Zdaj, načo vsega rekača od vsega lečenja, vsega najbolje vsega tudi vsega pridljamo ponkare, ponka re, in stroboscopična mapu, da vsegače dinamikovati sistem, in da vsegače poslednju kaos, In je to vse reprezentacije, ki njič ne se vse nekaj načinjali. Zato je spetno, tako je kaotic, da se povsem, da je povsem kaotic. Zato je povsem informacij, da je zelo lahko. Se svojte lahko vzaš z načinje, z witchjama, vsi je stajno počakvala. E點, ki je počakvala. Z travelần jazam that, našli sem genel vanen imam, da sem res vsem, našljenje squares, in tudi eno bo, da sem je to bo, neveda, ki so bo, neveda, našljenje squares, in je zelo, da se je vzelo, in da se je vzelo, da se je vzelo, še je ta stručenja, na kvama, na daj na rečenji, in je to dobro, zelo, da se je način, da se je način, da se je začniti, da se je začniti, in je to, da se je začniti, način, način, transformacije. In transformacije kononičnih je zelo vsega tehnika, kaj je zelo tehnika generacije. Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo da govorila do stran delom zelo to je vzelo, da je vzelo. Nisem zelo, da je to močnjega, ker je nezelo, nezelo bi ne bo zelo, ali ni se neglade. Ne začnevoj, da je zelo, ali je zelo. ... če še, da bi онежne izračiti izračati... ... rečne poživna, točne pa, ovo, da je pa... ... neko pač, ovo je, kot kde služeno. ¨ Here we go. ¨ ¨ These... this... this... ¨ This... this... this... the program is very simple in ... ... one dimension, mm? Because in one dimension for example If you have a... sestem, kaj presev te energije, danes lahko se počutite v tem, kako je potenšel. In, kako se počutite, kako je potenšel, in kako se počutite energije,こと, da je tukaj nače. Vse si speno se, da je ta temoč del, je ta temoč del, ki ti se, tu je tukaj nače, nije skorod naredo, tudi svega ideja si pripoče, v zelo tako svoj, jasno, je to inoč dob griza, pojad tukaj ne bom diaz, tukaj ne ima pripočen obiet, pojad nače, tukaj te ljubi ki bi, ki tej nače, tej nače, There you see, this is the liberation and rotation of this pendulum here at this level, this critical level, is called the separator, which separates the two kind of motions. It is very simple for a system in one dimension. What kind of motion is this? What kind of system is this? This is the same phase portrayed of this one. Znamo se, če je to? Znamo se? To je potenčnja, potenčnja. Partične vrste, vrste, vrste. To je... Vrste. To je postavno, partične vrste, vrste. Vrste. Vrste. Vrste. Vrste. Vrste. Znamo se. Što je? Quarter. Što je? Što je? kako je infinitiva, zelo to je objavljena objavljena.  More difficult in the situation to understanding the situation, the quality of the situation in more than one dimension. Two-dimension, three-dimension it is already affordable, but he higher dimension of course there is very difficult to be understood. For example in two-dimension what you can do is where is vector field and To je dela všechno-vettorku, je zelo vsezvanele, što se se odliče vsezna, bo je odliče vsezna. Kajวess, da boče, da se potrebe s pismo vseznjane, to je za učin, nekaj je však in izgledaj se in vsezna, da srezne vsezna. Zazvalo se, da so tako deločne, na vsega vsega, da se površila, načo je ta nekaj dinament, in vsega se ga tudi odrečiti, načo je, da bi je trajaktori, da se začne bi, da je to vsega trajaktori, bo trajaktori, da nekaj je trajaktori, načo je trajaktori, da nekaj je trajaktori, neskrič, ko je odstavljena. Však je, da je vsemoj počak, je to občatno, da je vsemoj način pa da je odstavljena z neko odstavljena, da je boj delarizena. In da je vsemoj način, da je začne odstavljenje. Prejdeš, da jošša izgleda noženja na konceptu nulklijenja? Nulklijenja je to, da je začala, da je začala, da je začala, da je začala, da je začala, da je začala, da je začala, da je začala. If you aren't able to make this analysis because your space is a higher dimension, you can, of course, you have to consider the fixed point that are equilibrium state of your of your of your motion, the fixed point satisfies this equation for flows and this equation for maps. It means that you remain in a fixed point, of vorever in that state. z vsej pridljaj, da jste priči. Ale pa, da jste vse vsej pridljaj, da vsej pridljajo, da nekaj možda je začal tega, začal tega, ki je zelo. Po izgledu, ne? Kaj je možda? Jeste poču, ki so priče vsej pridlj. Jeste poču, ki se priče vsej pridlj. Kaj je priča? Kaj je priča, ki je priča? To, ki izgovorite iz vseh domne, zelo jaz vse je nekaj nekaj nekaj nekaj, zelo jaz vse je na prezat, na radočni, matematikni, Aleksander Mikhailovich Ngliapunov. Osprej, izgledaj začetne stabilitve. Ako zafrešte stabilitve, nekaj ne nešte najbolje. Przelaš je tudi bolo stvari. Tudi je tudi stvari, to je bilo. Radjuz Delta. To je tudi stvari. X star. Tudi je stvari, delta. Zato, da se in tudi stvari izgleda, da se pravno vseče, da je stabil nekaj. Vseč, in to je druga sljeda. Vseče, da je. Vseče, da je stabil nekaj, Zdaj. Premaš, kako potražite, kako zdi expertsom prišla počke, čistmenje, representično, nešto, stranje, posledno, primizučno. OK, zato moram pokazati drugo lahko izpersim t파, kak الo so vignetite starring na vse epsilon, tako, da, kaj ta vse trajžaktor je vse zelo vzela v delta epsilon, tako, da moja moja moja moja moja moja moja vse epsilon. Zelo sem vzelo in tudi je srečen. Navrša. I to je tukaj. Zelo si, da echo, da si sem vzelo, da sem vzelo v termine delta, zelo se je, da se demeti. Moje objevak je, da sem vzelo, da si, da mi pomega, The fixed point, less than delta, will remain on the ball epsilon forever. The other notion that Ljaboljov introduced was the notion of asymptotic stability. Thus, you can define a delta, so a ball delta, such that if you start within delta, Je tudi nekaj razlogi, zato jao je, da je razlogi, zato je izstavno. Zelo je, da je izstavno. Zelo je zelo naredito. Zelo je asymtodist, zelo je. Zelo je stavno. Zelo je asymtodist. Zelo si, nekaj da so stabili, nekaj da so stabili, vse smo vsebežjene in stabilite. To je tebezmatematijne definicije, več hodnje definicije stabilitivi, ali zelo tudi bolj, da imajo našličenje, in da se našlič je operativne klj累erije in pa jaz sem način na stavo. Pasko. Ok. And the criterion is, for example, very simple for one dimension. No? The one dimensional system is very simple. In fact, this is your equation of mode. Your equation of mode. So, yes. You plot f of x and if you decide, if you look at the sign of f of x, you can always, also you can soon decide if your fixed point is stable or unstable. Of course, this f of x crossed the x-axis in the fixed points. In this case, f of x is positive, so the field goes this way toward this fixed point. Here is negative and the field is negative, so f of x is negative. So, I go this way and this point is stable, of course. In this sign, instead you have the field is negative because f of x is negative. So, I point this way. Here is positive, so it points this way and this fixed point is unstable and so on. But there is another criterion. Yes? Yes, please. Is it possible to visualize the notion of asymptotic stability with this kind of graph? Of this one, which kind of the asymptotic stability of this in one dimension, you mean? I mean, asymptotic stability, maybe you have, because you have point and something you should go, probably you should have something that's going, I don't know, I'm guessing, OK, I think on that better, but maybe something that is going like this. And I don't know. For the moment, I don't have any example, but I will think about that. Thank you for your, OK, I will think about that. Yes, so another, so, but the, OK, of course, this is of qualitative analysis, but the real analysis, the real mathematical analysis is done by considering the derivative, of course. In fact, the slope, you see, the slope says that this, if the slope is negative, the fixed point, the slope of f of x, if the slope is negative, the stable point is stable, so, if the slope is positive, of course, this is, the fixed point is unstable, OK. Now, what happens in two dimension? And I, in two dimension, the situation is a bit, is a little bit complicated. In fact, you have to study the linear stability analysis, what you do. Basically, you linearize your dynamics around the fixed point. And how can you do this one? You have your equation of motion. You have your equation of motion xi equal to fi x1 xn. You make a variation, so a linearization with respect to, linearization means that you are considering xy with respect to the, and this is called delta xy, no? You are considering this quantity. And you want to write the equation for this quantity. And the equation for this quantity is quite simple, because you have, make the derivative, so delta xy is this dot, sorry. Delta xy dot is something by, nothing but j, the f with respect to xj evaluated at the fixed point, multiplied by the variation xj, OK. This is the, OK. Basically, I do the differential, the differential of my equation of motion. OK. In terms, in matrix form, you can write this delta of x is equal to L of x, where L, of course, is the stability matrix, is the matrix with entries, with these entries, OK. So, L ij evaluated at the fixed point, is nothing but the derivative of f ij with respect to f ij with respect to j at x star, OK. This is the definition. And this is called stability matrix. Stability matrix, OK. Once you have the stability matrix, you know what you have to do probably, you saw in the courses. And, OK, it means that you have to diagonalize this matrix. So, you have to find again values and again vectors. That's a diagonalization. Once you have the stability matrix, your task now is very simple. The task is compute eigenvalues and eigenvectors. I'm supposing that you are familiar with eigenvalues and eigenvectors. Every one of you. Good. The eigenvalues could be, of course, real or complex conjugate, because the matrix is real. The problem is real, no? So, you can have either real eigenvector or at most complex, but complex conjugated. OK. If you want to just a little bit explanation of why this is how the diagonalization works, is that what you do generally in this case, you suppose that you have your matrix L. OK. Suppose that this matrix L has a set of independent dimension. You have a set of independent eigenvectors. OK. That means that they form a basis. No? They form a basis. In the independent eigenvectors. OK. Then you can, what can you do? You can, for example, expand your equation, your linear equation, your expand the solution of your equation that you are looking for. In this basis, no? In, for example, you have some k, ck, uk. You expand this one. And on the other side, of course, you have the same expansion. But since the matrix crosses all the scalars, you can write, sorry, this is c dot, because this is c dot. You are looking at this constant. This is the equation. This is expanded into the eigenvectors. On the other side, you have ck, l, uk. But luk, by definition, each uk is an eigenvector. So it satisfies this equation. And so you end in the simple equation k uk, lambda k, lambda k, ck. Now, since these guys are independent, you are allowed to identify the coefficient of this combination. And your equation for each ck is lambda k ck. Very simple. This is a linear equation. The solution is ck of t is equal to the exponential lambda t, lambda kt, ck with zero. Where this constant defines how you start nearby your fixed point. In fact, so the solution is this. And the constants are set by initial condition, how close you start from the fixed point. Uk determines the direction of approaching or of escaping from your fixed point. And luk determines the type of motion around your fixed point. Very simple. Maybe. OK, now you have the classification, very simple classification, two dimension. You consider the classifications. Very simple. If you plot your eigenvectors, eigenvalues, sorry, in the plane, in the complex plane. Because as I told you, they could be real or positive, real or complex conjugate. So this is the image of lambda, real part of lambda. And this is the imaginary part of lambda. Now suppose that lambda 1 and lambda 2 are real. Real means that they are on this axis. If both are negative, lambda 1 and lambda 2, what do you expect? You expect that your motion converge toward the fixed point. This is the eigenvector, for example, the direction of... And your motion is going... Sorry, let me... This is the vectors. The arrows are otherwise... I'm confusing you. So you have both going in this direction, in this situation. And this is called a node. But the node, if the eigenvalues stay here, so no more negative, but it becomes positive, the node becomes unstable. You run away along the direction defined by your eigenvectors. The other possibility is that lambda 1 and lambda 2 are real, but they lie in the opposite part of the imaginary axis. So this way, then what happens? The scenario is you have, of course, the direction defined by your eigenvectors. This one means you are unstable, you run away from the fixed point, but this one says that you are stable, so you run into the fixed point. And this is called saddle. The motion here happens... And this is saddle, because if you consider the surface, you fall from the saddle. So there is one part, your part you arrive this way, but the other side you go away from the saddle. It's exactly like a saddle. So the right picture is... Those are stable, and those directions are stable. In fact, it represents the saddle of a horse. Finally, if you have a complex conjugate, it means that you can have this situation, complex conjugate. The real part of the eigenvalues is negative, so you lie this part of the eigenvalue. So this is mu plus i omega, and this is mu minus i omega. If the real part of the eigenvalues is negative, so you lie this part of the axis. What you have? A spiral, because... Yes, a spiral. In fact, it means that the... Since mu is negative, you exponentially fall into the... You are attracted by the fixed point exponentially, and of course, there are also the rotation to the face. So you have spiraling around the fixed point. This is a spiral, or consider focus, it's generally called focus, but spiral is enough to visualize, to remember this. Of course, the spiral is stable. If you are spiraling this way, but if mu, of course, lies... If the eigenvalues lies in this part, mu becomes... mu is positive, so you spiral away from your fixed point. If finally the elliptic... If you are in this situation, where you stay here, so negative, the real part is negative, what you have is an elliptic point, so you stay around the fixed point, so you can circle the direction, for example, in values. You make a rotation around your fixed point, this is called elliptic point. Of course, the role of eigenvectors, as I told you, gives you the stable and unstable direction of the motion near the fixed point. There is another situation that sometimes in the book you don't see discussed, and when you have degeneracy, degeneracy means that some eigenvalues, in this case, since we are working in two-dimension, two eigenvalues are identical, lambda1 is identical to lambda2. Just a brief mathematical reminder, the situation when the matrix has degeneracy in eigenvalues is called defective matrix, and there are two notions that you have to be reminded. The algebraic multiplicity, so consider the characteristic polynomial, this is a polynomial. If your polynomial can be written in this way, plus another polynomial of degree n minus m lambda, m, so the generacy of the matrix, this is a root over the m, this quantity is called algebraic multiplicity, multiplicity, but to this algebraic multiplicity corresponds to geometrical multiplicity, and the geometrical multiplicity is the dimension, so the geometrical algebraic geometrical multiplicity, multiplicity is the dimension of the eigenspace, yes, the eigenspace spand by the eigenvector is differentiated to these eigenvalues, and there are two cases, of course, either the multiplicity are identical, so m, so generally you have this property that the geometrical multiplicity could be less than the algebraic multiplicity, the algebraic multiplicity. And what does it mean in terms of, so if you, for example, ok, let's discuss in this way, it's very clear. Consider the case of that, the dimension, the geometrical multiplicity is equal to the algebraic multiplicity. Then you can find, this means that you can find two independent vectors, because the dimension is two. You are still able to find two independent eigenvalues, but the solution, in this case, is written very simple in this way. Of course c1 and c2 are the initial conditions, and if lambda is positive, you have the star, the rate of approaching the fixed point is the same for both directions, because it's rude by this law, it's symmetric, but the situation could be even worse. The worst case is when the dimension is one, you are able to find only one eigenvector, and instead the multiplicity is two. And then you have only one eigenvector. What can you do in this case? In this case, you have one fundamental solution to construct your general solution, because the problem is linear. So the task, when you have a linear problem, you have to find the fundamental solution, combine them, and obtain the general solution. In this case, you have one fundamental solution that is simple e lambda t u1. The only eigenvector you can have at disposal. The other solution, you have to find the second fundamental solution. And what you do? You guess that your solution can be written in this way, e lambda t, a vector a to be determined, multiply by t, a vector b to be determined to. How do you determine this? You take this, and you substitute in the equation of motion. The equation of motion is this one, and I would like to leave you as an exercise to show that these two vectors, constant vector, satisfy this uA, b is nothing but u1. So the old eigenvectors, coincide with the old eigenvectors. The second one, instead b, sorry, a, instead a, satisfy the equation L minus lambda, u, y, u, sorry, a is equal to u1. This is called a generalized eigenvector problem. In this case, you can do the analysis, and you have, of course, a different behavior, because consider that now your solution, of course, the general solution in a combination, a linear combination of the two, and I wrote here this here, and the approaching to the fixed point if lambda is negative is this y, and of course, there is a deformation to the fact that you have also a linear part. You have not only the exponential behavior, but also a combination of t and the exponential behavior. And this changes the local topology of your flow around the fixed point. This is a scheme that I give you for, since you, I will lend you the presentations, you can find the summary of this. In the dimension, the situation is somehow similar. It's a bit tricky, but somehow similar, because, of course, you have to compute again the secular equation. The secular equation means that you have to diagonalize your stability matrix. And you have a situation like this. So, if this is the number, this is the set of eigenvectors, this is the set of eigenvalues, so you have a situation where you plot in the imaginary plane, your eigenvector. Of course, this condition corresponds to someone says this condition corresponds to all eigenvectors lies in this part of the plane. Stability, exactly. But there is the possibility that someone of these eigenvalues crosses the imaginary axis. And in this case, you have in stability a saddle in the dimension, a saddle in the dimension. But also there is a possibility that all eigenvectors or eigenvalues lies on the imaginary axis. In this case, you have a center or a marginal point. And this is a definition that I never understood very well, but in when the dimension is even, this center is called a elliptic point. As a elliptic point. But not a change. You have to, for example, to understand the stability, you have to compute your eigenvalues, put these eigenvalues on this imaginary plot, imaginary axis, imaginary plane, and complex plane, complex plane, and you have to see the position of the axis, to the imaginary axis. Finish, stop. Very simple. The same happens with maps, of course, the difference is that instead of considering the whole imaginary plane, you have to consider the unitary circle. In fact, you can see that basically if you solve the same equation, you have that the stability and stability depends on the sides of your eigenvalues with respect to the unitary circle in the complex plane. For example, here you have stable behaviors, stable points, but if you have these objects, you have a stable behavior. Inside the plane, inside the circle, you have stable behavior. Outside, you have stable behaviors. OK? But the classification is basically the same. The only difference that now if you have maps, you have to remember that the unitary circle is the criterion through which you can decide the unitary circle. OK? The unitary circle. I don't know that. You know that in this course there is a lot of time spent through symplatic maps. Because symplatic maps symplatic geometry, symplatic representation because of course symplatic means Hamiltonian. It's a synonym of Hamiltonian. Sorry? In the chart. Where? Yes? Of course. The maps, we use the circle. The circle in sense, yes, you have to put the eigenvalues with respect to you compare the eigenvalue with respect to the unitary circle. Unitary circle in complex plane. Yes, this is important. OK, again, for the symplatic map, you have to compute the stability. In this case, the stability satisfies again the properties of symplatic maps. The stability matrix is a symplatic matrix. That is something like nemonically you should remember that there is some strange no, some strange map some analogy with the orthogonal matrices. OK? What happens before symplatic maps? For symplatic maps I would like to make this statement. No? OK, the symplatic map you have a symplatic stability matrix. Simplaticity. Simplaticity. No, why? Is written. Simplaticity to use a constraint also on stability. And it is easy to prove that if lambda is an eigenvalue of the stability matrix even 1 over lambda is an eigenvalue 2. So, if lambda is an eigenvalue of L also 1 over lambda is an eigenvalue of L. OK? This is a general property of stability matrix of symplatic maps. How can we prove this one? Of course in dimension 2 is quite obvious because you know that the determinant of a symplatic matrix is 1. And the determinant generally is the product of the two eigenvalues. So, if you have lambda 1, lambda 2 is nothing but 1 over lambda 1. And it is simple in two dimension. In dimension of course the situation can be done in the proof can be done is very simple because you use the properties of the matrix the symplatic matrix you write you write a j a transpose j you can write this one multiply by this you have a j is equal to j a transpose minus 1 first you make first these algebraic passages you are multiplying by the inverse the inverse of that OK? And then you use properties of the transpose the matrices you know that if a matrix has eigenvalues a lambda transpose matrices has eigenvalues so if m is a matrix with eigenvalues lambda it's transpose which kind of eigenvalues has with respect to lambda exactly. And the proof is here in fact if you use the polynomial the fundamental polynomial here you make a transpose of that but the determinant of a matrix this matrix and the transpose is the same so at and a cannot have a different eigenvalues so they share the same set of eigenvalues OK? This is very simple now you can say that if one of your symplektive matrices admit eigenvalues eigenvectors e eigenvalues lambda with eigenvalues this one then you can write this multiplied by the inverse you can have one over lambda is equal to a transpose one to the inverse is e one over lambda e I didn't do anything just multiply by the inverse from here to here but then I use this identity in fact if now I multiply for j both members I can use sorry, I am doing a mess with my OK, if I multiply by j OK I can have and use this identity I can write this one in this way a j e is equal one over lambda j e OK what does it mean it means that if e is an eigenvector even j e is an eigenvector 2 but with eigenvalues one over lambda OK some question about this very simple proof is clear in the algebraic what is the implication of that now in geometry in geometry I think there is a lots of geometric geometry about symplectic manifolds the dimension should be only even so symplectic geometry works in even dimensions 2 in fact Hamiltonian dynamics lives in even in even dimension OK that's good question, thank you because baby I forgot to stress this point OK what's the implication of that so the question no what's the implication of that about the stability the implication is sorry I can't go the implication of that of the fact that you have a symplectic matrix is if the stability of a symplectic matrix is an if if you have a symplectic stability matrix that has a consequence that in 2n dimension of course even dimension there are 2n eigenvalues and eigenvalues are paired in this way if you have the eigenvalues here one here you also have after n eigenvalues you have one over lambda i OK and this is called pairing pairing means if you plot your eigenvalues into the complex plane and you consider the unitary circle because we are considering maps of course they are organized in a very strange way in a very symmetric way distributed in a very symmetric way in what is called what I mean what does it mean it means that suppose that you have eigenvalues lambda but you also complex of course a complex the red one but you have also the inverse also this inverse is eigenvalues and this inverse stay here stay here it is very simple to prove plot for example you have this here these eigenvalues stay here but if this is complex conjugate see this is complex you have also the complex conjugate and the complex conjugate of this stay here but there is of course another inverse of that conjugate eigenvalues should live here so this stability this stability of of simplicity maps is quite complicated for example because if you run soon you can run outside the unit circle for example if instead the eigenvalues is real then you have here inside the circle and one over over lambda is like this if the eigenvalues as rho equal to 1 you have just only the complex conjugate complex conjugate behavior of course you have to remember that for simplicity maps the stability the stability distribution as eigenvalues satisfy these strange rules and consider also these nice properties that these two eigenvalues lie exactly are symmetric with respect to unit circle you see also these two one are symmetric but be careful they are not the couple due to the pairing here and here the pairing is here her here but you need some this is a curiosity you saw that the distribution eigenvalues is somehow symmetric with respect to the circle in this way the property pairing is a general properties of Hamiltonian systems in fact Angelo Volpiani show you that the pairing properties if you compute the Lyapunov exponents Lyapunov exponents are indicator of chaos and they generalize the concept of instability from fixed point to a whole trajectory but this is the subject of Angelo Volpiani's lectures ok I will stop here just just with you I will stop how much ten minutes ok questions, comments any curiosity no linear stability next point is no linear stability in a pause


Plays the ball forward to Hoylund, who's watched him in space, but it's a heavy touch from him. And then Solanki trying to wrestle with Kambala, and he's done well, Dom Solanki running at Kambala. Solanki with a giant edge of the box, gets the shot, and scores! To the low drive! This is 16th or 17th goal of the season, I should say. Celebrates in front of the Bournemouth crowd. He had Kambala on toast there, Dom Solanki. Ran down the middle, shifted onto his right foot, shot towards the near corner. It's loose, Rashford on the edge of the area, still in possession for United. Tees up Garnaccio towards the bar line, pulls it back towards Fernandez. It's going to fall for Fernandez, who smashes into the roof of the net. It's going to see it back. There was one on the edge of the area of Rashford, and he tried to step in front of the Bournemouth defender to get there first, and they just falls into the path of Fernandez, who smashed it in. I think that might be fine, actually, without... The left-hand side of the box running at Kambala, shifts onto his right foot, shoots goals and scores. Basserback, Nippersen goal from Justin Kliber. Shifts the ball onto his right foot, shoots low, straight through O'Nana. And Justin Kliber has goal number six for the season, and restores form this lead. Now Maynard looks to try and bend one like he did against Liverpool, and the referee's giving a penalty, but just seeing a replay. His arm's by the side. Tony Harrington back in his spot, puts the whistle to his lips, Fernandez steps up, right foot in and scores. Four level turns. Bournemouth trying to win the second ball, and Christie left-hand side of the box, running at Kambala, goes one way, then the other. This is going to be a very, very big VAR check. Was that inside the box from Christie? What a huge moment in the game. The referee's giving a free kick.


Okay, and that's five after the hour. So this is Dan Kahn, Executive Director of CNCF, and I am actually in Paris today, closer I think to a lot of our regular participants. We have made good progress on this first chapter or preamble to the Cloud Native Telecom User Group white paper. So I want to thank everybody for their involvement so far and see if we could try and talk through what are some of the open issues that folks are aware of. I don't think Alex Salkever could make it on this call. So I'm curious if Lucina or someone else might be able to say what are some of the areas that are most contentious here. I'm actually, I think Alex is also going to be able to remove a lot of these comments the next time he goes through here. So that was the main thing that I was looking to do on the call today, our hope, if we can reach a rough consensus on this, is to be able to publish this white paper and preamble and announce it in time for KubeCon in two weeks. The other topics is that we do have this joint Tug and CNTT meeting that's there's two of them, two hours each occurring on both Monday, November 18th and Wednesday, November 20th. And I did want to encourage as many folks as possible on this call to attend. And then Lucina and Taylor, were there other, let's see this, yeah, so we, and then so one of the big areas that we need to discuss is the CNF definition. And then another one that I wanted to highlight in the white paper is the diagram of what used to be on past, present, future, and I just replaced that it's now at the top of page eight under the section involving the stack from BNFs to CNFs, where I tried to simplify it drastically and would like feedback on whether I succeeded and whether that new format makes sense or not. Lucina, could you find that? Yeah, it's that page eight diagram on the next tab. That's the footnotes. You're not looking at the right document. It is, yeah, exactly. And there we are. So I previously had a past, present, future version of this and it was specifically linked to some ideas about functionality in ONAP Amsterdam versus ONAP Casablanca, where it might be in the future. And so I really tried to simplify all of that away and just talk about this CNF architecture. And are there other, yeah, so we need to talk about agenda items for the joint Tug CNT team meeting. Taylor, were there other topics that you wanted to discuss on this call? I think this is probably enough for this call. I think if we wanted to take a quick view of the paper, the chapter one, and then maybe dive into the items that folks want to talk about, including the CNF definition. The request for comments, I guess I'll just repeating that we're wanting to get feedback before Wednesday and we're planning on setting the agenda. So this is a LF wiki. If you have an LF ID, then you should be able to update the wiki, add items to the brainstorming. And we have kind of a tentative skeleton. We're thinking lightning talks on the first day. And maybe some more in-depth working session on the second day and potentially a panel. But if you have ideas, please add them to that. Maybe one thing to follow up on the definition specifically, Dan, to try to help people that are in different parts of the world and may not be able to attend. Just a suggestion to have a follow-up to today's discussion of the chapter one white paper items and put out some suggestions for time. So if we can get those on and I'm willing to join an early morning to make it easier for people in Asia Pacific to join. Yeah. So if we scroll to that page, I believe it's page five is where we have different alternatives. And I wanted to throw out some principles to try and select it here. And I think we could probably do it fairly fast, which is I'm very interested to link to I believe it was Frederick who made the suggestion link to Etsy as a informative reference. That I think there's a huge amount of work that have been done in the past and a lot of valuable background that people should look at. But I would really like to try and avoid making normative references externally, where it requires you to be on top of all the Etsy material in order to understand the meaning of this document. And I'd like to try and minimize the normative references we made period at all. I do already have one exception to that, where we're making a normative reference to the cloud native definition. But because that documents three sentences are also controlled by CNCF, it's a I think a simpler process. And so I will come in and I think make my own all six suggestion here. But I guess I would love to see is there anyone and I was curious to hear on your perspective that has a perspective that a CNF can't be about whether it just a container can be a CNF or whether it has to be a pod or whether it can be a collection of pods. I think I think the fundamental question is what do you associate with the letter C if you associate containers and there is one answer. If you associate cloud native, then naturally containers or, you know, maybe even pods are not enough. Pods are certainly better because there is the assumption of using Kubernetes for orchestration containers, you know, you can use without Kubernetes and therefore could be heavily contested whether that could be considered cognitive. Sure, I will say that I helped coin the CNF term. And we were always clear that it was cloud native network function, not containerized network function to try and make things a little bit more specific. But that doesn't totally answer my question about trying to limit exactly what a CNF is. I would say not limiting it to containers or pods. Since I expect the platforms, including Kubernetes to use whatever underlying blocks make sense. I think the functionality may look similar to a pod or container. But there may be something that's not a container. And I don't think that's the definition of a container. And what it does is what I think you care about. And then more specifically, all the different parts in that would be the principles around being cloud native. And so you suggest that go on, go on. I would say that a container is an implementation of one part of that and just like pods, they're an implementation. And the technology that's implementing those, if we point to those and we need to grow beyond that, then we're not going to cover it. Or if someone came in and changed those and it's from someone's perspective of caring about the principles, maybe a pod is no longer cloud native. Unlackly, but that would be why I would say let's focus on the definitions of what they mean versus the implementation. I tend to approach the kind of cloud nativeness from the benefit and in the telco context, certainly the benefit is the desegregation, but also the unified lifecycle management via Kubernetes APIs. And so I would not link it to containers necessary because we might want to introduce new runtimes, handling VMs or Cata containers or whatnot, which are strictly speaking, not containers, not just Linux containers is something else. And therefore, if we limit things to containers, then we sort of shut down the cloud nativeness towards those technologies, which is certainly not something I would support. Just my two cents on containers. So containers are excellent for an example, but it cannot be only about containers. Exactly, that's it. Like it can be used as an example, but like either containers or bots, but I think like limiting it is not perfect. I think it's worth pointing out that the kind of five the five alternatives that we we've put into the chapter one document. I don't think any of them mention containers. I think that's important to note. There's a lot of talk of microservices and immutable infrastructure, but I don't think there's any mention of containers in those definitions. So this is Ramki. So I think it may be worth actually pointing out the distinction of the concrete example. So what I'm seeing is like, if you go to, for example, the RAM components, for example, CU, especially the DU, there the movement is just around containerization, but not really cloud native. I think I can, in fact, add that I can volunteer to add that. That really is a pretty concrete example, which is emerging. So then that draws like, hey, if you want to just go containers, but not truly cloud native embracing Kubernetes, and this is like a method to do this with an example versus just saying, you know, it's always cloud native, correct. So part of the way that I think of cloud native is if I if I have a piece of software and I stick it in in a cloud native orchestrated system, will my system will my application gain the benefits, the majority of the benefits that that people typically look for in such an environment? So, for example, can I scale it horizontally? Or can I get it to auto heal and recover? And is the configuration declarative so I can configure it in an easier and more holistic way? And so I think part of it is trying to try to drive the definition in that path. And I think part of the problem that we're going to run into is like, yes, you can build some like something that follows the cloud native definition, but and then stick it into something into an orchestrator that does not support cloud native principles or cloud native orchestration. And even though you've built it in this particular way, many of the benefits like you'll still get benefits out of it, but you won't get that. It's it's when you couple it with the orchestrator that you get that you that you maximize the benefits. So that's the heuristic guide that I tend to use myself. Frederick, can you go ahead and recommend one of the five options though, based on what you just said? Sure, let me read the five options. So and it looks like someone's accidentally deleted part of all one and all of there was all one dot one. I don't have access to the version history. Can someone put those back if you have like undue capabilities? I do. It'll just take me a minute to find it, though. But I don't think I'm the one who broke it. I just have suggest access. So I don't think I did. Well, that's ongoing. I think one of the big difference between the different alternatives is some of them are talking about cloud native network functions while others are talking about cloud native network applications. And I think it's worth pointing out that while, you know, typical telco functions are cloud native network functions. Network functions themselves are usually defined by standards like 3GPP, while application can be anything like an OSS or a BSS as well. So naturally, application is a much wider category than network function and a much less already defined category than network functions. So for that reason, maybe it would be better to talk about applications and not network functions. If then you are adamant not to link into any other normative definitions then network functions are very clearly defined by ACNFE and then the network functions themselves are defined by 3GPP. So maybe it's a better option to talk about applications. Sorry if this is unnecessary complication to the already complicated discussion, but I have to sort of say this. Yeah, this is Watson. As far as the definitions, one of the things that keeps coming up is the grouping of microservices or if you want to say containers or pods or VMs or whatever, and then the a single microservice pod or whatever. And so within cloud native, a group of microservices is an application. So within the Epidocs, it looks like there's a group of a group of a functionality can be a network function. So that's some one point of confusion. Yeah, I'd agree with that. And I think the what I tried to add in for one dot one and one of the others, I think was was to use the link it to the definition of a virtualized network function component which is the bit in Etsy, which decomposes that higher level function into those multiple bits of functionality potentially. Right. And so with a kind of cloud native network function, that cloud native network function component could be a microservice. Right. That's where the direction on the all four and all five we're going with pointing out there's a concept of component in Etsy. All right. And then hit undo or something and get the stuff back just from hitting undo. It might not look very nice. But if you if you click on see new changes at the top there, you'll be able to see what happened. It'll be crossed out, but you'll be able to read it. And I can't get to the version history. And I'm not saying that new changes. I don't know. I don't know who's sharing, but in seeing new changes at the top, it's grayed out next to help. That should let you see if we scroll down what the what was deleted. It's straight straight through, but you'll still be able to see the words. Oh, that'll work. Can you highlight those and we drop them back in? Thank you, listening. All right. One more time. You got to do it twice. It did a strike through and then you can do the end strike through after. OK, there we go. So I think that all one was trying to. Expand on what was in the one of the other white papers to have the Etsy. Definition that it was added. Yeah, and. It was calling out the specific terms from the Etsy definitions that were useful rather than just saying as per, you know, whatever the document reference was. So and then. I don't know if we want to break down these. What it doesn't have in all one is a. It doesn't define what well-defined external interfaces or well-defined functional behavior is. If those have definitions somewhere else. They leave it vague enough that that could be anything. What is what is a well-defined external interface? And this could I think there was someone mentioned benefits. As as a one way to look at things. The problem with having benefits is as your as your part of your definitions. It doesn't it leaves it open to interpretation. So then you could say if you want segregation, well, you could do segregation in many different ways. So that that would be the only thing there and in any case, this is just an expansion, I think, of the from the other white paper to make sure and include those parts. It doesn't include some of the things that folks have mentioned like the cloud native network components. It does mention part of the virtual network function, but then points over and I think all one dot one is trying to. Add in part of the cloud native. Definition or what what is that main out of the preambles and principles document and expand a little bit on the Etsy definition, it looks like. Yeah, so it's all dot one one dot one. Sorry, there was a comment on alt one that the CNCF cloud native definition wasn't wasn't kind of detailed enough about what we'd want to see from a cloud native network function. And so I added in that that bit about immutable infrastructure from the cloud native principles document that was being written. Because I felt it called out what we were looking for from a. You know, in terms of guidance for the software vendors, if you like. But the rest of it was the same. More or less sounds good. And can. Can you scroll down listening to all one dot two? So this one I was trying to build on this based on I think the alt four and five. I added this one in. So it's taking some of those ideas are the direction of all one dot one. And then linking it a little bit more towards the cloud native side. So this is mainly to do a mapping. So what are we looking at for folks that are doing cloud native already that are interested in? The net cloud native network functions. What would that mean? And then if you're going the other way, trying to say, how do we take something that we understand from the NFV world and move towards cloud native? So this was talked about a little bit earlier. It seems like a cloud native network function. If if we take actually, let me step over, if we take a virtual network function, it seems like a virtual network function is a cloud native or could be an application. And so a cloud native network function. Could be a cloud native application. So it's it's a specifically an application that focuses on the network domain. And that's that has a good definition. And and then this is saying it's similar to a virtual network function. And it's specifically saying similar because I think there's going to be pieces maybe like the virtualized and the virtualized hardware abstraction are probably not going to be part of a cloud native network function. That's going to be there's going to be something pushed out. So it won't actually be in the the the network function itself. That would be taken out in a virtual network function. It needs to know how to work with it. But that's potentially not going to be part of it. It may be part of it, but that's why I'm saying it's similar versus the same. And and then Reeves, the parts that you haven't heard about the immutable infrastructure, declarative APIs. What I added in was composed of microservices to your one dot one. I think that's important. It wouldn't be well and proposing that a cloud native application is composed of microservices. That's that's already a defined definition of what those applications are. So if a virtual network function is an application, then it seems like a cloud native network function would be an application and you would follow the standards. And then if we have that, if it's composed of microservices and other parts that allow it to be an application, will micro services are very similar to the virtualized network function components that are defined in Etsy. So now you have a mapping on both the application level, the network function and the components that it's built from. And I think that would allow people that are doing application level, but the development and deployment to know on both sides, if you're familiar with the networking world or the cloud native platforms, what you want to do. And then I think the rest of it is as one dot one. So that goes down to the saying what Etsy defines a virtual network function as and then going on from there. But what's not defined in here, but we could add, probably should be breaking down this as a paragraph would be what are virtual network function components? I think going back to one of your earlier comments as well about what is a well-defined external interface and well-defined functional behavior, we could add that they are defined by industry standard specifications. You know, Thomas mentioned 3GPP, for example. You know, that that I think that's how I would read it in that they are defined by an industry specification. And I think that would be a worthwhile addition just to quantify a little bit what we are referring to. Otherwise, I think that's a really good definition of the addition of similar and composed of microservices makes it more sense. And to me, I think clarifying the VNFC, mapping it out, I think makes perfect sense. I think the VNFC mapping would be to more of a part within a microservice. I think if you add it, I think that tells the story very clearly. So is do we want to add in the concept of having to tie the mechanism to talk to a CNF to a set of specifications? Because this is something that will probably happen organically, organically anyways through the market. One of the things that I want to be a bit careful with is that we don't we don't deny groups who have more esoteric needs or want to or want to experiment so they can try to push the the boundary further. Like perhaps what we could do is say for the purposes of certification or so on, then yeah, you absolutely must use a well-defined mechanism. It seems like we're mixing implementation along with the definitions for the the standards we're talking about where we're saying those standards have a definition for the well-defined external interfaces, for instance, and being able to point to a specific standard that we think is important. For example, I think we should make sure it is example. Yeah, that definitely has just an example because otherwise I think Frederick's concern is is met. From the standpoint of saying you're from a virtual network function, we're saying that those standards, like one of them, is you're going to use virtual hardware abstraction. So to be able to you're at some point, you're going to implement some type of connectivity to the network and hardware, but you're using some type of abstraction layer. And it seems like when you get down to the external interfaces and functional behavior, there's there's this edge between what the standard is and then how you're going to implement it in the virtual network function. I don't I don't know how far we need to go talking about implementation versus saying any type of behavior should be implemented following cloud native standards. If if you're if you have a standard that requires you to, let's say that maybe of either the functional behavior or the interface from the standard that you're wanting to integrate with would not allow you to do it in a cloud native way. I think that should we shouldn't make the definition of a CNF limited to a standard that's not cloud native. Instead, I think we should say this network function that's implementing that functionality that's required in the industry because it's already out there is not cloud native. And I think that would be OK. So we could say we have. Ninety five percent of our network functions are now cloud native. This set, which are critical to the business, are not cloud native. And here are the reasons why. And then you work on the standards to allow them to be implemented, if possible. But what we're talking about is a set of principles. And deaf and will not print definitions, but principles and they. That you can follow that will give some benefits. If the benefits do not meet your needs, then you select something else. What we're saying is if you follow the principles and they give the benefits, then someone should be able to say, I've followed all the criteria and meet those, the. What's required to get the benefits based on definitions. So that was one thing that that, you know, I I couldn't really place I couldn't really place. And then you refer to a standard, which is not cloud native. Which standard did you have in mind? Just curious, I'm you referring to three GPP or or or Etsy NFV? When you said it's a non cloud native standard. Well, I don't know why they're one of those. Well, I'll respond to it. Frederick, and then you can add. What what I was saying was if there's a standard out there. So this goes with. We need to be able to integrate with Brown. Phil, and there could be things I've done a lot of work with stuff that's legacy, you can have stuff that's been running for 20 plus years. And you need to keep integrations. So whatever it is, I'm not saying that three GPP is is legacy. What I'm saying is if there's something in there that you have a definition for some interface because it's something that's running and you can't implement that in a cloud native way, that's OK. It just happens to be one of one of the one particular application that you're going to implement that's not going to be able to follow all the standards. That's OK. Whatever it is. What what I'm saying is the cloud native network function definition should focus on being cloud. It should follow all the principles and definitions to be cloud native. And then and being able to implement as much of the existing applications that are needed in in the NFE world without. Without it being without compromising what it means to be cloud native. Sure. So let's say let's say if there is a standard that talks about an interface between two network functions that require, you know, layer two functionality. Would you claim that that that is something that is not cloud native? And it's not something that should be implemented in a cloud native way. Absolutely not on that particular example. That that example can be cloud native in my mind. I mean, that's specifically what stuff like network service mesh and other, you know, many other projects and are trying to solve how to do these layer two things in a cloud native way. I think that you can do all of if we say OSI layers in a cloud native way. If you go into very specific examples, I'm sure that we could go through and find something that we say this is not possible to do in a cloud native way. And you don't have to think about networking. There's enterprise applications and many other applications that you could say the way you've designed this does not allow you to build a cloud native version. You'd have to redesign it completely, which is OK. Not all principles are going to be beneficial to solve all problems. And this is Watson. Here's one thing also to keep in mind with the definition of cloud native microservices and so on and so forth. Even in this paper, if we were to say, you know, put with definition, we want here, if you have a monolith and then you say that it's cloud native communities just going to say, no, it's not regardless of what we put here. So there are certain certain patterns that we might consider anti pattern. And, you know, it's just we're not going to be fooling anyone. So it's something to be aware of. So some of the charge is going to be that some of these things that are ported over our monoliths and not microservices or not, they don't use microservices. I think I think fundamentally these standards that we're discussing very, very seldom define something that dictates a certain way of implementing a network function. So whether it's implemented as a network function or as a million of modules which are microservices, these questions are typically, you know, orthogonal to what these standards define, which is basically what is the protocol we are using for communicating and what are the state transition diagrams and so on that we need to adhere to. So I see in many cases that when it comes to telco standards, they sort of ignore whether the implementation fulfills the cloud native criteria or cloud native definition or not. They just look at how these functions communicate through their APIs or through their interfaces. So I don't see anything wrong here with the approach. I just wanted to understand what when you said, you know, non-cloud native standards, how do you mean that? Well, can I have a little bit on this particular one? Yeah, go ahead, Frederick. Yeah, so part of what I would, the reason I brought that particular thing up in the first place was first worst-case scenario, and I don't think anyone's doing this year, anymore is we don't we don't say it must be the host user, it must be Mehmet or, you know, here's the Blast, the Blast list. I in these standards, I'm I'm glad to hear like the other various standards don't push into the implementation to to that degree as well. So for me, that's that's a very good sign. The one thing that I was thinking of is we need to make sure that there's some form of flexibility for for operators to be able to choose what what they want within their their infrastructures and also the flexibility for vendors to pick and choose things that that work for them. And I want to be a little bit careful because we we we say that industry best best standards like that. You know, which standards are we talking about? Which ones are going to be the ones that are Blast or or not? Like for me, that that could be a best practice that we drive for saying we highly recommend you use industry, well known standards and and so on. But but I also want to be careful with not creating a definition that becomes dependent on an external on an external standard. And my preference would be that we that we have part of the kind of native path. The you say what you're high. You say how you communicate with the outside world.  And you specify with payload that you have, like my payload is going to be IP or Ethernet or MPLS or something else. And that way that you then give you're not created a declarative setup that allows your orchestrator to pair things together. You give it enough context that they can start pairing things or tell you when things will work while at the same time providing the flexibility for for future innovation and and for operators to be able to to standardize if they if they choose to do so as well. From the standpoint of implementation, when it comes down to whether it's vendors trying to help build things or the operators and either defining or the platforms that are being created, all of it, whoever is working on this. For all of the brownfield, folks are going to want to integrate with existing standards from wherever the standard bodies so that they can but utilize existing infrastructure, integrate with other groups. And I don't think that defining having a declarative manner for saying, here's what we want. They either a network function or platform is Frederick saying, I don't think that stops us from using existing standards. What the suggestion is is we don't have those external standards as part of the definition of a cloud native network function. Instead, we say, here's how a cloud native network function should behave. And here's how we expect it to be developed and the platform support that's expected. And you can talk about all the different layers, which are in other areas, not in the center. And then as far as implementation, you're saying, when you're following these principles, we want to be able to support existing standards. Well, that's that's would be developing an application to meet a specific standard. So I don't think there's anything that we're suggesting to define being cloud native that would say you're not able to meet a majority of the standards. I just put it as a note that there could be specific example application or I'll say, virtual network function or a NFV platform component that you wouldn't be able to implement in a way that someone would recognize it in a cloud native way. But as far as following, if we had a cloud native definition of a CNF that's fully cloud native and not saying it's using it's it's using an external standard and it must have those interfaces, then you're still open to implementing this doesn't block us in. And that seems like when you would say, what's an example of a cloud native 5G session border controller? Well, that's probably something that we could have an example section. How would you create that as a cloud native? And you'd probably break that down and say, here's how it could look. So it's not stopping either integration or implementation of those standards. It's as Frederick said, we're not locking ourselves into expanding. Someone could build a brand new Greenfield platform that doesn't maybe use this completely different protocols that gives you all the same benefits and and goals that that a network operator may want without using existing standards or they can integrate either way. OK, I think we're going to have a particular point then. So I think the plan is with this definition and we're coming up on the hour and what we're going to do here. We were going to schedule some follow up calls. Ideally have have it available for some of the folks that are in Asia Pacific that weren't able to join and give them a time to review. This call is recorded, so hopefully that can be sent out and folks can review, listen to these and add comments. We'll try to have the original one and all one back all one and all one dot one, sorry, which had comments and folks can add to this and and then get some, I guess, more feedback from hopefully more operators that are that are out from what I'm hearing it. It sounds like the. Part that is agreed is or maybe moving towards agreement is a cloud native network function. Seems to be we could say that it is an application, a cloud native application. I think there's agreement on that and it's based on components or pieces. And maybe we can say that's microservices, but there's I think we're saying. It's a the application level versus a CNF is a microservice. At least I'm hearing that there's more movement towards that. I think that would be all four if you look at the definitions to simplify it versus all three. So it's a network function is like a cloud native application, which is similar to a virtual network function. And that's made of a microservices, which are similar to virtualized network function components. And not to say we're going to use all four, but it sounds like at least that keeps coming back up that I at least from the NFV world, a virtual network function is composed of many little parts. And that's similar to a cloud native application. Yeah, I agree with that. And then we. Seemed to be in agreement that there needs to be something talked about, what are some of the principles? Some of them are pointing out immutable infrastructure, declarative APIs, microservices, and that repeatable deployment process. That's one of them that keeps coming up in reference to the cloud native principles, specifically saying if it's a CNCF definitions and we're talking about cloud native, those may be moved into a later chapter. So it may not be a different document. And then some type of mapping to the Etsy standards for for understanding. So get having an understanding of for folks going one direction or another would be desired, however the wording is going to be. And then maybe something regarding being able to implement existing standards and how we word that whatever that that portion needs to be looked at. That's that interfaces and functional behavior. Does anyone have anything else at about four minutes? When are we looking to have that follow up call to review the definitions? Do we think? Well, I'm guessing before Kubecon. Yeah, before Kubecon, I posted some suggestions in the chat, I think Alex is the only one that responded and I can drop them in here. But we're kind of at the end of the call. So let me drop it right in here in the action items. Well, I'll put it right here in there. So I'll put them in the document. Thank you. So I was suggesting five a.m. Eastern for Tuesday, that's tomorrow. And that would be the coincide with the normal talk on music group third Monday call time. If that's unless that's too soon. But thinking that we don't have a lot of time for. We're going to get feedback if we're trying to get this done before Kubecon and publish. There's just not a lot of time that we could spread it out. Maybe, you know, it just depends on what's the main thing I would like is if people would promote to get eyes on this and get some feedback. One last one last thing as well, unrelated to the to the white paper. So I just want to make a quick announcement. So there's an edge computing world events coming going on in Mountain View in early mid December. And I'm part of the program committee to help load it up with with people on the networking track. And there's still a couple slots open and I want to make sure we get some cloud native talks in that particular path. So if you're in the Bay Area or are OK with traveling in that time period, then get a hold of me on Slack or LinkedIn. I'll see about carrying you up with the with the organizers. You know, I just want to talk that out as well before we break. Thanks, Fredrick, for bringing it up. This is an empty cell. I'll touch base with you guys. Yeah, it's a good one. Thanks. Perfect. Thanks. I think you're welcome, too. So my, my, thank you. So the thanks, Frederick. Maybe post to the TELPA Music Group Melling List and the Slack channel about that. Yeah, that's a great idea. I'll send that off as well. So I'm not hearing anything other suggestions for a follow up. I will say that the main reason I was doing tomorrow at five a.m. Eastern was I know that there's some conferences going in Shanghai and I think that it ends on Wednesday and I could see folks traveling to trying to hopefully get some of those folks to join. So they'd see this in their evening, but not midnight. So that would be good. I guess I'll post and if I put another post in the Slack channel, but we can schedule those, maybe get them on the calendar for 5 a.m. Eastern and 8 a.m. Pacific on Wednesday. So two more calls to get some feedback. I think that the meeting, thanks, Lucina for posting this. I think the meeting on November 18th, which is during KubeCon, probably be canceled unless someone else is going to run that. I'm not going to be on that call. I don't know how many people would be available. Does anyone want to volunteer to run that and keep it going on November 18th or we can just cancel the call? We'll post about that. Taylor, this is Taylor from CNCF. Do you want me to go ahead and cancel that off the calendar then? Please. Okay. Thanks. Can you follow up with me after the call regarding those other two meetings? For November 5th and 6th. Yeah. Yes. Thanks, folks.


It's a great pleasure for me, these beautiful nations, these pearls of the ocean, which of the Caribbean nations are going for this fast, soil is not a separate subject. If we are interested in health, if we are interested in agriculture, if we are interested in the well-being of the citizens of today and the unborn child of tomorrow, attending to soil right now, attending to the soil biodiversity right now. This is a must-do thing in our life. We are at a crisp of time right now, if we act now, if we bring in the policies and start implementing this, in the next 12 to 15 years we can make a significant turnaround. So this memorandum of understanding that we are signing, it will also bring some technical assistance to the Caribbean region. This is a historic moment because here is the first step to turn around. Sar Guru, welcome to the Daily Show, wonderful. I know that we also are facing a soil crisis, where some people are estimating that if we do nothing in 50 years time, we may not be able to grow anything because we may run out of soil, which, forgive my ignorance, I did not know that soil is even something that we could run out of. The thing is, this has happened in the last 50 to 100 years of industrialized farming that the organic content in the soil is going away because there is no replenishment. Are you getting any signs that governments are enthusiastic to try and do something about this? See, in the last two years I've been talking to various country heads, various politicians, political parties. We have written to 730 political parties on the planet to make sure that they include soil and ecology as a part of their election manifestos as their political philosophy. Whatever they believe in, right, left, center, whatever they are, soil is one thing which is a common factor for all of us. We're always looking at what divides us. It's time if you don't understand the consciousness of this cosmos and stuff like that, at least you understand you come from the soil.  So this is something that you're very concerned about and rightly so, that there's a problem with the topsoil in America. Fifty percent of the American farmers have not seen a dollar in the last twelve years and the highest suicide rate among all professions is among the farming community in United States. I don't want to be that one who didn't care enough to do something which matters because right now we are consuming the soil and food that belongs to the unborn child. This is a crime against humanity. I hear you. Let's make it happen. Let's make it happen. His spiritualist techniques have gained him a following of millions and we welcome him, sir, if you'd like to. And you were interviewed by Trevor Noah, who is the darling of the left, and then you were interviewed by Joe Rogan, who is the beloved of the right recently. So is it Sadhguru who unites the left and the right, or the very important issue of… Sadhguru- Soil. Soil unites left, right, centre, everybody. Participant- Sadhguruji has said it's important for all, particularly leaders, whose decisions affect millions of lives to be conscious and inclusive. And this is where each one of us can help. Participant- How are we going to make farming more respectable and more… more profitable as well? Sadhguru- When I go to the farmer, we are only talking economics. How to enhance your income. And if you want next generation to go for agriculture, this is the only solution. That he must make more money than a lawyer or a doctor or a IT professional. Participant- Wish you a very successful, joyful, accelerating journey. Sadhguru- I will take care of the acceleration and enjoy myself. What I need is, next hundred days from 21st of March, we want the whole world to talk about soil. Participant- We're looking at how can we involve women as leaders in climate action.  Unfortunately, adults have come to this place where if I see him, I will see what country belongs to, what race is he, what religion is he, what caste is he, what creed is he, all this nonsense. But when they see a child, a girl child, I believe people's hearts will become little more tender. If you want to say, I love you to your child, you must just say, save soil, because it is a more committed way of saying, I love you. Participant- Sagar's gift of making a systemic issue so personal to each and one of us in this room and the generation of responsibilities so relevant for everyone. You know, it's shocking and unnerving. We need to take this up and save the soil and act more sensibly and responsibly. Participant- It's an extraordinary campaign. It echoes what we in the Commonwealth have been aspiring to do for a number of years. Participant- Hedge, Hedge, you win the toss. Participant- Ah, I'm batting your bowling. Participant- I wonder what the pebbling in. Participant- That gave me a bit of a feel about it. Participant- Who did you hear, right? Participant- Take a picture and actually copy it. Participant- It's a great pleasure to welcome you to London. Participant- You are probably one of the most famous Indians and what you're doing today is really important because people like you who have a wonderful following of millions and millions of people and what you're doing with that is so important with saving the soil. If only they could have seen this at COP26. So I think what you're doing is phenomenally important. Participant- You must slide to the frame and see the same soil, okay? Participant- All the children see the same soil.


I made a video the other day talking about using app images and having a problem using dmenu and having app images on my system because dmenu, by default, can't search for those app images. It actually can't search for anything that has a desktop file, which is typically how most menu systems figure out, you know, how to populate their menus with, you know, all your applications, but dmenu just simply looks in slash user slash men for binaries and that's all it displays. And people were like, well, why don't you use a better run launcher? Why don't you use Rofi? Because by default, Rofi searches the desktop files and it will actually display all these app images I have on the system. Now I have used Rofi in the past over the years. I've gone between both Rofi and dmenu many times and the early part of my channel's history. If you go back and look at all of my old videos about open box, especially in Qtile, you would see me using Rofi in those videos. If you go to my get lab and take a look at my dot files for open box, and I think in I3 as well, the default run launcher in my dot files for those particular window managers, I think is still Rofi. I don't think I ever changed it. So I've looked at Rofi in the past, but here in the last few days, because I've been getting into more app images, I thought, you know what, I'm going to switch back to Rofi for a little while. I want to play with something new anyway. And I really spent the last two or three days doing a lot of stuff with Rofi, neat stuff, theming Rofi, trying to get Rofi to work for me. And I've been enjoying it. And I wanted to show you guys a little bit of what I've been doing. So let me switch over to my desktop here. And I'm logged into Qtile today. So this is the Qtile window manager. Let me open up a terminal and I'm going to run this command dmenu underscore run, which is the standard dmenu command to actually search for programs. And by default, it's just going to search for binaries and user slash bin. Now I have hacked on it enough to where I can get it to display the app images that are installed in my home directory at slash applications. But if I try to launch one, will it actually launch? It actually does launch. Now that sometimes fills and I don't know why it fills. So it's kind of iffy. Dmenu is kind of flaky on these app images. That one actually worked. But sometimes I will actually hit enter on these and they don't work. You know, I'm just done with Dmenu. So what I have decided to do is instead of using Dmenu, I am now using Rofi. So when I hit my standard key combination to bring up a run prompt in my dot files, I typically use super shift enter. Now when I do super shift enter, this may look like Dmenu, but this is actually Rofi. I just themed it to kind of look like Dmenu. And you see the very first application there, Firefox. That is Firefox as an app image, because that's the only Firefox I have installed right now is the app image for Firefox. If I hit enter, it does always launch from Rofi. I've never tried to launch anything from Rofi that didn't actually launch. Now let me do super shift enter again to bring up that Rofi menu. Now this is not what Rofi typically looks like. Rofi typically is in a box in the center of the screen, but I kind of like the default horizontal bar of Dmenu. That's kind of why I've gravitated toward Dmenu these days is because it's just a simpler cleaner run launcher, right? I don't need anything flashy. So I kind of themed this to do what I want to do. Let me get into the configs for Rofi. So I'm going to CD into dot config slash Rofi. And if I do an LS, I created a couple of folders here, scripts for some scripts that I'm going to be playing around with with Rofi. I created a themes directory. That's where you need to stuff all of your themes that you create for Rofi. And I played around creating two themes here in the last couple of days. And then you need a config file. Typically, it needs to be config dot rassy. And if I open this with VM here, so you can see all it is is the word configuration. And then the squirrely braces. And then theme colon and then in quotes, the path to whatever Rofi theme that I want to use. And I'm using config dot config slash Rofi slash themes. And I created this thing called DT dash center. And let me quit out of that. If I just run a Rofi dash show run. And this will just be Rofi displaying the minaries on the system kind of like what D menu typically does. And it uses my theme that I call DT dash center. And you can see I get a run prompt that's got this blue background. It's got this border radius as this curved border around it. And then I've got some lines you can see the one that's selected has a red background. And then the others are dark, but they are alternating odd and even colors.  And I could start searching for something. Maybe I want to search for a lacquerities. And, you know, once I get the correct search, I just hint and are very similar to D menu. Now the difference with Rofi compared to D menu, obviously, you've got some more options as far as configuration and colors and everything. You can really make this thing look like whatever it is you want to make it. So let me CD into my themes directory here and do an LS. And let's actually take a look at that DT dash center dash recipe fall. So let's open that with VM. Let me zoom in a little further. And these config files, all they are really are just CSS cascading style sheets. If you've ever done anything with CSS and doing any kind of web development, it's it really accepts all the standard commands like background color, border color, text color, setting the font. You know, I created these variables here prompt font, prompt background, prompt foreground. And what I do is I just call upon these later. So I have a section down here where I define all the CSS or the prompt. And you can see the padding is at prompt padding. So I'm just referencing what I put up here toward the top of the document. So prompt padding is four pixels, etc. Pretty easy config file to work with if you're used to using cascading style sheets. Now let me quit out of that. And let me show you the D menu like config, which is this config here, where I get a very D menu like horizontal bar appearance. And this is not a very lengthy config like this was really simple to set up. All I do is you know, I define the background color, I define the prompt as far as the background and the foreground color of the prompt, the width and everything of the input box, you know, where you're typing, and then whatever I select, you know, I want to set a different background for, you know, what I'm selecting in the menu. And that's basically it. Rofi is really easy to configure to your heart's content. It's a lot easier to get Rofi to look the way you want it, then D menu. The other thing with Rofi is Rofi doesn't need patching. By default, D menu really doesn't do much out of the box. You don't have a whole lot of configuration options. You don't actually have to patch Rofi to do anything. All the features, all the stuff that Rofi can do is already built into it. Now some people may say, well, that's just bloat because there's going to be all kinds of functionality built into Rofi that maybe you'll never even use. You don't even know is there. I get that. But for most people, not having to patch a piece of software to do basic things is probably an advantage. So let me show you a little bit how to work with Rofi. I showed you the command earlier, Rofi space dash show and space run. You could also have done D run and this will actually do your dot desktop files. So instead of doing just the binaries, D run actually will give you the dot desktop files. It will give you the name. It will give you the generic name. It will give you a description out to the side that is in a little bit smaller font and it's a tele-sized and it will actually search all of this stuff. Like if I just search for browser, like it actually searches the descriptions as well. So it's great, especially if you're not sure of what the command, the binary command is for something like a library office writer. I know is L O writer all one word. But if you didn't know what that was, then having something like Rofi dash show D run is much better because now I can just search for, could I search for word processor? Yes. And you see it actually finds that because that's the description of library office writer. Some other possible things that you could use with Rofi Rofi dash show space SSH is another mode you can do. This is just some of the stuff I've SSHed into before. And I'm looking at the output here in the terminal. It's also telling me there is Rofi dash show space window. And this is showing me all of the programs that are currently opened. So if you needed a quick way to switch between programs, which in your traditional desktop environments, typically something like alt tab or super tab will bring up some kind of program switcher. And you don't have to use that. Or if you're using a standalone window manager that doesn't have that kind of functionality, you could actually bind that to do Rofi dash show window to switch between your open programs. Let me clear the screen here. And obviously the big thing with Rofi as compared to D menu is people are like, well, you know, typically with D menu, you're piping stuff into D menu, right? You're using D menu with your scripts. You can actually do that with Rofi to it works exactly the same way. A matter of fact, it's very easy to convert a D menu script over to a Rofi script, because whatever it was you were piping into D menu, all you have to do is just pipe it into Rofi instead. So it's like a quick 10 second change probably for for a script just converting it over from D menu to Rofi. For example, if I ran an LS command, and I'm going to type the full path to LS because my LS is actually aliased, but I'm going to type slash bin slash LS. And just to run it LS here in the current directory I'm in, and then I'm going to pipe that into Rofi. And then I'm going to do a space dash D menu, telling Rofi I want Rofi to act like D menu in this particular instance, which means I want to be able to pipe something into Rofi. So just letting Rofi know I'm piping something into you, then dash P for prompt.  And if I hit enter, it ran a LS in this directory, you can see the prompt run colon. So pretty cool. Let me CD in the home directory, because this will be more dramatic when I'm in a directory that actually has a lot of stuff that I could, you know, actually pipe LS into. So it's very easy to use Rofi exactly the same as D menu as far as just piping stuff into it. For now, though, I think, you know, I'm pretty satisfied just having my two configuration files because I only really ever want D menu or Rofi in two layouts. I either want it just one single horizontal bar at the top or the bottom of the screen, depending on what I happen to be preferring at that time. Or I want, you know, a menu centered in the middle of the screen. So I have two config files for each of those. And I'm pretty happy with this. I really like that that centered look there. I'm really happy with the theme there. And I'm also happy with this very minimal Rofi theme as far as just the horizontal bar at the top. And I think I'm going to just convert some of my config files over to just using Rofi for a little while, especially now that I'm focusing a little more on app images just because Rofi handles all of that stuff straight out of the box. And I don't have to fool with, you know, exporting, you know, new paths for the shell or sim linking anything or trying to patch D menu in any kind of weird way. Now, for those of you on Linux, Rofi should be available on every single Linux distribution. You should probably find Rofi in every standard repository for every GNU slash Linux distribution. And if you find D menu kind of a hassle, especially if you have to patch it for functionality, maybe give Rofi a try. Now, before I go, I need to thank a few special people. I need to thank the producers of the show, Devon Fran, Gabe Corbinian, Mitchell, Akami, R5530, Chris, Chuck, David, the other David, Donnie, Dylan, Gregory, Lewis, Paul, Pick, BM, Scott and Willie, they are the producers of the show. They are my highest tiered patrons over on Patreon. Without these guys, this episode about the Rofi run launcher would not have been possible. The show is also brought to you by each and every one of these ladies and gentlemen. These are all my supporters over on Patreon because the DistroTube channel is supported by you guys, the community. If you'd like to support my work, I hope you consider it. Look for DistroTube over on Patreon. Alright guys, peace. Yeah, enough of the patching already. That got old.


Hello and welcome to the session. I am Deepika here. Let's discuss the question where in the following form an AP if they form an AP? Find the common difference D and write three more terms A, A square, AQ, A raise to power 4 and so on. Let's start the solution. The given list of numbers is A, A square, AQ, A raise to power 4 and so on. We have A2 minus A1 is equal to A square minus A. Again A3 minus A2 is equal to AQ minus A square since A2 minus A1 is not equal to A3 minus A2. Therefore, the given list of numbers does not form an AP. The answer for the above list of numbers is no. I hope the question is clear to you. Bye and have a good day.


Ain't number 10 what a lineup water line up This is Dan beauty off the inside in red Jason Doyle out of gate number two in blue Martin Bachelick a winner last time out of gate number three in white and Ty Wolffenden a race winner last time as well off the outside in yellow Crikey, this is some race indeed something's got to give here could be action-packed Doyle out he'll have his elbows out Dan beauty's been making starts for fun Fantastic Ty Wolffenden was jet propelled out of gate number three last time This is what Grand Prix speedway is all about when the big guns come head-to-head Pressure beginning to build points on the line riders desperate to make the latter stages of a Grand Prix look at that crowd Basking in the sunshine beautiful day And so so good to see a healthy crowd here Speedway not a massive sport in this part of the world, but very well supported here in Riga this afternoon Here we go then heat number 10 water line up It is roar into the first corner and it's whooping them off the outside comes charging across doors there with him Down the back straight for the first time neck and neck Woffenden just about steals the march on Doyle beauty's back in third place running very wide that allows Martin Bachelick in the third place with Woffenden has done the business there Chris That's a terrific effort from him. Yeah, it was a drag race down that back straight on the opening lap one that tile Woffenden just and I emphasis just managed to win with Jason Doyle not backing out of the charge down the straight But fair play to Ty Wolfenden just got his bike over the top of Jason Doyle And that was important because if he hadn't have done he'd have been well He'd have been knocked for six and put out in the dirt And I think Bachelick Julie may have been up the inside of him But great opening lap and great ride from Ty Woffenden into the last lap then Ty Woffenden under pressure from door But his road perfectly impeccably out in front after a dynamite start away from the outside gate Woffenden back to his best here in the last two outings back-to-back heat victories moves on the seven points Looking good best Grand Prix of the season so far for Ty Woffenden Enjoy in Riga you make starts here then you're going to be in the big points Jason Doyle going along nicely as well He moves on the seven with a win and two second places. Julie missing out for the first time Woffenden out in front three points for him second place for Jason Doyle two points Martin Bachelick one point for him and Dan Buley after two wins fails to score in heat number 10 But for Ty Woffenden all of a sudden those adjustments he made after the first race when we heard with Scotty Nichols We need to get the bike off the start Gates three and four it's work to treat Calvin You're just talking about why do we see the inconsistencies within meetings from riders the unbeaten Dan Buley here missing out at the start And here's the drag race down the inside Jason Doyle He's just been rounded by Ty Woffenden and he makes a brave move to come across and and absolutely Stopped the run of Jason Doyle which he had to do and He doesn't clear off Jason Doyle keeps him on his keeps him under pressure Not a lot between the whole field, but Dan Buley at the back big disappointment for the Brit having had two race wins And we were just talking about it. Yeah, that happens. It does indeed and we just seen it highlighted Great effect there with Woffenden winning and Dan Buley who was unbeaten coming into heat number 10


Welcome to Toffy TV, it is the start and 11 show Wofford vs Everton, let's get into it Jordan Pickford in goal at right back Gible Sibody, for me I think he's, I think he's at start and right back now for me anyway, and I think I just think James Coleman's legs have gone good for coming off the bench not too bad playing at home maybe but playing away from home, I just don't think he's don't think he's up to it I don't listen and that's not me saying Cyddi B's miles ahead of him by the way Cyddi B has plenty of mistakes in him but I just think for me Cyddi B should start the game On the other side, Llew Cyddi'n obviously our number one left back so not much to say on that the two centre backs Mason Holgate for me and Yeri Mina alongside him I think they're our number one partnership again and other than Mason going into midfield I think they should be our start and play the manager has come out and said that you know the fact of Mason Holgate's come on so much is meant that he feels like he's got enough to go forward and even with Lewis Gibson going out on loan now which oddly makes Jalad Bramfritaf fourth choice centre back now I'd imagine with Morgan Feeney going out as well so interesting if one of them got injured yeah alongside Yeri Mina so that's the back forth for me into midfield let's start with the centre midfield interesting cos there's a lot of options available but none of them are like great, are they? and because of that change of formation you know the likes of a Wobie's back likes a Sigarton's back but it's just interesting because a Wobie he did play in the 4-4 too of course under Duncan Ferguson but I don't think certainly for this game he'd be ready to play 4-4 too so for me I'm going to stick with Morgan Snedlan and Febeen Delf because I actually don't think they were that bad against Newcastle you know if we'd walked off the pitch to Neil people would have been patting on the back saying how well they played I don't think Sigarton can walk back in the side especially when you know less than a month ago we were sitting here criticising him and Snedlan after the performance against Liverpool's kids so I'm going to continue with them too and now we've got the option of a Sigarton coming off the bench and a Wobie coming off the bench whether that sends me a feel for a Wobie I don't really think so but we'll see how that plays out so there the two I would play in the centre of the park right midfield I'm going to bring with Charleston in at right midfield cos I think for me I want as many good players on the pitch and I think with Charleston coming in at right midfield you know ideally I want them up front but I just look at the form of Theo Wockhart and just think well with Charleston on coming back he can be a major threat now of course Moise Keane coming off the bench could be a big asset as well and that's where maybe the manager will look but I'm going to put with Charleston on the right and on the left Bernard continue I think he's been playing well as last couple of games he's had a decent season although stop start on that injury he got away a bright and think it was wasn't he so to put the brakes on there he's come back and had a few subs appearances and he's played well of late so him on the left and up front Moise Keane obviously scored in his last game going to be full of confidence alongside Dominic Harvenlun who obviously scored in his last game as well it's hard to see how you can break that up although with Charleston coming back could do that but I just think the manager might just think well you know I'll put with Charleston on the right and we'll just see what we've got then we've got the likes of Theo on the bench and Woby on the bench Guilfee Sigarton on the bench and we'll look a lot more attack and come off the bench giving his options and the likes of Tom Davis so yeah that's my start at 11 let me know if you agree, disagree whatever your thoughts are on it in any of the positions, let me know thanks for watching Toffy TV don't forget to give this video a like subscribe if you haven't already and if you want more great videos join us over on page and see you later