the 1 half in this way and that it is called generally this fetuses is called at the 2 of right so we have not got to the the series The that will take a 3 on the uneven Wallstreet these the destinies are coming effect of deduced from the Jacoby the function right b so
09:40
this is for example this is a very well known example of a twodimensional lattice it is the aid to let this or exact below last so it is a name that examined because of the soul these are matters .period this is the basis of the lettuce so 1 0 4 and onehalf square of 3 0 with 2 which is the 2nd victim what we call the phone number of pirate up but we believe this kind of of of bodies right and then on the war in our region that is a set of points which are closer to some given that respond that to some other in this case it's a regular exactly OK so that this is a degenerate matrix all of these state to like it coming from this business because you can have of course infinitely many general possible during mattresses for letters and the the the series of 4 8 2 of 1 viable it can be computed by using only the power through the led the the freedom we should then White is a given by this expression and if we then operate that amenities to make a Q World Series development Of this suggestion is that we obtain this which means that you we have 1 point off norms III of 0 which is the 0 point of course then 6 bond off Norm 1 for example if it this is desirable and then I was sorry to hear awareness of this visit is this this other 6 points of someone as a truck 6 . of 3 6 for the etc. This is what it means but it have closets the treaty to and that OK no we have seen
11:36
some general this brought to the multidimensional carries the fall are going to the applications section so the 1st thing you see is how to compute the feeder series now for multidimensional areas because we need it just as a function of the Jacoby the functions OK OK so we basic rooms to do that in fact a 2nd legal yeah you just put I mean some elements of the on top of each other and then you can obtain what you so we need genes Japanese threeyear Russo to construct of 1st of all what this defeat a series of like this which is good that some of the worries OK so suppose that we know the fetuses of number 1 and the from that too and we have run out the direct some of under 1 arm that do what is the the the series of this 1 is simply the product of the 2 fetuses of thicknesses of underwhelmed by the series of London for solicitors I don't know what is where we need another rule it is the computation of the the destinies of the skull came soul if you multiply all points of random by some positive number I've thought what would be the the the series of I find it is simply that the fetuses from the evaluated the point of square block this simple can see just from the expression of views and finally something very important is the union of course sets suppose that it sought and that this is a group universal you can define the it's because it's come coming from groups from under the here so what is the union 2 processing 1 and 2 supposedly unless the union of 2 girls says he wants to and then simply the feeder series of of round that would be the fetuses of because that's the 1 blasted the destinies of the custody yet so let's see on 1 example what what we have they choose to examined we can in fact showed that it is the union of 2 calls that it is the effect of the events some of the and the squared off 3 times a year and you make the Union of there is a lot this because this is the but with is another call center which is Zepa last 1 there is some sort of 3 zebras onehalf OK so and that it would explain why we got you know all this stuff on level of
14:24
these decisions as just use
14:26
this 3 rooms we have here some skaters version we have direct sun and we have union of course and so we have ever seen we cannot a computer the citizens of the right signed for another example in a dimension it is a the gloss Atlantis This is the densest Madison dimension state and they president proven very very recently maybe some months ago that this latest was in fact that is could could be a good makes the best backing step in a dimension was not we knew that this was the best lattice backing but we will not be fit is the best package general yeah so Our OK so hot violated sorry to so you you understand now why it from this from halfway to we can obtain
15:28
this because it is we have here the products office to the test series corresponding to z and discredit the threetimes and then we have here the product of 2 fetuses corresponding on 2 zebras were half as part of threetimes zebras were off and then the union of 2 Cole said that means the sum of the 2 biggest focus everything is it's very simple he had admit more
15:56
complex itself is this construction Tokyo that is what we call on the binary construction aid so how does it work so in general by reconstruction aid is this kind of construction you take in fact the binary code of of France but when have coterie of friends and all right so end in fact you consider 1 post said I was being any but even valued the vector ndimensional vector right so it is and you added to all these bonds 1 given code words from coming from this binary code so it is only when zeros In the dilemmas of these good words are just a components of just 1 at once and zeros again and then you may do Hilliard Over the indict code all of all of the schools all right and you open your eyes land it if the city is thinner than it's very easy to do to prove that the mission of course that season so no let's on we know that choosy about the destinies of the the 3 of power why because the toughlove style is the the destinies of z and the twosies Justice version of the and so we know by the 2nd ruled the fetuses of tools disease the floor for cooking now dozy brass 1 because we have something like this press 1 coming from the 1 component here if you have 1 component of team which is a cost which across 1 then of course and the component of this this compromise this sector will be of the form to press 1 go to press 1 just 2 times zebras were and of course feeder series of this year as 1 of his 3 that you OK and I think that this series of 2 times the press 1 of its feet at so not by using the 1st through we know that because you have the best series feet of power the poll and and finally the Post said to the seed we have this he does serious which is the product of the dead too that to the power of their harming weight of seeing recalls the having weight of sees the number of once in the Cold War see and that's why in this case it would be so the debt to corresponds to the value of the property in question 1 and the necessary correspond to a value of that component was 0 so you have to get due to the ball W HFCs of harming their landing weight of the dime Cedar Street to the power of the number of 0 of of the good word which is an minus there having weight of soup cancel think the fetuses of from it's a uniform courses just add or distance and we will be so the hotly awaited the narrator other the coding OK so numbing make of of of binary code is just take so it said 2 viable reporting on your and the power that you can find on AIX will be the number of zeros of officals weren't the forgiving and the power of Why is the number of one's so I weight of a given called so just replace a by the industrial photo and 1 by the debt will focus on any obtained the the destinies of the lactose none of that we got by using this construction 8 you can so let's go back to welfare to all example
19:55
Yates this famous like
19:56
this OK so I recall you everything concerning the the Guthrie Theater OK and so this construction and so now Of course we get that by using the 3rd rule we get that the feeder series a friend is just the sum of all all concerts because yet prospects all of the the of the fetuses of this
20:25
said this is now it is something that we saw of the 4 most myopia
20:34
and that we know that To trapped in fact to construct 8 we needed this cold which is the extended the Hyman code of ranks 8th and dimension for and the ending and make of this code is this 1 which means that we have a code word of waves you all coming from 6 Apollo 8 so it is you go what was your coworkers who could ward off waited 8 which is all 1 work and finally 14 called also wait for cocaine and so just X and Y by the Guthrie of and the debt to fulltime respectively and the you get to the DM and these are the 1st terms and that series enumeration OK Justice developments and so and it's possible to generalize it will very launch family of medicines we can compute efficiently feeder series of freelancers in most interesting case OK In fact this lattice is not exactly 88 but it is skin version affiliated is quite of 2 years and so too have the fetuses of just divide by 2 or exponents here and you will fade so there's distinctions right so we have no but I mean by using this the Strickland said we have now a very nice way of computing the dismissed all of where when for example a lattices of being Beijing construction 8 by the reconstruction Of course there are many many other constructions and especially when the dimension of lettuce increases the memory of construction is no more in the house and In this case we need to consider some other kinds of construction but for which we have the same types of 2 so I will not go Of course it deeper into this direction because it's a very it's something which is it's very long to tutors to consider OK another way of
22:42
getting the focuses of the 8 is a factor but in fact what we have the exact we can use maybe we can use in fact that the the fury of modular forms and because the fetuses date is a very specific modular form 10 and that it is what we call an infected model form for go forward moderate progress said 2 of his use of the group of matrices of 2 by 2 matrices an integer values measures was determined course 1 and does so what we know is that the fetuses of Yates is just the Eisenstein's Series 4 and that which is given in this way worse 3 of em is the sum of the troops of the devices of an and so we have another way of computing the services of of of the year OK and we can check and that we have exactly the same coefficients here for Q 2 Q 4 succeeds it's a transfer yes OK so now let's
23:57
leave the constructions and that the computations of it as soon as it is good to them applications on the but so let's start by maybe the application which is which really needs a lot of these figures series Developpement it is to suffer the Guston white that China to support that you have analysts here which ones trust me to Walden through gas in Chamisal general resided Gulston noise and we support the the violence of the most users and 0 and there is an eavesdropper Eve which has which can each job the year the message I mean that sent the signal transmitted by analysts and through another dozen chairman was violence and 1 then we know that by ending
24:53
information theory says that the secrecy capacity that that means that the amount of information that that is contrast made reliably trusted to wall so that even in fact we have I mean no ideal for what would be the same information it is in fact that this this difference between the capacities of the channel between Alice and Bob the channel between I and Eve and of course this capacity secrecy harvesting been Nunzio if ends you always it's more than that and OK so if we have a advantage if lovers attending vintage compared to he and we know that by using ladders Co said according to that this capacity can be achieved up to some gaps to some smaller gap of 1 of its but I guess that's the small gap is just a technical gap the coming from the sum up along the which is not very tight OK so what is that this course
25:55
at college so let's say that we have support that we have a lot this land which is good for quality this program that this maybe can be shifted that means just translated by some given constantly and we have some alliance of land the which is on the EU which is good for secrecy we see what it means good for secrecy and that in fact as you can see it the being this mean this is being means that London B will be and that this for Boblett said and land like for you so this is the confusing letters from the among the views who would let his football In case as an example to dimension OK so we have so all these points you the lattice borrowings from London the From what it costs the finance and for the sunlight is in fact that we have in fact what we have is not exactly some itself from the beach but of 77 the shifted so you know you consider that the company shifted on number is shifted but 1 of the 2 the letter you shift and that to the summit is his to wide the lattices to fucking so that means that if we caution in London of the bite on that you will get a question group with 4 elements because this is the index "quotation mark land that the related to land at the and this fall and analysts suggest that these was that means stop socalled triangle His quite you can sell for example the squares all these big squares of warmer regions of land that use and that means that in this case you
27:55
know you can represent 1 given point of time that we in this way we say that it is 1 . 0 from there and in this case it is just that we have just to say that it is 1 of those squares right blasts 1 ball 2 of them the the overland that needs now we have to specify inside 1 of the squares if we have the start the 2nd training or the square look and that in fact that the idea this twoweekold the data inside that this cautioned OK so if we have the for example we have here to it's that we want to encode and so on if we want to sorry for example 0 0 always being encoded In truth start 0 1 into a soccer and 1 you into a triangle and 1 1 into a small square all right so and know
29:06
what to believe cancer even with song I mean what has been interested some knows but the problem is that
29:16
because of the the because the the data have been in inside this cautioned all right then the problem is that what even with if she's interested of course in calling the data where the National is not that safe
29:44
some folded Galston also will Wizard a PDF the sum of Golestan measures and in fact 1 happens is that in the variance often shunned ours is not to bigger then she can steer the normal which bond I mean she considered the cauldron someone money in the violence then becomes too big then what she wants you will be something almost flat and because it's almost flat it means that you will have no way to decode and in
30:19
fact that the information is encoded in this caution
30:23
4 and then you straighten this week for example suppose that even so that Alusuisse sense at this point as simple as that of all on these boats yes which are surrounded by pink areas correspond to the same Cole said right he needs that all of these points correspond to the same information secrecy secret information all right
30:52
so no what we want to do is that they can't even sees a viable distributed as a sum of Golestan measures and that we want that even I mean that the information in the cage that comes is Astros 2 0 as possible then it means that even has to see something I mean this kind of distribution that has to be as close as possible to the uniform OK so it means that we have to define a distance between this distribution and the uniform distribution yes and it is
31:29
what has been done and so in the end we have the sum of Golestan measures that he saw it is agreed that this family and singing and the final 1 we the threat this factor of which is in fact the sought the infinite distance between this Of from basing mile fixed and something uniform on the value of regional from there so it means that the it as a PDF equals to 1 of the volume of friendly so the volume is just that the determinant of sorry the absolute value of the determinant of degenerate matrix so which is also the volume of the following had region Of the matters and many can be proved that this the infinite distance is just going to just computing in this way so it is related to the feeder series offer alumni all right so it is very important now to mold of the of from Monday because if you want to have the information the cage are coming to evil witches the Astros as possible to 0 then we need to minimize this a series of more lattices we need to minimize it can and so now we
32:52
can show that the mutual information between the message transmitted by Eve as realities and what even CDs and the upper bound by this value which is on file an increasing function of X series Monday which is thought to this effect this factor yes and so we have seen we have shown that if this volume denies ratios so this is larger than 1 then there exists a fan of lactose is the main index soared and that's by the dimension and such that the sexy goes to 0 1 and goes to infinity and this that way this was the 1st step to prove that we had the a family of letters that could achieve at the secrecy capacity and in the wake in what
33:46
it is just something it is very important to other the full really series of directors and not on you some terms why I look at this latest Gunnarson institutes 72 dimension that is discovered recently this status this in fact is very special it's the densest window in the mentioned 72 and that this is the it's that the customers is that is the 1st time and so you know you see the number of points of Ukrainian nominates a few can wait 8 is very big and then of course this these numbers that includes a lot when the data when the weights so we want to consider a bigger and bigger so this is the fitness factor when you compute the war the decision OK and of course when the SNR ordered jurors this amount goes I mean it becomes a smaller than the distracted suckers goes to 0 and quite quickly in this case because we have dimensions 72 directions and it is normal it means that there and in the distance between all distribution the distribution seen by Eve and the uniform 1 goes was cooking and this is the same flatness fact about computed by using just the terms of 20 and they as you can see them it doesn't mean because the defenders activism is distances can be indicative of course so that's why it's very important to be able to complete the reported addresses not on the 1st another
35:33
example of the use of the facilities just considering coding letters calling on the managing white dust those Chung In this case there are probably at U.S. can never abound in this way so still did we have the to biggest series of the is the plants throughout the again OK another
36:02
and then there it's the obligation of traffic vessel is the 1 :colon probabilistic shaping up in what we have even in regular communication problems when we have is that Read we can transmit the fight nights subset of and in general would we do is that we use a fine at subset of ladies with smaller you can number of interests more energy cost more power right and so on there is no other way of doing them it is to use the 4 letters but wait each point of the matter but this probability OK so it means that it is just as much will bottom on PBS and in for the full of these Zenlike policies so is just this screen Golestan From the current so the problem is not what can be what can we do house Sorokin defined the power and the rate of such a signal that because these are the 2 most important things we have to move it we have to roll by another to trust me so the ball by dimension is just given by this so it is the sum although opponents of the nite itself in fact the Euclidean norm disquietingly none of the of the it's probably thing which is given by this wait and in fact we can prove that it is just it was just the "quotation mark Gould this look at me derivative of the feeder of the letters Monday also evaluated at "quotation mark by divided by 2 bicyclists OK so it is up so we have a very that the relation between the power of this consolation when using the standard but the destinies of the no the rates so hard to define the rate of surgical solution so jolly entrepreneur right so of course we normalized by the dimension to halve the rate per dimension and so it is this is the definition of course and control in the same way that it is given by people the overdosing squares of this be lost 1 of our and that the longest of the thesis OK so stayed there and we have a very we have a very direct characterization of power and rate in this case if we want to use Protestant to shipping by using the the destinies of the land is so it's very very very important to even in this case OK so there aren't many many other ways of using the series but of course it the stopped for the 1st part but here kid noticeable to the
39:02
2nd part so as vital as I told you I would not intervene to many things for this the 2nd part it is related to weight from the site and in this case we can show also that let citizens can be very important but in this case it will be and we have to use the tool viable fetuses not 100 as it it was the case for analysis OK so far as to all I just remind you some of the Japanese concerning communication over nonlinear in terrible channel OK and that the 1 we have to consider what we're doing in many years companies that an taken occasioned companies it is the nonlinear optical to maybe area are some years when we have to consider some of the more fancied challenged but for now it is the only 1 of interest for us and the as you can see this is for example what has been done up to now says he's not quite there on results the newest 1 that comes from 2012 but there are some better I mean some better results now of course OK so this is an optical communication so we have USA North here and we the specter efficiency we can achieve here so this is the socalled channel limits which is the log of 1 plus some of if we have if we wear in the literature Road but it's not the case we are in the nonlinear chairman and in this case so it you consider the money narratives are as I mean the result of these moderately America's just as noise so this is what you have so you will have a maximum here for the rate and then the money nowadays we become too important and then forgiveness and of course nothing will work character so and so in the single Malt fiber that promulgation it is isn't governed by this morning actually equation solids it's an approximation but it's a good very good approximation for 5th you and so the commercial communications systems up to now operate the questioning the regime and the constant the nonlinear effects houses most his just what he told you but there is another way To some the spending it is in fact so some of something that has been proposed recently to consider not only on a free transfer In order to not only to solve the money liquidation and that in this case the question as to be which is the case with the nonunion stranding equation so I would not good into the test for this integral equation 2nd because I don't know too much about it the and the soul but so in this case if we use the nonlinear free transform we would be able to not only To solve this nonlinear equations but also will be able to design some communications system in the same way as it has been going on for many years in once creations by using right so we have a kind of money no system all right so why
42:39
because of the money often inspections what happens is that we have a sorry we have a superposition principle that means that we can have the local boards which is not the case in the time domain of course and the and because it the question is not OK so just what it can do is just used nineyear free transfer she proposes or singers who want to transmit and then go back to the time you so
43:09
this is an example of nonlinear free transformers for rectangular plots so we have 2 components for the nonunion uh fully transformed we have the continuous Part 2 which is the more as equivalent to what we have about using the meaning of from and which is and the Pisa continues that means that it's it can be defined on all Real frequencies and then we have a discrete parts that response to something solutions and this is part of the barrels when the nonlinear regime becomes mitsui important OK for example here we have so the posts and capital it is the amplitude of the Richmond apart and so if a capital is not true begin then we have the famous seemed to function In the continuous part because it's it's a sport that is almost added to the regular free transformed and we have nothing in this respect no effect aid because bigger than that we have of course something here and the continuous spectrum which becomes different from there the same function you know and we have local board that appears in the industry spectrum Paul Okon then if a becomes even bigger than as against Evander did this even the speaker almost disappears the central peak and then we have a tool components into discrete spectrum which becomes a bigger friends OK so all right so
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what we have behind the here for transfer so 1st of all the money in Africa from depends on the nonlinear equations if we change the money immigration we have to change the 1 the which was it's not the same then as I told you the money free transformers ordinarily expect from contains a continues part and the district OK the only Dynamo lies the money located in which were in fact we can do by using the money have returned from a weekend right In fact the channel becomes but along with the same sense of when you consider that Libya the reduction look exactly the same sense but and maybe in this case the Chumley gains instead of being scoured can be matrices so it's a little bit different but it's the same kind of phenomenon that appeal so we have a superposition principle and the money every mainly can add the components which is not the case of course independently and it has been extensively investigated in the context of waterways because we can get this immigration In so we have some wastes of some ocean waves that in fact are governed by exactly the same equation as the propagation of flights inside the fiber which is the money trading your equation just exchange of the space and time complement and you have exactly the same kind of question no if talk at all not and into the face you but if you consider because you nodded to construct this the solutions based on money of which rose from we had to consider some boundaries which where in fact at the above is the miners said unprecedented boundaries but not if you want to consider something which is in my view this is easier to implement which is implement the book is about to consider something equivalent to 4 years the series instead of fully transform because everything is and digital you know how they're of course and that in this case we have some thing is equivalent here which of these generalized Fourier and what of his job as free citizens these are in fact the agreement functions so still the functions of bearing
47:38
OK so justifications very uh so really
47:45
the definitions as we John not only to analyze just signals to trust me OK so these this
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isn't really Monte de functions miscellaneous funded mind solutions that means that OK In this case all if you consider is a it's a generalized free series then that we only have a discrete spectrum components and it means that if we considered uninvited but solutions just grew into funded by OK so they can be given a chance of Fremont the definitions which are these series OK so we have a government which is symmetric matrix was positive definite Majorie parts so that the series can convert Shh and then a is a vector the all right and as you can see if you know case if we choose OK removed the rear part of Gomarsall suppose that the report is 0 so just considered that we have an imaginary but which is positive definite and then there just OK if you consider only these ones and that in this case is that because this symmetric is defined defiance affect the run matrix of a given Latins got remedies for those who don't know it's just a bit said the problem the transport of the generator matrix by the Jarrett matrix it's OK so this is the drumming From the lettuce and so in this case we can express visit this week on the disease of the tool viable fetuses are forgiven OK and in this case we know that the solution is given in this way so you have so here it is that so did every month the day function related to some of them are matrix here of this valuable or so feet minus 70 depressed just some shit vectors that depends on many things related to the propagation of trust and not even going to here as well so just it's just up to some of easy terms it is just the ratio between the next if you can use it some that this year but by doing it by using this trick is just the ratio between the 2 viable fetuses of shifted ladders and the same address shifted in another with "quotation mark so Standard & Exploration itself and the winner stop here all
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candidates some conclusion so however have seen that the decision I mean the destinies of the functions no powerful tool for analysis of Lecter school right we know many results about the destinies of union modular like this is and introduce war you know what I mean when your soul and modulators but you have to know that these letters is are the I among the best ones so no other letters of interest from us and even when the dimension becomes the highest then we know many things about the use of the is especially we know some concentration results about the disease so some would know that that this is in the same genius of Bertone to define but may have exactly the same way in terms of the customers when the dimension becomes high enough OK and make provides an elegant solution to the problem of communication on and on general and we stop here might to make presentation I got a lot of questions and result in the loss a ban on the sale of the yeah expressions of good will the invention use of force respondents in the look for this this is yes there is a paper bolted from portrait of the you see derived also exponents used for India for 4 1st for it said in the constellations in general and then specialized practices practices in India so that questions the sewage is the last thing you can otherwise In the current status of the West regions so as to make this thing special you can be arranged through if you think this is the story of the of one's own the these I the think I versions of this and I know that there are some works about from the medicines but I don't know too much it's but he has done and I know that our results in Baltimore and medicines some 1 of the worst thing about this because that it was too late for this fall for what we have to do yes have a right to know what the year old here but yes you might mean you use in delighted here yet I mean it's effect when I see random nite it's the choice of the letters which is run it's maybe it's not exactly what you mean kids so hopefully question I don't know but the question is so I have to question what is the reason I I wanted to do so this year and that is still in the game is over as far as you point out a new features that things on the chin you have risen but it's the the only thing that knowledge is power was the analysis with resolution of the findings and my number in the yes it's the same problem as a when you were implemented when implemented because additions as prison between 256 moral clarity constellations have exactly the same problem because in fact came as justified in part of the U to letters and so then when you have of course you can lose but the then it's a mean little when you criticize illegally and that this this is a variation of the results also the resolution to see those visions of the higher higher yes but but but as I said it's exactly the same problem was cured exactly in its and the fact that use ladders doesn't impact more than just a regular Cuenca which have exactly the same because In fact the that incident to construct for conditions purpose these lattices that can be built from the z z chew on India's the security conservation so basically they it said the is the problem of precision coming from user ATC insisted that we give you exactly the same Park if used as cause for each user just secure and blasts and only communication which is the solution that is implemented the 2nd question is is about the efficiency of the application of Article 2 of the year was always on the state of Louisiana has acknowledged that his life was in the fact that the future of the United States will see users would have call the only thing you have to check it is that the equations governing the diffuse when he won the Narnia regime of course isn't it if it is a case then you can use mean send Kansas to him but you have to find 1st I mean you have to find anything so what is the region's from this case there's a lot of work to do that the may be In general idea if you want to be very precise on model then the question is not going to but I guess that there is always something terrible roar equation not too far away from the the system you want to if you will be a few lines in some money from you there was a possibility the fact that the levels of debt and then I'll just have to some of UES is in no time to help Of course this is wilderness it's about the people who live here so that he is a good thing because it was thought that the entries last year as head of the house of difficult for all the processing including In fact itself and Litt said recently in a lot of progress it said that before maybe it a couple of years ago there we knew how to construct to some lowdimensional named in the Mitchell lattices and according was a really painful happened because it was approved but recently I mean thanks to some of highdimensional analysis or unique individual letters constructed by using a what is called construction the to it's not evil "quotation mark might even construction wielded by just using the multistage decoding so they could further difficult for the 1st then removed the components coming from this and go to the anyone's interest and factory can be proved that you can achieve the capacity of the ghost injury by between them and their timing even practiced works very well and that the complexity of the coding in this case Is there a resident orchestra no more than the 1 you have by using the example these bit into its communication the same than him this is the question the only way to reduce the risk of lessons it's over the because of related to the only have the stance of Silesia region "quotation mark In was Florida and then worked in the American food yeah you before construction of Europe yes the fact that the following is "quotation mark shipments will also be used to work for him (left parenthesis the used interim the reason for that so yes yes AlQaeda yesterday we are we would be very happy to have some handball From this point of view because of this I mean 1st of all I had and just know how for use on the software model from it can help to relieve water for hunting and so not I think you we could have some some cooperation and that because we have so many of which are not fulfilled and especially if we can have also where work for some of the problems of example from my communication probably a mean we needed to In stand To we need to understand that it has behavior of some quoting the user ,comma equally schizophrenia using and in this case said summer for us the fancy objects appear appearing at but we cannot analyze so for example rose and the instead of having their own it's it was a lettuce we can have for example case we could have lattice is defined on someone's community of order In this case the that of I think it's just a the city of giving way we have to consider this summer To determine timing the interim executive expression about you some of the determines will go to the polls something truly exactly and I guess it's probably it's half that it should be related to somewhat amorphous form of noise it president so it's it's an object that dialog manipulated so you have to be weakened In the the other questions I think we got everything please include