Okay, hello everyone, my name's Joel and this is Ralph. We're from a contemporary performance company called BenchCulture. So do I just tap this? We've got a little device that we've mocked up for a sort of read-the-room sort of device, but I won't explain too much. We'll just demonstrate how it works. So just tap this. Is this real good to go? So basically, the first trigger is just activated by a simple clap. We can try. Oh, do you? Are you going backwards, Ralph? All right, we'll try again. Yeah, we do a lot of this sort of thing. So clap. We might just have to skip that bit. Oh no, I can hear it. It's just really quiet. The third track is triggered by an absence of sound or gesture. Okay, here we go. So now we just need to keep quiet. The fourth track is triggered by a minimum 85% of the audience thinking of a pink elephant. The fifth and final track is triggered by the audience applauding riotously. Closing up. Well done. We have triggered all the tracks. Okay, so we're still ironing out a few of the kinks in that one, but maybe you get the idea. Over the past few years, Joel and I have been exploring different ways of making audio-based theatrical experiences for solo users. And in doing this, we've had to kind of learn to balance our desire to explore new opportunities offered by technology with a need to really dig into what's possible with the tools that we already have, such as an MP3 track. In other words, if this device were actually real, that'd be great, but it would be a long road for us to figure out how to make it work and what to do with it. And we found it's easy to overlook the journey that you need to take when you get cool new technology with and it's easy to forget about the potential of the tools that you already have. I'm going to grab the clicker there, Joel. So we've done a range of experiments with quite simple technology. This is a musical retelling of Pocahontas for your local supermarket. It's called Into the New World. This is a touristic experience for public parks this next one. And both of these are created just with MP3 tracks. But our first project was an unauthorized audio guide to Te Papa. And this was a single 40-minute audio track. And even with this simple technology, we spent a long time grappling with how to use it. Did a lot of testing and retesting in this space. And I'll add that we made this before I began. My current role as a writer here at Te Papa so hence the unauthorized tag. So what we'll play for you now is from the end of the tour and this is where the forces of evil are really closing in on you. So just outside of the toilet, we're going to head out towards the centre of the museum. Stop. Stop right there. Tie your shoes. Tie your shoes right now. You've been compromised. You need to do exactly what we say when we say it, okay? Okay, stop tying your shoes and stand up. See the walker. Hit toward it, but turn right down the corridor. Okay, past the house. Hurry, we've got to move. Okay, make like you're heading towards those trees. Duck left, duck left into the alcove. Wait here.  Okay, on my mark, go back out. Okay, five, four, three, two, one, now. Okay, out and left. Head towards the palm trees, past the walker thing. Okay, towards the trees. Keep that corned beef bowl to your left. To the left, at the trees. Okay, now slow, slow down. Head through this part of the museum casually. Turn slightly left and head toward the glowing boards with friendly faces on them. That woman, I think she's one of them. Middle-aged woman, negative. I think she's reaching for something. Not a threat. Stand the line you're on, do not deviate. Keep hitting toward those smiling glowing faces. Yeah, so that was the first one. And, you know, we found that we could do quite a lot with just a very limited sort of palette, so to speak, technological palette. Already we're sort of able to create an experience which creates a theatrical experience outside of a theatre that puts the audience in the role of the protagonist. And then we sort of, we did a few of those in different sort of styles and genres. I mean, we got really excited about doing something that would be kind of like outrageously huge over the whole city, GPS triggering, you know, people could kind of like, you know, interact with the whole city in a kind of fictional and blowing a fictional and reality in a kind of interesting way. And that had a lot of technical difficulties as you saw before. We struggled with operating simple technology. So we spent a lot of time grappling with that and trying to solve that. And then we actually found that if we just downsized not only were we able to achieve what we wanted to do a lot easier, but we also managed to having that restriction, restricted technology, so in this case it was more like a museum guide where you put in the numbers and stuff. When you get to the different parts of the city, you could get, we had that restriction which helped our creativity and helped our creative process. So we're just going to do a little bit, if we've got a little bit of time, since we're the last ones I think we can probably get away with it. I don't know. We'll just do a little, just a few seconds from this work in progress, which, so you come up to this parking meter, which we're generally on the plane. If you press a bit. Hello, Trevor.  So there's two kinds of shrines. Is the older kind, and they sort of look like a forked tree. And the others are more of an old log, almost like a coffin shape, the size of a child. There's a certain same function, that's just for the period that they were made. Okay, so what you can do is just walk up to the train and just bow once. And then you take out a coin. It can be any coin, but gold coins are usually considered to be the most lucky.  So you can do that now. And by doing that, what this does is it's seeing the coin kind of symbolic back to the center of the earth.  And just pray that we don't all die from climate change. And then just a little bow again. And that's it.


A from there we go to Oshrun Stad, where the governor Adegwegar O'Ytola is rejoising with Muslims as they congratulate or celebrate Idel Qaber. In a message from his office, the governor is encouraging faithful's to take cognizance of the symbolic significance of the festival. He is also asking them to imbibe the lessons of obedience and sacrifice as demonstrated by prophet Ibrahim. In addition, the governor O'Ytola wants Nigerians to use the celebration to pray for salvation gyda Covid-19.


we're gonna make it tip number one part a plan extra time when you got a little one oh my gosh all right we got to go pike speak cuz there is another mama hug such a great place they had to name it twice the big apple we made it and we got the catty back and we got the baby in the catty okay we are driving on the tecanic parkway they call them parkways out here not highways in New York we made it to New York to celebrate who's wedding are we celebrating Matthew we love you happen yes indeed so we survived the airport yeah with a baby with an infant five months old five months old little Michael and so we want to share with you some items that we think are essential for traveling with an infant all right to love what are they number one you're gonna need a catty that's what we call it it's basically a fray to click your car seat into and I think a lot of moms buy the car seat that clicks into the stroller but forget that that's way too bulky you need to just the thin little frame and it just pops up you can store a bag underneath then click your baby into it and you can push and maneuver all you want it's light it's collapsible it has brakes it's everything you need but it's not as bulky as actual stroller so I don't know one is what we call the catty is that really his name I think it's our brand is called the cat yeah it's actually called the catty and it's down below and what do you connect to the catty the mommy hook number two the mommy hook it's a giant carabiner and you can just hook in so it can hold so much weight and I bought one I got mine at Target we'll link it below but I would definitely recommend getting two so you have now it's way down here and strap there in the bags open right there or it just distributes the weight a little bit better just be careful the tipping oh yeah you have to balance out so if you have a heavy bag and you take the baby out boom the whole thing's gonna fall so balance that out but that is key to be hands-free and be able to maneuver and push for you to go and then as you probably know a lot of times when you're traveling you have to wake up before the crack of dawn and so you need your energy drink aka coffee in the morning but you can't carry the coffee through security so item number three that you connect to the mama what is it some type of insulated drink holder like a thermoflass or a yeti that has a hook so you can you can hook it into the carabiner and you can be hands-free you get through security then you go and get your coffee like what happened today we didn't bring those and we're at the gate and we have our water and our coffee and we're like huh we're so grateful it's gonna be and we're like oh wait because you have to check your carrier and your caddy and now you've got the baby in the air go in your bag and you don't have enough hands to carry all those you have to down them or toss them no pour them into for your hot coffee into the thermoflass or yeti and pour your water to your water bottle you can either shove them down into your bags and then your hands free again with the baby in your bags and you're getting through the aisles or you can hook them onto your bags or using the mommy hook but you need to be hands-free and you also want your coffee and your water and we flew frontier where you're gonna have to pay for all that once you get on might as well buy it at the airport might as well buy it at the airport and also we just wish we would have had those things so because I downed my coffee before we boarded because I didn't know how to carry it all so folks those are the three items the caddy the mama hook mommy hook in these flasks but make sure you get the ones that have the lid and then the lid with the hook thing basically you need your hands-free to manage the children or child no matter what and so that's that those three days helped us get to New York we kept saying all day what do we what do we kept saying like this is I can't believe how easy it was we left we didn't have traffic all the bus shuttle buses when we parked there was a shuttle bus waiting every single thing bing bing bing it's all lined up Michael slept on the plane but of course the hardest part was the coffee and the water because we forgot ours so as learn from our mistake so yeah learn from our mistakes in if you have any thoughts or questions or ideas or ideas just comment down below we love you guys all right almost to sucky close out the boxes in case oh yeah yep hey see beauty work hard and love each other


So, what we are going to do today is look at one particular technique that is used for non-linear systems and this is probably one of the first techniques that came in non-linear systems and it is amazing it is a frequency domain based technique, but subsequently of course, with the discovery of circle criterion and so on this particular criterion is more easier to understand, but how so this criterion is called Popov criterion and it was discovered by Popov in 1962 and how the techniques that Popov used of course, are one way to view the Popov criterion and in fact, the Popov criterion can be looked at from several viewpoints. So, what I would like to do is look at the Popov criterion in terms of loop transformations. So, for that we first recall what we did with loop transformations. So, if you recall so, we were always interested in a linear plant and a nonlinearity and they are interconnected in this feedback kind of situation and then of course, without giving any inputs we are interested in knowing whether this feedback circuit is asymptotically stable. Now, of course, this nonlinearity could belong to various kinds of classes and so, we had earlier already said like for example, the nonlinearity could belong to the k infinity class. Now, what we mean by that is that if you draw the nonlinearity characteristics. So, think of the input to the nonlinearity as sigma, output to the nonlinearity as psi, then if you plot sigma against psi, then here I have this line with slope k and then if the nonlinearity is in this sector, what it means is that the nonlinearity is something like that. So, essentially the nonlinearity is something that lies in the first quadrant in between this line and the zeta axis and similarly out here. So, what I am shading that is the area where the nonlinear characteristics lie. Now, going back to the loop if you think of this sigma as the input to the nonlinearity and psi as the output to the nonlinearity, then by the way this is drawn, the input to the linear plant is minus psi and the output of the linear plant is sigma. In other words, G s can be thought of as sigma divided by minus psi. Now, in the loop transformations what is done is this nonlinearity is changed. Now, for a nonlinearity in this class for example, we can convert this into a new nonlinearity in the 0 infinity class and the way we do it is the following. So, what we do is in this particular case we keep the output as it is. So, here we have the nonlinearity and let us say the output is psi, but we modify the input. So, we continue to have the input psi, but what we do is we use a feedback and what for example, we could think of a new input to a nonlinearity sigma 1 and the sigma 1 is given by let us say k times psi minus sigma. So, we think of the new input sigma 1 as being psi minus k sigma. Now, I am sorry. So, in this particular case because it is in the k infinity sector, it is not the psi that we change. So, let me use a new sheet. So, we have the linear plant and the nonlinearity and we have sigma and this is psi. So, this is minus psi, this is sigma and we are assuming that this nonlinearity is in the k infinity sector. That is like saying you have this line of slope k and so the nonlinearity lies in the hatched portion. So, what we do is we keep the input the same, but we change the output. So, how do we do that? We take the same nonlinearity, we keeping the input the same sigma therefore, the output here is psi, but we construct a new output psi 1 which is psi minus k sigma. So, if we have to construct the psi 1 which is psi minus k sigma, then we might as well have a loop. So, k times sigma which we add to psi with a minus sign and so what we have here is psi 1. Now, I am drawing this box here and whatever is inside the box, I call it a new nonlinearity. So, let me call this nonlinearity N L 1. Now, if you think of N L 1, this N L 1 has sigma as its input and psi 1 as its output. Now, if you now look at the characteristics of N L 1 which has sigma as the input and psi 1 as the output.  Now, the reason being that of course, the original one was lying in the hatched area. So, the psi could at most given a sigma, the psi could at most be k given a sigma, the psi could at most be k sigma and so if you are doing psi minus k sigma, it could come right down to 0 and so you have the 0 infinity sector. So, this nonlinearity that you have, this nonlinearity is in the 0 infinity sector. Now, the output of this and the input. So, the input is sigma and the output is minus psi and so we will try and maintain a similar configuration. So, this psi 1 is the one which is fed back. Now, if psi 1 is the one which is fed back and we are interested in this G s, we would of course, like the input of G s to be minus psi rather than minus psi 1. So, what we do is we modify the input and you see and we would like the output of the linear part, the output of the linear part is sigma and we would continue that sigma into the input of this new nonlinearity, but the linear plant we would modify in the following way. So, you have sigma, but what we do now is we take the sigma and multiply it by k and we would feed that back with a negative sign in here. So, now if you see and then we would give that input to G s.  So, that plus k sigma is cancelled and so what finally, you have in the input of the original linear plant is minus psi which is precisely what you had here. So, the linear plant can be as it was and so by modifying the nonlinearity we also modify the linear plant and therefore, what you have effectively is the same as what you had here. And so the way the, so the new linear plant that you get is what is contained inside this box and let me call it G 1 of s and if you want to look at the transfer function of G 1 of s, G 1 of s has as its output sigma and as its input minus psi 1, for psi 1 I substitute and I get sigma upon minus psi plus k sigma and this then above and below if I divide by minus psi 1, I end up with G upon 1 plus k G. So, this G 1, this new linear plant is related to the old linear plant in the following way. So, G 1 is equal to G 1 plus k G. So, this is the kind of thing that we did. So, I mean, so now we can talk about asymptotic stability of this is the same as the asymptotic stability of this loop, but in this loop because this new nonlinearity is in the zero infinity sector then we know that the linear plant must be a positive real plant. In other words, G 1 s, if G 1 s is positive real then this system is asymptotically stable, but G 1 of s being positive real is saying that as far as this thing is concerned it is asymptotically stable if G upon 1 plus k G is positive real. Now, if you look at it very carefully the original system what we have done in this system is in this loop with the in this arm with the nonlinearity we have fed back with a minus I mean it is a negative feedback with k. Of course, considering that the nonlinearity is going this way this is really a feed forward, but up there we do the same thing we take the sigma multiplied by k and feed it back. So, upstairs it is a negative feedback and here it is a negative feed forward and in this way we managed to convert a nonlinearity in the k infinity sector into a nonlinearity in the zero infinity sector. Now, in the same way we do these other loop transformations. So, for example, had the nonlinearity had the nonlinearity been in the 0 k sector. So, suppose the nonlinearity belong to the 0 k sector. So, that is like saying if you think of this as sigma and this as psi that is sigma and psi this is minus psi this is sigma then this nonlinearity has characteristics which lie in this hatch portion. This is the line with slope k. Now, again our endeavor would be to convert this nonlinearity into a nonlinearity in the zero infinity sector. So, how does one do that? So, in the earlier case we saw that we kept the input the same, but changed the output. In this case one approach that we could take is we could keep the output the same and change the input. So, let us do that. So, you could take this nonlinearity and its output is psi and so the new output will continue to be psi. What we would do is we will create this new nonlinearity and let me call this nonlinearity NL 1 now such that the input is changed. So, let me call the input to this nonlinearity sigma 1 and what is sigma 1? Well, let us take sigma 1 to be sigma minus 1 by k of psi. So, now if you take sigma 1 to be sigma minus 1 by k of psi. So, we feed back from here with a gain 1 by k in here and we add this. Then sigma 1 added to 1 by k times psi will end up in us getting just sigma here. So, the original nonlinearity its characteristics are sort of preserved out here, but what we have done is that the output we have fed back. We have fed back of course, it is a positive feedback such that it cancels this. Now, if we do this, then this nonlinearity. So, why is this nonlinearity now in the 0 infinity sector? Well, you can see that if sigma was the original input and psi was the output, but we modify the input to be sigma minus 1 by k psi, then sigma is here and 1 by k psi. So, suppose the point was let us say out here that means, if it was psi was k times sigma, then what we are saying is that sigma 1 the new input is going to be sigma minus 1 by k of k sigma. That means, sigma 1 is going to be 0. So, you are going to get that same output with 0 as the input. I mean close to 0 as the input. So, one can show that this nonlinearity, this new nonlinearity that you created in NL 1 is in the 0 infinity sector. Now, going back to the other portion, the linear part, if you have minus feedback, so what you have here is minus psi. So, we could put the linear plant there. So, if you put the linear plant here from here, you know that with minus psi as the input, sigma must be the output, but we want sigma 1 to be the input here. So, we have to modify this. So, what we do is we take this minus psi and multiply it by 1 by k and add it to the sigma. So, then effectively what you have here is sigma 1. So, this is the feedback that you will have. So, now, this here is the linear part here g 1 of s and so g 1 of s interconnected to this new nonlinearity NL 1 is equivalent to g of s interconnected to this original nonlinearity NL. Now, if you look at the arrangement that we have done, what we have done here is of course, here of course, the signal is flowing this way, but what we have done is we have taken the psi here and multiplied it by again 1 by k and put it back here with a positive sign and so here also we will do the same thing. So, just like in the last case, we do exactly similar things in the upper and the lower. Of course, what it means is here it is a positive feedback and here it is a positive feed forward. Now, in this case of course, this new g 1 of s turns out to be nothing but g of s plus 1 by k. And then therefore, for this system to be asymptotically stable, what we require is that this new plan g 1 of s because this nonlinearity is in the zero infinity sector, we want this g 1 of s to be positive real. In other words, g of s plus 1 by k, this whole thing must be positive real. Of course, this then you could interpret as saying that the Nyquist plot of g of s should lie to the right of minus 1 by k. Now, notice that in the last case and in this case, in both these cases, what you have done is some sort of feedback, feed forward that you have done and whatever you do in this arm which contains the nonlinearity, you do a similar thing up there with the linearity. And what it effectively does is it neutralizes the effect of that thing in some way so that the original nonlinearity and the original linearity still maintain the relation that they were supposed to maintain. Now, the point is that the gains that we used in both these situations were gains that were constant gains, but perhaps we could introduce gains maybe in such a way or maybe in series with the linear plant. For example, in series with the nonlinearity or in series with the nonlinearity and therefore, in series with the linear plant and so on and modify the loops. So, let us just try out one particular example. So, for example, if the original plant is like that with the nonlinearity like this, now let us not bother about where the nonlinearity lies, but let us do the following. So, the suppose the nonlinearity is here and let us assume that the output to the nonlinearity is kept the same. So, the output is kept as psi, but what we do is input. So, one would want the input here to be sigma, but maybe we put a gain here and maybe the gain that one puts here is let us say 1 by k. So, what that would mean is out here the signal that you have is k sigma and we could think of unity feedback. So, just like in the last case, but I have pushed the gain from 1 by k from the feedback loop in here. So, then we could think of the new input as sigma 1 and so in this case you have sigma 1 is k sigma minus psi and so this here is our nonlinearity. Then in order to find the equivalent thing to equivalent interconnection to this interconnection you have a gs here. So, this psi, so this minus psi, so the output of this is sigma, but you want to get sigma 1, sigma 1 is obtained this way. So, what we can do is multiply this by k. So, therefore you get k sigma and then you take this minus psi and feed it forward and therefore, what you will get is sigma 1 there. Now, the construction that I have done here is exactly the same as the earlier construction that I did. So, this construction and this construction are exactly the same. The only thing is that this gain I pushed it in here and so therefore, this gain has to get pushed in here, but when I do that here it is 1 by k there when it gets pushed it gets pushed to k and that is essentially because here it is in the feedback arm and there it is in the feed forward arm. So, when you put it in series it becomes k whereas, from the feedback arm it remains as it is and so what we have effectively done is we have done I mean we could just assume that this nonlinearity here is really a nonlinearity in the 0 k sector and therefore, this new nonlinearity NL 1. So, NL 1 would then be in the 0 infinity sector and this is the original plan G s and the new plan G 1 of s. The transfer function for G 1 of s is 1 plus k times G s which of course is roughly the same as the result that we had got earlier only it is this new G 1 of s is going to be k times that old G 1 of s, but you know by just multiplying by a scalar k you do not change anything. Now, instead of having gains here maybe we could have transfer functions. So, one kind of transfer function that you could possibly have is if you think you have G s here and you have a nonlinearity and let us again assume that this nonlinearity is in the 0 k sector and what we do is a construction similar to the earlier construction. So, what we do is this nonlinearity the k. So, this sigma this is psi therefore, you have the minus psi here sigma there and the nonlinearity is in the 0 k sector which essentially means this hatched area is where you have the nonlinearity this is sigma this is psi. So, what we do is this nonlinearity we multiply by first order transfer function like this. So, if you multiply the nonlinearity by a first order transfer function like this and of course this nonlinearity is the same as this nonlinearity. So, the signals across you want to maintain to be the signals across there. So, you have sigma here and psi here and so, if you use this psi in the feedback. So, you have minus psi here. So, let us assume you have a transfer function G s here. So, what you would get here is sigma and let me do the following. Let me assume that there is a gain here which is 1 by k of the psi and I feed it back that way. So, if you have something like this you should do a similar thing there. So, and I have this new input sigma 1 and so, this is my modified nonlinearity nL1.  So, what we do up there is I can again here I have a feedback positive feedback loop. So, there is a plus sign here and a gain of 1 by k. So, I take this with a gain of 1 by k and I am going to add it here with a plus sign and earlier we saw that if you had if you had 1 by k here then you have k here. So, we will just blindly do that and we will see what happens. So, let us blindly do that. So, the reciprocal of this is 1 plus AS. So, let us put that gain 1 plus AS out there and let me call this output that comes here sigma 1 and connect it up here. So, maybe this is looking all to clamped up. So, maybe I will use a new sheet with this particular diagram. So, we had the nonlinearity and then we used 1 upon 1 plus AS here and again a 1 plus k which we fed back in here with a positive sign and this whole thing is the new nonlinear system NL1 let me call it and so NL1's output is psi input is sigma 1 this is sigma here. We are assuming this is sigma here we do not yet know, but let us just do the construction and see whether we get something meaningful G s. So, with the input minus psi. So, the output here must be sigma and then we have gain like that and what we do is a similar construction. So, 1 by k through here again with a positive sign. Now, from the earlier situation here we know G of s is sigma by minus psi. So, let us just make this assumption. So, assuming that the output of the nonlinearity is still psi which is fed back here. So, you have minus psi here. So, this must be sigma. Now, if sigma passes through here what you get here is going to be a times sigma dot plus sigma and then when you have this thing. So, what you are going to get is minus 1 by k psi and this whole thing is sigma 1. Let us assume this whole thing is sigma 1. Now, if this whole thing is sigma 1. So, sigma 1 which is coming in here you have psi 1 by k times psi added to it. So, what you have here the signal here is going to be a times sigma dot plus sigma and if you have the signal a times sigma dot plus sigma here 1 upon 1 plus AS acting on that will in fact give you the signal sigma. So, this whole thing looks consistent. So, this whole linear plant that you have let us call that G 1 of s and this whole new nonlinearity N L 1. Then perhaps we could say that this interconnection is asymptotically stable and if we can say that this interconnection is asymptotically stable that would be equivalent to going back to the original plant and saying this interconnection is asymptotically stable. Now, in here if you look at this linear plant G 1 s this linear plant G 1 s is given by 1 plus AS times GS plus 1 by k. Now, if this linear plant is positive real. So, now let us make some assumptions. So, if this is passive and this is passive then the interconnection of two passive systems is passive. So, I mean this being passive and this being passive the interconnection is passive. So, because each one of them is passive one could find a Lyapunov function or a storage function for each one of them and the storage functions. So, recall that for passivity if you have a passive system then you have a storage function V such that V dot is less than input times output. This is how we sort of less than equal to input times output this is how we characterize passivity. So, what we can start doing is the following. So, let us assume that this whole plant this whole new linear plant G 1 of s is passive. Now, this whole thing being passive is like saying that G 1 of s which is 1 plus AS times GS plus 1 by k this is positive real. And then we have gone through several of these theorems earlier like the Kalman-Yakobovic lemma by which one can find for this part a storage function V 1 such that V 1 dot is less than equal to the input times the output which in this case is minus sigma times psi. So, if you can also show that this guy is passive then the sum of these two is going to be passive. So, how to show that this particular transfer function is also passive. Now, for that what we will do is we will look at the differential equation. So, the differential equation that one could look at is the following this whole nonlinearity is there. So, let me write it out in the following way. So, let me call the nonlinearity f such that if sigma is the input then psi is the output and what I am really saying is psi is equal to f of sigma. And then earlier we had 1 upon 1 plus AS here and then we also had this feedback gain of 1 by k. Now, if this is sigma then of course from the last slide we saw that this must be this signal here must be sigma plus a times sigma dot. And because of this we know that sigma 1 must be given as sigma plus a times sigma dot plus 1 by k times psi.  This thing is really like a differential equation with so think of this as a differential equation with sigma as the state variable and sigma 1 as the input. So, one could write something like a state space equation saying a sigma dot is equal to minus sigma minus 1 by k f of sigma plus sigma 1 where sigma 1 is thought of as the input and sigma is the state and then of course one could think of the output equation as psi is equal to f of sigma. So, take this equation this differential equation where sigma is a state sigma 1 is the input and this has the output equation. So, these two equations you have. Now, let us make one more assumption let us make the assumption that this nonlinearity this nonlinearity f is in the 0 k sector k is related to this k where the gain was 1 by k. So, the f the characteristics of f lie in the hatched area. Now, if this is the case then I sort of decide that I will take Lyapunov function in the following way you see if the characteristic of this f.  So, if this is the slope k sigma psi f is such that f looks something like that. Now, so this is the curve f of sigma then of course it would be clear that if you take integral of f of sigma d sigma going from 0 to some point x then this quantity is always going to be positive because if that x is somewhere here then this integral is positive. If it is somewhere here then this integral this integral is negative but of course the limits are going the other way. So, when you change the limit it becomes positive. So, you get something like this. So, this is going to be greater than 0 for all x. So, I have been using sigma as a state variable. So, maybe I could just rewrite this as x a x dot equal to minus x 1 by k f of x plus u let me call it u the input and then I am saying y is equal to f of x this is the state space equations is the same equations as before and we see that this function is such that if you take the integral from 0 to x this is true. So, I propose that we use a Lyapunov function v x given as this is equal to a times integral from 0 to x of f sigma d sigma. Therefore, this Lyapunov function or this Lyapunov function candidate has the property that this is greater than 0 for all x greater than 0. So, now if you evaluate v dot of x. So, v dot of x from this evaluation is going to be a times f of x times x dot. Now, a times x dot we can substitute in this.  Now, if you take any x x times f x f x is. So, for positive x f x is positive for negative x f x is negative. So, this quantity here x times f x is positive. So, minus of x times f x is negative. Similarly, this quantity here is also negative and therefore, one can conclude that v dot x is less than f x times u, but f x is y. So, u times y. So, this system that you have the state space system that you have with this particular Lyapunov function you can see or storage function you can see that this is passive. So, if you go back here this system that we have drawn here the system is passive with the storage function given as v is equal to a integral 0 to sigma f of r dr. Let me call this v 1 with this storage function and we already saw that in this interconnection that we were looking at this portion because, it is positive real has a storage function v 1 has a storage function v 1. So, is that v 1 dot is less than equal to minus sigma psi because the output here is sigma and the input is minus psi. And out here if you take this let me call it v 2 then we know v 2 dot is less than is less than equal to the input sigma 1 times the output psi sorry I have to correct one thing here. Here the output is sigma 1 and not sigma and so is minus sigma 1 psi. So, now if you take the Lyapunov function if you take the Lyapunov function of that net system to be v 1 plus v 2 which is equal to okay. So, that v 1 would have been given by because that that linear plant that effective linear plant that you had. So, this is the v 1 this effective linear plant you could look at the state space realization of this. And from that you could write in terms of the states of this thing as some x transpose p x where this p will satisfy all the Kalman Yakovovich Lemma conditions and plus a times integral 0 to x of f of r dr this as the net as the net Lyapunov function v then you have v dot equal to v 1 dot plus v 2 dot which of course we can calculate.  So, some of these two is 0.  And this quantity here is going to be positive because this quantity is going to be positive and this p has been obtained to be a positive definite matrix. Therefore, by Lyapunov theorem we have asymptotic stability. So, what we are effectively saying so this is what is called the Popov criterion and what the Popov criterion is saying is the following that if you had a linear plant and you had a nonlinearity and you had it interconnected in this way and this nonlinearity was in the 0 k sector and the linear plant was such that 1 plus A s times G s plus 1 by k is positive real then this interconnection is going to be asymptotically stable. Of course, this nonlinearity must be such that it is a memory less nonlinearity. The memory less Lipschitz nonlinearity, but this is the effective result of the Popov criterion. So, it looks like I am out of time now. So, let me stop for now.


Welcome to our unit on recursion. Our plan with this video and this unit is to empower you to enter your classroom ready to motivate and teach your students about the rich topic of recursion. You and your students may be asking why should we be interested in recursion? How would we use this in the real world? What are the most important elements that students should take from this unit? What kind of problems will recursion help us solve? To what other results or mathematics will our studies lead us? Recursion provides us with a method for creating models that capture phenomena that change at fixed time periods. Many of the mathematical models we create are continuous models. That is, they well model phenomena that are constantly changing. For example, a continuous model would be very appropriate for determining how many bacteria are left in a certain sample after different time periods, given the bacteria's doubling time. Suppose, however, we are modeling the amount of color left in a pair of genes after different numbers of washings. Since the decrease in color happens only at each washing, a discrete model is the more appropriate choice. In this case, the recursive model would be more useful than a continuous model. Studying recursive equations opens up a whole host of real world applications. In this unit, we explore questions such as how pollution spreads through the network of interconnected Great Lakes, and how two investment strategies can differ over time. The ideas we develop in these lessons could be further applied to problems, such as investigating the spread of a disease through a closed population, or determining how lead builds up in different areas of the body. Another motivation for studying recursion is to gain a better understanding of exponential functions. By deriving explicit exponential functions from recursive equations, students will get to actually see that exponential functions are the result of repeated multiplication. Along with gaining a better understanding of how basic exponential functions behave, this will allow students to understand when and why exponential functions will be appropriate models for phenomena. The following set of lessons on recursion helps us address many of the common core standards related to recursion, sequences, function interpretation, and exponential functions. A list of the relevant standards is provided at the end of each lesson. We hope you found the materials and lessons we provided here useful. We encourage you to send us any comments and feedback on other materials you would find helpful. Thank you.


Only one solution can keep Gates of Hell shut. Palestinian envoy, an international conference resulting in a limited time being given to Israelis and Palestinians to negotiate a two-state resolution of their conflict, is the best hope for reducing the violence in the Middle East. Abdel Hafez Naufal, the ambassador of the Palestinian Authority to Moscow, told RT, It's the only way. Otherwise, the Gates of Hell will be opened, he said, referring to the proposed creation of a Palestinian state which has been endorsed by the UN. We are not against the Jewish people. We are not against the Israeli state. We believe that we must live together with a two-state solution, where we can make a difference in our life, he added. The region is currently undergoing a dangerous surge of violence after the Palestinian militant group Hamas launched a deadly incursion from Gaza into southern Israel earlier this month. The group largely controls the coastal enclave while the Palestinian Authority is based in the West Bank. The Israeli military has retaliated with intensive bombardments of Gaza and an escalation of its blockade, which has been in place since 2007. The Jewish state said it was determined to obliterate Hamas. There are fears that Israel may launch a ground invasion of the Palestinian enclave, potentially triggering intervention by the Lebanese militant group, Hezbollah and other players.


Next I'd like to welcome to the stage Kelsey Long. Kelsey's from the ANU College of Physical and Mathematical Sciences and the title of Kelsey's presentation tonight is The Memory of a Fish Three Seconds or 20,000 Years. You are standing on the shoreline of a great lake, a dry, dusty, empty lake. As you walk along you feel something crunch under your foot. There is a cluster of charred fish bones, an ancient fireplace. You pick up one tiny bone, the key to unlocking the history of this lake and of the first Australians that walked along its shorelines. This is Lake Mungo. It's a World Heritage Site in New South Wales where human remains were excavated in the 1960s and 70s. Since then the lake has remained closed to all research until now. Mine is one of the first studies in over 30 years and that tiny bone you trot on is the focus of my work. This particular bone is called an otolith. It's a calcium carbonate structure that grows in the inner ears of most animals. In humans they are tiny pebbles that help us with balance and hearing. Without yours you'd fall over a lot. But in fish otoliths are so much more. They grow much bigger and they preserve a record of temperature and salinity that I'm analysing in my research. Each one of these rings represents a year of growth. Similar to what you get in tree rings and as each increment is laid down trace elements from the surrounding water are taken up and incorporated. Many of these change with the environment. So by studying the life of the humble fish through its otoliths we can learn about past environments. When it got hotter or colder more or less saline or whether it was big flood at Lake Mungo they provide us with a link between people eating fish and the environment.  But there's a catch. Not every element is affected by the environment alone. Some are affected by the fish's diet or its metabolism. So it is my job to figure out which elements are best suited for measuring temperature and salinity. I have been firing lasers at modern versions of these otoliths from across Australia. I have found evidence that people were living on the shorelines of Lake Mungo eating fish when the lake was drying out, which makes sense. As the lake is evaporating the salinity is increasing causing the fish to become sluggish and thus much easier for people to catch. And I've recently dated this to 20,000 years ago because a fish may have a memory of only three seconds but they are helping me recover memories that are 20,000 years old. So the next time you are sitting on the shorelines of Lake Burley Griffin eating your fish and chips just think about what that fish might say about you and your environment 20,000 years in the future.


In the previous module, we have studied the schematics of TCSPC or time-correlated single photon counting experiment. This is where we were, let us go a little further ahead today. And before we do that, let us just revise what we have learned. This here is the essence of TCSPC, essentially you are trying to record the difference in arrival times of a start and a stop signal and you are trying to plot a histogram. This is a histogram that we are talking about in the previous module. And what lies at the heart of TCSPC is that this histogram that you construct is actually of the same shape as this fluorescence decay that you are trying to construct. Now, let us get into a little bit of nitty-gritty of the instrument because the entire purpose of this is that we should be able to use the instrument when we do it. Of course, you might understand a little better when we see the instrument for ourselves, when you go to the lab, but we need some preparation for that as well. Now, there are certain terms that we will need to know. First one is tack range. Tack range is essentially the time for which a tack waits to get a stop signal before resetting and starting all over again. So you can think that the tack range is the maximum time measured in your experiment. So in this decay that you have, you can think that tack range is the full scale on x axis on time axis. Typical smallest value of tack range is that I know of is 26 nanosecond. 15 nanosecond is more common. What kind of tack range you will use depends on what kind of decay you are looking at because for a good analysis of the decay, it must be complete. So if your lifetime is say 5 nanoseconds, then well remember the decay law, right? It equal to i0 e to the power minus t by tau or it i at t equal to i at 0 divided by e to the power. So what happens when t equal to tau? t equal to tau, it becomes 1. So i at time t equal to tau is equal to i at time 0 divided by e. What is the value of e? Approximately 3 we can say 2.7 approximately 3. So very roughly we can say that a decay is in one lifetime the signal or population whatever you want to think decays to one third of its value at time 0, right? i at time t is equal to i at time 0 divided by approximately 3. Which means fluorescence intensity will decrease to about one third of its value in one lifetime. So to which value will it decrease in 3 lifetimes? Not 0 is correct but used as answer, yes?  Yes? 1 by 9, one third of one third, 3 lifetimes? 1 by 27, 4 lifetimes? 1 by 81, 5 lifetimes? Yeah, so it becomes very small. So typically you want to keep your tech range to about 5 times of the lifetime. So 50 nanosecond tech range is fine provided your lifetime is no longer than 10 nanosecond. At the same time you do not want to keep the tech range very large. Your lifetime is 1 nanosecond keep a tech range of 500 nanosecond. Then what will happen? 1 nanosecond, so 5 lifetimes it becomes practically 0. So for 200-5, 195 nanosecond out of 200 nanosecond you are going to be recording 0, it will be less time. So it is important to choose the correct tech range. When we say we are choosing correct tech range what are we actually doing? Setting the tech window, letting the tech wait for a bigger signal, right? Or we can say that see the maximum signal that we can usually get is 10 volt. We are saying that when tech range is 15 nanosecond then 10 volt signal is equivalent to 15 nanosecond. When tech range is 100 nanosecond then the same 10 volt signal is equivalent to 100 nanosecond. What does that mean? What is this 100 nanosecond? Delta T remember? Time of charging. So what we are saying is for different times of charging we are keeping delta V to be same. How do we do that? The voltage remains same of the capacitor. That means the capacitor is not the same. You have somehow changed the capacitance. How do you change the capacitance? Of course, nowadays everything is inside a chip. So God knows how they do it. But if I go back to the good old days of components, electronic components then what you actually have to use is a variable capacitor, right? Has anybody seen an old fashioned radio set where you actually turn a knob and go from one station to the other? What are you doing there? You are changing the capacitance in an LCR circuit. That is why the characteristic frequency changes. And in conventional capacitors what happens is you have these plates that go into each other like this, right? If you bring them out, of course the plates are not like my fingers. You can think that my fingers define the outer edge of some circle, okay? When they are like this then there is maximum overlap of areas. If you bring them out then overlap decreases and capacitance decreases. So you can use a device like that to change the capacitance and therefore tech range. The point I am trying to drive home that this is not magic, this is not voodoo. You can understand the principles by high school level modern physics that everybody has studied. Second thing that is important is number of channels. Remember tech range, it sounds foolish if I say it like that but a tech range is something that is associated with the tech. Sounds like stating the obvious. But the reason why I am saying it is that it is important to remember the number of channels is not associated with tech. It is associated with multi-channel analyzer, okay? It is very important to use the correct number of channels also, okay? Because suppose I have 50 nanosecond tech range and I have 1000 channels. You never have 1000 channels. It is always in some binary number but let us say for convenience let us say we have 1000 channels. Then what is the time resolution? What is the time per channel? One channel is, each channel is a point remember. So what is the resolution I have? 50 nanosecond full scale, 1000 points quickly, 50 picosecond. 1 nanosecond is 1000 picosecond so it is actually easy, okay? Now I am saying instead of 1000 channels I have say 5000 channels. Then what is the resolution? Earlier it was 50 picosecond per channel when number of channels was 1000. Now let us say I increase the number of channels to okay, 10,000. Then what is the resolution? 5 picosecond per channel, 1, 2, 3. Now the question is how many points will I use? Should I use 1000 points? Should I use 10,000 points? Again the answer to that depends on what kind of decay you are looking at. Suppose typically you should have 100 points per lifetime. So if you estimated lifetime is 1 nanosecond then 1 nanosecond divided by 100 how much is that? 10 picosecond? 10 picosecond per channel is a good resolution. But if your lifetime expected is 5 nanosecond there is no need to use 10 picosecond per channel. You can use 50 picosecond per channel, right? Why is this important? Because if you use a resolution that is not good enough then you do not get a good decay. If you use a resolution that is unnecessarily good then you waste time. Because remember how the experiment is happening in every point in every channel counts are being increased. So a poorer resolution essentially means 2 or 3 or 4 or 10 or 100 channels have been merged, okay. So it goes up faster. So it is important that you use not maximum not minimum but optimum number of channels for your experiments. Why? Because you do not want to spend an entire lifetime doing an experiment. But you do not want to do an experiment whose data is not reliable. How will you know exactly what number of channels to use? We discussed an example a little while ago. This comes from experience. And experience comes only when we do experiments with our brain switched on. If you do experiments with brain switched off there is no experience that is gathered even if you do the same thing over an entire lifetime, okay. So it is like driving a car or like swimming. Once you learn it you do not think you just do it, right? So that experience has to come. So initially when you do the experiment you have to do it consciously. And you have to do rough experiments in the beginning to understand what kind of time constants you expect. Because you are not going to know, right? Looking at the sample you cannot tell whether lifetime is 100 picoseconds or 5 nanoseconds. So many experiments are important. Thinking is important. Experience is important. Now we move on to a rather important question of data analysis. You got the data, fine. How do you fit it? How do you get the data? We have already discussed the fitting model. Most commonly used fitting model is single exponential or multi exponential. Later on we will talk about some more fitting models also. But now our problem is that we do not have the ideal situation where we are exciting with a delta pulse. We have a pulse of a finite width. So what we get Fd is really a convolution of the instrument function, laser pulse as the instrument sees it. You can think L for laser and the actual fluorescence decay F. Convolution means a mixture, a hopeless mixture that cannot be separated easily. We will see what it actually means graphically. It is important to understand that the graph we see is not the one we want. It is the one we want mixed with the laser pulse as the instrument sees it. So this is what is called the convolution integral. This is what we actually get. Fd at time t is F at time t-t dash multiplied by p at t dash integrated over values of t dash from 0 to infinity. This is convolution integral. And for the uninitiated, this I am sure means nothing. If you are very good in mathematics perhaps you can see this and make sense of it. But let us not take a chance. Let us try to see what it actually means. Let us think of a laser pulse of a finite width. So we can think that every point on this envelope is the tip of a delta pulse. And how many delta pulses would be there under this envelope? In principle infinite, in theory infinite, in practice the number will actually be finite. Because as we discussed already we worked with some picosecond per channel resolution. So depending on how many picoseconds are covered under this curve. Suppose we have 100 points here. Then we are going to get 100 delta pulses. So experimentally the number is not infinite. It is determined by what kind of time resolution we use. Let us see the effect of one such delta pulse. Let us take a small one at initial time. This delta pulse. Well what I have drawn is not a delta pulse. This is a delta pulse. What I am saying is that is going to give rise to a decay like this. What about the next delta pulse? That will also give rise to another decay like this. What about the next delta pulse? Same. And this goes on. This is what the meaning of deconvolution is. All these delta pulses under the instrument function gives rise to a decay. If you look at any time t here, the fluorescence intensity is given by a sum of all these fluorescence intensities from all these decays at that particular time. That is the meaning of the convolution integral. Let us see if you understand it a little better now. You see this arrow. This is the time t we are talking about. And we are saying that the delta pulse occurs at time t dashed. So what is the time for which the decay has actually taken place? We are measuring at some time absolute time t and the decay sorry the delta pulse is at some time earlier time t dashed. So what is the time for which the decay has actually taken place as a result of this delta pulse? T minus t dashed. That is very easy. So that is the time for this particular delta pulse. So if I took any other value of t dashed this t minus t dash would have changed. Now see this amplitude here or intensity at time t0 here we let us call it p t dash. So what will be the intensity at time t? F is the fitting function remember e to the power minus t. Instead of e to the power minus t now we have to write e to the power minus t minus t dash because the delta pulse is at t dash. So t minus t dash is the effective time. So if it is just for single exponential decay t e to the power minus t minus t dash and p t dash is the intensity at time well t equal to t dash here. So now we come back to this integral. What did we say? The intensity is going to be given by the sum of the intensities that occur as a result of all these delta pulses. So if you write the most general expression what is the range of t dash from 0 to infinity. So that is the integral. See f t minus t dash multiplied by p t dash. Summation is replaced by integration integral 0 to infinity but 0 to infinity for not t but t dash. So that gives us the intensity that we actually see at one particular point of time. Important to remember that at one particular point of time. I have not written this for the entire decay only at the value of t where the arrow presently is. To get the intensity at this value you have to work out this integral. This is the meaning of convolution but do not get scared by the integral sign and integral is just the summation. How do you actually do it? You do it by a method that is called iterative reconvolution. So anyway this is what it means. You see we are adding all the intensities due to all the decays coming from all the delta pulses and the sum is given by this integral. How do I do it? I do it by a method called iterative reconvolution. You see we are saying deconvolution and we are talking about we have to do deconvolution because here we have a mixture of the decay as well as the pulse as the instrument sees it. How do I deconvolute? The easier way of doing so-called deconvolution is to take a guess function. So you assume that the lifetime is 5 nanoseconds. The moment you assume that you know what f t – t dash is going to be okay. So what you do is with the guess value as it is called the assumed value you can easily construct well not easily if you have to do it manually it will take time for computer it is very fast. You can construct the fluorescence intensities at every point okay and then you compare the graph you have constructed and the graph you have experimented right. Of course you will not get it right the first time so you have to do it again that is why it is called iterative deconvolution. Deconvolution because you are knowingly convoluting the instrument response function with the decay law that you think is correct and iterative because you will never get it right the first time you have to do it in several rounds or several iterations okay. So I think we will stop here today and next day we are going to talk about goodness of fit because it is very easy for me to say that we will compare the graphs how will the computer compare computer does not have eyes. So next day we will learn first of all what are the eyes of the computer how does it know whether the fit is good or if it is bad and then we will talk about some more decay models.


Hey guys, it's Jess, and welcome back to my channel. I hope you're all having a very happy and productive start to your week. And for this week's video for brain food, I'm going to show you guys how to make some really delicious chocolate nice cream. Yes, nice cream because it's a really healthy take on traditional chocolate ice cream. And it's getting really warm in Australia, but if you guys aren't in Australia and it happens to be getting closer to winter, that's fine because who says that ice cream has to be restricted to hot weather? I mean, come on guys, really. OK, so it's a really, really simple recipe. There's only a couple of ingredients to include, so let's get started. All right, so I'm going to blend all of the ingredients together with my nutritional blender. So I'm going to put everything into this cup and to start, I'm going to take two bananas, peel them and just break them up into a little bit smaller pieces and put them in. So I love using bananas because they're a really natural way to sweeten the ice cream. And it doesn't actually taste like banana flavoured ice cream at all. It's really strange, but if anything, we're going to add some cacao and that seems to just overpower the banana taste. So if you guys don't really like bananas, I know not everybody does. I love them, but if you don't, don't worry, because you won't really taste them. So I'm just going to break them up and pop them in there. Now, also to naturally sweeten the ice cream, I'm going to add five pitted dates. So I've just taken out the pip and I'm just going to put them in like that. You don't even have to dice them because, like I said, it's all going to get blended up. And that's just a really good way to sweeten the ice cream. And then we're going to add two tablespoons of raw cacao. So this is going to make it all nice and chocolatey for all my chocolate lovers. Love chocolate. So two tablespoons. Why do I always spill things on the counter? OK, two tablespoons of cacao and then I'm using one tin of coconut cream. So there's a hundred and forty meals in this tin. So it's not that much, but this will make the ice cream nice and creamy and smooth. It'll give it that ice cream texture that we're all used to. And if I can just open that tin up, I feel like I need to get back into a gym and work my muscles. OK, I'm just going to put that into the mixture now. And then I'm also going to add just a little bit of milk as well because the cream isn't really enough to bind all of the ingredients, especially not in my blender. I think it's getting old. It just it doesn't like doing a lot of work. So I need to help it out there with just adding a bit more liquid. So I'm just going to add a quarter of a glass of rice milk. But if you have coconut milk or almond milk at home, that also works really well or dairy milk if you guys aren't vegan. And what is that instead? OK, and the milk magically appears from out of my fridge. I forgot to take it out. OK, so I'm going to put around a quarter of a glass in there. And then just see how you go. You might need to add a little bit more. Try not to add too much at the start because then it'll be really runny and it just takes a lot longer to harden. And now I'm just going to put the lid on and turn my blender on and mix it all together. So blocky ears, guys. That looks good to me. I'm just going to clean up my bench and then I'm going to put it into a Tupperware and pop it into the freezer. OK, so I've just got a glass Pyrex dish here and I'm going to pour the mixture in and I'm going to put it in the freezer. I think it needs a couple of hours to harden or get creamy enough that you can actually scoop it out. So it's not so runny. At the moment, it looks a lot like my chocolate mousse that I made recently. So I'd say a couple of hours and if I accidentally over-freeze it, then I can always just leave it out and it'll defrost a little bit so it's easier to scoop out. Oh, this looks so good. Oh, my gosh. Yum. All you Chocoholics out there, thumbs up this video for the love of chocolate. OK, now I am going to pop the lid on and I'm going to put it in the freezer. OK, let's see how this turned out. It's really cold. All righty, it's going to pop the lid off. This looks amazing. Wow, I haven't made this in a little while. So I hope I did it correctly. Oh, yeah, that's perfect. Yum, it just scoops up so nicely. I'm going to give myself a couple of scoops. I've been good this week. I can afford it. Yum, it is so rich and delicious and very chocolatey. I love it. Also, if you guys don't want to blend up the dates, you can actually dice them quite finely and then you get those little chewy bits inside as well. So I've done that before and it's really yummy as well.  No, this is so good. I don't want to put it down. And also, if you guys haven't subscribed to my channel already, click my face over there. So you can join in on the Jessica Pham and also you can check out previous videos by clicking the cards up there or also clicking the image over there. Also, I'll leave a link to my study skills book, the high school survival guide for you guys in the description box below. And if you guys recreate, please, please recreate this recipe because it's so young. If you guys do, then take a photo and upload it onto Instagram. Just tag me at studywithjesst or use the hashtag studywithjesst as well. I'll see you on Wednesday with another video. And until then, love you and bye.