Welcome to this lecture. In this lecture we are going to discuss about the non-homogeneous wave equation how to solve it using a very general principle called Duhamel principle. So the outline for the lecture is as follows. First we recall certain things from the lecture 4.1 and then we state the Cauchy problem for the non-homogeneous wave equation. And we introduce Duhamel principle after giving a motivation for it. Then we apply Duhamel principle to wave equation we do for 1, 2 and 3 space dimensions. So let us recall from lecture 4.1 certain things about the Cauchy problem for non-homogeneous wave equation. So given functions phi psi and f Cauchy problem is to find a solution to the wave equation this is the Duhamel operation operator equal to f on Rd cross 0 infinity such that u of x 0 is phi x ut of x 0 equal to psi x. Phi and psi are generally referred to as Cauchy data and f is referred to as a source term. The Duhamel operation operator is a linear operator since Duhamel applied to u plus v is Duhamel acting on u plus Duhamel acting on v. Therefore a solution to the Cauchy problem which was stated on the earlier slide that is the non-homogeneous wave equation with the Cauchy data that may be obtained as sum of two functions v and v tilde where v is the solution to the Cauchy problem where the wave equation is without the source term. It is called homogeneous wave equation. When there is no source term it is referred as homogeneous wave equation and v tilde is a solution to the non-homogeneous wave equation but with 0 initial conditions. We will state more clearly on the next slide what the problem precisely is. So u equal to v plus v tilde and v satisfies this homogeneous wave equation and the Cauchy data and v tilde satisfies non-homogeneous wave equation and 0 Cauchy data or 0 initial conditions. So when you add v plus v tilde by the linearity of the dilambation operator, dilambation of u will be dilambation of v plus dilambation of v tilde that will be 0 plus f therefore f. So you get the non-homogeneous wave equation satisfied by u and then look at the first initial condition v x 0 is phi x, v tilde x 0 is 0 therefore u of x 0 is phi x. Similarly ut of x 0 is vt of x 0 plus v tilde t x 0 one of them is psi and the other one is 0 therefore the addition will be psi, the sum will be psi. So therefore u indeed satisfies the non-homogeneous wave equation and recall that we have already analyzed these problems. The homogeneous wave equation and the Cauchy problem for it we have analyzed already therefore it is enough that we analyze this or know how to solve this problem then we know how to solve the non-homogeneous Cauchy problem for the wave equation. So let us look at the Cauchy problem for non-homogeneous wave equation with homogeneous data or 0 initial conditions or 0 Cauchy data and we want to solve this Cauchy problem in dimensions 1, 2, 3 because we are discussing only wave equation in 3 dimensions 1, 2 and 3. These are the 3 dimensions in which we are discussing wave equation so far. So a general principle called Duhamel principle will be used. So what is Duhamel principle? It says solutions to non-homogeneous linear differential equations that is also known as equations with source terms may be obtained as a superposition of solutions to the associated homogeneous differential equations. In other words what it says this is a rough statement, precise statement we are going to see later. If you want to solve non-homogeneous equations it is enough to solve homogeneous equations that is what it says. Of course the initial data will be tailored using the source terms because after all we want to solve the non-homogeneous equation that means source term is given to us. So we have to use that and make or manufacture this homogeneous differential equations in the corresponding Cauchy problem. Essentially the initial data will be tailored using the source terms. Motivation for Duhamel principle we will give using initial value problem for a second order linear ODE. So initial value problem for a second order linear ODE is given by this d2y by dt square plus mu square y equal to ft y of 0 is y0, y dash 0 is y1. So in this f y0 y1 are given and we are supposed to find a solution to this ordinary differential equation satisfying these two initial conditions. If you notice this looks like wave equation because in wave equation we have a u2t term that is like d2u by dt square but instead of mu square y we have minus Laplacian y equal to f of xt. Naturally wave equation being in partial differential equation you have both x and t variables but one can view wave equation also as an ordinary differential equation. Here in ordinary differential equation y is a function of t given any t y of t will be a real number. In the wave equation given any t u of t will be a function of x that is the difference we will come back to this later. So first let us see how Duhamel principle works in this example. So the solution to the initial value problem one can explicitly solve and write down as this y of t is y0 cos mu t plus y1 sin mu t by mu plus 1 by mu integral 0 to t sin of mu into t minus tau into f tau dt. This is a formula one can verify we are already used to this in a course on ODE we would have got this solution. Now we want to see each and every term closely and then we will see how Duhamel principle comes into play. So before that let us introduce this homogeneous equation. Remember Duhamel principle is something to do with the homogeneous equations now where the so we will make the equation homogeneous and the initial data has to be Tyler made using this source term. So with that view in mind let us discuss homogeneous ordinary differential equation with initial conditions where y of 0 is 0 that is initial position is 0 and initial velocity is y1, y prime 0 is y1. A solution to this we will denote it as S y1 of t. So S for solution y1 to remember that y prime 0 is y1 and solution for this ODE, ODE is in the background homogeneous and this map which maps given y1 a real number to this solution S y1 of t of this problem is called a source operator and of course we know the formula and that is given by y1 times sin mu t by mu. So S y1 of t is solution of this homogeneous ordinary differential equation and with y1 as the initial speed or initial velocity and initial displacement or initial position is 0. So in terms of the source operator the solution of the IVP is given by this yt equal to S y0 of t d by dt of that plus S y1 of t plus integral 0 to t S f tau t minus tau d tau. What is S f tau? It is a solution operator. What does it do? It maps f of tau as given as the initial velocity S f tau of t will be the solution of the given ODE. Let us define the integrand to be a function of t with the tau hanging around. So w of t and semicolon tau is S f tau of t minus tau. We will observe what happens to this w. What is that w satisfies? So where each tau positive the function w as a function of t is defined for t bigger than tau and it solves the ODE. So it is a solution to the homogeneous equation and it satisfies initial condition. When you put t equal to tau w will be 0 and wt of tau tau will be f tau. Remember that is how the source operator is defined. Source operator is defined like that. When t is equal to tau it is S f tau at 0. So that is why that is 0 because S f tau solves with 0 initial displacement and this is a velocity. So that is d by dt of this quantity at 0 will be f tau. Now if you observe this, this first two terms, some of the first two terms is a solution to the homogeneous ordinary differential equation and the Cauchy data is satisfied. That is y of 0 is y 0, y dash of 0 is y 1 is satisfied. Now this term essentially is solving the non-homogeneous ordinary differential equation with 0 initial conditions and how is it obtained as an integral? You can think integral is a certain sum therefore generalization of the sum therefore this is a superposition of the quantities inside. But what are the things inside? They are solution to the homogeneous equations. So this is the explanation or the motivation of Duhamel principle coming from initial value problem for a second order linear ODE. Now let us apply Duhamel principle to wave equation. So wave equation as a second order ODE, one may interpret the wave equation as an ODE in the variable t. In the case of ODE is for each fixed t, y of t is a real number or an element of Rd depending on whether we are dealing with a scalar equation or a system of equations. In the case of wave equation for each fixed t, u of xt is a function of x in Rd and thus takes value since space of functions defined on Rd. So let us compute the source operator in the context of wave equation and construct a solution to the Cauchy problem for non-homogeneous wave equation. So let us see what is the source operator for the wave equation. What is that Si of t for the ODE, right? For the wave equation Si of xt, what is the Si represents the initial velocity and initial displacement is 0 and you solve the homogeneous equation exactly as it was the case in ODE. So Si of xt let it denote the solution to this Cauchy problem for the homogeneous wave equation and 0 initial displacement and initial speed as Si. The source operator is well defined, we have to ask when is it well defined that means we have to ask when is that we have a classical solution to this. If D equal to 1, we need Si to be in C1 because we have a Dalambert formula which is applicable and then for D equal to 2 or 3, we need Si to be in C2 that is when the source operator is well defined. Now we expect the function, this is precisely that Sy is 0 dash, now it is y0 is phi and y1 is Si in the context of wave equation, this is Sy1 and this is integral 0 to t S f tau which we now wrote f subscript tau of t minus tau x dependence is there, so we write that. So this we expect to solve the non-homogeneous wave equation and the Cauchy data, so f tau of x stands for f of x, tau. So we expect that to solve this Cauchy problem, let us call this as a non-homogeneous Cauchy problem, so we have to check that. So in this formula, some of the first two terms on the RHS namely these two terms is a solution to the homogeneous wave equation and satisfies the initial conditions. What are those? Initial displacement is phi and initial velocity is Si, so these conditions are taken care by these two terms. Therefore what we expect is this term will be a solution to the non-homogeneous wave equation with a 0 initial data. Now remark on the first term, so we have to check that these are indeed solutions to the homogeneous wave equation, we are going to do that and they satisfy the Cauchy data, we will check that. So now let us concentrate on the first term, it is dou by dou t of S phi of x, t. It is a derivative with respect to t of what? S phi of x, t. What is S phi of x, t? It is a solution to the wave equation by definition, solution to the wave equation, homogeneous wave equation, initial displacement 0, initial velocity phi x. Now what we have the first term is a time derivative of a solution to the homogeneous wave equation. Therefore it is itself a solution, one can check that. If you have a homogeneous wave equation and let us say u is a solution to that then dou u by dou t is also a solution, dou k u by dou t k is also a solution. We have already mentioned this in one of the tutorials earlier, otherwise you can just check. So it is a solution that is not doubt. So it follows from derivative of a solution is a solution for the homogeneous wave equations. Now we have to check the initial data. So this first term satisfies initial displacement phi and initial velocity to be 0. So let us check initial displacement that is dou by dou t S phi of x, t at t equal to 0. We want to check this is phi x. This is by definition because what is S phi? S phi is solving homogeneous wave equation with initial velocity as phi. This is exactly that dou by dou t of that is initial velocity. So that is phi. So this is by definition. Now we have to check this condition. Initial velocity is it 0 or not? That means we have to take this term, differentiate this with respect to t. That means essentially dou 2 by dou t square of S phi and check that that is 0. So this is what we have to check. Dou 2 by dou t square of S phi of x, t at t equal to 0 compute what it is. By the equation because it satisfies homogeneous wave equation. So u t, t equal to c square Laplace in u. So that is what I have written. But now I will evaluate at t equal to 0. That is why both sides I put t equal to 0. Now what is Laplacian? It is a differential operator with respect to x. What am I doing here? I want to take the value at t equal to 0. Therefore, I can as well take the evaluate at t equal to 0 and then take the derivative with respect to the other variables, space variables or take the derivatives and then take t equal to 0. Both are same because the evaluation I am going to make is for t and the derivative that I have in the front is Laplacian that is with respect to x. Therefore, when I do that I get this. But what is S phi of x 0? By definition of S phi the initial displacement it is of the solution. So therefore that is 0. Therefore it is 0. Now if you notice instead of Laplacian you could have had any operator that does not depend on t. This is an interesting and important observation. So the Duhamel principle application is not limited to wave equation only. You can apply u t, t equal to any operator L of u as long as L does not involve the variable t. So therefore we check the first term satisfies homogeneous wave equation and it satisfies the initial conditions u of x 0 equal to phi, u t of x 0 equal to 0. Now let us move to the second one which is more straight forward because just the definition. The second term is by design or definition is a solution to homogeneous wave equation with initial displacement 0 and initial velocity as psi. That is the definition. Therefore the sum of the first two terms is going to solve homogeneous wave equation because both of them are solutions to the homogeneous wave equation and initial conditions will be now u x 0 equal to phi and u dx 0 equal to psi done. Now we have to concentrate on the third term now. What about the third term? So the integral term satisfies 0 initial conditions because if you check t equal to 0 integral collapses there is no integral so it will be 0 and check that the derivative is also 0 that I leave it as exercise very easy to check. So what remains to really check is that it satisfies the non-homogeneous wave equation. Then we would have proved that this is a solution to the non-homogeneous Cauchy problem. Let us do this checking. So first we have to compute the derivatives. If you want to check this solution to the wave equation you have to compute two time derivatives on two x derivatives. Let us compute the first derivative dou by dou t of this. Now we are going to use Leibniz rule. Whenever you have integrals which depend on the variable with respect to which you are differentiating there is a rule how to differentiate that particular integral. I am going to follow that. So therefore you get this derivative equal to the inside thing evaluated at tau equal to t that will give you this because tau becomes t Sf t x comma when tau equals t it is 0. So you get this plus you are going to differentiate inside the integral. So 0 to t remains as it is and dou by dou t of the inside quantity. So we have this. What is Sf t of x comma 0? It is 0 by definition that is how the source operator is defined and I have not changed the integrand as it is. So one at a time. Now let us differentiate once more. So we use Leibniz rule for differentiation of integrals here and it is actually a consequence of fundamental theorem of calculus and chain rule. Sf t of x 0 is 0 by definition of the source operator. So first derivative of the integral term is this. Now we will compute the second derivative. So this is being carried forward from the previous slide. This is the first derivative of the integral term. Therefore second derivative becomes first derivative of the first derivative. Now exactly again by the Leibniz rule this will be the integrand evaluated at tau equal to t plus differentiate inside with respect to t. This quantity is precisely f of x t. Once again from the definition of the source operator. Now here we see the dou 2 by dou t square of some solution. Sf tau after all is a solution to the wave equation. Therefore dou 2 by dou t square can be written in terms of the Laplacian. So it is c square Laplacian of whatever is here and Laplacian comes outside and we have this. Therefore we have proved that this integral term satisfies dou 2 by dou t square minus c square Laplacian equal to f x t. Take this term to the LHS then precisely we have this. That is integral term satisfies non-homogeneous wave equation. And rest of this lecture is devoted to writing down the source operator explicitly in each of the dimensions 1 to 3. And obtain explicit formulae of solutions to the non-homogeneous Cauchy problems.  Yeah we may call this thing as a particular solution. This is a nomenclature that we use even in ODE's. It is one particular solution that means it is one solution of the non-homogeneous equation. So as before we already introduced this in the context of ODE's. So integrand you call it as w of x t, tau then the function w satisfies homogeneous wave equation and it satisfies the initial conditions when t equal to tau w is 0 and the derivative of w when t equal to tau is f of x tau. So in other words you see this is a superposition of solutions to the homogeneous equation with initial data which is Tyler made using the source terms. This explains Duhamel principle. So let us obtain explicit expressions. We have already solved the homogeneous wave equation on Cauchy problem. So it is just a matter of writing down what source operator is. So by Dalamert formula the source operator Si is this 1 by 2c integral x minus c2 to x plus ct Si sds. So solution to the non-homogeneous wave equation is this dou by dou t of s phi plus Si plus the integral term 0 to t Si of some t minus tau extra dt which is there on the previous slide. Okay we can simplify this further and we get this expression because here it looks a very complicated expression. We can use the Leibniz rule and get this expression. So if you notice the first of two terms is precisely the Dalamert formula that means when f is 0 phi and Si that is solution of the homogeneous wave equation and the Cauchy problem for that. And this is the new term which we have added to get solution to the non-homogeneous Cauchy problem. Now of course we have to ask this question when is it a classical solution? What are the assumptions needed? Of course if you look at the formula it is meaningful. This is meaningful if phi is let us say continuous integral we need therefore Si should be continuous again some integral so f should be continuous. But that is not enough for classical solution we require phi in C2, Si in C1 we have already noted down this when we discussed the homogeneous problem for the one-dimensional wave equation. Now what about this? Remember this f this term how did we get? The f went into initial velocity in the definition of source operator therefore we need some first derivative of f to be continuous. So f should be continuous so that this makes sense in addition you need first derivative of f with respect to x to be continuous. That is what is needed to make sure that the source operator is well defined with this as the initial velocity done. Now the comment on the domain of integration in this integral what is this? It is a triangular region formed by x axis on the two characteristics which are passing through the point x t that is if you look at this take a point this is the point x, t. So this is a line which are going through this is x minus yeah we should have used x0 t0 then this is x minus ct equal to x0 minus ct0 that is the line and this line is x plus ct equal to x0 plus ct0 and this is x axis. So this is that triangular region if you see this is where we plot x and t here. So here this formula you just read with x0 t0 that is what I am reading x0 t0 this is also t0. So this point is t0 you fix any thing in between let us say t equal to tau okay then you are going from this point to this point this is precisely the two points here this and this okay. So this triangle is called characteristic triangle. Now let us move on to the two space dimensions by Poisson Kirchhoff formula the source operator is given by this formula and now this is the expression dou by dou t of s phi plus s psi plus that integral. So we get this formula using a change of variable it is convenient to write it on disc of radius one with centre zero the homogeneous equation part. Again same question under what hypothesis it is a classical solution no need to see look at the formula phi integral involved so phi continues grad phi integral is there therefore grad v continuous psi continuous f continuous we can put such conditions but then we want a solution for which we need higher kind of higher smoothness phi should be c3 psi should be c2 and f should be continuous and also gradient should be continuous with respect to x and the second derivative should also be continuous with respect to f reason is exactly the same as we gave for d equal to 1 this f is going to sit in according to the Duhamel principle this f the source term will go to the initial velocity and for initial velocity we need c2 s that is why this condition let us move on to three space dimensions again from Poisson Kirchhoff formula in three dimensions s psi has this expression and therefore by Duhamel principle u has this expression now which values of f are needed to determine u at the point x0 t0 that is a question we ask so we write the expression for u of x0 t0 of course f is not coming here but if you ask what values of psi and phi are needed it is clear that we need the values on the sphere of radius ct0 with centre x0 what are the values of f which are needed that is what we are asking here so set of all y tau size that tau is here y is here and tau between 0 and t0 so they are needed on this set now what is y is here y minus x0 norm is c times t0 minus tau c into t0 minus tau exactly that this is the sphere y should belong to that and tau should belong to 0 to t0 that is exactly what I have written here but what is this set this is a subset of r4 r3 cross 0 infinity this is the backward cone with vertex at x0 t0 we know the cone in three dimensions right so this is something similar we have to imagine in four dimensions I will show you a picture in other words the value of u at x0 t0 depends on the sources coming from the backward cone with vertex at x0 t0 so figure of a backward cone is on the next slide it is here this is a point x0 t0 what we are used to when it is r2 in the plane and you have a cone this is called backward cone or past cone is called forward cone or future cone we will discuss about this in a future lecture more on this but this is what is needed the values of f are needed on this cone therefore since y is in the sphere that means that norm y minus x is c into t minus tau the formula may be written like this now t minus norm y minus x by c and norm y minus x earlier it was written t minus tau but now that we can convert in terms of y and x and if you just combine what you get here that is precisely the ball of radius ct with center at x so the ball is written as a lot of spheres which are union take the union or yeah so that is what you have for each fixed tau you have a sphere and then take the union as tau varies from 0 to t so you get this so again the same question when is this a classical solution to the Cauchy problem of course formula is meaningful under some set of assumptions on phi psi and f but if you want a classical solution you need phi to be c3 psi to be c2 and f to be here continuous f a gradient to be continuous and second order to be continuous on r3 cross 0 infinity so the value of f of yt for any y in this ball of x comma ct does not play any role in determining the solution at the point xt on the other hand the values of f of yt at the retarded time t which is t minus norm y minus x by c that is what they place a role if you see the formula and the situation is totally different for d equal to 1 I will request you to check what is that difference from 3d in 1d and this integrand is called a retarded potential because instead of t we have t minus norm y minus x by c so justify the word retarded time after you learn about Newtonian potentials in the context of Laplace equation which will be done later on. So let us solve an example solve a problem Cauchy problem non-homogeneous Cauchy problem I have taken simple initial data because I want the computations to be simpler. Of course we have already said this is a formula f is a smooth function no problem so this will represent a classical solution because phi is smooth psi is 0 and f is definitely smooth you can compute directly by plugging in these values we do not do that what we do is we apply Duhamel principle directly. So for the given Cauchy problem psi is 0 and therefore the solution in terms of the source operator is given like this now we have to find the source operator and that is given by this by the Dallambert formula which we can compute that is xt s phi of s phi of x, t is xt. Now let us compute what is s f tau by definition it is this given by this formula and compute what is f tau of s that is s into e power tau compute these integrals you get t minus tau into x e power tau. Now find this integral because this is what will give you solution to the non-homogeneous wave equation and that is given by this integral on simplification we get this formula x e power t minus xt minus x. Now we have to add these two okay dou by dou t of s phi is required s phi was xt so dou by dou t will be simply x plus this we will plug in from the last slide solution is x into e power t minus t. So we have introduced Duhamel principle applied it to obtain solutions to non-homogeneous question problems we observed that the Duhamel principle works for any operators or the form ut t minus lu for wave equation l is the Laplacian for what operators it will work is the operator real can be any differential operator not involving the variable t in it. Thank you.


In this section, I will explain how the preferred portfolio is selected. So firstly, we need to understand the concept how the optimal combination of the risky assets are identified. So basically, we look at the capital allocation line and then we look at the efficiency frontier. So we have already discussed these two concepts. Capital allocation line explains all possible combinations of risky asset and risk-free asset which you can take from the amount of investment. Our efficiency frontier is a combination of all those points, those combinations of the assets which are most efficient at a given certain level of risk or volatility which we have defined as a certain risk level and the best of the best combinations we call their combination as an efficient frontier. So if we see that the capital allocation line is tangent to the efficiency frontier we call that particular point as the optimal combination of risky assets. We are talking about risky assets and we are not talking about risk-free assets. So we are assuming that we will be coming up, we will be developing a portfolio in which there are two risky assets and one risk-free asset and these three assets are based on our portfolio. So basically, if we look at this point of tangency of the capital allocation line and the efficiency frontier, it will give us the best risk to reward ratio and that will be considered as the optimal combination of the risky assets. So this is the definition. If we want to look at how we can calculate this through formula, we are assuming suppose if W1 is the proportion of risky asset 1 and W2 is the proportion of risky asset 2, then W1 can be calculated by taking into account the expected return from the risky asset 1 minus the risk-free return. We will multiply this with the variance of risky asset 2's return. Similarly, we will have to consider risky asset 2's expected return minus the risk-free asset and it will be multiplied by the variance of the standard deviation of risky asset 1's return, the standard deviation of risky asset 2's return and this will be further multiplied by the correlation coefficient between the risky asset 1 and risky asset 2. This whole thing will be divided by expected return from risky asset 1 minus risk-free asset multiplied by the variance of risk-free asset 2 plus expected return of risky return 2 minus RF is risk-free return, right? And this will be multiplied by the variance of risky asset 2, risky asset 1 minus, you will subtract the top term. I will add another term and that would be expected return from risky asset 1 minus risk-free asset plus expected return of risky asset 2 minus risk-free asset key return and this whole will be multiplied by, that is, row 1-2, the correlation multiplied by standard deviation of risky return 1, risky return 2 key standard deviation and if you multiply this, you will get a denominator. So, once you have estimated the value of the proportion of the total investment that will go to risky asset 1, the next thing you will be able to come up with is W2 which is the proportion of risky asset 2 and how much total investment you are going to invest in risky asset 2 and that can simply be obtained by subtracting W1 from the 1. So, you will get W2. From here, even with the help of W1 and W2 formula, you can remove the same point which we talked about the tangency point. So, if we want to see it graphically, that is again, for that matter, what we will have to do is, we will have to take into account the risk on the x-axis and the return on the vertical axis. So, then what we do is, we draw the capital allocation line. We have discussed it earlier, its slope, what will be its intercept? Its intercept is always your risk-free return and then we draw the efficiency frontier. So, suppose this is the efficiency frontier and the efficiency frontier point where the capital allocation line is tangent, we have just defined on that point that is that point gives you the optimal combination of the risky asset 1 and risky asset 2. I am labeling it as point t here. So, how much money do you have to invest in risky asset 1 and risky asset 2? That value comes out. Now, the formula of W1 and W2, you can take it out with the help of it mathematically by using the formula, but if you want to see it graphically, so that will correspond to this particular point where the efficiency frontier is tangent to the capital allocation line. Now, for the investor who has options to decide, we can see how he will invest because on the t-point, those combinations are coming that he is investing all his money in risky asset 1 and risky asset 2. But again, in order to properly develop a portfolio, sometimes it is recommended that you put some of your money in the risky asset also. So, for that, what will we do? For that, if we put this particular point, if we label this point as point f or label this point as point t, then any investor who has to decide the portfolio, they will go for any combination that fall on the line between point f and point t. All these points will be recommended. Now, which point or portfolio does one investor prefer? How that particular portfolio, the preferred, all these lines from point f to point t, all these are the preferred portfolios. But how we are going to select, which one of these portfolios we have to select? For that matter, there are certain other things that are taken into consideration by the investment managers, who are the investment advisors or financial advisors. They consider certain other factors also and in that, investors also take their input and to make a final product, what kind of products they like. So, for that matter, there are the other things that are considered in addition to the expected rate of return, the risk-free return, we discussed it as well. Then we counted the standard deviation values. Apart from this, it is important to consider what is your planning horizon. We have discussed the planning horizon. What is it? Next important cycle, important point is that you are sitting on the point of life cycle. You are close to retirement. You have retired. You have started a new job.  This will play a very important role that you, which is the different preferred portfolios between your entire effort and t, which one of the portfolios you are going to select? And similarly, another important factor that also plays an important role that what is your risk tolerance level? So, all these things, in addition to expected returns, risk volatility, we also have to consider all these things. Now, when we have to choose the optimal combination, now, between F and T, which of the combinations you are going to choose, the wealth of investors or their personal preferences do not count. You do not have to consider every detail of an investor when portfolios are being defined for them. The investors who have so much wealth, these details do not matter. What matters is that the financial advisors consider all these things, which we just discussed, what are the things people like in a different life cycle? What is their risk tolerance? And what combination is better for a high level of risk tolerance? What is the final product for people with low level of risk tolerance? But what they do is, they look at three important factors that you can see. If you look carefully, the formula that we just saw, for W1, W2, we used these three things. They consider the expected rate of returns from the risky return or from the risky assets and the risk pre-assets. The second important thing that we have considered, that is the standard deviations or the third important thing is the correlation between them. So, what is the extent to which they move together? They move together in the same direction or they move together. They move in opposite directions or if they move together in the same direction, then what is the strength of association? So, basically, what we are trying to do is, when we have to do the optimum allocation, we see three important things. We do not see the wealth of your investor, we do not see the basic details of personal characteristics. What we look at basically is primarily, what we see is the expected returns from different risky assets. What is the level of risk attached to it? You will get to know from Sigma. Then we wrote row one, two, symbol. It tells you the correlation that you are pooling the risky assets to make a portfolio. How is the correlation between them? So, these are the important factors that are considered when we go for the optimal allocation and we have a choice of different types of risky assets. And now, we assume for simplicity that we are dealing with a situation in which there are only two risky assets involved.


Welcome once again to The Breakfast on PLOS TV Africa. Our next conversation is going to be on the electricity tariff news. Sometime last week there were reports of a possible increase in the electricity tariff from the 1st of September. But the EKEDC and its spokesperson have come out to state that that might be false. And also in the news this morning, Festus K. Amel had stated that the government is not aware of any planned increment. We are speaking with Professor Kenneth Mouinke, he is a political analyst and professor of politics and governance at the Ignatius at Juru University of Education. Good morning Professor Ken. I hope you can hear us clearly. We are also speaking with Daga Tola from the Spokesman movement of Socialist Alternatives. Good morning Mr Tola. Good morning Africa PLOS Africa and good morning all Nigerians. Thanks for joining us. We hope we can get a better sound quality from Professor Kenneth. Let's start with Daga Tola. Of course a lot of Nigerians were worried last week. We've complained or Nigerians have complained about the high cost of leaving and of course continued increment in electricity tariff. So let me start with your thoughts. Do you think that these things are inevitable, seen the way that the country is currently moving? Would Nigerians need to always pay higher to get better electricity? Let me make it quite clear that I do not agree with that narrative. I don't think that increasing the pump price of electricity or new increase in tariff in electricity would resolve any of the contradictions plaguing this key sector. Interestingly, it's extremely very painful that the ruling class in this part of the world is not even learning any lessons from its own history and from the fact that all of these policies since they've been implemented all through these decades have not resolved any of the contradictions. The history of the increase of pump price of fuel for an example didn't just start now. Interestingly, the Bari regime, when it first came into power in 1984 to 1985, didn't follow these parts. All of the increases we have had through the austerity measure of the Shiv Shagari regime, through to the structural adjustment program of the Babangida regime and now to the policy of deregulations and privatizations which are the policies dictating all of these programs by ruling the leader has not in any way provided any increase for regular electricity for Nigerians. Interestingly, according to the World Bank statistics from the World Bank, Nigeria is still ranked 171 in a count of 190 countries. Access to electricity in Nigeria is still largely restricted to the urban areas in the population. Access of a country of 200 million plus, only 5 million Nigerians have access to water electricity. So the fact speaks otherwise in relation to these policies and programs. This singular narrative that there is no alternative to capitalism here, where the resources of society, the resources of this country can be muscled, can be employed to serve as the basis to develop the economy, to provide electricity, to drive power, to drive energy, to drive steel production, to drive the basis for industrialization of this country. Then the ruling at least on the basis of the own doctrines and philosophy will practically have no other choice than to continue with this destructive policy of further destroying the Nigerian economy, more further. Alright, Professor Kenneth Wake, you're welcome once again. Professor Wake, can you hear us? We can hear you. Okay, brilliant. Thanks for joining us once again. I want you to share your thoughts on the investments in electricity. Sometime in February 2021, there was the World Bank statement that we're assisting Nigerians with $500 million. We've also heard about the Siemens loan and Siemens, about $3 billion assistance to the Nigerian government with regards to electricity also. But this year, and in the last long while, we've had a few times than normal where the power grid collapses. Why do you think we still aren't getting it right with fixing Nigeria's electricity or power sector challenges? Yes, thank you. I get worried, especially when you talked about the Siemens contract, because that was one of the visits that our former chief of staff, Abakiri, traveled to Germany. And so when you talk about Siemens contract, talk about electricity and all that, and that reminds me of that on timely date of our chief of staff, because Nigeria needed power at all costs, and I think that's not necessarily the visit at all. So it clearly shows that Nigeria's investments in the power sector in Nigeria have been very, very slow in yielding the desired dividends. I know how much there was to sound into the power sector during the Abasanjo administration in Nigeria, between 1999 and 2007. And so, and then subsequently, a lot of investments have been made in the power sector in Nigeria, even given what we are also raising now. But power has been very, very epileptic in Nigeria, and then the issue of tariff is not about increasing the tariffs that is a problem. Nigerians are tired about these investments that are yielding little or no dividends. By now, we should be talking about having 24-hour electricity supply in Nigeria, given the normity of the funds sunk into the power sector. And so, and I know that the Siemens contract was going to last until 2024 or so, but this is 2022. And I believe that in that contract, it's still in place, and that the government is very, very serious about that contract, which will be getting updates regarding what the Siemens is doing and then what the funds that are coming from the World Bank, in order to assist Nigeria, get out of the woods. But I think that, irrespective of whatever funds that have been deployed into the power sector, if there is no political will on the part of the executive, including the literature, who does oversight on the power sector, then, and the Nigerians themselves, who should be doing what they should be doing in terms of checking the emphasis of those, or the witnesses of those who ought to do these things, then who will not realize anything. And that's why I think that this conversation today is something that should be a reminder to those who have been entrusted with this responsibility to raise up. All right, Dagatola, back to you. We, of course, have had, you know, the country has made its moves to fix the power sector. Privatization was one of them. You know, we now have distribution companies privately run. Why would you say that this might still not be working? And also, we've not still, over time, been able to generate as much as 10,000 megawatts of electricity in the country of 200 million people. Is that in any way embarrassing, you know, that we still are having these lapses, the power grid still collapses every, you know, for market days, and yet we're talking about increment in tariff? I'm extremely excited when you pose this manner of question. What we are confronting here is the contradiction in the capitalist system itself, because what is privatization? What is capitalism? It is run singularly for the purpose of making profit. And painfully for all of us, 80 to 90 percent of all of the entire resources that has been invested in electricity in Nigeria are public funds. But like Prof has pointed out, the ruling elites lack the appropriate, philosophical framework of the necessary goodwill with which to ensure that the electricity sector and indeed all sectors of the economy function appropriately. More additional loans have been procured. And you ask yourself, if we are running a private electricity-managed economy, what is the business of the state, employing the entirety of all of its trust and its resources to invest in electricity? And yet it lacks the moral will to protect that investment and ensure that the necessary electricity is what is provided. And this is why we will continue to say that there is no set-up of the Nigerian economy that is not plagued with crisis. And none of these issues can be resolved in isolation of a comprehensive interlink of all sectors of the economy. Electricity needs to be developed. Energy needs to be developed. The means of production, industrialization needs to take place. Steel production needs to take place. And none of these key vital sectors can be run effectively and efficiently without putting profit first by the private sector. And the more profit is placed first than foremost. It becomes a defeatist logic that ensures that it does not function effectively to allow the mass of the working people to assess all of these basic needs that are essential for economic development for the entirety of the mass of the people in the Nigerian society. Professor Ongweke, I'm going back to you now. We continue to have a problem. One of the things that I read up yesterday was the regulation and funding. Two major issues with the regulation, infrastructure and funding were two major issues with the Nigerian electricity sector. Quickly, let me know what you think about how the NERC can better regulate the sector and how we maybe can redirect the funding for electricity so we can achieve better results. The Nigerian electricity agency for me has not done well because it proposed that it has bad responsibility to give Nigerian government to do everything possible because it regrets the regulatory authority to ensure that power. And so what you hear every other time, the average man on the street is when you hear of the regulatory authority, you are talking about him presentary. That's what they are known for. Either they are giving a nod to him or they are not giving a nod to him. We are not interested as Nigerians about the tariff. For the average Nigerian, what we need is to see if you are showing me episodially evidence of power running, right? Through the bomb and all that. So we should have that 24 hours. I can still remember as far back as 2009 when I made the first trip to Togo in Olome. That was live, you know, 24 hours. In 2009, throughout one week I stayed in Olome. I didn't feel like, you know, the power blinked for one second and all that. In 2009. And then we are still talking about that 11 years or more down the ladder that we are still struggling with electricity. The best that we are having today is 10 hours, you know, I mean 10 hours after 24 hours and all that. That's the best. And that we are celebrating so much. And so the National Electricity Development Agency ought to see off. It's not the issue of the business tariff that matters. Nigerians are sick and tired of business tariff. What we want to see is power. We want to see power. We want to see, you know, a reflection of the normity. Oh, you are still bringing this bomb. That is showing that it's not.  You are showing me this to show evidence of 24 hour power in Nigeria that does not, you know, does not take us to anywhere. See, for me, the National Electricity Development Agency ought to see off, you know, and then ensure that the funding in the power sector is, you know, guaranteed. The collection of regulatory practices or policies in the power sector, you know, make it possible for private investors to take off the challenge of ensuring that power is deployed to Nigeria. It is not sitting down there and telling us whether tariff will happen in September or it will happen in June. And then even the National Assembly will come up to say, oh, no, no, slow down on the charge of tariff and all that. Yes, I know that they have a responsibility after oversight. But the National Assembly has every responsibility because that is the arm of the government that is charged with responsibility of oversizing what the government is doing. It is not about stopping them or not stopping them.  And then that your policies, you know, should not be anti or anti-investors and other private investors.  So what can we, you know, look at the magic that this of a firm in the telecommunication sector and apply same to the power sector. If you come, for instance, to give us the ICT protocol, you can privatize, you know, a whole lot of protocols for some, you know, local governments and other. Let the private individuals take the responsibility of distributing this power and they will let us know who they are and other. Not given the one company where you say it's the distribution company. The other one is the power generator and other. That is to the extent that Nigerians know. But if you privatize, you know, the power sector in such a way that, you know, a lot of investors will come in and then people will face different conditions and other. This competition will come in and so deregulate, you know, the power sector in such a way that we have in the telecommunication sector, including the deregulation industry, that is the short way of guaranteeing, you know, to support our power supply in Nigeria. And not the way we are going, you know, this way. And then I've said separately that it takes a political, I mean, the political wheel on the part of the executive arm of government, you know, to ensure that this thing comes before our power supply. If Nigerians are not even looking for power, we are even looking for just a thing, our sector. That will be OK for us as Nigerians because power sector drives the other sectors, you know, of the economy. Absolutely. Professor Kenneth Wynke is a political analyst and professor of politics and governance at the Ignatius Adjuru University of Education. We also spoke with Doug Dagatola, spokesman, movement for Socialist Alternative. Thank you both for being with us this Monday morning and always looking forward to having conversations with you. Have a great week ahead. Thank you. Thank you very much. Thank you. Absolutely. And this is where we will be wrapping up the breakfast this morning. Thank you so much for staying with us. If you missed out on any of these conversations, you know where to find us. It's pretty simple. At Plus TV Africa on Facebook and Instagram. And you can also find, subscribe to our YouTube channel at Plus TV Africa and Plus TV Africa Lifestyle. I am Osao Yee, Ogboa. See you tomorrow.


President of the United States, President of the United States. Thank you. Thank you very much. Thank you, John, and thank you all very much. It's a great pleasure to welcome you to Washington and an honor to address this assembly of some of the foremost members of our scientific and business community. I'm pleased to announce, though, I'm going to take a liberty here before getting on with the subject that brought us together to pause for a moment for an announcement, something that I think is of interest to all of us, that today our delegation to the Geneva discussions on intermediate-range nuclear missiles is putting forward a formal U.S. proposal that embodies two earlier suggestions made by the United States. In November of 1981, we promised elimination or eliminating the entire class of long-range intermediate missiles. On June 15th, we then formally offered to eliminate the shorter-range I&F weapons as well. As perhaps some of you know, until last week the Soviet Union had insisted on what could have been a major stumbling block to our double zero suggestion, the right to retain some of its missiles currently deployed in Asia. But last Wednesday, General Secretary Gorbachev announced that he was preparing to drop this demand. As I say, the proposal put forward today would reduce to zero the number of both longer-range and shorter-range I&F missiles, and we are pleased the Soviets have now expressed support for both concepts. It would make provisions for strict and effective verification and reject transfer of existing U.S. and Soviet I&F missiles and launchers to any third party. Two vital new elements are also included, the destruction of missiles and launchers covered by the treaty, and no conversion of these systems and launchers to other types of weapons. There is still much to do in Geneva, but I am heartened that the climate is now receptive to an eschoric proposal of this type. The United States is proud to be in a position to make this proposal. I can only add that there are other arms reduction negotiations going on as well, and these concerns strategic or longer range at nuclear forces. We have tabled our proposal in the form of a proposed START treaty, and we hope the Soviet Union will do likewise and formalize their views in this area. Our goal, as you can see, is not arms control, but arms reduction, and despite the skepticism when we first announced these plans, we are moving in this direction. Now before I begin our discussion of the vital subject that's brought us here today, there is one person I must mention, a moving force in getting this conference underway. He was a patriot, a fine cabinet officer, and a champion of American enterprise, and we will truly miss MacBodridge. But we're privileged to have here today with us many of those scientists whose pioneering work made this conference on superconductivity a possibility. Congratulations to you all, and it's a safe bet that this conference room also contains many of the minds and spirits who will carry this revolution forward, who will open up a whole new realm of here to for unimagined possibilities and practical applications. I've had a lot of experience in my own career of how technology can change things. I remember back in the 20s when somebody first told Harry Warner about talking pictures, and he said, who the heck wants to hear actors talk? Actually I don't think he said heck, but presidents aren't allowed the same license of studio executives. Of course when it comes to high tech, presidents often have trouble too, keeping up with the times. A favorite story of mine is about one of the first times the White House hosted a science and technology event. A demonstration of a recently invented device was put on for President Rutherford B. Hayes. That's an amazing invention, he said, but who would ever want to use one of them? He was talking about the telephone. It's hard to believe that it's been less than one year since we first heard news of the startling breakthrough in superconductivity by two scientists in the IBM labs in Zurich. And since then, it seemed as if the papers have had to struggle to keep up with the rapid advance up the Kelvin scale. You know it's been said that there are three stages of reaction to any new idea. One it won't work, two even if it works it's not useful, and three I said it was a great idea all along. Well, to most of us laymen, superconductivity was a completely new term, but it wasn't long before we learned of the great promise it held out to alter our world for the better. A quantum leap in energy efficiency that would bring with it a host of benefits, not least among them, a reduced dependence on foreign oil, a cleaner environment, and a stronger national economy. I've been accused of being an incurable optimist, but lately I've been playing catch-up ball with the usually more stage science profession. The other day I met with Dr. Graham and the members of the White House Science Council for a briefing on superconductivity. Edward Teller told me that we have seen discoveries in the laboratory these last eight months that the optimists who thought we wouldn't make for 200 years, one theorist was quoted as saying it shows all the dreams that we've had can come true. The sky is the limit. Well, there are predictions of high-speed trains levitated above their tracks, supercomputers on a single silicon chip, cheaper and more effective medical imaging devices. But I suspect that we haven't even begun to dream the possibilities or imagine the potential. Just as no one imagines 747s making transatlantic commercial flights when the Wright brothers first flew at Kitty Hawk, and no one dreamt of the computer or the communications satellite when Ben Franklin first captured electricity from a lightning bolt. It was Ben Franklin who wrote, I have sometimes almost wish it had been my destiny to be born two or three centuries hence, for invention and improvement are prolific and beget more of their kind. The present progress is rapid. Many of great importance now unthought of will before that period be produced, and then I might not only enjoy their advantages, but have my curiosity gratified in knowing what they are to be. Well, the present process is rapid, and it seems constantly accelerating. One can imagine how gratified Ben Franklin would be if he were alive today. But you know, I bet he would join me, and I'm sure most others in this room, in wishing that we had a window to the future, 10, 20, and 100 years hence, so that we too could see the marvels of the coming age. There's something universal in that sentiment. I'm sure, but I can't help but feel that there's something especially American in the optimism, the certainty of hope and faith with which we look to the future. Two hundred years ago Thomas Jefferson said, I like the dreams of the future better than the history of the past. Well, since that time we've built here something entirely new, a history in which Thomas Jefferson would have felt more at home. It is a history that is constructed of dreams, dreamt in freedom, and realized in a land of opportunity. Science tells us that the breakthroughs in superconductivity bring us to the threshold of a new age. It's our task at this conference to herald in that new age with a rush, if you will. It's our business to discover ways to turn our dreams into history as quickly as possible. The laboratory breakthroughs into high-temperature superconductivity are an historic achievement. But for the promise of superconductivity to become real, it must bridge the gap from the laboratory to the marketplace. It must make the transition from a scientific phenomenon to an everyday reality, from a specialty item to a commodity. And that's why we're here in this conference, so that business and science can cross fertilize, can begin at these early stages to dream and plan together. Because this new age of superconductivity is a new arena for the spirit of enterprise, one that we can't even guess the limits of. We also want to see how we in government can do our part in helping this process along. Now, I have to confess, and I'm one of those people who, when the government offers to help, gets very nervous. We've learned from experience that the helping hand of government too often has a crushing weight. I'm reminded of what Werner von Braun once said about America's space program. He said, we can lick gravity, but sometimes the paperwork is overwhelming. Well, this is not to say that government doesn't have an important role, especially at the basic research level. We must continue to support our outstanding researchers in all disciplines, giving them the freedom of resources and flexibility they need to dream their dreams and make their experiments. One key for doing this is the National Science Foundation, and we have proposed, as you've been told, to double its budget over the next five years. As you may know, the National Science Foundation and NASA provided funding for Dr. Paul Chu's landmark experiments at the University of Houston. Increasingly, in this last part of the 20th century, information is becoming the most valuable commodity. We have begun major initiatives at our national laboratories to improve communication. And last April, I signed an executive order ensuring that all federal agencies and laboratories become partners with the private sector in moving research from the laboratories to the marketplace. The message of government is simple. We have an open door policy to the private sector, cooperation wherever and whenever possible is the order of the day. We must also move to protect intellectual property and write protections into the Freedom of Information Act for scientific and technical information generated by government laboratories. We need to strengthen patent laws to increase protection for manufacturing processes and speed up the patent process so that it can keep pace with a fast-paced world of high technology. And of critical importance, we must modernize our antiquated anti-trust laws, laws designed for a previous century that only held America or hold America back and give our foreign competition an unfair advantage. If we're serious about improving American competitiveness, the way to do it isn't through protectionist trade legislation that closes markets and throws people out of work. One good place to start is bringing anti-trust laws up to speed with the modern world. This is no longer the era of the so-called robber barons. It's the age of high tech and global competition. Let's stop penalizing American business and treating it like an enemy. Let's give ourselves a fair shake in the world marketplace. I will soon send a legislative initiative to Congress addressing these three issues, anti-trust, patents, and the Freedom of Information Act, as they relate to the commercial application of superconductor technology. I hope you will lend your support to this legislation. Your opinions matter to your senators and representatives. Let them know how you feel individually and through your various trade organizations. This package of reforms will go a long way toward preserving the competitive advantage of U.S. industries in this field. These and other actions are part of an 11-point superconductivity initiative that will also include a wise men's advisory group on federal policies and regulations that affect superconductivity research and commercialization. The quick start grants for good ideas on processing superconducting materials into useful forms, the establishment of a number of superconductivity research centers, and a nearly $150 million research and development effort by the Department of Defense over three years. Funding basic research, sharing information, removing the impediments to commercialization. In this way, government can be a catalyst to the future. But we have an even more important role that we must be sure to accomplish. It is a fundamental role that not only looks forward to the 21st century, but harks back to the first principles enunciated in our Constitution 200 years ago. First, we must recognize that just as recent breakthroughs in superconductivity have outrun existing theories and brought us to the threshold of a new world of opportunities, so too the world around us is in the process of a radical transformation, a revolution of shattered paradigms and long-held certainties. This transformation, too, is opening for us new horizons of possibility. In a recent article, The New American Challenge, the economist George Gilder describes this new reality and how we must respond to it. The information age has only just begun, he says. Increasingly, we are moving from the economy of the Industrial Revolution, an economy tied to the Earth's natural resources, to an economy based on information, where that old cliche, knowledge is power, is truer than ever. It's estimated that raw materials account for 80% of the cost of pots and pans, 40% of automobiles, and less than 2% for an integrated circuit. The value of a silicon ship doesn't lie in the sand from which it comes, but in the microscopic architecture engraved upon it by ingenious human minds. The most promising superconductors are made from ceramics. Their value doesn't come from their material, but from the brilliant inspiration of a few scientists. It is the human imagination that is building the 21st century out of sand and clay. We're increasingly moving from an age of things to an age of thoughts, an age of mind over matter. In this new age it's the mind of man, free to invent, free to experiment that is our most precious resource. Gold, steel, oil, these were the treasures of the past that made people rich and nations strong. Today the premium is on the human heart and mind. They can't be locked in a vault, nationalized or expropriated. They can only be let free and then really the sky is the only limit. That is the American challenge. Will we continue the policies of economic freedom, of non-inflationary growth, of low and flatter tax rates and deregulation? Will we strengthen them with an economic bill of rights so that they will never again be lost? When I signed our tax reform legislation into law I noted that the last 20 years had witnessed an expansion of many of our civil liberties but that our economic liberties had been too often neglected, even abused. We protect the freedom of expression of the author as we should but what of the freedom of expression of the entrepreneur whose pen and paper are capital and profits whose book may be a new invention or small business? What are the creators of our economic life whose contributions made out only delight the mind but improve the condition of man by feeding the poor with new grains bringing hope to the sick with new cures vanquishing ignorance with wondrous new information technologies? When our forefathers wrote guarantees of life, liberty and property into our Constitution, they tapped a wellspring of hope and creativity that has transformed history. That basic blueprint they laid down, the Constitution, whose 200th birthday we celebrate this year, is if anything more vital than ever. That secular trinity of life, liberty and property is the key to the future, the key to meeting and winning the new American challenge. Archibald McLeish once said, there are those I know who will reply that the liberation of humanity, the freedom of man and mind is nothing but a dream. Then he said they're right. It is the American dream. You here today will be among the pioneers carrying on that American dream into the future. A day when this 20th century of ours may seem no more than a rough prototype of the 21st. I wish you well with the rest of your conference. You've captured the American, the imagination of the American people, and I'm sure I can speak for them all when I say, our hearts and hopes and best wishes go with you. Thank you all very much and God bless you all.