In the last lecture, we introduced the concept of a vector potential and we had seen that because of the fact that the divergence of B is always equal to 0, I can express B as a curl of a quantity which we call as the vector potential. This vector potential as we saw was not unique, but there was a choice in the sense that we could add to the vector potential the gradient of any scalar field. This was known as a gauge choice and we had seen that one of the most popular gauges that we use is a gauge in which del dot of a works out to be equal to 0. Curl of a gives you the magnetic field which has a physical interpretation. Since it is a vector field in addition to its curl, one could specify its divergence and in this particular case the divergence of a was a meaningless quantity and can be chosen as per our convenience and mathematical simplicity. We had also seen that vector potential is not merely a mathematical concept, there are actually experiments which tells us that we can find physical realization of the vector potential in certain interference experiments. What we wish to do next is to talk about what boundary conditions need to be applied in the interface between two media and suppose so this is something that we will be talking about today. So, here for example, I have two media called one and two and first what I will do is to find out what is the relationship or how do the components of the magnetic field compare at the interface between the two media. Now, for that what we do is to look at the Gauss's law which tells me integral b dot d s is equal to 0. Now, so what I have done here in this picture is to draw a Gaussian pillbox and this Gaussian pillbox is of a height h and this height is small compared to the other dimension namely the radius. The n here is a outward normal for from the on the surface between the medium one and the medium two. Now, so what happens is this that when you calculate the surface integral of the magnetic field b dot d s. Since h is very small compared to let us say the radius of the cap, the contribution from the curved surfaces of the pillbox is equal to 0. So, as a result I need to only worry about the contribution from the top surface and the bottom surface the two caps. So, here since b dotted with n and the normal n is outward. So, therefore, I get b dot n means b times s which is the s is the area of this cap and then I put a minus sign here because the direction of the normal on the second phase is opposite to the normal n there and since I have decided that this would be my direction of n. So, this is equal to s that is the surface area times b 2 n that is the normal component of the second surface minus the normal component on the first surface and that must be equal to 0. So, this tells me that the normal component of b is continuous. So, b 2 n is equal to b 1 n the second relation that you want is for the supposing I am working in Coulomb gauge. Now, we have seen that the Coulomb gauge means del dot of a is equal to 0 and just as del dot of b equal to 0 gave me integral b dot d s equal to 0. This will give me integral of a dot d s equal to 0 and just the way we had shown that integral b dot d s equal to 0 gives me b 1 n equal to b 2 n integral a dot d s equal to 0 will lead to the normal component of a namely a 2 n is equal to a 1 n. Having disposed of the normal component let us look at what happens to the tangential component this is a little tricky and so let us refer to this picture here. So, this is my interface between medium 1 and medium 2 and I am looking at what is the boundary condition first for the magnetic field of induction b. So, what I do is this I take a an amperian loop I take an amperian loop this way and the way I do it is this that this current you can see that I have provided a direction in which this current is flowing on the surface. So, I take an amperian loop which is consistent with this current which moves out from the plane of the paper on the surface this is surface current. So, what we do is this this amperian loop I know that integral of b dot d l. So, let me partly draw this picture here. So, we have a take an amperian loop this height will be taken to be h and as before I will take h to be much less than the length l of the loop. Let me first fix the coordinate system. So, what we have said is that outward from the page of the paper is what I will call as vector s the normal to the interface is by n and the if you look at a triad then of course, the tangential component will be just the other direction. So, in other words what I find is s t and n is a right handed system. So, let us look at this. So, what I am going to do is I am going to use the amperes law which tells me that integral of b dot d l is equal to mu 0 times the current i that is flowing through. Now, in this case I am looking at a surface current the reason for doing that is if you recall your electrostatics you notice that I had a discontinuity in the electric field when there were charges on the surface. Now, in this case I had integral b dot d s equal to 0 that gave me that the normal component of the magnetic field was continuous. Now, integral b dot d l is not equal to 0. So, this will lead to a discontinuity in the tangential component of the magnetic field and this tangential component will be related to the surface current density that is passing through. So, let us look at how does not do it. So, mu 0 times i now remember the current is flowing in the direction s. So, this will be mu 0 the if current density is j.  Now, I define a linear current density this will be simply j times h. So, the idea is that if just as ordinary current density is something which passes through an area in this case a linear current density a linear current is something which crosses a line. So, therefore, this quantity will be written as mu 0 times well I will take l out and I will write this as integral j d h dotted with s and this is my quantity which is the linear current density k this is an ampere per meter. So, this is mu 0 l k dotted with s. So, let us look at how this quantity is the left hand side. Now, notice the left hand side is I have this I have called as a loop 1 this is loop 2. So, I have taken this as the direction of the amperean loop. So, therefore, since this is along minus t and we have said s t and n form a triad. So, therefore, since s t and n form a triad what I have is t is equal to n crosses. You notice that my direction of the amperean loop was along minus t direction this was along minus t direction. So, as a result what I get is the if I look at the left hand side I get b 1 minus b 2 dotted with minus t and minus t is nothing but s cross n and this is equal to mu 0 k dot s. Now, the left hand side I simplify by using a dot b cross c equal to b dot c cross a. So, this will be s dot n cross b 1 minus b 2 and this is also dotted with s. So, as a result I will get n cross b 1 minus b 2 is equal to mu 0 k. I can rewrite this since I already know that normal component of b are continuous I can write this as b 1 minus b 2 is equal to mu 0 k cross n. This is simply obtained by trivial algebra because when you do a cross b cross c you get b a dot c minus c a dot c a dot b and I know that n dot n is equal to 1, but the n dot s and t become equal to 0. So, this is a complete statement of the boundary conditions that are applicable on b. What I want to do now is to use this to find out what will be the corresponding boundary condition on the vector potential a. We had seen because of use of Coulomb Geng's since del dot of a is equal to 0 the normal component of a was continuous. We will now derive the corresponding relation for the tangential component of a. So, here what we do is this we are using the fact that a dot del this goes to 0 as h goes to 0. So, I know that a t is continuous well this actually is fairly straight forward because I know that a is curl of sorry b is curl of a. So, curl a dot d s which is same as b dot d s is equal to 0 and by using Stokes theorem curl a dot d s is same as a dot d l the line integral. And since a dot d l is equal to 0 just as we had talked about b dot d l and obtained the from the Amperian loop here as I take h going to 0 this will give me a t the tangential component is continuous. So, what we have proved essential is that though the tangential component of b has a discontinuity because of current density on the surface the surface current density the potential a the vector potential a both its tangential and the normal component are continuous. In other words the vector potential a at the interface is a continuous quantity does not suffer any discontinuity. However, we are now going to show that though that is so the derivative the normal derivative of the vector potential suffers a discontinuity and this arises from the discontinuity that the tangential component of b has if there are surface currents. And that we had seen that b 2 minus b 1 can be written as equal to mu 0 n cross k this is what we had just now proved. Now, we write b 2 and b 1 in terms of the curl of the vector potential. So, this is curl of a 2 minus a 1 this is equal to mu 0 n cross k. So, notice what you could do is you could multiply n cross on both sides. So, that I will get n cross del cross a 2 minus a 1 equal to mu 0 times n cross n cross k. Now, I know that a cross b cross c is b a dot c minus c a dot b and that turns out to be equal to simply minus k. So, it is minus mu 0 k. Now, from this it is fairly simple to show that the normal derivative of a suffers a discontinuity. I am not going to explicitly prove it, but you could take this as an exercise. See basically what you do is this we had shown that T n and S they are a system of right handed coordinate system they are like Cartesian coordinate. So, therefore, you can treat them like your x y z and expand this out. Now, when we expand this out you will find that the derivative with respect to n the this is very similar to you have derivative with respect to x y z the definitions are very similar will turn out to have a discontinuity. So, what we say is that the normal derivative of a has a discontinuity. Remember that the vector potential a both the normal component and the tangential components are continuous. Now, I am going to do something which is rather unusual we had derived a vector potential and we had seen that for a magnetic field because del cross of b is mu 0 j I cannot define a scalar potential unlike in case of electrostatics. However, accepting where I have sources supposing I have a current distribution, but I am looking for solution or looking for the magnetic field in space other than where there is a current distribution. Now, then at the point of observation my current density j is equal to 0. Now, in the current density j is 0 at such points I must have del cross of b is equal to 0. Now, if del cross of b is equal to 0. So, j equal to 0 implies del cross of b which is equal to mu 0 j that is equal to 0. So, that enables me to derive define that b can be written as a gradient of a quantity which I call as a potential for dimensional reason I put in a mu 0 there and minus I retain to have similarity with the electrostatics where we have defined the electric field as minus gradient of the potential. So, therefore, del dot of b equal to 0 the Gauss's law implies that the scalar potential satisfies a Laplace equation del square phi m equal to 0. So, what we will do is to look at this is a useful thing in certain circumstances and what we will do is to talk about couple of examples of how to calculate magnetic scalar potential. So, let me take first a line current and we of course, know that if I have a line current for a line current the magnetic field. So, if I have a infinite current at a distance r the magnetic field is given by mu 0 i by 2 pi r r is the distance from the line and I know its direction is azimuthal direction. We had seen that what you have to do is to hold the wire with your thumb pointing in the direction of the current and then of course, the way your fingers curl they will give you the direction of the magnetic field there. Now, since we I have a cylindrical symmetry let me look at what does gradient look like in cylindrical coordinate system.  So, since the gradient of phi is b and b is only direct in the direction phi. So, I do not have to worry about the dependence of phi on r and z I only need dependence on phi because that is the only term which gives me term proportional to the unit vector phi. So, as a result what I have is 1 over r d by d phi of phi m that is equal to if you compare with this expression remember I had in my definition I had a minus sign. So, therefore, this quantity must be equal to minus i by 2 pi r r and r will cancel out and this gives me the scalar potential to be equal to minus i by 2 pi times the angle phi itself. Notice one thing that this potential is not really unique because every time you circle the origin the you know your angle phi increases by 2 pi that tells us that the curl does not have the vector potential does not have certain amount of liberty that we take. So, that is about line current little more complicated, but let me talk about a magnetic dipole. Remember that we had talked about that a current a small current carrying loop is an example of a magnetic dipole and the magnitude of the dipole is simply the area of the loop and the direction is outward normal and the outward normal of course is defined by how you circle the current loop. So, we had already derived an expression for the magnetic field for a current loop. So, let me write it as a magnetic dipole the magnetic field corresponding to a circular current carrying loop in the coordinate free representation was written in one of the earlier lectures to be minus m by r cube plus 3 m dot r r divided by r to the power 5. Actually all of them are inversely proportional to r to the power 3, but there are 2 r's in the numerator here. So, let us try to find out how to write this. So, this is mu 0 by 4 pi.  Now, that is fairly straight forward I take the magnetic moment m the angle between the magnetic moment m and the radial vector I take as theta. So, that this you can do it as a fairly simple exercise. So, basically what I do is I have a magnetic moment m there and a vector r there. So, this angle is theta and I know that the remember that this is the angle angle is measured from the direction of the magnetic moment which is taken to be along the z axis. As a result the unit vector theta is along this direction. You have to realize that we always define the direction of unit vector as a direction in which the quantity increases just as outward direction of r the direction in which r increases is the direction of the radial vector. Similarly, the direction in which theta will increase will be the direction of the theta vector just as we define unit vectors along x y and z as the directions in which x y and z increase.  Now, I want this quantity to be written as minus mu 0 times gradient of a quantity. Now, I claim that this can be trivially shown that this is mu 0 by 4 pi. I am simply writing down now what is the expression for gradient in the spherical polar system and that is r d by d r plus unit vector theta 1 over r d by d theta and just for completeness I will also write down unit vector phi 1 over r sin theta d by d phi. Now, notice since my magnetic field does not have any phi dependence it tells me that I could choose the vector potential or a potential here to only depend upon theta and phi. So, that this term will always give me 0 and it is fairly straight forward to see that if this quantity operates on minus m cos theta divided by r square you get the expression which is in the previous step. Now, you can see why this is d by d r.  So, that gives me 2 m cos theta by r cube unit vector r.  So, therefore, this is the expression from where I need to derive my potential. Now, I want this quantity then is identical to minus mu 0 I already have a mu 0 there. There is a 4 pi there. So, that I must absorb inside this minus I have taken care of. So, I must write it as a 4 pi r square times m cos theta I will write this as m dotted with unit vector r alternatively I could simply write the vector potential as equal to m dotted with full vector instead of the unit vector r divided by 4 pi times r cube. So, this is my vector potential corresponding to a magnetic dipole. Now, notice this this is now I want to write it in a slightly different way. Now, here is my current loop this is the current loop on the screen. This is the direction of the vector m and what I have done is this is the observation point. Now, suppose this observation point makes an angle makes a solid angle d omega here. Now, my expression for the vector potential is m cos theta by 4 pi r square. Now, I know that how to write down the area the projection of the area in terms of the solid angle. So, I write this as I times d a that is. So, I have a 4 pi r square is I times d a that is my magnetic moment. So, therefore, this quantity can be written as I by 4 pi because you have this d a by r square d a cos theta by r square is nothing but this solid angle d omega. So, as a result this phi m can also be written as equal to I by 4 pi I is the current passing in that loop times d omega. Now, this is an useful formula to have and let us look at what is the problem. Now, suppose I have a loop here. Now, notice that if this loop is encircled by a path which crosses this loop like for example, this one then the according to Ampere's law if I am going to integrate on this path then I will have a discontinuity because integral of b dot d l is equal to mu 0 times the current enclosed and in this path there is a current enclosed. But on the other hand suppose I make this line current loop as a rim of let us say a surfaces. Now, as long as I go from one point on the surface to any other point for example, this path along this path without crossing the loop then my potential will be single value. Now, if it does cross for example, this path then there will be a discontinuity. Now, this will enable me to write down what will happen to the potential. Now, you notice integral b dot d l let us do it on the screen on the paper integral b dot d l is equal to minus mu 0 del phi m dot d l. And by fundamental theorem of calculus this quantity here is nothing but the change in the potential. So, it is minus phi 0 I will use this sign delta to show this is the amount of change. And this must be equal to mu 0 I if this is intersecting a current loop. So, therefore, the discontinuity delta phi m is going to be simply given by minus I. Now, let us look at this then. So, what we have done is this that here I have a current loop arbitrary current loop. Now, and I am looking at what is the scalar potential at a point like this. So, firstly let me take this loop I have shown it here on this picture. Now, I can divide this current loop into large number of small loop each one of them behaves like a magnetic dipole. Now, you can see that if you have adjacent loops then of course, the direction of currents will cancel out from the adjacent loop leaving me only the outer boundary. Now, this is of course, a type of networking that we have done earlier. So, therefore, the contribution of this entire loop comes from the solid angle that this entire loop makes here because only the outside things on the boundary will continue. And so therefore, keeping in view that my discontinuity is supposed to be minus I if I make one loop which means if I change this if I go around it the solid angle increases by 4 pi. So, therefore, my discontinuity should be minus I and taking care of that it tells me my expression for the scalar potential phi of m should be given by minus I omega by 4 pi. Having done that let us switch over to a slightly different subject. If you recall when we discussed electrostatics we had seen that if I have a medium and it is subjected to an electric field it polarizes the medium. Now, in our case we have something very similar accepting I do not have an electric field, but I have a magnetic field. And we know magnetic field will exert a force on moving charges, but there are moving charges inside matter there are atoms. So, matter consists of atoms atoms have electrons. Now, the electrons provide me the orbital motion of the electrons provide me atomic currents. Now, the other thing is that as we know that each electron has an intrinsic magnetic moment because of its spin. Now, so as a result the atomic currents if you like have two parts one is because of the orbital motion of the electrons and the other one is not strictly a quantum mechanical language, but other one is because of the spin motion. That is a rather classical picture, but this will do for our purpose. Now, what happens in the presence of magnetic field? So, there are two major effects we will be talking about. One is something which will be talking about in a forthcoming lecture, but you must have already learnt about it from school that there are materials which are known as diamagnets. And what the diamagnets do is if there is a changing flux it opposes the change in the flux. As a result this leads to the magnetization or the current which will oppose the direction of the in which the current is flowing. There are materials which are known as paramagnetic where the macroscopic charge transport will be the additional thing will be in the same direction. So, what I do is this that my total current consists of two parts. One is the type of current if you like conduction current which we have been talking about that is what I am calling as the macroscopic charge transport. That is the bodily motion of the electrons that takes place for example, in a metal. The other one is the atomic currents, but of course the atomic currents are too tiny to talk about individually. So, what we do is to do an average atomic currents and I define a quantity called magnetization. The magnetization is defined as I take a collection of material and find out how much is the net magnetic moment. I vectorially add the magnetic moment of each tiny dipole and find out how much is the magnetic moment per unit volume and of course I give it the usual way in which I define by taking the volume to be small. So, delta v going to 0 sum over 1 over delta v sum over i m i. So, this is simply the magnetic moment per unit volume and this is given a name magnetization. This is analogous to what we talked about as polarization in our electrostatics lectures. Now, what we are going to do next is to calculate the vector potential of a magnetic material. So, we had seen the total contribution will be because of two things. One is that I have the charge transport and of course I have the average microscopic currents. So, let us look at this picture here. In this picture I have shown at a position r prime a magnetic moment m and I am looking at the field at the point p. Now, recall that for a magnetic moment m the vector potential at a position r taken from the position of the magnetic moment the tiny dipole is given by mu 0 by 4 pi m cross r by r cube. This is the expression which we had derived sometime back on the vector potential of a magnetic dipole. Now, what we have is I have a collection of magnetic dipoles. So, therefore, this a of r is given by. So, I take the location of the magnetic moment at r prime and I am looking at what is the field at the point p. So, with respect to this position it is at position r minus r prime. So, as a result by simply summing over I get vector potential at the position r is given by mu 0 by 4 pi. I will have to integrate over all volume. Magnetic moment this is actually the magnetization because I take a volume d v.  That is very analogous to the expression that I wrote down.  This is this into this is my magnetic moment and the rest of it is exactly what we had done earlier. Now, I want to write this in a slightly different way. If you recall when we did the electro scatics we had very similar situation and what we did is to resolve it into two parts. One which we called as the bound volume charge and another we called as the bio bound surface charge. The question that we are asking is is it possible to have a similar separation in case of the vector potential? The answer is yes and let us see how. So, what we have done is this. The first thing that we do is we will write this as mu 0 by 4 pi integral over the volume m of r prime cross.  Since my position r here is a fixed position I am going to take care of the negative sign by taking the gradient with respect to r prime and that I am denoting by a prime there and I will write it as 1 over r minus r prime d cube r prime. Now, I am going to be using several vector identities now, but the first identity that I will use is this that if I have a del cross of a scalar times a vector this is grad phi cross m plus phi times del cross m. Notice in this expression here I have got m cross gradient. So, this is very similar to what I have here, accepting that here the grad comes on this side m comes on that side. So, that is can be taken care of a by very minus sign minus sign. So, if you now use this relationship and plug it into this expression I will get as a quantity which is well I have a del prime. So, del prime cross m by r minus r prime and the minus sign that I have talked about. So, if you do that then what you get is the following. So, I am using this expression to write this del phi cross m as this quantity minus that quantity.   Now by argument very similar to what we had in case of Stokes theorem accepting that Stokes theorem usually was used to reduce the curl of a quantity the surface integral to a line integral. In this case because it is a volume integral we will reduce it only to a surface integral. So, this will give me minus sign is already there minus mu 0 by 4 pi integral over surface of n cross m of r prime divided by r minus r prime cube d s prime prime because I have always putting prime over that and the next one I simply write it the way it was mu 0 by 4 pi integral over volume del prime cross m r prime over r minus r prime d cube r prime. . So, notice that this is there is a surface integral this is taking the role this is taking the role of a current a surface current. So, I can define a surface current J s of well r prime or I am now changing it over to r as minus n cross m of r and this is my volume current density which is simply del cross. So, J of if you like magnetization current density J m of r is equal to del cross of m of r come back to this screen here this gives you that there is a volume current density due to magnetization and there is a surface current density due to magnetization. Now, what we are going to do next is to use this expression for the vector potential and try to see how does it affect or how does it give me the magnetic field how does it change the magnetic field due to the magnetized magnetization of the material. We will continue with this next time.


Oh, sorry. I should probably speak about what I'm doing. Self-instructions. Self-instructions can be over-to-covert. You just kind of witnessed the covert ones. I was pretending acting a little bit, but... So, you're just telling yourself what to do. You're working yourself through a chain or a sequence of tasks, right? So, this is what I'm going to do next. This is what I'm going to do next. Hi, Ryan. You need to speak about the video in this particular way. I don't know why I greet myself, but I do. So, hi, Ryan. Do this, okay? Then do this, and then do this, and this, and this. So, as long as you're telling yourself what to do, prompting yourself to engage in a particular response in a particular way at a particular time, that's called self-instruction. You're telling yourself what to do. It can be covert. Inside your head, right? Covert. The response is still going on. You're observing it. No one else can observe it, but you're observing it, so it makes it a good enough thing. Or you can do it overtly where you speak the instruction to yourself. So, I'm giving myself a countdown. Now, three, two, one, and it's time for an ending.


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Okay, okay, where's the guy at? Oh, that's the guy. He's it. Run away. What? You guys saw the computer screens? I want to go back in there. Oh my God. Okay, he left. Let's check out the computers. What is that? Oh my God. And then the little manatee and then big smoke over here. And the guy's hiding in there. Hey, what's up, butters? Oh my God. We just found the table. No, somebody help. Help me. I'll give you money if you help me. Oh, okay. No, no. I thought that was the guy in. It kind of looks like the same guy that was it. We got a lany face in the Oof head.  That boy is like rolling. I got you. Don't worry. Okay, let's just go after him, even though he's the one that's supposed to come after us. He just passed us. No, save him. No man is left behind. Hey, the guys are twins. Hey, don't get your brother. That's messed up, man. He's going after his twin. That is so sad. Okay, so that's the evil twin. Oh, wait. Where is he going? He's so quick. Hey, hold up, man. He's just staring at us. Oh, you're not going to get me, buddy. I guarantee you one thing. You will not freeze me. He just gave up.  Oh, there's a corn. No. Oh, how did he not get us?  Let's just follow him. I think we'll have fun and we just follow this guy. Run, butters. I got you, my boy. Oh, dude, you guys saw that skills. We just went straight past him. I got you. Oh my God, he almost got me there. Okay, no. Hey, this seems pretty hard for the guy that has to freeze everybody because we could just get each other. No, and he got me. Okay, you must be very proud of yourself. Round of applause for the man. Okay, wait. Hey, no, leave JG alone. Okay, seems like everybody's just trolling this guy. This is his chance to get us all here at once, but... Oh, let's look at the MLG horns. Someone's going off. Okay, watch it, man. You need some money. You want some money, boy? We're ready. We're ready to wait. Hey, come back here. Oh, he went around. Okay, where's he at? There he is. It was a blast, right? Oh, man. Oh, no. I got you. Oh, we won. Shrek music. The world is going to roll me. Oh my God. Okay, so let's go check out this side. We haven't really slipped around over here. Oh my God, really, butters? Really? Oh, is this shooting range? That's pretty cool. Practice your shooting. Jesus, hit your shots, kid. Okay, where is this over here? Hey, oh my God. Okay, this is too much for me to handle. Just talked the game already.  Revive falling players before you get any stabbed. So it's like zombies, but with memes, with dead memes. Oh, no. So there's zombie memes? Dude, I was just stepping on this guy. I'm sorry, man. That didn't mean to. Wait, why are they dressed the same? Everybody's dressed like the same here. What's up, Blastroid? Okay, let's get everybody. Oh my God, Shrek. Oh, watch out, buddy. Wait, he turned into, like, gnomes. What? Oh, no, there's a noob with a knife? Okay, that is not good, because you guys know these noobs are crazy, man. Okay, watch out, everybody, with the dead memes. Oh my God, look at Shrek. Why did he turn into memes? I mean, into gnomes. What does it say? Oh my God, I don't know, but what is he holding? Oh, boss spawns in three seconds. Oh, we even shot the doge. T-Po Shrek is the boss. Oh, he just quick sculpted him. Shrek, oh. Wow, help me. Somebody help me, help me, butters. Pick me up. I think you can pick me up, man. Just help. Oh, can we shoot on the ground? Oh, I don't think so. No, oh, we can. Provide me, JG, thank you. Provide me, please, no. We just took the owl. Bro, we just got quick sold by Shrek. Who would have ever thought that would ever happen? Okay, who is that? Is that CJ? I mean, Smoke. Yep, that's Smoke up there. Just chilling. Oh, nothing's like base wisdom. Oh my Lord, let's try to do this. Okay, uh-oh. This seems a little hard. Oh God, okay, we got it. Don't worry. Okay, we got this, man. Oh, okay, one more. I'm sorry, Rio, it's okay, man. Don't worry about it, buddy. Okay, oh, so that's it? I think we get points or something if you complete it. I don't know, man, but let's go see. Oh my God, a caveman's punch, Bob. Hey, who is that? I don't even call that. Hey, nice headphones, bro. Wow, Rio has become a joint. Move radio, play songs from the Roblox. Kind of like that. I want the radio. They get a little, oh, it's Kanye West and Lil Pump. Wow, it's both of them. Okay, stand feedback, training ground. Okay, so that's pretty much it. I mean, it has a lot of stuff like in the lobby. It's not a boring lobby like most games. This, can we shoot Shrek? My God, are people still alive? Let's back to them. Oh my God, what's up with this guy? King Jason, he's a noodle head, as you can see here. We got Azura looking cool. Oh, okay, I see y'all. I see y'all here. Put in, oh my God, what is up with that? Oh, oh, and Sannik is there too? Okay, uh-oh, this guy seems like he's in a ball right here. Get out of there, boy. Okay, he's cheating out of there. I guess Shrek, he's trying to like throw his knife. No run. Dude, that was so close. How did he not get him? Is he the last in the life? He is. You got this, man. Uh-oh, dude, he just jumped over like the whole crowd. Oh, you got this. I believe in you. Oh my God, Shrek is throwing knives at you, man. Watch it. Did you guys see Shrek in the back? He's like quick-scoping knives. How's that even possible, man? Wait, is he fighting the boss? Is that the boss? Uh-oh, watch out with the Shrek, man. The Shrek's are, I think the Shrek's are like the biggest danger in this game because they're the only ones that seem to throw the knife and they got him. Oh no, well, he had a good run, man. Okay, oh, look at Shrek. Oh my God. Dude, that looks creepy. Oh no. Okay, wait. He's the last one. The game has time left, 60 seconds. Are you kidding me? We have to wait till he's like, oops, there's something there. You got all the survivors were killed. No. We'll be here next time. Hey, hey, crafting. My God, there's a lot of crafting stuff here. Two menu shop. There's a lot of, wow. Look at how many stuff there is in the shop. Oh my Lord. There's a lot of stuff. Hood classic. This is a hood classic. Look at top players. 80 is gone. Level 69. Steven Gonzalez. Uncle David's 56. He's third place in his own game. That is not acceptable.  Okay, what is this? Infected your Sheriff. Okay, I guess. Where's the Sheriff's? Hey, this is a cool map. This map looks pretty epic. Avoid getting infected. Who's the infected? Oh, JG, are you trying to shield me? We're on the same team here, man. Hey, what is this place? It's pretty cool, man. Look at this. Whoa, can we go here? Okay, so I think there's like an infected person and he has to infect all of us. Oh, there you go. Wait, who is it? Oh, it's Shrek. Okay, watch out with Shrek. Oh, can we jump on these? Oh, we can. Okay, we got some parkour action going on. Oh no, watch out with Shrek. He's evil. Watch out for the yogurt. Okay, where is he at? Oh, hey, watch out, buddy. You kind of scared me there, man. Oh, okay, so the infected people turn into the memes. We got the dojo. Oh, okay. Watch out, Joy. You got to be very careful. These people are very sneaky. Oh, there he is. Boy, oh no, we got to help that guy. Oh, I'm trying to get him. Oh, I got him. I just saved his life. Okay, I'm going to stay up here, man. This is where I feel the safest. Uh-oh, exotic butters. Don't do it, buddy. I don't think he sees me. Well, oh, no, we got a Shrek income. JG's a Shrek. JG, no. They got JG. Oh, no. Oh, God. Oh, God. Oh, look at all these dodges. Oh, my God. He went just past us. Dude, did they not see me? Hey, no. Get away. Get away, Joy. I'm sorry, but I have to... I'm going to go hide, man. There's like a whole squad. Oh, there's someone up here with me. Oh, no, they're coming up here. Dude, watch it. Watch it. Oh, no. Oh, dude, we just went like in his face and we got him. Oh, gosh, no, there's too many of them. Oh, look. Oh, God. There's too many of them. We took the owl. Bro, there was like 10 people going at us. That was insane. All right, you guys. Well, this was attack murder on Roblox. If you guys enjoyed the video, make sure to slap the like button. If you're new to this channel, make sure to subscribe and hit that notification bell. And I will see you guys in the next one. For that, it will become a dead meme too.


Everybody force and jk ripping 2223 one-on-one basketball hobby ten bucks full case number 27 tiered You can find those results anytime at late in sports cards calm There's a Lincoln chat if you're watching live you want to check it out again Let's see what we got y'all our last case the very last case last boxes. I'm gonna lose boxes either Let's do it. I want to see Thomas. I want to see downtown See one of one one of one time's moments auto Steph Curry maybe run it back Steph Curry Start eating my jersey not biased for the Warriors by the way Just we pulled it one at one time was Steph Curry last year All right, here we go. That's a good start Very good start got a rookie Shet home grand number to 40 on the base nice chat okc That works That'll do Go okc. That's scratches on the surface of the mag there thunder and I start for you Eric M That's not a boxing wrapper. I don't know if this the card is miscut or what but it looks like it's Like it doesn't look like it's off-centered. It's like it's getting pinched over here Like the mag is too small for the card for the cards like just not cut small enough I said a box of rapper with that. It's a nice jet and we got a redemption here team Titan signatures blue card of three Deandre Aiton Deandre Aiton that's gonna go to the Suns he Is a blazer now, but he was a son all the flashers products, of course Suns in the Aiton Good photo of this guy's rookie to 25 Josh Menot for the Wolves This is a thunder RPA now J. Dubb Mmm, J. Dubb the 99 thunder on fire. There you go. Eric again Sweet hit awesome patch there. That team is Cruisin right now. They just whooped the magic last night. I imagine you have like half healthy team, but Still Get it pretty good. Clyde the glide out of 40 the blazers and this is time was timeless blue timeless boy Blue timeless it is out on Iverson a 35 Philly Andrew M. Wow, that's sick. That is so that is sick. Oh One oh one playoffs, right? Some damn. Yeah, dude. That's so sick the Iverson goes hard in the In the timeless that's such a cool card big one, too as Dude, he was he was the man in the early 2000s He was the man What up, Dan? Dude Dan was killing it on the next day. This man was on fire yesterday. I'm not alive Josh Green Mavericks number to 15. Oh, Ben caro RPA to 49 Orlando magic Orlando magic Big car Orlando Zijian Love to see it love to see it. Okay, I would say this case is going pretty well so far. I Think I think we're digging this one so far see we get some more teams involved now Anthony Simons out of 99 for the blazers next up is a Rockets Jalen Green out of the 35 nice second-year auto Jalen Green Good when they're Rockets That guy just find some consistency What was the MJ quote the floor is the ceiling Ceiling is the floor ceiling whatever But he has a high ceiling like Christian Coloco rookie to 40 for the Raptors President say it local for Christian Coloco. Yeah, I was debating Palo so I'm so glad the magic had Palo. All right redemption again Rookie jumbo Jersey autograph blue Jabari Smith junior for Houston Rockets in this break Andrew M hit the Rockets in the top tier six years in the second tier. Good Lord, dude guy Andrew Big hits for that guy Whoa, okay, we got the box this way with a sleeve is upside down. Oh boy. We're about to hit a monster Okay, good start good start gold Paul Pierce Number ten of ten the truth There's Caleb called him the answer He either called Paul Pierce the answer or Alan Iverson the truth. I forget which one it was It was like Sam with Frank Thomas Call me he called the big unit Boss dude. Yeah, come on Really bots. All right, this is a Pacers rookie autograph Andrew Neymar to 49 Pacers that team is the team scores a lot of points Ceiling is the roof.  The ceiling is the roof Whatever you said Chris Bosch at a 25 for Miami. Where's Bosch? Oh A timeless gold it is Obi-Toppin Two of ten of the gold timeless for the Knicks Tony with the Knicks that is really cool actually Thought that was the garden the ceiling of the garden. That's that's fire between the legs mid-game. It's pretty wild That's just Yeah, I love calling I love calling a bird man. This is an excellent game right there. So you got Bucks play Lions on Sunday. Okay Sounds good. Yeah, buccaneers play at the Lions on Sunday Lamello out of 99 for the Hornets. We should get Lamello to Orlando jazz on the RPA to 99 Walker Kessler go Well, I don't know if you're in here.  He's Canadian. I was on the Raptors Canadian MJ. He's killing it too All right, that'll do it for the break. Thank you very much everybody pretty nice case there the last one They get a downtown, but had some some very nice hits. Thank you very much going on the personals


Hi, I'm Zor, welcome to Unisor Education. We continue solving non-trivial, unusual problems, unusual in the sense that it's not like the problem which is supposed to check your theoretical knowledge, it's just something which trains your brain, basically. So the purpose of the whole course, which I called Mass Plus and Problems, is basically a training exercise for your logic, for your analytical abilities, for your mind, for your creativity. All the problems presented in this course are not really related in any kind of hierarchical way, you just solve them one after another in any order. So we assume the theoretical knowledge sufficient to solve these problems you do have and well if you don't you can always take the course Mass for Teens which precedes this one and that gives you the foundation for all the mathematics needed to solve these problems. Okay, today's problem, we will have only one problem today and I kind of like it because it's really, it involves, that's why I have only one problem for today, not like three, four, five, sometimes I have one problem but we will discuss it in detail. So here is a problem, consider you have a sum of fractions. So starting from one half, all the natural numbers are in denominators and we have some kind of fraction as a result, I mean if we will sum them up together, get to the lowest common denominator or everything. My question is, can it be an integer number as a result of all the manipulation, all the addition of all these fractions? Well, no it's not, it cannot be an integer number and we have to prove it today. So this is basically the problem, prove that this is not an integer number no matter what n actually is for any n, for any n. By the way, a different way to write it would be sum from n equal to 2 to n, 1 over n. Okay, that's just another recording of the same thing, more compact I would say. So it's not really important right now but it's important is how can we prove this particular, well theorem if you wish, that this is not an integer number. It's not really kind of a usual problem which you have in regular course of mathematics at school. So we need something unusual, some creative approach and that's basically the purpose of solving all these problems which I'm offering. You need some creative approach, you have to think about how to approach this problem and exactly the process of thinking and finding the solution which is not the usual kind of solution is what is supposed to help you in practical life to solve the practical problems which always exist in front of you. So this has absolutely nothing to do with real life.  So I will suggest two solutions. Okay so let's start, I mean first obviously you have to pause the video if you found this on the video somewhere on YouTube or whatever and think about this problem yourself and then continue listening to my solutions and you might actually come up with your own and I will gladly put it on a web in Unizor.com with your authorship so just send it to me as a solution. You have my email on every screen. Okay so here is my first solution. Among these numbers are numbers which have the form 1 over 2 to the power of K. Like one half, one fourth, one eighths, one sixteenths etc. So this is 1 over 2, 1 over 2 square, 1 over 2 to the cube, 1 over 2 to the fourth power etc. So let's choose the one which is the right most here somewhere here you have this. Now this is, well since this is the right most that's the minimum among all numbers, all fractions of this type this is the smallest one because it's to the right and our denominator is increasing. So the biggest K when 1 over 2K is still less, it's still greater than 1 over N. Okay now let's think about how do we add all these numbers together. Well we all remember something which is called least common denominator, right? So what is least common denominator? Well one of the ways basically to explain it is the following. Prime number has certain prime components because every number can be represented as a combination of prime numbers. Let's say number 18 can be represented as 2, 3 and 3, right? These are all prime numbers. Something like 65 is what, 5 times 13 again, prime numbers. Now then we take all the prime numbers from every number of these and compact them together without duplicating. So if you have already prime number here 2 then you don't have to really, one of these 2. Now there are 2 prime numbers too, right? 4 times 2. So we need at least 2, we have only 1. So we get 1 from here, then 1 from here, now from 4 we already have 2. So we take only 1, 2. 5 is a prime, 6 has 2 and 3 so we don't really need primes from 3, from 6 etc. So we combine them together and that would be our least common denominator. So this is the number which is divisible by each one of these and the smallest number which can be divided by any of these. Alright, let's consider a simple case. Simple case would be, let's say 10. Now for 10 the least common denominator, what is it? So we need 2, we need 3, 4 has 2 twos, right? So we need an additional 2, so it's 2 square, 5 we need 5 because we don't have 5 as a prime number, 6 we need 2 and 3, we already have 2 and 3, 7 okay, 8, 8 has 2, 2 and 2, right? So we need the power of 3 here. Then 9 has 3 and 3, we have only 1, 3 so we need 2, 3 and then 10 is 2 and 5, we already have 2 and 5. So this is the common denominator, least common denominator and it's 25, 20. Now how do we do the addition? Well, we assume that every one of these should be actually 25, 20 in denominator. So what would be numerator in this case? Well, we divide this by 2, so it would be 12, 60, 25, 20, 1, 3, we divide by 3, it's 20, 40, etc. And the last one would be divide by 10, it's 252 by 25, 20. I would like you to pay attention to one particular number here, the one which is 1, 8. One 8th would be here, 25, 20 and one 8th would be 315. So this is one 8th, this is one 10th, this is one half, this is one third and everything in between. Now I would like you to pay attention to one important factor that 315 is odd number. All others are even and here is why.  Now this is power, the k is equal to 3 in this case, one 8th is 1 over 2 cube and we have 3 twos in our list common denominator to satisfy one 8th. Now all other numbers have less number of twos in them, 2, 3, 4, 5, 6, they have 1 or 2, this guy has 2 twos but nobody except one 8th has 3. We specifically chose one 8th as the number with the greatest number of twos which means when we will divide this list common denominator by any one of these we will still have at least one 2 left. If we divide by 2 we will have 2 left so it's still even number as a result, that's why it's even. If we divide by 3 this, 2 is not involved so 2 will still be there and this is still an even number. By any other number we will divide except one 8th. We will have still some number of twos left but only if we divide it by the least number of this type we will have all twos exhausted and we will have everything else, all other prime numbers and obviously all other prime numbers are all odd so their product is odd and that's why we will have odd numerator in this case. So only in this case we will have odd numerator. But we have some other for example what if we will have to a hundred, right? What happens? If we will go to a hundred, obviously after one 8th we will have one 16th, we will have one 32nd and one 64th, right? So here we need 3 twos, here we need 4. So the one with the greatest number of twos will be 64 which is 2 to the 6th degree and they will have 2 to the 6th here and only this one will have all these twos basically cancel out when we divide by 64. Everything else would still have one or two or three or whatever number of twos left and will be even and only this one will be odd. So obviously this is a general rule that if we will take 1 over 2 to the power of k which is the right most to this boundary to the 1 over n we will have only one particular odd numerator after applying least common denominator and all others will be even. Now if you have all even numbers and only one odd, their sum will be odd, okay? So numerator of the entire thing when you add these numerators together will be odd but denominator obviously is even because there are twos in it. Now odd number cannot be divided by even number to get an integer number, right? So if you have odd number x divided by even number it cannot be some kind of integer number because integer multiplied by even will get even not odd. So that's the proof that this is not an integer number. But I would like to suggest another proof while I still have some time, okay, this is the wrong marker. All right, so another thing is instead of looking for one over 2 to the power of n or k whatever I will look for denominator which is a prime number. So there is a prime number which is the largest not exceeding n. So 1 over, so pn is the largest prime number which is less than equal to our n which is the end of our series. And this basically solution is very much analogous to the previous one. So again this is the right most fraction with the prime number in the denominator, okay? So now, now I would like to tell you that this prime number does not participate in any other denominator. Well, it does not participate in all those fractions which precede it because this is the largest prime number, right? And this is prime so it cannot be, it cannot divide any of these numbers so it's greater than these ones. So none of these is divisible by pn. How about those which are to the right of it? Well, that's very simple actually. The problem is what is to the right of pn? 1 over pn plus 1, 1 over pn plus 2, etc., right? Now what is the next number which is prime actually? Well the next prime number should be between this one and the next, right? So what I would like to say is that if this is the largest prime number there is no more numbers after that which are divisible by pn. Why? Very simple case. If you have 1 over pn what is the next number with denominator divisible by pn? Obviously 1 over pn times 2. That's the smallest number which is closest to this one with denominator still divisible by pn, right? Now, if this is the greatest prime number which does not exceed n, I'm saying that this is impossible, why? And here I would like to refer you to a so-called Bernhardt postulate. The Bernhardt postulate is as follows. If you have some prime number, if you have some any number, let's say m and the number 2m minus 2, there is always some prime number in between. Now this was not really proven by Bernhardt. He just proposed this and checked this for numbers from 2 to 5 million, whatever. But it was proven later on and it's not a trivial proof so that's why I would like to use this Berthrandt postulate as a given. We might address it in some other case but if this is true then it is impossible for pn to be the largest number, prime number less than n and still have this one. Why? Because between pn and 2pn, well between 2pn and 2pn minus 2, there is always another prime number. So that prime number will be bigger than this one and still less than this one. So if pn is the largest prime number, there is no other denominator divisible by pn. That's what's important. Obviously, there are no divisible by pn on the left but now I'm telling you that on the right from this number to 1 over n, there is also no numbers, no denominators divisible by pn because the smallest one, if it's still less than 1n, I mean this is actually greater than, the denominator is greater, then if this is the case then we will have another prime number and this will not be the largest prime number, not exceeding n. Okay, so being done with this, everything else is simple. If pn is the only, if 1 over pn is the only one which has prime number pn in its denominator, the least common denominator contains only one pn. So this least common denominator will be like 2 times 3 times blah blah times this pn times something else. Pn will be represented only once to cover this particular denominator. Now when we will multiply this to numerators and cancel out whatever is necessary, pn would cancel out and you will have all prime numbers here and we will have the pn for this one, we will have pn present for this one because it will not cancel out with anything else or even after 1 over pn. So all these will be divisible by pn except 1 and if all the numerators divisible by pn except 1 then we will not be able to divide, the result of division of some of the numerators divided by the common denominator will not be integer because in the numerator as a sum we will have everything divided by pn, so it will be like a times pn times something which is not divisible by pn divided by something which contains least common denominator which has many prime numbers including this one. So this one cannot be integer number because this is not divisible by pn, if this is integer number let's say k then you will have on the right you will have k times something which is divisible by pn and this is not divisible by pn. So again we have proven in some way it's similar to the first proof this one but it actually involves prime numbers and very important theorem which I was just talking about the Bert Rans postulate. I do suggest you to read the notes for this lecture they are more detailed and also there are maybe some other more precise information which I for instance I have given very brief kind of example of how to do the least common denominator I have elaborated this much more in the text for this particular lecture and to get to this lecture it's mass for mass plus and problems you choose arithmetic and this is a arithmetic O2 lecture. Okay that's it for today thank you very much and good luck.


I've got two. I think I'm... Am I working here? Hello, meaningful innovators. It's super exciting to be here. I'm... God, what am I? I was an artistic director of Coney. We made a massive interactive theatre for a while, which basically just means that the audience is the hero. So my job has always been to create frameworks in which audiences do amazing stuff. So it's just a pretty great thing to be able to do. So, for example, the last show I made was called Early Days for a Better Nation, and it basically meant that the audience created a new nation each night. So each night, different audiences across the UK, we toured it around the elections, created it, entered this fictional nation, it's in complete ruins, and they negotiate with each other with different factions and outside forces like the World Council, sort of UN meets US, and lots of different structures, and they create the fundamentals of a nation. So my job is to work with people like academics, experts, audiences in building these structures, so that every night audiences can come together, be absolutely awesome, and have brilliant new ideas. It's pretty good. So I'm now no longer that, but still a little bit. I've just left because Wyatt and The Space asked me to be their creative fellow. I don't know what it means. I have a very strong feeling that they don't quite know what that means, but it's pretty cool. So you then start pondering what you're going to do, and one of the things that I was really interested in, that I speak at these kind of things, mostly in the arts, this is quite businessy for me, so I'm very excited about that, but you speak at these sort of things and it's full of awesome people. I don't know about you, but just hearing the speakers this morning, I was at that speaker's dinner that was mentioned earlier, and everyone is incredibly cool and sort of changing the world, and it's super inspiring. But sort of thinking about this fellowship, I was thinking about who is making the future, because I'm utterly convinced as a theatre maker who makes structures in which awesome things happen that the future and many of our social structures are constructs that are made by someone in sometimes which awesome things and sometimes which awful things happen, and it's about how those are created. And I know, because I do this in theatre, that a lot of the time you don't really notice the structures you're functioning in, you're functioning within the boundaries that are set up by someone or something else that's well beyond you. So the future being sort of relatively important to most of us, and the idea that that's a construct, something that's made by people, or four people, it made me really interesting, so I had a bit of a look around. And what I found is that the future is being constructed before our very eyes, mostly by people like us in this room. Congratulations, you're very, very, very lucky people. But that also means that there is a very specific perspective. So I was interested in whose voices aren't there. So, because you've got the famous quote, the best way to predict your future is to create it, which we are all doing here, and again like awesome people here trying to change the nature of businesses run, but as one of the speakers said, for a lot of people this is just not the reality. So I decided that what would be really interesting is to look at people that might be on edges. So the thing I'm going to do over the next year is create the almanac of the future. It's going to be a practical guide to the next 50 years. It's going to be a practical guide for the next 50 years we want. So it's a relentlessly, I always say, to why it's radically optimistic, which they seem to like. But it's basically a worldwide intergenerational think tank to think of the future we want. So the idea is that I'm going to work with a group of 15-year-olds and 65-an-overs on each of the continents and we're going to dream up the next 50 years. What do we think is important? What do we want to do? What do we want to keep? What do we want to change? What do we want to progress? What do we want to be radically different? Each of the continents, not Antarctica, unfortunately, funding. I'm going to work with each of those groups individually. So 15, I am bringing the generations together on each of the continents and then we're going to link them up together to have a chat about what we all agreed on and what we didn't. I'm really interested to see where the juxtapositions are, like which continent is holding which continent back. Who wants something different that just would be completely opposite to something that's happening across the world. So I'm interested to see the differences geographically, intergenerationally. Where do we agree? What's easy and where are the wars for us? So after that, we're going to hook them up with experts and whoever wants to become involved with it. Just use the power of digital to extend the conversation to bring in people who can answer some of the questions that emerge from the initial process, but also to ask the other people, you, possibly, probably the people already more invested into the future, what they think about the agenda being set by these 15-year-olds with the support of 65-year-olds. So we're doing, because this is wired, a multi-platform publication creating an insight into possible future scenarios from a crowned world. You have to do something. So for me, what's really interesting, as I said, is the dreams we all share and the possible futures that clash. So what lives on the edges of these conversations where what are just the shared values that we have, what are just shared dreams we have, because those should be the easy ones, right? And then where are we clashing continuously? And where are some of the beautiful things that we're saying right here? How is that being received elsewhere in the world or in a different generation, and what does that mean? So I like to think of it as a bit of stargazing, but, you know, with a pattern. So I wanted to talk to you about a couple of things that I thought might transpose from this project and the way I'm thinking about it, because, you know, you have to make it useful. So the first one is that everyone's an expert. So because I'm looking at the next 50 years we want, literally everyone is an expert on that. I thought 15-year-olds would be like the Uber experts because, you know, it's their lifetime we're talking about. And I thought 65 plus would be the other sort of group that are really expert in this because they've really sentiently seen the last 50 years, so they sort of really get 50-year lifespans and what it means. So in my practice, whatever I do, it's very egalitarian. Everyone's an expert. It's just that you need to figure out what they're particularly expert at and what that brings. So I nick the sentence from Seth Godin, whose book this is, it's not an amazing book. It's all right if you're really interested. But I really, really like the title. And, you know, I stole it, so I might as well attribute. But I think it's really interesting that if you start treating everyone as an expert, and it's a bit about that investment of power in a weird way, you know, the thing that we've just talked about, if you take that seriously, if you talk weirdly, lucidly and semi-grown-uply to everyone, and people feel that you're talking in the same way to them as to people they think are different in social hierarchy and that can be higher or lower, they start thinking very differently about themselves within that conversation and start bringing in things that they normally keep back or are either afraid to share or they feel that position they're in doesn't allow them to share, it's really useful. It's also very useful, by the way, that I make art because it's useless, which has been the most useful thing in my career in the universe because it means that people can leave their agendas at the door a little bit. Just as soon as you bring people together, even with the best of intentions, a little bit of agendas come up.  So people are quite cleverly starting to manipulate and massage the conversation from the start. Being an artist means that I get access to the playful human underneath everyone, the playful expert. It becomes really generous. The other thing that I want to talk about is the long now. Again, stolen. The long now is a sentence that comes from the long now foundation which is a foundation that looks at the long now cunningly enough. I can't remember. I think they look at a thousand years. I just wheeled that back to 50. Apparently I've been recently taught by a futurist that I am part of a new trend called the no long trend. I'm very excited. He said I'm like Dyson's 40 year light bulb, the film Boyhood, which was 12 years in the making and Google's long now foundation. There is a real trend for people that are looking at stuff that lasts again because I think we've become quite bureaucratic about how we think about the future. It's quite incremental. It's making little changes. Again, I'm talking to the wrong audience, not you. Them are there. There's a lot of incremental change. A lot of short term is a lot of changing the status quo by incremental change. Thinking very long into the future allows you to take away a lot of the hobby horses, the politics, the things that were well rehearsed arguments. It allows to transcend politics, current institutions. You no longer have to talk about the NHS, which is awesome, but can talk about health and how we ensure health long term. Again, it frees off people from their rehearsed arguments and they're close by examples. Thinking 50 years from now opens up really quite radical new ideas. Then, like with the Almanac, you work your way back. What does that mean practically for next year then, if that's where we're going over there? I think this is becoming really, really important. Climate change, there's so many really drastic things, antibiotic resistance. There are some really big projects right now that we need to attend to collectively and we have to see as something that we do together. It's something that I was reading the other day. I love making random analogies. I was reading about cathedral builders and it struck me that that's where some of these projects are like that again. These were people who started something they knew they wouldn't finish in their lifetime. They just kicked it off and hoped that four generations from now it would be finished and cool. I always feel a bit sad for the second generation because the first one got to start it and the last one's got to finish it and the one before got to almost see it and the second generation is just like whatever. But I think that kind of thinking, that kind of multi-generational, how can we start things now that are not for us but for well beyond? How do we deal with things so big? Like again, climate change, the antibiotic resistance, there's so much going on that that is long term and we have to work together. We don't only have to work together because it's such a big problem. It's also about diversity and divergence. It was mentioned earlier I think by Miriam. None of us are as clever as we are together and there's a lot of scientific research and again you all know this but it's not being acted upon as much as you'd think that diverse teams in whatever way you look at it come up with better solutions than homogenous teams whether that's geographically, generationally, sex, socio-economical or ethnic diversity. Diversity is good, it's science dude. You can just sort of take this. So for me, I think he's currently on Team Hillary. He is a guy who thinks about social structure and for him it's just a logical next step from Moore's Law. So the power of creativity rise exponentially with the diversity and divergence of the group. So for me this is very much where the almanac comes from is let's get these very diverse voices together and see what they share and it's really important to understand but also see whether they're diverged because those are the things we need to work on. This whole thing is a backlash against instant gratification against getting people together who know already that we're sort of going to agree and self-congratulate about how cool and forward thinking we are. It's also against short-terminism. I think we'll see a bigger and bigger backlash against this. We're already seeing it in culture. I'm sure it will come as it is. When I say culture, I mean like the arts. I'm sure we'll see it in wider culture more and more as well. So I just wanted to break down the project. So think tank and to dream up. So again, the almanac is a think tank to dream up. So the first two cents for me are that they're both fun and challenging. I mean like doing things with meaning is fun. I think there is something about if you want to include different voices, different people, different constituencies that are, you know, we've been talking about the communities that are around whatever it is that you're doing that are so important, get them involved but make it exciting. We're all pioneers, we're all innovators. Everyone is, everyone's got great ideas but speak to that part of people. Not the feedback form. Like the more boring you make it, the more boring the feedback is that you're going to get, the more boring the ideas, the more... You know, I fill them in. I've never said anything interesting on them and I think very few people do. But once you bring people into a genuine conversation about innovation, like literally everyone and anyone I've ever talked to has got brilliant ideas. Also a lot of really, what's the nice word, bad, bad ideas. So it's not about building your entire structure, your entire idea set on everything that people have to say. I mean dear God, I make theatre and I do it iteratively and I get audiences to see unfinished work and they always have the most ludicrous, ludicrous, unfeasible suggestions basically which are film. So you don't listen to that but you do listen where it comes from.  Where does that come from? It's probably because I'm a little bit too boring right now. So I can solve the boring bit, probably not by the helicopter. But if you make it fun, I don't know if I can solve the boring bit on my own. Anyway, if you make it fun and challenging, if you genuinely push people to think beyond their own clichés because that's also the thing. We all need to get our clichés out of the way first. Then you can get access to an amazing think tank just from the people around you, from your customers, from your suppliers. Again, I loved the way Mary was talking about the ideas coming from loads of different directions. I think that happens, the unexpected happens if you give it space. Worldwide and intergenerational on my project is diversity. So we're back to cow's law. I think whatever you do, you need to make it as diverse as possible. Make sure that there are opinions that will clash. If you do not have clashing opinions in whatever you're doing or researching or being part of, it's not working. And then the future we want. So I think that working with diverse groups is about destination. It's not about road map, that's your job. Your job is to figure out how to get there. That's not, you know, there might be some good ideas, but that's where the experts, normal experts come back in again. But use your constituencies around you to create a destination point. Something to aim for because, again, this is a stolen quote, but I liked it. Our inability to plan for the future together is ruining our present. I'm well over time, so I'm going to leave that just for you to think about. So I would like to invite you with me to make the world a think tank for the best possible future. Let's do that together. And let's make sure it's fun. Thank you.


Everyone has a need for health care. For a person with a developmental disability, additional resources are often needed to ensure quality care. In California, the wellness initiative was created to address this need. With the initiative, the Department of Developmental Services is working to ensure the health and well-being of Californians with developmental disabilities. What's being done? Let's take a look. The Department of Developmental Services is committed to improving health care for persons with developmental disabilities. More people with developmental disabilities are living in the community than ever before. As the number of people needing health care increases, so does the challenge of providing quality health care services. In response to these emerging challenges, DDS launched the Wellness Initiative. The Wellness Initiative mission is to promote the health and well-being of all Californians with developmental disabilities. This initiative has since become a catalyst for innovative projects designed to promote quality medical, dental and mental health services for all Californians with developmental disabilities. DDS offers leadership for the Wellness Initiative by working consistently to create and facilitate partnerships with Wellness Initiative participants and stakeholders. Some of these participants include regional centers, universities and state and local agencies. These partners contribute to the Wellness Initiative by conducting special projects and trainings to address the diverse health issues of persons with developmental disabilities. California's 21 regional centers are non-profit corporations which coordinate services for individuals with developmental disabilities. Regional centers are developing programs that foster collaboration among parents, educators and administrators. These programs continuously strive to improve the physical health, mental health and dental health of persons with developmental disabilities. Regional centers focus on expanding local resources, such as training consumers how to use law enforcement services and understand that they are there to help. They also work to develop clinical expertise by providing training to healthcare professionals. Regional centers educate service providers about the healthcare needs of individuals with developmental disabilities and remove geographic barriers to healthcare by providing telemedicine clinics to allow persons with developmental disabilities access to professionals that can provide a diagnosis, treatment and close follow-up. Through partnerships with several California universities, the Wellness Initiative develops and funds multiple training opportunities for physicians and other healthcare professionals. The trainings focus on health issues pertaining to persons with developmental disabilities. Services include the physician assistants, consultative and training network. This service provides free consultation by telephone to physicians, nurses, pharmacists and physician assistants throughout California who are treating persons with developmental disabilities. In conjunction with stakeholders, the Wellness Initiative has produced a video that demonstrates appropriate self-administration of medication practices for service providers. The video is intended to teach best practice guidelines and help ensure the safety of consumers who are taking medications. Each year, DDS and ARCA, the Association of Regional Center Agencies, work together on the Health and Wellness Forum. This statewide forum provides an opportunity to share information on critical health issues affecting persons with developmental disabilities. The forum presents the most current information available. In addition to the Wellness Forum, information is also available on the Internet. This specialized health information website is an excellent resource for healthcare providers, caregivers and families. It offers quick access to a variety of information on medical care, healthcare provider training, resources for families and resources for healthcare providers. It is funded by DDS and coordinated by the University of California at San Diego School of Medicine and the Center for Health Improvement, a non-profit agency dedicated to improving the public's health. DDS Wellness Initiative staff produces several resources for persons with developmental disabilities and their service providers. The literature available ranges from a periodic publication highlighting health issues to an annual report of current and past wellness projects. All of these and other publications are available to the public by contacting DDS. The Wellness Initiative provides a warm line that families and persons with developmental disabilities can call if they are having trouble accessing appropriate healthcare. It is called the Warm Line for Health Access. A Wellness Initiative staff person will return the call to assist in directing the caller to appropriate services. The Wellness Homepage website can be found by going to the DDS homepage at www.dds.ca.gov, then selecting Initiatives and Projects and finally clicking on Wellness Initiative. The DDS Wellness Homepage provides the most current information on regional center projects, partnership activities, and publications. California is working to assist those with developmental disabilities in receiving timely and appropriate physical, dental, and mental healthcare services. For more information on the DDS Wellness Initiative, call the DDS Health and Wellness section or visit the DDS Wellness Homepage. The DDS Wellness Initiative, promoting the health and well-being of all Californians with developmental disabilities.


We're actually putting together a special program right now where we're going to take 15 kids over 3 months and we're going to use the same system that we're using. Wow. Yeah, definitely. We're being more accepted. I think it's because, you know, I'm using these frequencies every day. They know that this is revolutionary and the world needs to hear about it. They need to learn about this technology because people need it. These kids need it. It's an epidemic right now with so many kids having ADHD and autism.


So this is the start of the death road, but the weather up at this top section does not look very good. Hi, I'm Lavi. And I'm Oli. And this is our hero, Bumblebee. Together, we are attempting a Guinness World Record to become the youngest pair to circumnavigate the globe by motorcycle. Join us for season three, here in South America. Okay, this is what it looks like when you have to take down all of your equipment, all in one go. Oh, what an awesome room, hey? Bye-bye, most beautiful apartment in the world. This was our view for the last five days. See the church and the city skyline over there. We're up on the fifth floor. It's beautiful. We actually got an upgraded room because the previous room had a leak in the ceiling. And that was that room. And then they built a new roof. That room. And then they said, okay, we're going to move you to this room. And this one was a triple room with a better view. Beautiful. Hitting the roads of La Paz, let's do it. On a nice Sunday, empty roads. Yeah, this is definitely a good day to ride around the city. Oh, definitely. Good morning world, welcome back to the channel. It's day number 316 on our circumnavigation around the globe by motorcycle. We're here in the crazy busy center of La Paz in Bolivia. And we've already been refused petrol this morning, which is a bit strange, but we had heard that locals are quite hesitant to give fuel to foreigners. Locals actually have a discounted rate. It's a government subsidized rate. And for foreigners, the rate is like triple, the local rate. And so they have to like fill in loads of details in the machine and say, we're filling up for a foreign vehicle and a lot of petrol station attendants, they basically say, no, I'm not going to do that. We have spent actually the last five days here in La Paz. It wasn't our intention to spend that much time here, but we got food poisoning, both of us, with our nice vegetarian meal. So today was our death road day. But we're laying dead in the bed already because yesterday we had a meal outside and yeah, we came back and our bellies were really aching. And the whole night we were just like, you know, throwing up and going to the toilet. Yeah, can you believe it? No, I absolutely can't believe it. But anyway, we got food poisoning and then we were like in bed for like two days. We couldn't move. It was not a very good time. I can assure you. So yeah, it was pretty hectic, but we're feeling all right now. That's good. We are definitely ready to hit the road again. And we have a really, really, really exciting day today because we are heading to the famous death road here in Bolivia. Yeah, yeah, it's going to be absolutely awesome. So let me show you guys the route for today. We finally have a map of Bolivia. So we are here in La Paz. And today we are going to be making our way over to the mountains to take on the death road before returning to La Paz, heading back down to Patakamaya where we will be turning off and heading towards the border of Chile. Look at all these food places. Yeah, looks nice, eh? Oh man, look at this car. It just comes straight past me. Yeah. It's an incredibly busy morning in La Paz. And the entrance to the death road is 66 miles away. But man, I do not know how long it's going to take us to get out of the city. Probably the whole day. Oh my God. And it's a Sunday. We chose to leave on this day. We were like, wow, Sunday will be a great day because the roads will be nice and empty. It'll be good. But no, it's the same. Yeah, it's the same or maybe even worse. So we have another setback in our plans as well. And that is that the border between Bolivia and Peru that we were going to take at Lake Titicaca is actually closed at the moment. So Peru is going through some political turmoil at the moment and there's a lot of protests, there's a lot of violence and there's a lot of areas basically that you can't go. And one of those areas is right in front of the border between Bolivia and Peru. So basically the government has closed that border. So it means that our only option is to take the border between Bolivia and Chile. So back into Chile for like the 10th time and then take the border between Chile and Peru at Arica. But what it does mean is it's gonna add nearly a thousand miles onto our journey. So it is a big detour. Yeah. But before we left Bolivia, we still wanted to take on this death road. We'd been wanting to take it on since we got to La Paz. Yes, I remember when I was a child and there was a documentary on the TV about the death road when it was actually still used by people. So I had some flashbacks and I was like, that's so cool that we actually now can drive the death road. So it looks like it's gonna be a super busy but super exciting and quite scary day today. I'm ready for it. Yeah, and it's already 10.45. So better hit the road, let's go. And then stuck in the traffic. Better hit the road, let's go. Oh, look at this. But it's a really nice market here actually on the side of the road. You can buy anything and everything. Yes, she gave us fuel. Whoa, thank you. We basically said to her, oh, is it okay nine Bolivianos a litre, which is triple the local price? And she was just like, yeah, that's fine. And we don't even know if she put it through the system as a foreign price or whatever or if she took the difference. But anyway, good for her, that's fine. I'm happy for them if they can just keep it. And also at the beginning I said to her, oh, we don't need the receipt, it's fine. Exactly. Bolivianos triple the price, it's fine. Exactly, just try to encourage them. Just give us fuel, please. Because man, we can't keep riding to different petrol stations in the city. That's insane. Well, we're starting to get up and out of the city a little bit more now. We're heading up into the hills. Wow, the hills there look stunning around here. Absolutely stunning. But despite our illness over the last few days, we at least got to have one day to have a look around La Paz, which was nice. And when we rode in, we noticed they had the cable car system. So we said, oh my God, that is the way that we have to go around the city. And we did. Good. We saw the past from above. And it was absolutely spectacular.  I don't like riding in it, but seeing it from above in the comfort of a cable car is absolutely awesome. Yeah, it was really, really cool. But oh my god, we went as well to the witch market. There were like baby llama fetuses everywhere. It was absolutely insane. I mean, this place was crazy, but super interesting to see. I actually found out that they used the baby furters to build in their house. When they build a new house, they put it in there, in the foundation. No way.  Yeah. And it's like for protection. But it didn't protect the baby llama very much, did it? No, they are pretty crazy. Wow. We're out of the city. We've just passed the payage barrier heading out and into the mountains. And look how clear the roads are. Beautiful. It's like a dream. Yeah, it is a dream. And you have already massive and epic mountain views. So first we're heading to a town called Koryoko. And that is actually where the start of the death road is. Now you can start on either side of the death road. One entrance. Oh, look, they're waving at us. Hello. So you can either start at the top of the death road or at the bottom of the death road. But the interesting thing about the death road is that the traffic flow is the opposite to the normal road. So if you're heading downhill, instead of being on the right like we are now, you're actually on the left. Now the left side is the cliff side, the really, really, really crazy high and dangerous cliff side. So basically if you're heading downhill on the death road and you pass another car, you have to be on the cliff side and the car or motorbike or whatever coming up is on the inside. So for that reason we decided that we will take the death road from the bottom to the top. And that means we'll be on the left always against the cliff. And if we find a car coming up to us or our bicycles or motorbike, then we will be safely against the cliff edge. And there will be other dangerous outside. Exactly. Plus as well, going uphill, I believe you've got a bit more control than going downhill, going downhill. You basically have to keep the brakes on nearly the whole time. You've got the issue of maybe slipping and with this bike being as heavy as it is, we just can't risk that. So I mean we dropped Bumblebee already a few times and to drop it on the death road, it wouldn't be very wise I guess. No, this road is incredibly narrow at points and that drop is nothing to be taken lightly. That is a really, really dangerous drop on the side. So first we've got to make our way over to the small village called Corioco and then the party starts. Look at that. We can't even see the top of the mountain here. It's just in the clouds. Crazy man. Oh no, we've just come over the pass and it's raining on the other side. Oh, it's incredibly wet over here. This is terrible. It is actually terrible, yeah. And we can't see anything. Oh man, yeah I think we're going to have to get to this town and see how the weather is down there because this looks absolutely terrible conditions to take on the death roads. Yeah. I mean it was so sunny and nice on the other side and that's why we were like, oh it's a great day, let's go, let's go, let's do it. Yeah. I don't know anymore. Flash plate. So this is the start of the death road. So this is where you would start if you're heading from the top, heading downhill, but the weather up at this top section does not look very good. So still not 100% sure what to do. Ravi's trying to ask these guys over here if the weather will get better. We're actually considering now just starting at the top and rolling down because we're here. Hola. Yes. Today is not possible because the road is closed. The road is closed. There is the landscape. So you can't get to the bottom? Today is not possible. Today is not possible. Thank you so much for letting us know. You can come back for the new road? Yes. To Koryoko. When did the landslide happen? Approximately two days ago. Okay, thank you. Gracias. Thank you. The dream is over. Yeah, it looks really that all signs are against us going down or up this road, unfortunately, but instead we bought some crisps. So we're having a little snack break here and try to get over the big, big disappointment. I mean, we were so happy to go to the salt lakes. We were so happy to go here to the death road and like both things are not very recommendable in the wet season. So please guys, come here in the dry season. Sometimes you just can't go against Mother Nature and we just have to accept it. Yeah, we have to go with the flow and that means we actually will turn around and head to our real destination to head to the border to Chile. Yeah. And at least we can say that when we get back over that way that there's going to be no rain because we're heading towards the Atakama Desert. I'm actually excited for that. So that is it from the death road here in Bolivia. Disappointment. But I actually spoke to one of the workers that was coming up the road in a truck and he told me that this landslide is a massive landslide and he told me that it's still ongoing now and because of the wetness and the rain, the landslide is just continuing. They can't start working on it. They can't do anything basically. He basically said to us, I advise that you don't start going down because you're only going to have to turn around and come all the way back up again and even turning around on that road is going to be incredibly difficult. So anyway, we're going to make our way back over to La Paz now, back over to the sunny side of the mountains which I'm super excited about but at least we got to see the beginning of the death road and we can be sure that someone somewhere does not want us to ride that road. I can't see anything.  We're going to fill up our water here because this is beautiful, fresh rainforest mountain water and we can't say no to that. Oh my goodness. It's a good job I've got all my rain gear on, eh? I think the water went inside this glove. Ha! Yes! I am completely wet, hey? Some beautiful, fresh mountain water. We're back on the sunny side of the mountain. Yeah it's still very cold though but at least it's not raining anymore. Yeah I didn't realize until I saw the sign that this pass from La Paz over to the death road that we just crossed is actually 4,650 meters high which means it's the second highest pass we'll have ever taken. Absolutely crazy. Yeah it was pretty hectic and a little bit disappointing as well that we couldn't take the death road but I mean we had so many signs you know. There were so many obstacles for us it was like don't go to the death road don't go to the death road don't go to the death road we actually came to the start of the death road and right there it tells us a person literally comes and tells us don't do it. We are definitely going to come back for a series that's going to be called Bolivia in the dry season and we're going to go and sell out to uni and we're going to do the death road and we're going to stay dry the whole time. For sure. So we just stopped here for a snack in the town of Patakamaya. And this town is the turn off towards the Chilean border so we've looked on the high overlander and we found a hostel about 60 miles closer to the border so we decided that we're going to make one last push for today so tomorrow we're going to Chile but it's really hard to find like anything around here we just wanted to stop have a coffee have a sandwich but there's like no good options here so we just have to settle for a pack of crisps. It's a hard life as a vegetarian hey it's a hard life but there are always some nice potato chips also can you see some difference about Ollie? Oh the boy is getting a haircut so funny actually because it took him like 30 minutes and then I checked afterwards and I think he forgot like something in the back here his barber shop looked really cool so that's true he was really friendly and really nice I liked him a lot this is my first haircut in such a long time as well and I think you're going to have to get some scissors and touch it up later hey yeah can see there's a super mursado over here oh my helmet's so squeaky see what they've got quite a lot of stuff actually this is one of the biggest shops we've seen in Bolivia so far noodles yeah I can't go wrong with some of those and then maybe some cookies for dessert is that really five bolivianos? wow it's really cheap what else have they got? what else have they got? some juice can of peas to go with our noodles yes please that'll do for tonight beautiful gracias here we go I got us dinner and dessert I think I finally have my front camera working again because I couldn't get it to turn on all the way since we had that snack break anyway we had to go around in circles quite a few times but we finally found hostile cariwata which is right here and apparently we have a place to park bumblebee as well wow hola wow it's a puma yes he said in this area here hola buenas hola como estas? muy bien hola hola mucho gusto mucho gusto robert robert atrizia a mucho gusto looks like we found ourselves room for the night good evening guys we made it happy and alive yes yeah we were going round and round and round this town trying to find this hostile which at first we thought was like five k's down that way then we went back into town and then we reached like a military border post hostile ah aqui it's okay ah here no this way okay this way okay he was like no not this way but eventually we found it and we're here yeah it's a really really nice place to be here with two little cute kids running around they're trying to talk to us but we don't understand anything and there's a cute little dog and a little cat that they keep playing with yes yes and the dog's name is apple they're just preparing our room at the moment I don't know if they were expecting guests but it's good we at least have a room for tonight yes so we have decided to take some rooms for the next three to four days actually because we have to make a lot of distance 1000 miles detour exactly it's like a massive detour and we are so far behind schedule now that we said okay let's try to get up tomorrow morning very early let's try to smash out some k's let's not waste any time yeah because taking rooms we can just quickly get the stuff off the bike put it in the room next morning at sunrise put it back on the bike and hit the road and we have a lot of miles to cover so it's going to really benefit us so now we are about 60 miles from the Chilean border so we reckon that tomorrow we might be able to do a double border crossing crossing over to Chile and then up and crossing over at Arica to Peru as well we'll see if we can make it in one day it's going to be interesting to try and that's it from us today at the end we did 200 miles we hope you enjoyed the video if so please give us a thumbs up subscribe to the channel share the video with your friends and family comment below and if you really really really like our videos then you can join us on patreon the link is in the description below we will see you next time guys look at this nice room here everything's so nice and clean and a light and we have a little bathroom as well oh look at that it is as well just 50 bolivianos which is about I think it's about six pounds yeah amazing good price yes for this place and a place a secure place for bumblebee and guys I just want to point it out again that our travel kettle is absolutely our superstar because we can have noodles for dinner like super cheap and we can have coffee in the morning as well super cheap so if you want to travel cheap and you take some rooms get a kettle it's amazing and Ollie's working hard loading on the footage of the day yeah