 Should we switch? Ladies and gentlemen, it's my pleasure to introduce the promoter who will speak on relaxation, oscillation, limit cycles, and astronomy on your unit. Thank you. I'm very happy to be here to talk to the audience on the Poincaré. So I'm going to divide my talk into two parts. In the first part, I'm going to say a few words about... More or less, I will try to put what I'm going to do in the first, second part in perspective with what was done by Poincaré and others at the time of Poincaré a little bit later. And I'm going to explain what is written on the board. And afterwards, I'll switch to more recent things and to some work we have done with Martin, who is here. So this has been... Oh, she's not... I'm sorry for my poor handwriting. So everything basically will be concerned with limit cycles. And as probably everybody knows, the theory and the word limit cycle goes back to the second memoir of Poincaré sur les courbes définies par des équations différentielles. And this is a place where I introduce... This is chapter six, published in the Journal du Mathématique pure des appliqués in 1882. And in the same article, it defines the theory of limit cycles and what we call now Poincaré application. And the others have explained that before me. And what I'm going to say now is that this series of memoirs were about a purely mathematical problem. But afterwards, for reasons I'm going to explain, at the beginning of the 20th century, Poincaré became interested in a problem of modern science, modern science at the side. And even the most modern science of this time, which was the propagation of electromagnetic waves, what he called Hertzian waves, in a long distance. So the first, what triggered the interest of Poincaré in this was the success by Marconi in 1901. And Marconi managed to transmit Hertzian waves across the Atlantic. So this was between a place called Poldu, Poldu, I don't know to pronounce this, in the western part of the United Kingdom, to Canada, to Newfoundland. And this was a big success because given the method used at the time for emission of waves and for the reception of waves, I'm going to say a few words on that. So Poincaré became interested in what Marconi had done. And he wrote a paper in 1910, I'm going to comment a little bit afterward, about this paper of 1910. But for my topic, what is important is that in 1908 and 1909, Poincaré delivered a few lectures called the Telecommunication in France. And this lecture has been published in Revue d'Electricité, and this was discovered only recently in this publication by Giroud. And in this series of lectures, he solved, he considered two problems. One problem, which is the one connected with this paper in 1910, was the transmission of electromagnetic waves on long distances. And it was not the first one to be interested in that, but the big problem they had for the transatlantic crossing, but you had the emission in the UK here, the reception in Canada on the other side of the ocean, and of course we know that the earth is not flat, is round, is spherical, and so because of that, the inline transmission, direct transmission by ray, is impossible. So there is no link, you would have to cross the entire of the earth, which is not possible because the earth is a too good conductor. So the only possibility, supposing that outside of the earth, you have a uniform medium, which is not true. So the only possibility was to look for the transmission at the diffraction, the diffraction across perpendicular to the straight line to the geometrical optics. And what Poincare did compute in this paper published in 1910, was the amplitude of the diffracted wave. And this amplitude, this is a complicated problem, a difficult problem by the way, which was considered again by Fock, the Russian mathematician Fock in the 1940s. And what he found, Poincare, was that the amplitude of the diffraction of the diffracted wave was too small to explain the reception by Marconi in Newfoundland. So which was nevertheless, Marconi was successful. And at the end of his paper, Poincare suggests that the reception was made possible because there is a conducting layer in the upper atmosphere, and instead of having a straight propagation, actually the propagation is in between two conductors, the conductor in the upper atmosphere and the conductor on the ocean, and which is actually what happens. And because of this bouncing of the rays, so the direct, not so direct, but the emission and reception of the wave is possible on long distances. So this was the first part of this conference and on the paper in 1910. In the second part, which is very short by the way, it was considered the other problem posed by the radio waves, which is the generation of oscillations. You have to have electrical oscillation because by the equation of Maxwell, with a time-dependent current in an antenna, you radiate some energy outside of the antenna, which is basically the energy which is received at the other end in Canada. And to have an oscillating circuit, at this time it was in 1908. So the only way to have oscillating circuit was to have RLC circuit. This is a capacitor, an inductance, and the resistance R or R, by the way. And in between you need to have some amplifying system, a system that feeds energy into the oscillation. And the system used by Marconi and many others was an arc, electric arc. This electric arc has many instabilities. It's not well understood even yet. And because of the instabilities, you have an amplification. So the equation for this problem are written here. You have the RLC, which is actually RL, which is L, and C is H. I'm using the same notation as Poincare. And the non-linear element and the amplifying element is contained in this theta function, which is non-linear, this theta function as a linear and non-linear part. Actually it's not absolutely, Poincare probably mixed two papers, two things in his article in the conference, Téphonétien Field, because he writes this equation and then he writes another equation which is closer to the truth by connecting, coupling two equations, one equation for X, which is the charge of the capacitor and an equation for the temperature in the arc. So he has a system of two couplets equation and he claims without proof that this, which is, by the way, but this is true, that this set of two equations has actually stable oscillating solutions, which comes by, which it's not a system in a plane, for a plane flow, because there is a second derivative here. It's for three-dimensional system. And Poincare also gives a constraint on theta to have sustained oscillation. So, and this paper is apparently the first one, for the first paper where there is a connection made, this was pointed out by Giroud, there is a connection made between the limit cycle and the periodic solution, periodic in time, periodic solution of differential equation, nonlinear differential equation. Stable nonlinear. So, the same year, 1908, was the beginning of the modern era of radio waves, which is the year of the invention of triode by an American inventor called Leder Forest. And this triode is the element that replaces the arc in the Marconi and other experiments. So you have an amplifying element, which is much easier to use than the arc lamp. And the triode of Leder Forest was much improved by someone called George Abram, who was the director of the Laboratory of Physics of Eco-Normal, 100 meters from here. And so we are in 1908. So after 1908, before the war, because the war was getting close, basically all the research on all sides was becoming completely secret. So there is almost no publication on this topic between 1908 and after the war, after World War I. So after World War I, there is a first paper by Blondel, I think, André Blondel, who worked on this electrical circuit, the one I have written here, with the arc lamp substituted by a triode. And Blondel wrote an equation, I'm going to show again and again, which is called now a Van der Poel equation. So Van der Poel equation, a year later, wrote a paper on the theory of triode oscillations. So it's not very much a theoretical paper, it's only a paper where Van der Poel only writes the equation, and he says that this is a good equation for the oscillation, but it does not bring too much argument to prove his point. And his point was only proved in 1926, that the equation, which is called now the Van der Poel equation, has a self-associating solution, and this is a paper on relaxation oscillation. This is, as far as I'm aware, the first paper where you have this notion of relaxation oscillation that I'm going to talk about. And this was published in the London Journal of Science. And Leonhardt, in 1928, made all the theory of the Van der Poel equation in the two limits which are convenient, the limit of small amplitude and the limit of large amplitude. Although the paper by Balthazar, how to pronounce, Balthazar Van der Poel is a paper where the equation, which is a set of two coupled equations of first order, he solved the equation by a graphical method, geometrical method. It was not, he did not solve the equation by analytical method, you know, he used only this graphical method, which has also a long history and which is basically the kind of method which is used nowadays, you know, because when you solve differential equations with computers, you are following the same idea as Van der Poel. But only with some much big improvement in the technical method, but the idea is the same. And Leonhardt, you know, again, he solved analytically the problem that was considered by Van der Poel. So, this is basically the history and afterwards, you know, of course, you know, I should need 100 blackboards to say everything that's been done on this subject afterwards. But this is basically sets the stage of the early part of the history of the theory of relaxation oscillation in a simple dynamical system. So, let me try to do it well. So, I, oh, before, sorry, before we continue, I should say that this was also found by Giroud in 1933 in this building, this building where we are now, probably in this conference room. There was a conference on non-linear phenomena organized by Van der Poel and others by Russian. And the topic of this conference, it was years ago, the topics of this conference was precisely the relaxation oscillations, 1933. And one of the attendees of this conference was Yvrocar. And some of us, you know, probably have, we have, please, Martin and myself and others in this audience. I've heard talks by Yvrocar, you know, who wrote a paper on the dynamic, the vibration, which is still well known at least in France. By the way, I should say that this is something which is also pointed out by Giroud and others. You know, the paper after Poincaré, Nublondel, Van der Poel, etc., never referred to, in Yvrocar in particular, never referred to Poincaré. It was only after the Russian came in, Van der Sam, and his students that they began to refer back to Poincaré. You know, the French, or they had completely forgotten or they didn't want for some reason to refer to Poincaré. And from what I remember, you know, classes by Yvrocar, you know, he never mentioned the name of Poincaré, if I remember well. So, now, this is a quotation of a, well, there are many pure mathematicians here. So, this is a very good, very interesting piece by André Valle, l'avenir des mathématiques. And he says that the study of the equation of Van der Poel and of the oscillation called de relaxation is one of the very few questions, and a few interesting questions asked or posed by the contemporary physics because the study of nature, which was, which used to be one of the main source of big problems, big mathematical problems, seems lately to have been more borrowed than it has given, which was very much in the style of, so for those of you who had a chance to meet André Valle, you know, it was a bit critical, you know, a bit ironical, very much in the spirit. But it's true, you know, because this was written in 1946. And if you have a chance to find this piece of work by Valle, you know, it's very interesting because, yeah, the part devoted to the application is very small, you know, and he has a long development on the Riemann hypothesis, and he sees the Riemann hypothesis as the main, the biggest problem by far in mathematics, you know. And I think at this time it wasn't maybe a way which was original. Of course, nowadays, from what I understand, everybody believes that this is the biggest challenge for mathematics today, you know, the Riemann conjecture, but at this time, maybe it was not. So let's follow the advice of André Valle and try to study the Van der Poel equation. So by the way, this was in 1946 and the mathematician of the Van der Poel equation in the two interesting limits, small non-NIT, large non-NIT was done just a little bit later by Doronitsyn in Soviet Union. So I come back in the history much, much earlier. So the problem of limit cycle, if you wish, is following one. You want to describe mathematically an oscillating phenomena. So an oscillating phenomena, a time-periodic phenomena in nature, are very well known. They have been known since the Babylonian Western Hemisphere and Mesoamerican civilization in the Western Hemisphere and Babylonian in the Eastern Hemisphere. And it is known that you have regular trajectories and also not so regular trajectories which are due to the planets. But of course, mostly, and this was explained by Jacques Lascar yesterday, so mostly there is no dissipation in astronomical phenomena. Although if you want to make a clock, I mean a device which oscillates regularly in time, you have to bring energy in the system to balance dissipation. And surprisingly, after all, you know, from the very far away point, from the theoretical point of view, you know, oscillating circuits are just an example of an artificial oscillating circuit, but you have also many other naturally, not naturally, artificial oscillating systems. You have the clocks which have been made since middle age and even before, which are oscillating regularly in time and which might require a little bit of a mathematician. Actually, some great scientists were interested in the making of clocks. Classically, you have organs in the 7th century and Gahan was a British scientist less well known than organs. So what is the problem if you want to make a clock? To make a clock, in general, you have to have what is called an escape mechanism. You have to feed energy in the system and when feeding energy, you have to keep the system oscillating at a given frequency, which is not a trivial problem. So if I... So this is an escape mechanism invented by, I can say, perfected by Graham. So you have a pendulum oscillating at the end of this wire regularly, but you know that if you leave this pendulum oscillate by itself without doing anything else, in the plane of Poincare, vx, v being the velocity in x, the position, the angular position of the pendulum, you know that instead of having a closed curve, you will have a spiral going down to the center because of the friction. So what is the escape mechanism? So the escape mechanism is a way to give to the system some energy to balance this decay of energy. And this is done in the following way. I hope my explanation will be clear enough. So you have this piece is called the anchor. And there is a tooth wheel. You have this tooth wheel which is rotating because there is a weight going down. So the tooth wheel has some energy input coming from the motion of this weight in the gravity. So this anchor oscillates at the same frequency as the frequency of the pendulum. It is linked rigidly to the pendulum. And at each time these little pins hit the tooth wheel, the pendulum gets a kick from the tooth wheel because the tooth wheel is the one that receives the power. And when it does that, it does that at the end when the pendulum is at the extreme end of its motion, when it has no velocity. So what happens is that you see that the motion is the following one. It gets spiral inward and when the pendulum has no velocity, when it is here, it gets a kick. By getting this kick, it moves to the same, you know, it's an instantaneous, more or less instantaneous kick, you know. It moves from here to there and it gets back to the spiral. So, of course, you know, during the motion it will not visit any more this part of the spiral. So it will go from here to there, get back up again here and get another kick, etc. So this is the way you have a limit cycle which was more or less the invention of Agnes. It's a smart idea. So you have only a kick, you know, it's not something which is as regular as a non-linear term in the equation, but this is also a non-linear effect, but you have a kick only at a certain time. And so, let me continue. Clocks. There has been, by the way, very little interest in the year. In the general theory of clocks, you know, and even now atomic clocks, etc. And for instance, there is no general theory telling what is the ultimate accuracy, ultimate precision one can hope for a given clock, you know, as far as I'm aware. It's an interesting problem because it is connected with thermal fluctuations, quantum fluctuations, etc., which are certainly an incidence on the ultimate accuracy of a clock. So this is to recall what Poincare said, you know, it describes oscillating system with losses and energy input, even though this is seen with our eyes nowadays, you know, but basically what Poincare describes are oscillating systems from the point of view of physical system with losses and energy input by a theory of the energy system. So you have the limit cycle, Poincare map, this is something I already said. So this is to recall the Marconi transmitter, you know, he used again as an amplifier the arc lamp, which is not convenient at all. And see what is, this is also called the singing arc because, you know, it is oscillating, you know, including at the frequencies, audible frequencies. And this is the big device which was made by Marconi to generate the oscillation in the antenna, electric oscillation in the antenna. So now I'll switch to mathematics, but maybe not for the rest of the talk. So this is the equation which was derived by, actually there was a first, it was written for the first time by Rayleigh in another form, and as I said by Blondel and it is known now by Van der Poel, Van der Poel equation. So you have an equation of this form which is not exactly of the form of the equation by Poincare Ray, because in the nonlinear term you have a mixture of x dot and a function f of x, which is x square minus one. This equation is called the Van der Poel equation. So what is known that if eta is negative, so all the solution tends to rest, you know, to x equal x dot equals zero. And for eta positive, the only stable solution of this equation is a periodic solution, periodic in time. And there are two situations. One can do an analysis, detailed analysis of the behavior of the solution of this equation, which is a limit eta small, where we have this calculation of this, this is the limit of what is called the Poincare-Andronov bifurcation. And I'm not going to say anything on this. And what I'll be interested in is the limit eta large, which was the limit studied by Van der Poel, numerically again. So what you observe, what you observe without explanation, by the way, what you observe is that when eta is large, there are two stages in the solution of this equation. There is a first stage of decay of x as a function of time, and then a very quick jump for a positive to a negative value. This is, I think, if I remember, this is for eta equal 10, something like that. 100. For eta equal 100, for eta equal 100, so you have a very fast decay, then somehow a slow recovery of x, a big jump to a positive value, so decay, et cetera. So the oscillations are periodic, but with two very well-definite different stages, a stage of slow motion and a stage of very fast motion. In this image, eta large. So with Martin, we got interested into this problem because, to come back to, there is a, this, at least, this Van der Poel equation is very interesting because it is a simple model where you have a transition from, in the same equation, from something which is very slow, very slow dynamics, and a very fast dynamics. And there are very natural, very many natural phenomena where you have also this kind of transition. So the most remarkable one, which I've been studying recently, is, it's not periodic in time. It is an explosion of supernovae. In the explosion of supernovae, you have an explosion that lasts about 10 seconds, and the time of evolution, the slow time scale, is of the order of 1 billion years. So the ratio of time scale is 10 to the 14. So you have really a very large or a very small parameter depending on where you put the numerator and the numerator. And in this, this is probably the most extreme case where in the same physics, because there is no, you know, the physics remains the same, you have, in the same physical problem, you have a very, very different time scale. And you have also the other example where you have a very different time scale, and I will come back to that, is the case of the earthquake. In the earthquake, an earthquake lasts less than a minute, you know, it's about 10 seconds or 20 seconds, and the time scale, the time interval between major earthquakes is in the range of 100 or 200 years. There, the ratio is about 10 to 9, but it is also a very, very large time. So we were interested with Martin in this, the study of this problem, because the idea we had is that by understanding how do you, how does the transition from the slow phenomenon to the fast phenomenon occurs, my help in some cases, in some specific cases, particularly for earthquakes, to manage to predict the occurrence of the very fast phenomenon, the long time scale. Maybe there is something in between an acceleration that could be a way of predicting how does one do the transition. So maybe I'll jump to there. So this is the Lienard, the Lienard representation of the Van der Poel equation. So you have a slow, a stable manifold, a stable, no, a slow manifold, sorry it's not stable, a slow manifold which is this manifold here. Basically, this is the, y is an integral of, it doesn't matter too much, y is found by integration, by a first integration if you wish of this equation. So in the equation, the x derivative is very large. It is eta times, it is eta times the derivative of y, y dot. So you have a slow drift, and again in the limit eta large, you have a slow drift of the representative point where, again, in the realm of Poincare, you know, we are drawing everything in the phase place, in phase plane, y and x, y you can see as x dot. It is not exactly x dot, but it is the same quantity. So you have a slow drift along this curve. You see that this cubic here, which is a slow manifold, is attracting everywhere. This is attracting because the x derivative is much larger than the y derivative, in the limit eta large. So every point starting away from the cubic will be attracted to the cubic. But what happens is that you have a slow, the cubic is not a line of fixed point. It is. There is a small drift in the y direction because the y dot is not zero. There is a slow drift along the slow manifold. This slow drift is going upward on the right part of the plane and downward there. So you see that starting from this point, you drift on the slow manifold along this slow manifold here. But here you reach the extreme point on the slow manifold and jump out of the slow manifold because the drifting continues. And then the point is taken by the very large velocity in the horizontal direction. And it goes along this red arrow, the velocity in the x direction and you reach this point there and drift again along the slow manifold to this point and then to the very fast drift there from this point to that point and there again. So you have basically this explain this relaxation oscillation. The relaxation oscillation, you have the slow part which is a drift along the slow manifold, which is slow because the velocity in the x direction is much bigger. The velocity perpendicular or more or less perpendicular to the slow manifold is much larger than the velocity along the slow manifold. This is this part of the drift. But you reach at this point, the extreme point and jump from this extreme point to the other extreme point. This point is this point over there. And then you have the drift there from here to there, et cetera, et cetera. So this story was understood in terms of qualitatively understood by Liena in 1928 and there is other paper by Liena also. But the details of what happens when the point, representative point takes off from the slow manifold there and lands on the slow manifold here was understood by, as I said, in 1947 by Doronitsyn. This is a tricky problem to understand how things happen in detail. But to come back to this picture. So what you have here, and this is what we're very interested in. You have basically from the point of view of the stability in the transverse direction, here you have a point which is unstable because you know that the blue arrows point outward from the slow manifold. And here at this point, the black, the blue arrow point towards the slow manifold. And at this point, when you look at the dynamics in the x direction, you have the merging of an unstable fixed point and of a stable fixed point at this point, which is you have a kind of dynamical saddle node bifurcation. And a dynamical saddle node bifurcation, you can be represented by a one-dimensional equation which is written over there, x dot equals minus dv over the x. v is a cubic and you have a parameter b. When b, I think, when b is negative, you have two fixed points. You have this fixed point, which is the stable fixed point. You have the unstable fixed point. At b equals zero, you have a cubic. So there is a meta-stable point. And when b becomes slowly positive, you have no stable point anymore. So you have a merging of the two fixed points, stable and stable, and a disappearance of the fixed point, of any stable point. So by analyzing what happens, which was done by Dorotnitsyn, what Dorotnitsyn showed is that for the problem of transition of the slow-to-fast transition, in this equation you have to have a slowly-dependent, time-dependent b. So you have b depends on time, but very slowly so that you sweep again, you sweep across this picture from here to there. And by rescaling the equation, so near the transition, you have this universal equation, x dot equals x squared plus t. x squared is there because this is the derivative of the cubic force. And you look at the solution of this equation where at minus infinity you are in the stable fixed point. So x is like minus, I think there should be a minus square root here. Minus square root of minus t. And this equation can fortunately be solved by using the equation of every. So you have x as a function of time, which is x is what happens in the direction of x, in the original direction of the function of time, and there is a singularity at 2338, which is a time where the approximation of the cubic potential by, the approximation by the cubic potential there is not valid anymore, so you have to have higher order terms. So this is done by x dot, by adding to the potential a one for x to the fourth, and now you know the rolling down stops when the particle reaches the bottom of the potential. So with this, with this modellization with the cubic term now in the equation, we have something which is much, which has no singularity after a finite time. And so what is interesting from the initial point of view is that in this framework there is a long time scale, which as I said at the beginning for the application is very long time scale, which is the ripening time for the maturation, and there is a very quick time, oh sorry, this is a long time scale because A is large, and because A is small, there is a time scale, which is a short time scale, which is one, and there is an intermediate time scale, which was found by the way by Doronitsyn, which is A to the power minus one-third. So you have an intermediate in this phenomenon, at least the result of this analysis, that before the catastrophe you have a long time scale, which is intermediate between the short and long time. So what we did with Martin was, and maybe I won't tell too much on that, because you can detect, and this was done in an experiment which was done in the US, and this theory at least is successful in one experiment where you have, we observe this time, this kind of intermediate time scale. How much time do we have? No time? Five minutes. Now, maybe the most interesting part of this work for this audience is a stick slip model. So one could ask the following question. So we have this phenomenon of transition from slow to fast phenomenon, and which is well explained by the Van der Poel equation, the Van der Poel equation. So this is what we call the saddle node bifurcation. So saddle node bifurcation with a dynamical, it is dynamical because there is a slow sweeping from across the transition. So is this only possibility for having, in a differential equation, a transition from slow to fast? And the answer is no. We studied with Martin this model what is called stick slip behavior. So you have a mass which is drawn on a solid surface. So there is some friction, and the mass is drawn by a point moving with velocity v node and a spring of strength k. So this is a classical model for earthquake. And in this model, there is a parameter, there are many parameters, and here is a parameter gamma. The parameter gamma for earthquake is very large. This represents more or less the strength of the friction compared to the strength of the spring here. And what happens in this model? So this model is, as you may have noticed, is a system of three, a set of three ordinary differential equations which are written here. And in this, maybe, so what happens in this model is interesting. It is not at all what happens in this three-dimensional model. This is not at all what you have in the saddle node bifurcation of the van der Poel model. In this model, by restricting the equation to the slow manifold, because there is also a slow manifold, what one has is that the trajectory remains on the slow manifold, but on the slow manifold, the equation, the dynamical equation, have a finite time singularity. So after some time, the slow manifold is the point, representative point, get out of the slow manifold just because it accelerates too much. In some sense, the assumption of having something slow doesn't work anymore. So there is a transition to another kind of dynamics. And you see that this is what happens there. So you have this, the trajectory, which is very easy to draw, because it is in three dimensions. You have a surface, the slow manifold is a surface, which is in gray here, and the particle accelerates and accelerates and leaves the slow manifold in this direction, and makes a big turn, and come back to the slow manifold, make a large turn there, goes back, goes away from the slow manifold, and again and again. So in some sense, this is a second way for a system to have this relaxation oscillation. And so somehow it is kind of a point of mathematics, but it is not completely indifferent, because if... Or maybe I'm not going to explain that. But if one wants to, you know, the possible precursory phenomena for the transition are completely different in this case than they are in the case of the transition by the cellular node bifurcation. So it is a complete and another class, or another class of transition from slow to fast that you find in this model of friction. So I think I will stop here. Thank you for your attention. Thank you for the very nice talk. Other questions? Parle Four. Parle Four. Speak loud. Yeah, you didn't mention many people who studied this kind of phenomena using non-standard analysis. Could you comment on this? Which phenomena? The approach to the slow manifold and the times of exit and things like that. I mentioned the ordnance. I think I mentioned the ordnance. I have not used any. Yeah, I know that, of course, there are many followers, but what we are doing is used mostly only the ordnance. But this is the original part. As far as I am aware, these new Parle Four, let's go. Yes, thank you very much. You mentioned that Poincaré was very much interested in DSF wireless telegraphy following Marconi's exploit in 1902. That's true, but of course his interest predated that and there is his correspondence with Hertz from 1890 already. Yeah, but Hertz didn't make any transmission. Sure. The point I'm making is that you talk about long distance telegraphy, but certainly that's not really related to the work that he did on generating the waves from a mathematical standpoint. These are quite different, aren't they? You've got one problem generating the waves themselves and the other part is studying their transmission over long distances. So my question is, is there any relationship between these two problems other than the fact that they're both tangentially related to wireless? No, no, my point was to say more or less to describe what is in this 1908-1909 paper. The first part is about the working of an antenna. It computes various geometrical configurations for antenna and how does it define the geometry of the antenna, how does it define the geometry of the field at large distances. And in the second part, without, with no transition, he switched to this kind of equation. That's it.