And so the title of the talk is Millen of Thurston Determinant and the Royal Transfer Operator. And in some sense, it's sort of a kind of ending parenthesis on a long piece of historical work, starting with Millen of Thurston, and passing by a lot of intermediate persons, in particular the Royal and Baladine Royal and other people. And the setup is, in some sense, one of the most trivial, dynamical systems you can imagine. So I'll put some small symbols like this saying that this is the first part, and then it'll be a second. Then I'll come back a little bit to some of this. So the setup is what you call 1D, piecewise monotonic map. Should put continuous patrols into the continuous monotonic map or monotonic map. And then when you have a piecewise continuous monotonic map, so what can you then do about it? Well, you can study how the dynamics act upon your space. And let me give you just a drawing of some example. So you have an interval, A, B. And then you have some cutting points or critical points. You have a dynamic, which is a map of defined partially, because I'm likely not defined at the endpoints. And in particular, so the map might not be continuous like this. So you have some critical points inside. And then you will also be interested in denoting these something. So here we have two internal critical points. And out of this, we have three intervals, E plus 1 intervals. And on each of these intervals, so let me call them something like I0, I1, I2. On each of these intervals, you have a map, F, IK. IK is an open interval, and then you have a map, which is strictly monotonic and continuous. So the map is partially defined because it's not defined at these critical points. I don't define it at A and B either. But apart from that, you can start looking at what is the dynamics going to do for the interval. So we have a first Z1. We have just the three intervals, the collection of three open intervals, which form a partition for the domain of the function, say initial partition. And then you start looking at, if you iterate the map, then you have to refine your partitions in order to be able to define them. So you look at the refined partitions. So it would be, you might write like Z1.  So this will correspond to a refined partition. And then one of the, say, topological properties that you can associate with this dynamics is what happens with the number of partition elements as N goes to infinity. And it's not difficult to see that, for instance, it's very difficult. It's very easy to see that if you take the number in the N plus N's refinement, then it's not bigger than the number ZM times the number in ZN. So it means that if you take log, it's sub additive. So the topological entropy, which we may define as limit, 1 over N log number ZN, exists. So this is one of the first thing that Dynamics would consider studying in this context. And now the computing effort in calculating ZM is quite heavy. So you're computing effort for calculating ZN. Well, if you need to calculate ZN, what you do is that you need to keep track of all inverse images backward iterates of the critical points. So you see that when you look at the F squared, for instance, when it's defined, then you need to look at what happens when do you land on the critical point. So this needs to take care of all images of critical points. And they tend to grow in a number like ZN. So actually the computing effort is of the order of exponential of the number, which is N exponential of NH top. So this is just in a naive picture that you would like. If you want to calculate it, you would do like this. And it works fine in some cases. And in other cases, it works a little bit less. When you have a so-called finite partition, macro partition, then it's much easier in some sense. But I'll not talk about this. In the general case here, and that's sort of the part too, it is that Milner-Surston came up with some method for computing this topological entropy. And what it did was constructed a determinant. So I'll talk a little bit more about how you construct it actually a little bit later on. But for the moment being, let me just say it's a formal power series in some variable called T, say. And the formal power series dMt of T, mt for Milner-Surston, is expressed in terms of powers of T of course, but then of forward orbits of critical points. So this uses forward orbits of critical points. Now you would say to me that actually the forward orbit of critical points is not defined, because f was not defined at c1. But you have limits to the left and to the right. And what they do is that you look at what the left's limit point and right limit point are doing during the dynamics. And in particularly, you will only have up to order n. So if I look at computing effort, effort here is much less than in the first case, because it uses just the forward orbit. So if you want to get to order t to the n, say, in this determinant, the computing effort would be of the order simply of n. So instead of having a computing effort growing exponentially here, you just have a linear growth. And they would actually preside the same precision in the determination of the topological entropy. So now they have this determinant, but so what can you say about it? So their theorem, say it started in, so it's minus certain. And the first part of it actually started already in 77. But the proofs was published much later actually, only in, like, 88. And one of the reasons was that they had difficulties with a certain part of it. And then others, perhaps, they were a little bit less in publishing. But there are two parts of their theorem. So if I say that I take t star to be exponential of minus topological entropy, so I didn't state this. But obviously, this here is never negative, but it could be 0. So this here, actually, is just smaller than 1. And the first observation for the DMT is that DMT is not just a formal power series, but it actually becomes an analytic function in the unidisk. So it is analytic in the unidisk. So we have, like, d like this. So in the interior, it's an analytic function. So 0s, for instance, are particularly will be isolated points. And the first part of their theorem is that if t star is smaller than 1, so this quantity here, then it is actually a 0 of their determinant. And the second part is that if you take any number smaller than that, so if t is smaller than the t star, then the determinant is not 0. Now, the first part is proved by relating in a very ingenious way. So just say proved by relating forward and backward orbits. I should say in a very clever way, because it's non-trivial. But you can do it, say, if I talk about, it would be like half an hour to sketch it, but I won't do that. And I believe it was actually mostly Thurston who came up with this in the beginning. But then he didn't manage to see in the beginning why it was, say, the smallest 0 of the determinant. And then that's, I think, was where Milner came in. And they then added a proof, which was using what is called a theta function. Which essentially keeps track of periodic orbits. Now, you need to worry a little bit when you want to keep track of periodic orbit, because if the map is not expanding, then you might have a lot of them, more than you want. But you have to manipulate this in a slightly different way. But then they derive some magic identity, which is that if you take the Atin-Masur theta function here and then you multiply by their determinant, then it turns out that this has no 0s, no poles, 0s, no poles in the unit disk. So the structure of the 0s of the determinant, let's see, we have a unit disk like this. And then we have t star somewhere here, which is the leading, the smallest 0. And then in the circle here of that radius, you have no 0s in here, no other 0s. But you might have some 0s outside, which could actually like what we eventually would call resonances at all. So you might have some other 0s. And on the border of the circle, the function did not be defined. So you might have an, ah, sorry, there might be others. There could actually be others here. Yes. But then none inside. And this t star might actually even be a 0 of order bigger than 1. So it's not like in Vivian's talk that it was like simple eigenvalues or something like that. But the fact that you have no other 0s here means that, numerically, if you compute it, then you're sure you've got the right value. Now, I would call this a very indirect proof. And I think Milner-Sosten agrees with this, very indirect proof. And particularly, the proof actually uses the fact that for a particular map, you can verify it's true. And then you make continuous deformations of the map. And you verify that when you make the deformation, it remains true. So they obtain a little bit more than what I was saying here. With a very indirect proof. So you might say that's it. We have a nice theorem here. But there's still something nagging here, which is that how come that you get around this backward, forward orbits, and what are the relationships between two? And then there's actually a third thing, which is that you might describe the topological entropy in this setup by a real transfer operator. So there's the real transfer operator. There are much more general transfer operators on this. But in this particular context, it would simply be that you take L phi of y. And you say that it's the sum of phi of x, or all x which are primitives of y. You should act on some function space in order to start talking about this. So in the beginning, let's act in the beginning on, say, B, AB, which would simply be bounded functions with a uniform norm. And when you do this, then you realize that Ln1. So if you take the identity function, constant to get L1 equals 1. And you look at the value of y. It simply counts the number of primitives, of n's primitives.  You have to exclude. I mean, I should say x in the domain of f, such as this is true. So I don't write it, but here x in the domain of f. Because I should correctly say, I mean, it's not defined everywhere. But you do take functions, I mean, in the synthetic sense, or up to measure 0. No, this is really just bounded functions. So there's no measure theory here. So this counts the number of n's primitives of y. And so we're looking again at backward images. And it's not so surprising to see that if you look at the spectral radius of this operator acting on bounded functions, it will simply grow in the same way as the topological entropy, like Zn. So this will simply be e to the h-top. Now, the spectrum in itself on bounded functions is completely without interest, otherwise. Because the spectrum here would just be e to the h-top, which is sitting here. But all this disk here will actually belong to the essential spectrum. So you won't really see anything else. So this is for the spectrum of this operator acting on bounded functions with the uniform norm. Now, if you stay with the bounded functions, then you don't see much. So what you tend to do is that you respect to a small subspace. And then you see some structure emerge. Some spectral properties emerge from this. And in particular, what we do is that what tends to do in this context here is to look at bounded variation functions. So we restrict. Restricting. In the trivial case of the identity map, for example, nothing happens. So it's the identity function. Yes. And then the topological entropy is 0. Yeah, but the identity operator adjust spectrum, which is? 1. You're right. You're right. Thank you. So in general, actually, yeah, it's true. It's contained in the disk here. And in many cases, you will get the full disk. You would need some mixing or something, right, a full disk? Yeah, probably, yeah. At least in this space here, it's very difficult to get some interesting spectral structure out of it. So let me say what happens when you restrict to the space of bounded variation functions. Now, in that case, it turns out that the essential spectral radius will shrink, at least if the topological entropy is positive. So let me. So the essential spectral radius of L, but acting now on bounded variation functions, in my context, will just be 1, so in any case. And if the topological entropy is positive, strictly positive, then you have the following picture here. We have a disk of radius 1, which in general will be the essential spectrum of the operator. Then we have exponential of h top here. And that will be largest eigenvalue. As in the other case, there might actually be on the same radius. You might actually have other eigenvalues here. You might also have some eigenvalues in sitting inside. And when you look at this picture here, then you see that there's actually some quite big resemblance with this here, and said that you have just inverted t and the eigenvalue. So in this context here, Balladier and Ketter in a paper 1990, constructed actually for this space of bounded variation function. They constructed a theta function. I call it a real theta function for the operator L of this type. Actually, they do it in a more general context with weight, something like that, but it also works in this case. When the map is expanding, for expanding maps, then in fact, the real theta function here and the arc in measure of theta function will be pretty close to being the same. So usually, you can choose them to be the same, but otherwise, there might be some correct factor. But this here has no poles, no zeros in the unit disk. And furthermore, for the url operator, for this one here and for this theta function here, there's an intimate relationship between the zeros of the theta function or the inverse theta function. And the poles and the eigenvalues of L. So L lambda is an eigenvalue, I should say actually, for lambda bigger than 1. So again, outside the essential spectral radius. Lambda is going to be an eigenvalue of our operator acting on PV, if and only if 1 over lambda is a zero of the theta function here, which is the same as the theta function here. And therefore, it is going to be a zero of the minus-thousand determinant. So somehow, the auto multiplicities coincide with orders of the zero. So that's really a perfect relationship between the zeros of that minus-thousand determinant. And then the spectrum of this operator here. Now, so this obviously indicates that there's some relationship hidden here between what is this operator doing and then what is this minus-thousand determinant doing. The operator, I recall, is using, in principle, backward iterates of the map. And minus-thousand are using forward iterates of the map. So it's a bit strange, actually. So looking for dualities. So I think Davity, a couple of years after the first appearance of these theta functions, started thinking about this operator, L, which is using backward iterates, should somehow be looked as, one should look at the dual operator, some dual operator of this operator, which should correspond to looking at forward images. So he did this in, I think it's like 93 and then published a little bit later on, 96. And what, at least with this my interpretation, which might be not completely up to standard, but he looks at some operators, some even integral of operators, which acts on BV functions. And then there's a kind of dual operator. Dual operator, let me write L tilde, which in fact takes bounded functions in the uniform norm and maps it into itself. Now, this space is unfortunately not quite the banner dual of BV. So a priori, I mean the spectrum of this and then of this is the kind of dual operator, you can't really say, you have to be very careful of saying something about the identity between these spectral values. But he writes that something which is very ingenious, which is to introduce a trace, what you call this sharp trace, written like this. You could actually tell that eigenvalues of this operator and eigenvalues of this operator, which are outside of some essential spectral radius would be coinciding. Showing that, say, let's call it peripheral eigenvalues of L and L tilde coincides. Again, this is a little bit indirect approach, but still, I mean, you can actually obtain some information. If you know something about the essential spectral radius for this, for instance, you can deduce something about this without doing the first calculations. So then, Palladier and Creole. So I think that was probably 95 and published 96 also. The years might be a little bit. Then started directly the Milner-Surston determinant, but with the weights and showed that the theta function of Ruelle, say I'll call it a Milner-Surston light determinant, had the same peripheral structures, meaning in particular they had the same zeros. So again, there's a few things here, which is that it was through the sharp trace. So they used this construction here. And also they needed some assumption on the weights that it should be continuous. That could probably be improved. And it was already improved actually by Sebastian Gutsel in 2001. He managed to say that the weights needed them to be continuous at periodic points. So that's sort of a first thing, but still the proof is somewhat indirect. And they did not quite apply directly to the Milner-Surston determinant at first. So then what I started doing and it was in two years ago, I was sort of studying, I was supposed to give you some lectures about Milner-Surston determinant. So I was studying it quite hardly. And then I started together with my wife, Tenley, and we produced a paper on this, sort of the standard approach, you can say. But it was still nagging my head for a long time. If there's not a way to see directly that the Milner-Surston determinant and the Röhrl operator had a direct relationship. So then what I found finally was that, yes, indeed, that is a relationship. So that was something I discovered last year. So there's a subspace, it's called it X0, of boundary variation functions, which is invariant under the operator L. And then having a dual, so in this time it's a real Banach dual, so X0 prime, which then would contain this dual here, but be somewhat bigger. In particular, actually, it would be allowing for representation in terms of real, nice L infinity functions. So allowing for representation. So I'll call it as, say, B0, which corresponds to L infinity functions, but on a slightly different space, I'll return to this in a moment. So here the functions are strict functions or I have to measure zero? No measure, that's also why I write small L infinity but on an uncountable space. So what happens is then that sigma, so the spectrum of L acting on X0, is the same as the spectrum of the dual operator acting on X0 prime. And because of this isomorphism, I'll talk a little bit about how it works. That's the same as the spectrum of the operator, induced operator on this space here. Now the option is that this operator here has a very explicit representation, so there's a calculation, which shows that L, now the last one here, may be written as an operator S minus PS. So the two operators, bounded linear operators, acting on this last space, such that S has norm one and P is a finite rank and the rank actually is related to the dynamical system, which is E plus one. So why does this help you? Well, it does because if you remember actually, so this is for the unweighted trans-operator. Yeah, yeah, so this is related to the fact that weights are equal to one. What goes to the rank, you know? Well, it could be smaller. Ah, excuse me, yeah, no, no, you're right. Yeah, yeah, no, no, then it's much, yeah. So it should be, I mean, it works for constant weights, but yeah. So what we are aiming for is actually to describe not the essential spectrum here, which we can't really do anything about, but what's going on outside. And then if you take some value lambda bigger than one, then you notice that if lambda minus L is invertible. Now, because of this construction here, where you have real isomorphism between the dual and the isomorphism between the representation, that's really the same now as just L minus L hat being invertible. But L minus L hat being invertible. So what does it look like? It's lambda minus S plus PS invertible and when is it invertible? Well, lambda is not an eigenvalue of S because lambda is bigger than one and it's a spectral radius one. So I can put it outside the parenthesis and say one plus PS lambda minus S inverse. But this here should be invertible. And finally, this was invertible, so I can remove it. So it's the same as this here being invertible, but P is a finite rank. So here we have a genuine finite determinant or finite rank map. And if one plus this finite rank operator is invertible, that's even only if the determinant on the image of P of one plus PS lambda minus S is different from zero. So if we call this something here, D lambda, then this, as I said, is a finite rank. So this really involves just a computing of a D plus one, D plus one matrix. And how does it look like? Actually it looks like determinant image P and then you might write it like one plus P and sum of S to the K lambda to the K. So if I have the explicit representation, I can compute this. And then the revelation says, it took me like three days of computations where I did some sine error or something like that. But then finally it shows out that D of lambda here is in fact, mean or certain of one overland. So this gives the precise information I was looking for namely that zeros of the middle system determinant are the same as the zeros of this here determinant, which are then the same as lambda being an eigenvalue. And then there's a little more complication showing that the order of the zeros will be same as the multiplicity of the eigenvalue. So we get a perfect correspondence between the say the Huell operator spectrum and the zeros of the middle system determinant. So this of course doesn't tell you much about how did I then do the computation and how do I get this space B zero out of my setting. And the idea is really to choose the right subspace. There are two things which actually help in this aspect. So first of all, I was talking about the map was not defined on critical points, but it's defined when you take limits from the right hand or left hand side. So the idea is to, you should actually make systematic use of limits of right and left limits. Systematic use of, well you can call them, actually point germs is one word. You may also call them directed points. And if you, geometry, you might also say that we're looking at the unit times in bundle are different ways of looking at this. So the idea is that if you have a point on the real line, then you would like to associate to this two directions, X plus, X minus. So X, call it X hat, sort of a directed point, would be something with an X and then epsilon, or you might put them together like this. And then the epsilon would indicate whether you're coming from the right, from the left. You also naturally introduce an order, which is that if X small on a Y, then you would say that they're ordered like this. So I intertwine actually the two, defining an order between the two. If you have a map defined partially say, so this is like just one branch of these things up there. Then in this example here, for instance, the right endpoint will actually be mapped to F of V plus. And the left hand would be F of U minus. And then we would call this say the extended map.  So you see that every time you have a branch like this, you can actually extend it to looking at map of point germs. And then you get a nice map defined on sort of all the point germs between the two. In particular, if I look at my original map here, something like this, then from A to B, you actually have three intervals, which now become point germ intervals. So you have I zero hat, I one hat, I two hat. You have the point germs from B plus to B minus, which are simply becoming now partitioned into I zero hat, I D hat. How we found exceptions, we found. Exact, exact. So now this is actually becoming a perfect partition. And the reason for doing this is precisely that the extended map here is actually defined everywhere now. We also have these signs. So I didn't actually talk too much about this, but the signs mapping A plus to B minus. So that would be mapping to plus or minus one, which are sign of monotonicity of the branches. So here to plus one, minus one, minus one. And then we have the product along orbits, which are the product of these signs along these extended orbits. Now finally, and this is, maybe then we get some. Finally, the minocertan determinant is actually called a kneading determinant. And the reason is that they look at relative positions of orbits of the critical points to the critical points themselves. And there's something called kneading coordinates related to this. So if you take C somewhere among the critical points, C one up to C D. And then you would like to say that a point germ is either to the right or to the left of the critical point. And the way you write this is to write plus one half if X had this bigger than C, minus one half X had less than C. Now, one of the important things about this notation is that you never get zero. Now, when T is an auxiliary variable, then you construct for any initial point, you construct the kneading coordinate, which gives you rise to a kneading invariant of minocertan. Now, it's not quite that formulation, but it's equivalent, which is that you look at a generating function. You look along forward orbit, as I mentioned. You take this sign of monotonicity along the orbit and then you multiply the relative position of the last point, relative to the critical point in question. So this is their construction. And then they write a matrix, M i j of T, which is going to be the following. So I'm gonna write it down, C j plus T minus theta C i C j minus T. Now, we do that for J equal to one. So for the internal critical points, I define this. And I define for, oops, A plus T plus theta C i C minus T. For J equal to zero. So these look like some magical things just popping out of nowhere. And at present, that's the case. So this is, so that there are a certain number of critical points. So this is D plus one times D plus one matrix. And then it happens so that the minocertan determinant of T is the determinant of this matrix. Now, if you look at this as a formal power series, then the coefficients here will always be plus or minus one. Here we'll get plus or minus one half. If you look at this here, coefficient will be plus or minus one or zero. So coefficients all like plus minus one or zero. So these define nice analytic functions in the unit disk. And this here becomes an analytic function in the unit disk. Now this is the minocertan determinant, but in the version which I have developed reasonably with the 10 layers. So then you could go on to prove why the minocertan actually works, but that's not what I want to do. I want to say, what is the relation with this weird subspace that we're talking about? So here it's analytic, complex analytic, or what is the function you are? Coefficients are all real. So analytic in either sense, but here it's just analytic in the variable T, complex, smaller than one. I mean, these all have values plus or minus one half. So every coefficient in these powers here will stay bounded. And so you get an analytic function in the unit disk. I should say by the way that one thing that you easily proves that the value at zero is one. So indeed you will get that zeros will be isolated and things like that. So what is the function space that was interested in here and each dual? That's more interesting. So if you look at phi, which is of a function defined on the interval AB, then recall that the variation of phi, so that would be the supremum, sums of phi of x i, phi of x i plus one. And supremum speed taken over all possible finite points, choices of points. Between A and B. And you say that if phi is of bounded variation, even only if this number is finite. In that case, in particular limits from the left, right on the left exists. And you might define the property, the quantity which is phi of A plus and phi of B minus. It turns out then that a normal, a suitable norm on the space of functions of bounded variations is simply taking the variation and then plus absolute value of this boundary value here. Value from left, say right limit on the left hand side and left limit on the right hand side. And this is a norm on BV. What is then the small subspace that I'm looking for in BV? Because BV actually is too big. It's too big in order to do the direct analysis. And one reason being actually the dual is too small. So the small subspace I'm looking for is a space which is, first of all, is defined as functions which are simply locally constant except for finite number of points. So let me just give one example. One example would be to say that you take a function which is say one here, then it becomes minus one there at alpha. It becomes one at beta, and then it becomes two to the right of beta. So that would be one example of a phi in my space. And why do I give you an example like this? It's simply because when you look at these kind of functions here, then you realize that you may describe them in terms of the jumps. So you have a certain number of jumps here. If you look at this function phi, I mean describe it in terms of say elementary jumps. Let me define this for u hat in a plus b minus. I will define a jump function which is one half. If x is bigger than u hat, minus one half, x is smaller than u hat. Now if you look at this function, then you realize that actually it's very close to the one I had somewhere else over here, except that you have in fact inverted just the rows of the base point and the directed point. But the function phi that I have here for instance, may be written as say from a given value here you jump two values down. So it's like minus two sigma a minus. Then you jump two up here. So it will be sigma minus.  And then finally you add a constant value which is three halves. And I might write this as three sigma a plus. So this function is actually just the constant function one half times one. Now any phi in my space may in fact be written as omega i and then sigma u i hat for some final combinations of this. And I should still add omega zero times sigma a plus. Okay, so any simple function of this type can be written like this. And then you check that the norm of phi is simply the sum of the omega i's. So sum of omega i, i greater than two one. So this is like actually a small l one space. And what we do is then simply to let x zero be the closure of s in bounded variation. So the small subspace that I was looking for in order to do this trick here, this is a closure of these type of functions in bounded variation. And why is that good? Well, it's because it has a nice view. So x zero bar prime would be the Banach dual. And then given any l in x zero prime, I would define a function now on the space of point germs to be l acting on sigma u hat.  I don't have the time to prove this, but then there's a very simple proposition, which is that if I take l in x zero prime, map it to l hat in the space b zero. So there's one little thing which is that on a plus and b minus, you have to get just the same value but with opposite sign. So I'm looking at h in bounded functions on a plus b minus, such that h of a plus, h of b minus is equal to zero. This map is an isometric, an isometric isomorphism. The b here being equipped with uniform norm. Finally, so I should actually have stopped here but five minutes. We have to ask. So I'll just give the principal lines of what's going on because we were looking at the dynamical system where you had ik, which actually was from ck to ck plus one. We had a map fk, which then maps ik to ab. I have a sign of monotonicity. And then we define a transfer operator. So the transfer operator I defined in the beginning which was given by Ruel was in the case interval here, I can write it in a slightly more complicated way which is that I take phi, I compose for the inverse of the cased branch, except that I have to be in the image of the cased branch in order to do so. And now you look at how this actually acts on certain functions. In particular, if I act on a function which is in my space s, it suffices to look at what happens when I act on a simple base function my sigma u hat there. So let me give you a drawing. So sigma u, sigma u does the following. It looks like this. If I put say this point here, then this would actually be sigma u minus. And what is lk doing to this? Well, you simply look at the dynamics and you see that you will map the interval to f ck, say let's say f hat ck, f hat ck plus one minus. Now the part which is outside will be chopped off because it's not in the image. So here you'd get zero. Then the point u here will be mapped somewhere here and this would give rise to something like this if I have preserved the orientation.  So what happens is that lk acting on sigma u hat which after a long computation, you realize that when u is inside the interval here, you get sigma but at the image point but you get two corrections term because you have seen your chopped off twice.  So this took me, as I said, like three days to get the science right but after I did that, then I came to the say nice conclusion which is that if you look at this expression here, then here you have a function that depends on u. I mean there's a variable here y which I didn't write but this function here does depend on u. Actually this function here does not depend on u. It's just the same function all the time. This is the same function all the time. So this here will be the finite rank perturbation and this here will be an iteration by the map if hat. So let me just say that sk h of u hat I defined to be sk h of fk u hat and pk, well, I'll have omitted the precise forms but it's involving the correction terms there. This defined will be a rank two operator and then finally the lk hat which corresponds to how the lk acts in my dual space is simply given by sk minus pk sk. Finally, if I sum all the branches, I get an expression of the following form.  So very simple form for the part which is using iteration of the f hat and the p operator is a little bit more complicated but it's say h of u that will be sigma a u hat delta a of h plus and then there's a sum sigma j of u hat delta cj of h. And finally, what are these delta cj h that's h of cj plus minus h of cj minus. And there's one exception which is delta a h which is h of a plus plus h of b minus. So the upshot is that you get l is the sum of one operator here with by evidence has spectral radius one because this is just plus or minus one and you're working on l infinity. So iterating here doesn't matter. So this has spectral radius one actually norm equals to one. This here is well it's mapping to d plus one functions so it has rank d plus one. And then finally you do the computation and I won't do it but you will actually drop into the expression which is has now disappeared. Yeah that has disappeared but you will drop into the expression that if you calculate this determinant here you will simply get the same as the mean of certain determinant that defined before. So when you finally reach this point here what are then the conclusions you can draw? Well first of all it's nice to see that the intuition that you should have about the dual real operator being really related to mean of certain determinant is correct. And now if the top logical entropy is positive then just from the fact that we know that the real operator is positive implies that the exponential of h top is an eigenvalue of the real operator and it is the largest eigenvalue of the real operator which means that you get part one and part two of the mean of certain theorem for free without looking at any theta functions. So I mean the mean of certain theorem actually drops out as a simple corollary of the fact that this is a dual determinant of the real operator which is positive. And finally the fact that it is a regularized determinant of the dual real operator was just very nice to see when I came about it. So thank you very much for your attention. Time for some questions. Vivian and I tried to make an extension of this in the first in theory allowing varying coefficients. One other thing would be to try to find other extensions what one uses as a basis is really the order structure of the reals. Is there some order structure that one might use instead? One thing that sometimes works is to use the conformal structure involved in one. In the complex plane, yeah. Yeah, I remember you also had a paper with Sims I think, you and Vivian, Steven Sims. No, we had paper with Sims and Kitai and all these. But it didn't go very far perhaps. But the fact that, I mean in some sense the fact that working on the dual space, you also remember this paper of Levin Soutin and I forgot, yes. For the real determinant for hyperbolic Julia sets and what they did is for certain types of weight, they could actually, instead of looking at the sum of pre-images, they could look at images of the critical point and then make a determinant out of this. So somehow there's this duality between the operator and then the dual operator and the determinants you get from it is very intrinsic in some cases and sometimes you might find it and sometimes you may not. So really it's connected to David's question. I just want to point out that what you're doing here is you're really writing a regularized determinant. Sure. Because P is finite. Yeah, so it's a regularized determinant of the dual real operator. Which is the min assumption. And actually you seem to want to avoid the zeta function. But if you are interested in the zeta function, you can also study the zeta function by. You get it directly from this. In any dimension, and this is what you did with Tsuji actually. So you can write the regularized, I mean you write the transfer operator as a bounded press, an operator which has a power which is press class, and then you can mimic this quick action. But then you don't have this, you don't have any more than needing the determinant. But you have a shortcut to just point out that this computation can be used. It's used in other cases. Yeah, the question may be on the same line. Do you see or you don't see the periodic points appearing in the computation or at least in the definition? So in this approach, no. But are there still, can you get information on them? I would say the best information comes from Bellady Kehler. Which is that if you work on the real space of bounded variations, then they have this paper showing that you get a set of function which comes essentially from periodic points. And that has the same as eigenvalues as what are being shown here. And if you do that, you can also show that I work on this smaller space but I get the same eigenvalues. And the eigenfunctions that they would get in their space, this is in fact in my space. So that's consistency. I mean, I don't get more eigenvalues and they don't get more eigenvalues than I would do in this way. But the periodic points, I don't see it all here.


There are certain Super Nintendo games that are best described as hit and miss. They have a lot of positive aspects, but are met with an equal number of negatives. Lufia and the Fortress of Doom is like that, Breath of Fire 2 is like that, and Gradius 3 is definitely like that. Positive number one, this is a great looking game in every respect, with really cool enemies and bosses, and the backgrounds lend themselves to some really cool stage structuring. The corresponding negative is that this game has major problems with slowdown. The bubbles level here is probably the worst, it just gets cripplingly slow, to the point that the screen looks like it's gonna start glitching out. That's just not the sort of thing that ages particularly well. Positive number two, the power-up structure is a lot of fun. You can pick and choose from a list of how you want to set it up before you start the game. Then as you collect those red power pods, as they're called in the manual, you can apply them accordingly however you'd like. You can play with how you like your ship's speed, and whether or not you'd save your tokens for bigger stuff that protects your ship, or upgrade your weapon right away. The corresponding negative is that it's really easy to pick up two power pods right next to each other and completely miss your chance to activate that power-up. So then you gotta cycle all the way back around, that is really annoying. It can even happen when you're just caught up in the game and you don't have a chance to glance down and see where you're at.  It is awesomely overpowered and just mowing through the early levels like a hot knife through butter is the greatest. The negative about this is, again, the slowdown. Look at this, when I fire, the game slows way down. When I let go, the game goes back to normal. That's, uh, kind of a problem. Another positive is that Gradius III is a good game for newcomers to the shoot-em-up genre. It's a good, cheap, straightforward game to use for practice, but just please make sure to put that shit on easy first. This game is hard as hell on normal, and I don't even want to consider playing this game on hard. No frickin' thanks. But on easy, it's manageable and a lot of fun.  The inherent problem here, though, is that if you die, you start with nothing. Nothing!  That is a problem.  One other thing I gotta mention is the Konami Code.  Oh, okay. Actually, what you gotta do is replace the left and right on the D-pad with L and R.  So yeah, as you can see, Gradius III is a game of major strengths and major weaknesses. If you're a seasoned shoot-em-up veteran, you know you can find much better than this game to test your skill. Yeah, it's got its positives, but other games have similar strengths, without the debilitating negatives.  And if you end up liking Gradius III even a little bit, you gotta check out some of the better shoot-em-ups, like UN Squadron, Axel-A, and of course, Space Megaforce.


Welcome to module 9 of the software engineering course in which we will discuss model based engineering of software. So we have discussed initially that models can be used for a variety of things and we covered for example communication, you have a model and you discuss you for example explain to a client or to a fellow engineer what is going to happen, how things are going to implement, what kind of trade-offs you are having. We have seen in the architecture and design part that you can use models for analysis so you can use them to reason about the system, how is the performance going to be, how are we going to separate the system into sub-components for example. And there are a number of other things we might consider for example for certifications you might have to have models or at least they are encouraged in the automotive industry that is not uncommon. But you can also use models for I would say a bit more advanced purposes so you could use a model to actually simulate your system or part of the system or the environment of the system and finally you could actually use models to generate things. And here we are thinking about things like code, you can generate code, you can generate test cases, of course also code but it is slightly different usually but you can also generate documentation or a lot of other things. And the question is if you want to go down this road, if you want to do more things with a model what exactly do you need? And well the first thing you obviously need is a model that describes whatever you want to do for example if you want to generate code you need some kind of model that describes this.  You need to have some kind of format that is readable for example an XML file or some kind of other standardized way of reading in the model because in the end of the day you have to process it automatically and for that an image might not be enough so this is when we have to get away from the whiteboard or from the paper or from any drawing tool to actually use a real modeling tool. And then finally we need something that is called a transformation, a model transformation. So what we do is we read in our model, we know what the language is so we somehow understand it and we have some kind of program that takes the model and transforms it for example into running system code or into a markdown or HTML file that is used for documentation and well the transformations can generally be in a general purpose language so we can write this for example in Python or C++ we can just write a program that reads in the XML file and because we know that it is UML we can parse it in a certain way but there are dedicated transformation languages as well that make this a bit easier. So these are most likely languages you have not heard of yet because they're not that common but it's specific languages that can be used exactly for this purpose. Read in a model, push out something else, transform it into something else and they usually lack of course a lot of things that a general purpose language would have that are simply not needed in this context. So this is generally what we need and overall this process of using models and transform them into something else basically use model as a really important part of your overall software engineering process this is what is known as model-based engineering. So you base your software engineering on models and the abbreviation here is MBE. So model-based engineering there are a number of similar terms that you might sometimes have heard as for example model-driven software development, MDSD or model-driven engineering. The differences here are really not that large so it's usually for example model-driven engineering is considered to be a bit stricter so there are more models used that basically used for everything but in the end of the day people use these words interchangeably more or less. So why would we do this? Why would we write models instead of code? What do we do with that? The general idea is that we can improve our overall development and what we in particular would like to achieve is a higher quality. So quality goes up. We would like to reduce our costs. We would like to increase the reuse of the models and the software and in a way we would like to improve our engineering capabilities so we want to be able to use more engineering techniques to reason about things for example use simulation and so on we're not just coding and see what it works but we're doing this in a bit more systematic way. And the idea is basically the costs are reduced because you can generate things automatically because you can reuse the model for several purposes for example you could generate tests and documentation from the same model you don't have to write it. You only have to maintain the transformations and the models but you can generate a lot of things out of it. The quality goes up because you have a separation of concerns. You can focus on the model, you can focus on the transformation and if you for example have tested your transformation well enough then there will be less bugs in the development and that's essentially the reasoning around why we want to do model based engineering. So in the next video we go a bit into where is this used, what are the advantages and disadvantages about it and what are generally the challenges we are having with model based engineering and after that we dive a bit more into the details of how does this actually work.


Welcome back. Now it is time for us to look at one of the most important derivations in thermodynamics, perhaps something which is at the heart of the second law of thermodynamics and that is the Carnot theorem. What is this theorem about? This theorem is about the efficiency of 2 T engines or 2 T heat engines. It provides a special status for reversible 2 T heat engines. Such reversible 2 T heat engines are sometimes known as the Carnot engines. The Carnot theorem compares the efficiency of any 2 T heat engine with a comparable reversible 2 T heat engine. What is this theorem about? Let us sketch a figure to appreciate what the theorem is about before proving it. Let us say that we have two systems, two thermal energy reservoirs at distinct temperatures T1 and T2. It is given that in a thermodynamic sense T1 is greater than T2 and we consider two engines. One, any engine E working between T1 and T2 and let it absorb heat Q1 from the reservoir at T1, reject Q2 to the reservoir at T2 and produce work W. The other one is a reversible engine working between the same two reservoirs. Let it absorb heat Q1R from the high temperature reservoir. Let it reject heat Q2R to the low temperature reservoir and let it produce work WR. Now the efficiency of this engine by the definition of efficiency is W by Q1. The efficiency of this reversible heat engine will be let us call it eta R that will be WR by Q1R. Carnot theorem compares this efficiency and this efficiency for fixed values of T1 and T2 and the Carnot theorem says that if you have a fixed value of T1 and another fixed value of T2 then the efficiency of any engine E must be less than or at most equal to that of the reversible heat engine working between the same two fixed temperatures. Now it is important to note that here we are just talking about an engine and a reversible heat engine. The given or the fixed things are the two temperatures, one is T1 and one is T2. There are no other qualifications on the engines except that the one on the right has to be reversible. Otherwise the engines may be made up of any materials, any structure, they may use any fluid, they may use any set of processes. That is the importance of the Carnot theorem. In a few moments we will be proving the Carnot theorem. Thank you.


After all the countries that you've built, you decided to settle in this country? Yes. I've been to more than 21 countries in Africa, next time I'm going to beat 29 so that I can beat you. I say it again. I say it again for the day it's going to happen, so you think you're going to beat me? I'm better traveller than you. Okay. Anyway. Anyway. And I'm travelling all country in Africa. She's the biggest travel blogger in the whole of Africa. In so many countries in Africa, I've started at 18 years old with the first country Ghana. And the last two years I've been travelling alone, completely alone, in more than 15 countries. Okay. And I think that it's true, yes, that you're in Africa, you're in Abidjan. Why Abidjan? Because the first thing I realised when I was travelling was that travelling and living in a country is totally different. For example, you told me one day that your best country in Africa is Namibia, right? Exactly. But I don't know for you, but maybe you don't see yourself living in Namibia. Okay. It's different. You live in Acre and you know how it is in Acre and blah, blah, blah. And my favourite, my top destination in Africa may not be Abidjan, but living here, it's the perfect place for me. Why you chose to live here? First of all, I'm alone. I'm a solo burner. Abidjan in the Freckin' Fun Zone, to me, is like the number one when you're entrepreneur. Because there are so many people creating stuff like creative, entrepreneurial, rental, retail. Retail, retail business. Yeah, all those stuff. So you meet so many people that encourage you to do stuff and that motivate you to do stuff. And I was working in Paris, in La Défense, like big business centre in Paris. And I was in a company, so I had colleagues. So a lot of other people doing like me. But as a solopreneur, any other people doing like me in a big area, and this is Abidjan to me. And it's different to other capital cities in Africa, like the Don't Know War. I mean, I'm not saying that they're not entrepreneurs over there, but the energy of creating, of impacting is way used here in Abidjan. That's why I'm here. That's the number one. Also, as a woman, as a Caribbean woman, travelling solo in Africa, I've realised that you cannot dress the way you walk everywhere. And it's not about the religion, Muslim, Christian, Christian, it's really about the country. And the two capital cities in Africa where I could dress the way I wanted were Accra and Abidjan. So I was like, you've been travelling two years. It's been an amazing experience, but also it's been hard sometimes. Exactly. And I wanted my life to be solved. I wanted my life to be, like we say here, like solved. And I wanted to wear crop top if I want to wear crop top. But yeah, I wanted to be free to dress the way I wanted and Abidjan was the capital. It's not like the number one reason, but it was one of the reasons. Okay? Also, and it's something which is a little bit taboo to say when you're from Europe coming to Africa. But I wanted also the comforts. Is it English? Yeah. The comfort of travelling or living in a place? Living in a place. Yeah.  Internet collection is perfect because I'm also a digital nomad. Yeah. So you know, I need to get internet everywhere. Right. When I read my stories, when I read my video, when I read my life, I need to get internet everywhere. Some countries in Africa, like there is only this specific place. That has internet. Especially in France, the easiest country to start with is Africa. And how many countries have you been so far? How many countries have you been so far? The number is not important. Like I will be... No, no, no. No, just answer the question. The number is not important. The experience is good.  I will be to my 30 countries in some times, but for the moment, it's 21 countries. What really inspired you to start travelling with Africa? Absolutely. Okay. And when you are in France and you're from Caribbean Island, everybody, the school, your parents, your family, your friends, they're telling you, ah, you got a look for the Caribbean, but you're really ancestor there from Africa. So I grew up that way. Always, like always hearing stuff about Africa, but knowing nothing about it. Wow. Because on the TV, it was always like poor people, small children with big belly, with flies on the face and all. That was the only image I had from the media, from the school. They were saying that Africa is cool, blah, blah, blah, and that colonisation was not good, but it was okay, you see? And slavery too. So there were so many stuff I was hearing. That was what you were learning in school? In a lot of school in France. And even in high, like, you know, the school after high school, like super school and all. Yeah, the university. Yeah, university and all.  I knew it was wrong, but I didn't have the argument. I didn't know how to say that, because the only person in the world that was telling me Africa and not that, that was my father. He's never been to Africa, but he read a lot of books, and he knew that his people wasn't that big. So when I was 18, I told my dad, I wanted to go to Africa. Can you buy me the plane ticket, my mum and you? I didn't know nothing about Africa. So I just took a map, I took a ruler, and I chose the country that was closest to France, and a country where I was speaking English. So what I said to my daddy was, I don't just want to go to holiday, also I want to learn English. So I was telling him that way, like I want to see an African, and also I want you to improve my English. So the closest country was Ghana. So that means you're not a Ghanaian? I'm not a Ghanaian. What did you get the name up and off of? Because when I arrived, I was the only woman, like when I arrived, the only way to me as a young lady to travel to Africa without holidays was to do volunteer work. So I didn't want to do that with the French Association, saving the world in Africa. So I went on the internet and I looked for a Ghanaian association that was based in Ghana, that was 100% Ghanaian. So I contacted them and I said I was coming. So when I arrived in Accra, I was the only woman, there were 10 guys. I was the only woman, and I was the only woman under 20, all they were 30. So they saw that I was a little bit shy and all, and they wanted me to feel good. So they told me, when were you born? So I get on my birthdays. They say, no, no, no, no, no. The day of the week, when were you born? So I didn't know, so I checked on my phone and I said, okay, it was a Tuesday.  And doing all my thing in Ghana, my name was Abena. And a lot of people on the internet, they don't even know my name is Sandy. The thing is Abena. Your real name is Sandy. Yeah. But now you've adopted Abena as your name. Yeah. You are called Abena. I am called Abena. I'm also a Tuesday born. I'm a Tuesday born. So I'm called Abena. You are called Abena. Yeah. We both travel within Africa. Yeah. We both want to travel every country in Africa.  What country is your favorite country now? I can't say. I don't know how you can say in Namibia. You've been to so many countries. How can you say, okay, Namibia is number one? Because there are so many different experiences. No, listen. I always tell people, every country is so unique in their own way. But if you ask me to choose one country out of all the countries, I should be able to tell you because most of the countries that I've been to, everything that can be found in those countries can be found in Namibia. I'm telling you. So to me, my definition for Namibia is Africa. Okay. So can I do a top three? I can't say just one. Okay. Go for top three. Okay, top three. They're all on the same ranking. Cameroon. Have you been to Cameroon? No. But they think that all wanderers in Africa, they are in Cameroon. And I've been there and I can guarantee you that it's true. It's just that, communion people, some people, they think they are a little bit crazy, but that's okay. They're nice. You just need to, you know, to be nice with them. But there are everything over there. Mountains, beach. Even snow in the west of Cameroon, people have already seen snow. So Cameroon will be in the top three. Benin, of course, will be in the top three. For the authenticity, the spirituality, the food, everything. And the beach, they are so clean in Kota New. I won't target any country. But you live in Abidjan. Abidjan. There's no beach in Abidjan. You need to drive one hour to go to Assini. Come on, go to Assini. And you've got to enjoy one of the... I think it's amazing. One of the best... Ah, come on. So... It's not like that question. Every course is not on your top list, but you chose to live here. Yes. That's what they were saying. So now my next question is... I just feel like being a film and travel contentator in Africa. I think it's going to be maybe strange for me to say that, but it is so easy. I'm going to tell you why. I am alone. I am a woman. And I'm not from Africa. You know, I feel like I'm from Africa, but the first thing people ask me when they meet me in any country is where are you from. When I start talking about Guadalupe, Caribbean island, they're like, why are you coming from Guadalupe? Where it is? Are you our ancestors? There is something like... Also, we can say whatever we want about European countries, developed countries and all, the thing is that in Africa, any country, when you are alone, you're never alone. Any people, they're trying to help you. Somebody see you lost, they're going to help you. They see you alone, they're going to invite you to their wedding, lunch, dinner, family event, a lot of stuff. I've been travelling the entire world. It's just that I prefer the Afro world, but when I'm in other countries, I don't have those kind of warm welcome and I don't feel home, like I feel home in Africa, even if before, I knew nobody in this continent. Now, I have friends and families in every country I've been to. I feel like being a female travel content creator is dangerous though. Yeah. You don't get her out? I think there are two steps. The first step is that maybe now I'm used to travel, so I know what I should do and what I shouldn't do. For example, I'm in Ivory Coast and I know that I can dress that way in Ivory Coast, but I would never dress that way in Kotolou or in Duala or in other countries in Africa because I know it's not common to dress that way, so if I don't want people to talk to me badly, I would just dress another way. It's just that now, I know how to behave in each country in Africa. Even countries I've never been to, now you know how it's going to be, so that's the first thing. And the second thing is that, of course, some stories happen to me in Ethiopia, in Senegal, in maybe other countries, but it's the same level than what happened to me when I was in Paris, or New York, or Canada. It's the same. It's specifically Africa. Some stuff happened, but I don't think it's linked to the fact that I was in Africa. Yeah, these are the fans, you know? These are the fans, you know? Yeah? They said Sandy. Yeah, they said Sandy. They know your name. Oh, okay, but how did they know your real name? Because it's on Instagram, just like people they used to have now.  We travel out in Africa. The first one is that when you arrive to the airport, take your SIM card directly. A lot of people, because they're used to travel out of Africa, may depend on the country, okay? They think that it's okay, we're going to use the wifi of the restaurant, blah, blah, blah. But the thing is that it's... it's too difficult. You just take your SIM card, and you will have internet all the time. And compared to a lot of European countries, you just give them your passport and you get the SIM card very quickly in most countries in Africa. Okay, the second travel act is that when you go to holidays, most of the time you want to go to hotel, hotel, hotel, hotel. But the best experience you will have is when you sleep to local, to local house. The most famous website is Airbnb, but you can find also people on Facebook and other platforms. But really, my best experience is to give to people in the country. When you are in the hotel, you use a lot of other tools and you don't have the real experience of the country. The very last one is very important. Do not plan all your travel. A lot of people, when they come to any country, they know they're coming from day one to day seven, and they know exactly what they're going to do day one, day two, day three. And it's so boring. You just need to plan. You can do a checklist of all the things you really need to go before you leave the country. That's okay. You can even say, you can even plan maybe the first four days. But once you be in the country, you will meet people.  And if everything needs schedule day by day, hour by hour, then you won't be able to do anything. You won't be free. So, free yourself, plan half of the travel, half of the travel and the rest. Go with the flow. Go with the flow. That's it. I never plan my trips in Africa. All I do is I work out. I can confirm. I met you in Benin. Nothing has been planned. You have met you in here in Ivory Coast. Nothing planned. Why you see I need to be so real? That's too real. That's too real. Because I feel like whenever I plan, I feel. Yeah, I feel. You know, even I was supposed to leave today. Yesterday. Yeah. But now I've added two more days. You see, I'm leaving tomorrow. So, whenever you travel, we need Africa. Don't plan. That's why I don't buy my return ticket. Yeah. Because anytime I buy my return ticket, I end up changing and changing and changing. You know what? I'm gonna tell them something that they need to know about Africa. My friend, my family, is that when I came back to Africa, I find back my humanity. It's what I say to everybody. Because of the people I met, because of the people who helped me, because of all the connections, people here, I feel like they really care. When I'm lost, they really care. When I'm in trouble, they really care. When if somebody arrests me in the market, they really care. And this is not something that I've felt during my 29 years in Europe. I don't think that Europe is blue sheet that is not bad. I'm not saying that. I'm just saying that I have to better here in Africa.


if you went to open access, it's completely up to you. That's the beautiful thing about being an open access provider. You can choose your level of independence from the vision contribution plans. Understand. There's a lot to understand. Insurance companies of all types are known for being rather difficult to work with. And vision insurance is no exception. So the first thing we have to accept is that vision insurance plans are more like vision contribution plans. And honestly, if you take nothing else from what I'm telling you today, please, for the love of all things holy, stop calling them vision plans or vision insurance, right? Call them vision contribution plans and use that term every single time you talk to the patient or you're talking in your office about the plans. It really helps to, I guess, ready the mind of your team and of your patients, really, that the service is contributing towards their eyewear. So vision contribution plans. Okay, going back. Okay, so going back to understanding reimbursements, the plan will contribute towards the patients out of pocket for their eyewear, right? The better way to really understand reimbursements is to completely grasp, I guess, the different roles and functions of the main aspects that control the money distribution between the provider and the patient and the vision plan. So let's start with like the patient copay. Patient copay is what the patient pays out of pocket for that service or item. And this is rather confusing to patients because with other insurance companies, a copay is a single dollar amount. So the patients will go into any other provider to use any other insurance and pay a copay and walk out. And that's it. It's a single dollar amount and they're done. But with the vision contribution plans, the copays are added one on top of the other. If we use copays in the term that the vision contribution plans want us to use copays, they're added one on top of the other, which gives this patient the, I guess, idea of this nickel in dining them. The other thing to understand with the patient copay is that the patient copay is collected when the eyewear is ordered. Okay, when it's ordered. Here's the timeline when it's ordered. That's when we collect it. Now let's talk about charge backs or lab charges. Now this amount has taken out of the provider's check to pay the lab for the cost of the lenses. So this final charge backer or lab charge amount arrives on the remittance explanation of payment weeks after the patient's glasses have not only been ordered, but have been dispensed. And that's when the lab charge back actually comes in at the very, very end of this whole scenario. Now let's talk about material dispense fee. The material dispense fee is the amount that the vision contribution plans has designated the value of the base style of a lens. Sometimes it's called like a lens dispense fee. Now these dispense fees will be smaller for the single vision orders and they will be larger for the multifocal or progressive orders. Now amongst all of the single or smaller to larger differences, there are also different levels of plans that will have different values for this lens material dispense fee. So for example within a vision insurance plan, you will have different plans levels in there and each of those plan levels will value a single vision lens dispense fee at a different amount than you know the other plans that are on there. So it's really confusing. It gets even more confusing when the patient's material copay is larger than the lens dispense fee because the difference is taken out of the provider's check as a charge back or lab charge amount. So for example if the lens dispense fee is only valued at $15 but your patient had a $25 copay, guess who's taking that $10 difference out? So the last thing to really understand when it comes to reimbursements is the service fee or dispense amount. Now this amount is the amount that you should be making from that item. I say item because it could be that material or that AR treatment or a tent or any of the other add-ons that are on there. So the service fee is basically the patient copay minus the charge back and this is what you should have in the end. This is what it should be equal to as the service fee. However issues like incorrect collection of the copay when the patient is there can really cause a pickle. You won't get to your amount in the end or when changes have been made at the lab because of inability to like get a product or availability issues I should say will result in the service fee being inaccurate. Now basically what was entered in your EHR when you collected the patient's copay might be different to what was actually ordered from the lab. So when the charge back comes if that's different you don't make what you're supposed to on your service fee. That's a bummer. Now these issues are further intensified because the unfortunate timeline of when all of this is happening. So when the copay was collected versus when the charge back is actually taken out of the provider's check is weeks after the fact. So it's a real bummer going back to the patient and saying hey patient we were supposed to collect more from you. Doesn't look good right? Well there are actually two misconceptions that drive me absolutely crazy when I hear offices talk about them. The first is that the out of pocket that is collected from the patient is what the practice is actually making what they're actually earning for the job. You will not believe the amount of people on your team that probably think that that's what you end up earning and this is not the case. In fact a majority of what is collected by the practice for the eyewear is taken from the rider in the form of charge backs. Now many offices struggle I will say with understanding how charge backs work on the vision plan or remittance or explanation of payments. The confusion is further intensified with inconsistencies between the different plans. So for example you have all of these different line items that have copays, charge backs and service fees right? All these different ones that you have to try and remember. Well let's take for example Tribex. The Tribex will have a material copay, a charge back and a service fee of what you should be making at the end. Well these amounts are different amongst the different plans that are within that one vision insurance company. Making the ability to master the charge backs and lab charges you know understanding nearly impossible because of the unnecessary complexity of the reconciling right? So most go uh look at what we collected with the out-of-pocket because it's much easier to understand and it's much shinier and much prettier and so a lot of people really have a misconception that the out-of-pocket with the patient has paid is what the practice is earning on the job and that's simply not the case. The other misconception is when I hear independent opticles say you know what it's not that bad being a network because we don't have to pay a lab bill. Bro yeah you do. These charge backs are your lab bill and they're significant and it's not uncommon for an office to run numbers on a particular lens that they maybe just order through the vision contribution plan and their designated lab and take into account what you paid in charge backs. Then an independent optical could call up their independent lab and ask for what your optical would be charged for that exact same lens through the independent lab and then you can go a step further and inquire about a volume discount like if you happened to be sending all of your work there through that independent lab what could you be getting that lens for? You might be impressed it doesn't hurt to ask. It's already well known that we accept a lesser reimbursement for being on a panel with any insurance company vision medical whatever it is right the in-network reimbursement rates are typically about one half to one third for the patient with the vision contribution plans than the revenue that is made from a private pay patients so just depending on what your prices are listed at so the result is that the office must see more of these vision plan patients per day in order to make the necessary revenue to run an effective business. Now this is an unfortunate situation for for the vision plan patients as they are unknowingly having their appointments rushed right in order to make some profit in the office. Now when many independent optical set up their pricing for exams and eyewear they typically set their usual and customary fees knowing that they must inflate their private pay amounts to offset the dramatically low reimbursement rates that they're getting from being in-network with the vision contribution plans so really this makes an unfortunate situation over here for your private pay patients that are being unknowingly penalized for the cost of their eyewear to basically make up for you being shorted in your in-network agreement. Now the great news though is that both of these unfortunate situations can be remedied if the independent practice were able to control their own narrative so without the contractual amounts and mandates of the vision contribution plans offices are able to make more per exam which allows for more quality patient visits because the quantities aren't necessary to be profitable and the other benefit is that the independent offices do not have to charge as much on their eyewear when they are profitable on every pair sold rather than selling most of their eyewear for a fraction of their intended value it's a win all the way around. Open access is an office or provider that is not limited from seeing patients based upon the vision contribution plan that the patient happens to have so open access is a term I learned a long time ago from the founders of anagram and for those unfamiliar with anagram it's a service that allows practices to pull out of network benefits for patients this ultimately allows the practice to service all patients no matter their vision contribution plan and no matter the inner out out of network status of the provider so the opportunities that are found when optical begins to align themselves with an open access mindset are quite liberating the practice will begin to see hope and no longer being controlled by the vision contribution plans and a glimpse of this freedom from the plan mandates and from the rules of what products are utilized and and most importantly from the contractual reimbursements which are amazingly low to put it nicely so when an office is able to migrate towards an open access model they begin to see the the benefits of really controlling the lens choices and the prices from the labs where where they're charging the usual on customary pricing and still having the ability for the patient to utilize their vision contribution plans you know by utilizing their out of network benefits but with anagram you know it's much easier for the patient to be able to utilize that and still have the ability for the patient to utilize their their vision contribution plans you know at an adjusted rate when you have you know a service like like anagram that can help you pull those out of network numbers you can explain that to the patient and they're still able to use their benefit this is where things need to be done properly as communications with your patients is absolutely critical and in order to be brief I guess I'll answer with some key items for success and maybe the main things to avoid so dropping vision plans cold turkey do not recommend there is a process for for real success and I think that that should be done in a in a smart way not just dropping everything so you must teach the appropriate verbiage to the team you need to rehearse all of the objections and the questions that the the patient might have with your team so that they are well prepared with the proper responses and you need to practice practice practice invest in your team's learning invest in numerous team meetings with a lot of brain fruit and a high morale feed them and make them happy and they will learn so I also recommend that you incorporate a refined sales training ongoing education and depth of really understanding the execution of how to properly communicate and sell to your patients I mean this is what we teach at Spexy it's vitally important that your team be equipped with the tools to help your optical succeed in this process and an independent optical all around another thing to avoid do not tell patients you are going out of network patients think that out of network means that they'll no longer be able to come and see you so don't do that start implementing your open access alignment with your vision contribution plans that you're already out of network with is a good idea the most successful practices have utilized anagram for this process do not implement any changes without the entire team being on board with the plan you have to give it time for people to really understand what it is that you're doing if you guys make the call as leadership and then you go all right we're going to start on Monday everyone goes oh my gosh and you're not going to have people that have faith in what it is that you're doing the last thing if and when your office does decide to go deeper into your open access status and independence you'll want to start with the vision contribution plans that have the least influence in your optical and work your way up to the plans that have more influence um uh in your optical will you lose patience well uh potentially many many opticals lose them so but when handled properly the loss is minimal and the profitability is higher on the patients that you do keep uh in addition depending on how many vision plans your office is already out of network with um the open access model will allow for you to see an an increase in patients that you know have not been able to necessarily see you before because they weren't um in network with the plans that you were in network with and if you're able to properly market to these patients before making changes um you know with the more saturated plans it'll really help to offset the patients that you will potentially lose and to be honest most offices um see a dip you know after they started the open access thing once they pull their first um in network uh vision plan they often do see a dip but then that comes back up so something to think about well it's hard to just describe one because they're also different but for some offices open access gives the ability to see patients with you know vision contribution plans that they have never been a network with and this opens the door for opticals to reach for our patients while still maintaining their in-network status with the vision contribution plans um so that that's kind of like the the the smaller edge of it but for most offices open access looks like basically a road to adaptation it allows for the optical to create a mindset of being able to service all patients no matter the vision plan chosen by their employer right and um you know with the use of certain tools you know dropping anagram again but with with the use of you know things like anagram the patient is able to still utilize a majority of their benefits as well so in these offices the focus for the patient revolves around the level of service and of care for the patient and less around the narrative the vision contribution plans have limited providers to the focus for the team is on what is best for the patients and when it comes to eyewear rather than you know what is allowed and the focus for management and leadership is more about the patient experience and less about quantity right there there are some opticals that will implement an adjustment in an even lowering retail pricing of eyewear as the optical becomes more and more independent and makes you know much more profitability per job and funny enough when the price adapts you know even a small amount um is when offices really start to see uh increased returns with the amount of patients that that are coming out if you went to open access it's completely up to you that's the beautiful thing about being an open access provider you can choose your level of independence from the vision contribution plan so if you're one to typically fear change one to take things a little slower on the safer track then the open access principles and training allow for you to slowly adapt your mentality in your team to become less dependent on your in network status and this ultimately can build confidence on the idea of open access before making any dramatic adjustments but if your office is led by you know the forward thinkers the risk takers adopting the open access mindset for these practices allows for them to have a pathway um to the desire that they've been pondering for some time right these are the people who don't have to ask themselves if when it comes to change these are the people who ask themselves how not if they can get to you know real independence from vision contribution plans they ask how they can be freed from them and for these leaders implementing open access and their optical is how they're able to get it done


You've got, so if you can imagine, if you're more sensitive to noise and your world being some, there are autistics in Ukraine that we don't even know about, that their world has been turned upside down. And that's not just, that's talking about a minority within a community that's in that Ukrainian community's world has been turned upside down since February 24th. The not only acts of genocide and indiscriminate bombing and indiscriminate strikes with cruise missiles with 2,000 pound warheads to, and people being, you know, alive one minute.


What's going on guys welcome to Facebook Ads Friday week 22. This is a series where every single week I show you two winning drop shipping products. Let's not waste any time and let's jump straight into product number one. Let's take a look at product number one then it has 1.2 million views nearly 3,000 comments and 1,200 reactions. It's a toy for bath time for any new parents or grandparents that you can stick to the side of the bath using those suction cups and it allows you to put together different kind of slide designs and ultimately just make bath time a bit more enjoyable for the child. Any products that fit this niche of the kind of grandparents slash new parent niche in my mind toys like this that kind of fit that grandparent niche or that new parent niche are always the best products to choose when it comes to advertising on Facebook. It's no secret the average use age has started to creep up and up and up because younger demographics tend to migrate towards Instagram or TikTok and certainly in my experience even back when I first started in 2016 since then the strongest bias has always been have always been females over the age of 50. So how do I know it's the winning products? We've already covered the engagements 1,200, 3,000 comments and 1.2 million views. It's a relatively new ad created on Saturday 25th of February so it's only been going two maybe three weeks and it's already picked up quite a bit of traction so we're just going to go through the comments now from most relevant and ultimately what we're looking for is evidence of interest and even better evidence that people have actually bought this product. So Ellie's posted saying can you please ask me and daddy to buy this. Craig's bought great toy lots of fun we bought one for our 18 month to play with it sticks to the bath tiles fine and also sticks to the windows good for playing outside of the bath so on and so forth so there's one person who bought that that's three days ago. Craig Taylor from three days ago has says he's just ordered one. Lisa from four days ago has said that she's just ordered one. This person here posted a couple of weeks ago an image of their setup which is perfect this is marked in gold things like this comments like this from people that have said they've bought the product from you is brilliant for anybody who then comes across the ad because they're much more willing to trust you as a business if they've seen that other people are also trusting you on Facebook as well. So stuff like this is brilliant if you can get family and friends to leave comments and better yet image reviews like this on your posts it's going to help your ad results significantly. Natalie from one week ago says that she's just ordered one. This person here from four days ago have just ordered him it so lots and lots of people buying this product. I'm just going to double check the newest comments as well just to make sure it is still active.  So that just goes to show that this business is spending a significant amount of money on this ad and therefore it must be working. This is a winning product. Next up let's take a look at the business that's selling this product. So this is jackies-kids.uk so we can see they're selling the products for an extremely cheap price of £19.90 however it does say here that the offer ends on Wednesday which is today. Is that legit? I don't know I would have to check I'd have to come back tomorrow and see if they're still selling it at that same price but more than likely it's probably some sort of like scarcity offer trying to encourage people to pressure them into buying the product the day they see it. It's fairly kind of typical and standard kind of like textbook when it comes to drop shipping product description layouts so typically what you'll have is like a heading followed by a paragraph followed by a accompanying image like they've done here so heading paragraph gif heading paragraph image and then they finish off with the actual kind of finer details or the actual features bullet pointed the key points that consumer wants to know following that they go into looks review so again a very kind of typical and textbook product description layout and there's a reason why it's textbook and that's because it works. So I've gone ahead and found the product on AliExpress as well so we can see exactly what to expect in terms of profit margins. This is ASWJ toy store 94% positive and in business since 2018 so they're a pretty decent store it's quite rare you'll find suppliers that have been in business that long and AliExpress unless they do a particularly good job. The reason I like these guys is because they're competitive in terms of price and they also offer lots of different variants and options for this product which means when you get somebody onto your store then you can upsell into these bigger packages so you can get 48 pieces or 34 pieces get 66, 12, 24, so on and so forth. If you go middle of the road then you're looking at £10.30 delivered that does include the on-time guarantee as well within 15 days. If we head back to their site we can see they're selling it for £20 so it only leaves £10 room in there that leads me to think that they're probably working with some form of agent because they have that traction and therefore they're sourcing it a bit cheaper. They could even be running it very close to kind of break even there's kind of like a loss leader just kind of build up their email in lists knowing that they'll be able to kind of recoup those costs by upselling or cross selling further products down the line. Either way though a brilliant product that people want to buy and we're seeing the evidence that people are buying so definitely a product for consideration if you're still in the market looking for something to kind of invest your time and money into and build a store around. Moving on to product number two is this next stretcher so given the days and times that we're living in when more and more people are spending more and more time on mobile phones, on tablets, on computers, sacked desks, hunched on the sofa, I think it's fair to say that more and more people are getting hunched shoulders, more and more people are getting bad posture so there's an increase in demand for products like this and this is reflected in the engagements and the results that this business have been having. So if we jump into the engagements and have a look at when this was created, Wednesday of February 15th 2023, so what's that about a month? Since then they've had 1.2000 comments, 8000 reactions and over 3 million views so it goes to out saying it must be making the money because the post doesn't get this level of traction and eyeballs unless somebody's spending money to boost it and put it, push it out there that far. So we'll start with the most relevant comments, I have one of these and I love it, Doran I have one of these and it's absolutely amazing, Diane four days ago I love this, Beth what a difference this made in my neck, I love mine two weeks ago, Brenda I have one and it works great, three weeks ago, Cathy four days ago I bought one, Lisa two weeks ago love mine, Janice I love mine and Me Too I ordered hopefully will help with my problem one week ago, Cathy I ordered one weeks ago still haven't received it so just make sure if you do sell this job you have a more reliable supplier but it just goes to show people are definitely definitely buying this product. Jumping on to the Shopify store selling this then they're called gpmsign.com and if you want to check them out this looks like an actual Chinese supplier to me by the way that is kind of laid out and how kind of amateurish and basic it looks so it goes without saying that you've got yourself a winning product here however if you're going to make it a success then I would brand it one product store and just make it look so much more professional and then this actual Shopify store here. So at the top they have some scarcity to try and encourage people into making the sale, they have the Mother's Day sale obviously with Mother's Day coming up it's current it shows that active and relevant for the times the different variants and it does snap between the different imagery as well and they also have a buy two or buy three get two free so this is probably how they're making the money rather than somebody who spends $17 on one for $34 most people are probably going to buy two of these things to gift to a friend or family member and if we jump on to Aliexpress to see how much these things cost we can see that this business here with free shipping is selling it for 86 pence so unless this supplier so it's a top brand it's called LingPen official store if you want to check them out and buy from these guys if you're going to pursue this product I definitely would try and kind of milk it as long as possible 86 pence so if you can get somebody to buy two of these things for $34 it's going to cost you $2 delivered there's over $30 of profit margin in there even kind of worst case scenario if somebody buys one at $17 I'd probably have that as like a minimum of two that's what I would do in fact I would sell one for $20 or two for 35 and more often than not if I had to guess based on my own experience people are going to be more buying more than one of these things they probably have one in the downstairs and living room and probably one for upstairs in the bedroom so the profit margins aren't bad it's obviously products which is definitely in demand given the level of engagements and positive and encouraging comments that it's got in the space of a month so again if you're looking for a product to build a business around and I definitely think this is a good contender however I do want to be honest with this this is one of those products that has been done before but in a slightly different way so I've a slightly different design of neck pillow so it is quite a competitive market I would approach this with caution and I would only pursue it if this is something you have a passion in and you're willing to commit for the next six to 12 months to build up a legitimate brand behind and so with that being said guys I'm going to wrap up Facebook ads Friday week 22 I hope you've enjoyed this week I hope you've enjoyed the products you've seen if you want any help getting these launched just leave a comment down below I read every single one so we'll get back to you and I do have a free training as well which will help you launch you'll take you through the steps to launch your product in 21 days if you want to check that out just check out the links in the video description below don't forget to comment like and subscribe if you want to tune in next week for another two winners I'll see you then cheers