So, let's get to show the road here. So, we do have the next exam coming up, as I discussed, where we'll be taking an exam. And we'll be distributed both after the problem session next Friday. So, this Friday we'll have just a discussion section of posting all of the solutions. There's only one more solution to post problem set nine. And so, a couple of questions. All right. We'll discuss that. I will put it on the web immediately after this. If you're not there at the problem session, you can download it or email it to you. You can probably submit it. And as I said, we'll discuss a little bit more the rules of the game as time approaches. All right. So, last time we were talking about an important sort of class ingredient of the formula. Maybe there's one more, you might say, postulate upon mechanics that we didn't really discuss at the beginning. It's how do you deal with composite systems? And what that means, classically, we talk about the notion of a degree of freedom. So, in classical mechanics, to each degree of freedom, we assign a set of a coordinate and a conjugate momentum, right? And if I have multiple degrees of freedom, for example, I have a particle that's moving in two dimensions or I have two particles moving on a line. Those are two examples of systems with two degrees of freedom, okay? And we assign some canonical coordinates to each degree of freedom. I'm calling them generically A and B. And in classical mechanics, what we said is that the joint phase space for the system is given by that formally what we call the Cartesian product, okay? So the coordinates on phase space are B's, okay? And that's what we call the Cartesian product. Now, as we, I mentioned the division into the degrees of freedom is not unique necessarily. For example, as I mentioned last time, if I talk about a particle moving in a plane, I could define as my, cannot, I can have one degree of freedom be, say, motion along the x-axis and the other one be motion along the y-axis and then there's a conjugate px and py. And that's one way to define the degrees of freedom. Or alternatively, I might choose to have xad, the radial, we call it rho, and the other conjugate, I mean, xbd5, right? And then there's a conjugate pthi and prl. So the division into the degrees of freedom is not unique, right? And similarly, a particle, two particles in a line, I can talk about the coordinate of each particle. I can talk about their relative coordinate and their center of mass. And we'll come back to that. It's going to be important. So that's classical physics. In quantum physics, what we said was that if I had different degrees of freedom, what is true is that observables associated with those different degrees of freedom commute. And that means we can define sort of separate Hilbert spaces for degree of freedom one, for A and degree of freedom B with observables that act on each of those Hilbert spaces. And then the joint Hilbert space that describes states of the system that describes both degrees of freedom is given by the tensor product, OK? And the tensor product is such that we can define joint states of the system that are product states. And we can grab this tensor product in between sets. And what that physically means, because of course the state, in this case I'm talking about pure states, are uncorrelated. That is to say, if I have a product state, the joint probability distribution for degree of freedom A and B is a probability for A times a probability of B, right? And that means that there's no correlations between A and B, because the probability for both things to happen is just the product of one times the product of the other. Does that make sense to everybody? Why uncorrelated probabilities are products? But why, if that's true, why is it a product? I mean that's sort of a tautology. Yeah? So what aspect of probability? What do we say? Let me say it's at the cross terms. When you do the variance, you get just the square of the thing and you also get the cross. That's one way of looking at it? What we were talking about today is like, if I want to look at the probability of flipping two heads, is the chance of flipping one head times the chance of flipping the other head? That's exactly what I meant. So if you wanted to calculate the joint probability of two heads, why did you just multiply them? Because it seems intuitive to do that, right? That's like, so if you're doing tests, you want to use just the normal average, but if you're doing something like stocks, if you want to take an average, you use the other kind of average, like the geometric average, because they depend on each other. The way I look at it is the following. If something's uncorrelated, what it means is the probability of flipping this first point has nothing to do with what happens in the second point. So what that would mean is that the marginal probability distribution, which is the probability for A to happen, I can write without knowing anything about B. That's to say, if I want to know what the probability, you know, Z, you can have 1.9, I want to flip mine, I can calculate mine independently of yours. And what that says, we know from probability theory, if I have a joint probability for two things, A and B, then the marginal probability distribution on A is given by the sum of all possible V values, whatever they are. That's the marginal distribution. Now, if this is equal to a product state of probabilities, well, then it's obvious. This is just the probability of A. It's nothing. It's just given by that. So that's the key idea from logic. Two things that are uncorrelated for this and that to happen is just a product of their happening. There's no correlation between them. And in quantum mechanics, a state like that we call separable. And in this case, what it says is the probability of outcomes of observables of degree of freedom one, whatever that degree of freedom is, are not in any way correlated with the outcomes, they're independent random variables if the probability of distribution is of that form. Let me look at it this way as well. If we look at this in wave function land, so suppose that this was two particles moving on a line. So the wave function for this thing is given by this, where these are the position eigenvectors of particle A and particle B. If the state was a product state like this, this is the wave function. The wave function in this case is just the product of wave function A and wave function B. That means the joint probability distribution A is a function of random elements and probability distribution B, which is the square of each of these guys. Yeah? Can you try a particle's position and like stay in the same one half? Sure, that's fine too. So for every degree of freedom, if I have, you know, I can have as many degrees of freedom as I like. So if I have a tensor product, and we're going to talk about this more, we're going to get back to spin. We haven't forgotten it. We just put it on how I ate it. So every degree of freedom is in a tensor product. So for example, if I wanted to say I have one particle, I want to keep track of its position and its spin, so the system is L2, if it's moving say on a line, tensor two-dimensional complex numbers. That is the Hilbert space associated with particle having a certain position and a certain spin. That's how we perform the right thing. And we'll talk about it. They're called spinners. Yeah? So the parts we have, you know, psi AB is equal to psi of... If that's true. Yeah, if it's a product state. So that's not spanning a new space. It is not. That's one member of that new space. Correct. Indeed. And in fact, this is the definition of a tensor product space. A tensor product space is the linear span of all product states. And there are states in the tensor product space which are not product states. Okay? For example, I mean, I can write one down in terms of wave functions. Let's say, just as an example, let's say I had two particles in a harmonic well. They don't have to be in a harmonic well. If you look at the harmonic out of certain wave functions, I like to call them particle one and particle two. The joint wave function might be this. Alpha u0 x1 u1 x2 plus beta u0 or u1 x1 u0 x2. What is this? Well, this is the particle where one of them is in the ground state and the other is in the excited state. But I don't know which one it is. So there's some probability amplitude alpha that particle one is in the ground state and particle two is in the first excited state. And there's some probability amplitude here that particle one is in the first excited state and particle two is in the ground state. This is not a product state. There's no way. There is no way to write this as some psi x1, psi x2, or phi x2. This is not a product state. But it is in the tensor product space because it is spanned by the problem states. In the same way that I could say, you know, a particle or a vector in three-dimensional spaces, the span of vectors along x, y, and z. Every vector is along x, y, and z. It's the span. This kind of state is not a product state. It's not acceptable. It has another special name called the Nintendo state. Everyone knows about Nintendo. It sounds the coolest thing since Puff Weed. I don't know why Puff Weed is cool, but that's what people say. And we'll talk more about that at our semester. Okay. So, continuing on. So, formally speaking, as I said, the tensor product space is the span of all product states. And that is the space that describes the composite Hilbert space for multiple degrees of freedom. Okay? And we can always write a basis for the tensor product space as looking at the individual bases for, say, in this case, if this were just two degrees of freedom, if I have a basis for Hilbert space A, which has some vectors, and a basis for Hilbert space B, which has some vectors, then a basis for the tensor product space, there exists always a product basis. Again, not every basis needs to be a product basis, but I can always find a product basis by this construction. Okay? Question.  So the dimension of the joint Hilbert space is just in the product of the dimensions. Right? Because that means the dimension is growing if I have multiple degrees of freedom exponentially. If I have n degrees of freedom and each one of those degrees of freedom was associated with the Hilbert space of dimension d, what is the joint dimension of n in such things? In the end. Okay? What do you mean? If the dimensions of all of this space saves or is one, you understand it with one dimension? It's true, but I think you have to be careful about what you're thinking about here. This is the dimension of Hilbert space. It's not the dimension of physical space along which the particles move it. If a particle is moving along a line, what is the dimension of Hilbert space that describes all states of motion of the particle? Does it continue in? It's infinity. Okay? So if you deal with wave functions, you never see this exponentiation of the dimension. Because it started in infinity and the dimension is obvious, it's just infinity. It keeps infinity times infinity times infinity. It's only when you deal with finite dimension of Hilbert spaces that you ever see the exponentiation of the dimension of Hilbert space. Okay? Continue anyway. So another piece of the form that we talked about is, of course, we talked about the tensor product of observables. So I can define, if I have an observable, if I say O is observable, OA acts on Hilbert space, A maps vectors in this to vectors in that. And we have some other observable that acts on Hilbert space B. Then I can define an observable that acts on the composite Hilbert space, map vectors on vectors as the following that this acting on a product state equal to the tensor product of the individual action. That's quite definition. Now, what would you do if this tensor product acted on some general state and you don't know whether it's maybe it's not a product state? Well, it can always decompose it in a basis of product states. Okay? We can insert in here a resolution of the identity. Now, a resolution of the identity is, I'm going to write this, as I said, we have lots of different notations and sometimes when I write something like this, this is a shorthand for EI FJ algorithm with EI tensor FJ. Or I could think about this as the operator of the projector A tensor that's also a criminal. This is an operator that acts on Hilbert space A and this is an operator that acts on Hilbert space B. That's perfectly, they're all the same, they're just written to really in place. So this would be equal O A O B acting on, I'll call it alpha IJ EI tensor FJ where alpha IJ is a probability amplitude associated with that projection. And then I know how to act this. It's a linear operator so it comes inside the sum this guy acts on this basis vector and this guy acts on this basis vector. Let's talk about eigenvectors. Suppose I had a particular operator on a joint space which was of the following form. It was some operator A and it did nothing to B or it's kind of sum. A acts on Hilbert space A and D acts on Hilbert space B. It's a sum of two operators that act on this. What is an example of an eigenvector of this operator? Exactly.  So this gets confusing and this is sometimes I mean come back to that in a certain language here that I often confuse it but I'll come back to it. Alright. So this tells us something about if we're looking in particular if we're looking at let's go back to the time independent disorder equation. Okay? So we seek the eigenvectors so now there are a special class of cases where the Hamiltonian is this form. So suppose H is the Hamiltonian for multiple degrees of freedom. Particles, spatial dimensions as much as we like. Okay? So H acts on our composite open space. For the moment let's just focus on two degrees of freedom. Generally. Okay? So if, let's say EG2 if the Hamiltonian is of the form a Hamiltonian that acts on the degree of freedom A tensor one on D plus nothing, because nothing to A then we say the Hamiltonian is several. Now it's a little bit confusing that with respect to states a separable state was a product of a state on A and a state on B. Okay? But in Hamiltonian land, or in operator land this is a different notion of the term separable. Because this is not Hamiltonian 8 times Hamiltonian B. It's the identity of the other one, right? It's got this sum structure. But that, those words are used. If the Hamiltonian is separable that means that it's eigenstates our product states from the reason we just said it. Okay? So that's why it's called separable. Because you can separate the eigen vector as the product of eigenvector for Hamiltonian A and the Hamiltonian for Hamilton blah, blah, blah, blah, blah. Right? So that we can write energy B, A, B, B as some these are the eigenvectors. eigenvalues are just the sum of the eigenvalues associated with the two degrees of freedom. Now if we have a separable Hamiltonian the Hamiltonian can be separated as a Hamiltonian acting on a degree of freedom A and a Hamiltonian acting on a degree of freedom B then what about the unitary evolution, the time evolution? So let's look at that. But let's say we have the time evolution operator we call this HAB it's the Hamiltonian that acts. This is nothing over space, it's the Hamiltonian. It's got the hat on it, it's not the scripting age. So now I got my time evolution operator to say it's the time independent Hamiltonian and generally speaking we know right? But let's suppose H is separable. Then I can write this as a Hamiltonian acting on a degree of freedom A and I'm just going to write it out in its full pedantic glory T plus exactly. We can write that as the product of the exponentials because this commutes with this obviously it does and as I emphasize the whole point is that observables associated with different degrees of freedom commute and because they can use that means I can write this as a Hamiltonian and I'll be about the identity part and that product now is of this form. So this is the unitary operator acting on Hilbert space A generated by Hamiltonian HA and this is the unitary evolution UB generated by Hamiltonian UB and the joint unitary is the product of the unitary so this is where it gets confused. The Hamiltonian is a sum but the unitary is a product and we call this a separable unitary. Why is this interesting? Well, if this is true then what it says is that if I start with a product state, so if at time T equals 0 the joint state is the system state is a product state which means there's no correlation in space A and K in space B then what is what can we say about the state of the later time? Well, we applied that yet so I get UA acting on psi A tensor with UB tensor acting on phi B and this is just this tensor this it's always a product state so if my Hamiltonian is separable then if I start in a product state for all times I stay in a product state which means if there was no correlation between degree of freedom A and degree of freedom B at the initial time whenever are correlations generated between them which is to say that when the Hamiltonian is of this form the degrees of freedom don't interact so physically to say that the system is separable between A and B is to say that the two degrees of freedom don't talk to one another there's no interactions there's no correlations of freedom yeah I was expecting when you put UA under an eigenstate that you would just be summing the two energies in the same phase they each get their own little phase that's correct okay yeah so the point here is that when we have a separable Hamiltonian separable Hamiltonian implies no interaction physical interaction between degrees of freedom yeah yep we're about to do that indeed I just want to make one nice comment about if this is not true if the Hamiltonian is not separable okay if it didn't have this form and I started in a product state and I did the time evolution then it's not going to stay in a product state in that case we call the unitary entangling it's an entangling unitary because it creates entanglement it creates correlation quantum correlation yeah it would I mean it depends on the interaction between physical interaction so there's physics here there's enam and all that stuff quantum mechanics is not the void of physics right okay so now let's come to the kind of analysis here looking at a specific case let's consider suppose I have a free particle p squared over 2f if say this is in 3b 3 degrees of freedom correct is this separable it's separable and pxp1 is easy we have to say separable with respect to what degrees of freedom okay this is something that is almost always forgotten that what whether this state is separable or not depends on what kind of product structure we are imposing let's say which degrees of freedom are we choosing to analyze the system of respect what if we use p of r p of a that's the next step so we'll come back yeah let this one get at the moment so this of course what this really means is p sub x p on y tends to be the identity on z and this is the x tends to be the identity on z and this is blah blah blah blah why does it order form you have to decide if you're writing the state I'm saying that when I write this, tensor this tensor this, tensor this I have to talk about which Hilbert space I'm talking about so this is Hilbert space a or x you just have to keep track of it and then that's different if I went Hilbert space c, tensor Hilbert space a there are isomorphic Hilbert spaces but in like the same way when you write down a tone vector and you write down the components you just have to know what you agreed on the order I could order them anyway I like but once I write the order I've got to keep saying that that's the order it's just a bookie thing okay but I'm just emphasizing that fact just to say that this in fact is of the form I wrote and this is separable so there's a Hamiltonian that acts on the x degree of freedom there's a Hamiltonian that acts on the y degree of freedom and there's a Hamiltonian that acts on the z degree of freedom okay and the joint eigenstates of the system are just the product of the eigenstates associated with those degrees of freedom so I can write an energy eigenstate as px, tensor p1 tensor pz the eigenvalue associated with these three eigenvalues is the sum of the sums associated with each degree of freedom this is what we said I'm doing this in speuciating formalism you just sort of know this we've done this a million times I'm just showing you how it all fits together whereas I should just put a square root of qn and what is the eigenvector in position space it's a momentum eigenstate e to the i they're plane waves the momentum eigenfunctions are plane waves x px is with our usual delta normalization e to the i x px and so the three-dimensional wave function associated with the this xyz is x tensor y tensor z px dy pz px dy notice this is just normal multiplication multiplication by complex numbers and this is the i over h bar pxx plus pyy plus pzz we've got the h bar to the 3 halves or even the i over h bar p dot x so the wave function is a product state one for each degree of freedom because the Hamilton we've assembled let's get to he suggested we have different ways of assigning degrees of freedom to the system I can talk about r, theta and phi those are three degrees of freedom and then I could write I could try to say that to join Hilbert space of the system instead of having a Hilbert space and x degree of freedom y and z I could principle do this is my Hamiltonian for the free particles separable in that and yet it actually is because of course the momentum squared is equal to the artwork of p if you write if you thought about it as a Lafacian there's that act on the radio coordinates by coordinates and data coordinates of course this is just from you know from classical mechanics the kind of momentum squared over r squared yeah but I mean there's also a term in it though so that doesn't mean it's dependent on r though sure that is a good point here it is dependent on r in that way that is to say you can write eigenfunctions which are in this case separable so a free particle eigenstate I would say there is some radial part that depends on r tensored into some eigenvector that depends on theta times perfectly good because I can separate that these kinds of eigenvectors are called partial weights so separation doesn't necessarily mean that it's not dependent okay right so let's look at some example now with respect to the some potential so because the system is always separable of the kinetic energy for a particle moving in some number of spatial dimensions or some number of particles moving in some number of spatial dimensions whether or not the Hamiltonian is separable depends on the potential because we can always separate the kinetic energy we can do it in spherical coordinates we can do it in cartesian coordinates but the question is about the potential okay so when we have a Hamiltonian which is p squared over 2m plus the potential of the function position then separability depends alright so let's consider for example a particle in a box so what I mean here let's say a 2D box so let's just talk about two spatial dimensions both my potentials are the form that it's 0 if it's inside of the square and it's infinity outside so I have a box where the potential is 0 in here and everywhere out here it's infinite behind is this separable meaning can I write this as a sum of a potential on the x-coordinate and a potential on the y-coordinate as I can it is this is equal to v of x is 0 if x is between 0 outside and the same thing with so this is a separable potential okay what if I had a finite potential well suppose that this were true suppose I had something like that where everywhere outside here the height is v0 is this separable I mean what if I wrote this is that the product is this plus this that potential no it's not right that's the one that would be you know it would be v0 v0 2v0 2v0 I think right so that's not this potential is not the one I wanted the one where it's v0 and everywhere is not separable okay this is not separable now let's go back to the infinite case what are the eigenfunctions and eigenvalues of the Hamiltonian exactly so for each one of these guys we have now my energy eigenfunctions they're specified by two quantum numbers for each one of these guys it depends where I put this guy typically I write in terms of even an odd this is cosine and i xi over l I mean actually it's important I'm going to make this not a square let's say it's a rectangle let's start with that case so let's put that and then I have this with the sine one or two mean x or y for even parity in the odd parity the energy let's go to another photo energy is just the sum of the energies associated with each of those for the energy E and x E and y hr square over 2m k x of nx squared for k to meet the boundary conditions for the i coordinate equal to that's how it meets the boundary conditions as we know so this energy is hr square over 2m pi squared I'm going to factor out it lx squared nx squared plus ny squared lx squared so now one thing we see here is that whether or not there are degeneracies in the system that is to say if there are two different eigen functions that have the same eigenvalue depends on something about the ratio of these guys so for example if the ratio of lx and ly is rational then we have degeneracies ok so for example suppose that lx is y's ly suppose that were the case I had that kind of reaction right then e 1 1 here is the same thing as what well this is 2 squared right so if this is 1 and this is 2 that's the same thing if this is 1 right those two are the same they're both equal to 5 times that mass hr squared over 2m pi squared over lx squared because it's 1 plus this is 1 get that backwards well whenever nx is equal to twice ny and this is and it's it's hard to do that this is the case where I should look at my notes no I have to go the other way around so the other one is the same thing the way I've written it is this like ly is 2lx hold on a second you just do 2 2 on the left side and then it should be I think 2 1 2 1 which way do we want to do this ly is we want so let's say that ly is 2lx it doesn't matter which way I do this right then then we have that the eigenvalue squares here are proportional to and it's a constant times this divided by 4 right so if nx equals 1 and ny equals 2 then the energy in this case is this constant times 2 right now is there another case where that's true the other way around yeah n equals 2 nx equals 2 and y equals 1 why am I screwing this up so good I think a higher I'm just going to switch over algebra this looks better your journals can't submit anything you know I could see this here I'm going to come back because I can't see the algebra right now and obviously you can't do that so I'm going to just let this go for a moment and just say the following which is are degeneracies in this system I'm just screwing it up so now but the point is here that degeneracies are called accidental degeneracies although I can't seem to find an accident meaning that you know if it just happens to be that this were a rational number there can be an alignment of these two things now there's a very simple case where we can automatically see degeneracies and that is suppose that lx equals del y please 2, 2 and 4, 1 so nx equals 2 and ny equals 1 where ny equals 2 also and then the other one is 4 and 1 4 and 1 so in this case we have for this case 4 plus 4, yep that was thank you yay but that's an accident and I couldn't find it is that why they call it that? yeah because it's not in any way obvious it just happens to be that it just happens to line up but if this were true well then it's obvious if this were true then this is what we call if this is the case but this is a indication of is symmetry the reason for this kind of degeneracy is the fact that if I have a square potential then I have reflection symmetry that is to say that v of x, y is equal to v of y, x that is to say if I flip these coordinates the Hamiltonian is the same so if I have a symmetry operator let me just call it the reflection operator and it's I'll call it pi since x into y and y into x and px into p y and p y into p x the Hamiltonian is unchanged because the potential is reflection symmetry so if I have a reflection symmetric potential then the Hamiltonian is invariant under the reflection symmetry operation which means another way of saying that is the Hamiltonian commutes with reflection which means that the eigenfunctions there exist simultaneous eigenfunctions of the reflection operator and the Hamiltonian so um if nx and y is an eigenfunctor this way with the same eigenvalue that must be true yes just a tensor product so as I say often we just leave out the o times here it's just too annoying to write it all the time if I just write two kets to one another that's the tensor product is it kind of like if you were like to put in a projection into x you just have the two functions correct it's exactly what it is that is to say the joint eigenfunction of the system is equal to this eigenfunction and that's the eigenfunction that's what it means and so what is this eigenfunction see this acts on this which is x and y so that's u and x evaluated at y and u and y evaluated at x that's the reflective one now I claim that this these two functions are eigenfunctors simultaneously eigenfunctions of the Hamiltonian the same eigenvalue they are degenerate you see that because so now I'm just going to simplify notation even further just specify them by their eigenvalues the Hamiltonian acting on this with some energy what is acting on the reflected ket same energy but the right it's exactly the same energy because these commute because they commute I can move the Hamiltonian to the other side this then is the same energy eigenvalue so this is an important lesson the point is that if we have a symmetry in this case we have reflection symmetry if there is a symmetry of the Hamiltonian then there will be degeneracies symmetry and degeneracy are interrelated to one another because there is a symmetry operator that this commutes with there are a set of different eigen vectors which share the same energy eigenvalue that's sometimes instead of called an accidental degeneracy this kind of the degeneracy is called an essential degeneracy let's do another example before our time is up here let's talk about a particle in a simple harmonic oscillation well in 2D do you mean that it has a one half kx squared or a one half ky squared with a potential or is it a derivative? exactly so let me explain that so for example here is a simple which is a classical picture let's say I have two kinds of spring I have a particle that's attached by springs to a wall and there's a spring constant kx and the spring constant ky so I can pull it in either direction and there is a resonance frequency associated with that spring constant and a resonance frequency associated with that spring constant okay? what is the Hamiltonian for this? is this a separable Hamiltonian in x and y? sure is what are the energy eigenvalues and eigenfunctions of the system so the energy from omega x times n plus a half so there are two quantum numbers in n and y okay? again there will be there can be accidental degeneracies in this system but there's a special case that I just want to conclude the isotropic say in 2D let's say that the two frequencies were the same the springs were the same so it doesn't matter how I pull it it has a restoring force in that case which is just proportional to the distance from the origin it doesn't depend whether I'm pulling it along x or I'm pulling it along the y it just says as long as I pull it some distance r from the origin it feels the same force it's isotropic okay? in that case we have something special well first of all the energy in that case let's call that omega we see this guy has degeneracy right? so I have energies let's just make a little table here nx and ny so nx is 0 and ny is 0 the energy is I'll just call it 1 okay? when this is 0 this is equal to 2 etc when this is 1 and that's 0 f2 3 now when this is 1 and this is 1 then I have 3 when this is 1 this is 3 I'm sorry 4 5 etc so we see the degeneracies here there are many different combinations of nx and ny which in fact we could write this I mean they're always integers so the energy here is hr omega times sum n plus 1 n can be 0 1 2 3 and what is that degeneracy? n plus 1 so there's a degeneracy in this system which means that there's really another point on the m but this guy is completely independent of that point it only depends on n and not n what kind of degeneracy is this? is this an essential degeneracy? or an accidental degeneracy? it's essential it's associated with what kind of symmetry? it's true it has reflection symmetry but in fact this guy had even more symmetry than that let's look at the Hamiltonian again p squared over 2n plus 1 half n omega squared plus y squared this is my so let me coordinate in row this is p rho squared plus p phi squared right? what kind of symmetry does this have? remember this is just the rate this is just x squared plus y squared it's rotational symmetry the potential is independent it's out of a potential here it's independent of phi it's admiratively symmetric it's rotationally symmetric this has rotation symmetry it doesn't matter where how much I pull it from the origin at what angle it sees the same force okay? so this has rotational symmetry there's a rotation operator for the function of phi and my Hamiltonian is in varying respect to rotations around the z axis we see that classically because the potential is independent of phi the conjugate momentum is conserved right? remember that from Lagrangian mechanics so there's a conserved quantity what is that conserved quantity? it's this which is of course angular momentum around the z axis the angular momentum around the z axis is conserved because we have no torques because the potential is independent of phi which means that there is a symmetry conserved quantities and symmetries are one and the same and quantum mechanics are also related to degeneracies we will pick all that up and talk about angular momentum for many weeks to come


Hey there lads and ladies, it is Petrifying Pumpkins here with some breaking PSVR news that Pavlov Vior will indeed be a launch title. This was revealed via the Playstation store where you can now wishlist Pavlov and it reveals that the days is the 22nd of February 2023. This follows speculation that the game possibly had been pushed back thanks to the release date being removed from the video description of the launch trailer, but it seems that that's not the case and that it will indeed be a launch title. And that's your update from PSVR News. Be sure to like, subscribe, all that usual YouTube and Shise and we'll see you in the next one.


The Equitable Society presents this is your FBI. This is your FBI. An official broadcast from the files of the Federal Bureau of Investigation presented as a public service by the Equitable Life Assurance Society of the United States and the Equitable Society is representative in your community. Tonight's file, the sorrowful swindler. Before opening tonight's file, it is my pleasure to bring you season's greetings from the Equitable Society. This week at the Equitable Society in the lobby of our home office building, we have decorated one of the tallest Christmas trees in New York. This very afternoon as we gathered round this tree in the sound of the traditional carols echoed through the halls, there was one pleasant thought that kept coming to our minds. We thought of all the homes in this country that are celebrating Christmas more merrily, more securely. We thought of all the children to whom Santa Claus will be more real, because someone in that home had the forethought to purchase life insurance. And we of the Equitable Society and the Equitable Society representatives all over America are happy to have done our share in bringing that kind of happiness to so many American homes this Christmas time. And so to each of our three and a quarter million members and to the other millions of Americans who enjoy this radio program, we of the Equitable Life Assurance Society of the United States wish a very merry Christmas and a very happy new year. And now to the file on the sorrowful swindler. With America virtually on the eve of celebrating her first feast time Christmas in several years, the topic of crime seems hardly in keeping with the mood of the day. But then there is a negative kind of relationship between the two. Because it can be said truly that the doctrine of crime is the direct antithesis of the philosophy of Christmas. One is the religion of taking. The other, the religion of giving. And to the criminals, Christmas time is no more than just another time in which to ply his profession of cheating, high his profession of cheating, as demonstrated in tonight's case from the files of your FBI. Several years ago, during another Christmas season, a man using the alias of Colonel Weatherford and a companion in Larsonie were speeding eastward on a crack train headed for New York. You know, Colonel. Yeah? I still can't figure out how come we leave Chicago so quick. I think, Michael, we may sum it up in one word of two syllables. Like which? Police. You mean they were hepto's? They would have been Michael. Shortly after that check I cashed began to ricochet. Yeah, but suppose they get an idea we caught this train and they got the New York police waiting for us when we roll into Grand Central. Please, Michael. I'd rather not have to wrestle with that remote contingency for the moment. Allow me, if you will, to revel in a vision of the unbounded joy of my dear Valerie when I show her the fruits of this little mission to the West. You ain't going to give her the whole five grand. Valerie has demanded a mink coat of Santa Claus and Valerie, my dear Michael, knows who Santa Claus is. Have you got your tickets ready, please? I might get them right here, Colonel. May I check your ticket, please, Madam? Oh, yes. Here's my ticket, right here. Thank you. What time do we get to New York in the morning, Mr. Conductor? Nine o'clock. Well, I do hope my daughter is there to meet me.  Yeah. You may keep this part, Madam. Oh, thank you. Just a minute, please. Yes? I wonder if you'd help me. I have some stock certificates with me, which may be very valuable. Colonel, I'm listening. I'm kind of afraid to keep them in my birth with me tonight. Well, I'll be back directly, Madam, and we'll make some arrangements, I'm sure. Thank you very much. Michael, I think Donder and Blitzen and the other tiny reindeer are about to make a landing on our own roof. Since crime never takes a holiday, neither does your FBI. And at about the same moment that the pompous gentleman on the New York bound train became stock certificate-minded, Special Agent Barclay in the New York Office of the FBI was handed a teletype from Washington. What does it say, Ellen? Well, Jim, there goes my Christmas shopping push with Marjorie today. Oh. They want us to go to work on a swindler. Anybody we know? No, he's avoided federal violations up to now. Well, what's the up to now? He put over a fraudulent deal in Denver a few days ago by posing as a United States attorney. Uh-oh. He may have stopped over in Chicago, but they believe he's headed for New York. Oh, is this his home? He's got a record here. Who is he? Several persons, it seems. Colonel Josiah Weatherford and about six others. Well, here, look this over and let's get busy. Right. While you're digesting the teletype, I'll check with the New York police and also put a cover on railroad, plane, and bus terminals. I didn't quite catch the name. Weatherford. Colonel Josiah Weatherford. Oh, yes. Well, I'm Mrs. Greeley. How do you do? Uh, may you please sit down? Thank you. I came primarily to apologize for staring at you as I did. Oh, I didn't think anything about it.  Then I feel quite honored. She passed on last March. Oh, I'm so sorry to hear that. This isn't to be my first Christmas without her. It'll be most desolate. Yes, I know it will. You live in New York, Mrs. Greeley. Oh, I know.  I'm just going there to spend Christmas with my daughter. How fortunate for your daughter. I suppose that you live in New York. Well, I sort of divide my time between Chicago and New York. I have an investment business with offices with both. Investment business, did you say? Yes. Well, then maybe you'd know about my stock certificate. I beg your pardon. I mean, no further there any good or not. Well, I don't know. May I see them? Oh, dear. I've already let the conductor put them away in a safe place for the night. Oh. You see, I was going to have them looked into while I was in New York. Uh, I could do that for you. You see, my husband has been there about 10 years, and I didn't know he'd left anything like that till... To the other day, I was rummaging around in his old desk, and well, there it was, a thousand shares. All tied together. Thousand shares of what? A load-star mining company. What was that again? Load-star mining company. Loads? That's what I thought you said. Do you know something about it? Uh, well, it's not listed on the exchange anymore that I know. Oh, then, then you mean it's, uh, it's no good. I wouldn't say that. I want to look it up for you. You will let me serve you on this, won't you? Oh, I'd be very glad if you would. Especially since that's your business.  Oh, thanks. Oh, thank you so much. How'd you make out? Santa Claus is not merely knocking at our door, Michael. He's trying to break in with a pack full of gold. Special Agent Ferrell speaking. Morning, Jim. This is Alan. I say, where are you? Good. I was just going to have to hop over there myself. What's up? A weatherford's on that train. How do you know? I tell the type. Just came in from Chicago. He passed a bad check there yesterday. And the ticket agent at LaSalle station remembers selling weatherford and a man with him space on the Manhattan. Then I better run. It's about to pull in. Right. Yes, sir, Mr. Barclay. A man of that description occupied space in car 254. Then what happened to him, Porter? Well, he and the fellow with him got off at Harmon this morning. Uh-oh. I sure wish I'd have known earlier. Well, we didn't know ourselves in time to be prepared for that trick. Well, thanks anyway. Oh, say, uh, wait a minute. Yes? There's somebody might know something about him and maybe she's still in the station. Oh, a little old lady who had the space across the aisle from there. This man gave me a note when he got off at Harmon to give to her when she got up. What does she look like? Well, her name is Greeley. She's about five foot two, gray hair, and she's wearing... Michael. Yeah? Valerie's waiting in the apartment here for me. I prefer to see her alone. Yeah, I prefer the same thing. I'll wait for you downstairs. Splendid. Valerie? Valerie? I'm in here. Valerie, my darling, come here. Wait a minute. Not so fast. But my dear, aren't you glad to see me? I don't know. How was your trip? How was it? Look, my sweet. Yeah, $5,000. Let me see one of the bills. Yeah, genuine. Every last one of them. Oh, my darling, I am so glad to see you. I have missed you so much. And now I can go right down this very day and get my mink coat. Oh, well, you see, Valerie... What's the matter? Well, naturally, you're going to get the fair coat, Valerie. That's right. But tomorrow will be ample time. Worry, my sweet. I'm waiting for the hook. What is it? I merely want to retain possession of the money for the balance of the day. Go on. For $5 a share, darling, I can pick up a thousand shares of lodestar mining stock from a party who doesn't know their true value. Have they got any true value? Have they? Lodestar merged a few years ago with Rocky Mountain. Each share of lodestar is still exchangeable for one share of Rocky Mountain, with today $100 a share. You mean put out $5,000 and get back $100,000? Exactly. Look, a mink coat on the back is worth $40 in the window. Nothing doing. My darling, you can't buy it. You just dreamed it's up to keep him coming across. I swear I didn't, Valerie. She's going to call me any minute. What? Now, don't get excited, my dear. Don't get excited. It's a little old lady. A Mrs. Greeley I met on the train. She has the stock. Oh, yeah? That's probably Mrs. Greeley now. General Weatherford speaking. This is Mrs. Greeley. Oh, hello, Mrs. Greeley. Are you at your daughter's now? No. Well, Mrs. Greeley, I have some good news for you about your stock. Oh, you have? How would you like to have $5,000 in cash for a Christmas present? $5,000? That's right. Good gracious me. You just give me the name of your hotel and I'll be right over in a few minutes. You mean we'll be right over. Yes, Jim? Well, I must over as soon as I could. Good. I think this is our best prospect of getting a line on Weatherford. The Greeley woman checked her bag at the station, huh? Yeah. You got anything out of the conductor? Mrs. Greeley gave him what she said with some stock certificates to keep safe for her last night. Oh? Weatherford was across the aisle and saw it all. Let me see. 10 to 1, he's trying to pull a swindle on her for that stock. Well, I hope she comes back for her bag before the job's done. Yes, but she checked it two hours ago and a lot can happen on two hours. Colonel Weatherford. Yes? If the Low-Star Mining Company is out of existence, I don't see why the group this young lady represents wants to buy my stock. Oh, you should make that clearer, Colonel Weatherford. Oh, the group still controls the Low-Star Company's property, Mrs. Greeley. Oh. And they're going to start operating again and they're willing to pay $5,000 a share for all the old, outstanding stock. Well, maybe I'd better hold on to mine and maybe it'll be worth more after a five. Oh, explain it to her, Colonel. It may be years before it's worth a cent more, Mrs. Greeley, and after all, $5,000 is a lot of money. Well, I'll trust your judgment, Colonel Weatherford. Good, I'm sure you won't regret it. Do you have all that money with you? Yes, here it is. $5,000. $5,100 dollars. Gracious me. Now, if you have a stock for you. Oh, of course, yes. It's right here in my handbag. Landed. Yes. Yes, here we are. And now I've asked you both to put up your hand. What? What's the meaning of that gun? Oh, it merely means that I know as much about Low-Star as you do, you old swindler. And I wish I did have some of the stocks. Well, now, look here. You asked her if she didn't want $5,000 for a Christmas present, didn't you? Well, I'm not going to give up my main coat this easy. Be nice now. And back into that closet over there. Both sell yours. Yeah. Who knows? Maybe somebody will open it before Christmas. Crooks, don't qualify as men of good will, do they? So let's leave them for a moment while I tell you about someone you like. A man who is bubbling over with the contagious good humor that infects all good people this time of year. This week at the Equitable Society, I met a senior vice president coming out of the building. He was carrying a regular pyramid of packages in his arms. And just as I said hello to him, something went wrong with the middle of the pyramid and half of his packages fell out of his arms and slid to the floor. Serves me righty laughed as I helped him to pick him up. Just what I deserve for putting off my Christmas shopping to the last minute and then trying to do it all at once. He paused and chuckled. And I'm the man who's spent his life telling folks not to put things off. My business in life is telling folks not to put off buying the life insurance protection they need.  We smiled and answered saying, yes, there are a lot of people in this world who are much happier right now because someone from the Equitable Society kept urging a husband or father not to put off buying life insurance. Believe me, that's a pretty pleasant thought for a fellow to entertain this time of year. Well, as I said goodbye to him, the thought came to me that it'd be a very fine thing if all members of the Equitable Life Assurance Society of the United States had the same opportunity to know the officers of their society that I have. Could get to know the sincerity and human understanding that they put into their daily work managing the life insurance of three and a quarter million Equitable Society members. I've met all these men and I've yet to find any stuffing in any one of their shirts or any brass in any one of their hats. No matter how important their jobs are, their doors are always open, and their time is always at the disposal of members of this Life Assurance Society. You see the officers of the Equitable Society are the kind of men who take pride in the thought that this week and every week for 86 years, the Equitable Society has been building security for you, your home, and your country. And now back to the file on the sorrowful swindler.  Therefore, being denied recourse to the law, he usually takes matters into his own hands and generally with the same net result. Both are caught. At the moment, however, the little gray-haired confidence woman is trudging through the snow away from the hotel with $5,000 in $100 bills. While behind the locked door of the closet in the hotel room. But Valerie, darling.  Nagging me is not going to get us out of here. You just be glad it's a closet where there's not room enough to swing at you. Michael's waiting just outside the hotel. Sure, probably building a snowman. But he's surely seen the woman leaving by herself. Oh, of course, of course. He probably helped her across the street. You'd think he'd suspect something and come up here to see about us. Oh, no, that calls for thinking. Valerie, if you'll help me push against the door just once more, I'm sure we can force it.  Look, you're in here, too. You got us in, you get us out. $5,000 and get back $100,000. Can't miss, it's a sin.   Wait a minute, wait a minute. Now I'm warning you, I'm getting a mink coat tomorrow or else. Or else what? Or else the police are going to learn where you got that $5,000 in the first place. But Valerie, darling, you can't... You heard me. It's the mink or the clink. Well, it doesn't look like Mrs. Greeley is coming back for a bag. She only checked it a couple of hours ago, give her time. Yes, but in the meantime, this weather... Hey, wait a minute. This looks like our little lady now. Yeah, seems to fit her description all right. She's going over the baggage counter. Come on. No, wait a minute. What? For one thing, somebody's chaling her. Huh? Look over there. And according to the conductor's description, that would be Weatherford's pal. Yeah, you're right. And item number two, do you know who Mrs. Greeley really is? No. I had dealings with her a couple of years ago. That's an old-time operator who's known as Lars and the Annie. What? So far as Weatherford and Sheer are concerned, I'd say at this point it's a question of who has done what? To whom? Well, then let's pick them up and ask some questions. And Miss getting Weatherford? Hey, look, there she goes. And a shadow, too. Come on, Jim. Let's make it a convoy. What good is it going to do to come back here to the apartment? Michael wasn't in front of the hotel, was he? So what? Darling, please, patience and fortitude. Probably saw the woman leave in just a minute. Hello? This is Mike. Say, what's going on? Where the devil are you, Michael? I've seen the old lady leave the hotel by herself and I said something crazy about this. Where's she now? I'm in a telephone booth at the State National Bank. I said where is she?  We've been robbed.  If you slip up, Michael, it'll be a cheerless Christmas for you and me. What do you mean? Did you ever spend Christmas behind the bars, Michael? Not yet. Then do what I tell you or you will. And he shot her back from the phone. He must have contacted Weatherford. What do you make of all this? I'd say the Weatherford does now learn how it feels to be swindled himself. He took somebody for the money first and then Annie's taken him. Yes, but she puts it up to us to take them both. What's your idea?  And you? If Weatherford's pal saw me talking to Annie... What would he probably think? Well, that you were confederate at first, maybe. That's all I wanted to be sure of. Why? Maybe this'll do the trick, Jim. Come, darling, let me help you trim the Christmas tree.  I tell you, I'm waiting on a telephone call from Michael. I'm surprised he can even use a telephone. You should not disparage Michael's intelligence, my dear. Ha! Providence beat me to it. Michael is a simple soul, but a loyal one with a great amount of common sense. And on occasion displays a flash of superior intelligence. Maybe you ought to be working for him then. May I fix your cocktail? No, and don't try to soften me up because I'm... Yes? It's me, Colonel. Well? I've done what you told me. Where is she? You better... There are four corners, Michael, you know? Yeah, but I'll be standing on just one of them, boss. East side downtown. You better hurry. I'll be there right away. You mean we'll be there? Then come on quickly, my dear. Now you shall have that mink. Here you are, sir. Here you are, driver. Oh, thanks, miss. Where's the woman? I done a good job of trailing her. You ought to be proud of her. I said where is she? Right in that brown stone house. But what are we waiting for? Lead the way, Michael. Okay, come on. I went right in here to grant for... This better work. Please, my dear. Yes? Greetings, Mrs. Greeley. Well, come right in. Go ahead, darling. Michael, I dare say you're a trifle surprise to see us again. Well, as a matter of fact, Colonel Weatherford, I am a little surprised the way things turned out, but we rather expected you'd come here. Who's wait? Yes, what do you mean? What she means, Weatherford, is that you're all under arrest. What? What is this? We're special agents of the FBI. This is an apartment that we used on another case. It was also convenient to bring Mrs. Greeley here for questioning. We hoped you'd follow her. You mean she's working with you? Oh, not willingly, Colonel. But as you know, this is the Christmas season and it's full of surprises for everyone. And now your FBI would like to take this opportunity to wish for all of you a merry Christmas. And a happy, peaceful, and prosperous new year. And through your continued support and cooperation, it will go on protecting your right to enjoy them year after year. Before we tell you about next week's thrilling case from the files of your FBI, a word about a man worth knowing. To your FBI, you look for national security. And to the equitable society for the financial security of life insurance. In the past 86 years, the Equitable Life Assurance Society of the United States has weathered four wars and seven major depressions. During that time, over five and one half billion dollars have been paid to policy holders. This tower of strength, security, and stability is represented in your community by a man whom hundreds of your fellow citizens know as their good friend, the Equitable Society Agent, who like your FBI is dedicated to the protection of you, your home, and your country. Next week, we will bring you another colorful story from the files of the Federal Bureau of Investigation. Murder on the High Seas. The incidents used in tonight's Equitable Society's broadcast are taken from the files of the Federal Bureau of Investigation. However, all names used are fictitious, and any similarity, thereof, to the names of persons living or dead is accidental. Programs in this series of particular interest to servicemen and women are broadcast overseas through the worldwide facilities of the Armed Forces Radio Service. Tonight, the music was under the direction of Frederick Steiner, the author was Frank Ferris, and your narrator was Dean Carlton. This is your FBI is a Jerry Divine production. Now, this is Carl Frank speaking for the Equitable Life Assurance Society of the United States and the Equitable Society's representative in your community, and inviting you to tune in again next week at this same time, for this is your FBI. This is ABC, the American Broadcasting Company.


Hello and welcome to interview, production of the Government Information Service. I am Jolene B. Sett-Joseph. On the 3rd of October 2019, at 3.30pm, St Lucia will witness the unveiling of All-In, a statue designed by Jolene Yudevik. This historic event is being held as part of St Lucia's 40th Independence Celebrations, and to tell us a little bit more about himself, the event and the statue I am joined by none other than the sculptor Jolene Yudevik. Jolene, thank you so much for joining us today. Pleasure, Jolene. Okay, great. Now, as I said, I want to find out all about the sculpture and the event, but before we do, let me find out a bit about the sculptor. For example, I know that your father is none other than famed St Lucia sculptor Vincent Joseph Yudevik.  Very interesting question, Jolene. From the time my earliest memories of myself was always that of wanting to become a creator, wanting to become an artist. So I vacillated between sculpture, painting, architecture, you know, at a very early age of five. Even theater, I dab in theater, I dab in poetry, because I often say that the art world is sort of a universe, you know, and being an artist gives you the passport to navigate that space as freely as possible. So I still don't see myself as just being one thing. Sculpture is my core. That's what I've chosen as a career, but I've allowed myself the space to grow into other areas of creativity. But back to your question, yes, I have always felt from a very early age that I wanted to be a creator. That's what I am, essentially. Oh, brilliant. Now, as a creator, as a sculptor, I know that you actually have many pieces on show in different exhibitions throughout the world. Tell us a little bit about that, because as a young man, again, growing up in St Lucia and whatever, to have your art exhibited in so many far off lands to me, that's something to be very proud of. So what kind of pieces are actually on show globally and where are they located? Okay. But just to get to the genesis of how that happened, I was always sort of a daydreamer. Okay. And I dreamt of being far-filed in the world. I had the audacity of imagination and put it myself in those places. Because when you grow up on an island, that's the only way you could attain an international saviness and an international thinking. You literally have to deal with your imagination. So very early on, I took part in various exhibitions in St Lucia and eventually it led me to the MNC finance. I took part when I was in secondary school. I will reveal today that MNC finance was a post-secondary school exhibition. When I sort of cheated the process, I met in early. Because I was so eager to follow for the exposure. And I did well in MNC finance. I won a couple of prizes and so on. And that sort of put me on the national radar. And eventually I found myself on delegations, taking me to Canada, taking me to France, Africa, and then Martinique several times. I mean Martinique is right next door, but Martinique is really first world. And I learned a lot from that country. And on my own, I have been to many countries, including England, China. You name it. New York. I've had several exhibitions in New York City. Okay. Okay. All right. Now I guess it wouldn't be fair for me to ask, because again as an artist, it would be hard for you to answer which sculpture maybe stands out the most to you. But what experience, in that case, can you tell me what experience that maybe evolved around a sculpture that you did that remains quite significant in your own mind? Is it like the first time you exhibited internationally or is it when you went to a certain location and the reception you got, what stands out to you? Is there any specific event? For me, my most memorable sculpture or my most significant sculpture was actually my first sculpture that I did when I was five. Okay. All my sculptures are created equally. I love all of them and I put a similar amount of effort in all. But when I was five years old, my first sculpture I worked on tirelessly for about three years. I started when I was five and the sculpture probably finished when I was eight years old. Wow. And it was an abstract horse. I still have it. Okay. You know, and I could put on a horse and if my pillow, it was a two-legged horse that I put on my pillow every night and reached that it magically grew two legs. Two other legs, you know. So that sculpture, you know, represented who I was at the time. You know, it represented the epic shift I was making my learning curve, my understanding of the art world and so on. So all my aspirations and early desires at that time was encapsulated in these tiny horse sculptures, two legs. Okay. Brilliant. Okay. Well, Jelimi, it's time for us to have a break right now. However, we'll be back in a moment to find out a bit more about the event and the unveiling of all in. Stay with us. We'll be right back. One of the eight university-recognized rights of the consumer is the right to be heard. This means that every consumer who is dissatisfied with a good or service has the right to lodge a complaint to the provider of that good or that service. This should be the first point of lodging a complaint. Ensure that the receipt, as proof of the transaction, is available. Welcome back to Interview. I'm joined today by Jelimi Muldavek, who is a sculptor and who's here to tell us a little bit about all in his sculpture for the 40th Independent Celebrations. Now, Jelimi, let's talk about the sculpture. What was your inspiration behind it? Can you tell us? I mean, I don't want to take away anything from the unveiling, but can you tell us a little bit about what the concept is maybe? And also what was your inspiration behind you know, doing what you did? Okay. I'm St. Lucia and the people of St. Lucia is my muse. Okay. St. Lucia is an absolutely wonderful place. I've traveled globally and I've never seen an island fairer than St. Lucia, you know, according to Anne-Fam. And, you know, our attitudes towards life, simple but elegant, you know, and our mixed heritage, our cruel heritage, the foods that we eat, the conversations that we have, our past, our present and the way we project ourselves into our future, into the future is essentially what defines a St. Lucia, you know, and we do so in a very unique way. And so the sculpture speaks to the history of our people, the diversity of our people, because our people, as you know, is of mixed heritage. You know, there's four ethnic groups that mix up the population of this country. You have the Africans, you have the Caucasians, the East Indians and the Amaranians, you know, and so I want to say that St. Lucia and the Caribbean to a certain extent, at the parts of the Caribbean, is a sort of a social experiment or social miracle because we find a way to become, in a real sense of the word, intercultural, you know, and we really have time for multiculturalism because that doesn't necessarily mean that you're together. But we came in different circumstances and for different reasons, but time has amalgamated and united us all. So the sculpture really speaks to that, you know, but most importantly, it speaks to our united stride into the future and you know, where we project in ourselves.  We've given the world literature. We've given the world economic philosophy. We've given the world acts. We've given the world we have so much to be proud of. We have so much tolwats that has emanated from this small dot, you know, and I often see sandwiches like an atom, you know, small, but it's powerful.  You know, so I really believe that this country encapsulates all that and to ask questions, to speak more to your question. The sculpture was from an earlier collection, an idea called Kourme. And Kourme, as you know, means it's French for Koudemais, which means a lash of the hands, you know, and it's a French creole term which describes a part of our culture where people come together, specifically to be the house. So this is an act of love and had an opportunity to do an exhibition at the Steven Lawrence Center in London about eight years, nine years ago, in 2010. I was invited by Doreen Lawrence, Baroness Doreen Lawrence, and she wanted me to have the first ever exhibition at this renowned center. And so I conceptualized something that resonated with what she did with her life's work, which is creating architects out of the privileged inner city youth of England because her son had died in a hate-related crime and he wanted to be an architect before he died. So she wanted to replicate him, you know, for those kids. So I had to find something in my culture that was sort of architectural, but it was sort of of that love. And I discovered Kourme.  So people at home who are familiar with this collection, I know they must be grappling right now trying to find a rich one. So it's sort of a hint. But so when the opportunity for the 40th Independence anniversary came along, you know, this piece spoke very stridently and very loudly, you know, to this year's celebration. So it's a piece that I have been working on for, you know, before I created a year or two. And, you know, I've been tweaking it for many years now. And it was, I guess it was for a reason. Because it's finally going to be unveiled. That's great. Now we're going to run out of time, but before we go, I really want to know how do you feel those young sculptors being picked? I mean, I can hear it to be, I'm asking the question, but it's also as if I don't even have to ask the question because even in your answers beforehand, I could hear that, you know, what's coming through in your answers. But how do you actually feel about being the one chosen to do this, to have such a sculpture on display all the time? Well, great honor, but even greater humility. Right.  But I have grown broad shoulders because of my international, you know, undertakings. Because I have done, I have interpreted many societies around China. And, you know, I've been adopted by China and I do a lot of work for them, you know. And I've always yearned to do something for my country. So I always had that aspiration bubbling inside me that, you know, that I need to, as a Jamaican, say, dance a yard, you know. Okay. You know. So when the opportunity came, it's something that I received with much elation, but also with a great degree of reverence and, you know, seriousness, I would say. Okay. You know, to find a better word. Okay. For lack of a better word. Seriousness. Because this is epic. You know, our people need those beacons all the developed world. You have these iconic things. These iconic statues. Very true. You know, that break the monotony of your daily lives. That give inspiration, that serves as beacons for inspiration. And also sort of a mapping of a map of who you are and where you are and where you want to go. And people need these things. And also society needs these things to grow. This is very essential to our growth. You know, so hoping that my sculpture will not be the last. It's definitely not the first. But I hope that this is an act of perpetuity. Okay. This will continue into the future because we've been given a lot by God, given a beautiful landscape. But we need to put on stage on a platform our own talent, our own local talents. Brilliant, brilliant answer. Wonderful. Well, Jalim, we have actually run out of time. However, I know that you will join me in asking everybody to make sure that they come out on the 3rd of October at 3.30 for the unveiling for this wonderful event. Thank you so much for joining us today. And thank you for being in the audience. Stay tuned to NTN. However, for now, this is Jordan B. St. Joseph from Interview saying goodbye. Okay.


This programme contains some swearing, adult themes, moderate pain and outrageous tattoos. Hi, I'm Charlotte Crosby and we all know that some tattoos just shouldn't be for life. So here are my top tips to laser tattoo removal. If you do use false tan, you need to make sure that you haven't got any tan at all on that area of skin that you are getting lasered. What can happen is, whilst you're getting lasered, it can leave a dark stain around the lasered area and nobody wants that. Number two, I do highly recommend putting on some numbing cream before the treatment. It will completely dull down the pain and will be a lot more comfortable for yourself. Tip number three, do not pick the scabs or blisters. It will scar, it will leave a mark and you will be unhappy. Tip number four, if your tattoo is in an area which is quite visible, I highly recommend getting it done in winter. So you are mostly covered up so people never really have to see this mess, atrocious disaster on your skin while it's going through the stages because it looks pretty embarrassing. My last tip has got to be, think carefully about the reasons you want to get rid of this tattoo. Is it definitely something you want to do? I mean if it's an X, probably is for the best. I would think very carefully before you go through the stages because it's kind of irreversible.


The title of this presentation is Data Seller Package enables intuitive mobile data manipulation. Ah, hello. I'm Toshi. I'm developing a new package called Data Seller, which improves data cleaning experience in all. It's developed on GitHub. Before I'm going to talk about the detail of this package, I'm going to talk about why I started to develop this package. Usually, data analysis workflow is composed of some steps, such as making analysis plan, cleaning data, statistical analysis, and interpreting the results. I think most of you agree with these steps. In all, the boundary between data cleaning and the statistical analysis can be fake, because all is general purpose programming language and has much flexibility. So when you clean data, you can also do t-test at the same time. And even if you are doing regression analysis, you can manipulate the data at the same time. So it's really flexible. However, I would like to focus on data cleaning when I do data cleaning. Then I started to develop a new tool focusing on data manipulation, which is called DatSeller. What is DatSeller like? DatSeller does row by row data processing using Seller script. So Seller script is designed for this package. DatSeller does not replace R, of course. DatSeller just manipulate data using script called Seller script. How to install? As usual from C-LAN, install packages DatSeller, this installs tree system. Actually, I'm still submitting this package to C-LAN. So when you watch this video and if it's not still available, please visit this website. I'm hosting this website. Packages are available on the website. How to use? DatSeller has cell function. We can pass DatFrame and Seller script to this function. DF can accept data compatible with DatFrame. Code accepts Seller script in our string. So passing this DatFrame and Seller script and this function returns DatFrame processed. So this is an image passing DatFrame, passing Seller script, DatSeller calculate each row for each row and output result DatFrame. Then let's try DatSeller. In this example, I'm going to use R with DatSet. And only these two columns, petal.lengths and separate.lengths will be used. And I'm going to call this Iris-based DatFrame IrisX. So let's try to divide this column by this column. How can we do it? How can I do it? Yes, passing IrisX DatSet and passing Seller script. Maybe you can imagine what this Seller script does. Petal.lengths divided by separate.lengths is assigned to petal-separate ratio, which actually results in a new column, petal-separate ratio. So this is a variable. This is called variable, but variable names corresponds to column name and new column name. When we use a variable with new column name, the new column is created. So this example shows basic concepts of DatSeller. Variable names correspond to DatSet column names. This reduces the amount of typing, like non-standard evaluation in-out. And this idea is familiar to statisticians because they call column names as variables. Second, by Seller script, DatSet just instructs how each row or each record is processed. So we can just focus on how each row is processed and DatSet applies it to each row. I'm going to talk more about DatSet functionality. From here, I'm using another DatSet called MT-CarsX, and it's based on MT-Cars dataset and extracting only one column of HP. So this MT-CarsX has only one column, and the MT-Cars originally has row names, so MT-CarsX also keeps row names. So this functionality I'm going to talk variable, variable assignment types, operator splitting functions, affairs control flow, missing values, regular expressions. I'm going to talk about these things. First, variable and variable assignment. As shown in previous slides, each variable corresponds to column name. So HP, for example, HP variable corresponds to HP column. And assigning to a new column name. So yeah, so variable name corresponds to column name. So I will use variable and column name interchangeably. Assigning to a new column name. So assigning to h.power, this does not originally exist on MT-CarsX dataset. So assigning to a new column name creates a new column. So in this case, h.power is new column name and this column is new column is created. And this line does assigning HP times 2, assigning the value of HP times 2 to HP. So assigning to a new existing column name updates the column. So in this case, HP is updated with the original values of HP times 2. So let's look at the result. Yes, h.power holds original HP values and HP is updated with HP times 2. Next, cell types. In cellar script, there are only three types, integer, double, string, and compatible types are converted between R and cellar as follows. So if the column of data frame is integer vector, it's dealt as int from cellar. And if the column of the vector is double on data frame, it's dealt as double from cellar. And if the column vector of data frame is a character vector, it's dealt as string. When Boolean vector is dealt as 0 or 1, if factor is dealt as string. So yeah, you can see you can use integer, double, and string in cellar script. Yeah, you can see this result. About operators, assignment operator and arithmetic operators can be used. Assignment operator addition, subtraction, multiplication, division, power can be used. And also comparison operators equal to Rajasang, Rajasang y equal to Rajasang y equal to. And also regular expression operator can be used. So you can see HP times 2. And this variable is halfed. And this is power. And this is a squared root. You can see these results, the result of these operators. There are also built-in functions. Strings are usually manipulated using built-in functions. And these functions start with str underscore. And also there are other kind of functions. I'm not showing here. You can refer to documents. So in this example, this function, str underscore subset string. So I didn't talk about this variable, underscore onem underscore. But this special variable, this represents the column name. And so this function, what this function does is subtract from index 1 to index 3 of this variable, this string. So in this case, extracting index 1 to index 3. So first three letters, extracting first three letters of row name. And this is assigned to this first three column. Number 2 str converts number 2 string. And in this table, left aligned means its string. So this column string is converted successfully. You can also use affairs control flow. As usual, if condition statements, else if condition statements, else statements, this can be used. This structure can be used. And else clauses are optional. So in this example, if HP is larger than 145, power is assigned high. And if HP is between 0 to 145, power is assigned low. So this value is 175 is larger than 145. So power column has high. And in other cases, power column has low. You can also use missing values. That represents numeric missing values. And empty strings represent string missing values. So yeah, that means missing value in say that. So it can, when you compare with NA noun in R, it should return true. And yes, when you compare empty string with NA in character vector of R, it should return true. So this is example. I'm creating a new data frame containing missing value. And if x equals to that numeric missing value, x underscore NA is assigned 1. And else x underscore NA is assigned 0. So in this example, when x has NA, the x underscore NA has 1. And in other cases, this comes as 0. So it's working. And lastly, regular expression noun, this is a really powerful feature. Regular expression can be written like this RE, slash, pattern, and slash. And this is a regular expression literal in Sailor script. And also equal to that is regular expression matching operator, which is a value to be Boolean and should be used for if it's condition. So in this case, I'm creating a new regular expression pattern, and it's assigned to Germany variable. And Germany variable representing this regular expression is matched against the roll name. And if it's true, so it's matched, country, come has Germany. And also this function is executed. What is this function? This function is back reference function. This pattern includes groupings like this parentheses part. This is called grouping in regular expression. And the RSV underscore matched function can extract the matched strings. So in this case, when the roll name has Mercedes, milk is matched and milk part is extracted from string. So like this. So yes, if the roll name has milk, milk part is extracted. If it's Porsche, Porsche part is extracted. You can see. You can also see this is working. I also talk a little bit about that Sailor internal. Sailor script is passed to virtual machine code. DataFrame is extracted each row. And each row is processed like this on virtual machine using virtual machine code. And the result is redone. And these purple parts, this is that Sailor purpose itself. And these green and orange parts, this is implemented in this Sailor C or C++ library, which is created especially for this package. Thank you. Thank you for watching this video. This series is still in development. So feedbacks are welcome. This is really difficult time. So I hope everyone is safe and see you in the future.


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