So today, we have concluded the part for the moment, the part of first order equations. So we start now with a new kind of equation, which is, by the way, very, very useful in physics in particular. So what I would like to start today is to study the following PDE. So utt equal some constant, positive constant, plus of u. So let me introduce the symbols. So we are, our variables are t and x now, t and x into r1 plus n, which is split as follows. So t is a scalar variable, x is a vector. And we will, so if you want to compare with the previous notations, maybe we use this as, we use this capital N in the sense of 1 plus n. So t is real, x is a vector. We will be mainly interested in the case n equal 1, OK? Yes, we will study mainly. u is a function, and also for the region of interest is maybe not the wall space, but we will often restrict to this half space. So t maybe will be non-negative, positive times, say, and x will be in our n, usually. Even if we could also go back in time, but physically maybe is more meaningful to consider non-negative times. Our pictures will be usually putting x in the horizontal axis and t in the vertical axis, despite the fact that t is the first variable, as usual. Remember this strange convention that we took. So what does it mean, ut? Ut, for us, is du over dt. And therefore, so this is the partial derivative. And therefore, utt is the second derivative. So c is given, is a constant given. And this is called Laplacian of u and is defined as the sum from 1 to n du over dx i. Maybe these two we can put it here. So it's the trace of the action. It's the sum of the pure second derivatives. In one space dimension, our equation is utt equal c square uxx. This is our interest. Sometimes this can be written as follows, 1 over c square utt equal Laplace of u. And or if you wish, d over dt, d over. So sometimes the time variable is substituted by this constant c times the time variable. And very often, it happens that people take the normalization c equal 1. Very often, one normalize things so that c is equal to 1 normalization. So for many qualitative results, we can take c equal 1. Physically, maybe sometimes it is better to keep the c because c will be very large. So let us start. So this is a linear, first of all, which is the difference with respect to the previous equation. Well, this is a second order PD to the derivatives. So it is a completely different object with respect to first our quasi-linear PD that we have studied before. So this is, of course, linear, but first order, one derivative here, one time derivative, one first order spatial derivative, non-homogeneous. For the moment, this is, on the other hand, second order in time, second order in space, and homogeneous. So if the operator is written as follows, then this is the PD operator, and it is clearly homogeneous because there is no f on the right hand side. Don't make the confusion of this equation with another completely different equation, which I can write, for instance, like this. So don't confuse the sign. It is very important here that the sign is minus. Otherwise, if the sign would be plus, and if you take, for instance, c equal 1, then this is simply the Laplace operator in r1 plus n. So it's another equation. It's a so-called elliptic equation. So please keep in mind the sign. So the first arguments that we took for the study of first order linear PDs was to define the characteristic surface because that was the point on the question was, what does it mean given initial condition? And we discovered that we can give an initial condition for this equation provided that there is a transversality, say, between the vector field 1b and the sigma that we are considering where we want to give the initial condition, the hyperplane t equal to 0, for instance. Do you remember this? So if our initial condition was given at time 0, then our vector field b, say, at least in the simplest case, say, in the simplest case, when b is constant, for instance, then should not be tangential to sigma. Do you remember this? So now, and that was important for giving, because it's never immediate to say which are the natural boundary conditions for this PD or that PD. So any PD must be understood with this particular initial condition. So now we want to consider this free equation in the space, in time space. But we want to give also a notion of initial condition. So what does it mean initial condition in this case? That is the first question. And again, we need, as before, for these linear problems, we need the notion of characteristic surface. So it is interesting that also here, there is again the notion of characteristic surface. So let me give you some definitions. So definition. Let me call the operator, maybe the operator L. So let me call this L of u. So L is an operator taking, in this viewpoint, L take a function in some functional space that now, for the moment, I don't specify. So takes the function u into L of u, which is this combination of second derivatives. For instance, if u is c2, this is c0. So it is a linear operator from, say, for instance, c2 into c0. Define. So let me use this notation. So a vector, say, actually a covector xi into r1 plus n, let me split it into, so this actually would be a covector. But I don't want to insist in the geometrical meaning on this, but xi hat, maybe xi prime, xi prime. And so xi is in r, xi prime. So I split xi into the first component and the last components. xi prime are all the remaining components. And so xi 0 square equal c square xi square. So let me, for simplicity, in this argument, take c equal to 1, which is not, for the moment, important. So let me normalize constants so that c is equal to 1. And so let us consider this hypersurface in this space. And this is called, so a vector here is called the set of characteristic vectors for the operator L for L. So this is the set of characteristic vectors. So and I say that let sigma be a hypersurface into rn plus 1, hypersurface, c1 hypersurface, so that I can speak about its normal. And I say that x into sigma is characteristic if the, so I have maybe already used this notation, if the normal to the surface, one of the two unit normals, one of the two unit normals to the surface at x is a characteristic vector. I say that it is non-characteristic, non-characteristic. This point here, x is non-characteristic if sigma of x does not belong to car. So for instance, let us, so example, so first of all, maybe it is important to draw this set here. Can you tell me which kind of hypersurface is this, which kind of hypersurface is this hyperbolic? Do you think so? Hyperboloids, let us, for instance, consider we are interested mainly, as I said, n equal to 1. So let us try to understand what does it mean. Oh, sorry, x1 squared. What is this? Exactly, it's not a hyperbola. What is this? It's a cross. This is a cross. But more generally, if we are in three dimensions, what is this now? In three dimensions, this is the norm, by the way, the Euclidean norm. So this is simply, and what is this? Is that? And so what is this? Now, he says cylinder, no, it's not a cylinder, right? Only half cone, double cone, it's just the cone. So this is a cone in general. Now if you restore the C here, then you modify the aperture of the cone. Depending on C, then the cone change. But if C is equal to 1, this is exactly 45 degrees in two dimensions. If you change C, then maybe you can end up with some different cross like this or like this, depending on C. For C equal to 1, it's just at this 45 degree angles. This is the origin, by the way. This is the origin of this theory. The surface sigma is non-characteristic if all x in sigma are non-characteristic. So examples, exercises, say. So exercise one, take sigma equal this sigma. So you have to imagine this sigma here. Now is this sigma characteristic or not? Thank you. Take, well, then take only half of this, say without the origin. Or just a part of it. Just say this half line or just one line instead. Yeah, C equal 1 just for simplicity. Let us take C equal 1 for the moment. Or if you want to restore the current, well, take this without the 0, however. Remove the origin from the cone because it's a singular point. And the observation is that it's not in hyper-surface in the classical sense. OK, remove the origin from the cone. Take the cone without the origin. You see also here. So now what do we have to do? We have to compute. So this is sigma for the moment. Let us compute the normal to sigma. One of the two normals, say either this or this. One of the two. OK? How can I compute it? Well, let me introduce, for instance, one idea is to look at this as the level set of a function in 1 plus n variables. And then take the gradient. So if I look at this as the 0 level set of a function with non-vanishing gradient, at least out of the origin, then the gradient will be normal to the surface, right? Because the gradient is orthogonal to the level sets. So let me obviously introduce. Sorry, this is xi prime. So my notation was to consider the spatial part of xi, xi prime. So this is xi prime here. This is xi prime. And so two xi primes. OK, so now this is 1 half xi square minus xi prime square. It is clear that now sigma is the 0 level set of this. So the level set of sigma are, how are the level set of sigma? By the way, can you imagine? So this is the 0 level set. Yes, now yes. OK, but it doesn't matter. We are interested just only in the 0 level set. And so we know that now let us take the gradient of psi in the whole space. This is differentiable, of course. Let us take the gradient of psi everywhere. So psi is defined everywhere. This is the domain of psi. But psi vanishes here on the cone. But it is defined everywhere, also outside the cone, everywhere. Now this is xi 0 minus xi prime, right? Now this, of course, is 0 in the origin, at the origin. But out of the origin is not 0. So out of the origin, this vector is orthogonal to the cone. Is it clear? Is this clear? The cone is a level set. The gradient is not 0. The gradient is orthogonal to the level set, OK? Fine. Now, but the gradient, now take a point where, so for instance, in two dimensions, take this point here. This is sigma. And now what happens to the gradient? So this is psi 1, psi prime, this is psi 0. And now take the gradient of psi at this point, which means simply taking the gradient and restrict it at this point, evaluate it at this point. And where is it now? Now I change the sign of psi prime. So I take this psi prime and the same psi 0. And this is again, so this is the normal, which is here with our choice. This is the first example. Now take another example, is what we will always take, will be t equal to 0. Sigma is t equal to 0. And t equal to 0 is, again, this as usual. The normal is this. And in this case, the normal does not belong to the cone. This is more lucky. This surface now is such that the normal does not belong to the cone. In the previous case, the normal was belonging to the cone. As you see here, the normal was belonging to the cone. But now the normal does not belong to the cone. So this is OK. It's non-characteristic. The horizontal line is non-characteristic. The cone, on the other hand, is a problem because you see, take the normal here, then the normal belongs to the cone. So it's characteristic. This is not, is non-characteristic, and the normal is, of course, nu sigma at x is 1, 0, 0, which does not belong to the cone. So this suggests that maybe it is reasonable to take sigma as a surface where studying the Cauchy problems, where assigning initial conditions. So we will never assign initial conditions on surface sigma like this, because the normal to this belongs to the cone. But we will assign initial condition on sigma like this. So let me summarize. This set here, sigma equal t squared equal to x squared is characteristic. So let us forget the origin for the moment, which is not so important, but sigma t equal to 0 is non-characteristic. So we will consider having the experience of first order equation. So we will consider u t t equal, say, c squared laplace of u in, as I said, one could also work for backward in time. But let us consider for the moment just for simplicity. And then u 0 equal, say, 0 bar, u bar, u 0 bar, maybe, at t equal 0 on sigma. Now, the point is that one in this kind of PD, one has to assign also on sigma not only the initial position, the initial condition, but also the initial velocity. This is because the PD has the following structure. u t t equal something like this. Because here there are two times derivatives. The time derivatives, as we will see, requires a natural way to impose initial conditions, like for ODEs, in some sense. If you have a second order ODEs, you specify initial position and initial velocity, usually. Now you have not, this is not an OD, this is a PD, but it has two derivatives in time, and therefore it seems reasonable to specify not only the initial function at time 0, but also it's time derivative at time 0. So sigma now is non-characteristic. And I will assign u bar 0 and u bar 1. These are given, smooth enough. So now the point is to try to solve this, OK? But before doing this, I want to mention just one big theorem. Theorem. And for instance, you can look at the proof. For instance, in the book of Evans, and it is called Koshy Kovaleskaya. The theorem is called Koshy Kovaleskaya. And it says the following. Assume sigma non-characteristic. Actually, one could give this in a local version, but for simplicity, I don't want to state it locally around the point. I state it globally just for simplicity. So assume that sigma is non-characteristic and analytic. Assume u 0 bar, u 1 bar, analytic, real analytic. This means that around each point these functions coincide with their Taylor series around each point. And this means that you can look at this, for instance, in a parametric way or as a graph locally of an analytic function, as usual. So everything is analytic. And assume that you want to consider utt equal f of t x ut grad u, say, h of u, say space h, for instance. All second derivatives with respect to space. So the important is that here on the right hand side, you don't have a second time derivative here. And you don't have derivatives of u of the order higher than 2. For instance, in one space dimension, f, t, x, ut, ux, uxx, for instance. So on the right hand side, no third derivatives, not second derivative with respect to time. Just in place. Yes. And u equal u 0 bar and ut equal, say, for instance, say take this t equal to 0, for instance, ut equal u u 1 bar on sigma. So if everything is analytic, and f also is analytic, assume that you consider f analytic in its arguments, its arguments, utt equal this. Then, say, take this for simplicity. Then this problem, one, has a local in time solution, has a unique one, has a unique analytic solution u, solution u for short times. What does it mean? It means that sigma is clearly analytic in this case, because it is a hyperplane. If I give you this problem with initial condition, which is analytic, and initial velocity, which is analytic, and this is clear, this f is analytic in its argument, so then this has a local in time solution, unique which is analytic. I don't prove this theorem here. It's not easy, it's long, but just to let you know that, at least, when everything is analytic and sigma is non-characteristic, that we have a local solution. One can give this statement around the point of sigma without assuming this, but taking just sigma and non-characteristic, and so locally, you have a unique solution. But I don't want to insist on this, because I don't prove it. Just to let you know once that there exists this theorem, which is actually much more general than this. It is actually, it holds for a very large class of PDs, not only for kind of wave, sorry, this is called the wave equation, by the way, wave equation. This is called the wave equation. And this is a typical example of hyperbolic second order PDs. Hyperbolic second order linear PD. So this theorem, I was saying, this theorem is valid not only for hyperbolic type PDs, but it is much more general than this. But you need the analyticity of everything, initial conditions, sigma, f, and so on. If you don't have analyticity, then you cannot use this theorem. So this is just to let you know that, OK, even for PDs, in a large class of PDs, if everything is locally analytic, then you have a local solution analytic. Then you can expand in the Taylor series. It's not an easy theorem. And if you want, if you are interested just for curiosity, then, for instance, you can look at the book of evidence. t equals 0. So yes, thank you. What does it mean, existence for short times? Oh, it means that the PD is satisfied, for instance, in 0 t times rn, say, for some t, but not in principle for all t's, just in a strip. Ah, yes, but times is here. All i's, OK. So to study this PD, it is better to start the first one-dimensional space case. So we start with n equal 1. n equal 1 means this, OK? n equal 1 means this problem here. So now, for the moment, consider just the PD without conditions, and there is the very interesting structure of this PD is the following. In one space dimension, we have that this can be split into the composition of two operators, OK? And so composition of, so let me introduce v maybe. v, let me follow this notation v. v of tx equal du over dt minus c du over dx. So let me introduce this, and let us observe that dv over dt plus c dv over dx is equal to 0. So remark, assume that you have a solution, OK? Assume that you have a solution. You define v as follows. Then v satisfies this. Let us check this, d over dt. Indeed, d over dt u t minus c ux t plus c u t minus c ux plus c. So this is this. And so what happens? You see u dt. This is u dt minus c ux t plus c u tx. Well, we are talking about C2 solutions, OK? So the problem is also, let me write also the regularity class. So we look for a solution, which is u. Say in C2, 0 plus infinity times r, so that everything is smooth enough. Assume that u0 is maybe C1 or C2 also. Assume that this is C1. Let us take this. So let me go back here. So u dt minus c ux t plus c u tx minus c squared uxx. Since by assumption, u is a solution. And this C2, u tx and uxt are equal. And so this is equal to this. So this means that this operator, L of u, actually splits into the composition of two operators. So L of u splits into, first you do, say, this, and then you do this. So it is the composition of two operators. So L of u is, say, d over dt plus c d over dx of d over dt minus c d over dx of u. This is a little bit formal, but it is clear what it means. So first I do u t minus c ux here. First thing to do, for instance. And then to what I obtain that I have called v. Yes, this is v. And then to v, I apply this other first order operator. You see? So splitting property, this is an extremely important fact. So splitting property. So the operator L, remark, it's interesting to observe that L splits into the composition of two linear first order PDEs, operators, first order operators, first order operators. But which kind of operator? You see? This is linear transport operator with constant vector field. And this is, again, another linear transport operator, first order with the minus c, is now replaced by plus c. So first you apply a step in which you do a linear first order with minus c. And then to the result, you apply, again, linear first order with plus c. And this is very, very, very nice property of this second order operator. And this, in one space dimension, this is the trick that allows to find the solution. Now we will see how. So let me keep the blackboard. Let me keep this and this also. OK, this is not necessary anymore. We can start from studying this. And what do we know about this? We know the solution. So start, we start with this. And so now we know that v of tx, it is equal to what? Maybe we could write, say, v at time 0, just to be precise. And then we chart the, now, what is what? Now, exactly, x minus ct, do you agree? So let us call maybe the notation, maybe a, a of x, let us define it as v of 0x. Define this just to simplify notation. So this is actually simply a of x minus ct. Of course, this a is neither u0 bar nor u1 bar. You see? This is not. Because at the end, what is this? So I have to put, say, time 0 here. And so here you see there is 1 u1 bar minus c something. OK? So for the moment, let us just simply define this. So let me write it just not to forget it, the notation. So a of x is defined as v of 0x. And so v of tx is this. OK. Now, so we have solved this for v. And therefore, now, our problem is however this, which becomes ut minus cux equal a of x minus ct. Right? And what is this? Now, this is linear transport equation with constant coefficient, but, however, is non-homogeneous. So we need the solution. We have studied this with constant coefficients, but maybe it is better to, if we are able to remember it, maybe we should write it here. If I'm not wrong, so if I take ut plus bux, say v dot at u equal f of tx and u of 0, call it u and phi, then the solution was phi of x minus bt plus the integral from 0 to t. And then we have now f of tau. Let's please check this, the tau. Check, please, this formula. Is it correct? OK. Fine. Please check it, because otherwise, if it is not correct for some problems in the sign, then we don't arrive at the end of the reasoning. So this is linear transport with constant coefficients, but non-homogeneous. OK. Is it correct? OK. Fine. Now, therefore, now we can apply this to our problem. Our problem is this, exactly. And I simply have to change letters, symbols, and to keep a little bit, pay attention not to make mistakes. So this means that we know that the solution u of tx has the following expression. Is the initial condition, now b is, of course, minus c. OK, b here is. So when there is b here, I have to put it minus c. Look, b minus c. So change b into minus c in this formula. Take one dimension, space. And then, so phi is the initial condition. So now, finally, phi is actually u0 bar. So u0 bar. Now I have to take this. And then I have plus the integral from 0 to t. Then in place of f, I have, so let me write from here. I don't want to, I try not to make mistakes. So f of tau x plus b tau minus t is, in our case, is equal to what? Is equal to a of x plus b tau minus t. But b actually is minus c. So this is the new x. Is the new x is, and then, minus ct. This is our, if I'm not wrong. So let me check it. So our f is this, depending only on x, on the space variable. So f of tau, this is the object. So f of tx is equal to a of x minus ct. And therefore, f of tx is this plus this minus this. Ah, thank you. Thank you. Thank you. Thank you. Thank you. Let me check it if it is correct. Minus c tau. Yes, this is thank you. So now this is equal to a. And then I have x minus 2c tau plus ct. Therefore, our solution is a of x minus 2c tau plus ct in the tau. Now, let me change variable here. So let me now make a change of variable. And so x minus 2c tau plus ct, I call it for the moment, say, y. OK, I call it y. And therefore, u tx, now this integral, the integral from 0 to t of a x minus 2c tau plus ct d tau becomes a of y. Then d tau is minus 1 over 2c. So and then 1 dy. And then when tau is equal to 0, this is x. And then when tau is equal to t, this is x minus ct. And therefore, actually my solution u, read as follows. Now I raise this part. So I end up with u of tx equal to u0 bar of x plus ct plus. There is a minus here, so I change the integral. So 1 over 2c x minus ct x plus ct a of y dy. So what remains now? Now it remains exactly. It remains to express a in terms of u0 and u1. And therefore, what we know about a? So we know that a actually is what? Is u bar 1 minus c u0 x? Because you see we have u is equal to u0 bar on sigma. So on sigma, they agree. By smoothness, u actually is smooth up to sigma. So the derivatives of u in this direction is equal to derivative of u0 bar in the same direction. So our solution actually is smooth. And so a is this. Do you agree? By smoothness, u is c1 up to t equal to 0. And we can therefore almost, we are almost at the end. So utx actually is equal to what? To u0 bar of x plus ct plus 1 over 2c, the integral plus ct. And then I have u1 prime bar y dy. And this I keep alone. And then 1 over 2c. And then I have x minus ct, x plus ct. By the way, now we have, there is this minus c actually, OK? So I erase this. And then apparently there is also a minus. And then I have u0 x bar y dy. And then I end up with what? With 1 half u0 bar x plus ct. Because this 1 half cancels 1 half of this. Plus 1 half u0 bar x minus ct, plus 1 over 2c integral from x minus ct. This is a very, very interesting formula, dy. Let me check that I have, it's correct, surely correct. But yes, it is correct. So we have proven the following theorem. So we have proven the following theorem that it is called, no, well, this formula is called the Dallenbert formula. Dallenbert formula. Maybe Dallenbert, Dallenbert, Dallenbert. OK, the theorem that we have proven is the following theorem. So let say u0 bar in c2, u1 bar in c1 of sigma, define u of tx as 1 half u0 bar of x plus ct plus u0 bar of x minus ct, plus 1 over 2c, x minus ct, x plus ct, u bar 1, y dy. Then u is c2 times r. You see c2, because u0 bar is c2 by assumption. u1 is, u bar 1 is c1. So it's integral, actually, c2. And that's important, because we want to solve a second or the other PDE. So utt, uxx, we need c2, is a solution to our PDE. And say the limit as tx goes to 0x of utx is equal to u bar 0x limit as tx goes to 0x utt, tx equals u bar 1 of x. This is called the Dallenbert formula. It's remarkable to have such an explicit expression of the solution, rather remarkable. Remember, it is essential. We have used, in essential way, the one space dimension. The problem of having such a kind of formulas in two, three, four dimensions is another story. But in one space dimension, at least, we have this. Yes, it is unique. It is possible to prove that it is unique. That's right. You write here that it is a unique solution to our PDE. Yes. Indeed, you can prove the following. So the question was, is this solution unique for remark? So you see, what we have done up to now in this course is to look for more or less explicit formula, more or less explicit, maybe sometimes implicit formula for representing solutions. This is not possible, in general, for any PDE. But these are linear PDEs. And therefore, it's very interesting to start with a special solution, with an explicit solution, so that you can understand the qualitative properties. It's extremely important. Then, if you have a more difficult problem, for instance, assume that you want to solve this PDE, but with strange boundary conditions, or instead of the wall half space, just only in a piece, then maybe you don't have an explicit solution like this. And so you cannot hope to have such a kind of formula. But at least you know that in these special simple cases, you can understand a lot of things from this. And then you can hope that in more general cases, maybe qualitatively, there are the same properties, maybe. It's always important to have the largest possible number of explicit solutions at hand. This is very important. Remark concerning your question, then maybe this is some work is not so difficult. I think that assume u in C2 is a solution to utt minus C square uxx equal to 0, then necessarily in, say, 0 plus infinity times r, then necessarily there exists two real functions, fg of class C2 from r to r of class C2 such that u of tx is equal to f. Once you know this, this is not difficult, by the way. Why it is so? Because the operator splits into the composition of two operators of the form x minus Ct and x plus Ct. So this is homework. It's not difficult at all. This is very easy. But once you know this, then it turns out that necessarily f must be this. Now if you add initial conditions, now add your initial. This is in general without conditions on the boundary. This is in general, in the half space. So in the half space, a C2 solution must be the superposition of something traveling on the left. C is positive. And something traveling on the right. I will be more precise on this. Superposition of these two objects. Once you know this, if you want to solve your PD with u0 equal u0 bar at time 0, derivative at time 0 equal u1 bar, then necessarily you end up with this. This is uniqueness. This is uniqueness. Imposing boundary conditions, initial conditions, initial conditions implies that u is s. So now let us try to make some exercises on this expression. So I keep, maybe I keep, OK. So you see, this is of course of the form f of x plus ct plus g of x minus ct. No, not necessary. Not necessary. Not necessary. u0 x plus ct. So you claim that f of g are equal? No, I don't think so. Because you see? So who is f now? f equal 2. f is equal to, y is equal to? Yes. u0 bar, y plus 1 over 2c, integrals from x to y. So x now, maybe you have to just fix some point, 0, y.  Because this depends on y, you know? Ah, yes. OK. 0, y, say. u bar 1 of s in ds. OK. So this is f and g. It's the same you claim. Maybe you're right. Let's see. So if I take now f of x plus ct plus g of x minus ct, what do I find? I find. This is correct, of course. And then 1 over 2c equal to 0, y, sorry, 1x plus ct, plus integral 0x minus ct. But then there is a minus here. Minus x minus ct 0, u bar 1, u bar 1. And so I have just the problem of the sign here, right? There should be plus. Let me check one. So this is the part concerning this is this, from 0 to this. And then if I take plus 1 to c, 0x minus ct, u bar 1, y. Is this right? If I just sum them. So it seems to me that, well, the idea is OK. In order to construct f and g, you take this. This is OK. And then I take a primitive, or u 1 bar exactly. So having only f would, it is already interesting to take initial velocity equal to 0. It is already very interesting to start the problem with initial velocity equal to 0, which is only f, as you said, essentially. OK. So now, let us just argue. Let us just, so for the moment, let us just try to understand the proper qualitative properties of our wave knowing only that it has this particular structure. So the exercise, so let me write down the following exercise. So this is z equal g of x, z equal g of x, z equal f of x, for instance. And then, OK, ct. So c is positive. So this is the graph of capital G, say. And therefore, this is the graph of g of x minus ct. This is the graph of g of x minus ct, moving in this direction, on the right, with velocity ct. So this is a sort of wave. You can think of this. So you have the form of the graph of g. This is given. And then your solution, the part of your solution is something which travels on the right with constant velocity on the right, given by c, keeping the same shape. So it translates on the right, like this. This is, on the other hand, something which translates on the left with keeping the same form of graph of f. This is the graph of f. So it is the superposition. Our solution is the superposition of two objects traveling. For instance, if the initial speed is 0, which is more easy to understand, if the initial speed is 0, then this is not present anymore.  Essentially. And so you have u0 bar here and u0 bar here. Just only the superposition of u0 bar of x plus ct and u0 bar of x minus ct. It is, to understand the Dallenbert formula for me, it's interesting to analyze the case u1 bar equal to 0. So that these are, as he said, they are equal to u0 bar, 1 half, 1 or 1 half. So I leave you some homework for now. The next week I will not be here. So we will continue. So you have a lot of time. So maybe it is better to give you some exercise. So homework 1, u of t0 x0 depends on the values of u0 bar, u1 bar, only in x0 minus ct0 x0 plus ct0. It does not depend on the global properties. You have to think about this. This says, this is very easy to see from your solution. It says that your solution does not really depend on the global shape of the initial conditions, but the solution here does not depend on how u bar 0 and u1 bar are very, very far, but just only on an interval, which is this. So the value of u is insensitive of modification of the initial conditions out of this. If you change the initial conditions out of this, you don't see it at the level of the value of u at that point. Yes, I know. It is very easy, but it's immediate. Think about this. Write down the interval if you have here, say, t0 x0, write down here where is the interval. And you have to meditate on this, because this is an essential property of wave equations. It's immediate, I agree. Second immediate exercise. It's very easy. So fix an interval of the form x minus a less than or equal than r x-axis. Fix this interval. Draw the set of tx such that u of tx depends only on the values of u0 bar, u1 bar on the interval. So here, you have, I'm sorry, this is a homework. So third homework, draw and prove that this is equal to x minus a less than or equal to r minus ct. Maybe we can go also backward in time. More interesting exercise is the following. Take u, so assume u1 bar equal to 0. Take u0 bar of the following form x plus alpha if minus alpha is less than or equal. Minus x plus alpha if 0 less than or equal to x less than or equal to alpha and 0 else. Depict, draw, say depict, the graph of u tx at times u t dot at times t equal to alpha over 4c, alpha over 2c, 3 alpha over 4c, alpha over c, 5 alpha over c, and larger. And by the way, maybe a remark, this initial condition is not c2, as you see. It's just leapsheets. So this creates a little bit of a problem because our formulas for the moment work only for u bar 0 in c2. This is the corners in the graph, in these corners. By the way, the Dallenbert formula is OK also in this leapsheets case. So the exercise number three says, take the Dallenbert formula as it is also in this leapsheets case and draw the solution for these times. And then, suppose this is alpha 1, this is alpha 2, characteristic lines. These are called the characteristic lines of this x minus ct equal to constant. Draw the characteristic lines. Suppose u bar 0, in this case, again, is equal to 0 in alpha 1, alpha 2. Then divide this into six regions, time space into six regions, 5, 1, 3, 4. This is time space. Here, the initial condition is 0 in the complement of this. Otherwise, it is 3 in the complement of this. So here you have superposition of two ways. Try to look at that number three exercise. To interpret the number three exercise into the time space graph here, try to realize that here you have superposition of two waves. One going to the right, the other going to the left. Here in region two, here it is 0. Try to prove that the solution in region six is 0. Try to prove that the solution in region five is 0. Try to prove that the solution in region four is 0. Here, there is superposition of two waves. And here, there is just one wave. And here, there is just one other wave. So here, there is just direct wave. So called direct wave, just that one traveling to the right, and the other is called the inverse wave. So if you look in time space, there are these six regions, where here you have superposition of a direct and inverse wave. They sum up together. Here you see just in region two, you see just only one wave. Here you see the other wave only. And in the other regions, the solution is compass is 0. So this is interesting, this exercise, because the first part, you need to understand the shape of the solution. Given a time, then you depict the graph. And for that time, the graph in space, a fixed time. But this is a more, more, is another viewpoint. So the previous one is just the time slice of this.  Then the graph you're looking for there is just what you see from here at this time. But this is in time space, so it's more complicated. It's more global. Yes, u1 bar is 0. This is still exercise 3. This is a continuation of exercise 3. And u1 bar 1 is 0. Yes.


I've been serving in the church since I was 12 years old. I'm from Truman, Arkansas, one stoplight town next to Jones for Arkansas, maybe you've heard of that. And my grandmother paid for my mother's first three children to take piano lessons and I was the best of the three. So at 12 years old I was asked to be the organist of the church because they needed an organist. And I was a very serious 12-year-old and I took it very seriously and I started taking organ lessons and after one year I was on the official board at 13 years old.  I went back to that home church, took my six grandchildren there with me and one of my little granddaughters set up on that organ bench with me and saw the organ that I played for all the weddings, all the funerals, have a charm bracelet to prove it. That's how you got paid with the charms on your charm bracelet.  I remember when I very first started out my very, very first appointment. I was so anxious. I was a student at Perkins, I was anxious to get an appointment, I bugged my superintendent and so like the Wednesday before the Sunday the things were supposed to, he calls me that Wednesday and says, well, we have a church for you, oh, I was so excited and I said, well, how will they know that I'm the pastor?  And I've always thought about, well, that was probably some of the best advice I got in my entire ministry, put it in your tie, grab your Bible, show up, they'll know there was 20 folks in the congregation, it was in Sadler, Texas, you know, and they were so gracious, you know, I didn't know anything and they were so kind to me along the way, but you know, every appointment that I've served is its own unique gift. There's been in each one of them, there's a relationship that you share with them as a pastor where God is active within the context of the chapter that you write together and that to me has always been such a blessing. The one person who's been there and has been God's greatest blessing for me is my spouse, Julie, and she has just stood with me through everything, all the itineracy, you know, when I ask you the question, will you itinerate? That includes the spouse and the family in our case and I was the one who answered yes, but they were the ones who went also and I am grateful to her for just being such, she is such a gift of God to me, I'm so thankful for her. I grew up in the Northeast in a community that was on the shoreline and it was a place where the fresh water ran into the saltwater and they call it an estuary and an estuary, it's an ecosystem that just thrives for the vitality of life. It's very unique in terms of what happens in there and you know when I was coming along they used to sing the song Lord prepare me to be a sanctuary you know and I thought well that really ought to be Lord prepare me to be an estuary because really it's where it's in the coming together of where you know the people of the word meet the people of the world where they were the people who have a heart for Christ and that the heart of Christ that love of Christ are with folks who are we're all God's children you know whether in the church or not we are all God's children God's at work in the middle of all things. The most important thing is to listen to the needs first not make any assumptions of what is needed and to listen to people what you did last year it may change and you need to revisit what is now needed differently in ministry I know one year at Caldwell Elementary School our partner school we had been giving new clothes at the beginning of school year and communities and schools had to be honest with me and she says well really people can get inexpensive clothes now or hand me down clothes and that's not the focus school supplies have gotten so expensive and and socks and underwear are far more important and if she hadn't told me that we would have kept giving clothes and that would not truly meet the need and have done the wrong thing. I went to a seminar and they were talking about the church and kind of what the church is going through and they had young pastors up there and they talked about the future and when I saw that I thought praise God you know look there are there are people who are talking about the future and talking about the future with hope and and so I took hope from that that you know golly there's there folks who have a heart for what's ahead and and what a blessing that is. As often times in life you would think that the really tough things the disasters in life wouldn't be where you find hope but we are people of the resurrection and really that's where you find hope and really the intersection between despair and hope is where you find meaning in life. I had been putting flood buckets together for many years for UMCOR United Methodist Committee on Relief and knew that they were going to the UMCOR depot and and Baldwin Louisiana serving in Louisiana Conference but when Katrina hit and I evacuated and they let pastors back in before anybody sitting in the church hallway in the dark even before we got together for our first worship service floor to ceiling and the hallway as far as you could see were stacked UMCOR flood buckets they even beat me there I sat in the floor and I cried it was more than the gloves and the sponges and the things that were in the bucket and so it took on this whole new meaning of hope and when you survive something terrible for someone to offer you hope and it really gave me a new sense of ministry I will say in the tough times in life I've lived long enough as a retiree that I want people to hear the story we are people of hope and that flood bucket and our Connectionalism as United Methodists offering hope to people is such an important thing.


Hi, I'm Premier Christy Clark. This week, Finance Minister Mike D'Yang delivered a balanced budget for our province. Now that's a commitment that I made to British Columbians, and it's important that we delivered. Let's think about our shared values. The way we were raised, and how we hope to raise our own kids. We tell them, make sure you live within your means, do the best you can. And remember, don't saddle your own children with debt. That value is even more important when you're the person who's been entrusted with the hard-earned tax dollars of citizens. Because all around us, the world economies in real turmoil. We have worked too hard, too long, all of us together, to protect our home from that storm. And we can't risk throwing it all away. That's why we asked Tim O'Neill, who's a former chief economist at the Bank of Montreal, to take a hard independent look at our budget. So that you can have confidence in the work it took to reach balance. I've been guided by three principles in that work. One, we're all in this together, all of us. Two, government shouldn't ask taxpayers for a penny more without saving as much as possible by controlling its own spending. And three, that we have to protect families by making progress on childcare and helping them save for post-secondary education. That's the power of a balanced budget, a firm foundation to grow the size of our economy. Let's help families succeed, not by growing government, but by growing jobs.


Okay, here we go. So I started school again this week after a year long break from it. My last experience of school was in March, 2020. So I was trying not to give it too much thought. The great thing about Edinburgh is how close it is to nature. So it's been really nice just taking advantage of Hollywood Park and Arthur's seat. So Monday morning, I only had prerecorded lectures. Class is structured completely differently here. So you basically have lectures and then in-person discussion groups called tutorials. Two of my classes are pretty asynchronous and then I have one live lecture. But all of those little discussion groups will be in-person and they haven't started yet. So I wanted to get a jump on my work for the week so I did a lot of that on Monday. Okay, so I just took a break to go get a package that was delivered for my parents. I knew that it was coming because I left my ring code at home. Like I ordered a ring code but it didn't arrive in time so they didn't mail it to me. And he's been like telling me all week.  All your friends are gonna love you when this arrives. Me, I'm thinking like, oh, maybe he sent some candy corn because that's kind of like a normal, like funny thing to put in a package. Have all the people from the UK try candy corn because that's not a thing here. No, I was wrong. So he wrote out like 100 fortune cookies that you would find in a normal Chinese food restaurant. So now I have fortunes in my ring code pocket for my father for daily use, I guess.  It says he who throws dirt is losing ground. Then there's other ones that said help. I'm being helped prisoner in a fortune cookie factory. So anyway, that was funny. Thanks, dad. I just don't know how to explain this one to people. This is a very weird first day of school. Wait, wait, hold on. This one, this one is my favorite. It says a foolish man listens to his heart, but a wise man listens to cookies. I think that's good life advice. Tuesday morning, I started with a weird burst of energy. So I woke up and I went for a run slash hike in the Hollywood park. Honestly, wait, that's actually questionable. So basically I started running in one direction and then I saw this structure that looked like ruins or something on a building like it looked like it was on this hill. So I wanted to go see it because study abroad, but I ended up getting really lost and I went down this really steep side of the hill and I found myself in this little valley and then there was no way to get down except through this like very tiny and bushy trail. It was pretty fun. I recommend it, 10 out of 10. This day, I also have my first live lecture and it's been kind of weird doing real class for my bedroom and definitely a motivator to get my work done so I can spend some time, you know, out in the city. I did a lot of reading on Tuesday about medieval Europe, which was a real crash test to see if my brain still worked after 15 months out of academia. It was a little rough, but we got through it. I did the bulk of my work on Tuesday, so I did a lot of reading and I got to listen to this really cool talk about creative industries, which was just like right up my alley and in my course of study. It was really good to like tap into that side of myself that like really loves school. Wednesday, so on Wednesday, I signed up for a 720 AM Pilates class. It was my first workout class post pandemic and it was actually pretty fun. I did get a membership at the school's gym and it's a gigantic building and they offer a ton of classes that you can take. It was a small fee, but for a gym, it was pretty reasonable. Then I had a meeting with my tutor or I guess I would call like more of like an academic advisor and I went to go meet some friends at the library bar, which was really cute. These little moments. I wanna talk about how being here has weirdly made me hyper aware of the college experience. Because I'm here and I'm living with pretty much all freshmen and they're all like right at the beginning of their college journey and it's been kind of funny to like be here during like their first week. You know, I'm almost 22, so it's I'm here and I'm sort of going through my own adjustment, like trying to figure out life here because it's a new country, but like I've sort of already done a lot of what's happening right now. A lot of it is just sort of like reminding me, I guess that like I kind of will never get that like real like college experience that like we're all kind of sold before we go. Mine will just, it's done, it's gone. Like it's unsalvageable at this point. Like I will never have a true college experience if that even exists. Like I think that like we're sold that idea, but the perfect situation never exists. But obviously for those of us who, you know, had the pandemic disrupt all of it and you know, whether it was at the beginning or right in the middle or at the end, it's just like something that's maybe, maybe feels like something that we're lacking, but it's really been like I'm okay with it. Like I think I'm here and I'm sort of like watching the beginnings of everybody else going on this experience and it's just sort of reminding me that there's just like absolutely nothing I can do change the past. And it's like weirdly like grounding me a lot, just like saying yes to things and trying new things and not like taking myself too seriously, which I have a really, I have a habit of doing. So yeah, I don't know, this first week was good. It was so good to be in class again. I mean, even though it's virtual school, which is like still kind of weird, I'm really looking forward to like being in person next week and checking out like the tutorials and like what that means. So I'll definitely keep you guys updated next week. Am I doing dorm tour next week? I'm not really sure how it's gonna work here because I don't like, I feel like kind of shy. And I mean, like this is private. I keep making that joke to myself but like this is private. But like it's like, I literally don't tell anybody about this, but like I also tell everybody about it because it's like public, it's online. I don't know, it's very strange. But I guess I just wanna say like if you're starting college or if you're in the middle of it or I don't know, you're just like, life isn't going right for you. I mean, we all have this like collective experience that we're trying to grapple with. I don't know, it's just like a certain point where like we have to let it go. You know what I mean? Like it's done. The damage has like really been done, you know, in a real way. And I think that just looking forward is like if you're looking for a sign, this is it. It's time to move on, it's time to look forward and you know, just focus on good things and then more good things will come. I don't know, I'm being very sappy because it's late. They say the later it gets, the more honest you get. Who knows? Thank you for watching. Bye bye.


We were having good times on the sets. We were about to speak with Matas, Raghbi, Loka Land and International Land. We were going to speak with what happened on Thursday. We were at Coach Paul Fini and announced the squad for Los Angeles and Vancouver. That is the fifth and sixth leg of HSBC World Seven series joining us. Iwona Mai was hard touch and passion for Raghbi. Of course, overwhelming passion for Raghbi.  It was my first sports marketing job actually. So you developed passion for Raghbi since then? I learned from scratch and I think before I worked for the Kenya Raghbi Union, I had gone for Raghbi games. But I'll tell you for free, I did not remember who played, what the scores were, I did not know the rules. It was just a social, it made social sense to go for Raghbi before. Because I think myself and Osoro agree with you. We've been attending Raghbi games. A lot of ladies are coming, you know, very excited about the games, but when you ask them about the scores, they are not aware. No, you cannot blame them. It's because Raghbi is majorly a festival. What happens on the field is not just that that is happening on the field. There are other things that happen past what's happening at the field of play. Is that the same way Kenyan football administrators are supposed to package their round brand so that they can start attracting, you know, crowds? True, true. Our football has been majorly playground football. Just come watch, nothing else happens in the stadium. So people think that the stadium is all about going, sitting and watching the match. Yes, you have the 90 minutes of watching the match, but what else happens after that or before that? We were speaking about sports marketing and football specifically because right now no one is ready and very interested to go at Roaraka Sports Club or at Kamtoyoyo to watch Ushuru against Fortune Sarko because they say no value for their money. What do they mean when they say so? I would slightly disagree, but I believe the problem of Kenyan football is in the culture or lack of culture thereof. Culture is an association to something. For people to do what you want or for people to follow what you're following, then you should identify with something. I think since we started this at the backdrop of rugby, the difference is there is something that each rugby club stands for or the rugby players stand for. So when you ask somebody about rugby, they have different descriptions. The most common will be I can go to rugby with my child, I can go with my friends after doing my activities, I can hang out at rugby. It's something I can associate with. But when you ask about football, first of all, I think most people would start, I don't know where to watch the games, I don't know how to get where the games are. And even for those who know the people who are playing, I don't think I'll be so comfortable going to watch football. So I think it has to go back to defining what Kenyan football is because again you have to understand that the people who know the technical side of the game are not the people who will come and follow the game. That ratio is too little. So if you rely on the technicality of the games, then you'll remain alone with your games. That's why we have only footballers enjoying the games and going home. They only know what is happening. But the people who follow the game, they need much more than the technicality. Also I think because the culture that you said, a way of life, because you look at rugby, it's one spot that people love even from high school. You can follow them in that people know that high school rugby is big in Kenya. These are the same kids who will grow to university rugby, college rugby and these are the same kids who will grow to club rugby. And people can follow that. It's a true story. And it happens like in the world where look at Leblon, these kids have been followed from high school, college all the way to the NBA, NFL, the same thing. High school, college, NBA, then you turn professional. Rugby in Kenya is actually one spot that has done that and you can know that this kid actually went to Nairobi school. Then he went to University of Nairobi, then he went to Muamba and he went to the national team. But when you come to a football in Kenya, just one you will say that, eh, Osoro preda, I don't know for Maseno, then I don't know Western steamer. But that link between the narrative that they have for the people is really different. And without that narrative, without that storyline, then there is no way people can follow you. You will never have eyeballs. I mean a few forums of football where, you see there is a sharp disconnect in terms of funding football activities from the government and funding other, you know, sporting disciplines like, let's say golf, World Rally Championship is coming back to Kenya. We saw even President Kenyat himself being at Mu International Sports Centre Kasarana flagging off and even announcing sponsorship from the government. Kenya Open is slated for March 12th. And we will see a lot of government representatives there. But when it comes to football, no sort of, you know, government interests. And now that brings us to the question as to why several football journalists even have been questioning the government's lack of willingness to fund. But from marketing angle, why will governments prefer other entities at the expense of what we perceive popular sporting disciplines? From a marketing perspective, it still goes to what do you start for? What brand do you represent? Actually just a little bit of correction there. In terms of funding over the years, if we were to go at least two decades back, then football in Kenya has received the most amount of money of funding from the government. So that one is very clear. We can follow through. And ever since football has always been the only sport that has consist, those guys, they will go to the press and say, we can't go somewhere, the next minute they're given money of all the sports. They're usually very adamant. And they're usually the most, they're like that last born child. Whenever they throw a tantrum, things go. The government will rat. Exactly. So I think what do you start for? And we're getting into an age where everything has to be justified, even the expenditures within sport. And the only reason that Kenya is not allocating as much money to sport yet is because the sport has been able to account for all these fans that are being thrown to it. And football for us is the best example. We've seen countries, our neighbors Uganda, our drunkers, and their government still funds them. And it's very systematic, but its performance best is result best. They're playing better football than us, both at senior and junior level. So it comes down to what do you bring to the table? You've mentioned the Kenya Open. Yes. There's a huge diplomatic benefit of the Kenya Open that hadn't been actualized before. In times where sports tourism or sports is playing a crucial role in diplomacy, then you have to be cognizant of the political needs of the sitting government. So you are always, all the time, when you're looking to analyze such decisions, you have to get out of the sporting element and look at the bigger picture. So we've seen our neighbors like Rwanda invest money in clubs, in European clubs. And their president has been so much committed. Exactly. He's become the face, he's become the number one marketer for the country. He goes and is in the most important places, the right place at the right time, all the time. So instead of us, since we do not have a similar budget system, we do not have a similar funding system, instead of us doing the same of going out, then we had the one chance. And you have to remember the current Cabinet Secretary of Sports is a diplomat. Yes. She knows the importance of welcoming people and actually hosting them in a good way. She knows the importance of image. And right now that was the biggest and one of the first steps she did. If you look at some of the things that she's done just when she got into office, first was to fast truck, getting so far really back into the world rally calendar. She went out there to meet those people. And even if you look… There's a lot of lobbying. Lobbing that she does. Yeah. And in repackaging the Kenya Open, it is supposed to bring people together. You have to look at who plays golf, who watches golf. These are the decision makers. It means even as we are going as a country in the bigger picture that the country has as a government, then for us to capture the attention of the people who will spend money or pump money into the Kenyan economy, then we have to meet them where they are and most of them are where the golf is. Most of them are where the rally is because if you look at these two sports in terms of expenditure, when we are doing a marketing strategy, we look at consumer analysis, who consumes these particular things. So the people who consume golf and rugby, if you look at golf and rally, sorry, are these people who are able to make a significant impact economically in the long run. So for us, it's much bigger than just the sport. So the takeaway from this is that the current sports rights holders, the sports federations and the teams need to look at the bigger picture so that when you're talking to government, you're meeting them from their point of interest. So as a government, we know you're looking towards this. Like for instance, if you start with a vision 2030 approach towards government funding, you might get 10 times, even 100 times more funding than what you're getting as a sports federation. So look at what the government is doing and then look at how you fit in to them achieving their objective and the rally and Kenya Open have been able to articulate the same. So they are suiting the political and benefiting the sport. I know sports marketing is not our topic this particular afternoon, but just to wind up on that aspect, Yvonne is talking about a very important element of, you know, by the virtue that our CS is a diplomat, therefore, you know, she has to take us to the global front in terms of marketing as the same way a Paul Kagame Rwandis president is doing with now the number one marketer for that particular country. But now where does this leave the role of Brand Kenya and Kenya to what is in board? No, I think they are enjoying in that mostly with the Magiko Open, you will see Brand Kenya there and everything. I think Brand Kenya was raising awareness of Kenya all over and they are using golf to do that. I think it's because it's easy to use golf because it comes with many different terms and it being a European tour, it affects everybody without said Kenya. We are so in the world of golf because these are the people who follow the European tour. They want to know who is the best ranked Kenyan, who is the best seated Kenyan to come and play. And it will be major, people just don't realize this, but once a local or from the Magiko Open guests now to the European tour himself and you see our Kenyan playing in Europe, then people realize that this is actually big. But even away from that, look at the companies that are coming to sponsor that. You realize the company, the way you are saying it. Look at the companies that are coming to sponsor such an event. Look at Absa Group, look at EABL. And they know these are the consumers who are taking our product. Other sports, Muscle Football, they are still lagging behind. They don't want to embrace the way things are done at the moment. They are still lagging behind and saying that we need funding, we need to get this, we need to get this. They don't even know how to package to get money for themselves because last time the Kenya Open, last time itself, I think Marches gave them upwards of 450 million for broadcast rights of golf, which many Kenyans don't actually watch, don't even know that there's golf on TV. Even if they tune into a golf channel, they don't even realize that this is a big thing and that's where they lose the point of it all. And it is going to be big and it's going to be bigger. And these other sports, all the time they are going to be suffering because they don't want to embrace new ideas. They don't want to embrace new ideas. Now even let's speak about the squad naming of Shujabai Paul Finion Thursday ahead of Vancouver and Los Angeles 5th and 6th legs of Fethish Biswal 7 series respectively. Rotational aspect, why is the coach opting for the same? Because he's dropped a few players who featured in the last four legs and bringing in new faces, the likes of Oscar Uma wasn't played recently, making the cuts. It's an Olympic year. It is an Olympic year. And with the prestigiousness of the competition that is Olympics, then everyone wants to at least medal at the Olympics. If you remember Kenya's best showing in Global Rugby was during the 7th World Cup way back, it should be in 2013 or 2013. And we were able to make semis. I think Finlis, I've not had the opportunity to hear so much thoughts about that but I think he knows or he's realized the potential that Kenya has to actually medal at Global competitions. So he's been taking the past two years to create the squad that is going to play at the Olympics. So I would look at it as a strategy towards him picking his squad for the Olympics. For me, I think it's a positive. I didn't think people like Oscar Uma are going to come back to the 7th World Series because considering the conditioning required to play at that level, I thought that the absence from the 7th scene might not be a reality. Osoro Robert, the inclusion of those players notably Oscar Uma who has had, of course, his time with the national team Shujant with his vast experience he has played at the big stage most of the times. Sami Ullich also making a comeback. I don't know, how would Paul Finlis revolve around trying to blend those who featured during the previous four legs and those who are making the cut for the first time this year. I think in my opinion also he has to consider the performance of the previous legs because the previous legs we did not, like the last leg we did not, the sitting leg would not perform very well. We lost some, all the three matches that we had in the pool. So I think he has to consider that and going to Las Vegas is also considering that we are playing Las Vegas and Vancouver. Then the calendar changed because of the COVID virus and we are going to have the Asian legs coming a little bit later. He has to consider that I have to get my best players going for Las Vegas, rest some of my best players and bring them back into the tight end because now you realize that the format of the series also changed. We don't have the ball, we don't have the plate. It's all about the ranking number. Where are you going to get, where's your number going to be and that is actually what they are fighting for the moment. We are going back to the old format for this North American leg. I think it was a very, it was a commercial best decision which I did applaud. It's not good for the 7th rugby fan but when you come to an age where especially while rugby being the first global federation to actually remove the gender identity out of its competition then they are looking to give both men and women an equal opportunity and this was a very commercial decision in terms of ensuring that the 7th World Series incorporated as many women's competitions as possible and we went back to the ranking system and I think our coach Finley might have had a little bit of confidence bringing these players at this time because we are reverting back to the old system for this North American leg and we are going to have enough time to showcase or to test the fitness levels of Sami and Umma coming back and then I think also it was a very psychological move on the end of the coach because we've been having young lads we see them, we've seen them during the 7th circuit they do so well and then they get there and there's a little bit of... Complacency? Now performing at the world stage is different. So we need this blend actually is very good for us because now at least the young boys are going to actually have psychological confidence going into the game so they'll be able to execute I think it was the time of the former homeboys coach was also Paul Muruga the one who was put into the deep end baptism by fire bringing the young lads but when they get to that stage it's a bit different that's why even he has kept one captain at the moment, no changes in the captain because this is the guy who can bring the young guys come to the fold this is how we do it at the global stage and they can grow from there How instrumental is the experience tree of calling Sinjeru, Andrew Amond and Wilins Ambaka in that particular squad of course they have been evergreen, they have been there I think it's also they are getting into the toilet years of the game for them and now they have to give these players a chance they are calling Sinjeru but he is not performing the way he did in 2012 he is not performing the way he did in 2010 so I think he is there back from injury again this I think is third leg that he is getting into but now I think his major work at the moment is molding this young tux to take over and we've seen that during the half time talk the boys get it Sinjeru gets so personal he is like you know guys this is what we need to do he knows he is not as fast as he used to but he has taken that I think that's one of the things that made me think that Injera and Oscar might not come back because we saw them I noticed they were more inclined towards making things better than during their time I think sometimes I always think Andrew Amond is a superman he doesn't look like he is anywhere retiring of all the people we've had in the series our captain he doesn't look a day older than 25 I remember these players are the ones who brought us to the world stage they are the ones who brought us to the world stage and the narrative has been them at that stage but I think we are moving away from that now we are bringing in the young guys look at Bafa Meda I think is debut in Hong Kong 2016 and from there he has grown on to that stage 24 years but he has grown on to that stage and for now even everybody will be like I want to get on to that I think the role Andrew Amond has been playing giving his service to the nation needs to be applauded because this man I wonder which time does he rest because he is featuring in Safari 7 during Kenya Cup he is there for KCB KCB all the time you know for 15s I think what your team has done as a player during your prime just perform at your prime but he has been at prime for a decade Andrew Amond has been at prime for a decade the most interesting brother about Andrew Amond you remember when the heavyweight the experienced players were downing their tools over contractual Andrew Amond was there man I was captain he actually upheld I think Andrew was a brand I don't know I think at some point he gets branded as a betrayal because you remember when the likes of Injera Amond Injera, Uma Uma Adema, Lavin when there was that but I think for him he is shown the best leadership quality that would require any captain actually because he doesn't take sides all the time he doesn't take sides he is always almost neutral to try and fit in the needs of every party I think what goes on behind the scenes is maybe much important than what goes on in front and he knows that as a leader he knows he doesn't have to air his dirty laundry because to the public so he doesn't so he doesn't feel the need to look like oh let me do this so that I am seen like this he maintains that he is like our Waikibaki Okonobi was doing so well but anytime he was being bashed as president he never responded but things were going on the way so I think he is a very good leader and I think for all our players they responded the best to the strength and condition for the rugby player and he is the ultimate machine Ultimate machine of course as we wind up let's speak about Kenya Cup it's March day 14 and fixtures are on card of course KCB up again Oilers, oh my goodness Oilers, is it because of financial and concrete sponsorship they have been doing very well considering they are making their debut in Kenya Cup or it is the second time they have Gibson Wero it is also the people it is about the people and I think if you had a chance I didn't get to watch Gibson Wero as much but if you had a chance to go through his philosophy as a player then you'd understand why he's doing so well as a coach and think money not withstanding it's about the person behind the team it's about the people within the team and I think for him I think he makes the entire change in the Oilers team Gibson Wero must have played alongside the likes of Benjamin Nahim Yes, he was in fact he was the only kikuyu in the rugby team and just when people thought this line is not meant for people from Central but he has a very good work ethic and I think that's what he's instilled in his players and we've seen him handle situations I think when they made their debut during the seven circuit as a team we saw how he handled the team and his players have actually benefited a lot from his ethics Western Bulls, they are not doing very well and they are up against strong boys at the bull ring they lost much they attained they lost like the bull ring but they look like they will get relegated they actually relegation zone kizumu, relegation zone and kizumu is also playing blah blah blah blah that is at KU and they are pulled away from the zone and they are trying their best to stay there because you don't want to go for that relegation zone and you don't want to come back that looks like a derby it's been worrying actually to have both Western teams in relegation it's very very worrying generally what do you make of the kenya kapran this particular season how is it been like of course it's a battle between kcb and kabras kabras debut in the kenya kabran they are not stopping they are just maintained this season has been has been really good because if you look at the people who are fighting for the playoff slot they are also not so not so common people we've seen a season where heliquins are almost looking like it's their first season back in the kenya kapran so I think we are picking up the only worrying thing they are having the playoff slot we have got I think momba we have got nondis impala we have the first two kabras kcb but this one we have to fight for that position the only worrying thing is having sumu and western bulls at relegation worrying because initially playing in the kenya kabran playing in the premier competition is a sign of a result of the work of growing the participation because if you do not have all these regional representatives in your premier league then it means we are doing something really wrong so for me this season that will be the only worrying thing the national championships are amazing and I think sometimes if that one day and in a deal situation we had TV for that I think people would understand why it means so much to have regional TV I think if there is a spot in kenya any corporate to come on board and sponsor it is rugby that overwhelming passion from rugby fans just like you indicated even from players themselves they deliver to the fullest I think what people have failed to understand is I was told at the moment is being failed by decision makers most of our decision makers more so in a traditional TV is that they don't understand rugby is a big sport everybody thinks football is a big thing athletics is a big thing but they don't know the value of rugby on TV look at the Super Rugby everybody watches Super Rugby everybody wants to watch when they go on TV everybody watches on TV everybody goes to TV and says I want to watch Super Rugby look at 6 nations people are following it all over the series comes everybody is on TV to watch and it's us as TV which I am personally in to make that concrete decision to cover up that's the open because you look at even these sports like the NBA, the NFL which are big on TV they not just that a boom to TV they took there was a president said let's go cover basketball even when you go back to the history of the English Premier League it's all driven by the media largely driven by the media and you go to the UK you will not see anything to do with any other league or federation until they cover everything to do with the UK you watch BBC you watch Sky Sports you will know up to that 6th league before you hear about any other league so there is a huge media influence in terms of Kenya I think we have to be cognizant of the fact that the sports ecosystem is not fully developed so it might not make economical sense right now for broadcasters to pick up such properties but I think we should we should empower our current government to understand the economical sense that sports brings and that way then it will have a ripple effect if you notice in anything that is done if it's government backed the ripple effect always ends up being a lot if we just go back to the Kenya open within sports marketing and sponsorship we have something we call category inclusivity if I have this water then I can't have another water in the same event and you'll see it's replicated across various sports properties but look at Kenya open how many banks give money to it all our Kenyan banks give money to the Kenya open why because of the government factor so anything that government backs for instance right now if government was to take a strategic direction to invest in grassroots sports you'll find that all these branches of our nini will give something to support those grassroots projects so we are moving away from playing blame games but what we can do right now is to ensure that we expose especially people like me in sports marketing and in sports business we are working towards ensuring that we expose our knowledge because when we do that then they understand what role they play now like for instance some of us are trying to lobby to have the government include sports as a key strategy for achieving the vision 2030 because we understand how that can influence or rather accelerate the government achieving this vision so it's about now sharing the stakeholder within sports you speak to it's about lack of knowledge most of them do not understand what role they play so right now we are moving away from also and so should have that we are saying the table is so big so you can take this seat at the table no game of musical chairs this time the table is set the best advice for the federation that everybody in sports is give a chance to someone who has landed to help you out because if you remain your traditional role you are going to fail completely quality conversation it's been with Yvonne Amaya reputable sports marketer and someone who has overwhelming passion for Agdi as well talking about what has been happening in the world of sports marketing of course this sports marketing topic is a huge conversation that we need to create a two hour block show just to talk about it because even as we speak now at some point one of the Kenyan leading betting firms donated their contracts with Irish football body they have also left sponsoring Everton and though they continue with other franchise Simba and Yunga in Tanzania alongside which entity in South African also they continue with sponsorship of the same so Yvonne it's been a pleasure having you on board thank you for coming through I can't go without saying that you forgot to say about the challenges series and they have qualified to the Olympics they qualified to the Olympics and if you remember the last Olympics they did not get so many chances to actually play more games before the Olympics so we did not do so well in the Olympics so I think I've heard Lenis on the show in all fairness just as we close out also the Coach Findlis because you have to remember Findlis so part of his program included increasing the exposure for exactly especially much fitness heading to the Olympics so they are heading out to the challenges series also for them in Cape Town and I think we're going to get a few more games than we had before the previous Olympics thank you for that clarification and of course looking for sparkling performance from Kenya Linus we have qualified so two Kenyan women sports team sports they have already qualified of course Suja too have qualified so we are waiting for other representatives to see whether they can also qualify we've seen noq trying to intensify their preparations in terms of even urging the government to fund all these sporting disciplines to prepare very well ahead of the summer games slated for Tokyo that brings us to the end of this conversation we're going to take a short break of about two minutes before we come back with the funds and where we will give a focus on international football thank you for sticking to the touch line we will be back shortly


You know we're very fortunate to have someone like Rob, I think it's very difficult at the Division III level that no one is awarded necessarily a full-time assistant and you know as you can see from the results of what he does throughout every day and as a teacher in our community we're just very honored to be able to have him here at Stevenson as well because he does put in full-time hours here as well and I really appreciate that. He's in charge of our offense this year, he's taken on some more administrative roles for us, I'm very lucky. You know to be able to rely on him, if you know ever there's a time that I can't be at practice I have 100% faith that he'll be able to run practice as if I was there. So I couldn't be more grateful to everything that he does for us and I think that he's an integral part of the success that we've been able to have over the past couple years. Coach Slokak, can you maybe even guess maybe while we're here? One million dollars. No, close. Alright everyone, so Coach Slokak has been named the WBCA Assistant Division 3 Coach of the Year. I think it's just a great award for all of us, I mean you know being with Coach Boswell and what she's allowed me to do to allow me to keep coaching and do what I do from from Centennial and bring it to the college level and to be able to help the other coaches and have that freedom within the athletic department and the support of everyone else it makes it a lot easier when you're allowed to go in some place and just coach and not worry about stepping on people's toes and not saying this or not saying that and being able to agree or disagree. So it's a great environment and I think that's why I've been able to be successful and win an award like this because again of the people that we've had around us and the support, so it's quite an honor. I think the other asset that Rob really brings to women's basketball is just his involvement at the national level and that he sits on the WBCA board representing the assistant coaches for Division 3 and just being able to give up his time to be able to make sure that the WBCA at the national level is doing good things as well. He's had the opportunity to present at our national convention over the past couple years so he's always looking to give back to the entire sports for women's Division 3 basketball. I think it's a really big honor to be one of 430 Division 3 schools in the country I think it's a really big honor for our program and Stevenson himself to put us, you know, our name out there again nationally of what we're doing and the support that I've had with the players that we've had, the coaches and administration. So I think it's pretty neat, it's exciting to be one of those people. Well first I'd like to start off by saying congratulations coach Rob, well deserved. You're the hardest working person here.  You always are like the most goofiest person in our circle and I wouldn't have it any other way in the past 4 years. We've gone really close and I really appreciate everything you've done for me and this program. First congratulations coach Rob and coach of the year award. I really think you deserve it. I don't know anyone else that would be as deserving as you. You really put a lot of time and effort into making sure that we reach our goals for the season and really capitalize on all of our talent and stuff and it wouldn't be the same without you. To me you're the one that recruited me, the whole reason I'm here at Stephenson today and I got to learn from you and I can't wait to see how far you go. This is only the beginning for you and I just, everything you've taught me I've applied it in basketball and I know my fellow teammates have as well and I just can't congratulate you enough and the skills that you, the coach that you've taught me, I hopefully one day I can use them with my, if I ever coach one day and hopefully I can be as successful as you are.