OK, oh yeah, it sounds much better. Sound OK. Ça va vous m'entendez? Je ne peux pas trop le mettre plus bas là. Je vais le mettre sur le... OK, cool. I'll wait for the mic. All right, so let's continue with the... Here is Kami Mal, welcome Kami. Thank you. He's a CNLS researcher in the University of Bordeaux. He's a specialist of free probability. So we go more from the let's say to the actual mathematics side. And welcome and we're looking forward to know about free probability. Thank you very much and thank you for the invitation. So I will present to you a subject which is at the intersection of free probability and random matrix theory. Free probability can be developed a part of random matrix theory and I will introduce you the general context to put the setting. So the kind of problem you should... where you should consider free probability to solve a random matrix problem is the following. Imagine you are given a matrix that is written as a polynomial in several matrices. A matrix A1N, ANN. For me, big N is the size of the matrices. They are all of size N by N. So this size is going to tend to infinity. Okay. So you assume that you have a matrix of this form where P is a fixed function. And let's say a non commutative is a fixed non commutative polynomial of rational function if you want to generalize this. So what is a non commutative polynomial? You just make a product of matrices and you know that the order in the product matters. This is why we say non commutative and you do linear combination of this. Okay. And the kind of assumption we want to make on the AA is not really specifying a model but having a generic position. And addressing the question of what does that mean being generic position is quite rich and there are different levels that we are going to consider. But what is important for us is that the matrices A, L, N these matrices, we assume they are independent. This is really what we want to do. Independent random matrices. So this is several, sometimes called the several matrices problem because several matrices are involved. What we want to understand is a spectrum of matrix matrix. When the size goes to infinity. So maybe you are used. This is common for you, but if not let me remind you the notion of empirical Eigen value distribution of a matrix HN. So this is a probability measure. I will denote it L index HN. And so it is the sum of direct mass with weight 1 over N on the Eigen values. So delta is the direct mass and lambda I of HN is the Nth Eigen value of HN. In all the talk, I will assume that the matrices, I want to compute the spectrum Hermitian matrices because otherwise it's a different problem, much more complicated. So we just consider this. Moreover, this is a random probability measure if HN is a random matrix. I would rather consider the expectation of this guy. And if you're interested in the random guy, you just build on this construction and try to find concentration. And just what is the meaning of that? If you evaluate a function, it's just the expectation of one other end, the sum of F applied on the Eigen values. OK. Here it is. So what will be the strategy? The strategy is given by the theory of free probability by Voie-Coulet school. We start constructing this setting, the abstract setting in the 80s and apply it 10 years later in random matrices. Just to say that the theory exists before. And the idea is to see to consider a notion of limit for the matrices when the size goes to infinity. And to consider a random matrix, not like a collection of a huge number of random variables, but just as a single random variable of a different nature than what you do classically in probability. A random variable which is non-commutative. OK. This is just the construction of the spirit. But it comes with a notion of non-commutative probability. The concept of non-commutative probability is an analog of independence in a very deep sense. So I don't have a lot of space in this tiny black box. So let me just organize this like this. So everyone is, I suppose, familiar with the notion of classical probability where we have the concept of law of random variable which is important. Voie-Coulet school created the concept that you know here have analogs in this place. And analogs not in an obvious way which have their own mathematics. And sometimes there are some things that you, a problem where it is difficult to compute on that side but much easier in this side. This is what we discover with this Eigenvalue problem. But what is important to note here is that in this context you know you have classical independence the expectation of the product is the product of expectation. Here we have a notion that we called free-ness or free-independence. The definition is much more complicated that the expectation of the product is the product of expectation. But there is a definition. This notion is as strong as this one is found in probability. There is a notion of Gaussian random variable as you know, or Gaussian processes. Let me say random variable. Here there is an analog that we will discover during this first lecture today with classical GU or Vignard matrices. In the limit when the side goes to infinity there is an abstract object which replace the Gaussian process and which describes the Eigenvalues of these matrices when we consider such a generic problem. This is what we call a semi-circular system or circular system for non Hermitian matrices. So we will discover this. What can we mention also? Something we Marc mentionned this this morning. You have the convolution of random variable which explains the distribution of the sum of independent random variable. If you have a major mu on a major mu you know what is the convolution of this and you know what is described. In this context you have the free convolution which is a different rule but which plays an a log role. We will denote this operation differently because it's a different operation. It is a boxed convolution an operation between two measures that give you one measure such that if you have two matrices two random matrices which are in the context of this topic, let's say a rotational invariant and you are interested in the distribution of the sum of these two matrices in a generic position and in the limit when the size goes to infinity this is the free convolution that will give you this answer. This is just a notion for us right now, how to compute this and how to compute this when we are not just in this context will be the subject of this lecture also. Okay, we have some notions of subordination property this morning this is what we are going to explore. Okay, but this work this context of freeness under kind of robust on a kind of strong assumptions this context is no longer applicable it happens from kind of classical matrices that you can construct easily but if you consider the distribution of the sum it is not given by the free convolution and this is the kind of ensembles that I want to focus on during this week. So this is applicable when for the matrices under consideration the eigenvector vector basis the random matrices are asymptotically uniform. This is heuristic and this is not a precise definition what does that mean being asymptotically uniform I'm not going to precise this but your random matrices if you look at the eigenvectors they are very they should be to apply this theory very close to what happens if you sample it uniformly independently of the eigenvalue. I will introduce some natural example of matrix model for which this is not true and so we cannot apply this and to solve this question I will talk about a third generation of probability theory I call it traffic probability and this guy comes with another notion of independence and this notion of independence is much richer than what we do in these two contexts because the notion of law of distribution is much richer we are not just considering moments we are considering something that we introduce that take much more information on the matrices and if you take much more information of the matrices you are able to describe much more a good variety of phenomena what happen is that this notion of independence actually encode both freedom and classical independence because it is a richer notion and it also encode other notion of non commutative independence that appear in the zero ok and we will see how to compute things like this the interest of what has been discovered since about 20 years in these free probability techniques is that we just we don't have just a conceptual framework we have analytic tools with this the subordination property and all the numerical methods that you may use in random matrix theory have a great theory of analysis in this non commutative setting ok so this is what we call free harmonic analysis free harmonic analysis maybe is a world afraid it's strange but think about the stiges transform the air transform on what you heard about this is just a world to sum up this ok so ten years ago I developed this theory to deal with this kind of weird ensembles and I was asking can we develop the analog of free harmonic analysis and it's not just a question for fun just because the theory that we had 5 or 6 years was just combinatorial in the sense that we had a theory where we were able to compute the spectrum in more cases but computing means having moments of the empirical distribution and if you know the moments of the distribution you don't know a lot of its qualitative properties you don't know what is the behavior of if there is a density or something like this and with the team that joined me on this question we address this problem of finding analytic tools and what we discover and I will introduce this in the second lecture is that if we take this theory it has something that I can mention is the analog of conditional independence ok if you have a function of x and z a function of y and z x z x y and z are independent these two functions are not independent because of the z but they are independent over z conditionally on this variable ok this has an analog invented 40 years ago by Vocule school which has a different name it's called amalgamation over an algebra amalgamation is just because it's coming from group theory where it has a sense ok just say it's the analog of conditional independence it exists in this area and what we discover is that traffic independence implies a certain notion of conditional independence in this non commutative way and the result of that is that we can use the analytic tools the free harmonic analysis in this setting and we get algorithms that allow to compute eigenvalue distribution in this situation ok so this is an overview this is maybe a lot of information at the moment so we will discover this in detail today I will talk about some aspects which are classical in the series of free probability we will see how to compute what is called the non commutative distribution of several normaticies for this it will be the opportunity actually to introduce the traffic tools in the background but I won't put in the front series the second lecture I will focus a bit more on the problem that we solve with these tools and go to this notion of freeness over the diagonal which is this analog of conditional independence we use to solve this question ok these two topics will be very combinatorial I will be dealing with moments under using these combinatorial techniques of traffic probability in the last lecture I will introduce more analytical problem how to deal with this still just transform for these ensembles and I will focus on these outliers problems when we have finite trunk perturbation so if the program is clear and everyone is happy to start let's go and if you have any question you have the opportunity to ask it so what is a non commutative polynomial is the question that typically you take a b plus b a something like this you can put a complex coefficient ok so it's a linear combination of words in the matrices I mean the word a b is not the word b a the other matter let's start with some basic notions in M specifically what I want to do right now is to give you a short zoology of random matrices zoology M is a short cut for random matrices for us today ok so what kind of random matrices do you know you know these covariance matrices with i, i, d and 3 we are about this so I won't go back but I want to define the definition of a vigno matrix because we will compare with different so if you miss this definition it will be a problem a vigno matrix what is it I will call it x so it's a real complex matrix we assume that the entries of the form x a j over square root of n implicitly we have a noise which is quite small we divide it by a very small amplitude ok so we have a small fluctuation ok we assume that matrix is a mission it is written like this we assume that the upper diagonal entries are i, i, d independent of the diagonal entries which are i, i, d as well of course if your matrix is complex you have different distribution in general and we assume that let's say the entries are centred it is not very important but let's say that and we have a finite variance and I will assume for us for using these techniques that we have finite moments of any order all orders but the really important assumption is that this is finite for k equals 2 otherwise with truncation argument you will manage to deal with that ok so you may know the Wigner theorem and we will give a proof of that which tells us that the empirical eigenvalue distribution of a Wigner matrix xn converges to what is called the semicircular distribution but if you have a complex matrix you have complex entries there and real entries there and you don't really care about the detail of the diagonal entries actually what is important is an easy variance of the diagonal entries for this theorem you have universality this result does not depend on the exact distribution we put on the entries ok, the semicircular distribution if the variance is 2 as a support minus 2, 2 your eigenvalue will be most of the eigenvalue will be around this interval and if you draw a finite by large size matrix on histogram it will be very close to the semicircular ok so the second matrix that I want to consider looks very similar to this one but the behavior is different we call it a Bernoulli matrix we consider a real symmetric matrix where the xn aj iid but we don't assume that they are normalized like this with a random variable whose distribution is independent on n and such that we assume that these guys are Bernoulli random variables of very small parameter let's fix an integer p and take a Bernoulli random variable parameter p over n which means that with probability p over n, which is very small this guy is 1 otherwise it is 0 ok so this is a rousse matrix if you want you center the matrix by removing all entries on normalize the variance but it is good like this this guy so let maybe I should use another symbol like y the empirical and value distribution of this guy has a limit which is very different from the semicircular distribution it converge to a guy that I will call L of y just a symbol I prefer to the limit of matrices but no free probability will give a meaning for this way which depends on this parameter p ok and this measure is very singular it's not we don't have an expression there is no density in general and knowing the properties of this distribution is complicated having an expression for moments is quite easy but knowing really what this distribution is complicated and in particular if you take a generic random matrix or a deterministic matrix and you consider this matrix plus such a matrix you won't have the same phenomenon as what you do with a vignard matrix a deterministic plus a vignard matrix in general under the good assumption we'll have an eigenvalue distribution which converge to the free convolution of the two eigenvalue distributions but for this kind of matrices it's true but we'll see that freeness gives a good solution not only the solution but also the algorithm on the numerical methods to compute concretely the spectrum yes why the universality results like of Tao and Wu does not apply to Bernoulli matrices because it's not a vignard matrix in this situation you have a field of smaller random variable where you have some entries most of entries are 0 and some entries are 1 you know it's actually there is a model which is similar to this one which are vignard like matrices with heavy tailed entries that is you remove the assumption that the second moment is finite you have a Couchi distribution you must normalise such a matrix in a different way and then you will see the same phenomenon there is no universality you will depend on the heavy tailed distribution and if you look at the sum of such a matrix with a deterministic matrix it will be the convolution over the diagonal that gives a solution there is a question somewhere yeah the question is if we replace the Bernoulli bar Hanmarreur is a symmetric Bernoulli but it's just if I'm not wrong it's just shifting the matrix yes you can do that what is important is to have a matrix which is not uniform in the noise you have some entries which are very big compared to the others and here big is just being 1 compared to 0 but for heavy tailed entries that is clear that you have some entries which are huge but then Rademacher I think it would be like a vignard matrix at that point if you divide your Hanmarreur by square root of n Rademacher would have all the entries ah c'est symmetric plus or minus 1 and then you divide by square root of n ok you will probably it depends on kind of if you think of the symmetric Hanmarreur you will need to normalise it like this but you can cook something which is a bit weird but you have to have something which is really not symmetric so it's maybe not what Bernoulli Rademacher where you have many zeroes the entries are minus 1 you must cook something which is very unbalanced ok ok we will talk about Hanmarreur until wise product but it will be another subject absolutely and more generally so there is other kind of matrices which are heavy tailed other kinds of graph we start adjacency matrices of sparse graphs ok if you like this kind of graphs and look at the eigenvalue of polynomials of the adjacency matrices or something else you are in this kind of bad ensembles or weird ensembles for which we will need freeness over the diagonal or not just the plane notion by your equation ok ok so this is just to give so the sample of matrices so matrices of graph was in my list I want to conceptualize something a bit more general but about these two notions as you know there is as you maybe know there is one vigno matrix which is very specific which is a GU matrix which is very important in the theory and it is important because it is a unitary invariant matrix this morning we said rotationalia invariant this is the same notion but here in the complex case we have when we have a matrix A n it is unitary invariant if it is equal in distribution to the matrix U A n U star and this for any U which is a unitary matrix again a matrix which sees property will be a nice matrix that will consider with a theory of Euclides and the matrix that we consider with the theory they have a weaker assumption which is the following that we call permutation invariant matrices permutation I mean invariant by conjugation by any permutation matrices that's a matrix that is invariant a load definition is quite simple when we have this one but this is assumed for all permutation matrix ok so if you just know this assumption you don't know a priori if very cool school theory will work it may happen but not necessarily but you can be confident that a freeness over the diagonal and traffic independence should say something interesting about this ensemble ok and if you take, you consider the matrices which were on this blackboard the Ardoche phonograph, the Bernoulli matrix they satisfy this assumption ok this assumption is not really satisfied for Wigner matrices but we don't really need matrices to strictly be unitary invariant we just need that assumption to please the eigenvectors are more or less uniform and actually this is what happened for Wigner matrices they are not exactly unitary invariant but we don't care it's most of the same ok and just to mention that this assumption is much more cold than this in the sense that if you consider t vectors there is much more chance that it is invariant when you exchange the vectors than invariant when you apply a nortagonal matrix that will mix all the entries so for the point of view of modellisation it is clear that it will be much easier to justify this kind of assumption than this one of course if this one works you can start safely assuming this and it's more difficult to be in this situation with this unitary invariant yes sure of course so what is the permutation matrix it's a matrix so you have sigma permutation of one n so for me one n I will denote it with a bracket like this so it's just a b-jection of this assumption ok the permutation v is the matrix associated to sigma if vaj is indicator then I think that the definition will be this one but you choose a convention I think that this is the one I use ok so it's just a way to encode the permutation in the matrix this will result in the matrix where in each row on colon you have one entries which is 0 for instance this block tells us that you exchange one and two no it says that these two guys are fixed point and these guys say you exchange three and four ok so just a way to encode the permutation to have a more basic way to see that if you are given a matrix an what is this matrix it's a matrix which describes the same operator as an but you just exchange the order of the vector of the basis when you express the operator ok this is a different way to see that other question ok let's go we have 10 minutes to give a little bit of free probability then in the second part of the lecture we will consider the distribution of ignore matrix ok, so now that we have some characters to tell a story let's consider a good framework so what is an encumutative ? random variable donc je vais essayer d'évoiler une lecture très abstracte, mais je vais vous donner quelques ingrédients que nous utilisons en practice. Le formalisme est assez simple comparé à la probabilité classique. Une probabilité non commutative est le data of a couple, a phi, a is a space and phi is the expectation. The space is an algebra, just the field where you can do products. And we assume that it is a non commutative algebra, typically an algebra generated by random matrices, matrix algebra. And phi is a linear map, a linear formative, it is a linear map from a to the field of complex numbers. It tells you what is the expectation of a, if a is a non commutative or non variable. In classical probability you define sigma field, you define some very difficult actions, here it is much more accurate. There are a little bit of assumptions that we can assume that I will not want to tell so much, but we also assume that these guys are endowed with an application star, which is from a to a, which is a non-tilinear involution, such as the complex conjugate of matrices, anti-linear involution, such that when you take the star of a product, you exchange the order of the product as for the complex transpose of matrices. Okay? Why we want a star? Because we want a notion of positivity. I will not explain all the details about. But let's consider examples of these kinds of ensembles of spaces which are related to random matrices. So first before random matrices are random variables. Example one, you have omega, Fp, a classical probability space. How to see that the classical probability space is a non commutative probability space. That way, we consider as an algebra, the algebra of boondid, non commutative, non boondid commutative random variable, define omega with value and complex number. And this symbol just means that the random variable, I assume they have moments of all orders. All right? And for five, I will consider the expectation. It's a linear form and it's always well-defined on this algebra. And the star is just the complex conjugate. Okay, just to say that this setting of non commutative probability is an extension of the notion of probability space of complex random variable. There is of course a restriction with this. We want to develop all the probability setting. So the second example is the example of matrices or random matrices. Let's take for A, an algebra generated by some matrices, by certain matrices. You can take the full matrix algebra, but you can restrict to matrices you are given and you want to study. Okay? And five, I will put n as for the size of matrices. And n by n matrices. And this expectation that we will consider in this non commutative setting is one of our answer traits. Okay? Why it is an expectation? We'll see just in a moment. But it plays a role of an expectation. And if you're considering random matrices or certain n by n random matrices, you can combine these two settings and just you will put an expectation here. And this will be our setting. I did not mention, but this guy is complex conjugate, complex transpose. I don't know how you call it. Right? Okay, we have defined the objects, but as in classical probability, this is not what matters. What matters is the notion of distribution. In probability, this is really the notion we should focus on how you realize random variable is not really important. So let's talk about the notion of non commutative distribution. Okay, definition. Given a family of non commutative random variable, so I will denote this family with this bold symbol, just mean that it is a collection of guys and x j. Okay? So we consider this guy in our setting. So this is a collection of an element of a. This is what we call non commutative random variables. And we call the distribution, we call it the star distribution to distinguish from the usual notion of distribution. And because it involves a star, that's just a trick. The star distribution of a is, so it is a linear map, which gives you all the complex number, you can compute from a, given this setting. So if you are given a family of variable like this, because you are in algebra, what you can do is making products, some, and multiplying by complex numbers. That is considering a non commutative polynomial in the variable. In the variable and in the adjoint, because we have this star. Okay? So this gives you variables. And then you can apply phi. And it gives you complex numbers. And we can sum up like this. This map, we will note it phi and x a, because it's given by phi. It takes p. It is a non commutative polynomial. I should, okay, I should introduce some symbols. The set of non commutative polynomials in some family x will be denoted like this with this bracket. And here it's x and x star because we will have a family and the variable with the star. Just x is just a family of undeterminates. They are symbols, which are indexed by the same ensemble. So this is just a word in symbols, where there is an obvious way to replace this symbol by the variable under consideration. Okay? And you associate phi of this polynomial. So maybe I have chosen a way very obscure to define it. But an example of this is just taking phi of a1, a2 plus a2 star, a1 to the third, which is a third power, a2. Okay? And if you evaluate your linear form in a non commutative polynomial, you get a complex number. And this is what we call a non commutative moment or star moment called, sure, just a notation for the space of non commutative polynomials in some variables. So here I just denoted the family of variables xj in xj's and xj's star. We should have a void to use this notation. It is quite confusing. We will not use it anymore later. Okay? It's just the notion of non commutative distribution is the data of non commutative moments or star moments. For the moment, it's kind of abstract. But it's evaluating for us. And we start with this, the second part of the lecture, the expectation of the normalized trace of any polynomial in the matrix. So this is the end of the first part of the lecture. We have a 15 minutes break. We illustrate more concretely what is this definition and what we are going to compute with that. Thank you. Okay? So let's resume this. So we finish the session with this kind of abstract definition. And we're going to illustrate in the two examples that we have considered before. So you know what is the notion of distribution in classical probability. The distribution is a map where for every let's say boondi, the measurable function, you associate the expectation of f of x. x is the collection of a variable. And this is the data of these numbers which characterize the distribution. You also know that you have other ways to characterize the distribution, which may be very convenient, such as the characteristic function. If it is allowed on CT or accumulative functions for one variable and so on. Okay? But here, this is one notion. What do we have here? We have something which is almost this, but a bit more restrictive. Here, so just, right? It's a classical notion of distribution. The notion of distribution on where f is a measurable boondi. Okay? Versus, the non-commutative notion, the restriction, it is, it is more algebraic. The non-commutative distribution or star distribution will be the collection where for each polynomial on, in this context, it is not relevant to consider a non-commutative polynomial. The order doesn't matter when you evaluate to complex random variables. So it's just a classical polynomial that you denote maybe in C, x, x bar. Okay? A polynomial in the variable on their conjugate. And you associate the expectation of the polynomial in these variables. What, what's the name for this? This is called the moments. Okay? So, there are the moments, the joint moments, if you have several variables of the variables. Okay? So, does the moment is really the distribution of variable? Not always. If you have a very weird distribution, maybe it has no moment. Maybe it is not characterized by its moment. If you take the log normal distribution, the moment are defined, but there are different distribution with the same moment. But they are very weird situation. If you have a Gaussian variable or a sub-Gaussian variable, a boondi don variable, it is entirely characterized by its moments. So, just to sum up, there is the main part of a classical probability, where if you know moments, it's the same as knowing the distribution. And there are some random variables which are not, where this is not true. And this is maybe a problem if you want to see free probability as an extension of classical probability, but that's a life. We just restrict to this algebraic notion. Okay? But we, we don't really care about this. We care about random matrices. So, what we must know is, what is the notion of non-commutative distribution for matrices? And why do we care? Okay? Given the family of matrices, xn, let's call them xn, xn, xn, those questions? Okay. The non-commutative distribution is the data of the expectation of one over n. The trace of, if you take one matrix, l1, let me remove the dependence on n on the matrices, just a question of clarifying the notation, times a second matrix, times another matrix, and you take either the matrix or it's a joint that I will encode with this index here. And this is so for all l1, so for all big l.


I have some new plastic bags inside which are capacitors. What I'm going to do today is to recap my old Amiga power supply, which is this repulsive piece of yellowed plastic. Because it is old and repulsive and I've already had to repair something inside. Really I should get a new power supply but I can't be bothered. So I am just going to replace all the capacitors in it, which should hopefully make it a little less. Well you'll see when I open it. So this is probably going to be a rather boring workbench video, but I'm recording it anyway just in case something interestingly hilarious happens. This is for an Amiga 600 as I got a while back. It's shown up in some videos. I actually have a full capacitor replacement kit for that, which I'm going to be doing at some point, but there's a lot of surface mount stuff that I'm a bit nervous about, so I'm taking my time preparing there. But this is boring simple through-hole stuff with big thick capacitors and you may notice that the board is somewhat warped and has some heat marks on it. I have discharged this by the way, so it is safe. So what we've got is... I should actually unbag my capacitors. I've been through this and should hopefully have all the right ones. This is a 400 volt 47 microfarad that is replacing this one. You can see that the new one is significantly smaller. These are 4,716 volts and these are replacing these ones. 4,716, just double checking everything. These are from a different manufacturer, 22 microfarads 25 volts. These should be replacing these little ones here, which are very difficult to... 22 microfarad 25 volts, so that's good. And then we've got this one and this one, which is a 4,716 volts and another 4,716 volts, which should be these. And there's actually three in here because I'm using one for something else. That might be the A600 actually. So, let us start removing them. Soldering iron is hot, so this should be relatively straightforward, even though desoldering is one of my least favourite workbench activities. Doing that quite badly. There we go. And the negative side with the black stripes is marked, but the type is not, so let us write that down. Is this pen writing on this thing? Not very well, but that is readable just 385. Okay, so let's try and do this one next. Lovely through-hole stuff, so easy. Negative marks in the right place, 4,716. I have to remove the fluff though. Interestingly, this is also a capacitor, 6.8 nanofarad ceramic job. As far as I can tell, that does not need to be replaced. Okay, so now these two grounding plane means that you can actually get the thing hot, which is nice. Is that way round? Yep. That is a 22. And I'm always checking to make sure the capacitor is actually the same way round that the board says it should be, because otherwise, well, sometimes they print the board markings incorrectly. And sometimes they put the capacitors on backwards. But as this is a power supply, I would have expected it to explode if that was going to happen that way round. This is also a 22. You can see that this board has actually seen rather better days and all the corners have broken off. I really don't know what's happened to this, why it's in such a bad state. Maybe it's seen a lot of heavy use. Okay, come on, someone doesn't want to move. Okay, and that one, negative down 4,700. I've already replaced one resistor more or less on spec. And when it was behaving oddly and that seemed to have made it better. But it's not in particularly great shape. I don't normally touch mains voltage stuff due to knowing my limits, but a simple recap should be okay. It's also 4,700. And this one, heating up one leg at a time and slowly rocking it out. That way around, yes, and this is a 470. Last one, two more to go. There's also a 4,700 and the last one. This might be a rather quick job, which would be nice. This is almost certainly soldered with lead-free solder, which is a higher melting point than leaded solder. Okay, and that is also a 4,700. All right, so that would be all the caps. So let us now apply some fresh solder to the solder joints and then clean them. This will make it much easier to work with. These joints are one of the other capacitors. This is the mains input side. You can see from the big, thick tracks that this is the high voltage side of things. This board actually has decent separation between the high voltage and the low voltage side. Was that one there? No, that's actually a missing component. These two pads here, you see there's a little square there, but nothing in it. Resistor maybe? This is high voltage. This is low voltage with much thinner tracks. This is, what is this? Yeah, this is the electronics behind the switch mode power supply. There's a chip here that spans the high and low sides. So what this is doing is it's measuring the voltage of the low side in order to control the transistor output from the high side. There should be, if I understand things correctly, a capacitor that spans high and low. But I don't see one. Maybe it doesn't work like that. Anyway, this is the last. Okay. So now we get the solder sucker out and we take all that solder off again, hopefully freeing up the holes. This will allow us to get the new capacitors in. I haven't cleaned these ones. It seems a waste to put fresh solder on just to take it off again, but it does make a big difference. See, this is leaving nice clean solder holes. And the new solder is noticeably shinier. So a single pin there. Ah, there's one side of, seems to be an inductor. Inductors have always been a bit of a black art to me. I don't really know what they do. I just don't know why they do it. Or why one would want them to do it. Good evidence this thing is easy to work on. It's not having a ground plane that makes a difference. This is a sheet of copper down the middle of PCBs that's used to, well, for grounding and to prevent RF interference.  And this one doesn't have one, which means that I can heat up a pad very quickly. Okay, well, that seems to be all the joints clean. So let's start placing some. Let me see. This is a 70. That goes here. This is another 470 that goes here. I've bend the legs slightly so they don't fall out. These are the 4700s. Okay, the leg spacing is slightly different, but it will work. We also have to be careful not to accidentally place one backwards. So I'm double checking as I go. This one, this one. Finally, this big one. Oh, hang on. There's the two little ones I haven't done. I'm just thinking these two little ones. Just thinking the best order in which to do them. Normally small first. That is in 2225. So here that goes here. Yeah, let's do those, I think. They are sufficiently solid. They don't need support. So I can just solder those straight on. The others will all need a bit of support. Those are adequate joints. Okay, let's take that out because it's huge. Okay, this is not staying in firmly. So I'm going to have to push it with one finger. That means that it's not going... As I need three hands, one for the solder, one for the soldering iron and one to push the component. We'll just tack this on for now. But the capacitor is in the wrong place. Now I can push from this side and remelt this. Like so. Now the solder joint will hold this component in place while I solder this side. Perfect. Okay, so let's do the other 4700. This one, which will need to do the same way. So the end is shaking. I'll get it into place. The other leg, like so. So now we can do these ones. Now the legs are not quite spaced as well.  But the plus side is that this will actually wedge them into place. It should make it easier to solder. That is, wedge them into place while unsoldered. Like so. That one's a bit looser. Yeah, one of the legs has bent so that it's now sort of straighter. So let's just do the tack thing. Okay. Just position like so. And do the other leg. Let's just re-solder this one, I think. Okay. Only two left to do, both of which will need tacking. And do the legs. Oh, yes, and this one, the big one. So that goes in this way around. This will also need tacking. I thought there was enough solder left on the leg, left on the pad not to need more. Do it like that. Adjust the position. Do this leg. Do that leg again. And we should be finished. Okay, so now I remove the stray bits of wire. Actually, before I remove the stray bits of wire, just want a last double check to make sure they're all in the right way round, which it looks like it. Looks reasonable. And also when removing these bits of wire, you do want to make sure that you get them all rather than leaving stray bits of wire sitting on the board. Particularly with the main stuff, that would be bad. And these bits of wire all end up in a little pot, which are used for useful things later. So I think that is now done. So let's reassemble it. This is the... Hang on, this is the main side. Wait a minute. The small tracks are the main side. The big tracks are the computer side, the low-voltage side. That's an interesting way round of doing it. Maybe they're using the big tracks because the low-voltage side has higher current. That would make sense. But that is not how Switch Mode supplies I've seen before work. Well, these power supplies do have a certain reputation for not being particularly good. So maybe there's a reason why it's like that. Okay. Actually, before doing anything else, is my tester still working? Yes, it is. Let's just... The wires are long enough. Let's just see if I can measure some of these. Okay. That's telling me it's a... 47-microfaric capacitor. Looks okay. 5188 microfarad. Well, that's kind of out of spec. It's supposed to be 4700. These are taking a while to test as well. 49, that's a little better. And by the way, you're going to throw these away. I'm not going to try and keep them for anything else. That one's better. Some of these numbers, I don't really know how to interpret such as V loss and the resistance figure. I believe that the resistance, the ESR, gets too high that indicates the capacitor's not working. I don't know about V loss. Yeah, it looks like these aren't terrible, but they are rather elderly, so new ones will be definitely worth it. That's a high ESR. And the value of 25, it's supposed to be a 22, so that one I think is suspect. They're both the same. Okay, well, I now have figures on record, so let's get the Amiga out and then we will smoke test this. Okay, I have the Amiga out. I'm using this phone to record the video and by the magic of compositing, that will appear. Yeah, smoke test time. Power is not plugged into the Amiga. Let's just turn it on and see what happens. Well, no funny noises, no disturbing smells, none of my fuses have blown. That sounds good. Let's just take a quick look at the voltages. The meter is somewhere where it can be seen. Now, I believe that this power supply needs to be pinged by the computer before it turns on, but let's just see what we get. DC, not a lot, not a lot, not much, very little, and nothing. Okay, so, nothing seems terrible or seems to have happened, so let's turn that off again, plug it into the computer and see what happens. Okay, moment of truth time. Go. Got power lights. I believe it's coming to life. It's trying to boot off floppy. There is no floppy in it. Okay, that looks good. I need to reinstall this. I'll probably make a video about that sometime. I have not plugged the mouse in, which was intelligent. Can I move the mouse around? Using key combinations. No, I can't. Okay, well, this seems very plausible. The screen image looks decently solid. The drive is working. Let's just stick a random disk in and see what happens. Okay, and reboot. This one now tried to boot off the floppy. I actually forget the precise status of this machine, which workbench is in it, so which kickstart is in it. That's not so great. Okay, well, as power supplies go, I think that is working. The floppy drive when it was operating seemed decently positive, but I think that is a success. Actually getting the thing up and running again is going to be a different project. So here is the brick itself. It smells warm, nothing more than that. Yeah, I think that's a success. Excellent. It's nice to have a project that actually hasn't turned out to be more complicated and or more failing than I was actually expecting. Also, I now notice that I actually forgot to press the record button on the phone, so the video you're seeing is not captured on the phone. It is in fact just zoomed in from the phone screen, which is why the quality will look terrible. I mean, I could just reshoot this section again, but then it would be fake. Anyway, I hope you enjoyed this video. Please let me know what you think in the comments.


Welcome into the show. Essentially look at that yield story lowering the odds both in terms of yields lower but also the bets on twice more by the Fed this year seem pretty distant now, do they not? Good afternoon Cass and thanks for having me. That's right, we are seeing the expectation of this June hike pretty much fully priced in it's at 90% but the real interesting thing here would be to see if there are another one or two hikes after that and that's really what we'll be listening to on the 14th to find out where we go from here. Goldman Sachs on Friday actually priced they had an expectation for a September hike and they've actually now pushed that back and priced that in for December so it seems to be that this expectation is waning a bit and it seems to be it's going to be either pushed out to next year if at all. Yeah the the call it from BK asset management is that they won't provide strong guidance on future tightening following the June decision are you hearing differently? Yeah I actually do agree with that and I think the reason for that is that they'll just wait and see what what happens from here they've very much talked about this June this June rate hike and prior to that the data had actually been quite strong it's only really been the last few months or so that we have had some negative prints come through and that seems to be an ongoing trend so I think they will pull back a bit and just wait to see what the those further prints are data prints are to then you know come to market and make a statement about the future of it. The dollar of the looks decidedly unloved because let's face it in order to keep the optimism in play you really have to have progress on tax cuts and fiscal spending neither of which seem likely the side of Christmas. Yeah that's right I think Trump certainly did talk up his budget with the tax cuts and the infrastructure spend but he's been a bit distracted with some other you know political uncertainty and the likes of what is going on over with the Comey meeting Congress and the likes of that and so the expectation actually is that all of those the core infrastructure spend and all that will be pushed out until 2018 and so that's really been priced out of the market now and we're seeing that come through and in yields. Yeah we're looking at the 10 year UST and then a comparison with the Australian equivalent the 18 bit differential how much more of a tightening story might we expect even post-RBA this week. Yeah I think there's another another good you know up to five basis points to go in that in in this week and then whatever we'll see next week come through with the rate hike expected out of the Fed on on the 14th so I think with the RBA if anything there's a 20% chance priced in for a rate cut from the RBA later this year and then this rate hike from the out of the Fed next week I think we will see that compress even tighter again over the next coming week. Yeah I was going to say is that if anything a misprice the 20% given that we could well get a negative GDP print come Wednesday so might we perhaps be bracing for more like a 45% move on perhaps further easing. Yeah Carlson I think you're actually really right there this prior to let's say a couple of weeks ago that was actually priced in at 10% it's only recently gone to 20% but I do think there is I think that will increase the chances of that I think with the the fall in house prices that we've seen a first couple of prints come through May was the first month that that had decreased in 18 months and I think now that pricing the house pricing seems to be cooling in Sydney in particular it was down 1.3% I think it now does give a bit of leeway for the RBA to cut where previously we hadn't seen that. Technically do they judge it though to be transit tree and therefore they might just well sit on their hands till Q2 say let's get validation of this and not jump the gun you know in other words the old argument of being ahead of the curve or being passive but ready to be there for the cleanup. Yeah I think you're right that the RBA certainly does like to wait and see that generally has been their stance in a lot of situations and they do like to see a lot of hard data come through before making any decisions I think they're one central bank that certainly doesn't want to have to make a decision and then go back on it like some others have a New Zealand in particular so I think they definitely will wait to see some more prints as you mentioned GDP that's due out which is expected to be quite weak so I think they will wait for to get a fuller picture before they really do make any big decisions. Jessica thank you as always have a good week. Lovely thanks Carson. All the best Jessica Russet live at FIG for


And to the show, it's me, JP, it's time for another weekly edition of JP's product pick of the week. See, it's right there in the name. Welcome and my apologies if you got tricked by my typo in the blog post, which was linking to last week's product. So I've updated that in case you want to refresh and make everything good again. You can head to this URL right there. Look, I got it right there. Presumably, so is that QR code. Heading to product ID 5740, not 5766, which was last week's, 5740, 5740, that is this week's product pick of the week. Head on over there, thanks to DJ Devin3 in the chat over in our Discord for letting me know that I had the wrong link there. So thank you. That's our chat, by the way. If you're somewhere like Twitch or Facebook and you're wondering, hey, where is all the chat going on? We have a chat that we are checking out over in YouTube and then we have our Discord server, which is adafru.it, slash discord. Look for the live broadcast chat channel right there, that one right there. That's where this conversation is going on. So head on over there. So before I go any further, however, what I'd like to do is have Lady Aida go back just a little bit in time and tell us about this week's product pick. So please take it away, Lady Aida. Okay, with the ANO Rotary, so that's the name of the Rotary Encoder, it's the part number is called ANO, so that's why it's called ANO. The Rotary Encoder with Seesaw, so this is, you know, we've had this Encoder for a bit. I love it because it's basically iPod Classic Rotary Encoder where you've got the little click wheel in the center, it's a Rotary Encoder, and then up, down, left, right, and select. But the thing that's a little annoying about it is there's five buttons and two encoder pins, and it's a lot of pins, and before you know it, you're like, I've just used half of my microcontroller, GPIO, and, you know, you're constantly pulling the Encoder, and so why not make it a lot easier for people by making a breakout board where on the back is an AT Tiny 816, a little AVR chip, and, you know, the code is up on GitHub, if you'd like to take a look at how it does this, it can keep track of the Encoder, so it just keeps a little counter of how many clicks left or right, it can read the buttons, and it can also have four different address selects, which is, you can directly have 16 of these, all over I squared C, three or five volts, so it's very easy for you to connect this up to your Raspberry Pi, which doesn't even have a encoder support, or other single board computer, you can connect it up to an Arduino, you can connect it up to a Pico, you can connect it up to pretty much any microcontroller, as many as you like, you can even connect it to a computer by going through one of our dev boards or one of our USB to GPIO converters that once took a Python, and on board is a little CSOT chip, and there is an IRQ output, if you'd like, but it also works just great over I squared C, as long as you don't mind asking it every once in a while, hey, are there any buttons pressed or any loader encoder clicks going on. And that's it, that's the product I can, you can show the little demo video, so this is just the hand, and you can see it's plugged into a seven segment display, and then, you know, it just counts up and down as you click the loader encoder. That's your products. Yes, indeed it is. This, look right here, I have one. That beauty right there, that is our product. Pick the week this week, it is the ANO Rotary Encoder breakout board with Stema QT, that's the exciting new part here. So we've had a very, very similar board previously, where you had to solder on pins to use the little pin header there, plug it into a breadboard and wire it up to your project. Now we can string these together with Stema QT cables, just like that, and that'll plug into pretty much any board you can imagine, you can use Stema QT to Stema QT. We also have adapters that go Stema QT to Stema, which is the largest sized one, or Stema QT to just wires that you can plug into things. These go hand in hand with this ANO click encoder wheel. I'll do the dramatic peel in fact. Let's get that, oh yeah, get that off of there. That's nice and shiny.  But these right here have, as Lady has said, an unusual pin layout. You can't really fit that onto any kind of breadboard. So you're stuck having to try to wire this in the first place. So that's one use of the breakout board. However, besides just getting this wired, it's interfacing with it. It has a lot of pins. It uses five different pins just for the up, down, left, right, and center buttons. And then we have a few pins used for the rotary encoder as well. And so what we are able to do with this little AVR chip right here is put seesaw on it so that it can run its own code. So all of the interpretation of those buttons as well as the rotary encoder that's all done in Stema on the chip. And then those messages can be sent out over I square C to your microcontroller board, which makes it really easy to work with. Let's take a look. In fact, if you go to the main page here, so 5740 is the product ID there. You can see we have about 80 of them in stock, at least when I refreshed a moment ago. And these are half off, so you don't need a coupon code or anything. Just throw these in your cart. You'll get them for half off. You will want to also pick up one of these, the ANO directional navigation and scroll wheel rotary encoder. That's a long product name. Looks like we have about 25 of those in stock right now. And we tend to get those pretty regularly. But that is what you're going to want to use with this. If you take a look and scroll down here, you will find some other videos on it as well as links to projects and learn guides. Here is the main learn guide. So it's by Liz Clark. This takes you through setting it up all of the pinouts. If you're going to use the header pins on it, normally, I think with this when you'll want to use I-square-C stem of QT cable. So you can see the pinouts here. This has four jumpers on it for adjusting the address, which means you can do 16 of these, 16 unique I-square-C addresses. Put them all on a single bus so you can just chain them together if you want to make a bandolier of interface. I don't know what you're up to, but that could be pretty cool. You can do that. You can set link, link, link, link them up to one microcontroller or a single board computer or what have you and then be able to pull all of them and ask them to send updates as you change the scroll wheels, as you change the buttons. And if you take a look a little further down, we have code examples in Circuit Python and Arduino, and I'll show you an example of a little demo project I did using Circuit Python. There are also some really neat projects. Actually, both of these are from Liz. Liz loves this rotary encoder. Click wheel rotary encoder. So she did a wireless BLE scroll wheel. Allows you to both scroll webpages by making the mouse scroll wheel. You can USB HID and USB mouse with it. And she also has it playing Doom. And then this is a recent one, Liz, that uses five of these. Again, the wiring was super simple. If you take a look at her, let's see, circuit diagram here. You can see they're just chained together.  If you take a look at the wiring page here, you can see some of the... Oh, that's just the amplifier wiring, sorry. Assembly. Oh, no, demo wiring. There it is. She made it easy. I just ignored it.  And you can do many, many more, thanks to the unique I2C addresses. Here is a little demo I did. Let's, nope, not that. Let's go to... How about this view of the world here? So let me get this out of the way. And I'm going to bring up the video editor I use. This is Adobe Premiere. And what I've got is I've built a little Lego enclosure for this, just kind of temporarily hold it in place. But there you can see I've got my A&O rotary scroll wheel click encoder navigational thingamajig. Gosh, I can't remember its name. The click wheel. I've got the click wheel there. I've got our breakout board. You can see I have a little STEM acute T cable running to a QT pie, and then that's off to USB on this very computer. So it's very dangerous. It's plugged right into the computer I'm broadcasting from. And what I have it doing is sending USB HID key shortcuts, which can be used to work with my editor. So first of all, you can see here, if I click in my timeline here, I can do some scrolling through as I try to find that exact point where I want to make an edit, let's say. I can press the center to play. I can left click multiple times to go fast reverse. I can go fast forward, like so. If I find a spot that I want to place an edit, let's say I want to set the end point here. I'm just pressing up on my nav wheel and scroll over here and press down to get the out point. So this is usually used when I'm preparing media and bringing it into the timelines, taking clips and bringing them into my sequence. But that's a really simple way to set this up, just using Circuit Python and some USB HID commands because, like a lot of software, you can control it with USB shortcuts. The scrolling, in fact, is just sending the period key rapidly or the comma key when I want to go backwards. So that's a really simple way to use that in Premiere. If you want to take a look at how this is set up, what I'll do is hide Premiere for a moment. And now that this is plugged in, let me come over to Sublime Text and open up Code.py that's on this Circuitpy device, a little Qtpy. That's an RP2040 there. So you can see here in my code, the sort of important thing I'm doing is this right here, from A to Fruits seesaw, import seesaw, rotary IO and digital IO. So rotary IO allows us to read the encoder. Digital IO allows us to read the five switches that are these different navigational buttons that I can press, the North, South, East, West as well as the center button. This is running on the default I2C address. I'm setting my pins to be input with pull-up resistor on them for the five buttons, setting those up as seesaw, digital IOs in their relative pins. And then the encoder is set up as an object using rotary IO, incremental encoder, seesaw. So all of that is running on the chip right on the breakout board. And then we have some button states that I'm using, button names, I've set up the key codes I want to use as space bar, I for setting an input, O for setting an out, out point, in point and out point I should say. J is the rewind and L is the fast forward. And then in the main body of the code, all I have to do is say, okay, position equals encoder dot position that pulls the seesaw chip there to find out what is going on with the rotary encoder. If there's a change, I'm just taking the difference between the last time I stored the state and the current, and that just helps if I rotate it super fast, we can store up a number of clicks. And then depending on if that is going up or down, if the delta is greater or less than zero, I will run through and hit the right arrow key for scrolling to the right. And the left arrow key, if I've scrolled to the left, sends those over HID just as if I typed them on the keyboard. And then we check through all of these buttons to see has anything been pressed? Has anything been released? If so, I'm gonna press these keyboard shortcuts that I listed there. So if I go down here and do disco tool, just so you can see the output, I'll scroll. You can see that relative position as I scroll the scroll wheels changing. I think I'm combating it because I'm also sending, there we go. I don't know why that's not updating. What happened? I've broken it. Oh, funny. If we go up here, we should be able to see me typing essentially. So those LI, O, J, space, those are the things that get typed. And then this is sending these arrow key commands. Yeah, that arrow key stuff was battling my output in the REPL here for some reason. So let me get rid of that stuff there. So that's how easy it is to use that. You can use it also under Arduino. It's very, very similar setup and there's some sample code there. And let's see, questions. So looks like a jog wheel for sure. Yeah, it's a really nice jog wheel. It's also small enough to use in little sort of handheld devices, which is nice, a little thumb wheel here. You can navigate everything just with your thumb. Kind of like the old iPods. That was so nice when they had a click wheel, a real physical click wheel on them. Aza, or Aza, sorry, asks in the Discord, can you use the rotary with a Adafruit macro pad using its STEMAQT connector? Yeah, I don't think, I've tried it, but I will say I am 99.999% sure you can because you can plug in the STEMAQT device and code that in either CircuitPython or Arduino to read that. And now you've got a second encoder. We've got one push encoder on the macro pad and then this would give you a second one depending on your needs. So that would be kind of cool. I like that idea. Let me know if anyone has any other questions. Just Jules, it's time to make myself a my pod. Yeah, you can make a very personalized iPod with this. What else have we got to say about it? I think that covers everything, yeah? So head on over here to product ID 5740. Let me see if any are in stock. I'll do a little refresh here. 54 in stock, yeah. So we've sold quite a few, but you can still grab some. You can grab up to 10 of these if you have big, big plans. Just make sure you get yourself the click wheel encoders and wire them up and make your project. It's begging to be made into a kind of modernist etch-a-sketch, maybe a little OLED display. That could be a kind of fun thing to do. You can wire up lots and lots of these and make some pretty cool projects. So I think we're gonna wrap it up right there. That, that's my product pick of the week this week. Wait, before I say this, I wanna get it right. So this is going in a video, what is your name? Okay, now it's even right side up. That's my product pick of the week this week. It is the ANO Rotary Navigation Encoder with iSquare C over StemaQT. And that is your product pick of the week for Adafruit Industries, I'm John Park. And this has been JV's product pick of the week. Guy, we'll see you next time. Bye-bye.


Hello and welcome to the session. In this session we will discuss the following question and the question says construct a triangle ABC with AB is equal to 6.2 centimeters, BC is equal to 5.8 centimeters, AC is equal to 4.5 centimeters. So we have to construct a triangle in which the lengths of three sides that is AB, BC and AC are given. Let's start the solution now. We shall write down the steps of construction and step one of construction is draw a line segment BC which is equal to 5.8 centimeters. So we have drawn a line segment BC of length 5.8 centimeters. Now we are given that AB is equal to 6.2 centimeters. So in the second step we will draw an arc with DS center and radius equal to 6.2 centimeters. So step two is with BS center and radius equal to 6.2 centimeters draw an arc. So with BS center and radius equal to 6.2 centimeters we have drawn this arc. Also we are given that AC is equal to 4.5 centimeters. So in the third step we will draw an arc of length 4.5 centimeters with CS center. So step three is with CS center and radius equal to 4.5 centimeters draw another arc such that it cuts the previous arc at the point A that is with CS center and radius equal to 4.5 centimeters. We have drawn this arc and it cuts the previous arc at this point A. Step four is join AB and AC. So now we will join the points A and B and A and C. Thus we have constructed the triangle ABC in which AB is equal to 6.2 centimeters, BC is equal to 5.8 centimeters and AC is equal to 4.5 centimeters. Therefore triangle ABC is the required triangle. With this we end our session. Hope you enjoyed the session.


You are paralyzed. It's very difficult to get recovery after three months. You need to accept your disability. What do patients need to do? Just sit at home, do nothing. That's not what we want. We want to have a better recovery, better independence. When we talk to another patient, and then he mentioned that he's active before, you know, and now becomes an impair, and then one side of the body, you know, couldn't move, and then he may bond with viewers. He's not happy. And then can we make some change? With technology nowadays, with the understanding of new science, it's become possible. We know about the brain have what we call the new plasticity. So can we use the robotic system to train the brain again to learn how to walk? So walkings, more and more, actually, the brain will learn it just like how we ride the bicycle. So with the robotic system, with the intelligent system, that may be possible. Existing robots are already doing such a thing. In the clinical trial, we can see some of the large equipment and repeat the walking. And then we can show the data that improve walking independence. But you can see those equipment have limitations, bulky, heavy, and expensive. And all the movement is maybe triggered by the joystick. This is not natural. With the advancing technology in robotic and the intelligence, actually, we can make the things more lightweight, like we can use carbon fiber and also with some of the powerful motors. So we can make the things much smarter and more easy to access. The other important thing is the intelligence. We don't want to walk with a joystick. We just want naturally walking. So nowadays, with those embedded motion sensors and force sensors, we can detect intentions. So when they move their hips, we know they want to make a step, then we will power the motor. And then they can go upstairs, downstairs. They don't need to change the algorithm. Everything is automatic. And then see in our hands, this is a robotic system. We use powerful motors. We have a new design of the gear mechanisms that can transfer the powers, amplify the power, and also all the sensors we embed in the insole. So it's easy, simple for the user to use it at home, and even in the Kinect. For the knee, it's a bit tricky because you need more power. So nowadays, we can use what we call the electric mechanical lock system to lock the knee during the stand phase and then provide the support. During the swing phase, we unlock it and then use the motor to drive it. So all the things need to be synchronized with all the intelligence. But everything can be automatic nowadays. You can see in these lights, everything can be synchronized. All the things we need is with a powerful microprocessor. Now we can do it. So everything embed together with the sensor, with the microprocessor. And the last thing is the battery. The battery to power the motor, power the lock. Is it sufficient enough? You can see now that the battery is very small and powerful. One battery that you know left, you can see it can last for two days. So you can use it for the whole day, work with the device. That's what the patient needs. More practice, better. They can't. So the technology now advances to the stage where we can do it. So let's look at how we can do the training. So you can see the patient nowadays is not limited. They can do the training indoor, outdoor, anytime they want. It's a new experience for the user and also for the clinicians. So more practice, actions have to do it. So this is the beauty of the robotic and the intelligent system. Let's look at the results. And we have a pilot test and then after 20 sessions of one-hour robotic training, we can see 30% of the lower limb functions have been improved in the patients. And 40% of the patients have been improved in the working speed. So they regain the independence. So this is important for them in the quality of life. Let's look at how the patients are feeling about those robotic systems. After training, they don't need the robotic system. They can enjoy the outdoor activity. They can go hiking. They can regain the power, strength and balance. So they don't feel tired. So this is good. The brain learned it again. And when we look at another patients, now he can walk independently. They can go stairs with confidence. They don't offer anymore. They can stand steadily after those robotic training. They can go outside. They can go to our lab anytime by himself. That's great. So what we can see now, this is with those robotic system and the intelligent system, they can give a new experience to the user and also a new experience to the stroke bear petitions. Thank you very much.