MRS Factorisations and Applications

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MRS Factorisations and Applications
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We review simultaneously the essential steps to establish the equation bridging the algebraic structures of converging polyzetas, via their noncommutative generating series put in factorised form MRS. This equation then allows us to describe polynomial relations, homogenous in weight, among these polyzetas, via an identification of local coordinates.
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[Music] thank you very much so i continued with mrs factorization already presented by gerard and but particularly i will give some application of this mris factorization this application conserves polynomials polynomial data which i try to to explain in the recent report english in google and best mathematics c now i would like to present it in i will give some return concerns political capitalism and some in which i can derive some property about policing [Music] series so i will refer also some important result analytical results of that before to throw out some consequences of on the future i will give some computational example about this formula in particular the preached equation the first equation give operation among effectually true of the visitor on the description of the image immersion execution of the polymorphism okay i will explain more why is the origin first as all uh what is policing what is the policies it's just special values of the data function on several variables is defined by this but it's positive intuition and when is this conversion policy data uh can be considered as a limit of affinity of athleticism and the meter of that go to one of political ammunition is considered eyes are considered as a payload coefficient of the poly logarithm over one minus five and the polygon characterism is defined by well defined on the skin and now sorry uh police ether the agar power police is spanish by the values of political f1 here the media disease are interconnected with sr 51 okay in this case when the milky and disease are integer we can see that this mediation can be considered as a sequence seconds of some monolith generated by individuals which can be considered as conviction rejection with wealth over the alphabet y again it is can be considered as a objection with words ending with x1 x1 over the alpha finite alphabet x0 and s1 okay now we can we can see that the zeta is sum but and in the case of uh with the uh eternity and uh disease when the government can be encoded by wilson and in with x-ray because this can be decoded by iterated and regarded and they definitely can't in this interactive we can observe the order of the words and with respect to the different form of years ago and only have one so in this we already see the role of x and y at least to consider the lingerie of x x and the words of a white x win the kid the author of the excellent christian x0 and y1 fits on the etc okay and here the caligraphical is red in the slide is impressive x of y and we will consider of course polynomial and formal power series of x with coefficient on some computative thing containing q and we will consider also the biagapa concatenation and go cooperate after shuffle of the bianca part of alphaline y they keep it the concatenation and the dual law of okay uh she had already gave her the definition of the software and was in software products of the stuff so i don't need to read one more okay in the follows i will consider the rings often consider the ring is some differential ring of polymorphic function as a separately connected anomaly equipment the different differentiation differential function can be extended over formal power services of x with coefficient on the polymorphic function by derivation step by the coefficient by coefficient now completed the definition of polyurethane i put polyurethane extra okay i directed with the path 1 to that associated to the world x 0k it's just a logarithm power k
over 4 k and by this one the maps and into that now become the morphism of anger from anger of politicalism it is a morphism of anger and it is adjective so a this map this anger from politicalism the political racism alexis by the words i agree basis now if modify we mean modify the politicalism by taking voting volcanism over one minus side this function is exponent as tail of expansion and the coefficient of the logarithm is nothing else harmonics okay and this mark again is the office of agatha to the uh stefan again to the adama shepherd [Music] worse from the aggregate biases for the other product there is an equivalent between modified body organism and their dialogue expansion so the map architecture from uh stefan agatha to the hangar of africa yeah i consequence history once [Music] the zeta now is really the polynomial from the um so the polymorphism zeta structure i would say polymorphism already gave the reason why we adopt the name zeta for a policy card now we have second reason that it is polymorphism this we just need them to have their value after the agatha the leader was a follower i have action this function is actually defined so we can extend it as a character [Music] foreign localism one minus set in this [Music] cooperation scale is zero the same for ammunition the asymptotic expansion on this comparison scale gives the whole eyes a constant term we have a definition of a set module generated by policy okay we can also introduce the express queue vector space of course of longer gate or off-way gate and we can see that that k is q transfer over set of a k and z here have a conjecture of that the d motion of ak satisfies this regression equation it needs to release leads to problem the problem to know if this map this arrow is adjective oh that is a character so let us see now to study these two competitors i now to go to record the apple lighter frame by considering the diagonal series on the concatenation suffered viagra this year some of the views are earned out of you by the unity uh resolution of unity it also is the sum of the universe sw pw pw is the faculty because with dinner worlds after enveloping your heart and sw is explored basis these two your bases are multiplicative so it's the same way some of the coca nations is a sigma l answer sum of all sigma works on over all words of x y star sigma l pi yes sigma preview conserved by duplicate here again by one is what can be complete by this offer the enveloping agreement and pn here is the pl for basis of the lee now user we use two jaguars to follow at first the objective series of bullying practices this isn't just the image [Music] is factorization we can properly lie at zeta and f n in z y and we can the definition that shuffle either the value of one of l regular here's the factorization of conversion because all the coefficients on local coordination coordinator of this series are policy congress there are no diversity there is in the local government the same for the sexual [Music]
are conversion and this 3l satisfy the functions say this new function okay and satisfy also the asymptotic condition and with the change after our variables x over equal a over 2 by i and x 1 again minus by b over 2 by i is nothing isn't the equation cassette tree and the series that shuffle is nothing as the internet associated with now let's just consider serve and last for me the negotiator is that gamma where here's the question gamma just the final part of the asymptotic expansion of the hbw on this now is in the comparison scan so we just need to have value you just need to have a loose absolute aggressive basis here we take the basis and so so gammas this series is a group live series and we can be photographed by mrs stefan modify mfs cultivation and we do up there on the left of this line series the factor exponential of gamma y1 the coefficient of exponential the coefficient of the exponential by one interval but in this series the coefficient is gamma the left [Music] series of pi y one upper n is the one variable series with coefficient on holographic function and it's the the log condition of h of b y one s and number one and is a upper sum so we i can consider this no competition this is a ordinary genuine series and using the definition of p1 p y upper one is nothing else you can get x one over k over uh one minus z with either this expression so we get the munozetta is this function and using the newton zero identity of equivalently this star q star d y k is exponential of this exponential this goes on in this formula the finite part on the left and the right number of the symbol acorn we get here b1 b y1 is nothing yes the function 1 over gamma one plus y one here you see that the coefficient of y one is gamma again is gamma but for the prime y one is the model under of a curve the coefficient of y one is zero okay we will use this local digital series letter now the number of digital series of integrated and all iterated and associated to work of the purpose is nothing else the gen series of the difference of objective and object one along the path zero two three and this series is good line this is also differential uh solution of the new complete function equation now we take the transformation option z mapped to one minus zero this change available will map on by pullback on the default zone of objective one by g star feasible equal minus omega 1 and we start omega 1 equal minus omega0 so the change series associated to along the path general to is by definition is a different sentence of this article yeah using the change of variables is nothing is the substitution on the cancer reason along the path that the road happened by the marxism sigma x0 a1 minus x1 and sigma x1 equal minus x2 and now the tensorism of this path is rely uh leading to [Music] look at this ring of polyorganism by this formula it's the sum from along the pathway v that's the whole two preset okay and use the asymptotic we have here of n at zero we can deduce that when that 0 goes to the whole this tensor is is equivalent to sigma and exponential x1 logarithmic at zero and which can be looks when z go [Music] sigma of l that suffer this morphism is evolutive so we can put again and this connection now use the customization mls of the new vacations and output mechanism we get the second identity so we can deduct so we have here asymptotic we have here s1
f and z is equal to x minus x1 logarithm 1 minus z and taking the taylor expression we can also divide the behavior infinity of us and equivalent two coast and by one another we get this available okay especially the light meter of this polyurethane now now use in the the last quality we take the finite path and we get each equation this equation make the relation between the surface and the stiffener of polycell okay use the factorization of mfs it is equivalent to this formula okay here is that gamma here is that stuff in a second idol tdd the regulation that use b time why why the first equation user b y one in the b y one trigger gamma but not so after that when we identify the local coordinate of this equation we obtained a variation among what is it [Music] [Music] okay the coefficient of y1 is l okay i i can remark that in this asymptotic expansion we can say that uh zeta shuffle of the review is the coefficient of zetas fern it is a finite part of the singular expansion of an i in this scan the same way theta shuffle w is the coefficient of the partial form the final part of ash omega in this comparison scale and at the present i think this is an only case that we can justify how to realize diversion police data simultaneously at x1 and y1 for stuff and what is only one case we can justify analytically and accurately now let us back to the differential equation va that was just the first of the function equation the galway function equation is nothing else the half loss of a is a group of exponential c of where c is some least series so we can clone solution of a by multiplying on the left on the right of l by sub exponential of b series and we so we can plot the secant letter by multiply you know on the right by the exponent of c and we have again the asymptotic expansion after the plant solution at one and their dialogue creation [Music] and the other light still man is again is a sensor available now if we introduce them the dna is a curve obtained by multiplying on the left of exponent c the c is some lead series on the coefficient a and the coefficient of x0 and each one on exponent c is equal to zero okay in this case here again we obtained the equivalent on the clone of jetta gamma as a clone of the passion by this formula called chris equation again and it is equivalent to this we have already saw that the local coordinator of the stefan zekstaffer and don of that polynomial on congratulations with coefficient in a when we identify the local coordinator we get correlation among political polynomial connection with coefficient in a but this coefficient of this collision are free of constant gamma if gamma is not belong to the ring of our coefficient let us see how to use this formula for example for the first equation this single equation okay if we i don't think the coefficient identifies the question uh y one of k w we get the generalization of the gamma y1 okay this is close formula to give polynomial polygeta cohesion for single zeta values gamma y1 okay the formula is more complicated but we can implement data in the world and to obtain their expression by exactly cohesion by cohesion here again we can see they are on and but when we use this sum formula this equation we make the polynomial to identify the local you get the polynomial correlation among politicians okay these equations are always a must equate since all the coordinates are operating and exist by in the world and living words are totally operated and we can recruit them by weight you get this relationship and i insist again that all polynomial correlation among political acts
equation now since the zeta is the mafism when we can put the depth on the left hand of the of the equality we get and we do appear here the polynomials of q a and q are generating inside the kernel of zero this each polynomial uh alex inspired in the universe and we can take the idiom generated by qn f f y for the stubborn and fx for the shuffle we went here you can see the inclusion now replace the symbol equal by zero we get here we can see that for instance and on the different words of sigma l previous polynomial on other c zeta and x is by limbo okay and on the right of these refining room they are the irritable policy because all this stuff if i took self they don't cannot be relatable so they are heroes and we fought again weight by weight as a writer you can see in this formula that they fought against generator system of the part of policy okay now we take the image of this image because now zeta we already see that data is subjective so we can take the image of the section we get the reflecting system over sigma okay here and variable is obtained by the ever inverse image of the section of sigma we have to get again the chain between and irreligible [Music] himself is this equivalent to say that to qn a q a so let's exclude a summary summary like that the identification of lukan cardinale gives us the two family the first family is a irritable [Music] a verse image by the section after the path so that zeta excited on q the action of q by an irritable to that this amount this restriction is rejected basically and it gives also the second okay in which for adm the qn is homogeneous in weight the weight is nothing else the way after the leader was a so we have to see that there is equivalent to say that q is equal to 0 of that sigma n refers to sigma m of pi division sigma n is irrelevant in the case of qr is not negative sigma l sigma l is a element of identification is not belong to an irreligible so it is a transcendent on the anger heart generated by so that ql is equal sigma n plus epsilon l epsilon and l is a polynomial on q and eric okay so we can say that again write it by sl for l progression can be decomposed by fx plus q a derivative this sounds like a direct sum is a direct sum of vector space of course so we have already seen that fx is included in the kernel of zeta and now if we take any polynomial on k theta not three of those uh constructor no with outputs of that so q can be decomposed on q one this is existing coefficient and bioreaction by fx q1 belongs to fx so you can see now the image of azita is nothing is that it's really presented by irreligible police and the cat name of the zipper is exactly the ideal fx by science we can conclude that that is a curtain can we could conclude [Music] is obtained as a quotient of this by the cutter individual okay so that is carbon and now if we take each polynomial c n is of different way because we have this equation so c could not satisfy the anchor factor solution with coefficient in q but for any asset in irritable s is homogeneous in way it satisfies the previous scheme so we can say that theta s is constant over thank you very much thank you for interesting talk
questions comments yes i have a [Music] question yes if if you go through a yes then you say that at the middle of point one you say that the map going from the algebra generated by linden irreducible to zed is a big active yes can you explain a little bit that's the expression of zeta because the zeta is subjective okay yes the element and r are obtained as in my adverse after section so the extinction of the [Music] between them there are some illegal zeta please i just take the image adverse image by the section and the system and r is advanced offset irreducible to software this media bijection and so you constructed a section yeah did you construct it formally or is it a reasonable intuition yeah the contraction is like that okay from the relation among police sata okay yes data is the morpheme so i can regroup on the left-hand collision and after that i use the fact that theta is subjective yes i just subtract the relation each relation and exits by this volumin okay i just forget everything to take the annexation by qn the section you are describing as a unique uh unique section yeah okay let's take one this one is um for my convenience i think this one but from every transaction you can always build the section section yeah i can find the section to just detect by this way but if it is your objective it you you you need this section to be unique at the moment i don't need this hypothesis that it is unique i just take this step i don't suppose that is unique it is a good track it is a good proposal to with many many things to check together of course and with people who would like to because it is a easy to implement as i see and if you consider the relations you have by identifying coordinates yes did you recover the relations up to order i i don't know up to a certain order yes recover the relations that you have with double shuffle yes of course and formally to prove that the chris equation the relation obtained by in identification of this equation implies the perception relation and at the weight graph we have already verified that the own relation obtained by software satisfy the and we can say that if all are correct we obtain the set of okay i am not a specialist of the subject but the zaggy paper in the zaggy paper i don't remember which one which year but the the the direct sum was formal direction the paper of the year is this one in this paper in black house oh yes yes yes direct some but it was very descriptive in style so as he didn't say that it was formal i think it was formal at the time because it is still conjectural yes i i i don't know i don't ask yet because this direction is some something is abstract and formal direction why that here is something is closely closely [Music] that like no it is direct because it is formal so you take the slices ak and you put them in the direct sum uh in the insecure paper because who knows the subject much more than me yeah so maybe a very can because in two minutes it has to be stopped by natalia okay so maybe we can discuss this later yeah okay okay thank you thank you again you