So, we will take a look at Laplace's equation, last class we ended just short of my assigning a problem on Laplace's equation, right. So, we will get back to that maybe along the way. We will take a look at Laplace's equation and see what we are trying to do, okay. Laplace's equation, we have written it in terms of phi nabla squared phi equals 0, right. So, in 2D maybe I will write 2D Laplace's equation, we will restrict ourselves to 2D right now, 3D will follow in a similar fashion. So, in two dimensions Cartesian coordinates, this turns out to be dou squared phi dou x squared dou squared phi dou y squared equals 0 and we wrote for an equally spaced mesh and I am writing only for one set right now, we wrote for an equally spaced mesh at the point i, j in terms of i plus 1j i minus 1j ij minus 1 ij plus 1, we wrote this differential equation representation that we could use on the computer as phi i plus 1j plus phi i j plus 1 either way plus phi ij plus phi i 4 times phi ij phi i minus 1j plus phi ij minus 1 equals 0. So, this equation representation at this point ij right, we have written in this fashion and we got a truncation error, there is a representation error, we got the truncation error for that representation, for this representation, okay. Now, right in the beginning minus 4, minus 4 very important, minus 4, right in the beginning I had mentioned that this equation actually gives us, allows us when you are given a differential equation, in the sense you are given a question, find phi and if you find the phi, if somebody gives you a candidate phi, somebody says I have a solution, you would verify it by substituting into this equation and checking whether they have the solution or not, right. And if they give you something that is the solution, the right hand side will turn out to be 0, left hand side will be 0, it will equal the right hand side. On the other hand, if they give you something that is not the solution, it will leave what is called a residue, okay. So, if I give you a phi which is not a solution, if I give you a function which is not a solution, you substitute it into this equation, it will not give you 0, it will leave a residue, okay, that I am repeating again, this is something I said earlier in the semester, it will leave a residue, the residue is what you get if you substitute a candidate function, right, into our differential operator here and it should be 0 but it is not 0, what you end up with is the residue, fine, we write our equation in this fashion, something equals 0, left hand side should be 0, it is not 0, what is left is called the residue. Well, clearly if on a mesh, I give you discrete points on a mesh, like I had mentioned in the last class, you cannot substitute back into the original equation but you can substitute back into this, right. So, if I were to give you various values of phi ij at the various grid points, you could actually substitute that into this equation and find out whether this algebraic equation is satisfied. If the left hand side is not 0, that is it leaves a residue, that means it does not satisfy, am I making sense? So, if you end up, if you substitute, you have a candidate solution, you give me a candidate solution saying here is a solution. So, the way I verify it is I substitute it and I find out, I try to find out what I get and if I get a residue which I will call rij because it is the residue at the point ij and that residue is non-zero. So, now I change it a little, every time you give me a candidate solution, I will substitute it into this equation and see evaluate the residue and I will ask the question, is the residue 0? If the residue is 0, you have given me a solution at that point. If the residue is not 0, you have not given me a solution at that point. What you have given me is not a solution at that point. So, we have these 5 values, the algorithm that I proposed in the last class which was the phi at ij is phi at i plus 1j plus phi at ij plus 1 plus phi at ij minus 1 plus phi at i minus 1j times, I am going to say divided by, but times 0.25, 1 fourth. So, what we could do is, if you give me something so that the residue is not 0, what I could do is, I could reset the value at ij. I can evaluate a value at ij as the average of these 4 values. And obviously, if I substitute back in it will be 0, you understand. So, in a sense I have adjusted the value at ij so that it satisfies Laplace's equation, the discrete form at that point. This is called relaxing. It is called relaxing, the process is called relaxation. I have relaxed the value of ij, I have relaxed the value, it is as though it is in tension and I have relieved that tension, you understand. I have relaxed the value of ij, I have relaxed the value of phi ij by taking the average and substituting the average value there, it is satisfied. Equation, it is obvious because I got it from the same equation, it should be satisfied. So, what is the algorithm that we are talking about now? So yesterday, this is the problem that I started off with at the unit square, the unit square at the origin. So, I have the unit square at the origin. We can divide it up into, if I remember right, I divide it up into 9, divide it up so that there are 9 interior grid points. We could of course choose more. And what you do is, so at any of the interior grid points, we do not have the values. The problem is defined so that boundary conditions are given on all 4 sides. So, the boundary condition is prescribed on all 4 sides. So, in theory right now you do not have, I have not assigned the problem so you could just assume any boundary condition. You can just make assumptions on values of phi. When I say any boundary condition, values of phi on the 4 sides of this unit square. You can make an assumption on the values of phi at the 4 sides of this unit square. So, that sets your problem. So, you can take phi equals 100 here, maybe phi equals 100 here, phi equals 200 there, phi equals 200. You can pick values either constant or you can pick something varying as a quadratic curve. You can pick values. You can pick values along on the boundary, only on the boundary. And the way Laplace's equation, the problem is specified now that I have given you the boundary conditions. The question is to determine the values at the interior. And we propose to use the averaging process that we just talked about right now. So the first candidate solution that I propose. The first candidate solution that I propose. So I will give it a superscript 0 to indicate that right. It is the first guess that we have got at ij and ij are interior points. I will say that it equals 0 i,j interior points. This is the first candidate solution that I propose. So if you were to substitute it into the equation, you would be left with the residue. At every point if you check the residue, you will find that in fact because these values are not necessarily 0, you may not have guessed them to be 0. Whereas these are 0, you will find that a 0 here is not the average value. So this has to be relaxed. So you set this value to the average of the 4 neighboring values. Am I making sense? So we come up with the algorithm. I repeat that again but this time with the superscript.  Is that fine? And of course wherever it is on the boundary, so wherever it is on the boundary, you take the boundary values. Wherever it is on the boundary, you will take boundary values. So for the very first one, j minus 1 will be on the boundary, i minus 1 will be on the boundary for the very first one. Everybody is with me? So in general, what we could write is, in general we can write phi ijn or n plus 1 is, this is the relaxation scheme that we are talking about at the nth one.  Is that fine? So I have a mechanism, I have an automate on. I have a way by which I can now automate it. I have something given the value, I have an equation, given the values at n, I can find the values at n plus 1. So the mechanism that I have presumably, which we hope, the mechanism that I have is a mechanism that if you give me a guess, I hope I can get a better guess. And given a guess, if I can always get you a better guess, eventually I will converge to a solution. That is the hope. So what am I generating? What am I proposing to generate here? I am saying you give me phi ij, so I will drop the i because it is for all the interior ij's, phi 0. And then you are going to generate phi 1, phi 2, phi 3, phi n. You understand what I am saying? You are going to generate these values, you are going to generate a sequence of solutions. How do you check whether a sequence converges or not? There are various tests that you are familiar with. You could do, we could use the equivalent of a Cauchy test, equivalent of a Cauchy test. Cauchy test basically says, you remember Cauchy test? Cauchy test basically says phi n minus phi n minus phi m will be less than epsilon.  Once you have passed this capital N, there is an epsilon, so that this will only be that close, will be close enough. Well, we cannot actually do this. So what we will do is we will do an engineering approximation. So we will basically ask ourselves the question what happens to phi n plus 1 minus phi n. I will write it as a norm because given these nodal points you can actually come up with functions. And we want this to be less than some prescribed epsilon prescribed. So you will prescribe this epsilon. So now it is clear, we are generating a sequence of fees and all we have to do is check whether the fees converge. Is that fine? The other way to do it, the other way to do it since we are on convergence and then I will get back to the other way to do it is to look at Rij at each of the points. You will have similarly a sequence of Rs. You have a sequence of Rijs. In fact, I will drop the ij just like I did here. So you have a sequence of Rs, which are the residues and what do you want to happen to the residue? It should go to 0. So we want the norm of the residue to go to 0. So the other possibility is from here, we can say we want the norm of this to be less than epsilon prescribed. Is that fine? So you can test whether this actually tells you whether it satisfies the equation or not. So you can either say I am generating a sequence of fees. Does the fees converge? I am generating a sequence of Rs to the Rs converge. This will do for a first order. For now, for you to start writing some code or whatever it is, this is enough. We have to actually make this a little more precise. And right now this test will work for you. You can test to see whether your code works or not. Are there any questions? So if there are no questions, we will just get back here. So what we have is you will notice that the iteration, each pass through this is called an iteration. The fee that you get out of it, this is the jargon. You have to learn this. The fee that you get out of it is called an iterate. And the process that you are going through is called an iteration. Through each iteration, through each relaxation, it is called a relaxation also. So it is a relaxation sweep. So you will hear people use the term relaxation sweep. So it could be, you could call it a relaxation sweep. So each relaxation sweep or each iteration will give you a new improved fee at n plus 1, from the nth iterate. Because this is the n plus 1st iterate from the nth iterate, I will go back there to the, because it is the n plus 1st iterate from the nth iterate, it is called simultaneous relaxation. You are basically saying that fee at n plus 1 comes from fee at n. So in fact, someone had pointed out last time, we look at this in greater detail. It is actually a linear combination. So it is some p times, some matrix times fee at n. In fact, that is what it is. Now there is another possibility. So if I have fee i, j, n plus 1, what I have done is, I have done, if you look at, if you come here, if you come here and you look at this, when I do, when I relax this point, I take the average of these 4 points and I get a value here. So what I can do is, I can replace the value here, I can replace the value at that point by what I have just calculated. Because it satisfies Laplace's equation at that point now, my approximation to the Laplace's equation. So when I come here, I can use the latest value that I have got. I do not have to use the old value. I can use the latest value as and when I get it. Do you understand what I am saying? So then we are not relaxing. They are not at the same level. It is not simultaneous relaxation. It is called successive relaxation. So when I come to this point, I can use the latest value there and consequently get the latest value at that point. And repeat that, get the latest value at that point. So as I progress through taking the averages, you will see that I am using the latest value at any given point. So when I come here, what happens? The i-1 value is the latest value in general and the j-1 value is the latest value in general.  Is that fine? And this is called successive relaxation. This is called successive relaxation. It is just a name. But when people talk about successive relaxation, you should understand that they are talking about using the latest value. Fine. The first one simultaneous relaxation is also called Jacobi. See the depending on which direction you are coming from, they have different names and they are attributed to the people that have brought up the thought of the algorithm, Jacobi iteration. Simultaneous relaxation is also called Jacobi iteration. Is that fine? Okay. Now, how do we figure out whether the code is working or not? How do you find out whether the code works or not? How do you test to see whether the code works or not? You know how to check convergence. If it converges, does that mean that if it converges, does that mean we have a solution? See we have the following questions. How do we know our code is working? How do you know if it is converging that you are getting the right answer? So we need to answer these questions. How do you know if two of you get two different answers, if two of you get two different answers, you run the program and the two of you get two different answers. Which answer is the right answer? Is there a way for us to decide, right? Is it possible that two people get two different answers? Am I making sense? These are natural questions that you have to ask yourself because if you go out and you start solving this problem, right? And two different people are solving the same problem. Is it possible that somehow they end up with two different answers, right? Why would you say it is not possible? But how do we know that? Is there a way we can show that? So you are saying Laplace, the solution to Laplace's equation is unique, okay. Solutions, but what about the discretization? To our approximation, is the solution to our approximation unique?  So in your partial differential equations course, you may have learned that the solution to Laplace's equation is unique, okay. That the solution exists and that it is unique. But how about for our discretization, right?  Because we are doing, we are only approximating it. So is it possible that we have already seen when we talked about machine epsilon that there are a bunch of numbers that are represented by one number. So we have to have a little anxiety saying that are we going to get the same answer? Are all of us going to get the same answer? Okay. So we have these set of questions that need to be answered. So first we look at is there a way for us when you are given an equation, when you are given an equation, nabla squared phi equals 0, phi equals 0 and we have a discrete representation for this equation. Is there a way for us to set up the problem so that we can check whether the program that we write is generating? So we need an answer, we need a solution to this. If you have a solution to this, we are set, okay. So Laplace equation fortunately is easy. That is why I have picked Laplace equation as the first problem that we look at, right? Any analytic function is a solution to Laplace's equation, right? So the simplest thing is to take z or z squared. z is not, z being a complex number is not that interesting. z squared, yes. So we can take either real or imaginary of z squared as a start. I want to keep it simple. So a solution is x squared-y squared and I invariably start with something like this because I want to keep it easy. So if I were to solve this problem, I will pick a simple, so is this a solution to Laplace's equation? You can substitute and see and it is indeed a solution to Laplace's equation. So on the boundaries here, we can pick the boundary condition as it comes from x squared-y squared. So on the bottom boundary you can set phi of x, 0 equals x squared. On the right boundary here, you can set phi of 1, y as 1-y squared. On the top boundary, we can prescribe the boundary phi of x, 1 is x squared-1, sorry, x squared-1. And on the left, phi of 0, y is-y squared. And then you would expect that these points would be samples from x squared-y squared. They would actually be sampling the function x squared-y squared. The solutions here should correspond to x squared-y squared. So this is something that you can try out, okay. And now this is something that you can try out. You have an answer to the problem and therefore you can test to see whether your code actually converges to that answer. Is that fine? Okay. And you can see how well it, how well does it, if it converges, how well does it converge? Does it actually go to the answer? What does it go to? You can find that out. Second question, is everybody going to get x squared-y squared? Is it possible that the numerics goes somewhere else? Okay. So we will do something that mirrors, we will do something that mirrors the typical, the continuous function, the derivation that you would have seen in continuous function. I will just draw this here. So if you have, if you have these 4 points, what are we doing here? What is at the point ij, at any given point ij at the interior? What is phi ij? Phi ij is the average of its neighbors. So what is an average? The average is not larger than the largest neighbor nor is it smaller than the smallest neighbor. So any given point here, any given point that we take the average for, actually I guess I could have done it just here, any given point that you take the average for, any given point for which you take the average is not larger than its largest neighbor nor smaller than its smallest neighbor. So this point is not larger than its larger neighbor smaller and this is true of all the interior points okay and therefore can I conclude that the maximum and minimum will actually occur on the boundary the only values that I do not touch that I do not change are on the boundary every other point I ensure every other interior of every a point on the interior I ensure is larger than the smallest neighbor and smaller than the largest neighbor you understand. And all of these satisfy that condition they are all larger than the smallest neighbor and smaller than the largest neighbor consequently the maximum and minimum to the solution must occur on the boundary fine okay. So if the maximum and minimum solution occur on the boundary what can we say the maximum and minimum occur on the boundary, what can we say? We will use this argument now, this is called the maximum principle. I am just saying this just for okay, I will just say maximum principle okay fine. So what we are saying is if I instead of writing the system of equation, so if I have, I will still write it as nabla square phi equals 0. So I have replaced this by some, what shall I call it? L of h phi equals 0 right, so this is some system of equations, that is what we called it right, h is the grid size okay. And the discussion that we are having now is, is it possible for two people, is it possible for two different students to get two different answers and it is important that L is a linear okay that we get a system of equations, this is possible now for two people to get two different answers. So let us say two people, two different students get two different answers okay. So the two answers, two different students get two different answers, one gets the answer capital phi 1, so I am making it capital phi so that we do not confuse it with our iterates okay and the other gets the answer capital phi 2 okay. So we do the usual mathematical trick because we know we have a linear equation, what are we going to do? We are going to look at what is the difference between the two answers phi 1-phi 2 because the equation is linear, because the equation is linear, this is also a solution to Laplace equation right, because the equation is linear, this is also a solution. So if this is some phi, this is the difference Lh phi is Lh phi 1-Lh phi 2, you understand because it is linear, because it is linear the L actually distributes across it, because it is linear, this is just a system of equation, linear system of equations in this case and therefore this satisfies Laplace's equation right. What are the boundary conditions that it satisfies? Remember both phi 1 and phi 2 satisfy my x squared minus y squared boundary condition, therefore phi 1-phi 2 the boundary condition is 0. Now we use the maximum principle, the maximum and minimum occur on the boundary and that is 0 right, so the only function for maximum and minimum occur on the boundary and that is 0, this phi is identically 0, is that clear? So as I said this sort of mirrors the proof that you would have seen possibly for the differential equation itself, so yes it will turn out that if you do, if you solve this problem if two different students get two different answers the possibilities are both of you have the wrong answer right, that is one obvious possibility or one of you has the right answer right and one of you has the wrong answer okay right. After you have been programming enough you start with the feeling that you do not start with the confidence my answer is right, your feeling always is that likely that your answer is wrong, you have to start with the assumption, the presumption is my answer is wrong and you have to work to show to yourself, convince yourself that the answer is right, is that okay, fine right, so here we have what should I say, we have shown that the solution the answer is unique, we need to show now whether the scheme always converges you look at that right, but before we go there I have just quietly written this as Lh, let us figure out what is this L, what is the nature of this L, what is the nature of this operator L okay fine. So back here, so there are two possible numbering schemes that you can, the different ways by which you can number these okay, intuitively if you look at the way I have written this, if you look at the way I have made these pink colored dots, the assumption is that I have numbered this, this possibly is 11 or 00 or whatever it is and you know I have numbered them in increasing fashion of Y and an increasing fashion of J right, in fact I do not even have tell you, instinctively a lot of you will assume that that is what I am doing okay, so we number this in this sequential fashion right, that is what we have done, but is it possible for me having gone through this process, I now understand that it is only the interior points that are unknown that need to be determined, so possibly I can number the interior points in a sequential fashion and then number the boundary points okay fine, so one way to do it is you say you use two subscripts, you use two subscripts, the other way to do it is that you basically number the boundary points after this, so that is 9, so and this becomes 10, 11, 12, 13, 14 and so on, you could also number them in a random fashion, would that make a difference you think, so this is something for you to think about, if I would, should I do this sequentially, would it matter if I started my iterations from the top, would it matter if I iterated, I relax this point, this point, this point and did it in the reverse fashion okay or can I just do it in random, can I just do it at random, we should always get the answer, the question that still there, we should always, we would expect that we will get the answer, we will look at, we will see whether we can throw some light on that idea, right now intuitively I am getting a lot of head shakes saying that you do not expect, I do not expect it either, you do not expect from what we have seen, we do not expect that the answer will change, we are taking averages, we do not expect that the answer will change, so there are different ways by which you can sweep through, sweep through the domain, not necessarily from left to right, bottom to top, not necessarily in that direction, that you can sweep in any direction and instinctively the way we number it is the way we tend to sweep it, therefore the way we number it becomes important, okay. So we will definitely get to the answer, then the second question is how fast we get to the answer, right, okay, so we will look at those issues, right now let us stick to this, so I have numbered it in this fashion 1, 2, 3, 4, 5, 6 and we can write for each one of these we can write an equation and as a consequence we will get a linear system of equations, so for each one of those we write an equation, for the first one I get phi at 2 plus phi at 4 or phi at 4 plus phi at 2 it does not matter, minus phi at 1, if you want minus 4 I keep forgetting, I will start the other way around, I will do it in a sequential fashion, minus 4 phi 1 plus phi 2 plus phi 4 plus phi 25 plus phi 11 equals 0, is that fine, what is the next one turn out to be, phi 1 minus 4 phi 2 plus phi 3 plus phi 6 or phi 5 minus phi 12 plus phi 12 equals 0, am I making sense, we will write one more, so there is no phi 1, phi 2 minus 4 phi 3 plus phi 6 and then you get two boundary points plus phi 15 plus phi 13 equals 0, of course the boundary points we are going to take over to the right hand side, okay. In general if I write for a general point in between as I go down to an interior point which is only surrounded by interior points, I will get phi, if I am at the point I remember I have only one subscript now, okay, phi i minus 1 plus phi i minus 4 phi i plus phi i plus 1, that is the easy one, so you can see the second derivative i plus 1 i minus 1, then what do you get, plus phi i plus n where n is the number of points in that row in a given row, so phi 1, there were 3 in the row, 1 plus 3 gives me 4, 2 plus 3 gives me 5, 3 plus 3 gives me 6, okay, there are 3 in that row and subsequently I will get a phi i minus n now and this should equals 0, this is the general situation, okay. Coming back here if you look at it, if I have a general point, if there are n of them in this row, this is minus 1, this is plus 1, this is minus n, there will be, you will have to go back n, this is minus n, that is plus n, okay, so with a single subscript we can actually go i minus 1, i plus 1, i minus n, i plus n, is that fine, so we can really write this as a system of equations, you can really write this as a system of equations, maybe I will write it in a bigger fashion in a system of equations, I will write it there in that first, you can really write it as a system of equations, which will be minus 4, 1 and what, whole bunch of zeros, so this is the first one, then you have to go to i plus 1, i plus n, so when you get to i plus n, there will be a 1 there, lots of zeros, this is the structure of the matrix that you get, what is the next one, 1 minus 4, 1, lots of zeros, till you get to 1 on the diagonal there, lots of zeros, is that fine, you get 0, 1 minus 4, 1, lots of zeros, 1 on the diagonal, so the equation keeps on shifting to the right, lots of zeros after that and this will keep on shifting to the right till you get to a point where a 1 will appear, till you get to a point where you have gone to the second row, now from the first row something is contributing, while you are in the first row, 1 below goes to the right hand side, so you get lots of zeros, 1 minus 4, 1, lots of zeros, 1, lots of zeros, in case this does not make sense, I am going to erase this, I will write it as a, I will just draw lines, so you get a minus 4 on the diagonal and we are all agreed that that minus 4 is there for all the points, right, so you get a minus 4 on the diagonal, there is nothing to the left of this but you will get a plus 1 on the super diagonal, on the first diagonal above the diagonal, fine and you will get a plus 1, so there is a minus 4 here, you will get a 1 here on the sub diagonal, fine, so that takes care of the second derivative in the x direction, in our primary coordinate direction, then what happens, then you have n away from this, you have a 1, all of these in between are zeros, they are all zeros, okay and you are going to have a bunch of ones, you are going to have a bunch of ones till you come to the right boundary, when you get to the right boundary or the top boundary, right, when you get to the top boundary or the right boundary, at each of the boundaries you have to be a bit careful, so here n away you will get a 1 there, am I making sense, okay, there is 1 at the bottom and you will get 1's along this, 1's along this and 1's along this, this corresponds to the top boundary, this corresponds to the bottom boundary, there are zeros here which correspond to the bottom boundary, there are zeros here that correspond to the top boundary, here as you go along, just like you did not have any entry here, as you go along, as you come to a left entry or a right entry, you will get a 0, as you come to the left boundary or the right boundary you will get a 0. Does that make sense? I still see a few faces that are confused. When you have, when you have, when you are at the boundary, when you go, when you traverse along this, when you come to this, there is no, the entry here is known. It is not an unknown. So that goes to the right hand side. So you do not get a plus 1. That will be a 0. Just like that, with the very first one, there will be no i-1 entry. That will also be, right, there will be no i-1 entry. It will also be 0. When you go to the top, you go to the top, there is no i plus n entry. When you go to the bottom, there is no i minus n entry, okay. The points right next to the right boundary have no i plus 1 entry. The points right next to the left boundary have no i minus 1 entry. The ones at, at the bottom will have no i-n entry, no i-n entry, okay. So the no i-n entry, no i-n entry and then embedded in here there will be 0s with the frequency of n. Every time you go through, you come to the end, there will be a 0 because there is no right hand, there is no point to the right, it is a known. And all of those points will come to the right hand side. So this equation, this matrix A, what does it multiply? It multiplies phi 1, phi 2, phi 3, phi subscript 1, phi subscript 2, phi i, phi, I will make it capital M because I made n the number of values in the row, phi m, m interior grid points. And this equals, these will be the values from the boundary conditions. Known values from boundary, okay, these are prescribed, fine. So we actually get something that looks like A5 or Ax equals b, A5 equals b, you have a system of equations, okay. Some observations, the equations are symmetric about the diagonal, the equation that we get, the matrix that we get A is symmetric about the diagonal. The diagonal entries are negative, the off-diagonal entries are positive. So it has some nice structure to it, okay. You can just go look it up, I am not going to do this. It falls into a category of something that is known as an M matrix, we will just check that out. So the diagonal entries are negative, the off-diagonal entries are positive, it is symmetric, right. So there must be, maybe there is something neat that we can do with this, we can check to see whether, what kind of, right, what it is that we are, but we have a system of equations. We have a system of equations. Let us now see whether what it is that we are actually doing when we solve the system of equations, okay. What is it that we are actually doing when we solve the system of equations? So when we do Jacobi iteration, okay, what are we doing? The matrix A can be partitioned as the diagonal matrix D plus a lower triangular matrix plus an upper triangular matrix. Just to give you an example, the diagonal matrix D is minus 4 times i, right. The diagonal matrix D is minus 4 times i, fine. The remaining part lower and upper triangular you can extract out, I think that you can do. So when we do Jacobi iteration, what we are actually doing is, we are saying phi n plus 1 equals, you can check this out, D inverse phi n minus, well maybe we will work this out next time. I think we are coming close to. So what you can do is, you can just take the equation phi n 1 plus 1 equals, we are trying to find, just to give you an idea as to where we are going. We are trying to find out what is this P, okay. We are trying to find out what is this P so that we can say something about how this equation converges. So far what we have shown is that if you get a solution, it will be the solution and it is unique. You understand what I am saying? Now what we want to do is, we want to see whether does it actually converge, how fast does it converge, how fast does it get there. Now we are talking about the operational part. It is nice to know that if two people will get two different solutions, right? I mean we know that everything works, two people should not get two different solutions. The solution is unique. Now we want to know that we want to answer the question, how fast do I get it? I would like to get it today. I would like to get it in a few minutes. I do not want to take forever to get the answers. How fast? What can I do? The issue is if I can figure out, if I can answer the question, how fast will it, how fast does it, how long does it take to get it. Then I can, if there is a systematic way by which I can do it, then I can ask the question, why is it taking so much time? Is there something that I can do to make it faster? Remember always we always ask, how good is it and if you are able to determine to answer that question, right? The objective of asking how good is it is not just find out how good is it, but if we understand the question in the way we answered it, it may help us improve, right? Whatever it is that we are asking, how well does it converge and we can figure out, right, a way to improve the convergence if you are unhappy with the convergence, is that fine? Okay, so I will see you in tomorrow's class. We will look at this Jacobi iteration and see what is happening tomorrow's class. Thank you.


In my YouTube comments someone had mentioned that TP link maker of network equipment made an inexpensive managed switch And I thought you know I'll give it a try so I just you know threw it in the Amazon cart Got it here and said I'm curious about this now if you're not favorite TP link They make a lot of We'll call it lesser price network equipment we've used in the past We've actually never had a problem with their equipment their switches and that generally worked really well But I've never tried Any of their managed switches, so I thought you know why not order one now this box if you can't notice is Super shiny, and I'm carefully touching it because I didn't want to get fingerprints on it We even made a joke that we would only touch it wearing gloves that we would not sure why we have these what look like mine gloves But this switch is got such a shiny box. I'll put the gloves on real quick to touch it here That it's just super reflective, and I don't know I mean ten out of ten on super shiny box But of course that's not what really matters. We really want to know does it work? Well, I haven't opened it yet I think we took the outer plastic off, and I'm just and I managed to finger print it up So let's open it up and see what's inside, and then we'll get to the software testing that you know the real part, but Box wise ten out of ten definitely cool. I mean, I don't know why you throw a box I don't really keep them unless I plan to resell things. So yeah, we can take the gloves off now. That was that was my Yeah, the these super shiny and pretty box So this actually kind of heavy got an instruction manual one of those mini CDs Hopefully I don't need that to install anything the Five slash eight port gigabit smart switch. I'm assuming that's manual for either the five or the eight port these come in a couple varieties The switch itself is nice metal. Well, it looks like it's got a Kensington lock and a mount Over here. So we got that going for it What else is in the box here? Now I do know they make a Powered POE version of this as well. So that's an that's an option. So I like the fact that we have on the back here the screw hole mounts to mount it like on a flush mount and we also have the Screws here, or I'm sorry that screws the feet here So when you've went on a bottom so it doesn't slide around so you got some sticky feet a little basic power adapter Wow, that power brick is small That is really tiny But I guess it doesn't take a lot. It must be super efficient. This is only 9 volt at point 6 amps. So yeah, wow, that's it doesn't take much to run up a boy That is that is a intriguingly small adapter for this but Obviously, let's take a look inside and see how it looks inside and then we'll actually get to the software part So only two screws here on the bottom And it just slides apart here You can't get much more basic on the inside just a couple chips a heat sink. It's completely passively cool Not a lot in here a little reset switch So yeah, the reset switch is pretty easy access right there. Nice little quick quick switch to master reset this But yeah, not much inside. See is that glued on good? Yeah, there's no wiggle to this So they use they use good glue. They didn't use just like a sticky pad that might fall off later This seems really glued well to the board here But yeah, not many circuits on here So I like the simple design so that we can move on from that and actually get into the software and Does it route does it actually manage to VLANs properly? So now that I have the switch hooked up. I updated the firmware on it The default IP address for this is <IP_ADDRESS> I have logged in and changed that I think that's kind of a dumb IP address It should be something I don't know Let's say 20 at the end and the reason I say that is if a lot of your networking And I think some of the networking devices from TP link come at a default IP address of <IP_ADDRESS> for like your routing devices. It would immediately conflict So before plugging anything else into this I plug my laptop in you set it static because this doesn't assign any addresses It does not have a DHCP server nor really should it And set my IP to be in the same range to change your IP address this is currently set to <IP_ADDRESS> and My pfSense box over here is <IP_ADDRESS> And we're going to walk through first the physical layer of how we get it set up Then I'll walk you through the software a little bit here I have on the first port here the purple goes into IGB 1 on the pfSense so that is my LAN port and it's coming into port 1 on here And that's going to matter because this is going to have VLANs coming over So this is really what we want to do is test the VLANs. So this is Kind of like a trunk port which means all the trunks come in here All the VLANs are going to come in here no matter which ones we create We're going to create one for this demo so you get an idea of how the switch works and that it works And we're going to talk about little issues with the switch port 2 is my laptop and that is plugged in right now To the second port right to my laptop now my laptop is going to get an IP address assignment based on port 2 Being tied to the main LAN and we're going to leave ports 7 and 8 are open at the moment But we're going to move my laptop over there and I'll show you how to do an untagged port Now we have another tagged port right here Which is port 4 and we have a unify plugged in there because the goal is going to be to get the unify To pick up the the VLAN tag for both LAN and the second One we called crap LAN And that's going to pull over to there So we're going to go through this is the physical layer So we have only one cable coming out of the pfSense box One physical cable going into port one and then we'll split out everything else to where it needs to go using VLANs and tagging in here and we'll try to hop the VLAN to that's part of the demo here Is it something a switch shouldn't let you do and that's an important aspect and something we're going to talk about Okay, we're going to start at the pfSense side so you can see what is feeding into this switch So right here. We have the VLAN. We have the Vlanlies edit it real quick. I've covered other videos on how to do the VLANs In pfSense so IGB one we have it tied to the LAN VLAN tag of 30 VLAN 30 for craptastic devices Interfaces assignments. There's our LAN 30, which is the craptastic VLAN for devices Once again, like I said tied to IGB one to the same as the LAN It has an assignment of <IP_ADDRESS> and we're going to go to your services DGP server So our LAN being the 192.168.11 and LAN 30 is <IP_ADDRESS> and both of these are being pushed out of that purple cable and into the TP link So that's all the configuration We really need to do on the pfSense side to have two separate lands and just so you know in terms of the firewall rules I do have whoops rules I do have the LAN and LAN 30 allowed to talk to each other So I have not set up a restriction between them That's just for the demo of this because most I want to show you that the VLAN tags come through and it assigns the proper DHCP ranges toward for them So we're going to go over here and to our switch now. This is the assignment I can set out of the box is <IP_ADDRESS> and we change it to 1.2. This is the TL SG108E It's got the hardware revision of 3.0 and the firmware is 2017-12-14 this is the latest firmware and it's weird because it starts to 2017 when you go to the website It says it was released in January of 2018 was a release date. Maybe it's just a weird numbering scheme They have I don't know but I did update it to the latest firmware IP settings are pretty straightforward. It does support DHCP But you may not may or may not want to do that. So here I have the IP address set pretty straightforward there User account by default it's admin and admin, but you can change both the username and password, which is great because You may have noticed Connection is not secure. That's because this doesn't support HTTPS I found that to be interesting. It doesn't for whatever reason even have an option to switch to it So definitely a concern there backup restore system reboot save config on reboot And let's talk about how this switch works now. This is common with a lot of your commercial switches Any effect anything you change will apply, but may not apply on restart There is a running config and a read any config on restart and what that means is anytime you make changes You want to go over here to save config? Are you sure you want to save to config config sec? It says that's fully saved. So even though you may change it all these are running now But if you lose power to this without having clicked that save this system will lose those settings And this has caused confusion before with some switches I've run to this for people who I thought I had set up right and had a power outage and it all went wrong What did I do? And I'm like, well, you probably didn't save the running config You had it configured to work right this also will save you From oopses if you accidentally are plugged into a port and you tagged that port wrong so you can't access it anymore Then unpower power back in and because you weren't as long as you didn't save that setting It'll just reboot and have the last known working config. So save often So as you're doing things and once you verify that thing you're doing is working go ahead and save Switching port settings will go through this real quick. It does have options for status duplex and flow control So you can force things to certain settings on here. So configed actual configed actual for the flow control Supports IGMP snooping If you don't know it is go ahead and read about it. I didn't want to test this because I have time But it does support lag which is cool. So it'll create up to two lag groups So that's a link aggregation for ports. That's kind of novel and what this will allow you to do it I'm nice. I'm not going to actually apply it. You can aggregate ports together And as long as you have something and we're going to talk like say free NAS for to be an example that supports it Actually, so does pfSense and you can tie the ports together to act as one almost like one physical port and it's a good It's a bandwidth enhancement. So I can only get a gigabit If I have a gigabit card, but it has two ports on there I can take both of those gigabit ones plug it into the lag interface Lag them together and now I have a two-gigabit connection and then I can have another If I have another computer that supports lag I can tie that together and then it has a two gig connection or four gig if you have Four ports enabled and so on and so forth. So that is a Supporting feature in here, which is pretty cool. Like I said, I didn't test it, but I'm sure it works Monitoring Port statistics. So if there's bad packets transmit Receive good packet bad packet. So if there's any problems, it does let you know And it has a couple bad packets on the receive because let's do it some testing with it Port mirroring this is a popular feature if you want to just listen on a port and what a port mirror is is You take one port and you just say replicate all the data that goes this port to this other port Sometimes that's used when you're doing like wire Using a tool like wire shark to do packet sniffing and things like that you want a mirror port So the same data comes over here and maybe you want it for logging purposes and things like that Now this is weird this I thought was an odd feature This is cable testing and it doesn't just support cable testing. It supports and I'll I don't play moving things So I'll show you real quick. I took a screenshot of when I plugged in a hundred foot cable Well a 31 meter cable 30.5 meters to be exact. It just rounds up. It doesn't do decimals if you have an open-ended cable And you plug it in it will tell you the length of the cable odd feature. I think for Being inside of a switch the fact that it has that it seems to not understand the cable length if it's plugged in though So it only works when the cable is open-ended. So if you plug an open-ended cable in it will display the length of that cable I don't I didn't test exactly how record but it did measure my hundred foot one quite well it measured it to be the hundred foot or 30.5 meters for the rest of us who aren't inside the US So that is a interesting feature that it does support so that you can see that and It's showing right now cable fault distance of one and zero So we let you know which ports are open which ports are normal as in in use. So kind of novel Loop prevention and it's off by default But loop prevention is spanning tree protocol, but they've just called a loop prevention If you're not familiar with that that means don't plug a loop into a network look up spanning tree protocol And now here's the V lands Now it supports a couple things this works the the multi-tenant V land which is this I'm not as familiar with that It creates its own V lands if you use port beast V land where you can just say segment these ports and these ports So you can kind of divide it into a couple logical switches. It's got that so if you don't have a managed or a Router like PF sense that supports V lands. You can just create your own network segments with it Which is kind of neat. We're gonna use 801 to Q V lands on here and show you how that works Last couple features it does is bandwidth control So you can individually set the ingress and egress rates of each port And it does support storm control as well. So kind of neat that it has that and some real basic QOS I didn't really test any of this. I don't know how great the QOS is going to be but you can't set some priorities in this So neat that it has it so let's focus though because this is what a lot of people ask is, you know What's an inexpensive router for what are doing V lands? And let's show how this works. This is a little bit weird. I found to set up So we have here Vlan ID one once you enable 80 Q 80 to one Q Vlan Enable it it creates this one here and you can't do anything with it Vlan tag one is default Vlan on here I've seen some people complaining about this by the way Each one of these if we switch over to like an MTU Vlan or a port based Vlan it Disables the other option so MTU each one of these is exclusive MTU is exclusive port Vlan or 802 1Q Vlan Each one are exclusive and disable the others when you turn them on so If we go here to one and this is what this is why I said about being a little weird It's not as intuitive we put the one here and it fills in and You can now start messing with all the ports and changing things So we're gonna leave the port one at default and we're gonna put 30 here Now when I type it in it pulls over what we had so this is how this works And like I said, this is maybe a little confusing The tagged ports Mean allow all the Vlan traffic to come through so port one is plugged into our PF sense And that's where the Vlan traffic comes from so we have to tag that port for port 30 to work We have port two and port three. They're ignoring anything Vlan 30 I just just for testing purposes. I did that you can change it Port four is where we have the P the unify plugged in So we want that to pull the tags and bring them over to here and send them out Then we have port seven and eight now What we're gonna do with seven and eight is we want them part of the members was it is a member But it's an untagged member what that means is we strip off the Vlan tag and forward over So if I plug in a my laptop and we'll do this in the demo here to port seven or eight Untagged my laptop will get an IP address because it's gonna come in here It's gonna have the tag of 30 and it's gonna get an IP address from port seven or eight of the 192 168 dot 30 network versus when I plug it in here because everything else is a member of this Network the other ports will get that now There's one more step to being able to do that and also if you want to create a new port Let's say if we had another Vlan you actually just type it in and hit add and it shows up down here Kind of weird how that works, but to make the ports work You have to go over here then and then assign a PV ID setting Now this is a port Vlan ID and what we're doing here These ports all have the default of one, but we took port seven and eight and assigned them up 30 And with that that is the final follow-through so ports seven and eight will get to dot 30 address So it's kind of weird because they should I think this could be done in one menu, but they've chosen to do it too So we're gonna go here. We put in 30 We have an untagged port up here because we want the tags removed because we want physical devices plugged into those Land ports versus when you have it tagged here that's because we want both Vlands the main Vlan the main land and then the Vlan 30 to be forwarded to the unify so we have to have this one tagged But these are the untagged ones because if these were tagged that means they could switch back in for three either so you only want to plug in devices that are smart enough to Delegate Vlands where they're supposed to go to be on a tagged port and untag port is for dumb devices that are just going to Get an IP address based on the Vlan tag. So there's our setup port seven at 30 port eight at 30 So they're going to pull the 192 168 dot 30 network So pretty straightforward. I have tested the Unify and we'll pull that up real quick here just to show you the unify settings So here's that unify plugged in it's at 192 168 102 We're gonna go over here to settings Wireless networks. Here's our craft Wi-Fi Vlan 30 and I've covered this before in other videos You just create it pull that same Vlan and now the tagged port brings over all the Vlands I've tested this it works perfectly fine. So if I kept to the crap things it gets that But this is where I found kind of a flaw in the switch. So we're gonna take my laptop and open up the terminal So here's my adapter that's plugged into the network And right here is my 192 168 1.100 Like I said, this is plugged into port 2 on this. So if we look at what port 2 is set at Port 2 carries all the traffic and gives me the 192 168 1 and then we're gonna go switch over to port 8 here which carry which should change my IP address to match the dot 30 network and now we look and I've got a 30 dot 100 address So it does work, but that's only part of it. Let's talk about Vlan hopping real quick So the question is and I'm not gonna do a full-blown attack. There's lots of different attacks You can read about this for brevity here. We're just gonna show that Normally I should not be able to Get back to the other Vlan without a firewall rule that allows me to do so And I do have a firewall rule enable that does allow me to get back to the PF sense box But I shouldn't be able to hop this Vlan to a different IP address So by default I have via DHCP one nine two six eight thirty dot one hundred from plugged in there But let's go ahead and change my static IP address settings Now what I'm doing here is assigning my system one nine two one six eight one dot nine But leaving it plugged into port eight, which is D dot 30 network now a smart switch should protect me from doing this And let's check it out here. So I am at one nine two one six seven nine I have forced my way on to the network and this is what an attack vector is to Try different IP blocks, and then I'm gonna try and ping one two one six one dot one So this would be going in through port eight out through port one Where that's located and I can't ping it. So good. I'm not Jumping out of the Vlan now. I like I didn't try all kinds of different attack vectors I'm not gonna think that this switch is gonna really hold up to anything really scrutinous here But I will point out this I Can ping the management interface and I should not be able to do that So I it would appear the management interface. I can only speculate without having a design Spec of this that the management interface is simply tied to the backplane as a virtual extra port on this and Doesn't look through the Vlan rules before allowing me to get to it So by simply forcing my computer to be the same address as whatever the management address Ranges by putting myself in that same network. I'm able to get to it and I can prove that Here so it does recognize me as a different IP address. It doesn't make me login again But I'm in so that's kind of interesting to me And I point this out to a security friend and he asked well, how are you gonna threaten me to get this? I'm like I'm gonna tell people not to put 30 dot $39 switches in their corporate network but but if you want to get a switch that's affordable and We'll get Vlan started or be able to do lag and port mirroring and especially if you're if you're into digging into how networks work Great thing to start playing with I wish it had enough to say trouble I can get him behind the scenes and really start playing with it. I didn't find anything open If I find anything later, I'll update this but so far I didn't find I didn't spend a whole lot of time trying to hack it Maybe there's someone out there who did if I do find that I will do and follow up video on it But I would say this is not a bad for the price If you're a home user and you want to start understanding how VLANs and tags work and start experimenting with networks It's still like I said, I've recommended by I would definitely Think it's a great place to start if you plan to use it for your own little storage network and use some lag interfaces on there I may do some testing with that later I just don't have anything handy that I can take apart to do that with at the moment But I'm sure it would be good for that as well So like I said, maybe you want to do a failover or you need that kind of manage and set up the lag on there Might be fun. It's a fun thing to play with. The price is very reasonable on this But the security you know less than reasonable if you have a good password on there I probably still could be brute force them like I said it doesn't support any HTTPS it's a HTTP only to get to it So they don't even have a self-sign certificate in this thing But hopefully this was helpful and like I said if I don't think it's bad for the price But it's certainly not an advanced piece of networking equipment and certainly don't install this in your corporate network It will probably cause some headache at some point down the road. All right. Thanks Thanks for watching if you liked this video go ahead and click the thumbs up Leave us some feedback below to let us know any details what you like and didn't like as well because we love hearing a feedback Or if you just want to say thanks leave a comment If you wanted to be notified of new videos as they come out go ahead and subscribe and the bell icon that lets YouTube know That you're interested in notifications. Hopefully they send them as we've learned with YouTube Anyways, if you want to contract us for consulting services You go ahead and hit Lawrence systems calm and you can reach out to us for all the projects that we can do and help you We work with a lot of small businesses it companies even some large companies and you can farm different work out to us Or just hire us as a consultant to help design your network Also, if you want to help the channel in other ways, we have a patreon We have affiliate links you'll find them in the description You'll also find recommendations to other affiliate links and things you can sign up for on Lawrence systems calm once again Thanks for watching and I'll see you in the next video


Hi, it's Darren Marlar! Welcome to the Darren Marlar Radio Show, and coming up today – junk food and monkeys! It's not a new flavor combination of Ben & Jerry, it's a sneak peek at our weird holidays. Nobody likes going to the dentist, but we redheads. Well, we have more reason than the rest of you. Don't believe me? Well, I will explain coming up here in a few minutes. Got our birthday wrap up, question impossible on the way as well, along with our joke of the day, and a Norwegian doctor he called in for military service that would have made Corporal Klinger of Mash proud, our moment of duh later on in the show. And where in the world do you think it's actually appropriate for your wife to be a minimum weight? We're going to find out later on in the show, and if you drink tea and you already seen your citizen, congratulations, you're helping your brain. Our brain-on-drug story delivering sweets to the police department lands one teenager in a lot of trouble. And not all heroes wear capes. Some of them are even over 70 years old. I'll tell you that story, which is amazingly inspiring. I loved reading that story and had to share it with you. Oh, by the way, you know, it's important here, you know, now that we're in the hot weather, you need to remember your brain. It works, you know, slower in warm temperatures. Your response times are slower. You're watching McCollet.  Yeah. Well, I might not know how to tango, but I know how to man dance. I think you're a little confused here. Nobody likes going to the dentist, but well, we Red Heads, we have more reason than the rest of you. I will explain why coming up here in a few minutes. Hey, by the way, if you want to be a part of the show, you can visit the contact page at DarrenMarlar.com. You can send me an email for my email bag. Right now, let's take a look at today's weird, wacky, strange, zany, odd, bizarre, quirky, unusual holidays. Today is Friday, July 21st. You have 156 shopping days until Christmas and today is National Junk Food Day. Or you can just celebrate it every day like I do. You know, that way you don't miss it. Today is Monkey Appreciation Day. You know, I don't have a lot of experience with monkeys, so I'm not really sure what to do to show my appreciation for them. I mean, even not kidding, my sea monkeys. They even died after a couple of days. And today is Don Knott's Day, marking his birth on this day back in 1924 in Morgantown, West Virginia. He played his most famous role, of course, Barney Fife on the Andy Griffith Show, but he also played Ralph Furley on Threes Company, Les Ace Calhoun on Matlock. He was also a regular on Search for Tomorrow, believe it or not. The Steve Allen Show, the Don Knott Show, yeah, he had his own show there for a while. What a country, believe it or not, he has won five Emmy Awards, an amazing actor, an awesome comedian. See, he was on a soap opera, Search for Tomorrow. I thought soap operas only hired bad actors. All I'm saying is that there are some things beyond the can of mortal man that shouldn't be tampered with. We don't know everything, Andy. There's plenty going on right now in the twilight zone that we don't know anything about, and I think we ought to stay clear. You're pretty far out there, even for the guys who do a lot of this stuff. Today's question impossible. What U.S. state is surrounded by the most water? I'll have the answer for you here in just a few minutes. Hey, if you would like to like me, poke me, tweet me, follow me, or stalk me, you can find links to all of my social media at DarrenMarlar.com. While few people enjoy a trip to the dentist, it is particularly bad for us redheads. Scientists are even saying this is true. Carrot tops are twice as likely as blondes or brunettes to avoid the dentist because of fear-related anxiety. According to a report in the Journal of the American Dental Association, researchers believe it is because the gene, a melanocortin-1 receptor, which causes red hair, may be present in redhead's brains, making them more susceptible to pain. Yeah, so lay off, it's not my fault that I'm a woozy man. Hi, I'm Darren Marlar, and you can keep up with everything I do by signing up for my newsletter, absolutely free. It's the Marlar Sheet, and you can sign up for it at DarrenMarlar.com. Doing so will also automatically enter you into contests that I have on a regular basis. We'll take a look at today's celebrity birthdays coming up with our birthday wrap-up. But first, question impossible. The question again was, what U.S. state is surrounded by the most water? What U.S. state is surrounded by the most water? Well, it really isn't that difficult of a question. Hawaii, it's an island, it's surrounded by water. Most people though, amazingly enough, they get this question wrong. They either say Maine, Florida, Alaska, or California. They just completely forget about Hawaii. How about you? How did you do on it? Our joke of the day is coming up here in just a bit. If you missed any part of today's show, you can catch up on it at DarrenMarlar.com. Celebrating birthdays today from the perfect man? Are You Afraid of the Dark? Actress Vanessa Legges is 32. From Pearl Harbor, Black Hawk Down, and Sin City, actor Josh Hartnett is 39. Actress model Allie Landry from the TV show Felicity She's 44 today. And actor-comedian from Saturday Night Live, News Radio, The League of Their Own, John Lovex is the big 6-0. Hi, I'm Darren Marlar. Welcome to the show. If you want to listen to any of my past shows, you can get them on my free mobile app. Just search for Marlar House in your phone app store. It's time for today's joke of the day. A man was walking through a forest pondering life. He walked, pondered, walked, pondered. He felt very close to nature, and he even felt close to God. But he felt so close to God, in fact, that he felt that if he spoke to God, God would reply. So he asked, God, are you listening? And God replied, Yes, my son, I am here. Well, the man stopped and pondered some more. He looked towards the sky and said, God, what is a million years to you?  So the man continued to walk and continued to ponder, walk and ponder, walk and ponder. Then he looked to the sky again and said, God, What is a million dollars to you?  It means almost nothing to me. It does not even have a value. It is so little. The man looked down, pondered a bit and then looked up to the sky and said, God, can I have a million dollars? And God replied, In a second. Today's moment of duh is coming up here in just a bit. A Norwegian doctor, he was called in for military service and he would have made Corporal Klinger of the TV show Mash Proud. Our moment of duh on the way. Hey, if you like scary stories, well, I actually have a show on the weekends that I do called Weird Darkness. If you want to check it out, you can find it at WeirdDarkness.com. British politicians are being urged to eat gray squirrels as part of a radical plan to prevent the native red squirrels from being eliminated by the gray imports from North America. As if English food isn't bad enough.  We're in the world, do you think it's actually appropriate for your wife to be a minimum weight? Well, we're going to find out in a few moments. You can hear the show anytime by the way, just go to DarrenMarlar.com. Today's moment of duh, a Norwegian doctor, he called in for military service and he would have made Corporal Klinger of Mash Proud. The doctor rubbed sour cream in his hair, poured sticky liquor in his shoes, spilled beer on his clothes, and sat in a closet smoking 40 cigarettes at once in an attempt to convince the military that he was not fit mentally for service. Well, and just to be sure he looked and felt his worst, he stayed awake for two days before his physical. He was so convincing that the military doctor alerted the National Health Authority about the man that they had licensed to work as a physician. Well, an analysis of his records, however, showed that not only was he not insane, but he had received high marks from his patients. So now, the doctor is facing likely disciplinary action from the military and the medical board. If you'd like a few more laughs after the DarrenMarlar radio show, you can check out dailydoseofweirdnews.com for hundreds of the videos I've created over the last couple of years.  I'll tell you how right around the corner. So, where in the world do you think it's actually appropriate for your wife to be a minimum weight? I was always taught that asking a woman's weight was a very impolite thing to do, but in Finland, it is required. That is, if you want to compete in the annual World Wife Carrying Championship, the Finnish village of Sankt-Jarvi, and I'm sure I'm totally destroying that name, but anyway, a Finnish village has set a minimum weight for women taking part in its annual World Wife Carrying Championship to stop men with the very tiny petite spouses having an advantage. By the way, if you have no idea what this is all about and there's really no reason that you should, well the championships they have men carry women on their backs over an obstacle course. Wives must now weigh at least 92.5 pounds.  It does, oh good, that means we're ready to compete. It must be true, because I heard it on the radio. It must be true, it must be true. Ladies and gentlemen, the Darren Marlar Radio Show. This is important. The most significant event on the internet today. Who may I ask, are you? Darren Marlar. Delivering sweets to the police department. It lands one teenager in a lot of trouble coming up in a story that I'll have for you in just a couple of minutes. If you'd like to be a part of the show, you can visit the contact page at DarrenMarlar.com. You can send me an email and maybe end up in my email bag. Well, drinking tea as a senior citizen, especially those at a higher risk of diseases like Alzheimer's, that could help prevent serious mental decline. Researchers at the National University of Singapore looked at 957 Chinese individuals who were 55 years of age or older in this longitudinal study, and found that those who drank tea daily reduced their risk of cognitive decline by 50%. Participants who carried the gene associated with Alzheimer's disease saw a diminished risk of cognitive impairment by up to 86%. This finding held regardless of the type of tea that somebody drank. As long as the tea leaves had been freshly brewed, consuming tea was found to help protect the brain. If you want to read a little bit more about that, I do have a link to that article in my blog, which you can get to at DarrenMarlar.com. You know, not all heroes wear capes. Some of them are over 70 years old. That story is on the way. If you'd like to like me, tweet me, whatever you want to do on social media, you can find me there at DarrenMarlar.com. I have links to all my social media there on my site. An 18-year-old suspected of delivering drug-laced cookies to a dozen police stations in Texas is fulfilling court-ordered community service work for mothers against drunk driving. Christian Phillips told police he was a representing, that is, MAD, M-A-D-D, when he dropped off the baskets of treats that were loaded with LSD. A few officers ate the cookies, but no illnesses had been reported. Phillips was arrested at the Lake Worth Police Department after officers who had received a call from M-A-D-D, MAD, Mothers Against Drunk Driving, in case you didn't know what that represented. The officers said the cookies taken to another station may have been tainted. While they accepted the basket, and after he was busted, police discovered Phillips had visited 13 of 24 departments. Ironically, Phillips was on probation for an assault last summer after officers responded to a loud party at his house. Several teams fled, but Phillips tried to karate-chop in officers' arms. Thanks for listening to The Darren Marlar Radio Show! If you want to keep up with everything I do, you can sign up for my newsletter, The Marlar Sheet, at DarrenMarlar.com. Doing so will also automatically enter you into contests that I hold on my site. Well, not all heroes wear capes. Some wear aprons and oven mitts. After Leo Kellner's wife of 72 years died from complications related to dementia in 2012, the Hastings, Nebraska man began baking desserts for people in need in his community. The 98-year-old told today, I didn't know what to do with myself. I was moaning and moping, and I said, I gotta have something to do. And that's how it started. If you want to see a video of this guy in action, a great interview with him, and very inspiring, I was smiling the entire way through it, you can find that video in my blog at DarrenMarlar.com. Definitely worth checking out. Hey, if you missed any part of today's show, you can catch up at DarrenMarlar.com. I had a blast today. I hope you did too. The DarrenMarlar radio show was recorded before a large cardboard cutout of a live studio audience. Good night, ladies.


I'm going to show you three basic Wing Chun exercises that you can do at home by yourself to get stronger, to get better Wing Chun techniques and to be even stronger in whatever you do. So the first thing we do is let's get into our A stance. From here we're going to buy a top. So what am I looking at?  One is at your elbow position, one is out here. Elbows low, not completely extended, but slightly extended. Do not extend completely like this. Okay, so this is your five zone position. Toes are pointing in and hands are on your center line.