I think it's time to start for the last lecture of Professor Otto. Thank you. So let me recap. So that's the main theme of my course, which is this conditional convergence of the thresholding scheme, which is written here, to mean curvature flow. And we have two types. I mean, Tim Lauches and I have two types of results. One is more in the spirit of Lukhausen-Sturznecker, but does not contain even the global dissipation inequality. And there is the second result, which contains not just the global dissipation inequality, but the family of local dissipation inequalities, which according to Bracke, are sufficient to characterize mean curvature flow. At least in the sense that if you knew that your limiting evolution was coming from a smooth evolution, then this condition is equivalent to mean curvature flow. But of course, in general, as you know, there might be non-uniqueness and singularities. So this is just a convenient, weak notion of solution. So it's a notion of solution that's built on the dissipation inequality. And since the global dissipation inequality is just a single inequality, it could never be enough to characterize an infinite dimensional evolution equation, but it turns out this localized version does. But of course, it is for it to really characterize mean curvature flow in the sense, so normative velocity is equal to 1 half of the mean curvature, you of course need everywhere the right constants. So this constant 1 half is exactly the right constant, comes from the two here. And that's the constant which you need there. Otherwise, it would fail to characterize it. It would be completely floppy condition. So keeping track of the constants is the right thing. And as I mentioned at the very beginning of my lecture, whenever you have a minimizing movement scheme, you have an easy a priori estimate or something which almost looks like the dissipation inequality, which is the basis for all a priori estimates, which tells you that e h of chi n or capital N plus the sum, little n from 1 to infinity, 1 over 2 h d h square chi n, chi n minus 1 is less or equal than e h of chi 0. So that you get always for free for a minimizing movement scheme. And as I mentioned in the first hour, you just get it from taking the previous step as a competitor. But this does not turn in the limit into the right dissipation inequality. It misses the dissipation inequality by a factor of 2 or 1 half. And therefore, life is not that easy. I mean, this is extremely useful. And we use it a lot to gain a priori estimates, to gain that the curvature is square integrable, that the normal velocity is square integrable, that the parameter stays bounded or in fact is decaying. You get all of these a priori informations from that simple estimate. So that's, of course, a big power of a minimizing movement scheme that you get an important a priori estimate for free. But by a factor of 1 half, it fails to capture the right dissipation inequality. So that's in a certain sense the life motif that you need to do more. And it's the, I'm going backwards, yes. And it's really the merit of the work of the Georgie that he showed a path how, in a completely abstract framework, how to go, in a certain sense, from this only seemingly optimal estimate to a truly optimal estimate. How to recover the missing second half in the dissipation inequality. And he does that by introducing this variational interpolation, which is kind of a very general tool.  So that was an elementary, elegant argument. And that's precisely the main abstract tool we're using here to get the right inequality. So that's the, yes. Several glasses of wine. I can even hear you worse. So I just heard Angrintella Wang. And so the Angrintella Wang scheme, which let me erase this here. So I think that that work was a very influential work. And I think at least Luigi Ambrosio told me that this was probably kind of had inspired the Georgie to think about this in more general terms. And the Angrintella Wang scheme runs as follows. So you say that your set at time step n minimizes the interfacial area of the boundary plus a squared distance to the previous set. And here, the notion of distance they proposed written in a kind of elementary way is the following. You look at the symmetric difference between your nominal set and the previous set. And you integrate the unsigned distance function to the previous set. And because we're looking here at 2v is equal to h, I need to put a 4 here, I think. So that's their scheme. And you can convince yourself that in the graph case, and if you have a little slope, this turns into the L2 gradient flow of the Dirichlet integral as it's supposed to be. So what's known for this scheme is that, I mean, the same type of conditional convergence result, which Tim Lauches and I had before, which I named theorem 2. To my knowledge, there is no analog of the theorem which I'm presenting now known for this scheme. I think that's a genuine advantage of the thresholding scheme. I mean, first of all, what's the difference? So of course, I think this scheme is historically very influent, but it doesn't have much of a numerical significance. This scheme has an numerical significance. And I think theoretically one's even better off in the sense that here, at least I see how to get also the dissipation inequalities and even the local ones, where I wouldn't know how to do that for this one. I mean, other people may, but I don't. It's not so clear, because here I'm not, I mean, that's not so clear to me. I'm not localizing a variational problem. I'm giving a different, I mean, the thresholding scheme a priori doesn't have any variational structure. It's kind of this point-wise scheme. And therefore, it turns out that it satisfies kind of a whole family of variational principles. Whereas here, it's not so clear. I mean, when you start out with a variational principle, it's not so clear how to localize that in a suitable way. So I haven't given you the proof of Lemma 6. I'm not doing that by starting from this variational problem and modifying the variational problem. I'm really starting from here. So I mean, like in Lemma 2, I derived this from this. And Lemma 6 is not derived from the outcome of Lemma 2, but it's derived as in Lemma 2 from this here. So I wouldn't, I mean, that's really, in a sense, what I think is an advantage of the thresholding scheme that it's, in a certain sense, so natural that it does fit into this variational framework, but it fits into the variational framework in many different ways. And here we picked one which is convenient for us. OK, so now you have the choice. So I can either kind of finish the proof of Lemma 6. I can go to the things of the first. So yeah, let me remind. So let's continue with the recap. OK, so right. So the main, the main merit of, that gives you the, I'm not so sure whether I understand. So in a certain sense, I mean, this, the term which you're, the term which you're adding here, I mean, it has to, it has to, the important thing about the metric term is such that from an infinitesimal point of view, it has to act like the metric tensor which you would like to, which you would like to write down. And now I think I need the factor, I need the factor of 4 because of this 2v here. So it has to act like this tensor. So in other words, if in a certain sense your omega can be seen as, let me write it in a very sloppy way, as perturbing this set with a normal velocity v, then this expression has to behave like with a 1 over 2h. That's the, that would be the Euclidean analog of, I mean that tells you, it's this property of this expression which tells you that you shouldn't be surprised that this converges to mean curvature flow. I don't know what that answers your question. Perhaps I didn't understand your questions. OK, so of course, you know, I mean, one could write down many different expressions here. It's in a certain sense, I mean, if I were to use a kind of a more abstract point of view, so you have a Romanian manifold, infinite dimension Romanian manifold with the metric tensor, and the metric tensor gives rise to an induced distance in the large. OK, so this terribly fails in our case, but let's for a moment ignore that this fails. So whenever you have this structure, you can write down this natural time discretization of a gradient flow which assumes this form, right? I mean, it's gotten quite popular in cases where this is the Wasserstein distance and so on, but it started earlier, right? And so that's what you would like to write down. But it's clear that you're not forced to put the induced distance here. You can put anything there which has the same local behavior, which quadratically, from a quadratic point of view, has the same behavior. And that's exactly what they're doing. They're writing down because they couldn't write, I mean, because of the observation of Michaud and Mumford, which came later, but which they must have been aware of. They cannot write down the induced distance anyway. It would be terribly non-explicit. And so they come up with a proxy. And because all they need is that close, I mean, in the regime when these two points are close, what they write down could radically simmer. I mean, I could even make it more explicit. Think of the Euclidean case, where here you would just put the square of the Euclidean norm, whether you add a cubic term, of course, affects the scheme, but doesn't affect the final outcome, the limit, because it's a higher order term. And therefore, I mean, that's perhaps the easiest way to put something, which is nicer. That's the easiest way to see that that's not really, it doesn't really matter what exactly you put there, provided it has the right quadratic behavior. And that's what this term achieves. But also, you know, thresholding does that. But it does that in a kind of, because it doesn't start out variationally, it does that in a multitude of ways. So there is a lemma, which then must be lemma 9 or so, which reads as follows. So in our notation, provided we have, so provided we have a sequence, now you can fix time, which converges strongly in L1, and for which you have this convergence of the parameters. So under these assumptions, provided this holds, so provided that we have, first of all, that the, and remember that this here essentially was up to the factor C0, the parameter, we have that the localized energy functionals. So that would be just seeing this term because of these two others vanish. This converges to C0 zeta times grad chi. But more importantly, the first variations of such a configuration in direction of vector field converges to the first variation of the limiting configuration, which is a classical object in differential geometry. And I wrote it down already many times. It's the integrating the, I don't think probably you can see that, which is C0, the tangential divergence integrated over the interface, which is, I mean, the weak formulation of mean curvature. So this is a type of lemma that's well appreciated in kind of the phase field theory. So kind of, if it were not this type of approximation coming from minimizing movements, but if it were the Ginsburg-Lander functional, then this could be credited to Lukhaus and Modika. And they essentially name always say that this is a Reshetniak type argument. So there is this statement, and that's exactly the statement for which you need the convergence assumption. The convergence assumption, if you have it globally, it also localizes. That's not so surprising. That's just kind of additivity and the fact that you always have lower semi-continuity. But more importantly, you also have convergence of the first variation. And that's, in a certain sense, the only way we use this assumption, because it guarantees convergence of the first variation. And the two lemmas, which I showed towards the end of this morning, exactly, I mean, made it clear that the problem reduces to studying the first variation. So that would be the only place where we need the, essentially, the only place where we need the convergence assumption. More questions? OK, so let's recap. So OK, so I'm repeating myself, but I was told that it's good teaching to repeat things. So this completely abstract observation by the Georgie kind of gets you the right, gets you the missing term in the dissipation inequality at the expense of having to introduce a somewhat non-standard interpolation between the time steps. This what's called variational interpolation. And at the expense of having to introduce a non-Euclidean, non-Riemannian notion of the square of a slope of a function, but which in the end is completely natural from the point of view of difference questions. So that's really kind of the key theoretical idea, and it was completely elementary and tremendously elegant. And here's the result, I think I already mentioned that. So the goal is getting the localized energy inequalities, the family of localized energy inequalities, with the right constant. And for this, we show on the basis of the thresholding formulation itself that the thresholding scheme doesn't just satisfy the standard minimizing movement formulation, but also satisfies kind of the localized minimizing movement formulation. And that allows us to use the Georgie's abstract result in a slightly more general framework than he originally conceived, namely where this energy functional also depends on the previous time step. But there's no problem with this. And then summing up, one gets in a certain sense already the exact discrete version of what you want to get in the end. So in a certain sense, I mean, you may see this in a certain sense is really like numerical analysis. You're showing that thresholding is a geometric integral. I mean, it has a good geometric properties, right? I mean, that's what people in numerical analysis are after. They want to find discretizations that preserve as much the underlying properties of the continuum equations. And that's then called geometric integrators when it's about kind of conservative dynamics. And now this is showing that from the point of view of gradient flow dynamics, the thresholding method actually has this nice method. And it's a bit, I mean, as I said, I mean, for many years, it was realized and used that thresholding is wonderfully compatible with the comparison principle and thus kind of nicely, very elegantly connects with viscosity solutions. But I would claim it as elegantly connects with the gradient flow structure and kind of the underlying variational principles as this result shows. So now it's really just a matter of passing to the limits, term by term, black into black, green into green, and red into red from the discretized version of the dissipation inequality to the right version of the dissipation inequality. And so in the black terms, that's just sitting here. And for the initial data anyway, it's the assumption of well-prepared initial data if you want. And now we have to worry about the green and the red terms. And that was what lemma 7 was for. So lemma 7, so there is an abstract feature to lemma 7, namely that the metric slope, this general metric notion by de Georgie, in a certain sense, controls a norm of the total variation with respect to this infinitesimal part of the metric, which looks a little bit like this, which again is kind of the main advantage of this de Georgie formulation that it lends itself to lower semi-continuity methods to this type of arguments. So that's what we use. So that makes the connection between the metric slope or the square of the metric slope and the classical first variation, so the infinitesimal change if you'd flow your configuration by a vector field psi. And then on the level of the infinitesimal formulation, the main insight is that this localization and taking the first variation commute. So whether you take the first variation of your localized energy functional or you take the first variation of your non-localized energy functional but localizing the vector field, it gives you the same result up to an error that goes to 0 as h goes to 0. And you saw the proof and essentially was completely elementary. And slightly more subtle is the passage to the limit in the term that gives you this transport term in the Braque formulation. So the term which arises from the fact that your energy functional has now two entries, not just the usual one but also the one which comes from the placeholder from the previous time steps. And that the difference quotient of this expression can be related to the first variation of the metric, the non-localized metric in direction of the gradient of the localization function. And then you can use the Euler-Legrange equation of the unlocalized variation principle to also kind of bring back this term to the first variation of the energy. So in the end, in order to understand both the term which comes from the metric slope and the term which comes from here, you have to understand, you have to pass to the limit in the first variation. And that's what I wrote down here. We can pass to, thanks to the convergence assumption, we can pass to the limit in the first variation. So these two lemmas reduce everything to the passage to the limit in the first variation. OK, so that's the story now. OK, so what do you want me to do? I can either, so one option would be lemma 6, one option would be lemma 8, one option would be going back to what I did on the first day. So who's in favor of lemma 8? We, who's in favor of lemma 6? Most of the people have given up. So three in favor of lemma 8. OK, so that was lemma 8, yes. So did I get it correctly? I first asked for lemma 8, sorry? No, no, no, but I first asked for lemma 8, right? Then let's do lemma 8. And so, yeah, that's what we have to do. And essentially, let me start with this. It relies on a higher order expansion of the commutator. That sounds fancy, but it's elementary. And so what we need is that if you look at the, I was right at the other way around, if you look at the commutator between multiplication with the cutoff function zeta and the convolution with the heat kernel, and you apply it to some test function v, then this is equal to minus h gradient g h convolved with gradient zeta, let me write like this, v gradient zeta. That's the first term, and the second term is plus h over 2 g h times identity plus h times the Hessian of the heat kernel convolved with v second derivatives of zeta plus a term which is of order h to the power of 3 halves, the L infinity norm of the third derivatives of the cutoff function and the L infinity norm of v. And we need something to some order less. But where we organize this differently by pulling the gradient of zeta out of the convolution, so this is like this. And then obviously, we get an error which is of the order h second derivatives v infinity. So that's elementary calculus, very much like what we did before when looking at the commutator writing this as writing it out. And the only additional, so the argument for one, the only additional feature is that because we're dealing with the heat kernel, we have some explicit formulas. Namely, so let's recall that the heat kernel is of the form, is of Gaussian form. And therefore, the derivative of the heat kernel gives you the first moment with a minus sign. And the Hessian of the heat kernel gives you essentially the second moment. And now you can use this to write g h times or z times g h as minus h times g h of z. And this is the origin of this term. Because from expansion, from Taylor expanding zeta, you would get, from the first order term, you would get z. But then you rewrite it in terms of the gradient. And here you do a similar thing. From the second order term in the Taylor expansion of zeta, you would get this term here, which according to this formula, you can write as h times g of h z times identity plus the Hessian of g h z. And that's at the origin of this term. So essentially it's just Taylor's expansion on zeta, based on the fact that what we used before lunch, that the commutator has this convenient integral representation. And now you use Taylor on this. And in fact, you use it in the form you write this as x minus z plus z minus x minus z. So you do Taylor around x minus z. And that gives you the gradient of zeta inside the convolution. But on the other hand, if you do Taylor around x, that gives you the gradient of zeta outside of the convolution, like here. So that's just an elementary real analysis that you can deal with these commutators. And now from this, in the second step, you get something which is in between a representation and an error estimate plus 1 half times the first variation of the non-localized metric in the configuration u in direction of gradient of zeta. So that's what we want to be small. And the representation is that this can be written as such a good term. And the term which comes from here, gh times identity plus h times Hessian, convolved with 1 minus u times the Hessian of the cutoff, plus something which is of the order of the third derivatives of the cutoff function times the one distance of u and psi plus or minus, if my notes are correct, 1 quarter gh over 2 minus chi square Laplacian zeta plus this error term. OK, so that's the second claim. And from this claim, we then will quickly get the estimate. So that's the main part. It uses the definitions, and it uses this expansion. So let's see how to get this. OK, so let's look at the difference quotient of this object in the second variable. So clearly, this term drops out. What we would think of the leading order term drops out, because it doesn't depend on chi. On the other hand, the contributions of the second term, when these two things are equal, these things are not there. So what we're left with is just these two contributions. And I'm going to put the 1 over h inside.  So that's this expression. And now for the second expression, so that's taking the first variation of this thing here. So as we see from here, this is a symmetric bilinear expression. So if we take the derivative, we get a factor of 2. And we can put the derivative in here. But there is one half in front of it, so that the factor of 2 goes away. So we get a plus. Then we get there is still the 1 over h in front of everything. Plus u minus chi gh convolved with minus psi grad u. And let me put this minus here. So that's what we get from just plugging in the definitions. And this is not zeta but psi because the zeta is sitting there. And now I claim, and you tell me why, that I can substitute the chi by u. Why can I do that? It looks like cheating. Why can I replace the chi by u there? So in other words, why does the contribution from u minus chi 1 over h zeta gh u minus chi, why is the contribution of this 1 0? Because that's the difference between the two terms. So it's an argument we already had before the break. So we have a term of this form. Why does this integral vanish? Exactly. Because this expression, the abstract formulation is a and b are two symmetric operators. So the dual of the commutator. Here I've just plugged in the definition of the commutator. Now I use the fact that dualization changes the order. Now I'm using that individually these operators are symmetric. And then you see that this is equal to that, which just tells you that the commutator of two symmetric operators is anti-symmetric. So in particular, this expression vanishes. And of course, multiplication with a function is symmetric. And convolution with our Gaussian is symmetric because the Gaussian is an even function. So therefore, I can do that. OK, so that simplifies it a little bit because now I can put 1 minus u here. And I want also to put 1 minus u here. Since there is a gradient, I can do that at no expense provided I change the sign here. Hopefully I will have the right signs in the end. Because now I can combine this first and the last term. So these two terms combined are u minus chi. And then I get 1 over h, the commutator, plus gh xi gradient. And both of this apply to 1 minus u. And that's the first bid in this expansion with my doing something. Ah, sorry. Yeah, so I made. So I want to take the first variation, not in direction of a general vector field xi, but in direction of the gradient zeta. So we have this expression here. And now you see that this term exactly is these two things here with v playing the role of 1 minus u. So this term here is equal to h over 2, gh identity plus the hessian gh convolved with 1 minus u times the hessian of the cutoff function zeta plus an error which is of the order h3 over 2, the l infinity norm of the third derivatives of the cutoff function, and the l infinity norm of 1 minus u, which is equal to 1.  So u minus chi is here, and there we have the third derivatives of the cutoff function. And this term here gives rise to this term, and I made a mistake here. That should be h over 2 because this is not exactly this term, but I'm using the semi-group property of writing this term as g of h over 2 convolved with g of h over 2 identity plus h gradient squared h over 2. So I can factorize this convolution operator into two convolution operators by the semi-group property. And the first factor I bring on the first factor I bring on the first factor in this product, and that's sitting here. So we also have this term. And now we're still left with this one here, and that's where we use the second somewhat easier relation here. So now I'm looking at this term here, and one of a square root of h u minus chi, 1 over h commutator of zeta g h over 2, h over 2, excuse me, u minus chi. So that's the blue term which I've copied. And now I'm using this formula. So this is equal to minus the gradient of zeta. So here I have to use this for h over 2. Nothing easier than that, h over 2, h over 2, 2h. Doesn't matter. So here we have minus 2 times g h over 2, convolved with whatever I have there, g h over 2, u minus chi. OK. Am I happy with this? Perhaps not quite. I would have preferred to have this in the other place, after all not changing this here. But that I was, I mean, if I do it, let's for a moment think I did it like this one here. Then this would read minus 2 gradient g h over 2, gradient zeta h over 2, u minus zeta. And now I could bring, yeah, that's better. So now I can bring this here on the first factor, and then I can use Leibniz's rule. So that would have been slightly smarter. I mean, shorter, more elegant, anyway, to keep it like the upper one. So then I'm using this. So minus 2, the 1 over h kills the h, gradient g over 2. I'm not sure whether I'm just confusing myself, h over 2, u minus chi. That plays the role of v. Then we have gradient of zeta. No. No, it was better before. Grad zeta, gradient g over h, v minus 2, red zeta, red g over h. Or perhaps I'm missing a small additional argument. I thought I could do it without any further work.  I shouldn't write it like this. I should write it. I should put the gradient first, and then the second convolution. So I should write it like this. So I'm confusing myself with my efficient notation without taking records. So that would be the correct interpretation of this here. And now I can move this on the other side, minus 2 over square root of h. Gradient gh over 2 convolved with u minus chi. Gradient of zeta, gh over 2, u minus chi. And now I can combine these two things by Leibniz's rule to the gradient of 1 half gh over 2, u minus chi squared. And then I can do another integration by parts where I put this gradient on to that gradient getting Laplacian. So this is equal to 1 over square root of h Laplacian zeta gh over 2 convolved with u minus chi squared. So up to the strange factor of 1 quarter, which I got there, and up to a wrong sign, which I'm not quite sure whether. So the sign is not wrong because this kernel is anti-symmetric. So there is a change of sign if I bring it on the other side. So there is a minus sign here. So that means this minus sign is correct. But now the 1 quarter, why did I put 2h here? Should have been h over 2. So because I'm just stupidly substituting h by h over 2, then I'm dividing by h. So there's 1 half. So I get 1 half here and another 1 half. So that's perfectly correct. OK, so that's correct. And then there is this error term plus order h times second derivatives of the cata function times the l infinity norm of this guy, which is certainly controlled by 1. So that gets multiplied before doing this and I'm just going to say it in words. It's better to change the order here. So put this thing with a minus sign on the first factor so that you can free this beneficial term, which still has part this convolution, so that you get the l infinity norm of this times not the l1 norm of u minus chi, but the convolution of u minus chi. So that requires one more using symmetry or anti-symmetry. And the factors of h are correct because this one's gone because of 1 over h. This one's gone. So there's no h here. And there's one of a square root of h, which is still sitting here. OK, so that's the argument for 2. And so now we don't need this anymore. And now the estimate is easy. So third step is the conclusion. So how are we going to estimate these terms? So for the first term, we use Cauchy-Schwarz in order to get the metric here. And this second term is certainly estimated in Ellen. So this is because thanks to the h here, this is an l1 kernel with an l1 norm that's of order 1. This is an l infinity. So we get the l2 norm of the Cauchy function which of the second derivative. So we take the l infinity norm. So that term is fine and gives rise. The first term would give rise to the metric term, square root. There is still a 1 over square root of h in front of it. And then we have the l infinity norm of the Cauchy function where we're being very generous. That's exactly the same term which we anyway have here already. And now we have to look at this term. But in this term, we just pull out the l infinity norm. This time, we need the l infinity norm of the second derivative of zeta. So there is 1 over square root of h g h over 2 u minus chi squared times the infinity norm of the second derivative zeta. And then there is this term here, plus 1 over no. Sorry, there is this term without any bad h, but no convolution and third derivatives of zeta. So this here is just, by definition, 1 over h. By definition, d square u chi over h, perhaps with a factor of 2. That's just the definition. So that's this error term, which is kind of small. And this term here then goes like 1 over square root of h is sitting here. And then here you have another 1 over, so this thing here goes like 1 over or is equal to 1 over 2 square of h d square u chi, but then I'm taking the square root. That's the first term. And that gives 1 over h 3 over 4, which up there I wrote as d over h times h 1 quarter to highlight that this also goes to 0. And now for this term, we're using our very first lemma 1. No, it was lemma 1, but it was this estimate here, which controls such a, so here we use, let's say we use Cauchy-Schwarz at first. And then we use lemma for a second point. And that gives rise to another metric term, which is smaller and this energetic term, or rather the square root of this energetic term. And that's this term here. OK. So there is an h 1 quarter that might be typo. So anyway, so these things go to 0. OK, so that shows the argument, perhaps that's the most, I would say this is the, I mean, in terms of length, this is the most involved argument in the paper, proving this estimate. How much time do I still have? Am I done? 10 minutes. OK, so in principle, so I could finish the proof of lemma 6, but I could also answer questions. I'm happier answering questions than doing lemma 6. I mean, you get it anyway, but if you don't have questions, then I can either stop, which I wouldn't be unhappy about, or I can do lemma 6. Yes. I think there's recent work by Aaron Yip and his former PhD student, Drew Schwartz, who showed that if mean curvature flow, I think that, but they looked at the single phase case, as long as mean curvature flow is smooth, thresholding converges nicely to that smooth solution. So this type of result is, I mean, this type of, let's say, more classical result now seems to be around, too. I don't know what that's what you were referring to. OK, so if the flow is regular, then just the global energy inequality, I mean, for a smoothly evolving surface, which is a priori unrelated to mean curvature flow, the global energy inequality cannot characterize the evolution, because in a sense, it's just a single scalar equation. And so just by dimensional arguments, it couldn't give you that information. Perhaps you're familiar with these rate independent evolutions, where the situation is slightly different. You have kind of local minimization problem. And in this case, because you have additional information, global energy identity can characterize, at least formally, the evolution. But here it's clear that it wouldn't be enough.  But then it's a, I mean, it's an easy formal consideration that provided the evolution is smooth, but a priori unrelated to mean curvature, and it satisfies this family of inequalities, then it must be mean curvature. That's, I mean, a three line observation. So the short answer is no. And in principle, that's a question which I have been thinking about for many years. I mean, you must be familiar with the fact that grain growth in some multi-phase mean curvature flow, network flow in, let's say, the extended space generically is expected to be statistically self-similar, where the number density in two dimensions, the number density of grains decays like one of a square root of t, the typical size of the grains, one of a t, the typical size of the grains goes up like t one-half. So you would expect these to see, and in numerical simulations, and I guess in experiments, you see these robust scaling laws through this cascade of grains vanishing exchange of neighborhoods. And with Bob Cohen 15 years ago or so, we came up with a method which in principle can give upper bounds for coarsening rates. And based on the gradient flow structure and based on kind of a global property of the energy landscape. And we worked this out, I mean, worked out successfully for Caen-Hilliard and the kind of sharp interface version, which is Mao and Cikurka, and for surface diffusion, so degenerate Caen-Hilliard. But we never got to work it here. So that's something I would be very interested in kind of saying, giving kind of these soft types of statistical statements on generic evolutions also here, but I haven't made, recently I haven't made the connection between these results and what we were thinking of before. So the short answer is I said it was no. OK, I'm looking at Nicola, Nicola. I mean, yeah, I agree with Nicola. So I mean, here I was very happy that this abstract theory kind of nicely applies. And I think this abstract theory of doing analysis in metric spaces is interesting by itself, of course. But also, in a certain sense, sometimes it's interesting to develop analysis on potential limiting objects, if you want to give convenient indirect proofs, like we're used in geometric analysis blow-up. And so then it's good to understand the objects which could arise as potential blow-ups. And often you lose some structure, but if you have certain curvature bounds, you perhaps don't lose all structures, and you certainly retain sometimes something which is better characterized as a metric structure. And so I think in these more pure areas of geometry or geometric analysis, it certainly is very smart to work in such a general form. Now, from the point of view of applied analysis, it's a bit less clear, but as Nicola said, sometimes it gives the proofs from the book, because you have so little objects that you have to find the most efficient proof. More questions? That could be conceivable, but in a certain sense, I mean, from an numerical point of view, of course, I would not necessarily advocate using the minimizing movement scheme with the Wasserstein distance as a smart way to solve the parabolic equation. I don't think that this is, I mean, they're very efficient way, I mean, personally, of course, I like this, but I don't think that this is necessarily from an numerical point of view the best way to do it. But in general, of course, you're right. I mean, when you want to solve, I mean, in certain sense, you may say that the numerical idea of preconditioning goes a little bit in that direction, kind of taking, when you do steepest descent, and you have a steepest descent algorithm, and now you want to do that in a numerical efficient way, you can choose your metric in a way that suits you, which numerical people would rather address to as a smart way of preconditioning. So there are certainly numerical situations where you play with this. So fourth order equations, of course, I mean, if you think now of the thin film equation, have their special problems because near the contact line, they have very intrinsic singular behavior. So again, I'm not sure whether these variational schemes are kind of the best. But yeah, why not? Short answer is, I don't know. Interesting question.


How does a free blockchain work? In Bitcoin, the miners are rewarded, but in a blockchain to keep contracts, who will do the validation? Can a blockchain exist without miners? Yes, a blockchain can exist without miners. A blockchain can exist without proof of work. A blockchain can exist without a game theoretical, competitive consensus algorithm. However, it won't be a decentralized blockchain. It will require trusted third parties. These trusted third parties will be the ones doing the validation. Alternatively, you can consider a mechanism whereby the validation is done based on an alternative consensus algorithm, such as proof of stake. But in that case, you still have reward. It's reward in the form of fees for the risk taken for validating the rules and the cost of money locked up in that validation. You either have a reward system that allows you to have competition, proof of stake, proof of work, and the variance of that. Or you have a system without reward. But in that case, the validators must be trusted. Then you introduce centralization, trusted third parties. That kind of blockchain can exist, but it's not a blockchain. It's a database with digital signatures. A lot of banks are trying to do exactly that, to create a centralized database, and then call it a blockchain, so it appears very innovative, even though it isn't. Larissa asks, are blocks really an advantage in the blockchain? Do the blocks that have formed in the blockchain need to exist? If every single transaction was validated by itself, wouldn't that solve the block size problem? Wouldn't it be faster to validate just a single transaction? Is it possible to create a cryptocurrency without blocks? Yes, in fact, you can. In fact, many of the systems that use a decentralized signing algorithm, instead of a proof-of-work mining algorithm, don't really need blocks. Blocks are needed for proof-of-work. Because doing the proof-of-work on the granularity of a single transaction essentially means increasing the rates at which blocks are found. Let's say you did it. Each transaction was its own block, had its own proof-of-work, was changed to the previous transaction, minus just selected a single transaction, calculated proof-of-work on it, and issued that transaction. That is perfectly possible. The problem with it is that the rate of orphaned blocks increases dramatically. Right now, you can have a fork that occurs in the chain when a block is orphaned, because two blocks were found more or less simultaneously within a ten-minute window. If you make that ten-minute window five-minute, the number of orphaned blocks increases quite significantly. You have to account for that in the algorithm. There have been some attempts to shrink that all the way down to about 15 seconds, which is what Ethereum does. But Ethereum, in order to do that, has a special mechanism for accounting for orphaned blocks, and sharing the reward between multiple winners of the proof-of-work, if you like. If you went all the way down to a singular transaction, and you increased the throughput rate of transactions, you are now talking about doing a thousand blocks per second, each one, just one transaction, then the problems with forking the network and synchronizing it become insurmountable. The only way you can do that is if you no longer care about competition between miners, if you no longer care about proof-of-work, if the consensus algorithm is fundamentally different. We see that in distributed ledger technology, where it is signing instead of mining, where the energy requirement is zero, and where you can now essentially issue blocks as fast as you want, at which point you don't need blocks anymore. You can just chain transactions together, which is why a lot of people don't consider DLT's distributed ledger technology to be blockchains. They are not blockchains because they don't require blocks, and they don't require chains. Can the cost of energy be minimized by green energy? What about the economy of the whole world mining in such a case? Yes, in fact, green energy plays a very important role in mining, because one of the characteristics of mining is that it can occur anywhere, especially now that we have the ability to receive blocks by satellite. You can pretty much do mining anywhere in the world where you have energy, and that energy can come from any source. It can come from wind, solar, hydroelectric, dirty coal, nuclear energy, or any of these sources. But energy has different properties, and energy does not have energy, it is fungible, but energy production facilities have different properties. For example, solar produces the most amount of energy during the midday noon sun, and the least amount of energy at night, which is zero. Often, the amount of energy that is produced does not match the amount of energy that is consumed at those times. In fact, the most energy that is consumed in a city is usually around 8 or 9 pm in the evening, whereas the production of solar energy is not peaking at that time, it is near zero. How do you reconcile these two mismatches? Essentially, you have energy that is being produced for free once you have the capacity in place. The capital cost is in place. There is no operating cost for a solar energy system, other than a small maintenance cost. Therefore, if you are producing it in the middle of the day, nobody needs that energy, that energy is wasted. Mining can actually take that wasted energy and use it to produce income. That income allows you to very rapidly depreciate the capital expense of building the solar plant in the first place. You can take a solar plant that would otherwise be depreciated over a period of, say, 5 or 10 years, and you can depreciate it over a period of 1 or 2 years, which leads to an enormous investment in solar. If you can depreciate the cost of the underlying capital, that makes the deployment of solar much cheaper. For the miners who are mining on this energy, the cost of that energy can be much lower. The profitability can be much higher. Because mining can happen anywhere, mining will happen in the places where electricity is cheapest, where the difference between capacity and demand is the greatest, where the ability to distribute the electricity by other means, such as high-voltage distribution networks, exists less or doesn't exist at all, and all of those represent opportunities. In fact, that means that the greatest opportunities for mining are from sources of alternative energy, such as wind, solar, and hydroelectric, even geothermal, because those are the sources that are remote, often from populated areas, that are difficult to distribute because of the cost of distribution networks, and where there is a large mismatch between capacity and demand. Which means that Bitcoin is currently underwriting massive investments and alternative energy around the world. John asks, while I understand that the computing power used to generate the proof of work derives value from the very fact that it secures the Bitcoin blockchain, I was wondering if there are any workable solutions for diverting that computer power for bonus value creation, tackling problems requiring big computing powers. Are applications like protein-folding or finding sets of primes workable, or anything else workable in this space? Yes, in fact, there have been a number of systems that attempted to exploit the mining algorithm, the consensus algorithm, to do things that people consider useful, such as protein-folding or finding primes, etc. However, Bitcoin doesn't do that. Bitcoin uses an algorithm whose only use is for Bitcoin, really, and has no other practical use other than proving to the rest of the world that the miners have committed a certain amount of energy and therefore expenditure to back the security of the network. It is essentially the promise each miner makes. Here is a block. I have validated all of the transactions according to the consensus rules. I swear this under penalty of losing the energy that I have put in. Here is the proof of work demonstrating my commitment to securing and validating these transactions and obeying the consensus rules. That promise is in the form of energy consumed for the proof of work algorithm. Part of the problem with doing something else, something that other people might consider useful, is that that splits the reward. It means that the miners have two reasons for which they are mining. One is to secure the network, and the other is to produce protein-folding signatures or large primes. What happens if the production of primes becomes more valuable than the Bitcoin security? What happens if there is a disruption in that network? For example, suddenly we find a new application for specific types of protein-folding, or specific types of prime numbers that make it extremely valuable to produce these numbers or proteins. The problem with that, of course, is that it is no longer worthwhile doing the security on the Bitcoin blockchain. It is only worthwhile using that equipment to do the protein-folding. It is more worthwhile to do the secondary function than it is to do the primary function. Bitcoin security ceases being the primary function, at which point you can no longer trust the promise. The promise being, I committed this energy to prove that I did the security of validating the transactions. If you can't trust that promise, then Bitcoin securities fail. The whole idea of only doing this to secure Bitcoin is that you can trust that promise. It has no other incentive. There is no other reason you would be doing this calculation other than proving your commitment to doing the security of validating the transactions correctly. Bitcoin mining is useful in proving the security of the Bitcoin network. The more energy that is committed to proving that security, the more secure the network is, and the harder it is to attack or compromise the security of the Bitcoin network. Bitcoin's proof-of-work algorithm produces useful work. It produces the security that backs a global currency worth almost $15 billion, and it produces the security that ensures that that global currency cannot be attacked by any actor, however large they may be, without colluding and investing a lot of money in hardware and energy. That becomes very difficult, the more energy and the more hardware that backs Bitcoin security. That is the useful application of Bitcoin's mining. We only have one, because we want that to always be the dominant use of Bitcoin's mining, and never become second place to something else. Can I explain the most important differences between proof-of-work and proof-of-stake? Very simply, proof-of-work requires the investment of energy, which is outside of the system, in the form of work. That energy is a scarce resource. Energy has cost everywhere, anywhere, in any form. Proof-of-work forces miners to deposit that energy in the form of work, and prove that they have deposited that energy by producing proof-of-work in order to validate and claim a probability of a reward. Proof-of-stake does a similar function, only instead of requiring the validators to deposit energy in the form of work, it requires them to deposit cryptocurrency within the system in the form of a stake. They basically bet cryptocurrency. They get locked up for a certain number of blocks, and they put that behind their claim to validate the rules. If everybody agrees with the validation of their rules, they remain on the majority chain, and they get a small reward back in the form of fees in return for the stake that they put in, proportional to the stake they put in. If instead they find themselves on the wrong chain, depending on how proof-of-stake is implemented, they may lose some or all of their stake, or they may simply end up with their stake locked up for a while without being able to gain anything from it.


This is Doctomer 11080A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used SUV headquarters. Today we are checking out this super clean 2011 GMC Yukon Denali. This vehicle has the 6.2 liter V8 motor, which pumps out 403 horsepower. This vehicle has been fully safe and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off and this vehicle is 100% ready to go. Onyx Black is the color. We shoot all of our videos in 1080p, 60 frames per second. So if you have HD capabilities on your computer, tablet, or smartphone device, turn them on right now because it is definitely your best way to check out the quality and condition of the vehicle before seeing it in person. And if you'd like to check out all the photos on this vehicle in the upper right hand part of your screen is a link to our website, click that and check us out. There it comes with the 20 inch chromed alloy rims. Goodyear Wrangler SRA 275 55 hour 20 tires. These tires have right around half the tread left on them. Front fender, no major tensor dings on that. Looks pretty good. Headlight lenses for being a 2011 are in nice shape. Comes with the chrome trimmed grille. Front bumper is in pretty nice shape. It does have factory fog lights. The hood is in pretty nice shape. Didn't see just a couple of dings on there and nothing too bad. Passener side front fender is in excellent shape as well and the passener side rim no major scuffs or scrapes. As you go down this side the vehicle take note of how clean the body is, how reflective and mirror like that paint is. We take these HD videos so if you are far away or even if you're close by and you just can't make the trip down but you're still interested in purchasing the vehicle. You can see the vehicle, hear the vehicle and have confidence in what you're looking at before you even get here. So that when you do get here there's absolutely no surprises. And that is why we offer the Summit Auto Fastpass option in the upper right hand part of your screen. A link to our website where you can apply for financing, get pre-approved, see a payment code, even appraise a trade all from the comfort of your own home. So when you get here all you got to do is drive the vehicle, love it, be all set to go. Back tires have just as much tread as the front tires and the back rim no scuffs or scrapes on there. Pretty decent condition. This vehicle does have 100,000 miles on it and it is nine years old. So pretty nice shape considering the age and miles on it. Rear bumper is in great shape. It does have a full towing package which includes a receiver hitch and seven pin wiring. The rear gate is in nice shape as well. This is a power lift gate. We do have the cover for the receiver hitch there and that's in really nice shape too. Those seats are removable and you get a husky all-wether format back here.  It does have the flipper glass right here. Now it works like so. Didn't see any major dents or dings on that rear gate and as you go down the driver's side just as clean as that passenger side. Rear quarter is in nice shape no major dents or dings on there and for full disclosure this back rim pretty nice shape as well. Down the rest of this side looks really good. It has power folding mirrors with built-in directional signals and blind spot monitoring.  There are no rips or tears on the seat here. Just a little bit aware on that edge both these seats are heated and cooled. Husky all-wether formats throughout power windows, power locks, power mirrors, memory driver's seat, auto headlamps and power tilt steering wheel. It has power folding mirrors which I always like showing both sides so that you know that both sides are working good. As we hop inside the vehicle here you can see that this one has 100,018 miles. Instrument cluster is very nice and clean. It comes with the heated and wood grain trim steering wheel, Bluetooth audio controls on the right, heated steering wheel and cruise controls on the left. It comes with the six speed automatic transmission with the optional tap shift and it does come with the factory navigation system. You have your information center controls there as well AM, FM and Serious XM radio capabilities. Dual climate control, power pedals, parking sensors you can turn those off and it does have stability control. Get the nice wood grain trim on the dash and doors there. Passenger side seat is in excellent condition. No rips or tears on there does come with the power sunroof. Home link buttons for your garage door security systems and lighting systems and you do get on-star capabilities in that rear view mirror. We'll take a quick look at the back seats and then we'll start it up and take a quick look under the hood. It does have the second row quad seating. These second row seats are heated. You get the Husky all-weather format for the second row as well. There's your rear air, audio and heated seat buttons. These seats fold down and then up like so. See the carpetings in really nice shape and those third row seats are very clean too. If you get child safety locks on the back doors inside the doors look pretty good and we'll start it up and take a quick look under the hood. See there's no check engine lights or anything. I would personally like to thank you for checking out the video today and hopefully from this HD video you've been able to tell just how clean this vehicle is all the way around inside and out. Under the hood we have the 6.2 liter of V8 motor. Engine based, very clean, runs very smooth. Once again this Yukon has been fully safety and inspected by our service shop has a fresh oil and filter change. All the fluids have been checked and topped off and this one is 100% ready to go. And I'd highly recommend this vehicle from a quality and condition standpoint considering the age of miles. And to see more pictures of this Yukon or one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers, you name it we got it go to that website right there www.summitauto.com. Full pictures and descriptions of every single vehicle from two locations all at summitauto.com and if you'd like to check out more HD videos you can go to youtube.com slash summit auto remember to like subscribe and share on this video and all the videos that you see there. In fact any second you will see a link to subscribe to our YouTube channel on your left. A link to more Yukon SUV videos like this one on your right if you have not been to our website on the bottom link to this vehicle on our website click those check us out and we look forward to helping you with this super clean 2011 GMC Yukon Denali in Onyx Black Clear Coat. Thanks again for checking out the video.


The study examined the effect of zeolite on the compaction properties and California bearing ratio, CBR, of cemented sand. It was found that the addition of zeolite reduced maximum dry density and increased optimum moisture content of the cemented sand. Additionally, the highest CBR values were observed when zeolite was added up to 30% of cement content. Correlations between CBR and maximum dry density were also identified. This article was authored by Geizem Noraznejad, Issa Shushpasha, Said Muhammad Mirasini and others.


The following is a presentation of TFNN The Trader's Edge with Steve Rhodes. Toll free at 1-877-927-6648 or internationally at<PHONE_NUMBER>. The Trader's Edge. Now Steve Rhodes. Good morning folks. Welcome to the September 20th. A terrific Tuesday edition of today's Trader's Edge show. I'm your host, Steve. Perseverance Rhodes who absolutely knows that each of us should always be pioneers of our future versus prisoners of our past. Hope everyone out there's having a great day. And let's make sure we have an extraordinary one. The easiest way to do that, well, it's to always remember that life is happening for us, not to us. That's right. When you and I make that one little two by four shift well means we can find the gift in every set of circumstance that life is going to toss at us. Now today you and I are going to go check out the circumstance of these markets. We're going to go figure out what those bowls and bears, what those buyers and sellers are communicating to you and I at just past 11 o'clock in the morning. I do want you to know I'm absolutely grateful for your presence here, but more important than that. And that's this during this next 30 minutes, 30 minutes, this next 53 minutes. I'm here to serve you. So feel free to give us a call<PHONE_NUMBER> would love to hear from you. But if you can't call in, you still have a question. I'd be happy to feel that to send me an email. Send it to Steve at TFNN.com inside that subject heading. If you'd be kind enough to put radio show question. And of course, inside our Tigers didn't will any in every ping will do. So let's go ahead and get this show started on terrific Tuesday. This is Tiger Financial News Network. I'm Steve Rhodes. Welcome to the show right now. We got all the U.S. and C's trading to the downside.  Gold's off 3 bucks. Silver's down 12 cents. Light's recruit off a buck 60. Natural gas up 3 cents. The 30 Treasury trade down to 130, 01. That's down 1.11 ticks. Lead in the charge. Dollar wise, the upside. You've got booking holdings of 20 bucks. Humana up 10. Arjenix SE up 7.5. Oxford Industries 6. Karuna Therapeutics is up 6. To the downside, it's regenerate pharmaceuticals. Off 15, Buccaroony. Zepam systems down 14. Repligen down 10. Lenox International off about 10. Black Rock down nearly 9. Buccaroony. So let's begin with the ESMini out here. Let's go change some screens out here. We'll take a look at some multi-time frames. We'll take a look at some TAS market profile, market breadth out here. We'll do the whole kit and the Kaboodle. We begin by taking a look at the ESMini. You know what we'll do here, because it's easier to see. We'll switch back and forth between panels. So let's do this. Change windows. And now we're back. Okay, so now we have the four daily equity future contracts out here. Just simply the good place to start. And as a reminder, where are we at? Here are completed A to B equal CD patterns. If you take a look at the ESMini, the actual A to B that we're taking a look at here is going to be the high, is going to be the same. That's going to be the August 16th high. The B point for the first A to B equal CD is the August 24th low. And the C point is the high from August of 26th, about a 48% retracement. That led to a 1 to 1.618 A to B equal CD that was confirmed on Friday when we formed a bullish hammer candle. Now, if that bullish... And then yesterday was a bullish engulfing. So it is the low of yesterday's session that is the key area of support, that if that gets broken, then odds favor what the ESMini will have is a larger A to B equal CD to the downside. And that would look like this. And the 1 to 1 price projection on that would be a 3729, 1 to 1.272 at 3608. So the key area is also a brand new daily profile that is attempting to form. I see it tempting. Of course, we were using that advanced Doppler tool. It's an advanced detection tool. Let's just know where buyers and sellers are attempting to take their positions. In the case of the buyers, they're at the 3879 level. We're trading at 3883 right now. Resistance where the sellers are at is at 4010. Now, this profile will not be confirmed until this evening at 601. That's the same with the NQ. That's the same with the Dow. That's the same... Well, the Dow has actually been confirmed. That was a goody-future contract. But not the case with the ES, the NQ, and the Russell 2000. I won't go through the small A to B equal CD to the downside, but we had that same pattern form inside of the NQ. Now, it's key level of support that if broken will set up the larger A to B equal CD is going to be the low of that hammer candle from last Friday. And that price point is 11778.50. So if we were to see a close below that, 11161.50 is the area. What price has done so far today is pulled back and tested the area where buyers are located. That's the bottom of that profile. That's 11899. The low so far today, 11887. If you look at the Dow equity future contract, again, a similar setup to the ES and the NQ. You can see the smaller A to B equal CD pattern. Let's go ahead and get rid of that. It too has a new profile that is a temp uniform. No, it did form because it did form. We had... Look, it was a temp uniform yesterday. So in the case of the Dow equity future contract, 3809 is its key level of support. We're trading right now at 3821. Resistance is 31469. So if the support level hole fails, that's going to be yesterday's bullish reversal candle versus the hammer candle from Friday. So the key area there to watch is 3590. Now if price bus through that area, that'll set up the larger A to B equal CD to the downside. That larger A to B equal CD to the downside will get us into the 29537 to 28652 level. But remember, as we speak right now, we have a buy the D point pattern that is in effect for each of the daily equity future contracts. In the case of the Russell 2000, you can see it's buy the D point pattern. It actually really formed out here on September 7th. That buy the D point pattern has not been negated. There's another one that formed yesterday because of its bullish reversal signal. So it really sets up the key area of support for it being 1779-30. It too is trying to form a new profile with support or buyers located in 1794-87 in resistance in 1841-58. So we have established here on a daily timeframe new profiles that either have formed like in the Dow, attempting to form in the ES, the NQ, and the Russell 2000, and each of them with confirmed daily buy the D point patterns. What does that mean? That means that we should see price at least target their oscillator and change line for the daily timeframe or the top of their new profiles, which we will not have confirmation of until tomorrow. So now let's go take a look at the short-term timeframe chart for the ES mini. This way, at least we know where we're at on the daily timeframe. Now we take a look at the ES mini charts out here on a five-hour timeframe. I do not have any kind of a bottom signal out here. What we do have this morning is price pulling back and testing support. That's the bottom of this profile at 3865. The 240 or the four-hour timeframe chart does have a confirmed Rosemont Dominicator bottom. Price also testing support the bottom of that profile.  That took place at four o'clock this morning. Price has pulled back to test its Bola structured profile. That's an area of support. Below that, there's additional support at 3851. As we take a look at the 60-minute timeframe chart, if, as we come into the 12 o'clock hour, the ES mini closes below 3898.50, the 60-minute chart will then have a confirmed TD nine-count bottom. What that should then lead to is a test of its oscillator and change line. That's currently at 3897. 3901 is the top of its profile. A close above that would suggest a further rally. The 30-minute chart has confirmed a buy-the-de-point pattern. That suggests that price should go target its oscillator and change line. That's currently priced at 3889. Steve Rhodes with TFNN will be right back to finish off the ES mini. For booming inflation, we are purchasing powers eroded. There's no better place to protect your hard-earned money than in gold. This-the-goals flagship asset is the Monk Todd Gold Project in the Northern Territory of Australia. This is Australia's largest undeveloped gold project. We are talking a world-class gold project in a Tier 1 mining district. This is a large-scale, low-cost project with significant existing infrastructure and a politically safe and friendly mining jurisdiction. This-the-goal just completed the Monk Todd Feasibility Study, which resulted in a 7 million-ounce gold reserve in a 16-year monlife. All of this combined with the approvals of all major operational, as well as environmental permits. This-the-slingwiches Monk Todd as an attractive, devious party, ready-development stage gold project. This-the-goal trades on the New York Stock Exchange under the symbol VGZ. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible after all. For daily market overviews that give you direction on the key indices, selective stocks, and commodities, subscribe to the Opening Call newsletter at tfnn.com. The Opening Call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices. Get the Opening Call newsletter by Basil Chapman in your inbox every day. First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know, and you'll get a full refund within 30 days of signing up. tfnn.com Educating investors Steve Rhodes started his trading career as a student almost 20 years ago, and the student has now become the master. Steve won the prestigious Timer of the Year award in 2018 and barely missed that mark again in 2019, finishing at number two for the year. An amazing accomplishment. Steve Rhodes is committed to sharing his techniques and knowledge with anyone who wants to learn, and he shares his vast amount of trading knowledge every day in his Mastering Probability newsletter. Steve's award-winning newsletter, Mastering Probability, is delivered every trading day with updates throughout the afternoon. Sign up for Steve's Market newsletter, Mastering Probability, and you'll receive access to seven of Steve's educational webinars absolutely free. At tfnn, all our newsletters come with a 30-day money-back guarantee, so you have absolutely nothing to worry about. Visit tfnn.com and try Mastering Probability 30 days risk-free today. tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors tfnn Educating investors How could that possibly be? Right now, by a show of hands out there, we'll go to Mr. Bill. Mr. Bill, are you seeing the charts for ET? Three panel charts out here right now. Should be what is up on your screen out there. And I hope that's the case. So I don't know how people were seeing the Dow charts. But I tell you what I can do. Yes, OK, great. OK, so perfect. So no request out here. So let's go to what do we want to go take a look at? Yes, you see it, ET and the 10. OK, thank you. Thank you, guys. Let me see what is moving out here. Let's go take a look at Apple. AAPL, it's up to 15 right now. And the Apple, really much like the NQ, I know that on Friday it formed a bullish hammer candle out there. That was really a confirmation of the smaller A to B equal CD to the downside out here. So the key low, the key area to watch is going to be Friday's low. The markets are going to move lower and get back towards those June lows. You're going to need to see a close below the low of that bullish hammer candle. That was 148.37. That was a trading low from Friday. Now, because we have an A to B equal CD to the downside, whenever you get a confirming bottoming pattern, could be a rogment of indicator bottom, could be a TD9 count bottom, could be a buy the D point pattern, could be a three drive to a bottom pattern, a number of patterns that are out there. What then typically unfolds, as we're seeing here right now, in the case of Apple, is to move up to its oscillator and change line. The oscillator and change line on the daily base for Apple is at 156.93. The actual high today so far is 156.89. Price right now is trading with inside a bullish structure profile. If price can clear that red oscillator and change line, again, that's 156.92, what Apple will do is make a move to the top of that profile. That's at 159.60. So you've got to confirm by the D point pattern, price right now testing resistance, that's the weekly timeframe. If price does clear that daily oscillator and change line, it will then have some potential sellers at about 157.63. That's the weekly oscillator and change line. So therefore, if price can clear that level, then that's a move up to 159.60. And if price can clear that, then we're looking to move to 164.6, maybe 164.38. Steve Rhodes with TFNN, we'll be right back. If you want to take advantage of this sector now is the time to subscribe to my Gold Report. The Gold Report is a comprehensive look at the metal sector as well as the markets that move gold, which is the currency and bond markets. New subscribers get a 30-day money back guarantee so you have nothing to lose. Every Monday morning, I publish the Gold Report with coverage of gold, silver, bonds, the XAU, HUI, GDX, as well as more than 30 different mining equities. To see for yourself the types of profitable trades that are recommended within the Gold Report, sign up now by visiting TFNN.com. Don't miss out on the next great gold trade. Sign up today. TFNN is excited about our new software charting program, The Art of Timing the Trade Charts. In collaboration with Tom O'Brien and using his best-selling book, The Art of Timing the Trade, your ultimate trading mastery system, David White has programmed an outstanding piece of software that will complement any trader's methodology. Using this first-of-its-kind program, The Art of Timing the Trade Charts allows you to scan thousands of stocks for Fibonacci formation setups, including guardleafs, ABCs, butterflies, and much more. The Art of Timing the Trade Charts is designed to help you when scouring the markets for stocks just beginning to form the trading patterns that many investors spend days, weeks, or even months searching to find. And right now, we're offering licenses available at only $79 a month. We are so confident that you're gonna love this new charting software that will even give you a 30-day, unconditional money-back guarantee. Don't miss out on this incredible new piece of software. Get your copy of The Art of Timing the Trade Charts today by visiting TFNN.com. Sharpening your skills as an investor is like getting better at playing a musical instrument. You have to practice, sure, but you also need excellent instruction from experts. At TFNN, you'll get advice and guidance from the authority in technical market analysis. And it's not just dry, tedious text either. TFNN airs live financial content streamed live on TFNN.com and TFNN's YouTube channel with Tiger TV, live every market day from 8.30 a.m. to 4.00 p.m. Eastern for free. Each host is an experienced trader and gives their take on the market while taking calls and questions live from around the world. From the moment the market opens until the closing bell sounds, Tiger TV has eight different shows with expert hosts to help you make the right moves with your money. Watch online at TFNN.com or on TFNN's YouTube channel and become the investor you were born to be. TFNN Educating Investors. This segment is brought to you by Think or Swim. For more information, just click the Think or Swim banner on the front page of TFNN.com. Welcome back, folks. Let's get to a couple of DEN requests out here. The first one's coming in from Sat P inside the target. So let's take a look at the XLF. So we take a look at the XLF at top with the TD9 count top, that was on August the 16th. Bottoms with the TD9 count bottom, that was on September 1st, also formed a buy the D point pattern. That completed on September the 7th out there. Yesterday was a test of two prior swing points. The first one being September 1st, that did 40 million shares. The second one that was on September 6th, that did 45 million shares. Yesterday was a test rejection of that swing point on lighter volume, 36 million shares. So the bottom seems to be pretty solid here. Now what I mean by that is just simply the XLF should go target that oscillator and change line. That's up at the 33.49 level. Above that, resistance would be at 33.95. The weekly timeframe chart says, hey, hold your horses here, guys. I'm trading below my red oscillator and change line. Trading right at about the bottom of its new weekly profile that formed yesterday, just slightly above it. So just to suggest on a weekly basis, caution. But you do have a confirmed two bottoms out there for the XLF and the third one, which was a test of a swing point, a test rejection on lighter volume. You wanted to look at BAC as well. So I thought we would do the sector first just to see what it was signaling to us. And now let's go check on the health of Bank of America. It'll take a moment here for the Bank of America charts here to populate. But once it does, it will give us a what. So this did not have a TD9 account bottom that formed out here because it was not, the low was not on the bar of eight, nine, or the bar following nine, but there certainly was a buy the D point pattern. And that was confirmed with that bullish hammer candle on September 1st. That was confirmed with the bullish piercing candle on September 7th out there. Bank of America a little bit stronger than the XLF, but it's really dealing with the consolidation with inside its daily profile, 3351 to 3463. So you're up at resistance. I wouldn't tell you to buy here if you're looking to buy your active position inside the XLF. I would say one area to look would be 3351, the bottom of that daily profile. As I take a look at the 30 minute timeframe chart here for Bank of America, just see if there's any kind of signals there. The answer is not really what this is signaling to us on a 30 minute basis is price trading with inside its profile. It's fairly large, 3375 and $34. That is the support zone, resistance at 3475, which I was hit and rejected at 10 o'clock. Now that was 10 o'clock yesterday, at 10 o'clock yesterday morning. Price to blow the oscillator and chains on. So Bank of America should pull back in that 3375, 34 level out there. So I hope that helps you out, Sat with regard to what the XLF and Bank of America are doing. Thank you very much for your request. Dan, inside the Tiger's Den, I want to take a look at Casaba Sciences. S-A-V-A is the ticker symbol there. Casaba Sciences right now is taking on resistance. Dan, that is the top of its daily profile. The top of that daily profile, as you can see is at 3446. So that's where your sellers are lined up. Casaba Sciences doesn't have a confirmed sell the D point pattern out here. I don't think so. It looks like it's a much larger A to B equal C, but maybe there's a smaller one. No, that's not even it. Yes, I don't see the A to B equal C D pattern out here. And the reason I don't see that, well, here I'll just draw on the A to B line. Whoops, that's a hit the wrong button and you'll get the wrong result. That's for sure. So let's draw in the A to B line. So that's at that wave seven bottom out here back on July 27th. Then I go all the way up that shooting star. Now, if we simply take that A to B line, move it to where the C point is, you'll see Casaba Sciences has got a A to B equal C D or could have one. If price can clear that shooting star, that's the high from the trading day of August 17th. And that would then take us up into the $45 ish area 44 in change. So what you've got a price is moving up today. It's done 1.8 million shares versus on September 13th 2.7 million shares, but doesn't really matter from a volume perspective. Yes, price is pushing higher. It does have accelerating volume. But what we know is that the sellers are lined up at 34.46 and Dan, that's the number you are looking for price to close above for two consecutive sessions. If we get that, we likely then get the fulfillment of that A to B equal C D to the upside. The monthly chart out here has a nice TD nine count bottom. That suggests that over time, price should go target 38.71. Well, if we get a close above or two closes above 34.46, really the level that price really needs to close above to confirm that A to B equal C D is going to be 35.58. If you can close above 35.58, you'll go target 38.71 to 41 bucks. And if you get above that, then you'll get off into that $44 area out there. So right now you've just got a good old fashion consolidation. It's easy to draw in inside of the weekly timeframe chart. We'll just use our rectangular tool. Bottom is very clear. And I think the top is very clear as well out here with regard to Cassava sciences. So it's somewhere right around that range. Dan, thanks so much for the request. I do hope that that helps you out. And let's go to our next request. That next request coming in from, let's see here. Bob and Spokane. Let's take a look at McDonald's. MCD is the ticker symbol. Let me get a little quench of water here. Give me a moment here folks. Actually, before we go take a look at McDonald's, we've got a caller and it's John and Philly. Hey, John, thanks for calling. Thanks for holding. How are you today? I'm doing very well, but based upon the stories you told yesterday about the preceding week and not nearly as well as you. Well, John, I've never had a fishing day like we had on Friday last week. The most amazing thing. You've done a lot of fishing. The only thing comparable to it, the only time I've ever had anything similar to it, but it's pretty normal was perch fishing in Lake Michigan. And you know there, you take barrels, right? Garbage barrels out there. You can throw, you know, right? You can throw your line over with 10 hooks and you'll come up with, you know, nine or 10 perch out there. But that's a different type of fishing. You know that you kind of typically experience out there. You don't, you know, experience that in the, you don't get three out of first, the first two casts and pull, before I could throw the second line in, we were hit on the first rod. So that was a beautiful thing. But I know you want to take a look at Adobe out here, ticker symbol ADBE. Or do you have any fishing stories like that? Well, Steve, I do not want to bore your audience and the TFNN crowd. So I will not share my personal best fishing story, but sometime we'll work together. You can tell me more about Massachusetts fishing and I'll tell you about Maine fishing. Ah, perfect. Perfect, perfect. That sounds good. Well, buddy was not up. Steve, yes. Steve, I wanted to ask, please help me with your tools assessing Adobe. Just for what it's worth, literally in the past hour, I took a partial position. This is an unleveraged portfolio position in Adobe at 292. So that is where I'm standing. I'm taking a shot here. The stock has come off of that 2021 peak up at 700. And most recent dump from, you know, 390. This occurred as Adobe management spent just a huge chunk of change buying another software company. And clearly owners of Adobe are just voting with their feet saying Adobe management paid far, far too much for they just entered into an agreement with last week. Anyway, so, but I have known this company obviously for 20, 25 years. Yep. And I'm taking a starter position. I'll do a little scale down if that occurs. So that's why we're going to a hard break, John. We're going to a break here, John. Hold that thought, John. If you were going to a break, we'll come back, we'll take a look at Adobe with John and Philly, just a few. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible after all. For daily market overviews that give you direction on the key indices, selective stocks and commodities, subscribe to the opening call newsletter at tfnn.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices.  First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. tfnn.com, educating investors. The technology around us is changing every day. With so much happening, it can seem impossible to keep up with all the information. David White's investment newsletter, the technology insider, is designed to give you all the information you need to understand the technology that shapes today's markets and tomorrow's future. David White has made his living staying on the cutting edge of technology. His weekly newsletter will give you specific recommendations for valued tech stocks as well as entry prices, target prices and stops to set for each trade. Dave delivers his weekly newsletters every Friday with updates throughout the week. You can get the technology insider at tfnn.com for only $37.50. Sign up for Dave's newsletter, the technology insider and get an inside look at everything the technology sector has to offer. Try it risk-free today with our 30-day money-back guarantee. tfnn, educating investors. Are China A shares hot or not? If you trade China A shares, now may be time to take a closer look. Trade CHAU or CHAD, directions daily CSI 300 China A share bull and bear ETFs. China A shares in either direction. Visit directioninvestments.com today. An investor should consider the investment objectives, risks, charges and expenses of the direction shares carefully before investing. The prospectus and summary prospectus contain this and other information about direction shares. To obtain a prospectus or summary prospectus, please contact direction shares at<PHONE_NUMBER>. The prospectus or summary prospectus should be read carefully before investing. An investment in the funds is subject to risk including the possible loss of principal. The funds are designed to be utilized only by sophisticated investors such as traders and active investors. Distributor, four-side fund services, LLC. This program is brought to you by Vista Gold, traded on the NYSE American and TSX under the symbol VGZ. Welcome back, folks. We're on the line with John and Philly. Hey, John, I apologize for having to cut you off there as we're going into the hard break. But I know you want to take a look at Adobe. You've taken that long position there. Is there anything else that you wanted to share with me before I take a look at the charts as I was cutting you off, so to speak? Thank you, Steve. Specifically, I asked you in observing your monthly weekly daily charts. Yeah. If you can share with us what those say to you in attempting to identify potentially where and when the trade will bottom is likely to form. Excellent, okay. So easy question to answer from this standpoint. I'm taking a look at a A to B equal CD pattern. There's several of them potentially within sight of Adobe. The one that I'm looking at, John, is when this formed a wave number seven top back on August 15th, I'm sure you have noticed that. And then the A to B equal CD to the downside. Now this completed more than a one to one A to B equal CD to the downside, I would say. And it did that when we had a bullish hammer candle that formed on September 16th. The problem is that that bullish hammer candle is also a gap to the downside or a falling window. So the question is, was it a bullish candle or a bearish candle? Well, the answer to that could come today. If price were to close below 292.14, the low of that bullish hammer candle, then that would tell us that a key level of sport had failed and that price is likely to head lower. If price holds at 292.14 level, that by the D point pattern, then you could see a move up towards its oscillator and change on a 337.32. So the key area I believe to watch today, tomorrow, 292.14. The reason why I think that's a key level to watch is because my eyes gravitate over to the monthly timeframe chart, the very right hand panel. And what we see here, John, is that this month, the month of September, will be the bar following bar number nine. It is the low of the pattern. And that suggests that a teeny nine count bottom should form this month. Now it's breakout level where prices may target, it doesn't have to target, is 255.13. So I would say, odd's favor, if price closed below 292.14 on a daily timeframe, then Adobe should go target at 255.13 area. Does not mean that it needs to get down there, but that should be its price target. And the weekly timeframe chart has a Rogement Dominicator signal that's been triggered. So there, if you get the early entry, such as let's say the daily, the position that you took earlier, you had to close above 292.14, that A to B equal CD pattern holds, and then you eventually get a bullish reversal candle on the weekly, you'd have then two confirmations to follow up with the teeny nine count on the monthly. So it's 292.14 is the key area to watch out here. You get a close below that. And odd's favor, it's gonna make, or price gonna make its way back towards that 255.13 level. Does that make sense to you? What I'm looking at, what I've shared with you. In that month, which the bar after bar nine on the teeny nine count, that's very, very fascinating. My only, just my observation over the years and years and years is I have seen many, many times in which a company does a big dilutive deal. Yes. And holders just give up and just get me out, get me out, get me out. Oftentimes those horrible declines and this, both the monthly, the weekly and the daily charts are absolutely horrible. No doubt that I've seen situations like that where it's capitulation and price doesn't go up for a period of time, but it doesn't go down very much further. And so that's terrific set of parameters. That's all I asked for. So I thank you, Steve. Hey, John, thanks so much for calling. You have a great Tuesday. That was John and Philly and that was Adobe. Let's go to our next request out here. I wanna make sure that I get to it. I believe that is going to, we took a look at BAC and the XLF. So I believe the next question out here is for Tessa, TSLA, and that is for MKC inside of the Tiger's Den. There we go, we got rid of that. So that's helpful. So I'm just trying to go back and take a look at Steve. McDonald's, oh, McDonald's. Do we need to take a look at McDonald's, MCD? So there we go. We'll go back and take a look at that Tesla momentarily, but their first question came from Bob and Spokane. And he says, McDonald's has been stable through this last downturn. Thoughts, please, may start a position soon. So we take a look at McDonald's out here. You do have that nice TD-9 cal bottom that formed on September 1st. That swing point had volume of 2.5 million shares. That was tested with 4.3 million shares. So McDonald's isn't out of the woods just yet on a daily basis. Price did find resistance. It's really consolidating with inside the daily profile. So Bob and Spokane, you're looking to add to a position. I would say it would be between the levels right now of 251.76 to 252.92. 252.92 is the bottom of the daily profile. The first figure I gave was the TD-9 count bottom low. That's a swing point from September the 1st out there. As we take a look at the weekly timeframe chart, we can see that there was a TD-9 count bottom, forms March 11th, and that takes price all the way up to its breakdown level of 266.72. So we know that's a key area of resistance. With price below its green weekly asset or in chains line, that pull back into that 259 is not to unwarranted out there. In fact, if I look at a 30-minute timeframe chart, what this is signaling to us is that price closing below 254.05, that's the bottom of its current profile. You should see it move to 252.98. So on the 30-minute chart, price is at support. If that support level fails, that suggests lower price out here. On a monthly timeframe, you've got a TD-9 count top, that is like really nothing more than a test of support at the bottom of that monthly profile. You're right, this has held up very well up towards its highs out there. If those are the all-time highs, let's pull this back. Yeah, I'd just favor that. I'm not only back to 2009, but it looks like McDonald's is very near the all-time high out there. So you're looking to add to this or perhaps get a starter position. Again, watch the bottom of that profile or the low of those swing points from September out there. So I do hope that helps you out. Thanks much for the request. The next request was from MKC who wanted to take a Tesla and Coin. Let's put up Coin, Coinbase first. COIN is the ticker symbol. Now let's see what we can find here. And with regard to Tesla, let me get that up on a different screen. So with regard to Coinbase, what is DB Show? Daily timeframe, not a whole lot. Price is consolidating with inside its daily profile between 61, 66, and 73, 80. Nice rogment of indicator bottom. On the weekly, that took price right up to where the sellers were located at 101.73, the top of its profile. You're consolidating with inside that profile, 56.05 would be a potential area to consider a long position, monthly not helping me out. So if you're looking for a long position, you just say you just have to take a look at it, I believe. If you're looking to go long, I'd wait for about the 56.05 area, and I'd like to see some kind of bottom signal on an intraday chart. Tesla, TSLA, that's trading right now at 312. It's trying to take out a prior high. The prior high it's trying to take out, takes you back to August 16. There were 88 million shares traded on that day so far in the first two hours of trading you've done about 30 million shares. So this very well could generate the type of volume to take out that swing point. That swing point being right here, trading day of August the 16th, that high is at 314.67. In fact, you may have a confirmed A to B will see it in the upside, September 12th. That did 48 million shares. You crossed that yesterday with 60 million shares. So Tesla has a confirmed A to B equal CD to the upside.  Steve Rhodes with GFN, we'll be right back. Vista Gold owns and operates the largest undeveloped gold project in Australia, the Mount Todd Gold Project. Vista Gold just completed their feasibility study resulting in a 7 million ounce gold reserve. Vista Gold has all major permits approved and has retained CIBC capital market assistance in evaluating alternatives and in completing an accretive transaction. Vista Gold trades on the NYSE American and TSX under the ticker symbol VGZ. Vista Gold executing a strategy to create shareholder value. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible after all. For daily market overviews that give you direction on the key indices, selective stocks and commodities, subscribe to the opening call newsletter at tfnn.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices.  First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. tfnn.com, educating investors. Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at tfnn.com. When you subscribe, you'll get a weekly report from veteran day trader Larry Pezzavento on stocks you need to pay attention to and you can trust Larry's analysis. After all, he's got 45 years experience as a day trader. Larry will also provide daily charts, videos, and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. First-time subscribers also get a 30-day money back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. Subscribe to the Fibonacci 24-7 newsletter today. tfnn.com, educating investors. Tfnn has launched the Tiger's Den. Hosted at Discord, Tfnn has been educating traders for more than 20 years with live programming hosted by a variety of professional traders during market hours. The Tiger's Den, available to all Tigers and Tigresses for just $1 for the year. There's no cash or added costs when you join our community of traders. Sign up today and become a part of this educational community of traders. Just visit the front page of tfnn.com. Back up, folks. So we got a request out here all the way from Poland, Mike in Poland, and he wants to take like a nasty headache, so I do want a telephone, no problem there. Once take a look at three ticker symbols, we only have really time for one, so we'll choose Franco-Nevada, F-N-V, you want to take a look at Qualcomm and Tigger Symbol, D-A-N. So we take a look at Franco-Nevada out here. We can see that it has triggered a roadsman to indicator signal, but what we don't have is a bullish reversal candle, and that ideally is what you want. You can't see on a daily basis yesterday, price had a nice move, ran right into resistance, that resistance being its red oscillator and change line. That's currently printed at 1,1986. Ideally today, or ideally maybe tomorrow, you get some type of bullish reversal candle to confirm that roadsman to indicator bottom. Then you need to see a close above 1,1986 to suggest a rally up in the 124, 126 level out there. On a weekly timeframe, what I don't see, I don't believe what we see out here, is a buy the D point pattern, but let's take a look at the A to B level. Let's move that over to our C to D area. That would take us to about right there. Yeah, so we don't have a completed weekly A to B equal CD pattern. The monthly does not have a completed pattern. So what that is suggesting on those longer-term charts is likely lower price out there. Well, we do have a chance. Maybe let's go take a look at Qualcomm's QCOM out here. They're on your watch list for long-term buys. So Franco-Nevada, keep the powder dry there. Let's wait for some other better patterns to show up before you take a long position there. Qualcomm does have a wave seven bottom. That was confirmed yesterday. Now, what you need to see here is for price to close above its oscillator and change line. That would need a close above around 126.93. That number's gonna change by pennies or so as price moves up and down. But if you do get a close above that, you should see a move to 128.63. It's a close above 128.63 where you would have a change in trend signal. That's the top of its daily profile. Otherwise, the weekly chart says 12103 might be a better buying area out there. There is no bottom signal yet on the monthly timeframe. Folks, thanks so much for all the requests out there. Have a terrific Tuesday. I'll see you back here at 11 o'clock sharp on wonderful Wednesday. Thanks so much for joining us. Have a great day.