So, yeah, so today we will basically summarize what we have been doing in the last 40 odd classes. I will start by before we recollect, collectively recollect, right, what we have done so far. I will just sort of make an observation on what we have been doing so far. So, though it is supposed to be computational fluid dynamics, introduction to computational fluid dynamics, you realize that basically what we have done is we have looked at ways by which we can solve differential equations, right, and variational problems, how to represent them on the computer and how to get solutions of a numerical nature. So, I want to put that in some kind of a context. In the context, in the sense now I want to make it a very broad context. So far, we have been acting as though it is fluid dynamics, but if you think about it, I have not really mentioned fluids that much. Maybe I have mentioned a little gas dynamics along the way, right, I mean, but we have not really spent, we have not really, the equations are not like the full flow, full fledged fluid equations or whatever. We have looked at Euler equations and so on, okay, right. So, just to put that in context, I want to draw, graphically show you something about the nature of how these things work, okay, just a big picture, just a big picture. So, on the one hand, okay, I have, we will, I will maybe even color this, I will color it, I will fill it inside. On the one hand, that picture is supposed to represent reality as it is, right, what we are trying to model as it is. And if I am able to reproduce it, but if I am not able to reproduce it exactly, it does not matter, okay. So, I will say that makes sense. And from here to here, I am going to write processes, okay. This is perception, this is what we see. It is perception, that is what we see, whether you use a microscope to see or whether you use, right, whether you just look. When we talked about, when we talked about driving the governing equations in one of the classes, I sort of may have waved my hands around and said molecules, what molecules, right, I do not, I do not see the molecules. Am I making sense? We may be able to infer that they are there, but a phenomenological attitude would be continue, it is continuous media around, okay. So, that would be perception, this is what we, this is what we see. And the trouble with perception is there is a lot of detail, I mean benches, carpeting, I mean there is a lot of detail, lot of fine grain detail, we cannot handle that much detail, okay. So, even from what we, so there are things that we do not see, that there are things that we just cannot see, I mean we are unlikely to see. So, locations of every molecule at every point at a given instant, you know, it is not going to work. So, we come up with, we throw away what we think is the detail, we throw away the details, we throw away the details and come out with a mathematical model. And for a mathematical model, of course, what am I going to choose is the geometrical shape as close to the circle as I can get, right. So, we come up with a mathematical model, okay. So, this is a big picture of what we are doing. As the reality, this is what we see, cannot handle the detail mathematical model. And very often, as in the case of the Navier-Stokes equations and so on, cannot handle the mathematical model, okay. So, then we go, where do we go? Then we go to the computer, okay. So, this is deliberately of course, it is clearly geometrically supposed to represent the fact that this has an infinite number of sides that has only a finite number of sides. So, one of the things that we have looked at is, for instance, we ask the question of consistency. So, if the number of sides keep increasing, does this become the circle, right? Does the solution go, does the model go? There are questions that we have asked. So, there is an issue of verifying that this is indeed this. You have written a program, right. So, actually I just say computer model. This itself you can add levels. There is the chalk dust that we write, right, of the finite difference scheme or whatever. Then there is the actual program, which is representation of that algorithm, actual representation of the algorithm on that particular computer. And then there is the issue of whether does that represent the algorithm that you set out to implement? Does this become this?  Or does it, you may want to verify whether that is correct. So, you can verify whether this is correct. This is called, very often this is called validation, okay, right. From here to here, of course, we actually called it consistency. You understand what I am saying? So, this is sort of the big picture. This is what we have. This is what we see. This is what we are hoping we can do. This is a mathematical language. This is the abstraction from this. Abstract. Abstract basically means to extract, right.  And if by chance you throw away something that is an essential, that is a different story. But you throw away, you abstract out the essentials and you do not worry because that is how science works. Science works on generality. Science works on things that are common. The particular is of no interest to science. We want to make a general statement, right. We are not, the particular is of no interest to us, right. So, we abstract out the general. And very often we find that this, we cannot solve this, right. This is where we are. So, actual fluid, what we see Navier-Stokes equation finite difference method. That is one way to look at it. Is that fine? Okay. So, as I said, I just wanted to talk about this so that we had this, we have this general context in which we can look at the course and possibly some of the material that you have seen elsewhere. So, why do not we see if we are able to do a review of the material that we have covered so far? Okay, fine. Where did we start? We have about 40 odd minutes in which we can cover the one minute per day. It is essentially where here, right. So, where did we start? What is the, what is the start? Yeah, so we had, we basically said that we started off with the idea that the problems that we are solving have solutions which are functions. Remember, the problems that we are solving have solutions which are functions. So, we need to be able to represent functions on the computer, right. So, we sort of realize that oh, we have to be able to represent things on the computer. So, towards that end, we basically said that let us see how things are represented on the computer, okay. That is where we, that is where we, so that was the motivation. All the equations that we solved typically end up being, the solutions end up being what we mean when we solve. It is not like x squared equals 2, therefore x is square root of 2, right. So, you get a nice number whereas what we have are more like functional equations, equations for which the solutions are functions. And we basically said oh, if I want to solve that on a computer, then I need to be able to represent the solution on the computer. I need to be able to represent the problem on the computer. So, how does one represent things on the computer, right. And we started by saying how do we represent? Well, integers. Computers are good at representing integers. How do you represent the real line? And computers are not good at representing the real line. That is what we found, right. At the real line, we found that the computers are not good at representing the real line, right. So, the real line was the first time that we encountered a form of representation error called floating point, round off error, right. So, of course, there is fixed point and floating point. I will just mention that since I inadvertently said floating point. So, you have fixed point representation and floating point representation. Fixed point representation is just like dealing with integers. It is just like having an integer and assuming the decimal point is at set point. Floating point, of course, the decimal point in theory moves because you have an exponent, right. So, the representation errors, one of the things that you have to take from this course, the representation error in numbers has a special name. It is called round off error. So, anytime you are basically asked, so what is the error that you get? And the round off error, what is the round off error? Round off error is the difference between the number and its representation on that computer. So, that computer could be a piece of paper. You decide that you are going to represent a number using 3 decimal places. Then you are the computer, right. And then that is 3 decimal places, right, is what you are going to have. And therefore, that would be the difference. Now, if you read, I smile, but if you read old books and they say the computer should be careful, they really mean you, right. So, you go back and get a numerical analysis book from 1940s or whatever and they are saying the computer should be careful. They really do mean you, you as the individual doing the computing, okay, fine, right. So, that is as far as the, as far as representing numbers goes. And once we said we can represent numbers, obviously we look at other mathematical entities. And an obvious mathematical entity that we have is a vector or an array. An array is a general idea in computers. So, whether you can represent vectors and matrices, okay. You can represent matrices, but if you want, most programming languages will not do the matrix algebra. They will not do the matrix addition. You have to do it, okay. Most programming languages will not do that for you. There may be a few that do it, but most will not do it for you, okay, right. So, it is possible and how did we represent, how did we represent matrices? So, we represented matrices and vectors by basically, well vectors we represented by basically locating in memory in a sequential fashion, right. We were able to say that there are, so the 1, 2, 3, 4, 5, 6, 7 elements, right, placed adjacent to each other, could represent a vector, right, in a one-dimensional array. Inherently, so we are using the term array, right. And of course, if you wanted to go make, get a matrix, you could then stack one on top of the other, okay. That was the idea, right. So, it is actually possible to get a single subscript object Ai or a double subscript object Aij, okay. Or even a triple subscripted object and so on. And then what you do is, you take whatever it is that you have stacked in the second dimension, you would stack it in the third dimension. So, then you know that you can actually create multi-dimensional arrays. It is actually possible to create multi-dimensional arrays. Is that fine, okay. So, by now what we have done is, we have managed to represent integers, numbers, real line. We are able to represent, then using those integers and real numbers, so to speak, real line. We are able to represent matrices or arrays, right. One-dimensional arrays, two-dimensional arrays, multi-dimensional arrays. So, the second thing to do is now we have to state where we are talking about representing functions, right. So, what did we do after that? What did we go about that? We did a little, we set up the, we looked for function basis. So, we did a little review of vector algebra, right. We showed how we constructed vectors in the normal vector algebra. When I say vector algebra, the algebra that you learned in your college, we did a little vector algebra and showed that well if you had the dot product, it was enough for you to actually construct, okay. It was enough for you to actually construct a set of basis vectors and so on, given linearly independent set of vectors that span that space that you are interested in, okay. So, what we did was, we constructed box functions and here, this was the first thing that we did. We constructed box functions. So, the box functions were two kinds. We constructed a box function f and g. I am not going to do all the details and we said go, it is a quick recollect, right. We constructed box functions f and g and basically said that on non-overlapping intervals, if the box function is 0, right, in this case, it is 0 here, 0 here and the intervals do not overlap. Remember, wherever the function is non-zero, it is called the domain, part of the domain where the function is non-zero is called the support of the domain, okay. So, the support of the function, the supports of the two functions are non-overlapping and the functions turn out to be orthogonal. That is one way to get orthogonality. The other way that you get orthogonality is that the functions somehow change signs appropriately, right, electric functions, sin x and cosine x, right, occupy the same interval. They are defined on the same interval 0 to 2 pi but they are still orthogonal to each function, okay, right. So, we were able to show that we could represent functions using function basis, using the box function and we saw again that there is a representation error, okay. We again saw that there was a representation error. For the box function, we asked ourselves the question, what is the nature of the representation error. The reason why we ask ourselves what is the error? The question is important, is can I get an idea of what is the error? Because if you know what is the error, you can then try to get an idea as to where the error is coming from and then there is scope for improvement, okay. So, as I keep repeating, it is not just enough that what we do, that those are skills but you should also see there is a higher level skill that you have aimed at. You have to look at the process that we are going through, right. So, you always ask, so you do something, then you have to ask the question, how good is it? Is there a way that I can get an idea as to how good is it? Because the minute you are able to quantify it, then there is scope for improvement, okay. That is very important. So, we asked the question this representation error and then our analysis basically showed me these are constants. So, you can represent constant functions exactly and nothing else, okay. So, basically what we then did was, we said we will go to higher order polynomials. Of course, one of the disadvantages this had was that representation using box function, what was the bad thing that it had? You had a lot of jumps. There are a lot of jumps. So, in fact, I think I used a straight line. You get a lot of jumps. They are non-overlapping. So, I have to make sure that you get a lot of jumps. It gets, the function is very jumpy. You can get closer and closer to the solution but the function just gets jumpier and jumpier, okay. The alternative to this was saying that well, we got orthogonality this way but we know that using something like the Gram-Schmidt process or whatever, right. We could make functions. You can get come up with an independent set even if they are overlapping. So, we looked at the set of polynomials 1x, x squared and so on, okay. On an interval, we have to specify the interval, on an interval. So, on –1 to –1 for instance, 1 and x are orthogonality each other. On 0 to 1, they are not. If they are not orthogonality each other, you can subtract from x, it is projection on 1 and get an orthogonal set. You understand what I am saying and then you can build up an orthogonal set using. That is what we saw, right. However, just like sine x, cosine x and so on, this has a problem. So, on the whole interval on which you are trying to represent a function, you represent the function using a series of 1x, x squared and so on or an orthogonal set formed from it, any change in coefficient will change the function on the whole domain. Whereas, box functions have something called locality, right. So, I want to recollect that locality. Box functions have something called locality which allows you to locally change the value, locally move the function value, the representation without affecting anything else where, okay, right. So, now we are able to represent functions. We got, we basically combined this idea of this with the non-overlapping, what should I say, intervals or supports to construct hat functions. To construct functions that look like this, just call it f, to construct hat functions, okay. Hat function was 1 at the node at which it is defined and 0 at the adjacent grid points, okay. 0 at the adjacent grid points. Is that fine? Everyone, okay. So, we were then able to get linear interpolants. We were able to get linear interpolants. However, what was the disadvantage? That you have overlap of adjacent, right, which can create interesting things like in the last but one class. It can create interesting situation, okay. But you have overlap and it is always a problem. So, orthogonality, that is the reason why we seek orthogonality because otherwise you do not have, right. But on the other hand, you get linear and we saw that you could do not only linear, you could do quadratic, you could do cubic. Am I making sense, right?  There are many ways by which we can use it, okay. So, I think I did, I mentioned quadratic and I did cubic in class, right. If I remember right, okay. What else could you do? What did we do after that? We got an estimate of the representation error here. Then we basically turned around and said, well, if I know the function, I can use this for the representation. But what if I do not know the function I had of time? That was the motivation that I used, okay. If I do not know the function I had of time, what do I do, right? So, then we used Taylor series and we came up with a finite different representation. So, I am not, the finite difference representation of course, basically made use of the fact, I am not going to use i's and j's as the indices. I am going to just make a general statement made use of the fact that if you have the function value f of b and you have the function value f of a, then what is fb – fb – fa divided by b – a? This looks like something for mean value theorem, right. So, mean value theorem says that there is some point in between some point unknown location psi at which this is the exact derivative, right for a continuous function there. Now, what we know is that if I represent this using Taylor series, if I represent this, if I say, see fb – fa divided by b – a is a number, right. It is a number. Just say it is 2. So, you calculate this and you get the answer 2. What it basically says is if you say 2 is the derivative there, the representation is first order representation. The error is proportional to b – a. If you assume that it is the derivative there, if you say 2 is the derivative there, that is the approximation that you are making, then the representation is first order, right. The representation is still first order, the sign of the error changes and if you assume that it is the representation of the derivative of the same number 2, if you assume that it is the derivative at the midpoint, then the error becomes second order b – a whole square, okay. Is that fine? So, here it turns out that the derivative corresponds to the function being represented by linear and there it is actually quadratic, okay. You look at the term. It is as though the function is represented by a quadratic, even though it is actually, you have drawn a straight line, okay. So, the same number and as I said, unfortunately, mean value theorem says that at some unknown point it is exact but we do not know. That is the whole deal, right. So, at some unknown point in between, it is exactly the right, it is and you know, we are looking at it graphically, you can just basically say what is most probably at that point, right. Graphically, yes. But in reality, we do not know what it is. So, that is basically the idea of, so finite differences and we found the truncation error using Taylor series, right. Truncation error. So, I introduced that term. Truncation error is if you have an infinite series, you truncate, you throw away, you eliminate all the higher order terms and retain only the leading terms. The error that you do, you make doing that is called truncation error. What else did we do after that? What was the, once we have finite difference represented, so we can represent first derivatives, second derivatives, we figured out, we showed that we could represent higher order derivatives, right. Then we turned around and said, okay, why do not we apply it to solving an equation and we took Laplace's equation as the start, right. And Laplace's equation was relatively easy. Laplace's equation is relatively easy and we basically showed that Laplace's equation, nabla squared phi equals 0. This is an equation that I have written many times in this class. So, you could do, right, phi at tq was a quarter of basically the average of the neighbors, okay. Summation neighbors, right, neighboring fees. So, it is the average of the neighbors. Laplace's equation turned out to be averaging of the neighbors. And we use that fact that it was averaging to actually show that, what did we show? We showed that the solution is unique, you do not understand that two of you cannot get two different answers and so on, right, that the solution is unique. We use that averaging to show that there is a principle called maximum principle that it satisfies that the maximum and minimum in this case actually occurs on the boundary, okay. We proved the consequence that maximum and minimum occur on the boundary. Am I making sense? And use that then show that the solution was unique. For the numerical case, you may have seen for the continuous case, for the discrete case. We showed it only for the discrete case, okay. So, what else did we do with Laplace's equation? What did we do next? We looked at some acceleration schemes. In particular we looked at SOR. So, we basically said that well, if you have a phi that comes from this, instead of calling it phi n plus 1, we called it phi star. And it is possible to take a linear combination of the proposed solution and the current solution. So, to get phi n plus 1, you could get omega times phi star plus 1 minus omega times the old one, okay. And we looked for ways by which we could find the optimal omega. Like we said, this is a situation where we are looking for non-uniform convergence. If it is uniform convergence then omega, it does not help. We want as a function of, we want the convergence rate to change as omega and we found that there is actually a way to find the optimal omega, okay. By systematically hunting, there is no analytic method by which you can do it but you can systematically hunt for the optimal omega. Is that fine? We also showed that solving this was the same as solving a system of equations. Solving this is the same as solving a system of equations. So, we had what? We showed that it was the same as solving x tilde equals b tilde, right, which was the same as minimizing a functional q of x tilde equals minus 1 half or 1 half. It depends which way you want to take it, x tilde transpose a x tilde minus b x tilde b x transpose b because I think I did that at least twice. So, we went through some interesting errors there but anyway it is fine, okay. So, we basically showed that this was minimizing this, right. It is the same. Minimizing this is the same as solving that a, remember is symmetric. That is an important property here. a is symmetric, a equals a transpose, a is symmetric, okay. a is symmetric. And there are some interesting properties but anyway we would not worry about it. And we showed for SOR as a consequence why omega has to be in the interval in the range 0, 2. So, that we got an idea that we have to pick omega within the range for this linear equation. Omega should be within 0 and 2. And we looked at hunting, we looked at a few demos, you have to write it out, okay, right. And we found that for Laplace's equation at least that omega is very close to 2. What was the other thing that we did? Anything else? Laplace's equation. Then we looked at what is the convergence rate basically, right. We looked at how quickly does it converge. And indeed we found that because again using the same principle we found the keyword that you used with error. We found out what is the error. And we basically, once we got the expression for the error, we asked the question how quickly is it decaying. And that gave us an idea of convergence rate. In a funny fashion I introduced the ideas of eigenvalues and eigenvectors and the spectral radius, largest eigenvalue, right. So, rho of a or rho of I think if I remember it I call the iteration matrix P sub j for Jacobi iteration. Rho of P sub j is the value of the largest eigenvalue or the largest eigenvalue, okay. Magnitude of the largest eigenvalue. It is called the spectral radius of that matrix. And we basically showed that you want the spectral radius of the matrix to be less than 1 for convergence. That is fine. But that is not enough. That is okay if you are saying does it converge eventually. But eventually is not good enough, right. And we found that actually for this class of problems rho j, rho of Pj is extremely large. It is very close to 1. It is very close to 1. So there are some issues there, okay. There are some issues there. SOR is one of those things that sort of helped us fix that issue. But there are some issues there. Then what did we do? We looked at, we changed gears and looked at linear wave equation, right. We looked at linear wave equation. Linear wave equation, Laplace's equation is averaging, right. Averaging basically means if there are differences, eliminate differences. Linear wave equation was different. Linear wave equation is propagating. Whatever there is is carried. It is like a stream. In the example, I gave a stream of water flowing at Choctus. Choctus is carried at that speed, okay. So linear wave equation basically looks like and we tried a variety of schemes for this. Just like things work for Laplace's equation. We just tried a variety of schemes for it. That was sort of deliberately set up because I wanted to make sure that since Laplace's equation everything worked, I wanted to do the same thing and I mean, we obviously know. I obviously know that they are not going to work. We tried forward time central space which was unconditionally unstable. So the stability analysis, we looked at the von Neumann stability analysis, right. We basically linearized stability analysis. Anyway, it is already linear. So it does not matter. But we added, right. We looked at the same kinds of exponentials that we looked at when we are looking at the error term here and showed that forward time central space was unconditionally unstable. There is nothing that you could do, right. Forward time central space was unconditionally unstable. In all of this discretization, of course, we came up with a parameter which we called sigma which was the Courant-Friedrich-Lewy number, okay. And this number turned out to be pretty critical, right. So anyway, maybe I am getting a little ahead of myself here. We also did FTFS. I should not forget that. It is also unconditionally. And then finally we found, we did FTBS forward time central backward space and basically showed that if the Courant-Friedrich-Lewy number is less than 1, we had a condition. We got a condition. FTFS in fact basically gave us the idea. And from here we got the idea of upwinding. That is, FTFS did not work but FTBS worked because A is positive. And if the sign of A changed, if A became negative, then FTFS would work and FTBS would not work, right. So the equation is propagating U in a certain direction. I generally use some information but the equation is propagating U in a certain direction and your scheme also it looked like needed to propagate U in the same direction. Is that fine? To which we asked the question, so what is the difference between FTBS and FTCS? Well, I guess before we did that, what did we do after that? We looked at the modified equation. We asked ourselves the question, what is the problem that we are actually solving, right. And we got from that the modified equation. The modified equation looked something like this. It had terms which were of that form, higher order derivatives. So we asked ourselves the question what each one of these things do. In fact, we looked at it up to the fourth derivative and we concluded that the odd derivatives are dispersive in nature for this equation and even derivatives are dissipative in nature. Whether the U decays or is amplified depends on the sign of U and the nature of the derivative. So in this case, mu 2 has to be positive. In this case, mu 4 has to be negative. That is what we conclude. And this is just dispersive. So we saw decay and we saw dispersion. Dispersion basically means high frequencies and low frequencies travel at different speeds. So which is a very critical, very important phenomena. Various frequencies travel at different speeds. So then we came back and asked ourselves the question, what is the difference between FTCS and FTBS? How come this works and this does not work? And we realized the difference was that in one case, the mu 2 was negative and in the other case, the mu 2 was positive for the modified equation. So the natural question was, cannot I just use FTCS and add a little amount of artificial viscosity, which we have tried, it actually works. And that FTCS, the relationship between, so any scheme literally can be written as FTCS plus an appropriate correction. If you choose to do it. So that is as far as, so from here, did we do anything else with linear wave equation? Then we looked at the quasi-linear wave equation and showed that and of course the general form, where f is a function of u. And we showed that in this case, in this case, where there were no discontinuities to start with and the original function is smooth, the initial condition is smooth, the discontinuity can form. So it is an interesting combination. On the other hand, you have diffusion or Laplace's equation kind of which is averaging out and eliminating discontinuities, high frequencies are decaying faster than low frequencies. That is one of the outcomes of this which I forgot to recollect, important outcome, high frequencies decay faster than low frequencies. So that being the effect of the right hand side, this is just interesting that this was creating the high frequency. And I pointed out that if you have in a different context, I pointed out that if you have u dou u dou x and if you substitute at sin theta, u is like sin theta, then dou u dou x is like cos theta. So sin theta cos theta is like sin 2 theta. So there is an increasing, there is a mechanism that increases the frequency, that doubles the frequency, which is very critical. So this quasi-linear, this term seems to do that, it seems to increase the frequency. On the other hand, this seems to try to decay and that combination is what of course makes fluid mechanics so interesting. So then from here, what else did we do? Of course, we derived the, if you get a discontinuity, it is called a shock and we derived the Rankine-Huguenau conditions for the shock speed and so on. Then what else did we do? We looked at one-dimensional flow, we derived the governing equations, looked at one-dimensional flow. We tried the FTCS plus dissipation for it. We tried first of all to make the equation look like this linear wave equation. So we got it in a, so we wrote it in two forms, we wrote it in a conservative form and a non-conservative form.  These variables are said to be the conservative variables written in a divergence-free form, mathematically said to be written in a conservative form. For us, of course, from gas dynamics, we know that across the shock, these quantities are conserved. So you can give any reason you want as to why you want to call it conservative form, but these are called conservative variables as a consequence, even if you write it in a non-conservative form. So you can use, you can still write this in a non-conservative form and sort of a desperate attempt to make it look like the wave equation, linear wave equation. Where this term, the flux Jacobian is dou e dou q, okay. The flux Jacobian is dou e dou q and then we say rho at the system is coupled, isn't there something that we can do to decouple it. We tried to do, change the variables from q to q tilde which was rho up and that did not help either. That still gave a couple of system of equations, dou q dou t plus a tilde dou q tilde dou x equals 0 in terms of q tilde which are not non-conservative variables, certain combination. Of course, we could have used rho u t also, right. So and it turned out this was still coupled. Then we said wait a minute, I can relate these two to each other with a similarity transformation. I can transform this equation to this equation and the a and a tilde will be related to a similarity transformation. We asked ourselves the question, is there a transformation that will diagonalize the matrix, right. So if you use the modal matrix away or you use the modal matrix away tilde, the matrix made up of eigenvectors away, then it is actually possible to diagonalize it and we got the characteristic form, okay, where the system of equations was diagonalized and it was decoupled, right. So we got three equations that were basically propagating, they were propagating like the wave equation, okay, right, but not linear wave equation because they happened to, it was a quasi-linear wave equation. But at least our analysis would work and the CFL condition for this, if we do a discretization and we got the Courant number at least, the Courant-Radix-Levy number would be of the form u delta t by delta x, that would be sigma, I guess you could say u plus a or mod u plus a or whatever, that would be the largest Courant number, am I making sense, okay. And the typical stability condition would require that this is less than 1, is that fine, okay. We are using mod u because u can be positive or negative, a is the speed of sound, okay. Are there any questions? Anything else, what did we do after that? We looked at the critical part of applying boundary conditions and applying boundary conditions we decided to use the fact that this equation is propagating in a certain direction, determined by a. So at a subsonic inlet, at a subsonic inlet, so the eigenvalues for the lambdas in that equation will u, u plus a, u minus a, raise the speed of sound. So at a subsonic inlet, these two are positive, that is negative. So at the inlet, these two are propagating into the domain, this is propagating out of the domain and we use that to determine boundary conditions. So typically at the inlet we applied, we applied for flow through a pipe, P0 and T0 at the inlet and P ambient at the exit because it is a subsonic exit, this is negative, this is positive, these two are positive. So these are propagating out of our domain at the exit whereas this is propagating back into the domain, am I making sense, is that okay, fine. So we needed three quantities, we have developed, we have got three quantities but actually at each point we need to get three each, so we have to get three more, right, which we did by extrapolating. So if I actually write the domain and show the grid points, the first grid point and the penultimate grid point and the last grid point, then we basically said that there are certain quantities that have to be extrapolated. We explored various things that you could use to extrapolate to the boundary, right and this is very important. So the emphasis here is there are boundary conditions that you recollect, there are boundary conditions that are required by the physics of the problem. You have a pressure vessel, if the air in it has a certain P0, it has a certain T0, these are measurable quantities. You have a pipe, you have a valve, there is an ambient pressure, open the valve, the ambient pressure and the P0 basically determine what is the speed with which the air is going to flow, am I making sense and the T0 will help you determine the other parameter. That is what the physics requires. The mathematics basically says that you need three quantities because plus an initial condition because you have a single time derivative and you have a single spatial derivative, first order spatial, first degree spatial and we have three quantities but the numerical algorithm insists that you need Q here and Q there and therefore you have to generate, the numeric requires more boundary condition. We have to actually generate those boundary conditions, we have no choice, is that fine, okay. So then of course we also wrote, one important thing that we did was we wrote this in the delta form and this is a, we wrote the equation in delta form, I will just write one, none of these, these are just to recollect, R is the residue, I have not defined in the recollection, I have not defined the residue so far. The residue is if you have an equation and you substitute a potential solution into that equation, if the right hand side does not match whatever is left over is called the residue, okay. So if you have LU equals 0 and you substitute a candidate U and you do not get a 0 but you get some value that is called the residue, whatever it leaves is called the residue, that is the residue from this steady state Euler equation and this is the delta form, this is the correction to the current state in Q that you have, current candidate solution that you have. The big thing that we want to take from here is that given this R will determine if you reach the solution or not, if the residue is 0, your current state is correct, if the residue is not 0, your current state has a problem, needs to be corrected, you can use the solve the system of equations to determine the correction, if the residue is 0, the correction will be 0, if the residue is 0, the correction is 0 as long as this is not singular, it can be anything and therefore it is possible for us to choose something that will make the convergence faster, that was the other deal that we have. So we do this carefully because this determines when you reach the solution, you do this carefully because that will determine how fast you get to the solution, is that fine, okay, right. And then we did a little more involved boundary conditions, application of boundary conditions here, right, using converting it to characteristic coordinates and so on, okay, is that all? Did we do anything in the, then I wrote out the equations for the quasi 1D, right, quasi 1D and that was basically about it, right. What did we do after that? We looked at unsteady flow, what did we do after, we looked at, I think we spent a little time on unsteady flow, right, we looked at the preconditioning the unsteady term, is that what you said? So we use this, see there is a certain freedom that this gives, this argument gives, that is the reason why I would not emphasize this, that is this determines that you have the solution and therefore if this goes to 0, the correction goes to 0 and therefore whatever multiplies it goes to, see that gives you a certain freedom and we look at it and say if I am looking for the steady state solution, dou E dou x goes to 0, right, dou q, if you look at steady state solution, dou q dou t goes to 0 and therefore if I multiply dou q dou t by some matrix, it does not matter. So why would I do that?  So by pre-multiplying this by gamma, it is actually possible for us to precondition the problem and if I am, since I am looking only for the steady state, this pre-multiplying by gamma dou q dou t does not really affect the steady state but affects the rate at which you are going to reach, same idea as that, right. There is something going to 0, I can multiply it by whatever I want as long as it changes the algorithm so that I get there faster but where I go is the same destination, that is the key, okay, that is the key, is that fine, right. Then of course we looked at unsteady flows by adding a pseudo-time term, right, of course you can solve this directly using, I think post my demo I did Runge-Kutta method, right and you can solve this equation without the gamma of course if you are looking for the, if you are looking for the unsteady solution, you can solve this directly. So if you are looking only for the steady state solution, this is called a time marching scheme. If you are looking for the transient, we are looking at a time accurate computation, you could use Runge-Kutta or whatever it is but what we said is we have built up so much machinery for steady state solution that if you add a pseudo-time term, dou q dou tau and you could then because that is going to 0 multiply that by a preconditioning, right, you could actually converge to the steady state and tau and get the unsteady solution in real time that you want, okay. So we were then talking about acceleration schemes, right and then we went back and basically remembered that high frequencies decay faster than low frequency, we recollected all of these, recollected all of these things and of course in my review, I think I have left out representation of functions, the demo that, the critical demo that I did of high frequencies and low frequencies but it does not matter, right, that high frequencies decay faster than low frequencies, right and we basically came up with the multigrid scheme. So the demo that I am talking about is where we represented sin x using, the demo that I recollect the demo, I will go back to the demo, we talked about representing sin x using hat function, that was a critical demo for us because it showed that for a given grid there was a highest wave number that we can represent, that is very important or turning it around whether the wave number is high or low depends on the grid that you are talking about and if you say high frequencies decay faster than low frequencies then you are interested in using a coarse grid because the wave numbers then become higher frequencies, what are low frequencies on a large grid become higher wave numbers, right on a coarse grid hence the multigrid scheme. So what you basically do is you use grids of different sizes, right, you use grids of different sizes h, 2h, 4h, 8h and you transfer the residue, this is a critical part, you transfer the residue from the fine grid to the coarse grid and you transfer the correction from the coarse grid to the fine grid, okay. So it is possible for you to actually go through this process of transferring the problem in a sense by you transfer the residue or transferring the problem, transfer the correction you are transferring the solution in a sense if you think about it, okay, so it is actually possible for you to run your program on a very coarse grid, you run your problem on a very coarse grid to eliminate to so that you convert what seem to be low frequencies on h to a relatively high frequency on 8h, is that fine, okay. So multigrid schemes of course and then we said though there is possible actually in fact why would we do that you just basically start with the coarse grid and transfer in fact when we talked about Laplace equation in beginning that is one of the things that we suggested that compute the solution on a coarse grid and transfer it as a initial condition on the finer grid, finer grid, so you could actually do this here, you can start on the coarse grid, get iterate a few times for Laplace equation or take 10 time steps for Euler equation or whatever and transfer it to the next finer grid, so you could actually go from 8h, 4h, 2h, 2h and then start your go through this process and typically the critical thing that you want here is what is the thing that we have to recollect here, work units that is the that is what you always work units that one sweep on the finest grid in one dimension is equivalent to two sweeps on the coarser grid, equivalent to four sweeps and if you go to multiple dimensions of course that gets even better, so you would expect that in three dimensions multigrid will work extremely well, the actual convergence rates that you will get in wall clock time, how quickly you get to the solution will be much better, what did we do after that after multigrid methods, calculus of variation, we looked at, so calculus of variations basically comes back and says so we are back to this business I am representing a function, the only thing is now you actually have a measure of something, so you say that I have, if I think the example that they gave is if you are coming from your dining hall to its classroom what is the shortest path that you have, so you can come up with some kind of a measure or a metric to see which is a function that you want, you are looking for the, you have a functional, you have a measure and you want to minimize that measure and from there we derived the Euler Lagrange equations, we showed then that the relationship between the variational problem and the differential equation because it involves optimization, there is a differentiation and setting equal to 0 kind of a process that you do to get the Euler Lagrange equation, so in that sense the Euler Lagrange equation is like a derivative, right, the Euler Lagrange equation is like a derivative, right, so you differentiate, find the first variation and you get this derivative and we showed that Laplace's equation, we found and looked at the variational form of Laplace's equation and showed that you could get Laplace's equation directly, which was analogous to the earlier ax equals b being a minimization of a quadratic q, so it is analogous to that, okay and we basically said that yes, so it is possible for us sometimes to solve it in the variational form, if you get the variational form the amount of smoothness required of the function is not as much, smoothness required of the function is not as much, you do not need as many derivatives, right, so it is possible for us to reduce the derivative requirement, fine and very often the expression in the variational form is simpler than in the differential form, very often that is so, but it is not always easy to get the variational form given the differential form, okay, you do not get something for nothing, right, so there is an element, there is a difficulty in doing that, okay and finally of course we end with today's class, so I will just recap, put it back to the big picture, so the big picture possibly is that this is the reality as we see it, so you have, right and of course I have put this little brown to indicate and from here reality you get, this is the perception, this is what you see, so this is what it is, I just draw some, I can draw anything there because we cannot see it, this is what I am saying, this is the perception and from this perception we come abstract out a model, it would be ideally in a figure that I drew to do an exact circle, right because we have mathematical precision, we abstract out a model and then of course we cannot even solve the model, we cannot even get this model, so we end up doing a computer program which is discrete, the idea that is, that is, that is pictorially that is what I am trying to represent here, so this is modeling, this is discretization if you want, you can check consistency whether this model is consistent with this as the number of sides that polygon increase does it become the circle, its consistency and you can do validation, you can check, perform an experiment and you can check whether the result is, fine, cannot do that, the scientists and everybody constantly trying to do that, am I making sense and of course here there is, so there is a test that you do here also, you can add terms, remove terms, all the flow is inviscid, the flow is viscous, right, so all of that, all of that stuff is happening here but once you decide to do the Euler equation you go back here, so there are two possibilities, either you say this solves this and therefore I validate there but you look at this, you know they are not, so validation is validation, that is it, right and no more than that, fine, okay, thank you very much.


The St. Lucian Nurses Association recently presented care packages to the nurses at the Millennium Heights Medical Complex expressing appreciation for their commitment to the profession. Permanent Secretary and the Ministry of Health, Benson Emill, expressed heartfelt gratitude to the nurses for the services they extend to the public. St. Lucian's is grateful as an institution in the Millennium Heights Medical Complex. You know, grateful as a Ministry of Health and Wellness. Returnally grateful for the services that you provide and the services that you continue to provide. As much as possible, we will support you in your endeavor. But we know that you are the ones on the ground. You are the ones making the commitment. You know, you are the ones making the self-sacrifice every day to ensure that we care for our citizens of St. Lucian. Chief Executive Officer of the Millennium Heights Medical Complex, Siobhan James Alexander says she's extremely proud of the nurses and will continue working with them to bring about success. It's indeed a pleasure that we're standing here and that I am standing here and recognizing your efforts. Last year I am sure has been overwhelming and we must recognize your dedication, your loyalty, your selflessness. And I think sometimes these things go unheard of and unspoken about but we need to always celebrate our key wins and for us to actually achieve what we've done over the year despite all odds is significant. And so for now I want us to celebrate that win. Just give ourselves a tap on the back and say thank you. The nurses have been phenomenal. President of the St. Lucian Nurses Association, Alicia Baptiste, saluted other nurses and asked them to continue being indispensable. Reporting from the Communications Unit of the Ministry of Health and Wellness, I am Fennel Neptune.


The power-serving myth that anti-war protests make no difference. Thousands of people from across the political spectrum gathered at the Lincoln Memorial in Washington, D.C. to protest U.S. militarism, proxy warfare, and nuclear brinkmanship in Ukraine on Sunday. I've been seeing some people try to minimize the numbers on social media, but footage from Rage Against the War machine makes it clear that attendance was in the thousands. People who were there placed the number at around 3,000. This is significantly better attendance than any other American anti-war demonstration in recent years that I'm aware of. It's nowhere remotely close to the historic numbers people demonstrated in to protest the war in Iraq, and it's nowhere remotely close to what it should be for an issue of such existential importance. But it's a start. Maybe the start of something good. The Answer Coalition has a march on Washington scheduled for March 18th for the 20th anniversary of the Iraq invasion, demanding negotiations, not escalation in Ukraine, and an end to the U.S. militarism abroad. We shall see if this thing continues to pick up steam. One criticism I hear of anti-war demonstrations is that they don't make a difference. Millions of us marched in opposition to the Iraq invasion, and they did it anyway as a common sentiment. While it's true that demonstrations failed to stop the invasion of Iraq, if you look at the U.S. war machine's actual behavior following that war, it has clearly been reacting defensively to public opposition. If anti-war protests made no difference, the U.S. Empire wouldn't have completely abandoned full-scale ground invasions after 2003 and switched to sneakier less effective means of warfare while launching unprecedented narrative management systems to suppress anti-war sentiments. They abandoned Bush-era Hulk smash ground invasions in favor of drones, proxy warfare, covert ops, and sanctions because enough people rose up and said no to make them afraid of the masses beginning to wake up and begin turning against them in their institutions. And now people are even beginning to protest the proxy warfare. I guarantee you that's making our rulers nervous about the possibility of losing the ability to effectively dominate the world with violence and coercion, and even losing the ability to continue to rule us. These things very clearly and obviously make a difference. The only reason Syria and Iran remain sovereign, unabsorbed governments, and the only reason the imperial body count isn't much higher than it is today is because enough people put their foot down and said no to that kind of war. Our rulers pour so much energy into manufacturing consent because they absolutely require that consent in order to rule. Their absolute nightmare scenario is the emergence of a large, robust movement of people saying no to the imperial war machine because military violence and the threat thereof is the glue that holds the empire together. It's bringing public consciousness to the very most important aspect of the empire, which also happens to be the least defensible. Noam Chomsky said propaganda is to a democracy what the bludgeon is to a totalitarian state because over the centuries those who seek large-scale power over other humans have discovered that dominating people psychologically is much more energy efficient than dominating them with brute force, and is far less likely to see them wind up on the business end of a guillotine blade. If you can simply trick a profoundly unfree populace into thinking that they are free, you don't have to waste any further energy wrestling their freedom away from them. But what that means is that the entire power structure which rules over us is entirely dependent on the ability to successfully administer propaganda, and maintain the illusion of freedom. If it can't manufacture consent for the things it wants to do, it has to either refrain from doing those things until it can manufacture the necessary consent, or do those things anyway without the consent of the public. If they do that, public trust in the ruling institutions will immediately begin to disintegrate, and they won't be able to propagandize people anymore because propaganda only works if people trust its source. Our rulers could of course then switch to the direct bludgeon of totalitarianism if they want to, but then they're up against an angry populace, and in America's case a very heavily armed one. And the narrative management that holds the US centralized power structure together would lose trust around the world because the freedom-loving good guys versus tyrannical bad guys framing of imperial propaganda would no longer be believable. The US centralized empire will crumble if it cannot preserve the illusion that it is accountable and responsive to the public. Of course a few thousand people at one protest is not going to bring peace to our world. Even a few million wouldn't be enough. But public demonstrations are one of the many ways in which our society can be drawn toward awareness of what's really going on in our world, what our rulers are really up to, and how much we've been lied to all our lives. From there health can follow because with enough awareness people will cease consenting to things that they've come to recognize as being against their interests. All positive changes in human behavior are always preceded by an expansion of consciousness. Demonstrations are one of the many different tools ordinary people have to spread awareness so that positive change can occur. That's why they've featured so prominently in civil rights movements throughout history, and that's why the US empire has been pouring so much energy into preventing the reemergence of a lively anti-war movement in the western world ever since Vietnam. Other tools we can use to spread awareness include new media like online videos, blogs, tweets and memes, and old media like pamphlets, public speaking, and graffiti. Anything we can do to get people opening their eyes to the dangers of imperial war-mongering and start bringing some actual movement into the anti-war movement will help. Our survival may very well depend on it.


Think Tech Hawaii, civil engagement lives here. Welcome back to The Prince of Investing, right here with your host, The Prince of Investing, Prince Dyke, coming all the way live from the beautiful state of Denver, Colorado, via Haluulu, Hawaii. But as always, I don't have a lot of time, and I definitely know you guys and girls don't have a lot of time, so we're definitely going to jump straight into it. So this is a topic that I picked. We're going to talk about the Robin Hood app. If you don't know what the Robin Hood app is, the Robin Hood app is a app, essentially it is an app that you can trade by and sell, stop, right? I get a lot of reviews about it, people asking about it, and other things like that, right? Prince, what do you think about the Robin Hood app? What's the good? What's the bad? It's things like that. So what I'm going to do in this video, and what I'm going to do, and the people that catch the audio version of podcasts across the globe, what we're going to talk about today, we're going to talk about the pros of the Robin Hood app, the kinds of Robin Hood app, and that's pretty much it. We're going to keep it short, sweet, and simple, and to the point, and we're also going to talk about, of course, what it is. So first, let's talk about what is the app. The app itself is the app you can download to your phone, and you can use it to purchase stops. So people who are just starting to buy stocks, people who want to get into stocks, the first thing you must do, if you want individual stocks, the old school days, you can call a broker, and I'm pretty sure there are brokers out there now that you can call and buy stocks. But most people, most of the millennials, people that are, you know, in my age range of 33 or whatnot, is mostly be used on online brokers, someone like the E-Trade, a T-Emeritrade, a Scottrade, a place where you go, set up an account, fund it, put money into it, and buy a particular stock. So Robin Hood is one of those, and it has the app. Just about every desktop form of an online broker, there is an app version. So Robin Hood mostly focuses on the app. You can just pick up your handy-dandy phone, download the app, and boom, start purchasing stocks.  Now we're going to talk about one of the, I want to say about three good things about the Robin Hood app. The first thing is an easy convenient, you can download it right now while we're talking, and by the time I finish this video, you can probably already have requested an account, right? So that's a good thing, the convenience is right there on your phone. One of the second things is it's great for people that are just starting to invest in because it has no account minimum, no minimum account. You don't have to start with $500 or $300 or whatever the case may be, or whatever the case may be, and you can just start right away. You can start funding it with $10, $20, $30, and now you have an account.  Now the third thing is, commissioners, think about it. How do stock brokers make money? That's a question you should ask anybody in the financial industry because you might discover some things. How do you make money? When we say a broker, for example, how does a broker make money? When you call up a broker, you want to buy 10 shares of Amazon stock. When it goes in and processes it, it adds a little fee on it. When you buy insurance policy, there's a commission for an insurance agent. When you buy a car from a car dealer, there's an insurance. There's a commission. Just like when you buy a house from a real estate agent, and so on and so on, right? So robber hood, it doesn't have any commissioners. Don't ask me how they do this. I don't know. I don't know if they make their money through advertising or whatever the case may be, but they have no commission. Right now, we live in a very, very low-fee market with like TD Ameritrade. TD Ameritrade only costs us about $5 or $6, so does E-Trade. Or the lowest one, I think, on the Marks, Charles Swab at $4 or $5. But anyway, robber hood lets you do it for free. When you go to buy four stocks, you can buy it particularly for free. So look at the top three things, right? Easy to start, easy to buy, easy to start, no account minimum. And you have no commissions. You don't have to pay any fees. So that's the biggest thing, the no commission, because they started with no commission. Everybody else was charging about $10. So that's definitely a big plus, because we know those fees, they add up over time. Now, but with everything that's good, there also are some bad things. Some of the bad things about robbing hood. One of the bad things is that the limits that it has.  It doesn't have every mutual fund. It doesn't have every ETF. It doesn't allow you to do everything. Now, I know at one time, they have upgraded to options. That was very cool. But you can't do futures. You also don't have access to certain mutual funds, for example, right? And also with that, it brought in cryptocurrencies, not too long ago. But when it brought in the cryptocurrencies, you can monitor Bitcoin and cryptocurrencies. But it only will let you buy them if you were like in certain states or something like that. So it has a little bit limit. It has some limits on what you can buy and what's open to the market. But it has all the basic stuff. You could go in and buy the majority of your stocks, majority ETFs, stuff like that. But there are limits. You don't get every single thing, right? Now, another thing is when you look at the, another downside to it is the platform itself. You know, there's only so much analytical tools I can do with my particular phone. Yes, I can look at the price. I can look at the highs. I can look at the lows. I can look at a simple chart. But if I want to do some comparisons and things like that, I want to use these tools to compare fees and to compare performance and to compare things on charts, Robinhood is just not there from what I've seen on the app. That's when things like E-Trade and TD Ameritrade and other charge swap platforms start to shine because I can pull up on my nice little laptop or iPad or whatnot, compare, contrast, add a fee, see how things perform, look at nice little pie charts. Because I like nice little pie charts and bar charts because it makes it smart for me. But with Robinhood, you just don't have that, right? So you don't have the analytical tools and things like that. You kind of go to that notion, you kind of get what you're paid for. But TD Ameritrade and E-Trade and you, all these other brokers, they have an app version as well. They have an app version and you have more news and things like that. But the gist of it, the apps are about the same. But when you look at doing those deep down analyticals and like a lot of tutorials you see here on the show, and on the YouTube channel, you don't have those with Robinhood. So those are the pros and cons of the Robinhood app. It's very popular right now. Now, granted, you have other apps out there like Stash, you have other apps out there like Acorn, where they're great. If you are investing, you're doing good, right? But can you do better? And can you do, you know, it's always good, better investing, good, better and best in investing. So when you are investing, that's great. I mean, that's good, but could it be better? Yes, could it be the best? Of course. But when I looked at the Stash app, what I did like about it is some of the mutual funds that it was throwing PBNs that was riddled with high fees that they, you know, I didn't see them offering a broad-based index. I think that a lot of people, a lot of amateur and new investors should put the bulk of their funds until they learn more and things like that. So that's the cool, that's the great part about it. So, you know, we went over the good parts about the Robin Hood app and went over the bad part. Me personally, I don't use the Robin Hood app. Why I don't use the Robin Hood app? Because I can get all of that same information with E-Trader TD Ameritrade. And the bulk of the stuff that I buy are no commission ETFs. And what I mean by no commission ETF? No commission. So if it has no commission, one of the biggest things that drew me to Robin Hood was trading without no commissions. Whoa, that stuck out. But if I can get no commissions over at TD Ameritrade with some of the index funds that I purchased? Huh? Or what not? But is it a good app? Of course it is. It's a great app because it's charging you no commission. That's the biggest thing. That's the biggest plus. One of the biggest downfalls, the analytics, right? But if you're someone that are new, that just want to buy some Nike stock or you just want to buy some Amazon stock or whatever the case may be, you can download the app, fill out an application. Once you have your application, you fund it from your check-in bank and have you want to fund your account. You put the money in your account and boom, you can start investing the day. So that's my review of the Robin Hood app. People ask me if it's a good thing or a bad thing. I don't personally use it, but it is a good app. And I hope they help you guys out. But as always, my name is Prince Dax. That's your review of the Robin Hood app. Don't forget to hit the like, subscribe, comment, share button and follow us and also check out us out and check out some of our cool stuff in the description box. Until the next video, podcast, cartoon on the next episode of Silver on the Globe, peace, be safe, I'm out and thank you.


Today's question is from Switzerland. Michael asks, I had a penalty and I fixed it. Will Google hold a grudge? First off, at Google, we call them manual actions, not penalties, since they're generally applied manually by a team here and they don't always have a negative effect on a site overall. In Search Console, we inform sites about any manual actions their site might have. If you receive such a notification, you can take action on that, resolve the issue, submit a reconsideration request. The web spam team processes these and if they can confirm that the issue is fixed, they'll lift the manual action. It might take a bit of time for everything to be reprocessed, but Google's algorithms won't hold that issue against the site in the long term. However, it's possible that a site temporarily had an unnatural advantage before. By fixing this issue, your site will return to its natural location in our search results. Additionally, things change on the web and in our search results all the time. A site's visibility in search can change over time, even if nothing on the website changes. So with that in mind, it can be normal that a site doesn't return to exactly the same place as before manual action. So in short, no, Google's algorithms don't hold a grudge. However, visibility in search can change over time, regardless of any manual action. You can join the discussion in the webmaster forum plus check out the links below to get even more helpful information.


looks like it's 701 so we will go ahead and get going and hopefully oh I just see Commissioner Kennedy just joined so perfect with that I will call the June second 2022 City of Santa Cruz Planning Commission meeting to order close clerk can we please have a roll call Commissioner Conway here Greenberg here Kennedy here Maxwell is here the CD Miller here different here here Dawson here are there any statements of disqualification for tonight's agenda items from anyone okay seeing none we will move on to oral communications this is a time for the public to address the Commission for items that are not on the agenda for this evening so if you would like to address the Commission for an item that we will not be discussing this evening you can go ahead and this is a time for you to raise your hand you can go ahead and press star 9 to raise your hand or just hit the raise hand function you will have three minutes and again this is only for items that are not on the agenda tonight so I'll give a second for the lag see if we see any hands for oral communications sir I am seeing no hands can you confirm you are seeing the same that's correct okay thank you very much so we will go ahead and move on to approval of the minutes for the May 19th meeting and I would be happy to entertain a motion for their approval unless there's any questions or corrections I'll move the minutes move by commissioner shifrin is there a second all second thank you seconded by commissioner Kennedy can we just try a voice vote and we'll just say I and nay so also if you could just raise your hand when you say I so the clerk will call real quick sorry chair but I was absent I believe we both were absent so we did same right I was getting to that and commissioner Dawson and commissioner Maxwell will not be voting so calling for approval of the May 19th minutes with the voice vote clerk can you call for us and mays could I ask a point of procedure I want to make sure that's legal I'd like to ask staff if we're able to have these voice votes as opposed to roll call votes yeah that would be great to get clarification I looked in the bylaws and didn't see anything but it would be great to get clarification from staff just thought we would be able to move things a little quicker but let's go ahead and hear from staff their thoughts happy to stay with the roll call system if that's just easier for everyone I good afternoon leave all their planning and community development director I do not know that it is a problem I would ask our clerk here tests do you feel that there's an issue with that I think that most boards have tended to do roll call votes with the zoom just because of the technology being what it is it's just more clear for the public who may not be viewing it but maybe listening to it yeah let's let's just stay with that then let's go ahead and go with a roll call vote and stay with let's work before so clerk can we please have a roll call vote for approval of the minutes commissioner Dawson and commissioner Maxwell be abstaining yeah I'll try and make it quick commissioner Conway hi Greenberg hi Kennedy hi the CD Miller hi different hi okay thank you very much approval of the minutes for the May 19th meeting passes let's move on to the consent agenda we have two items on the consent agenda and I would like to say that I have heard from staff and staff is requesting that we continue the 109 South Rapetta Road project to a date certain so I would like to pull that from the consent agenda and at this time I would entertain a motion from any commissioners to continue to open a public hearing to make sure that there's nobody from the public who wants to run number two okay so we've pulled the agenda item off the consent agenda and then now I'll go ahead and open a public hearing to if anybody would like to speak about this project staff has said that they have received numerous information from new information that they would like time to process and allow plenty of time for discussion of this project so their request is to move this forward to a date certain two weeks so any members of the public like to speak on the 109 South Rapetta project at this time if so you can press star nine on your phone or you can raise your hand okay I'm seeing no hands move that we continue this item was there a particular date certain that was given by staff yes two weeks please item for two weeks and second that okay so we have a motion by Commissioner Schifrin second by Maxwell to move item number two that has been removed from the consent agenda forward two weeks can we have a roll call vote before we could I just say a under discussion of the motion let me just say I I appreciate that this is being continued it it's much as far as I'm concerned it's more complicated than it first appeared so I appreciate that the staff is taking more time to particularly review the original project I hope and and provide additional information on them on the recommendation okay just for clarity could we just say that that it's to the June 16th regular meeting that's fine they're expectable to the make of the motion okay great so back to the vote moving it moving this item the 109 South Rapetta Road project to June 16th can we have a roll call vote please Commissioner Conway hi green up Berg I Kennedy right Maxwell hi the CD Miller hi different hi your Dawson hi you're muted Cindy all right thank you we motion passes we will continue item three the 109 South Rapetta Road project to the next planning commission meeting on June 16th back to item number two which is on the consent agenda I will entertain a motion at this time to move the consent agenda or if any commissioners would like to remove that item from the consent agenda up up to the commission at this point I think you also have to if anybody from the public wants to speak on this can clarify that do for consent agenda items the staff that the public hearing needs to be opened for consent agenda items individually or as a group just her future reference as a group the the item should be opens and then the members of public have an opportunity to request that the item be pulled for additional consideration so during the consent calendar you would offer up opportunities for the public to speak yeah okay so I let's go ahead and hear from commissioners and then we'll go ahead and go to the public commissioner Conway and then Mr. Kennedy I'm prepared to move the consent agenda but after we get public comment okay great let's go to the let's go ahead and open the public hearing are they any members of the public who would like to speak on item number two five five five Pacific project this is an ordinance for the second amendment of the development agreement to convert 94 small ownership units to 100% rental projects subject to the city's current inclusionary requirement of 20% in perpetuity any members of the public like to speak on this item or request that it be removed from the consent agenda this is the time to hit star nine or praise the press the raise hand function okay I'm seeing none so we'll bring it back to the commission now to entertain a motion I'll go back to commissioner Conway I'd like to move the consent agenda okay okay moved by commissioner Conway seconded by commissioner Kennedy are there commissioner shifrin usually we don't have discussion on consent items so discuss any but it's any member of the commission can discuss any motion before it's voted on so I'm I support the motion I just wanted to thanks staff for responding to the questions that I raised and for putting the the agreement that we're recommending approval on tonight I think it's


Mate, to be honest with you, you're talking to the wrong one here, all of that has to go through the owner, Pat, he's not here at the moment, but if you want to give him a call back tomorrow, you can have a chance for me for sure. What about you, what if I sell you the pig, I can sell a very good price, half price what I normally sell and then you can sell the pig to butcher make three, four hundred dollar maybe five hundred dollar, what you think of the pig, it's a very large pig, nearly I'd rather make more money on the pokies, mate, I'm all good, thank you. What is the pokies, what is that, is that a pig, type of pig? No, no, no, it's the rattle machines with the flash of light, you chuck your money in and press the button and it could be down. Okay, so you say the rattlesnake you buy, but not the pig? Yeah, yeah, definitely. Okay, well, maybe I have a friend who also sells rattlesnake, maybe I give him your number? Yeah, go for it, I'll snap them up. Okay, what's your number? What's your number, write down. I'm number one, mate, always number one. Number one, and what is the rest of the number? That's it, I'm just number one, buddy. They make no sense, huh? Your number one? Your number one, they make no sense, huh? Do not lie to me, do not waste my time. Excuse me, I'm talking now, I'm not fine. I am talking, do not waste my time. Your number is not one. Why you laugh, huh?  I need to sell the pig, okay, to buy a medication.      You're calling the rest of my time. I tell all other butcher, you waveless fucks. Yeah, you just tell them, Snippy says, oh, yeah. Fresh, fresh pig. You all fresh pig. I'm fine when he's fired up. It's so cute.


One, two, three. Buddy, good morning. Welcome back to my channel. Today I have a very exciting DIYing Your DMs episode, which is a little series I do here on my channel inspired by The Sorry Girls. Now they did the series a long time ago and I absolutely loved it. And it's essentially where I take your guys's DMs or emails or requests for projects and I try to interpret them and create them here on the channel. So I have four really cute projects that you guys have sent me over on Instagram and you can always totally Instagram DM me. Make sure to follow me as well if you're not already. It is Lone Fox home on Instagram. And you guys, I burnt my finger making one of the projects but it was totally worth it. It wasn't really my fault. I mean, it was my fault, but it wasn't really because I didn't drip the hot glue on my finger. I actually set my hand on top of the tip of the hot glue gun, which was my bad. However, I would do it again for the project honestly. It turned out really, really good. But let's go ahead and jump on into our first project and I'll tell you guys who sent this one over. Now the first project was sent in by Lone Fox family member Alejandro. And he says, hi Drew, I absolutely love your channel. Thank you so much by the way. I'll be moving to a new place soon and I definitely love your style. I wanna decorate my walls and give them some more depth. And I would love to know if you can create these clay knots which I've seen these a couple of times before on Pinterest. I'm not sure if a specific company makes them or if it's just like people that create them but I figured it seems pretty simple to just roll out some, you know, long pieces of clay and tie them into a knot like this and create a cute little piece of decor which is great for like a coffee table or a paperweight or just something to style on top of a set of books or anywhere you just need a little bit of texture. So let's go ahead and see if we can create this little clay knot decor. For our first project, I'm going to be starting off with a large block of clay and I actually just cut that right in half and started smooshing it up just to make sure it was nice and ready to be rolled out into a long log. Now whenever I'm rolling out any form of clay I suggest using your fingertips kind of as I'm doing here. So what I'll do is I'll kind of start rolling it with my fingertips and I'll actually spread them as rolling it which I feel like widens it but it evenly widens the roll at the same time. So you don't have skinnier sections and then wider sections in some areas. That's kind of like my tip for rolling out clay when you want to have a consistent roll but you don't have something like an extruder to make it perfect. So I'm going down and I'm just eyeballing how much I think I need to create a knot and I'm going to fold that in half as shown here and then what I'm going to do is start actually creating the knot. So it's very simple.  So it's pretty simple to work with without having it stick together. So I maneuvered my knot into the kind of location I wanted it to be in and the shape that I liked and then I cut off the tails just to make sure that it was nice and clean and I brought this into the oven and baked it for about 30 minutes or so just because it was on the thicker side and once it came out of the oven you can totally leave it as is if you want like a more natural look or you can spray it like a stone texture but I ended up opting for my favorite terracotta spray paint. So I went ahead and I gave it two full coats of this on the front and backside, let it fully dry and that finished off this little knot to core piece. Now our next project is the one I actually burnt my hand on. This was sent over by Courtney Jackson. So Courtney, I'm blaming you for the burn on my hand. I'm just kidding. I'm totally kidding by the way. She said, hey Drew, I bought this planter second hand and I haven't been able to find another one similar. I would love your take on it mainly because I want more than one in my home and it's a really cute planter. I love the kind of slotted detail of the wood and I also love how the legs are just a continuous piece of wood that kind of coordinates with the sliding but not too much. Now I'm actually going to be using a court size paint can to create mine. So I'm doing a little bit of a smaller version but if you want a large one like what Courtney has in her home, I suggest just using a gallon size paint can. So if you have an old can of paint anywhere or anything that kind of is that cylindrical shape but I'm gonna go ahead and use a court of pink paint that I ended up going through quite a while back. So I just had a little bit left. I cleaned it out and this is what I did with it. You know it's moving about. Before this next project, I ended up repurposing a court of paint. So I'm just using the can and then I got these wooden dowels at Joann's and I'm actually gonna chop them up into the sizes that are shown here. So I have a majority of my pieces at five inches and then I have nine pieces at six and a quarter inch. Now I'm also gonna go ahead and spray the can black just to kind of camouflage it a little bit. We're gonna be covering the full can. However, I just wanted to make sure that none of the silver popped through anywhere. So starting off with my five inch dowels all you have to do for this process is use a strong bond hot glue like the one I'm using here. This is the Gorilla hot glue sticks, which I love. And I'm gonna go ahead and glue the five inch strips. These are the exact height of our can. So I'm gonna glue these around the exterior just making sure that they're perfectly straight and in alignment. And when you feel like you wanna go in and add a leg, all you have to do is grab one of your longer dowels and then just place one of the shorter ones next to it so you know how much glue to add to the dowel itself and then place three of them next to each other which will create the leg and just make sure that the top is super flush with the five inch sections and then continue adding your five inch sections until you wanna add another leg section. Now I will tell you guys I got lucky with this and if you do use a quart can of paint as well, I used eight of the five inch sections and then I did three legs and then eight five inch sections, three legs and it ended up working out perfectly in the end. Everything was nice and snug and this is what the planter ended up looking like once all of the wood was glued on and now you have a choice to make. You can either go in and stain this, you can leave it as is, you can give it a top coat, you can use acrylic paint on it, but whatever you wanna do, I went in with my favorite early American wood stain just because I thought it was a warm touch and very reminiscent of the one that was sent over so I added a coat of this and that finished off my mini planter. Now our next project was sent in by Lone Fox family member Mia and she actually wanted me to create a large cabinet and for this video I was like, oh my gosh, a large cabinet, I don't even know how to create this. It was a $2,000 cabinet. She's like, I think you'll be able to conquer this but she also sent along a pillow which said or this project and it was literally a link. I went ahead, I copied and pasted the link in and I was able to find that it was this really cute kind of tufted pillow and I wanted to go ahead and do a little bit of needle punch anyway so I figured this would be the perfect project to go ahead and share with you guys how to create a really interesting pillow like this and this one actually retailed for I believe like $130 so we're gonna be making that for a fraction of the price and let's get started on this project. Now I have done punch needling a couple times in the past on the channel and what I always suggest doing to create the frame for your punch needle is just to go out and get an artist canvas at your local craft store, Joanne's or Michael's wherever you are, use a coupon on it and then remove the canvas and use the wooden frame as the base for your project. So next what I'm doing here is I'm cutting out a piece of monk's cloth which you can also find this at Joanne's as well. That's where I got this one and this is the proper fabric for punch needling. It's just more of a almost like loosely tightly woven material that allows you to really nicely punch needle on top of it. So I'm going in with the ruler here and just marking out the square shape that I wanna make sure that fits for my pillow. I believe I was able to get a 16 by 16 inch pillow on this particular frame here and then I'm going in based off the reference photo and kind of just free handing in the design there. This particular pattern is super organic so I just went ahead and kind of traced out my pattern there and I'm using my punch needle tool. I got this one on Amazon. I'll link it below for you guys, super affordable and punch needling is very, very simple.  So basically on the other side of the fabric which you can't currently see, we are creating loops and on this front side here we're just creating these little tiny little stitches of yarn. Now this is going to be the proper side here so the white that I'm currently doing this is going to be the front side of the pillow. The back side is going to have a ton of loops on it but you're gonna see what I'm kind of doing with this pillow. I'm gonna mix and match the textures which I think looks really nice when it comes to punch needling. So this white section here probably took me about an hour and a half to complete and then I'm going in with the softer green color and I'm going to be doing the same exact technique. So I'm doing all of our kind of flat patches first which are the ones that kind of just have the stitching on them. And when you have a color I do suggest just trying to go as close as you can to the stitch next to it just to eliminate as much white as possible but at the same time I don't really mind having that white pop through because it does almost add a handmade element to this piece which I do really like as well. So I wasn't too concerned about that. So going in here, finishing off our green section you just have to snip it and then we're going in with this kind of like brownish umber color which I'm going to be doing the circular shapes in. So all of our flat punch needling as I'm calling it is done. So as you can see the other side we have a ton of loops, a lot of texture on this side which is really fun but we want to add texture to the front side of our pillow as well. However, I really wanted to mix it up. So as you can see I actually flipped my frame over and I'm punching through the opposite side. That way on the front side of our pillow we have a loopy section of this dark green color which I'm going to add to two different sections of the pillow just to add a little bit of visual interest. So I'm going to go ahead and punch through here that I'm also doing this bottom left corner as well with that dark green color creating a ton of really fun loops. And last but not least I'm also using just a solid gray tone to go in and fill in this larger patch here which is going to add a ton of texture. And after about four hours my punch needling is complete. Now this is a great project to do in front of like a movie or something like that. It is not hard at all. It's just a little bit time consuming but it's totally worth it. So next step is going to be removing our little punch needle section from the frame itself. So I cut that out making sure to leave some seam allowance on all of the edges and on the backside of our punch needling I'm actually going to iron down some fusible interfacing. And if you've never used fusible interfacing before it is essentially a lightweight fabric where one side has glue on it and when you place it against the fabric and iron it the glue gets hot and of course adheres to the fabric. So that's going to hold all of the yarn in place on the backside there. Now I'm going in with just a little bit of lightweight muslin. I'm going to use this as the backer fabric for the pillow, cutting it out to the same exact size and using some pins to pin together the right sides of our pillow. So essentially the sides that you wanna be on the outside or the sides that are shown you're gonna wanna pin those together. And then I'm using a straight stitch on my sewing machine to sew all the way around the edge making sure that you're sewing right up to the edge of that yarn that you went ahead and tufted.  I just had some excess from an old throw pillow that I'm going to repurpose. So I stuffed it as much as I wanted to and then I hand sewed off the open edge and that finishes off the pillow. And our last project you guys this was sent in by Noel. I'm hoping that I'm pronouncing her name right. She says, hi Drew, I would love you to recreate this decor these sculptural spheres and maybe even do them in a different color. Now I believe these are actually from West Elm because when I used to work at West Elm we would sell these like hotcakes. And I've really never even thought about recreating or DIYing my own. However, when I saw this photo I instantly thought of how I can easily do this because I think these are pretty pricey. They're like 50, $60 a piece and we're gonna be creating them for a fraction of the price and they actually turned out almost identical. Let me show you how I did it. And I think this last project was my favorite one. I just love how they ended up turning out. So I got these metal rings at the craft store. I believe these ones are 10 inches and then the smaller ones are five inches. So I'm grabbing two of my 10 inch rings and I'm going to interlock them kind of in a cross shape as shown here so that they're kind of going over the top of each other and then using a little bit of wire this is going to be totally hidden in the next step. So it doesn't matter what color you have but we're gonna wanna bind that joint together to make sure it's nice and sturdy. So as long as you're wrapping it in literally every single direction going diagonally, across, however many times you want you're going to wrap this so that the joint is nice and fastened and attached and these pieces are not coming apart. I also slipped my third ring over the middle sections of those other two. It was a little bit challenging and it's very, very snug. However, I like that because it makes it feel like it's sturdy.  So wherever any of the rings cross over another ring you're just gonna wanna wrap those so that this doesn't come apart in any way in the future.  Now we're gonna be covering the entire sphere with clay. Now you can totally leave it as is. I think it's actually a really fun kind of metal object or maybe even could be some kind of light fixture or light pendant but I wanted to go in and make it as similar to the reference photo as possible. So just using some polymer clay I am going in and applying a thin layer onto every single section over the top of our wired joints and also over the top of all of the ring sections. Once all of your clay has been applied I just went back through with my finger and smoothed out any creases or cracks or just anything that didn't look perfect. So that way once we do go and bake it and we could paint it afterwards it looks pretty perfect.  It is the same exact process of wirewrapping all of the joints together and then just kind of adding all the rings and a figuration that you personally like. And then you're gonna go through and cover all of it with the clay as well. Once your little clay sculptures have reached your desired final result you're gonna pop them on top of a baking sheet with some parchment paper and I'm just gonna go ahead and put this into a preheated oven and I'm going to bake this for about 25 minutes or so. The clay really isn't that thick on the ring itself and I know that the ring is gonna heat up too. So once that was fully baked I just went ahead and let it cool down and then I'm going in with some gold leaf Rub-n-Buff. I have been loving using Rub-n-Buff as a paint lately. I just feel like it's the prettiest gold that you get a lot better than any spray paint or any other paint that I've used that's kind of a metallic paint. I just love the way that this one looks and I'll link the particular one I used below. So I went through and I painted both of our spheres and once they're fully done that finishes off this project. And that guys finishes off today's DIYing at your DMs episode. Now I have multiple more of these videos on my channel so if you are curious I'll link a couple more below if you like this concept and do not forget to also follow me over on Instagram it is lonefoxhome I'll put it on the screen right there for you guys. That's a great way to just stay in contact with me see new posts and also of course you could send in your DIY requests. Now I had so much fun DIYing these projects and I hope that you guys did as well. Do not forget to give this video a thumbs up if you did enjoy and I'm so excited for the next one. I actually don't even know what I'm filming next but I'm just always excited to film for you guys. So in case you didn't know I'm excited but I'll catch you all on my next one have an amazing amazing rest of your day and enjoy the weather outside if it's hot where you are it's finally getting a little bit more warm in LA which I'm very excited about and I will catch you all in my next one. Bye you guys.