2/4 Trace functions over finite fields
5 views
Formal Metadata
Title 
2/4 Trace functions over finite fields

Title of Series  
Number of Parts 
36

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. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

Content Metadata
Subject Area 
00:00
Number
Computer animation
Lecture/Conference
Function (mathematics)
Multiplication sign
Theory
Normal (geometry)
Right angle
Analytic set
Freezing
Field (mathematics)
00:57
Geometry
Point (geometry)
Complex (psychology)
Group action
State of matter
Multiplication sign
Direction (geometry)
Connectivity (graph theory)
1 (number)
Sheaf (mathematics)
Parameter (computer programming)
Student's ttest
Mereology
Theory
Grothendieck topology
Number
Group representation
Conjugacy class
Lecture/Conference
Average
Wellformed formula
Term (mathematics)
Square number
Cuboid
Subtraction
Representation theory
Condition number
Process (computing)
Independence (probability theory)
Nominal number
Mortality rate
Price index
Cartesian coordinate system
Functional (mathematics)
Bilinear form
Estimator
Field extension
Proof theory
Arithmetic mean
Vector space
Hausdorff dimension
Discrepancy theory
Family
Local ring
Resultant
Directed graph
12:22
Complex (psychology)
Group action
Multiplication sign
Parameter (computer programming)
Complete metric space
Solid geometry
Mereology
Food energy
Frequency
Goodness of fit
Lecture/Conference
Term (mathematics)
Wellformed formula
Oval
Square number
Product (category theory)
Independence (probability theory)
Line (geometry)
Functional (mathematics)
Hand fan
Field extension
Summation
Voting
Right angle
Coefficient
Units of measurement
Arithmetic progression
20:17
Point (geometry)
Sequel
Multiplication sign
Infinity
Expandierender Graph
Line (geometry)
Average
Food energy
Bilinear form
Explosion
Arithmetic mean
Mathematics
Summation
Sign (mathematics)
Lecture/Conference
Wellformed formula
Stress (mechanics)
Hausdorff dimension
Reduction of order
Physical system
28:58
Axiom of choice
Complex (psychology)
Group action
State of matter
Direction (geometry)
Multiplication sign
1 (number)
Parameter (computer programming)
Mereology
Subset
Group representation
Mathematics
Matrix (mathematics)
Meeting/Interview
Square number
Product (category theory)
Moment (mathematics)
Infinity
Functional (mathematics)
Bilinear form
Maxima and minima
Shooting method
Arithmetic mean
Lattice (order)
Hausdorff dimension
Right angle
Mathematical optimization
Thermal conductivity
Resultant
Point (geometry)
Geometry
Finitismus
Presentation of a group
Observational study
Divisor
Image resolution
Similarity (geometry)
Number
Frequency
Thermodynamisches System
Lecture/Conference
Helmholtz decomposition
Term (mathematics)
Operator (mathematics)
Subtraction
Condition number
Addition
Eigenvalues and eigenvectors
Total S.A.
Cartesian coordinate system
Field extension
Formal power series
Unitäre Matrix
Summation
Network topology
Discrepancy theory
Film editing
Local ring
Freezing
46:32
Complex (psychology)
Group action
Musical ensemble
State of matter
Multiplication sign
1 (number)
Parameter (computer programming)
Food energy
Grothendieck topology
Mathematics
Square number
Ranking
Moment (mathematics)
Nominal number
Price index
Functional (mathematics)
Bilinear form
Category of being
Arithmetic mean
Lattice (order)
Mathematical singularity
Normal (geometry)
Right angle
Thermal conductivity
Pole (complex analysis)
Point (geometry)
Divisor
Transformation (genetics)
Variety (linguistics)
Heat transfer
Rule of inference
2 (number)
Number
Thermodynamisches System
Lecture/Conference
Energy level
Integer
Rational function
Addition
Series (mathematics)
Focus (optics)
Standard deviation
Projective plane
Element (mathematics)
Mathematical model
Set (mathematics)
Cartesian coordinate system
Field extension
Doubling the cube
Object (grammar)
Pressure
00:02
at the time and money this so
00:15
I wrote down again here the the main strands of it in the way freezing yesterday he recalled for example and return and I would die along with normalization where defined something is the the right you book I want to do that as a 1st step now true find argue another example of trying to apply this as really as pure black box and see where we get into a little bit of difficulties and then
00:57
introduced following Bayern's witches soldiers this is going to be hard to this I want to take a lot more comments on this condition of juridical distributed the richness of the comes into the into this so fast weapons among examples so so examples long I'll always dramatically reduced most of this I mentioned yesterday because the underlying vector spaces mentioned 1 and onedimensional conditions always irreducible 2 1 can insurance is dramatically reduced vault this follows either from the construction and all bided their function Gretchen mutual state as to the 2nd part of market examples no geometrically reducible accepted as far as the wants to add a few days of putting a mail with which is not constant I want to treason dramatically reduced this is the degree of that to or simply because there always the kind the average value Of these representation number is always want and that correspond to the fact that there is a crucial component which 1 can think of and so care of and wants to win defined this way also its function the yield to maturity so if you remember the simplest example was when the point is excluded then Kevin minus 1 is just a little symbol and the and this is an example of case 1 so it's human years I should say maybe in that case the dimension of the underlying it is the degree of the OK and case is like most families submitted :colon the genetic theories that the example Greece examples and trying to see which ones argument .period so you can try to apply this year and then too many explanations and you just on this example during get many statements most of which would not be able to pull directly without applying the remaining but it so this code cancelation and you should take it said if someone some times example for conjugate this is not something you can talk to people look at so 2nd remark is what is the meaning so there is often a very convenient I functions the interpretation of geometric disability that can be used at least again that sings reducible quite often that you can be used to prove it so it's also a consequence of the imminent and it was his 1st stated explicitly by cats and states the following the only the opposition in a way which is convenient for larger number services as a more general version which is more convenient source what right questions so as to we have for every trace function KP with complexity from about so imagine numbers here it is often the situation so for every prime gain a different rates from fun and the complexity of the airline does not growers then sold 110 for his final leave for the application I want to out it's more interesting in the direction of putting geometric disability from something so understated if component so if you compute the have to Of of Over and this is equivalent to pee when the restriction so this is something you might open a number of cases to compute an educator and if you have anything but a formula that begins with Wednesday to introduce it means that on average these things at once also because they're abound and the size of the summer's then came the geometric purity is also hoping that anyone can be more precise but I think this is a rough indication and in Natick applications enough to get the keys that something is terrific adjustable you might not
06:39
be able to get a synthetic form of it might have sticks to guess that the terms yeah In the year of the U.S. is made from where I think that the U.S. and that that's a good remarks the 1st address the remark and then go to my example so that often says that so something I have not said but the underlying proof of this gives more information and 1 component of this information is that so if want to insure so if you can find some exponent of positive independent of his such that this too long it is the 1st to do this then in fact it would have to be accused of sites that it won't tell you that is equivalent to people because it could be more than 1 but is would get that then on gets 4 3 that this With these conditions so I'm still working on the assumption that the complexity bound so this form follows from the underlying type of results of dealing with the specialized to state this year and decency to the 1 case I know where this is used is that it as a paper of the Hungarian border "quotation mark paper on applications to opponents of the Canaries Kansas tougher putting so it's not really necessary because they felt the pull my cats estimated the need and then their approved the same estimate kind of more but directly by using the fact that get a tiny compensation and then you get his coat of so it's more in the opposite direction so here I could actually Cesarewitch Yukon this process goes both ways you can bounce when I guess and would not be in front of the principle of this discrepancy decided that the explanations and to jump between onehalf and 1 with no gaps in in the proposed is using these people wandering the holiday he made the underline just so as I said yesterday so the complexity is among other things so we have stated that subtlest the to but fact it also the 2 dimension of so by the end of the section we have almost all the components of the actual definition of complexity and then on the next week I have to introduce the last fall which is necessary to understand cases what so as an application by saying that you can so you can still check disability for geometric distributed for example well 1 word said but if you know that there is an underlying vector space so the nice thing about these pretenders and some sense you don't need to know that there is an underlying vector spaces just a purely die statement you might be able to be verified without knowing anything about any underlying so you gang case 1 effect is a fix for a nominal fee social function this kind of acute because then the moderates is 1 except that not many .period words he wrote his solitary but you can check easily can so it's for I will to to but the menu square off some there is actually a fully transforms its profit from the phylogeny streets society local media competition the student so when you get equivalent to please for K Street you can do so in in case so good state and the family this would not allow you to deduce that it's not Jennifer irreducible but that's not most interesting application and for peaceful I I guess it would work and new states in the society 105 1 the main square of the number of ports on the families actually use computer would get the details should be of the same complexity as Christensen took so now we have made so we have a bad box at least some of the town in the black box we can interpret very concretely even if we don't care at all about the argument so geometric distributed you can replace To some extent by discrediting and geometric Asimov is only can just try and replace it by proportionality of function so let's try with these 2 poor unable to extensions honest men and see whether there so this
12:24
example comes from the work the London so in studying and Kennedy devise function innercity progression production line explained that was straying all the week so they found that there were saying could be reduced to good estimated for the following financial sums of life did fans watched with the writers of the different origins to Greece to parameters and being so a sponsor 0 my and and and so it's not actually exactly the way stated but that's the way it comes in the pool so there's a natural way of writing and that's the way that there wouldn't back so I can that so vital commonsense reliable to viable then but because he shifted Wong killed 3 of eggs here and there because they have to use completion techniques to someone on the .period vise you demanded energy character so you can you be estimating the sum was to push .period for them and it basically you are screwed cancelation so the way they're actually write it down 1 which down as expanding the 2 instruments also informed as that which is not actually the best way to think about it there is some of the work well I really mean it seems often to get but all well and expand the data and after the right to dissuade it's true but so I think that this is only appearance appearing in the paper his brow rights to explicitly OK so if 1 expands the concerns of the Protestants and I know this is 5 viable but it the bomb it's called cancelation so if you write it as a father I will go to some just shifted the intended do coefficients Cinderella more than you would get a genuinely irreducibly 5 I will cut the sums and basically you would be in most cases while stock even now so at the time this was little before the ideas behind such original versions were completely available in the literature there was the necessary assistance and was given by Burton born independence from England on it so they produce required cancer so In the case of white makes sense of what how the obvious affection and so the only completely obvious Navy problem that can occur competitive again is when a is equal to 1 then there's the square here and voters here In that case between moderate square killed 3 and that cannot be but the cannot be cancelation so if it is different from 1 of these New 0 so a equal to what is permitted be used and you then you have uniform "quotation mark cancelation estimates 4 times and all so uniformly in terms of a so why do I say so cancelation remember the chemistry of the normalized by dividing by besides equal to 3 so the fact that these abounded in nation's already tells you that the kids bounded by the which is already "quotation mark cancelation at a bike industry of so it's discovered cancelation is included in the chemistry and so what we want is good cancelation in December about that volatile but when you explain everything into a found rival Santos would be for the psychologist some people His according to the final 4 the 5 so full code cancelation and I think they don't show exactly how much they can do about that document attended bid but not so you really need solid Peace Square would not work if only because this would be military abound from soldiers here which corresponds to the square Beatrice so let's try and get hold of this using what know and see whether there is a problem OK so we want to write this as a K 1 and K 2 so you can find Q 1 of the Act for instance the history of 2 it is clear from a clear nuclear accident considered 2nd so this 1 is a tree from so the 2nd 1 is part of a relatively easy farther from his chair will stage in more general versions and that this is also true function it's very difficult to hold and most cases the complexity of the uniform rebounded and was a and basically because they were taking the product of 2 trace functions and each of the as bonded complexity estuaries by the by Caesar overstated or by his following is and all their complexity .period but I mean is to some extent so this complexity of overnight so it would be a mistake of action stock would be like them all nite versions of the actual all that occurs in the exact formula sold for 180 publicist for bound enough to override the quantity but when you want to do really write things you might want to the nation that more from but here it doesn't really matter so it's OK what is dramatically reduced bald find just checked in and out by using the buffeting criteria can really be checked with honoring
20:18
instead also I would just like to dine with something of modest 1 so it doesn't change the cemetery the benefits of at a school for exit by ejection on he also doesn't have seem so now the question is whether they are intermittently isomorphic on so can want all the underlying all want to Q 1 and Q 2 us German nuclear plants will think that rather than the but because the estimated so do you want question grantees enough abandonment of constant this is exactly what now this can said so I noticed always going to faint we think of the abortion criterion is if a sequel to 1 there this of course and we want to show these are the only case so is that it so it was reduced to a priori simplelooking problem concrete problem has that kind far such that killed 2 of his own way the move was 1 of 5 times appeal might be cares for him so it is not I'm leaving concrete this is something solitary by computer for small farms side likely to be true but but it's not so obvious really out to prove that so last time when we have to do can evaluate quality from a bygone we used the fact that the is of the concerns some 0 at a point where are the other salmon which Wisconsin 1 was not here there's no when the obvious point where things are strangers the but it's the same on both sides began trying to take the money Of course Chelsfield of expiation of that distinguishes the skills of a speed things no 1 but you have to prove that stuff so simple so I think you can probably for some kind of cool by some means but I want to use this as a pretext to introduce more of the underlying they so back different ways of doing this is 1 which starts with a new trade which as he also good to see if there's nothing that says that we cannot try to use knighted techniques to study summoned to actually use a little bit of native so you can try to flee tries followed by handling can just expand of restaurants under Nunavut more carefully than writing justifiable
24:18
some immunity to reduce stress without character sum by using a lot of energy so expensive cares his applying stability and we're somewhere you obtain a different formula which is actually a way to return to the end of the year in the appendix of broken bones that actually right in this western some overtures and you have no to look at systems at 1 key were to so I don't put the predicted and you know you have to that the nonzero an obstacle to minors between you can simply some of the same thing and just use the fact that the nation that killed 2 at infinity it is hoped the election this is reasonable thing to do so what we have been here is so full fault but was important because expand revival character songs actually succeeded in applying the work for fire dimension sounds to estimated get screwed cancelation but you can also use this form to get a quick reports through the blood books notion of dramatic What is this the point being that here to attend a special board which is equals 0 but because pharmaceuticals miners beyond so you can't immediately guessed that the is being 0 distrustful cancelation OK so applied to now we do another change of revival which is again permitted by the farmers and recall again because adjusted by ejection on matters of the protective line is always the that case care to have a 1st also about so assume that still carry on his claim to adhere to make for instance by day but from the criteria and K 1 2 that is not proportional to cater to that history there is no there the value so that people can do not nothing sign sold 1 that because there is in India so it is not so this is geared toward a so when there is no
28:58
reason for control always we actually achieved using about it's not obvious to actually pull something so I should say ,comma so that's the point is that I mean the Sony's suggested but because we don't have a simple cases like the veil case where no of Minnesota 1 of the role it's easy to say that the constitutional was but it would be but action calling it might request some the argument so actually going there consonant sounds on 0 because it occurred to you can do by hand but you ignored complicated cases instead of trying to use just proportionality will give you completely off the top of very concrete looking very obvious looking question but not a situation where you can do something so we needed to be done extra from user 2 and elected so I will now go into the direction but I would like to ask consolidated prevent another application where just about books versions in Oakland would have done so and later next week have the opportunity so we need some more from them and then will come back to this as financial services and there will always remain the case will be is 0 0 but we have to prove that when it is not 1 it's not a part of that solution of so so formalism so we already so some of it comes naturally in in defining who is a special cases so 1 was something that was mentioned already indicates the decomposition intermittently irreducible pieces still to be disasters does not quite work on in the JIG I'm going to stay there is a slight technical issue between representation which could be adjusted a global editions of the book by called by 1 where you material distributed by this suggested indigenous studies so there could be a discrepancy it's not up to you is just a technical so I'm going to state something that is beautiful budding papers consider the actual version the correct so and care which is trace functions can be read as a of Atmore the complexity of geometrically registrable Siemens the complexity of KIII complex here so if I don't do geometric irreducible this would be entirely correct but genentech what you need to have the form of the so this is technical issue which in applications as never Archer ,comma concretely sold at its it's worse trying to think about it at the moment but so payment more interesting things so if you have representation just 2 of them the candidate directs some of the most obvious way of constructing new presentation that the old ones and the corresponding operation is just as some of the trees functions so as not to trace function is trees function so typically if you want to or not this year that this can never be geometrically just for this is returned by and of course in any of such operations the 1st thing is to know that you can do the operation for function but you should always immediately and think of the complexity change without control the complexities operations are useless for applications and what happened in that case simple whatever the definition the complexity of world to is the sum of the complexities of the cement so it's a very busy easy operational stories so there's a twisting operation which only be denoted by far a 10th of the where the fire was so whatever that is this an operation in which mobs Kato this because I cents during the hearing will to dramatically reduce the will and the complexity of the then if you go to the jury representation the intro the the corresponding operation just published condition this is actually quite deep at the meeting points to check that the congressional debate is the right thing is defied due to affordable but this 1 out and the complexity doesn't In addition 1 could work with optimism .period interest them afterward so that the fact that is had in fact is very convenient the 1 doesn't have to make changes when taking conjugate but he not be a massive problem is worse change so supportive of the demining PGA Tour which acts on the 1 of its but then came it is infrastructure so this to the fact that so there is a natural extension of the truce function on to the 1 so 1 can always speak for any given trace function of what is scary infinity in completely chemical waste so the same way that you I said that killed 2 of injured 0 this is not just a convention the right in other cases it might be something so natural extensions to define their and the complexity of of him it is the the same so in particular tale of a X been for initiatives and we in the euro results I want to do other operations you can also change viable voted for on unjustified moving him from moment the warmth of With yeah and you can prove it with the definition criteria and when you change viable this summer is basically the same said you might ask reaching shooting and the primitive infants so the sound was changed by about that amount between and it's also a video viewers want 1 most of the deformities and it's just because precomposition by the enough to become so much they also increase the visibility of the so they could establish means also frees not to mention distributed but it's not something that's just at the time of the day I guess it's
37:48
6 so what other products sold products has been authenticated because of what you want to do a sure to observation to market to support the so the characters from the time I however just like the product of 2 usually characters which are primitive might not be permitted to think of the shown his true character so if you get something which should be to but is not quite so so this is again a technical issue which has to do with the the choice of Liza you want to amend Jewish character on of them please usually got both of them advantageous I prefer the primitive Walden swerved to change that sometimes the product so there exists said In the both sides bounded by the sum of the complexity of the 2 plans to put such that ended In the case for 1st such that the history of X is K1 K2 outside of peace but there were the complexity of cases through his I remember something like a kid in the right Kansas you want sequestered square and the former it's correct that it's almost so it's not about in terms of complexity of 1 and 2 and out because K Street bounded by the quantity of hidden arms bound by this quantity K1 and K2 is bounded by she wanted to the resolution responded by complexity but this voluntary .period where the 2 functions on coincided differences also soulful acts in history of X minus Q 1 of its will be bounded by 1 depending on who wants a ride on the qualities that come and so far promises that means that the fact that it's not quite assumption is just a technical musicians just like the products of usually characters and words by now something which is not quite from a Jerusalem the deeper in explaining what is the complexity is the issue of running condition so additional character as a conductor an idea I said the complexities 90 conduct also at some future similar to the fact that the and conductor 4 additional factor is places where something strange happened anywhere from 2 senior so warned defines I finite that sad because this is a scientist at the the representation I should say that we don't want to give additional there the finite subset of socalled modification or really couldn't hold similarity is this is the most natural thing in a subset of the 1 of the few bombs to to Kaiser of the so it's unnecessary and moderate could be something which lies in extension or it could be continue the search that so 1 way of freezing is the 1st decisive as is again bounded by the complexity so that's the 2nd ingredient in the complexities of the of the antigen or if you want the dimension of and there is the 3rd of single .period and the sum of these 2 Alliance was enough to buy all the complexities would be great for total of the cable will see that next week and what is the meaning of this single Ponce so 1 way of saying this is that for fact not a single also notified I'm going to face in this way he alleged that the return of the trace of a unitary matrix of size dimensions of replace the trees functions of 2 so it Texas fight so it is in place not from the bank is the trend of unitary matrix of size dimensions became so in particular and of course this week over the fact that you can always bothered by the dimension which is bounded by the complexity at least for this fight lies the value of effects "quotation mark mention of the and so this would not be so useful but things from this more precisely so that is the truth of matrix 12 more the restrictions .period and was eigenvalues all of size at most 1 good of so the eigenvalues dropped action all the things that you know something so long we don't want to so dropping you subjective values it is you know a bit about the fungus is to the fact that the running fight crimes associated parameters drop images surround a bit of conjecture says produces 1 at the fall local parameters unrefined plants and other artifacts crimes it is known that actually drops images produced in this case because it's not necessary to support it could be that some eigenvalues women of what but that's the main point is this and from this all from artifacts but it's easy to show them that this is an environment of all of 2 German figures amorphous and therefore these gives us way sometimes to say that 2 things are not genetically isomorphic simply because we know what is this final and the 2 said some of the same damaging this is mentioned but is something I so in a way we define the complexities so wretched defined the complexity of the sum of 3 times the dimension of the condition the number of single points and invited the onearmed tradition policy not just the if you give him a lot problems so we should just a gadget conductor than from the time was the so to make a final IPO geometric she's and the sense of cuts would have the
46:34
same kind and some of these unbounded Rankins so you're going to do it would depend what you mean by energy conduct so it's not that the conduct of 1 is a little bit more information so I don't know that that's 1 of the that this year we have time right now elections so we've have been told by referring that so there's a bit of seemed to take which would be a special stated later because I need something else so they are more monitored right versions of the definition of the conductor but it's American not just right 1 papers for the moment so we're keeping the old 1 of 2 and append to the point where possible to ignore that addition inviting number series this is not an issue so people can still to remain a member to become the so we have 1 explained set single point when it goes with examples and say what we can say about this thing up the case was sold for additive character this includes so there could be some issues for more private including the state of play on the floor if you have a formidable catches it also poses and 1 can be more precise and say exactly which ones which runs on so to see an example where it's not acquitted him discovered other porn and title holder too so double pole major effect on it's a oneover square and this is not from fight for the project the 2 issues you can more presided meeting of the minds of full to focus on and so on it's 0 and the focus the case 3 Cale before counting function of the most solutions again I think there might be some things up and which don't remember exactly but it's not included in the set of critical values of so critical values means you get dizzy with the dignity and the image of Africa is that of of zeros Of course so I said could be to be cleared the .period analyst agreed that will by P something you could expect strange things that what can be more precise is just begin with his hands folded I don't care of soldiers killed in which Council of points on the company to carry out the want to which depicted here who seduces the situation worsens given restraint so you want should take any calls you all 1 of double want any future seems to to be aware of this stranger introduced not always I should have so you have a rational function which is I like to extend this is not the singularity of your so with this guy we can't go back flight what I was trying to do is that we don't even have time to transform into some of the Back in the clean so this was the son of 2 transformation and clear the kid to rule of the kitchen to keep this and we can use this transformation to see that the army is known K 1 of these was located along yeah right side but not too so for articles your executor ingenuity and came to its ingenuity and finds example to and for articles that we have a over which in that case is just an element which is nice as you anointed disparities 2 so there remains a key this is equal to 0 and different from 1 where we expect them not to mention the endured some but it's not obvious that case both horrified as you want but I think the people that don't meet that so far variety for some reason we have to check I think it's enough for for the fate the Union's argument to deal the case where business 2 so next week will following violence which allow you to distinguish care to avoid what he was going to have to ensure that when there is not 1 that know Asimov so this is still not clear with the amount of information we have but nonetheless remains to introduced 1 piece of foam ozone which is much cheaper than what I would describe it as indicate some other applications naturally this is number of 7 it's it now this is extremely deep if only because there's no 1 out of this at all fall transfer model forms or other forms of nominations so this is a corporation that makes sense trace functions and which has an incarnation of the level of the underlying representations but for which there are no honor or number of so it's really did construction which is again due to the need and which was studied extensively by no more but kept fuel so that can lead to function In the days year and all of which he was I guess that's the way to do it that I'm not there yet so if you can check is being is there actually 0 or you're going to compute despite bipartisan harmonized and thanks on the soldiers this true when in for so this is if you replaced by 1 of the changes will be transferred for transform of characters onedimensional saying inorganics executive someone when musicals to perfection with on the environment which will but which again exemplified the fact that we should not just blindly tried to buy from us and I mean you I things and United knowledge can always or often still be used even in situations where the bike into the future and that of someone can also protect the right of course but 1 because in the middle of sanction against KBR function .period we need an assumption and we need that isolates you interview adjustable and not negative character all I could say that we know In fact so when I'm factor in some of the type has 5 times plea of Ericsson P In a statement so in case you might appear to support me that is not proportional to all it's not going to get some of the 2 proposed became disaster the statement at the level of the underlying representation but false more complexity it's the same thing that the critical but as a some office functions the judgment here mystical does not which is just proportional to energy and the reason we need this is simply that the 4 tries from on that is characterized it as a function and functions are not well handled by the former is amended From you can do it but it's even more abstract and and prevented falling stance on any of the examples of the property including history which is not entirely that it is the degree of actions that led to the provided an indication of the degree so all the examples here work except this 1 after degree 1 sometimes the splendid started letters so it's a generic assumptions then defined before a transformer has demand the function as the recession true so we normalized cost would have been the most natural the scariest thing at 1st using just nice salmon and Partners pointed out there was much better to be there in this case and so please just the usual for a transfer taken aback the time of writing your aim of peace of mind and that that's from cats because that's what users books that endangers and a so eject the just before transformed with this normalization and so the meaning of the normalization is that when this code cancelation this would be bound so does the chance of being a traitor function of something responded complexities and this is 1 of the soldiers this year it's really did it also uses the reminder what fellow from fall so it is also said there exists a representation such that the trip functions is exactly that about .period doesn't much so this is the leading a minor additions and at the institution that was not in the literature but was not offered to check based on the work of room on butterflies that as I said this would be nice to ridicule but not very useful finding mystery unless we control the complexity and from the work of remorse which says a lot about the kinds of from it's not out to check that complexity is controlled and what we got was that complexity of Square which is probably not the best possible company is enough occupation OK so that's a lot of the is normal to them because I didn't do it for the Science it must be the most probable defining of minors OK so examples life well an H 1 a trace homage wanted us to be an ultimate success yes I know that I have not mentioned the word ,comma adjusting the country's like 2 seconds but some audio do I want to see that by state of an immediate corner of this which actually principle goes practice more of this is that you can have put in about a month for any trace function so that they can satisfy trace function which has returned home a sensible i model please and so meaning projection would appeal to all integers alliance between the lesser of 2 people and what you did use funding standard competition at the was explained by the for stance is that what you get when the conductor the rich and strong so would be complexity of case when I go to Q and over 3 times where the paperwork that was the biggest but and incessant remark that you could replace the but less than a quarter of the fury less than lucky that he doesn't like it comes but so NBC comedies straightforward you just apply your ideas as you apply pressure from up so this be the standard for a transformer in Denver justification demonstrators some of K you bound for a transfer of K bytes Infantino which is bounded by the complexity which is most in the square and then use under which entomology and of which we have normalization at most code of Putin's freedoms and as OK so I'll "quotation mark In the highest so it's his case completely abuse from the day criterion for its transformed the way findings you time and therefore the as to quantity that you want to go on what this so if studio adjustable so and of course it can also be differently but from the point of view there is no mystery in this fact it's a very useful facts because would transform of something all right 1 can be quite complicated and Robert ran from another Tory rank objectives sometimes I want to check into just about any of this would have right ones on the site In the last year and now I don't think so the yes I at considered degrees coupons to the degree of that consistently assigned no