So let me sort of recall a little bit what we have been doing last time. So here is just a little summary. So we're talking about this Markov evolution equation, right? F0 is a density. And it's a density on the sphere Sn minus 1 with radius squared of n, probability density. And then you evolve the F0 with this operator where the Q is given by this average over scatterings. And the scatterings are done with the Rijf, where you replace the i's and the j's variable by the post-collisional velocities. And you average with respect to probability measure, rho theta over the angles, right? So it's an extremely simple model. And last time we talked about the entropy. So we looked at, in a way, we showed last time that there is an entropy structure for this problem in the sense Laurent de Willet explained. So we introduced dissipation, which is given by the integral of 1 minus Qnf log f, and n in front. And always, yeah, maybe I should say, unless I say anything else, an integral is always an integral over Sn minus 1 squared of n d sigma n. And this d sigma n is uniform, normalized. We also introduced the following quantity, gamma n. So by the way, so what is this? I should maybe say. This dissipation comes out by taking the entropy, integral f log f. And what do we do? We differentiate the entropy back to t at t equals 0.  And I forgot there's a minus sign. So that's what we showed last time. And then we introduced what is called the entropy production, gamma n, which is d of f divided by s of f. But this time, we take the infimum over all densities with integral equals 1. And then we talked about Villain's theorem. You see, what you would like to have is, you would like to in some sense compute this number, and then you sort of hope for the best, right? You hope that this number is not terrible in the sense that it goes to 0 as n goes to infinity or something like that, right? And this story is really bleak, because first of all, we have Villain's theorem, which says that gamma n is greater equals 2 over n minus 1 when rho is equals just 1 over 2 pi. And we have also Inov's theorem for any theta positive, there exists theta so that gamma n less or equals c theta divided by n to the 1 minus a. So in some sense, this was actually, Villain conjectured that this is more or less the right behavior in n, and more or less that's proofs. Anyway, but let's try, what I wanted to show you is how to prove that theorem, because it raises a bunch of interesting mathematics. We did last time, we showed, we started out with an induction argument here on this problem in terms of n. We inducted in n. Induction worked like this. We were able to estimate it, sorry. Now, by gamma n minus 1. So it should maybe put an n here. This would be better, I think. Because it really depends on n, that's of course the point. And here we got n over n minus 1, the entropy, and then comes unfortunately another term, which is gamma n minus 1, 1 over n minus 1.  What is p k f? There are two ways of thinking about this. On the one hand, you can just integrate f over the normalized measure on the sphere s n minus 2 square root of n minus v k squared. So in other words, what you do is you fix the variable v k.  And that's a projection. Another way of thinking about the p k f is by just thinking, what do I do? I take my function and I average it over all rotations which fix the v k axis. That's the same thing. All right, so in a way another third way of saying what this p k f is, in a certain sense it is a marginal. You take your density, and you take the marginal with respect to all variable except the variable v k. But it's a little bit complicated because these variables are dependent. OK, so now the theorem, so in some sense what is our goal? Our goal is to find an estimate on the ratio of this number divided by the entropy. So we have to estimate this right hand side in terms of the entropy. This is a theorem. So this is theorem. Well, let's leave it here. So again, f in L1 of the sphere as n minus 1 squared of n, integral f equals 1.  OK, so that's what this result is. And this is a result which was proved by Carlin. And so let me make a few comments about this result here. It's very easy to see that the number two is actually sharp. That's easy. And the way you think about this the following, let me draw it by pictures. And of course, it's very hard to put pictures in n dimensions when n is 10 to the 26. But morally, it works out. You see, what you're going to do is you take a function, so here's the north pole, if you like. That's the one axis. And now what you do is you make a very high characteristic function over a tiny little set, but you integrate this to be 1. So this function has a certain height h. And this has a certain area, a, which you call epsilon to the power n minus 1, because the epsilon is roughly the dimension of this patch. Remember, this here is Sn minus 1. So what do we need? We need that the integral of this function f should be 1, so that forces you, remember, epsilon is really, really tiny. So this is flat, basically. So you get that h is of the order epsilon minus n plus 1, and the a, as I said, is of the order epsilon n minus 1. And you notice this function is, of course, radially symmetric. In other words, when I apply the P1 to it, it just gives me back the same function. Remember, this is the one axis. So when I rotate this function, it stays invariant, so in average, it stays invariant. Now what happens when you take Pkf, when k is not equals to 1? Well, you notice that, so you have an axis which goes now out this way, but what you know is that when you average this thing, this stays always a good portion away from the north pole, and it stays a good portion away from the south pole. And this has nothing to do with the dimension. That's just the way it is. So what you get is that this Pkf is a function which, essentially, when this is very, very small, is a characteristic function now. But what it is, it's a band on the Sn minus 1. Band, right? And what is, so Pkf is, again, a function of height h area a, little a. What is little a? Well, this time, the little a is of the order of epsilon, because this is a band, right? And therefore, because the whole integral of this function should be still 1, the h must be of the order epsilon to the minus 1. So that's what this Pkf is. So now let's go on and check out these terms here. So you notice that the integral of P1f log p1f, well, that's, of course, just integral f log f. And you see that eats up half of this 2 here. And now you have to ask yourself what happens with the Pkf's. Well, the Pkf's, what is the entropy? Yeah, by the way, what is the entropy here? We should actually compute it. Well, that's easy. Why? Because when f is equal to 0, the entropy is 0, right? f log f is 0 when f is equal to 0. And on the top here, this is just log h times the integral f, which is 1. So that's just the logarithm of epsilon to the minus n plus 1, which is minus n minus 1 log epsilon. I think you would agree with that, right? That's what this entropy is. And now, what is the entropy of integral pkf log pkf? What do you get? Well, this is, again, same reasoning. It's more or less a characteristic function with tiny little arrows, maybe. But what you get is that this is equals to, I should say, yeah. Well, the height is epsilon minus 1. So you get minus log epsilon times the integral of pkf, which is 1. So now you see what happens. How many such terms do you have? n minus 1. And so when you add this up, you remember the log epsilon gets very, very negative because epsilon is small, right? So this behaves roughly like minus 2n minus 1 log epsilon, which is twice this f log f, right?  This is very different when you go, for example, into rn. And let's make this point here. It's a very elementary point. Suppose you have a density f on rn. And you define the marginal mi of f.  So this is a density, so in other words, the f over rn. And mi f is simply defined as the integral of f of all its variables, v1. And now the vi variable, well, I don't even have to write this down. I'm going to write dv1, dvi hat dvn. And what does this mean? It means that it did not integrate over the ith variable. Yeah, no, no, I mean, this would have to do with the proof of that inequality. So physically, what this means is we have to maybe ask ourselves, what are the bad configurations for the Katz model? And the bad configurations are precisely this, where you have a substantial part of the mass centered at the poles. And that takes a very long time. Imagine what this means. Suppose you stick all the mass at v1. This means that only one particle has all the mass and all the others. Yeah, exactly. That's right, that's right. That's precisely the intuition, right? Because you concentrate too much into a very small system, and the other system which has no energy, and then it takes forever for the collisions, right? Yes, that's right, exactly. And this is what I'm going to talk about. OK, yeah. I mean, this isn't, in some sense, the question, what kind of reasonable states can you write down which avoids these problems? That's going to be the exact. Very good. Now, thank you for making this remark. This is good. OK, so let me now finish this elementary example. You see, I claim now, and this is totally elementary to show, when you take MIF log MIF, and the sum on i, i equals 1 to n, this is always less than the integral f log f. And the proof is extremely elementary, right? Because what you do is you write this thing out in the following way. You take here the product of the MIFs, log f divided by the product of the MIFs, here, product MIF, then dv1 up to dvn. OK, this is what you start. And you notice what I've done. I mean, in some sense, I've written down the relative entropy of f with respect to this product, but I'm writing this by dividing here and multiplying here. Now, why do I multiply? The reason why I multiply is because this product here is a probability measure. And that's, of course, extremely elementary, right? Because when you integrate, these guys are independent now. So what you get is just n times the integral f. But the integral f is 1. So this is a probability measure, OK? Good. So therefore, what can you do? Because this function, x log x, is a convex function. You can use Rensen's inequality. And you get that this quantity is what? It's greater or equals the integral of this quantity against this measure, which is just integral f, which is 1, times the logarithm of this quantity integrated against this measure, which is again log f, integral f. But the integral f is 1, so this is equals to 0, right? It's extremely elementary. And unfortunately, such kind of arguments, it would be lovely to apply to this inequality. But we are not quite there because these two, we don't really understand. I mean, this example explains that it's good. But how does it come in? All right? Great. So let me just check that I have. Yeah, let me also mention that Cedric has a particular conjecture about this. And this conjecture says, if you assume that the initial condition has fourth moments of order n. So let me write this here in this corner here. So suppose the integral of f0, sum of the vi fourth, is less or equals to constant times n. So what does this mean? You see, what is vi? Suppose you concentrate on a pole, then the vi squared is of order n, right? Because you see that you're on the sphere of radius squared of n, the vi, when I concentrate on the pole, is squared of n. When I raise it to the fourth power, I get n squared. So what this does, in a certain sense, it depresses the function f0 near the poles.  You know that, right? That's his conjecture. So you should have exponential decay in entropy. And by the way, let me mention this argument which I gave so far is not going to work, because you would think that you could improve this inequality under this assumption here. You cannot. That's unfortunate.  Still, the entropy would be huge, would still be log epsilon. Because you notice these powers just go along for the right. So you cannot beat this inequality. And it improved that. All right. Good. So let's go on and start thinking about how to prove such kind of problems. Yeah, let me finish, of course, the argument. Sorry. So we have now this inequality here. So let me go on and finish the proof of Vellani's theorem. You see, we have now an estimate on this which says that this is less or equals twice s of f. So therefore, we get that this gadget is greater or equals gamma n minus 1. Here you have an n. Here you have a 2. You have the same denominator. So you get n minus 2 over n minus 1 times the entropy s of n. And now you just iterate this. And you notice I computed this last time that gamma 2 is equals 2. And you'll learn that gamma n is greater equals 2 n minus 1. That's easy. All right. OK, good. So now let's go back to this kind of, I would like to call them entropy inequalities. These kind of inequalities show up very, very often. And therefore, I would like to sort of expand a little bit on it, because that's kind of useful. Let me first start with a very general remark. Again, it's completely elementary, but quite useful. Let me note this here. In this little example here, I used Jensen's inequality. And Jensen's inequality you can actually use to prove the following lemma. So x mu is just a measure space. And you have a density, f density d mu. Then the integral of f log f of x d mu is equals 2. The supremum over all phi's of the integral of f phi over x minus the log of the integral e to the phi d mu. So this is what you call a Legendre transform, right? In some sense, this is the dual function. This is the function dual to that function here. Of course, I have to assume that the integral of f is 1. And the proof is quite elementary, because what you do is you pick any phi, and you form e to the phi divided by the integral e to the phi. Of course, you have to make sure that these things are integrable, right? Otherwise, it doesn't make any sense anyway. And let's call this function psi. And then you do precisely what I did before. You compute f divided by psi log f divided by psi mu. Notice again, this is a probability measure. That's because the way it's cooked it up. Therefore, this is positive. Can you still see? And now you just work this thing out what it is, right? When you pull things apart, one term is integral f log f. Then you get, you see this psi cancels out. And then you get from this one over psi, you get a minus integral f phi. And then you should not forget the normalization here. The normalization comes up. That comes with a plus. And that's positive, right? So that shows that this quantity here is always greater or equal to that quantity here. Now we have to check that the supremum is actually equal. That's maybe not important, but it's useful to do. Well, what do you do? You choose to phi cleverly. And when you choose phi to be the logarithm of f, you get equality. So this is a useful device, because in some sense, it reduces proof of any kind of entropy and equality always to the computation of such an integral. And there's a vast technique behind computation of such integrals here. So let me erase this now because the next few, which shall I say the next half hour, I'm not going to talk now about cats. I'm talking more generally about inequalities of a certain type. They are known as brass-cumblebe inequalities. Now let me just write down an inequality which is fundamental and will actually give us this result when we just apply this lemma. Inequality is the following. Again, Carlman. And it's completely disgustingly easy. When you write it down, the proof is a little bit more tricky. So imagine you have j functions, and you evaluate each function at the coordinate vj. So you have to imagine, right? You have this sphere n minus 1, n minus 1 dimension. But you have given n functions. And these are functions of a single variable. And now what you do is you take the product and you integrate it over the sphere. So let me again write it here. It doesn't really matter what the radius of the sphere is. Point is that the measure should be always normalized. OK. So now what would you like to do? You would like to estimate in terms of some LP norms. What you can show is that this is less or equals the product, j equals 1 to n. Here you put the fj 2 norm on the sphere. This is on the sphere. You see, each function is a function on the sphere. You can look at it this way because the vj, right? There's a coordinate on the sphere. So you integrate this function, you square it, and you integrate it on the sphere. The constant 1 is sharp. You cannot do better. And the 2 is also sharp in the sense that you can replace the 2 by something bigger, but you can never replace it by something smaller. And why? Because if you could, this inequality couldn't be true. It would be true for a smaller number. Anyway. So these are what is called a new version of the brass complete inequality. We'll explain it later what this is. So let's try to understand how we get the entropy inequality from that. And you see, this is precisely the point. Because what you do? What are we going to do? So proof, what was it? Let me call this theorem 1. Let me call this theorem 2. So what you do is you're going to choose the fj to be vj, our choice. So we're going to prove right this better, OK? And now what you do is, how are you going to apply this lemma? But you have to ask yourself, what should the phi be? Let me think. I want to check that it didn't screw up the computation. So look, that's right. You're going to read this inequality in a slightly different way. You're going to put it here as a square root. Why do you put the square root? Because then you get the integral of the product of the square roots of the pjf is less or equals the product of the L2 norm of the square roots, which is nothing but the integral of pjf to the power 1 half. But these guys are 1 anyway, you agree? So this guy is equals to 1. So what you know automatically is that this is a probability measure, right? OK? Good. So therefore what we're going to do is now, we're going to choose phi to be what? Phi is going to be, so sorry, e to the phi is going to be the product of the square roots pjf. And now you go to this lemma. What do you know? Well, you know that e to the phi, the mu, we just showed it. Well, it's maybe not a probability measure, but I know that the integral is less or equals 1, right? That's what this inequality tells me. Therefore the logarithm is negative with a minus sign is positive, so when I drop this, I get a lower bound. Agreed? But this here is less or equals 1. This is negative with this minus sign is positive. When I drop it, I get that this quantity is greater than or equals that quantity. Forget about the sum, OK? So now what is that? So all I have to do is I have to compute the integral xf phi. So what is it? Well, phi is the sum with a half in front of the logarithms of pjf. So therefore we know by this lemma, lemma, what was it called? Just lemma. The integral of f phi is equals the integral of f pk f sum on k with a half in front. Sorry, log pk. OK, so we are therefore at the stage that we know, because of this inequality there, that the integral of f log f rate equals the sum with a half in front, k equals 1 to n. Integral f log. And now you realize, because this function depends only on the variable vk, you can actually average this f freely over all rotations which keep the vk axis fixed. So you're free to put here a pkf. But you see, you have this one half in front. And that you push over here, and that proves the theory. This is how these things work. So in some sense, that's this lemma which helps in this way. I mean, you can also do it simpler by some elementary Jensen, but I would like to really push this lemma a little bit. This is an idea which was used heavily by Carlin and Cordero Arauskin proving entropy inequalities. OK, good. So we have now this theorem 1. That's great. So the question comes how you prove something like that. And here things get a little bit hairy. Let me tell you what is the idea. So you see, very often, what do we do? We use entropy inequalities and all these kinds of things to show that certain flows converge to equilibrium. And however, there is also a way. So what is it? It's the idea of a Lyapunov function, right? Because the Lyapunov functions tells in general something about the flow near the equilibrium. And what you do now here, you can also do the opposite. You can take an inequality which you try to prove, and you try to invent the flow for which this inequality is a Lyapunov function. That's quite a fruitful idea. This has been done in many, many circumstances. And you can also do it in this connection. So let me explain this. Now, of course, you can always say, my goodness, how you got to these ideas. And of course, this is simply experience, right? You try whatever you can. And sometimes it works, and sometimes it doesn't. So let me explain. So how to prove theorem 2. So what you're going to do is the following. You're going to replace, first of all, let me rewrite this inequality in the following way. I would like to put here square root of fj. I hope you don't mind. I just replaced the fj by square root. And then this inequality just looks this way. Product integral fj to the power 1 half, right? So here's the idea. We take fj, which is a function of vj, and we replace it by e to the Laplacian t f of vj t Laplacian t fj of vj. And the Laplacian is just the Laplacian spectrometer of the sphere as n minus 1. Well, how do you think about it?   This is, you see, these vector fields, so to speak, are tangent to the sphere. That's the point, right? Because when you take a radial function and you differentiate, then you always pull down a v beta, and here a v alpha, these things cancel out. Or put differently, these are just angular derivatives in the alpha beta plane. f of vj, and I should say an fj here. It's f, it's just f, there's nothing else, OK? Yeah, no, no, this is good, this is good. So by the way, who has seen this kind of notation? Nobody. I mean, this is what physicists do. And it's a very useful way of thinking about, for computations, OK? Really useful. OK, so now, so the Laplacian is this gadget here, right? And now you have a semi-group that's very nice. I mean, there's no problem in defining that. And what you do is you apply this e to the Laplacian t to this fj. And now you notice that this is still a function only of vj. Why? Well, because the Laplacian commutes with rotations. So therefore, when you take any rotation which fixes the vj, you can commute it through the e to the Laplacian t. And here it doesn't do anything. So therefore, this function can only be a function of vj. Good. So that's number one. Number two, why do I choose it? Well, it turns out, when you look at this integral over the sphere, the integral of f of vj t over the sphere is just the integral of fj. Why? Because the e to the Laplacian is a self-adjoint operator. And when you push it on the constant function, it preserves constant functions. So you see, you have already gained something. Namely, you have a flow which preserves the right-hand side. The right-hand side doesn't go anywhere. It's fixed. And what about the left-hand side? Well, let's analyze the problem as t goes to infinity. As this guy goes to infinity e to the Laplacian t fj, what does it do as t goes to infinity? Well, as t goes to infinity, all the higher modes get damned out. You see, the eigenvalues of Laplacian are all negative. The first one is 0, and all the others are negative.  And therefore, what you left over is just the integral of fj of vj. That converges to fj of vj. So now you see, if you look at this, if you replace these fj's by these time-dependent functions, what happens to this in the limit as t goes to infinity? Well, the fj inside, you have to replace by the integral. Then the product of these square roots, this is just a constant. This integral is normalized. You get precisely the right-hand side. OK? All right. So how then can I prove this inequality? All I have to show is that the left side is an increasing function in time. And when I have done this, when I've proved this, I'm done. But to show, hence, we have to show the left side is an increasing, or shall I say, let's put it this way, non-decreasing function in time, a non. All right, so what you have to do is, how do you check this? You just take the derivative and check that it's non-negative, right? So that's a computation. I don't need this lemma anymore. So we have to compute d dt product j vj. Good. So now, OK, there are many ways to compute it. But the most convenient one is, notice, this operator here, e to the power of t, the fj is a non-negative function. In fact, this semi-group is actually monotone, sorry, positivity-improving, right? The function is not going to be 0 anywhere when you start. This is essentially, what is it? Harnack or something, right? It's fairly obvious, OK? Standard, let's put it this way. It's not obvious, but standard. All right, so now, what you do is, write the fj as e to the phi j. And now, you compute. So now, we know dfj dt Laplacian fj, right? This then implies that d phi j dt is Laplacian phi j. But then you get the correction term, which is the gradient of phi j squared. And you have to remember what these things are. This Laplacian is the sum of this l alpha beta squared. By the way, alpha and beta are just any indices, right? With a half in front. And what is this guy here? This one is just l alpha beta phi j squared sum alpha beta with a half in front. That's what it is. OK, so now, you compute that, right? I think I shouldn't do this on the board because I surely make some mistakes. Yeah, that's right. So when you differentiate this, remember, when you plug these guys in, you have here a half in front, right? When you plug these guys in here. And then you have to sum on the fj. So when you compute this derivative, what you get, you get that this is equals to 1 half the sum on j, the integral grad phi j squared minus 1 quarter sum j and m, the integral grad phi j dot grad phi m. And here, I put here a d mu and a d mu.  So I just absorb d e to this exponential here, which is an exponential. I just absorb it into the measure because now, at this moment, I don't do any differentiation anymore. What I have done, however, is that this laplacian, I have made an integration by parts, which gives me the second term. So all you do is you try to get down to first order terms. Good. And now, notice something. When you have an L alpha beta on phi j, remember, this function phi j, what does it depend? It depends only on the variable vj. So the only L alpha beta is where the alpha is either j or the beta is j.  And now, you use this very carefully. And what you notice is it's not a miracle. It's just due to computation that this gadget is a full square. Namely, what this is, it's 1 half. Yeah, no, I get 1 eighth because there's no point in doing this computation because I'm anyway behind writing this way. It's a full square. Oh, you cannot see it. Let me write it up here. 1 eighth, sorry, sum of beta. And that's, of course, clearly positive. So therefore, this function as a function of time is increasing. In the limit, it converges precisely to the right-hand side. And you already understand what are the optimizers. The optimizers are the constant functions. So this way, you can prove such type of inequalities relatively easily. And you see, you think this is a trick. Well, not really. You see, what you have to imagine in this business is always, what could the optimizers be? And the constant functions are a reasonable assumption that they're going to be the optimizers. And then you say, aha, so what kind of flow converges to a constant function? Then, of course, the easiest one to take is, of course, the heat equation. So now next time, I'm going to expand on this. Namely, there is a whole class of inequalities which we're going to use, actually. They are called the brass-combleep inequalities. And they will take only a special case of those. So let me sort of give you a sort of a general view of such kind of inequalities.    And this is a subspace of Rm. And I'm just saying, you see, I don't want to specify that it's Rk or whatever. They can be crooked, right? They can just hang in there. These are subspaces, OK? So these are subspaces of Rm. All right, so now you can ask yourself, well, suppose I take a product of these functions evaluated Bjx. X is in Rm. And I would like to bound this by, say, the product of certain LP norms of the functions in their own variable, H, I, P norm, so LP, OK? Can you prove something like this? So what would you have to do? And then, of course, there's a constant, right? And this constant will depend on the B's and everything. So what you would like to do is you would like to actually optimize this problem. J equals 1. I mean, this is K. It doesn't matter. Fj, I forgot to say, K here, LP. And what you would like to find is what is the supremum of that gadget over all such functions, OK? So that's the problem. Now, you agree. I mean, this is unsolvable, right? And what Braskham and Lieb, actually, it was later Lieb mostly who realized this, said, well, you see, instead of the theorem is, instead of optimizing this overall functions, it suffices to optimize only over Gaussians. In other words, all you have to do is you take the Fj to be Gaussian functions with some, you know, what is a Gaussian function? A Gaussian function is a quadratic form in the exponent, yeah? No shift's nothing. You just plug these Gaussians in. We know how to compute the integrals with Gaussians. And then you optimize. But of course, when you compute this integral with these Gaussians, you get, of course, extremely, exceedingly complicated functions in terms of the Bi and the variances of these Gaussians. These are usually unsolvable problems. But this is this theorem, and that's a fantastic theorem, because we know how to compute with Gaussians, OK? Now, let me give you now a rigorous version of that theorem where you actually can compute the constant. And that's a theorem which comes from convex analysis, believe it or not. This all that stuff has beautiful applications to volumes of convex sets. So let me write down this theorem. Now, this is Raskamp-Lieb. I should also mention Keith Ball. By the way, at the end of the course, I give you a list of references about all that stuff, right? I think that's presumably the most useful thing about the course at the end. And also Frank Bachte. So now what is the theorem? Let me make sure that I don't make too many mistakes. So for i equals 1 up to k, that h i in Rm p subspaces of dimension, let me call the dimensions, p i from Rm to h i, linear maps with the property p i transpose the identity on h i. So that denotes the identity map on h i. So it's not the identity matrix, right? Because the identity matrix would always refer to the basis standard basic of Rm. I just call it the identity on the Hilbert space, OK? Next, assume further there exist numbers, non-positive numbers, you should say, i equals 1 up to k. So that is this important relation, sum c i, i transpose bi. Notice now, i, here is bi transpose bi. This, by the way, says that bi is an isometry. And here we have bi transpose bi, OK? That's reversed. What should the sum be? Let's assume that the sum is now the identity on M. So that's really the identity matrix in Rm. Any collection of non-negative functions, i, from, I should say, h i into the reals, i equals 1 up to k, the integral, let me write it this way, product i equals 1 to k. The integral over Rm, fi of bi x, x, and here erase it to the power c i, is lesser equals product i equals 1 to k. The integral over fi, now over its own Hilbert space, to the power c i. And you notice here now, I have really a constant. The constant is 1. And that's sharp, as best possible. What are the optimizers? Well, I don't have time anymore. But next time, I will show you that the optimizers are Gaussian. That's very easy to see. That follows immediately from this relation. And in fact, I will also try to convince you that this is an inequality which you surely have seen in certain versions, namely, it goes back to an inequality by Loomis and Whitney from 1948, which you by the way use in the proof of the Sobolev inequality, namely, the L1 norm of the grade in this grade equals the n minus 1 norm of f. You use that inequality implicitly. So I show this because it's a very simple example. And then I try to give you a proof, and I would like to apply this now to a new topic in the Katz model, namely to thermostats. It raises some interesting mathematical issues. Good. So then I'll see you tomorrow. Thank you.


With action and suspense out of the Old West comes the most famous hero of them all Huffleong Cassidy starring William Boyd The ring of the Silver Spurs heralds the most amazing man ever to ride the prairies of the early West Huffleong Cassidy the same hoppy you cheer in motion pictures and the same California You've laughed at a million times Raw courage and quick shooting have built a legend around this famous hero Huffleong is a name to be feared respected and admired for this great cowboy rides the trails of adventure and excitement William Boyd as Huffleong Cassidy and and declined as California Here's Hoppy now with another new story This one is titled the man in the yellow mask California and I were on our way to Brownsville to pick up a herd of coasters little Gulf of Mexico cattle locally known as sea lions At the time I was carrying a site draft for $7,000 We meant to get a pickup crew and trail the herd back to the bar 20 for fattening But then something happened that drove Brownsville sea lions and everything else right out of our minds Shots California sounded to me like rifle shots hoppy here. What's your figure made them somebody out hunt? That's right California out hunting men look at that horse at saddle What's happened to the rider figure he gets shot off in his saddle could be and I'm willing to wager He did come on. He must be down somewhere in that draw. Let's go topper here boy. Get up There is hoppy. I see him right whoo-hoo boy. Oh, there's whoa Come on. He was shot. All right Look there hoppy here comes somebody else. He couldn't be the father that done it or I doubt it He's not carrying a rifle. Maybe he'd get rid of it. You could have in that case. I doubt it He'd ride up while we're here. Hi there It is a yeah, you did when we got here. Yeah, I Thought it'd be okay here, mr. If you're thinking we don't know. No, I know you didn't I was up over there when the shots As far as seen the fella did it. Oh by the way, my name is Stokes Jared Stokes on the circle S over yonder. I'm pleased to meet you. This is California Carlson. I'm Bill Cassidy. Oh Hop along Cassidy. That's right. Yeah, I've seen you when you rode through town You know this fellow do you never saw him before? Well, I did I knew him well He's Sam Wilkins. He's a neighbor of mine. Let me see here Yeah, I thought so. It's gone. What was you looking for, mr. See if Sam still had his money on him. He just sold off some of his cattle, but I see it's gone Well, this is one more we can check off again that ombre in the other mask the man in the yellow mask He did this sure so I'm riding away Heard of him. Have you yeah, I've heard of him seems he's been making new fellas quite a bit of trouble up this way trouble and killings About time somebody put him away That's a job you could do Cassidy if all I heard told got any truth in it Oh, why me because you've laid umbra's like this and by the heels of four I Sure wish you'd stay and give it a try. I wouldn't know where to began Couldn't trailing mobile this stuff's mostly mouth. I well now what if I told you where to start looking for that'd be fine But that's just the difficulty from what I've been told no one has the slightest idea who he is or where he hides out Yeah, I Can tell you both you can how come well, it's easy mister. You see He's my son Now back to hop along Cassidy and our story the man in the yellow mask Poppy in California hearing rifle shots investigated and found the body of a local rancher Sam Wilkins They were presently joined by a second local rancher Jared Stokes Who identified Wilkins killer as the notorious outlaw known as the man in the yellow mask and now goes on to identify the man In the yellow mask as his son That's right Cassidy. He's my own boy Terry. I reckon you can guess how rotten I feel having to tell anyone this but It's so all right. He's my boy. You mean the man in the yellow mask. Yeah, I got a look at him once with that mask off What it really wears, you know is a hood covers all his head right down to shoulders and call it yellow We heard that well, I really don't know what to say Stokes if we caught him You know we'd have to turn him over the law, don't you I'd expect that he'd hang, you know hangings what he's got coming to him I raised him right All I can figure is he's Somehow gone kill a local Now own boy or not he ought to hang all right We'll go after him our business can wait, but you'll have to tell us where to look for him. I'll do that See what I have in mind is this ever since he was just a little button He's had his own private hideout didn't even know I knew about it, and it's a good one. It's a cave He found once back Topper boy This is where we get off and walk California leave the horses here Think we're getting close to that cave hoppy well of Stokes directions. We're right It should be at the base of that beaut you see up there head come on, California Now let's see how good you can make out on foot me I'll do all right Keep your head down Hey, don't move ahead so fast the way you were walking ain't walking at all.  Can you stand up? I'm scared to try. Oh, what is it hurt? I feel too doggone numb to tell then what are you groaning for? I ain't got nothing to do with that hoppy it it just come out of me How we got to get out of here and get out as fast as we can That yell you were made was enough to wake up the dead. We're like to have the whole county down on us here I'll give you a hand see if you can stand up. Well, I Go gone of camp that's good. How do you think you can find that rope? I got attached to a rock up above You're pretty well shook up up here. I'll go ahead and try anyhow. I'll come up behind you I'll give you a hand if you need it, but hurry all right hoppy Here goes There you go.  You need some hard work. No, all right. All right. I'm getting it Hoppy what's the matter? I'm almost at the top of her. I can't quite make it in my hands are slipping There you go. Oh There They've done it hoppy Thanks. No way to check it out. I'll I'll give you a hand here grab a hold you got it All right, pull away up with you Thanks see anybody No, no, and I've been looking all around. Oh, maybe he didn't hear you Or maybe isn't that the cave just now in that case will give us a chance to do what? What had nothing foolish careful? I had you both covered the man in the yellow mask I thought you looked all around Well, I did hoppy. He must have just didn't look in the right place All right move ahead of me with your hands high both of you Looks as if you're gonna be my guest for a bit This is the cave you've been using as a hideout a that's right Rumored up in there for you and a dozen more so keep right on going Dog gone you even get it right it in here. I have a lantern over there Okay, this is all right. Stop right where you are And what do you think you're gonna do with us?  What'd you do that for your Terry Stokes you?  He knows that's how we found our way to this cave But there's something else that interests me in your talk Terry What's that you try to sound so tough, but you don't sound convincing I'm beginning to wonder if you are the man in the yellow mask what what a man's a killer it gets into everything He does his voice his manners everything you don't look or sound like a killer to me What do you do with it? Get back? And if I don't what will you do? I'll shoot. I'll shoot. Oh, you won't give me that I Can try to play the part of a killer unless you mean to go through with it Terry There's always someone just fooling up to call your bluff.  You're no killer Terry.  You can trust hop along Cassidy Your Cassidy you really are I usually am now. Let's hear what you got to say Well, I Planned always hoping to get the real man in the yellow mask. Oh He killed Buck Swanson my best friend. I swore I'd get him. Well, how was this get up gonna help you? I figured if I posed as the man in the yellow mask and let myself be seen once in a while The real one would hear about it and get to wondering. I Figured after he'd wondered long enough. He'd come looking for me to see what was up. I Had a trap set for him. I figured if he did come I'd have a good chance of nabbing him that trap you had set up Was it that deadfall California stumbled into that's right? I put a lot of work into that deadfall Cassidy I don't think you'd have ever noticed us. I sure didn't and I get aches and pains to prove it Hey, you weren't interested in looking all you were interested in was showing how fast you could walk on those high-heeled boots of yours Well, it wasn't a bad plan Terry. I'd have been proud of it myself There's the gent in the yellow mask quite right Might have fallen into your trap Terry But unfortunately this fool is strong before I had a chance and very fortunate for me Now I don't get any notions Cassidy. I'd advise you just relax let that gun drop Now what? Well, I'm not just sure yet Cassidy, but to court our young friend here. I'll probably kill you Now back to hop along Cassidy and our story the man in the yellow mask Young Terry Stokes had conceived the clever plan for the capture of the man in the yellow mask There was only one thing wrong with it Terry hadn't realized that his trap might be prematurely sprung by California Carlson and now in the cave that he's been using as a hideout Terry and California and hoppy find themselves the outlaws prisoners As I say I haven't decided just what to do with you yet, but I'll probably kill you That is well I have certain sources of information Cassidy I happen to know for instance that you were heading for Brownsville to buy a herd of cattle and have a site draft for $7,000 in your possession now If you just put your signature to that draft then you'd kill me and have it over with oh no no not at all And that case I might be persuaded to let you go Sorry, but I can't accommodate you my friend that money isn't mine And I don't give away what doesn't belong to me. I could take it from your course and it wouldn't be any good without my signature Watch yourself Terry. I'm gonna give you a chance to get away. What yeah, that'll be enough of that Cassidy What'd you say to this fella? Why I was just asking the time friend.   Oh, I wasn't worrying about that The way I figured you'd like to keep us alive till I sign that site draft Well, you'd better sign it soon have much patience Cassidy then you'd better cultivate it if you're thinking Terry You'll come back to help you. Oh, he will he will all right. Let him you won't be here.  Yeah, I think so Plenty of water from that stream over there good grass plenty of fuel No one ever comes here. I suppose you would have the tires here, but you have to use raw hide I like raw hide when it's green like this. It's nice and flexible. I want to dry this Well when it dries it shrinks and grows good and hard. Yeah, I don't think you and your friend will be leaving in a hurry Cassidy Oh, we like it here. Well, you'd better because you're gonna stay here a while Now let's see that site draft you have Cassidy. I think I'd better take care of it Yeah, it is 7000 When I have your signature on this I'm gonna be a considerably richer man Which will be some considerable time from now This where you always hide out you say mister. Yeah, of course Why'd he ask that? I wondered usually when a man spends all his time at one place. There's more signs alive Doesn't look to me like you hear much. Where do you think I stay? I don't know. That's something I'd like to find out. Yeah, you dream We'll dream all you like I gotta leave you two here for a few hours. Let's see. Are you a tight, California? I reckon you'll hold all right See that you behave yourselves while I'm gone. You don't you may be sorry for it Haven't left us much choice. Have you know no record. I haven't See you later. Yeah, boy.  I'd like to do more than that Hmm Oh gone. I'm getting thirsty already. I reckon it's cause I can't move none Anyway, you figure we could dip our beaks in that cricket under if you should think of a way you let me know Yeah You think we're done for this time happy I do not You're gonna sign that site right for you not at all. Then how are we gonna get away? We'll just wait for Terry to get help and come after but he don't know where we are No, don't worry. He'll find her but how we I left a trail for him. Oh, uh-huh. You don't see any cartridges in my gun belt to You see that's right. I dropped one every once in a while on the way over here I figure Terry's reaching the circle this just about now. He should find us in an hour Whoa boy, whoa Stay there boy Listen dad. Don't ask me any explanations just now just do as I say Casting in California are in trouble. They've been captured by that ombre in the yellow mask I kind of get help we ought to get about three or four the boys and Hey dad, what's the idea of that gun or the man in the yellow mask son? No, dad. No, you don't understand. I just I Swore if I had the chance I'd take you in son or no son and I am I'm taking you into town to hang Happy if Terry's ever gonna get here. Don't you think it's about time he was getting here now? They should have been here long before this I like it Maybe he missed those bullets after all he shouldn't have any luck with that wrong hide he put on you Canary a bit dog on stuff just getting tighter. So was mine. I wait a second This might be Terry now can't see who it is. It's still on the other side of those rocks. You'll see him in a minute. He Yeah, our friend in the yellow mask again So you got back here and we kind of hoped you'd break your neck or Enjoy dying of thirst would you and speaking of thirst how's our chance of getting a drink out of that creek? I've returned to town. I thought I'd better see to you and that site trap first Untie you wanted at time, but I'll leave your hands as they are They got nothing to lose giving you a drink. I guess You'll be first Cassidy make it fresh dog. Go on here. I'm Sidrayim getting thirsty. What are you going in the town for another little job on tap? Hey, it's something I think will interest you. Oh There you are Now you can walk to the creek just a minute. What's this you say will interest me? Oh that? Well, if you're waiting for Terry to free a year in for a disappointment He's been delivered to the sheriff for trial. What do you know? I would interest you It occurred to me it might when I saw those cartridges you dropped on the trail You know about that. Do you of course? Dispose of them.  Hey stop Cassidy. What do you think you're doing stop? Hold on I just need to get away from your foot your hand side well I could try couldn't I I see you're not to be trusted. That's right. I'm deceitful as all get out All back to hop along Cassidy Well, hello, Jared Terry higher this ain't a social call sheriff I Brought you the man in the yellow mask the man in the yellow mask or where is he right here? My own son your son. Well, you can't right? I suspected it for a long time But when I was sure I figured it was my duty to bring him in yeah, but somehow I can't see Terry is the man in the yellow mask Well, what do you got to say for yourself son? I wore a yellow mask sure But all I did it for was to trap the real outlaw and by holding me like this you're probably condemning Cassidy to death Well, I wouldn't hurt to check on your story.  He must have friends While you're away they break him from jail Well, if you say so mr.  I've been inspecting some property over north Naturally had to put a little muddy, but as I was saying, that's right, mister as you were saying Come on California. You were saying what about the little muddy? I was simply explaining how I got my trousers wet I've forwarded the little muddy myself that mud on your boots doesn't look to me like anything you got there Well, what do you look to me more like the mud you might have got from the bottom of a certain little creek I know about back up in the hills Here's your man. Sheriff. You're looking for the man on the yellow mask here. He is arresting There's any doubt about who you are, mr. I'd suggest we search for a certain site draft. I Wait a minute Searching Terry find that draft. Holy Oh still friend you aren't going any place here. It is hoppy. It was in his side pocket What is all this your man and the yellow mask sheriff and proof that I know what I'm talking about The man in the yellow mask took that draft from me and now Bailey or whatever he calls himself has it How'd you ever get free? Oh, that was easy, mister What did you think I took that run through the creek for boy as you said braw had shrinks when it dries But it stretches when you wet it so I wet it and here I am why you all right Bailey get going I've been saving herself with the man in the yellow mask This time I figure I got it. Oh, you can't do this to me sheriff. Well, what do you think about your son now stokes? They'll think he's an outlaw Well, I I feel like an agent Son if you can ever see your way to kind of Excuse me while forget it dad. It wasn't your fault I wanted folks to think I was the man in the yellow mask can't complain when they start to believe me Well, you had a good idea Terry, but next time don't try it. Huh playing outlaws again That's played for keeps and next time you might not be so lucky Mighty lucky for Terry and mighty clever of hoppy to think of taking that run through the creek and outwitting the man in the yellow mask Trickery being something hoppy just can't tolerate leads him in California into a thrill packed story When hoppy gets the trickster on the run in next week's tale of action run sheep run Hop along Cassidy starring William Boyd is transcribed and produced in the West by Walter white junior The man in the yellow mask was written by Gibson Scott Fox with original music under the personal direction of Albert Glasser All stories are based upon the characters created by Clarence Mulford. This is a Commodore production


You need ten about marriage. I make it an assumption to succeed in marriage. If you know two, will your marriage succeed? Tell me, will your marriage succeed? The issue about ignorance is not knowing anything at all. We all know some things. But what we know is so low, is so minimal that it cannot take as far


I mean bro have you never played Bunch before in your life ever never have played Bunch in your life Yo, this that work on bazooka Larry work on Earl. I swear to God. I Don't know how I feel about our boys staffer boys. Who are you guys using a quarterback now? I mean, I don't know. I mean you guys know I love Josh Allen. He hasn't missed a girl yet though So oh, this is Earl. Oh my god Earl. Oh I'm nervous. I'm nervous It's a game you can never lose Alright staffs. He's just too slow to get out there Yo, I can't lie Calvin got to put his hands on the football. It's sad to say, but Bro, thank you. I don't like staffer again. I didn't think he would like really let that look Yo, that's the one thing when I run that play I never think somebody is not one cover Y'all saw how I swerved out and came back. Y'all saw y'all saw Oh Come on. I got a possession catcher man. I'm really getting bagged staffer this bro. He's a He's just wait. He's way too slow to like get out and make that bro like It's going to get the quarterback man. Look at this. What is this? It's so bad. How is nobody here? Is this the best in the game? I'll be the first to tell y'all 3 3 5 is against the run now What was that dude? And he can like what the? Bro, give me off this game, bro. Yo, 35 is terrible against the run, dude. It's terrible All right, let me play real offense, man. Just trying to bomb his I should have a bunch of bombs and Calvin is the softest in the world, dude Like how does how does that's what I mean? How does cam not get him right there? Like he's spied and he just like outruns him It's really I really just need to get a more mobile quarterback.  No, Josh Allen is it bro I mean Stafford hasn't missed any throws though. Let's just be honest Just keep it real. What's that maddie staff was Oh my f**k, it's just no f**king way dude. Come on dubby He was doing somewhere near me, bro I mean he was doing somewhere near me.  It's no way he catches that dog I mean Reeves just stood out about man. God like bro. I just couldn't work I like bumped into my mid read right there. Oh, he had all f**king day for that play, dude Ah, just put these on 30 dubby man Two for 12. How is this game close? How are we dropping that dude? Come on, man This quarterback off the field, bro. Good catch at read. I love you, buddy. I love you I was just no way, dude. Can we catch a pick please? Thank you, bro. Oh my goodness Calvin It's my new play. It's my new play. It's my new play It's not the play. It's not the play now. It's gonna lay this game was close, bro Three for 16. He's three for 16 Dub need me. Dub need flex. Dub need flex. Dub need time. Dub need time. Try this play again Played well, bro. GG's buddy. GG's. He needs to get rid of quarterback, bro. Oh my god I really didn't think he would like expect me to run covered to two plays in a row Idiot. I'll be forgetting. I'm playing idiots, bro I'll be trying to be like one step ahead of one step ahead But these the guys that just go look oh, we ran out. Let me run my play for this The needed gaming post True font. Come on. Dumb it. What are you doing, bro?  Give it up Give it up, bro. Give it up. He's gonna be like current gen game for a thousand current gen current gen I mean, bro, have you never played bunch before in your life ever never have played bunch in your life Yo, the s*** that work on bazooka larry work on Earl. I swear to god


Good evening friends. As we had told on the earlier occasions, this has been a passion towards sharing knowledge can be seen on the social media like anything. And we are all eagerly waiting for the book which he's writing and we are quite set to send that that book will be not only hit, it will be a mega hit. Or we can say it will be a blockbuster for the legal knowledge as such and even the common persons can understand that. That's the way he illustrates things. You will ask at the end of the session what type of things can people look upon in that book. Today's session is also quite fascinating for the purposes of the lawyers and common man. That when does the trial start in a summons case? For today's topic, we will not discuss the warrants case. Some brief difference between the summons case and warrant case will ask sir to explain. So it's not one of the parts of the slave bus if we ask typically because four questions have been given to me, which have to be read out. So we will take one indulgence, the extra fifth question difference between a summons case and a warrant case. And as we all know, Justice Biram Kumar is not only an erudite speaker, but also person with immense knowledge and the way he unblocks the blocks, that is the most fascinating part. And Justice Biram Kumar is a former judge of Kerala High Court and his passion to take legal knowledge started when he was in the director judicial academy. That was to the next level. And thereafter, because of the COVID, we all got connected. I do not know that what positives others have learned, but at least we have been able to do persons and resources like this is Biram Kumar. His knowledge has gone viral in the social media. And we can also understand that Justice Ramkumar's today health is slightly down, but they say that he can be down, but he cannot be out. He will continue to share his knowledge and reminded of Mr. Anil Kumbley when he was injured, he came back and played an innings, which is remembered for all times to come. What do you say? Thank you. Thank you, Mr. Vikas. Good evening friends. You can read the first question. What do you mean by a summons case, sir? Summons case, very often we come across the expression summons case, warrant case, etc. What exactly is meant by a summons case? Now, going by section 2 clause WCRPC, a summons case means a case relating to an offense not being a warrant case. You are not taken anywhere. Summons case is not a warrant case. That is what the CRPC says section 2W. So, you necessarily, you have to go and find out what exactly is a warrant case. One has to find out what exactly is a warrant case from section 2 clause X of section 2 CRPC, which means a warrant case is a case relating to an offense punishable with death, imprisonment for life or imprisonment for a term exceeding 2 years. So, if the offense is punishable with death, imprisonment for life or imprisonment for a term, a case. So, summons case is a case which is not a warrant case. So, summons case can ordinarily be a case where imprisonment is the extent of imprisonment can be up to 2 years and below. In imprisonment can be awarded up to 2 years and below. Above 2 years means warrant case. This is the conjoined effect of reading section 2W index of section of CRPC. Yes. Yes. Sir, the next question which normally will come in the mind of anyone is, when does the trial start in a summons case? This is a very, very interesting question. Courts have generally taken the view that trial in a summons case starts as soon as the accused in response to the process issued against him either appears or is produced before the magistrate and the substance of accusation is stated to him by the magistrate. Under section 251 CRPC, the substance of accusation as soon as the accused appears or is brought before the magistrate, magistrate complies with section 207 or 208 as the case may be. The prosecution records will have to be given to him, copies of the prosecution records. So, as soon as the prosecution records copies are given to him, the magistrate will read the substance of accusation. That is the particulars of the offense will have to be read over to the accused and explained to him in the vernacular language or in the language known to the accused. This is what is called that the courts have taken to view that as soon as the accused appears and the substance of accusation is stated to him by the magistrate, trial has started. This is the general view taken. In fact, way back in 1980s Kerala, this is the Kerala decision. Shantamma Rathamani Amma versus Kunjupillai, 1980 KLT 393, a decision rendered by Justice SK Kathar, who was a very well-versed judge in criminal law. The corresponding citation is 1981 criminal law journal 247, 1981 criminal law journal 247. It was taken, it was held that as soon as the accused appears and the substance of accusation is read over to him, trial has started. The reading of substance of accusation or particulars of offense is done by the magistrate in exercise of the power under section 251 CRPC. Again in Para 16 of Subramaniam Sethuraman versus Tito Maharashtra. Here 2004 Supreme Court 4711, three judges. Sandosh Agade, the author of the judgment, same view was taken. Again in SV Enterprises versus Rajasekar and Nair. 2006, three KLT 930, a judgment rendered by Justice M. Shashidar and Nambia. Same view has been taken in that ruling also. But I would hasten to add that the trial in the real sense of the term cannot start merely on the stating of the substance of accusation to the accused under section 251 CRPC. This is because the accused may voluntarily plead guilty, in which case there is no trial. He may voluntarily plead guilty under section 252 CRPC. So no trial in that case. So even when the accused pleads guilty, voluntarily pleads guilty, section 252 CRPC does not mandate that the accused should be convicted on his plea of guilt. The magistrate can still in his discretion call upon the, instead of convicting the accused, call upon the prosecution to prove its case. Therefore, according to me, trial in a summons case can really start when the accused either pleads not guilty or when even after he pleaded guilty, the magistrate in his discretion has not convicted the accused. Thereupon the magistrate can proceed and take all evidence as may be produced by the prosecution under section 254 CRPC. So I would respectfully add, of course, nothing wrong in saying that trial starts when the particulars of the offence are read over to the accused. Or trial starts when the substance of accusation is read over to the accused. But then, supposing he pleads guilty, voluntarily pleads guilty, there is no trial. So the trial can really start when after reading this particulars of the offence, he pleads not guilty. Or even after he pleads guilty, the magistrate in his discretion does not convict him but calls upon the prosecution to prove his case. In both these situations, then only the trial can start in a summons case. Yes, you may read question number 3. Yes, sir. Can the magistrate after the commencement of the trial, instead of completing the trial by either convicting the accused or acquitting him, brought the proceedings against him midway? Yes. This was a controversial area. In fact, the decision rendered by the Kerala High Court was overruled by a two-judge Benjavati Supreme Court in KM Matthew. Of course, I will refer to that. Now, that decision said that in the absence of any prohibition in the CRPC, nothing wrong if the magistrate after the commencement of trial drops the proceedings against the accused. Now, this KM Matthew citation, I will give you the KM Matthew versus the State of Kerala, a year 1992 Supreme Court double 206, a year 1992 Supreme Court double 206. The judges were Jagannatha Shetty, who was the author of the Shetty Commission, whereby judicial officers in the whole country were given a lot of benefits, a lot of purchases. Just as Jagannatha Shetty and Yogeshwara Dayal, Jagannatha Shetty being the author of the judgment. Now, here, in fact, it was after reversing the judgment of the single judge of the Kerala High Court, Justice S. Patmanabhan, that the Supreme Court held that, in the absence of any prohibition in the CRPC, nothing wrong in the magistrate dropping the proceedings after the appearance of the accused. That was the view taken. In fact, after this view, what happened was the magistrates all over the country, throughout the length and breadth of the country, used to drop the proceedings or discharge the accused for one reason or the other, following KM Matthew. And it took nearly 12 years for the Supreme Court to realize its mistake. KM Matthew was overruled by a three-judge bench in Adal Prasad versus Luplal Jindal. Citation is A.R. 2004 Supreme Court, 4674. A.R. 2007 Supreme Court, 4674, three judges, the author of the judgment being Justice Sandosh Hegde. The other judges are Justice S.B. Sinha and A.K. Mathur. In fact, this was reiterated by another three-judge bench, Subramaniam Sethiraman versus Taito Maharastra. A.R. 2004 Supreme Court, 4711. A.R. 2004 Supreme Court, 4711. Again, judgment rendered by the same judge, Justice Sandosh Hegde. Their lawsuit said that, you know, KM Matthew cannot be upheld because the, once the substance of accusation have been read over to the accused, once the process has been issued to the accused, and the accused has appeared, and the substance of accusation have been read over to the accused, there's no provision in the CRPC to recall the process issued, and to allow the accused to come to the pre-process stage and plead for a termination of the proceeding, premature termination of the proceeding. No, it will amount to review. If such a, if such a view is taken, it will amount to review of the earlier, of the order issuing process, of the order of the order reading the substance of accusation, etc. And no, there's no power of review in the CRPC for a criminal court, unlike in the case of a civil court. Civil court, there is a power of review, whereas CRPC, there is no power of review in the, for the magistrate. There is a view taken in Azalat Prasad case. So after 12 years of K. Matthew, it was overruled by the 3D bench, virtually resurrecting the view of justice as Patminavan of the Kerala Icon. May I read question number four? Immediately after his appearance in a summons trial case, the accused filed an application requesting the magistrate to drop the proceeding since the ingredients of the alleged offence were absent in the substance of the accusation proposed to be stated. Even though the magistrate was convinced that the ingredients of the offence were absent, he dismissed the application filed by the accused by relying on the dictum, and Azalat Prasad v. Rupal Jental, AIA 2004, Supreme Court 4674. Will the accused be able to successfully challenge the order of the magistrate? See, summons trial case, warrant was issued to the accused. The accused entered appearance. Then the copies of prosecution records were furnished to him. He is appearing through a council. There is a council appearing for him. The council raised a condention that please terminate the proceedings against my client because the materials produced furnished to me do not make out any offence at all. The ingredients of the offence are absent, totally absent. In such a case, why should the accused suffer the trauma of a trial? Why should he suffer the ordeal of a trial? So please drop the proceedings. The magistrate said nothing doing. I am sorry, Azalat Prasad v. Rupal Jental prohibits me from dropping the proceedings. That is the view of the magistrate. The question is whether the magistrate was right or wrong. In fact, the magistrate was wrong because what was laid down in Azalat Prasad and what was laid down in Azalat Prasad was that after the commencement of trial, when does the trial come in? With the rating of the substance of accusation. Now after the commencement of trial, the case should go on and the trial should go on and it can end only in a conviction or an acquittal. Now the question here is whether trial has started. After the appearance of the accused, the council appearing for the accused was sharp enough to notice that the materials sought to be read over to him do not contain the ingredients of the offence. No offence is made out. That is why he made a submission to the magistrate. Please don't read the substance of accusation to the accused. Please drop the proceedings so that the accused need not undergo the ordeal of trial. This was not exceeded by the magistrate who thought that he was bound by Azalat Prasad. Now this is a via media, in fact a very tightrope walking done by the Kerala High Court in two decisions of the Kerala High Court. One by the Sambal Self. In fact you may, there is one decision of the Supreme Court that is Sebi versus Gaurav Vashnev. Gaurav Vashnev 2016, Volume 14, SEC 430 where Justice Kehar has, speaking for the bench, has observed that material facts constituting the offence for which an accused is being charged must mandatorily be put to the accused. Material facts constituting the offence would mandatorily be put to the accused. And a lack of material facts which are vital to establish the ingredients of the offence cannot be viewed as a procedural omission. The above requirement is not a procedural omission but it is a substantive requirement. Therefore if the material facts are not there in the substance of accusation to be read over to the accused that is not a proper compliance of Section 251, CRPC. So it was noticing that the council raised that please drop the proceedings. The ministry was unwilling. But this is a case where the trial is yet to start. This is a case where the substantive accusation is yet to be read over to the accused. Therefore the bar under Adalat Prasad and Subramaniam Sethuraman may not apply here because the trial has not started. The bar will apply only if the trial has started by the magistrate reading out these particulars of the offence to the accused. Before reading out the particulars of the offence, the council very vigilantly raised a point. See the materials do not make out an offence. So please don't start the trial. The magistrate found that he was of the view that he was bound by Adalat Prasad. The magistrate was wrong. In fact, the Kerala High Court in there are two decisions. One is by Anand Tawail versus Fuddin Specter. 2010 Vol. 3 KLT 449. 2010 Vol. 3 KLT 49. That's a decision rendered by the humble selves where the high court held that. Then the trial is yet to commence. The accused will be within his right to plead for a discharge or a rally. Technically it is not a discharge in a summons game. The accused can say that the proceedings against him may be dropped because there is nothing to be read over to him. The ingredients of the offence are not made out. So nothing to be read over to him. So the only thing is after the appearance of the accused and before the commencement of trial, this has to be done. So the council has to be very vigilant. The other day I referred to the Latin maxim. Vigilante bus at Normandy bus, Juras of Veen. The law helps only those who are vigilant and does not help those who report their rights. Here is a case where the council was very vigilant. And therefore before the commencement of trial, he raised this point. And the magistrate, if the magistrate had cared to prove the records he would have been convinced that the, in fact, my state is convinced there. My state was really convinced that there is no, nothing made out against the accused. But he's wrongly found himself bound by Adhaal Prasad and rejected the application. So the accused have to go to the high court or the session judge to meddle in his grievances. In a, in a, in another case, police charge sheeted case, police charge sheeted someone's case. Also yet another Leonard judge justice Basant of the Kerala High Court in Dr. Kamala Rajaram versus D.Y.S.B. 2006 criminal law journal 1447 corresponding to 2005-03 KLT 617. It was held that the magistrate can either drop the proceedings in exercise of his power under section 251. But since it is a police charge in someone's case, he can also terminate the proceedings under section 258. If the principal witness has been examined, it will amount to a acquittal. If the principal witness has not been examined, it will be a discharge, technically a discharge. Therefore, 258 can be pressed into service by the magistrate. These are the two rulings according to me, to my knowledge, which have taken a wire media. Before the commencement of trial, if the accused is able to make out a case that there's no offense made out in the, in the particular offense, the case can be terminated without the commencement of trial. The accused need not undergo the or, for my conclusion is that in all cases where there is want of sufficient allegation or incriminating material, making out the offense in a someone's case, instituter either on a private complaint or on a police report, if the counsel appearing for the accused is vigilant enough or sharp enough to notice the lack of material, he can plead for a, for dropping the proceedings so that the accused need not undergo the trauma of a trial. With that, we conclude today's webinar. Thank you. You are not audible. You are not audible. I'm not transmuting myself. So I am also controlled by the computer only. Yes. What is the fundamental difference between a someone's and a summary trial in terms of taking evidence? Summary trial, it is a substance of evidence, the substance of evidence need to be recorded. It is a very simple procedure in summary trial. In a particular case, the accused can even plead guilty through post. He need not even engage a counsel. So we are, procedure is someone's procedure only. 250 instead of 250, this is 260. The procedure is same, but the long drawn out procedure of trial, etc, he need not be complied with in his summary. Proceeding the summary in nature. We will ask, this is Ramakrishnan to share his insights. Is it not necessary for a magistrate before reading out the particulars of funds to make out his mind as to whether any ingredients are sufficient for the purpose of reading out itself? How many magistrates do that? It is done as an empty formality. Not because it is an issue. Once it is read over automatically, the accused has no option to say whether the charges are not. He can only plead or not plead guilty. Yes. Is it not better for the magistrate to find out as to... Apply his mind. Yes. It is always, it should be. The magistrate should always apply, but the docket before them, the docket explosion, its pendency of cases before them is so prohibitive that they will find time to apply their mind even. No, that was the case. I think it was one of the decisions of the Kerala High Court where the Kerala High Court itself, even if you plead guilty, appeal can be filed against it because normally when the case is there, it is not there. He cannot file an appeal. He cannot challenge the conviction. He can only challenge the sentence. Under section... I think it is section 375. 375 also. 375. Let me check. Is it not section 375? No, no appeal can be filed except number 272. 272 is the appeal provision. I think there comes... 375. Interpreting that section, 376. No appeal in petty cases. 375, no appeal instead of... 375. No appeal if the accused pleases guilty except on the question of sentence. Their Justice Subramanian Poti of the Kerala High Court had taken the view that even if in a case where he has pleaded guilty, if there was nothing to plead guilty, he can even question the conviction. Because there... it was a ritualistic formality done, undergone by the magistrate as well as by the accused. Without knowing what exactly is the material, he simply pleaded guilty. But actually there was no incriminating material against him. So the Kerala High Court interpreting section 375 CRPC held that. Notwithstanding his plea of guilt, he can challenge the conviction also because there was nothing to be convicted. There was no material to plead guilty. Because the ingredients of the offense have not been made out of it. Even he is admitting the facts. The commission of offense. It may not amount to commission of offense. I think it is 1978 KLT by Justice Subramanian Poti. Very, very shrewd and intelligent judge. No, really it's an area where always there will be doubt. Whether the charge will out of the frame, etc., etc. It's nice to hear Justice Ramakumar always. Thank you. I thought that let us have a flurry of knowledge from SKP. Now this was a very inodate class. I cannot add anything. No. But just for the purpose of learning myself, as of course Justice Ramakumar has put in, the world has not been defined in the CRPC. Maybe rather purposefully. But half late in the ruling by Justice Sungresh, Satyendra Punga Anthony was a CBI. He was a businessman, which is supporting AIR 2022 at page double three eight six. Justice Sungresh goes on to hold that the world trial, even though it's not explained or defined under the CRPC by way of a judicial interpretation and extended meaning has to be given to the world trial even for the purpose of enlarging that accused on bail at the stage of investigation. Now I don't know whether now whatever is said by the Supreme Court, of course it's in orderly. And of course as Justice Ramakumar has put in, the trial of someone's case it is dealt with in chapter 20 starting from sections 251 and ending at 259. And as we all know that under the summons case, it is not necessary to claim a charge in writing. Because taping the substance of the offense to the accused is enough under 251. Now this is a grey area which has created confusion. Now Justice Ramakumar has both cited K.M. Matthew the two judgments which was overruled in Adarat Prasad, the three judgments in 2004. And within a month thereafter another three judgments now the first three judgments was Justice Santoshakya, Justice Esmi Sinha and Justice A.K. Mathur. Whereas in the next three judgments the solitary substitution is that of Justice A.K. Madhuri, Tarun Chakram, Justice Tarun Now Justice Santoshakya and Justice Esmi Sinha who are there in the last three judgments they were the parties to this three judgments. That is, I am talking about Subramanian Sivaram. Now Ramakumar sir, I just have one doubt because K.M. Matthew was a case, if we read closely K.M. Matthew was a case after issuance of summons on a company after entering the appearance accused he pleaded not guilty. Now before evidence was recorded he request the ministry to draw the proceedings the ministry the ministry brought the proceedings the revision went on to the Kerala High Court that was allowed against with the accused practitioners of Supreme Court and the Supreme Court held that if the ministry is satisfied then of course the ministry can draw the proceedings this was K.M. Matthew Now 12 years thereafter comes Adalat Prasad Now if we closely read Adalat Prasad the complaint was filed against accused optimizing taken someone's issue accused then moved the Delhi High Court in the case against all of issuance of process which was allowed then in the revision it was remanded for fresh consideration Now of course one way to the magistrate we called the summons which again was telling to go to the High Court on the ground that the magistrate had absolutely yet no jurisdiction to recall the summons Supreme Court of course K.M. Matthew is not a good guy and K.V.C. what the Supreme Court says Supreme Court says that the remanding in such a case lies only in 482 CRP Now K.V.C. the Supreme Court proceeded on the basis that Adalat Prasad was a summons case but maybe Adalat Prasad was not a summons case it was a viral case which was covered under Chapter 90 and please see the Supreme Court in the judgment has made it clear that it was not necessary to go into the post whether a summoning order amounts to an interim order or not then of course comes the third judgment Supreme Court said that the summons in the complaint that was challenged by the accused company the company was an accused company challenges the Supreme Court that the Supreme Court had taken cognisance of the summons because of it defective statutory notice which was issued and since it was in discharge application was rejected second application for discharge was filed and please see that was allowed for all K.M.A taken in revision before the session 2 allowed K.V.C. for the Supreme Court after 10 days first application filed by the executive director of the accused company on the very same grounds that was taken in the first application which was allowed by the magistrate to read in the statutory notice which was issued prior to filing of the complaint that was not in accordance with law again why even K.M.A things were done that was K.M.A penetrating its earlier view and the Supreme Court was approached and the Supreme Court penetrates adalaprasad empathically holding that the sole remedy is to invoke section 480 to CRS now the Supreme Court said so for the reasoning that it was a challenge to an order in an interoperatory stage kindly see the Supreme Court goes into a pointy that he shows a challenge made in an interoperatory stage in my limited knowledge I would see that this was a wrong proposition now even prior to analaprasad it should be known in Supreme Court the Supreme Court in a brief time of precedence considered the maintenance of revision besides just one case or one way I will just say K.K. Patel v. Sita Gujarat which is A.R. 2000 Supreme Court W346 which held that in deciding whether an order which is challenged is interoperatory or not when it was created in 97 the sole test it is not whether the order was passed would be Indian states the three semesters for the Supreme Court came was whether according to the objections which would have been linked to a party and after little time of time whether that would result in culmination now the principle was made for analap in 1977 not even in my 1978 how this principle is so powerful we see Shukla we see Shukla exactly, 80 Supreme Court that's it then how great we can see that is the 2009 decision 2009 to SEC 370 which held that issuance of summons is not an interoperatory order within the meaning of this so revision is perfectly mediable not really forwarding to ascending the Adalya Prasad as I already said the supreme court in Adalya Prasad goes on to say it is not necessary for this court to go into the question whether the order which issuance that announced an interoperatory order Adalya Prasad did not even decide that issue I am sorry another decision comes in during 2002 which is Nishankumar a year 2020 supreme court 1747 2 judgements decision by Justice Sarasivam and the second judge of the reclamation which went as an absolutely divergent view as it is both these 3 judges which are saying that it is inherent in section 251 CRP so inherent to what extent that when an accused tried to see the difference closer to his summons then the violent duty of the what the violent duty as Justice Jankumar has put in in Anubhav Vail as well as in the Kamna Rajan 2005 magistrate he has to carefully go through the allegations they rival in the police support on the company and then consider to report him to come to a conclusion whether or not commission of offense is disclosed and if only the answer is in affirmation he should explain the substance of the accusation whether you are pleading guilty or otherwise anyways he is bound to permeate the proceeding but kindly see Bhushan Kumar use of the word this exception for Bhushan Kumar since it is absolutely correct because if there is nothing against the accused but Bhushan Kumar says if it is an affirmation then I will say state the substance of the accusation otherwise Bhushan Kumar gives you the option to discharge the accused in section 2, 29, 0 pieces there is both of them and the power of their side and that can be understood on the police work may be the principle is what may be one of the learned judges we don't know and so one more thing I would say not what capable said was and what Adarya Prasad and Subramanian said was disagreeing was may be this would amount to a review under 362 Bhushan Kumar we see that the power to incur because suppose a court has done here it is the accusations actually the accusations were wrong or inherently improper but nothing could be spread out it was a right hander the court then applying the maxim act as theory then normally wouldn't the court have a power to recall the judgment because I had made by hands upon a decision of the Supreme Court in 1995 1999 Indian man was a Supreme Court where on the Supreme Court after recording exhaustive news to the Indian judgment like large numbers of states so people were about to say whether it's a civil court or a criminal court most courts do have the inherent power to recall the judgment that was a case of fraud still sir applying this particular principle act as theory in Eminem drama it was done here wrong it was not that particular court got the particular power it was to the extent of doing the undoing the fraud there was no sufficient inherent power should be given as in the case of the as in the case of the civil court section 151 CPC why reserve it to the I court alone why can't the why can't the magistrate rectify take rectify a wrong step taken rectify a mistake committed by him like as in the case of the civil court inherent power should not be confined to the I court alone and many of the 482 petitions can be avoided because justice can be done at the level of the magistrate himself but unfortunately that is not the and the observation that 482 is the only remedy is also not correct 397 is also not correct 397 nice 397 would lie right with 401 will lie but see unfortunately three of us think of Supreme Court once but twice diversion this is what is happening and we have I think a number one in our constitution after confusions that was an absolute thank you thank you we are very dedicated to justice for sharing the knowledge thank you everyone stay safe and also be tuned with us on our social media for the further updates of webinars like speakers of immense knowledge like this is Ramakrishna thank you once again


Is there been a thought about how much you love these young weapons because that became like a narrative almost. Aaron would like for the young weapons to prove that they could be somebody he can grow with and that will affect his decision going forward. Is that real? My decision will come down to mentally where I'm at, physically how I'm feeling and then encompassing the enormity of another grind and then committing to that and moving forward or stepping away. Listen, I mean obviously you look into the team and who's going to be there but you know I feel like that you know as long as I've been there we've been able to do things with a number of different players and make things work and there's excitement around any time young players start to step into their own and find their confidence but for me it's just going to be my own personal decision and where I'm at mentally and physically.


I'm the developer of the next cloud news app for Android, and I'm excited to be here today and to talk about a new feature That we've been working on which is called the next cloud single sign-on for Android Some of you might be wondering what is single sign-on? Why do we need it for Android? So a couple of weeks ago we received on our issue track on to that the comment there's someone said that every time he tries to Convince new people to use next cloud and all the apps that go with it There's a point when they try to install and set up all the applications on their phone And then they have to you know, like lock in in every single app set up username password type in the URL Then there's like always a different lock-in dialogue So it's really a hassle to get everything up and running and that's when they just be like on all just gonna use Google or whatever Which is like easier to set up So yeah user experience is Not there right now Yeah, users need to look into every single app Every app has its own lock-in dialogue with own custom design custom buttons. Everything is like differently on the layout Every app has the username password and URL So there might be some security concerns that app might misuse them and send them away, for example Then there's also the new login web login app tokens Which most third-party apps don't support yet And the arrow handling for example when VPN Connections out breaking or self-sign certificate support is not working They're poor error message handling. Just mostly just has like another pointer exception or some other random exceptions Which user don't understand usually For developers It's always a hassle to start developing for Android because there's a lot of custom Yeah, connection handling because everyone has its own server setup. You need to handle as a self-sign certificate Properly you need to support HTTP to now Proxies you need to store credentials securely on the device Then you also need to implement the new to factor authentication or the app tokens And you need to store the login data Yeah, and many more things to consider and this is something that every developer has to do over and over again So we were thinking what can we do to make this easier for developers to start working? So we created a centralized lock-in method Which makes it extremely fast lock into all the different apps that you install on your device And for example when you change your password in your next cloud, you have a single place You only go to the next but fast app and change your password there, and then all the other apps will automatically use the new password and As a good thing for developers. We don't need to worry about storing the credentials anymore Because we're gonna leave that task to to be us in the past We also have some integration for retrofit and higher extra. Um, yeah, so let me show you a quick demo how that looks So when you start the new set for example, which will be coming out in the next couple of days, hopefully Then you can click on the new single sign-on button and Then it will ask you which account you want to use from your files app That you've all the accounts that you've set up in your files app Then you can just select one click on okay, then you click on sign in and then we'll just sign it So that's as simple as that as Setting up a new account in another app So summarize The new features we have a unified user experience across all the different apps now because we have one the lock-in screens Provided by our library then We have improved security also and for developers. You don't need to Implement the whole network stack again worry about self-sacrificates and all those different edge cases that can occur and yeah Contact is yeah to be us and me And there's the link to the top page where there's a lot of documentation and how it all works and how to set it up Yeah, huge. Thanks to everyone who contributed their ideas and to everyone who was involved and we're still looking for developers to try out the New library and to give us some feedback